Ray tracing in concentric gradient-index media: optical Binet equation

Wanguo Liu; Wanguo Liu

doi:10.1364/JOSAA.456203

Journal of the Optical Society of America A
Vol. 39,
Issue 6,
pp. 1025-1033
(2022)
•https://doi.org/10.1364/JOSAA.456203

Ray tracing in concentric gradient-index media: optical Binet equation

Wanguo Liu

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Wanguo Liu, "Ray tracing in concentric gradient-index media: optical Binet equation," J. Opt. Soc. Am. A 39, 1025-1033 (2022)

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Abstract

The Binet equation in mechanics describes the orbital geometry of a moving particle under a central force field. In this paper, as its counterpart in optics, we show this formula can be similarly utilized in ray tracing of a gradient-index (GRIN) medium with a concentric field. As an inference of Fermat’s principle, this generalization is called the optical Binet equation (OBE). A remarkable advantage of OBE is that it can not only determine the ray trace or concentric GRIN field once one of them is given, but also derive the propagation time inside the medium. As examples, we apply OBE to rays passing through a Maxwell fish-eye lens, Luneburg lens, Eaton lens, concentrator, and hyperbolic deflector, the time delay of which can be calculated once the GRIN field or ray trace equation is solved. The results are well matched with simulations, proving it to be an effective tool in solving problems of the concentric GRIN field.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

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Equations (38)

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Name	GRIN Field	Incident Point	Incident Direction	Ray Equation	Time Delay (Inside the Media)	Range
Power-law	$n (r) = A r^{k}$	–	–	$r^{k + 1} \cos ((k + 1) θ) = C, k \neq - 1 r = C^{'} e^{C θ}, k = - 1$	$\frac{C A}{c} [\tan ((k + 1) θ_{t}) - \tan ((k + 1) θ_{0})]$	–
Maxwell fish-eye lens	$n (r) = \frac{2 R^{2}}{R^{2} + r^{2}}$	$(R, 0)$	Incident angle $φ_{0}$	$\frac{R}{r} - \frac{r}{R} = - 2 \cot φ_{0} \sin θ$	$\frac{π R}{c}$	$\| φ_{0} \| \leq \frac{π}{2}$ , $\| θ \| \leq π$
Luneburg lens	$n (r) = \sqrt{2 - \frac{r^{2}}{R^{2}}}$	$(R, θ_{0})$	Parallel to polar axis	$r = \frac{R \| \sin θ_{0} \|}{\sqrt{1 - \cos θ_{0} \cos (2 θ - θ_{0})}}$	$\frac{R}{c} (\frac{π}{2} + \cos θ_{0})$	$\begin{matrix} \| θ_{0} \| < \frac{π}{2} \\ \| θ_{0} \| \leq \| θ \| \leq π \end{matrix}$
Eaton lens (retro-reflector)	$n (r) = \sqrt{\frac{2 R}{r} - 1}$	$(R, θ_{0})$	Parallel to polar axis	$r = \frac{R \sin^{2} θ_{0}}{1 - \cos θ_{0} \cos θ}$	$\frac{R}{c} (π + 2 \cos θ_{0})$	$0 \leq θ_{0} < \frac{π}{2}$ $θ_{0} \leq θ \leq 2 π - θ_{0}$
Concentrator	$n (r) = \frac{R}{r}$	$(R, θ_{0})$	Parallel to polar axis	$r = R e^{\cot θ_{0} \cdot (θ_{0} - θ)}$	$\frac{R \ln 2}{c \cos θ_{0}} (\| θ_{0} \| < \frac{π}{2})$	$\| θ_{0} \| \leq \frac{π}{2}$
					(“half-life” period)	$\| θ \| \geq \| θ_{0} \|$
					$\frac{π R}{c} (θ_{0} = \pm \frac{π}{2})$ (another type of retro-reflector)
Hyperbolic deflector	$n (r) = \frac{r}{R}$	$(a, \frac{π}{2})$	Parallel to polar axis	$r^{2} \cos 2 θ = a^{2}$	$\frac{R}{2 c} \tan 2 θ_{d}$ ( $θ_{d}$ : deflecting angle)	$\| a \| \geq R$ $\frac{π}{2} \leq θ < π$ $0 < θ_{d} < \frac{π}{4}$

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Journal of the Optical Society of America A