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Lecture 2 - Wave propagation
Maxwell wave equation
We already saw that :
dt
P
J M
dt
E
C
B
B
dt
B
E
E
total
v
v r v
v
v v
v v
v
v v
v
+ + ∇× +
∇× =
∇× =−
2 0 0 0
0
μ μ μ
ε
ρ
for M=0, ρf , J^ f =0, we have:
( ) ⎟ ⎠
⎞ ⎜ ⎝
⎛ ∂
∂ =− ∂
∂ ∇ × ∇× =−∇× 2
2 2
2 0 0 dt
P dt
E t
B E
r v v v v v v μ ε
or,
( ) ⎟ ⎠
⎞ ⎜ ⎝
⎛ ∂
∂ − ∇ +∇∇⋅ =− 2
2 2
2 0 0
2 dt
P dt
E E E
v v v v v μ ε
Now, E^ (^ )^ P
free ∇⋅ = free + bound = − ∇⋅ 0 0 0
ε ε
ρ ρ ρ ε
r
, so we cannot set E
r
∇ ⋅ to
zero in general. (As is often done.) In a moment, we will show that this is OK for plane waves, but first, we will look at the instructive (and often valid to a good approximation) example of an instantaneous response.
(i) Wave Equation in material with instantaneous response
For a material with an instantaneous polarization response, χ(t) ≅ χ (^) δ δ(t)
Hence P(t) = ε 0 χ δ E(t). Since ∇ ⋅ D = 0
r
, then ∇ ⋅( 0 ( 1 + ) E = 0
r
ε χ , so that,
0 E^ ⋅^ ∇(^1 + )+( 0 (^1 + )∇⋅ E =^0
r r ε χ ε χ
For homogeneous media, ∇χ = 0 , so that, ∇ ⋅ E = 0
r .
Hence, (^2)
2 2 2
2 2 0
2 2
t
E
t c
P
t
E
c
E
v v
v
,
or,
( ) 0
1 2
2
2
2
∂
E
c
E t
v v^ χ
.
We recognize this as a wave equation with plane wave solutions:
E r t E e c c i k r t ( , ). ( ) = 0 +
-^ r−ω r (^) r r .
Now, ( , ) ( , ) 2 2 E r t k E r t
r (^) r r r ∇ =− , and E t
E (^) 2 2
2 =− ω ∂
∂
v
,
0 0
2 2
2
2
2 2
2 or,
( 1 ) 1
k c
k
c c
k
η λ
π η
ηω
η η χ
ω χ
ω δ δ
= = =
∴ = + = ⇒ = +
η is generally the complex refractive index, η = n +i κ , but in this case χ (^) δ
is real and independent of frequency, hence so must η be.
Free - space λ, k
P ( ω )= ε 0 χ( ω) E ( ω).
Clearly, χ(ω) is generally complex, so we write,
χ ( ω)= χ'( ω)+ i χ''( ω).
Reality of Electric Field, Polarization
The electric field is a REAL, observable quantity. (Same is true for Polarization, current, etc.). Hence our mathematical description of these quantities should not be complex,
e.g. E(t,z) = E 0 cos( ω t-kz+ φ ).
This may sometimes be written in the form:
E(t,z) = ½E0ei(^ ω t-kz+^ φ )^ + c.c. = ½A0e i(^ ω t-kz)^ + c.c,
Where, A 0 = E0ei^ φ^ is now a complex amplitude that contains magnitude and phase information. Since A 0 is a Fourier amplitude of the Fourier component at frequency ω, it is only in the Fourier transforms of the field that we find complex numbers cropping up. In the time domain, observable quantities like field and polarization should always be real. Likewise, optical properties that relate polarization to field, or input filed to transmitted field in the time domain must also be real, but in the frequency domain, such properties may be complex. Thinking of the high frequencies involved and the way in which measurements of optical properties are measured, it is natural that it is preferred to express parameters that describe the optical properties of materials in the frequency domain.
Polarization response of a material
Look at linear material first - “Linear” means
3
P r t 0 d r dt r r t t E r t
v v v v v v
∞ −∞
∞ −∞
normally ( r^ r ')
v v
χ ∝δ − - entirely local response.
P ( r , t ) 0 dt ' ( r , t t ') : E ( r , t ')
v v v v v
∞
where χ is the response function
e.g. if ( )^
t
t e
γ
−
= 0 - use scalar E , P
'
P t 0 0 e E t dt
t t
∞
−∞
− −
γ
for E ( ) t ' = E 0 δ( ) t ' ; impulse response is
t
P t E e
γ
δ ε^ χ
−
then for E ( ) t =
We have P ( ) t =
So that: 2 {^1 ( )}
2 2 χ ω
ω = + c
k (^) ,
and η^ (ω)=^1 +χ(^ ω)
Therefore, n (^ ω^ )+^ i κ(ω)=^1 +χ'(ω)+ i χ"(^ ω).
This leads to the relationships:
n 2 (ω) - κ 2 (ω) = 1+χ’(ω) = ε’(ω)/ε 0
2n(ω)κ(ω) =χ”(ω) = ε”(ω)/ε 0
Example: magnitude of κ
If α = 1 cm-1^ = 10 2 m-1^ , then κ = α/2k 0. And k 0 is about 10 7 m-1^ for visible light, so κ ~ 10 -5, << n.
Hence, n 2 = 1+χ’(ω) is a good approximation, provided α < 10 4 cm-1.
Then we can say that refraction is solely related to χ’ away from resonances.
