Aspects of contemporary optical design

Aspects of contemporary optical design P.J. ROGERS

Some aspects of contemporary optical design are discussed. The more commonly encountered modern design methods and also some of the more unusual techniques are briefly described. Particular emphasis is placed on the role of the computer as this has become an essential in today’s optical design. Examples of recently designed optical systems are given to illustrate the results obtained using modern design methods.

All fields of technology have seen rapid advances over the la8t twenty years. One of the major reasons has been the increasing use of computers which has revolutionized the design process in many parts in industry and research. Optical design is as good an example as any: only a few decades ago the majority of the time involved in the design of an optical system was that required for repetitive ray tracing by hand. Admittedly, many clever methods of extracting the maximum of information from the minimum of ray tracing were developed, but the days are not long gone when tracing a skew ray was regarded as-a luxury only to be considered when an exacting lens was being designed. Nowadays by comparison, optical designers appear to be spoilt by having available to them powerful, relatively inexpensive computers and versatile design programs capable of handling most types of optical system. This is only partially true in that the level of requirement has increased at the same rate as the means by which that requirement can be fulfilled. No doubt relatively untrained personnel can now design optical systems using a monkey/typewriter approach but this only applies to the most straightforward optics. Today’s difficult design tasks require not only a fast computer and a good optimization program but, even more importantly, a flexible, creative approach by an experienced designer. The relative ease with which manufacturing tolerances can now be estimated also gives less excuse for designers who produce ‘paper’ designs which are difficult to make. Less subjective, better quantified methods of evaluating image quality such as the MTF are now widely accepted and easy to calculate. Agreement between theory and practice is closer than ever before although inevitably the good old fashioned gremlin still appears on occasions, even in the best regulated circles. Computer-aided

optical design

Speed of computation

To put the following section into context. It is easy for a designer new to the profession to be patronizing about the The author is at Pilkington P.E. Limited, Received 15 February 1979.

St. Asaph, Clwyd,

UK.

0030-3992/79/040203-09 OPTICS AND LASER TECHNOLOGY. AUGUST 1979

optical design methods and the slow computers of only twenty years ago. It should be remembered that the designers of that period had the same feelings when they looked back at their predecessors with their log tables and, later, Marchant calculators. Each age of a technology develops to its own high level of sophistication. Designers of previous generations were forced by circumstance to understand thoroughly the how and why of lens design and used many ingenious methods of getting round their lack of computing power.’ Our generation of designers has reason to be grateful to them for passing on their knowledge, as a similar understanding is still essential if a modem optimization program is to be used efficiently. The increase in computing power has indeed been amazing over the past decade. An article written by the author in 19692 quoted a typical time of 20 minutes for one cycle of lens optimization on our computer of that time, a single station IBM 1130 (Fig. 1): a similar cycle on our present PDP 1 l/55 (Fig. 2), a machine that costs about the same as the 1130 allowing for the change in the value of money, would take 30 seconds. In addition to this, the PDP has six times the working storage, ten times the disc space and allows two designers to work interactively with the optimization program at the same time. By contrast with their predecessors who had to regard skew rays as a luxury, today’s optical designers are much less restricted. For example, the PPE computer program has the capability of tracing up to 60 skew rays from a skew object through a series of non-rotationally symmetrical aspheric optical surfaces: though this was a formidable mathematical problem to sort out initially. There has been much discussion in the past on the subject of whether optical design is most efficiently performed on a small dedicated computer or on a time-sharing basis on a larger machine. The latter was the fashion until fairly recently3 and has some advantages particularly those of speed and accuracy of computation; but getting access to a larger computer can be difficult when it is in heavy demand. Smaller dedicated machines have become much more attractive in the last few years, as a computer having a memory and

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Fig. 1 IBM 1130 computer -- a photograph o f the PPE optical design facility taken in 1967

speed sufficient for semi-automatic optical design can now be purchased relatively cheaply: in addition there are the obvious advantages of easy access and greater suitability for the interactive type of optimization program. Facilities of modern optimization programs and examples of optical design The following are just a few examples of the facilities of some modem optimization programs and the types of optical system made possible by their use.

Multi-configuration optimization.

