Figure 2.                                                       



         














                                 Figure 3.




Normal and parallel incidence and Brewster’s angle

Reflection coefficients are measured at the incidence angles between 0º (normal incidence) and 90º (parallel to the surface). At normal incidence q_i = q_r = 0 reflection co-efficients of all polarizations are equal:

The above-mentioned case represents fused silica represents (n₁ = 1; n₂ = 1.40) R_P = R_S = R₄₅ = 2.78 % at normal incidence conditions, while sapphire represents (n₁ = 1; n₂ = 1.785) R_P = R_S = R₄₅ = 7.94 %.

For the parallel incidence (q_i = 90º) all light is reflected regardless of refraction indices n₁ and n₂ and R_P = R_S = R₄₅ = 100%.

Another Special Scenario

Let's say at a certain incidence angle the P-polarized component of a light wave is not reflected from the surface and the resulting light beam is getting strictly S-polarized. This angle is called Brewster’s angle, in memory of Scottish physicist David Brewster:

The above-mentioned case represents fused silica (n₁ = 1; n₂ = 1.40) Brewster’s angle q_B = 54.46º (see Fig.2), while the sapphire represents (n₁ = 1; n₂ = 1.785) q_B = 60.74º (see Fig.3).

Total internal reflection (TIR)

When light moves from the medium with a higher refraction index to the medium with a lower refraction index (n₁ > n₂), the above equations have no solution above a certain angle of incidence which known as the critical angle q_CR. Above this angle all light is reflected back to the first medium and no transmission takes place. This effect underlies the principle of fiber optics design. The critical angle can be derived from the equation:

Again, we have calculated reflection coefficients R_P, R_S and R₄₅ for the incidence angles from 0 to 90º for the monochromatic NIR light beam with wavelength l = 3700 nm entering air (n = 1) from the fused silica (n = 1.40). As you can see from Fig.4 all reflection co-efficients reach 100% at the critical angle q_CR = 45.58º. All incidence angles higher than this will result in total internal reflection. Note that Brewster’s angle is still present in this situation and equals 35.54º.

For the visible monochromatic light with wavelength l = 400 nm entering sapphire (n = 1.785) from the air (n = 1) the critical angle q_CR = 34.07º. Brewster’s angle q_B = 29.26º.

                            Figure 4.    







                                                  










                             Figure 5.


Mirror and diffuse reflection

In the real world almost every surface or media interface has structural irregularities. When the size of these irregularities is smaller than the light wavelength l, there is a mirror reflection also called a "specular reflection." When the mirror reflection occurs, the reflected beam follows the laws of optics described below:

- reflected beam lays in the same plane with incident beam and the normal to the surface;

- angle of reflection equals the angle of incidence;

- Fresnel’s equations for reflection coefficients are correct and so on.

Therefore, in practical terms, we will know exactly were to expect the maximum intensity of the reflected beam and what the intensity is going to be.

The diffuse reflection occurs when arbitrarily situated irregularities of the surface are larger than the light wavelength l. In this case each structural irregularity acts as an independent micro-surface. The multitude of such irregularities reflects light in all possible directions. Truly diffuse reflection is described by the Lambert’s law according to which the intensity of light from a diffusely reflecting surface is uniform in all directions. Objects following this law are called "Lambertians." A good example of a Lambertian is a matte piece of paper; however, most real-life objects exhibit both types of reflection.