Optical Properties of Thin Films on Transparent Surfaces by Ellipsometry; Internal Reflection for Film Covered Surfaces Near the Critical Angle

Abstract

The application of ellipsometry to the determination of the optical properties of thin films on transparent substrates by the use of internal reflection and angles of incidence near the critical angle for total reflection is described and illustrated. Four cases are considered: 1. the angle of incidence, θ_i, is less than either the critical angle for total reflection between the substrate and the film, ${\theta }_c^{1,2},$, and the critical angle between the substrate and the surrounding medium, ${\theta }_c^{1,3};2.{\theta }_c^{1,3}>{\theta }_i>{\theta }_c^{1,2};3.{\theta }_c^{1,3}<{\theta }_i<{\theta }_c^{1,2};$ and $4.{\theta }_c^{1,3}<{\theta }_{ı}>{\theta }_c^{1,2}.$ For case 1, at certain critical values of film thickness d and refractive index n₂, the reflected light may be polarized with its electric vector either entirely in or normal to the plane of incidence. Near these conditions the sensitivity of ellipsometric measurements is extremely high, but the intensity of the reflected light is very low. Except under these conditions the intensity is adequate for experimental measurements, as it is also in case 2. For case 1 it is always possible to determine n₂ and d by a single measurement; for case 2 this is possible only tor thin films. For cases 3 and 4, and case 2 for thick films, only one of these may be determined. Under these conditions, however, the reflectivity is 100 percent.

1. Introduction

The determination of the optical constants of surfaces and the thickness and refractive index of thin films by measurement of the changes in the state of polarization of polarized light upon reflection is based on the well-known equations of Drude [1].¹ The technique for carrying out these measurements is frequently referred to as ellipsometry and has been described in detail elsewhere [e.g., 2, 3, 4].

Most work with ellipsometry has been concerned with studies on metallic surfaces [2, 3], either because of their intrinsic interest or because they form highly reflecting substrates for the study of thin films. However, it is sometimes desirable to measure the properties of thin films on transparent surfaces, in which case the low intensity of the reflected light makes accurate analysis of its state of polarization difficult.

There are two possible systems which would, in general, increase the intensity of the reflected beam over that obtained by direct external reflection from a transparent surface. In one, a thin layer of the transparent material which is to be used as a substrate for the film is placed in intimate contact with a metal, which is used as the reflecting substrate. The transparent material may be placed as a film on a metallic plate, for example by evaporating a film of silica on a metallic substrate in a vacuum evaporator. It is necessary, of course, to determine the optical properties of the metal prior to deposition of the film, and these properties must not change as a result of this deposition. The metal may also be evaporated onto the transparent material. In this case it should be possible to determine the optical properties of the bare metallic surface directly, and, from the other side, the properties of the transparent plate. For this technique the faces of the plate must be sufficiently flat and parallel. This system of a transparent film backed in some manner by a metallic reflecting surface does not differ from the usual system of a metallic reflector with multiple films [5], and will not be discussed further.

A second possible system is one in which it is arranged to have internal reflection at the boundary of the transparent material in question and to use angles near or above the critical angle for total internal reflection. This can be accomplished by constructing a prism of the substrate material or, if other considerations permit, by placing the material in a liquid of higher refractive index. At angles of incidence greater than the critical angle, all the light is reflected and intensity problems are obviated. Although there is total reflection at angles of incidence greater than the critical angle, the beam does penetrate the medium of lower refractive index; this effect has been used to study the infrared spectra of opaque materials [6, 7]. In ellipsometric applications, total reflection permits the use of multiple reflections to increase sensitivity [8], but does lead to the serious limitation that the refractive index and thickness of a thin film may not be determined simultaneously.

At angles between the polarizing angle and the critical angle the intensity of light reflected rises very sharply as the critical angle is approached, and this region may be used quite effectively for ellipsometric studies. For example, the intensity of reflected light as a function of angle of incidence is shown in figure 1 (critical angle=46.68 deg). For an angle of incidence within 0.2 deg of the critical angle, the reflectivity is 60 percent. Although the reflectivity will vary greatly with the thickness and refractive index of any film on the surface, it will be shown that the reflectivity remains quite high for relatively thin films and hence this is a useful region for carrying out ellipsometric measurements on thin films on transparent substrates.

Figure 1. Reflectivity as a function of angle of incidence for internal reflection in the neighborhood of the critical angle.

The calculations were made using the refractive indexes given in table 1.

This paper will illustrate the application of ellipsometry to this type of system by presenting detailed calculations of the changes in the state of polarization of light reflected from a transparent surface covered by a transparent film for certain specific values of angles of incidence and refractive indexes. The calculations were carried out by means of the program described by McCrackin and Colson [9] using the IBM 7094 computer.

