Freeform Imaging Optical System Design: Theories, Development, and Applications

Tong Yang; Yingzhe Duan; Dewen Cheng; Yongtian Wang

doi:10.3788/AOS202141.0108001

[1] Thompson K P, Rolland J P. Freeform optical surfaces: a revolution in imaging optical design[J]. Optics & Photonics News, 23, 30-35(2012).

[2] Wills S. Freeform optics: notes from the revolution[J]. Optics & Photonics News, 28, 34-41(2017). http://adsabs.harvard.edu/abs/2017OptPN..28...34W

[3] Reimers J, Bauer A, Thompson K P et al. Freeform spectrometer enabling increased compactness[J]. Light: Science & Applications, 6, e17026(2017).

[4] Wu R M, Feng Z X, Zheng Z R et al. Design of freeform illumination optics[J]. Laser & Photonics Reviews, 12, 1700310(2018). http://onlinelibrary.wiley.com/doi/10.1002/lpor.201700310/abstract

[5] Wu R M, Xu L, Liu P et al. Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation[J]. Optics Letters, 38, 229-231(2013).

[6] Liu Z J, Liu P, Yu F H. Parametric optimization method for the design of high-efficiency free-form illumination system with a LED source[J]. Chinese Optics Letters, 10, 112201(2012). http://www.opticsjournal.net/Articles/Abstract?aid=OJ120914000005bIeKhN

[7] Wang K, Chen F, Liu Z Y et al. Design of compact freeform lens for application specific light-emitting diode packaging[J]. Optics Express, 18, 413-425(2010).

[8] Feng Z X, Cheng D W, Wang Y T. Transferring freeform lens design into phase retrieval through intermediate irradiance transport[J]. Optics Letters, 44, 5501-5504(2019). http://www.ncbi.nlm.nih.gov/pubmed/31730093

[9] Wu R, Yang L, Ding Z et al. Precise light control in highly tilted geometry by freeform illumination optics[J]. Optics Letters, 44, 2887-2890(2019).

[10] Fang F Z, Zhang X D, Weckenmann A et al. Manufacturing and measurement of freeform optics[J]. CIRP Annals, 62, 823-846(2013). http://www.sciencedirect.com/science/article/pii/S0007850613001935

[11] Lee R W B, To S S, Cheung B C F[M]. Design, machining and measurement technologies of ultra-precision freeform optics(2015).

[12] Wang Y T. Ray-tracing formulae for optical surfaces of unusual shape[J]. Optical Technique, 16, 2-8(1990).

[13] Cheng D W. Study on design methods of freeform imaging systems and their application in head-mounted displays[D]. Beijing: Beijing Institute of Technology(2011).

[14] Zernike F. Inflection theory of the cutting method and its improved form, the phase contrast method[J]. Physica, 1, 689-704(1934).

[15] Forbes G W. Characterizing the shape of freeform optics[J]. Optics Express, 20, 2483-2499(2012).

[16] Forbes G W. Shape specification for axially symmetric optical surfaces[J]. Optics Express, 15, 5218-5226(2007).

[17] Broemel A, Lippmann U, Gross H. Freeform surface descriptions. Part I: mathematical representations[J]. Advanced Optical Technologies, 6, 327-336(2017). http://adsabs.harvard.edu/abs/2017AdOT....6..327B

[18] Ye J F, Chen L, Li X H et al. Review of optical freeform surface representation technique and its application[J]. Optical Engineering, 56, 110901(2017). http://proceedings.spiedigitallibrary.org/journals/optical-engineering/volume-56/issue-11/110901/Review-of-optical-freeform-surface-representation-technique-and-its-application/10.1117/1.OE.56.11.110901.full

[19] Cakmakci O, Moore B, Foroosh H et al. Optimal local shape description for rotationally non-symmetric optical surface design and analysis[J]. Optics Express, 16, 1583-1589(2008).

[20] Piegl L, Tiller W. The NURBS book[M]. 2nd ed. Berlin: Springer(1997).

