The Ultimate Guide to Lens Design Forms:
The types of optical systems in a lens designer’s toolbox
There are so many optical systems that are out there.
Why is one lens type used over the other?
What lens design for do I need for this system?
What lens design alternatives should I consider?
Can I use this lens somewhere else?
What if we don’t know where to start with the lens design, when only given a specification sheet?
Overview
This is an Ultimate Guide of lens design forms, the optical systems that are used in our world.
The basic lens design forms are in here, and we can take a deep look into the development of lens design. But the not all the lens designs are simple lenses, we will look at newer and important lens design forms as well.
You’d be surprised to see what lenses are related to one another, and how we can break down seemingly complex lens design into parts from different lens forms.
I can’t catch all of the design forms, but let me know in the comments if you want to know more about a subject, or if you feel there is a lens form missing.
N.B. A few primers before you read on:
I may repeat the same explanations from time to time. This is because of the web and ebook format, where I feel it isn’t quite as easy to go back a few pages and reread the material and immediately come back to the place you left off. I’ll provide links within the Guide, but I’ll also try to make it an easier read without saying “I said this already lookie over here”
There is a lot in here, so feel free to navigate around with the table of contents below.
If you’re interested, you can get the PDF version too.
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Table of Contents
 Motivation and introduction to the General Theme
 Background

Imaging lenses: Classic imaging lenses and the dawn of lens design
 Coffee break: Higher order aberrations

Imaging lenses: The evolution of lens design that leads to diversification
 The Ernostar lens: Evolution from the triplet still used today
 The Sonnar lens: Bertele’s genius in lens design
 The Double Gauss lens: The winner of the standard lens on a photographic camera
 The symmetric wide angle lens: The quest for Field of View
 The Telephoto lens: The telephoto lens: The term that is confusing for photographers(but not lens designers)
 The Retrofocus / Reverse telephoto lens: The practical solution to wide angle lenses on SLRs and digital sensors
 Imaging lenses: Specific use lens systems
 The Fisheye lens: Testing the limits of Field of View
 Zoom lenses: How we can get many focal lengths in one lens system
 Afocal lens systems: manipulating rays to get them to behave the way we want
 Telecentric lens systems: when and where we need straight rays
 Reflective optic lenses: Changing the direction of light
 Coffee break: Some cool lens design names

Imaging lenses: The eye and the lens design forms that use our vision
 Imaging lenses: Modern use cases based on new lens design forms
 The stepper lens: A mastership in lens design
 Laser beam printers and laser scanners: ftheta lenses and their seagull like shape
 The Aspherical lens: an addition to the spherical shape that opens up possibilities
 The freeform lens: Thinking of optical lens design in three dimensions
 The mobile phone lens: Taking aspherical lenses to the extreme
 Laser applications: The new age of optical lens design
 Diffractive optics:
 Illumination lenses: A totally different approach to optical lens design
 Bonus: A list of interesting applications of lens designs not mentioned thusfar
 Lens design forms and the principles of optical lens design
 References
 Comments
Motivation and introduction to the General Theme
As an optical lens designer, there were times I used to think that “I can get the performance that I need for this system with the software, but I want to understand what I am doing”.
Nowadays, we are required to deliver high performance (not necessarily high quality) lens designs in a short amount of time. Nonintuitive lens design systems like aspherical lenses and offaxis lenses too. Lens design with computational software like Zemax and CodeV are the norm now.
Sometimes, I see myself punching in the parameters that are needed for a lens design, and I can press a button to get the desired performance of the system. In an extreme sense, I can turn my brain off and use the software to get the desired outcome. I try to catch myself every time I do so. I’m a scientist, researcher, engineer, and genuinely interested in the process of lens design. From my own experience, the time I realized that I was merely punching out lens designs like a machine, I realized that I had a hard time designing different systems when I needed to.
This did not make me a wellrounded lens designer, and frankly, it was a lot less fun!
So I hit the books. Thankfully, there are a multitude of books on lens design, starting with the optics theory, optical system design, manufacturing technology, and yes, lens design methods too.
I got mentors in different fields to help me with lens designs that I wasn’t familiar with.
I quickly found that knowledge from books is applicable in many situations, but to use the information in a meaningful way, I had to figure out how to apply them to different situations on my own.
In order to truly understand the material, I needed to connect the dots of the knowledge I gained from reading all of those books, and create a web of lens design knowledge to be able to catch everything that is thrown at me.
What I’ve done over the years is distilled the text book material from my favourite books into the usable concepts, in a logical format that shows each piece of the lens design puzzle and process.
This logically lead to writing this Guide, and it’s what I would have wanted at my fingertips when doing a lens design early in my career or when I was learning lens design. But make no mistake, I myself will be looking back at this Guide often as a reference for designing different lens designs as I go forward in my lens design career.
About this guide
This Guide provides the lens design forms of various lens designs from simple lenses to complex lenses, and is intended to provide many examples of the lens designs that we use today. By becoming familiar with the essential lens design forms, we can use them to our advantage during lens design.
The examples in this Guide provide a birdseye view of the various lens design forms and why certain lens combinations are used, to help visualize typical lens designs and even complex lens designs.
This helps us become more efficient in our lens design process. I call this training your Pattern Recognition skills for lens design.
What you will learn in this guide
We will look into lens design forms and see when and where to use specific lens designs, techniques of lens designs, with plenty of examples.
Who this guide is for
 People who want an overview of the essential lens design forms
 People who want applicable lens design forms for reference for their lens designs
 People who are starting out in lens design as a reference for lens design forms
 People who are selftaught in optics to a certain degree, and want to get more familiar with lens design forms
Who this guide is NOT for
 People who want a quick fix to a specific optics and lens design problems
 People that don’t need to be wellrounded as a lens designer
 People that are happy with letting the computer software do the work for them
 People that are okay with becoming a lens design zombie
Who am I?
Hi, I’m Kats Ikeda, Ph. D, and my expertise is optical lens design, nonimaging / illumination lens design. I have enjoyed a lot of product development based on optics and lens design. I love nerding out on optics and lens design talk.
Background
Optical lens design is made up of many disciplines, one of which is imaging lenses. Depending on the desired outcome, which are the specifications of the lens, there are similarities and differences for each lens design. The similarities can be grouped up as a Lens Design Form.
A lens design form can be a combination of positive optical power and negative optical power lens elements that share characteristics with each other.
Part of the lens design form is from the configuration, as it is important to be able to see the configuration of the lens and decipher what it means. That includes what lenses to use where, the spacing of the lenses, the number of lenses, the material of the lenses, the optical power of the lenses, and so on.
Another part of lens design form is from the history of lens design, as there are different needs in different eras, and the technology associated with each era is different. The needs are fulfilled by different lens designers of the day, with the technology available to them at the time. New lens design forms are therefore approved by other lens designers if they use them in their lens designs.
Each topic will have the following format:
 History and background of each topic: For example, in the Tessar chapter, I talk about how the lens design can be viewed as a Protar and Unar or that optically we can think of the Tessar as an evolution of the triplet.
 Essentials for lens design, what you need to know: to proceed with the lens design. For example, for the Cooke Triplet, we need to know how the aberrations are controlled.
 Where you can use knowledge used in other lens designs: For example, there is a link between Double Gauss lenses and stepper lenses. Also, there is a relationship with Zoom lenses and Retrofocus lenses and Telephoto lenses.
 Tips and tricks: We look into useful techniques we can use to facilitate the lens design process.
 Master the specsheet: Clues in the specification sheet (or specsheet) to figure out when to use a lens design form.
 Realworld examples: Actual lens design specs designed by me or from patents. Lots of images and ray diagrams of the lens design form. (The patented examples belong to the inventor and assignee of the patent, and my design is for educational purposes only. I can’t be responsible for any legal ramifications if you use any of these designs in a product you’re going to sell)
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Pattern recognition
On a basic level, when we can understand the lens design form of a lens design, we can look at the various properties of the lens and we can figure out the if the performance is good or not, or if the rays are behaving the way that we want.
Some of the lens design properties I’m talking about are the lens diagram, the type of glass used, and the rays passing through the lens system.
For the lens diagram, we can get an idea of the shape of the lens, even without the hard numbers for the curvature of the thickness of the lenses and the spacing between the lenses. We can see if a lens is a strong convex and causing unneeded problems, or if a combination of lenses is needlessly either too close or too far away to be meaningful in the lens design.
For the type of glass used, we can get an idea if the combination of glass types are beneficial to decreasing the aberrations if they are cancelling out to give an overall good performance.
For each surface, we can see if the rays passing through the surface is bending back and forth in a needless way, or if there are strong refractive surfaces that hinder the performance of the lens.
With a good eye for lens design, we can think “Hmmm, this lens is soandso, that surface is soandso” and really get a feel for the lens design simply by looking at the lens diagram and the rays, and with an idea of the refractive indices and dispersion of the glass. No complex calculations, no expensive software, no building and testing the lens performance.
With this “lens design pattern recognition”, it is possible to decipher more complex lens surfaces, even with aspherical lenses! It’s like a secret weapon.
With a good eye for evaluation of the aberration correction of any lens design form, we can use that knowledge to improve the lens design by further improving the aberrations or decreasing the aberrations and achieve a better lens design.
Looking at the above statements, you can see that I really value the ability to be able to “look” at a lens and figure out its good parts and bad parts. The lens diagram and ray paths are necessary to do this. Before computeraided lens design, the lens design giants would rely on their intuition and eye for lens design. For some historic lens designer, sometimes pattern recognition trumps even aberration theory.
For example, lens design genius and lens design hero of mine, Ludwig Bertele knew about aberration theory, but supposedly never used the theory for his lens designs. Bertele relied on ray tracing the lens system, looking at the performance, and changing the shape/index/thickness of the lens design to get better performance. The fact that Bertele invented many innovative lens designs during his time (in the 1930s) with this method is nothing short of extraordinary and speaks to his lens design intuition and lens design ability. This is called the ray tracing method or the change table method, the latter because the performance of the lens after raytracing would be displayed in a table, and the lens designer would look at how the little changes to the lens design affect this performance table.
Another lens design hero of mine, Nikon’s Zenji Wakimoto, also used ray tracing and pattern recognition for his many lens designs. In his Nikon days, he designed the Nikkor 50mm F1.4 lens and other lenses like the NikkorN Auto 24mm F2.8, NikkorSC 8.5cm F1.5, and NikkorPC 10.5 cm f/2.5. Wakimoto eventually designed the Ultra Micro Nikkor, while at Nikon. The Ultra Micro Nikkor had extremely high resolution and was the start of stepper lens design. Zenji Wakimoto also invented many innovative lens designs without computeraided automatic lens design optimization and used ray tracing and the change table. Much like Bertele, he would change the lens design slightly, raytrace the optical lens system, look at the results of raytracing and change the lens again.
As an aside, both Bertele and Wakimoto didn’t write any books or academic papers and document their findings. They seemed to be more interested in actually doing lens design than writing about it. Shame for lens design nerds like me.
In any case, these lens design geniuses don’t use software optimization and produced many many innovative lens designs. That’s not to say we shouldn’t use computers for our lens design, that would be ridiculous. But I do think that we can incorporate their philosophy of pattern recognition and “looking” at a lens design to make the process easier for us. Maybe we can’t get to the level of Bertele or Wakimoto as far as intuitive lens design, but we have the history of their lens designs and lens designs inspired by the many lens designers since then, on our side.
It seems as though pattern recognition, an ability that humans and not machines possess, is a good way to pursue lens design, as demonstrated by my lens design heroes. In spite of that, a lot of lens design books and textbooks that I have read rely heavily on the derivations of mathematical equations, without actual lens design data and figures and graphs, especially lens design diagrams. A schematic diagram is not good enough in my opinion, I want to see the rays passing through the system.
It’s a lot of work to put data together, and I’ve tried to do that in this guide. I list many lens design forms, illustrate the lenses and show the rays passing through the lenses.
Let’s get to the meat of the Guide!
Imaging lenses: Classic imaging lenses and the dawn of lens design
To start off, we look at classic lens design forms in the dawn of lens design. These classic lenses may look simple given the more complex lens designs we have today.
Lens design was more conceptual in the early days of lens design since computational lens design had not been developed. Ray tracing and aberration theory was invented during this period. By examining the examples of relatively simple lens designs, it is actually easier to dissect why the lens designs are the way they are.
The history of lens design is an evolution of new lens designs given the concepts and advancements in technology, and it’s great to start where it all began.
The singlet lens: The first lens that deserves your attention
1. History and background
In all honesty, a singlet lens looks really simple.
You may be thinking,
“What? Designing a singlet lens? A piece of cake!”
which is true, of course, but I want to dig a bit deeper since everything is simplified in a singlet lens. So much so that the merits and demerits of the lens are clear, and we can use this knowledge to our advantage in the bigger picture of lens design, and by proxy more complex systems. After all, a multilens system is a string of singlets when you think about it.
“Lens”, named from lentils, can be traced back to the 7th century, may or may not have been used as a burning lens, may or may not have been used as a reading lens, but by the 13th century spectacles were made, and in the 16th century optical microscopes and telescopes used lenses.
Truthfully, a singlet it the simplest lens form there can be, and it doesn’t need any explanation for even a novice lens designer. But to truly understand the singlet, and its limitations is the first step to understanding lens design.
Let’s take a look at the concepts.
2. Essentials for lens design
You might have seen the lensmaker’s equation early as high school, and this is the essence of the performance of the lens.
For a thin lens,
$$
\frac{1}{f} = (n1) \left[ \frac{1}{R_1}\frac{1}{R_2} \right].
$$
Where \(f\) is the focal length, \(n\) is the index of refraction, \(R\) is the radius of curvature of the lens (enumerated by surface).
For a thick lens with some thickness \(d\),
$$
\frac{1}{f} = (n1) \left[ \frac{1}{R_1}\frac{1}{R_2}+\frac{(n1)d}{n R_1 R_2} \right].
$$
Where \(f\) is the focal length, \(n\) is the index of refraction, \(R\) is the radius of curvature of the lens (enumerated by surface).
For imaging properties, we can use an even simpler equation like the following:
$$
\frac{1}{f} = \frac{1}{d_1} + \frac{1}{d_2}
$$
Where \(f\) is the focal length, \(d_1\) is the distance from the object to the lens, and \(d_2\) is the distance from the lens to the image.
As simple as the singlet is, there are multiple lens forms associated with the singlet.
From left to right: Positive rear meniscus lens, positive planoconvex lens, biconvex lens, positive convexplano lens, positive front meniscus lens.
From left to right: Negative front meniscus lens, negative planoconcave lens, biconcave lens, negative concaveplano lens, negative rear meniscus lens.
3. Where you can use knowledge used in other lens designs
Since every other lens design form is a combination of multiple singlets, there isn’t too much to say here. What you know about the singlet applies everywhere. For example, you may see any number of combinations of the positive or negative lenses in a lens system.
It’s good to know the limits of a singlet, because we can then know when a singlet isn’t enough in a lens design.
4. Tips and tricks
The singlet is a lens system with a single positive lens, and the stop is on the surface of the lens. As simple as this lens is, it has characteristics that teach us the advantages and disadvantages of a single lens.
 Although the spherical aberration cannot be fully corrected to zero, the smallest possible spherical If the radius of curvature ratio of the front radius to the rear radius is 1:6. This means that any other configuration that you are thinking of is at the expense of the spherical aberration.
 The coma is also close to zero.
 The longitudinal chromatic aberration cannot be corrected. The positive lens only has positive power, which causes the chromatic aberration depending on the index of refraction of the material.
 Astigmatism can only be controlled by changing the size of the aperture stop. Thus the speed of the lens is determined by how much Astigmatism we can allow.
 The Petzval sum, i.e. the field curvature, cannot be fully corrected. The Petzval sum is dependent on the sum of the lens power divided by the index of refraction. A singlet lens has only one lens power and one index of refraction, so the Petzval sum is always non zero.
 The distortion can’t be corrected because there is no way to achieve symmetry about the stop in the design with one lens.
 The transverse chromatic aberration can’t be corrected because there is no correction with one material.
5. Master the specsheet
Again, there isn’t too much to say here.
\(R_1\), \(R_2\), \(t\).
If you want to go a level deeper, I recommend trying to draw the lens by hand. Don’t underestimate this step, you can learn so much from the application, even if it is as simple as a handdrawing.
6. Realworld examples
Camera Obscura
[Camera obscura(https://en.wikipedia.org/wiki/Camera_obscura) the first camera system.
The Camera Obscura literally means “dark room”, and is said to be named by Johannes Kepler.
I made one during summer vacation one time, and I used a pinhole instead of a lens made of glass.
Kodak Hawkeye
The famous lens for a camera that I know is the Kodak Hawkeye, and it was riddled with aberrations.
FujiFilm Quicksnap
A more modern example, is Fujifilm’s QuickSnap(写ルンです).
(via Fujifilm)
The QuickSnap is interesting because if we look at the innards of this camera, the film is curved a bit on the image plane (far left). This accounts for the field curvature, since it cannot be corrected with a landscape lens, we curve the image instead.
(via Fujifilm)
Bonus: Two types of singlets, the telescope objective and landscape lens
There are basically two types of singlet lenses.
One, the telescope type objective, that I explain in detail in my Ultimate Guide to Spreadsheet Lens Design.
(Look at 4. Tips and tricks above for the rundown on the advantages and disadvantages)
Two, there is the landscape lens, used for photography. For a landscape lens, things are a bit different.
Probably the first real camera lens, used in the Camera Obscura, which is basically a box with a lens that formed an image. It was first used for sketching and painting.
There are two different lens forms for the landscape lens. The rear meniscus form in the image above, and a front meniscus form is shown below.
Landscape lenses are solved by determining the minimum field curvature while the coma is zero.
What we can expect is:
 The positive lens and the stop are separated, and the meniscus lens corrects astigmatism and coma.
 However, the spherical aberration is completely out of whack, and the only way to minimize it is to make the aperture smaller.
 The distortion can’t be corrected because of the asymmetry about the stop.
The landscape lens is an excellent example to illustrate that there are multiple solutions to a lens design. Even a lens as simple as the landscape lens has two solutions, called the rear meniscus form and the front meniscus form. If you optimize with a planoconvex lens with the stop in front of the lens Zemax will give you the rear meniscus form. If you optimize with a planoconvex lens with the stop in behind the lens, Zemax will give you the front meniscus form. Optimization from a flat surface, and it can fall to either lens form, depending on the optimization parameters we set in the form of a merit function.
The basic lens design method for a landscape lens is as follows:
 Make a rough meniscus lens and choose the position of the aperture stop
 Take an Fnumber about twice the speed than needed (set F8 if we’re designing an F16 lens)
 Raytrace the FOV at about 70%, use bending to make minimize the aberrations
 Set the optical systems to the desired Fnumber
 Move the stop position to find the best location that minimizes the aberrations
 Finish the design off with a few more points in the FOV, and we’re done
Although the design method is straightforward, it covers the basics and a good rule of thumb to follow, and will be useful when we look at more complex systems.
A few things to note in the design:
 To minimize astigmatism and coma, the stop and the lens are separated.
 The spherical aberration is impossible to correct given this situation. Since the spherical aberration is dependent to the Fnumber of the system, the only way to correct the spherical aberration is to make the aperture smaller and decrease the spherical aberration.
 In both the front or rear meniscus case, since there is only one lens on either side of the stop, the distortion and transverse chromatic aberration cannot be corrected.
Let’s take a qualitative look at the performance of the lens design for the two meniscus forms.
As far as optical performance, the rear meniscus is a bit better.
 The MTF is better since the spherical aberration is smaller
 The surface curvature is weaker
 The distortion is barrel distortion, perhaps more unobtrusive to the human eye
But the front meniscus form is dominant in the singlelens camera for ages.
So what gives? Why are we choosing an optically inferior lens?
 The overall length of the lens is shorter, due to the higher curvature of the lenses. In our example, it is about 20% shorter. Remember, this is the balance of the smallest field curvature at zero coma.
 With the lens on the outside, the stop (and therefore the shutter mechanism) is protected from any outside dirt from the lens itself.
 Aesthetically, the camera has a lens in front, which is more appealing than the peculiar shutter and aperture stop sticking out. We see an aperture stop, normal people see a hole.
 Since plastic lenses were invented, the stronger meniscus curvature for the front meniscus form is no more expensive to manufacture than a weaker curvature of the rear meniscus form, unlike with glass lenses.
The dominance of the front meniscus lens form for inexpensive cameras is a lesson that good optical performance is not always the be all end all of lens design.
Once you can design a singlet telescope objective and a singlet meniscus landscape lens, you’ve entered the gate as a lens designer, in my opinion.
If you want more information on landscape lenses, I have more information with a blogpost I wrote about the history of the landscape lens.
Do you want to design this lens? I have more information with my Ultimate Guide to Lens Design Using Spreadsheets with complete calculations on how to calculate the performance of a lens without complex software, but spreadsheet programs such as Excel.
So much to write about for a simple singlet lens system. To me, this is why lens design is fascinating 🙂
The doublet lens: More to it than meets the eye
1. History and background
There are many types of doublet lenses, but the predominant doublets are achromatic doublets, and they correct the chromatic aberration.
From early design of spectacles and magnifiers, the lenses soon change to telescope objectives.
When we change from magnifiers, which are closerange, and manageable focal magnification, to telescopes, that have long focal lengths, there was a new problem to be solved.
The chromatic aberration.
Sir Isaac Newton famously stated that chromatic aberration correction was impossible for a refractive lens, so much so that the reflective type of telescope, which does not have chromatic aberration, is now called the Newtonian telescope.
Newton is right of course, if we only think of single lenses.
We can think of changing a singlet into a compounded doublet in two different ways:
 as a substitute for nonexisting glass type
 as a method to introduce a surface with a smaller refractive index change to control the ray paths
In the 18th century, the achromatic doublet saw a lot of development from George Bass), John Dolland, and his son Peter Dolland as well.
2. Essentials for lens design
For achromatic doublets, we need to choose lens material wisely.
A good place to start is K1 and F1.
Depending on the system, we may need a more thorough examination of the lens choices.
Hans Harting studied the different combinations for achromatic lenses and even has a table of his studies.
Summarized below.
The Harting method of choosing materials for zero chromatic shift
We have two materials \(a\) and \(b\), with an index of refraction \(n_a\) and \(n_b\), and index difference with wavelength as \(\Delta_a\) and \(\Delta_b\).
We make a twolens optical system with spherical refractive surfaces that are close together, like a telescope.
This optical system has a focal length of 1.0 to make things easier to calculate. The lens should have rays at low angles coming from an infinite conjugate, without spherical aberration and coma.
In order to have zero chromatic shift, the equations for the four surfaces. Two surfaces for each lens, where lens \(a\) has \(R_1\) and \(R_2\), and lens \(b\) has \(R_3\) and \(R_4\). Note that \(R_2 = R_3\) for a cemented doublet.
$$
\frac{1}{R_1} = \frac{n_a}{2} \left[ \frac{\chi_1}{\psi_1} + \frac{\psi_1}{n_a – 1} \right] \\
\frac{1}{R_2} = \frac{n_a}{2} \left[ \frac{\chi_1}{\psi_1} – \frac{\psi_1}{n_a – 1} + \psi_1 \right] \\
\frac{1}{R_3} = \frac{n_b}{2} \left[ \frac{\chi_2}{\psi_2} + \frac{\psi_2}{n_b – 1} + \psi_1 \right] \\
\frac{1}{R_4} = \frac{n_b}{2} \left[ \frac{\chi_2}{\psi_2} – \frac{\psi_2}{n_b – 1} + 1 \right]
$$
where
$$
\psi_1 = \frac{\nu_a}{\nu_a – \nu_b} \\
\psi_2 = 1 – \psi_1
$$
and
$$
\nu_a = \frac{n_a – 1}{\Delta n_a} \\
\nu_b = \frac{n_b – 1}{\Delta n_b} .
$$
\(\psi_2\) is a quadratic function
$$
\alpha {\psi_2}^2 + \beta \psi_2 + \gamma = 0
$$
and
$$
\psi_1 = \frac{1 – (n_b + 1) \psi_2}{n_a + 1}
$$
with
$$
\alpha = (n_a +1)^2 \cdot C + (n_b +1)^2 \cdot A \\
\beta = (n_a +1)^2 \cdot E – (n_a +1)(n_b +1) \cdot D – (n_b +1) \cdot A \\
\gamma = (n_a +1)^2 \cdot F + (n_a +1) \cdot D + A \\
A = \frac{n_a (n_a +2)}{\psi_1} \\
C = \frac{n_b (n_b +2)}{\psi_2} \\
D = 2 n_a \cdot \psi_1 \\
E = 2 n_b (2 – \psi_2) \\
F = \left[ \frac{n_a}{(n_a – 1)} \right] ^2 \cdot {\psi_1}^3 + \left[ \frac{n_b}{(n_b – 1)} \right] ^2 \cdot {\psi_2}^3
$$
A piece of cake, right?