2. Reflection at an Interface

As is well known, ρ, the ratio of the reflection coefficient for light polarized with the electric vector in the plane of incidence, r^p, to the reflection coefficient for light polarized with the electric vector normal to the plane of incidence, r^s, may be represented by the equation

$$\rho =\frac{r^p}{r^s}=\text{tan}\psi e^{i\mathrm{\Delta }}$$ (1)

where tan ψ is the ratio of the magnitude of the two coefficients and Δ is the difference in their phases. For a bare surface ρ is simply the ratio of the Fresnel coefficients. For a semi-infinite medium (which we call the “medium”) of refractive n₃ separated by a film of refractive index n₂ from another semi-infinite medium of refractive index n₁ (which we call the “substrate”), the ratio of reflection coefficients for light incident at the n₁−n₂ interface from the n₁ side is given by the exact Drude equation

$$\text{tan}\psi e^{i\mathrm{\Delta }=}=\frac{\frac{r_{12}^p+r_{23}^p\text{exp}D}{1+r_{12}^p{r}_{23}^p\text{exp}D}}{\frac{r_{12}^s+r_{23}^s\text{exp}D}{1+r_{12}^s{r}_{s3}^2\text{exp}D}}$$ (2)

Where r₁₂ and r₂₃ are the Fresnel coefficients at the substrate-film and film-medium interface respectively. D is given by

$$D=-4\pi i{n}_2\text{cos}{\theta }_{2}d/\lambda$$

Where θ₂ is the angle of incidence at the film-medium interface, λ is the wavelength of the light in vacuum, and $i=\sqrt{-1}$. It is clear that if all the parameters in this equation are known but two, say n₂ and d (provided n₂ is real), then a measurement of Δ and ψ permits the determination of these two, and the methods for doing this have been described elsewhere [2, 3].

If total reflection occurs, however, r^p and r^s both have a magnitude of unity, so that ψ is always equal to π/4. This gives only one experimental parameter (Δ) and permits the calculation of only one of the quantities n₂ or d; the other must be determined by other means. This is the most serious limitation imposed by total internal reflection.

3. Parameters of Systems Discussed

We consider only the case in which n₁ is greater than both n₂ and n₃ and all indexes are real. Since light is considered incident at the n₁−n₂ interface from the higher refractive index side, internal reflection occurs. We call the angle of incidence θ_i, and for such a system two critical angles must be considered. These are ${\theta }_c^{1,2},$ the critical angle for total reflection between the substrate and the film, and ${\theta }_c^{1,3},$ the critical angle for total reflection between the substrate and the surrounding medium. Four cases can then be considered: one in which the angle of incidence, θ_i, is less than either ${\theta }_c^{1,2}$ or ${\theta }_c^{1,3}$, two cases in which θ_i is greater than one of these but less than the other, and the case in which θ_i exceeds both these critical angles. Note that the critical angle for reflection at the film-medium interface, ${\theta }_c^{2,3}$, need not be specifically considered, for when θ_i=${\theta }_c^{2,3}$, the angle of incidence at the film-medium interface, θ_i=${\theta }_c^{2,3}$ is equal to ${\theta }_c^{2,3}$, and if θ_i is greater or less than ${\theta }_c^{1,3},$${\theta }_i^{2,3}$ will automatically be greater or less than ${\theta }_c^{2,3}$.

The parameters of the system chosen for calculation are given in table 1. The values for n₁ and n₃ arose out of an experimental study reported elsewhere [10]. The index n₁ pertains to a leaded glass prism immersed in a very dilute solution of an organic compound in water (n = 1.337). The adsorption of this compound onto the glass prism was studied by ellipsometry. Calculations have been carried out for the values of film refractive indexes n₂ shown in the table. The value of 1.20, while not easy to achieve experimentally, was chosen merely to illustrate the behavior when n₂<n₃.

Table 1.

Parameters of system chosen for calculation λ = 5461 Å

n1=1.8376 θc1,3=46.68 deg
n3=1.3370 θp1,3=36.05 deg*
Case No.		θi, deg	n2	θc1,2, deg	θp1,2, deg
1	θi<θc1,2	46.00	1.60–1.36	60.7–47.8	41.1–36.5
	<θc1,3	46.60	1.38–1.36	48.6–47.8	36.9–36.5
2	θi>θc1,2	46.00	1.200,	40.8	33.3
	<θc1,3		1.300	45.0
			2.00
3	θi<θc1,2	46.70	1.60	60.7	41.l
	>θc1,3		1.40	49.6	37.3
			1.35	47.2	36.3
4	θi>θc1,2	46.70	1.20	40.8	33.2
	>θc1,3	50.00	l.40	49.6	37.3
		50.00	1.20	40.8	33.3

^*θ_p denotes the polarizing (Brewester’s) angle.

In all the cases shown we shall be interested in computing the experimentally obtainable quantities ψ and Δ as a function of film thickness. We shall also be interested in the amount of light reflected, and will consider each case in turn.

