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

Tong Yang; Yingzhe Duan; Dewen Cheng; Yongtian Wang

doi:10.3788/AOS202141.0108001

Journals >Acta Optica Sinica >Volume 41 >Issue 1 >Page 0108001 > Article

Acta Optica Sinica
Vol. 41, Issue 1, 0108001 (2021)

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

Tong Yang^1,2, Yingzhe Duan^1,2, Dewen Cheng^1,2, and Yongtian Wang^1,2,*

Author Affiliations

¹School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China

²Beijing Engineering Research Center of Mixed Reality and Advanced Display, Beijing 100081, China

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DOI: 10.3788/AOS202141.0108001 Cite this Article Set citation alerts

Tong Yang, Yingzhe Duan, Dewen Cheng, Yongtian Wang. Freeform Imaging Optical System Design: Theories, Development, and Applications[J]. Acta Optica Sinica, 2021, 41(1): 0108001 Copy Citation Text

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Locations of footprints for central field and marginal field on different kinds of surfaces (considering there is even number of intermediate images between the surface and exit pupil). (a) Light beams use central area of the stop surface. The footprints of central and marginal fields are the same;(b) light beams use central area of the surface away from the stop;(c) light beams use off-axis section of stop surface. The footprints of central and marginal fields are the same; (d) light beams use off-axis section of the surface away from the stop

Fig. 1. Locations of footprints for central field and marginal field on different kinds of surfaces (considering there is even number of intermediate images between the surface and exit pupil). (a) Light beams use central area of the stop surface. The footprints of central and marginal fields are the same;(b) light beams use central area of the surface away from the stop;(c) light beams use off-axis section of stop surface. The footprints of central and marginal fields are the same; (d) light beams use off-axis section of the surface away from the stop

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Fig. 2. Generation methods of starting point for freeform imaging system design

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$Propagation of a ray at different types of surface. (a) Refractive surface; (b) reflective surface$

Fig. 3. Propagation of a ray at different types of surface. (a) Refractive surface; (b) reflective surface

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Fig. 4. Design examples of differential equation method and SMS method. (a) Design example using single freeform surface differential equation method;(b) design example using W-W method; (c) design example using two freeform surface differential equation method^[42]; (d) design example using SMS method^[48]; (e) design example using analytic method related to SMS method to realize the design of two freeform surfaces which control three wavefronts^[51]

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Fig. 5. Design of good starting point generated from initial planar structure for optimization using CI method

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Fig. 6. Sketch of process for starting point design of freeform imaging system using CI method

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Fig. 7. Design framework for the generation of starting points of freeform off-axis reflective system using neural network based machine-learning and related design examples^[79]

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Fig. 8. Examples of optimization strategies of freeform imaging systems

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Fig. 9. Early applications of freeform surface in imaging system. (a) Surface shape of progressive ophthalmic lenses; (b) Alvarez lens;(c) Polaroid SX-70 camera^[98-99]

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Fig. 10. Design examples of freeform off-axis reflective imaging systems. (a) Freeform off-axis three-mirror system with small F-number and single integrated primary-tertiary mirror element;(b) freeform off-axis four-mirror afocal telescope^[103]; (c) freeform off-axis three-mirror system with integrated primary-tertiary mirror element using the same surface equation^[81]; (d) freeform off-axis three-mirror system with special spherical package^[84]

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Fig. 11. Design examples of freeform head-mounted displayer using prisms. (a) Freeform head-mounted displayer with small F-number and large field of view^[80];(b) freeform head-mounted displayer using stitching prisms^[121]; (c) freeform head-mounted displayer with dual focal planes^[122]; (d) freeform light-field head-mounted displayer^[124]

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Fig. 12. Freeform see-through head-mounted displayer with combiner. (a) Design in Ref. [126]; (b) design in Ref. [128]

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$Waveguide type head-mounted displayer with freeform surface. (a) Geometric waveguide[131]; (b) diffractive waveguide[133]$

Fig. 13. Waveguide type head-mounted displayer with freeform surface. (a) Geometric waveguide^[131]; (b) diffractive waveguide^[133]

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Fig. 14. Examples of freeform imaging system. (a) Freeform varifocal panoramic objective^[138]; (b) freeform projection system with ultrashort throw ratio^[141]; (c) freeform deep ultraviolet lithography objective^[144]; (d) freeform prism for transverse image translation^[62]

