Abstract

Conventional optical designs with gradient index (GRIN) use rotationally-invariant GRIN profiles described by polynomials with no orthogonality. These GRIN profiles have limited effectiveness at correcting aberrations from tilted/decentered or freeform systems. In this paper, a three-dimensional orthogonal polynomial basis set (the FGRIN basis) is proposed, which enables the design of GRIN profiles with both rotational and axial variations. The FGRIN basis is then demonstrated via the design of a 3D GRIN corrector plate targeted to correct the rotationally-variant aberrations induced from a tilted spherical mirror. A sample corrector is manufactured and tested, showing significant correction of astigmatism. The FGRIN basis opens a new design space of 3D rotational variant GRIN profiles, which has the potential of replacing multiple freeform surfaces and simplifying complex systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

2018 (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

2017 (1)

A. Broemel, U. Lippmann, and H. Gross, “Freeform surface descriptions. part i: mathematical representations,” Adv. Opt. Technol. 6(5), 327–336 (2017).
[Crossref]

2016 (4)

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

J. A. Easum, S. D. Campbell, J. Nagar, and D. H. Werner, “Analytical surrogate model for the aberrations of an arbitrary grin lens,” Opt. Express 24(16), 17805–17818 (2016).
[Crossref]

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]

2015 (3)

2014 (1)

2013 (2)

C. Menke and G. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. 2(1), 97–109 (2013).
[Crossref]

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]

2012 (1)

2011 (2)

2007 (1)

1994 (1)

1982 (1)

1981 (1)

1980 (1)

1971 (3)

1970 (1)

1965 (1)

W. Heller, “Remarks on refractive index mixture rules,” J. Phys. Chem. 69(4), 1123–1129 (1965).
[Crossref]

1934 (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Ajiboye, O.

S. Isehunwa, E. Olanisebe, O. Ajiboye, and S. Akintola, “Estimation of the refractive indices of some binary mixtures,” Afr. J. Pure Appl. Chem. 10(4), 58–64 (2015).
[Crossref]

Akintola, S.

S. Isehunwa, E. Olanisebe, O. Ajiboye, and S. Akintola, “Estimation of the refractive indices of some binary mixtures,” Afr. J. Pure Appl. Chem. 10(4), 58–64 (2015).
[Crossref]

Atkinson, L. G.

L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” (1994). US Patent 5,361,166.

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]

Bauer, A.

Beadie, G.

G. Beadie, J. Mait, R. Flynn, and P. Milojkovic, Ternary versus binary material systems for gradient index optics, Advanced Optics for Defense Applications: UV through LWIR II, vol. 10181 (International Society for Optics and Photonics, 2017), p. 1018108.

Beier, M.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

Benítez, P.

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

Bentley, J. L.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Bodell, S. Y.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Bray, M.

M. Bray, “Orthogonal polynomials: a set for square areas,” in Optical Fabrication, Testing, and Metrology, vol. 5252 (International Society for Optics and Photonics, 2004), pp. 314–321.

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.

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]

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, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Broemel, A.

A. Broemel, U. Lippmann, and H. Gross, “Freeform surface descriptions. part i: mathematical representations,” Adv. Opt. Technol. 6(5), 327–336 (2017).
[Crossref]

Brömel, A.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

Buchdahl, H. A.

H. A. Buchdahl, An introduction to Hamiltonian optics (Courier Corporation, 1993).

Campbell, S. D.

J. A. Easum, S. D. Campbell, J. Nagar, and D. H. Werner, “Analytical surrogate model for the aberrations of an arbitrary grin lens,” Opt. Express 24(16), 17805–17818 (2016).
[Crossref]

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]

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Chow, W. W.

Conkey, D.

D. Conkey, “Freeform optics for optical payloads with reduced size and weight,” [retrieved 14 Feb 2020], https://nasasitebuilder-prod-cdn.nasawestprime.com/wp-content/uploads/sites/42/2018/02/30193206/31-Voxtel-Freeform-Optics-for-Optical-Payloads.pdf .

Dudukovic, N.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Dylla-Spears, R.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

Easum, J. A.

Ferreira, C.

Flynn, R.

G. Beadie, J. Mait, R. Flynn, and P. Milojkovic, Ternary versus binary material systems for gradient index optics, Advanced Optics for Defense Applications: UV through LWIR II, vol. 10181 (International Society for Optics and Photonics, 2017), p. 1018108.

Forbes, G.

C. Menke and G. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. 2(1), 97–109 (2013).
[Crossref]

G. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20(3), 2483–2499 (2012).
[Crossref]

G. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
[Crossref]

Fuerschbach, K.

Ghatak, A. K.

Grabovickic, D.

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

Gross, H.

A. Broemel, U. Lippmann, and H. Gross, “Freeform surface descriptions. part i: mathematical representations,” Adv. Opt. Technol. 6(5), 327–336 (2017).
[Crossref]

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Hartung, J.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

Heller, W.

W. Heller, “Remarks on refractive index mixture rules,” J. Phys. Chem. 69(4), 1123–1129 (1965).
[Crossref]

Herrera, O. D.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

Isehunwa, S.

S. Isehunwa, E. Olanisebe, O. Ajiboye, and S. Akintola, “Estimation of the refractive indices of some binary mixtures,” Afr. J. Pure Appl. Chem. 10(4), 58–64 (2015).
[Crossref]

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]

Johnson, M. A.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

Kindred, D. S.

L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” (1994). US Patent 5,361,166.

Kumar, D. V.

Li, L.

Lippmann, U.

A. Broemel, U. Lippmann, and H. Gross, “Freeform surface descriptions. part i: mathematical representations,” Adv. Opt. Technol. 6(5), 327–336 (2017).
[Crossref]

Liu, J.

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

López, J. L.

Ma, B.

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.

Mait, J.

G. Beadie, J. Mait, R. Flynn, and P. Milojkovic, Ternary versus binary material systems for gradient index optics, Advanced Optics for Defense Applications: UV through LWIR II, vol. 10181 (International Society for Optics and Photonics, 2017), p. 1018108.

Marchand, E.

E. Marchand, Gradient index optics (Academic Press, 1978).

Menke, C.

C. Menke and G. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. 2(1), 97–109 (2013).
[Crossref]

Milojkovic, P.

G. Beadie, J. Mait, R. Flynn, and P. Milojkovic, Ternary versus binary material systems for gradient index optics, Advanced Optics for Defense Applications: UV through LWIR II, vol. 10181 (International Society for Optics and Photonics, 2017), p. 1018108.

Miñano, J. C.

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

Moore, D. T.

D. T. Moore, “Gradient-index optics: a review,” Appl. Opt. 19(7), 1035–1038 (1980).
[Crossref]

D. T. Moore, “Design of singlets with continuously varying indices of refraction,” J. Opt. Soc. Am. 61(7), 886–894 (1971).
[Crossref]

L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” (1994). US Patent 5,361,166.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Nagar, J.

Narasimhan, B. A.

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

Navarro, R.

Nguyen, D. T.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

Ni, Y.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Nikolic, M. I.

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

Ochse, D.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” in Freeform Optics, (Optical Society of America, 2015), pp. FT2B–4.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

Olanisebe, E.

S. Isehunwa, E. Olanisebe, O. Ajiboye, and S. Akintola, “Estimation of the refractive indices of some binary mixtures,” Afr. J. Pure Appl. Chem. 10(4), 58–64 (2015).
[Crossref]

Oleszko, M.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

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 injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

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]

Reichmann, L.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” in Freeform Optics, (Optical Society of America, 2015), pp. FT2B–4.

Rolland, J. P.

Sands, P.

Sasan, K.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

Schiesser, E. M.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

Sharma, A.

Sinusía, E. P.

Song, W.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Steinkopf, R.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

Swantner, W.

Takaki, N.

N. Takaki, A. Bauer, and J. P. Rolland, “On-the-fly surface manufacturability constraints for freeform optical design enabled by orthogonal polynomials,” Opt. Express 27(5), 6129–6146 (2019).
[Crossref]

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Thompson, K. P.

Uhlendorf, K.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” in Freeform Optics, (Optical Society of America, 2015), pp. FT2B–4.

Weisstein, E. W.

E. W. Weisstein, “Legendre polynomial (from mathworld - a wolfram web resource),” [retrieved 25 Feb 2020], http://mathworld.wolfram.com/LegendrePolynomial.html .