4. Case 1, ${\theta }_c^{1,3}>{\theta }_i<{\theta }_c^{1,2}$

In this case, when the angle of incidence is less than either of the two critical angles, some interesting polarization effects occur. This is due to the fact that for certain critical values of thickness and refractive index of the film r^s or r^p may go to zero. When this occurs the reflected light is polarized with its electric vector in the plane of incidence (r^s = 0) or normal to the plane of incidence (r^p = 0). The value of tan ψ may therefore be anything from zero (r^p = 0) to infinity (r^s = 0). The vanishing of r^p for dielectric films on dielectric substrates is well known [11, 12], and the vanishing of r^p and r^s for dielectric films on metal surfaces has been discussed by Hartman [13] and Mat tuck [14]. The conditions for the vanishing of r^s for dielectric films on dielectric substrates have not, to our knowledge, been previously studied. In the appendix there are derived the conditions for the vanishing of r^s and r^p. It is shown there that for each case the optical thickness of the film must be such that the phase retardation in the film is an odd multiple of π, i.e.,

$$\frac{2n_{2}d\text{cos}{\theta }_2}{\lambda }=\frac{k}{2}$$ (3)

where k is an odd integer. We shall call the thickness necessary to fulfill this condition the “quarter wave thickness” For this thickness Δ is zero or π, and if the incident light is linearly polarized, the reflected is also.

In addition to this condition, there is a condition on n₂ for the vanishing of r^s or r^p. We shall discuss the former case first.

For the vanishing of r^s, it is shown in the appendix that n₂ must assume a critical value, n_2s, given by the equation

$$n_{2s}^2=n_1{n}_3\text{cos}{\theta }_1\text{cos}{\theta }_3+n_1^2{\text{sin}}^2{\theta }_1$$ (4)

where the symbols have the obvious meanings. For an angle of incidence of 46.00 deg and the other parameters as in table 1, n_2s= 1.41542, and the corresponding thickness from eq (3) is 2700 Å. For these values of the film refractive index and thickness the reflected light will be linearly polarized, but, unlike the condition at Brewster’s angle, the electric vector is in the plane of incidence, not normal to it. Thus tan ψ becomes infinite. We shall now discuss the behavior of ρ, the ratio of reflection coefficients (see eq (1)) with film thickness for two values of refractive index near the critical index n_2s.

A polar plot of ρ is given in figure 2 for n₂ values of 1.41485 and 1.41600, i.e., slightly lower and slightly higher than n_2s. The numbers on the curves correspond to film thicknesses. While the curve for n₂=1.41600 lies entirely to right of the origin, the curve for n₂= 1.41485 goes around the origin. (The very large values of tan ψ for these cases make this somewhat hard to see, and therefore, an enlargement of the region around the origin is shown separately. The value of ψ for these curves at zero thickness is, of course, a constant determined by θ_i, n₁ and n₃.) That is to say, for n₂ values above the critical value, as film thickness increases Δ oscillates between two extreme values, Δ_max and Δ_min, symmetrically disposed with respect to the value of Δ at zero film thickness. For values of n₂ less than n₂, Δ assumes all values from zero to 2π.

Figure 2. Case 1. A polar plot of ρ for n₂ values slightly lower and slightly higher than n_2s (see text).

The insert is an enlargement of the rectangular area at the center. The numbers on the curves correspond to film thickness.

○, n₂ = 1.41485 ●, n₂ = 1.41600

Other parameters as in table 1.

At the quarter wave thickness, the waves reflected from the 2, 3 interface and transmitted into medium 1 are π radians out of phase with both the incident wave and the wave reflected from the 1, 2 interface. For n₂>n_2s the amplitude of the sum of the s waves reflected from the 2, 3 interface and transmitted into medium 1 is less than the amplitude of the s wave reflected at the 1, 2 interface, and hence the total reflected s wave is in phase with the incident wave. For n₂<n_2S the amplitude of the sum of the reflected and retransmitted s waves is greater than the amplitude of the s wave reflected at the 1, 2 interface, and the total reflected s wave is π radians out of phase with the incident wave. Thus Δ at the quarter wave thickness changes from 0 to π as n₂ goes from above n_2s to below.

There is an extremely great dependence of Δ upon film thickness for values of n₂ near the critical refractive index n_2s and for film thicknesses near the quarter-wave thickness, i.e., when Δ = 0 for n₂>n_2s or π for n₂<n_2s. Thus, for example, from the curve for n₂ = 1.41600, a change of thickness from 2688 to 2690 Å produces a change in Δ of 22 deg. This is a change of 11 deg per angstrom. Clearly, ellipsometric measurements under these conditions are exceedingly sensitive to thickness.

