What is the “logic” inside lens design? There has to be a process, an approach, a method to which this all works, right? I took a deep look into my own lens design process and methods to craft what I think is a guideline of sorts when doing the nitty gritty of lens design.

## The agony of the starting point of a lens design

The optimization of a lens design starts with what we call a starting point. This starting point affects all processes afterwards and determines the final performance of the lens. Therefore, it is important to make the starting point of the lens design clear.

One such way to determine the starting point is by determining the lens design form. What is the performance needed? Does it match with a certain lens design form? Another point to determine the starting point is the parameters of the performance to be achieved. Although the lens type determines the somewhat the specifications of the final lens design, there are many lens design forms and one can agonize over the results with thoughts like “what would have happened with a different starting point” or “what if I changed a parameter during the optimization” entering the mind all the time.

These questions arise because there is no singular path towards the “correct” lens design, as there are various solutions to the same lens design problem. In some ways, the theory that gives us the “best lens design” is not established, as ten lens designers can produce ten different lens designs. Problems in calculus, chemistry, and physics all lead to one correct answer. Does that mean that lens design as a science is not developed enough?

It is good practice to have a starting point data that has a somewhat good aberration correction. However, that starting point alone does not mean that a good or great lens can be designed. On the other hand, there are cases where the starting point lens data has a somewhat large aberration, but it is still possible to design a final lens that has low aberration when finished.

## The potential of a lens design

The power of modern lens design is because almost all lens design today is done with a computer, and a starting point design may be taken from an old design that did not have the advantages of spot diagrams, wavefront analysis, and in some cases, more than 3rd order aberration correction. This means that it can be trivial to get better performance from a classic lens design, because the criteria for the performance evaluation is different. We have the merit function today and not for old designs.

I think that the non-computational part of lens design is also important, the intuition, the educated guesses, the experience level of the lens designer, and the logical thinking that comes along with making decisions during lens design. The lens design form can teach us a lot of things about why a certain lens combination is useful for the intended performance. This also means that on the flip-side, just having computational power or aberration correction, or wavefront analysis isn’t enough. We can’t just evaluate the aberrations of the current lens design, but we also need to determine the aberration correction *potential* of a lens design.

The *potential* of a lens design isn’t as easy to see, because numerically there is nothing that shows a lens has potential. However, all skilled lens designers use their experience and educated guesses on how to determine a lens design form to start off with.

## How to optimize and improve the lens design through logic

What are some ways to improve the lens design through logic? What is at the bottom of this process? I have a few theories…

### 1. The historical perspective of lens design

Most of what we lens designers do is start off with a starting lens design, and make improvements based on that starting point. As such, this means that over 99% of lens design is based on a past design. Also, since technology has also advanced with time, some of the past lens design form can be improved with the simple introduction of a technology that wasn’t available at the time of the past lens design.

Sure, it’s possible to design a lens without knowing the history, the process, the theory behind the lens design. I contend that knowing the history and the steps that the lens design took to get to where it is today, and analyzing the inner workings of the lens design, all help with understanding why a certain lens design looks and performs the way that it does today, and it’s potential today.

By Paul1513 at en.wikipedia, CC BY-SA 3.0, Link

#### The evolution of lens design

Looking back to my Ultimate Guide to Lens Design Forms, I think there are a few ways to think about the evolution of lens design:

- The requirement of systems that have new specifications, and new performance goals mean there is a need for that type of lens in the market, and therefore will lead the development of said lens and/or similar performing lens designs.
- A new lens type can be born from summarization of the historical lenses that preceded it, and therefore has new information as a new lens design form.
- There can be new technical progress that either introduces a cancelling out of aberrations, or introduces a decrease in the aberrations, or both.
- We can discover new methods of lens design, while referencing existing lens designs.
- Some new methods may improve some of the performance, but at the same time decrease other parts of the performance. By extracting only the good solutions with as few bad solutions as possible, is the challenge.

### 2. Opposing properties and contradictions in the lens design

Along with looking at a lens design from a historical point of view, it’s important to look at a lens design from a structural point of view. One point is to decipher the contradiction or opposing properties within the lens design.

