## Abstract

Gradient-index filters from Lys & Optik were developed on the basis of an inverse Fourier transformation of the desired spectral characteristics. Filter designs with steep skirts and high reflection are possible because of a new correction method where analytical calculations for rugate structures are used as a basis for the initial corrections of the input characteristics. Overlaying of the quintic matching layers and successive approximation methods are used for the suppression of ripples on the transmission curve resulting from misadaptation between the gradient-index thin film and the surrounding media.

© 1992 Optical Society of America

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### Equations (22)

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(1)
$${\int}_{-\infty}^{\infty}\frac{\text{d}n}{\text{d}x}\frac{1}{2n}\text{exp}(ikx)\text{d}x=Q(k)\text{exp}[i\mathrm{\Phi}(k)],$$
(2)
$$k=2\mathrm{\pi}/\mathrm{\lambda},$$
(3)
$$x=2{\int}_{0}^{z}n(u)\text{d}u,$$
(4)
$$Q(k)={\{1/2[1/T(k)-T(k)]\}}^{1/2}.$$
(5)
$$Q(k)={\{-\text{ln}[T(k)]\}}^{1/2}.$$
(6)
$$\mathrm{\Phi}(k)=\frac{\mathrm{\pi}k}{{k}_{\text{min}}+{k}_{\text{max}}}-\frac{\mathrm{\pi}}{2}\text{sin}\left(N\mathrm{\pi}\frac{k-{k}_{\text{min}}}{{k}_{\text{max}}-{k}_{\text{min}}}\right),$$
(7)
$$n(x)=\text{exp}\left\{\frac{2}{\mathrm{\pi}}{\int}_{0}^{\infty}\frac{Q(k)}{k}\text{sin}[\mathrm{\Phi}(k)-kx]\text{d}k\right\}.$$
(8)
$$Q{(k)}_{j}=Q{(k)}_{j-1}+[{Q}_{o}(k)-{Q}_{a}(k)],$$
(9)
$$\text{if}\hspace{0.17em}Q{(k)}_{j}<0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{then}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}Q{(k)}_{j}=0.$$
(10)
$${n}_{i}={n}_{a}+\frac{{n}_{p}}{2}\text{sin}({k}_{i}x-{\mathrm{\Phi}}_{i}),$$
(11)
$${k}_{i}=2\mathrm{\pi}\times {n}_{a}/{\mathrm{\lambda}}_{i}.$$
(12)
$${n}_{p}({\mathrm{\lambda}}_{1})=\frac{2{n}_{a}{\mathrm{\lambda}}_{1}}{\mathrm{\pi}\hspace{0.17em}OT}\text{arctanh}\{{[1-T({\mathrm{\lambda}}_{1})]}^{1/2}\},$$
(13)
$${n}_{a}={({n}_{\text{H}}\xb7{n}_{\text{L}})}^{1/2},$$
(14)
$$R(\mathrm{\lambda})=\frac{{\mathrm{\beta}}^{2}\hspace{0.17em}{\text{sinh}}^{2}(sL)}{{s}^{2}\hspace{0.17em}{\text{cosh}}^{2}(sL)+{(\mathrm{\alpha}/2)}^{2}{\text{sinh}}^{2}(sL)},$$
(15)
$$\mathrm{\beta}=\mathrm{\pi}{n}_{p}/(2\mathrm{\lambda}),$$
(16)
$$\mathrm{\alpha}=4\mathrm{\pi}{n}_{a}(1/\mathrm{\lambda}-1/{\mathrm{\lambda}}_{1}),$$
(18)
$$s={[{\mathrm{\beta}}^{2}-{(\mathrm{\alpha}/2)}^{2}]}^{1/2},$$
(19)
$$\text{d}\mathrm{\lambda}/\mathrm{\lambda}={n}_{p}/(2{n}_{a}).$$
(20)
$$n(x)={n}_{s}+[n(x)-{n}_{s}](6{t}^{5}-15{t}^{4}+10{t}^{3}),$$
(21)
$$t=(x-{x}_{\text{min}})/T\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}x-{x}_{\text{min}}\leqq T,$$
(22)
$$t=({x}_{\text{max}}-x)/T\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{x}_{\text{max}}-x\leqq T,$$