Method of confocal mirror design José Sasián José Sasián, “Method of confocal mirror design,” Opt. Eng. 58(1), 015101 (2019), doi: 10.1117/1.OE.58.1.015101. Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 13 Aug 2019 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Optical Engineering 58(1), 015101 (January 2019) Method of confocal mirror design José Sasián* University of Arizona, College of Optical Sciences, Tucson, Arizona, United States Abstract. We provide an overview of the method of confocal mirror design and report advances with respect to pupil imagery. Two real ray-based conditions, Y þ ¼ −Y − and ΔY ¼ 0, for the absence of linear astigmatism and field tilt are presented. One example illustrates the design of a system confocal of the object and image, and another illustrates the design of a system confocal of the pupils. Stop shifting formulas are provided. Three three- mirror anastigmatic systems further illustrate the method. © 2019 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.58.1.015101] Keywords: unsymmetrical systems; freeform surfaces; pupil aberrations; confocal imaging; plane symmetry; linear astigmatism; field tilt; three-mirror anastigmatic systems. Paper 181506 received Oct. 21, 2018; accepted for publication Dec. 6, 2018; published online Jan. 5, 2019. 1 Introduction receiving recently some attention3–6 as they provide useful The word confocal means having a common focus or foci. starting design points. In this paper, we review their theory, The design method of confocal mirror systems, and confocal report some advances pertaining to pupil imagery, present refractive systems, is very old. This method was used by a real ray-based condition for the absence of linear astigma- Cassegrain and Mersenne in their famous telescope mirror tism, discuss an ad hoc optical surface, and summarize configurations. However, little or nothing has been done a systematic design approach. Some design examples are provided. A reduced version of this paper has been previ- to advance the confocal design method when used for axially 7 symmetric systems. In the case of refractive systems, one ously published. advancement would be to combine lens elements where each lens element is aplanatic. Fourth-order theory shows 2 Review of Aberrations of a Plane Symmetric that the astigmatism of each element does not change System upon stop shifting. If all the lenses are made of the same The aberration function Wð~i; H;~ ρ~Þ for a plane symmetric glass then field curvature is corrected when astigmatism is system1 can be written as corrected. An aplanatic and confocal lens concatenation is conceptually simple and useful to lay out a starting lens ~ ~ ~ EQ-TARGET;temp:intralink-;sec2;326;397Wði; H;ρÞ design. X∞ In the case of nonaxially symmetric systems, the method k m n p q ¼ W 2kþnþp; ðH~ H~ Þ ðρ~ ρ~Þ ðH~ ρ~Þ ð~i H~ Þ ð~i ρ~Þ ; 2mþnþq; · · · · · of confocal design turns out to be a powerful method. One k;m;n;p;q n;p;q reason is that the method provides excellent starting design points. Another is that the method can be applied in a very systematic manner. Still another reason is that the surfaces where W2kþnþp;2mþnþq;n;p;q is the coefficient of a particular required have as a base surface an off-axis conic surface, aberration form defined by the integers k, m, n, p,andq. or off-axis Cartesian ovoid for refractive systems, which has The lower indices in the coefficients indicate the algebraic two foci where it can be optically tested. powers of H, ρ, cosðϕÞ, cosðχÞ, and cosðχ þ ϕÞ in a given Because of the potential innovation in optical systems, aberration term. The angle χ is between the vectors ~i and H~ , nonaxially symmetric systems using freeform surfaces are and the angle χ þ ϕ is between the vectors~i and ~ρ. By setting enjoying attention from the optical design community. the sum of the integers to 0, 1, 2,. , groups of aberrations Important classes of systems are those that are reflective are defined as shown in Table 3. The vector ~i defines the and have a plane of symmetry, as fabrication and alignment ~ρ are feasible. The question is how to design such reflective direction of plane symmetry, is the aperture vector at ~ plane symmetric systems. Although there are several the exit pupil, and H is the field vector at the object approaches to design optical systems, we highlight what plane. A ray in the system is uniquely defined by H~ and ~ρ, we call the method of confocal mirror design.1,2 In the and the aberration function provides the optical path for present context, a confocal system is defined as a mirror sys- every ray. tem that provides point images, surface after surface, along A study of Table 1 provides useful insights. The aberra- a selected ray called the optical axis ray (OAR). The field tion terms are divided into groups and in turn in subgroups center, the center of