Reflectivity from a homogeneous layer

In the previous section the equations for reflectivity $r$ and transmission $t$ of an infinite thickness have been derived. The situation is sketched in the figure a. A wave propagating in medium 0 is reflected at an infinitely thick medium 1. The transmitted wave can be described by the wave vector $n$k. Due to boundary conditions, the waves (reflected and transmitted) have to be continuous at the interface and due to the fact that $n$ is smaller than unity, the angle $\alpha^{'}$ will be smaller than $\alpha$.

Figure: Reflection and transmission from an infinitely thick layer (a) and from a layer with finite thickness $\Delta$. The total reflectivity from the finite layer is the sum of infinite numbers of reflections and transmissions. The pathways for waves are shown at the right side.

The same happens in the case of a layer with a finite thickness (medium 1) deposited on a substrate (medium 2). The additional interface between medium 1 and 2 leads again to a reflection of the wave transmitted through medium 1 and a transmission of a wave through medium 2. The reflected wave may interfere with the wave reflected at the interface between medium 0 and 1. On the other hand the wave reflected at the interface 1-2 can be ``backscattered'' at the interface 1-0 and again transmitted and reflected at interface 1-2. This consideration leads to an infinite number of transmitted and reflected waves. Three possible pathways are shown in . The total reflectivity is the sum of all possible pathways. Each arrow in the pathway can be described either as a reflectivity on an interface A-B $r_{AB}$ or as a transmission from medium A to medium B $t_{AB}$. Each addend in the total reflectivity of the layer is then the product of the individual transmissions and reflectivities. The second path in figure demonstrates the following situation: the wave is transmitted through the interface 0-1 ($t_{01}$),reflected from the interface 1-2 ($r_{12}$) and transmitted through the interface 1-0 ($t_{10}$). The resulting contribution to the total reflectivity of the layer will be $t_{01}r_{12}t_{10}p^{2}$. This contribution will interfere with the wave which is reflected at the uppermost interface ($r_{01}$). Due to this the phase factor $p=\exp{\frac {iQ}{\Delta}}$ appears. Taking into account all possible pathways of the wave, the total reflectivity of the slab is

$$r_{slab} = r_{01}+t_{01}r_{12}t_{10}p^{2}+t_{01}t_{10}r_{10}r_{12}^{2}p^{4}+t_{01}t_{10}r_{10}^{2}r_{12}^{3}p^{6}+ ...$$

$$= r_{01}+t_{01}r_{12}t_{10}p^{2}\left\{1+r_{10}r_{12}p^{2}+r_{10}^{2}r_{12}^{2}p^{4}+...\right\}$$

$$= r_{01}+t_{01}r_{12}t_{10}p^{2}\sum^{\infty}_{m=0}(r_{10}r_{12}p^{2})^{m}$$

$$= r_{01}+t_{01}r_{12}t_{10}p^{2}\frac{1}{1-r_{10}r_{12}p^{2}}$$

$$= \frac{r_{01}+r_{12}p^{2}}{1+r_{01}r_{12}p^{2}}$$ (25)

As mentioned the idea of the calculation of the total reflectivity is to let interfere all possible waves with different pathways in the layer. One could imagine that at certain incident angle $\alpha$ the wave reflected at the interface 0-1 is in or out of phase with the wave transmitted through medium 1 and reflected from interface 1-2. This simple consideration explains the typical shape of the reflectivity from a single layer. The reflectivity consists of intensity oscillations known as Kiessig oscillations [23] with a distinct period as shown in figure . The maxima correspond to incident angles where all the waves considered in equation are in phase. The period is first of all defined by the thickness of the layer $\Delta$:

$$\Delta Q=\frac{2\pi}{\Delta}.$$ (26)

Figure: Typical shape of the total reflectivity of a single layer. The reflectivity oscillations (Kiessig oscillations [23]) are the result of the interference of waves reflected from the upper (e.g. vacuum - layer) and lower (e.g. layer - substrate) interface. The periodicity of the oscillations is determined by the wavelength of the reflected radiation $\lambda$ and the thickness of the layer $\Delta$ (equation and ).

Using equation one can receive the angular periodicity (in radians) which is defined by the thickness $\Delta$ of the layer and the wavelength $\lambda$

$$\Delta \alpha=\frac{\lambda}{2\Delta}$$ (27)