Abstract

Imaging systems consisting of flat phase elements can realize the same functions and applications of conventional geometric optical systems, as well as the ones using aspherical or freeform optics, but can achieve more compactness, lighter-weight and easier-alignment. In addition, it is easy to integrate multiple phase elements into a single flat element. Here we propose a novel design method and realize the design of off-axis nonsymmetric imaging systems consisting of multiple flat phase elements. Compared with other traditional design methods of phase elements, the whole design process starts from an initial system using simple true geometric planes. The phase profiles or functions are generated point-by-point directly based on the given system specifications and configuration. In comparison with other direct or point-by-point design methods of flat phase elements, the rays of multiple fields and pupil positions are employed in the design framework. Closed-form phase functions of multiple flat elements are designed quickly and effectively by connecting and integrating the real three-dimensional space and the phase function space. This method can be taken as a fast phase retrieval method to some degree. To demonstrate the feasibility of the proposed design method, we present a high-performance compact system as design example. The design method and framework depicted in this paper can be applied in many areas, such as virtual reality (VR) and augmented reality (AR), miniature cameras, high-performance telescopy, microscopy, and illumination design.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

M. Martínez-Corral and B. Javidi, “Fundamentals of 3D imaging and displays: a tutorial on integral imaging, light-field, and plenoptic systems,” Adv. Opt. Photonics 10(3), 512–566 (2018).
[Crossref]

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] [PubMed]

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

T. Yang, D. Cheng, and Y. Wang, “Aberration analysis for freeform surface terms overlay on general decentered and tilted optical surfaces,” Opt. Express 26(6), 7751–7770 (2018).
[Crossref] [PubMed]

2017 (5)

2016 (3)

2015 (6)

J. Han, J. Liu, X. Yao, and Y. Wang, “Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms,” Opt. Express 23(3), 3534–3549 (2015).
[Crossref] [PubMed]

M. Beier, J. Hartung, T. Peschel, C. Damm, A. Gebhardt, S. Scheiding, D. Stumpf, U. D. Zeitner, S. Risse, R. Eberhardt, and A. Tünnermann, “Development, fabrication, and testing of an anamorphic imaging snap-together freeform telescope,” Appl. Opt. 54(12), 3530–3542 (2015).
[Crossref]

T. Yang, J. Zhu, X. Wu, and G. Jin, “Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method,” Opt. Express 23(8), 10233–10246 (2015).
[Crossref] [PubMed]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[Crossref] [PubMed]

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

2014 (4)

2013 (2)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

2010 (1)

D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE 7849, 78490Q (2010).
[Crossref]

2009 (1)

2000 (1)

J. P. Rolland, “Wide-angle, off-axis, see-through head-mounted display,” Opt. Eng. 39(7), 1760–1767 (2000).
[Crossref]

1995 (1)

A. D. Kathman, D. H. Hochmuth, and D. R. Brown, “Efficiency considerations for diffractive optical elements,” Proc. SPIE 2577, 114–122 (1995).
[Crossref]

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–242 (1972).

1950 (1)

Aieta, F.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Arbabi, A.

Arbabi, E.

Bai, B.

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

Bauer, A.

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] [PubMed]

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

Beier, M.

Benítez, P.

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

Brown, D. R.

A. D. Kathman, D. H. Hochmuth, and D. R. Brown, “Efficiency considerations for diffractive optical elements,” Proc. SPIE 2577, 114–122 (1995).
[Crossref]

Capasso, F.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[Crossref] [PubMed]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Cheng, D.

Colburn, S.

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref] [PubMed]

C. Hong, S. Colburn, and A. Majumdar, “Flat metaform near-eye visor,” Appl. Opt. 56(31), 8822–8827 (2017).
[Crossref] [PubMed]

Damm, C.

Dodson, C. M.

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref] [PubMed]

Dong, J.

Eberhardt, R.

Evans, C.

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Fang, F.

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Faraon, A.

Fienup, J. R.

