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`OPTICAL
`SYSTEM
`DESIGN
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`DOI: 10.1036/0071472487
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`9CHAPTER 9
`The Optical
`Design Process
`
`The optical design process includes a myriad of tasks that the designer
`must perform and consider in the process of optimizing the perfor-
`mance of an imaging optical system. While we often think primarily of
`the robustness of the optimization algorithm, reduction of aberrations,
`and the like, there is much more to do. The designer must be at what we
`sometimes call “mental and technical equilibrium with the task at
`hand.” This means that he or she needs to be fully confident that all of
`the following are understood and under control:
`
`All first-order parameters and specifications such as magnification,
`focal length, ƒ/number, full field of view, spectral band and relative
`weightings, and others.
`Assure that the optical performance is being met, including image
`quality, distortion, vignetting, and others.
`Assure that the packaging and other physical requirements,
`including the thermal environment, is being taken into account.
`Assure that the design is manufacturable at a reasonable cost based
`on a fabrication, assembly, and alignment tolerance analysis and
`performance error budget.
`Consider all possible problems such as polarization effects,
`including birefringence, coating feasibility, ghost images and stray
`light, and any other possible problems.
`
`167
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`Once every one of these items has been addressed and is at least rec-
`ognized and understood, we start with the sketch of the system. First,
`the system is divided into subsystems if possible, and the first-order
`parameters are determined for each subsystem. For example, if we are to
`design a telescope with a given magnification, the entrance pupil diame-
`ter should be chosen such that the exit pupil size matches the eye pupil.
`A focal length of the objective and the eyepiece should be chosen such
`that the eyepiece can have a sufficiently large eye relief. Now, when the
`specs for each subsystem are defined, it is time to use the computer-
`aided design algorithms and associated software to optimize the system,
`which will be discussed in the rest of this chapter. Each subsystem can
`be designed and optimized individually, and the modules joined together
`or, more often, some subsystems are optimized separately and some as an
`integral part of the whole system.
`
`What Do We Do When We
`Optimize a Lens System?
`
`Present-day computer hardware and software have significantly changed
`the process of lens design. A simple lens with several elements has nearly
`an infinite number of possible solutions. Each surface can take on an
`infinite number of specific radii, ranging from steeply curved concave,
`through flat, and on to steeply curved convex. There are a near infinite
`number of possible design permutations for even the simplest lenses.
`How does one optimize the performance with so many possible permu-
`tations? Computers have made what was once a tedious and time-
`consuming task at least manageable.
`The essence of most lens design computer programs is as follows:
`First, the designer has to enter in the program the starting optical
`system. Then, each variable is changed a small amount, called an
`increment, and the effect to performance is then computed. For
`example, the first thickness may be changed by 0.05 mm as its
`increment. Once this increment in thickness is made, the overall
`performance, including image quality as well as physical
`constraints, are computed. The results are stored, and the second
`thickness is now changed by 0.05 mm and so on for all variables
`that the user has designated. Variables include radii, airspaces,
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`element thicknesses, glass refractive index, and Abbe number. If
`you are using aspheric or diffractive surfaces, then the appropriate
`coefficients are also variables.
`The measure of performance as used here is a quantitative
`characterization of the optical performance combined with a
`measure of how well the system meets its first-order constraints set
`by the user such as focal length, packaging constraints, center and
`edge thickness violations, and others. The result of the computation
`is a single number called an error function or merit function. The
`lower the number, the better the performance. One typical error
`function criteria is the rms blur radius, which, in effect, is the radius
`of a circle containing 68% of the energy. Other criteria include
`optical path difference, and even MTF, as described in Chap. 15.
`The result is a series of derivatives relating the change in
`performance (P) versus the change in the first variable (V1), the
`change in performance (P) versus the change in the second variable
`(V2), and so on. This takes on the following form:
`∂Pᎏ
`∂Pᎏ
`∂Pᎏ
`∂Pᎏ
`…
`∂V1
`∂V2
`∂V3
`∂V4
`This set of partial derivatives tells in which direction each parameter
`has to change to reduce the value of the sum of the squares of the
`performance residuals. This process of simultaneous parameter
`changes is repeated until an optimum solution is reached.
`
`,
`
`,
`
`,
`
`A lens system consists of a nearly infinite number of possible solu-
`tions in a highly multidimensional space, and it is the job of the designer
`to determine the optimum solution.
`Designers have used the following analogy to describe just how a lens
`design program works:
`
`Assume that you cannot see and you are placed in a three-
`dimensional terrain with randomly changing hills and valleys. Your
`goal is to locate the lowest elevation or altitude, which in our analogy
`equates to the lowest error function or merit function. The lower the
`error function, the better the image quality, with the “goodness” of
`performance being inversely proportional to the elevation.
`You are given a stick about 2 m long, and you first stand in place
`and turn around tapping the stick on the ground trying to find
`which direction to walk so as to go down in elevation.
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`Once you determine the azimuth resulting in the greatest drop in
`elevation, you step forward in that direction by 2 m.
`You now repeat this process until in every direction the elevation
`goes up or is level, in which case you have located the lowest
`elevation.
`But what if just over a nearby hill is an even lower valley than you
`are now in? How can you find this region of solution? You could
`use a longer stick, or you could step forward a distance several
`times as long as the length of your stick. If you knew that the
`derivative or slope downward is linear or at least will continue to
`proceed downward, this may be a viable approach. This is clearly a
`nontrivial mathematical problem for which many complex and
`innovative algorithms have been derived over the years. But the
`problem is so nontrivial as well as nonlinear that software
`algorithms to locate the so-called global minimum in the error
`function are still elusive. Needless to say, the true global minimum
`in the error function may be quite different or distant from the
`current location in our n-dimensional terrain.
`
`Figure 9.1 shows a two-dimensional representation of solution space as
`discussed previously. The ordinate is the error function or merit func-
`tion, which is a measure of image quality, and the abscissa is, in effect,
`solution space. We may initiate a design on the left and the initial
`
`Figure 9.1
`Illustration of Solution
`Space in Lens Design
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`optimization brings the error function to the first minimum called a
`local minimum in the error function. We then change glasses and/or
`make other changes to the design and ultimately are able to move the
`design to the next lower local minimum. Finally, we add additional ele-
`ments and make other changes and we may be able to reach the local
`minimum on the right. But how do we know that we are at, or even
`close to, a global minimum? Here lies the challenge as well as the excite-
`ment of lens design!
`It is important here to note that reaching global minimum in the
`error function is not necessarily the end goal for a design. Factors
`including tolerance sensitivity, packaging, viability of materials, number
`of elements, and many other factors influence the overall assessment or
`“goodness” of a design. Learning how to optimize a lens system is, of
`course, quite critical to the overall effort, and learning how to reach a
`viable local or near-global minimum in the error function is very
`important to the overall success of a project.
`
`How Does the Designer Approach
`the Optical Design Task?
`
`The following are the basic steps generally followed by an experienced
`optical designer in performing a given design task. Needless to say, due
`to the inherent complexity of optical design, the processes often
`become far more involved and time consuming. Figure 9.2 outlines these
`basic steps:
`
`1. The first step in the design process is to acquire and review all of the
`specifications. This includes all optical specifications including focal
`length, ƒ/number, full field of view, packaging constraints,
`performance goal, environmental requirements, and others.
`2. Then we select a representative viable starting point. The starting point
`should, wherever possible, be a configuration which is inherently
`capable of meeting the specifications for the design. For example,
`if the specifications are for an ƒ/10 monochromatic lens covering
`a very small field of view and having an entrance pupil diameter
`of 5 mm, then the lens may very well be a single element. However,
`if the requirements call for an ƒ/1.2 lens over a wide spectral band
`covering a 40° full field of view, then the solution may very well
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`Figure 9.2
`Lens Design and Optimization Procedure
`
`be a very complex six- to seven-element double Gauss lens form.
`If we were to use a single element for this latter starting point,
`there would be no hope for a viable solution. Finding a good
`starting point is very important in obtaining a viable solution.
`The following are viable sources for starting points:
`You can use a patent as a starting point. There are many sources
`for lens patents including Warren Smith’s excellent book Modern
`Lens Design. There is also a CD-ROM called “LensView,” which
`contains over 20,000 designs from patents. These are all searchable
`by a host of key parameters. While the authors of this book are
`not patent attorneys, we can say with confidence that you may
`legally enter design data from a patent into your computer and
`work with it in any way that you would like to. If your resulting
`design is sold on the market, and if the design infringes on the
`patent you used (or any other for that matter), you could be cited
`for patent infringement. It is interesting to note that the purpose
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`of our patent system in this country is to promote inventions and
`innovation. This is done by offering an inventor a 17-year exclusive
`right to his or her invention in exchange for teaching in the patent
`how to implement the invention. Thus, you are, in effect, invited
`to use the design data and work with it with the goal of coming
`up with a better design, which you can then go out and patent.
`By this philosophy, inventors are constantly challenged to improve
`upon an invention, which, in effect, advances technology, which is
`what the patent process is all about. Needless to say, we urge you to
`be careful in your use of patents.
