|Thin Film Design & Applications|
THEORY (at oblique incidence)
This section is similar to the normal incidence section, but of course it is for oblique incidence (or light coming in at an angle to the surface). When light hits an optical surface at an angle there are two polarization states of the light reflecting and transmitting that have to be accounted for. This section will review Transverse Magnetic (TM) waves of light, or P-polarized light.
Solving for Amplitude Reflectance at Oblique Incidence (P-polarization)
Let's consider a 3-dimensional Axis...
Where the symbols:
¤ = positive y (coming out of page)
Now let us consider light coming and hitting a surface at oblique incidence...
The z axis represents the Optical Axis
In some textbooks, it is easy to remember the direction of the Electric vector by naming P-polarized light as "Plunging" because the Electric vector looks like it is plunging into the surface.
The Magnetic vector can be represented by the equation:
And, we can relate the Electric (E) and Magnetic (H) vectors to their amplitude components by the following relationship:
If we consider the amplitudes of the Electric (E) and Magnetic (H) fields then,
If we combine equations, substituting in for Et and solving for the
and we can see from the diagram at the top of the page that
Similarly, you can solve for the amplitude transmission coefficient for s-polarized light (tp) with:
where we get:
So, we solved for r and t in terms of the optical admittance of both mediums that the light ray comes in contact. We rarely describe films or substrates in terms of the optical admittance, therefore, we must solve for r and t in terms of common values, such as the complex index of refraction (N).
Now, the optical admittance (y) is related to N by:
where Y is the admittance of free space. So, substituting the above
The complex index of refraction can be broken down into a real and imaginary component:
where, n = index of refraction and k = extinction coefficient. k is also the component of the complex index of refraction (N) that represents the inherent absorption in a material.
Now, if we assume that there is no absorption in our films we can solve for the index of refraction (n) by:
and substituting for y in the equation for t we get:
Calculating Reflection & Transmission Coefficients
Solving for Rp and Tp we get: