1. Wave reflection and refraction1

2. Plane harmonic wave at boundaryki kr kt z
z=0
must hold for any point (x, y) at plane z=0, hence or
the three wave vectors have identical projection
on the boundary plane
Feature 1：The incident wave vector can always be selected in a plane (y=0) perpendicular to the boundary plane (z=0). As such, the reflected and refracted wave vectors must be in the same plane (y=0), since kiy=0 requires
kry= kty=0 as well. This plane (y=0) is called the incident plane.
Feature 2: or
i.e.,
Snell’s law

3. Plane harmonic wave at boundaryki kr kt z
z=0 E H
E-field parallel to the boundary (s-wave): or
Hence

4. Plane harmonic wave at boundary
H-field parallel to the boundary (p-wave): z ki kr
z=0 or kt H E
Hence

5. Dielectric-dielectric boundary
For non-magnetic materials
s-wave
p-wave

6. Dielectric-dielectric boundary-1 90 1 s p

7. Total internal reflection
Under the internal
reflection scheme
for
total reflection happens. Since
is purely imaginary, the refracted wave vector becomes:
i.e., the refracted wave is propagating along the boundary, decaying in the direction
perpendicular to the boundary.
Therefore, the refracted wave under TIR is reduced to a surface wave propagating
along the boundary only, formed by the projection of the incident and reflected
wave vectors on the boundary plane.

8. TIR as an all-pass filter
Once
we find
s-wave
p-wave

9. Applications of TIR
APF – for polarization splitting and conversion (example: wave plate)
Stop the traveling wave – for waveguide
Under TIR, refracted wave is traveling along the surface – for surface wave excitation
TIR is not wavelength sensitive, however, its combination (multiple TIR) is wavelength sensitive! – for construction of BPF

10. Total refraction If we have or Hence
For p-wave, if the incident angle hits the Brewster angle, total refraction happens.
For s-wave, total refraction is impossible, since for
we always have

11. Total reflection and total refraction
Total reflection happens to both s- and p- waves for internal reflection (when incident light from high refractive index medium); it doesn’t happen to external reflection (when incident light from low refractive index medium); hence the name total internal reflection.
Total refraction, however, only happens to p-wave, regardless of internal or external incidence; it doesn’t happen to s-wave.
Why total refraction can happen?
What will happen at the boundary with identical permittivity but different permeability?

12. Applications of total refraction
Polarization splitting
Filtering (Once the medium is dispersive, the Brewster angle becomes wavelength dependent. Otherwise, a front stage, e.g., a prism or a diffractional grating, is needed to convert the wavelength change into the angle change)