ゴンザレスの論文を発見
https://pdfs.semanticscholar.org/54fb/e0cc15627647688f16a38a7b349fbeb308ab.pdf?_ga=2.33575969.1419438563.1562672304-https://pdfs.semanticscholar.org/54fb/e0cc15627647688f16a38a7b349fbeb308ab.pdf?_ga=2.33575969.1419438563.1562672304-98427616.1549287961
解析式を発見したということは、そもそもシミュレーションなら色収差はできたとも行っているのだろうか？
Rigorous analytical formula for freeform singlet lens design free of spherical aberration
Rafael G. Gonza ́lez-Acun ̃a1,∗, H ́ector A. Chaparro-Romo2 and Julio C. Guti ́errez-Vega1
1 Photonics and Mathematical Optics Group,
Tecnol ́ogico de Monterrey, Monterrey 64849, M ́exico 2UNAM, Av. Universidad 3000, Cd. Universitaria, Coyoaca ́n, 04510 Ciudad de M ́exico, M ́exico.
email * rafael123.90@hotmail.com Abstract
An analytical closed-form formula for the design of freeform lenses free of spherical aberration is presented. Given the equation of the freeform input surface, the formula gives the equation of the second surface in order to correct the spherical aberration. The derivation is based on the formal application of the variational Fermat principle under the standard geometrical optics approximation.
1 Introduction
Freeform optics involves the design of optical elements with at least one surface which has no translational or rotational symmetry about a propagation axis. In recent years, the topic has gained increasing popularity in the optics commu- nity, partly because of the rapid development of new computing technologies and the emergence of potential applications. In general, the design of freeform elements has combined theoretical approximation methods with brute-force optimiza- tion techniques leading to a diversity of results and methodologies which have proved to be useful for particular cases [Bauer et al., 2018, Yang et al., 2017]. For instance, G. W. Forbes [Forbes, 2012, Forbes, 2013, Forbes, 2010, Forbes, 2007] described freeform surfaces based on a set of characteristic polynomials for non rotationally symmetric systems. Re- cently the generation of freeform mirrors have been studied [Muslimov et al., 2017, Bauer and Rolland, 2015], which con- siderably reduce the optical aberrations. The theory of aberration of freeform optics has been developed by several au- thors [Ochse, 2018, Fuerschbach et al., 2014, Zhong and Gross, 2018, Zhong and Gross, 2017] applying numerical optimiza- tion schemes.
In this paper, we introduce a closed-form expression for the design of freeform singlets lenses free of spherical aberration, which is a continuation of our work [Gonz ́alez-Acun ̃a and Chaparro-Romo, 2018, Gonz ́alez-Acun ̃a and Guiti ́errez-Vega, 2018]. The formula gives the exact analytical equation of the output surface given the arbitrary freeform expresion of the input surface in order to correct the spherical aberration introduced by the first surface. The derivation is fully analytical based on the formal application of the variational Fermat principle under the standard geometrical optics approximation. In the process of deriving the formula, we apply a design methodology free of numerical optimization strategies. We illustrate the applicability and robustness of the formula by showing some representative design examples using very sophisticated input functions that have no been used before in optical design. As far as we know, this exact formula has not been reported before in the optical design literature.
1 Introduction
Freeform optics involves the design of optical elements with at least one surface which has no translational or rotational symmetry about a propagation axis. In recent years, the topic has gained increasing popularity in the optics commu- nity, partly because of the rapid development of new computing technologies and the emergence of potential applications. In general, the design of freeform elements has combined theoretical approximation methods with brute-force optimiza- tion techniques leading to a diversity of results and methodologies which have proved to be useful for particular cases [Bauer et al., 2018, Yang et al., 2017]. For instance, G. W. Forbes [Forbes, 2012, Forbes, 2013, Forbes, 2010, Forbes, 2007] described freeform surfaces based on a set of characteristic polynomials for non rotationally symmetric systems. Re- cently the generation of freeform mirrors have been studied [Muslimov et al., 2017, Bauer and Rolland, 2015], which con- siderably reduce the optical aberrations. The theory of aberration of freeform optics has been developed by several au- thors [Ochse, 2018, Fuerschbach et al., 2014, Zhong and Gross, 2018, Zhong and Gross, 2017] applying numerical optimiza- tion schemes.
In this paper, we introduce a closed-form expression for the design of freeform singlets lenses free of spherical aberration, which is a continuation of our work [Gonz ́alez-Acun ̃a and Chaparro-Romo, 2018, Gonz ́alez-Acun ̃a and Guiti ́errez-Vega, 2018]. The formula gives the exact analytical equation of the output surface given the arbitrary freeform expresion of the input surface in order to correct the spherical aberration introduced by the first surface. The derivation is fully analytical based on the formal application of the variational Fermat principle under the standard geometrical optics approximation. In the process of deriving the formula, we apply a design methodology free of numerical optimization strategies. We illustrate the applicability and robustness of the formula by showing some representative design examples using very sophisticated input functions that have no been used before in optical design. As far as we know, this exact formula has not been reported before in the optical design literature.
2 Analytical design of freeform singlet free of spherical aberration
We assume that the singlet lens is a lossless and homogeneous optical element with relative refractive index n and axial thickness T, see Fig. 1. Its input surface is known and it is described by the freeform function za(xa,ya), where the subindex a refers to the coordinates on the input surface. The shape of the output surface is unknown and it is described by the function zb(xb,yb) to be determined, where the subindex b refers to the coordinates on the output surface. We will further assume that the normal vector of the input surface at the optical axis points out in direction z, i.e. the normal is perpendicular to the tangent plane of the input surface at the origin.
無損失の材料はないのでこれがネックになるか？