Download Electromagnetic waves in vacuum.

Electromagnetic waves in vacuum.
The discovery of displacement currents entails a peculiar class of solutions
of Maxwell equations: travelling waves of electric and magnetic fields in
vacuum. In the absence of currents and charges, the equations governing
electric and magnetic field are:
∇⋅B = 0
∇⋅E = 0
∇×E = −
1 ∂∂B
B
c ∂t
∇×B =
1 ∂∂E
E
c ∂t
Taking the curl of (3) and using (4):
1 ∂ 2E
∇ × ∇ × E = ∇(∇ ⋅ E) − ∇ E = − 2 2
c ∂t
2
Using (1) we get D’Alambert equation:
1 ∂ 2E
∇ E− 2 2 = 0
c ∂t
2
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Electromagnetic waves in vacuum.
Each of the two equation pairs lead to the D’Alambert eq. (respectively
for Ex and By, and Ey and Bx). For example, the first pair combines into:
∂ 2 Ex 1 ∂ 2 Ex
− 2
=0
2
2
c ∂t
∂z
∂ 2 By
E x = E x ( z − ct )
B y = B y ( z − ct )
∂z
2
2
1 ∂ By
− 2
=0
c ∂t 2
(E ⊥ B ! )
The linearity of D’Alambert eq. entails that the generic EM plane wave
can be seen as the superposition of sinusoidal waves. Let us the assume
that each component has the form
u = u0 exp[i (kz − ω t + φu )]
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Electromagnetic waves in vacuum.
The pair of equations for (Ex,By) yields
kE0 x exp[i (kz − ω t + φ E )] =
kB0 y exp[i (kz − ω t + φ B )] =
ω
c
ω
c
B0 y exp[i (kz − ω t + φ B )]
E0 x exp[i (kz − ω t + φ E )]
These equations must be verified for any t and z. Therefore
E0 x = B0 y
φE = φB
The same conditions are true for (Ey,Bx). At each (t,z) the magnitude of the
electric and magnetic field is the same (in CGS-units) .
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Polarization of EM waves
The two classes of solutions (Ex,By) and (Ey,Bx) are independent: they
represent the two polarization modes of EM radiation. As the E and B
fields lie on a plane, these modes correspond to linear polarizations.
A generic (unpolarized) EM wave is a superposition of the two modes,
with different phases and amplitudes. For example, introducing the unit
vectors along x and y:
E( z , t ) = E0 x cos(kz − ω t + φ x )]xˆ + E0 y cos(kz − ω t + φ y )]yˆ
Instead of the linearly polarized modes, one could use two circularly
polarized modes:
Ex = E y
φ y = φx +
π
2
Left Circular Polarization (LCP)
and
Ex = E y
φ y = φx −
π
2
Right Circular Polarization (RCP)
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Maxwell Equations in a Plasma
The wave propagation is governed by Maxwell equations (CGS-Gauss):
∇ ⋅ E = 4πρ
∇⋅B = 0
1 ∂B
∇×E = −
c ∂t
∇×B =
4π
1 ∂E
J+
c
c ∂t
We will solve the combined set of Maxwell equations and electron
equation of motion using 1) a perturbative approach (to first order) and
2) normal modes.
1) All quantities are the sum of an unperturbed (background) value and
a small perturbation:
n = n0 + n1
v=
v1
(quiescent plasma)
B = B 0 + B1
E=
E1
(no external electric field)
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Maxwell Equations in a Plasma.
2) All perturbations are periodic and depend on space and time as
u = u0 exp[i (k ⋅ r − ω t )
The surfaces of constant phase move in space at the phase velocity
vφ =
ω
(suggestion: consider the surfaces with null phase and compute r/t)
k
The set of differential equations becomes now a set of algebraic equations;
the differential operators transform as
∇ → ik
∇⋅ → ik ⋅
∇× → ik ×
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
∂
→ − iω
∂t
AA 2007-2008
The equations of motion.
Navier-Stokes equation for the electrons (collisionless plasma):
1
 ∂v



me ne  + ( v ⋅ ∇) v  = −∇p − ene  E + v × B 
c
 ∂t



The space charge density ρ = -ene and current density J = -enev fullfil the
equation of continuity:
∂ρ
+∇⋅J = 0
∂t
We assume that the plasma the plasma is magnetized: a constant and
homogeneous magnetic field is superimposed to the magnetic field of the
wave. The plasma is supposed to be isothermal, so we do not need to
consider the energy equation. (Not only: we will see soon that the plasma
may be safely supposed to be cold, with T = 0!)
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
The linearized, normal mode equations.
The Maxwell equations, charge conservation and electron equation
of motion become:
ik ⋅ E1 = 4π en1
k × E1 =
ω
c
B1
eω n1 = k ⋅ J
(1)
k ⋅ B1 = 0
(3)
k × B1 = −i
(5)
(2)
4π
ω
J − E1
c
c
(4)
1


− ime n0ω v1 = −en0  E1 + v1 × B 0 
c


(6)
Note that now the first two Maxwell equations (1-2) follow from the last pair (3-4) and
charge conservation (5): (2) follows by taking the scalar product of (3) by k; (1) follows
by taking the scalar product of (4) by k and using (5).
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
The conductivity tensor.
Eq. (6) entails a linear relation between current density and electric field.
e 2 n0
e
iω J = −
E1 +
J × B0
me
me c
or, writing B0=B0n and introducing the plasma and cyclotron frequencies,
ω p2
Ω
J =i
E1 − i c J × n
4πω
ω
Expanding the vector product, one gets the generalized Ohm’s law and the
conductivity tensor σ:
J = σ ⋅E
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
The conductivity tensor.


