Appendix
In this Appendix, we will explain transfer equations and refraction equations used for skew ray tracing in an optical system containing a freeform surface and the derivation of their differentiations.
At the end of skew ray tracing, the various physical quantities defined at the beginning of Sect. 3.1 are known. The following relations exist among them:
$$x_{k + 1} = x_{k} + L_{k} X_{k} ,$$
(105)
$$y_{k + 1} = y_{k} + L_{k} Y_{k} ,$$
(106)
$$z_{k + 1} = z_{k} + L_{k} Z_{k} - t_{k} ,$$
(107)
$$\varepsilon_{zk + 1} = \left\{ {1 + \left( {\partial f_{k + 1} /\partial x_{k + 1} } \right)^{2} + \left( {\partial f_{k + 1} /\partial y_{k + 1} } \right)^{2} } \right\}^{ - 1/2} ,$$
(108)
$$\varepsilon_{xk + 1} = - \left( {\partial f_{k + 1} /\partial x_{k + 1} } \right)\varepsilon_{zk + 1} ,$$
(109)
$$\varepsilon_{yk + 1} = - \left( {\partial f_{k + 1} /\partial y_{k + 1} } \right)\varepsilon_{zk + 1} ,$$
(110)
$$\xi_{k + 1} = X_{k} \varepsilon_{xk + 1} + Y_{k} \varepsilon_{yk + 1} + Z_{k} \varepsilon_{zk + 1} ,$$
(111)
$$\mu_{k + 1} = n_{k} /n_{k + 1} ,$$
(112)
$$\xi '_{k + 1} = \left\{ {1 - \mu_{k + 1}^{2} \left( {1 - \xi_{k + 1}^{2} } \right)} \right\}^{1/2} ,$$
(113)
$$g_{k + 1} = \xi '_{k + 1} - \mu_{k + 1} \xi_{k + 1} ,$$
(114)
$$X_{k + 1} = \mu_{k + 1} X_{k} + g_{k + 1} \varepsilon_{xk + 1} ,$$
(115)
$$Y_{k + 1} = \mu_{k + 1} Y_{k} + g_{k + 1} \varepsilon_{yk + 1} ,$$
(116)
$$Z_{k + 1} = \mu_{k + 1} Z_{k} + g_{k + 1} \varepsilon_{zk + 1} .$$
(117)
Equations (105)–(107) are called transfer equations, and Eqs. (115)–(117) are called refraction equations. Those equations mean that an intersection point (x
k+1, y
k+1, z
k+1) of a ray and surface k+1 and the exit direction cosines (X
k+1, Y
k+1, Z
k+1) from surface k+1 are expressed as functions of the following values: the intersection point (x
k
, y
k
, z
k
) of the ray and surface k; direction cosines (X
k
, Y
k
, Z
k
); distance between surfaces t
k
; refractive indices n
k
and n
k+1 of media anterior and posterior to surface k+1, respectively; and the group of parameters c
k
, p
k
, c
k+1, and p
k+1 that express the shapes of surfaces k and k+1. However, not all those values are independent variables; the following relations must be met, as well:
$$z_{k} = f_{k} \left( {x_{k} ,y_{k} ;c_{k} ,\varvec{p}_{k} } \right),$$
(118)
$$z_{k + 1} = f_{k + 1} \left( {x_{k + 1} ,y_{k + 1} ;c_{k + 1} ,\varvec{p}_{k + 1} } \right),$$
(119)
$$X_{k}^{2} + Y_{k}^{2} + Z_{k}^{2} = 1,$$
(120)
$$X_{k + 1}^{2} + Y_{k + 1}^{2} + Z_{k + 1}^{2} = 1.$$
(121)
If we eliminate z
k
, z
k+1, Z
k
, and Z
k+1, using the above relational expressions, we can eventually express x
k+1, y
k+1, X
k+1, and Y
k+1, as follows:
$$x_{k + 1} = x_{k + 1} \left( {x_{k} ,y_{k} ,X_{k} ,Y_{k} ;c_{k} ,\varvec{p}_{k} ;t_{k} ;c_{k + 1} ,\varvec{p}_{k + 1} } \right),$$
(122)
$$y_{k + 1} = y_{k + 1} \left( {x_{k} ,y_{k} ,X_{k} ,Y_{k} ;c_{k} ,\varvec{p}_{k} ;t_{k} ;c_{k + 1} ,\varvec{p}_{k + 1} } \right),$$
(123)
$$X_{k + 1} = X_{k + 1} \left( {x_{k + 1} ,y_{k + 1} ,X_{k} ,Y_{k} ;n_{k} ;c_{k + 1} ,\varvec{p}_{k + 1} ;n_{k + 1} } \right),$$
(124)
$$Y_{k + 1} = Y_{k + 1} \left( {x_{k + 1} ,y_{k + 1} ,X_{k} ,Y_{k} ;n_{k} ;c_{k + 1} ,\varvec{p}_{k + 1} ;n_{k + 1} } \right).$$
(125)
Equations (122) and (123) and Eqs. (124) and (125) are symbolic expressions of transfer equations and refraction equations, respectively.
