Appendix A<!--RH>Appendix<!--\RH>
For the case of zero rotation, the coordinate transformation between coordinate frames \( (\text{x}^{\prime}y^{\prime}\text{z}^{\prime})_{\text{n}} \) and \( (xyz)_{\text{n}} \) can be obtained simply by setting \( \mu = 0 \) in Eq. (6.13). The MTF is therefore denoted as \( {\text{MTF}}(\nu ,0) \). Since \( {\text{MTF}}^{2} (\nu ,0) = {\text{L}}_{\text{c}}^{2} (\nu ,0) + {\text{L}}_{\text{s}}^{2} (\nu ,0) \), the gradient of \( {\text{MTF}}^{2} (\nu ,0) \) with respect to \( {\text{x}}_{\text{n/chief}} \) can be obtained from
$$ \begin{aligned} \frac{{\partial {\text{MTF}}^{2} \left( {\nu ,0} \right)}}{{\partial {\text{x}}_{\text{n/chief}} }} &= 2{\text{MTF}}\left( {\nu ,0} \right)\frac{{\partial {\text{MTF}}\left( {\nu ,0} \right)}}{{\partial {\text{x}}_{\text{n/chief}} }} = \frac{{\partial {\text{MTF}}^{2} \left( {\nu ,0} \right)}}{{\partial \text{x}^{\prime}_{\text{n}} }}\frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} + \frac{{\partial {\text{MTF}}^{2} \left( {\nu ,0} \right)}}{{\partial \text{z}^{\prime}_{\text{n}} }}\frac{{\partial \text{z}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} \\ &= 2\left( {{\text{L}}_{\text{c}} \left( {\nu ,0} \right)\frac{{\partial {\text{L}}_{\text{c}} \left( {\nu ,0} \right)}}{{\partial \text{x}^{\prime}_{\text{n}} }} + {\text{L}}_{\text{s}} \left( {\nu ,0} \right)\frac{{\partial {\text{L}}_{\text{s}} \left( {\nu ,0} \right)}}{{\partial \text{x}^{\prime}_{\text{n}} }}} \right)\frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} \\ \, & + 2\left( {{\text{L}}_{\text{c}} \left( {\nu ,0} \right)\frac{{\partial {\text{L}}_{\text{c}} \left( {\nu ,0} \right)}}{{\partial \text{z}^{\prime}_{\text{n}} }} + {\text{L}}_{\text{s}} \left( {\nu ,0} \right)\frac{{\partial {\text{L}}_{\text{s}} \left( {\nu ,0} \right)}}{{\partial \text{z}^{\prime}_{\text{n}} }}} \right)\frac{{\partial \text{z}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} \\ &= 0 \, . \\ \end{aligned} $$
(A.1)
Similarly, one has
$$ \frac{{\partial {\text{MTF}}^{2} \left( {\nu ,0} \right)}}{{\partial {\text{z}}_{\text{n/chief}} }} = 2{\text{MTF}}\left( {\nu ,0} \right)\frac{{\partial {\text{MTF}}\left( {\nu ,0} \right)}}{{\partial {\text{z}}_{\text{n/chief}} }} = \frac{{\partial {\text{MTF}}^{2} \left( {\nu ,\mu } \right)}}{{\partial \text{x}^{\prime}_{\text{n}} }}\frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{z}}_{\text{n/chief}} }} + \frac{{\partial {\text{MTF}}^{2} \left( {\nu ,\mu } \right)}}{{\partial \text{z}^{\prime}_{\text{n}} }}\frac{{\partial \text{z}^{\prime}_{\text{n}} }}{{\partial {\text{z}}_{\text{n/chief}} }} = 0. $$
(A.2)
Equations (A.1) and (A.2) show \( {{\partial {\text{MTF}}(\nu ,0)} \mathord{\left/ {\vphantom {{\partial {\text{MTF}}(\nu ,0)} {\partial {\text{x}}_{\text{n/chief}} }}} \right. \kern-0pt} {\partial {\text{x}}_{\text{n/chief}} }} = {{\partial {\text{MTF}}(\nu ,0)} \mathord{\left/ {\vphantom {{\partial {\text{MTF}}(\nu ,0)} {\partial {\text{z}}_{\text{n/chief}} }}} \right. \kern-0pt} {\partial {\text{z}}_{\text{n/chief}} }} = 0 \), indicating that the \( {\text{MTF}}(\nu ,0) \) value is stationary in the neighborhood of \( \left[ {\begin{array}{*{20}c} {{\text{x}}_{\text{n/chief}} } & 0& {{\text{z}}_{\text{n/chief}} } & 1\\ \end{array} } \right]^{\text{T}} \), proving Theorem 6.1.