In practice, it is easier now with a pocket calculator or a spreadsheet.
3. Where you can use knowledge used in other lens designs
This lens is mostly used in long focal length systems with longitudinal chromatic aberration.
Here are some examples:
 The telescope objective
 Telephoto lenses with longer focal lengths. They are typically placed in the front
 Zoom lenses that cover a wide focal range including a long focal length on the tele side, or a telezoom
 Laser applications where focal length changes due to the wavelength need to be corrected
4. Tips and tricks
Splitting the cemented doublet
When you split the cemented doublet to a separated doublet, be careful of the rays in between the two lenses, as you may see an abrupt change in the refraction angles.
In some cases, the rays may even have total internal reflection (TIR) if we’re not careful. In this case, we need to decrease the angle of incidence by manually changing the curvature of the lens.
The nonachromatic doublet
As stated above, we can think of changing a singlet into a compounded doublet in two different ways:
 as a substitute for nonexisting glass type (which is the achromatic doublet)
 as a method to introduce a surface with a smaller refractive index change to control the ray paths
There is a secret doublet besides the achromatic doublet that has nothing to do with colour. For example, the cemented doublets in a Tessar lens can correct astigmatism, field curvature, and other fieldrelated aberrations.
The doublet here is far away from the stop, and the rays pass through obliquely far from the optical axis.
These oblique rays have different angles of incidence at the cemented surface, depending on where the rays are. The rays in the upper part of the lens are refracted more than if it were a single lens, and the effect of the cemented surface is asymmetrical to the oblique beam.
This doublet for the Tessar:
 corrects the zonal spherical aberration
 corrects astigmatism at higher fields of view
 corrects the field curvature
More information on the Tessar is available at a future part of the Guide.
5. Master the specsheet
A simple rule of thumb is when you realize that you need an unrealistic glass to achieve the desired performance, consider a twolens system. We can use this principle for multilens systems as well, since we can break down the lens system into components and look at one lens element and change it to a doublet.
An achromatic doublet for a telescope is used when the focal length is relatively long, and the longitudinal chromatic aberrations become a problem. A singlet cannot correct the colour.
6. Realworld examples
Edmund Optics
There are a lot of achromatic doublet examples by Edmund Optics. Have fun and plugandplay with my spreadsheet for achromats. They also have tips on why we use an achromatic lens.
Rear element of Tessar and other objectives
Front element for telescope objectives
Here are some examples of some telescope objectives and their optical performance. All have very good colour correction, and good spherical aberration correction. The field curvature and distortion are nonexistent since the FOV angle is 1~2 degrees.
Telescope Gauss Objective
Telescope Fraunhofer Objective
Telescope Steinheil Objective
For more information on the telescope, I have a link to it at a future point in this guide.
For more information on the colourcorrection process of doublets, I have a complete blogpost on the topic here.
If you want more information about lens design process of the doublet, I suggest you check out My Ultimate Guide to Lens Design Using Spreadsheets, where I decode the lens design process of achromatic doublets and other lenses with Excel/spreadsheets.
Petzval lens
Although the Petzval lens on its own isn’t as used much today, we can find the lens type inside a more complex lens system like a zoom lens, so knowing the properties of this lens design form is important, in my opinion.
1. History and background
By Szőcs TamásTamasflex – Own work by uploader – Lens Photo by Андрей АМ, CC BYSA 3.0, Link
The Petzval lens was invented by Austrian mathematician/physicist Joseph Petzval, in 1840 for Voigtländer.
In a conference in Paris in 1839, Vienna University professor Andreas von Ettingshausen was in attendance.
François Arago presented photography techniques based on Louis Daguerre’s Daguerreotype lens , a lens designed by Charles Chevalier . This is now called the landscape lens for wide angle photography. In contrast, Petzval lens is used as a fast portrait lens.
Until the invention of the Petzval lens, people sat in the blistering sun, waited for a song to finish during the exposure. The exposure time is 30 minutes, enough to make you cry, although that wouldn’t be caught on the photo.
Ettingshausen went back to Vienna and told Petzval, an associate professor he knew could design telescope objective lenses, about the need for a fast lens.
Petzval obliged, and with the help of the Austrian army, he made them do calculations for the lens design.
The calculations involved a very scientific approach, with refractive data from multiple wavelengths. The previous generation had a trialbyerror approach, making lenses and then measuring them.
There is a lot of mystery as to how Petzval was able to design such a lens, as the manuscripts are lost. But the fact is, Petzval was able to design a lens that had an fnumber of F3.5, in an era where F8 was the norm.
Since making a fast Fnumber lens was of great importance and had a tremendous military application, Voigtlander and Petzval had the Austrian army at their disposal. Legend has it that there was an army (pun intended) of military personnel that did calculations on Petzval’s behalf. This military personnel dealt with calculating the projectiles of missiles and bombs on the spot in the battlefield, so their prowess was put to good use for ray tracing calculations.
The aftermath story of this lens is just as intriguing as its birth. Voigtlander and Petzval had a nasty legal battle over the rights of this lens, and Petzval actually won. But the patents were only valid in Austria, and when Voigtlander moved their head office to Germany, Petzval’s patents were not valid there. Thus, the majority of Petzval lenses produced didn’t bring in a cent for Petzval himself. He was largely forgotten later in life, and didn’t end up extremely wealthy despite the booming sales of his namesake lens.
In 1859, his manuscripts which documented many years of research were destroyed due to a breakin at his home. This is a shame since the Petzval lens is one of the first pure “lens designs”, as it was made by precise mathematical calculations. My personal opinion is that this is a loss of knowledge on how lens design progressed.
2. Essentials for lens design
 The image of a typical Petzval lens is above, which has two positive colour correction doublet and a large air space between.
 If the front doublet power is \(\phi_1\), and the rear doublet power is \(\phi_2\), the total power \(\phi\) is
$$
\phi = \phi_1 + \frac{h_2}{h_1} \phi_2.
$$  To keep the total power constant, just having the first doublet group gives \(\phi = \phi_1\), but the second doublet group is a positive number, so \(\phi_1\) can be small to get the same overall power. To get the same Fnumber after the addition of the second doublet group, the beam entering the first doublet group is wider, but the overall focal length doesn’t change. Since the doublets are far apart, the ray height \(h_2\) is small, so ray height \({h_1}^4\), which governs the spherical aberration, doesn’t affect the system much.
 The Petzval lens itself is not an anastigmat, but can be designed as one, in order to compare with say, a triplet lens.
 The Petzval type lens has a large Petzval sum (the irony!), so the disadvantage is that astigmatism and the field curvature can’t be corrected at the same time.
 However, this disadvantage is directly linked to the fact that the spherical aberration and the longitudinal chromatic aberration can be corrected very well.
 The lens is relatively symmetrical about the stop, so the distortion can be corrected, as well as the coma and transverse chromatic aberration.
 The stop is far away enough from the lens so that astigmatism can be corrected, Therefore, the field curvature can’t be corrected, meaning that the field of view can’t be increased too much. Thus a narrow field of view and portrait lens usage is recommended.
3. Where you can use knowledge used in other lens designs
I feel that the Petzval lens is one of the first lenses to incorporate the distance between lenses as a feature to correct the performance of the lens design. That is why I think that the Petzval lens is an important lens to study, even though the lens design form itself may be outdated, the concept can be found in many places, and these concepts can still be used today.
Zoom lenses
Some Petzval lens principals, separating lens groups in zoom lens design or telephoto design.
4. Tips and tricks
 The spherical aberration is corrected with the front lenses, and the longitudinal chromatic aberration can be corrected by the rear lenses.
 The coma and distortion can be corrected by having opposite magnitude in the front and back groups, balancing about the stop.
 The lateral chromatic aberration can be corrected with the front and back groups.
 The Petzval sum, and incidentally the field curvature cannot be corrected since the positive element and negative element in each of the lens groups are not separated.
5. Master the specsheet
A Petzval lens design is an extremely good choice for when the FOV is manageable. The lens gives low spherical aberration, low coma, low chromatic aberration, at a very fast speed (large aperture, small Fnumber).
Also, make sure that the field curvature does not play a large part in the performance.
6. Realworld examples
Satellite optics
Satellite optics still use Petzval types with a field flattener lens. The high manufacturability and relatively low cost along with the high contrast of this lens make it a preferable lens type.
Microscope lenses
Some low magnification microscope lenses use the Petzval type.
Lomography revival of Petzval lens
Wow, that is some swirly bokeh.
Lomography has a fun site on the Petzval lens.
I took a look at the US patent 2500046A and plugged it into Zemax for fun. What do you see?
(click to enlarge)
(Top, from left to right: Lens layout, Ray fan, Spot diagram)
(Bottom, from left to right: Field curvature, Distortion, Longitudinal aberration, Lateral colour)
 The spherical aberration is well corrected
 The coma is well corrected
 The chromatic aberration is well corrected
 The field curvature and distortion are not noticeable because the FOV is relatively small
If you want to look at photos of this lens, you can find them here.
Cooke triplet and anastigmat lenses
1. History and background
Designed by H. D. Talyor in 1893 (GB 22,607), this lens is the perfect lens, from a 3rd order aberration theory perspective. Although the spherical aberration and Coma were corrected with other lens forms, the triplet was the first lens to correct the other monochromatic aberrations like astigmatism, field curvature (Petzval sum), and distortion. Also, the chromatic aberrations like longitudinal chromatic aberration and transverse chromatic aberration are also corrected.
The name “Anastigmat” literally means “nonastigmatic” lens, since it corrects astigmatism and its cousin the field curvature. Which is funny, because astigmatism means “nonstigmatic”), where stigmatism is “an imageformation property of an optical system which focuses a single point source in object space into a single point in image space”.
I love the word Anastigmat. That means that “Anastigmat” can be translated as a “nonnonstigmatic”. I guess “Stigmat” didn’t really have a good ring to it. Nowadays, almost all lenses have the basic aberrations corrected, so we don’t have the need to call them Anastigmats anymore. Sad.
Back to the triplet, three lenses are the absolute minimum that can correct the 3rd order aberrations, namely the 5 monochromatic 3rd order aberrations and the 2 chromatic aberrations. Gauss had already shown some interesting solutions with three lenses, but it was H. D. Talyor that designed a flat field lens design with conventional glass. At the time, anastigmats (as they were called) were thought to only be correctable with the newer glass of the time. Petzval showed that spherical aberration and chromatic aberrations can be well corrected with conventional glass, and the potential was there to make the lens an anastigmat.
It’s interesting that Taylor was led to this design by thinking about how to make the Petzval sum zero. We can do this with a positive lens and a negative lens of equal power. But the asymmetry in this system would lead to lateral chromatic aberration and distortion. So he split the positive element in two and sandwiched a negative lens in between. It’s fun to think he also tried other combinations, like negativepositivenegative lens combinations.
2. Essentials for lens design
 The second lens (negative power) and the third lens (positive power) are separated, so the Petzval sum can be corrected.
 By adding the first lens (positive power), there is sufficient symmetry about the stop, so distortion and transverse chromatic aberration can be corrected.
 The first lens (positive power) and the second lens (negative power) correct the longitudinal chromatic aberration.
 Spherical aberration, coma, and astigmatism, can all be corrected by changing the radius of curvature of the 3 lenses and 6 surfaces.
The triplet can balance all 3rd order aberrations, but the balance is tricky. Since we are using the minimum lens surfaces for aberration correction, a change in any surface affects every aberration. A predesign step balancing these aberrations helps quantify the process.
More indepth information available on my epic piece on The Ultimate Guide to Lens Design using the Classic Spreadsheet Method. I go into a detailed but simple calculation to make this happen.
3. Where you can use knowledge used in other lens designs
I think the triplet is the perfect lens to explore lens design. Although it is not the first photographic lens designed, almost all modern lenses can be traced back to the triplet. Therefore, studying the triplet carefully can provide the basis for most modern photographic lens design. By understanding the shortcomings of the Cooke triplet we can make strategic improvements to out lens design.
The triplet design was revolutionary and spawned a lot of lens designs afterward. Most notable is the Tessar lens design form. Historically, the next evolution was the Ernostar lens design form, and its direct evolution, the Sonnar lens design form. Perhaps most notable, Double Gauss type can even be traced back to the triplet. Even some retrofocus types can be looked at as a triplet lens with a wide angle converter in the front.
4. Tips and tricks
 Choosing glass: Glass choice is the most important part in lens design, and the triplet is no different. The choice of glass has a direct impact on the optical performance, and namely, the choice of the crown lenses determines the Petzval sum. We can have expensive glass for highperformance triplet lenses. Oftentimes we need to come back to choosing glass after hitting a roadblock in the lens design.
 Schwarzschild solution: We set a fixed focal length, and correct the longitudinal chromatic aberration, and construct three equations for the Petzval sum, and solve for the three values of power. The three equations are total power, Petzval sum, and longitudinal chromatic aberration.
 3rd order aberrations can be corrected for thin lenses at first: It is possible to correct the 5 monochromatic 3rd order aberrations and the 2 chromatic aberrations, and then make the lenses thick.
More detail in my Ultimate Guide to Lens Design using the Classic Spreadsheet Method.
5. Master the specsheet
This is the most basic lens that corrects the chromatic aberration, the spherical aberration, the coma, astigmatism, the field curvature, and the distortion in a reasonable manner.
 Usual focal lengths are 35mm or to 60mm or so, for fullframe cameras, which is about a field of view of 60 degrees to 40 degrees.
 The usual Fnumber is anywhere from F6.3 to F3.5 or so.
 Anything within this range may be perfect for the triplet.
Below is an example of Taylor’s design.
We can see in the bottom middle graph that the spherical aberration looks very well controlled, and the chromatic aberration is also well controlled. The field curvature (bottom left most graph) and the distortion (second to the left graph on the bottom) are also reasonable.
6. Realworld examples
Triplet lenses were almost the default lenses in old cameras. Back in the large format days, the triplet was the standard lens for moderately wide to moderately tele focal length ranges.
In more modern times, compact cameras usually have a Triplet or a Tessar lens.
Even today, some highperformance lenses with extremely good colour correction can be the form of a triplet, like in the US Patent 3486805.
Bonus: A brief story of H. D. Taylor, inventor of the Cooke Triplet
Taylor had developed other optical systems besides the Cooke Triplet, such as telescope objectives, various eyepieces, and the first lens coatings. In 1906, Taylor’s major written work, A System of Applied Optics, was published, followed later by a German translation. This is a very dense, 300page development of the algebraic formulae of Airy and Coddington into a system of optical design.
Taylor was quite negative on the concept of ray tracing, and even presented a talk at the Optical Society titled Optical designing as an art. Taylor contends that the time it takes for ray tracing is so long, and not that different from manufactured testing. He considers this a waste of time for lens designers, who should do more philosophical thinking process than just brute force calculations.
Upon presenting in detail the process of designing the Cooke photographic lens, he uses an algebraic process to illustrate his philosophy of lens design. At the time, the great success of German designers with trigonometrical ray tracing triggered widespread use of these methods. With no computers, this was a very laborious chore for the designer, tracing rays on paper through various oblique angles and positions over many surfaces. Until then, the typical lens design method was iterative, design > assembly of a prototype > shop testing > redesign.
I suppose that Ernst Abbe of Zeiss wanted to save money on prototyping by doing ray tracing, but I can understand Taylor’s point of view that although ray tracing is more exact, it reveals nothing more than physical testing. Since the trigonometric ray tracing process took a huge amount of the designer’s time, Taylor probably felt that conserving the time of the highly skilled lens designer was more important than saving workshop time.
Other speakers noted that only the most skilled designers could utilize algebraic methods to good effect, and that most would have to rely on the slow, mechanical process of ray tracing.
Fastforward to today, we have complex lens systems that are hard to manufacture and test, in combination with the extremely fast computational powers that we have now, the calculations on software dominate the lens design process.
I agree with Taylor in spirit. We need to be more analytical and use our brains in lens design. I hope that more lens designers (including myself) can detach from plowing through the computational nature of optical lens design software and really do some deep thinking, at least from time to time.
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The Tessar lens: A commercial success that started an era
1. History and background
The Tessar is one of the most used lenses in photography. If you’ve been photographing in the film days as I have, then you were bound to have come across the Tessar, even if you didn’t know it.
This lens was developed by Paul Rudolph in 1902, and it is a 3 group 4 element lens. The last two lenses are cemented together.
Like the image above, we have a positive lens, a negative lens, and then a cemented doublet that is a negativepositive combo of lenses. In the original Tessar, the aperture stop is placed between the second lens and the third lens group.
The first Tessars were conservative in their Fnumber, as it started at about F6.3, which is a bit slow by modern standards. However, after incremental upgrades and lens design evolution, the Tessar soon became F4.5, F3.5, and F2.8.
I think that the arrival of the Tessar ushered in the Golden age of Carl Zeiss, and with it a slew of lowcost cameras that had great imaging quality that could be made cheap and compact. I would go as far to say that the Tessar documented most of the 20th century. This is because the patents ran out in 1920, and many many camera makers and lens makers took this design and made variations of their own.
The SchneiderKreuznach Xenar, Rodenstock Ysar, Voigtländer Skopar, are just a few examples of Tessar copies along with minor variations in Kodak Ektar, Agfa Solinar, Leitz Elmar, and Steinheil Culminar.
It’s been over a century that the Tessar was invented, and we can still see Tessar types today.
2. Essentials for lens design
Contrary to popular belief, the Tessar is not an evolution of the Cooke triplet, but a combination of two of Paul Rudolph’s previous design works in the Unar and Protar. Carl Zeiss even says so, that the Tessar has a front group that shares properties with the front group of the Unar, and the back group of the Protar.
This came at a time where glass manufacturing had a revolution (in 1886), and various “new” glasses were produced. In the old ages of glass, high index negative lenses and low index positive lenses were the only choices for a cemented doublet. The new glass made it possible to have low index negative lenses and high index positive lens combinations in the cemented doublet.
Rudolph used this “new” glass at the time to design a newly cemented doublet for the Protar lens which was eventually used for the Tessar as well. The doublet is a combination of a high index positive lens and a low index negative lens, that corrected astigmatism and the Petzval sum, while increasing the spherical aberration somewhat.
Having said that, optically speaking, the triplet > Tessar evolution makes the aberration correction an easier to understand. In particular, if we treat the doublet as a single lens that doesn’t exist as a glass type, it’s essentially a triplet. Looks like I’m going have to ignore Zeiss’ press release copy above 😉
Here’s a summary of what the doublet for the Tessar is for:
 corrects the zonal spherical aberration
 corrects astigmatism at higher fields of view
 corrects the field curvature
3. Where you can use knowledge used in other lens designs
The Tessar in its purest form is an anastigmat, so any lens that needs correction of astigmatism and field curvature lands here. Plus, the Tessar doublet makes it a better performer than a Cooke triplet.
The doublet in the Tessar corrects the field curvature and astigmatism. Strategic use of this doublet is useful in any lens design, but is particularly useful for and retrofocus lenses, telephoto lenses, and zoom lenses.
4. Tips and tricks
The doublet of the Tessar is not for colour correction
I had this preconceived notion that a cemented doublet, boom, that means colour correction. Wrong. This last doublet is not for chromatic aberration correction, nor is it for spherical aberration correction, that we see in the telescope objective.
Let’s take a look at the lens data for the Tessar.
This is the glass for a typical Tessar lens:
 Lens 1 (positive): n=1.59, v=61.3
 Lens 2 (negative): n=1.58, v=46.5
 Lens 3 (negative): n=1.52, v=51.5
 Lens 4 (positive): n=1.61, v=56.7
The lens data shows that the Abbe number for the doublet is 51.5 and 56.7, which are too close to effectively correct chromatic aberration. From thin lens design and colour correction studies of the achromatic doublet, we find that for a typical doublet used for a telescope objective this won’t give us colour correction.
For spherical aberration correction, the index of refraction of the negative element of the cemented doublet lens should be larger than the positive element. In fact, the Tessar has worse correction of the spherical aberration in the center of the lens compared to the Cooke Triplet.
Generally speaking, there are two solutions for the Tessar lens. High index glass in the doublet for good zonal field curvature correction, and lower index for a more Cooke triplet like spherical aberration correction.
The Tessar patent by Paul Rudolph
The patent of the Tessar USP721240 has the stop in between the negative lens and the doublet. Zeiss had a large monopoly on this type of construction, because Rudolph’s patent was very general. He only had one claim.
The Tessar was heralded as a superb lens compared to the triplet, with lower distortion, beautiful bokeh, and sharp image at the focus. With the marketing copy of “Das Adlerauge Ihrer Kamera – The Eagle Eye of your Camera”, many photographers worldwide used this lens.
5. Master the specsheet
Let’s take a look at the Tessar again.
The doublet is far away from the stop, and the rays pass through obliquely far from the optical axis.
These oblique rays have different angles of incidence at the cemented surface, depending on where the rays are. The rays in the upper part of the lens are refracted more than if it were a single lens, and the effect of the cemented surface is asymmetrical to the oblique beam.
Any application that needs this oblique ray correction whether it is the higher angles in a retrofocus lenses, or the field curvature of a telephoto lenses, or the variator and/or other lenses in a zoom lenses is up to us to decide, upon close observation of the system.
6. Realworld examples
We can see the Tessar for a lot of older cameras, typically of fixed lenses. These fixed lenses were relatively cheap to make and still offered high performance. Perfect for the compact camera.
The Zeiss Ikon Contessa – A classic compact film camera. Simple to use, like a pointandshoot of the day.
The Rollei 35 – This was an extremely small camera, the film seems so large compared to the camera, and the lens would collapse down to this compact form. Truly a pocketable camera. There were the triplet and Sonnar variants for this camera, but the Tessar was the most popular.
Minox Camera with a Tessar – Similarly a pocket camera, but with a little more luxury. Looks like the Contax T, but that one has a Sonnar lens.
Kyocera borrowed the Contax brand name, and hence had some lenses with the Carl Zeiss branding, like the Tessar, Sonnar, Planar, Biogon, and Hologon. This lens for their compact camera the Tproof, mounts the Tessar.
Teletessar 4/85 – A tele lens for Leica M mount cameras. Modern lenses still use the Tessar form, and Zeiss is really the only one who can use the name. This lens is a tele lens for rangefinder cameras, and the contrast is said to be superb. It is a wonder that classic designs can do with the advancements in modern technology like new glass types and advanced antireflective coatings.
(via Zeiss)
The Original Olympus Pen – Although not a Tessar per se, the same design as the Tessar was used in other cameras as well, the Olympus Pen (old version) was a halfcamera, meaning the size of the film was half of fullframe. The Tessar lens variant was perfect for this camera, there is high performance in a small package that was the concept of this camera.
(via Olympus)
Below is a camera that I found at my grandfather’s house. It is about the size of a large format but the film size and proportions aren’t sold today. I’d like to be able to use the lens for photography some day. I’m thinking maybe stick a roll film adapter to it, or to take the lens and fix it on a 4×5 large format camera.
Quick note about Paul Rudolph
Paul Rudolph) was the man behind the Tessar lens. He was a lens designer for Carl Zeiss, and is also famous for designing the Double Gauss lens. Later in life, he joined Hugo Meyer and designed the Plasmat variations of cine lenses.
Coffee break: Higher order aberrations
So far, we only had to deal with the third order aberrations. However, past the Aplanatic lenses and Triplet lenses, we need to think about higher orders of aberration than simply the 3rd order aberrations.
There are ways we can use the higher order aberrations to balance the overall performance of the lens.
What are higherorder aberrations?
The Taylor expression of the sine of an angle is as follows:
$$
\sin u = u – u^3/6 + u^5/120 …
$$
Taking the first term, or \(\sin u = u\), first order or Gaussian optics like focal length, and has no aberration. Taking the second term, \(\sin u = u – u^3/6\), is the basis of 3rd order aberration theory. Taking the third term, up to \(u^5\), is 5th order aberration theory. 7th order… you get the picture. Take the entire sine function, and you’ve considered the aberration completely.
Take a look at the spherical aberration, for example.
If we consider the 5th order aberration, we can balance it with the 3rd order aberration.
If we consider the 7th order aberration, we can balance it with the 3rd and 5th order aberrations.
The question becomes, how to generate these higher order aberrations to optimize performance. Here are some examples:
 Use cemented surfaces
 Use a concave surface where the marginal rays are closing in on the optical axis
 Use an “air” lens (LuftLinsen)
 Place a low curvature surface where the rays enter at a large angle
 Use aspherical surfaces
Schematically, if we fully correct the spherical aberration with 3rd and 5th order terms, we get the maximum spherical aberration at 70% of the marginal rays. Likewise, for full correction using 7th order terms, the maximum is at 90%.