`You could use a so-called hybrid design. We mean a hybrid to be the
`combining of two or more otherwise viable design approaches
`so as to yield a new system configuration. For example, a moderate
`field-of-view Tessar lens design form can be combined with one
`or more strongly negatively powered elements in the front to
`create an extremely wide-angle lens. In effect, the Tessar is now
`used over a field of view similar to its designed field, and the
`negative element or elements bend or “horse” the rays around
`to cover the wider field of view. An original design can, of course,
`be a viable starting point. As your experience continues to mature,
`you will eventually become comfortable with “starting from
`scratch.” With today’s computer-aided design software, this works
`most of the time with simple systems such as doublets and
`triplets; however, with more complex systems, you may have
`problems and will likely be better off resorting to a patent or
`other source for a starting point.
`3. Once you have entered your starting point into the software
`package you are using, it is time to establish the variables and
`constraints. The system variables include the following: radii,
`thicknesses, airspaces, surface tilts and decenters, glass
`characteristics (refractive index and Abbe number), and aspheric
`and/or other surface variables, including aspheric coefficients. The
`constraints include items such as focal length, ƒ/number,
`packaging-related parameters (length, diameter, etc.), specific airspaces,
`specific ray angles, and virtually any other system requirement.
`Wavelength and field weights are also required to be input. It
`is important to note that it is not imperative (nor is it advisable)
`to vary every conceivable variable in a lens, especially early in the
`design phase. For example, your initial design optimization should
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`probably be done using the glasses from the starting point, in
`other words do not vary glass characteristics initially. This will
`come later once the design begins to take shape and becomes
`viable. You may also want to restrict the radii or thicknesses you
`vary as well, at least initially. For example, if adjacent elements
`have a very small airspace in the starting design, this may be for a
`good reason, and you should probably leave them fixed. Also,
`element thicknesses are very often not of great value as variables, at
`least initially, in a design task, so it is usually best to keep element
`thicknesses set to values which will be viable for the manufacturer.
`4. You now will set the performance error function and enter the constraints.
`Most programs allow the user to define a fully “canned” or
`automatically generated error function, which, as discussed earlier,
`may be the rms blur radius weighted over the input wavelengths
`and the fields of view. In the Zemax program the user selects the
`number of rings and arms for which rays will be traced into the
`entrance pupil (rays are traced at the respective intersection points
`of the designated number of rings and arms). Chapter 22 shows a
`detailed example of how we work with the error function.
`5. It is now time to initiate the optimization. The optimization will
`run anywhere from a few seconds for simple systems to many
`hours, depending on just how complex your system is and how
`many rays, fields of view, wavelengths, and other criteria are in the
`system. Today, a state-of-the-art PC optimizing a six- to seven-
`element double Gauss lens with five fields of view will take in the
`order of 5 to 10 s per optimization cycle. Once the computer has
`done as much as it can and reaches a local minimum in the error
`function, it stops and you are automatically exited from the
`optimization routine.
`6. You now evaluate the performanceusing whatever criteria were
`specified for the lens. This may include MTF, encircled energy,
`rms spot radius, distortion, and others.
`7. You now repeat steps 3 and 5 until the desired performance is met.Step 3
`was to establish the variables and constraints, and step 5 was to
`run the optimization, and these steps are repeated as many times
`as necessary to meet the performance goals. You will often reach
`a solution that simply does not meet your performance
`requirements. This is very common during the design evolution,
`so do not be surprised, depressed, or embarrassed if it happens
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`to you… it happens to the best of us. When it does happen, you
`may need to add or split the optical power of one or more of the lens
`elements and/or to modify glass characteristics. As we have discussed
`previously, splitting optical power is extremely valuable in
`minimizing the aberrations of a lens.
`8. There is a really simple way of splitting an element in two, and
`while it is not “technically robust,” it does work most of the time.
`What you do is insert two surfaces in the middle of the current
`element, the first of which will be air and the second is the
`material of your initial element. The thickness of each “new”
`element is one-half of the initial element and the airspace should
`be small, like 0.1, for example. Now simply enter twice the radius
`of the original element for both s1 and s 2 of the new elements.
`You will end up with two elements whose net power sum
`is nearly the same as your initial element. You can now proceed
`and vary their radii, the airspace, and, as required, the thicknesses.
`9. If you still cannot reach a viable design, then at this point you
`will need to return to step 2 and select a new starting point.
`10. Your final task in the design process is to perform a tolerance analysis
`and performance error budget. We will be discussing tolerancing
`in more depth in Chap. 16. In reality, you should be monitoring
`your tolerance sensitivities throughout the design process so that
`if the tolerances appear too tight, you can take action early in the
`design phase and perhaps select a less sensitive design form.
