Open Access
Add to BibSonomyAbstract
This paper proposes a new design paradigm which allows for a modular approach to replacing a homogeneous optical lens system with a higher-performance GRadient-INdex (GRIN) lens system using a WaveFront Matching (WFM) method. In multi-lens GRIN systems, a full-system-optimization approach can be challenging due to the large number of design variables. The proposed WFM design paradigm enables optimization of each component independently by explicitly matching the WaveFront Error (WFE) of the original homogeneous component at the exit pupil, resulting in an efficient design procedure for complex multi-lens systems.
© 2016 Optical Society of America
1. Introduction
Traditional homogeneous lenses require curvature to provide optical power and sometimes difficult-to-manufacture surfaces, such as aspheres, to provide aberration correction. Polychromatic correction requires the use of additional lenses with different dispersions to eliminate focal drift at different wavelengths [1,2]. The resulting doublet systems often utilize a thin layer of air between the lenses, which are then subject to loss of alignment and strict manufacturing tolerances. In contrast, a single GRIN lens can provide improved monochromatic and polychromatic performance due to the additional degrees of freedom provided by a spatially-varying refractive index [3,4]. For example, a radial GRIN profile can provide optical power to a flat lens and axial terms can provide aberration corrections equivalent to those of aspherical surfaces [5]. A single GRIN lens with curvature can provide polychromatic correction by balancing the dispersion of the surface with the dispersion of the GRIN distribution itself [1,2]. In the past few decades there have been major advances in the field of GRIN lens manufacturing [6,7], leading to a push to replace homogeneous components in optical systems with GRIN components.
While the performance of traditional homogeneous lenses can be analyzed by employing Snell’s law on each surface, analytically evaluating a GRIN lens often can only be accomplished for the simplest index profiles, such as radial-only or axial-only [4,5,8]. However, high performance optics requiring, for example, significant field-of-view necessitate sufficient degrees of freedom in the design space for simultaneous control of all Seidel aberrations. In [9], this was accomplished by introducing cross-terms to the GRIN profile, which requires the capability to evaluate a completely general GRIN medium. To do this, a ray tracing computer program that numerically solves a coupled set of first-order Ordinary Differential Equations (ODEs) describing the ray trajectory must be employed [10]. This lack of an analytical model for a general distribution implies the need for new GRIN lens design paradigms. One powerful design approach recently introduced is Transformation Optics (TO) [11], a method which provides a mapping between a geometric transformation and a spatially varying, generally anisotropic, material profile. Then, Quasi-conformal Transformation Optics (qTO) was introduced as a numerical technique to eliminate anisotropy [12]. For example, developing a qTO-derived GRIN from a single homogeneous aspherical lens has proven to be very successful [13], but extending qTO to transform multiple air-spaced lenses into a single continuous GRIN which is color-corrected and bounded within a certain refractive index range is a very difficult procedure. Alternatively, the refractive index can be directly optimized using a polynomial basis, which is the method opted for in this work.
Furthermore, instead of using a traditional cost function based on arbitrarily minimizing RMS spot size or RMS WFE relative to an ideal spherical phase profile, a method based on WFM provides the flexibility to match the WFE of an existing homogeneous lens system over the entire exit pupil. Subsequently, each component of an optical system can be viewed as a black-box phase mapping, or transfer function, facilitating individual lens tuning within a multi-lens system—as opposed to highly-variable system-level optimization. Additionally, this approach works for a wide variety of optical elements, including singlets and achromatic or apochromatic doublets where air gaps may be included in between elements. That is, if the WFEs of the GRIN and original optical element match exactly, then the GRIN lens is optically identical to the original element, and the original element can be replaced with identical system-level performance.