Imaging lenses: The evolution of lens design that leads to diversification
Next, we look at the evolution of lens design forms. In actuality, there isn’t a real boundary line between “classic” lenses mentioned before compared to the lenses to follow, but I’ve grouped the lens design loosely with preanastigmats to postanastigmats. Anastigmats can be translated as lenses that correct all of the aberrations. (I realize that the Cooke triplet and the Tessar are anastigmats, so my groupings fail in that sense)
The following use cases are based on the “needs” that the lenses need to fulfill.
For example, the mirrorflap of Single lens reflex (SLR) cameras, the chase for bright Fnumber lenses, the need for resolution due to better film emulsions, wider field of view (FOV), compact but long focal length lenses, and even a combination in wideangle lenses with the mirror of an SLR.
The Ernostar lens: Evolution from the triplet still used today
1. History and background
The Ernostar is an evolved triplet lens that was designed by Ludwig Bertele. The name Ernostar is partly taken from the company that Bertele worked at the time, Ernemann. Ernemann made cameras and lenses until Carl Zeiss formed Zeiss Ikon by merging Ernemann and other notable camera and lens companies like ICA, Goerz, and ContessaNettel AG in 1926.
2. Essentials for lens design
In one of Ernemann’s last photographic lenses, Bertele, in 1925, observed the performance and shortcomings of the triplet lens and tried to increase the aperture and speed of the lens. Let’s take a look at the Ernostar, break it down piece by piece, and try to look into Bertele’s mind when he was innovating photographic lenses.
First thing, Bertele took a clever method to make the entrance pupil larger, and thus making the lens speed faster. Bertele added a positive lens in the front of the triplet. The resulting lens had an Fnumber of F2, which for 1920’s standards was a super fast lens. Although the speed is fast, spherical aberration is well corrected.
What happens when a positive lens is added to the front of the triplet? We see a bit of imbalance about the stop, as there are now two positive lenses in front of the stop as opposed to one positive lens behind the stop. This increases the overall optical power of the lens system about the stop, and the balance of distortion is broken, making distortion correction more difficult.
Also, perhaps less obvious, the large asymmetry about the stop, which has more positive power in the front of the stop compared to the back of stop, caused coma aberration.
So there we have it, this lens is faster than its predecessors, while retaining good spherical aberration correction, but cannot correct for coma distortion quite as easily. This means that for the field of view for a normal lens, let’s say about 40 to 60 degrees FOV (35mm to 55mm or so focal length for 35mm format), coma and distortion could not be ignored, so it did not become the defacto standard lens of its time.
However, for longer focal lengths, the coma and distortion subside for a very usable lens. I have examples of several Ernostar types with moderately long focal lengths below.
3. Where you can use knowledge used in other lens designs
Longer focal length photographic lenses. Be careful, though, after a certain focal length, maybe 135mm or so (for 35mm format), the longitudinal chromatic aberration will start to kick in.
4. Tips and tricks
The main improvement of the Ernostar compared to the triplet is the faster speed caused by the addition of a positive front element. By adding the positive lens to the front element, the second positive lens, the lens behind the first positive lens is changed from a conventional positive lens to an Aplanatic shape to minimize the spherical aberration.
In order to correct the distortion as much as possible, the positive lens behind the stop and last element of the lens system is moved yet further away from the stop so that the power of the lenses about the stop is balanced a bit. By balancing the power about the stop, it is possible to correct for distortion.
Without antireflective coating technology in the 1920s, the Fresnel lossreflectionandtransmissioncoefficients) of the lens surface, namely the glasstoair and airtoglass surfaces caused a loss in transmission and the contrast decreases. Also, the reflections caused more flare as well, further decreasing the contrast of the image. With today’s technology and superb antireflective coatings, the Ernostar type can be made with very high contrast.
5. Master the specsheet
Let’s take a look at the performance of a typical Ernostar lens:
(click to enlarge)
(Top, from left to right: Lens layout, Ray fan, Spot diagram)
(Bottom, from left to right: Field curvature, Distortion, Longitudinal aberration, Lateral colour)
Very small distortion, good spherical aberration correction, and a manageable field curvature since the FOV is not too large.
6. Realworld examples
A slightly old lens from Nikon, built in the late 1940s, after World War 2, in “occupied” Japan.
(via Nikon)
This lens comes at a time when there were very few glass types post WW2, and there were great advancements to make the lenses just as good as the German lenses prewar, that had access to many more glass types.
(via Nikon)
Another more recent example of this lens is a lens design by Miyazakisan, of MSOptical, the Aporis 135mm F2.4.
Miyazakisan took this classic lens design and souped it up with the highest performing glass we have now, as evidenced by the selection of fluorite lens (CaF2) in the front element. This decreases the longitudinal chromatic aberration while still retaining the focusing power, and is easier to correct with the following elements.
Sometimes simple lenses get the job done. You don’t necessarily need your 24 element lens with extremely low dispersion glass topped with aspherical lenses to enjoy photography. It’s a word of caution to for what a lens designer thinks is a “good” lens design.
The Sonnar lens: Bertele’s genius in lens design
1. History and background
Before the Sonnar, the Tessar type lens was the standard lens of its time. Many minor variations of this lens existed for photography lenses. Typical Fnumbers could be F4.5, F4, F3.5. Very rarely F2.8.
People always wanted more speed in their lenses. This was because film emulsions at the time required longer shutter times for exposure because the ISO of the film was low, something like ISO25 or so, sometimes less. Since cameras were evolving from large format large box cameras on a tripod to handheld cameras, faster lenses made for shorter shutter times which helped with handheld blur.
The Ernostar gave improved spherical aberration for larger apertures, but there was large asymmetry that caused coma and distortion. To keep the performance high with the Ernostar, it was used more for longer focal lengths, long enough that the lenses were not normal lenses.
The Ernostar also had four separate lens elements, which have 8 glasstoair surfaces, and caused transmission loss and contrast degradation of the image. Antireflective coatings as we know them today were invented in 1935, and weren’t available to the public until after WWII, since it was largely a military secret then.
Bertele improved upon his Ernostar, and tried to decrease the number of glasstoair surfaces by replacing the air between the second positive lens and the negative lens with glass.
Take a look at the Sonnar on the right. It has 6 lens elements, two more elements than the Tessar, and two more lens surfaces as well. However, the number of glasstoair surfaces are the same. This means the contrast is essentially the same from a transmission standpoint.
Bertele later evolved this design into the F1.5 Sonnar, which has a cemented triplet group after the stop.
Personally, I really like the allaround performance of the F2 Sonnar, and the F1.5 seems more like a specialty fast lens to me (which I also love, BTW).
After the emergence of SLRs, the Double Gauss made the Sonnar obsolete due to the short back focal length of the Sonnar, and the advancements in antireflective coatings decreased the advantages of the Sonnar.
2. Essentials for lens design
First, the Sonnar generates higher order aberrations to correct the aberrations at higher field.
For the F2 Sonnar, Bertele added a doublet in the final group, much like the Tessar. The doublet in the Tessar allows for a better image at larger fields of view, since it corrects the field curvature and astigmatism. Higher field of view means the focal length is shorter, which made the Sonnar a more adequate normal lens.
3. Where you can use knowledge used in other lens designs
The Sonnar is actually a good performing lens with very few expensive glass choices. Sure, the manufacturing of all the cemented surfaces is difficult, but it provided a lens with an F1.5 aperture in a time when this was not possible for Double Gauss lenses due to lack of technological advancements (high index low dispersion glass, antireflective coatings, for example).
4. Tips and tricks
The Sonnar uses 7th order aberration generation for aberration correction.
I compare the Double Gauss lens and the Sonnar in a separate blog post. Check it out.
5. Master the specsheet
The modern variant of this lens can be found at Cosina, a lens they designed for their rangefinder series. The lens is designed by the people at Cosina, but go under strict Quality Control of Zeiss standards.
(via Zeiss)
6. Realworld examples
Japanese postwar Sonnartype Nikkor lenses
Three lenses in postwar Japan are the Nikkor H.C. 5cm F2, Nikkor S.C. 8cm F1.5, and the Nikkor P.C. 10.5cm F2.5. We can see that the 5cm F2 is the classic Sonnar type, with the doublet as the last group of lenses, and the 8.5cm F1.5 a triplet lens at the end. Finally, the 10.5cm has a singlet at the end, all signifying the different needs of the optical system, depending on the specifications. In this case, most of it comes down to field of view (focal length), and the aperture (Fnumber).
The 5cm F2 lens was one of the first postwar Nikon lenses, and served important for the company.
A little historical context. The NikkorH.C. 5cm F/2 was made not only for the Nikon S series, but also for Leica Thread Mount ( LTM or M39). Leica thread mounts not only could be fit onto Leica cameras, but also to many other cameras like Nicca, Tower, Leotax, etc.
At some point, I want to compare the different Sonnar variations. I want to especially look at the original F2 Sonnar and compare it to the F2 Nikkor. There were a lot of discussions at the time (and maybe even now) about how the early Nikkor lenses were German ripoffs. How did Nikon come to manufacture a copy of a Zeiss design in the first place? Well, at the end of World War II, German patents were nullified and the Americans ordered the Japanese to make these lenses. There is a total of 6 lenses that were manufactured in Nikon (Nippon Kogaku at the time), all based on German designs.
But from what I understand, the postwar Nikon did not have as many glass variations as the prewar Zeiss did. That means that the lens designs, although inspired from Bertele, needed to be redone from scratch. I have stated before that the first step for lens design is choosing the glass, so it really is a different starting point, and I’d like to explore that someday by looking at the aberrations closely.
The Sonnetar by Miyazakisan
One of my favourite lenses is a Sonnar type lens. It is made by a solo lensmaker, Miyazakisan. He does the lens design and the manufacturing of the lenses on his own.
The lens is the Sonnetar from MSOptical. It has a lot of spherical aberration at full aperture, but decreases as the aperture is closed.
Miyazakisan, much like myself, contends that the designer Ludwig Bertele was a genius, and it has been over 80 years since the Sonnar was made in 1932. Miyazakisan finds that the current normal lens market is dominated by the Double Gauss lens form, and like myself, finds this a little dull. Ever the challenger, Miyazakisan developed this lens after making an F1.3 version. To make this a faster lens, the first two lenses are high index glass. The result is the Sonnetar 50mm F1.1, a fast but compact lens.
He also made another feature, the first of its kind: Manual Coma Correction.
This lens was the first to incorporate a “manual coma correction” feature, where you can move the rear two elements to correct the coma at various focus distances. For example, if you like shooting this lens at 4 metres or so, you can make the coma corrected for that distance. You can handcorrect the coma from 1m to infinity. In addition, this feature can correct the field curvature of the lens as well.
In my use of this lens, I find that wide open, we get very high resolution and good contrast. There is a hint of spherical aberration, so you may feel that the lens isn’t sharp, but that’s the optics playing tricks and the resolution is quite high. Stop it a little bit to F1.25 to F2 and the performance improves significantly. past F4, the performance is high throughout.
The way I use this lens is in daylight, at F5.6 or so to get a highperformance lens at very little size and weight. Indoors or in darker situations, I like to have the F1.1. Portraits are beautiful with this lens.
This lens is multicoated on all surfaces. You may have noticed that there are five lens elements and four groups, compared to the seven elements in three groups, like the original F1.5 Sonnar. This means that the three lenses in front of the stop are all individual lenses. My guess is that air, with an index of refraction of 1.0, is the best solution. Also, the cemented threelens group is a doublet.
Sonnar example
Focal length: \(f = 100\)
Aperture ratio: 1:1.5
Back focal length: 59.21
Field angle: \(\pm 22.5 \deg\)
no.  \(R\)  \(t\)  \(n\)  \(\nu\) 

1  +69.21  9.33  1.6710  47.2 
2  +433.84  0.38  1.0  – 
3  +35.86  11.81  1.6710  47.2 
4  +85.87  7.05  1.4892  70.1 
5  646.31  1.90  1.7394  28.2 
6  +23.51  13.00  1.0  – 
7(stop)  \(\infty\)  2.24  1.0  – 
8  \(\infty\)  2.48  1.5232  50.9 
9  51.09  19.81  1.6578  51.2 
10  22.12  4.57  1.5894  61.2 
11  103.13  44.90  1.0  – 
(Where no. is the surface number, \(R\) is the radius of curvature, \(n\) is the index of refraction, and \(\nu\) is the Abbe number)
Sonnetar example
no.  \(R\)  \(t\)  \(n\)  \(\nu\) 

1  +95.14  16.00  1.73  54.7 
2  +458.91  1.00  1.0  – 
3  +51.67  16.00  1.77  49.6 
4  +99.72  8.00  1.0  – 
5  453.14  6.00  1.78  25.7 
6  +35.90  16.00  1.0  – 
7(stop)  \(\infty\)  0.20  1.0  – 
8  100.09  24.00  1.76  47.8 
9  30.06  6.00  1.72  47.9 
10  127.93  43.04  1.0  – 
(Where no. is the surface number, \(R\) is the radius of curvature, \(n\) is the index of refraction, and \(\nu\) is the Abbe number)
Other Sonnar lenses:
 NikkorH.C 5cm F2
 Nikkor 5cm F1.4
 NikkorP.C 8.5cm F2
 NikkorS.C 8.5cm F1.5
 NikkorP.C 10.5cm F2.5
 AI Nikkor 105mm F2.5
 Cosina Carl Zeiss Sonnar
 Rollei 35 Sonnar
 Contax T2 Sonnar
The Double Gauss lens: The winner of the standard lens on a photographic camera
1. History and background
By Paul1513 at en.wikipedia, CC BYSA 3.0, Link
Before the Double Gauss lens, there was the Gauss lens, which got its name from the mathematician Carl Friedrich Gauss.
The original Gauss designed lens is a two meniscus telescope objective lens. It corrected the spherical aberration and the longitudinal chromatic aberration. I suspect that Gauss would flip out if he saw the Double Gauss lens, and say “Why is this named after me?”
A. G. Clark took the Gauss lens in 1888 and flipped it about the stop to correct coma.
Paul Rudolph noticed that making the negative meniscus lenses thicker in this configuration lead to an improvement in field curvature (Petzval sum) but that there was an increase in chromatic aberration. Rudolph took the negative meniscus lenses and split them into a cemented doublet that had the same index with different Abbe number in order to correct the chromatic aberration. Thus the Planar (Carl Zeiss’ name for the Double Gauss) was born.
The innovations in the Double Gauss lens were made possible due to the innovation in glass types, in particular, the high index glass, and the increase in the computational power of computers.
2. Essentials for lens design
Rudolph designed lenses in a time when there was a surge in glass types. Also, he was able to calculate the chromatic aberrations using first order optics, making calculations easier without ray tracing. For modern lens design, the cemented lenses serve more of a purpose than chromatic aberration correction, but it is useful to keep this concept in mind.
There are a few ways to look at the Double Gauss.
 The Double Gauss as an evolution of the triplet, with a positive lens, negative lens, positive lens layout
 We can think of the 1st lens and the front of the first doublet as three surfaces for correction, likewise with the last lens and the second doublet, which has more degrees of freedom than a single positive lens.
 The middle two surfaces are concentric negative lenses relatively symmetric about the stop.
 The Double Gauss as a symmetrical lens
 We can think of a positive lens with a thick meniscus lens to correct the Petzval sum, and by using a similar but opposite configuration opposite of the aperture stop, we can correct the coma, distortion, and the transverse chromatic aberration due to symmetry. This is more in line with the history of lens design.
3. Where you can use knowledge used in other lens designs
Double Gauss lenses are used everywhere. If you look closely, you can find them at the rear elements within some retrofocus lenses, more complex lens forms like stepper lenses, and zoom lens systems.
4. Tips and tricks
 The Double Gauss uses the cemented surface to generate higher order aberration for better spherical aberration correction.
 The front cemented surface, the negative lens (lens #4) index of refraction can have a higher index to generate higher order aberrations.
 The second cemented surface, the positive lens (lens #6) index of refraction can have a higher index to balance astigmatism and field curvature, much like the Tessar lens.
To summarize, the front half of the Double Gauss is used to correct the aberrations related to the aperture size, like the Fnumber. The rear half of the Double Gauss is used to correct the aberrations related to the field of view. This is similar in concept to the Triplet, Petzval lens, and most objectives.
The negative lenses of the Double Gauss lens correct the spherical aberration and the field curvature, but overuse of these corrections cause coma. The development of the Double Gauss lens design is in lieu with how to take the good part of the design (spherical aberration correction, field curvature correction), while minimizing the bad part of the design (coma flare).
Therefore, the Double Gauss lens has its advantages and disadvantages.
 A. Warmischam used strong curvature on the cemented surfaces
 M. Berek strategically broke the symmetry
 H. W. Lee split the last lens
 R. Tomita combined the Double Gauss with a Sonner type lens group but has a rather large Petzval sum
… but none of them succeeded in solving the Double Gauss lens puzzle.
The second phase innovation of the Double Gauss lens is postWWII, with the help of high index low dispersion glass and other new material innovations. Using high index glass for the positive lenses decreases the Petzval sum, and the higher index glass leads to lower curvature surfaces which help decrease the spherical aberration and the coma.
Interestingly, attempting to make the Double Gauss lens compact and thinner leads to the same result, the positive lenses need to have a higher index of refraction. Keeping the power the same and making the negative meniscus lenses thinner will increase the Petzval sum, so the positive lenses with a higher index will decrease the Petzval sum.
Splitting the cemented group makes the negative surface just left of the stop smaller in lens curvature leading to less coma.
Finally, I compare the Double Gauss lens and the Sonnar in a blogpost. Check it out.
5. Master the specsheet
The Double Gauss is useful for most modern normal focal length lenses. If you look at older 50mm equivalent normal lenses, you will definitely see a recognizable Double Gauss lens form. Even for more modern 50mm high performance lenses, look closely and you can see the Double Gauss in there.
Can you see it in the above lens design diagram? (Hint: in between the blue and yellow coloured lenses)
6. Realworld examples
By Paul1513 at English Wikipedia, CC BYSA 3.0, Link
By Paul1513 at English Wikipedia, CC BYSA 3.0, Link
By Paul1513 at English Wikipedia, CC BYSA 3.0, Link
Bonus: Kubrick’s Planar 50mm F0.7 lens
Have you heard of the Carl Zeiss Planar 50mm F0.7 lens? It is a stellar optical lens design (in more ways than one).
NASA had Carl Zeiss make 10 lenes for their satellites, and Kubrick got a hold of three of those lenses.
Since it wasn’t possible to use these lenses as is, he had a lens mount made to put on his cine camera.
By Gbentinck – Own work, CC BYSA 4.0, Link
The lens was used in the famous scene in Barry Lyndon, where only natural light was used to film the candlelight scene.
Fair use, Link
The biggest problem was the last lens and film separation, this was extremely close together, something like 2 to 3 mm (the backfocal length was short). It must have been a hard task to focus the lens throughout the scene as well!
Looking at the lens, we can see that it follows the familiar pattern of a Double Gauss lens. At least the first 6 lenses (first 4 groups) are a typical Double Gauss type. The Planar 0.7 then adds two more lenses, a large and thick positive lens and a thin negative lens. The positive lens looks like something a microscope objective would use.
References: OMAGGIO ALL’IMMORTALE KUBRICK ED AL MITICO PLANAR 50mm f/0,7
The symmetric wide angle lens: The quest for Field of View
1. History and background
Wide angle lenses were developed to achieve a very wide field of view across the image plane. The first design was the Hypergon, by Goerz).
By Tamasflex – Own work, CC BYSA 3.0, Link
Other famous variations of the symmetric wide angle lens are the Topogon, Biogon, Aviogon, and Hologon, all by Carl Zeiss.
In particular, I find the Hologon very interesting, since usually, the evolution of lens design makes the lens configurations more complex. More combinations of glass, more lenses added, etc. But the Hologon is as simplistic as can be, after the 8 element Biogon it was now a 3 element Hologon. However, this is a good lesson that the evolution of lens design is the subtraction of unneeded lenses, not only the addition of more complex lenses.
2. Essentials for lens design
For wide angle lenses, the emphasis is obviously placed on the aberration correction at wider angles, or wider fields of view.
The three largest contributors are astigmatism, field curvature, and distortion.
To correct astigmatism, the lens must be concentric about the stop.
The field curvature depends on the Petzval sum, which based on the lens power and index of refraction.
As we can see with landscape lenses, high symmetry about the stop improves distortion.
3. Where you can use knowledge used in other lens designs
To be honest, there aren’t many cases where the symmetric wide angle lens can be used other than wideangle imaging. However, imaging is not only excluded to photography, but it can also be imaging for sensors, scanning, and other uses where a flat field with minimal distortion can be accepted.
This lens is good for:
 Wide angles
 Deep depth of field
 Low distortion)
 High center resolution
 Compact lens size
This lens is NOT good for:
 Relative illumination (it has dark corners)
 Long back focal length (length of the last lens to the image plane or sensor)
 Small CRA or chief ray angle, the angle the light hits the image plane or sensor (image sensors require a minimum CRA)
4. Tips and tricks
Think about the system as a triplet lens, with a positive element followed by a negative element, and then a positive element. Group the powers together and solve the thin lens equation to get a feel for where the each of the lenses elements should be.
Gradually change the FOV angle in the design because most software can’t handle extreme jumps in FOV and can cause calculation mistakes.
5. Master the specsheet
Wide angles larger than 90 degrees FOV, most often 120 degrees or 130 degrees. When you see this specification, you have to decide in your mind if you need to use the symmetric wide angle (with short back focal length and low distortion), or the retrofocus lens (long back focal length and large distortion).
6. Realworld examples
Below are some wideangle lenses, in historical order:
Hypergon – 1900
By Tamasflex – Own work, CC BYSA 3.0, Link
Designed by Emil von Höegh, Goerz
Orthometar – 1926
Designed by Willy Walter Merté, Carl Zeiss
Topogon – 1933
Designed by Robert Richter, Carl Zeiss
Biogon – 1951
Designed by Ludwig Bertele, Carl Zeiss
The SuperAngulon was the Leitz version.
Hologon – 1966
Designed by Erhard Glatzel, Carl Zeiss
The telephoto lens: The term that is confusing for photographers(but not lens designers)
1. History and background
The history of the telephoto lens is not clear, although the concept was developed by Johannes Kepler and Peter Barlow, and later simultaneously designed by T. Dallmeyer and Adolf Miethe.
Pre WWII telephoto lenses looked relatively simple, and for a time the distortion correction was thought to be impossible.
This changed with a lot of research, but the correction of distortion made it a staple for the SLR era of long focal length lenses.
In the rangefinder era, focal lengths of 135mm were maximum, whereas 200mm, 300mm, and 500mm were possible with the SLR.
2. Essentials for lens design
The essential point of the telephoto design is that its physical length is shorter than its focal length. A lot of photographers classify a telephoto lens merely has a long focal length, but you and I know that’s not true.
The lens design form is a positive front group, a large space in between, and a negative rear group.
Depending on powers of the positive lens group and the negative lens group, and how they are placed with respect to one another, multiple configurations of the telephoto lens are possible.
The length \(L\) of the lens divided by the focal length of the lens \(f\), is usually denoted as the telephoto ratio, \(T = L / f\). The telephoto ratio is a measure of the compactness of the telephoto lens.
3. Where you can use knowledge used in other lens designs
The tele side of wide range zoom lenses share the same properties as a telephoto lens, as we have to take care of the longitudinal chromatic aberration and the spherical aberration.
The concept can be used if you are designing a teleconverter.
Perhaps counterintuitively, since the telephoto lens form, power placement of the lens groups, and even distortion, are all the exact opposite of a retrofocus lens, they are complementary to each other.
4. Tips and tricks
Telephoto ratio tips
If we make the telephoto ratio small, the longitudinal chromatic aberration increases, and the Petzval sum becomes negative. for a full frame 35mm size film or sensor camera, this means that the longer the focal length, the telephoto ratio can be made small. Likewise, if the Fnumber is slow, the telephoto ratio can be made small.
A rule of thumb for the telephoto ratio is 300mm, \(T = 0.7 \sim 0.8\) (full frame).
Chromatic aberration and extraordinary low dispersion lenses
The rear group has a power that with a large magnification, but unfortunately, this negative rear element magnifies the aberrations caused by the front element. Aberrations in the transverse direction are magnified by the square of the magnification.
In order to decrease the longitudinal chromatic aberration of the entire system, the front lens group needs to have low chromatic aberration, to begin with.
In order to achieve low chromatic aberration, extralow dispersion lenses are needed for the primary positive power lens(es) in the positive front group. This is why we see expensive lenses that are very large in most telephoto systems and highend zoom systems.
Distortion in a telephoto lens
The telephoto lens has negative distortion, or pincushion distortion. This lens is one of the most popular formats for long focal length photographic lenses, so there had to be a solution for distortion.
