Abstract

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

Full Article  |  PDF Article
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References

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  1. A. I. Hernandez-Serrano, M. Weidenbach, S. F. Busch, M. Koch, and E. Castro-Camus, “Fabrication of gradient-refractive-index lenses for terahertz applications by three-dimensional printing,” J. Opt. Soc. Am. B 33(5), 928–931 (2016).
    [Crossref]
  2. 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]
  3. S. D. Campbell, D. E. Brocker, J. Nagar, and D. H. Werner, “Size, weight, and power reduction regimes in achromatic gradient-index singlets,” Appl. Opt. 55(13), 3594–3598 (2016).
    [Crossref] [PubMed]
  4. D. T. Moore, “Gradient-index optics: a review,” Appl. Opt. 19(7), 1035–1038 (1980).
    [Crossref] [PubMed]
  5. J. Nagar, D. E. Brocker, S. D. Campbell, J. A. Easum, and D. H. Werner, “Modularization of gradient-index optical design using wavefront matching enabled optimization,” Opt. Express 24(9), 9359–9368 (2016).
    [Crossref] [PubMed]
  6. S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
    [Crossref]
  7. S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).
  8. P. J. Sands, “Third-order aberrations of inhomogeneous lenses,” J. Opt. Soc. Am. 60(11), 1436–1443 (1970).
    [Crossref]
  9. P. J. Sands, “Inhomogeneous lenses, IV. Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. 61(8), 1086–1091 (1971).
    [Crossref]
  10. D. T. Moore and P. J. Sands, “Third-order aberrations of inhomogeneous lenses with cylindrical index distributions,” J. Opt. Soc. Am. 61(9), 1195–1201 (1971).
    [Crossref]
  11. F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A 13(6), 1277–1284 (1996).
    [Crossref]
  12. D. Y. Wang and D. T. Moore, “Third-order aberration theory for weak gradient-index lenses,” Appl. Opt. 29(28), 4016–4025 (1990).
    [Crossref] [PubMed]
  13. E. W. Marchand, Gradient Index Optics (Academic, 1978), Chap. 8.
  14. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
    [Crossref]
  15. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575–578 (1979).
    [Crossref]
  16. M. Gu, Advanced Optical Imaging Theory (Springer, 2000), Chap 2.
  17. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
    [Crossref] [PubMed]
  18. S. Guha, “Validity of the paraxial approximation in the focal region of a small-f-number lens,” Opt. Lett. 26(20), 1598–1600 (2001).
    [Crossref] [PubMed]
  19. D. E. Brocker, S. D. Campbell, and D. H. Werner, “Color-correcting gradient-index infrared singlet based on silicon and germanium mixing,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).
    [Crossref]
  20. 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. 18(4), 044019 (2016).
    [Crossref]
  21. D. R. Jones, “A taxonomy of global optimization methods based on response surfaces,” J. Glob. Optim. 21(4), 345–383 (2001).
    [Crossref]
  22. A. Forrester, A. Sóbester, and A. Keane, Engineering Design via Surrogate Modelling: A Practical Guide (John Wiley & Sons, 2008).
  23. M. Rafiq, G. Bugmann, and D. Easterbrook, “Neural network design for engineering applications,” Comput. Struc. 79(17), 1541–1552 (2001).
    [Crossref]
  24. D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).
  25. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, 2001), Chap. 7.
  26. E. Masry, “Multivariate regression estimation: local polynomial fitting for time series,” Stoch. Proc. Appl. 65(1), 81–101 (1996).
    [Crossref]
  27. A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
    [Crossref]
  28. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering11 (Academic Press, 1992), Chap. 8, pp. 2–53.
  29. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976).
    [Crossref]
  30. M. Born and E. Wolf, “The circle of polynomials of Zernike,” in Principles of Optics6 (Pergamon Press, 1993), Appendix VII, pp. 767–772.
  31. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980).
    [Crossref] [PubMed]
  32. N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation,” in Proceedings of IEEE Conference on Evolutionary Computation (IEEE, 1996), pp. 312–317.
    [Crossref]
  33. M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2010).
    [Crossref]
  34. F. Schäffler, Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe (John Wiley & Sons, 2001), Chap. 6.
  35. J. C. Helton and F. J. Davis, “Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems,” Reliab. Eng. Syst. Saf. 81(1), 23–69 (2003).
    [Crossref]
  36. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72(9), 1258–1266 (1982).
    [Crossref]

2016 (4)

2015 (1)

2014 (1)

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
[Crossref]

2013 (1)

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

2010 (3)

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[Crossref] [PubMed]

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2010).
[Crossref]

2003 (1)

J. C. Helton and F. J. Davis, “Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems,” Reliab. Eng. Syst. Saf. 81(1), 23–69 (2003).
[Crossref]

2001 (3)

S. Guha, “Validity of the paraxial approximation in the focal region of a small-f-number lens,” Opt. Lett. 26(20), 1598–1600 (2001).
[Crossref] [PubMed]

D. R. Jones, “A taxonomy of global optimization methods based on response surfaces,” J. Glob. Optim. 21(4), 345–383 (2001).
[Crossref]

M. Rafiq, G. Bugmann, and D. Easterbrook, “Neural network design for engineering applications,” Comput. Struc. 79(17), 1541–1552 (2001).
[Crossref]

1996 (2)

E. Masry, “Multivariate regression estimation: local polynomial fitting for time series,” Stoch. Proc. Appl. 65(1), 81–101 (1996).
[Crossref]

F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A 13(6), 1277–1284 (1996).
[Crossref]

1990 (1)

1982 (1)

1980 (2)

1979 (1)

1976 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

1971 (2)

1970 (1)

Agrawal, G. P.