Appendix B
The gradients of phase shift \( \varpi (\nu ,0) \) with respect to \( {\text{x}}_{\text{n/chief}} \) and \( {\text{z}}_{\text{n/chief}} \) can be obtained respectively as
$$ \begin{aligned} \frac{\partial \varpi (\nu ,0)}{{\partial {\text{x}}_{\text{n/chief}} }} = & \frac{\partial \varpi (\nu ,0)}{{\partial \text{x}^{\prime}_{\text{n}} }}\frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} + \frac{\partial \varpi (\nu ,0)}{{\partial \text{z}^{\prime}_{\text{n}} }}\frac{{\partial \text{z}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} \\ = & \frac{{{\text{L}}_{\text{c}} (\nu ,0)}}{{{\text{L}}_{\text{c}} (\nu ,0)^{2} + {\text{L}}_{\text{s}} (\nu ,0)^{2} }}\frac{{\partial {\text{L}}_{\text{s}} (\nu ,0)}}{{\partial \text{x}^{\prime}_{\text{n}} }}\frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} - \frac{{{\text{L}}_{\text{s}} (\nu ,0)}}{{{\text{L}}_{\text{c}} (\nu ,0)^{2} + {\text{L}}_{\text{s}} (\nu ,0)^{2} }}\frac{{\partial {\text{L}}_{\text{c}} (\nu ,0)}}{{\partial \text{x}^{\prime}_{\text{n}} }} \, \frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{n/chief}} }} \\ = & \, \frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{x}}_{\text{nc}} }} = - 1 \, , \\ \end{aligned} $$
(B.1)
$$ \begin{aligned} \frac{\partial \varpi (\nu ,0)}{{\partial {\text{z}}_{\text{n/chief}} }} = & \frac{\partial \varpi (\nu ,0)}{{\partial \text{x}^{\prime}_{\text{n}} }}\frac{{\partial \text{x}^{\prime}}}{{\partial {\text{z}}_{\text{n/chief}} }} + \frac{\partial \varpi (\nu ,0)}{{\partial \text{z}^{\prime}_{\text{n}} }}\frac{{\partial \text{z}^{\prime}_{\text{n}} }}{{\partial {\text{z}}_{\text{n/chief}} }} \\ = & \frac{{{\text{L}}_{\text{c}} (\nu ,0)}}{{{\text{L}}_{\text{c}} (\nu ,0)^{2} + {\text{L}}_{\text{s}} (\nu ,0)^{2} }}\frac{{\partial {\text{L}}_{\text{s}} (\nu ,0)}}{{\partial \text{x}^{\prime}_{\text{n}} }}\frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{z}}_{\text{n/chief}} }} - \frac{{L_{s} (\nu ,0)}}{{{\text{L}}_{\text{c}} (\nu ,0)^{2} + {\text{L}}_{\text{s}} (\nu ,0)^{2} }}\frac{{\partial L_{c} (\nu ,0)}}{{\partial \text{x}^{\prime}_{\text{n}} }} \, \frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{z}}_{\text{n/chief}} }} \\ = & \, \frac{{\partial \text{x}^{\prime}_{\text{n}} }}{{\partial {\text{z}}_{\text{n/chief}} }} = 0. \\ \end{aligned} $$
(B.2)
Equation (B.1) shows that \( \varpi (\nu ,0) \) changes with a change of \( {\text{x}}_{\text{n/chief}} \). However, Eq. (B.2) shows that \( \varpi (\nu ,0) \) is unchanged with small changes of \( {\text{z}}_{\text{n/chief}} \). As discussed in Sect. 3.6, a phase shift of \( \varpi = 180^{ \circ } \) yields a reversal of contrast.
Appendix C
$$ \begin{aligned}
&{\text{L}}_{\text{c}} (\nu ,{{\upmu}} + 180^{ \circ } )\\
&= \frac{1}{{{{\uppsi}}_{0} }}\iint {\text{C}\left[ {2\uppi \nu
({\text{x}}_{\text{n}} - {\text{x}}_{\text{n/chief}}
){\text{C}}({{\upmu}} + 180^{ \circ } ) + 2\uppi \nu
({\text{z}}_{\text{n}} - {\text{z}}_{\text{n/chief}}
){\text{S}}({{\upmu}} + 180^{ \circ } )} \right]{\text{C}}\beta_{0}
}{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\
&= \frac{1}{{{{\uppsi}}_{0} }}\iint {{\text{C}}\left[ {2\uppi \nu
({\text{x}}_{\text{n}} - {\text{x}}_{\text{n/chief}} ){\text{C}}\mu
+ 2\uppi \nu ({\text{z}}_{\text{n}} - {\text{z}}_{\text{n/chief}}
){\text{S}}\mu } \right]{\text{C}}\beta_{0} }{\text{d}}\alpha_{0}
{\text{d}}\beta_{0} = {\text{L}}_{\text{c}} (\nu ,\mu ). \\
\end{aligned} $$
(C.1)
$$ \begin{aligned}
&{\text{L}}_{\text{s}} (\nu ,{{\upmu}} + 180^{ \circ } ) \\
&= \frac{1}{{{{\uppsi}}_{0} }}\iint {{\text{S}}\left[ {2\uppi \nu
({\text{x}}_{\text{n}} - {\text{x}}_{\text{n/chief}}
){\text{C}}({{\upmu}} + 180^{ \circ } ) + 2\uppi \nu
({\text{z}}_{\text{n}} - {\text{z}}_{\text{n/chief}}
){\text{S}}({{\upmu}} + 180^{ \circ } )} \right]{\text{C}}\beta_{0}
}{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\
&= \frac{ - 1}{{{{\uppsi}}_{0} }}\iint {{\text{S}}\left[ {2\uppi \nu
({\text{x}}_{\text{n}} - {\text{x}}_{\text{n/chief}}
){\text{C}}({{\upmu}}) + 2\uppi \nu ({\text{z}}_{\text{n}} -
{\text{z}}_{\text{n/chief}} ){\text{S}}\mu }
\right]{\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0}
= - {\text{L}}_{\text{s}} (\nu ,{{\upmu}}). \\ \end{aligned} $$
(C.2)
As a result, one has the following two equations from Eqs. (C.1) and (C.2):
$$ {\text{MTF}}(\nu ,{{\upmu}} + 180^{ \circ } ) = \sqrt {[{\text{L}}_{\text{c}} (\nu ,{{\upmu}} + 180^{ \circ } )]^{2} + [{\text{L}}_{\text{s}} (\nu ,{{\upmu}} + 180^{ \circ } )]^{2} } = \sqrt {[{\text{L}}_{\text{c}} (\nu ,{{\upmu}})]^{2} + [{\text{L}}_{\text{s}} (\nu ,\mu )]^{2} } = {\text{MTF}}(\nu ,\mu ), $$
$$ \varpi (\nu ,{{\upmu}} + 180^{ \circ } ) = {\text{atan}}2({\text{L}}_{\text{s}} (\nu ,{{\upmu}} + 180^{ \circ } ),{\text{L}}_{\text{c}} (\nu ,{{\upmu}} + 180^{ \circ } )) = {\text{atan}}2( - {\text{L}}_{\text{s}} (\nu ,{{\upmu}}),{\text{L}}_{\text{c}} (\nu ,\mu )) = - \varpi (\nu ,\mu ). $$
Appendix D
Recall that in the irradiance method of Sect. 6.1, we indicated that a general source ray \( {\bar{\text{R}}}_{0} \) intersects the image plane at coordinates \( \left[ {\begin{array}{*{20}c} {{\text{x}}_{\text{n}} } & 0& {{\text{z}}_{\text{n}} } & 1\\ \end{array} } \right]^{T} \), where \( {\text{x}}_{\text{n}} = {\text{x}}_{\text{n}} ({{\upalpha}}_{0} ,{{\upbeta}}_{0} ) \) and \( {\text{z}}_{\text{n}} = {\text{z}}_{\text{n}} ({{\upalpha}}_{0} ,{{\upbeta}}_{0} ) \). As shown in Fig. 6.5, and with no loss in generality, the general source point can always be defined as lying on the \( y_{0} z_{0} \) plane and expressed as \( {\bar{\text{P}}}_{0} = \left[ {\begin{array}{*{20}c} 0& {{\text{P}}_{{ 0 {\text{y}}}} } & {{\text{P}}_{{ 0 {\text{z}}}} } & 1\\ \end{array} } \right]^{\text{T}} \) due to symmetry of the optical system. As a result, its chief ray intersects \( z_{n} \) axis of the image coordinate frame \( (xyz)_{\text{n}} \), thus
$$ {\text{x}}_{\text{n/chief}} = {\text{y}}_{\text{n/chief}} = 0. $$
(D.1)