A feature of major importance in a modem program is a multi-configuration (or split) facility,4 the purpose of this being to enable a designer to optimize several different aspects of an optical system simultaneously. In the PPE program, this facility has been structured to be as versatile as possible by defining variable system parameters - curvatures, separations etc in three ways: global (affecting all configurations); independent (affecting each configuration separately); and/or linked (affecting several parameters in any configuration in the same or opposite sense). Fig. 3 shows an example of parameter specification. By the use of this approach, up to four optical systems, which may be anywhere between identical or totally independent, can be optimized at the same time. No-one would really want to use the two extremes: optimizing the same system four times is pointless and it would require a very flexible mind to handle four completdy different systems simultaneously. The possibilities inbetween these two extremes however are numerous. Some of the many example are: 1. Zoom lenses - either optically or mechanically compensated with any number of moving groups of lenses, their motions being linked or independent. 2. Lenses requiring a large depth of focus - achieved by specifying the same system at two or more foci and then optimizing these together. 3. Reducing the sensitivity of a design - this can be treated in the same manner as 2 except that the system is specified up to four times, ie basic design plus three others in which small changes are made to particularly sensitive parameters. The original and perturbed systems can then be optimized together. This is really only a fining technique, the design of an optical system that will be as easy as possible to manufacture is more a question of 204

Fig. 2 Part of the current Pilkington P.E. PDP 11/55 optical design computer facility

working out an initial scheme which has no inherently

sensitive parts such as high individual lens powers giving large aberration contributions (see for example the last part of the section entitled 'Estimation of manufacturing tolerances'). 4. Multi-channel optical systems with some common and some independent components. This category extends

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Fig. 3 An example o f multi-configuration optimization where the parameter specification is given in terms o f / , JIK1, J2K2 etc. I - parameter code; J -- configuration code; K - surface number. * Global parameter -- vary first curvature, all systemS. 01, 1 0 1 , 2 0 1 , 3 0 1 @Independent parameter -- vary air gap, system 1. 02, 104 @ Linked parameters -- vary curve on moving element. 0 1 , 2 0 5 , --306

OPTICS A N D LASER T E C H N O L O G Y . A U G U S T 1979

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over all types of multi-channel system including switchable and also those covering more than one spectral bandwidth. Anamorphic optics having an image scale that varies with orientation. Multi-conjugate fLxed focal length lenses required to give good imagery over a range of magnifications. Elimination of narcissus effects (self-imaging of the detector elements) in infra-red optics. Part of the system can be run on the program alongside the overall system. The part system, defined from the detector to the offending surface and then back to the detector, can be deliberately 'de-optimi."zed' by def'ming large target aberrations (including the effects of defocus). Optical systems containing scanning elements. For example the afocal plus scanning plus detector optics combination found in some infra-red systems. The afocal and detector optics can be optimized together at four different positions of the scanner drum (given that collimation is approximately maintained between afocal and detector optics and the aberration transfer is not too great).

9. Single optical systems requiring control over more than just the usual imagery. For example, the good pupil and intermediate image quality often required in non-Galilean afocals, Fourier transform lenses and daysight telescopes. An example of type 9 is shown in Fig. 4. This is a compact infra-red (8 to 13 ttm bandwidth) afocal telescope having an internal bend and covering wide angular fields of 28 and 70 degrees in object and scanner space respectively. By the use of a multi-configuration facility this extreme system was optimized in terms of both good real world and intermediate image quality and also fidelity of pupil imagery.

Specialized designs. Other features built into the PPE program allow control over individual and overall lengths,

lens diameters and the positions of both pupils; angle and height solves for specific rays; and also ray tracing through Fresnel lenses. Between them these features will cope with many unusual optical design problems. Examples are telecentric optics (telecentdc in either or both conjugates); systems with internal collimated sections; periscopes; and glass lens plus Fresnel lens combinations, An external exit pupil head-down display optical system using the latter combination is shown in Fig. 5. The Fresnel was optimized in conjunction with the transfer lens to give a well-controlled exit pupil shape by the use of height solves at the exit pupil plane. The high aperture transfer lens was designed to have two air-spaces suitable for the injection of data in addition to that of the main channel.

