## Abstract

We investigate the origin of low-level scattering from high-quality coatings produced by ion-assisted deposition and ion plating. For this purpose we use the polarization ratio of light scattering to separate surface and bulk effects that characterize the intrinsic action of the thin-film materials. In the first step the method is tested and validated at scattering levels greater than 10^{−5}. In the second step it is applied at low levels, and the results reveal some anomalies. To conclude, we perform a detailed analysis of scattering resulting from the presence of a few localized defects in the coatings.

© 1996 Optical Society of America

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### Equations (16)

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(1)
$$I(\mathrm{\theta},\mathrm{\varphi})=\sum _{i,j}{C}_{ij}(\mathrm{\theta},\mathrm{\varphi})\hspace{0.17em}{\mathrm{\alpha}}_{ij}(\mathrm{\theta},\mathrm{\varphi}){\mathrm{\gamma}}_{jj}(\mathrm{\theta},\mathrm{\varphi}),$$
(2)
$${\mathrm{\gamma}}_{jj}=\frac{4{\mathrm{\pi}}^{2}}{S}\mid {\widehat{h}}_{j}(\mathbf{\sigma}){\mid}^{2},$$
(3)
$${\mathrm{\gamma}}_{jj}=\frac{4{\mathrm{\pi}}^{2}}{S}\mid {\widehat{p}}_{j}(\mathbf{\sigma}){\mid}^{2}.$$
(4)
$${\mathrm{\gamma}}_{jj}(\mathrm{\sigma})=\text{TF}\left\{{{\mathrm{\delta}}_{e,j}}^{2}\hspace{0.17em}\text{exp}\left(-\left|\frac{\mathrm{\tau}}{{L}_{e,j}}\right|\right)+{{\mathrm{\delta}}_{g,j}}^{2}\hspace{0.17em}\text{exp}\left[-{\left(\frac{\mathrm{\tau}}{{L}_{g,j}}\right)}^{2}\right]\right\}\Rightarrow {\mathrm{\gamma}}_{jj}(\mathrm{\sigma})=\frac{1}{2\mathrm{\pi}}\frac{{{\mathrm{\delta}}_{e,j}}^{2}{{L}_{e,j}}^{2}}{{(1+{\mathrm{\sigma}}^{2}{{L}_{e,j}}^{2})}^{3/2}}+\frac{1}{4\mathrm{\pi}}{{\mathrm{\delta}}_{g,j}}^{2}{{L}_{g,j}}^{2}\hspace{0.17em}\text{exp}\left[-{\left(\frac{\mathrm{\sigma}{L}_{g,j}}{2}\right)}^{2}\right].$$
(5)
$$\mathrm{\mu}=\frac{{I}_{PP}}{{I}_{SS}}=\frac{{\displaystyle \sum _{i,j}}{C}_{ij}(P){\mathrm{\alpha}}_{ij}{\mathrm{\gamma}}_{jj}}{{\displaystyle \sum _{i,j}}{C}_{ij}(S){\mathrm{\alpha}}_{ij}{\mathrm{\gamma}}_{jj}}.$$
(6)
$$\begin{array}{c}\mathrm{\mu}(\mathrm{\theta})=\frac{{\displaystyle \sum _{i=0}^{p}}\mid {C}_{ii}(P){\mid}^{2}}{{\displaystyle \sum _{i=0}^{p}}\mid {C}_{ii}(S){\mid}^{2}}\\ \text{for}\hspace{0.17em}\text{roughness}\hspace{0.17em}\text{brought}\hspace{0.17em}\text{by}\hspace{0.17em}\text{materials},\end{array}$$