The dependence of ψ on thickness is less easy to see from the figure. The calculations indicate that for a film refractive index of 1.41540, i.e., very close to n_2s, a change in thickness from 2694 to 2698 Å changes tan ψ from 80 to 1250. This extreme dependence of tan ψ on thickness indicates that the production of accurately linearly polarized light by this method would be a difficult experimental feat. Moreover, the intensity of light reflected under these polarizing conditions is very low, but we shall defer discussion of this until later in this paper.

We shall now discuss the behavior of ρ when r^p vanishes. For this to occur, eq (3) must be obeyed, and in addition there is a condition on n₂. As shown in the appendix, the values of n₂ necessary to have the r^p vanish, which we call n_2p, are given by

$$n_{2p}^2=\frac{1}{2}\frac{n_1{n}_3}{\text{cos}{\theta }_1\text{cos}{\theta }_3}{\left[1\pm \left(1-\text{sin}2\theta \text{sin}2{\theta }_3\right)\right]}^{1/2}$$ (5)

There are two values of n_2p, corresponding to the positive and negative signs, and we call these $n_{2p}^{+}$ and $n_{2p}^{-}$. For an angle of incidence of 46.00° and n₁ and n₃ as shown in the table, $n_{2p}^{+}$ is 4.654 and out of the range of our consideration. However, $n_{2p}^{-}$ is 1.3786, and the corresponding thickness from eq (3) is 3487 Å. For these values of thickness and refractive index the reflected light is polarized entirely with its electric vector normal to the plane of incidence, and tan ψ = 0.

Figure 3 shows the behavior of ρ with film thickness for η₂ = $n_{2p}^{-}$ and two other values near this value (1.36 and 1.3942). For values of n₂ less than $n_{2p}^{-}$ (but of course greater than n₃) the ρ curve lies between the origin and the value of tan ψ for the bare surface (fig. 3, n₂ = 1.36). Again Δ oscillates between two values, Δ_max and Δ_min, symmetrically disposed with respect to the value for the bare surface. For n₂ equal to $n_{2p}^{-}$, the curve of ρ passes through the origin (tan ψ = 0) and Δ_max and Δ_min are π/2 and −π/2 respectively (fig. 3, n₂ = 1.3786). For values of n₂ larger than $n_{2p}^{-}$ (but smaller than n_2s) Δ takes on all values between 0 and 2π (fig. 3, n₂ = 1.3942).

Figure 3. Case 1. Polar plot of ρ for several values of n₂ between n₁ and n₃.

The three curves drawn with dashed lines represent values of n₂ equal to and near $n_{2p}^{-}$. (See text.) The three curves drawn with solid lines represent values of n₂ between $n_{2p}^{-}$ and n₃. The symbols on the curves correspond to film thicknesses.

O,1000 Å ●, 2000Å □, 2500Å ◐, 3000Å ◑, 4000 Å ◒, 5000Å ◓, 7000Å

For the quarter wave thickness and values of n² between $n_{2p}^{-}$ and n_2s the sum of the amplitudes of the waves reflected from the 2,3 interface and transmitted back into medium 1 is greater than the amplitude of the wave reflected from the 1,2 interface for s waves, but less for p waves. Thus the total reflected s wave is π radians out of phase with the incident wave. For angles of incidence greater than the polarizing angle the reflected p wave however is in phase [2] and Δ is therefore π. For n₂<n^-2p, however, the sum of the amplitudes of the waves reflected from the 2,3 interface and transmitted back into medium 1 is greater than the amplitude of the wave reflected from the 1,2 interface for both the s and p components. Thus the reflected s and p waves are both π radians out of phase with the incident wave and Δ is zero.

It is now easy to reconstruct the behavior of the reflection coefficient ratio for all values of n₂ between n₁ and n₃. For values of n₂ slightly less than n_1, such as the n₂ =1.60 curve in figure 3, the ρ curve lies entirely to the right of the bar e-surf ace point. In this region tan ψ oscillates between a minimum value for the bare surface and a maximum value at the quarter-wave thickness (at which Δ = 0) and Δ in turn oscillates between two values, Δ_max and Δ_min. As the refractive index decreases, the maximum value of tan ψ increases, as do Δ_max and Δ_min. (fig. 3, n₂ = 1.50 and n₂ = 1.46), approaching infinity and ± π/2, respectively as n₂ approaches n_2s (fig. 2, n₂= 1.416). For values of n₂ less than n_2s but greater than $n_{2p}^{-}$, Δ takes on all values between zero and 2π, but the maximum value of tan ψ (occurring when Δ = π) decreases as n₂ becomes progressively lower (fig. 2, n₂ = 1.41485 and fig. 3, n₂=1.3942). At some refractive index (equal to 1.3942 in the present case) between $n_{2p}^{-}$ and n_2s the value of tan ψ for Δ = π is equal to the value for the bare surface. As n₂ decreases from this value, the value of tan ψ when Δ = π is now less than the value for the bare surface, and approaches zero as n₂ approaches $n_{2p}^{-}$. At this point Δ begins to oscillate between a maximum and minimum value, which are π/2 for n₂=$n_{2p}^{-}$ Finally, for n₂ between $n_{2p}^{-}$ and n₃ curves such as those for n₂ = 1.36 in figure 3 are obtained, becoming smaller and smaller as n₂ approaches n₃.

