518
H Maxwell’s Equations for 2-d Billiards
∇
2
tr
˜
E z,i (r, φ, ω) = γ i ˜
E z,i (r, φ, ω)
(H.11)
must be solved numerically and will yield a set of eigenmodes and eigenvalues, γ i ,
for the transverse component of the electric field. This yields a secular equation for
the allowed frequencies of the electric field modes in the cavity,
ω
2
i,n = c
2
nπ
d
2
+ γ i
for n = 0, 1, 2, . . . .
(H.12)
H.2 Transverse Electric Modes
For transverse electric modes, ˜
E z (r, φ, ω) = 0 everywhere inside the cavity. Since
the component of the magnetic field perpendicular to each surface must be zero at
the surface, we must choose
B z (r, t) = ˜
B z (r, φ, ω)sin
nπ z
d
e
iωt for n = 1, 2, . . .,
(H.13)
and we automatically satisfy the boundary conditions. If the cavity walls are irregularly shaped, then the eigenvalue problem ∇ 2
tr
˜
B z,i (r, φ, ω) = β i ˜
B z,i (r, φ, ω) must
again be solved numerically and will yield a set of eigenmodes and eigenvalues, β i .
This yields a secular equation for the allowed frequencies of the electric field modes
in the cavity,
ω
2
i,n = c
2
nπ
d
2
+ β i
for n = 1, 2, . . . .
(H.14)
The eigenvalues γ i and β i in general will be different but will be of the same
order of magnitude. Thus, if the spacing, d, is made small enough relative to the
average radius in the x − y plane, the lowest TE mode (n=1) will have a frequency
well above the lowest frequency for the TM modes (n=0). There will be a frequency
interval in which only the TM modes can exist in the cavity. For these modes, the
electric field ˜
E z,i (r, φ, ω) will satisfy the two-dimensional Schrödinger equation in
Eq. (H.11), and the eigenfrequencies of the cavity will be given by ω 2
i,0 = γ i c 2 .
Reference
Jackson JD (1975) Classical electrodynamics. Wiley, New York
H Maxwell’s Equations for 2-d Billiards
∇
2
tr
˜
E z,i (r, φ, ω) = γ i ˜
E z,i (r, φ, ω)
(H.11)
must be solved numerically and will yield a set of eigenmodes and eigenvalues, γ i ,
for the transverse component of the electric field. This yields a secular equation for
the allowed frequencies of the electric field modes in the cavity,
ω
2
i,n = c
2
nπ
d
2
+ γ i
for n = 0, 1, 2, . . . .
(H.12)
H.2 Transverse Electric Modes
For transverse electric modes, ˜
E z (r, φ, ω) = 0 everywhere inside the cavity. Since
the component of the magnetic field perpendicular to each surface must be zero at
the surface, we must choose
B z (r, t) = ˜
B z (r, φ, ω)sin
nπ z
d
e
iωt for n = 1, 2, . . .,
(H.13)
and we automatically satisfy the boundary conditions. If the cavity walls are irregularly shaped, then the eigenvalue problem ∇ 2
tr
˜
B z,i (r, φ, ω) = β i ˜
B z,i (r, φ, ω) must
again be solved numerically and will yield a set of eigenmodes and eigenvalues, β i .
This yields a secular equation for the allowed frequencies of the electric field modes
in the cavity,
ω
2
i,n = c
2
nπ
d
2
+ β i
for n = 1, 2, . . . .
(H.14)
The eigenvalues γ i and β i in general will be different but will be of the same
order of magnitude. Thus, if the spacing, d, is made small enough relative to the
average radius in the x − y plane, the lowest TE mode (n=1) will have a frequency
well above the lowest frequency for the TM modes (n=0). There will be a frequency
interval in which only the TM modes can exist in the cavity. For these modes, the
electric field ˜
E z,i (r, φ, ω) will satisfy the two-dimensional Schrödinger equation in
Eq. (H.11), and the eigenfrequencies of the cavity will be given by ω 2
i,0 = γ i c 2 .
Reference
Jackson JD (1975) Classical electrodynamics. Wiley, New York