Differentiation with respect to the parameters c, p, t, and n of an optical system may be the objects of interest in differential ray tracing, but here those parameters are assumed to be fixed. Then differentiation of Eqs. (122) and (123) with respect to x
k, y
k
, X
k
, and Y
k
, and of Eqs. (124) and (125) with respect to x
k+1, y
k+1, X
k
, and Y
k
will be calculated.
When Eqs. (105), (106), (107), (118), (119), and (120) are differentiated,
$$\delta x_{k + 1} = \delta x_{k} + L_{k} \delta X_{k} + X_{k} \delta L_{k} ,$$
(126)
$$\delta y_{k + 1} = \delta y_{k} + L_{k} \delta Y_{k} + Y_{k} \delta L_{k} ,$$
(127)
$$\delta z_{k + 1} = \delta z_{k} + L_{k} \delta Z_{k} + Z_{k} \delta L_{k} ,$$
(128)
$$\delta z_{k} = \left( {\partial f_{k} /\partial x_{k} } \right)\delta x_{k} + \left( {\partial f_{k} /\partial y_{k} } \right)\delta y_{k} ,$$
(129)
$$\delta z_{k + 1} = \left( {\partial f_{k + 1} /\partial x_{k + 1} } \right)\delta x_{k + 1} + \left( {\partial f_{k + 1} /\partial y_{k + 1} } \right)\delta y_{k + 1} ,$$
(130)
$$X_{k} \delta X_{k} + Y_{k} \delta Y_{k} + Z_{k} \delta Z_{k} = 0.$$
(131)
After eliminating δz
k
, δz
k+1, δZ
k
, δL
k
from Eqs. (126)–(131), making variable substitutions and modifying the presentation of the resulted equations using Eqs. (109)–(111), the following equations will be derived:
$$\begin{aligned} \delta x_{k + 1} &= \left[ { 1 + X_{k} \xi_{k + 1}^{ - 1} \varepsilon_{zk}^{ - 1} \left( {\varepsilon_{zk + 1} \varepsilon_{xk} - \varepsilon_{xk + 1} \varepsilon_{zk} } \right) } \right] \delta x_{k} \hfill \\ &\quad+ \left[ { X_{k} \xi_{k + 1}^{ - 1} \varepsilon_{zk}^{ - 1} \left( {\varepsilon_{zk + 1} \varepsilon_{yk} - \varepsilon_{yk + 1} \varepsilon_{zk} } \right) } \right] \delta y_{k} \hfill \\ &\quad+ \left[ { L_{k} \left\{ {1 + X_{k} \xi_{k + 1}^{ - 1} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right)} \right\} } \right] \delta X_{k} \hfill \\ &\quad+ \left[ { L_{k} X_{k} \xi_{k + 1}^{ - 1} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right) } \right] \delta Y_{k} , \hfill \\ \end{aligned}$$
(132)
$$\begin{aligned} \delta y_{k + 1} &= \left[ { Y_{k} \xi_{k + 1}^{ - 1} \varepsilon_{zk}^{ - 1} \left( {\varepsilon_{zk + 1} \varepsilon_{xk} - \varepsilon_{xk + 1} \varepsilon_{zk} } \right) } \right] \delta x_{k} \hfill \\ &\quad+ \left[ { 1 + Y_{k} \xi_{k + 1}^{ - 1} \varepsilon_{zk}^{ - 1} \left( {\varepsilon_{zk + 1} \varepsilon_{yk} - \varepsilon_{yk + 1} \varepsilon_{zk} } \right) } \right] \delta y_{k} \hfill \\ &\quad+ \left[ { L_{k} Y_{k} \xi_{k + 1}^{ - 1} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right) } \right] \delta X_{k} \hfill \\ &\quad+ \left[ { L_{k} \left\{ {1 + Y_{k} \xi_{k + 1}^{ - 1} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right)} \right\} } \right] \delta Y_{k} . \hfill \\ \end{aligned}$$
(133)
As (ε
xk+1, ε
yk+1, ε
zk+1) is a unit normal vector,
$$e_{xk + 1}^{2} + e_{yk + 1}^{2} + e_{zk + 1}^{2} = 1.$$
(134)
When Eqs. (108), (109), (110), (111), (113), (114), (115), (116), and (134) are differentiated,