E. W. Weisstein, “Chebyshev polynomial of the first kind (from mathworld - a wolfram web resource),” [retrieved 25 Feb 2020], http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .

Werner, D. H.

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]

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. A. Easum, S. D. Campbell, J. Nagar, and D. H. Werner, “Analytical surrogate model for the aberrations of an arbitrary grin lens,” Opt. Express 24(16), 17805–17818 (2016).
[Crossref]

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

Wong, L.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

Yang, T.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Yee, A. J.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Yee, T. D.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

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]

Youngworth, R. N.

R. N. Youngworth, “Tolerancing forbes aspheres: advantages of an orthogonal basis,” in Optical System Alignment, Tolerancing, and Verification III, vol. 7433 (International Society for Optics and Photonics, 2009), p. 74330H.

Zernike, V. F.

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Zhao, Y.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

Zhong, Y.

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

Zinter, J. R.

L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” (1994). US Patent 5,361,166.

Adv. Opt. Technol. (2)

C. Menke and G. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. 2(1), 97–109 (2013).
[Crossref]

A. Broemel, U. Lippmann, and H. Gross, “Freeform surface descriptions. part i: mathematical representations,” Adv. Opt. Technol. 6(5), 327–336 (2017).
[Crossref]

Afr. J. Pure Appl. Chem. (1)

S. Isehunwa, E. Olanisebe, O. Ajiboye, and S. Akintola, “Estimation of the refractive indices of some binary mixtures,” Afr. J. Pure Appl. Chem. 10(4), 58–64 (2015).
[Crossref]

Appl. Opt. (4)

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. (5)

J. Phys. Chem. (1)

W. Heller, “Remarks on refractive index mixture rules,” J. Phys. Chem. 69(4), 1123–1129 (1965).
[Crossref]

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

Opt. Eng. (2)

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]

M. I. Nikolic, P. Benítez, B. A. Narasimhan, D. Grabovickic, J. Liu, and J. C. Miñano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55(7), 071204 (2016).
[Crossref]

Opt. Express (9)

G. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
[Crossref]

G. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20(3), 2483–2499 (2012).
[Crossref]

J. A. Easum, S. D. Campbell, J. Nagar, and D. H. Werner, “Analytical surrogate model for the aberrations of an arbitrary grin lens,” Opt. Express 24(16), 17805–17818 (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]

N. Takaki, A. Bauer, and J. P. Rolland, “On-the-fly surface manufacturability constraints for freeform optical design enabled by orthogonal polynomials,” Opt. Express 27(5), 6129–6146 (2019).
[Crossref]

B. Ma, L. Li, K. P. Thompson, and J. P. Rolland, “Applying slope constrained q-type aspheres to develop higher performance lenses,” Opt. Express 19(22), 21174–21179 (2011).
[Crossref]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
[Crossref]

A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23(22), 28141–28153 (2015).
[Crossref]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref]

Physica (1)

V. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[Crossref]

Other (16)

E. W. Weisstein, “Legendre polynomial (from mathworld - a wolfram web resource),” [retrieved 25 Feb 2020], http://mathworld.wolfram.com/LegendrePolynomial.html .

E. W. Weisstein, “Chebyshev polynomial of the first kind (from mathworld - a wolfram web resource),” [retrieved 25 Feb 2020], http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .

M. Bray, “Orthogonal polynomials: a set for square areas,” in Optical Fabrication, Testing, and Metrology, vol. 5252 (International Society for Optics and Photonics, 2004), pp. 314–321.

D. Ochse, K. Uhlendorf, and L. Reichmann, “Describing freeform surfaces with orthogonal functions,” in Freeform Optics, (Optical Society of America, 2015), pp. FT2B–4.

S. D. Campbell, D. E. Brocker, D. H. Werner, C. Dupuy, S.-K. Park, and P. Harmon, “Three-dimensional gradient-index optics via injketaided additive manufacturing techniques,” in 2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2015), pp. 605–606.

