Wavefront Aberration and Wavefront Shape

Abstract

The study of optical systems may be considered from two different perspectives, namely systems analysis and systems design, respectively.

Appendices

Appendix 1

The explicit expression of ${\partial } {}^{2}\uplambda_{\text{i}} /{\partial } \bar{\text{R}}_{{\text{i} - 1}}^{2}$ when a ray hits a flat boundary surface $\bar{\text{r}}_{\text{i}}$ has the form (see Eq. (7.5))

$$ \begin{aligned} \frac{{{\partial } {}^{2}\uplambda_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}}^{2} }} & = \frac{1}{{\text{E}_{\text{i}}^{2} }}\frac{{{\partial } \text{E}_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i - 1}}} }} }}\frac{{{\partial } \text{D}_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} + \frac{1}{{\text{E}_{\text{i}}^{2} }}\frac{{{\partial } \text{D}_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i - 1}}} }} }}\frac{{{\partial } \text{E}_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} - \frac{{2\text{D}_{\text{i}} }}{{\text{E}_{\text{i}}^{3} }}\frac{{{\partial } \text{E}_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i - 1}}} }} }}\frac{{{\partial } \text{E}_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} \\ & = \frac{1}{{\text{E}_{\text{i}}^{2} }}\left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & 0 \hfill & 0 \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & 0 \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & {} \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & {\text{symm}.} \hfill & {} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & { \, 0} \hfill \\ \end{array} } \right] \\ & \quad - \frac{{2\text{D}_{\text{i}} }}{{\text{E}_{\text{i}}^{3} }}\left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & {\text{symm}.} \hfill & {} \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{iz}}} } \hfill \\ \end{array} } \right]. \\ \end{aligned} $$

(15.60)

When $\bar{\mathcal{{\ell} }}_{\text{i}}$ is the unit directional vector of the reflected ray at $\bar{\text{r}}_{\text{i}}$, ${\partial } {}^{2}\bar{{\ell} }_{\text{i}} /{\partial } \bar{\text{R}}_{{\text{i} - 1}}^{2}$ is given by

$$ \frac{{{\partial } {}^{2}\bar{{\ell} }_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} }} = \bar{0}_{4 \times 6 \times 6} . $$

(15.61)

When $\bar{\mathcal{{\ell} }}_{\text{i}}$ is the unit directional vector of the refracted ray at $\bar{\text{r}}_{\text{i}}$, ${\partial } {}^{2}\bar{{\ell} }_{\text{i}} /{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}}$ can be obtained by differentiating Eq. (7.10), to give

$$ \frac{{{\partial } {}^{2}\bar{{\ell} }_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} }} = \left[ {\begin{array}{*{20}c} {{\partial } {}^{2}{\ell}_{{\text{ix}}} /{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} } \\ {{\partial } {}^{2}{\ell}_{{\text{iy}}} /{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} } \\ {{\partial } {}^{2}{\ell}_{{\text{iz}}} /{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} } \\ {\bar{0}} \\ \end{array} } \right], $$

(15.62)

where

$$ \frac{{{\partial } {}^{2}{\ell}_{{\text{ix}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} }} = \frac{{\text{s}_{\text{i}} \text{N}_{\text{i}}^{2} (1 - \text{N}_{\text{i}}^{2} )\text{J}_{{\text{ix}}} }}{{\sqrt {\left( {1 - \text{N}_{\text{i}}^{2} + \text{N}_{\text{i}}^{2} \text{E}_{\text{i}}^{2} } \right)^{3} } }}\left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {\text{symm}.} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{iz}}} } \hfill \\ \end{array} } \right], $$

(15.63)

$$ \frac{{{\partial } {}^{2}{\ell}_{{\text{iy}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} }} = \frac{{\text{s}_{\text{i}} \text{N}_{\text{i}}^{2} (1 - \text{N}_{\text{i}}^{2} )\text{J}_{{\text{iy}}} }}{{\sqrt {\left( {1 - \text{N}_{\text{i}}^{2} + \text{N}_{\text{i}}^{2} \text{E}_{\text{i}}^{2} } \right)^{3} } }}\left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {\text{symm}.} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{iz}}} } \hfill \\ \end{array} } \right], $$

