Third-Order Susceptibility 195
terms up to the first temporal derivative lead to:
i
µ
0
k
d
2
dt
2
P
(3)
=
−i
n
2
k
n
0
|
˜
E|
2
˜
E − β
∂
∂t
|
˜
E|
2
˜
E
+...
e
i(ω
t−k
z)
+ c. c. (3.144)
where
β =
n
2
c
2 −
ω
χ
(3)
∂χ
(3)
∂
ω
. (3.145)
It should be remembered that the temporal derivative in Eq. (3.144) becomes
important if the light period is not negligibly short compared to the pulse duration.
Equations (3.144) and (3.145) indicate that the first-order correction to the SVEA
and the finite response time of the nonlinear susceptibility (the two summands
forming β) have the same action on pulse propagation. Corrections to the SVEA
may also be important in the spatial (transverse) propagation of the beam as
indicated in Eq. (3.97).
Pulse propagation through transparent media which is affected by SPM and
dispersion has played a crucial role in fiber optics. For a review we refer to the
monograph by Agrawal [48]. In this chapter we will discuss the physics behind
SPM and neglect GVD. In Chapter 8 we will describe effects associated with the
interplay of SPM and GVD.
3.6.2. Short Samples with Instantaneous Response
For interaction lengths much shorter than the dispersion length L
D
, and for an
instantaneous nonlinearity, the wave equation (3.92) with the source term (3.144)
simplifies to
∂
∂z
˜
E(z , t) =−i
3ω
2
χ
(3)
8c
2
k
|
˜
E
2
|
˜
E =−i
n
2
k
n
0
|
˜
E|
2
˜
E (3.146)
after applying the SVEA to the polarization term. For real χ
(3)
, substituting
˜
E = E exp(iϕ) into Eq. (3.146) and separating the real and imaginary parts result
in an equation for the pulse envelope
∂
∂z
E = 0 (3.147)
196 Light–Matter Interaction
and for the pulse phase
∂ϕ
∂z
=−
n
2
k
n
0
|E|
2
. (3.148)
Obviously the pulse amplitude E is constant in the coordinate system travel-
ing with the group velocity, that is, the pulse envelope remains unchanged,
E(t, z) = E(t,0)= E
0
(t). Taking this into account, we can integrate Eq. (3.148)
to obtain for the phase
ϕ(t, z) = ϕ
0
(t) −
k
n
2
n
0
zE
2
0
(t) (3.149)
which results in a phase modulation given by
∂ϕ
∂t
=
dϕ
0
dt
−
n
2
k
n
0
z
d
dt
E
2
0
(t). (3.150)
This result can be interpreted as follows. The refractive index change follows the
pulse intensity instantaneously. Thus, different parts of the pulse “feel” different
refractive indices, leading to a phase change across the pulse. Unlike the phase
modulation associated with GVD, this SPM produces new frequency components
and broadens the pulse spectrum. To characterize the SPM it is convenient to
introduce a nonlinear interaction length
L
NL
=
n
0
n
2
k
E
2
0m
(3.151)
where E
0m
is the peak amplitude of the pulse. The quantity z/L
NL
represents
the maximum phase shift which occurs at the pulse peak, as can be seen from
Eq. (3.150). Figure 3.17 shows some examples of the chirp and spectrum of self-
phase modulated pulses. Because n
2
is mostly positive far from resonances (see
Problem 4 at the end of this chapter), upchirp occurs in the pulse center. We also
see that SPM can introduce a considerable spectral broadening. This process as
well as some nonlinear processes of higher order can be used to generate a white
light continuum, as discussed later in this chapter.
For an order of magnitude estimate let us determine the frequency change δω
over the FWHM of a Gaussian pulse. Substituting E
0m
e
−2ln2(t/τ
p
)
2
for E
0
(t)in
Eq. (3.150) yields
δωτ
p
=
8ln2
√
2
z
L
NL
. (3.152)
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.
