Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`Global view of optical design space
`
`Doron Sturlesi
`Donald C. O'Shea, FELLOW SPIE
`Georgia Institute of Technology
`School of Physics
`Center of Optical Science
`and Engineering
`Atlanta, Georgia 30332-0430
`
`Abstract. The optical design space of some simple lenses is investigated
`systematically. Typical space topographies are visualized with 3-D graph-
`ics, where the complete set of available solutions is clearly identified. The
`space characteristics are then studied and compared through the use of
`several merit functions with differing degrees of complexity. A two-phase
`search algorithm, based on global optimization techniques, is proposed
`here. In the first phase, using a coarse sampling approach, the program
`finds the favorable regions that correspond to potentially promising con-
`figurations. In the second phase, conventional optimization routines are
`used to find the best solutions in each region. Then an optimum solution
`is determined according to the application at hand. The proposed algo-
`rithm is analyzed and compared to more conventional design approaches.
`A further refinement of the algorithm excludes from the systematic search
`some unfavorable configuration regions through the use of a simple expert
`system. Search times are further reduced through parallel-processing
`methods. The algorithm provides overall information about a given design
`space and offers a selection of "best" solutions to choose from. As an
`example, it is applied to a triplet objective.
`
`Subject terms: optical design; lens design; parameter space; global optimization;
`sampling techniques; expert systems; parallel processing; Cooke triplet.
`
`Optical Engineering 30(2), 207-218 (February 1991).
`
`CONTENTS
`Introduction
`1.
`2. Global optimization
`3. The optical design space
`3. 1 . Merit function spaces
`3.2. Design space examples
`3.2. 1 . Cemented doublet (CD)
`3.2.2. Air-spaced doublet (ASD)
`3.3. Badly formed merit functions
`3.4.. The general characteristics of the design space
`3 .5 . The characteristics of the various merit functions
`4. The proposed search algorithm
`4. 1 . Algorithm description
`4.2. Limits to the sampling algorithm
`4.3. Benchmark results
`4.4. The triplet objective
`5. Conclusions
`6. References
`
`1. INTRODUCTION
`The design space of optical systems is typically a complicated
`multidimensional parameter space. By constructing a merit func-
`tion space that expresses the departure of individual configura-
`tions from ideal required performance, it is possible to determine
`which of all the possible configurations in this space will yield
`the best solution to the design problem. One of the known
`characteristics' of the merit function space is the large number
`of stable configurations that are contained therein. These con-
`figurations correspond to a large number of local minima (of a
`
`Paper 2857 received Jan. 2, 1990; revised manuscript received July 26, 1990;
`accepted for publication Aug. 7, 1990. This paper is a revision of paper 1168-
`13, presented at the SPIE conference Current Developments in Optical Engi-
`neering and Commercial Optics, Aug. 7, 10—11, 1989, San Diego, Calif. The
`paper presented there appears (unrefereed) in SPIE Proc. 1168.
`1991 Society of Photo-Optical Instrumentation Engineers.
`
`merit function defined over the space), many of them showing
`comparable performance.
`In a conventional design process, an optical designer chooses
`an initial configuration—a point in the design space—and with
`the help of an automatic optimization algorithm moves it around
`in the design space to find a solution that gives the best, or at
`least a good workable performance. The search for the ''ultimate
`
`best,' ' or ''global optimum,' ' is perhaps the most difficult part
`
`in the lens design art. Even the best optimization routines cannot,
`in general, find it. Most of the popular optimization routines
`like damped least squares (DLS), and others related to it, are
`based on systematic descent principles that accept only steps
`which decrease the merit function (improve the performance).
`Thus the designer's solution almost always gets trapped in a
`local minimum 2
`The limited success of the conventional design methods in
`reaching the global minimum is mainly due to the fact that the
`search is confined to narrow corridors along the (downhill) op-
`timization paths. The solution is therefore crucially dependent
`on the initial configuration. As a result, conventional design
`methods work satisfactorily and converge quickly to a good
`solution only if a good starting point has been chosen. Most
`often, the routine ends up in an unfavorable configuration that
`does not satisfy the design requirements. It is not unusual for
`this to happen several times during the first design sessions.
