## Abstract

Tracing rays through arbitrary diffraction gratings (including holographic gratings of the second generation fabricated on a curved substrate) by the vector form is somewhat complicated. Vector ray tracing utilizes the local groove density, the calculation of which highly depends on how the grooves are generated. Characterizing a grating by its groove function, available for almost arbitrary gratings, is much simpler than doing so by its groove density, essentially being a vector. Applying the concept of Riemann geometry, we give an expression of the groove density by the groove function. The groove function description of a grating can thus be incorporated into vector ray tracing, which is beneficial especially at the design stage. A unified explicit grating ray-tracing formalism is given as well.

© 2005 Optical Society of America

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### Equations (16)

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(1)
$$({g}_{\mathit{ij}})=\left[\begin{array}{cc}1+\mathit{\xi}_{w}{}^{2}& {\mathit{\xi}}_{w}{\mathit{\xi}}_{l}\\ {\mathit{\xi}}_{w}{\mathit{\xi}}_{l}& 1+\mathit{\xi}_{l}{}^{2}\end{array}\right].$$
(2)
$$({g}^{\mathit{ij}})=\left[\begin{array}{cc}1+\mathit{\xi}_{l}{}^{2}& -{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}\\ -{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}& 1+\mathit{\xi}_{w}{}^{2}\end{array}\right]/(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}).$$
(3)
$$\mathit{\sigma}=-\left\{[(1+\mathit{\xi}_{l}{}^{2}){n}_{w}-{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}{n}_{l}]\frac{\partial}{\partial w}+[(1+\mathit{\xi}_{w}{}^{2}){n}_{l}-{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}{n}_{w}]\frac{\partial}{\partial l}\right\}/(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}).$$
(4)
$$\mathit{\sigma}=-(u{\mathit{\xi}}_{w}+v{\mathit{\xi}}_{l},u,v)/(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}),$$
(5)
$$u=(1+\mathit{\xi}_{l}{}^{2}){n}_{w}-{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}{n}_{l},v=(1+\mathit{\xi}_{w}{}^{2}){n}_{l}-{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}{n}_{w},$$
(6)
$$|\mathit{\sigma}|={\left[\frac{n_{w}{}^{2}+n_{l}{}^{2}+({n}_{l}{\mathit{\xi}}_{w}-{n}_{w}{\mathit{\xi}}_{l}{)}^{2}}{1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}}\right]}^{1/2}.$$
(7)
$$|\mathit{\sigma}|\mathbf{p}\times \mathbf{s}=-\mathit{\sigma}.$$
(8)
$$\mathbf{s}=(-1,{\mathit{\xi}}_{w},{\mathit{\xi}}_{l})/[(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}){]}^{1/2}.$$
(9)
$${\mathbf{r}}_{i}=(L,M,N),{\mathbf{r}}_{o}=({L}^{\prime},{M}^{\prime},{N}^{\prime}),{\mathbf{r}}_{o}-\mathit{\mu}{\mathbf{r}}_{i}=(a,b,c).$$
(10)
$${\mathbf{r}}_{o}\times \mathbf{s}=\mathit{\mu}{\mathbf{r}}_{i}\times \mathbf{s}+m\frac{\mathit{\lambda}}{{\mathit{\nu}}_{2}}|\mathit{\sigma}|\mathbf{p},$$
(11)
$$({\mathbf{r}}_{o}-\mathit{\mu}{\mathbf{r}}_{i})\times \mathbf{s}\times \mathbf{s}=m\frac{\mathit{\lambda}}{{\mathit{\nu}}_{2}}|\mathit{\sigma}|\mathbf{p}\times \mathbf{s}=-m\frac{\mathit{\lambda}}{{\mathit{\nu}}_{2}}\mathit{\sigma}.$$
(12)
$$\left[\frac{-{\mathit{\xi}}_{l}c-a\mathit{\xi}_{l}{}^{2}-a\mathit{\xi}_{w}{}^{2}-{\mathit{\xi}}_{w}b}{1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}},\frac{-a{\mathit{\xi}}_{w}-b-b\mathit{\xi}_{l}{}^{2}+{\mathit{\xi}}_{l}c{\mathit{\xi}}_{w}}{1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}},\frac{{\mathit{\xi}}_{w}b{\mathit{\xi}}_{l}-c\mathit{\xi}_{w}{}^{2}-c-a{\mathit{\xi}}_{l}}{1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2}}\right]=\left[\frac{m\mathit{\lambda}({\mathit{\xi}}_{w}{n}_{w}+{\mathit{\xi}}_{l}{n}_{l})}{{\mathit{\nu}}_{2}(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2})},-\frac{m\mathit{\lambda}(-{n}_{w}-{n}_{w}\mathit{\xi}_{l}{}^{2}+{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}{n}_{l})}{{\mathit{\nu}}_{2}(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2})},\frac{m\mathit{\lambda}({n}_{l}+{n}_{l}\mathit{\xi}_{w}{}^{2}-{\mathit{\xi}}_{w}{\mathit{\xi}}_{l}{n}_{w})}{{\mathit{\nu}}_{2}(1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2})}\right].$$
(13)
$$b=\frac{m\mathit{\lambda}{n}_{w}}{{\mathit{\nu}}_{2}}-a{\mathit{\xi}}_{w},c=\frac{m\mathit{\lambda}{n}_{l}}{{\mathit{\nu}}_{2}}-a{\mathit{\xi}}_{l}.$$
(14)
$${L}^{\prime}=\mathit{\mu}L+a,{M}^{\prime}=\mathit{\mu}M+b,{N}^{\prime}=\mathit{\mu}N+c.$$
(15)
$$q-2\mathit{pa}+{\mathit{ea}}^{2}=0\to a=\frac{p\pm ({p}^{2}-\mathit{eq}{)}^{1/2}}{e},$$
(16)
$$e=1+\mathit{\xi}_{w}{}^{2}+\mathit{\xi}_{l}{}^{2},p=\left(\frac{m\mathit{\lambda}{n}_{l}}{{\mathit{\nu}}_{2}}+N\mathit{\mu}\right){\mathit{\xi}}_{l}+\left(\frac{m\mathit{\lambda}{n}_{w}}{{\mathit{\nu}}_{2}}+M\mathit{\mu}\right){\mathit{\xi}}_{w}-L\mathit{\mu},q={\left(\frac{m\mathit{\lambda}{n}_{w}}{{\mathit{\nu}}_{2}}+M\mathit{\mu}\right)}^{2}-1+{L}^{2}{\mathit{\mu}}^{2}+{\left(\frac{m\mathit{\lambda}{n}_{l}}{{\mathit{\nu}}_{2}}+N\mathit{\mu}\right)}^{2}.$$