## Electromagnetic waves

Review Maxwell's equations!
In lih materials in regions with ρf = 0 and jf = 0 both E and B satisfy the wave equation.
2E - με∂2E/∂t2 = 0,  2B - με∂2B/∂t2 = 0.
Plane wave solutions E = E((k/k)∙r) - vt,  B = B(k/k)∙r - vt) exist.
Here  k/k is the unit vector pointing in the direction of propagation.
We have  EB,  Ek,  Bk,  B = (με)½(k/k)×E.
Sinusoidal plane waves are of the form  E = E0ei(k∙r - ωt)B = B0ei(k∙r - ωt),
with k2 = μεω2.

In regions with ρf = 0 and jf = σcE both E and B satisfy the damped wave equation.
2E - μσcE/∂t - με∂2E/∂t2 = 0,  ∇2B - μσcB/∂t - με∂2B/∂t2 = 0.
Plane wave solutions exist.   For sinusoidal plane waves we now have
k2 = iμσcω + μεω2 = με(ω)ω2, i.e.  k is complex, the wave damps out.
ε(ω) and the index of refraction are complex numbers.

The skin depth is defined as the distance it takes to reduce the amplitude by a factor of 1/e.

### Plane waves at dielectric-dielectric boundaries

(D2 - D1n2 = σf,
(B2 - B1n2 = 0,
(E2 - E1t = 0,
(H2 - H1t = kf∙n.

These are the boundary conditions the electromagnetic fields have to satisfy on any boundary.
For waves propagating across a dielectric-dielectric boundary or from a dielectric into a conductor we use that the tangential components of E and H are continuous across the boundary.
From these boundary conditions we derive the law or reflection, θi = θr, and the law of refraction, (Snell's law), n1sinθi = n2sinθt, for a wave propagating from a medium with index of refraction n1 into a medium with index of refraction n2.  Here θi, θr, and θt are the angles the incident, reflected, and transmitted wave vectors make with the normal to the boundary.
The plane of incidence is the plane containing the wave vector ki and the normal to the boundary.
A linear polarized wave has p-polarization if E lies in the plane of incidence, and it has s-polarization if E is perpendicular to the plane of incidence.
The reflection coefficients for s and p polarization are
r12s = sin(θ1 - θ2)/sin(θ1 + θ2),   r12p = tan(θ1 - θ2)/tan(θ1 + θ2).
The reflectance is  R = |r12|2.  The transmittance is T = 1 - R if there is no absorption (energy conservation).

### Polarization by reflection

When θi = θB, the Brewster angle, then only light with s-polarization is reflected.
We have tan θB = n2/n1,  θt + θi = π/2.

### Total internal reflection

If n2 < n1, then for  θi > θc = sin-1(n2/n1)  we have total internal reflection.  The wave vector kt then has a component parallel to the interface which is real and a component perpendicular to the interface which is imaginary.

### Reflection by a good conductor

For good conductors R ≈ 1, T ≈ 0.  The wave does not penetrate into the conductor.  We then use as the boundary conditions that the tangential component of E and the normal component of B are continuous.

### Waveguides

Electromagnetic waves can propagate in a conducting channel.
If we have a rectangular channel as shown in the figure, with its axis in the z-direction, than any wave propagating into the z-direction can be viewed as a superposition of waves with either the electric or the magnetic field vector lying in the x-y plane.  These waves are called transverse electric (TE) or transverse magnetic (TM) waves.
E and B have to satisfy boundary conditions on the conducting walls of the channel.
E∙t = 0 and B∙n = 0 on the walls.
A TE wave is a superposition of waves of the form
E
= E0ysinkxx exp(i(kzz - ωt)) j + E0xsinkyy exp(i(kzz - ωt)) i,
with  kxa = nπ,  kyb = mπ, with m and n positive integers.  One of these integers, n or m, may be zero.
These waves are called the TEmn modes of the waveguide.
For each mode there exists a cutoff frequency ωc = ((nπc/a)2 + (mπc/b)2)½
Waves with ω < ωc do not propagate in the waveguide.
If a > b then the cutoff frequency for all TE modes is ωc = nπc/a.

Phase velocity:
The phase velocity vp is the velocity with which planes of constant phase move in the z-direction.
vp = ω/kz > c.
Group velocity:  The group velocity vg is the velocity with which the energy moves in the z-direction.
We have vg < c,  vpvg = c2.

A TM wave is a superposition of waves of the form
B
= B0xsinkxx exp(i(kzz - ωt)) j + B0ysinkyy exp(i(kzz- - ωt)) i,
with  kxa = nπ,  kyb = mπ, with m and n nonzero, positive integers.
These waves is called the TMmn modes of the waveguide.
TM10 or TM01 guided waves do not exist.  They cannot satisfy ∇∙B = 0.

### Electromagnetic waves formulas

 Energy density in EM fields: u = ½ε0E2 + (1/(2μ0))B2 EM waves, energy flux: S = (1/μ0)E×B Radiation pressure: P = (1/c)S Polarizers: Itransmitted = I0cos2θ. Geometrical Optics Law of refraction: n1sinθ1 = n2sinθ2 Mirror equation: 1/xo + 1/xi = 1/f,  M = -xi/xo,  f = R/2 Lens equation: 1/xo + 1/xi = 1/f,  M = -xi/xo Wave Optics Double slit, diffraction grating (maxima): dsinθ = mλ, m = 0, 1, 2, ...,  λ ≈ zd/(mL) Single slit, (minima): wsinθ = mλ,  λ ≈ zw/(mL) Resolving power: θmin = 1.22 λ/D Constructive interference (thin oil film on water): 2noilt cosθt = (m + ½)λ, m = 0,1,2,...