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Surface shape	Mathematical expression
Toroid (X toroid for example)	z= $\frac{c_{y} y^{2} + S (2 - c_{y} S)}{1 + \sqrt[]{(1 - c_{y} {S)}^{2} - (c_{y} {y)}^{2}}}$ , where S= $\frac{c_{x} x^{2}}{1 + \sqrt[]{1 - (1 + k_{x}) c_{x}^{2} x^{2}}}$ + $\overset{p}{\sum_{i = 2}}$ A₂_ix²ⁱ
Anamorphic asphere	z= $\frac{c_{x} x^{2} + c_{y} y^{2}}{1 + \sqrt[]{1 - (1 + k_{x}) c_{x}^{2} x^{2} - (1 + k_{y}) c_{y}^{2} y^{2}}}$ + $\overset{p}{\sum_{i = 2}} A_{2 i} {[(1 - B_{2 i}) x^{2} + (1 + B_{2 i}) y^{2}]}^{i}$
XY polynomials surface	z= $\frac{c (x^{2} + y^{2})}{1 + \sqrt[]{1 - (1 + k) c^{2} (x^{2} + y^{2})}}$ + $\overset{p}{\sum_{i = 0}} \overset{p}{\sum_{j = 0}}$ A_i_,_jxⁱy^j, 1≤i+j≤p
Zernike polynomials surface	z= $\frac{c (x^{2} + y^{2})}{1 + \sqrt[]{1 - (1 + k) c^{2} (x^{2} + y^{2})}}$ + $\overset{p}{\sum_{j = 1}}$ C_jZ_j, where Z_j is the jth Zernike term
Q polynomials surface	z(ρ,θ)= $\frac{c ρ^{2}}{1 + \sqrt[]{1 - c^{2} ρ^{2}}}$ + $\frac{1}{\sqrt[]{1 - c^{2} ρ^{2}}} \{u^{2} (1 - u^{2}) \overset{N}{\sum_{n = 0}} a_{n}^{0} Q_{n}^{0} (u^{2})$ + $\overset{M}{\sum_{m = 1}} u^{m} \overset{M}{\sum_{m = 1}} [a_{n}^{m} \cos (mθ) + b_{n}^{m} \sin (mθ)] Q_{n}^{m} (u^{2})\}$ , where $Q_{n}^{m}$ (u²) can be found in Ref. [16]
Radial basis functionfreeform surface	z= $\frac{c (x^{2} + y^{2})}{1 + \sqrt[]{1 - (1 + k) c^{2} (x^{2} + y^{2})}}$ + $\overset{N}{\sum_{i = 1}}$ w_iφ(‖x-C_i‖)
NURBS	S(u,v)= $\frac{\overset{n}{\sum_{i = 0}} \overset{m}{\sum_{j = 0}} N_{i, p} (u) N_{j, q} (v) w_{i, j} P_{i, j}}{\overset{n}{\sum_{i = 0}} \overset{m}{\sum_{j = 0}} N_{i, p} (u) N_{j, q} (v) w_{i, j}}$ ,where $\{\begin{array}{l} N_{i, 0} (u) = \{\begin{array}{l} 1, u_{i} \leq u \leq u_{i + 1} \\ 0, other \end{array} \\ N_{i, p} (u) = \frac{u - u_{i}}{u_{i + p} - u_{i}} N_{i, p - 1} (u) + \frac{u_{i + p + 1} - u}{u_{i + p + 1} - u_{i + 1}} N_{i + 1, p - 1} (u), p \geq 1 \end{array}$

Table 1. Mathematical expressions of common freeform surface shape types

Surface type	Even number of immediate images betweenthe freeform surface and the exit pupil	Odd number of immediate images betweenthe freeform surface and the exit pupil
Reflective surfacelocated at the stop	W=- $\frac{2 n}{λ}$ C·Z(ρ)	W=- $\frac{2 n}{λ}$ C·Z(-ρ)
Refractive surfacelocated at the stop	W= $\frac{(n_{2} - n_{1})}{λ}$ C·Z(ρ)	W= $\frac{(n_{2} - n_{1})}{λ}$ C·Z(-ρ)
Reflective surface locatedaway from the stop	W=- $\frac{2 n}{λ}$ C·Z(ρ+Δh)	W=- $\frac{2 n}{λ}$ C·Z(-ρ+Δh)
Refractive surface locatedaway from the stop	W= $\frac{(n_{2} - n_{1})}{λ}$ C·Z(ρ+Δh)	W= $\frac{(n_{2} - n_{1})}{λ}$ C·Z(-ρ+Δh)

Table 2. Aberrations introduced by Zernike freeform surface term overlaid on different types of surfaces (light beams use central section of surface)

Design method		Pros	Cons
Partialdifferentialequation(PDE) method	Design method for singlefreeform surface^[34]	Easy to use, capable ofdesigning nonsymmetricsystems	Only one freeform surface can bedesigned and only one fieldpoint is considered
	Methods proposed byAndrew Hicks et al.^[35,44]and Hou et al.^[36]	Easy to use, capable ofdesigning nonsymmetricsystems	Only one or two freeform surfacescan be designed and only the chiefrays of different field pointsare considered
	W-W method^[38-41]	Easy to use, capable ofdesigningnonsymmetric systems	Only two freeform surfaces canbe designed and only a verysmall field-of-view is considered
	Methods proposed by Volatieret al.^[42-43]	Easy to use, capable ofdesigningnonsymmetric systems	Only two freeform surfacescan be designed and only a verysmall field-of-viewis considered
SimultaneousMultiple Surface(SMS) design method	SMS3D^[48]	Excellent control ofthe rays in M fieldsusing M surfaces	Having a restriction onthe number of fields consideredin the design process, mainlyused for coaxial systems
	Method proposed byDuerr et al.^[51-52]	Excellent control of therays in three fieldsusing two surfaces	Having a restrictionon the number of fields consideredin the design process, mainlyused for coaxial systems
	Method proposed byNie et al.^[53-54]	Considering multiple fields anddifferent pupil coordinates,working for nonsymmetricoptical systems	Only two surfacesare considered
Construction-Iteration(CI) method^[55-56]		Considering multiple fields,different pupil coordinates,and multiple surfaces,working for different kinds ofnonsymmetric optical systems	Numerical calculation will bevery difficult if there are a largenumber of surfaces betweenthe unknown freeform surface andthe image plane, time consuming

Table 3. Comparison of numerical solving methods based on direct point-by-point control of light rays

Tong Yang, Yingzhe Duan, Dewen Cheng, Yongtian Wang. Freeform Imaging Optical System Design: Theories, Development, and Applications[J]. Acta Optica Sinica, 2021, 41(1): 0108001

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