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Current analytical expressions between Gradient-Index (GRIN) lens parameters and optical aberrations are limited to paraxial approximations, which are not suitable for realizing GRIN lenses with wide fields of view or small f-numbers. Here, an analytical surrogate model of an arbitrary GRIN lens ray-trace evaluation is formulated using multivariate polynomial regressions to correlate input GRIN lens parameters with output Zernike coefficients, without the need for approximations. The time needed to compute the resulting surrogate model is over one order-of-magnitude faster than traditional ray trace simulations with very little losses in accuracy, which can enable previously infeasible design studies to be completed.
© 2016 Optical Society of America
1. Introduction
Recent advances in both manufacturing and design techniques of GRadient-INdex (GRIN) lenses have dramatically enhanced the feasibility of implementing GRIN lenses in high performance optical systems [1,2]. The ability to spatially vary the refractive index of glass throughout the bulk volume of a lens is extremely promising for reducing the size and weight of existing and future optical systems [3]. This additional control of refraction can yield enhanced performance in optical systems and has the potential to reduce the number of elements in an optical system [4,5]. However, the additional degrees of freedom to be considered with GRIN materials make lens design a unique challenge. With the advancement of modern additive manufacturing techniques, such as polymeric nanolayered GRIN lenses [6,7], arbitrary index distributions can be realized, which has enabled substantially larger design spaces to be considered for lenses.
Previous work has been conducted to formulate analytical relationships between GRIN parameters and first through third order optical Seidel aberrations [8–13]. However, these analytical relationships are formulated based on paraxial approximations which may not provide enough accuracy for all lens scenarios [8,9], particularly for wide angle lens design where imaged rays originate at significant angles away from the optical axis [14–17], or for lenses with small f-numbers [18]. Furthermore, these formulations only consider purely radial or axial GRIN distributions; there is no analytical theory for GRIN lenses of arbitrary refractive index prescription. Specifically, previous formulations do not include cross-term GRIN distributions (i.e. multiplicative combinations of radial and axial terms), which have been shown to improve off axis optical performance [5,19,20]. Unfortunately, evaluating the performance of an arbitrary GRIN lens without approximations requires a numerical ray trace, which can be computationally intensive, especially for lenses which possess radical gradients. Therefore, formulating an analytical expression to relate arbitrary input GRIN lens parameters with optical performance while remaining general enough to extend beyond the traditional paraxial approximation based approach would be a powerful tool for rapid high performance GRIN lens design.
Surrogate models are a practical method for estimating the outcome of a simulation, particularly when directly computing the solution is time consuming or computationally intensive, and are commonly used in engineering design. The simplest surrogate models are one-dimensional linear or non-linear regressions, which create a simple analytical equation to relate input and output values. More advanced surrogate models are synthesized using processes such as the Kriging model [20–22] or artificial neural networks [23,24]. The goal of these methods is to correlate input values to an output value, often by creating an analytical equation which relates these parameters. The resulting correlations make a coarse statistical approximation of the underlying physics of the problem, and are computationally cheaper to evaluate than the original simulation. In real-world engineering design problems, there is often a need to perform costly parameter studies or optimization procedures, which can be infeasible if the simulations involved are computationally intensive. By properly setting up a surrogate model, computation time can be drastically reduced with only minor losses in accuracy. This tradeoff can enable previously infeasible parameter surveys to become practical, which can then be utilized to communicate critical information about the design space to the engineer.
The study here will create a set of analytical equations, i.e. a surrogate model, via multivariate polynomial regressions [25–27] to relate several input parameters of a plano-convex Silicon-Germanium (Si-Ge) GRIN lens to a set of output Zernike polynomials [28] which describe the lens’ optical aberrations. The Zernike polynomials are commonly employed in aberration analysis in modern optical systems and are mathematically convenient, as they are orthogonal and rotationally invariant, where each Zernike term represents a different optical aberration [28]. Various normalization and indexing schemes are used to define Zernike polynomials; however, for clarity we will implement the normalization method described by Born and Wolfe [29] and the single term indexing described by Noll [30]. Zernike polynomials can be readily computed by interpolating values from simulated ray traces [31], where no paraxial approximations are made. For this reason, decomposing a lens’ wavefront error into Zernike polynomials is an effective method for analyzing optical wavefront aberrations. Here, the validity of using a surrogate model approach will be demonstrated by performing a brief perturbation study of GRIN lens parameters to show that a smooth and continuous responses surface exists between the input lens parameters and the resulting wavefront aberrations of the lens. Next, the surrogate model will be formulated by utilizing multivariate polynomial regressions, which are capable of capturing the highly coupled nature that exists between input and output lens parameters. A study to investigate the number of samples needed to train a reasonably accurate model will then be conducted. Finally, the surrogate model will be applied to several optimization problems to demonstrate the power and versatility of this technique.