`The conventional design methods are, in principle, local search
`methods and do not provide any global information on the design
`space. The only exception is simulated annealing (SA). This
`method, which was recently revived from an older statistical
`cooling algorithm,3 is the first global optimization algorithm to
`be applied to optical design. This is a stochastic search algorithm
`that follows the principles of the annealing process, where the
`goal is to release residual stresses and to bring a substrate to a
`low energy state. The algorithm can accept uphill steps with a
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`OPTICAL ENGINEERING / February 1991 ,' Vol. 30 No. 2 / 207
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 1
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`STURLESI, O'SHEA
`
`certain finite probability, and thus can theoretically get away
`from local traps. But even this algorithm, according to results
`published recently,48 seldom converges to the global minimum,
`since it is too sensitive to the tuning of its internal constants and
`becomes increasingly inefficient as the number of degrees of
`freedom rises. There are efforts to overcome the poor internal
`constant tuning by dynamic compensation. Thus a number of
`approaches are being used to attack the problem of the global
`optimum in lens design. This paper represents a report on some
`initial results using another technique.
`
`2. GLOBAL OPTIMIZATION
`Global optimization (GO) is a separate area in mathematical
`optimization. Although the fundamental problem is not new,
`most of the work in this area had been done in the past two
`decades. Because of the availability of powerful computers, a
`considerable number of algorithms have been introduced as part
`of the general effort to solve the GO problem.9"° These minimum-
`seeking algorithms were designed to deal with multiminima spaces
`and are, in principle, different from the local optimization al-
`gorithms.
`Since the work in this area is still going on, it is obvious that
`there is no general deterministic algorithm that can locate global
`optimum (or minimum) for every general multidimensional func-
`1 The usefulness of the algorithms depends crucially on
`the type of (objective) function that forms the merit function
`space. The most difficult situation, which is unfortunately typical
`to many engineering problems , is the ''blackbox ' ' situation ,
`2,13
`where the objective function (merit function), which is multi-
`dimensional, cannot be expressed in closed form and its eval-
`uation requires massive numerical computation. As such, func-
`tion evaluation, which is the only way to get information on the
`design space, is a time-consuming operation. Optical design is
`an example of just such a case.
`The problem can be stated most generally as follows; Let R"
`be the n-dimensional design space. A point in that space xER'1
`is characterized by the vector, x = (Xi , x2, X3, .. . , x) where
`{Xi , X2, X3, .. . , x} are the set of coordinates or design parameter
`values that specifies the configuration x. The merit function is
`defined by F(x), where xER'1 and F: R'—>R1 over a compact
`set SC R. The GO problem is to find y' minEs{F(x)}, for
`every xES.13
`This problem is one of a class of NP-hard problems. This is
`a class for which no algorithm is known to give an exact solution
`within polynomial time (i.e. , the computation time increases at
`least exponentially with the complexity of the problem).3
`The simplest global search method is a systematic sampling
`of the function on a multidimensional grid. This deterministic
`method was originally called the factorial' method. In this method
`each design parameter (factor) is divided to a number of levels,
`and for each combination of levels of the set of the design
`parameters a sampling is performed. Sampling, in this context,
`means simply evaluation of the merit function. There are some
`modifications to this method, such as the fractional factorial
`method,'4 in which a systematic deletion of some parameter
`combinations is performed before the function evaluation begins.
`The main problem with this systematic sampling method is that
`the number of function evaluations increases exponentially with
`space dimension which sets some practical dimension limits to
`this method.
`Probabilistic approaches have been introduced to GO in order
`to overcome this practical limit. In most of these stochastic
`
`208 / OPTICAL ENGINEERING / February 1991 / Vol. 30 No. 2
`
`methods there are two
`13 a global phase, during which
`the function is evaluated at randomly sampled points, and a local
`phase, during which the sample points are manipulated by means
`of some local searches to yield a candidate global minimum.