Large pupil visual optics. Visual optical systems that provide a large exit pupil which is sampled by both eyes - biocular magnifiers, external pupil displays (as in the above paragraph) - have always been difficult to evaluate and optimize, largely because the situation is dynamic. Each eye may see an acceptable image but every part of each image may move both laterally and longitudinally at different rates and/or in a different direction as the eyes move inside the exit pupil. Computer techniques have been evolved s that will calculate the resolution, geometry and vergence values corresponding to a pair of eyes viewing the image(s) from any part of the exit pupil. These values can then be used as criteria to optimize the overall pupil/overall picture quality. Biocular magnifiers, often in the form of large aspheric plastic lenses, have been in use for years (there is also the novel expanded-pupil Dynascope~) but these were of low magnifying power, typically x 2½. A few years ago it was found to be possible to design angled biocular magnifiers of 80 millimetres width, up to x 5~ magnification and 60 degrees field of view: a relative aperture of about F/0.55 being necessary to achieve this (Fig. 6). These

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A wide angle infra-red afocal optimized using a multi-configuration facility to give good real world and intermediateimage quality and low pupil aberrations (Pat. applied for)

OPTICS AND LASER TECHNOLOGY. AUGUST 1979

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Fresnel lens with image on faceted surface

v Transfer lens Fig. 5

Transfer plus Fresnet field lens combination used in an external pupil head-down display (Pat. applied for)

biocular magnifiers, for use in night vision equipment, were made possible by the optimization techniques mentioned above that gave the correct balance to the large residual aberrations inevitable in such extreme systems, s A more recent and alternative approach to the problem of designing a single lens system that will provide a magnified, two-eye view of a small object is shown in Fig. 7. This uses Inten~fier tube

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a combination of a high aperture, wide angle collimating lens and two pairs of cemented rhomboid prisms; the latter having variable ratio semi-reflecting/transmitting coatings at their cemented interfaces. 7 A continuous, wide-angle image of even brightness is presented to each eye by careful optimization of prism dimensions and variable ratio coatings, each eye seeing the image through a series of overlapping, multi-reflected vertical pupil strips. The resultant device can provide both eyes with a high quality, x 10 magnified image of the output of a small image intensifier tube. If a full width biocular magnifier is required, the central 'V' between the prisms could be tidied by another prism and semireflecting/transmitting coatings substituted for the inner totally reflecting coatings: the drawback is that the system would then be sensitive to bright lights behind the observer.

Estimation of manufacturing tolerances. Many modern optical design computer programs include some means of manufacturing tolerance estimation; these ranging from a simple change table to a fully automated process. The fully automatic methods need to be treated with caution as the assumptions often made, for example that all the manufacturing errors will follow a rectangular frequency distribution within the specified tolerance band, only approach the truth for very high volume production levels. Nevertheless, if used intelligently, an automatic method gives a good indication of likely manufacturing tolerances: it should also be excellent at establishing the comparative sensitivities of alternative optical designs. Smith s has gone one stage further and extended a tolerancing program into a manufacturing cost estimation program that also takes into account material

OPTICS AND LASER TECHNOLOGY. AUGUST 1979

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aberrations and this perhaps explains their attraction in academic circles. Conrady's warning of 50 years ago is worth recalling - 'There is a deplorable tendency among students to concentrate on the mathematical equations and their proofs. These, whilst necessary and highly useful, are merely the dry bones of applied optics'. 1 Wave aberrations are indeed highly useful in certain circumstances (see later) but other measures of image defect can also be valuable. Transverse ray aberrations (TRA) can represent a good, easy-to-appreciate criterion of performance (namely the geometric point spread function) for optical systems far from the diffraction limit.

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Even when a high level of aberration correction is desired, intelligent use of TRA can result in a diffraction limited lens; although it must be admitted that, in themselves, transverse ray aberrations are of dubious validity near the diffraction limit.

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cost and workability, blocking-up quantities etc. In an attempt to aid manufacture, Grey9 has described ways of decreasing the sensitivity to tilt of individual lens components during the optimization of the overall lens system performance.