[21] Hopkins H H[M]. Wave theory of aberrations(1950).

[22] Buchroeder R A. Tilted-component telescopes. Part I: theory[J]. Applied Optics, 9, 2169-2171(1970). http://www.ncbi.nlm.nih.gov/pubmed/20094216

[23] Thompson K P, Schmid T, Cakmakci O et al. Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry[J]. Journal of The Optical Society of America A-optics Image Science and Vision, 26, 1503-1517(2009).

[24] Thompson K. Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry[J]. Journal of the Optical Society of America A, 22, 1389-1401(2005).

[25] Thompson K P. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration[J]. Journal of the Optical Society of America A, 26, 1090-1100(2009).

[26] Thompson K P. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations[J]. Journal of the Optical Society of America A, 27, 1490-1504(2010).

[27] Thompson K P. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations[J]. Journal of the Optical Society of America A, 28, 821-836(2011).

[28] Schmid T, Rolland J P, Rakich A et al. Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT)[J]. Optics Express, 18, 17433-17447(2010).

[29] Fuerschbach K, Rolland J P, Thompson K P. Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces[J]. Optics Express, 20, 20139-20155(2012).

[30] Fuerschbach K, Rolland J P, Thompson K P. Theory of aberration fields for general optical systems with freeform surfaces[J]. Optics Express, 22, 26585-26606(2014). http://europepmc.org/abstract/med/25401809

[31] Yang T, Zhu J, Jin G F. Nodal aberration properties of coaxial imaging systems using Zernike polynomial surfaces[J]. Journal of the Optical Society of America A, 32, 822-836(2015).

[32] Yang T, Cheng D W, Wang Y T. Aberration analysis for freeform surface terms overlay on general decentered and tilted optical surfaces[J]. Optics Express, 26, 7751-7770(2018).

[33] Ju G H, Yan C X, Gu Z Y et al. Computation of astigmatic and trefoil figure errors and misalignments for two-mirror telescopes using nodal-aberration theory[J]. Applied Optics, 55, 3373-3386(2016).

[34] Zhu J, Wu X F, Yang T et al. Generating optical freeform surfaces considering both coordinates and normals of discrete data points[J]. Journal of the Optical Society of America A, 31, 2401-2408(2014).

[35] Andrew Hicks R. Controlling a ray bundle with a free-form reflector[J]. Optics Letters, 33, 1672-1674(2008).

[36] Hou J, Li H F, Zheng Z R et al. Distortion correction for imaging on non-planar surface using freeform lens[J]. Optics Communications, 285, 986-991(2012). http://www.sciencedirect.com/science/article/pii/S0030401811013769

[37] Wassermann G D, Wolf E. On the theory of aplanatic aspheric systems[J]. Proceedings of the Physical Society Section B, 62, 2-8(1949). http://www.onacademic.com/detail/journal_1000038180094410_1f2a.html

[38] Mahajan V N[M]. Optical imaging and aberrations(1998).

[39] Vaskas E M. Note on the Wasserman-Wolf method for designing aspheric surfaces[J]. Journal of the Optical Society of America, 47, 669-670(1957).

[40] Knapp D J. Conformal optical design[D]. Tucson: The University of Arizona(2002).

[41] Cheng D W, Wang Y T, Hua H. Free form optical system design with differential equations[J]. Proceedings of SPIE, 7849, 78490Q(2010). http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1348136

[42] Volatier J, Druart G. Differential method for freeform optics applied to two-mirror off-axis telescope design[J]. Optics Letters, 44, 1174-1177(2019). http://www.researchgate.net/publication/331285284_Differential_method_for_freeform_optics_applied_to_two-mirror_off-axis_telescope_design

[43] Volatier J B, Duveau L, Druart G. An exploration of the freeform two-mirror off-axis solution space[J]. Journal of Physics: Photonics, 2, 014004(2019). http://www.researchgate.net/publication/337566181_An_exploration_of_the_freeform_two-mirror_off-axis_solution_space

[44] Andrew Hicks R, Croke C. Designing coupled free-form surfaces[J]. Journal of the Optical Society of America A, 27, 2132-2137(2010).

