## Abstract

Measurements of angular scattering due to surface roughness were taken from a 24-layer dielectric mirror and compared to theory. In addition, the top surface roughness of the multilayer stack is analyzed from Talystep profilometer measurements. These roughness data are used to obtain a roughness spectral density function to be used in a vector multilayer scattering theory. The theory uses three multilayer stack models to incorporate possible effects of different degrees of correlation between interfaces of the stack. It was found that the angular scattering calculated using the experimentally obtained roughness spectral density function agreed remarkably well with the measured angular scattering data. This is especially true if care is taken to differentiate between particulate and roughness scattering. For the sake of comparison, the angular scattering from an aluminum film is also given, and differences from scattering from the multilayer mirror are noted.

© 1980 Optical Society of America

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### Equations (20)

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(1)
$$\mathbf{E}(\mathbf{r},t)=\sum _{l=1}^{L+1}\int {d}^{2}k{\zeta}_{l}(\mathbf{k}-{\mathbf{k}}_{0}){\u220a}_{l}(\mathbf{k},{\mathbf{k}}_{0})\hspace{0.17em}\text{exp}[i(\mathbf{k}\xb7\mathit{\rho}+qz)],$$
(2)
$${\zeta}_{l}(\mathbf{k}-{\mathbf{k}}_{0})=\int {d}^{2}\rho {\zeta}_{l}(\mathit{\rho})\hspace{0.17em}\text{exp}[i(\mathbf{k}-{\mathbf{k}}_{0})\xb7\mathit{\rho}].$$
(3)
$$\u3008dP/d\mathrm{\Omega}\u3009/{P}_{0}=\sum _{m=1}^{L+1}\sum _{n=1}^{L+1}{\mathbf{F}}_{m}(\mathbf{k},{\mathbf{k}}_{0})\xb7{\mathbf{F}}_{n}^{*}(\mathbf{k},{\mathbf{k}}_{0})\times \frac{\u3008{\zeta}_{m}(\mathbf{k}-{\mathbf{k}}_{0}){\zeta}_{n}^{*}(\mathbf{k}-{\mathbf{k}}_{0})\u3009}{A},$$
(4)
$$\u3008{\zeta}_{m}(\mathbf{k}-{\mathbf{k}}_{0}){\zeta}_{n}^{*}(\mathbf{k}-{\mathbf{k}}_{0})\u3009=\iint {d}^{2}\rho {d}^{2}\tau \times \text{exp}[i(\mathbf{k}-{\mathbf{k}}_{0})\xb7\mathit{\tau}]\u3008{\zeta}_{m}(\mathit{\rho})\hspace{0.17em}{\zeta}_{n}(\mathit{\rho}+\mathit{\tau})\u3009.$$
(5)
$${g}_{mn}(\mathbf{k}-{\mathbf{k}}_{0})=\int {d}^{2}\tau \hspace{0.17em}\text{exp}[i(\mathbf{k}-{\mathbf{k}}_{0})\xb7\mathit{\tau}]{G}_{mn}(\mathit{\tau}),$$
(6)
$${g}_{mn}(\mid \mathbf{k}-{\mathbf{k}}_{0}\mid )=2{\int}_{0}^{\infty}d\tau \tau {J}_{0}(\mid \mathbf{k}-{\mathbf{k}}_{0}\mid \tau ){G}_{mn}(\tau ),$$
(7)
$$\u3008dP/d\mathrm{\Omega}\u3009/{P}_{0}=g(\mathbf{k}-{\mathbf{k}}_{0}){\left|\sum _{l=1}^{L+1}{\mathbf{F}}_{l}(\mathbf{k},{\mathbf{k}}_{0})\right|}^{2}.$$
(8)
$${\zeta}_{m}(\mathit{\rho})={\mu}_{m}(\mathit{\rho}).$$
(9)
$$\u3008dP/d\mathrm{\Omega}\u3009/{P}_{0}=g(\mathbf{k}-{\mathbf{k}}_{0})\sum _{l=1}^{L+1}\mid \mathbf{F}(\mathbf{k},{\mathbf{k}}_{0}){\mid}^{2}.$$
(10)
$${\zeta}_{m}(\mathit{\rho})={\zeta}_{1}(\mathit{\rho})+\sum _{j=2}^{m}{\mu}_{l}(\mathit{\rho}),$$
(11)
$${G}_{mn}(\mathit{\tau})==\u3008{\zeta}_{m}(\mathit{\rho}){\zeta}_{n}(\mathit{\rho}+\mathit{\tau})\u3009=\sum _{l=1}^{p}{G}_{l}(\mathit{\tau}),$$
(12)
$${G}_{1}(\mathit{\tau})=\u3008{\zeta}_{1}(\mathit{\rho}){\zeta}_{1}(\mathit{\rho}+\mathit{\tau})\u3009=G(\mathit{\tau}),$$
(13)
$${G}_{l}(\mathit{\tau})=\u3008{\mu}_{l}(\mathit{\rho}){\mu}_{l}(\mathit{\rho}+\mathit{\tau})\u3009\hspace{0.17em}l\ge 2.$$
(14)
$${\delta}_{\text{rms}}^{(m)}={\left(\sum _{l=1}^{m}{\delta}_{l}^{2}\right)}^{1/2},$$
(15)
$${g}_{mn}(\mathbf{k}-{\mathbf{k}}_{0})=\sum _{l=1}^{p}{g}_{l}(\mathbf{k}-{\mathbf{k}}_{0}),$$
(16)
$${G}_{l}(\tau )={\delta}_{l1}^{2}\hspace{0.17em}\text{exp}(-{\tau}^{2}/{\sigma}_{l1}^{2})+{\delta}_{l2}^{2}\hspace{0.17em}\text{exp}(-{\tau}^{2}/{\sigma}_{l2}^{2}).$$
(17)
$${\delta}_{\text{rms}}^{(m)}={\left[\sum _{l=1}^{m}({\delta}_{l1}^{2}+{\delta}_{l2}^{2})\right]}^{1/2}.$$
(18)
$${g}_{l}(\mid \mathbf{k}-{\mathbf{k}}_{0}\mid )=\pi [{\delta}_{l1}^{2}{\sigma}_{l1}^{2}\hspace{0.17em}\text{exp}(-\mid \mathbf{k}-{\mathbf{k}}_{0}{\mid}^{2}{\sigma}_{l1}^{2}/4)+{\delta}_{l2}^{2}{\sigma}_{l2}^{2}\hspace{0.17em}\text{exp}(-\mid \mathbf{k}-{\mathbf{k}}_{0}{\mid}^{2}{\sigma}_{l3}^{2}/4)],$$
(19)
$$G(\tau )={\delta}_{11}^{2}\hspace{0.17em}\text{exp}(-{\tau}^{2}/{\sigma}_{11}^{2})+{\delta}_{12}^{2}\hspace{0.17em}\text{exp}(-{\tau}^{2}/{\sigma}_{12}^{2}).$$
(20)
$$\frac{1}{{P}_{0}}\frac{dP}{d\mathrm{\Omega}}=\frac{{V}_{m}(\theta )}{{V}_{c}}\frac{S}{\pi},$$