The overall lens configuration is a positive lens group in the front, and a negative lens group in the back. We split the negative lens in two, as a positive and negative lens group. The positive lens has relatively low power, but it serves an important purpose.
The positive lens has a large effect on the rays, while the negative lens does not. Conversely, in the center of the lens (FOV = 0), the positive lens doesn’t change the rays too much while the negative lens does. This positive lens, therefore, causes a change in the higher angles of the field while having little effect on the center, thus the distortion can be corrected.
Additionally, the front lens group can also be split into two, for example, we can have a colour correction cemented doublet and a positive lens followed by a negative lens a little bit away from the others. This negative lens also contributes to the telephoto ratio, while also decreasing the spherical aberration and spherochromatism, without affecting the distortion too much.
The lens can have an additional field flattener at the back, to correct for field curvature (the last lens in the lens design diagram shown below).
( Nikon imaging )
Inner focusing
A popular way to achieve focus for a telephoto lens with long focal lengths is to use an inner element focusing system. A small inner focus lens makes for fast focus, especially for autofocus. The conventional method of focus is moving the entire lens, which is not practical for very large telephoto lenses, in particular for fast systems with a large (read: heavy) front element.
5. Master the specsheet
Simple. Use this lens design for and its principles whenever you need a focal length that is long, but the specification says that the total length of the lens must be shorter than the focal length.
6. Realworld examples
 NikkorT 10.5cm F4
 NikkorP・C 10.5cm F2.5
 AI Nikkor 105mm F2.5
 AI Nikkor 105mm f/1.8S
 NikkorQ Auto 200mm f/4
 AI Nikkor ED 180mm F2.8S
 NikkorH 300mm F2.8
Further reading:
 Kazamaki, T. and Kondo, F., “New Series of Distortionless Telephoto Lenses,” J. Opt. Soc. Am. 46, 2231 (1956).
 Kingslake, R., “Telephoto vs. Ordinary Lenses: A Tutorial Paper,” in Journal of the SMPTE, vol. 75, no. 12, pp. 11651168, (1966)
The Retrofocus / Reverse telephoto lens: The practical solution to wide angle lenses on SLRs and digital sensors
1. History and background
Historically, the retrofocus lens is a relatively new design. The telephoto lens was made before it.
Produced in the 1950s by French camera company Pierre Angénieux, the name retrofocus was actually a product name for a lens that we identify as the retrofocus type today. The optical configuration is similar to what we see in most retrofocus type lenses. Truthfully, this lens should be called a reverse telephoto lens or an inverted telephoto lens if we were to respect the trademark of the name by Angénieux, but since many people identify the lens form with the name retrofocus, that’s what we’ll call it from here on.
The basic concept of the lens is to achieve a wide angle lens while having a large back focal length. It was originally applied to 16mm cine lenses that had a rotating shutter mechanism and required a large back focal length.
Public Domain, Link
Remember the telephoto lens? The telephoto lens has a long focal length and its physical length was shorter by using a positive lens and a negative lens. Reverse telephoto uses a similar but opposite concept, the lens has a short (therefore wide) effective focal length, but a longer physical length, usually a longer back focal length. The lens configuration is also the opposite to the telephoto lens, it has a negative lens and then a positive lens. The need for longer back focal length is because of the popularity of Single Reflex Lens (SLR) cameras, which has a mirror in front of the imaging surface to reflect the image through the viewfinder.
The first retrofocus lenses had about 35mm focal length, but considering the back focal length had to be larger than 40mm for most cameras, the concept was revolutionary.
2. Essentials for lens design
The essential point of the retrofocus design is that its physical length is longer than its focal length, and more importantly, that its back focal length is longer than the focal length. This was particularly useful after the single lens reflex camera became popular around the 1960s.
The lens design form is a negative front group, a large space in between, and a positive rear group. It can almost look like a normal lens with a wide angle converter attached to it.
We usually use a few lenses for both the negative group and the positive group, but there are many possible variations depending on how many lenses and what type of glass we use.
A retrofocus wide angle lens requires a longer back focus, consequently may have some distortion, and a large front lens with large asymmetry.
The early retro focus lenses look relatively simple and are easy to understand. This one has a Gausslike or Tessarlike positive lens group in the back, with a negative lens in the front to widen the overall FOV and increase the back focus.
This lens is good for:
 Wide angles
 Deep depth of field
 Long back focal length (length of the last lens to the image plane or sensor)
 Relative illumination
 Small CRA or chief ray angle
This lens is NOT good for:
 Compact lens size (the backfocus is large)
 Low distortion (the asymmetry in the system makes it difficult to correct distortion)
3. Where you can use knowledge used in other lens designs
The basic concepts like distortion are linked to wide angle zoom systems.
The concept can be used if you are designing a wide angle converter.
Perhaps counterintuitively, since the retrofocus lens form, power placement of the lens groups, and even the distortion are all the exact opposite of a telephoto lens, they are complementary to each other.
4. Tips and tricks
Tips to improve the performance of a retrofocus design
 With a powerful negative lens in the front, wide angles are possible, and relative illumination can be kept high
 With a powerful negative lens in the front, pincushion distortion appears easily and is difficult to correct
 Since the positive rear lens group receives diverging rays, it is difficult to make the lens system a fast speed fnumber
 The field curvature in the corners of the image changes dramatically with close focus
5. Master the specsheet
If we need a wide field of view lens, but we need a long back focal length and/or telecentricity in the image plane.
Be mindful of the amount of space you can have, both in the length of the lenses and the diameter of the lenses.
6. Realworld examples
from NikkorS Auto 35mm f/2.8 to “New” Nikkor 35mm f/2.8
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Imaging lenses: Specific use lens systems
As lens design forms evolve, the lens use cases get more and more specialized. To meet these demands, lens designers developed more specific use cases for these systems.
Although the lenses get more specialized, the foundation of these lens designs is built on previous lens design forms.
For example, the fisheye lens is an evolution of the retrofocus lens. Zoom lenses use both retrofocus and telephoto lens properties, and afocal lenses are also conceptually a part of zoom lenses.
The Fisheye lens: Testing the limits of Field of View
1. History and background
A typical lens will image a flat surface onto a flat surface, but a fisheye lens images a spherical surface onto a flat surface.
The basic form of a fisheye lens is similar to the retrofocus lens, in that it has a negative front group and a positive rear group of lenses.
Commercial fisheye lenses have complex negative and positive lens groups in order to maximize the resolution.
As a photographic lens, the fisheye lens is usually used in special circumstances, and more emphasis is placed on whether a FOV of 180 degrees (or more) is achieved, rather than what type of projection system (listed below) is used.
A 180degree field of view lens imaging a circular image is sometimes called a circular fisheye, and a lens that images the corner FOV to be 180 degrees is sometimes called a diagonal fisheye lens.
There are people who prefer the fisheye “look” for artistic purposes.
2. Essentials for lens design
There are different ways to image a sphere onto a plane. I will list some examples below, and \(y\) is the image height, \(f\) is the focal length, and \(\theta\) is the half angle of view.
Equidistant projection
Equidistant projection, is the most typical, also called ftheta imaging.
By Strebe – Own work, CC BYSA 3.0, Link
The definition of an ftheta lens is:
$$
y = f \cdot \theta
$$
Although used for fisheye systems, the ftheta systems are used more in laser scanning systems like the ftheta lenses for laser beam printers.
Orthographic projection
Orthographic projection, is like taking the cross section of a sphere, and projecting that crosssection directly on a plane.
By Strebe – Own work, CC BYSA 3.0, Link
The definition for orthographic projection is:
$$
y = f \sin{\theta}
$$
One characteristic of the orthogonal projection is that the illuminance of the image is constant across the field of view.
Thinking about this way, we can say that the fisheye lens has a larger distortion than most traditional imaging systems. In this case, the cosine fourth law does not hold, and the edges of the image are less prone to a drop in relative illumination.
Stereographic projection
In a stereographic projection image, the imaging at the equator is twice as close than in an orthogonal projection.
By Strebe – Own work, CC BYSA 3.0, Link
The definition for stereographic projection is:
$$
y = 2f \tan{ \theta / 2}
$$
Equisolid angle projection
The equisolid angle projection is where the area of the image is proportional to the solid angle of the object. When calculating the area of the image we can calculate the solid angle of the object, so it is used a lot in metrology.
By Strebe – Own work, CC BYSA 3.0, Link
The definition for equisolid angle projection is:
$$
y = 2f \sin{ \theta / 2}
$$
There is a nice summary of the projection systems here.
3. Where you can use knowledge used in other lens designs
Equidistant is used more in laser scanning systems like the ftheta lenses for laser beam printers.
Bonus: Full Sphere imaging
R. Miyamoto, Fish Eye Lens, J. Opt. Soc. Am, Vol. 54, No 8
Zoom lenses: How we can get many focal lengths in one lens system
(One of the first zoom lenses, the Cooke varo 40mm120mm F3.5F8 lens)
1. History and background
The often referenced documentation is the patent by H. Gramatzki in 1936, which showed several zoom systems with several equations. He started with afocal lens attachments, and used this concept for zoom lenses as well.
The name zoom lens comes from one of the early lens companies Zoomar which has now become the definition of a variable focal length lens. I wonder if varifocal lens would have caught on?
Systems like Gaussian brackets were introduced to mathematically calculate zoom systems.
Over the years, K. Yamaji, the tobe president of Canon published a paper on zoom lenses that I think still stands the test of time today. Ironically, Yamaji later became a proponent of keeping technical information inhouse in favour of disclosing them as patents, and good papers such like his own became scarce to this day.
Optically speaking, zoom lenses can be traced back to the converter. We see the wide converter and teleconverter lenses that convert the focal length of a system as an attachment. The Fuji X100 series comes to mind, which is a fixed lens digital camera that has both a wide angle conversion lens and a tele conversion lens to change the focal length of the lens.
(via Fujifilm)
Mathematically, this means that the original master focal length \(f_m\) is converted to the new focal length \(f_t\) via the magnification \(\beta\) of the converter.
$$
f_t = \beta \times f_m
$$
This is actually a simple way to think about the zoom lens, because the zoom lens has a number of fixed focal length groups in the lens system that change position to give a different magnification. Just like if you were to move a magnifying lens from your eye front to back, you'd see the image get bigger or smaller.
There are two types of zooms, optically compensated zooms and mechanically compensated zooms.
The optical compensation zoom system has one linearly moving part and essentially does not use a cam system for movement. The focus is within the depth of field by using clever positioning of the lenses.
The mechanical compensation zoom system has at least two moving parts with a cam system, usually a variator lens and a compensator lens.
Nowadays, all zoom systems have very complex movements and it’s hard to tell which lens has what function. Let’s take a look and break the zoom lens system down.
2. Essentials for lens design
What defines a zoom lens?
 It can change magnification
 The focus point is fixed
 The performance is above a certain level at any zoom point
A lens that doesn’t satisfy (2) is called a varifocal lens. That means that a lot of photography lenses are not zoom lenses, they are varifocal lenses, because we usually need to refocus the lens.
For a video system, staying in focus while zooming is critical as the picture would go out of focus as we zoom in or out. For a still image photography camera, this isn’t a big deal, we just need to refocus the image. This makes zoom lenses (sorry, varifocal lenses) for photography have a higher degree of freedom, and more compact and large zoom ratios can be made.
The main components of a zoom lens are:
 Focusing lens – used for focusing from infinity to focusing at a finite distance
 Variator – the lens that moves over the largest distance and changes the magnification of the system
 Compensator – compensates the focal point of the system from the magnification change induced by the variator.
 Master lens – the main imaging lens of the system, and is responsible for the focal length, Fnumber, and back focal length of the lens. Most often there is higher order aberration generated by this group to counteract the aberrations from the preceding components. Also called the relay lens.
Some zoom systems have a compensator that doubles as a focusing lens, and some zoom systems have two variators with one variator doubling as a compensator. But even if a lens group doubles functions, the components are still there. It may be a little tricky to decode these lenses, but if we know the properties of each component it is easy to find them in the system.
3. Where you can use knowledge used in other lens designs
A wideangle zoom lens shares a lot of properties with the retrofocus lens.
A telezoom lens shares a lot of properties with the telephoto lens.
And not coincidentally, widerange focal length zoom shares the properties of the retrofocus lens on the wide angle side, and also shares the properties of the telephoto lens on the tele side.
In most cases, a challenging retrofocus or telephoto lens is more difficult than the single focal length of a wide angle zoom or a tele zoom. The difficulty of the zoom lens is the balance between all of the focal lengths, and often the individual focal lengths are not too difficult on their own.
4. Tips and tricks
Focusing
Making the front element the focusing lens, or making the lens an inner focusing system. Figure out which system you need for the focus system.
The front focus is most straightforward and robust. It is also the optically logistic way to focus a lens. However, if the front element is large, this can mean moving a large chunk of glass to focus, which is harder to do mechanically, requires more precision, and can cause focus backlash (overshooting the focus point).
Also, some focusing systems do not work well when the distance of the object to the front of the lens cannot change. Some systems require that the total length of the lens from the first element to the last element not change.
Which brings us to inner focus, which seems like the perfect solution, since the focus is in the inner location of the lens. The lens can be made small so that movement is smoother and more precise, and the total length of the lens doesn’t change.
However, inner focus is harder to achieve optically, since the focus must work for all zoom positions, and the focus is likely on the same pathway as the variator or the compensator.
Choose the focusing mechanism accordingly to the application.
Using Gaussian brackets
Gaussian brackets themselves are a mathematical tool, and when used for lens systems they can be extremely powerful. Gauss came up with the mathematical algorithm, and it can be used to evaluate the focal length, magnification, and back focus of a lens system with some given parameters like the thickness or lens power.
The Gaussian brackets were introduced in optics by M. Herzerberger in 1943, and then expanded upon by K. Tanaka in the 1980s.
Gaussian brackets are too involved to get into here, but defining a zoom system with Gaussian brackets and is useful. I might have a lot of fun trying to explain this someday. (Maybe an epic post on zoom lenses?)
5. Master the specsheet
First, we have to see what kind of zoom we need.
 Is it a wide angle only?
 Is it a telezoom?
 Does it have a large zoom range in both the wide and the tele?
 What is the Fnumber requirement across the zoom range?
If we can answer these questions, then we can get a better idea of the requirements of the system.
Conjugates
There are four conjugates of the zoom lens.
 Finitefinite conjugate system: The object is at a finite location. Examples such as projector systems, factory inspection systems, and scanner lenses. A projector projects the LCD (object at a finite distance) to a screen (image) at a finite distance, and can often zoom to change the size of the projection on the screen.
 Infinitefinite conjugate system: The object is at infinity, and the image is at a finite distance. Photographic lenses take a scene (object at infinity) and images onto a sensor/film plane, which is at a finite distance.
 Infiniteinfinite conjugate system: The object is at infinity, but the image is at infinity as well, so it is an afocal system. Zoom finders, binoculars, telescopes. A zoom finder is an optical viewfinder that changes FOV when you zoom. The scene you view (object at infinity) is imaged to your eye via the exit pupil of the system, making it an infinite conjugate system.
 Telecentric conjugate system: The object and image side are both telecentric, and zooms as well!
6. Realworld examples
The zoom systems fall into a few general systems.
 4group zoom system
 2group zoom system
 3group zoom system
 Multigroup zoom
The 4group zoom
The 4 group zoom system is the most basic, and has the four components – focus, variator, compensator, master lenses clearly defined in the system.
The zoom range is typically a relatively wide angle (FOV 7080 degrees) to relatively long focal length (FOV 2530deg).
Close observation of the lenses shows that the variator covers the most distance, and is close to the focusing lens for the wide end and moves ever so close to the master lens in the tele end. The variator is usually a negative lens group, mainly because the zoom system can be made more compact.
The master lens group usually has an afocal component, and the firstsecondthird group is the zooming part.
The 2group zoom
The 2 group zoom system has two main forms, a negativepositive group and a positivenegative group form. The picture below is a negativepositive zoom since the first group has negative power and the second group has positive power. The concept is much like the retrofocus lens.
For a wideangle zoom, a 2 group zoom lens form is typical, with a negative focusing lens, and a positive variator. Therefore, the 2 group zoom is harder to cover a large focal length range, and the Fnumber can’t be made too fast either.
The other 2 group zoom, a positive first group and a negative second group is mostly used for compact zoom lenses.
The 3group zoom
The 3 group zoom system is a variation of the 2 group zoom, where the first group is split in two for a higher degree of freedom. By using a 3 group zoom, it is possible to achieve a larger zoom range in a compact form.
Multigroup zooms and modern zoom lenses
Multigroup zooms are zooms that many groups that move for zooming. These multigroup zooms typically are used for a large zoom range.
As far as modern zoom lenses are concerned, just go to any famous lens maker (Canon, Nikon, Olympus, Sigma, FujiFilm, Panasonic, Cooke, etc) and you will find plenty of zoom lenses. Nowadays, there are so many variations that it is hard to keep up. But the essence of lens design is still there, and the four components in focusing lens, variator, compensator, and the master lens are used.
I think I have an epic post on zoom lens design in me, waiting to come out. I have too many ideas bouncing around in my head, so stay tuned while I try to figure things out.
Afocal lens systems: manipulating rays to get them to behave the way we want
In this subchapter, I thought I would take a deeper look at an often forgotten lens design technique that can be quite useful, and that’s afocal lenses.
You might be thinking, “What? How can I use those?” Well, it turns out that they are within all sorts of lens designs more than we think.
Afocal systems alone may not dazzle us too much, but they are used in many situations, and making a system afocal or making a portion of the system afocal has many benefits like making the system simpler to understand, or to simpler to systematically build the optical system.
1. History and background
A lens that has parallel incident rays and has parallel exit rays is called an afocal system. Simple enough, right? Well, there are a few variations on how we achieve this.
In an afocal system, the object is at infinity, and the focal length is also infinity. That makes things interesting because an afocal system doesn’t have a focal length, per se.
In an afocal system, the object is at infinity, and the focal length is also infinity.
What to do? Well, the most typical way to express an afocal system is by magnitude, not focal length. We’ll get into the specifics below.
2. Essentials for lens design
The most general afocal system is made up of two lenses, and the focal point of the image of the first lens is placed at the focal point of the object of the second lens. We can actually have a positivenegative lens combination and a negativepositive lens combination.
The former, the positivenegative afocal system shrinks the beam width (if it were a beam expander), while the negativepositive afocal system expands the beam.
Since the focal length of afocal systems is infinity, we use the angular magnification as a unit of measure. Let’s call it \(\gamma\).
When the parallel ray enters the system at an angle \(u_1\), and exits the system at an angle \(u_k\),
$$
\gamma = \frac{u_k}{u_1}
$$
is the angular magnification.
Also, if there are two lenses in the afocal system, and their focal lengths are \(f_1\) and \(f_2\), the angular magnification is
$$
\gamma = \frac{f_1}{f_2}.
$$
And finally, the parallel beam entering the system at a height of \(h_1\) and exiting the system at \(h_k\) has an angular magnification of
$$
\gamma = \frac{h_1}{h_k}.
$$
When the system is a positive lens and negative lens combination, the magnification is positive, while for a positive lens and positive lens system the angular magnification is negative.
We can even have positive – positive combinations to get an afocal system.
3. Where you can use knowledge used in other lens designs
 Telescopes: Those above images already looked like telescopes, so I think you already guessed this one. Both Galilean and Keplerian telescopes are afocal systems. Even the Newtonian telescope is an afocal system with a reflective lens rather than a refractive lens.
 Optical viewfinders: Optical viewfinders change the magnification of the field of view of our eyes. It’s typical to use a wideish viewfinder when using a wideangle lens. This is before SLR cameras, by the way.
 The zooming component of the 4group zoom lens: Note that it doesn’t have to be a 4group lens, but the 4group lens is the easiest to understand schematically, and therefore the best choice here.
 Laser beam expanders
These all are either afocal systems or contain afocal properties within them.
An example of a zooming optical view finder below.
Next, we take the closer look at the zooming portion of the 4 group zoom lens.
For both examples, we can see the parallel rays incoming to the front lens, and it is parallel going out, for all zoom positions.
Below is a laser beam expander, and it takes a narrow beam and widens it significantly. If we look closely, it is the same as the above examples and the parallel rays incoming to the front lens, and it is parallel going out also.
4. Tips and tricks
Afocal systems in finite space
When an afocal system is used in a finite system with an object and is used as an imaging system, the magnification \(\beta\) is
$$
\beta = \frac{1}{\gamma} =\text{constant},
$$
and this relationship does not change with the position of the object whatsoever.
And FYI, this afocal imaging system where the principal ray is parallel to the optical axis is a telecentric system.
Afocal systems combined with other lenses
If an afocal system is followed by an imaging lens with a focal length of \(f_0\), the focal point of the combined system is at the same point as the focal point of the imaging lens.
wide / tele converters and zoom lenses use this property of afocal systems.
5. Master the specsheet
Afocal systems are used for a specific purpose, so instead of figuring out how to spec out an afocal system, it is more important to think about where an afocal system is useful within the lens design.
One thing to be careful of with an afocal system is the resulting aberration change it induces.
If the rays on the optical axis to the marginal rays change angles, that means that the system induces spherical aberration.
The difference to the actual ray \(\tan(u_k)\) and the ideal ray is \(\gamma \tan(u_1)\) is the distortion.
6. Realworld examples
In an actual optical lens design, unless we are using it with our eye, the simplest way to express an afocal system is with an imaging lens.
Take the zoom lens above for example. We can see the parallel rays incoming to the front lens, and it is parallel going out, regardless of zoom position. (In actual use, there is another group of lenses after the exiting parallel beams forming an image to a sensor)
Comment on conversion lenses
 If it is possible to easily mount and dismount an afocal lens on the front of a lens system to a master lens, it is a conversion lens.
 If the magnification is larger than 1, it is a tele conversion lens
 If the magnification is smaller than 1, it is a wide conversion lens
 If the magnification is variable, it is a zoom conversion lens.
 The resulting focal length is the product of the focal length of the master lens and the magnification.
A conversion lens that is mounted on the back of the lens system is a rear conversion lens or alternatively called a rear converter. Rear converters that lengthen the back focal length are common.
Rear converters that shorten the back focal length did not exist for a long time, because it was not practical to shorten the back focal length to widen the focal length, we would much rather use a front converter for that.
For mirrorless camera systems, there are rear converters that take legacy lenses, widen the focal length, and by proxy shorten the back focal length. (Thanks for the tip, Hans!)
When a rear conversion lens is mounted, the resulting Fnumber is the product of the magnification and the Fnumber of the master lens. Note that the Fnumber when using a front conversion lens does not change.
Below is an example of a conversion lens for a compact camera lens.
Teleconverter
Rear converter
Usually, the Petzval sum of the master lens is small, and this rear converter has a negative Petzval sum, so the resulting Petzval sum is negative as well. Therefore, the positive lenses should have as low of an index of refraction as possible, and the negative lenses should have as high as an index of refraction as possible.
The aberration correction of rear converter lenses is difficult since the rays usually pass above the optical axis, or most of the ray bundles pass through one side of the optical axis.
Telecentric lens systems: when and where we need straight rays
1. History and background
A lens system that has its focal point at either the entrance pupil or the exit pupil is called a telecentric system.
For the former, a telecentric system with the focal point at the entrance pupil, the principal ray is parallel on the image side. This is an image side telecentric system.
For the latter, a telecentric system with the focal point at the exit pupil, the principal ray is parallel on the object side. This is an object side telecentric system.
When both the object side and image side are telecentric, this is an afocal lens system. Therefore, the magnification is always constant, and the displacement or tilt of the object or the image does not affect the system.
2. Essentials for lens design
3. Where you can use knowledge used in other lens designs
Since the LSI fabrication process requires extreme precision the image side telecentricity is important.
Systems that require optical fiber bundles, such as endoscope imaging require image side telecentricity. The fiber bundle requires that the rays are not oblique, since this will reduce the transmission. Therefore an image side telecentric optical system with parallel rays to the fiber bundle works nicely.
Some projection systems have object side telecentricity. This is useful since the object can move a bit and the resulting image will still be good.
A microscope objective is designed with a very small object, so designing it with object side telecentricity is useful as well.
Super LSI optics require extremely balanced optics that can withstand some minute displacements or tilt of the object plane and the image plane, and bitelecentric lenses are very useful here.
4. Tips and tricks
Image side telecentric systems have zero chief ray angles for all fields. This means that \(\cos^{4}u=1\), and the image illumination is constant without a drop in relative illumination.
The easiest way to achieve object side telecentricity is to flip the image side telecentric system around. For optical design software settings, it is sometimes easier to design an image side telecentric system and flip it around.