It will be clear that the ρ curves completely cover the plane, those with n₂>n_2s covering the half plane to the right of the bare-surface point and those with n₂<n_2s covering the half plane to the left of the bare-surface point. Thus every conceivable state of polarization of the reflected light is possible when the incident light is linearly polarized at an azimuth of 45 deg. Linearly polarized light at all possible azimuths is represented by the intercepts on the horizontal axis, and this occurs for the quarter-wave thickness. Circularly polarized light is obtained for that refractive index and thickness for which tan ψ = 1 and Δ = ± π/2, and so forth. The efficacy of any of these conditions must, however, be determined by calculations such as those carried out for the special cases n_2s and $n_{2p}^{-}$.

The curves of figure 2 and 3, while illustrating the dependence of ρ upon thickness, do not show clearly the dependence of tan ψ and Δ on thickness. In figure 4 we show how tan ψ varies with d for several values of n₂. All these curves are periodic in 2n₂d cos θ₂/λ. However, since we have chosen certain critical values of n₂, the curves are not as symmetric as those previously described [2]. The values of n₂ chosen are as follows: two (1.41485 and 1.41600) close to n_2s, one (1.3786) at $n_{2p}^{-}$, one at the value of n₂ (1.3942) for which tan ψ at Δ = π is equal to tan ψ for the bare surface, and one value of n₂ (1.50) greater than n_2s. Considering first the case where n₂=$n_{2p}^{-}$, it will be seen that at d = 3500 Å, tan ψ = 0, and hence the reflected light is polarized with the electric vector in the plane of the surface. Similarly, for n₂ values slightly lower and slightly higher than n_2s (1.41485 and 1.41600, respectively) tan ψ becomes extremely large when the thickness is the quarter-wave thickness. Since the sensitivity (which may be defined as the change of ψ with thickness [3,15]) is the slope of these curves, it will be recognized that n₂ values near n_2s and thickness values around the quarter-wave thickness give the highest sensitivity. Moreover, the curves for n₂ = 1.3786 and 1.41485 and 1.41600 again show the difficulty of constructing a polarizer using films of these critical refractive indexes; the change of tan ψ with thickness near the proper thickness is almost discontinuous, so that to make a worthwhile polarizer using these effects would be very difficult indeed.

Figure 4. Case 1. Dependence of tan ψ on film thickness for several values of n₂ shown in figures 2 and 3.

In figure 4 we have also shown the curve for the value of n₂ for which tan ψ has the least variation with thickness, with the exception of values of n₂ very close to n₁or n₃. This is the value of 1.3942, and for this value, tan ψ when Δ = π (the quarter-wave thickness) is equal to the value of tan ψ for the bare surface. As can be seen from figure 3 this ρ curve is almost circular and centered at the origin. The total variation in tan ψ is from 0.743 to 0.650, or, in degrees, from 36.66 to 33.0; this change of 3.7 deg is to be compared with the change of 45 deg obtained for n₂=n_2s or n₂ =$n_{2p}^{-}$.

In figures 5 and 6 we illustrate the behavior of Δ with thickness for the same values of n₂ shown in figure 4. For n₂>n_2s or n₂<$n_{2p}^{-}$, Δ oscillates between two values. This behavior is typified by the curve for n₂ = 1.50 (which is, however, somewhat more asymmetric than previously shown curves) and has been illustrated many times [2]. As n₂ approaches n_2s from above or $n_{2p}^{-}$ from below, the curve of Δ versus thickness becomes more and more asymmetric, finally approaching a discontinuous curve where n₂=$n_{2p}^{-}$ or n_2s. This is illustrated by the curves for n₂ = 1.3786 and 1.41600 in figure 5. In these curves as the “half-wave” thickness is passed, Δ suffers a very sharp change from −π/2 to π/2, and since ψ under these conditions is close to either zero or π/2, the ellipse which represents the polarization of the reflected light goes very rapidly from a very thin ellipse with one sense of rotation to a similar ellipse with the opposite sense of rotation.

Figure 5. Case 1. Dependence of Δ on film thickness for the three values of n₂ shown in figure 4 for which n₂ ≥ n_2s or n₂ ≤ n⁻2p.

Figure 6. Case 1. Dependence of Δ on film thickness for the two values of n₂ shown in figure 4 for which $n_{2p}^{-}$ <n₂<n_2s.