`The major drawback of the stochastic method is that the pos-
`sibility of an absolute guarantee of success is sacrificed in favor
`of limiting the effort. The global phase can, however, yield an
`asymptotic guarantee in the stochastic sense.
`Most successful methods for GO involve local searches from
`some or all of the sample points. This presupposes the avail-
`ability of some local search (LS) procedure. LS is assumed to
`such that if LS is started from any point
`be strictly
`in the xES and converges to a local minimum x , there exists
`a path from x to x along which the function values are non-
`increasin. A common feature in GO methods is domain par-
`titioning. This is basically a grid sampling of the design space
`that creates a collection of cells that can be analyzed later by
`LS procedures.
`The efficiency of GO algorithms can, in principle, be mea-
`sured by the probability of a specific algorithm to find the global
`minimum in a certain number of steps. However, in practice,
`the convergence of a certain algorithm is highly dependent on
`a number of external factors such as the properties of the ob-
`jective function, the dimension of the space, the programming
`approach, and hardware characteristics. So it is difficult to con-
`struct general objective tests for the purpose of comparing meth-
`ods, even if we were to attempt to apply them to a standard set
`of test
`Although the GO methods are commonly classified to
`deterministic/probabilistic classes , a more profitable approach is
`constructed by combining elements of these two classes. In gen-
`eral, a pure deterministic approach has advantages up to certain
`design space dimensions, where the time required to perform a
`systematic search is reasonable. Above that dimension, intro-
`duction of some probabilistic elements is necessary to overcome
`the exponential growth in the number offunction evaluations,9"0
`as noted above.
`Another interesting feature of the optical design space is that
`the space dimension is itself a design parameter. Thus, the ideal
`search algorithm should allow for the addition and subtraction
`of surfaces and lenses dynamically during the optimization pro-
`cess, according to some previously stated criteria. To our best
`knowledge, the only attempt to approach this ''dynamical space"
`property was recently reported in Refs. 16 and 17. In this work,
`a change of dimension from R" to Rm (m>n) occurs automati-
`cally if the algorithm cannot reach a certain figure of merit within
`R. This automatic dimension change is done by a special pro-
`cedure called a sequential cluster algorithm.
`
`3. THE OPTICAL DESIGN SPACE
`The above analysis of GO gives rise to a fundamental postulate
`regarding the successful application of global search to the op-
`tical design procedure (and, in fact, to any other applied prob-
`lem). To apply GO techniques efficiently, there needs to be an
`investigation of the general properties of the design space and
`With this information
`a formulation of its basic
`in hand GO techniques can be applied more efficiently. Con-
`sequently, we may expect considerable improvement in the search
`for the ultimate (optical) design solution. The basic topics for
`the investigation are (1) the type of merit function to be used,
`(2) the existence of discontinuities, and (3) the topographical
`characteristics of the merit function spaces.
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 2
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`GLOBAL VIEW OF OPTICAL DESIGN SPACE
`
`The design space, or parameter space, is a multidimensional
`space over which a single merit function is defined in terms of
`the degrees of freedom that are being made available to the
`specific design task. The merit function is a combination (usually
`sum of squares) of departures from the criteria that are considered
`important characteristics for a specific design task and that are
`to be minimized during the design process. A point in this space
`represents, therefore, a specific configuration of the optical sys-
`tern; and the merit function, a measure of its deviation from the
`required optomechanical performance. To this design space is
`added one more dimension, the merit function. It is this space
`consisting of n independent variables representing the design
`parameters and one dependent variable, the merit function, that
`is explored in the optimization process . Aconfiguration is math-
`ematically stable, for small changes in the degrees of freedom
`values, at the local minimum of this merit function space. How-
`ever, the local minimum may not necessarily be acceptable as
`a good enough solution to the design task considered. A practical
`definition of a minimum in terms of optical design is a (stable)
`configuration that cannot further be improved by conventional
`optimization (i.e. , DLS, etc.).