5. Master the specsheet
 The system requires the angle to the image to be parallel to the optical axis: image side telecentric system
 The system requires uniform illumination of the image: image side telecentric system
 The system requires large magnification, and the object will be magnified a lot: object side telecentric system
 The system needs to be robust to the movements and tilt of the object: object side telecentric system
 The system requires two of the properties for image side telecentricity and object side telecentricity: bitelecentric system
6. Realworld examples
 Optical measurement techniques with telecentric lenses – A course on Gaussian optics of telecentric systems from Schneider Kreuznach
 Technical description of telecentric effect – Edmund Optics
 Telecentric lenses tutorial – Opto Engineering
Tandem lens example
When two positive lenses groups have parallel rays between them, it is called a tandem system.
This means that the object is placed at the focal point of the first positive lens group, and the image is placed at the imaging surface of the second positive lens group.
Usually, the same positive lenses are used, just opposing each other. Also, most tandem lens systems are the same scale in most cases.
Tandem lenses can work as relay lenses, but relay lenses don’t have to have parallel rays in between.
Tandem lenses can work as telecentric systems, but tandem lenses don’t necessarily have to be telecentric. But if a tandem lens is telecentric, it is certainly bitelecentric.
Since tandem systems have parallel rays in between the two lens groups, there are a few characteristics:
 Changing the separation of the lenses does not change the system. The object to the lens distance does not change, and the lens and image distance does not change. The magnification does not change.
 We can place a prism or beam splitter in between the parallel rays without changing the magnification or the position of the object and the image.
 Likewise, a parallel plate can be placed in the system without changing anything.
 If we place an afocal system in between the tandem system, we can change the magnification without changing the object and image position.
To design a tandem system, we first design with the object at infinity, with the aperture stop in front of the lens, and make an image. We then take this lens to duplicate it and flip it around the aperture stop.
Reflective optic lenses: Changing the direction of light
1. History and background
The importance of reflective optics
Since ancient times, mirrors were used as reflective optics. There is a hypothesis that Archmides used collective mirrors as a heat ray to burn ships with the sunrise.
Since sir Isaac Newton, reflective telescopes were used to observe the stars.
In Japan, for example, optics made its development via photographic lenses, so it was mainly refractive optics. If you are more familiar with photographic lenses like me, you’d know that for photographic lenses, the number of reflective optics are few, like a 500mm F8 lens and beyond.
In Physics, reflection can be explained in a line or two, but from a lens design standpoint, there are so many angles to look at lens design.
There are plane mirrors, spherical mirrors, aspherical mirrors, ellipsoidal mirrors, parabolic mirrors, hyperbolic mirrors, and toric mirrors. These mirrors can be further classified into convex shaped mirrors and concave shaped mirrors.
There are several advantages of a mirror lens compared to a refractive lens.
 The possibility of making large optics compared to refractive lenses. For example, a refractive lens is difficult to make over 500mm in diameter, but a reflective system can be made much larger.
 Reflective lenses do not have chromatic aberration.
 The power of lens being equal, a reflective lens has approximately 1/4 the radius of curvature compared to a refractive lens (if we assume \(n = 1/5\)). This means mirror lenses have a smaller spherical aberration.
Perhaps the last point, if the radius of curvature is decided with the power of the reflective optic, there is no room for bending the surface to balance the aberrations.
2. Essentials for lens design
The reflective surfaces are lapped and polished for high surface precision, and coated with reflective materials. Aluminum is common, but some applications use silver, gold and other materials.
As lens designers, we need to keep in mind the reflective properties of the material and make sure that it is usable in the wavelength range and the manufacturing.
The focal length of a spherical mirror is half of the radius of curvature, or \(f = r/2\).
3. Where you can use knowledge used in other lens designs
Not only Newtonian telescopes that use mirrors exclusively, but catadioptric systems use reflection and refraction in the optical system.
Prisms and reflectors are used in laser applications, binocular system, and foldable optics as well.
TIR properties are used in the prism of an HMD lens, and an HUD uses a reflector to image a display into our field of view.
4. Tips and tricks
How to proceed with the lens design
In a reflective system, there can be one surface that reflects light twice, or reflects light to a refractive lens for the second time. In the lens design, we number the surfaces in the order that the ray hits the surface, so some surfaces will have two numbers associated with it, depending on the number of times the ray hits the surface.
Keeping all of the sign conventions consistent, a reflected ray will move through the system negatively, and subsequent surfaces that it hits will reverse in sign.
This way, we can trace the rays in the software with the shape intact.
When there is a hole in the center of the mirror
Some reflective systems have a hole in the center of the mirror. We will still trace the rays near the optical axis for paraxial calculations. As far as the Fnumber, it is the ratio of the reflective area with the entire system.
By Krishnavedala – Own work, CC BYSA 4.0, Link
Protect the reflective surface
Also, reflective mirrors deprecate with time, as the reflection percentage can change with time, and dirt, fingerprints, dust, scratches all contribute to the reflection loss of the mirror.
If possible, the best way to solve this problem is to place the reflective surface behind a sheet of glass, as a means to protect the reflective surface. This is called a Mangin mirror.
Interestingly, the spherical aberration of a Mangin mirror is much smaller than a simple reflective mirror.
Total internal reflection (TIR)
Total internal reflection occurs when a ray passes through a higher index of refraction material to a lower index of refraction material. There is a critical angle at which this happens, and any angle larger than the critical angle continues to have internal reflection.
Since Snell’s law is
$$
n_1 \sin{\theta_1} = n_2 \sin{\theta_2}
$$
For a ray that passes through a material to air will have a critical angle of
$$
n_1 \sin{\theta_1} = 1.0 \sin{90^o}, \\
\theta_1 = \sin^{1}\frac{1}{n_1} = \theta_c
$$
Below is a table for various materials and their critical angle with respect to air.
material  \(n\)  \(\theta_c\) 

Water  1.3345  48.5336 
BK  1.5  48.5336 
SK  1.6  41.8103 
LaK  1.7  38.6822 
LaSK  1.88  36.0319 
Ge  4.0  14.4775 
BK7  1.51633  41.2608 
BaK4  1.56883  39.5995 
SK4  1.61272  38.3213 
LaK14  1.6968  36.1105 
SF2  1.61769  38.1823 
PMMA  1.4918  42.0926 
PC  1.5855  39.1030 
5. Master the specsheet
Pretty straight forward, when we are concerned with the chromatic aberration it is good to consider a reflective system. We have to be aware of the tolerances of the system and see if it makes sense as an overall system.
6. Realworld examples
 Laser applications: The power density of the laser beam might not be suitable for some materials, so reflective optics with special coatings are used.
 Stepper lenses: In order to achieve the resolution needed, the wavelength used for stepper lenses need to be shorter, and even go to the ultraviolet wavelength range. Refractive optics can lose transmission at these wavelengths. Also, the chromatic aberration may become larger with refractive optics. A well coated reflective optic can solve both of these problems.
 Astronomical lenses and surveillance: For longer wavelengths in the far infrared range, likewise there are few materials that can be used for refractive optics.
 Large telescopes: For extremely large optics, the refractive lenses would be so heavy that they would not be practical.
 Projector systems: LCD projection systems, dichroic mirrors
 Prisms for HMDs as viewfinders: require freeform lenses and reflective optics
 Illumination systems: by using reflective optics we can fold the optics to a more compact form.
 Catadioptric systems: This system is a combination of reflective properties and refractive properties in one optical system. They are sometimes called reflectiverefractive optical systems and mirror lens systems. Typical catadioptric systems are astronomical telescopes, and some mirrored longfocal length photographic lenses. Optical systems with only reflective components have no chromatic aberration, and catadioptric lenses have refractive components so there is chromatic aberration. However, catadioptric lenses can make for compact optics and wide FOV that are difficult with reflectiveonly optics.
 Optical fibers: Although a little off topic, optical fibers are essentially TIR insude of the fibler core, and can be considered as a reflective system.
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Coffee break: Some cool lens names from yesteryear.
Along with names like the Tessar and Sonnar, some of the most creative lens names belong to the symmetric wide angle lenses.
Carl Zeiss named their lenses after lens forms,
 Tessar = 4 element lens (from tetra)
 Sonnar (either from the Carl Zeiss head plant in Sonthofen, or from the German word for Sun
 Sonne)
 Planar = Double Gauss (which is named after the mathematician Gauss), Biogon/Hologon = symmetric wide angle
 Distagon = retrofocus.
Voigtländer also named their lenses after lens forms but also after the performance. Lens forms are:
 Skopar (Tessar type)
 Heliar (a Tessar type with a cemented first element as well)
 ApoLanthar (a variant of the Heliar but with an opposite front group)
 Ultron (Gauss type).
Voigtländer also had a lens called the Nokton, which is a high speed lens for dark environments.
On the other hand, Leica named their lenses after lens speed.
 Elmar = F3.5
 Elmarit = F2.8
 Summicron = F2
 Summilux = F1.4
 Noctilux = F0.951.2
Although they had some early lenses named by lens form like Hektor (Berek’s dog’s name), Summar, Summitar, Summarit, and Thambar. The modern Leica lenses use Summarit for F2.5 lenses. (oldlens.com )
The Japanese lens makers didn’t do either, they named their lenses
as one name altogether, but have interesting historical reasons for each.
 Nikon = Nikkor (shorthand from their company name NIpponKOgaku and a play on Sonnar,
Hektor, Planar\ldots{})  Pentax = Takumar (named after the sibling of an executive at Pentax named Takuma, and a play on the Japanese phrase
\href{https://en.wiktionary.org/wiki/%E5%88%87%E7%A3%8B%E7%90%A2%E7%A3%A8}{sessatakuma} = apply oneself and study hard to improve oneself)  Olympus = Zuiko (named after the optical research facility, and a play on the word zuiko = auspicious light)
 As an aside, Olypus made a play from their original company name Takachiho Seisakujo, and the city Takachiho, which is the land where the first Gods landed in Japanese mythology. A more international version of the land of Gods is Mounnt Olympus, hence the name change.
 Mamiya = Sekor (named after their optical research center SEtagaya KOki)
 FujiFilm = Fujinon
 Minolta = Rokkor (named after the mountains near their headquarters, \href{https://en.wikipedia.org/wiki/Mount_Rokk%C5%8D}{Mount Rokko})
 Yashika = Yashinon
 Konica = Hexanon (Konika was called Konishiroku, after founder Konishiya Rokube, and since roku = the number 6 in Japanese, this led to Hexa, the Greek numeral for 6)
 Topcon = Topkor
 Ricoh = Rikenon
Some of these may sound familiar, others less so.
Nikon would name their early lenses with Q, P, H and S, presumably for the Latin/Greek prefixes numerals like Quadr (4 lens elements), Penta (5 lens elements), Hexa (6 lens elements), and Septua (7 lens elements).
Also, there are more functional names like Telephoto, Retrofocus, 2group zoom.
It’s a little sad that lenses aren’t named more creatively today, but with the lens forms becoming so complex and diverse for performance purposes and IP purposes, I guess naming lenses is a little pointless now.
Imaging lenses: The lens forms that helps us see more than the naked eye
Our eyes work a little differently than photographic lenses with sensors, so we have to take a different approach in thinking about the lens design form.
In particular, optical lenses for the eye are afocal since the eye itself has focus properties, whereas photographic lenses focus on a plane, either film in the old days or a digital sensor today.
Let’s take a look into how the lens design forms for the eye are constructed.
The eyepiece: imaging the world on our retinas
1. History and background
Optical systems that are made for direct viewing with our eyes, such as Loupes (a simple magnifier), viewfinders, telescopes, microscopes, and many others, all share issues with the characteristics of the eye.
The visual angle:
The angle between the center of the entrance pupil of the eye and the object, in terms of the principal ray. The size of what we see is proportional to the tangent of the angle u. This angle increases as the object is moved closer to the eye, and there is a limit to how close we can get this to our eyes, but it depends on person to person, and their nearpoint.
Note: the entrance pupil of the eye is located about 3.3mm behind the cornea.
2. Essentials for lens design
The nearpoint and farpoint:
Our eyes focus by moving the muscles around the lens to change the shape of the lens and increase the optical power. There is contraction that makes the curvature of the lens larger and nearpoint is achieved. The eye continuously focuses on the retina so we see things in focus. Although there are some differences from person to person, in general younger people can accomodate for a larger range between their nearpoint and farpoint, and this range gets shorter as we get older. For most, it’s the nearpoint that gets longer and thus the range becomes shorter.
Usually, we use the reciprocal of metre as the unit of measure of how far we can see. This is called the diopter.
The distance of distinct vision is the closest nearpoint we can get while being able to see the details of the object. This distance is 4 diopters, which is 1/250mm.
The “diopter” in an eyepiece is how much the rays from the eyepiece either converge or diverge, and it is a measure of the distance from the eye to the image. Usually we use the distance 4 diopters for lens design, but for a camera viewfinder we may use 1 diopters instead.
The magnification of a loupe or simple magnifier is the ratio between the tangent of the viewing angle of the object through the magnifier and the tangent of the viewing angle of the object without the magnifier.
The magnification of the loupe/magnifier is expressed as
$$
\beta = \frac{250}{f}
$$
where the magnification is \(\beta\), and the focal length is \(f\). Your eye has to be at the focal point of the magnifier, though 😉
In exact optical ray path terms, the focal point of an eyepiece and the entrance pupil of our eye does not match up perfectly, but this is the equation used to get the numbers for magnification in general.
I think I have talked about a lot of different types of magnification like transverse magnification and longitudianl magnification, and even angular magnification, but this magnification is a little different.
The other three magnifications are all conjugate systems, while the optical system for the eye is not a conjugate system.
Most eyepiece optics are afocal. The aberrations for an eyepiece and optics using an eyepiece allow about 3 minutes of arc in general. This means that a perfect eyepiece designed into the eyes have parallel rays, but we can afford less than 3 minutes of arc from parallel rays and still be fine, in most cases.
This is because it is said that the human eye has a resolution of one minute of arc, and for observational optical systems 3 minutes of arc is an acceptable compromise.
3. Where you can use knowledge used in other lens designs
Telescopes use a telescope objective and an eyepiece.
Periscopes are also optically similar to telescopes.
Binoculars are optically similar to telescopes, but use two lens systems together (and a Porro prism).
Microscopes use a microscope objective and an eyepiece.
Optical view finders and prisms for SLR camera objectives are optically similar to the microscope. They have the image of the film surface on a ground glass, and the eyepiece takes the image of the groundglass into our eye.
1. Lens assembly, 2. Mirror, 3. Focalplane shutter, 4. Sensor/Film, 5. Focusing screen, 6. Condensing lens, 7. Pentaprism, 8. Eyepiece. By en:User:Cburnett – Own work with Inkscape based on Image:Slrcrosssection.png, CC BYSA 3.0, Link
For that matter, electronic view finders are also similar to microscope systems and OVF systems, since there is an LCD in the digital camera that shows the image on the sensor. This image on the LCD is then imaged to the eye, which is exactly the same thing.
Riflescopes are a longrange finite to finite conjugates, and there are zooms as well.
Headup displays (HUD) and Head mount displays (HMD) are glorified loupes, in the optical sense. The shape may be very complex, with freeform surfaces and offaxial surfaces, but the essence of the lens design is the same as the loupe / magnifier.
And of course, normal eyeglasses whether they be nearsighted or farsighted, use the same optical principles for lens design.
4. Tips and tricks
When we design an eyepiece, it is often useful to flip the lens design around and trace the rays from infinity (where our eye will eventually be) to the lens, and the focal point is where the object will be.
Set the first surface as the entrance pupil of the eye, and build the lenses from there. The diameter of the entrance pupil is usually about 3~4mm in diameter, but we can increase it to a diameter of 7mm (the maximum diameter of a dilated pupil) just in case.
The distance between the eye point and the first lens is the eye relief distance, and this is usually about 15~20mm for a normal eye piece. For a riflescope, where the kickback from the rifle is large after shooting, a longer eye relief of about 90mm might be more suitable.
5. Master the specsheet
For an indepth document of the history of the eyepiece, check out this report, it’s more than I could write about the subject. Enjoy!
6. Realworld examples
I have some examples of typical eyepiece lens design forms below. As a preface, the focal lengths are all 100mm.
6. Realworld examples
I have some examples of typical eyepiece lens design forms below. As a preface, the focal lengths are all 100mm.
Singlet eyepiece
Huygens eyepiece
The Huygens eyepiece is originally two planoconvex lenses with the convex side facing the objective lens. The first Huygens eyepieces were made of more simpler glass, like BK7. The objective side lens is called the field lens, and the eye side lens is called the eye lens. Interestingly, the Huygens eyepiece has a virtual image in between the two lenses, so in a sense, the field lens can optically be part of the objective lens. It is possible to correct the transverse chromatic aberration very well. Later, the Huygens eyepiece used meniscus lenses, which slightly improves the performance. The most important point of the Huygens eyepiece is its ease in manufacturing, especially with two planoconvex lenses. It is bright but can’t have a large magnification. Some disadvantages of the Huygens eyepiece is the longitudinal chromatic aberration and large field curvature.
The Ramsden eyepiece is two planoconvex lenses with the convex surface facing each other. There is a significant amount of chromatic aberration which does not make it ideal for telescopes. Compared to the Huygens eyepiece, the focal point of the Ramsden eyepiece is outside of the two lenses, so placing patterns like a scale of a scope is easier.
Kellner eyepiece
The Kellner eyepiece tried to solve the chromatic aberration problem by adding a doublet, and can be used for a wider field of view. Although there is significant astigmatism, field curvature, and distortion, it can still be used for a fairly wide field of view. The Kellner eyepiece was popular in the old days but we don’t see them much anymore.
Plossl eyepiece
Plossl eyepieces are a lowcost solution because of the symmetrical shape, and we only have to design one doublet, flip one, and then put them together. The cemented lenses are used for colour correction, and can be many variations of glass materials. Of the more classic eyepiece lens design forms, the Plossl is still used in many eyepieces today.
Abbe orthoscopic eyepiece
The Orthoscopic eyepiece, also called the Abbe eyepiece, since Abbe presented this lens design for a microscope eyepiece. Although expensive to make due to the threecemented lenses, it is very well corrected chromatically and has low distortion. Although the eyepoint is relatively close and the brightness is lower than other lens designs, the Orthoscopic eyepiece still has more advantages that make it one of choice for even today’s eyepieces.
Erfle eyepiece
The Erfle eyepiece is a wide angle eyepiece that was first designed for binoculars but is also used for astronomical eyepieces.
Bertele eyepiece
Bertele eyepiece (wide)
Astronomical eyepiece
The telescope lens: The first steps to the dream of space travel
1. History and background
Telescopes are one of the oldest lens design forms, a lot earlier than photographic lenses. This is probably because film emulsion technology developed much later, and the camera obscura only had limited applications compared to a telescope. The telescope had scientific applications for studying distant objects in astronomy, and military applications as well.
There are two major telescope objectives:
 The Keplerian type telescope
 The Galilean type telescope
By Szőcs Tamás – Own work, CC BYSA 3.0, Link
Johannes Kepler developed the Keplerian telescope, and it is a positive objective lens and positive eyepiece lens combination. Since it is two positive lenses, the image is flipped around toptobottom and also lefttoright. This is okay as far as astronomical observation, but not practical for landscape viewing, especially in a military situation.
By Tamasflex – Own work, CC BYSA 3.0, Link
Galileo Galilei developed a different configuration that solved the image flipping issue, he used a positive objective lens and a negative eyepiece lens. Although the image is upright in this configuration, there is no place we can place a physical aperture stop, since the exit pupil is a virtual image that is inside the lens. The field of view of this configuration diminishes considerably. You can see that the pupil position play a great role in telescopes. Interestingly, the Galilean telescope is not invented by Galilei, but was filed for a patent a year earlier by Hans Lippershey and Jacob Metius independently from each other. Zacharias Janssen’s name is also mentioned making it debatable who the actual inventor is.
Most of the telescope analysis in the rest of this chapter focuses on the Keplerian telescope, which is more complicated with a wide use case, and frankly more interesting than the Galiean telescope configuration (Sorry Gallileo).
2. Essentials for lens design
Along with the eyepiece, the telescope is also an important lens form that uses stops and pupils in order to work the way that they do.
The typial telescope is shown below. If you remember, I talked about the achromatic doublet as a telescope objective. The telescope has two components to it, the objective and the eyepiece.
The objective creates an enlarged image of the object at infinity, and the eyepiece takes that intermediate image and creates a virtual image at infinity that we can see. To illustrate the system, the image above shows the object and virtual image at a finite distance. I think this is a better conceptual description.
In order for the eyepiece to function properly, the focal plane of the eyepiece is matched to the focal plane of the objective lens.
To describe the performance of a telescope, we generally refer to the angular magnification \(\gamma\), which is
$$
\gamma = \frac{h_i}{h_o} = \frac{L \tan{\theta}}{L \tan{\theta_o}}= – \frac{f_o}{f_e}.
$$
(where \(h_o\) is the height of the object, \(h_i\) is the height of the image, for convenience \(L\) is the distance of the object to the lens, \(\theta_o\) is the half field of view of the objective lens, \(\theta\) is the half field of view of the eyepiece, \(f_o\) is the focal length of the objective lens, and \(f_e\) is the focal length of the eyepiece)
In Gaussian optics, the \(\tan{\theta}\) and \(\sin{\theta}\) is the same as \(\theta\), but in actual lens design is it a good idea to use the full trigonometric functions.
3. Where you can use knowledge used in other lens designs
The telescope is an objective lens combined with an eyepiece. These are two lenses with distinct focal lengths each.
A similar configuration with different focal lengths is a microscope system, which enlarges a closefocus object with an eyepiece. The only optical difference with the telescope is the focal length of the objective lens.
Some large range zoom systems have an intermediate image to help with the magnification. In this case, the second lens is not an eyepiece, but a focus lens, and a similar type of logic with an intermediate image applies.
Riflescopes, and binoculars are also similar optical systems.
Some projector systems create an intermediate image (of the LCD, for example) with one lens, and then project the intermediate image onto a screen with a projection lens, another similar optical system.
4. Tips and tricks
Stops and pupils in telescopes
Most telescopes and binoculars have the entrance pupil on the objective lens, and the exit pupil is after the eyepiece. This distance of the eyepiece to the entrance pupil is called the eye relief. The aperture stop is usually located at the intermediate image.
Take a look at the image above, it’s a schematic diagram but represents the system well. The blue rays are the onaxis, and the green rays are at an angle.
The blue rays focus at the intermediate focal plane which is the rear focal plane of the objective lens. This focal plane is also the front focal plane of the eyepiece, so the blue rays then become parallel rays after exiting the eyepiece lens.
The green rays also focus at the intermediate focal plane which is the rear focal plane of the objective lens. Again, this focal plane is also the front focal plane of the eyepiece, but this time the green rays exit the eyepiece at an angle, but also parallel.
If you have a chance to look at a pair of binoculars or a telescope, try to look at the exit pupil of the lens. You can see the exit pupil floating in the air if you get the angle right.
The mistake I made with aberration correction of a telescope system
I do a lot of practice lens designs during my work time. It’s not like Google’s policy of having 20% of your work time to do anything you want, but I am able to take a little bit of time here and there, in the name of educating myself, to work on lens designs that may not be related to any of our products. I have a lens design mentor who is an advisor to our company, and he regularly gives me problems to solve. Doing these exercises has really upped my lens design game.
The telescope system I was tasked to design was not a immensly difficult design, and I proceeded to design the system in two parts: The objective lens, and the eyepiece.
I improved the objective the best I could, which is not difficult because it is a long focal length system and all I had to worry about was the spherical aberration and the chromatic aberration. And a little bit of coma.
I then proceeded to design the eyepiece, as I tested a few eyepiece design forms that was the best. I made sure the chromatic aberration, spherical aberration, and field curvature were as small as possible.
This is typical for commercial eyepices and telescope objectives, because we have to have multiple objectives that work with multiple eyepieces, mix and match for different magnifications.
All I had to do now was to dock the two systems together, right? I flipped the eyepiece design around in the software (this is a useful tool when we design lenses in the opposite configuration of actual use), and added it to the objective lens. I didn’t forget to change the system into an afocal system since there is no “point of focus”, and the evaluation of the system would now be in diopters, not millimeters.
Since both systems had the minimum error possible, I was convinced that I had a winner of a design. There was just no way in my mind, because both were maximally optimized! Right?
My mentor took a look at my lens design and said: “Did you design these separately?” He knew exactly what I did by looking at the various parameters in the system.
I replied, “Yep, they’re both optimized really well!” beaming with pride.
His next question: “So how do you know that your telescope is good?”
I replied, “Because I used an achromatic doublet, which cancels out the colour…” and I stopped as I realized that the same thing could be done with the objective lens and the eyepiece.
We can leave a little bit of chromatic aberration in the objective lens and then correct it with the eyepiece. We can intentionally put in a bit of overcorrected field curvature to cancel the field curvature that can’t be corrected by the eyepiece alone.