Furthermore, two general rays \( \left[ {\begin{array}{*{20}c} {{\bar{\text{P}}}_{ 0} } & {\bar{\ell }{}_{0/1}} \\ \end{array} } \right]^{\text{T}} \) and \( \left[ {\begin{array}{*{20}c} {{\bar{\text{P}}}_{ 0} } & {\bar{\ell }{}_{0/2}} \\ \end{array} } \right]^{\text{T}} \), where \( \bar{\ell }{}_{0/1} = \left[ {\begin{array}{*{20}c} {{\text{C}}\beta_{0} {\text{C}}(90^{ \circ } - {{\upalpha}}_{0} )} & {{\text{C}}\beta_{0} {\text{S}}(90^{ \circ } - {{\upalpha}}_{0} )} & {{\text{S}}\beta_{0} } & 0 \\ \end{array} } \right]^{\text{T}} \) and \( \bar{\ell }{}_{0/2} = \left[ {\begin{array}{*{20}c} {{\text{C}}\beta_{0} {\text{C}}(90^{ \circ } + {{\upalpha}}_{0} )} & {{\text{C}}\beta_{0} {\text{S}}(90^{ \circ } + {{\upalpha}}_{0} )} & {{\text{S}}\beta_{0} } & 0 \\ \end{array} } \right]^{\text{T}} \), always have identical \( {\text{z}}_{\text{n}} \) intersections and oppositely signed \( {\text{x}}_{\text{n}} \) values in this axis-symmetrical optical system, i.e.
$$ {\text{x}}_{\text{n}} (90^{ \circ } - {{\upalpha}}_{0} ,{{\upbeta}}_{0} ) = - {\text{x}}_{\text{n}} (90^{ \circ } + {{\upalpha}}_{0} ,{{\upbeta}}_{0} ), $$
(D.2)
$$ {\text{z}}_{\text{n}} \left( {90^{ \circ } - {{\upalpha}}_{0} ,{{\upbeta}}_{0} } \right) = {\text{z}}_{\text{n}} \left( {90^{ \circ } + {{\upalpha}}_{0} ,{{\upbeta}}_{0} } \right). $$
(D.3)
Now we consider the calculation of \( {\text{L}}_{\text{c}} (\nu ,90^{ \circ } + {{\upmu}}) \) by substituting Eq. (D.1) into Eq. (6.28), yielding
$$\begin{aligned} {\text{L}}_{\text{c}} \left( {\nu ,90^{ \circ } + {{\upmu}}}\right) & =\frac{1}{{{{\uppsi}}_{0} }}\iint {{\text{C}}\left({2\uppi \nu \left[ {{\text{x}}_{\text{n}} {\text{C}}\left( {90^{\circ } + \mu } \right) + \left( {{\text{z}}_{\text{n}} -{\text{z}}_{\text{n/chief}} } \right){\text{S}}\left( {90^{ \circ }+ \mu } \right)} \right]} \right){\text{C}}\beta_{0}}{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\ &=\frac{{{\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n/chief}}{\text{C}}\mu } \right)}}{{{{\uppsi}}_{0} }}\iint {{\text{C}}\left({2\uppi \nu \left( {{\text{x}}_{\text{n}} {\text{S}}\mu -{\text{z}}_{\text{n}} {\text{C}}\mu } \right)}\right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0}\\ &\quad - \frac{{{\text{S}}\left( {2\uppi \nu{\text{z}}_{\text{n/chief}} {\text{C}}\mu }\right)}}{{{{\uppsi}}_{0} }}\iint {{\text{S}}\left( {2\uppi \nu\left( {{\text{x}}_{\text{n}} {\text{S}}\mu - {\text{z}}_{\text{n}}{\text{C}}\mu } \right)} \right){\text{C}}\beta_{0}}{\text{d}}\alpha_{0} {\text{d}}\beta_{0}\\ & =\frac{{{\text{C}}\left( {2\uppi \nu{\text{z}}_{\text{n/chief}} {\text{C}}\mu }\right)}}{{{{\uppsi}}_{0} }}\left[ {\iint {{\text{C}}\left( {2\uppi\nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left({2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu }\right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0}} \right.\left. { + \iint {{\text{S}}\left( {2\uppi \nu{\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{S}}\left({2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu }\right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0}} \right] \\ & \quad - \frac{{{\text{S}}\left( {2\uppi \nu{\text{z}}_{\text{n/chief}} {\text{C}}\mu }\right)}}{{{{\uppsi}}_{0} }}\left[ {\iint {{\text{S}}\left( {2\uppi\nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left({2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu }\right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0}} \right.\left. { - \iint {{\text{C}}\left( {2\uppi \nu{\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{S}}\left({2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu }\right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0}} \right]. \end{aligned} $$
(D.4)
If the upper and lower integration limits of \( {{\upalpha}}_{ 0} \) are given by \( 90^{ \circ } + \alpha_{{0/{\text{limit}}}} \) and \( 90^{ \circ } - \alpha_{{0/{\text{limit}}}} \), respectively, the integrand of the first term in Eq. (D.4) can be written in the form shown in Eq. (D.5) when Eqs. (D.2) and (D.3) are used.
