## Abstract

Recently we performed a numerical investigation of antireflection coatings that reduce significantly the reflection over a wide range of wavelengths and angles of incidence, and we proposed some experiments to demonstrate their feasibility. We provide a theoretical description of omnidirectional antireflection coatings that are effective over a wide range of wavelengths.

© 2004 Optical Society of America

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### Equations (12)

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(1)
$$\frac{\mathrm{d}}{\mathrm{d}z}\left[n\left(z\right)sin\mathrm{\theta}\left(z\right)\right]=0,$$
(2)
$${\mathrm{\eta}}_{s}=n\left(z\right)cos\mathrm{\theta}\left(z\right),{\mathrm{\eta}}_{p}=n\left(z\right)/cos\mathrm{\theta}\left(z\right).$$
(3)
$${\left[n\left(z\right)\mathrm{d}z\right]}_{\mathrm{\theta}}=n\left(z\right)\mathrm{d}zcos\mathrm{\theta}\left(z\right),$$
(4)
$${\mathrm{\theta}}_{c}=\mathrm{arcsin}\left({n}_{i-1}/{n}_{i}\right).$$
(5)
$${\mathrm{\theta}}_{\mathrm{Cinc}}=\mathrm{arcsin}\left({n}_{i-1}/{n}_{a}\right).$$
(6)
$${n}_{Q}\left(z\right)={n}_{max}-\left({n}_{max}-{n}_{min}\right)\left[10{\left(\frac{z}{d}\right)}^{3}-15{\left(\frac{z}{d}\right)}^{4}+6{\left(\frac{z}{d}\right)}^{5}\right]$$
(7)
$${n}_{\mathrm{ES}}\left(x\right)={\mathrm{\eta}}_{max}exp\left(\frac{1}{2}ln\left({\mathrm{\eta}}_{max}/{\mathrm{\eta}}_{min}\right)\times \left\{sin\left[\mathrm{\pi}\left(\frac{x}{{x}_{\mathrm{tot}}}\right)+\mathrm{\pi}/2\right]-sin\left(\mathrm{\pi}/2\right)\right\}\right),$$
(8)
$$R\sim {\left(\frac{\mathrm{\lambda}}{2\mathrm{\pi}d}\right)}^{2m},$$
(9)
$${\mathrm{\rho}}_{s}=\frac{1}{2{\mathrm{\eta}}_{s}}\frac{\mathrm{d}{\mathrm{\eta}}_{s}}{\mathrm{d}z}=\frac{n\prime}{2n}-\frac{\mathrm{\theta}\prime}{2}tan\mathrm{\theta},{\mathrm{\rho}}_{p}=\frac{1}{2{\mathrm{\eta}}_{p}}\frac{\mathrm{d}{\mathrm{\eta}}_{p}}{\mathrm{d}z}=\frac{n\prime}{2n}+\frac{\mathrm{\theta}\prime}{2}tan\mathrm{\theta}.$$
(10)
$$\mathrm{d}{z}_{\mathrm{new}}=\frac{\mathrm{d}z}{{\left[1-\frac{n_{M}{}^{2}}{n{\left(z\right)}^{2}}sin{\left({\mathrm{\theta}}_{0max}\right)}^{2}\right]}^{1/2}}.$$
(11)
$$R\approx {\left|\frac{\mathrm{\eta}\left(d\right)-{\mathrm{\eta}}_{a}}{\mathrm{\eta}\left(d\right)+{\mathrm{\eta}}_{a}}\right|}^{2}.$$
(12)
$${Y}_{j}=\frac{i{\mathrm{\eta}}_{j}sin{\mathrm{\phi}}_{j}+{Y}_{j-1}cos{\mathrm{\phi}}_{j}}{cos{\mathrm{\phi}}_{j}+\frac{i}{{\mathrm{\eta}}_{j}}{Y}_{j-1}sin{\mathrm{\phi}}_{j}},$$