(7)
$$\begin{array}{c}\mathrm{\mu}(\mathrm{\theta})=\frac{{\displaystyle \sum _{i,j}}{C}_{ij}(P){\mathrm{\alpha}}_{ij}{a}_{js}}{{\displaystyle \sum _{i,j}}{C}_{ij}(S){\mathrm{\alpha}}_{ij}{a}_{js}}\\ \text{for}\hspace{0.17em}\text{substrate}\hspace{0.17em}\text{roughness}\hspace{0.17em}\text{replication},\end{array}$$
(8)
$$I(\mathrm{\theta},\mathrm{\varphi})=\sum _{i,j}{C}_{ij}{\mathrm{\alpha}}_{ij}{\mathrm{\gamma}}_{jj}.$$
(9)
$${\mathrm{\gamma}}_{jj}=\frac{{{A}_{j}}^{2}}{S}{{a}_{j}}^{4}{\left|\frac{{J}_{1}(\mathrm{\sigma}{a}_{j})}{\mathrm{\sigma}{a}_{j}}\right|}^{2}.$$
(10)
$${\mathrm{\gamma}}_{jj}=\frac{l}{S}{\left(2\frac{\mathrm{\Delta}{n}_{j}}{{n}_{j}}\right)}^{2}{{a}_{j}}^{4}{\left|\frac{{J}_{1}(\mathrm{\sigma}{a}_{j})}{\mathrm{\sigma}{a}_{j}}\right|}^{2}.$$
(11)
$${\mathrm{\sigma}}_{m}a\approx m\mathrm{\pi}\Rightarrow \text{sin}\hspace{0.17em}{\mathrm{\theta}}_{m}\approx m\frac{\mathrm{\lambda}}{2a}.$$
(12)
$${\mathrm{\gamma}}_{kk}={\mathrm{\gamma}}_{\text{unc}}+{\mathrm{\gamma}}_{\text{coh}},$$
(13)
$${\mathrm{\gamma}}_{\text{unc}}=\frac{1}{S}\sum _{i=1}^{N}{{a}_{i}}^{4}{\left|2{\left(\frac{\mathrm{\Delta}n}{n}\right)}_{i}\frac{{J}_{1}(\mathrm{\sigma}{a}_{i})}{\mathrm{\sigma}{a}_{i}}\right|}^{2}$$
(14)
$${\mathrm{\gamma}}_{\text{unc}}=\frac{1}{S}\sum _{i=1}^{N}{{a}_{i}}^{4}{{A}_{i}}^{2}{\left|\frac{{J}_{1}(\mathrm{\sigma}{a}_{i})}{\mathrm{\sigma}{a}_{i}}\right|}^{2}$$
(15)
$${\mathrm{\gamma}}_{\text{coh}}=\frac{1}{S}\sum _{i=1}^{N-1}\sum _{j>i}^{N}{{a}_{i}}^{2}{{a}_{j}}^{2}\left|4{\left(\frac{\mathrm{\Delta}n}{n}\right)}_{i}{\left(\frac{\mathrm{\Delta}n}{n}\right)}_{j}\frac{{J}_{1}(\mathrm{\sigma}{a}_{i})}{\mathrm{\sigma}{a}_{i}}\frac{{J}_{1}(\mathrm{\sigma}{a}_{j})}{\mathrm{\sigma}{a}_{j}}\right|\times 2\hspace{0.17em}\text{cos}[\mathbf{\sigma}\xb7({\mathbf{r}}_{j}-{\mathbf{r}}_{i})]$$
(16)
$${\mathrm{\gamma}}_{\text{coh}}=\frac{1}{S}\sum _{i=1}^{N-1}\sum _{j>i}^{N}{{a}_{i}}^{2}{{a}_{j}}^{2}{A}_{i}{A}_{j}\left|\frac{{J}_{1}(\mathrm{\sigma}{a}_{i})}{\mathrm{\sigma}{a}_{i}}\frac{{J}_{1}(\mathrm{\sigma}{a}_{j})}{\mathrm{\sigma}{a}_{j}}\right|\times 2\hspace{0.17em}\text{cos}[\mathbf{\sigma}\xb7({\mathbf{r}}_{j}-{\mathbf{r}}_{i})]$$