Analytical solution of an afocal two freeform mirror design problem

We investigate a new afocal two freeform mirror design problem in first order optics. The resulting first-order partial differential equations for the freeform two mirror system have an analytic solution with the sole condition that the x-y and x'-y' axes are parallel. Two selected solutions are presented. One of them is semiaplanatic (fulfilling the aplanatic condition only for the x-coordinates), while the other is, to our knowledge, the first example of an aplanatic two-mirror system without rotational symmetry. OCIS codes: (080.2720) Mathematical methods (general); (080.2740) Geometric optical design; (080.3620) Lens system design; (080.4225) Nonspherical lens design; (080.4298) Nonimaging optics. References 1. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999). 2. K. Schwarzschild, Untersuchungenzur geometrischen optik I-III (W. Fr. Kaestner, 1905-1906), Vols. 1-3. 3. A. K. Head, "The two-mirror aplanat," Proc. Phys. Soc. London Sec. B 70(10), 945-949 (1957). 4. J. J. Braat and P. F. Greve, "Aplanatic optical system containing two aspheric surfaces," Appl. Opt. 18(13), 2187-2191 (1979). 5. D. Lynden-Bell, "Exact optics: a unification of optical telescope design," Mon. Not. R. Astron. Soc. 334(4), 787-796 (2002). 6. R. V. Willstrop and D. Lynden-Bell, "Exact optics II. Exploration of designs onand off-axis," Mon. Not. R. Astron. Soc. 342(1), 33-49 (2003). 7. W. T. Welford, "The most general isoplanatism theorem," Opt. Commun. 3(1), 1-6 (1971). 8. J. C. Miflano, P. Benitez, and B. Narasimhan, "Freeform aplanatic systems as a limiting case of SMS," Opt. Express 24(12), 13173-13178 (2016). 9. B. Narasimhan, P. Benitez, J. C. Miflano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, "Design of three freeform mirror aplanat," Proc. SPIE 9579, 95790K (2015). 10. M. Herzberger, Modern Geometrical Optics (Interscience, 1958) pp. 152. 11. V. Oliker, "Optical design of freeform two-mirror beam-shaping systems," J. Opt. Soc. Am. 24(12), 3741-3752 (2007). 12. R. J. Koshel, Illumination engineering: Design with Nonimaging Optics (Wiley-IEEE Press, 2013) pp. 83. 13. B. D. Stone and G. W. Forbes, "Second-order design methods for definitive studies of plane-symmetric, twomirror systems," J. Opt. Soc. Am. A 11(12), 3292-3307 (1994).


Introduction
Optical systems that are free from spherical aberration and circular coma are known as aplanatic systems [1].Aplanatic here refers to full aplanatic, i.e. those aberrations vanish for all orders, and therefore they satisfy the Abbe sine condition.The first rotationally symmetric two mirror aplanatic systems were described by Schwarzschild in 1905 [2] and were later developed and analytically deduced for telescopes which required superior aberration-free imaging [3][4][5][6].One particular aplanatic design is the afocal case, in which the null spherical aberration implies that the plane input wavefront WF in is coupled to a plane output wavefront WF out stigmatically.The Abbe sine condition in its general form [7] states that for this afocal case the mapping of the rays between the two wavefronts is not arbitrary, rather that the transformation between Cartesian coordinates x-y defined on WF in and Cartesian coordinates x '-y' defined on WF ouh given by: x' = a 00 (x,y) y' = b 00 (x,y) must be linear (the sub-indices are used for consistency with equations below).
We will use here the term semi-aplanatic for optical systems in which only one of the functions in Eq. ( 1) is linear.Since the coordinate systems on the wavefronts can be rotated, without loss of generality it can be considered that the semi-aplanatic case corresponds to the case in which a 00 is a linear function of x only, that is,x' = m x x + l x (where m x and l x are real numbers), which can be viewed as a one-dimensional Abbe sine condition.
In this paper we will first address a more general afocal design problem in first order optics, in which the ray-to-ray mapping between WF in and WF out is such that a 00 is a function only of x, and not necessarily linear.Section 2 is devoted to deriving the first-order angular magnification of the afocal system for this case, using the étendue conservation theorem of bi-parametric ray bundles in phase space.In Section 3 the design problem of two afocal freeform mirrors fulfilling such a ray-to-ray mapping is stated and the resulting first-order partial differential equations are solved in a closed-form.Two specific design examples are given in Section 4, one of them being, to our knowledge, the first example of an aplanatic freeform (i.e, non-rotationally symmetric) two-mirror system.The apparent contradiction with the general rule that three freeform surfaces are needed [8][9] is discussed in Section 5.

