## Abstract

A new approach to inhomogeneous layer synthesis is proposed that differs from the classical Fourier transform methods. This approach, based on the wave-number discretization of the continuous *Q* function, is described together with the relationship between the smoothness of the inhomogeneous layer’s refractive-index profile, the complexity of the target reflectance specification, and the total optical thickness of the layer. An iterative procedure is introduced to overcome the inaccuracy connected with the approximate *Q*-function representation. Some control features of the *Q*-function phase are studied, and numerical examples are given.

© 1994 Optical Society of America

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### Equations (39)

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(1)
$$Q(k)={\int}_{0}^{{x}_{a}}p(x)\text{exp}[ik(2x-{x}_{a})]\text{d}x,$$
(2)
$$p(x)=\frac{1}{2}\frac{{n}^{\prime}(x)}{n(x)}.$$
(4)
$${x}_{j}=h(j+\xbd),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}j=0,1,\dots ,N-1$$
(5)
$${p}_{j}=p({x}_{j}).$$
(6)
$$\begin{array}{l}Q(k)\approx h\sum _{j=0}^{N-1}{p}_{j}\hspace{0.17em}\text{exp}[ik(2{x}_{j}-{x}_{a})]\\ =h\hspace{0.17em}\text{exp}[ik(h-{x}_{a})]\sum _{j=0}^{N-1}{p}_{j}\hspace{0.17em}\text{exp}(i2jk{x}_{a}/N).\end{array}$$
(7)
$$f(k)={h}^{-1}\hspace{0.17em}\text{exp}[-ik(h-{x}_{a})]Q(k),$$
(8)
$$f(k)=\sum _{j=0}^{N-1}{p}_{j}\hspace{0.17em}\text{exp}(2ijk{x}_{a}/N).$$
(9)
$${k}_{l}=l(\mathrm{\pi}/{x}_{a}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=0,1,\dots ,N-1.$$
(10)
$${f}_{l}=f({k}_{l})={h}^{-1}\hspace{0.17em}\text{exp}[i\mathrm{\pi}l(1-h/{x}_{a})]Q({k}_{l}),$$
(11)
$${f}_{l}=\sum _{j=0}^{N-1}{p}_{j}\hspace{0.17em}\text{exp}(2\mathrm{\pi}ilj/N),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=0,1,\dots ,N-1.$$
(12)
$${p}_{j}={N}^{-1}\sum _{l=0}^{N-1}{f}_{l}\hspace{0.17em}\text{exp}(-2\mathrm{\pi}ilj/N),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}j=0,1,\dots ,N-1.$$
(13)
$${p}_{j}={n}^{\prime}({x}_{j})/2n({x}_{j}).$$
(14)
$$n(x)=n(jh)\text{exp}[2{p}_{j}(x-jh)].$$
(15)
$${f}_{N-l}={f}_{l}^{*},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=1,2,\dots ,N/2-1.$$
(16)
$$Q(k)\approx {f}_{2}(k)=r(k)/\mathrm{\tau}(k),$$
(17)
$$Q(k)=\mid Q(k)\mid \text{exp}[i\mathrm{\phi}(k)],$$
(18)
$$\mid Q(k){\mid}^{2}=\frac{R(k)}{T(k)}=\frac{R(k)}{1-R(k)}.$$
(19)
$$n(0)={n}_{s},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}n({x}_{a})={n}_{a}.$$
(20)
$${k}_{l}=0.1\mathrm{\pi}l,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=0,1,\dots ,N-1.$$
(21)
$${k}_{l}=l(\mathrm{\pi}/{x}_{a}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}l=0,1,\dots .$$
(22)
$$\mathrm{\Delta}k=\mathrm{\pi}/{x}_{a}.$$
(23)
$$Q(0)=\frac{{n}_{a}-{n}_{s}}{2{({n}_{a}{n}_{s})}^{1/2}}.$$
(24)
$$\mathrm{\nu}=k+i\mathrm{\sigma},$$
(25)
$$R(k)=\mathfrak{A}[n(x)].$$
(26)
$$n(x)={\mathfrak{A}}^{-1}[R(k)].$$
(27)
$$n(x)\approx \mathfrak{B}[R(k)].$$
(28)
$$\mathfrak{B}\approx {\mathfrak{A}}^{-1}.$$
(29)
$$\mathfrak{A}[n(x)]={R}_{0}(k).$$
(30)
$${n}_{0}(x)=\mathfrak{B}[{R}_{0}(k)].$$
(31)
$${R}_{1}(k)=\mathfrak{A}[{n}_{0}(x)]=\mathfrak{A}\mathfrak{B}[{R}_{0}(k)].$$
(32)
$$\mathfrak{A}\mathfrak{B}[s(k){R}_{0}(k)]=\mathfrak{A}\mathfrak{B}\tilde{R}(k)={R}_{0}(k),$$
(33)
$$n(x)=\mathfrak{B}[\tilde{R}(k)].$$
(34)
$$s(k)=\frac{\tilde{R}(k)}{{R}_{0}(k)}.$$
(35)
$$s(k)\approx {s}_{1}(k)=\frac{{R}_{0}(k)}{{R}_{1}(k)}.$$
(36)
$${s}_{i+1}(k)=\frac{{\tilde{R}}_{i}(k)}{{R}_{i+1}(k)}.$$
(37)
$$s(k)-{s}_{1}(k)=\frac{{(\mathfrak{A}\mathfrak{B})}^{-1}[{R}_{0}(k)]\mathfrak{A}\mathfrak{B}[{R}_{0}(k)]-{{R}_{0}}^{2}(k)}{{R}_{0}(k)\mathfrak{A}\mathfrak{B}[{R}_{0}(k)]}.$$
(38)
$$\mathfrak{A}\mathfrak{B}\approx \mathfrak{I}+\mathfrak{E},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{(\mathfrak{A}\mathfrak{B})}^{-1}\approx \mathfrak{I}-\mathfrak{E},$$
(39)
$$s(k)-{s}_{1}(k)=\frac{{\{\mathfrak{E}[{R}_{0}(k)]\}}^{2}}{{R}_{0}{(k)}^{2}+{R}_{0}(k)\mathfrak{E}[{R}_{0}(k)]}.$$