2. Perturbation study of GRIN parameters
Surrogate models are most effective for modeling systems that have a smooth and continuous response surface, which is a function that maps the input parameters to their corresponding output values. The GRIN lens parameters are a series of polynomial terms which describe the index of refraction throughout the bulk volume of the lens. The index of refraction throughout the lens, n(r,z), is defined by the superposition the individual GRIN terms in the form of the general polynomial GRIN equation given in Eq. (1).
where ci,j is the coefficent of the corresponding GRIN term, r is the radial coordinate, D is lens diameter, z is the axial coordinate, T is the lens thickness, and the index limit p is the maximum order GRIN term allowed in the system. The lens is defined to be rotationally symmetric about the optical axis, therefore, establishing an index of refraction over a single radial-axial cross section will completely define a three-dimensional GRIN profile.Traditional GRIN lens design is frequently conducted by using a truncated set of polynomials to define the refractive index distribution within a lens. The coefficients of the polynomials defining the GRIN lens are optimized so that the GRIN distribution yielding the best optical performance can be obtained. Optimizing the index distribution with a polynomial basis function may not be ideal, since the lens designer must have a priori knowledge of what polynomial GRIN terms are best to include in the optimization; however, there are several polynomial distributions which have well understood effects on focusing power [8–13]. Furthermore, a particular set of GRIN terms may be suitable for one scenario, but not-suitable for a different scenario. This means that an optical engineer must frequently try many sets of time-consuming optimizations with different polynomial terms in order to design a single lens that simultaneously meets all performance requirements. This is the case even when using advanced global optimizers, such as the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) [32,33].
In order to identify if GRIN lens parameters have smooth and continuous relationships with the Zernike coefficients of the wavefront aberrations, a series of perturbation studies were completed on a plano-convex lens with 25% max curvature. Here, 100% max curvature is defined as the amount of curvature required for the front convex surface to just intersect the back flat surface of the lens whereas 0% max curvature defines a flat lens. Aberrations were analyzed on a flat f/5 observation plane for all cases. The first set of these studies vary the coefficient magnitude of a single GRIN parameter at a time, and it is noted that nonlinear relationships appear for some aberrations, as shown in Fig. 1. If a first order analysis is completed on this simple, one term perturbation study, then the effects that a single GRIN term has on specific aberrations can be analyzed. As depicted in Fig. 2, perturbing any single GRIN parameter will cause a corresponding change in all Zernike coefficients, which givesrise to the notion that this problem has strong coupling between the inputs and outputs. From Fig. 2, it can also be noted that perturbations of radial and cross-terms generally yield larger responses to aberrations than purely axial terms, i.e. it would require a large axial gradient to have the same affects as the radial and cross-term parameters. This axial behavior agrees with the previous findings of [9]. Finally, a multi-term perturbation study was completed, where several GRIN parameters were simultaneously perturbed. Expected values of aberrations were constructed assuming that a linear superposition of responses from the single term perturbation study could be utilized to estimate the amount of aberrations in the system. As shown in Fig. 3, it is clear the superposition assumption does not hold for the simple case of simultaneously varying multiple GRIN terms, furthering the notion that coupled responses exist between the input GRIN parameters and output aberrations. Interestingly, all perturbation studies appear to show that the response between inputs and outputs are indeed smooth and continuous. This suggests that a simple polynomial-based surrogate model may be formulated to relate the inputs and outputs; however, it is to be expected that higher order polynomials may be needed to effectively model the nonlinearities and coupled behavior of the system.