(15.64)

$$ \frac{{{\partial } {}^{2}{\ell}_{{\text{iz}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}} }} = \frac{{\text{s}_{\text{i}} \text{N}_{\text{i}}^{2} (1 - \text{N}_{\text{i}}^{2} )\text{J}_{{\text{iz}}} }}{{\sqrt {\left( {1 - \text{N}_{\text{i}}^{2} + \text{N}_{\text{i}}^{2} \text{E}_{\text{i}}^{2} } \right)^{3} } }}\left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & 0 \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & 0 \hfill & { \, 0} \hfill & { \, 0} \hfill & { \, 0} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{ix}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{ix}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {\text{symm}.} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iy}}} } \hfill & {\text{J}_{{\text{iy}}} \text{J}_{{\text{iz}}} } \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & {\text{J}_{{\text{iz}}} \text{J}_{{\text{iz}}} } \hfill \\ \end{array} } \right]. $$

(15.65)

Appendix 2

The Hessian matrix ${\partial }^{2} {{\bar{\text{R}}}}_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ of the reflected/refracted ray $\bar{\text{R}}_{\text{i}} = \left[ {\begin{array}{*{20}c} {\bar{\text{P}}_{\text{i}} } & {\bar{{\ell} }_{\text{i}} } \\ \end{array} } \right]^{\text{T}}$ at a spherical boundary surface $\bar{\text{r}}_{\text{i}}$ with respect to the incoming ray ${{\bar{\text{R}}}}_{{\text{i} - 1}}$ is composed of Eqs. (15.6), (15.10) and (15.11). However, the following terms are required before ${\partial }^{2} {{\bar{\text{R}}}}_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ can be determined.

• (1) Determination of ${\partial }^{2}\upsigma_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$, ${\partial }^{2}\uprho_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ and ${\partial }^{2}\uptau_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$:

  Differentiating Eq. (7.57) of Appendix 1 in Chap. 7 with respect to incoming ray $\bar{\text{R}}_{{\text{i} - 1}}$ yields

  $$ \frac{{{\partial }^{2} {{\bar{\text{P}}}}_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} = {}^{0}\bar{\text{A}}_{\text{i}} \left[ {\begin{array}{*{20}c} {{\partial }^{2}\upsigma_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {{\partial }^{2}\uprho_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {{\partial }^{2}\uptau_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {\bar{0}} \\ \end{array} } \right], $$

  (15.66)

  where ${\partial } {}^{2}\bar{\text{P}}_{\text{i}} /{\partial } \bar{\text{R}}_{{\text{i} - \text{1}}}^{\text{2}}$ is given in Eq. (15.6). The terms ${\partial } {}^{2}\upsigma_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$, ${\partial } {}^{2}\uprho_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ and ${\partial } {}^{2}\uptau_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ in Eq. (15.66) can then be obtained as

  $$ \left[ {\begin{array}{*{20}c} {{\partial }^{2}\upsigma_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} \, } \\ {{\partial }^{2}\uprho_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {{\partial }^{2}\uptau_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {\bar{0}} \\ \end{array} } \right] = \left( {{}^{0}\bar{\text{A}}_{\text{i}} } \right)^{ - 1} \, \frac{{{\partial }^{2} {{\bar{\text{P}}}}_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }}. $$

  (15.67)

• (2) Determination of ${\partial } {}^{2}\upalpha_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ and ${\partial } {}^{2}\upbeta_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$:

  ${\partial } {}^{2}\upalpha_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ can be obtained by differentiating Eq. (7.59) of Appendix 1 in Chap. 7 with respect to $\bar{\text{R}}_{{\text{i} - 1}}$, to give

  $$ \begin{aligned} \frac{{{\partial } {}^{2}\upalpha_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} & = \frac{1}{{\upsigma_{\text{i}}^{ 2} +\uprho_{\text{i}}^{ 2} }}\left( {\frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} +\upsigma_{\text{i}} \frac{{{\partial } {}^{2}\uprho_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} - \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} -\uprho_{\text{i}} \frac{{{\partial } {}^{2}\upsigma_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }}} \right) \\ & \quad - \frac{2}{{\left( {\upsigma_{\text{i}}^{ 2} +\uprho_{\text{i}}^{ 2} } \right)^{2} }}\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}} \right)\left( {\upsigma_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} -\uprho_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}} \right). \\ \end{aligned} $$

  (15.68)

  Similarly, ${\partial } {}^{2}\upbeta_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ can be determined by differentiating Eq. (7.60) of Appendix 1 in Chap. 7 with respect to $\bar{\text{R}}_{{\text{i} - 1}}$, to give

  $$ \begin{aligned} \frac{{{\partial }^{2}\upbeta_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} & = \frac{{\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )}}\frac{{{\partial }^{2}\uptau_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} + \frac{1}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}} \right)\frac{{{\partial }\uptau_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} \\ & \quad - \frac{{2 \, \sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )^{2} }}\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uptau_{\text{i}} \frac{{{\partial }\uptau_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}} \right)\frac{{{\partial }\uptau_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} \\ & \quad - \frac{1}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}\frac{{{\partial }\uptau_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}} \right) \\ & \quad - \frac{{\uptau_{\text{i}} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )\sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}\left( {\frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} +\upsigma_{\text{i}} \frac{{{\partial }^{2}\upsigma_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} + \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} +\uprho_{\text{i}} \frac{{{\partial }^{2}\uprho_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }}} \right) \\ & \quad + \frac{{2\uptau_{\text{i}} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )^{2} \sqrt {\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)} }}\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uptau_{\text{i}} \frac{{{\partial }\uptau_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}} \right)\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}} \right) \\ & \quad + \frac{{\uptau_{\text{i}} }}{{(\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} +\uptau_{\text{i}}^{2} )\left( {\upsigma_{\text{i}}^{\text{2}} +\uprho_{\text{i}}^{2} } \right)^{3/2} }}\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}} \right)\left( {\upsigma_{\text{i}} \frac{{{\partial }\upsigma_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} +\uprho_{\text{i}} \frac{{{\partial }\uprho_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}} \right). \\ \end{aligned} $$

  (15.69)

• (3) Determination of ${\partial } {}^{2}\bar{\text{n}}_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$:

  One approach to compute ${\partial } {}^{2}\bar{\text{n}}_{\text{i}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ is to differentiate Eq. (7.61) of Appendix 1 in Chap. 7 with respect to $\bar{\text{R}}_{{\text{i} - 1}}$, to give

  $$ \frac{{{\partial } {}^{2}\bar{\text{n}}_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} = \left[ {\begin{array}{*{20}c} {{\partial } {}^{2}\text{n}_{{\text{ix}}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {{\partial } {}^{2}\text{n}_{{\text{iy}}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {{\partial } {}^{2}\text{n}_{{\text{iz}}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} } \\ {\bar{0}} \\ \end{array} } \right] = {}^{0}\bar{\text{A}}_{\text{i}} \frac{{{\partial } {}^{2}({}^{\text{i}}\bar{\text{n}}_{\text{i}} )}}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }}, $$

  (15.70)

  where ${\partial } {}^{2}({}^{\text{i}}\bar{\text{n}}_{\text{i}} )/{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ is obtained by differentiating Eq. (7.62) of Appendix 1 in Chap. 7 with respect to $\bar{\text{R}}_{{\text{i} - 1}}$, to give