Phase space considerations
Let us consider the plane wavefronts, WF in and WF out , which are in general non-parallel, and two Cartesian coordinate systems, x-y-z and x '-y '-z', such that WF in and WF out coincide with the planes z = 0 and z' = 0, respectively and propagate towards positive z and z' values.
Applying étendue conservation for bi-parametric bundles as done in [8], we can deduce that the coefficients a 00 , b (M , c 10 , c M , d\ 0 and d 0 \ fulfill: where the subindices x and y denote partial derivation.Both matrices on the left hand side of Eq. ( 3) must therefore be non-singular, in particular: We are interested in analyzing the implications of imposing the condition that a 00 is a function only of x, i.e. a 00 (x, y) = a 00 (x).Thus, a 00y = 0 and from Eq. ( 4) it follows that b 00y ^ 0. Hence, expressing Eq. ( 3) as a set of 4 separate equations, we obtain: á 00 (x)c 0l (x,y) + KA x >y).
b oo y ( x ,y) (5) where the dot on top denotes derivation of a one-variable function.Equation ( 2) now becomes: p+ á 00 (x)b 00y (x,y) -q + .Koy^y) Particularly, we conclude that if ÜQQ IS 3. function of x only, a 00 (x), then q' must be independent on the first order of p (since d l0 = 0), and that when q = 0, the ratio of the direction cosines p' and p (which indicates the first-order angular magnification) equals the inverse of the derivative of a 00 (x).The ray-to-ray mapping between WF in and WF out is visualized by making/» = q = 0 in Eq. ( 6), so we obtain that also p' = q' = 0 and that: -Q 00 \X) (7) y' = b 00 (x,y) If we apply these results to the semi-aplanatic case, in which a 00 = m x x + l x , we obtain, as expected, that such magnification is the constant \lm x , independent of x.To simplify the nomenclature, in what follows we will call a 0 o(i) = a{x) and b (M (x,y) = Y(x,y).

Particular two-mirror solution
Consider now the particular case where the two coordinate systems have parallel axes as shown in Fig. 1.We are going to design an afocal two-mirror system such that the ray of WF in that impinge on mirror S\ at the point P will be reflected towards point Q of mirror S 2 , and reflected there again to become a ray of WF out , such that the x-coordinate of Q will only depend on the x-coordinate of P. Therefore, in the coordinate system x-y-z we can write the equations of the two mirror surfaces in parametric form as P = (x, y, z(x,y)) and Q = (a(x), Y(x,y), Z(x,y)), where x and>> act as parameters for both surfaces.First, the optical path length between WF in and WF out must be constant, and thus: where D is the constant distance between WF in and WF out in Fig. 1 and Let us now apply a reflection law on the first surface.It can be computed in the standard vector form or, alternatively, as the derivative of the optical path length with respect to two independent parameters.The latter, with Herzberger notation [10], is written as: where vectors s and s' are unit vectors, the primes denoting output space, n, n' are the refractive indices, r, r are the position vectors, E is the optical path length between the position vectors, and u is the parameter.Thus, for the reflection on the first mirror, taking r' = (x,y, z), r = (x, y, 0), E = z and parameter x first we have: ) Eliminating L from Eq. ( 8) and Eq. ( 11), the partial derivative of z with respect to x is: The analog to Eq. ( 11) for y as a parameter is given by: Eliminating L again with Eq. ( 8), the partial derivative of z with respect to y is: Equations ( 8), ( 12) and (14) form the system to be solved with unknown functions z, Y and Z. Integrating Eq. ( 12): and z 0 (y) is an arbitrary function.Note that z(0,y) = z 0 (y), so this 0 function describes the profile of the cross section of mirror Si with the plane x = 0, and can be seen as a boundary condition for the PDE system.Substituting Eq. (15) in Eq. ( 14): x, y) is a function of y only, which we call b(y).Finally, from Eq. ( 8)-( 9) we get: and substituting Eq. ( 16): Equations ( 15), ( 16) and (18) describe the two mirror analytic solution to the problem in which the ray mapping between WF in and WF out satisfies x' = a(x), independently of y, with the only condition that the x-y axes and x'-y' axes are parallel.We have obtained in the process that this solution also satisfies y' = b(y), independently of x (while in the general case of Eq. ( 7), y' may depend on x too).Therefore this afocal two-mirror system has a decoupled set of ray-to-ray mapping equations of x-variables and ^-variables, with full symmetry in between them (although our initial condition was asymmetric).
This resulting two-mirror system of Eq. ( 15), ( 16) and ( 18) also has relevant nonimaging properties.When an irradiance distribution is defined on the wavefronts, the energy transfer is controlled by the Jacobian matrix of the mapping function Eq. ( 1), which for this two mirrors is a diagonal matrix with a \x) and b '(y) as diagonal elements.This nonimaging problem was in fact treated in [11] for irradiance beam shaping when the prescription has rectangular symmetry (although restricted to the case in which the diagonal elements of the Jacobian matrix have equal sign).Notice that the energy transfer problem leads to a second-order Monge-Ampére PDE, while the mapping-contrained problem stated here is of first order only.
-x + - The parabolic dependence of Eq. ( 19) with respect to x implies that in this semi-aplanatic case Si is plane-symmetric with respect to the x = constant plane passing through the parabola vertex (coinciding with x = 0 when l x = 0), and it is easy to see that S 2 has the same symmetry.

