Oblique incidence

Now let us consider oblique incidence, still retaining our absorption-free media.

For any general direction of the vector amplitude of the incident wave we quickly

find that the application of the boundary conditions leads us into complicated and difficult expressions for the vector amplitudes of the reflected and transmitted waves. Fortunately there are two orientations of the incident wave which lead to reasonably straightforward calculations: the vector electrical amplitudes aligned in the plane of incidence (i.e. the x y plane of figure 2.1) and the vector electrical amplitudes aligned normal to the plane of incidence (i.e. parallel to the y axis in figure 2.1). In each of these cases, the orientations of the transmitted and reflected vector amplitudes are the same as for the incident wave. Any incident wave of arbitrary polarisation can therefore be split into two components having these simple orientations. The transmitted and reflected components can be calculated for each orientation separately and then combined to yield the resultant. Since, therefore, it is necessary to consider two orientations only, they have been given special names. A wave with the electric vector in the plane of incidence is known as p-polarised or, sometimes, asTM(for transverse magnetic) and a wave with the electric vector normal to the plane of incidence as s-polarised or, sometimes,TE

(for transverse electric). The p and s are derived from the German parallel and senkrecht (perpendicular). Before we can actually proceed to the calculation of the reflected and transmitted amplitudes, we must choose the various reference directions of the vectors from which any phase differences will be calculated.

We have, once again, complete freedom of choice, but once we have established the convention we must adhere to it, just as in the normal incidence case. The conventions which we will use in this book are illustrated in figure 2.3. They have been chosen to be compatible with those for normal incidence already established. In some works, an opposite convention for the p-polarised reflected beam has been adopted, but this leads to an incompatibility with results derived for normal incidence, and we prefer to avoid this situation. (Note that for reasons connected with consistency of reference directions for elliptically polarised light, the normal convention in ellipsometric calculations is opposite to that of figure 2.3 for reflected p-polarised light. When ellipsometric parameters are compared with the results of the expressions we shall use, it will usually be necessary to introduce a shift of 180^◦in the p-polarised reflected results.)

We can now apply the boundary conditions. Since we have already ensured that the phase factors will be correct, we need only consider the vector amplitudes.

2.2.2.1 p-polarised light

(a) Electric component parallel to the boundary, continuous across it:

Eicosϑ0+^Ercosϑ0=^Etcosϑ1. (2.50) (b) Magnetic component parallel to the boundary and continuous across it. Here we need to calculate the magnetic vector amplitudes, and we can do this either by using equation (2.28) to operate on equation (2.50) directly, or, since the magnetic vectors are already parallel to the boundary we can use figure 2.3 and then convert,

Figure 2.3. (a) Convention defining the positive directions of the electric and magnetic vectors for p-polarised light (TMwaves). (b) Convention defining the positive directions of the electric and magnetic vectors for s-polarised light (TEwaves).

since^H= y^E:

y₀^E_i− y0^Er= y1^Et. (2.51) At first sight it seems logical just to eliminate first^Etand then^Erfrom these two equations to obtain^Er/^Eiand^Et/^Ei

but when we calculate the expressions which result, we find that R+ T = 1. In fact, there is no mistake in the calculations. We have computed the irradiances measured along the direction of propagation of the waves and the transmitted wave is inclined at an angle which differs from that of the incident wave. This leaves us with the problem that to adopt these definitions will involve the rejection of the(R + T = 1) rule.

We could correct this situation by modifying the definition of T to include this angular dependence, but an alternative, preferable and generally adopted approach is to use the components of the energy flows which are normal to the boundary. The^E and^H vectors that are involved in these calculations are then parallel to the boundary. Since these are those that enter directly into the boundary it seems appropriate to concentrate on them when we are dealing with the amplitudes of the waves.

The thin-film approach to all this, then, is to use the components of ^E and ^H parallel to the boundary, what are called the tangential components, in the expressions ρ and τ that involve amplitudes. Note that the normal approach in other areas of optics is to use the full components of ^E and ^H in amplitude expressions but to use the components of irradiance in reflectance and transmittance. The amplitude coefficients are then known as the Fresnel coefficients. The thin-film coefficients are not the Fresnel coefficients except at normal incidence, although the only coefficient that actually has a different value is the amplitude transmission coefficient for p-polarisation.

The tangential components of^Eand^H, that is, the components parallel to the boundary, have already been calculated for use in equations (2.50) and (2.51).

However, it is convenient to introduce special symbols for them: E and H . Then we can write

The orientations of these vectors are exactly the same as for normally incident light.

Equations (2.50) and (2.51) can then be written as follows.

(a) Electric field parallel to the boundary:

Ei+Er=Et

(b) Magnetic field parallel to the boundary:

y₀

giving us, by a process exactly similar to that we have already used for normal incidence,

where y0= n0^Yand y1= n1^Yand the(R + T = 1) rule is retained. The suffix p has been used in the above expressions to denote p-polarisation.

It should be noted that the expression for τp is now different from that in equation (2.52), the form of the Fresnel amplitude transmission coefficient.

Fortunately, the reflection coefficients in equations (2.52) and (2.58) are identical, and since much more use is made of reflection coefficients confusion is rare.

2.2.2.2 s-polarised light

In the case of s-polarisation the amplitudes of the components of the waves parallel to the boundary are

Ei=^Ei Hi=^Hicosϑ0= y0cosϑ0Ei

Er=^Er Hr=^Hrcosϑ0= y0cosϑ0Er

Et=^Et Ht= y1cosϑ1Et

and here we have again an orientation of the tangential components exactly as for normally incident light, and so a similar analysis leads to

ρs= Er

Ei = (y0cosϑ0− y1cosϑ1)/(y0cosϑ0+ y1cosϑ1) (2.60) τs= Et

Ei = (2y0cosϑ0)/(y0cosϑ0+ y1cosϑ1) (2.61) R_s= [(y0cosϑ0− y1cosϑ1)/(y0cosϑ0+ y1cosϑ1)]² (2.62) T_s= (4y0cosϑ0y₁cosϑ1)/(y0cosϑ0+ y1cosϑ1)² (2.63) where once again y₀= n0^Yand y₁= n1^Yand the(R + T = 1) rule is retained.

The suffix s has been used in the above expressions to denote s-polarisation.