`Typically the merit function is not an analytical function, and
`it is not expressible in closed form in terms of its independent
`variables. Its components may have been obtained from the
`various orders of the geometrical aberration theory or from nu-
`merical ray-tracing calculation. The ultimate goal of the design
`is to find the optimum among all the minima, which is the global
`optimum.
`This design space is the playground of the optical designer.
`The correct construction and understanding of this space is es-
`sential for a successful design task. What follows is a report on
`investigations of the properties of this design space to provide
`an understanding of how one might attack the problem of finding
`the global optimum.
`
`3.1. Merit function spaces
`The merit function depends differently on the different design
`parameters and on the numerical scaling of the essentially non-
`metrical space. For example, lens curvatures, as a class, are
`usually more effective than the surface separations in manipu-
`lating this function. An artificial metric is therefore essential in
`'balancing'' the design space to unify the dynamic range of
`'
`these numerical values. Even so, some dominant parameters
`always exist. As a result, the design parameters can be sorted
`according to their effect on the merit function. This property
`will be used in the sampling algorithm to be described here.
`We can only visualize a three-dimensional space, sometimes
`resorting to stereo-viewing techniques to assist us. Multidimen-
`sional spaces are beyond our abilities to depict. However, a
`good deal of information can be obtained by looking at three-
`dimensional slices of these spaces. We can use a two-dimen-
`sional design space to visualize the distribution of our config-
`urations, reserving the third dimension to represent the merit
`function value at each point. With today's computer graphics
`this space can be drawn in perspective on a sheet of paper or a
`computer monitor. In higher dimension spaces, we can represent
`the essential physics of the system by selecting the most sig-
`nificant pair of parameters to be plotted and freezing all others
`at some intermediate value. In this manner, we can develop an
`intuition for cases having three, four, and more degrees of free-
`dom.
`The exact composition of the merit function has a key role
`
`to play in the success or failure of a design task. We have chosen
`to highlight our approach by using three types of such merit
`functions, which we believe to be representative of three major
`types: an aberration-based merit function, a selective rays merit
`function, and a full-beam-analysis merit function.
`The easiest of these to evaluate is one that seeks to minimize
`the third-order Seidel and color aberrations:
`
`(1)
`
`MF = [±s2 + (Cf)2]
`where S, = spherical aberration, coma, astigmatism, Petzval
`curvature, and distortion (i = 1 to 5), respectively, and C1 =
`longitudinal color and C2 = transverse color. The Seidel ab-
`erration coefficients,'9 represent the third-order terms in the de-
`velopment of the aberration function of a rotationally symmetric
`optical system. They can be weighted, of course, when the sum
`of squares is formed, to reflect the designer's view of a particular
`task, enhancing or excluding some of them.
`A merit function, derived from the exact trace of a pair of
`meridional rays, can be used to provide a more realistic eval-
`uation of the system:
`
`MF =
`
`(2)
`
`where A1 = spherical aberration, offense against the sine con-
`dition (OSC), Conrady color, sagittal and tangential curvatures
`(Coddington), and distortion (i = 1 to 6), respectively.
`All that is required is a marginal ray passing through the rim
`of the entrance pupil (or a fraction of it) and a chief ray starting
`from the extreme point of the object (or, again, from a fraction
`of it). In this way a finite spherical aberration, the offense against
`the sine condition (OSC), the color according to Conrady (d-D
`method), the astigmatism (tangential and sagittal, using a Cod-
`dington trace), and a finite distortion measure can be included,
`weighted to reflect their relative importance.20
`A third merit function uses the blur spot size for three points:
`an axial point, a 70% field point, and a point at full field:
`
`(3)
`
`MF = [SP2]"2
`where SP, = polychromatic blur spot radius at three fields of
`view, corresponding to i = 1 to 3. The mean blur spot is
`measured as the root mean square of ray hits around their center
`of gravity obtained from a fully polychromatic (3 colors) exact
`ray trace, at three selected field points, at its best position along
`the optical axis. The number of rays traced per field point per
`color depends on the lens type and varies between 1 5 and 100.