Flügel-Paul, T.

Foster, L. V.

Fuerschbach, K.

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gebhardt, A.

Genevet, P.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–242 (1972).

Gross, H.

Han, J.

Hartung, J.

Hochmuth, D. H.

A. D. Kathman, D. H. Hochmuth, and D. R. Brown, “Efficiency considerations for diffractive optical elements,” Proc. SPIE 2577, 114–122 (1995).
[Crossref]

Hong, C.

Horie, Y.

Hou, W.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Hua, H.

D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE 7849, 78490Q (2010).
[Crossref]

Huang, L.

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

Infante, J.

Javidi, B.

M. Martínez-Corral and B. Javidi, “Fundamentals of 3D imaging and displays: a tutorial on integral imaging, light-field, and plenoptic systems,” Adv. Opt. Photonics 10(3), 512–566 (2018).
[Crossref]

Ji, Z.

Jin, G.

Kamali, S. M.

Kathman, A. D.

A. D. Kathman, D. H. Hochmuth, and D. R. Brown, “Efficiency considerations for diffractive optical elements,” Proc. SPIE 2577, 114–122 (1995).
[Crossref]

Kats, M. A.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Kildishev, A. V.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

Li, X.

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

Lin, W.

Liu, C.

Liu, J.

Ma, H.

Majumdar, A.

C. Hong, S. Colburn, and A. Majumdar, “Flat metaform near-eye visor,” Appl. Opt. 56(31), 8822–8827 (2017).
[Crossref] [PubMed]

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref] [PubMed]

Martínez-Corral, M.

M. Martínez-Corral and B. Javidi, “Fundamentals of 3D imaging and displays: a tutorial on integral imaging, light-field, and plenoptic systems,” Adv. Opt. Photonics 10(3), 512–566 (2018).
[Crossref]

Mendes-Lopes, J.

Meng, Q.

Miñano, J. C.

Mühlenbernd, H.

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

Muñoz, F.

Peschel, T.

Reimers, J.

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

Reshidko, D.

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

Risse, S.

Rolland, J. P.

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] [PubMed]

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

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] [PubMed]

J. P. Rolland, “Wide-angle, off-axis, see-through head-mounted display,” Opt. Eng. 39(7), 1760–1767 (2000).
[Crossref]

Santamaría, A.

Sasian, J.

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–242 (1972).

Scheiding, S.

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] [PubMed]

Shalaev, V. M.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

Song, X.

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

Straif, C.

Stumpf, D.

Surman, P.

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Thibault, S.

Thompson, K. P.

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

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] [PubMed]

Tünnermann, A.

Wang, D.

Wang, H.

Wang, K.

Wang, W.

Wang, Y.

Weckenmann, A.

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Wu, X.

Yang, T.

Yao, X.

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Zeitner, U. D.

Zentgraf, T.

L. Huang, H. Mühlenbernd, X. Li, X. Song, B. Bai, Y. Wang, and T. Zentgraf, “Broadband hybrid holographic multiplexing with geometric metasurfaces,” Adv. Mater. 27(41), 6444–6449 (2015).
[Crossref] [PubMed]

Zhan, A.

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref] [PubMed]

Zhang, G.

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Figures (7)