Fig. 1 Single GRIN term perturbations of a homogenous plano-convex lens with 25% max curvature affecting (left) astigmatism Z5, (center) coma Z7, and (right) secondary spherical aberration Z22 at a 4 degree field angle. Rf denotes the front radius of curvature while the other parameters correspond to the coefficient of the respective GRIN terms.
Fig. 2 First order derivatives of single GRIN term perturbations affecting (left) astigmatism Z5, (center) coma Z7, and (right) secondary spherical aberration Z22 at a 4 degree field angle.
Fig. 3 Multiple, simultaneous GRIN term perturbations affecting (left) astigmatism Z5, (center) coma Z7, and (right) secondary spherical aberration Z22 at a 4 degree field angle. (Top) For radial terms only. (Middle) For radial and axial terms. (Bottom) For radial, axial, and cross-terms. Estimated aberration values are computed assuming that superposition is valid and that no multi-term coupling exists.
3. Development of an empirically-derived surrogate model
Based on the previous perturbation study, the relationships between the input lens parameters P1, P2, …, PN and the output Zernike coefficients Z1, Z2, …, ZM are known to be non-linear, but the lens designer does not know a priori what the order, δ, of the relationship is. To accomplish appropriate curve fitting between the input and output data, several multivariable regressions are completed in an iterative manner. The designer specifies a maximum order, w, to consider, and a polynomial multivariable regression [25,26] is iteratively performed on each successive order from δ = 1, δ = 2, …, δ = w . To determine which order of fit is best (without overfitting), the regression that results in the smallest Coefficient of Variation of the Mean Absolute Error (CVMAE) is selected and used in the surrogate model. This process is described in Fig. 4. It is important to note that this regression process is conducted for each individual aberration, i.e. an analytical equation is generated to relate all the input parameters to Astigmatism at 45 degrees (Z5), and another analytical equation is generated to relate all the input parameters to y-Coma (Z7). This process continues until each aberration has an analytical equation which correlates its value with the input GRIN parameters. The surrogate equation for the qth Zernike coefficient takes the form of a series of summations of the product of coefficients and each parameter included in the training process P1, P2, …, PN. The coefficients are determined from the best fitting multivariable regression. Equation (2) explicitly shows the form of the surrogate model representation for first order (the first summation), second order (the second set of summations), and third order relationships (the third set of summations).
Fig. 4 Overview of the iterative multivariable regression process. For each Zernike coefficient at each angle of incidence, multivariate regressions of varying order are computed. After evaluating the CVMAE statistic for each regression, the best fitting order of regression can be selected and used in the surrogate model.
Here, only the first and second order Zernike coefficients will be considered (Z1, Z2, …, Z22). This includes major aberrations and alignment errors such as Piston (Z1), Defocus (Z4), Astigmatism at 45 degrees (Z5), y-Coma (Z7), Primary Spherical (Z11), and Secondary Spherical (Z22) Zernike coefficients, as well as other minor (but still significant) aberration terms. The Zernike coefficients are computed by performing a ray trace of the GRIN lens at a specified angle of incidence and the difference between the desired and actual wavefronts—also known as “Optical Path Difference” (OPD)—is interpolated from the ray trace [31]. TheOPD is then decomposed into an orthogonal Zernike polynomial basis as described in [28], where the coefficients of these Zernike polynomials (with units of wavelengths) are used as the wavefront aberration Zernike coefficients. The surrogate model here is formed by testing numerous sample designs and evaluating their performance with a custom ray tracing engine, reTORT (Transformation Optics Ray Tracer), which was developed at Penn State for analyzing arbitrary three-dimensional GRIN lenses. It should be noted that the commercial simulator CODE V can trace GRIN lenses, albeit with a specialized add-on module, and the accuracy of the custom reTORT simulation tool has been previously verified against the CODE V module. The results from these training designs, combined with the multivariate regression scheme described above, effectively train the surrogate model to statistically mimic the underlying physics of the GRIN lens problem. Once the training process has been completed, the aberrations from other arbitrary GRIN lenses can quickly be estimated using the newly formed analytical expression described in Eq. (2). This surrogate model is an effectively “empirical-derived arbitrary GRIN theory” that can be used to rapidly optimize GRIN lenses which have improved optical performance over their homogenous lens counterparts by utilizing numerical global optimization techniques, such as the CMA-ES [32,33]. A high-level overview of the entire surrogate training process is provided in Fig. 5.
