Light Rays in Lens-Like Media

Yoshinao Aoki

doi:10.1364/JOSA.56.001648

Light Rays in Lens-Like Media

Yoshinao Aoki

Journal of the Optical Society of America
Vol. 56,
Issue 12,
pp. 1648-1651
(1966)
•https://doi.org/10.1364/JOSA.56.001648

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Yoshinao Aoki, "Light Rays in Lens-Like Media," J. Opt. Soc. Am. 56, 1648-1651 (1966)

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History
- Original Manuscript: March 21, 1966
- Published: December 1, 1966

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Abstract

In this paper, the geometrical light rays in lens-like media are discussed. The eikonal equation curlK = 0, written in terms of the wave vector, is used and the characteristic differential equations of the light ray, h₁du/k_u = h₂dυ/k_υ = h₃dw/k_w, are obtained. The quantities which are conserved along the ray are also discussed and the equations of the rays in idealized lens-like media whose refractive indexes vary along one direction, are obtained to integrate the differential equations. For a practical example, the trajectory of the ray in a lens-like medium with square-law index variation is discussed.

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Lens-like media	Directions of the gradient of the refractive index	Quantities which are conserved along the ray	Equation of light rays
Fig. 1.	ρ	k_z = const. = c₁	$\begin{array}{l} z = \int_{}^{ρ} c_{1} {[k^{2} - {c_{1}}^{2} - {(\frac{c_{2}}{ρ})}^{2}]}^{- \frac{1}{2}} d ρ \\ φ = \int_{}^{ρ} c_{2} ρ^{- 2} {[k^{2} - {c_{1}}^{2} - {(\frac{c_{2}}{ρ})}^{2}]}^{- \frac{1}{2}} d ρ \\ z = \int_{}^{φ} \frac{c_{1}}{c_{2}} {[ρ (φ)]}^{2} d φ \end{array}$
Fig. 1.	ρ	k_φρ = const. = c₂

Fig. 2	z	k_z = const. = c₁	$\begin{array}{l} x = {\int_{}^{z} c_{1} [k^{2} - {c_{1}}^{2} - {c_{2}}^{2}]}^{- \frac{1}{2}} d z \\ y = {\int_{}^{z} c_{2} [k^{2} - {c_{1}}^{2} - {c_{2}}^{2}]}^{- \frac{1}{2}} d z \\ x = \int_{}^{y} \frac{c_{1}}{c_{2}} d y \end{array}$
Fig. 2	z	k_y = const. = c₂

Fig. 3	x	k_y = const. = c₁	$\begin{array}{l} z = \int_{}^{x} c_{2} R {(R + x)}^{- 2} {[k^{2} - {c_{1}}^{2} - {(\frac{c_{2}}{R + x})}^{2}]}^{- \frac{1}{2}} d x \\ y = \int_{}^{x} c_{1} {[k^{2} - {c_{1}}^{2} - {(\frac{c_{2}}{R + x})}^{2}]}^{- \frac{1}{2}} d x \\ y = \int_{}^{z} \frac{c_{1}}{c_{2} R} {[R + x (z)]}^{2} d z \end{array}$
Fig. 3	x	k_z(R + x) = const. = c₂

Fig. 4	$r$	$k_{θ} r = const . = c_{1}$	(a) $\begin{array}{l} θ = \int_{}^{r} c_{1} r^{- 1} {[{(r k)}^{2} - {c_{1}}^{2} - {c_{2}}^{2}]}^{- \frac{1}{2}} d r \\ φ = \int_{}^{r} \frac{c_{2}}{r sin θ (r)} {[{(r k)}^{2} - {c_{1}}^{2} - {c_{2}}^{2}]}^{- \frac{1}{2}} d r \\ φ = \int_{}^{θ} \frac{c_{2}}{c_{1} sin θ} d θ \end{array}$
Fig. 4	$r$	$k_{φ} r = const . = c_{2} .$

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