R. Dylla-Spears, N. Dudukovic, K. Sasan, O. D. Herrera, M. A. Johnson, T. D. Yee, D. T. Nguyen, and L. Wong, “Tailored glass optics using 3d printing (conference presentation),” in Computational Imaging IV, vol. 10990 (International Society for Optics and Photonics, 2019), p. 109900M.

G. Beadie, J. Mait, R. Flynn, and P. Milojkovic, Ternary versus binary material systems for gradient index optics, Advanced Optics for Defense Applications: UV through LWIR II, vol. 10181 (International Society for Optics and Photonics, 2017), p. 1018108.

L. G. Atkinson, D. S. Kindred, D. T. Moore, and J. R. Zinter, “Negative abbe number radial gradient index relay and use of same,” (1994). US Patent 5,361,166.

A. J. Yee, W. Song, N. Takaki, T. Yang, Y. Zhao, Y. Ni, S. Y. Bodell, J. P. Rolland, J. L. Bentley, and D. T. Moore, “Design of a freeform gradient-index prism for mixed reality head mounted display,” in Digital Optics for Immersive Displays, vol. 10676 (International Society for Optics and Photonics, 2018), p. 106760S.

“Defining selfoc gradients,” in CODE V Lens System Setup Reference Manual, vol. II (Synopsys Inc., 2019), chap. 8.

“Defining university of rochester gradients,” in CODE V Lens System Setup Reference Manual, vol. II (Synopsys Inc., 2019), chap. 8.

H. A. Buchdahl, An introduction to Hamiltonian optics (Courier Corporation, 1993).

E. Marchand, Gradient index optics (Academic Press, 1978).

H. Gross, A. Brömel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” in Optical Systems Design 2015: Optical Design and Engineering VI, vol. 9626 (International Society for Optics and Photonics, 2015), p. 96260U.

R. N. Youngworth, “Tolerancing forbes aspheres: advantages of an orthogonal basis,” in Optical System Alignment, Tolerancing, and Verification III, vol. 7433 (International Society for Optics and Photonics, 2009), p. 74330H.

D. Conkey, “Freeform optics for optical payloads with reduced size and weight,” [retrieved 14 Feb 2020], https://nasasitebuilder-prod-cdn.nasawestprime.com/wp-content/uploads/sites/42/2018/02/30193206/31-Voxtel-Freeform-Optics-for-Optical-Payloads.pdf .

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Figures (16)