Optimization to the diffraction limit. In recent years, some university courses in applied optics have concentrated on wavefront aberrations and diffraction-based criteria to the exclusion of other methods of image analysis. Wave aberrations have greater theoretical elegance than finite ray

The following are two example of recent optical designs having a performance close to the diffraction limit that were optimized on a transverse ray aberration basis. On the face of it, this method may seem the antithesis of a modem optimization technique but, for the first example at least, the results obtained represent the current state-of-the-art. 1. Aerial camera lens: Fig. 8 shows a comparison between a recent 1 metre focal length F/8 aerial camera lens design and an equivalent design of 25 years ago. The increase in MTF and consequent system performance are considerable,

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OPTICS AND LASER TECHNOLOGY. AUGUST 1979

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Infra-red zoom lens: 130--45.5 mm EFL; F/2.5; 27.4°-7.6 ° FOV

OPTICS AND LASER TECHNOLOGY.

AUGUST 1979

although it is only fair to say that the modern design is more complex. It was made possible by the fact that, using an optimization program, the modern designer can handle a large number of parameters more easily and also obtain a better corrected solution than a designer of 25 years ago. The performance level achieved by the new design is worth emphasizing: in free-air it corresponds to a resolution of one arc second over at least 14 degrees of the field of view.

(Fig. 10). The hologram in combination with a relay lens produces an infinity image of the CRT and a real exit pupil at the observer's head position. The current drawback to this system, apart from chromatic effects, is that the relay lens contains several non-rotationally symmetrical components in order to compensate for the asymmetric aberrations of the holographic combiner.

Unusual modern optical design techniques 2. Infra-red zoom lens (8 to 13/am bandwidth): It has been suggested that the design of infra-red optics is made more difficult by the lack of suitable materials. This is true in a way but if a designer of a previous generation had been told that he was constrained to using a material having a refractive index of 4.0 and almost no dispersion, he would have thought that it was a particularly bad joke. The real problem in the design of the majority of infra-red optics is not so much that of aberration correction - the high refractive index of germanium largely takes care of that but more those of size, weight, thermal effects, transmission, cost and the characteristics demanded by the current methods of detection. Medium to large diameter optics are inevitable at far infra-red wavelengths simply because of the limitation of resolution caused by diffraction. Germanium has a high specific gravity and dn/dt coefficient, a poor transmission and is relatively expensive: zero vignetting and a high quality of pupil imagery are also essential in most systems. Some of these requirements indicate an optical system with the minimum number of lens elements. Aspheric surfaces appear to be the obvious solution and in many cases this is certainly correct. There is no such thing as a universal panacea however, and aspheric surfaces do not always have a major advantage over spherical surfaces. 1° Figure 9 shows an infra-red zoom lens continuously variable over a 130 to 455 mm focal length range: it has an aperture of F/2.5 and fields of view of 27.4 and 7.6 degrees at the extremes of the zoom range. The number of elements in the zoom lens are a minimum given the three requirements of continuous operation; the ability to focus at the same object distance for all zoom positions; and complete achromatism. H

Holographic optics. Holographic optical elements are gaining popularity now that reflective holograms can be produced that are highly efficient (90-100%) at least over small wavelength and incidence angle ranges. Large lightweight reflective holograms that are the equivalent of conventional optical elements of complex aspheric shape can be used in conjunction with normal glass lens elements to give a combined system of considerable potential. The design of composite holographic/conventional optical systems has, however, many difficulties, particularly those of making allowance for the unusual parameters involved (emulsion shrinkage, off-axis operation etc) and also of correcting the inherent non-rotationally symmetrical aberrations introduced by a hologram. Development work on holographic optics is being carried out in the UK although a lot of the progress in holographic plus conventional optical systems has been in the USA. Withrington 12 has disclosed a holographic head-up display module that employs a large hologram as a powered combiner

OPTICS AND LASER T E C H N O L O G Y . AUGUST 1979

Several optical design techniques that appear at first sight to be fundamentally different from the classic methods have been put forward. A number of these techniques involve the use of a more elaborate merit function than the conventional sum of the squares of weighted values of selected ray aberrations (Jamieson 13 gives detailed analysis of all but the most recent techniques). It may be relevant to quote R.E. Hopkins' perceptive remark that some merit functions are just an expression of a particular designer's prejudices (author's version of Hopkins' remark heard at the 1975 Haverford Lens Design Conference). Doubtless this is true, but the following examples have advantages to recommend them in certain circumstances. The author apologises in advance for any misrepresentation of the techniques instanced or the omission of other worthy examples.