[45] Minano J C, Gonzalez J C. New method of design of nonimaging concentrators[J]. Applied Optics, 31, 3051-3060(1992).

[46] Minano J C, Benitez P, Gonzalez J C. RX: a nonimaging concentrator[J]. Applied Optics, 34, 2226-2235(1995).

[47] Winston R, Miñano J C, Benitez P G. Nonimaging optics[M]. Amsterdam: Elsevier(2005).

[48] Miñano J C, Benítez P, Lin W et al. An application of the SMS method for imaging designs[J]. Optics Express, 17, 24036-24044(2009).

[49] Lin W, Benítez P, Miñano J C et al. Advances in the SMS design method for imaging optics[J]. Proceedings of SPIE, 8167, 81670M(2011). http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1266902

[50] Duerr F, Benitez P, Minano J C et al. Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles[J]. Optics Express, 20, 5576-5585(2012).

[51] Duerr F, Benitez P, Minano J C et al. Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces[J]. Optics Express, 20, 10839-10846(2012). http://www.ncbi.nlm.nih.gov/pubmed/22565708

[52] Duerr F, Meuret Y, Thienpont H. Potential benefits of free-form optics in on-axis imaging applications with high aspect ratio[J]. Optics Express, 21, 31072-31081(2013).

[53] Nie Y F, Thienpont H, Duerr F. Multi-fields direct design approach in 3D: calculating a two-surface freeform lens with an entrance pupil for line imaging systems[J]. Optics Express, 23, 34042-34054(2015).

[54] Nie Y F, Mohedano R, Benitez P et al. Multifield direct design method for ultrashort throw ratio projection optics with two tailored mirrors[J]. Applied Optics, 55, 3794-3800(2016).

[55] Yang T, Zhu J, Hou W et al. Design method of freeform off-axis reflective imaging systems with a direct construction process[J]. Optics Express, 22, 9193-9205(2014).

[56] Yang T, Zhu J, Wu X F et al. Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method[J]. Optics Express, 23, 10233-10246(2015).

[57] Yang T, Zhu J, Jin G F. Starting configuration design method of freeform imaging and afocal systems with a real exit pupil[J]. Applied Optics, 55, 345-353(2016).

[58] Hou W, Zhu J, Yang T et al. Construction method through forward and reverse ray tracing for a design of ultra-wide linear field-of-view off-axis freeform imaging systems[J]. Journal of Optics, 17, 055603(2015). http://www.ingentaconnect.com/content/iop/jopt2/2015/00000017/00000005/art055603

[59] Yang T, Jin G F, Zhu J. Design of image-side telecentric freeform imaging systems based on a point-by-point construction-iteration process[J]. Chinese Optics Letters, 15, 062202(2017). http://www.opticsjournal.net/Articles/Abstract?aid=OJ1703230001341w4z7C

[60] Yang T, Zhu J, Jin A G. Design of a freeform, dual fields-of-view, dual focal lengths, off-axis three-mirror imaging system with a point-by-point construction-iteration process[J]. Chinese Optics Letters, 14, 100801(2016). http://www.opticsjournal.net/Articles/Abstract?aid=OJ161008000005z7C0Fb

[61] Wu X F, Jin G F, Zhu J. Freeform illumination design model for multiple light sources simultaneously[J]. Applied Optics, 56, 2405-2411(2017).

[62] Wu X F, Zhu J, Yang T et al. Transverse image translation using an optical freeform single lens[J]. Applied Optics, 54, E55-E62(2015). http://www.osapublishing.org/ao/abstract.cfm?uri=ao-54-28-E55

[63] Yang T, Cheng D W, Wang Y T. Freeform imaging spectrometer design using a point-by-point design method[J]. Applied Optics, 57, 4718-4727(2018).

