## Abstract

The intensity of high-order rainbows for normally incident light
and certain rotation angles of a cylinder with an elliptical cross
section is greatly amplified with respect to the intensity for a
circular cross-sectional cylinder. The amplification is due to a
number of the internal reflections occurring past the critical angle
for total internal reflection, and the effect is especially strong for
odd-order rainbows, beginning with the third order. Experimentally,
the fourth- and the fifth-order rainbows of a nearly elliptical
cross-sectional glass rod were observed and analyzed.

© 1998 Optical Society of America

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### Equations (10)

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(1)
$$\frac{x{\prime}^{2}}{{a}^{2}}+\frac{y{\prime}^{2}}{b_{1}{}^{2}}=1\mathrm{for}y\prime \ge 0,\frac{x{\prime}^{2}}{{a}^{2}}+\frac{y{\prime}^{2}}{b_{2}{}^{2}}=1\mathrm{for}y\prime 0,$$
(2)
$$tan{\mathrm{\gamma}}_{p}=\frac{\left(b_{j}{}^{2}/{a}^{2}\right){sin}^{2}\mathrm{\xi}+{cos}^{2}\mathrm{\xi}-x_{p}{}^{2}}{\left(b_{j}{}^{2}/{a}^{2}-1\right)sin\mathrm{\xi}cos\mathrm{\xi}+{x}_{p}{y}_{p}}$$
(3)
$$\mathrm{\theta}_{0}{}^{i}=\mathrm{\pi}/2-{\mathrm{\gamma}}_{0},$$
(4)
$$\mathrm{\theta}_{0}{}^{t}=\mathrm{arcsin}\left[\left(sin\mathrm{\theta}_{0}{}^{i}\right)/n\right].$$
(5)
$${\mathrm{\delta}}_{0}={\mathrm{\gamma}}_{0}+\mathrm{\theta}_{0}{}^{t},{\mathrm{\delta}}_{p}=2{\mathrm{\gamma}}_{p}-{\mathrm{\delta}}_{p-1}\mathrm{if}p\ge 1,p=\mathrm{odd},=2{\mathrm{\gamma}}_{p}-{\mathrm{\delta}}_{p-1}-\mathrm{\pi}\mathrm{if}p\ge 1,p=\mathrm{even},$$
(6)
$$\mathrm{\theta}_{p}{}^{i}={\mathrm{\gamma}}_{p}-{\mathrm{\delta}}_{p-1}.$$
(7)
$${\mathrm{\theta}}_{P}\left(\mathrm{\xi}\right)=\begin{array}{l}P\mathrm{\pi}/2-{\mathrm{\gamma}}_{P}+\mathrm{\theta}_{P}{}^{t}\\ \left(P+1\right)\mathrm{\pi}/2-{\mathrm{\gamma}}_{P}+\mathrm{\theta}_{P}{}^{t}\end{array}\hspace{1em}\begin{array}{l}\mathrm{if}P=\mathrm{odd}\\ \mathrm{if}P=\mathrm{even}\end{array},$$
(8)
$$\mathrm{\theta}_{P}{}^{t}=\mathrm{arcsin}\left(nsin\mathrm{\theta}_{P}{}^{i}\right).$$
(9)
$$tan{\mathrm{\delta}}_{p}=\left({y}_{p+1}-{y}_{p}\right)/\left({x}_{p+1}-{x}_{p}\right).$$
(10)
$${I}_{P}=\frac{1}{2}{\left[{{t}_{0}}^{\mathrm{TE}}\left(\prod _{p=1}^{P-1}{{r}_{p}}^{\mathrm{TE}}\right){{t}_{P}}^{\mathrm{TE}}\right]}^{2}+\frac{1}{2}{\left[{{t}_{0}}^{\mathrm{TM}}\left(\prod _{p=1}^{P-1}{{r}_{p}}^{\mathrm{TM}}\right){{t}_{P}}^{\mathrm{TM}}\right]}^{2},$$