YY diagram. A first order design technique due to Delano 14 in which the paraxial image and principal ray intersection heights are plotted against each other as ordinate and abscissa respectively. A great deal of information on the Gaussian properties of a given optical system can be gained from the YY diagram. Alternatively, a complex optical system may be deduced quite easily in terms of the spaced thin lens powers from a YY diagram constructed from the first order constraints of a system. It is claimed that the first order layout of an optical system can be derived more easily by using the YY method than by using conventional techniques. In spite of the latter, and the undoubted elegance of the method, it does not seem to have gained wide acceptance to date. Caustic surface analysis. A geometric evaluation technique described by Shealy ts involving the analysis of the caustic surfaces (loci of points of high energy concentration) produced in the focal region of an aberrated lens. It has the advantage that, as with the much earlier Diapoint analysis due to Hertzberger, it is independent of focus position. The caustic surfaces, one per field position, indicate the state of

Fig. 10 Schematic of a holographic head up display module (reproduced by permission of Hughes Aircraft Co.)

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correction of an optical system, if rather indirectly. The compactness of the caustic surfaces may be used in combination as a merit function in that, as the aberration residuals reduce to zero, the caustics reduce to geometric point images in the paraxial image plane. The technique would seem to be of the most advantage in systems where longitudinal aberrations are relatively large and qualitative information is desired with regard to the performance in the general focal region rather that at a specific focus. Stravroudis and Fronczek 16 have recently published a paper demonstrating two sheets of each of the various caustic surfaces given by a simple piano-convex lens, each sheet being shown as a stereo pair. Fig. 11 gives an example.

Geometric MTF optimi';atiorL This has been suggested by Gostik (in an unpublish d paper) l? as a way of achieving a final balance of aberrations in any optical system, even one in which the optimized performance is expected to be close to the diffraction limit. Gostik submitted that performance could be improved by maximizing the real cosine part of the geometric MTF equation and ignoring the imaginary sine part (although the latter would seem to be rather dangerous, particularly off-axis). Obviously the geometric MTF becomes inaccurate as the diffraction limit is approached but the claim was made that it remains a useful criterion in that an increase in geometric modulation implies an increase in performance.

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For small errors (RMS wave aberration ~< 0.07 X), the wavefront variance can be related to the Strehl Ratio using Marechal's equation, 21 on axis at least. As the Strehl Ratio approximately represents the integrated spatial frequency response, ie the area under the diffraction-based MTF curve, the wavefront variance is a good overall criterion of high quality lens performance. Hopkins' method optimizes the response at a specific spatial frequency, sometimes to the detriment of other frequencies) 9 This is not serious if the frequency at which maximum response is required is low with respect to the cut-off (see following paragraph) as only the higher spatial frequencies would be affected.

MTF optimization at low spat&l frequencies. Optimization of the Strehl Ratio improves overall MTF response; in most cases, however, a maximum response is required at spatial frequencies much less than that of the diffraction limited cut-off. Offner 22 has described a merit function that enables the diffraction-based MTF response to be optimized for a system with an arbitrarily shaped pupil at normalized

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Wave aberration based merit functions. Optimization of wavefront aberration variance or variance of a wave aberration difference function has been suggested by Barakat and Houston 18 and also Meiron 19 in the former case and Hopkins 2° in the latter.

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OPTICS AND LASER T E C H N O L O G Y . AUGUST 1979

