Review Maxwell's equations!
In lih materials in regions with ρ_{f} =
0 and j_{f} = 0 both
E and
B satisfy the wave
equation.
∇^{2}E - με∂^{2}E/∂t^{2}
= 0, ∇^{2}B - με∂^{2}B/∂t^{2}
= 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 E ⊥ B,
E
⊥ k, B ⊥
k,
B = (με)^{½}(k/k)×E.
Sinusoidal plane waves are of the form E =
E_{0}e^{i(k∙r
- ωt)}. B = B_{0}e^{i(k∙r
- ωt)},
with
k^{2} = μεω^{2}.
In regions with ρ_{f }= 0 and j_{f
}= σ_{c}E both
E and
B satisfy the damped wave equation.
∇^{2}E - μσ_{c}∂E/∂t
- με∂^{2}E/∂t^{2}
= 0, ∇^{2}B - μσ_{c}∂B/∂t
- με∂^{2}B/∂t^{2}
= 0.
Plane wave solutions exist. For sinusoidal plane waves we now have
k^{2} = iμσ_{c}ω
+ μεω^{2}
= με(ω)ω^{2}, 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.
(D_{2} -
D_{1})·n_{2} = σ_{f},
(B_{2} -
B_{1})·n_{2} = 0,
(E_{2} - E_{1})·t = 0,
(H_{2} - H_{1})·t =
k_{f}∙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, θ_{i}
= θ_{r}, and the law of
refraction, (Snell's law), n_{1}sinθ_{i}
= n_{2}sinθ_{t}, for a wave propagating from a medium with index of refraction n_{1} into
a medium with index of refraction n_{2}. Here θ_{i},
θ_{r}, and θ_{t} 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 k_{i}
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
r_{12s
}= sin(θ_{1 }- θ_{2})/sin(θ_{1 }+ θ_{2}),
r_{12p }= tan(θ_{1 }- θ_{2})/tan(θ_{1 }+ θ_{2}).
The reflectance is R = |r_{12}|^{2}.
The transmittance is T = 1 - R if there is no absorption (energy conservation).
When θ_{i} = θ_{B},
the Brewster angle, then only light with s-polarization is reflected.
We
have tan θ_{B} = n_{2}/n_{1},
θ_{t} + θ_{i}
= π/2.
If n_{2 }< n_{1}, then for θ_{i} > θ_{c} = sin^{-1}(n_{2}/n_{1})_{ } we have total internal reflection. The wave vector k_{t} then has a component parallel to the interface which is real and a component perpendicular to the interface which is imaginary.
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.
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 = E_{0y}sink_{x}x
exp(i(k_{z}z - ωt)) j + E_{0x}sink_{y}y
exp(i(k_{z}z - ωt)) i,
with k_{x}a = nπ,
k_{y}b = mπ,
with m and n positive integers. One of these integers, n or m, may be
zero.
These waves are called the TE_{mn} modes of the waveguide.
For each mode there exists a cutoff frequency ω_{c}
= ((nπc/a)^{2} + (mπc/b)^{2})^{½}.
Waves with ω
< ω_{c}
do not propagate in the waveguide.
If a > b then the cutoff frequency for
all TE modes is ω_{c} = nπc/a.
Phase velocity: The phase velocity v_{p} is the velocity
with which planes of constant phase move in the z-direction.
v_{p} = ω/k_{z} > c.
Group velocity: The group velocity v_{g} is the velocity
with which the energy moves in the z-direction.
We have v_{g} < c, v_{p}v_{g} = c^{2}.
A TM wave is a superposition of waves of the form
B = B_{0x}sink_{x}x
exp(i(k_{z}z - ωt)) j + B_{0y}sink_{y}y
exp(i(k_{z}z- - ωt)) i,
with k_{x}a = nπ,
k_{y}b = mπ,
with m and n nonzero, positive integers.
These waves is called the TM_{mn} modes of the
waveguide.
TM_{10} or TM_{01} guided waves do not exist.
They cannot satisfy ∇∙B
= 0.
Energy density in EM fields: |
u = ½ε_{0}E^{2 }+ (1/(2μ_{0}))B^{2} |
EM waves, energy flux: |
S = (1/μ_{0})E×B |
Radiation pressure: |
P = (1/c)S |
Polarizers: |
I_{transmitted }= I_{0}cos^{2}θ. |
Geometrical Optics | |
Law of refraction: |
n_{1}sinθ_{1 }= n_{2}sinθ_{2} |
Mirror equation: |
1/x_{o }+ 1/x_{i }= 1/f, M = -x_{i}/x_{o}^{, } f = R/2 |
Lens equation: |
1/x_{o }+ 1/x_{i }= 1/f, M =- x_{i}/x_{o} |
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: |
θ_{min }= 1.22 λ/D |
Constructive interference (thin oil film on water): |
2n_{oil}t cosθ_{t }= (m + ½λ, m = 0,1,2,... |