#### Chromatic aberration in a doublet

One example is colour correction in an achromatic doublet. Sure, the colour is well corrected in the end, but let’s look closely at each lens. These two lenses, individually, each have worse colour dispersion on their own than what we started off with! Isn’t that counterintuitive? By generating opposing aberrations, we were able to eliminate the overall aberration.

#### Aplanatic lenses

A second example is an aplanatic lens in the Ernostar lens, which is the second lens in the diagram below. First, the residual aberrations in a triplet lens were corrected with the addition of an aplanatic meniscus lens in between the first and second lenses, and by decreasing the power of the 3rd lens.

However, by doing so, the pincushion distortion which used to be a minor problem became large. The power of the last lens was decreased, and the correction to the principal ray also decreased. The pincushion distortion was finally resolved by placing the final lens further away from the stop, so that the marginal ray is at a higher point of the lens.

#### Double Gauss lenses

A further example is the strong concave surfaces about the stop of a Double Gauss lens, which decreases the Petzval sum but increases the coma.

#### Retrofocus lenses

Another example is the retrofocus lens, which introduces a strong negative lens to increase the field of view, but needs correction of the barrel distortion caused by this strong negative lens.

#### Telephoto lenses

Still another example is the telephoto lens lens which has a negative lens that shortens the length of the lens compared to its focal length, but increases the positive power of the positive lens as well.

I could go on and on, but you can see that depending on the system, there is diversity, individuality, and integration, with differing cases for each.

### 3. Phenomena and essence within the lens design

Photographic lens design, a subset of optical lens design, starts with understanding the aberrations in the system by using ray tracing. The common information available is the curvature of the lens, the spacing between the lens surfaces, and the index of refraction of the lens material. These properties are independent of each other, and don’t follow any particular pattern or system.

To understand the phenomenon within the lens design is to understand its essence.

For example, I have talked about the coma that is the key in Double Gauss lens design. The coma in a Double Gauss comes from the F-number, the radius of curvature, the spacing, the index of refraction, so it is different in each case but is still universal in this lens design form. The *essence* of the Double Gauss lens shows up as a *phenomenon* in the form of coma. We don’t want to choose the classic Double Gauss lens design form if we want to have a lens with low coma flare. We either need to improve upon the Double Gauss design, or we need to choose a different lens design form.

Optical lens design deals with many types of numerical analysis, so it is easy to get wrapped up in the numbers and lose sight of the essence of the lens design. To understand the essence of the lens design form is the first step to correctly approach the aberration correction process, and the progress and evolution are how new lens designs are born.

### 4. Uniqueness and universality of the lens design

Let’s start with the widely known Cooke triplet lens. The first lens and the second lens, together, are characteristic and unique.

Although the exact focal length differences or lens power relationship of these two lenses can differ depending on the triplet design, all triplets have this relationship.

Further, the first group and second group of a typical zoom lens also have a unique relationship, and the first lens and second lens in a telephoto lens also has a unique relationship.

This unique relationship can be viewed as a universal property in lens design. Within the triplet, there was a uniqueness in the lens design form, and this was universal and will help us understand more complex lens designs like the telephoto lens and the zoom lens designs that came later on.

Looking for the uniqueness in a lens design form, and recognizing the universal aspects of the lens design form, goes a long way to understanding the more complex lens design forms. If we can find the universal aspects within the lens design, as we did for the triplet and the telephoto/zoom lens case, we can use that information to understand the lens design and know where to improve it. Equally as important, we can use the same information to figure out where *not* to improve a lens design.

### 5. The details and format of the lens design

In general, the details of the lens design has a format that is suitable for that particular lens design. On the other hand, if we are given a *format* of lens design, this format can lead to understanding the details of the lens *design*. Let me show you what I mean.

For a photographic lens, the *details* are the specifications and the optical performance of the lens, while the *format* is the lens design form. Human nature would say that the details choose the format, but a totally new lens specification (the details) has no lens design form (the format) to reference. If we are only using existing formats, there is bound to be a limit to the lens design.

This is especially true for zoom lens design because there are two formats, one being the lens design form, and the other format being the zoom system itself. There are more possibilities for exploration of lens design forms even for similar specifications or lens systems.