4. Setup of lens parameters
All possible input parameters used to describe the GRIN lens must be clearly defined to study a plano-convex GRIN with a surrogate model. For this study, one parameter will describe the front radius of curvature of the plano-convex lens, and a total of 10 parameters will describe the index of refraction within the lens—here these will be called “GRIN parameters”. The GRIN parameters include 0th to 6th order radial and cross-terms. Radial terms describe purely radial distributions of the index of refraction, while cross-terms describe index distributions that change in both radial and axial directions simultaneously. Axial terms, which describe distributions in the index of refraction along the optical axis, are often included in GRIN optimizations, but axial terms will be omitted from training this surrogate model since the study in Section 2 indicated that axial terms had the smallest affects on optical aberrations. In previous studies, cross-terms have been omitted from describing GRIN lenses due to their mathematical complexity. However, recent findings have shown that the inclusion of cross- terms in lens optimizations can lead to enhanced off-axis optical performance [5,19,20]. Thecomplete list of selected lens parameters used in this study are depicted in Fig. 6.
Fig. 6 Parameterization of a plano-convex GRIN singlet with radial and cross-terms determining the distribution of index of refraction within the lens.
Two constants in this design study are the thickness and diameter of the lens. These parameters are often fixed in a realistic scenario based on size constraints. The thickness will be defined as the center to center distance between the front and back surfaces, and this value is fixed at 10 mm. The diameter of the lens is fixed to 50 mm, giving the lenses here a 5:1 aspect ratio. For all surrogate training and optimizations, the front radius of curvature is bounded to stay within realizable values, so that total internal reflection (TIR) is avoided in all cases. Based on previous experimentation, restricting the radius of curvature to be between 0% max curvature and 66.7% max curvature will give a reasonably high success rate, i.e. no TIR occurs. The lens will be optimized to operate at an f/5 focal plane for a wavelength of 4 microns—a central wavelength in the mid-wave infrared regime. The indices of refraction will be scaled to be within 3.425-4.025 to simulate a Silicon-Germanium (Si-Ge) mixture based material system [34], which is commonly employed in the mid-wave infrared regime and is notable for its exceptionally large range in the index of refraction, Δn.
5. Training procedure
To train the analytical surrogate model, the design space must first be surveyed with a sufficient number of training points. To accomplish this, Latin Hypercube Sampling is utilized to effectively generate sample points that are uniformly distributed in the hyper-dimensional search space [35]. Each of these sample points are used to construct a GRIN lens that is then simulated with reTORT. If the simulation is completed without error, the resulting wavefront from the ray-trace is then decomposed into the Zernike polynomial basis. The input lens parameters and output aberrations are recorded only for successful ray traces. Several of these training lenses are shown in Fig. 7.
Fig. 7 (Left) Ray traces of several training lenses created from the Latin Hypercube Sampling method. (Right) corresponding Zernike coefficients of the resulting wavefront at a focal plane of z = 250 mm for light incident 4 degrees from the optical axis.
5.1 Number of samples needed to train a surrogate model
One of the major difficulties in forming an effective surrogate model is in selecting an appropriate number of training samples. If too few samples are used to train the surrogate model, then the model will fail to completely capture the relationship between input and output parameters. Conversely, it would be expected that an extremely large numbers of training samples would eventually become computationally inefficient, i.e. an infinite amount of time could be spent training a surrogate model with negligible improvements in accuracy. To investigate how many samples are needed to train a reasonably accurate surrogate model, several models were trained with an increasing number of training samples. The accuracy of each surrogate model was gauged by comparing the estimated aberrations computed by the surrogate model with a set of 175 randomly sampled lenses simulated with reTORT. The average percent error between the expected and actual aberrations were compared, as well as the time needed to serially perform the multivariable regressions on a Xeon E5645 processor. In practice, the regression process could be parallelized on a multi-core processor to reduce the time need to train the surrogate model. As shown in Table 1, a considerable degree of accuracy can be obtained with relatively few training samples (one thousand or more samples). Increasing the number of training samples past this point dramatically increases thetime needed to train a surrogate model, but with little gains in the accuracy of the model. For the remainder of this paper, the surrogate model with 10,000 initial training samples will be utilized, since this sample size has an acceptable tradeoff between accuracy and training time.