  $$ \begin{aligned} \frac{{{\partial } {}^{2}({}^{\text{i}}\bar{\text{n}}_{\text{i}} )}}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} & = \text{s}_{\text{i}} \left[ {\begin{array}{*{20}c} { - \text{S}\upbeta_{\text{i}} \text{C}\upalpha_{\text{i}} } \\ { - \text{S}\upbeta_{\text{i}} \text{S}\upalpha_{\text{i}} } \\ {\text{C}\upbeta_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{{\partial } {}^{2}\upbeta_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} + \text{s}_{\text{i}} \left[ {\begin{array}{*{20}c} { - \text{C}\upbeta_{\text{i}} \text{S}\upalpha_{\text{i}} } \\ {\text{C}\upbeta_{\text{i}} \text{C}\upalpha_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{{\partial } {}^{2}\upalpha_{\text{i}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} + \text{s}_{\text{i}} \left[ {\begin{array}{*{20}c} { - \text{C}\upbeta_{\text{i}} \text{C}\upalpha_{\text{i}} } \\ { - \text{C}\upbeta_{\text{i}} \text{S}\upalpha_{\text{i}} } \\ { - \text{S}\upbeta_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{{\partial }\upbeta_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\upbeta_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} \\ & \quad + \text{s}_{\text{i}} \left[ {\begin{array}{*{20}c} {\text{S}\upbeta_{\text{i}} \text{S}\upalpha_{\text{i}} } \\ { - \text{S}\upbeta_{\text{i}} \text{C}\upalpha_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{{\partial }\upbeta_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\upalpha_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} + \text{s}_{\text{i}} \left[ {\begin{array}{*{20}c} {\text{S}\upbeta_{\text{i}} \text{S}\upalpha_{\text{i}} } \\ { - \text{S}\upbeta_{\text{i}} \text{C}\upalpha_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{{\partial }\upalpha_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\upbeta_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} \\ & \quad + \text{s}_{\text{i}} \left[ {\begin{array}{*{20}c} { - \text{C}\upbeta_{\text{i}} \text{C}\upalpha_{\text{i}} } \\ { - \text{C}\upbeta_{\text{i}} \text{S}\upalpha_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{{\partial }\upalpha_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial }\upalpha_{\text{i}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}. \\ \end{aligned} $$

  (15.71)

• (4) Determination of ${\partial } {}^{2}(\text{C}\uptheta_{\text{i}} )/{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$:

  The term ${\partial } {}^{2}(\text{C}\uptheta_{\text{i}} )/{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2}$ is computed from Eq. (7.63) of Appendix 1 in Chap. 7 by noting that ${\partial } {}^{2}{\ell}_{{\text{i - 1x}}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} = {\partial } {}^{2}{\ell}_{{\text{i - 1y}}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} = {\partial } {}^{2}{\ell}_{{\text{i - 1z}}} /{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} = \bar{0}$, to give

  $$ \begin{aligned} \frac{{{\partial } {}^{2}(\text{C}\uptheta_{\text{i}} )}}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} & = - \left( {{\ell}_{{\text{i - 1x}}} \frac{{{\partial } {}^{2}\text{n}_{{\text{ix}}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} + {\ell}_{{\text{i - 1y}}} \frac{{{\partial } {}^{2}\text{n}_{{\text{iy}}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }} + {\ell}_{{\text{i - 1z}}} \frac{{{\partial } {}^{2}\text{n}_{{\text{iz}}} }}{{{\partial } {{\bar{\text{R}}}}_{{\text{i} - 1}}^{ 2} }}} \right) \\ & \quad - \left( {\frac{{{\partial } {\ell}_{{\text{i - 1x}}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial } \text{n}_{{\text{ix}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} + \frac{{{\partial } {\ell}_{{\text{i - 1y}}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial } \text{n}_{{\text{iy}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }} + \frac{{{\partial } {\ell}_{{\text{i - 1z}}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}\frac{{{\partial } \text{n}_{{\text{iz}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}} \right) \, \\ & \quad - \left( {\frac{{{\partial } {\ell}_{{\text{i - 1x}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}\frac{{{\partial } \text{n}_{{\text{ix}}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} + \frac{{{\partial } {\ell}_{{\text{i - 1y}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}\frac{{{\partial } \text{n}_{{\text{iy}}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }} + \frac{{{\partial } {\ell}_{{\text{i - 1z}}} }}{{{\partial } \bar{\text{R}}_{{\text{i} - 1}} }}\frac{{{\partial } \text{n}_{{\text{iz}}} }}{{{\partial } \bar{\text{R}}_{{\underline{{\text{i} - 1}} }} }}} \right) .\\ \end{aligned} $$

  (15.72)