[64] Duan Y Z, Yang T, Cheng D W et al. Design method for nonsymmetric imaging optics consisting of freeform-surface-substrate phase elements[J]. Optics Express, 28, 1603-1620(2020). http://www.researchgate.net/publication/338155744_Design_method_for_nonsymmetric_imaging_optics_consisting_of_freeform-surface-substrate_phase_elements

[65] Zhong Y, Gross H. Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and nodal aberration theory[J]. Optics Express, 25, 10016-10030(2017).

[66] Cao C, Liao S, Liao Z et al. Initial configuration design method for off-axis reflective optical systems using nodal aberration theory and genetic algorithm[J]. Optical Engineering, 58, 105101(2019). http://www.researchgate.net/publication/336616010_Initial_configuration_design_method_for_off-axis_reflective_optical_systems_using_nodal_aberration_theory_and_genetic_algorithm

[67] Papa J C, Howard J M, Rolland J P. Three-mirror freeform imagers[J]. Proceedings of SPIE, 1069, 106901D(2018). http://www.researchgate.net/publication/325916860_Three-mirror_freeform_imagers

[68] Papa J C, Howard J M, Rolland J P. Starting point designs for freeform four-mirror systems[J]. Optical Engineering, 57, 101705(2018).

[69] Sasian J M. How to approach the design of a bilateral symmetric optical system[J]. Optical Engineering, 33, 2045-2062(1994).

[70] Chang S. Linear astigmatism of confocal off-axis reflective imaging systems with N-conic mirrors and its elimination[J]. Journal of the Optical Society of America A, 32, 852-859(2015).

[71] Papa J C, Howard J M, Rolland J P. Automatic solution space exploration for freeform optical design. [C]∥Optical Design and Fabrication 2019 (Freeform, OFT), June 10-12, 2019, Washington, D.C.: OSA, FM4B, 1(2019).

[72] Bauer A, Schiesser E M, Rolland J P. Starting geometry creation and design method for freeform optics[J]. Nature Communications, 9, 1756(2018). http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5931519/

[73] Wetherell W B. -12-23[P]. Womble D A. All-reflective three element objective: US4240707.(1980).

[74] Horisaki R, Takagi R, Tanida J. Deep-learning-generated holography[J]. Applied Optics, 57, 3859-3863(2018).

[75] Rivenson Y, Zhang Y B, Günaydın H et al. Phase recovery and holographic image reconstruction using deep learning in neural networks[J]. Light: Science & Applications, 7, 17141(2018). http://www.zhangqiaokeyan.com/academic-journal-cn_light-science-applications-english_thesis/0201232260981.html

[76] Jin K H. McCann M T, Froustey E, et al. Deep convolutional neural network for inverse problems in imaging[J]. IEEE Transactions on Image Processing, 26, 4509-4522(2017).

[77] Cao Z Y, Guo N, Li M H et al. Back propagation neutral network based signal acquisition for Brillouin distributed optical fiber sensors[J]. Optics Express, 27, 4549-4561(2019). http://www.researchgate.net/publication/330978794_Back_propagation_neutral_network_based_signal_acquisition_for_Brillouin_distributed_optical_fiber_sensors

[78] Zuo C, Feng S J, Zhang X et al. Deep learning based computational imaging: status, challenges, and future[J]. Acta Optica Sinica, 40, 0111003(2020).

[79] Yang T, Cheng D W, Wang Y T. Direct generation of starting points for freeform off-axis three-mirror imaging system design using neural network based deep-learning[J]. Optics Express, 27, 17228-17238(2019). http://www.researchgate.net/publication/333632295_Direct_generation_of_starting_points_for_freeform_off-axis_three-mirror_imaging_system_design_using_neural_network_based_deep-learning

[80] Cheng D W, Wang Y T, Hua H et al. Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism[J]. Applied Optics, 48, 2655-2668(2009). http://www.ncbi.nlm.nih.gov/pubmed/19424386

[81] Yang T, Zhu J, Jin A G. Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface[J]. Chinese Optics Letters, 14, 060801(2016). http://www.opticsjournal.net/Articles/Abstract?aid=OJ160523000063fLhOkR

[82] Fischer R E, Tadic-Galeb B, Yoder P R. Optical system design[M]. New York: SPIE(2008).

