30 Fundamentals
Substitution into Maxwell’s Eq. (1.101):
e
iω
s
2iω
∂
∂r
+
∂
2
∂s∂r
+
cσ
2n
∂
∂s
+
∂
∂r
+ 2iω
1
2
˜
E
F
+ e
iω
r
2iω
∂
∂s
+
∂
2
∂s∂r
+
cσ
2n
∂
∂s
+
∂
∂r
+ 2iω
1
2
˜
E
B
=−
µ
0
c
2
4n
2
∂
∂s
+
∂
∂r
2
˜
P, (1.105)
which we rewrite in an abbreviated way using the differential operators L and
M for the forward and backward propagating waves, respectively:
L
˜
E
F
e
iω
s
+ M
˜
E
B
e
iω
r
=−
µ
0
c
2
4n
2
∂
∂s
+
∂
∂r
2
˜
P. (1.106)
In the case of a linear medium, the forward and backward wave travel indepen-
dently. If, as initial condition, we choose
˜
E
B
= 0 along the line r +s = 0(t = 0),
there will be no back scattered wave. If the polarization is written as a slowly
varying amplitude:
˜
P =
1
2
˜
P
F
e
iω
s
+
1
2
˜
P
B
e
iω
r
, (1.107)
the equations for the forward and backward propagating wave also separate if
˜
P
F
is only a function of
˜
E
F
, and
˜
P
B
only a function of
˜
E
B
. This is because a source
term for
˜
P
B
can only be formed by a “grating” term, which involves a product
of
˜
E
B
˜
E
F
. It applies to a polarization created by near resonant interaction with a
two-level system, using the semiclassical approximation, as will be considered
in Chapters 3 and 4. The separation between forward and backward traveling
waves has been demonstrated by Eilbeck [17, 18] outside of the slowly varying
approximation. Within the slowly varying approximation, we generally write
the second derivative with respect to time of the polarization as −ω
2
˜
P, and
therefore, the forward and backward propagating waves are still uncoupled, even
when
˜
P =
˜
P(
˜
E
F
,
˜
E
B
), provided there is only a forward propagating beam as
initial condition.
1.2.3. Dispersion
For nonzero GVD (k
= 0) the propagation problem (1.93) can be solved
either directly in the time or in the frequency domain. In the first case, the
Pulse Propagation 31
solution is given by a Poisson integral [19] which here reads
˜
E(t, z) =
1
2πik
z
t
−∞
˜
E(t
, z = 0) exp
i
(t − t
)
2
2k
z
dt
. (1.108)
As we will see in subsequent chapters, it is generally more convenient to
treat linear pulse propagation through transparent linear media in the frequency
domain, because only the phase factor of the envelope
˜
E() is affected by
propagation.
It follows directly from the solution of Maxwell’s equations in the frequency
domain [for instance Eqs. (1.74) and (1.79)] that the spectral envelope after
propagation through a thickness z of a linear transparent material is given by:
˜
E(, z) =
˜
E(,0)exp
−
i
2
k
2
z −
i
3!
k
3
z −···
. (1.109)
Thus we have for the temporal envelope
˜
E(t, z) = F
−1
˜
E(,0)exp
−
i
2
k
2
z −
i
3!
k
3
z −···
. (1.110)
If we limit the Taylor expansion of k to the GVD term k
, we find that an initially
bandwidth-limited pulse develops a spectral phase with a quadratic frequency
dependence, resulting in chirp.
We had defined a “chirp coefficient”
κ
c
= 1 +
M
4
4t
2
2
0
dφ
d
ω
2
when considering in Section 1.1.4 the influence of quadratic chirp on the
uncertainty relation Eq. (1.64) based on the successive moments of the field
distribution. In the present case, we can identify the phase modulation:
dφ
d
ω
=−k
z. (1.111)
Because the spectrum (in amplitude) of the pulse |
˜
E(, z)|
2
remains constant
[as shown for instance in Eq. (1.109)], the spectral components responsible for
chirp must appear at the expense of the envelope shape, which has to become
broader.
32 Fundamentals
At this point we want to introduce some useful relations for the characterization
of the dispersion. The dependence of a dispersive parameter can be given as
a function of either the frequency or the vacuum wavelength λ. The first-,
second-, and third-order derivatives are related to each other by
d
d
=−
λ
2
2πc
d
dλ
(1.112)
d
2
d
2
=
λ
2
(2πc)
2
λ
2
d
2
dλ
2
+ 2λ
d
dλ
(1.113)
d
3
d
3
=−
λ
3
(2πc)
3
λ
3
d
3
dλ
3
+ 6λ
2
d
2
dλ
2
+ 6λ
d
dλ
. (1.114)
The dispersion of the material is described by either the frequency dependence
n() or the wavelength dependence n(λ) of the index of refraction. The deriva-
tives of the propagation constant used most often in pulse propagation problems,
expressed in terms of the index n, are:
dk
d
=
n
c
+
c
dn
d
=
1
c
n − λ
dn
dλ
(1.115)
d
2
k
d
2
=
2
c
dn
d
+
c
d
2
n
d
2
=
λ
2πc
1
c
λ
2
d
2
n
dλ
2
(1.116)
d
3
k
d
3
=
3
c
d
2
n
d
2
+
c
d
3
n
d
3
=−
λ
2πc
2
1
c
3λ
2
d
2
n
dλ
2
+ λ
3
d
3
n
dλ
3
. (1.117)
The second equation, Eq. (1.116), defining the GVD is the frequency deriva-
tiveof1/ν
g
. Multiplied by the propagation length L, it describes the frequency
dependence of the group delay. It is sometimes expressed in fs
2
µm
−1
.
A positive GVD corresponds to
d
2
k
d
2
> 0. (1.118)
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