Sometimes it’s important to have fully optimized lens components on their own. Sometimes that becomes an unnecessary constraint on our lens design. In commercial systems with interchangeable lenses, it may make sense to make two perfect lens designs and just stick them together. But for a custom system, it may be better to balance the performance as a system. Even for a commercial system, if two or three objectives share the same aberration properties, it would be sensible to make the eyepieces match those aberrations to counteract the aberrations and make a nice image overall.
And my mentor knew the exact questions to ask that would lead me to the conclusion ON MY OWN. Another lesson in lens design (all part of the journey).
5. Master the specsheet
A few terms to note when designing a telescope:
 The focal length of the objective lens
 The focal length of the eyepiece
 The FOV of the system
 The magnification of the system
 The eye relief
6. Realworld examples
Besides the obvious telescope, binoculars are a worthwhile example to explore.
Typical binoculars are double Keplerian telescopes in an optical sense.
Binoculars fix the image flipping issue with a porro prism to make the virtual image upright to the eye.
By Antilived – Own work based on: Binocularoptics.png, CC BYSA 3.0, Link
In modern binoculars there are a lot of innovations to make the prism as small as possible, for better handling in use.
The microscope lens: The quest to enlarge the microscopic world
1. History and background
(Just to be clear, I’m going to be talking about optical microscopes, not AFMs and SEMs and other nonoptical microscopes)
Photographic objectives are generally reduction optical systems, but in contrast, microscopes are enlargement optical systems. In that sense, it may seem like they are opposites of each other, but the method to approach lens design is fundamentally the same.
That doesn’t mean that there aren’t unique points about the microscope, we have to think of the system and metrology to get a good lens design. It’s all about specifying the system.
Historically, people would try to enlarge an object with a single lens, usually a loupe. The shorter the focal length, the larger the magnification, but at a certain point, the distance between the object and the lens (working distance) will be extremely close, and therefore the distance between the lens and your eye (eye relief) will also be small.
This means simple magnifiers like a magnifying lens can’t make the magnification too large, and it is typically 10x or a few 10x.
In the early 1600s, someone had the bright idea (the inventor is disputed) to place an objective lens very close to the object, but instead of sticking the eye close to the image, another magnifier lens was used to look at the image the first magnifier produced.
This other magnifier is called the eyepiece. This entire system is called a compound microscope.
The invention of the compound microscope is brilliant. The objective lens magnifies the object to an image, but the eyepiece gives an additional magnification so that the image can be enlarged with high magnification while the distance from the object to the eye is far away.
2. Essentials for lens design
For a microscope, if we think about the back focal point of the objective lens being the stop, we can place the image where the light source is going to be.
The chief ray goes through the stop, so the chief ray at the object is parallel to the optical axis. The entrance pupil is at infinity, and we call this situation telecentric.
Why do we want telecentric rays for a microscope?
If the chief rays are not perpendicular to the image (parallel to the optical axis), the defocus blur during focusing will be different across the screen, an is a problem for usability.
Here’s an example of a microscope objective.
According to the diffraction limit, the resolution of an optical system is
$$0.61 \lambda \div NA$$
For optical microscopes, the wavelength \(\lambda\) we use is 546nm, or green.
The NA is the numerical aperture and represents how much light enters the system, and is usually represented with the equation
\[NA = n \sin{\alpha}\].
You might recognize this as similar to the Fnumber, which is \(F/\# = 1/2 NA\). NA and Fnumber are interchangeable, but from my experience, I usually like to define NA as an object side parameter and Fnumber as an image side parameter. (Not exclusively, though)
3. Where you can use knowledge used in other lens designs
The microscope is an objective lens combined with an eyepiece. These are two lenses with distinct focal lengths each.
A similar configuration with different focal lengths is a telescope system, which reduces a far object (like a star in the sky) with an eyepiece. The only optical difference with the microscope is the focal length of the objective lens.
Some projector systems create an intermediate image (of the LCD, for example) with one lens, and then project the intermediate image onto a screen with a projection lens, another similar optical system.
4. Tips and tricks
The aberrations of a microscope objective
The field of view of a microscope is small, so the aberrations that we concentrate on are the spherical aberration, the coma, and the chromatic aberration. Although the focal length is short, since the microscope is an enlargement optical system the longitudinal aberrations become pronounced.
It is good practice to design the lens as if it is a reduction optical system, in the opposite direction of actual use.
Like for the image above, the object to be magnified clearly should be at the right. But if I were using optical design software, the rays are going left to right, so in fact I have the setup that has starts opposite of what it should.
Regardless of the configuration, it is important to eliminate the aberration. Also, the final design should be based on wavefront aberration rather than ray aberration, although ray aberration is faster and therefore better to use in the early stages of the lens design.
As the magnification increases, the lens power increases, and the magnification affects the transverse chromatic aberration. In general, the transverse chromatic aberration is difficult to correct with the objective lens alone, so Abbe took the method of leaving the opposite transverse chromatic aberration in the eyepiece to cancel out the aberrations as an entire system. This is called the compensation method.
The compensation method does have a disadvantage in that the objective lens needs to be paired with the eyepiece if they correct each other. A wellcorrected eyepiece used with a microscope objective that has aberrations will not work well, and conversely, an eyepiece that has the opposite aberration of a certain microscope objective can’t be used with an objective with a more modest NA that has wellcorrected aberrations.
The obvious thing to do was to make a chromatic aberration free method. (shown below in “Realworld examples”)
Index of refraction of the microscope slide
If you’re designing a microscope used in a typical biology lab, we put the sample in a microscope slide. Don’t forget to include the thickness of the glass and the index of refraction of the glass. Also, if there are any oils or other liquids used for immersion, then we have to take into account the index of refraction of those liquids as well.
5. Master the specsheet
The objective lens
There are several microscope objective lens designs, which have varying performance depending on the use.
 Achromat objective: the spherical aberration, coma, astigmatism is well corrected, but has residual field curvature. The colour correction is for two wavelengths, usually the C line (656nm) and the F line (486nm).
 Semiachromat objective: The secondary spectrum is smaller than an achromat, but larger than an apochromat. Since the chromatic aberration correction is better, it is possible to make the NA larger.
 Apochromat objective: Has great correction over the entire visual spectrum, and the NA is large.
 Plan objective: The image of this objective is flat and good across the field, ideal for microscopes. In exact terms, most Plan objectives are Plan achromat objectives, so they have the highest of performances for a microscope objective.
6. Realworld examples
Lens design forms of the microscope objective
There are two types of lens design forms for the typical microscope objective, the Lister objective and the Amici objective. A microscope that requires a higher NA has an aplanatic lens or a field flattener on top of either of the two lens types.
By https://wellcomeimages.org/indexplus/obf_images/b1/93/6417ef8dff8aa74fdcd7a9529c3d.jpg
Gallery: https://wellcomeimages.org/indexplus/image/M0010611.html
Wellcome Collection gallery (20180331): https://wellcomecollection.org/works/qvuszbsv CCBY4.0, CC BY 4.0, Link
By Fondo Antiguo de la Biblioteca de la Universidad de Sevilla from Sevilla, España – “Microscopio horizontal de Amici”., CC BY 2.0, Link
Chromatic aberration free method
The chromatic aberrationfree microscope objective has three components, the front, middle and rear groups.
The front group is used for undercorrection, the middle group is used for overcorrection, and the rear group corrects the chromatic aberration.
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Imaging lenses: Modern use cases based on new lens design forms
I classify “modern” after the invention of lasers in 1960.
After the invention of lasers, it opened up a world of new possibilities in terms of lens design and optical systems. For example, there are lens designs that can be used with lasers, and there are fabrication technologies that were made possible with lasers and similar technology.
Stepper lenses
Lens design in Ultraviolet
Short wavelengths in Ultraviolet (UV) are the opposite in properties to the long wavelengths of infrared (IR).
Usually, IR in the 850nm to 1300nm range still has the same difficulties as the visible spectrum of 450nm to 680nm (or so). But when we get into far IR, with wavelengths of 5 microns, 10 microns, 20 microns, the index of refraction issue is smaller, the roughness of the lens is less of an issue, and in general, the tolerances are looser and the aberrations are less of a problem.
Given that UV is the opposite of IR, that means that the tolerances tighter for UV than the visual spectrum. The aberrations must be better corrected in UV than visual, much much better than IR.
Another issue with UV is that there are very few optical glass materials that are usable below 400nm. Some examples are fused quartz (SiO2) for as short as 250nm, and Fluorite (CaF2) for as short as 150nm.
Stepper lenses for Microlithographic process
Stepper lenses are lenses are used to produce integrated circuits and computer chips with a lithography process and requires ultra precision lenses. The lenses need to be of very high image quality, highresolution, and the image has to be in the order of a few hundred nanometers, which is the size of circuits today.
Since the diffraction limit is proportional to the wavelength (\(0.61 \lambda \div NA\)), in order to image small sizes the wavelength needs to be short, thus UV. Also, we see that the larger the NA, the smaller the image, so very fast lenses are also needed. A quick calculation shows that for a wavelength of 250 nm and an NA of 0.5 (or Fnumber 1.0), the diffraction limit is about 0.31 microns, so we get an idea for what size of circuits are fabricated.
Other issues we need to note are the antireflective coating of the material, as the transmission should be 100% for microlithography.
As far as chromatic aberration, this can be corrected with a combination of SiO2 and CaF2, but we can also rely on monochromatic illumination so that there is very little chromatic aberration, to begin with.
Lens design is nearly afocal and bitelecentric, and the Petzval sum needs to be small since any field curvature in the image will degrade the microlithography patterning.
A stepper lens needs to have high resolution so the Fnumber is fast, and the aberrations must be as close to an ideal lens as possible. Correcting the spherical aberration for resolution and the field curvature (Petzval sum) for image flatness is done with many lenses to eliminate the aberrations.
Compound the Fnumber, the resolution expectation, the correction of aberration, along with a high field of view, and the optical lens design is extremely difficult. Couple all of that with a bitelecentric lens design and we have one the highest level lens designs in a stepper lens.
Since the resolution is achieved with shorter wavelengths, there are fewer glass types that become usable due to transmission concerns. Fewer glass choices also make the lens design process more difficult, as field curvature, which is the sum of the optical power divided by the index of refraction. With few glass choices, we essentially are giving up a degree of freedom in the Petzval sum.
$$
P_z = \phi_1/n_1 + \phi_2/n_2 +… \phi_k/n_k \\
\phi_t = y_1 \phi_1 + y_2 \phi_2 +… y_k \phi_k
$$
The concept of splitting lenses to decrease the aberration is in full effect in the image above. Splitting lenses are done to decrease the curvature of each surface while keeping the overall optical power the same. Having two lenses with less curvature with an equivalent focal length to one lens with high curvature is better for the decrease of aberrations.
We can see that many lenses in the image above are an aplanatic shape so that the aberrations are as small as possible during refraction.
Marginal ray paths show the large ray heights at positive elements and small ray heights at the negative elements correct the Petzval sum. This is because the lens materials are largely similar, and using the equation for Petzval sum (\(\sum_{j=0}^m n_j / R_j\)) is approximately zero.
This basic idea we have seen in photographic lenses, but there are so many more lenses for the stepper lens because of the level of correction that we need.
For the above example, the meniscus lenses at the end correct field curvature. This can be seen for microscope objective lens design and the Planar F0.7 lens. These meniscus lenses near the object/image used to achieve telecentricity.
Manufacturing stepper lenses
Boy, these lenses are large! Large lenses are expensive, and the level of precision needed to make these lenses only adds to the price. A perfect lens design needs perfect fabrication, or the lens cannot function as we need it to. The tolerances is to the order of a wavelength, and even the assembly is done with lithography. The final cost of a stepper lens for microlithography is in the millions of dollars.
These stepper lens manufacturers have lens polishing experts, who do an extraordinary job in creating these microlithographic lenses. You can say it is at an artisanship level.
Factors affecting the quality of the surface can be the mechanical properties of the material like hardness, to temperature affect during polishing, the flow of the abrasive material on the surface of the glass, and manufacturing issues of that nature.
One important thing to note is that the surface roughness can’t be measured with conventional measuring tools. At the same time that the surface needs to be the designed shape, the roughness has to be to the order of a few Angstroms. Yes, you read that right, lower than a nanometer in measure.
The difficulty of manufacturing a microlithography lens is extraordinary. Even more than huge reflective telescope objectives like the Subaru telescope in Hawaii
All of these issues are not directly related to the lens designer, but knowing about manufacturing properties and effectively transferring them to the lens design is the difference between an amateur lens designer and a professional lens designer. Professional lens designers know more about the manufacturing process, and take into account how the lenses are made, and how they are assembled together. I sometimes wonder how many lens designers truly take into account the manufacturability of their lens designs.
There is a nice paper on the evolution of the stepper lens for microlithography (by Nikon). For further reading.
The Lithographic Lens: its history and evolution – Proc. of SPIE Vol. 6154 6154031
Laser beam printers and laser scanners: ftheta lenses and their seagull like shape
A laser beam printer or laser scanner has a rotary polygon mirror that rotates to change the angle of entry into the optical system, and has field property of ftheta (instead of f tangent theta, as in most imaging systems).
Sometimes MEMS mirrors are also used.
As an optical system, the wavelength of the laser is fixed.
The mirror requires high precision for good reflection, so we don’t want it to be too big.
Also, the mirror is going to spin at very high speeds like 30,000 RPM, so if it’s too big it can be damaged or break easily.
The aperture of the system needs to be on the mirror. That means that the aperture of the lens is outside of the lenses. Also, we need space for the rotary polygon mirror, and space for the laser beam, so the aperture is way out of the lenses.
For a wide angle scan, we can see from the image below that the lens is extremely large compared to the fnumber, which is very slow compared to photographic objectives.
An ftheta lens has an image height y that is proportional to the ray angle \(\theta\). Most systems are ftangenttheta lenses, but the definition of an ftheta lens is:
$$
y = f \cdot \theta
$$
This is also known as an equidistance projection, and ftheta lenses are commonly used in laser beam printer and laser scanner systems, and fisheye lenses.
For a laser beam printer, a laser beam is collimated onto a rotating polygon mirror, and then through the ftheta lenses. The angle of the beam rotates at a constant speed, and the beam reflects from one side of the ftheta lenses to the other, in effect “scanning” through the optical system.
Since rotational speed is constant, the change in angle is constant with time, and therefore the optics need to be in an ftheta configuration in order for the scanning speed to be constant across the plane.
The lenses can be spherical, but modern scanner lenses are both aspherical and toric. This can make the lenses more compact, and a multilens system can be simplified to a two or one lens system.
The location of the reflection at the mirror is the entrance pupil. The fact that the pupil is in front of the lens and the ftheta properties makes the fnumber of the system quite slow compared to other systems. However, a laser beam has enough power that the slow optics causes no disadvantage.
In order to make the system compact while keeping the same scanning distance \(y\), we need to shorten the focal length \(f\) so that the angle \(\theta\) increases.
Since the perpendicular direction does not require scanning, the precision of the mirror is important. Any vertical tilt in the polygon mirror will cause the scanning beam to move in the vertical direction, causing irregularities in the print. One way to mitigate this is to use a cylindrical lens and focus the perpendicular side onto the polygon mirror. In this case, we need to make sure the fnumber is consistent with the cylindrical lens compared to the image.
For these realworld examples, we can tell that the aperture of the image and the entrance pupil are important for lens design.
The Aspherical lens: an addition to the spherical shape that opens up possibilities
Who doesn’t love aspherical lenses? Pop one into a lens design and it can do wonders. Seemingly, all the aberrations go away, and you can make do with one lens instead of many!
Not so fast. We can’t blindly use aspherical lenses without knowing what they can do and their limitations. Let’s dive in!
1. History and background
Aspherical lenses are popular use cases in compact disc lenses, reflective astronomical telescopes, Fresnel loupes, autofocus condenser lenses, compact camera zoom lenses, mobile phone lenses, highend photographic zoom and prime lenses, television cine zoom lenses, stepper lenses, and various illumination lenses just to name a few.
Unlike spherical lenses, aspherical lenses can’t be polished in the traditional way, and require very precise tooling and metrology. This is bound to pull the industry forward from a technical perspective.
Today, there are not only injection moulded plastic lenses, but also glass moulded aspherical lenses as well, so the selection of lenses for an optical lens designer is abundant.
Let’s take a mathematical look at aspherical lenses for just a second, and get into a more conceptual analysis of the aspherical lens. I don’t like to say this too often, but math can help understand the concepts in most situations.
A ray passing through a surface with a spherical shape with a refractive index is advanced through the system by refraction. The laws of refraction behave as Snell’s law,
$$
n sin{u} = n’ sin{u’}
$$
where \(n\) is the index of refraction before the refracting surface, \(u\) is the ray angle before the surface, and \(n’\) is the index of refraction after the refracting surface, and \(u’\) is the ray angle after the surface.
Simple enough, right? Okay, if we introduce a surface shape to this equation, we might express Snell’s law like this:
$$
n sin{\xi – u} = n’ sin{\xi – u’}
$$
where \(\xi\) is the normal angle to the surface at some angle.
From the above equation and diagram, we can see that it is possible to change \(u’\) while keeping \(u\) the same when we change \(\xi\). For a spherical surface, \(\xi\)determined by the ray height of the lens. By using aspherical surfaces, it’s possible to change the incident angle to the surface without changing the curvature, and therefore the focal length. By controlling the refraction angle at key points in the pencil of rays, it can be possible to create an aberrationfree lens design.
2. Essentials for lens design
An aspherical lens, by definition, is a lens with at least two surfaces, with one or both surfaces are not spherical, including plane surfaces. Cylindrical and toroidal surfaces, which are not rotationally symmetric and are not spherical but have spherical crosssections are not considered aspherical surfaces.
The terms \(A_4\), \(A_6\), so on are called aspherical coefficients, and along with the conic constant \(k\) is part of the aspherical surface. We can tell by the equation that as the coefficients increase in power, the coefficients have a larger effect on the shape. \(A_2\) has a smaller effect than \(A_{12}\) on the surface shape. If you’re crazy you can use as many terms as you like, \(A_{20}\), even.
A surface that is aspherical, in a literal sense, is a surface that is not a simple sphere. However, when we say that a surface is aspherical in lens design, we usually mean the following equation:
$$
z = \frac{cr^2}{1+\sqrt{1(1+k)c^2r^2}} + A_{2}r^2 + A_{4}r^4 + A_{6}r^6 + A_{8}r^8 + \cdots
$$
where \(z\) is the lens sag, \(c\) is the curvature of the lens, \(k\) is the conic constant, and \(r\) is the radius. \(A_n\) is the aspherical coefficient.
The first term in the equation above is the same as a spherical lens, which we call the base surface. Any changes to the surface are caused by the aspherical coefficients that follow. By changing \(k\), \(A_4\), \(A_6\) appropriately, we can change \(\xi\) and therefore gives us many extra degrees of freedom. In modern lens designs, we are using these extra degrees of freedom to not only decrease the aberrations of a given lens surface but to also counteract aberrations from other lens surfaces. Also, aspherical surfaces for modern lens design are used to achieve a certain performance or to raise the overall performance not necessarily linked to aberration correction.
To get the most out of our aspherical surfaces, it is important to understand the effects of the base shape of the surface or the underlying spherical shape. In general, the spherical shapes should have a small an aberration as possible, before the addition of the aspherical terms. This is because in general, if the aberrations of the spherical shape are small to begin with, and the subsequent adding of the aspherical terms are small as well, and help the ease of manufacturing.
On the other hand, we may find some lens design concepts that treat the base surface essentially as a dummy surface and lend the aberration correction to the aspherical terms. I personally don’t like this method, but for some optical systems it may be the only choice we have.
One thing you may notice about the equation above is that the aspherical terms are even powers. There are more sophisticated aspherical surfaces that have the odd terms as well. I won’t get into it in too much detail here, but these odd terms are useful to correct the higher order aberrations.
For the discussion of the aspherical lens going forward, the vertex of the surface is the point of reference, and it is rotationally symmetric.
3. Where you can use knowledge used in other lens designs
Aspherical surfaces are the basis for more complex surfaces. Biconic surfaces, extended polynomial surfaces, toric surfaces, odd aspherical surfaces share properties with aspherical surfaces.
Aspherical lenses are useful in many situations, but it is important to know their advantages and disadvantages, and even their limitations, for effective use. For example, there are things that even aspherical lenses can’t do. Let’s take a look.
If we take a look at the aspherical portion of the 3rd order aberrations, we have something like this:
 Spherical aberration \(= y^4 \cdot \psi\)
 Coma \(= y^3 \cdot y_p \cdot \psi\)
 Astigmatism \(= y^2 \cdot {y_p}^2 \cdot \psi\)
 Petzval sum \(= 0\)
 Distortion \(= y \cdot {y_p}^4 \cdot \psi\)
Where \(\psi\) is some constant, \(y\) is the marginal ray height, and \(y_p\) is the principal ray height.
1: Gaussian optics are not affected by the aspherical contributions. Only the spherical surfaces affect the Gaussian parameters, which means that the following cannot be changed or corrected with aspherical surfaces:
 the focal length
 the back focal length
 Petzval sum (and therefore field curvature)
 chromatic aberrations (we’ll still need our extralow dispersion glass 😉 )
These seemingly fundamental properties are unaffected by the aspherical surface. In a sense, that’s a huge constraint in our design. It means that if we didn’t understand or stop to think about the above restrictions, we could mistakenly use an aspherical surface hoping for the best, when in fact, we do not get the result we want.
Let’s say we wanted to add an aspherical lens in the hopes that we could decrease two lenses into one, but we could end up with a lens that curves so much that the edge thickness is less than zero, which is a fictional lens. The only way to fix such a lens is to make it thick enough to accommodate the thin edge, which again may be an unreasonable thickness for manufacturing.
A prime example is for certain lens systems, using an aspherical surface doesn’t mean we can eliminate an expensive extralow dispersion (ED) glass lens.
2: We cannot correct the coma and astigmatism independently. Since \(y\) and \(y_p\) are in both aberrations if you fix one, the other changes as well. This means that there is a limit to the performance in terms of the coma and astigmatism.
3: To control the spherical aberration independently, use an aspherical surface at \(y_p = 0\), or near the entrance pupil. This we can see from the distortion equation.
4: Conversely, to control the distortion independently, use an aspherical surface near the image plane, or in other words, near \(y = 0\). This acts as a field flattener close to the image plane.
5: The spherical aberration can be corrected to zero with one aspherical surface. This can be very useful for zoom lenses where the lens that moves can be made compact. This obviously makes the lens lighter for easier movement, and the thinner lens can move further within the zooming space. As a side note, it is possible to correct the spherical aberration to zero with only the conic constant if the rays entering the lens are parallel.
6: The distortion and coma have the opposite sign about the stop. This is because \(y\) is an odd power for both. By using this property, we can:
 Correct the spherical aberration independently with an aspherical surface near the stop, as we’ve seen above.
 Correct the distortion with an aspherical surface near the image plane, like a field flattener, as we’ve seen above.
 Correct astigmatism and the coma with two aspherical surfaces on both sides of the stop.
Note that any chromatic aberration and field curvature (Petzval sum) has to be corrected with the curvature of the lenses (exceptions later).
7: For a thin lens, since \(y\) is not that different on either surface of the lens, it doesn’t matter too much which side the aspherical surface is placed. This is useful for a meniscus lens, we can make the aspherical surface on whichever side is easier to fabricate or measure.
All in all, aspherical lenses are very useful, and most times essential for modern lens design, but for maximum effect, it’s good to know their properties so we can take full advantage of them.
4. Tips and tricks
The conic constant
k is the conic constant of the surface. Without any aspherical coefficients,
 If \(k = 0\), the surface is spherical
 If \(k = 1\), the surface is a paraboloid
 If \(k \lt 1\), the surface is a hyperboloid
 If \(0 \gt k \gt 1\), the surface is an ellipsoid
 If \(k > 0\), the surface is an oblate ellipsoid This is a useful way to handle the conic constant, and use it to our advantage in lens design.
For example, if we set \(k = 1\), and if the radius of curvature is zero, the aspherical equation simply becomes
$$
z = A_{2}r^2 + A_{4}r^4 + A_{6}r^6 + A_{8}r^8 + \cdots
$$
which is a flat surface with aspherical components. That means that the Gaussian optics of this surface has no power, and the surface can be used to correct aberrations slightly.
For most imaging applications, aspherical coefficients to the \(r^{10}\) term are plenty, but as optical lenses become more and more complex, \(r^{12}\), \(r^{14}\), and \(r^{16}\) are becoming more common.
The shape of an aspherical surface
For a spherical lens, the shape deviation from a spherical surface is measured with interference, but an aspherical surface needs a few more parameters to effectively evaluate the lens design.
 Figure: This is equivalent to the spherical surface measurement, where the overall shape of the aspherical lens surface is leastsquare fitted to an equation, and the differences are compared to the designed aspherical surface.