the stop, and the pupils are centered according to symmetry characteristics. Thus, the third group about the OAR. As shown in Fig. 1 for a two-mirror system, contains the primary aberrations of axially symmetric sys- point imaging along the OAR requires the surfaces to be tems as a subgroup, contains the aberrations of double- on-axis or off-axis conic sections. Confocal systems are plane symmetric systems as a subgroup, and a subgroup of aberrations for plane symmetric systems that are not *Address all correspondence to José Sasián, E-mail: jose.sasian@optics .arizona.edu 0091-3286/2019/$25.00 © 2019 SPIE Optical Engineering 015101-1 January 2019 • Vol. 58(1) Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 13 Aug 2019 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Sasián: Method of confocal mirror design axially or double-plane symmetric. Thus, the aberration properties of a plane symmetric system can be thought of as the superposition of the properties of axial, double plane, and plane symmetric systems. The correction of the aberra- tions of a given subgroup can be carried out using system properties according to subgroup symmetry. For example, using a freeform surface having axial Zα, double plane Zβ, or plane symmetry Zγ. Mathematically, these aspheric terms are Z ¼ αðρ~ ρ~Þ2; EQ-TARGET;temp:intralink-;sec2;326;653 α · Z ¼ βð~i ρ~Þ2; EQ-TARGET;temp:intralink-;sec2;326;623 β · Z ¼ γð~i ~ρÞð~ρ ~ρÞ; Fig. 1 (a) A two mirror confocal system consisting of two off-axis EQ-TARGET;temp:intralink-;sec2;326;594 γ · · elliptical mirrors. (b) The central ray is the OAR. where α, β, and γ are the aspheric coefficients. The coefficients of the first and second groups in Table 1 Table 1 Aberration terms of a plane symmetric system. are set to zero because piston terms do not degrade the image, and second-order optics predicts the image size and location. First group Because Table 1 has as subgroups the aberrations of axially symmetric systems, double-plane symmetric sys- W 00000 Piston tems, and plane symmetric systems, we can treat a plane symmetric system as the superposition of three systems Second group according to symmetry. In particular, we can associate an axially symmetric system to a plane symmetric system. ~ W 01001i · ~ρ Field displacement Then, the second-order properties of the plane symmetric ~ ~ system are given by the second-order properties of the asso- W 10010i · H Linear piston ciated axially symmetric system. Using the oblique power ~ ~ W 02000ρ · ρ Defocus 2 cosðIÞ ϕ ¼ ; W H~ ~ρ EQ-TARGET;temp:intralink-;sec2;326;411 oblique 11100 · Magnification rs W H~ H~ 20000 · Quadratic piston for each surface in the system, we then can use any of the standard methods for axially symmetric systems to calculate Third group second-order properties of the plane symmetric system. For ~ 2 example, the focal lengths, and the image position and size. W 02002ði · ρ~Þ Uniform astigmatism In the oblique power, I is the angle of incidence of the OAR ~ ~ ~ W 11011ði · HÞði · ~ρÞ Anamorphic distortion in the surface and rs is the sagittal radius of the surface at the 2 OAR intersection. In what follows, we will not discuss the W ð~i H~ Þ 20020 · Quadratic piston subgroup of aberrations that belong to axially symmetric ~ systems as these are well understood. These aberrations W 03001ði · ~ρÞð~ρ · ~ρÞ Uniform coma depend only on the rotational invariants ðH~ · H~ Þ, ðH~ · ρ~Þ, ~ ~ ~ ~ W 12101ði · ρÞðH · ρÞ Linear astigmatism and ðρ~ · ρ~Þ. ~ ~ For the case of a confocal reflective system that is plane W 12010ði · HÞð~ρ · ~ρÞ Field tilt symmetric,1 some of the coefficients in the third group are ~ ~ ~ W 21001ði · ~ρÞðH · HÞ Quadratic distortion I zero as shown in Table 2. Field tilt is proportional to linear W ð~i H~ ÞðH~ ~ρÞ astigmatism, and provided there is enough surface optical 21110 · · Quadratic distortion II power, one surface tilt can be used to correct for both linear ~ ~ ~ ~ W 30010ði · HÞðH · HÞ Cubic piston astigmatism and field tilt simultaneously. Furthermore, the coefficient for quadratic distortion II is zero. Except for 2 W 04000ðρ~ · ρ~Þ Spherical aberration quadratic distortion I, if left uncorrected, a confocal system may act as an axially symmetric system to the fourth-order of W ðH~ ~ρÞð~ρ ~ρÞ 13100 · · Linear coma approximation. Thus, a confocal reflective system provides ~ 2 W 22200ðH · ρ~Þ Quadratic astigmatism an excellent starting design point. In a similar manner to the derivation of the image aberra- ~ ~ ~ ~ W 22000ðH · HÞðρ · ρÞ Field curvature tion function, and since the entrance and exit pupils are opti- ~ ~ ~ W 31100ðH · HÞðH · ~ρÞ Cubic distortion cally conjugated, we can derive a pupil aberration function ¯ ~ ~ 2 Wði; H;~ρÞ for a plane symmetric system and write it to W ðH~ H~ Þ 40000 · Quartic piston fourth-order as Optical Engineering 015101-2 January 2019 • Vol.