`
`3.2. Merit function space examples
`With these three merit functions as tools, we chose two simple
`lenses to explore techniques for searching for the global opti-
`mum.
`
`3.2.1. Cemented doublet (CD)
`A crown-first, cemented, two-glass, fl4 achromat, working at a
`10° full field and infinite object distance, was investigated and
`is shown in Fig. 1. This is perhaps the simplest example of two-
`
`OPTICAL ENGINEERING / February 1991 / Vol. 30 No. 2 / 209
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 3
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`STURLESI, O'SHEA
`
`Fig. 1. Cemented doublet, the first example studied here.
`
`2A
`
`0
`
`C,,
`
`Fig. 3. Cemented doublet design space, with finite-ray-aberrations
`merit function.
`
`0•
`
`0 0
`
`1
`
`Fig. 2. Cemented doublet design space, with Seidel-aberrations merit
`function.
`
`dimensional design space, if the four curvatures only are con-
`sidered as free degrees of freedom. The last surface curvature
`(C5) is solved to maintain a constant effective focal length (EFL);
`the design parameters are C2,C3 (C1 is the stop and C4 = C3).
`The two component thicknesses were kept constant and the im-
`age distance was adjusted to the ''best''
`average blur spot po-
`sition. Figures 2 through 4 show the merit function spaces of
`the three merit functions described above.
`It can clearly be seen, for all three different merit functions,
`that within the permitted region of the variation of the variables
`(C2,C3), the landscape is fairly smooth and there is a well-
`defined minimum that corresponds to the single optimum so-
`lution.
`
`3.2.2. Air-spaced doublet (ASD)
`This crown-first, fl5, infinite-conjugate, air-spaced achromatic
`doublet with a 2° full field was extensively explored. The space
`was found to contain at least nine discrete stable configurations,
`shown in Fig. 5. Two of these are traditionally identified as
`Fraunhofer (A) and Gauss (D) configurations,4'2 1-23 bearing the
`names of their creators. In the ASD case, if we let only the
`curvatures vary and solve for the last one (C5) to keep the EFL
`at a constant value, we need three dimensions to specify all
`possible configurations and four to represent it:C2, C3, C4, and
`the merit function (C1 is the stop). Practice has shown that the
`
`210 / OPTICAL ENGINEERING / February 1991 / Vol. 30 No. 2
`
`Fig. 4. Cemented doublet design space, with three object points,
`polychromatic blur-spot merit function.
`
`first curvature (C2) is the least significant in terms of its effect
`on the merit functions used, so that an instructive general view
`can be achieved by projecting the design space on the two re-
`maining dimensions, C3 and C4, and keeping C2 at an inter-
`mediate constant value. Part of the merit function spaces so
`obtained, one for each type of the three merit functions, are
`shown in Figs. 6 through 8. By inspection, and by following
`their evolution from the simple to the complex, one may acquire
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 4
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`GLOBAL VIEW OF OPTICAL DESIGN SPACE
`
`Fraunhofer group
`
`Gauss group
`
`("Fraunhofer")
`
`A
`
`Negative first group
`
`Negative Fraunhofer-Ir
`
`Fl
`
`Fig. 5. Nine different configurations for the air-spaced doublet design space.
`
`5O\
`
`Fig. 6. Air-spaced doublet design space, with Seidel-aberrations merit
`function.
`
`an overall view of the ASD design domain, a view that is not
`obtainable through local analysis, which is used by the current
`design methods. Five of the configurations mentioned above,
`namely A,C,D,F, and H, are seen in Fig. 7, distributed in the
`shallow area. The whole solution set can be classified into four
`major groups, each group occupying a different shallow valley,
`as follows:
`Fraunhofer group: Configurations A ,B ,C;
`Gauss group: Configuration D;
`Negative Fraunhofer group: Configurations E,F,H,I;
`Negative first group: Configuration G.