$$ \begin{aligned} & \iint {{\text{C}}\left( {2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu } \right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\ & \quad= \int {\left[ {\int_{{90^{ \circ } - \alpha_{{0/{\text{limit}}}} }}^{{90^{ \circ } + \alpha_{{0/{\text{limit}}}} }} {{\text{C}}\left( {2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu } \right){\text{d}}\left( {{{\upalpha}}_{0} - 90^{ \circ } } \right)} } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} \\ & \quad= \int {\left[ {\int_{{ - \alpha_{{0/{\text{limit}}}} }}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}\left( {2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu } \right){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} \\ & \quad= \int {\left[ {\int_{{ - \alpha_{{0/{\text{limit}}}} }}^{0} {{\text{C}}\left( {2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu } \right){\text{d}}\alpha_{0} + \int_{0}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}\left( {2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu } \right){\text{d}}\alpha_{0} } } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} \\ & \quad= 2\int {\left[ {\int_{0}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}\left( {2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu } \right){\text{C}}\left( {2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu } \right){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} . \\ \end{aligned} $$
(D.5)
Similarly, the second, third, and fourth terms in Eq. (D.4) can be rewritten in the forms shown in Eqs. (D.6), (D.7) and (D.8), respectively, i.e.
$$ \iint {{\text{S}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} = 0, $$
(D.6)
$$ \iint {{\text{S}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{C}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} = 0, $$
(D.7)
$$ \begin{aligned} & \iint {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\ & \quad = 2\int {\left[ {\int_{0}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} . \\ \end{aligned} $$
(D.8)
Substituting Eqs. (D.5), (D.6), (D.7) and (D.8) into Eq. (D.4) gives
$$ \begin{aligned} {\text{L}}_{\text{c}} (\nu ,90^{ \circ } + {{\upmu}}) = & \frac{{2{\text{C}}(2\uppi \nu {\text{z}}_{\text{n/chief}} {\text{C}}\mu )}}{{{{\uppsi}}_{0} }}\int {\left[ {\int_{0}^{{\alpha_{{0{\text{limit}}}} }} {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{C}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} \\ & + \frac{{2{\text{S}}(2\uppi \nu {\text{z}}_{\text{n/chief}} {\text{C}}\mu )}}{{{{\uppsi}}_{0} }}\int {\left[ {\int_{0}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} . \\ \end{aligned} $$
(D.9)
Since Eq. (D.9) indicates that \( {\text{L}}_{\text{c}} (\nu ,90^{ \circ } + {{\upmu}}) \) is an even function with respect to \( \mu \), we have
$$ {\text{L}}_{\text{c}} (\nu ,90^{ \circ } + {{\upmu}}) = {\text{L}}_{\text{c}} (\nu ,90^{ \circ } - {{\upmu}}). $$
(D.10)
Similarly, by substituting \( {{\upmu}} = 90^{ \circ } + {{\upmu}} \) and Eq. (D.1) into Eq. (6.29), \( {\text{L}}_{\text{s}} (\nu ,90^{ \circ } + {{\upmu}}) \) can be rewritten as
