Ultrashort Laser Pulse Phenomena, 2nd Edition

172 Light–Matter Interaction

there will be at least two differential equations of the form Eq. (3.93), with a

third-order susceptibility, corresponding to the fundamental and third harmonic

fields. More equations have to be added if more frequencies are generated.

It is beyond the scope of the book to give a detailed description of the

various possible nonlinear effects and excitation schemes. The reader is referred

to the standard texts on nonlinear optics, for example Schubert and Wilhelmi [2],

Boyd [8], Bloembergen [9], and Shen [11]. Here we shall restrict ourselves to

a nonlinearity of second order that is responsible for second harmonic genera-

tion, optical parametric amplification (OPA), and to a nonlinearity of third order

describing (self-) phase modulation [(S)PM].

The tensor character of the nonlinear susceptibility describes the symmetry

properties of the material. For all substances with inversion symmetry, χ

(2n)

= 0

(n = 1,2...)holds,andtherefore no second harmonic processes can be observed

in isotropic materials and centrosymmetric crystals for example. In contrast,

third-order effects are always symmetry allowed. However, even in isotropic

materials, the tensor character of the nonlinear susceptibility should not be

ignored. The electric field of the light itself can break the symmetry, leading

to interesting polarization rotation effects.

In the following sections we will discuss various examples of nonlinear optical

processes with short light pulses. The propagation of the corresponding wave

packets at carrier frequency ω

i

is described by a group velocity ν

i

for which

1

ν

i

=

n(ω

i

)

c

+

ω

i

c

dn

d









ω

i

(3.99)

holds. Sometimes it will also be necessary to specify the polarization direction,

ˆe

j

, of the waves participating in the nonlinear process.

Unless stated otherwise we will assume that the nonlinear susceptibility is

much faster than the time scale of interest (pulse duration). This will allow us

to simplify the derivations by applying the concept of an instantaneous material

response. Also, to simplify the discussion on effects typical for the conversion

of short light pulses, we will usually neglect any change in intensity because of

focusing effects; an approximation, which generally holds for nonlinear materials

shorter than the Rayleigh range. An exception is when self-focusing occurs, a

nonlinear effect discussed in Section 3.8.

3.4. SECOND HARMONIC GENERATION (SHG)

Second harmonic generation has gained particular importance in ultrashort

pulse physics as a means for frequency conversion and nonlinear optical