Fig. 1
Fig. 1 The polar ray grid for defining the feature rays of each field.
Fig. 2
Fig. 2 Propagation of one ray through the phase element. (a) Refractive flat surface. (b) Reflective flat surface.
Fig. 3
Fig. 3 Design steps of the preliminary construction stage of phase function for jth element. (a) Find the first data point P1 in the real 3D space and its projected point P1,�� in the phase function space. (b) When Pi and Pi,�� have been obtained, the “surface normal” Ni,�� at Pi,�� can be calculated. Calculate the distances from Pi to the intersections of the remaining rays with the phase element in the real 3D space. Find the shortest distance and the corresponding feature ray and intersection are defined as Ri+1 and Pi+1. (c) Find the nearest point Qi among all the used data points in real 3D space to Pi. The corresponding projected point in phase function space is Qi,��. Find the projected point Pi+1,�� of Pi+1 on the tangent plane of Qi,�� in the phase function space. (d) When all the data points as well as their surface normals in the phase function space have been calculated, fit the data points into a closed-form phase function considering both the coordinates and surface normals. The graph of the phase function is sketched here.
Fig. 4
Fig. 4 The schematic view of the intersections of one feature ray Ri with the phase plates in real 3D space.
Fig. 5
Fig. 5 Flowchart of the design process for the construction and iteration stage.
Fig. 6
Fig. 6 Preliminary construction stage and iteration stage of the design example. (a) Initial system using simple geometric planes. (b) System layout after the preliminary construction stage. (c) The change of σRMS with iterations. (d) Distortion grid of the system after iterations. (e) System layout after iterations.
Fig. 7
Fig. 7 Final design result of the example. (a) System layout. (b) MTF plot. (c) Distortion grid.

Equations (15)

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OPL( S i , E i )=n| P i S i |+n'| E i P i |+ mλ 2π ϕ( P i ),
[ x i local y i local z i local ]=[ 1 0 0 0 cosα sinα 0 sinα cosα ][ x i y i z i ].
OPL x =0, OPL y =0.
n ( x i S i,x ) | P i S i | +n ( z i S i,z ) | P i S i | z i ( x i , y i ) x i +n' ( x i E i,x ) | E i P i | +n' ( z i E i,z ) | E i P i | z i ( x i , y i ) x i mλ 2π [ ϕ( P i ) x i local ϕ( P i ) y i local z i ( x i , y i ) x i sinα]=0.
[n' r i,z 'n r i,z + mλ 2π ϕ( P i ) y i local sinα] z i ( x i , y i ) x i +[n' r i,x 'n r i,x mλ 2π ϕ( P i ) x i local ]=0.
[n' r i,z 'n r i,z + mλ 2π ϕ( P i ) y i local sinα] z i ( x i , y i ) y i +[n' r i,y 'n r i,y mλ 2π ϕ( P i ) y i local cosα]=0.
[n' r i,y 'n r i,y mλ 2π ϕ( P i ) y i local cosα] z i ( x i , y i ) x i [n' r i,x 'n r i,x mλ 2π ϕ( P i ) x i local ] z( x i , y i ) y i =0.
n'( N i × r i ')=n( N i × r i )+ mλ 2π [ N i ×Tϕ( P i )],
T=[ 1 0 0 0 cosα sinα 0 sinα cosα ].
n' r i 'n r i mλ 2π Tϕ( P i )=k N i ,
{ n' r i,x 'n r i,x mλ 2π ϕ x | P i =0 n' r i,y 'n r i,y mλ 2π ϕ y | P i cosα=ksinα n' r i,z 'n r i,z + mλ 2π ϕ y | P i sinα=kcosα .
[ mλ 2π 0 0 0 mλ 2π cosα sinα 0 mλ 2π sinα cosα ][ ϕ x | P i ϕ y | P i k ]=[ n' r i,x '+n r i,x n' r i,y '+n r i,y n' r i,z '+n r i,z ]
OPL( P i , I i,ideal )=n'| E i,1 * P i |+ n q | I i,ideal E i,q * |+ ξ=1 q1 n i | E i,ξ+1 * E i,ξ * | + mλ 2π ξ=1 q ϕ ξ ( E i,ξ * ) .
σ RMS = ζ=1 TR σ ζ 2 TR ,
ϕ(x,y)= A 2 y+ A 3 x 2 + A 5 y 2 + A 7 x 2 y+ A 9 y 3 + A 10 x 4 + A 12 x 2 y 2 + A 14 y 4 + A 16 x 4 y+ A 18 x 2 y 3 + A 20 y 5 + A 21 x 6 + A 23 x 4 y 2 + A 25 x 2 y 4 + A 27 y 6 ,

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