 Accuracy: This is simply the largest difference of the actual surface compared to the designed surface. Some terms used are PV, standing for “peaktovalley”.
 Smoothness: We also look at sections of the lens for differences in tilt from the designed surface for small areas.
How to scale an aspherical lens
Scaling a spherical surface is rather straightforward. If we want to make the lens twice as large, we multiply the radius of curvature by 2. If we want to make the lens half as small, we multiply the radius of curvature by 0.5.
It’s not as simple with an aspherical surface. Let me explain.
The conic constant is simple, it does not change with the scaling of the lens. If a surface has a conic constant of 1, it is 1 regardless of the surface scaled 2x or 0.5x.
The even aspherical coefficient is the linear sum of \(A_4\) and \(A_6\) (and so on…), we can think in terms of \(z = A_j y^j\).
Let’s say we scaled the aspherical surface \(z = A_j y^j\) by \(m\), we got a surface in the form of \(z = B_j y^j\). The coordinate on the aspherical surface is scaled from \((z, y)\) to \((mz, my)\), so in exact terms, \(mz = B_j (my)^j\) and \(z = A_j y^j\) are coupled to each other, and we can solve the two to get
$$
B_j \frac{m^j}{m} = A_j \\
B_j = \frac{A_j}{m^{j1}}
$$
This means that the scaling of the aspherical coefficients are different for each coefficient, and we can scale them by \(1/m^{j1}\) for each.
Here’s an example of an aspherical lens scaling:
1x  3x  

Radius, R  10  30 
Conic constant, k  1  1 
Coefficient for \(r^8\), \(A_8\)  0.12  \(0.12 \div 3^7 = 5.49 \times 10^5\) 
The optimization of an aspherical surface
Let’s think about the optimization of an aspherical surface.
A finished sphericalonly lens design is computationally at a valley in terms of the performance. That’s to say that the performance can’t be improved any further by brute force. In this case, it’s possible that a simple addition of aspherical surfaces won’t improve the system any further. The rule of thumb is to modify the lens design slightly or to add another target in the computation.
It’s not advisable to use aspherical surfaces in a system with a ton of aberrations. Since the effect of the optimization of aspherical surfaces is large to the system, we may get a surface heavy in the aspherical without optimization in the spherical. In a system with large amounts of aberration, my suggestion is to use the conic constant only or the lowest aspherical coefficient \(r^4\) only, and slowly add higher order coefficients to the mix.
In general lens design, I usually don’t use more than the \(r^{10}\) coefficient for the optimization. Some large NA objectives for collimating systems may need more than the 10th order aspherical coefficient, and complex shaped mobile phone lenses.
One thing to keep in mind is that although the aberrations can be corrected and the spot diagram will be smaller using higher aspherical coefficients, the aberration curve and the aspherical surface shape itself can become undulatory and cause unwanted difficulties in the lens fabrication process.
5. Master the specsheet
Any time there is a confinement of space, aspherical lenses can be considered. Be careful of the properties of the aspherical lens, as some aberrations can’t be corrected by simply adding an aspherical surface.
6. Realworld examples
Below are some examples of the realworld use of aspherical lenses.
Condenser lenses
Although the calculations and theory of the usefulness of aspherical lenses were known for a long time, the precision of aspherical lens tooling had not caught up to be effective for mass production. The very first use cases of aspherical surfaces were therefore on systems that did not require a high order of precision tooling, for example, the condenser lens on a microscope illumination system.
The Köhler illumination is used for microscope illumination systems, that require a bright NA matching the NA of the microscope objective. The resulting illumination is uniform and bright over the illuminated space.
Köhler illumination images the light source onto the entrance pupil of the condenser lens, and the rays passing through the condenser lens are parallel as a result. On the other hand, the image of the entrance pupil of the light source lens is imaged to the sample, so the sample surfaces have the image of the light source entrance pupil. Any uniformity of the light source, like LEDs or halogen lamps and even incandescent light sources are eliminated and the sample surface has a uniform illumination as a result.
The Köhler illumination method is a very useful method to achieve uniform distribution. The best part about the Köhler illumination is that extreme precision is not needed to achieve the desired result, and the aspherical terms are used to help with achieving a high as NA as possible in the system, which matches the NA of the microscope objective.
Photographic lenses
For illumination systems, in most cases, there isn’t a need for super precision. But if we were to use an aspherical surface for photographic applications, we would need high precision to have the desired effect without detrimental effects. One reason why the theoretical advantages of aspherical surfaces were developed well before the application of these aspherical surfaces is that it took a long time to be able to make aspherical surfaces with enough precision to be used in most conventional optical applications.
Even still, the Noctilux 50mm F1.2 asph. caused the lens community to call them crazy in the 1960s, and legend has it that this lens was ground one by one by a lapping meister to achieve the precision needed for Leica quality lenses.
The more wide application use of aspherical lenses in photographic lenses came during the 1980s when injection moulding of plastic lenses was developed for mass production of disposable cameras. These cameras had only one lens, but an aspherical lens, which was small enough to be mass produced easily, unlike SLR camera lenses, for example.
Traditionally, large plastic lenses are difficult to make due to the large change in the thermal coefficient of expansion (TCE) and its absorbency to water in the air. Both cause unwanted problems, so the wide use of aspherical surfaces for photographic lenses happened when glass moulded lenses were manufacturable on a large scale.
Other methods are the hybrid type of lens, which is a spherical glass lens with a thin aspherical plastic layer. This has the benefits of a cheaper alternative to plastic aspherical lenses while retaining the advantages of glass over plastic.
Semiconductor laser collimator lenses
For the collimation of semiconductor lasers, the Fnumber of the lens has to match the NA of the laser. With the development of the laser and changes to the NA, the lenses developed along with it. On the other hand, most semiconductor lasers are essentially single wavelength optics, so chromatic aberration is not an issue. The size of the illumination point of the laser is small, so the field size is also very small, and the field curvature is not an issue. These two issues are precisely the issues that aspherical surfaces can’t solve, so semiconductor lenses are ideal for aspherical surfaces. Older applications are CD/DVD/Bluray collimator lenses.
Other exmaples in bullet point form:
 CD/DVD/Bluray lens. A singlet lens that focuses laser light onto the surface of the CD/DVD/Bluray disc.
 Si or Ge extreme high index for farinfrared applications
 Extreme aspherical lenses: Mobile phone lenses, ftheta lenses
 Reflective optics: Telescope lenses like the Ritchey–Chrétien telescope
 Single or multiple lenses within a lens group: video camera zoom lenses, compact camera zoom lenses, photography lenses, stepper lenses
 Fastaxis collimator for high power diode laser
 Fresnel loupes
 Auto Focus condenser lenses
 Rearprojection TVs
 Schmidt camera
 Digital projectors
 Automotive headlights and other illuminations
(Do you want some more reading? how about a link to the Edmud Optics page on apherical lenses? Click here)
Bonus: Big mistake I made when using aspherical lenses
For one of my early lens designs, I was designing a lens with aspherical lenses. The lens systems was a surveillance lens for IR applications, the wavelength was fixed, and there were three lenses, all surfaces aspherical, at my disposal.
I happily went about using the aspherical lens equation below, which was the equation
$$
z = \frac{cr^2}{1+\sqrt{1(1+k)c^2r^2}} + A_{2}r^2 + A_{4}r^4 + A_{6}r^6 + A_{8}r^8 + \cdots
$$
Mistake #1: Since I was a newbie, I thought, “Gotta use as many aspherical parameters as I can”, I mean, more is always better right? So I ramped up to the 16th power or so.
I used all the parameters I could. The more variables the more precise my result would be, right?
WRONG! (I’m yelling at my past self)
Well, that was too many factors for the application. Depending on the specification goals, we might not need the \(r^{14}\) or \(r^{16}\) parameter, as it is overkill. Had I known a bit more about which aspherical parameters affect which property, I may have been able to use my aspherical parameters more diligently.
More parameters give more degrees of freedom, but for manufacturing purposes, it gives more room for error. The slightest twitch in the aspherical constants for a higher order aspherical will likely cause more errors in the lens in production. This is what we refer to as the tolerance of the system. Also, for the fabrication process of the lens, more parameters mean that the computer program that tools these lenses will have a harder time accounting for shrinkage of the (plastic) lens, and accounting for tooling errors.
Mistake #2: In typical novice fashion, I added the A2 parameter, for the \(r^2\) component.
This is a big nono for imaging lenses or lenses that rely on a focal length (and other Gaussian optics parameters).
As you can see from the first half of the equation above, the spherical shape with the conic constant is already using \(r^2\). It’s right there in my face, \(c r^2\).
All calculations that use the focal length, like Fnumber, relative illumination, etc., all use this number. As I was optimizing for the spherical component (which is proportional to \(r^2\)), and \(r^2\) itself, I was messing up the numbers I was getting. For example, I was getting a focal length that looks good as a number, but didn’t accurately represent the shape, because it had an \(r^2\) aspherical component. I got the wrong focal length! And all other calculations based on the focal length were wrong too!
Sigh
Don’t make the mistake I made, look at your aspherical lenses carefully when you’re implementing them. More is not better, I found out the hard way 🙂
The freeform lens: Thinking of optical lens design in three dimensions
Freeform surfaces
Have you designed freeform optics before? Or have you run away from a freeform before? I’ve actually done a lot of freeform lens design, ranging from HMD (head mount display) prisms, HUD (headup display) reflective optics, illumination optics, laser scanners, and even freeform micro lens arrays.
The main benefit of freeform optics is the fact that they are not limited by having the axis at its center. Also, if the optical system you are designing is asymmetrical in some way, the aberrations can arise that can only be corrected by a freeform.
Anamorphic optical system
An anamorphic optical system has two different perpendicular surfaces. For example, a surface that has the xdirection flat and the ydirection as a spherical surface is an anamorphic lens.
Anamorphic optical systems that have afocal properties are called anamorphic afocal optical systems, and are used in combination with other optics. They can be used in combination with prisms, toric lenses, and other lenses in an optical system.
To design an anamorphic optical system, we take the meridian slice and design a normal lens system, and then take the perpendicular plane and also design the lens curvature while keeping the thickness and lens separation distances consistently.
toric lenses, toroidal lenses, cylindrical lenses
A Toric surface is a surface that is a circle rotated about an axis, also called a toroidal surface.
By Dnu72 – Own work, CC BYSA 4.0, Link
The crosssection of a cylindrical lens is spherical on one side, and flat on the perpendicular crosssection. Therefore, the optical power of the surface is different on one cross section compared to its perpendicular. One simple way to use the toroidal lens is to correct the astigmatism.
A toroidal lens can correct the distortion of an image in toptobottom and lefttoright separately, can correct the beam shape of a semiconductor laser, they can be used for a laser beam scanner. Toroidal lenses have a lot of unique use cases.
A cylindrical lens has a spherical cross section, and is the most basic of toroidal lenses. If we use a cylindrical lens on a parallel beam, one side will focus to a point while the perpendicular side will stay parallel, giving a linelike focus distribution. So the beam is not focused to a point, but focused to a line.
To design a toroidal lens, we think of two systems, and correct the aberrations in each direction separately.
A cylindrical surface is a toric surface with a rotational axis radius of infinity.
Here are some examples of cylindrical lenses and toric lenses in action.
Laser beam collimators using cylindrical lenses.
Toric lenses are used in combination to form an ftheta lens.
Zemax has a nice webinar on the design of freeform surfaces. Here is the video of the webinar. If you prefer the transcript version, you can find It here.
Some examples of freeform surfaces (equations)
The lenses used are aspherical surfaces in both the X and Y direction, and not just the crosssections. Let’s take a look at some useful equations for freeform surfaces.
XY polynomial surface
An XY polynomial surface that has an aspherical surface as its base, and then has terms multiplied by \(x\) and \(y\) to represent a shape.
$$
z = \frac{cr^2}{1+\sqrt{1(1+k)c^2r^2}} + \sum^l_i \sum^m_j A_{ij} x^i y^j
$$
Zernike polynomial surface
A Zernike polynomial surface is an aspherical surface with Zernike coefficients.
$$
z = \frac{cr^2}{1+\sqrt{1(1+k)c^2r^2}} + \sum^8_{i=1} A_{2i}r^{2i} + \sum^N_{i=1} A_i Z_i (\rho , \phi), \\
Z_1 = 1 \\
Z_2 = \sqrt{4} \rho \cos{\phi} \\
Z_3 = \sqrt{4} \rho \sin{\phi} \\
Z_4 = \sqrt{3} (2 \rho^2 – 1) \\
Z_5 = \sqrt{6}(\rho^2 \sin{2 \phi}) \cdots
$$
There are more Zernike terms, but you get the idea.
I don’t like dissecting equations, and I like to give a qualitative view on equations and how they are used. For the Zernike surface, I like to think of it as a complex freeform surface that can replicate various freeform surfaces but also little surface irregularities, expressed in radial terms.
Chebyshev polynomial surface
A Chebyshev polynomial surface is a surface defined by Chebyshev polynomials. In the optical design software Zemax Optics Studio, there is a new surface called the Chebyshev surface, defined below.
$$
z = \frac{c(x^2 + y^2)}{1+\sqrt{1c^2(x^2 + y^2)}} + \sum^N_{i=0} \sum^M_{j=0} T_i(x) T_j(y), \\
T_n(x) = \cos ( n \cos^{1}x ) \\
T_0(x) = 1 \\
T_1(x) = x \\
T_2(x) = 2x^2 – 1 \\
T_3(x) = 4x^3 – 3x \\
T_4(x) = 8x^4 8x^2 + 1 \\
T_5(x) = 16x^5 – 20^3 + 5x \cdots
$$
There are more Chebyshev terms, but we can see that with both \(x\) and \(y\) terms, the polynomials get complex quickly.
The advantages of the Zernike polynomial and the Chebyshev polynomial are that they are orthogonal surfaces, and the coefficients are comparable to each other.
A qualitative view on the Chebyshev equation is similar to the Zernike, it is a complex freeform surface that can replicate various freeform surfaces but also surface irregularities. Compared to the Zernike surface, the surface is expressed in rectangular terms, as everything is \(x\) and \(y\).
A radialbased surface like the Zernike can be useful for some situations, and the rectangularbased surface like the Chebyshev surface can be useful in other situations.
For further reading, Zemax offers a Knowledge Base article on how to implement the Chebyshev surface , and a PDF of a paper explaining results between the Chebyshev polynomial compared to other freeform surfaces.
The mobile phone lens: Taking aspherical lenses to the extreme
I have two examples of mobile phone lenses. One is a relatively simple three lens system, and the other is a more complex lens systems with more lenses. We can break down the lens design by looking at the lens the rays, and looking at the material (index of refraction and Abbe number).
Decoding the three element mobile phone lens
Okay, first the three element mobile lens. A lot of three piece mobile phone lenses take this shape. It is the easiest to explain optically.
The front lens has most of the optical power, as you can see from the blue rays in the center of the field of view (or zero degrees). The material is a relatively low index with a low dispersion, since most of the optical power is in this lens and we do not have many lenses for correction, we want to keep the chromatic aberration as small as possible with this first lens.
The second lens is a high index high dispersion lens, which is there to offset the chromatic aberration and the spherical aberration caused by the first lens. The different material also helps correct the field curvature.
The third and last lens is the largest and has the most complex shape. This lens is close to the image plane, and corrects the distortion of the image. From the what we know about aspherical lens, we know that the lens closer to the image plane, which is \(y = 0\), to control the distortion independently of the other aberrations. As an aside, the distortion near the entrance pupil is zero due to \(y_p = 0\), so to control the spherical aberration independently, use an aspherical surface on the front lens for maximum effect.
The thrid lens also corrects the field curvature. Now, I said before that the field curvature cannot be corrected with an aspherical surface, but that was for aspherical surfaces that were close to a spherical shape. What this third lens does is it changes the ray path of the lenses so that the field curvature flattens as a result. Really bruteforce, in a sense.
Decoding the multi (6) element mobile phone lens, in an iPhone
Let’s look at another mobile lens, more complicated than three lenses. Below is a lens design recreated from a patent from Apple. This lens is presumably the iPhone camera from a few years ago, perhaps the iPhone 7, timeline wise. Regardless, the camera is about the same today.
This is obviously a much more complicated configuration, we have a whopping 6 lens elements. If we look closely at the patent tables, we get this:
Element  \(n\)(index)  \(\nu\) (Abbe) 

Lens 1  1.545  56.0 
Lens 2  1.640  23.5 
Lens 3  1.545  56.0 
Lens 4  1.545  56.0 
Lens 5  1.545  56.0 
Lens 6  1.545  56.0 
Hmmm…, what can this mean?
First, I see that a lot of the lenses are the same material, and a moderate index of refraction with a relatively low Abbe number, so not too much dispersion. Similarly to the three lens mobile phone lens, the front lens has most of the optical power, as we can see from the blue rays in the center of the field of view (or zero degrees). The material is a relatively low index with a low dispersion, most of the optical power here.
The second lens, Lens 2, is a high index high dispersion lens, which is there to offset the chromatic aberration and the spherical aberration caused by the first lens. The different material also helps correct the field curvature. Lens 2 seems to be the negative lens that works as the high index high dispersion lens that we see in the Cooke Triplet Anastigmat.
The higher the field of view, the harder it is to correct the distortion. The faster the Fnumber, the harder it is to correct the spherical aberration.
Take a look at the last four lenses of this camera. These lenses all have extreme aspherical properties and all the same material. These lenses try to correct the distortion and the field curvature. The dispersion is kept low so that no dramatic increase of chromatic aberration happens. The residual spherical aberration is corrected with say, Lens 3 and Lens 4 (maybe even Lens 5).
Just like for the three lens system, the field curvature is flattened by bruteforce change of the optical ray paths of due to the lens.
Break the lens down, and it becomes a little simpler to the eye. Still, this is a whopping complex lens and really difficult to design.
The material choices of a mobile phone lens
Below is the glass map for glass (and plastic). Pretty straightforward, and although there are some exotic glass at the extremes of the graph, it’s a typical glass catalog we can use.
Okay, now I’m going to show you the glass map for plastic only. Care to take a guess how many we can have from the list above?
It’s okay, I’ll wait…
Plastics are in two regions, the \(n = 1.50\) and Abbe \(= 57\) region, and the \(n = 1.59 \sim 1.65\) and Abbe \(= 22 \sim 28\) region. Although there are many types of plastics in lens design, as far as refractive index and Abbe number, this is a huge disadvantage compared to glass. To be perfectly honest, this is fewer choices than glass in the preWWII era, some in the 1940s years ago.
There have got to be a lot of sensor advancements compared to the lens design advancements, but I still think there are things to consider in lens design and what can make the lens performance improve. For example, over 150 years of lens design has given us a lot of knowledge of lens design and how different materials with different refractive index are used in combination to improve the aberrations. More specifically, the chromatic aberrations can only be corrected with two or more lenses with a different abbe number. We know that there is very little choice of refractive index for plastic materials that are required to make compact, highperformance lenses. One choice is COP/COC materials which have a refractive index of about 1.5 and an Abbe number of 50 or so (close to PMMA), and a variety of polycarbonate with a refractive index of about 1.6 and an Abbe number of 2030 or so. There really isn’t that many to choose from.
If I could only choose another type of reasonable refractive index and Abbe number it would probably be an index of 1.55 and Abbe of 45 or so, right in the middle of the current two. If I were being a little more unreasonable, I would want a refractive index of over 1.75 with an Abbe number of 40 or larger. I know that’s more difficult. Equally difficult is the low index low dispersion of 1.48 and over 80.
The more the technical specifications push the envelope, the more we have to find the base technologies that help with our task. Sometimes it’s as simple as investigating new refractive materials that may work with your system.
Laser applications: The new age of optical lens design
How to think about laser applications in this guide
The physics of lasers is fascinating. From the physical properties of a laser to how the pulse lasers are made, all interesting stuff. However, for this guide I want to concentrate on how to design optical lenses for the laser as a light source. That means that although the properties of the laser beam are important, I won’t get into how those beams are created, but more about how to use them in practice, in terms of lens design. Most of the discussions will be consistent to the discussions above.
The optical lens design of laser applications
The spectral width is very narrow compared to photography lenses. The wavelength width can be a few nm. This is convenient since the wavelength can often be a singular wavelength. Most cases do not need chromatic aberration correction. Even with multiple light sources, the design can be done individually.
The power of a laser is usually a Gaussian distribution, and can be approximated as such. However, with laser systems becoming more and more complex and requiring precise designs, there are some cases where approximating a Gaussian distribution can lead to errors. In this case, be sure to measure the distribution of the laser or get someone to measure it for you, so you can put it in the lens design.
Most lasers have polarization, and this has to be taken into account in the lens design. For some laser applications, the polarization is random, like VCSELs.
Since the coherency of the light is high, any defects in the lens will cause errors in the distribution. Defects include scratches, dirt/dust on the surface, air bubbles in the lens, shape irregularities of the optical surface. For example, for high power lasers, any dust can burn the surface of the lens and cause damage, even as severe as cracking the lens. Cemented lenses use adhesives, which may be sensitive to some lasers, so most lens forms will be separated.
The laser light going back to the source is called retroreflection. When retroreflection occurs, this can damage the source or cause the source to be unstable. In order to accommodate the retroreflection, we can use antireflective coatings, isolators, and tilting the optical system a little bit so that light does not go back.
Gaussian beams
The distribution of a laser beam is usually a Gaussian. The intensity of a laser beam \(I\) is the exponential function as a function of \(x\) multiplied by the center intensity \(I_0\). Since the intensity is decreasing the exponential function is negative. The parameter \(x\) in the exponential is squared, and theoretically the beam has intensity to infinity. Also, the distribution is symmetrical in the positive and negative directions.
\(e\) is equal to 2.7182818, and represents a mathematical constant. Since it’s not possible to calculate a width of an infinitely wide laser beam, the width is measured at \(1/e^2\) of the maximum power. \(1/e^2\) is about 13.5%, and let’s call the width \(d\). The point at which the beam is smallest is the beam waist.
\(1/e^2\) is used since the standard deviation in \(2 \sigma\), and the area underneath accounts for 95% of the laser beam, and is deemed sufficient for this case.
Conversely, if we’re thinking of crafting an optical system where more than 95% of the beam is used, we need to think about how far we want to widen the beam, and use the appropriate intensity that comes with it. For example, a width of \(3 \sigma\) is \(1/e^3\) and accounts for more than 99% of the beam power.
There are some cases where the width of the beam is the full width at half maximum (FWHM), and both \(1/e^2\) are common. As a lens designer, one of our jobs is to figure out what the data of the laser source gives us in terms of width, and then use what is appropriate for the system. For example, the laser data sheet may give the width in FWHM, but the client may want the results in \(1/e^2\) because that is appropriate for the application. It is important to make these things clear.
The Gaussian beam can be manipulated as rays from a source. The distribution can be calculated by assigning the power of the rays to each, and re calculating them after the rays go through the system.
A beam expander is a combination of two lenses, where the image side focal point of the first lens matches the object side focal point of the second lens. The magnification is \(M = f_1 / f_2\). The beam expander also expands the beam waist by \(M\), and the divergence of the laser beam is \(1/M\). A typical semiconductor laser has a 10 degree slow axis and a 30 degrees fast axis, so we can control the beam divergence with cylindrical lenses on one side.
Also, we can design aspherical beam expanders that not only change the beam divergence but can change the distribution from a Gaussian distribution to a uniform distribution. Of course, the distribution need not be uniform, and it is possible to make different variations of the distribution.
Lasers are an important part of a lens designer’s toolbox. I plan to get into laser applications more indepth laser in the future, I promise!
Bonus: How the Gaussian distribution was almost not called the “Gaussian” distribution
New Year’s Day 1801, Giuseppe Piazzi discovered the dwarf planet Ceres. He quickly lost it to the glare of the sun, and could not calculate the orbit with the existing methods at the time. Gauss quickly developed and published a new method for calculating orbit, and Ceres was soon found again. Gauss also provided the corrections for the orbit as well. The methods Gauss used were we based on linear regression and least squared analysis, but didn’t publish his methods. French mathematician AdrienMarie Legendre also developed this method and published it before Gauss could get to it. For a long time this method was credited to Legendre, an equally superb mathematician, until historical evidence in letters and notes proved that Gauss was the inventor of the method, thus called the “Gaussian distribution” that we call it today. Funny to think that maybe we would be calling the Gaussian distribution the “Legendre distribution” had history played out differently.
Diffractive optics: Harnessing the phase and wavelength properties of light
Basics, how to use diffractive optics
Although a typical lens refers to a refractive lens, a lens that uses diffractive properties is called a diffractive lens.
Diffractive Optical Elements (abbreviated as DOE), and even the old name, the diffractive surface, is sometimes wrapped with the term DOE today.