$$ \begin{aligned} {\text{L}}_{\text{s}} (\nu ,90^{ \circ } + {{\upmu}}) =&\,\frac{1}{{{{\uppsi}}_{0} }}\iint {{\text{S}}\left( {2\uppi \nu \left[ {{\text{x}}_{\text{n}} {\text{C}}(90^{ \circ } + {{\upmu}}) + ({\text{z}}_{\text{n}} {\text{ - z}}_{\text{n/chief}} ){\text{C}}(90^{ \circ } + {{\upmu}})} \right]} \right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\ =&\,\frac{{ - {\text{S}}(2\uppi \nu {\text{C}}\mu {\text{z}}_{\text{n/chief}} )}}{{{{\uppsi}}_{0} }}\iint {{\text{C}}\left( {2\uppi \nu ({\text{x}}_{\text{n}} {\text{S}}\mu - {\text{z}}_{\text{n}} {\text{C}}\mu )} \right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\ \, & - \frac{{{\text{C}}(2\uppi \nu {\text{z}}_{\text{n/chief}} {\text{C}}\mu )}}{{{{\uppsi}}_{0} }}\iint {{\text{S}}\left( {2\uppi \nu ({\text{x}}_{\text{n}} {\text{S}}\mu - {\text{z}}_{\text{n}} {\text{C}}\mu )} \right){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} \\ =&\,\frac{{ - {\text{S}}(2\uppi \nu {\text{z}}_{\text{n/chief}} {\text{C}}\mu )}}{{{{\uppsi}}_{0} }}\left[ {\iint {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{C}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} } \right.\left. { + \iint {{\text{S}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} } \right] \\ & - \frac{{{\text{C}}(2\uppi \nu z_{\text{n/chief}} C\mu )}}{{\uppsi_{0} }}\left[ {\iint {{\text{S}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{C}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} } \right.\left. { - \iint {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{C}}\beta_{0} }{\text{d}}\alpha_{0} {\text{d}}\beta_{0} } \right] \\ =&\,\frac{{ - 2{\text{S}}(2\uppi \nu {\text{z}}_{\text{n/chief}} {\text{C}}\mu )}}{{{{\uppsi}}_{0} }}\int {\left[ {\int_{0}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{C}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} \\ \, & + \frac{{2{\text{C}}(2\uppi \nu {\text{z}}_{\text{n/chief}} {\text{C}}\mu )}}{{{{\uppsi}}_{0} }}\int {\left[ {\int_{0}^{{\alpha_{{0/{\text{limit}}}} }} {{\text{C}}(2\uppi \nu {\text{x}}_{\text{n}} {\text{S}}\mu ){\text{S}}(2\uppi \nu {\text{z}}_{\text{n}} {\text{C}}\mu ){\text{d}}\alpha_{0} } } \right]} {\text{C}}\beta_{0} {\text{d}}\beta_{0} . \\ \end{aligned} $$
(D.11)
Again, since \( {\text{L}}_{\text{s}} (\nu ,90^{ \circ } + {{\upmu}}) \) is an even function with respect to \( \mu \), one has
$$ {\text{L}}_{\text{s}} (\nu ,90^{ \circ } + {{\upmu}}) = {\text{L}}_{\text{s}} (\nu ,90^{ \circ } - {{\upmu}}). $$
(D.12)
Then, from Eqs. (D.10) and (D.12), we have
$$ \begin{aligned} {\text{MTF}}(\nu ,90^{ \circ } + {{\upmu}}) &= \sqrt {[{\text{L}}_{\text{c}} (\nu ,90^{ \circ } + {{\upmu}})]^{2} + [{\text{L}}_{\text{s}} (\nu ,90^{ \circ } + {{\upmu}})]^{2} } \\ &= \sqrt {[{\text{L}}_{\text{c}} (\nu ,90^{ \circ } - {{\upmu}})]^{2} + [{\text{L}}_{\text{s}} (\nu ,90^{ \circ } - {{\upmu}})]^{2} } = {\text{MTF}}(\nu ,90^{ \circ } - {{\upmu}}), \\ \end{aligned} $$
(D.13)
and
$$ \begin{aligned} \varpi (\nu ,90^{ \circ } + {{\upmu}}) &={\text{atan}}2({\text{L}}_{\text{s}} (\nu ,90^{ \circ } + {{\upmu}}),{\text{L}}_{\text{c}} (\nu ,90^{ \circ } + {{\upmu}})) \\
&={\text{atan}}2({\text{L}}_{\text{s}} (\nu ,90^{ \circ } -
{{\upmu}}),{\text{L}}_{\text{c}} (\nu ,90^{ \circ } - {{\upmu}})) =
\varpi (\nu ,90^{ \circ } - {{\upmu}}). \\ \end{aligned} $$
(D.14)