Table 1. Number of Samples to Train an Effective Surrogate Model
6. Applying the surrogate model to lens optimization
Once the training process has been completed, the aberrations of an arbitrary input lens can be quickly estimated by evaluating the analytical surrogate model using Eq. (2). A traditional GRIN lens optimization which implements a ray trace simulation in each function evaluation will be compared to the same optimization which utilizes the surrogate model in lieu of the ray trace. A plano-convex lens with the eleven variables listed in Fig. 6 will be optimized using the CMA-ES [32,33]. Since CMA-ES is a stochastic global optimizer, three optimization trials will be completed and the performance will be averaged in order to accurately compare the convergence of the optimizations.
The cost function will be defined as the sum of the absolute value of the Zernike coefficients at normal incidence, 2 degrees oblique incidence, and 4 degrees oblique incidence—yielding a lens with a full field of view of 8 degrees. The minimum possible cost value is zero, which would correspond to a lens with no aberrations at 0, 2, and 4 degrees of incidence from the optical axis. Only the first 22 Zernike terms will be considered in this optimization. This cost function formulation is defined in Eq. (3).
On average, the surrogate-assisted optimization was able to complete 100,000 function evaluations 64 times faster than the optimization with ray trace simulations while achieving a design with similar optical performance. Furthermore, the surrogate model was computed serially while the ray tracing optimization was computed in parallel. This means that a single optimization could be further enhanced with parallel computing, or alternatively, numerous surrogate-assisted optimizations could be computed simultaneously in parallel over the same period of time. This dramatic increase in speed and computational efficiency enables previously infeasible design studies to now become tractable problems. The results from the optimization procedure are shown in Fig. 8 and Fig. 9. In these figures, the cost of each surrogate assisted design during the optimization was verified with a traditional ray trace simulation, and it can be seen that there is good agreement between the estimated cost from the surrogate model and the cost realized from the ray trace throughout the optimization procedure. The optimized lens parameters are summarized in Table 2.
Fig. 8 Comparison of average convergence between the ray tracing and the surrogate model optimizations per function evaluation on dual Xeon E5-2680 v3 processors. The inset image shows the optimized lens from the surrogate-assisted optimization, where cyan, pale-green, and magenta depict rays incident at 0, 2, and 4 degrees from the optical axis, respectively. The optimizer appeared to converge to an optimized solution after roughly 40,000 function evaluations.
Fig. 9 Comparison of average convergence between the ray tracing and the surrogate model optimizations over time on dual Xeon E5-2680 v3 processors. A total of 100,000 function evaluations were completed in each case.
Table 2. Optimized Lens Parameters from the Surrogate Assisted Solution
7. Applying the surrogate model to lens optimization
One immediate difficulty facing a GRIN lens designer is the decision of which GRIN terms to include in a lens optimization, as choosing the best set of GRIN terms for a given set of design objectives is non-trivial. Completing a thorough investigation of which terms to include for a given scenario requires an exceptionally large number of nonlinear optimizations to be completed, which is not feasible when using traditional ray tracing. For example, using dual Xeon E5-2680 processors to optimize all possible GRIN term permutations listed in Fig. 6 (a total of 512 different combinations) in parallel would take on the order of 1 year to compute. However, by utilizing the computational efficiency offered by a surrogate model, this time can be reduced to several days when optimizing in parallel on the same dual Xeon E5-2680 processors.
To illustrate this idea, the same surrogate model employed previously is used to optimize plano-convex Si-Ge GRIN lenses with all possible combinations of GRIN terms (parameters P3 through P11). Since the GRIN terms have highly non-linear and coupled effects on lens performance, it is necessary to perform a global optimization procedure for each case. Each optimization used CMA-ES with restarts enabled and a maximum of 300,000 function evaluations. As shown in Fig. 10, it can be seen that there is a general trend where introducing additional GRIN terms to the optimization will yield better optical performance. In some cases, the GRIN term permutations yield poorer performance than the homogenous lens and this is likely due to the imposed requirement that the entire range of index of refraction is to be used in the GRIN lens; a restriction that can be relaxed in future studies. Table 3 lists the combinations of terms that resulted in the best performance—all of which outperformed the homogenous case. From this table, it is also clear that cross-terms play a critical role in improving the GRIN lens’ optical performance over the 8 degree full field-of-view. To better understand how the performance of the optimized surrogate-assisted GRIN lenses compare to a traditional homogeneous lens that is optimized using the same metrics, the Strehl ratios [36] of several of the best performing GRIN lenses are compared to the homogeneous lens atvarying angles of incidence. As illustrated in Fig. 11, there is improvement in optical performance for all the optimized GRIN lenses at all angles of incidence. Interestingly, the performance of the best GRIN lenses did not dramatically improve as more GRIN terms were added. This demonstrates that if the proper GRIN terms are chosen, then relatively simple GRIN lenses can lead to enhanced optical performance. This is similar to eliminating functions from an over-complete basis.
Fig. 10 Results from optimizing plano-convex lenses with all permutations of GRIN parameters. Each point represents an optimized plano-convex GRIN lens with a unique set of GRIN terms. The horizontal dashed black line shows the cost of an optimized homogeneous plano-convex lens.
Table 3. Best Performing GRIN Term Permutations
8. Applying the surrogate model to lens optimization
Further illustrating the power of using surrogate-assisted optimization, the same surrogate model will be used to design a set of lenses with different f-numbers. Since a lens that focuses light without aberrations onto an improper focal plane can be represented as a lens with significant Zernike Piston and Defocus aberrations [28], the cost function can be modified to design a lens with a different focusing power. Here, the cost function is modified so that
where Ti is the target aberration value at the observation plane used to train the surrogate model. In this case, the f/5 lens that the surrogate model was initially trained for a focal plane located at z = 250 mm, and this will serve as the observation plane for the lenses with shifted focal planes. Shifting the focal plane will be approximated by incrementally changing the target aberration values of Zernike Piston (T1) and Defocus (T4). All other target aberrations will remain zero in this case. Each case here will be optimized using CMA-ES with restarts enabled and a maximum number of function evaluations of 300,000 for each trial. Here the surrogate-assisted optimizations will be run in parallel to simultaneously design lenses with different focal planes by utilizing a multi-core processor.As shown in Table 4, the homogeneous lenses have excellent performance for large f-numbers, but have dramatically reduced performance for cases with small f-numbers. The GRIN lenses have slightly reduced performance for cases with large f-numbers (likely due to the imposed requirement that the entire Δn from Si-Ge is utilized), but the GRIN lenses are significantly better performing at low f-numbers. In all cases, better optical performance was realized with the GRIN lenses for rays at the edge of the field of view. Discrepancies in the resulting focal plane position for given target aberration values are likely due to variation in the location of the exit pupil of the GRIN lenses. Future surrogate models could be trained toinclude the location of the entrance and exit pupils as additional outputs to address these discrepancies. Overall, the ability to quickly and simultaneously optimize lenses with different focusing power promises to be a powerful tool for reducing the size of optical systems. A diagram of a GRIN lens with a shifted focal plane is shown in Fig. 12.
Table 4. Resulting Strehl Ratios from Shifting the Focal Plane of a Homogeneous and GRIN Lens Using a Traditional Ray Tracing Optimization and Surrogate-assisted Optimization
Fig. 12 Ray trace of an optimized plano-convex GRIN lens with + 100 wavelengths of Zernike Piston and + 100 wavelengths of Zernike Defocus at the observation plane (z = 250 mm).
9. Conclusion
Here we demonstrated a method for creating an analytical surrogate model to correlate input GRIN lens parameters with lens aberrations through the use of Zernike decomposition and non-linear multivariable polynomial regressions. The surrogate model provides insight into which GRIN polynomial terms have the most impact on optical performance. In addition, use of this surrogate model can lead to substantially faster GRIN lens optimizations. We have also demonstrated that a single surrogate model can be used to rapidly design lenses at varying focal planes by optimizing the lens to include alignment aberrations of Piston and Defocus. Future work with the surrogate approach to GRIN lens design may include higher order GRIN terms, the use of other surrogate model formulation techniques, and/or to include entrance and exit pupil information in the training procedure. It also would be beneficial to have a surrogate model that could account for varying thicknesses, diameters, or axial terms. Overall, this surrogate-assisted approach represents a powerful tool for rapidly designing and studying the properties of GRIN lenses.
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