[83] Liu C, Gross H. Numerical optimization strategy for multi-lens imaging systems containing freeform surfaces[J]. Applied Optics, 57, 5758-5768(2018). http://www.researchgate.net/publication/326233074_Numerical_optimization_strategy_for_multi-lens_imaging_systems_containing_freeform_surfaces

[84] Fuerschbach K, Rolland J P, Thompson K P. A new family of optical systems employing φ-polynomial surfaces[J]. Optics Express, 19, 21919-21928(2011).

[85] Bauer A, Rolland J P. Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics[J]. Optics Express, 23, 28141-28153(2015).

[86] Bauer A, Pesch M, Muschaweck J et al. All-reflective electronic viewfinder enabled by freeform optics[J]. Optics Express, 27, 30597-30605(2019). http://www.researchgate.net/publication/336351301_All-reflective_electronic_viewfinder_enabled_by_freeform_optics

[87] Cheng D W, Wang Y T, Hua H. Automatic image performance balancing in lens optimization[J]. Optics Express, 18, 11574-11588(2010).

[88] Houllier T, Lepine T. Comparing optimization algorithms for conventional and freeform optical design[J]. Optics Express, 27, 18940-18957(2019). http://www.ncbi.nlm.nih.gov/pubmed/31684590

[89] Chrisp M. Method of. -10-08[P]. system for optimizing NURBS surfaces for an imaging system: US-10437943.(2019).

[90] Chrisp M P, Primeau B, Echter M A. Imaging freeform optical systems designed with NURBS surfaces[J]. Optical Engineering, 55, 071208(2016).

[91] Yang T, Jin G F, Zhu J. Automated design of freeform imaging systems[J]. Light: Science & Applications, 6, e17081(2017).

[92] Xu C, Cheng D W, Wang Y T. Automatic obscuration elimination for off-axis mirror systems[J]. Applied Optics, 56, 9014-9022(2017). http://europepmc.org/abstract/MED/29131187

[93] Cai D Y, Gross H. Obscuration elimination in three-dimensional nonsymmetrical optical systems[J]. Journal of Physics: Photonics, 1, 044002(2019). http://www.researchgate.net/publication/334752592_obscuration_elimination_in_three-dimensional_nonsymmetrical_optical_systems

[94] Xu C, Lai X M, Cheng D W et al. Automatic optical path configuration variation in off-axis mirror system design[J]. Optics Express, 27, 15251-15261(2019). http://www.ncbi.nlm.nih.gov/pubmed/31163723

[95] Trumper I, Aftab M, Kim D W. Freeform surface selection based on parametric fitness function using modal wavefront fitting[J]. Optics Express, 27, 6815-6831(2019).

[98] Plummer W T. Free-form optical components in some early commercial products[J]. Proceedings of SPIE, 5865, 586509(2005).

[99] Plummer W T. Unusual optics of the Polaroid SX-70 Land camera[J]. Applied Optics, 21, 196-202(1982).

[100] Rogers J R. Aberrations of unobscured reflective optical systems[D]. Tucson: The University of Arizona(1983).

[101] Bottema M. Reflective correctors for the Hubble Space Telescope axial instruments[J]. Applied Optics, 32, 1768-1774(1993). http://www.opticsinfobase.org/ao/abstract.cfm?uri=ao-32-10-1768

[102] Zhu J, Hou W, Zhang X D et al. Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view[J]. Journal of Optics, 17, 015605(2015).

[103] Beier M, Hartung J, Peschel T et al. Development, fabrication, and testing of an anamorphic imaging snap-together freeform telescope[J]. Applied Optics, 54, 3530-3542(2015). http://www.opticsinfobase.org/ao/abstract.cfm?uri=ao-54-12-3530

[104] Fuerschbach K, Thompson K P, Rolland J P. Interferometric measurement of a concave, φ-polynomial, Zernike mirror[J]. Optics Letters, 39, 18-21(2014).