`
`Fig. 7. Air-spaced doublet design space, with finite-ray-aberrations
`merit function. The position of five of the configurations in Fig. 5
`are shown here distributed in three solution regions: Fraunhofer (A
`and C), Gauss (D) and negative Fraunhofer (F and H).
`
`The nine are shown and compared for performance and sensi-
`tivity in Table 1. The column labeled Sensitivity provides an
`objective evaluation of the lenses on another criterion. All of
`the configurations are evaluated to determine the RMS of the
`partial derivatives of the merit function with respect to each of
`the design parameters at the minimum. This provides some as-
`sessment of the lens to manufacturing tolerances. (The inclusion
`of tilt and decenter in the sensitivity analysis did not change the
`relative position of the configurations in the table.)
`
`OPTICAL ENGINEERING / February 1991 / Vol. 30 No. 2 / 211
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 5
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`STURLESI, O'SHEA
`
`25
`
`Fig. 9. Merit-function-space topography of the cemented doublet
`generated from blur-spot evaluations with 10 rays per coordinate.
`
`0 0
`
`1
`
`Fig. 10. Merit-function-space topography of the cemented doublet
`generated from blur-spot evaluations with 250 rays per coordinate.
`Note the smoother variation of the surface in the upper left hand
`corner compared to that in Fig. 9.
`
`Insufficientrays: An insufficient number of rays in the blur-spot-
`forming beam may cause false topography near the extreme
`values of the curvatures. The merit function plot evaluated with
`10 rays per coordinate for the cemented doublet is plotted in
`Fig. 9. The same system with 250 rays per coordinate is plotted
`
`0
`
`0
`Fig. 8. Air-spaced doublet design space, with three object points,
`polychromatic blur-spot merit function.
`
`Table 1. Nine possible configurations for the air-spaced doublet. The
`configurations are shown in Fig. 5.
`No. Configuration Blur Spot Size
`A
`I
`I
`2
`1.6
`B
`C
`2.2
`3
`D
`4
`2.6
`E
`5
`3.6
`F
`6
`3.9
`7
`4.1
`G
`H
`4.3
`8
`9
`5.8
`I
`
`Sensitivity
`5.4
`9.1
`6.9
`1
`16.3
`2.3
`1.6
`2.2
`2.3
`
`The general topography for the three different merit functions
`and the respective coordinates of the related minima are re-
`markably similar in the ASD case. The configurations of the
`negative Fraunhofer group are not clearly identified on the Seidel
`and on the blur spot plots, though being true local minima,
`because of the particular value of C2 chosen for these projections.
`
`3.3. Badly formed merit functions
`One important advantage of the parametric survey approach to
`the design of optical systems is that badly composed merit
`functions2 can be readily detected. Such improper merit func-
`tions may destroy or erode the physical information that is rep-
`resented by the space topography. They can lead to false mdi-
`cations of minima and sometimes saturate the whole space with
`an uninformative constant value. Five causes that may poten-
`tially bring about bad compositions are listed below, and their
`effects are shown in Figs. 9 through 14. All of these were
`encountered during this investigation.
`
`212 / OPTICAL ENGINEERING / February 1991 / Vol. 30 No. 2
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 6
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`GLOBAL VIEW OF OPTICAL DESIGN SPACE
`
`Fig. 11. Merit-function-space topography of the cemented doublet
`due to the effect of improper scaling.
`
`Fig. 13. Merit-function-space topography of the air-spaced doublet
`with 60% vignetting at the edge of the field. Note the additional
`features introduced in the center rear of the plot.
`
`too-
`
`0
`
`0
`
`/
`
`Fig. 12. Merit-function-space topography of the air-spaced doublet
`with no vignetting.
