## Abstract

The ratio *ρ*_{t} = *T*_{p}/*T*_{s} of the complex amplitude transmission coefficients for the *p* and *s* polarizations of a transparent unbacked or embedded thin film is examined as a function of the film thickness-to-wavelength ratio *d*/λ and the angle of incidence *ϕ* for a given film refractive index *N*. The maximum value of the differential transmission phase shift (or retardance), Δ_{t} = arg*ρ*_{t}, is determined, for given *N* and *ϕ*, by a simple geometrical construction that involves the iso–*ϕ* circle locus of *ρ*_{t} in the complex plane. The upper bound on this maximum equals arctan{[*N* − (1/*N*)]/2} and is attained in the limit of grazing incidence. An analytical noniterative method is developed for determining *N* and *d* of the film from *ρ*_{t} measured by transmission ellipsometry (TELL) at *ϕ* = 45°. An explicit expression for Δ_{t} of an ultrathin film, *d*/λ ≪ 1, is derived in product form that shows the dependence of Δ_{t} on *N*, *ϕ*, and *d*/λ separately. The angular dependence is given by an obliquity factor, *f*_{o}(*ϕ*) = 2^{1/2} sin*ϕ* tan*ϕ*, which is verified experimentally by TELL measurements on a stable planar soap film in air at λ = 633 nm. The singularity of *f*_{o} at *ϕ* = 90° is resolved; Δ_{t} is shown to have a maximum just short of grazing incidence and drops to 0 at *ϕ* = 90°. Because *N* and *d*/λ are inseparable for an ultrathin film, *N* is determined by a Brewster angle measurement and *d*/λ is subsequently obtained from Δ_{t}. Finally, the ellipsometric function in reflection *ρ*_{r} is related to that in transmission *ρ*_{t}.

© 1991 Optical Society of America

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### Equations (36)

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(1)
$${\rho}_{t}=[(1-{r}_{p}^{2})/(1-{r}_{s}^{2})][(1-{r}_{s}^{2}X)/(1-{r}_{p}^{2}X)],$$
(2)
$$X=\text{exp}[-j4\pi (d/\lambda ){S}_{1}],$$
(3)
$${S}_{1}={({N}_{1}^{2}-{N}_{0}^{2}\hspace{0.17em}{\text{sin}}^{2}\varphi )}^{1/2}.$$
(4)
$${N}_{1}>{N}_{0}\hspace{0.17em}\text{sin}\varphi ,$$
(5)
$$X=\text{exp}(-j2\pi \zeta ).$$
(6)
$$\zeta =d/{D}_{\varphi},$$
(7)
$${D}_{\varphi}=\lambda /2{S}_{1}$$
(8)
$$N={N}_{1}/{N}_{0},$$
(9)
$${\rho}_{t}^{+}=1,$$
(10)
$${\rho}_{t}^{-}=[(1-{r}_{p}^{2})/(1+{r}_{p}^{2})]/[(1-{r}_{s}^{2})/(1+{r}_{s}^{2})].$$
(11)
$$\text{tan}{\mathrm{\Delta}}_{t\text{max}}=\hspace{0.17em}\mid CT\mid /\mid OT\mid \hspace{0.17em}=\hspace{0.17em}\mid CT\mid /{(\mid OA\mid \hspace{0.17em}\mid OB\mid )}^{1/2}=\hspace{0.17em}\mid CT\mid /\mid OB{\mid}^{1/2},$$
(12)
$${\mathrm{\Delta}}_{t\text{max}}=\text{arctan}[({r}_{s}^{2}-{r}_{p}^{2})/{(1-{r}_{s}^{4})}^{1/2}{(1-{r}_{p}^{4})}^{1/2}].$$
(13)
$${\widehat{\mathrm{\Delta}}}_{t\text{max}}=\text{arctan}\{[N-(1/N)]/2\}.$$
(14)
$$\zeta =(1/2\pi )\hspace{0.17em}\text{arc}\hspace{0.17em}\text{cos}[({r}_{p}^{2}+{r}_{s}^{2})/(1+{r}_{p}^{2}{r}_{s}^{2})]$$
(15)
$$X=[{\rho}_{t}(1-{r}_{s}^{2})-(1-{r}_{p}^{2})]/[{\rho}_{t}{r}_{p}^{2}(1-{r}_{s}^{2})-{r}_{s}^{2}(1-{r}_{p}^{2})]$$
(16)
$$\mid X\mid \hspace{0.17em}=1,$$
(17)
$$\mid {\rho}_{t}-(1+{R}_{s})\mid \hspace{0.17em}=\hspace{0.17em}\mid {R}_{s}^{2}{\rho}_{t}-{R}_{s}(1+{R}_{s})\mid ,$$
(18)
$${R}_{s}={r}_{s}^{2},$$
(19)
$${\rho}_{t}=a+jb,$$
(20)
$$=\text{tan}{\psi}_{t}(\text{cos}{\mathrm{\Delta}}_{t}+j\hspace{0.17em}\text{sin}{\mathrm{\Delta}}_{t}),$$
(21)
$${R}_{s}^{2}-2\mu {R}_{s}+1=0,$$
(22)
$$\mu =(a-1)/[{(a-1)}^{2}+{b}^{2}].$$
(23)
$${R}_{s}=\mu \pm {({\mu}^{2}-1)}^{1/2},$$
(24)
$$N={(1+{R}_{s})}^{1/2}/(1+{R}_{s}^{1/2}).$$
(25)
$$d=\{[-\text{arg}X/(2\pi )]+m\}{D}_{\varphi},$$
(26)
$${\mathrm{\Delta}}_{t}=2\pi \zeta \{[{r}_{s}^{2}/(1-{r}_{s}^{2})]-[{r}_{p}^{2}/(1-{r}_{p}^{2})]\},$$
(27)
$${\mathrm{\Delta}}_{t}/(2\pi )=[{({N}_{1}^{2}-{N}_{0}^{2})}^{2}/2{N}_{0}{N}_{1}^{2}](\text{sin}\varphi \hspace{0.17em}\text{tan}\varphi )(d/\lambda ).$$
(28)
$${f}_{o}(\varphi )={\mathrm{\Delta}}_{t}(\varphi )/{\mathrm{\Delta}}_{t}(45\xb0),$$
(29)
$$={2}^{1/2}\hspace{0.17em}\text{sin}\varphi \hspace{0.17em}\text{tan}\varphi .$$
(30)
$${f}_{n}(N)=[{\mathrm{\Delta}}_{t}(45\xb0)/2\pi ]/(d/\lambda ),$$
(31)
$$=(1/{2}^{1.5}){[N-(1/N)]}^{2},$$
(32)
$${\rho}_{r}=G(N,\varphi ){\rho}_{t},$$
(33)
$$G(N,\varphi )=({r}_{p}/{r}_{s})[(1-{r}_{s}^{2})/(1-{r}_{p}^{2})]$$
(34)
$${\mathrm{\Delta}}_{r}={\mathrm{\Delta}}_{t}\pm \pi ,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\varphi <{\varphi}_{B},$$
(35)
$${\mathrm{\Delta}}_{r}={\mathrm{\Delta}}_{t},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\varphi >{\varphi}_{B}.$$
(36)
$$\text{tan}{\psi}_{r}=\hspace{0.17em}\mid G\mid \text{tan}{\psi}_{t},$$