## Abstract

The ratio *ρ* = *R*_{p}/*R*_{s} of the complex amplitude-reflection coefficients *R*_{p} and *R*_{s} for light polarized parallel (*p*) and perpendicular (*s*) to the plane of incidence, reflected from an optically isotropic film–substrate system, is investigated as a function of the angle of incidence *ϕ* and the film thickness *d*. Both constant-angle-of-incidence contours (CAIC) and constant-thickness contours (CTC) of the ellipsometric function *ρ*(*ϕ*,*d*) in the complex *ρ* plane are examined. For transparent films, *ρ*(*ϕ*,*d*) is a periodic function of *d* with period *D*_{ϕ} that is a function of *ϕ*. For a given angle of incidence *ϕ* and film thickness *d* (0 ≤ *ϕ* ≤ 90, 0 ≤ *d* < *D*_{ϕ}), the equispaced linear array of points (*ϕ*,*d* + *mD*_{ϕ}) (*m* = 0, 1, 2,…) in the real (*ϕ*,*d*) plane is mapped by the periodic function *ρ*(*ϕ*,*d*) into one distinct point in the complex *ρ* plane. Conversely, for a given film–substrate system, any value of the ellipsometric function *ρ* can be realized at one particular angle of incidence *ϕ* and an associated infinite series of film thicknesses *d*, *d* + *D*_{ϕ}, *d* + 2*D*_{ϕ}, …. This analysis leads to (1) a unified scheme for the design of all reflection-type optical devices, such as polarizers and retarders, (2) a novel null ellipsometer without a compensator, for the measurement of films whose thicknesses are within certain permissible ranges, and (3) an inversion procedure for the nonlinear equations of reflection ellipsometry that separates the determination of the optical constants (refractive indices and extinction coefficients) of the film and substrate from the film thickness. Extension of the work to absorbing films is discussed.

© 1975 Optical Society of America

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### Equations (34)