$$\begin{aligned} \delta \varepsilon_{zk + 1} &= \varepsilon_{zk + 1}^{2} \left\{ {\varepsilon_{xk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1}^{2} } \right) + \varepsilon_{yk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right)} \right\}\delta x_{k + 1} \hfill \\ &\quad+ \varepsilon_{zk + 1}^{2} \left\{ {\varepsilon_{xk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right) + \varepsilon_{yk + 1} \left( {\partial^{2} f_{k + 1} /\partial y_{k + 1}^{2} } \right)} \right\}\delta y_{k + 1} , \hfill \\ \end{aligned}$$
(135)
$$\begin{aligned} \delta \varepsilon_{xk + 1} &= - \left( {\partial f_{k + 1} /\partial x_{k + 1} } \right)\delta \varepsilon_{zk + 1} - \varepsilon_{zk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1}^{2} } \right)\delta x_{k + 1} \hfill \\ &\quad - \varepsilon_{zk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right)\delta y_{k + 1} , \hfill \\ \end{aligned}$$
(136)
$$\begin{aligned} \delta \varepsilon_{yk + 1} &= - \left( {\partial f_{k + 1} /\partial y_{k + 1} } \right)\delta \varepsilon_{zk + 1} - \varepsilon_{zk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right)\delta x_{k + 1} \hfill \\ &\quad - \varepsilon_{zk + 1} \left( {\partial^{2} f_{k + 1} /\partial y_{k + 1}^{2} } \right)\delta y_{k + 1} , \hfill \\ \end{aligned}$$
(137)
$$\delta \xi_{k + 1} = X_{k} \delta \varepsilon_{xk + 1} + Y_{k} \delta \varepsilon_{yk + 1} + Z_{k} \delta \varepsilon_{zk + 1} + \varepsilon_{xk + 1} \delta X_{k} + \varepsilon_{yk + 1} \delta Y_{k} + \varepsilon_{zk + 1} \delta Z_{k} ,$$
(138)
$$\delta \xi_{k + 1}^{{\prime }} = \xi_{k + 1}^{{{\prime } - 1}} \mu_{k + 1}^{2} \xi_{k + 1} \delta \xi_{k + 1} ,$$
(139)
$$\delta g_{k + 1} = \delta \xi_{k + 1}^{{\prime }} - \mu_{k + 1} \delta \xi_{k + 1} ,$$
(140)
$$\delta X_{k + 1} = \mu_{k + 1} \delta X_{k} + g_{k + 1} \delta \varepsilon_{xk + 1} + \varepsilon_{xk + 1} \delta g_{k + 1} ,$$
(141)
$$\delta Y_{k + 1} = \mu_{k + 1} \delta Y_{k} + g_{k + 1} \delta \varepsilon_{yk + 1} + \varepsilon_{yk + 1} \delta g_{k + 1} ,$$
(142)
$$\varepsilon_{xk + 1} \delta \varepsilon_{xk + 1} + \varepsilon_{yk + 1} \delta \varepsilon_{yk + 1} + \varepsilon_{zk + 1} \delta \varepsilon_{zk + 1} = 0.$$
(143)
After eliminating δε
xk+1, δε
yk+1, δε
zk+1, δg
k+1, δξ
k+1, \(\delta \xi_{k + 1}^{{\prime }}\), δZ
k
from Eqs. (131) and (135)–(143), then variable substitution and modification of the presentation of the resulting equations using the following equations,
$$\rho_{xxk + 1} = \left( {\varepsilon_{yk + 1}^{2} + \varepsilon_{zk + 1}^{2} } \right)\left( {\partial^{2} f_{k + 1} /\partial x_{k + 1}^{2} } \right) - \varepsilon_{xk + 1} \varepsilon_{yk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right),$$
(144)
$$\rho_{xyk + 1} = \left( {\varepsilon_{xk + 1}^{2} + \varepsilon_{zk + 1}^{2} } \right)\left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right) - \varepsilon_{xk + 1} \varepsilon_{yk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1}^{2} } \right),$$
(145)