The behavior is decidedly different for n^-_2p<n₂<n_2s. Under these conditions Δ takes on all values from 2 π to 0. This is illustrated in figure 6 for the refractive indices 1.41485 and 1.3942. The value of 1.41485 is slightly less than n_2s, and the curve of Δ versus thickness for this value is to be compared with than for n₂ = 1.41600 in figure 5. In the present case, Δ starts from 2π, approaches 3π/2 as the thickness approaches the quarter-wave value, but then drops almost discontinuously to π/2 as the thickness becomes greater than the quarter-wave value. The curve for n₂=1.3942 illustrates the behavior for intermediate values of n₂ between $n_{2p}^{-}$ and n_2s. As n₂ approaches $n_{2p}^{-}$, again the curve of Δ versus thickness becomes more and more asymmetric, approaching a discontinuous curve as n₂ approaches $n_{2p}^{-}$.

We have finally to discuss the intensity of the reflected light for these various situations. This is shown in figure 7 as a function of thickness for the same refractive indexes used for figures 2 to 6. The reflectivity shown here is calculated for “ellipsometric” conditions, i.e., the incident light is linearly polarized at an azimuth of 45 deg to the plane of incidence, and is given by (r_p²+r_s²)/2. Starting from a bare-surface value of 41 percent, the reflectivity drops to a minimum value at the quarter-wave thickness. For n₂ dose to n_2s or n_2p this minimum value is very low indeed, being of the order of 2 to 3 percent. With the present values of refractive index for n₁ and n₃, production of polarized light by such dielectric films would be an inefficient process. More propitious values of n₁ and n₃ do, however, give higher reflectivity, at least for $n_{2p}^{+}$ [12]. Increasing the angle of incidence to 46.60 deg (only 0.08 deg less than ${\theta }_c^{1,3}$) gave n_2s = 1.3679, increased the bare-surface reflectivity to 72.8 percent, but increased the minimum reflectivity to only 3.3 per-cent. Nevertheless, for films less than about 1000 Å the reflectivity for an angle of incidence of 46.00 deg is 30 percent or greater, and hence the intensity of reflected light is sufficient for experimental measurements under these conditions.

Figure 7. Case 1. Dependence of reflectivity on film thickness for several values of n₂ shown in figures 2 to 6.

The reflectivities are calculated for incident light at 45 deg azimuth with respect to the plane of incidence.

5. Case 2, ${\theta }_c^{1,3}>{\theta }_i>{\theta }_c^{1,2}$

This case is known as “Frustrated Total Reflection” and has already been extensively treated [16]. For this case n₂ must be less than n₃, and Δ and ψ are shown in figure 8 as a function of film thickness for the two values of n₂ given in table 1. For large film thicknesses tan ψ must approach unity, and Δ approaches some value determined by the angle of incidence and the refractive indexes in question. For n₂ = 1.20 this value of Δ is 44 deg, while for n₂ = 1.30 it is 19.5 deg for this angle of incidence (46.0 deg). For very thin films whose refractive index is such as to make the present conditions apply, it is possible to measure thickness and refractive index by a single measurement of Δ and ψ. For the present conditions the minimum reflectivity (occurring when the films are of zero thickness) is 41 percent, and this rises as the film increases in thickness.

Figure 8. Case 2. Frustrated total reflection.

Variation of Δ and ψ with film thickness.

6. Case 3, ${\theta }_c^{1,3}<{\theta }_i<{\theta }_c^{1,2}$

For this case n₂>n₃, and tan ψ is always unity. The reflectivity is therefore always 100 percent. However, the only experimental parameter which varies is Δ, and this permits only the evaluation of n₂ or d, but not both; one must be known from other considerations. The behavior of Δ with film thickness is shown in figure 9 for several values of n₂. As in case 1, Δ oscillates between maximum and minimum values with a periodicity of (2 n₂d cos θ)/λ. However, since tan ψ is always unity none of the interesting polarization effects that occurred in case 1 occur here. As is evident from figure 9, Δ is more sensitive to thickness for those refractive indexes which make θ_c^1,2 close to the angle of incidence. The curves of Δ against thickness for these refractive indexes (1.40 and 1.35 in fig. 9) are asymmetric as were the curves in figure 5. We have calculated for this case the behavior of Δ with film thickness for one value of n₂>n₁ (n₂ = 2.0), and except for the fact that the “phase” of this curve is different from those with n₂<n₁, the behavior is very similar.

Figure 9. Case 3. Dependence of Δ on film thickness for several values of n₂, including the condition n₂ > n₁.

7. Case 4, ${\theta }_c^{1,3}<{\theta }_i>{\theta }_c^{1,2}$

This is the case which is easiest to achieve in practice [10], for all that it necessitates is that θ_i be sufficiently high. As in case 3 only Δ varies with film thickness and thus from a single determination of ψ and Δ only the refractive index or thickness of the film may be determined; one must be known from other considerations. Figure 10 shows the result of computations of Δ as a function of film thick-ness for the three cases shown in table 1. As expected, the curves level off and show none of the periodicity present in cases 1 and 3. It is to be noticed that Δ rises with film thickness for n₂<n₃ and falls for n₂>n₃. Thus it is not strictly correct to say that only thickness or refractive index can be determined; by comparing the value of Δ for the film-covered surface with that for the bare surface one can determine whether the film has a refractive index which is greater or less than the refractive index of the medium 3. Beyond this one cannot in general say from a single measurement.