`11. Finally, you will need to generate optical element prints, contact a viable
`lens manufacturer, and have your elements produced. You will also need
`to work with a qualified mechanical designer who will design the
`cell or housing as well as any required interfaces. It is important
`to note that while we list the mechanical design as taking place
`at this point after the lens design is complete, it is extremely
`important to work with your mechanical designer throughout the
`lens design process so as to reach an optimum for both the optics
`as well as the mechanics. Similarly, you should establish a dialog
`with the optical shop prior to completing the design so as to have
`time to modify parameters which the shop feels needs attention
`such as element thicknesses, glass types, and other parameters.
`12. Once the components are in house, you will need to have the lens
`assembled and tested. Assembly should be done to a level of
`precision and cleanliness commensurate with the overall
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`performance goals. Similarly, testing should be to a criterion which
`matches or can be correlated with your system specifications and
`requirements. We discuss testing in Chap. 15.
`
`Sample Lens Design Problem
`
`There was a very interesting sample lens design problem presented at
`the 1980 International Lens Design Conference. The optimized design
`for an ƒ/2.0, 100-mm focal length, 30° full field-of-view double Gauss
`lens similar to a 35-mm camera lens was sent out to the lens design com-
`munity. One of the tasks was to redesign the lens to be ƒ/5 covering a
`55° full field with 50% vignetting permitted. Figure 9.3 shows the origi-
`nal starting design, as well as the design after changing the ƒ/number
`and field of view, without any optimization.
`Sixteen designers submitted their results, and they spent from 2 to 80 h
`working on the problem. We will present here three representative solu-
`tions in Fig. 9.4. The design in Fig. 9.4a is what we often call a happy lens.
`What we mean is that the lens is quite well behaved with no steep bend-
`ing or severe angles of incidence. The rays seem to “meander” nicely
`through the lens. It is a comfortable design. We show to the right of the
`layout a plot of the MTF. MTF will be discussed in detail in Chap. 10. For
`the purpose of this discussion, consider the MTF to be contrast plotted
`in the ordinate as a function of the number of line pairs per millimeter
`
`Figure 9.3
`Starting Design for
`Sample Lens Design
`Problem
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`Figure 9.4
`Representative Solutions to Sample Lens Design Problem
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`in the abscissa. The different curves represent different positions in the
`field of view and different orientations of the resolution patterns. The
`higher the curves, the better the contrast and the overall performance.
`The MTF is reasonable for most of the field positions. As will be dis-
`cussed in Chap. 22, a good rule of thumb for the MTF of a 35-mm cam-
`era lens is an MTF of 0.3 at 50 line pairs/mm and 0.5 at 30 line pairs/mm.
`The design in Fig. 9.4b has a serious problem; the rays entering the last
`element are at near-grazing angles of incidence. Notice that the exit
`pupil at full field is to the right of the lens (since the ray cone is
`descending toward the axis to the right), and at 70% of the field the exit
`pupil is to the left of the lens (since the ray cone is ascending to
`the right and therefore appears to have crossed the axis to the left of the
`lens). This is a direct result of the steep angles of incidence of rays enter-
`ing the last element. The variation in exit pupil location described here
`would not itself be an issue unless this lens were used in conjunction
`with another optical system following it to the right; however, it does
`indicate clearly the presence of the severe ray bending which will
`inevitably lead to tight manufacturing and assembly tolerances. Further,
`the last element has a near-zero edge thickness which would need to be
`increased. The lens is large, bulky, and heavy. And finally, the MTF of
`this design is the lowest of the three designs presented.
`Finally, the design in Fig. 9.4c is somewhat of a compromise of the two
`prior designs in that it is somewhat spread out from the design in Fig. 9.4a
`but does not have the problems of the design in Fig. 9.4b. The MTF of
`the design in Fig. 9.4c is the best of the three designs.
`Comparing the three designs is very instructive as it shows the
`extreme variability of results to the same problem by three designers.
`The question to ask yourself is what would you do if you subcontracted
`the design for such a lens, and after a week or two the designer brought
`you a stack of paper 200-mm thick with the results of the design in Fig. 9.4b.
`And what if he or she said “wow, what a difficult design! But I have this
`fabulous solution for you!” Prior to reading this book, you might have
`been inclined to congratulate the designer on a job well done, only to
`have problems later on during manufacturing and assembly. Now, how-
`ever, you know that there may be alternate solutions offering superior
`performance with looser tolerances and improved packaging. Remember
`that even a simple lens has a near infinite number of possible solutions
`in a multidimensional space.
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