There are more and more terms that identify as diffractive optics, DOE being one of them, and others like HOE (Holographic optical elements) and BOE (Binary optical elements). Holography technology is used for HOEs, and lithography technology is used for BOEs, but in principle, they are all using diffractive properties instead of refractive properties to change the ray angles of the light.
A binary shape diffractive surface loses efficiency while a blazedshape diffractive surface has 100% diffractive efficiency, and the technology has evolved a lot since the early days of DOEs.
The holography technique is used with two laser beams causing an interference and patterning a sinusoidal shape. Usually, the holography is done on a master lens, and a replica of the lens is taken to make the mould for mass production. Since the tooling method is optical, the shapes and periods of the diffractive grating are much more consistent than machine tooling. The precision and consistent shapes due to the holography method results in higher diffractive efficiency and less stray light.
Nowadays, computational design of the diffractive element and photolithography is often used to make diffractive elements. A computer is used to make a mask pattern on the substrate, and photolithographic etching is performed to make the desired pattern for diffraction.
The simplest conceptual method for making a diffractive optical element is by mechanical tooling, for example, diamond turning a tooling bit to make precise microscopic blazed structures.
With the development of DOEs, they are used in diverse applications for many optical surfaces and optical systems.
Basics, how diffractive optics work
A simple equation of the diffractive surface is as follows:
$$
\sin \theta ‘ – \sin{\theta} = \frac{m \lambda}{d}
$$
where \(\theta\) is the incident angle, \(\theta ‘\) is the diffracted angle, \(m\) is the diffractive order, \(\lambda\) is the wavelength, and \(d\) is the period of the diffractive grating.
Just to recap, the Abbe number of a refractive lens is
$$
\nu_d = \frac{n_d – 1}{n_F – n_C},
$$
and the average glass lens can have an Abbe number of anywhere between 20~95.
The DOE has an Abbe number that is only dependent on the wavelength, as follows:
$$
\nu_d = \frac{\lambda_d}{\lambda_F – \lambda_C} = \frac{587.56nm}{486.13nm – 656.27nm} = 3.453.
$$
We immediately see that the Abbe number for a DOE is negative, and therefore opposite to the refractive lens. This gives us an interesting property to play with.
Therefore, the refractive lens bends shorter wavelengths like blue light more than longer wavelengths like red light, while the diffractive lens is the opposite, bending longer wavelengths like red light more than shorter wavelengths like blue light.
We can use the two properties together to get a single lens that has very good colour correction. This is particularly useful when only a single element is possible, for something compact like a DVD or Bluray disc lens.
If we change the period of the diffractive grating at different points in the overall surface, we can change the angle of diffraction at different points, thus making it similar to the aspherical surface in terms of being able to control the angle of diffraction/refraction in any point along the lens surface.
The problem with a DOE is the efficiency. Theoretically, 100% efficiency can only be achieved at one wavelength and one angle, and all other wavelengths and incident angles and lose efficiency.
How to design diffractive optics
The way that I suggest to design a diffractive surface is to use the phase function method or the high index method.
For the phase function method, we add a phase function to any surface, it can be a plane or a spherical surface, or even an aspherical surface.
$$
\psi (r) = \frac{2\pi}{\lambda} \Psi(r) \\
\Psi(r) = \alpha_{1}r^2 + \alpha_{2}r^4 + \alpha_{3}r^6 + \alpha_{4}r^8 + \cdots
$$
where \(\psi\) is the phase function and \(\Psi\) is the ray path function.
Optical design software such as Zemax and Code V have the phase function built in, and we can take the phase values and convert them to actual physical shapes for manufacturing.
Just to be thorough, the high index method is like a thin film, and we use a high index like 500 to 10,000. If the index is too small, there can be errors that become problematic, and if the index is too large it calculation accuracy can drop depending on the effective digits we’re using.
$$
n(\lambda) = m \lambda \times C + 1
$$
where \(C\) is some constant.
The index acts as a surface that changes the angle of the rays depending on the diffractive order and the wavelength.
Either way, we have to change the calculation we get from the phase or the ray path difference into an actual surface.
$$
L = \frac{\Psi(r) + j \lambda}{n1}
$$
where \(L\) is the amount tooled for each orbicular zone \(i\).
The surfaces can be fabricated by electron beam lithography and other etching processes.
It is also possible to tool a mould, and for materials like Germanium and Silicon, we may tool the material directly.
Some realworld examples of DOE lenses.
Diffractive optics can be used in examples where the diffraction is used in optical components other than a replacement lens application.
The following are just a few examples of where diffractive optics can be used:
 Lens application
 Splitting of light beams
 Wavelength filters
Some specific uses case examples:
Holographic scanner:
A hologram is either moved or spun to control a laser beam. Depending on where the laser is hitting the diffractive element, the laser spot will move to a different location, thus enabling highspeed scanning across a given area.
Holographic scanners at IEEE GlobalSpec
Headup display HOE:
Using the wavelength filtering properties of an HOE, we can show different images in the vision. This can be used in wearable displays.
Laser beam splitter:
For example, these are used in laser scanner applications like laser beam printers. Also, they are used in laser tooling applications where one parallel beam of light can converge to several points on the tooling surface.
Laser beam homogenizer:
Usually, laser beams have a Gaussian distribution. There are applications where a uniform distribution or a tophat type distribution is needed. Uniformity is useful in illumination optics, and for laser tooling applications. A common term is a “beam shaper”, which shapes the beam into an arbitrary shape. To achieve this, a nonaxiallysymmetric DOE is needed, and photolithography is used.
Diffuser:
For a long time, diffusers were used with a groundglass plate or a glass plate with diffractive material laced inside. With holography, we can make a smoothly diffused surface by using the higher orders of diffraction effectively. Also, unlike groundglass, the diffusing pattern can be shaped into oval or linear shapes rather than simply round diffusion. These holographic diffuser plates are compatible with coherent light and incoherent light, do not have wavelength dependency and polarization dependency and can be used effectively in many situations.
Diffractive optics have properties that are different from conventional lenses and prisms, so it is good to acknowledge these differences and keep in mind when they will be useful.
On the other hand, there are disadvantages to diffractive optics as well, so it is best to acknowledge those weaknesses as well.
Advantages of diffractive optics:
 We can control the wavefront of the outgoing light from an incoming wavefront
 Diffractive optic lenses can be very thin, as the diffractive grooves are small and shallow
 It is possible to combine both converging properties and diverging properties in one diffractive element
 It is possible to utilize multiple diffractive orders to our advantage
 The shape is replicable in metallic moulds, and therefore it is possible to mass produce diffractive optics with high accuracy
 The dispersion properties of the wavelength are the opposite to refractive lenses, and we can use this effectively for chromatic control
 It is possible to make the optical systems more compact using diffractive optics
Disadvantages of diffractive optics:
 Since the dispersion of diffractive optics is dependent on the wavelength, it is not easily used in white light applications where there is a continuous band of wavelength for the light
 It is imperative that all of the diffractive orders are taken into account, as some higher orders can cause stray light or ghosting problems
 The diffractive efficiency, wavelength properties, and angular properties all have to be taken into account in use
Light Shaping Diffusers by Luminit
Plastics for diffractive optic applications: submicron structures
Diffractive optics are mostly submicron plastic lenses, since the structures need to manipulate light on a wavelength level. It is definitely different than our imaging lens designs, because we need to calculate the wavelike properties of the light, and not simply treat them as rays.
Having the ability to design diffractive structures is useful for any lens designer’s toolbox. You can combine it with a normal lens design for a more complex system, and your range of techniques will certainly expand.
I thought I’d take a closer look at our favourite diffractive optic in the iPhone X series, which makes FaceID possible.
To get a dotstyled projection system we need to do it by using diffractive optical elements. Also, DOEs can be used in applications such as spectroscopy, beam shapers, athermalization, and correction of chromatic aberrations. Learning how to use diffractive optics is key as a lens designer.
The difficulty of submicron plastic lenses is the moulding process, which has to be done for structures that are 10 – 50 microns in size.
The Face ID patent in the iPhone X series smartphone.
I did some research on the FaceID technology of the iPhone X. It is basically similar to a motion capture system, and records depth data from an infrared dot projector. The FaceID algorithm is a bit of machine learning technology in itself, but lens design wise the technology is in the dot projector.
There is a whole list of patents that can be found on the web, but it’s not that fun to read the language. I found some interesting images and information, though.
If you read the claims, I have to agree with you, the language is unbearable. But we need this language to make sure there aren’t any misunderstandings (like Oxford commas). In any case, patents are a wealth of information and you can use the patents as research for technology in a company or field of interest. Check out some of the patent documentation for yourself, you can dig into them if you have the time. If you put the work in to be able to effectively and efficiently read patents, I guarantee you it will help you develop as a lens designer.
US9740019B2
US20170314763A1
US20150193657A1
US9554122B2
US9400177B2
Keeping on topic of the FaceID technology, my notsooriginalguess to the future of Face ID is that it is coming to the Mac. Imagine the Macbook unlocking itself with Face ID as you open the clamshell. There would no longer be that little wait to the login screen and then the typing of your password. I know the Apple watch already provides this capability, but to own an Apple Watch you need an iPhone. With Face ID the same handiness is embedded into the hardware itself.
Also, this patent was filed in 2010, so you get an idea of how far Apple (and other tech companies) think things through. At the same time, just because something is patented doesn’t necessarily mean that there is a product behind it.
If you’re interested in diffractive optic design, Zemax Optics Studio has a webinar on diffractive optics tools.
Wrapup on diffractive optical elements
We can see that diffractive optics are not only useful for lens type applications but all sorts of optical systems. As a lens designer, we are not only restricted to refractive and reflective optics, but also more complicated optical properties like diffraction. Diffractive optics will continue to be used in many optical systems like optical communication, laser applications, displays, and many many more.
Although understanding diffractive optics and using them effectively can seem more difficult than our conventional refractive and reflective lenses, it is mostly because of the concepts like wavefront, phase control, and other properties that don’t come up too often in conventional lenses. I also feel that there are few textbooks and examples that cover both the theory and the application in one wide swoop.
However, my feeling is that a more efficient optical element at the smallest size possible is the highest motivation for the development of diffractive optics.
Illumination lenses: A totally different approach to optical lens design
Illumination optics is totally different than geometrical lens design, as there is less emphasis on mathematics and more brute force simulation. Some intuition and experience apply, and innovative lens configurations can help the optical design.
With optics having so much applications in industry, engineers and companies that used to have nothing to do with optics now need optical components. Most optics books deal with the classical ray tracing method mostly used in photographic lenses, and the documentation is more specialized for certain areas.
Further, since more complex illumination lens design cannot be done exclusively by geometric ray tracing, as there are large and complex light sources, mirrors, projection lenses, all in the optical system. Since there is so much that affects the optical system, there are a few things that I’d like to cover to get a feel for illumination.
The two things you need to absolutely be familiar with to design an illumination system
Here they are:
 The various terms of photometry and radiometery
 Étendue
Photometry and radiometry
Photometry and radiometry are the same terms described for the visual spectrum (photometry) and all wavelengths (radiometry).
We distinguish between the two because there are illumination systems like automobile headlamp lighting which are visible to the eye, and infrared lighting which is not visible to the eye.
There are a lot of terms, but I will show the most basic terms.
Radiometric  Photopic)  

Flux  Power (Watts = W)  Luminous flus (lumens = lm) 
Flux/area  Irradiance (W/m2)  Illuminance (lm/m2 or lux) 
Flux/solid angle  Radiant Intensity (W/sr)  Intensity (lm/sr or candela = cd) 
Flux/areasolid angle  Radiance (W/m2sr)  Luminance (lm/m2sr or nit) 
There are equations and calculations for all of these terms that I won’t get into here, but if you can familiarize yourself with this table, it will be easy to follow over 90% of all illumination designs.
If you like visual representations of units of measure like I do, below is my schematic diagram of photometric units.
For more information on how to spec out an illumination system, check out the illumination section to my specification cheat sheet, and it will give you a good idea on what to look for in an illumination design.
Étendue
Étendue may be the single most basic yet important term when designing an illumination lens system.
First, étendue explains the flux transfer of the optical system. Second, étendue is a measure to how we can shape the distribution of the illuminated target.
There are papers, textbooks, equations, and derivations that explain and prove the mathematical relationship of étendue. I like to simplify étendue to the solid angle times the area, and this has to be conserved.
Conservation means that beam diameter or area multiplied by the beam solid angle is a constant value. In simple terms, in an optical system where the étendue is conserved, the amount of light that can pass through the system is determined by the product of the solid angle and the area.
I like to use étendue when I’m explaining to a client how their large LED can’t be collimated. The area of the LED is large (finite). They want a perfect collimator. That means, from an étendue standpoint, we have the product of the area of the LED \(A\) and the angle of the beams from the LED \(u\). Both are going to be finite numbers. They want the collimator to have perfect collimation, which means the refracted rays \(u’\) is zero. \(A \times u = A’ \times u’\) can’t be conserved if \(u’\) needs to be zero. We’d need an infinitely large lens (essentially, \(A’\)) to do that. No go.
I have more examples, but you get the idea. Use the laws of physics to debunk unachievable specifications that you may notice. No fault to the client, it’s what they need for their system. It’s up to us to figure an optical workaround to get them what they want with minimal compromise.
Étendue is a French word which literally means extent.
Köhler Illumination
Köhler illumination is the illumination method where an objective lens is used to image the light source onto a plane. If we place the aperture on the object, in this case the source, the it can be imaged onto a surface.
Designing a Köhler illumination system we do not raytrace the rays from the light source, but we use the opposite idea where we pretend as if there is an uniform light distribution at infinity. If we trace these rays at infinity to the source, we should get uniform light when using the source as a light source. This method is useful for many illumination applications.
The field stop is used to limit the area of illumination on the illumination area. Limiting the amount of light helps eliminate flare and ghosts, and sharper images are possible.
On the other hand, aperture stop limits the projection of the light source, in this case the filament of our light bulb, which enables adjustment of brightness of the field of view.
There is a handy link where you can roughly calculate the paraxial optics of a Köhler illumination system at Edmund optics, click here. Note that although the final design needs some optimization from optical design software like Zemax, This gives us a great starting point of the design based on the specifications and limitations of the system.
For high end illumination systems like a stepper system, the uniformity must be below a few percent. In order to make the distribution uniform, there is an integrator in the middle of the system, usually multiple rods that act as multiple light pipes. Each portion of the integrator acts as a Köhler illumination system.
Microlens arrays (MLAs) can used as multiple Köhler illumination systems to get a small NA for each lens but a good brightness and uniformity overall, since the illumination of each microlens is superimposed on the illuminated image plane.
In the two images below, we can see that each microlens acts like the two images above, if we break it down.
A term that is useful in illumination optics is the coherence ratio. The coherence ratio is the NA of the illumination divided by the NA of the objective lens. If this coherence ratio is zero, the system is coherent and similar to ray optics used in photography. If this coherence ratio is infinity, the system is incoherent and an illumination system. When the coherence ratio is somewhere in between, there is a partial coherence to the system, and the smaller this ratio is, The contrast is better, but if it is too small the system may not be too bright.
Where illumination optical lens design is useful
In general, the light source used for illumination is an ideal light source with a circular or rectangular shape, with a uniform light source. In this case, the main aberration to be careful of are the spherical aberration and the coma. This is sufficient in some cases but as the illumination system becomes more complex or the illumination system requires precise simulations, a more precise representation of the light source is needed.
Illumination optics are used in the following cases:
 Projection lenses for projectors and/or measurement apparatus
 Microscope systems
 Steppers
 LCD back lights for TV, smartphone screens
 Lightguides
 Solar concentrators
 Lamps such as LED lamps, automotive headlights
 Collimators and beam shapers
There are more and more components that diverge from a simple photograph, and the lens design need to accommodate for it. Here are a few examples of light sources:
 Lasers
 VCSELs
 LEDs
 Incandescent light sources
 Halide lamps
 Fiber optic sources
Plastic lenses for illumination applications: lighting, LEDs, and a host of applications
Like I have stated before, illumination optics is totally different from imaging optics, as the lens design is done with nonsequential raytraces rather than sequential raytracing. In sequential raytracing the ray passes through the surfaces one by one (in sequence), while for nonsequential analysis the rays can hit any object in any order. Accurate 3D models are also usually used in nonsequential analysis.
The single most important thing to consider in illumination lens design, in my mind, is the light source. The more accurate the light source is set in the lens design, the more the result will represent real life. We can approximate some applications with a point source, but most of the time we need to either model the light source, or use ray data.
Below is an example of how different an illumination lens can look. It is an LED collimator lens.
Since the light from an LED is spreading in all directions, the angle is very large. There are two optical components to this lens, a refractive surface in the center of the lens for the narrow angles coming from the light source, and a Total Internal Reflection (TIR) surface for the larger angles.
The way we would go about the optical design is different than a normal photographic objective. First of all, there are no thirdorder aberrations that we can calculate to decrease. Second, the rays that are refracted and the rays that are reflected cover different paths, and have to be optimized separately.
Other illumination methods
 Critical illumination: Imaging the light source on the object surface of a projecting lens.
One method of doing critical illumination is using a light source, a elliptcal mirror, and a projection lens. The advantages are that it is bright and the efficiency is very good. The disadvantage is that if the light source is not uniform it can be hard to get a uniform distribution, and light sources like incandescent lamps do not work well. Also, the projected surface becomes very hot. Finally, it can’t be made too wide.  Transmitted illumination: method to observe a sample through transmission much like in a microscope.
 Reflected illumination: A method to observe a sample by shining reflective light onto the sample.
 Dark field illumination: method of illumination that shines light offaxis so that the light does not enter the observer directly, and uses diffusion of diffraction to observe the sample.
 Telecentric illumination: An illumination method where the illuminated light is perpendicular to the surface of the illumination.
Bonus: Real world calculation of luminance, illuminance, and luminous intensity.
Recently at work, I used the basic equations for intensity, luminance, and illuminance, to figure out a problem I was having at work.
We were comparing two illumination devices for their performance, and they have similar functions but have some significant differences as well. They both illuminate a certain area, so we wanted to know how they perform against each other. One is a recent lens design, and not a product yet, but we had the design specs on hand. The other device was a module we took apart and measured. We wanted to compare the performance of the product that we were going to make, to the one in the market. Oh, the battle is on!
Anyway, the lens design (my lens design, ahem) had a specification in luminance (candela per square metre). Here’s the setup for the system:
It’s an illumination system that is placed diagonally, hits a surface (presumably the floor or ground) and we observe it nearby. The specification sheet said, “Put a luminance meter above the illuminated area and make sure to get this luminance [cd/m^2]”. The product and prototypes were going to be measured with a luminance meter, so the target was luminance.
On the other hand, this is the existing module:
We didn’t know the configuration, but since we had the existing product on hand, we dismantled from the module and decided to measure it to see the performance. The configuration above was measured by our metrology team with an intensity meter, so the units are for luminous intensity or candela.
Now, if you are confused by all the terms sounding the same, you aren’t alone. There are plenty of people who say illuminance but mean luminance, use the term brightness incorrectly, and confuse how luminance is a bitch to calculate in lens design software accurately. It’s worth it to write the terms schematically to get a feeling for what they mean.
So what did I do? I decided to reverse calculate and convert the specification of luminance in our design into the luminous intensity of the measurement. So I set up the comparison system on paper:
Our design shines light onto a surface, and the specification is measuring the luminance (\(L\)). That means that there is some illuminance hitting the surface, denoted \(E_h\). That came from the module at an angle theta, so we converted the cosine component to get \(E\) (illuminance) from the source. The illuminance from the source was emitted as luminous intensity I, which is calculated by multiplying the illuminance with the square of the distance.
Even though the design required luminance as a measure for performance, it was possible to estimate the measurements of the existing product to compare. We just needed to calculate the perceived intensity of the light source and the lens of our design as if we were measuring it.
In any case, this is an example of how equations we found in a book directly helped a product. Exciting day at the office!
There is much much more to illumination lens design. I will dedicate a complete blog post to it, I promise.
Bonus: A list of interesting applications of lens designs not mentioned thusfar
 Telecommunication lenses and optical fibers
 Optical fibers
 Optical wireless communication
 GRIN (gradient index) lenses
 Selfoc lenses
 IRcorrected lenses
 Lenses for XRay imaging
 Lenses for fluorescent imaging
 Endoscopes, other medical applications
 Lens Arrays
 Hyperspectral lenses
 Thermographic lenses (Germanium, ZnSe, …)
Lens design forms and the principles of optical lens design
The principles and methods of lens design
The scientific method states that we take an assumption or hypothesis and then test that hypothesis. The lens design is a similar process. To increase the performance of the lens, we sometimes change a section of the lens design, or we contrive a change in the method of aberration correction, but these are all testing the hypothesis. The correction of the aberrations and the change in the performance determines the outcome of our hypothesis.
The hypothesis that we can try is also very diverse. Trying a lens type is a hypothesis in itself, and changing a section of the lens design or providing targets and goals for the aberrations, providing the parameter or variables to change within the lens design are all hypotheses that we try on route to completing the lens design.
Hypotheses are based on thought experiments like experience, knowledge, or hypotheses are based on intuition. If we base our hypotheses on experience and prior knowledge too much, we can fall into traps like stereotypes, and may not be able to think outside of the box for some innovative ideas. Sometimes not knowing can be a strength. We can go for trialbyerror and set up multiple hypotheses one at a time, and end up with the distilled essence of a final lens design.
The identity of a lens designer
The characteristic of optical lens design techniques is determined by the trial and error process. The trial and error process can give rise to great diversity in the lens design, and this is common in lens design. The diversity of lens design shows the difficulty of lens design, but also the potential of lens design, and the fact that lens designers have been selfreliant of their outcomes. The day may come at some point where human beings aren’t needed to complete a lens design, but I think that this diversity tells us that there is hope for a while.
Unfortunately, subjective hopes and wishes of a lens design result must match the objective reality of the laws and principles of physics. This is a fact. But there are instances where the qualitative desires of a product can open up new solutions to problems that we may not have realized before.
For example, if lens designers of the past persisted on continuing to use the triplet format for all lens designs, if we wanted a faster lens with a larger aperture (qualitative desire), we would have been blocked by the properties of the triplet (objective reality), and there would be major compromises. But the desire for faster lenses helped develop a triplet with an aplanatic lens called the Ernostar, and this was the new objective reality. The Ernostar produced the Sonnar, which is the next of the objective reality, and so on and so on. The geniuses of lens design came up with solutions to unique problems to further the development of lens design.
The fact that there are many solutions to a single problem means that a hundred lens designers can come up with a hundred different lens designs. This means that each lens designer’s choices, ideas, thoughts, and even daily life can affect the outcome of a lens design. Yikes! For example, if your day by day is dominated by authoritarian thoughts, you may come up with safe but a lens without individuality. Of course, learning the essence of past lens designs are equally important, but we have to keep in mind this aspect of lens design.
Like I’ve stated before, automatic and computational lens design are powerful tools, but if we have the same specification or target for a lens design with the same starting point, and the same lens design form, chances are that two different people can arrive at a similar solution. In these kinds of cases, we can differentiate by making our design more compact, or our design with a faster Fnumber. This requires the addition of a different idea in the starting lens design so that a difference emerges. Automatic optimization in lens design alleviated the calculations and solutions to linear and nonlinear equations, but on the flip side, made the specifications more complex and forces the output of more and more ideas to more difficult problems.
If you’ve ever done lens design, you know that there is always the possibility of a better lens design. In a sense, there is no end to lens design, and I think that this Guide proves that. As a lens designer, it is our challenge to take these new problems headon and to find creative solutions to unique problems to better our knowledge on lens design.
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References
 Lens Design Methods, Yoshiya Matsui, Kyoritsu Press 1972 (Japanese)
 Lens design Fundamentals, Rudolf Kingslake, Academic Press 1978
 Lens Design Engineering, Jihei Nakagawa, Tokai University Press 1986 (Japanese)
 A History of the photographic lens’ R. Kingslake, Academic Press Inc. 1989
 Introduction to Lens Design: With Practical Zemax Examples, Joseph Geary, WillmannBell 2002
 Modern Lens Design, Warren J. Smith, McGrawHill Education 2004
 Field Guide to Geometrical Optics, John E. Greivenkamp, SPIE Press 2004
 Optical System Design, Robert Fischer, McGrawHill Education 2008
 Applied Optics and Optical Design Part 1 and Part 2, A. E. Conrady, Dover Publications 2011
 Field Guide to Lens Design, Julie Bentley, Craig Olson, SPIE Press 2012
 Kazamaki, T. and Kondo, F., “New Series of Distortionless Telephoto Lenses,” J. Opt. Soc. Am. 46, 2231 (1956).
 Kingslake, R., “Telephoto vs. Ordinary Lenses: A Tutorial Paper,” in Journal of the SMPTE, vol. 75, no. 12, pp. 11651168, (1966)
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