Figure 10. Case 4. Dependence of Δ on film thickness for n₂ < n₃ and n₂ > η₃.

8. Sensitivity

We have spoken of the desirability in cases 1 and 2 of using an angle of incidence near ${\theta }_c^{1,3}$ in order to increase reflectivity. There are other criteria, however, which influence the choice of angle of incidence. In case 3 and case 4, in particular, where total reflection occurs, there is the question of experimental sensitivity. This may be defined as the change in Δ for a given change in film thickness [3, 15], and, for very thin films, a determination of this quantity as a function of angle of incidence leads to an unequivocal result. In figure 11 we show the change in Δ for a 100 Å film as a function of angle of incidence for n₂ values of 1.20, 1.40, and 1.60. Two of the curves span cases 1, 3, and 4, and one spans cases 2 and 4. For each of the curves it is seen that the sensitivity rises as the angle of incidence is lowered. Without considerations of intensity one would conclude that the lowest angle of incidence should be used. However, with an angle of incidence less than ${\theta }_c^{1,3}$ the conditions are those of case 1 or 2, and reflectivity falls off very rapidly as the angle of incidence is lowered. (The minimum angle of 44° in this graph was chosen to give a reflectivity of about 17 percent for zero film thickness.) Balancing reflectivity against sensitivity one is led to the conclusion that it is always best to work as close to ${\theta }_c^{1,3}$ as possible. If θ_i is slightly less than ${\theta }_c^{1,3}$ then, for thin films such as these, one has the added advantage of having ψ also vary with film thickness. When n₂>n₃, choosing θ_i slightly above ${\theta }_c^{1,3}$ gives optimum sensitivity and complete reflectivity, although only Δ varies. While it is not our purpose to pursue the effect of n₂ on sensitivity, it will be clear that sensitivity will be zero for n₂ equal to n₁ or n₃, be a maximum for some value of n₂ between n₁ and n₃, and again increase for n₂>n₁ or n₂<n₃. This is the reason that the curves for n₂ of 1.20 and 1.60 in figure 11 are higher than the curve for an n₂ of 1.40.

Figure 11. Sensitivity in Δ as a function of θ_i for three values of n₂.

Each of the curves spans at least two cases. Solid line: case 4 case 3; case 2; … case 1.

9. Appendix

It will be shown that for reflection from a film covered surface either r^p or r^s may be zero, and hence tan ψ may be either zero or infinity.

The reflection coefficient r^v, where v represents either s or p is given by

$$r^{\nu }=\frac{r_{12}^{\nu }+r_{23}^{\nu }\text{exp}D}{1+r_{12}^{\nu }{r}_{23}^{\nu }\text{exp}D}$$ (A1)

where the terms have the same meaning as in eq (2). In order for this reflection coefficient to vanish, two conditions must be met:

$$D=-i\pi ;\frac{2n_{2}d\text{cos}{\theta }_2}{\lambda }=\frac{k}{2}$$ (A2)

$$r_{12}^{\nu }=r_{23}^{\nu }$$ (A3)

where k is an odd integer. For these conditions satisfied and v = p, tan ψ = 0; for v = s, tan ψ is infinite. The first state has been studied already [11, 12] but we shall be interested in more detail. We shall discuss each of the two states in turn.

State 1—vanishing 0f component with electric vector in plane of incidence

For this state, v = p, and the Fresnel coefficients are given by

$$r_{ij}^p=\frac{n_j\text{cos}{\theta }_i-n_i\text{cos}{\theta }_j}{n_j\text{cos}{\theta }_i+n_i\text{cos}{\theta }_j}$$ (A4)

where θ_j is the angle between the normal to the i, j interface and the ray in medium j. Substituting this expression into eq (A3) and simplifying leads to the following equation for n_2p, the critical value of n₂ necessary to make the p component vanish:

$$n_{2p}^2=\frac{1}{2}\frac{n_1{n}_3}{\text{cos}{\theta }_1\text{cos}{\theta }_3}\left[1\pm {\left(1-\text{sin}2{\theta }_1\text{sin}2{\theta }_3\right)}^{1/2}\right]$$ (A5)

This equation has already been derived by Schröder [12] in slightly different form. There are two solutions, which we call $n_{2p}^{+}$ and $n_{2p}^{-}$, and we shall investigate certain of their properties.