[105] Fuerschbach K, Davis G E, Thompson K P et al. Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors[J]. Optics Letters, 39, 2896-2899(2014). http://adsabs.harvard.edu/abs/2014OptL...39.2896F%201

[106] Muslimov E, Hugot E, Jahn W et al. Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture[J]. Optics Express, 25, 14598-14610(2017). http://www.onacademic.com/detail/journal_1000040493390610_cacc.html

[107] Zhang X J, Xue D L, Li M et al. Designing, fabricating, and testing freeform surfaces for space optics[J]. Proceedings of SPIE, 8838, 88380N(2013).

[108] Meng Q Y, Wang H Y, Wang K J et al. Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror[J]. Applied Optics, 55, 8962-8970(2016).

[109] Meng Q Y, Wang H Y, Liang W J et al. Design of off-axis three-mirror systems with ultrawide field of view based on an expansion process of surface freeform and field of view[J]. Applied Optics, 58, 609-615(2019).

[110] Wu W C, Jin G F, Zhu J. Optical design of the freeform reflective imaging system with wide rectangular FOV and low F-number[J]. Results in Physics, 15, 102688(2019). http://www.sciencedirect.com/science/article/pii/S2211379719323551

[111] Jahn W, Ferrari M, Hugot E. Innovative focal plane design for large space telescope using freeform mirrors[J]. Optica, 4, 1188-1195(2017).

[112] Chen L, Gao Z S, Ye J F et al. Construction method through multiple off-axis parabolic surfaces expansion and mixing to design an easy-aligned freeform spectrometer[J]. Optics Express, 27, 25994-26013(2019). http://www.ncbi.nlm.nih.gov/pubmed/31510461

[113] Howard J, West G, Trumper I et al. -09-07)[2020-05-09]. https:∥www.zhangqiaokeyan.com/ntis-science-report_other_thesis/020717857.html.(2015).

[114] Feng L, Zhou J S, Wei L D et al. Design of a compact wide-spectrum double-channel prism imaging spectrometer with freeform surface[J]. Applied Optics, 57, 9512-9522(2018). http://www.ncbi.nlm.nih.gov/pubmed/30462000

[115] Zhang J L, Lin C, Ji Z H et al. Design of a compact hyperspectral imaging spectrometer with a freeform surface based on anastigmatism[J]. Applied Optics, 59, 1715-1725(2020). http://www.researchgate.net/publication/338441684_Design_of_a_compact_hyperspectral_imagingspectrometer_with_freeform_surface_based_onanastigmatism

[116] Xu C, Cheng D W, Chen J J et al. Design of all-reflective dual-channel foveated imaging systems based on freeform optics[J]. Applied Optics, 55, 2353-2362(2016). http://www.opticsinfobase.org/ao/upcoming_pdf.cfm?id=253717

[117] Zhu J, Zhang B Q, Hou W et al. Design of an oblique camera based on a field-dependent parameter[J]. Applied Optics, 58, 5650-5655(2019). http://www.researchgate.net/publication/334521687_design_of_an_oblique_camera_based_on_a_field-dependent_parameter

[118] Liu X Y, Gong T T, Jin G F et al. Design method for assembly-insensitive freeform reflective optical systems[J]. Optics Express, 26, 27798-27811(2018).

[119] Deng Y T, Jin G F, Zhu J. Design method for freeform reflective-imaging systems with low surface-figure-error sensitivity[J]. Chinese Optics Letters, 17, 092201(2019). http://www.opticsjournal.net/Articles/Abstract?aid=OJ6bf5850d28450cc0

[120] Hua H, Hu X D, Gao C Y. A high-resolution optical see-through head-mounted display with eyetracking capability[J]. Optics Express, 21, 30993-30998(2013).