As long as we stick to old lens design forms, there is a limit to the achievable lens performance and specifications. A format (lens design form) that is fitting of the details leads to the creation of new lens designs and the discovery of new lens designs.

To be honest, the uncovering of a new lens design form is one of the most enjoyable parts of lens design for me.

### 6. Sections and entirety of the lens design

Relatively complex lens design forms like zoom lens groups, inner focusing lenses in a telephoto lens, or a floating focus system, all benefit from the lens designer looking at the different *subsections* of the lens and simultaneously looking at the *entirety* of the lens. This is because the subsections have a position and role within the entirety of the lens design, and with each other, and understanding this helps move the lens design forward while it helps build the strategy for aberration correction.

For example, a 4 group zoom lens has a zoom section and a focus section within its entirety. The zoom section has the role to minimize the aberration and aberration change while zooming. The focus section has the role of correcting the remaining aberrations from the zoom section. Each section has its own role within the big picture, and the lens choices and shapes have to reflect these properties while performing the lens design. At the time, computers can’t judge this correctly.

### 7. Ideal circumstances and limiting of the lens design

The idealization and/or limits are used all the time in science. When we calculate the speed of a ball rolling down a hill in physics class, there is no friction between the ball and the hill, the hill has no bumps, and the ball is a perfect sphere. These are all ideal conditions that enable us to calculate the physics behind the ball affected by gravity. How many times have we limited a value to zero if it was small enough to be negligible? All the time, in my experience.

#### Ray tracing and thin lenses are ideal circumstances

In lens design, a prime example of an ideal circumstance is has got to be ray tracing. A bundle of light is idealized into straight lines refracting on surfaces. The range that is being photographed is idealized to a few points in the field of view.

Also, thin lenses is a form of an ideal lens and a limit at the same time. By making the thin lens an ideal lens, we can see the pure relationship of the lens to the system.

In actual lens design, it is often useful to use a thin lens (an ideal circumstance) to uncover the true nature of the lens design form.

#### Limiting the system to a few wavelengths

Let’s think about how the separation of the achromatic doublet leads to an increase in chromatic aberration if we separate the lenses too much. We can simplify the problem and use only the d-line (587.6nm) and the g-line (435.8nm) wavelengths. The d-line and the g-line do not change drastically after passing through the positive first lens of the doublet, but as the rays extend further from the first lens, the difference becomes larger. If the d-line ray and the g-line ray pass through the second lens at different ray heights, the chromatic aberration has no choice but to be large.

On the other hand, if the lenses are closer together, the d-line and g-line ray height differences are negligible and we can concentrate on changing the ray angles without worrying about the ray height differences.

(My Ultimate Guide to Lens Design Using Spreadsheets has a section dedicated to doublet colour correction that will beautifully help to understand this, and I also have a post called doublet that goes in depth into the colour correction of doublet lenses)

Choosing two or three wavelengths for the doublet colour correction is an example of how our thought process can be guided by simplifying the systems with ideal circumstances (thin lenses) and limiting the system (taking only two wavelengths). This thought process can also be used in the lens design of long focal length and telephoto lens designs as well.

Another example of limiting the system is starting the lens design with a single wavelength so we don’t have to worry about the Abbe number of the lenses. This way we can concentrate on the aberration correction, and we can add the wavelengths later to correct the chromatic aberration. Of course, we can always do the opposite too.

## The importance of logic in lens design

What do you think? I hope I have made my case that although the computational calculations of optimizing lens design lead to the evolution of lens design as we see it today, it is equally important to have an analytical way to do lens design as well.

For more information on lens design forms, feel free to check out my Ultimate Guide to Lens Design Forms. It is an in-depth discussion on different lens design forms, both classic and modern, breaking down the lens design unlike anything you can find online (be warned that it is over 44,000 words long).

Alternatively, for more information on lens design that is not dependent on expensive proprietary optical design software, feel free to check out my Ultimate Guide to Spreadsheet Lens Design. This is an alternative method to perform lens design, with simple tools in spreadsheet software like Excel.

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