Fig. 1.
Fig. 1. The configuration used to analyze aberrations generated by FGRIN basis terms. The 3D GRIN profile is prescribed inside the corrector. The perfect thin lens cannot be seen in the figure, but is at the back surface of the corrector.
Fig. 2.
Fig. 2. The non-rotationally symmetric aberrations induced by axially constant FGRIN basis terms. Circular symbols represent defocus-like aberrations ( $Z_4$ ), sticks representing astigmatism-like aberrations ( $Z_{5/6},Z_{10/11}$ , etc.) and arrowed circles representing coma-like aberrations $(Z_{7/8})$ . The FGRIN terms are shown in the left column, and the top row shows the field dependence of FRINGE Zernike aberrations. The aberration terms come in pairs because of the property of Zernike polynomial: a complete description of any asymmetric aberrations usually requires a pair of Zernike polynomials with orthogonal directions (e.g. $Z_{5/6}$ works together to describe Zernike astigmatism).
Fig. 3.
Fig. 3. The field dependence of aberrations induced by axially-linear FGRIN basis terms $(U_xV_1)$ . Only the terms with lateral index profiles described by Zernike terms of indices higher than or equal to $U_{07/08}$ generate nonzero aberrations.
Fig. 4.
Fig. 4. The field dependence of aberrations induced by axially-quadratic $(U_xV_2)$ FGRIN basis terms. Only the FGRIN terms with lateral order higher than or equal to $U_{12/13}V_2$ generate nonzero aberrations.
Fig. 5.
Fig. 5. The YZ cross-sectional view of (a) the starting design without a corrector and (b) the final design with the corrector. A zoom-in view (circles) on the initial design shows the rays near the image are highly aberrated due to the tilted mirror. In final design the back focal distance and image plane tilt are both optimized to achieve the best performance, but the small amount of change shows the majority of correction is done by the FGRIN corrector.
Fig. 6.
Fig. 6. FFDs of uncorrected wavefront aberrations from the tilted spherical mirror with maximum magnitude greater than $1\lambda$ . The dominant aberrations is the field-constant $Z_{5/6}$ . The three figures are under the same scale.
Fig. 7.
Fig. 7. FFDs with maximum magnitude greater than $0.5\lambda$ after $U_{05}V_0$ is added to the corrector. The four dominant aberrations shown here are field-linear $Z_4$ , field-asymmetric field-linear $Z_{5/6}$ , field-constant $Z_{7/8}$ and field-constant $Z_{10/11}$ . Note the scale change from Fig. 6.
Fig. 8.
Fig. 8. Zernike components of wavefront aberrations with maximum magnitude greater than $0.1\lambda$ after $U_{05}V_0$ , $U_{08}V_0$ , $U_{11}V_0$ are added to the corrector. The dominant aberration is field-asymmetric field-linear $Z_{5/6}$ .
Fig. 9.
Fig. 9. Four most significant residual wavefront aberrations after field-asymmetric field-linear $Z_{5/6}$ is corrected.
Fig. 10.
Fig. 10. Final RMS wavefront error across the field of view of the system with the 3D GRIN corrector.
Fig. 11.
Fig. 11. (top) The 3D refractive index profile. (bottom) The XY and YZ slice of the 3D index profile. The positions where slices are taken are indicated on top. The index color scales, indicated by the colorbar, are the same for all three figures.
Fig. 12.
Fig. 12. The RMS wavefront aberration FFDs of (a) a system with only a freeform mirror and no corrector (b) a system with a single-side freeform corrector at stop and a spherical mirror and (c) a system with both a single-sided freeform corrector at stop and a freeform mirror. To achieve comparable performances as a single 3D GRIN corrector, at least two freeform surfaces need to be used in the system.
Fig. 13.
Fig. 13. A photo of the FGRIN corrector. The numbered ruling is in centimeters and the clear aperture is shown in the dotted red circle.
Fig. 14.
Fig. 14. (left) Schematic diagram of the double-pass Twyman-Green interferometer to test the 3D FGRIN corrector. The designed tilt angles and spacings between elements are listed in the figure. (right) A photo of the real experimental setup. The sample is denoted by the yellow rectangle. The optical axes are marked with red arrows, and the major elements are noted in yellow.
Fig. 15.
Fig. 15. (top) The interferograms of the (a) simulated and (b) measured wavefront of the system without the corrector. The clear apertures are marked by the red circles. The interferograms show that the system is dominated by astigmatism. (bottom) The (c) simulated and (d) measured interferograms of the wavefront after correction. The astigmatism is significantly corrected.
Fig. 16.
Fig. 16. The 3D GRIN profile modeled as a stack of 2D GRIN phase plates. The model is comprised of $N$ phase plates, each with a thickness $\Delta z$ . The distance of the $p^{\textrm {th}}$ element from stop is $z_p$ and the chief ray height on the element is denoted as $\Delta h_p$ .

Tables (3)

Tables Icon

Table 1. The lateral and axial basis polynomials up to 6 th order

Tables Icon

Table 2. Design specifications.

Tables Icon

Table 3. Coefficients of FGRIN terms used in the final design.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