 1

2
n e
 Ω
e
σ=i
i c
2  ω

Ω




ω m 1 −  c   


  ω   0



−i
Ω
c
ω
1
0



0


0


  Ω  2 
  c  
1 − 
 
ω
 
 


*
The conductivity tensor is therefore anti-hermitean σ ij = −σ ji . Therefore no
heat is dissipated, as the real part of σ is skew and Re( Ei ) Im( E j ) = 0
Q = Re(E) ⋅ Re(J ) = 0
Re( J i ) = Re(σ ij ) Re( E j ) − Im(σ ij ) Im( E j )
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Plane EM waves in a plasma.
The refractive index of a medium is defined as the ratio between c and the
phase velocity
nr =
c ck
=
vφ ω
a × (b × c) = (a ⋅ c)b − (a ⋅ b)c
By combining Maxwell eq.
k × E1 =
ω
c
k × B1 = −i
B1
2
2
r
(1 − nr ) Ei + n
ki k j
k2
E j = −i
4π
ω
4π
ω
J − E1
c
c
Ji
one gets:
(7)
which gives the current as a function of the electric field. Making use of the
relationship J = σ ⋅ E one obtains
2
2
r
(1 − nr ) Ei + n
ki k j
k
2
Ej = i
4π
ω
σ ij E j
kk


4π
2
2 i j
δ
σ
(1
−
n
)
+
n
−
i

r
ij
r
ij  E j = 0
2
k
ω


Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Plane EM waves in a plasma.
The homogeneous set of equations has non-trivial solutions iff
kk


4π
2
2 i j
det  (1 − nr )δ ij + nr 2 − i
σ ij  = 0
k
ω


This is the dispersion equation of the plasma.
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
EM waves in a plasma: B=0
If B=0, the conductivity tensor reduces to a scalar:
ω p2
σ = −i
4πω
Maxwell eq. allow two modes of propagation:
(1) k || E1
and
(2) k ⊥ E1
2
(1) iω E + 4π J = iω E + 4π σE = i (ω − ω p / ω ) E = 0
⇒ ω 2 = ω p2
Longitudinal plasma waves occur at the well known plasma frequency.
The wavenumber is free!
2
r
(2) (1 − n )E = i
4π
ω
J
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
EM waves in a plasma: B=0
(2) k ⊥ E1
(1 − nr2 )E = i
4π
ω
J
ω p2
Substituting J = σ E = −i
E one gets
4πω
2


ω
p
2
1 − nr −
E = 0
2 

ω 

ω p2
nr = 1 − 2
ω
The index of refraction of a plasma is smaller than unity. This implies that the
phase velocity of e.m. waves is larger than c:
vφ =
c ω
= >c
nr k
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
EM waves in a plasma: B=0
However, information does not travel at the phase velocity: a continuous,
monochromatic, plane wave (starting at t=-∞) cannot be used to convey
information. Information (e.g. electromagnetic pulses) travels at the group
velocity (always smaller or equal to c):
vg =
dω
dk
c ω
vφ = = > c
nr k
In an unmagnetized plasma:
ω p2
dω
nr =
= 1 − 2 → ω 2 = c 2 k 2 − ω p2 → v g =
= cnr
ω
ω
dk
ck
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Group velocity
Let us consider a superposition of two EM waves with slightly
different frequency and wavenumber:
u1 = cos[(k + ∆k ) z − (ω + ∆ω )t ]
u2 = cos[(k − ∆k ) z − (ω − ∆ω )t ]
The superposition of the two waves is:
u1 + u 2 = cos[(k + ∆k ) z − (ω + ∆ω )t ] + cos[(k − ∆k ) z − (ω − ∆ω )t ] =
cos[(kz − ωt ) + (∆kz − ∆ωt )] + cos[(kz − ωt ) − (∆kz − ∆ωt )] =
cos(kz − ωt ) cos(∆kz − ∆ωt )] − sin(kz − ωt ) sin(∆kz − ∆ωt )] +
cos(kz − ωt ) cos(∆kz − ∆ωt )] + sin(kz − ωt ) sin(∆kz − ∆ωt )] =
2 cos(kz − ωt ) cos(∆kz − ∆ωt )]
The low frequency modulation travels at the speed
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
vg =
∆ω
∆k
AA 2007-2008
Group velocity
For ∆ω , ∆k → 0 v g = dω / dk
Let us consider now a “wave packet”, i.e. a more general superposition of
EM waves with different frequency and wavenumber:
k 0 + ∆k
Ψ( z, t ) =
∫ A(k ) exp i[kz − ωt ] dk
k 0 − ∆k

1  d2A 2
 dω 
≅ ∫  A(k0 ) +  2  ξ  exp i[(k0 + ξ ) z − ω (k0 )t − 
 ξ t ] dξ
2  dk  k 
 dk  k0
k 0 − ∆k 
0


k 0 + ∆k
k 0 + ∆k
 dω 
exp
i
[
z
−
ξ

 ξ t ] dξ
∫
 dk  k0
k 0 − ∆k
≅ A(k 0 ) exp i[k0 z − ω (k0 )t ]
 dω 
sin[ z − 
 t]
 dk  k0
≅ 2 A(k0 ) exp i[k0 z − ω (k0 )t ]
 dω 
z −
 t
 dk  k0
Ambiente Spaziale e Strumentazione Spaziale- Prof. L.Iess
AA 2007-2008
Group velocity
This “pulse” travels at the speed v g =
The superposition
the two
waves is:SpazialeAmbienteof
Spaziale
e Strumentazione
dω
dk
Prof. L.Iess
AA 2007-2008