The critical value n_2p is symmetrically dependent upon n₁, n₂, θ₁, and θ₃. At normal incidence, (θ₁ = 0, θ₃ = 0), we obtain the well-known solution [11]

$$n_{2p}^{+}={\left(n_1{n}_3\right)}^{1/2}$$ (A6)

for the positive solution, and $n_{2p}^{-}$ =0 for the negative solution. The latter is, of course, inadmissible. For n₁<n₂ may take values up to π/2, and as θ₁ approaches π/2, $n_{2p}^{+}$ increases without bound. For n₁>n₃, θ₃ approaches π/2 as θ₁ approaches ${\theta }_c^{1,3}$, the critical angle for total internal reflection, and $n_{2p}^{+}$ increases without limit as θ₁ approaches ${\theta }_c^{1,3}$. It may be shown [12] that the behavior of $n_{2p}^{+}$ as a function of θ₁ is monotonic. Hence, for the positive solution

$$n_{2p}^{+}\ge {\left(n_1{n}_3\right)}^{1/2}$$ (A7)

the equality sign holding at normal incidence.

For the parameters studied here, i.e., θ₁ = 46°, n₁=1.8376, n₃ = 1.337, $n_{2p}^{+}$= 4.654. For this value of $n_{2p}^{+}$, d is calculated from eq (A1) as 304 Å for λ = 5461 Å.

For the negative solution, $n_{2p}^{-}$=0 at normal incidence, and becomes (n₁ n₃ sin θ₃)^1/2 at grazing incidence. The value of $n_{2p}^{-}$ is always less than (n₁n₃)^1/2, which in many cases of interest is too low to be realized experimentally. For the present case, at an angle θ₁ of 46°, n₁ = 1.8376 and n₃ = 1.337, $n_{2p}^{-}$ = 1.3786, which is easily attainable. For this refractive index, the critical thickness is 3490 Å for λ = 5461 Å, and r^s = 0.25.

It is of interest to calculate the value of θ_i for which $n_{2p}^{-}$ is unity. This, of course, represents the minimum angle of incidence for a given n₁ and n₃ for which the value of $n_{2p}^{-}$ is admissible. A certain amount of algebraic manipulation gives

$${\text{sin}}^2{\theta }_1=\frac{n_3^2+n_1^2-2n_1^4{n}_3^4}{2n_1^2\left[1-n_1^4{n}_3^4\right]}-\frac{{\left[{\left(n_3^2-n_1^2\right)}^2+4n_1^4{n}_3^4\left(n_1^2-1\right)\left(n_3^2-1\right)\right]}^{1/2}}{2n_1^2\left[1-n_1^4{n}_3^4\right]}.$$ (A8)

Except for the presence of the term $n_1^2$ in the denominator this expression is symmetrical in n₁ and n₃. Angles of incidence less than the value given by this equation cannot produce linearly polarized light with the electric vector normal to the plane of incidence for the negative solution to eq (A5).

State 2—vanishing of component with electric vector normal to plane of incidence

Although mentioned by Schröder [12], this case does not appear to have been studied. Indeed, it has been considered impossible [11]. For this case the Fresnel coefficient is

$$r_{ij}^s=\frac{n_i\text{cos}{\theta }_i-n_j\text{cos}{\theta }_j}{n_i\text{cos}{\theta }_i+n_j\text{cos}{\theta }_j}.$$ (A9)

Substituting this in eq (A3) leads to the following equation for the critical value of n₂ for the reflected light to be polarized entirely with its electric vector in the plane of incidence (n_2s):

$$n_{2s}^2=n_1{n}_3\text{cos}{\theta }_1\text{cos}{\theta }_3+n_1^2{\text{sin}}^2{\theta }_1.$$ (A10)

In this case there is only one value of n_2s which is admissible. At normal incidence we again obtain n_2s = (n₁n₃)^1/2.

Not all combinations of n₁ and n₂ are possible in this case. To show this we recast eq (A9) into the following form

$$\frac{n_2^2}{n_1^2}={\text{sin}}^2{\theta }_1\left[\frac{1}{\text{tan}{\theta }_1\text{tan}{\theta }_3}+1\right].$$ (A11)

Now, if n₂> n₁, then

$${\text{sin}}^2{\theta }_1\left[\frac{1}{\text{tan}{\theta }_1\text{tan}{\theta }_3}+1\right]>1$$

or

$$\text{tan}{\theta }_3<\text{tan}{\theta }_1$$ (A12)

and hence n₃> n₁.

A similar calculation for n₂<n₁ leads to the condition n₃<n₁. An obvious way to satisfy these conditions is for n₁, n₂, and n₃ either to increase or decrease in order. For the case of n₁ = 1.337, n₃= 1.8376, θ₁ = 46° (which is the opposite of the direction of propagation treated in the text), direct computation gives n₂ = 1.543 and d₂=1130 for λ=5461. For these values the reflected light is polarized entirely with its electric vector in the plane of incidence. The value of r^p is, however, only 0.032. For the opposite direction of propagation (n₁ = 1.8376, n₃ = 1.337) n₂ = 1.41542.