[121] Cheng D W, Wang Y T, Hua H et al. Design of a wide-angle, lightweight head-mounted display using free-form optics tiling[J]. Optics Letters, 36, 2098-2100(2011).

[122] Cheng D W, Wang Q F, Wang Y T et al. Lightweight spatial-multiplexed dual focal-plane head-mounted display using two freeform prisms[J]. Chinese Optics Letters, 11, 031201(2013). http://www.opticsjournal.net/Articles/Abstract?aid=OJ130224000001cJfLiO

[123] Hu X D, Hua H. High-resolution optical see-through multi-focal-plane head-mounted display using freeform optics[J]. Optics Express, 22, 13896-13903(2014).

[124] Song W T, Wang Y T, Cheng D W et al. Light field head-mounted display with correct focus cue using micro structure array[J]. Chinese Optics Letters, 12, 060010(2014). http://www.opticsjournal.net/Articles/Abstract?aid=OJ140530000013qXtZw3

[125] Huang H K, Hua H. High-performance integral-imaging-based light field augmented reality display using freeform optics[J]. Optics Express, 26, 17578-17590(2018).

[126] Zheng Z R, Liu X, Li H F et al. Design and fabrication of an off-axis see-through head-mounted display with an x-y polynomial surface[J]. Applied Optics, 49, 3661-3668(2010). http://www.ncbi.nlm.nih.gov/pubmed/20648131

[127] Li H, Zhang X, Wang C et al. Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions[J]. Optics Communications, 309, 121-126(2013). http://www.sciencedirect.com/science/article/pii/S0030401813006184

[128] Wilson A, Hua H. Design and demonstration of a vari-focal optical see-through head-mounted display using freeform Alvarez lenses[J]. Optics Express, 27, 15627-15637(2019).

[129] Pan J W. Che-Wen C A, Huang K D, et al. Demonstration of a broad band spectral head-mounted display with freeform mirrors[J]. Optics Express, 22, 12785-12798(2014).

[130] Bauer A, Rolland J P. Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays[J]. Optics Express, 22, 13155-13163(2014).

[131] Cheng D W, Wang Y T, Xu C et al. Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics[J]. Optics Express, 22, 20705-20719(2014).

[132] Wang Q W, Cheng D W, Hou Q C et al. Stray light and tolerance analysis of an ultrathin waveguide display[J]. Applied Optics, 54, 8354-8362(2015). http://www.ncbi.nlm.nih.gov/pubmed/26479609

[133] Han J, Liu J, Yao X C et al. Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms[J]. Optics Express, 23, 3534-3549(2015).

[134] Yang J M, Twardowski P, Gerard P et al. Design of a large field-of-view see-through near to eye display with two geometrical waveguides[J]. Optics Letters, 41, 5426-5429(2016).

[135] Wei S L, Fan Z C, Zhu Z B et al. Design of a head-up display based on freeform reflective systems for automotive applications[J]. Applied Optics, 58, 1675-1681(2019). http://www.researchgate.net/publication/331285439_Design_of_a_head-up_display_based_on_freeform_reflective_systems_for_automotive_applications?_sg=faFSzv_rjncLBB87vfCQxQs6ZQNIzzlmWT5vbx4HKwLH12BhXtaWs9Gy0WW7O2wHcVN4QWRyT4psnPo

[136] Qin Z, Lin S, Luo K et al. Dual-focal-plane augmented reality head-up display using a single picture generation unit and a single freeform mirror[J]. Applied Optics, 58, 5366-5374(2019). http://www.researchgate.net/publication/334205598_Dual-focal-plane_augmented_reality_head-up_display_using_a_single_picture_generation_unit_and_a_single_freeform_mirror

[137] Gu L, Cheng D W, Wang Q W et al. Design of a uniform-illumination two-dimensional waveguide head-up display with thin plate compensator[J]. Optics Express, 27, 12692-12709(2019). http://www.ncbi.nlm.nih.gov/pubmed/31052807

[138] Ma T, Yu J C, Liang P et al. Design of a freeform varifocal panoramic optical system with specified annular center of field of view[J]. Optics Express, 19, 3843-3853(2011).