`
`in Fig. 10. Note the merit function surface in the upper left-
`hand corner is considerably smoother in the plot of the merit
`function with the larger number of rays.
`
`Improper scaling (or improper metric): The different coordinates
`composing a design space should be scaled properly. It is usually
`not correct to assume that a unit of length exists in these spaces.
`Sometimes the dynamic ranges of the merit numerical values
`may change by several orders of magnitude as witnessed by the
`
`Fig. 14. Merit-function-space topography of the air-spaced doublet
`due to effect of improper weighting among the targets composing
`the merit function (saturation effect).
`
`presence of very high ridges in the permitted, or feasible, region
`of design. In these cases the depths of some minima are too
`shallow to be detected by gradient-type optimization or below
`the precision limit of the computer. Much help can be derived
`in such cases from a first general inspection of the topography
`and subsequent "scaling-up" of the space by eliminating whole
`slices of the high areas. In Fig. lithe merit function surface
`
`OPTICAL ENGINEERING / February 1991 / Vol. 30 No. 2 / 213
`
`Downloaded from SPIE Digital Library on 14 Oct 2010 to 66.165.46.178. Terms of Use: http://spiedl.org/terms
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2007
`Page 7
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`STURLESI, O'SHEA
`
`shown in Figs. 9 and 10 is displayed with an improper scaling
`applied.
`Vignetting: Vignetting of rays, as an intended means of im-
`proving the image quality at the field edge, is a potential troub-
`lemaker in merit function behavior. For example, if the ASD is
`evaluated with no vignetting, the merit function surface shows
`a smooth variation in the major valley at the center of the plot,
`as shown in Fig. 12. In contrast, if this system is vignetted at
`60% at the edge of the field, phony minima are created that have
`no physical existence. These are evident in the valley at the back
`of the merit function surface in Fig. 13 . They are caused by the
`discrete nature of the ray-tracing process. Thus, vignetting should
`not be included in the first stages of a design.
`
`Improper weighting: Correct weighting of the different targets
`composing the single merit function is important. It was found
`that the values of all of the targets that make up the merit function
`should be within one or two orders of magnitude. This is nec-
`essary to ensure that the individual effect of each of the targets
`will be seen in the combined merit function. "Saturation ef-
`fects' ' can occur in the configuration space if one (or more) of
`the targets has a substantially large value relative to the others
`(high dc level). In Fig. 14 the merit function surface for the
`system with an improperly weighted function is a weak version
`of the properly weighted system shown in Fig. 12.
`
`Nonphysical regions: Care should be taken to work only with
`realistic curvatures of the optics . The merit function may vary
`wildly outside its intended region of definition. Such parameter
`ranges may be significantly smaller than their geometrical limits.
`This behavior of the merit function is mainly due to rays that
`fail to hit the surfaces or that are internally totally reflected. This
`situation, which usually occurs at the ends of the parameter
`ranges, can be identified and put out of bounds on a first general
`inspection of the merit function space, avoiding costly unpro-
`ductive searching later. Note the negative curvature end of the
`middle curvature axis in Fig. 4 for an example of such resulting
`rdugh terrain.
`
`3.4. The general characteristics of the design space
`In the examples we have shown, and in others we have studied,
`the solution regions in which the minima are located in groups
`form large, slightly inclined valleys bordered by high, steep
`ridges. The merit function deteriorates rapidly away from these
`areas.
`Another typical characteristic is that in most nontrivial cases
`there may exist a number of regions of solutions, each dotted
`by many local minima that have essentially comparable optical
`quality. Singularities and discontinuities appear at edges of these
`regions when pathological events, like total internal reflection
`or no-hit situations, occur. The third-order space, which is based
`on the aberration contribution calculated analytically from the
`paraxial equations, has no discontinuities and is usually smooth.
`The minima themselves are shallow relative to their imme-
`diate surroundings. Nevertheless, there is almost no chance for
`a point moving under the guidance of a conventional descending
`algorithm to escape the attraction of such a hole, once within
`the convergence region, and