2. Motivating example
Consider the focusing lens system consisting of two achromatic doublets shown in Fig. 1. The design was inspired by both the classic Dallmeyer telephoto and the Petzval portrait lens [1] and consists of two Fraunhofer doublets. The system was designed to be diffraction-limited in the Mid-Wave InfraRed (MWIR) at wavelengths of 3, 4 and 5 microns for incident angles of 0, 1 and 2 degrees. The doublets are composed of a Silicon-Germanium pair, very popular material choices in the MWIR [14]. The final optimized design is shown in Fig. 1 and the resulting Strehl ratios [1] are given in Table 1.
Table 1. Strehl ratios for homogeneous focusing lens system.
Next, both homogeneous doublets of the focusing lens system were simultaneously replaced with GRIN singlets using CMA-ES, a powerful global optimizer [15,16]. There were 23 total design variables controlling the geometries and the GRINs. Due to the stochastic nature of CMA-ES, five randomly seeded optimizations were run and the results are shown in Fig. 2. The goal is the diffraction limited RMS spot size, marked with a black dashed line. Table 2 shows the Strehl ratios of the best design after 20,000 function evaluations.
Table 2. Strehl ratios for GRIN focusing lens system using simple optimization.
As can be seen, the convergence was poor and the resulting Strehl ratios are unacceptably low. The poor convergence is likely due to the large number of optimization variables, since the number of function evaluations required for convergence is on the order of Mn2, where M is the population size and n is the dimensionality of the problem [17]. Therefore, convergence can be substantially improved if the number of dimensions is reduced. The most obvious way to do this is to optimize each GRIN lens individually, essentially cutting the system in half thereby halving the number of dimensions in each case. Often, however, a single element in a high-performing optical system will not be free of aberrations; it is the other elements of the system that correct these aberrations. In fact, it’s been found from experience that attempting to design the all-GRIN system by minimizing the RMS WFE of each optic individually doesn’t lead to good system-level performance. For the homogeneous focusing lens system, the first doublet in isolation yields the on-axis spot diagram and WFE at λ = 3 µm shown in Fig. 3(a). This is a poor focusing lens, but an examination of the spot diagram and WFE of the entire system, Fig. 3(b), shows that the system-level aberrations are almost completely eliminated.
Fig. 3 Spot diagram and WFE on the exit pupil at λ = 3 μm and on-axis illumination for (a) the first doublet and (b) the focusing lens system.
Therefore, it makes sense when attempting to optimize the GRIN lenses to reproduce the behavior of each homogeneous optic explicitly, including their aberrations, allowing for direct lens replacement. To achieve this, the GRIN lens needs to match the WFE at the exit pupil of the original homogenous optic, and therefore the design paradigm is called “wavefront matching”. This strategy treats optical elements as black-box phase filters, allowing for design of equivalent GRIN singlets.
3. Theory
The WFE is the deviation of an optical element’s wavefront from a desired phase profile [2]. Figure 4(a) shows an example of an aberrated wavefront generated by a homogeneous doublet. The goal is to replace the doublet system with a GRIN singlet, as shown in Fig. 4(b). The dashed lines show the locations of the exit pupils, which should be aligned in both systems. Note that this is a theoretical system and that the location of the exit pupil has been shifted from the lens for visual clarity. The depicted GRIN singlet is flat, but parameters such as the thickness and radii of curvature can be constrained or used as optimization variables. For example, depending on the material system employed, curvature may be required for color correction [14].
Fig. 4 Depiction of the wavefront matching principle. The exit pupils, shown by the dashed lines, are aligned in both systems, and this is where the wavefront errors need to match for the systems to be identical. (a) Original homogeneous lens. (b) GRIN replacement.
Thus, when optimizing the GRIN singlet replacements the cost function is the maximum difference in WFE between the original element and the GRIN replacement, represented by the following equation:
Hereandare normalized pupil coordinates,andare the WFEs of the original system and GRIN lens, respectively, and the maximum operator is performed over the exit pupil. Then, the Optical Transfer Function (OTF) of the lens is directly related to the WFE through an autocorrelation of the pupil function as shown in the following equation [18]:Here H is the OTF,andare spatial frequencies in the x and y directions, and ψ is the pupil function. Due to the direct correspondence between WFE and OTF, recreating the WFE of the original lens leads to an identical OTF and optical performance as well. Therefore, if a GRIN lens design achieves a WFE identical to the original homogeneous lens then the outputs of the two are equivalent.4. Results
Again, consider the focusing lens system in Fig. 1. By applying the WFM method, the first doublet can be replaced separately and independently from the second doublet. As a result, the system will be optimized using two independent sub-system optimizations done in parallel, each with one-half of the variables compared to the full system, converging at least four times faster. Finally, when the two optimized GRIN singlets are brought together into the full system the resulting system behavior will be compared to the original homogeneous focusing lens system. It has been found experimentally that a maximum wavefront error of ~0.2 λ is necessary to match the original lens accurately and maintain the system-level performance. This is close to the approximate theoretical diffraction-limited max WFE of ~0.25 λ [2]. A GRIN singlet made of a Si-Ge volume-fraction-based mixture with respect to the two materials indices [19] was optimized to match the WFE of the first doublet. For the sake of generality, the spatial distribution of the refractive index will be represented with an 8-term polynomial:
Attempts to wavefront match the first homogeneous doublet were made with polynomials of increasing order. The resulting GRIN profile for each case was constrained to be within the limits of the Si-Ge mixture at each wavelength and to have corresponding dispersion curves. Furthermore, surface curvatures were included as optimization variables to compensate for the dispersive effects of the Si-Ge GRIN material and minimize chromatic aberrations [1]. Table 3 shows the maximum difference in the wavefront errors (in terms of wavelength) at the three incident angles for various GRIN profiles.Table 3. Error in WFE match for first doublet.
A smaller error in the WFE match means that the behavior of the original doublet is more accurately reproduced. A purely radial GRIN can match the on-axis wavefront error fairly accurately, but cannot fully capture the oblique-incidence aberration correction of the doublet. Adding axial terms helps, but it is not until cross-terms are introduced that the wavefronts can be accurately replicated at all angles. The spot diagram and WFE for the full 8-term GRIN is shown in Fig. 5.
Fig. 5 Spot diagram and wavefront error on the exit pupil for the first GRIN singlet for λ = 3 μm and on-axis illumination.
The results are nearly identical to those shown in Fig. 3(a). To further demonstrate that this lens recreates the behavior of the original doublet, the GRIN singlet was used as a replacement for the first optic in the original focusing lens system, making sure the exit pupil remains at the same location. Figure 6 shows the refractive index profile, which is rotationally symmetric around the optical axis, and a ray trace. The resulting refractive index profile is mostly axial, but there are strong cross-term components as well. The Strehl ratios compare extremely well to the original homogeneous design, with the largest deviation being only 0.004.
The convergence curves of five randomly seeded optimization runs are shown in Fig. 7. The goal was 0.2 λ WFE deviation, and as can be seen this value was reached in at most 4,000 function evaluations.
Next, the second optic was replaced with a GRIN profile defined by (3). The error in the wavefront match for the second doublet is shown in Table 4, where the maximum error was 0.0768 λ.
Table 4. Error in WFE match for second doublet.
The convergence curves for five randomly seeded optimization runs are shown in Fig. 8. The goal was again 0.2 λ, which was reached in about 6,000 function evaluations for four of the five cases. The fifth case required 10,000 function evaluations. This is much improved over the full system optimization, which failed to converge in 20,000 function evaluations. The full ray trace of the all-GRIN system is shown in Fig. 9.
The Strehl ratio for the system including both GRIN lenses is given in Table 5. The maximum degradation in Strehl ratio from the original design occurs at θ = 2 degrees and λ = 5 μm, however only by 0.009. The resulting Strehl ratios of the final system are orders of magnitude better than those of the full-system optimization shown in Table 2. Figure 10 shows the actual and diffraction-limited modulation transfer functions (MTF) at this worst-case angle and wavelength for both the tangential and radial cuts, clearly demonstrating how well the GRIN system reproduces the original Si-Ge doublet focusing lens system.
Table 5. Strehl ratios for all-GRIN focusing lens system using WFM.
Fig. 10 MTF at θ = 2° and λ = 3 μm along the tangential and radial cuts for (left) the homogeneous and (right) the all-GRIN system.
The major advantage of this design paradigm is that each component can be designed independently and in parallel, leading to a smaller number of parameters for CMA-ES, or any optimizer, to handle and, subsequently, a faster convergence and higher success rate.
5. Conclusion
This letter introduced a powerful new method of GRIN system design that enables individual optimization of each component in a multi-lens system. The modularity of the approach dramatically reduces the degrees of freedom of the problem, leading to a simpler and therefore faster-converging and more stable system-level optimization. The approach was demonstrated by considering a focusing lens system and replacing each homogeneous doublet with an equivalent GRIN singlet. When combined, the resulting GRIN system replicated the original doublet system’s performance with two optical elements instead of four. Convergence studies were performed to show that the traditional paradigm of optimizing the full system leads to an intractable optimization problem. Conversely, the wavefront matching paradigm resulted in fast convergence to a desired solution.
Acknowledgements
This work was supported in part by DARPA/AFRL under contract number FA8650-12-C-7225.
References and links
1. M. J. Kidger, Fundamental Optical Design (SPIE Publications, 2001).
2. R. Fischer, Optical System Design (McGraw-Hill Education, 2008).
3. E. W. Marchand, Gradient Index Optics (Academic Press, 1978).
4. G. I. Greisukh, S. T. Bobrov, and S. A. Stepanov, Optics of Diffractive and Gradient Index Elements and Systems (SPIE Publications, 1997).
5. P. J. Sands, “Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. 61(8), 1086–10917 (1971). [CrossRef]
6. D. J. Fischer, C. J. Harkrider, and D. T. Moore, “Design and manufacture of a gradient-index axicon,” Appl. Opt. 39(16), 2687–2694 (2000). [CrossRef] [PubMed]
7. A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. McLeod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015). [CrossRef] [PubMed]
8. F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A 13(6), 1277–1284 (1996). [CrossRef]
9. S. D. Campbell, J. Nagar, D. E. Brocker, and D. H. Werner, “On the use of surrogate models in the analytical decompositions of refractive index gradients obtained through quasiconformal transformation optics,” J. Opt. Soc. Am.in press.
10. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21(6), 984–987 (1982). [CrossRef] [PubMed]
11. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18(20), 21238–21251 (2010). [CrossRef] [PubMed]
12. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
13. D. E. Brocker, J. P. Turpin, P. L. Werner, and D. H. Werner, “Optimization of gradient index lenses using quasi-conformal contour transformations,” IEEE Antennas Wirel. Propag. Lett. 13, 1787–1791 (2014). [CrossRef]
14. J. A. Corsetti, P. McCarthy, and D. T. Moore, “Color correction in the infrared using gradient-index materials,” Opt. Eng. 52(11), 112109 (2013). [CrossRef]
15. N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation,” in Proceedings of the 1996 IEEE Intern. Conf. on Evolutionary Computation (IEEE, 1996), pp. 312–317. [CrossRef]
16. M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetics design problems through the CMA evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2011). [CrossRef]
17. N. Hansen and A. Ostermeier, “Convergence properties of evolution strategies with the derandomized covariance matrix adaptation: the (μ/μ_I,λ)-CMA-ES,” in 5th European Congress on Intelligent Techniques and Soft Computing (Verlag Mainz, 1997), pp. 650–654.
18. C. P. Grover and H. M. van Driel, “Autocorrelation method for measuring the transfer function of optical systems,” Appl. Opt. 19(6), 900–904 (1980). [CrossRef] [PubMed]
19. J. Moon, D. Lu, B. VanSaders, T. K. Kim, S. D. Kong, S. Jin, R. Chen, and Z. Liu, “High performance multi-scaled nanostructured selective coating for concentrating solar power,” Nano Energy 8, 238–246 (2014). [CrossRef]