$$\rho_{yxk + 1} = \left( {\varepsilon_{yk + 1}^{2} + \varepsilon_{zk + 1}^{2} } \right)\left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right) - \varepsilon_{xk + 1} \varepsilon_{yk + 1} \left( {\partial^{2} f_{k + 1} /\partial y_{k + 1}^{2} } \right),$$
(146)
$$\rho_{yyk + 1} = \left( {\varepsilon_{xk + 1}^{2} + \varepsilon_{zk + 1}^{2} } \right)\left( {\partial^{2} f_{k + 1} /\partial y_{k + 1}^{2} } \right) - \varepsilon_{xk + 1} \varepsilon_{yk + 1} \left( {\partial^{2} f_{k + 1} /\partial x_{k + 1} \partial y_{k + 1} } \right),$$
(147)
these equations will be derived:
$$\begin{aligned} \delta X_{k + 1} &= [ g_{k + 1} \left\{ {\mu_{k + 1} \varepsilon_{xk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right) - \varepsilon_{zk + 1} } \right\}\rho_{xxk + 1} \hfill \\ &\quad+ g_{k + 1} \mu_{k + 1} \varepsilon_{xk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right)\rho_{xyk + 1} ] \delta x_{k + 1} \hfill \\ &\quad+ [ g_{k + 1} \left\{ {\mu_{k + 1} \varepsilon_{xk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right) - \varepsilon_{zk + 1} } \right\}\rho_{yxk + 1} \hfill \\ &\quad + g_{k + 1} \mu_{k + 1} \varepsilon_{xk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right)\rho_{yyk + 1} ] \delta y_{k + 1} \hfill \\ &\quad + \left[ { \mu_{k + 1} \left\{ {1 + g_{k + 1} \varepsilon_{xk + 1} \xi_{k + 1}^{{{\prime - 1}}} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right)} \right\} } \right] \delta X_{k} \hfill \\ &\quad + \left[ { \mu_{k + 1} g_{k + 1} \varepsilon_{xk + 1} \xi_{k + 1}^{{{\prime - 1}}} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right) } \right] \delta Y_{k} , \hfill \\ \end{aligned}$$
(148)
$$\begin{aligned} \delta Y_{k + 1} &= [ g_{k + 1} \left\{ {\mu_{k + 1} \varepsilon_{yk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right) - \varepsilon_{zk + 1} } \right\}\rho_{xyk + 1} \hfill \\ &\quad + g_{k + 1} \mu_{k + 1} \varepsilon_{yk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right)\rho_{xxk + 1} ] \delta x_{k + 1} \hfill \\ &\quad + [ g_{k + 1} \left\{ {\mu_{k + 1} \varepsilon_{yk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right) - \varepsilon_{zk + 1} } \right\}\rho_{yyk + 1} \hfill \\ &\quad + g_{k + 1} \mu_{k + 1} \varepsilon_{yk + 1} \xi_{k + 1}^{{{\prime - 1}}} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right)\rho_{yxk + 1} ] \delta y_{k + 1} \hfill \\ &\quad + \left[ { \mu_{k + 1} g_{k + 1} \varepsilon_{yk + 1} \xi_{k + 1}^{{{\prime - 1}}} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} X_{k} - \varepsilon_{xk + 1} Z_{k} } \right) } \right] \delta X_{k} \hfill \\ &\quad + \left[ { \mu_{k + 1} \left\{ {1 + g_{k + 1} \varepsilon_{yk + 1} \xi_{k + 1}^{{{\prime - 1}}} Z_{k}^{ - 1} \left( {\varepsilon_{zk + 1} Y_{k} - \varepsilon_{yk + 1} Z_{k} } \right)} \right\} } \right] \delta Y_{k} . \hfill \\ \end{aligned}$$
(149)
Equations (132) and (133) are the differentiations of the transfer equations and Eqs. (148) and (149) are the differentiations of the refraction equations. From these, the partial differential coefficients expressed by Eqs. (44)–(59) in Sect. 3.1 are derived.