Anderson, K.

Baer, E.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Bayraktar, Z.

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2010).
[Crossref]

Bociort, F.

Brister, A.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Brocker, D. E.

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. 18(4), 044019 (2016).
[Crossref]

J. Nagar, D. E. Brocker, S. D. Campbell, J. A. Easum, and D. H. Werner, “Modularization of gradient-index optical design using wavefront matching enabled optimization,” Opt. Express 24(9), 9359–9368 (2016).
[Crossref] [PubMed]

S. D. Campbell, D. E. Brocker, J. Nagar, and D. H. Werner, “Size, weight, and power reduction regimes in achromatic gradient-index singlets,” Appl. Opt. 55(13), 3594–3598 (2016).
[Crossref] [PubMed]

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

D. E. Brocker, S. D. Campbell, and D. H. Werner, “Color-correcting gradient-index infrared singlet based on silicon and germanium mixing,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).
[Crossref]

Bugmann, G.

M. Rafiq, G. Bugmann, and D. Easterbrook, “Neural network design for engineering applications,” Comput. Struc. 79(17), 1541–1552 (2001).
[Crossref]

Busch, S. F.

Campbell, S. D.

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. 18(4), 044019 (2016).
[Crossref]

J. Nagar, D. E. Brocker, S. D. Campbell, J. A. Easum, and D. H. Werner, “Modularization of gradient-index optical design using wavefront matching enabled optimization,” Opt. Express 24(9), 9359–9368 (2016).
[Crossref] [PubMed]

S. D. Campbell, D. E. Brocker, J. Nagar, and D. H. Werner, “Size, weight, and power reduction regimes in achromatic gradient-index singlets,” Appl. Opt. 55(13), 3594–3598 (2016).
[Crossref] [PubMed]

D. E. Brocker, S. D. Campbell, and D. H. Werner, “Color-correcting gradient-index infrared singlet based on silicon and germanium mixing,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).
[Crossref]

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

Castro-Camus, E.

Çeçen, A.

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
[Crossref]

Couckuyt, I.

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

Crombecq, K.

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

Davis, F. J.

J. C. Helton and F. J. Davis, “Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems,” Reliab. Eng. Syst. Saf. 81(1), 23–69 (2003).
[Crossref]

Demeester, P.

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

Dhaene, T.

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

Dupuy, C.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

Easterbrook, D.

M. Rafiq, G. Bugmann, and D. Easterbrook, “Neural network design for engineering applications,” Comput. Struc. 79(17), 1541–1552 (2001).
[Crossref]

Easum, J. A.

Fast, T.

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
[Crossref]

Gorissen, D.

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

Gregory, M. D.

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2010).
[Crossref]

Guha, S.

Hansen, N.

N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation,” in Proceedings of IEEE Conference on Evolutionary Computation (IEEE, 1996), pp. 312–317.
[Crossref]

Harmon, P.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

Helton, J. C.

J. C. Helton and F. J. Davis, “Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems,” Reliab. Eng. Syst. Saf. 81(1), 23–69 (2003).
[Crossref]

Hernandez-Serrano, A. I.

Ji, S.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Jones, D. R.

D. R. Jones, “A taxonomy of global optimization methods based on response surfaces,” J. Glob. Optim. 21(4), 345–383 (2001).
[Crossref]

Kalidindi, S. R.

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
[Crossref]

Koch, M.

Kumbur, E. C.

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
[Crossref]

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[Crossref] [PubMed]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Mackey, M.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Mahajan, V. N.

Masry, E.

E. Masry, “Multivariate regression estimation: local polynomial fitting for time series,” Stoch. Proc. Appl. 65(1), 81–101 (1996).
[Crossref]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

McLeod, R. R.

Moore, D. T.

Nagar, J.

Noll, R. J.

Ostermeier, A.

N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation,” in Proceedings of IEEE Conference on Evolutionary Computation (IEEE, 1996), pp. 312–317.
[Crossref]

Park, S.-K.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

Pattanayak, D. N.

Ponting, M.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Rafiq, M.

M. Rafiq, G. Bugmann, and D. Easterbrook, “Neural network design for engineering applications,” Comput. Struc. 79(17), 1541–1552 (2001).
[Crossref]

Sands, P. J.

Silva, D. E.

Smith, D. R.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[Crossref] [PubMed]

Urness, A. C.

Wang, D. Y.

Wang, J. Y.

Weidenbach, M.

Werner, D. H.

