Wave reflection and refraction 1

Plane harmonic wave at boundary ki 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

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

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

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

Dielectric-dielectric boundary -1 90 1 s p

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.

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

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

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

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?

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)