Design examples
Figure 2 shows as a first example an afocal freeform semi-aplanatic system that performs the ray-to-ray mappingX= 2x/3 and Y = -0A6y 3 + 2.2y -4.2.Integrating Eq. ( 16) we get: Surfaces Si and S 2 are given by Eq. ( 19) and (20).The specific design in Fig. 2 has the parameter C-D = 3 and the arbitrary integration constant K = -\.In the previous example, the boundary condition z 0 (y) given in Eq. ( 21) is a fourth order polynomial, which leads through Eq. ( 16) to a cubic mapping of the y coordinates.If we consider next a parabolic boundary condition z 0 (y), Eq. ( 16) will imply a linear mapping of the y coordinates.In Section 3, we found that in this afocal two-mirror y' = b(y), independently of x, so that linear mapping is fulfilled not only along the cross section x = 0, but for all x and y values.Thus, the semi-aplanatic system with parabolic boundary condition is a full aplanatic system.Taking y' = m y y + l y (where m y and l y are real numbers), then: When m x = m y it is easy to see that the mirrors are two confocal paraboloids of revolution with axis parallel to the z axis.However, the magnification m x and m y can be different, as was already discussed in [8] and in the design of afocal étendue-squeezing microlens arrays in [12], and the resulting mirrors are parabolic-type quadrics without rotational symmetry.From the parabolic dependence of z 0 (y) in Eq. ( 23) with respect to y it is easy to deduce that in this aplanatic case Si and S 2 are not only plane-symmetric with respect to a plane x = constant but also with respect to a plane y = constant (coinciding withy = 0 when l y = 0).Plane-symmetric two-mirror systems were already studied in [13], restricted to off-axis rotationally symmetric conies with sharp imagery through second order in both spatial and angular coordinates (while the spatial coordinates are not limited is the present work).
Figure 3 shows a specific example of such an afocal aplanatic two mirror design for parameters C -D = 10, K = 3, l x = 0, l y = 0 and magnifications m x = -2 and m y = 2/3.The mirrors are hyperbolic paraboloids with axes parallel to the z axis.

Discussion
The second selected design shown in Section 4 (Fig. 3) is, to our knowledge, the first example of an aplanatic system made of only two freeform (non-rotationally symmetric) mirrors.This seems to contradict that in general at least three freeform optical surfaces are necessary to design an aplanatic system [8].However, the necessity of three freeform surfaces for the general case does not impede that there are particular cases in which fewer surfaces can provide an aplanatic design.In fact, apart from the trivial example of the flat mirror, two other particular cases are already known: the two-mirror rotational case mentioned in the introduction and the aplanatic points of the refractive sphere (using a single surface) [1].
Although in general forcing that an afocal system has a mapping such that x' is a function of x only does not impose any condition on>-', when the x-y axes are parallel to the x'-y' axes and two mirrors are used, y' results to be a function of y only.As a consequence, the twomirror semi-aplanatic system was automatically aplanatic.If there are other semi-aplanatic configurations in which this also occurs it is an open problem that deserves further research.

Conclusions
An analytic solution of first-order partial differential equations for an afocal two freeform mirror design problem was presented.If the ray-to-ray mapping between a selected input and output stigmatic wavefront pair is such that the output x-coordinate of a ray is a function of only the x-coordinate of its corresponding input ray (independently of its ^-coordinate), the ycoordinate of the output ray also results in being a function of only the ^-coordinate of the output ray in these two-mirror systems.Furthermore, for a semi-aplanatic case, choosing a specific boundary condition, a full freeform two-mirror aplanat is achieved, which is to our knowledge the first reported full aplanatic two-mirror system without rotational symmetry.The systems found in this paper have multiple applications; as beam shaping [11], étendue squeezing [12] and anamorphic imaging systems.

Fig. 1 .
Fig. 1.Two mirror afocal system that transforms WF^to WF C

Fig. 2 .
Fig. 2. Semi-aplanatic afocal two freeform mirror system obtained analytically presented in 3D (a) and the 2D cross section with the plane x = 0 (b).