•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Maxwell’s equations
in differential form
r
r
∂B
∇×E = −
r∂t
r ∂D r
∇× H =
+J
∂t
r
∇⋅B = 0
r
∇⋅ D = ρ
E
H
D
B
J
ρ
∇×
∇⋅
Faraday’s law
Ampere’s law
Electric field
Magnetic field
Electric flux density
Magnetic flux density
Electric current density
Electric charge density
Curl
Divergence
Gauss’ laws
[V/m]
[A/m]
[C/m2]
[Wb/m2]
[A/m2]
[C/m3]
[1/m]
[1/m]
If you can’t transform these to integral form in 30 seconds, go review.
Robert R. McLeod, University of Colorado
12
•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Constitutive relations
Interaction with matter
t
r
r
D = ε0 ∫ ε(t − τ) ⋅ E (τ )dτ
Dispersive & anisotropic
−∞
r
→ ε0 ε ⋅ E (t )
r
ε=ε I

→ ε0 ε E (t )
ε ≠ f(t)
Anisotropic
Isotropic
t
r
r
r
Nonmagnetic
B = µ0 ∫ µ(t − τ) ⋅ H (τ )dτ   
→ µ0 H
−∞
r
r
J =σ⋅ E
ε0
ε
µ0
µ
σ
Ohm’s Law
Permittivity of free space 8.854… 10-12
Dielectric constant
Permeability of free space 4 π 10-7
Relative permeability
Conductivity
Robert R. McLeod, University of Colorado
[F/m]
[H/m]
[Ω/m]
13
•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Boundary conditions
Fields at sharp change of material
These are derived from Maxwell’s equations.
In the absence of surface charge or current…
Et1 = Et 2
H t1 = H t 2
Conservation of transverse
electric and magnetic fields
Dn1 = Dn 2
Conservation of normal electric
and magnetic flux densities
Bn1 = Bn 2
n̂
tˆ
Medium 1
Medium 2
n̂
tˆ
Unit vector normal to boundary
Unit vector transverse (or tangential) to boundary
Robert R. McLeod, University of Colorado
14
•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Monochromatic fields
Expand all variables in temporal eigenfunction basis
1
f (t ) =
2π
+∞
f (ω ) =
∫
+∞
∫
f (ω ) e jωt dω
−∞
f (t ) e − jωt dt
Fourier Transform.
Note factor of 2π which
can be placed in different
locations.
−∞
E (t ) = Re(Ee
jω t
)
d
→ jω
dt
r
r
∇ × E = − jω B
r
r r
∇ × H = + jω D + J
r
∇⋅ B = 0
r
∇⋅ D = ρ
Robert R. McLeod, University of Colorado
Monochromatic fields E
transform like timedomain fields E for linear
operators
Removes all time-derivates.
Monochromatic
Maxwell’s equations.
15
•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Monochromatic
constitutive relations
The reason for using the monochromatic assumption
Convolution
t
r
r
D = ε0 ∫ ε(t − τ) ⋅ E (τ )dτ
−∞
t
r
r
B = µ0 ∫ µ(t − τ) ⋅ H (τ )dτ
Multiplication
r
r
D = ε 0 ε (ω ) ⋅ E
r
r
B = µ 0 µ (ω ) ⋅ H
−∞
∞
ε (ω ) = ∫ ε (t )e − jωt dt
0
∞
µ (ω ) = ∫ µ (t )e − jωt dt
Inverse Fourier Transform.
Note that ε is now f(ω) & not f(t).
If ε is not constant in ω,
it causes “dispersion” of pulses.
0
+
ε =ε
+
µ =µ
Robert R. McLeod, University of Colorado
Conditions for lossless materials derived
from Poynting vector (next)
ε+ is the Hermitian conjugate:
ε ji → ε ij*
16
•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Complex dielectric tensor
For conductive materials
r
r r
∇ × H = + jω D + J
r
r
= jωε 0 ε ⋅ E + σ ⋅ E
= jωε 0 (ε −
j
ωε 0
r
σ )⋅ E
Ampere’s law
Constitutive relations
Group terms
Complex dielectric tensor
From this point on the dielectric tensor will be taken to be
complex via this definition.
Robert R. McLeod, University of Colorado
17
•Background
–Maxell’s equations
ECE 6006 Numerical Methods in Photonics
Poynting vector
Power flow
r r r
P = E ×H
T r
r
1
P = ∫ P dt
T 0
r r∗
1
= Re E × H
2
r
= Re P
(
()
r 1 r r∗
P = E×H
2
P
Power per unit area
Robert R. McLeod, University of Colorado
Instantaneous power flow
Time-averaged power flow
Algebra left out here…
)
Define complex vector
r
Real part of P is <P >
[W/m2]
18
•Background
– Waves in ∞ space
ECE 6006 Numerical Methods in Photonics
Wave equation
Eliminate all fields but E
r
r
∇ × ∇ × E = − jω ∇ × B
Take curl of Faraday’s law
r
= − jωµ 0∇ × H
r
2
= ω µ0 D
r
2
= ω ε 0 µ0 ε ⋅ E
Magnetic constitutive
Ampere’s law
Electric constitutive
r
r
2
∇ × ∇ × E − k0 ε ⋅ E = 0
Monochromatic WE
∇2E + k 2E = 0
Scalar simplification
k0
c
Wave number of free space
Speed of light in vacuum
Robert R. McLeod, University of Colorado
ω/c = 2π/λ0 [1/m]
[m/s]
1 µε
0 0
19