n GRIN ( λ , x , y , z ) = μ ( x , y , z ) n lo ( λ ) + ( 1 μ ( x , y , z ) ) n hi ( λ )
n GRIN ( λ , x , y , z ) = μ ( x , y , z ) n lo ( λ ) + ( 1 μ ( x , y , z ) ) n hi ( λ ) = 1 μ ( x , y , z ) 2 n lo ( λ ) + 1 + μ ( x , y , z ) 2 n hi ( λ ) = 1 2 ( n hi ( λ ) + n lo ( λ ) ) + 1 2 μ ( x , y , z ) ( n hi ( λ ) n lo ( λ ) ) = n 0 ( λ ) + Δ n ( λ ) μ ( x , y , z ) .
x = x ρ max y = y ρ max z = 2 z z max 1
n ( λ , x , y , z ) = n 0 ( λ ) + Δ n ( λ ) m , n c m , n U m ( x , y ) V n ( z )
G m n ( x , y , z ) = U m ( x , y ) V n ( z )
D G m n ( x , y , z ) G m n ( x , y , z ) d x d y d z = x 2 + y 2 1 U m ( x , y ) U m ( x , y ) d x d y 1 1 V n ( z ) V n ( z ) d z = 0 unless m = m and n = n .
Δ n s s 2 = m , n c m , n 2 ξ m , n
n ( λ , x , y , z p ) = n 0 ( λ ) + Δ n ( λ ) m , n c m n U m ( x , y ) V n ( z p )
W p ( z p ) = Δ n ( λ ) V n ( 2 z p 1 ) Δ z [ c n ( ρ + Δ h p ) ] [ ( ρ + Δ h p ) ( ρ + Δ h p ) ]
W = p W p ( z p ) = p Δ n ( λ ) V n ( 2 z p 1 ) Δ z [ ( c n ( ρ + Δ h p ) ) ] [ ( ρ + Δ h p ) ( ρ + Δ h p ) ] .
Δ h p = z p tan ( θ ) h ^ = p Δ z tan ( θ ) h ^ ,
W = p Δ n ( λ ) V n ( 2 z p 1 ) Δ z [ ( c n ( ρ + z p tan ( θ ) h ^ ) ) ] [ ( ρ + z p tan ( θ ) h ^ ) ( ρ + z p tan ( θ ) h ^ ) ] .
W = lim N p = 1 N W p = p = 1 N Δ n ( λ ) V n ( 2 z p 1 ) Δ z [ ( c n ( ρ + z p tan ( θ ) h ^ ) ) ] [ ( ρ + z p tan ( θ ) h ^ ) ( ρ + z p tan ( θ ) h ^ ) ] = Δ n ( λ ) [ 1 1 V n ( z ) ( c n ρ ) ( ρ ρ ) d z + 1 1 V n ( z ) ( z + 1 ) tan ( θ ) ( c n h ^ ) ( ρ ρ ) d z + 1 2 1 1 V n ( z ) ( z + 1 ) tan ( θ ) ( c n h ^ ρ 2 ) d z + 1 2 1 1 V n ( z ) ( z + 1 ) 2 tan 2 ( θ ) ( c n ρ ) ( h ^ h ^ ) d z + 1 4 1 1 V n ( z ) ( z + 1 ) 2 tan 2 ( θ ) ( c n h 2 ^ ρ ) d z + 1 8 1 1 V n ( z ) ( z + 1 ) 3 tan 3 ( θ ) ( c n h ^ ) ( h ^ h ^ ) d z ] ,
W Z 7 / 8 = 1 1 V n ( z ) ( c n ρ ) ( ρ ρ ) d z W Z 5 / 6 = 1 1 V n ( z ) ( z + 1 ) tan ( θ ) ( 2 c n h ^ ) ( ρ ρ ) d z W Z 4 = 1 1 V n ( z ) ( z + 1 ) tan ( θ ) ( c n h ^ ρ 2 ) d z .
W Z 7 / 8 = ( c n ρ ) ( ρ ρ ) 1 1 d z 0 W Z 5 / 6 = ( 2 c n h ^ ) ( ρ ρ ) 1 1 ( z + 1 ) d z 0 W Z 4 = ( c n h ^ ρ 2 ) 1 1 ( z + 1 ) tan ( θ ) d z 0.
W Z 7 / 8 = ( c n ρ ) ( ρ ρ ) 1 1 z d z = 0 W Z 5 / 6 = ( 2 c n h ^ ) ( ρ ρ ) 1 1 z ( z + 1 ) tan ( θ ) d z 0 W Z 4 = ( c n h ^ ρ 2 ) 1 1 z ( z + 1 ) tan ( θ ) d z 0.
W Z 7 / 8 = ( c n ρ ) ( ρ ρ ) 1 1 1 2 ( 3 z 2 1 ) d z = 0 W Z 5 / 6 = ( 2 c n h ^ ) ( ρ ρ ) 1 1 1 2 ( 3 z 2 1 ) ( z + 1 ) tan ( θ ) d z = 0 W Z 4 = ( c n h ^ ρ 2 ) 1 1 1 2 ( 3 z 2 1 ) ( z + 1 ) tan ( θ ) d z = 0.

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