Second Harmonic Generation (SHG) 173
correlation. Owing to the characteristics of ultrashort pulses, a number of new
features unknown in the conversion of cw light have to be considered [12–16].
We will examine first the relatively simple case of type I SHG, in which the
fundamental wave propagates as an ordinary (o) or extraordinary (e) wave, pro-
ducing an extraordinary or ordinary second harmonic (SH) wave, respectively.
We will briefly discuss at the end of this section the more complex case of type
II SHG, in which the nonlinear polarization, responsible for the generation of a
second harmonic propagating as an e wave, is proportional to the product of the e
and o components of the fundamental. We will see that group velocity mismatch
between the fundamental and the SH leads generally to a reduced conversion effi-
ciency and pulse broadening. Under certain circumstances, however, it is possible
to have simultaneously high conversion efficiency and efficient compression of
the second harmonic in presence of group velocity mismatch. Second harmonic
is only a particular case of sum frequency generation. Therefore, in some of the
subsections to follow, we will treat in parallel SHG and the more general case of
sum frequency generation.
3.4.1. Type I Second Harmonic Generation
Let us assume a light pulse incident on a second harmonic generating crystal.
The electric field propagating inside the material consists of the original (funda-
mental) field (subscript i = 1) and the second harmonic field (subscript i = 2).
The total field obeys a wave equation similar to Eq. (3.92) with a nonlinear
polarization of second order as source term:
∂
∂z
+
1
ν
1
∂
∂t
−
ik
1
2
∂
2
∂t
2
˜
E
1
+ D
1
e
i(ω
1
t−k
1
z)
+
k
2
k
1
∂
∂z
+
1
ν
2
∂
∂t
−
ik
2
2
∂
2
∂t
2
˜
E
2
+ D
2
e
i(ω
2
t−k
2
z)
+ c. c. = i
µ
0
k
1
∂
2
∂t
2
P
(2)
(3.100)
where the second-order polarization can be written as
P
(2)
=
0
χ
(2)
1
4
˜
E
1
e
i(ω
1
t−k
1
z)
+
˜
E
2
e
i(ω
2
t−k
2
z)
+ c. c.
2
. (3.101)
Because the group velocities ν
1
and ν
2
are not necessarily equal there is no
coordinate frame in which both the fundamental and SH pulses are at rest.
Therefore z and t are the (normal) coordinates in the laboratory frame. With
the simplifications introduced above for the polarization, we obtain two coupled
174 Light–Matter Interaction
differential equations for the amplitude of the fundamental wave
∂
∂z
+
1
ν
1
∂
∂t
−
ik
1
2
∂
2
∂t
2
˜
E
1
+ D
1
=−iχ
(2)
ω
2
1
4c
2
k
1
˜
E
∗
1
˜
E
2
e
ikz
(3.102)
and for the SH wave
∂
∂z
+
1
ν
2
∂
∂t
−
ik
2
2
∂
2
∂t
2
˜
E
2
+ D
2
=−iχ
(2)
ω
2
2
4c
2
k
2
˜
E
2
1
e
−ikz
,
(3.103)
where k = 2k
1
(ω
1
) − k
2
(ω
2
) is the wave vector mismatch calculated with
the wave vector values at the carrier frequency of the fundamental and second
harmonic. Because k
1
, k
2
are functions of the orientation of the wave vector
with respect to the crystallographic axis, it is often possible to find crystals,
beam geometry and beam polarizations, for which k = 0 (phase matching) is
achieved [2,8,9]. Note that in the case of ultrashort pulses the wave vectors vary
over the bandwidth of the pulse. This variation caused by the linear polarization
has already been taken into account by the time derivatives on the left-hand sides
of Eqs. (3.102) and (3.103), cf. Eq. (1.89).
Type I—Small Conversion Efficiencies
Small conversion efficiencies occur at low input intensities and/or small length
of the nonlinear medium and nonlinear susceptibility. Under these circumstances
we may assume that the fundamental pulse does not suffer losses. If we assume in
addition that k
1
= k
2
= D
1
= D
2
≈ 0 we find for the fundamental pulse, using
Eq. (3.102),
˜
E
1
(t, z) =
˜
E
1
(
t − z/ν
1
)
. The fundamental pulse travels distortionless
in a frame moving with the group velocity ν
1
. This expression can be inserted
into the generating equation for the SH, Eq. (3.103). Integration with respect to
the propagation coordinate yields for the SH at z = L:
˜
E
2
t −
L
ν
2
, L
=−i
χ
(2)
ω
2
2
4c
2
k
2
L
0
˜
E
2
1
t −
z
ν
2
+
1
ν
2
−
1
ν
1
z
e
−ikz
dz.
(3.104)
Using the correlation theorem, Eq. (3.104) can be transformed into the frequency
domain:
˜
E
2
(, L) =−i
χ
(2)
ω
2
2
4c
2
k
2
˜
E
1
(
)
˜
E
1
( −
)d
L
0
e
i
(ν
−1
2
−ν
−1
1
)−k
z
dz.
(3.105)
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