Pulse Propagation 21
field vector E from Maxwell equations (see for instance [12]) which in Cartesian
coordinates reads
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
−
1
c
2
∂
2
∂t
2
E(x, y, z, t) = µ
0
∂
2
∂t
2
P(x, y, z, t) , (1.67)
where µ
0
is the magnetic permeability of free space. The source term of Eq. (1.67)
contains the polarization P and describes the influence of the medium on the field
as well as the response of the medium. Usually the polarization is decomposed
into two parts:
P = P
L
+ P
NL
. (1.68)
The decomposition of Eq. (1.68) is intended to distinguish a polarization that
varies linearly (P
L
) from one that varies nonlinearly (P
NL
) with the field.
Historically, P
L
represents the medium response in the frame of “ordinary” optics,
e.g., classical optics [13] and is responsible for effects such as diffraction, disper-
sion, refraction, linear losses and linear gain. Frequently, these processes can be
attributed to the action of a host material which in turn may contain sources of a
nonlinear polarization P
NL
. The latter is responsible for nonlinear optics [14–16]
which includes, for instance, saturable absorption and gain, harmonic generation
and Raman processes.
As will be seen in Chapters 3 and 4, both P
L
and in particular P
NL
are often
related to the electric field by complicated differential equations. One reason is
that no physical phenomenon can be truly instantaneous. In this chapter we will
omit P
NL
. Depending on the actual problem under consideration, P
NL
will have
to be specified and added to the wave equation as a source term.
1.2.1. The Reduced Wave Equation
Equation (1.67) is of rather complicated structure and in general can solely
be solved by numerical methods. However, by means of suitable approximations
and simplifications, one can derive a “reduced wave equation” that will enable
us to deal with many practical pulse propagation problems in a rather simple
way. We assume the electric field to be linearly polarized and propagating in
the z-direction as a plane wave, i.e., the field is uniform in the transverse x, y
direction. The wave equation has now been simplified to:
∂
2
∂z
2
−
1
c
2
∂
2
∂t
2
E(z, t) = µ
0
∂
2
∂t
2
P
L
(z, t). (1.69)
22 Fundamentals
As known from classical electrodynamics [12] the linear polarization of a medium
is related to the field through the dielectric susceptibility χ. In the frequency
domain we have
˜
P
L
(, z) =
0
χ()
˜
E(, z), (1.70)
which is equivalent to a convolution integral in the time domain
P
L
(t, z) =
0
t
−∞
dt
χ(t
)E(z, t − t
). (1.71)
Here
0
is the permittivity of free space. The finite upper integration limit, t,
expresses the fact that the response of the medium must be causal. For a
nondispersive medium (which implies an infinite bandwidth for the suscepti-
bility, χ() = const.) the medium response is instantaneous, i.e., memory free.
In general, χ(t) describes a finite response time of the medium, which in the
frequency domain, means nonzero dispersion. This simple fact has important
implications for the propagation of short pulses and time varying radiation in
general. We will refer to this point several times in later chapters—in particular
when dealing with coherent interaction.
The Fourier transform of (1.69) together with (1.70) yields
∂
2
∂z
2
+
2
()µ
0
˜
E(z, ) = 0
(1.72)
where we have introduced the dielectric constant
() =[1 + χ()]
0
. (1.73)
For now we will assume a real susceptibility and dielectric constant. Later we will
discuss effects associated with complex quantities. The general solution of (1.72)
for the propagation in the +z direction is
˜
E(, z) =
˜
E(,0)e
−ik()z
, (1.74)
where the propagation constant k() is determined by the dispersion relation of
linear optics
k
2
() =
2
()µ
0
=
2
c
2
n
2
(), (1.75)
Pulse Propagation 23
and n() is the refractive index of the material. For further consideration we
expand k() about the carrier frequency ω
k() = k(ω
) + δk, (1.76)
where
δk =
dk
d
ω
( − ω
) +
1
2
d
2
k
d
2
ω
( − ω
)
2
+··· (1.77)
and write Eq. (1.74) as
˜
E(, z) =
˜
E(,0)e
−ik
z
e
−iδkz
, (1.78)
where k
2
= ω
2
(ω
)µ
0
= ω
2
n
2
(ω
)/c
2
. In most practical cases of interest, the
Fourier amplitude will be centered on a mean wave vector k
, and will have
appreciable values only in an interval k small compared to k
. In analogy
to the introduction of an envelope function slowly varying in time, after the
separation of a rapidly oscillating term, cf. Eqs. (1.11)–(1.14), we can define
now an amplitude which is slowly varying in the spatial coordinate
˜
E(, z) =
˜
E( + ω
,0)e
−iδkz
. (1.79)
Again, for this concept to be useful we must require that
d
dz
˜
E(, z)
k
˜
E(, z)
(1.80)
which implies a sufficiently small wave number spectrum
k
k
1. (1.81)
In other words, the pulse envelope must not change significantly while travel-
ling through a distance comparable with the wavelength λ
= 2π/ω
. Fourier
transforming of Eq. (1.78) into the time domain gives
˜
E(t, z) =
1
2
1
π
∞
−∞
d
˜
E(,0)e
−iδkz
e
i(−ω
)t
e
i(ω
t−k
z)
(1.82)
Get Ultrashort Laser Pulse Phenomena, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