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. 18(4), 044019 (2016).
[Crossref]

S. D. Campbell, D. E. Brocker, J. Nagar, and D. H. Werner, “Size, weight, and power reduction regimes in achromatic gradient-index singlets,” Appl. Opt. 55(13), 3594–3598 (2016).
[Crossref] [PubMed]

J. Nagar, D. E. Brocker, S. D. Campbell, J. A. Easum, and D. H. Werner, “Modularization of gradient-index optical design using wavefront matching enabled optimization,” Opt. Express 24(9), 9359–9368 (2016).
[Crossref] [PubMed]

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2010).
[Crossref]

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

D. E. Brocker, S. D. Campbell, and D. H. Werner, “Color-correcting gradient-index infrared singlet based on silicon and germanium mixing,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).
[Crossref]

Wilson, W. L.

Ye, C.

Yin, K.

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Appl. Opt. (4)

Comput. Struc. (1)

M. Rafiq, G. Bugmann, and D. Easterbrook, “Neural network design for engineering applications,” Comput. Struc. 79(17), 1541–1552 (2001).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Fast optimization of electromagnetic design problems using the covariance matrix adaptation evolutionary strategy,” IEEE Trans. Antenn. Propag. 59(4), 1275–1285 (2010).
[Crossref]

J. Glob. Optim. (1)

D. R. Jones, “A taxonomy of global optimization methods based on response surfaces,” J. Glob. Optim. 21(4), 345–383 (2001).
[Crossref]

J. Mach. Learn. Res. (1)

D. Gorissen, K. Crombecq, I. Couckuyt, T. Dhaene, and P. Demeester, “A surrogate modeling and adaptive sampling toolbox for computer based design,” J. Mach. Learn. Res. 11, 2051–2055 (2010).

J. Opt. (1)

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. 18(4), 044019 (2016).
[Crossref]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Power Sources (1)

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, “A data-driven approach to establishing microstructure–property relationships in porous transport layers of polymer electrolyte fuel cells,” J. Power Sources 245(1), 144–153 (2014).
[Crossref]

Nat. Mater. (1)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[Crossref] [PubMed]

Opt. Eng. (1)

S. Ji, K. Yin, M. Mackey, A. Brister, M. Ponting, and E. Baer, “Polymeric nanolayered gradient refractive index lenses: technology review and introduction of spherical gradient refractive index ball lenses,” Opt. Eng. 52(11), 112105 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Reliab. Eng. Syst. Saf. (1)

J. C. Helton and F. J. Davis, “Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems,” Reliab. Eng. Syst. Saf. 81(1), 23–69 (2003).
[Crossref]

Stoch. Proc. Appl. (1)

E. Masry, “Multivariate regression estimation: local polynomial fitting for time series,” Stoch. Proc. Appl. 65(1), 81–101 (1996).
[Crossref]

Other (10)

F. Schäffler, Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe (John Wiley & Sons, 2001), Chap. 6.

M. Born and E. Wolf, “The circle of polynomials of Zernike,” in Principles of Optics6 (Pergamon Press, 1993), Appendix VII, pp. 767–772.

N. Hansen and A. Ostermeier, “Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation,” in Proceedings of IEEE Conference on Evolutionary Computation (IEEE, 1996), pp. 312–317.
[Crossref]

A. Forrester, A. Sóbester, and A. Keane, Engineering Design via Surrogate Modelling: A Practical Guide (John Wiley & Sons, 2008).

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering11 (Academic Press, 1992), Chap. 8, pp. 2–53.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, 2001), Chap. 7.

D. E. Brocker, S. D. Campbell, and D. H. Werner, “Color-correcting gradient-index infrared singlet based on silicon and germanium mixing,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).
[Crossref]

M. Gu, Advanced Optical Imaging Theory (Springer, 2000), Chap 2.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via inkjet-aided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, Vancouver, BC, 788–789 (2015).

E. W. Marchand, Gradient Index Optics (Academic, 1978), Chap. 8.

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Figures (12)

Fig. 1
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
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
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.
Fig. 4
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.
Fig. 5
Fig. 5 Overview of the surrogate model training process.
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.
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.
Fig. 8
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
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.
Fig. 10
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.
Fig. 11
Fig. 11 Strehl ratios over various angles for each of the best performing GRIN lenses from Fig. 10.
Fig. 12
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).

Tables (4)

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Table 1 Number of Samples to Train an Effective Surrogate Model

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Table 2 Optimized Lens Parameters from the Surrogate Assisted Solution

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Table 3 Best Performing GRIN Term Permutations

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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

Equations (4)

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n(r,z)= j=0 j=p i=0 i=p c i,j ( r D/2 ) i ( z T ) j
Z q = i=1 i=N A i (q) P i + j=1 j=N i=1 i=N B i,j (q) P i P j + k=1 k=N j=1 j=N i=1 i=N C i,j,k (q) P i P j P k +...    for q=1,2,...,M.
cost= i=1 i=22 | Z i θ=0 | + i=1 i=22 | Z i θ=2 | + i=1 i=22 | Z i θ=4 |
cost= i=1 i=22 | Z i θ=0 T i θ=0 | + i=1 i=22 | Z i θ=2 T i θ=0 | + i=1 i=22 | Z i θ=4 T i θ=0 |

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