frequencies of typically S = 0.1. Offner's merit function equates to zero for a diffraction-limited response but, as it is represented by the difference between two functions, it can give an incorrect solution if used initially on a highly aberrated system. It is most useful as a method of achieving a better aberration balance in a fairly well-corrected system. An alternative approach to the same problem, applicable to an initial system in any state of correction, would be to optimize transverse ray aberrations using a grid of rays equally spaced over the entrance pupil for each field point, all rays having a unity weight. Adaptive correction method An optimization technique due to Glatzel and Wilson. 23 To quote from their paper 'It is fundamental to this method that the number of aberration being corrected at any one time should be less than the number of available optimizing parameters. The optical designer chooses a small number of aberrations and establishes target values. For the other (generally higher order) aberrations limits are set. In the course of correction both the target values and the limits are reduced'. A major claim of the adaptive method is that it enables the designer to learn about the inherent correction potientials of a given basic type of system. It is possible that a similar optimization may be carried out using a damped-least-squares program, if rather less efficiently. Lower weights could be applied initially to the more non-linear aberrations thus allowing larger steps to be taken in parameter space. When a high state of correction of the more linear aberrations has been achieved, the weights on the other aberrations could be increased. Simplex optimization. An optimization technique advocated for optical design by Blandford. 24 Any merit function may be used, whether simple, complex or discontinuous. The value of the merit function is computed for the base system and for systems with each variable parameter incremented in turn. The 'worst' point, that having the highest merit function, is identified. The co-ordinates (values o f all variables in multi-dimensional parameter space) o f the centroid of all the points except the worst are found, and a new point is identified by extrapolating along the line joining the worst point to the centroid. The worst point is then dropped, the new point added, and the process repeated until a minimum value of the merit function is reached. Simplex optimization has the advantages that it is easy to program and can be used on a very small computer. It probably has no major advantage when a larger machine is available, but nevertheless it is a very neat technique. Conclusions Ten years ago my conclusions on the state o f optical design at that time were: 'To sum up, the job of a lens designer is

OPTICS AND LASER TECHNOLOGY . AUGUST 1979

at a very interesting stage, the tedium of the job has been removed by the computer but the reasoning and intuitive work still remain'. 2 The major changes since the above was written are that, although little different in concept, the majority of optical design computer programs have increased considerably in sophistication and some of these programs can be operated on relatively inexpensive machines. Commercial program packages are widely available and fairly small organizations can now equip themselves with some form of optical design facility. Fortunately perhaps for professional optical designers, there is still no substitute for skill and experience. The design of the complex, heavily constrained optics required for the majority of electro-optical systems nowadays demands: a reasonably sized computer; a large, versatile optimization program - preferably interactive in operation - and, most important of all, an experienced designer capable of indulging in the occasional bit of lateral thinking. Acknowledgements

The author thanks his colleagues in the PPE Optical Design Department for their helpful comments: also the directors of PPE for permission to publish.

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Conrady, A.E. Applied optics and optical design, Vol 1 (Constable, 1929) Rogers, P.J. Man Opt lnt 22 (1969) 163-167 Proceedings of lens design with large computers, University of Rochester (1966) Gustafson, D.E. International lens design conference, Haverford (1975) unpublished Rogers, P.J. Proceedings of EO Int, Brighton (1972) 37-43 Freeman, R. British Patent No. 1 275 917 (1971) Rogers, P.J. British Patent No. 1 506 614 (1978) Smith, W.J. International lens design conference, Haverford (1975) unpublished Grey, D.S. Appl Opt 9 (1970) 523-526 Rogers, P.J. Proceedings of computer-aided optical design, San Diego, SPIE 147 (1978) 141-148 Rogers, P.J., Andrews, G.N. Proceedings of the third European EO conference, Geneva, SPIE 99 (1976) 163-175 Withrington, R.J. US Patent No. 3 940 204 (1976) Jamieson, T.]. Optimization techniques in lens design (Hilger, 1971) Delano, E. Appl Opt 2 (1963) 1251-1256 Shealy, D.L. Appl Opt 15 (1976) 2588-2596 Stavroudis, O.N., Fronezek, R.C. Opt and Laser Tech 10 (1978) 185-191 Gostik, R.W. Colloquium on optical design, Imperial College (1977) unpublished Barakat, R., Houston, A. JOSA 55 (1965) 1142-1147 Meiron,J. Appl Opt 7 (1968) 667-672 Hopkins, H.H. Optica Acta 13 (1966) 343-369 Marechal,A. Revue d'Optique 26 (1947) 257 Offner, A. Appl Opt 8 (1969) 2545-2547 Glatzel, E., Wilson, R.L. Appl Opt 7 (1968) 265-276 Blandford, B. Institute of Physics optical group conference, Bath (1978) unpublished

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