[139] laser scanning unit: US7852566[P]. -12-14. Shih B Y. Single F-theta lens used for micro-electro mechanical system, MEMS(2010).

[140] Hirata K. -03-02[P]. Yatsu M. Projection-type image display apparatus: US7670009.(2010).

[141] Yu B H, Tian Z H, Su D Q et al. Design and engineering verification of an ultrashort throw ratio projection system with a freeform mirror[J]. Applied Optics, 58, 3575-3581(2019). http://www.ncbi.nlm.nih.gov/pubmed/31044857

[142] Mann H J. -12-17[P]. Shafer D. Imaging optical system, projection exposure installation for microlithography with an imaging optical system of this type: US8610877.(2013).

[143] Liu Y, Li Y, Cao Z. Design method of off-axis extreme ultraviolet lithographic objective system with a direct tilt process[J]. Optical Engineering, 54, 075102(2015). http://proceedings.spiedigitallibrary.org/journals/OE/volume-54/issue-07/075102/Design-method-of-off-axis-extreme-ultraviolet-lithographic-objective-system/10.1117/1.OE.54.7.075102.full

[144] Mao S S, Li Y Q, Jiang J H et al. Design of a hyper-numerical-aperture deep ultraviolet lithography objective with freeform surfaces[J]. Chinese Optics Letters, 16, 030801(2018). http://www.opticsjournal.net/Articles/Abstract?aid=OJ180315000077gMjPlS

[145] Mao S S, Li Y Q, Liu K et al. Optical design of high numerical aperture extreme ultraviolet lithography objective with freeform surfaces[J]. Infrared and Laser Engineering, 48, 0814002(2019).

[146] Brückner A[M]. Multiaperture cameras, 191-250(2013).

[147] Dunkel J, Wippermann F, Bruckner A et al. Laser lithographic approach to micro-optical freeform elements with extremely large sag heights[J]. Optics Express, 20, 4763-4775(2012).

[148] Li L, Yi A Y. Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera[J]. Applied Optics, 51, 1843-1852(2012). http://www.researchgate.net/publication/224846352_Design_and_fabrication_of_a_freeform_microlens_array_for_a_compact_large-field-of-view_compound-eye_camera

[149] Pang K, Fang F Z, Song L et al. Bionic compound eye for 3D motion detection using an optical freeform surface[J]. Journal of the Optical Society of America B, 34, B28(2017).

[150] Pang K. Study on design and application of optical imaging systems based on micro lens arrays[D]. Tianjin: Tianjin University(2017).

[151] Li H, Naples N J, Zhao X et al. An integrated approach to design and fabrication of a miniature endoscope using freeform optics[J]. Advanced Optical Technologies, 5, 335-342(2016).

[152] Chang C W, Sun H Y, Horng C T et al. Progressive rear-view mirror for motorcycles[J]. Optics Express, 24, 29283-29294(2016).

[153] Ohde H, Nagata T. Optical design, fabrication, and evaluation of optical systems using free-shaped prism[J]. Proceedings of SPIE, 6834, 68340K(2007). http://cpfd.cnki.com.cn/Article/CPFDTOTAL-ZGGY200711013020.htm

[154] Nie Y F, Gross H, Zhong Y et al. Freeform optical design for a nonscanning corneal imaging system with a convexly curved image[J]. Applied Optics, 56, 5630-5638(2017).

[155] He S F, Meng Y, Gong M L. Freeform lens design to eliminate retroreflection for optical systems[J]. Applied Optics, 57, 1218-1224(2018). http://www.ncbi.nlm.nih.gov/pubmed/29469867

[156] Yoon C, Bauer A, Xu D et al. Absolute linear-in-k spectrometer designs enabled by freeform optics[J]. Optics Express, 27, 34593-34602(2019). http://www.researchgate.net/publication/337191360_Absolute_linear-in-k_spectrometer_designs_enabled_by_freeform_optics