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(1)
$$\begin{array}{ll}{R}_{\nu}=\frac{{\mathcal{r}}_{01\nu}+{\mathcal{r}}_{12\nu}{e}^{-j2\beta}}{1+{\mathcal{r}}_{01\nu}{\mathcal{r}}_{12\nu}{e}^{-j2\beta}},\hfill & \nu =p,s\hfill \end{array}$$
(2)
$$\beta =2\pi \left(d/\mathrm{\lambda}\right){\left({N}_{1}^{2}-{N}_{0}^{2}{\text{sin}}^{2}{\varphi}_{0}\right)}^{1/2},$$
(3)
$$\rho ={R}_{p}/{R}_{s},$$
(4)
$$\rho =\text{tan}\psi {e}^{j\Delta},$$
(5)
$$\rho =\frac{{\mathcal{r}}_{01p}+{\mathcal{r}}_{12p}{e}^{-j2\beta}}{1+{\mathcal{r}}_{01p}{\mathcal{r}}_{12p}{e}^{-j2\beta}}\times \frac{1+{\mathcal{r}}_{01s}{\mathcal{r}}_{12s}{e}^{-j2\beta}}{{\mathcal{r}}_{01s}+{\mathcal{r}}_{12s}{e}^{-j2\beta}}.$$
(6)
$$\begin{array}{ll}{R}_{p}=\frac{a+bX}{1+abX},\hfill & {R}_{s}=\frac{c+dX}{1+cdX},\hfill \end{array}$$
(7)
$$\rho =\frac{\left(a+bX\right)\phantom{\rule{0.2em}{0ex}}\left(1+cdX\right)}{\left(1+abX\right)\phantom{\rule{0.2em}{0ex}}\left(c+dX\right)},$$
(8)
$$\rho =\frac{A+BX+C{X}^{2}}{D+EX+F{X}^{2}}.$$
(9)
$$X={e}^{-j2\beta},$$
(10)
$$\begin{array}{ll}\left(a,b\right)=\left({\mathcal{r}}_{01p},{\mathcal{r}}_{12p}\right),\hfill & \left(c,d\right)=\left({\mathcal{r}}_{01s},{\mathcal{r}}_{12s}\right),\hfill \end{array}$$
(11)
$$\begin{array}{lll}A=a,\hfill & B=\left(b+acd\right),\hfill & C=bcd,\hfill \\ D=c,\hfill & E=\left(d+abc\right),\hfill & F=abd.\hfill \end{array}$$
(12)
$$X=\text{exp}\left[-j4\pi \left(d/\mathrm{\lambda}\right)\phantom{\rule{0.2em}{0ex}}{\left({N}_{1}^{2}-{\text{sin}}^{2}\varphi \right)}^{1/2}\right],$$
(13)
$$X=\text{exp}\left[-j2\pi \left(d/{D}_{\varphi}\right)\right],$$
(14)
$${D}_{\varphi}=\frac{\mathrm{\lambda}}{2}{\left({N}_{1}^{2}-{\text{sin}}^{2}\varphi \right)}^{-1/2}.$$
(15)
$$0\le d<{D}_{\varphi}.$$
(16)
$$0\le \varphi \le 90\xb0,$$
(17)
$$0<\varphi <{\varphi}_{s},$$
(18)
$${\varphi}_{s}<\varphi <90\xb0,$$
(19)
$$\rho ={\rho}_{0}={\mathcal{r}}_{02p}/{\mathcal{r}}_{02s},$$
(22)
$$\begin{array}{l}{\mathcal{r}}_{01p}+{\mathcal{r}}_{12p}X=0,\\ X=-\left({\mathcal{r}}_{01p}/{\mathcal{r}}_{12p}\right).\end{array}$$
(23)
$$\left|X\right|=1,$$
(24)
$$\begin{array}{ll}{\mathcal{r}}_{01p}=\left|{\mathcal{r}}_{01p}\right|{e}^{j{\delta}_{01p}},\hfill & {\mathcal{r}}_{12p}=\left|{\mathcal{r}}_{12p}\right|{e}^{j{\delta}_{12p}},\hfill \end{array}$$
(25)
$$-{e}^{-j2\pi \left(d/{D}_{\varphi}\right)}=\left(\left|{\mathcal{r}}_{01p}\right|/\left|{\mathcal{r}}_{12p}\right|\right){e}^{j\left({\delta}_{01p}-{\delta}_{12p}\right)}.$$
(26)
$$-{e}^{-j2\pi \left({d}_{p}/{D}_{\varphi p}\right)}={e}^{j{\left({\delta}_{01p}-{\delta}_{12p}\right)}_{\varphi p}}.$$
(27)
$$\begin{array}{l}-2\pi \left({d}_{p}/{D}_{{\varphi}_{p}}\right)+2\pi \left(m+\frac{1}{2}\right)=\left({\delta}_{01p}-{\delta}_{12p}\right){\varphi}_{p},\\ \left({d}_{p}/{D}_{{\varphi}_{p}}\right)=\frac{1}{2\pi}{\left({\delta}_{12p}-{\delta}_{01p}\right)}_{{\varphi}_{p}}+\left(m+\frac{1}{2}\right),\end{array}$$
(28)
$${d}_{p}=\frac{1}{2\pi}{\left({\delta}_{12p}-{\delta}_{01p}\right)}_{{\varphi}_{p}}{D}_{{\varphi}_{p}}+\left(m+\frac{1}{2}\right){D}_{{\varphi}_{p}}.$$
(29)
$$X=\frac{-\left(B-\rho E\right)\pm {\left[{\left(B-\rho E\right)}^{2}-4\left(C-\rho F\right)\phantom{\rule{0.2em}{0ex}}\left(A-\rho D\right)\right]}^{1/2}}{2\left(C-\rho F\right)},$$
(30)
$$-2\pi \left(d/{D}_{\varphi}\right)+m2\pi =\alpha ,$$
(31)
$$d=-\frac{\alpha}{2\pi}{D}_{\varphi}+m{D}_{\varphi},$$
(32)
$$\rho ={e}^{j\Delta},$$
(33)
$$\text{ln}\left(X\right)=-j2\pi \left(d/{D}_{\varphi}\right),$$
(34)
$$d=j\left(\mathrm{\lambda}/4\pi \right){\left({N}_{1}^{2}-{\text{sin}}^{2}\varphi \right)}^{-1/2}\text{ln}\left(X\right)+\frac{1}{2}m\mathrm{\lambda}{\left({N}_{1}^{2}-{\text{sin}}^{2}\varphi \right)}^{-1/2},$$