180 Light–Matter Interaction
3.4.2. Second Harmonic Type II: Equations for
Arbitrary Phase Mismatch and
Conversion Efficiencies
Treatment in the Time Domain
As pointed out before there is no analytical solution to the general problem of
SH generation. Numerical procedures have to be used to describe the propagation
of the fundamental and the SH pulses under the combined action of (linear)
dispersion and nonlinear effects. The possible effects are particularly complex
and interesting in type II SH generation.
Type II SH generation involves the interaction of three waves, the SH, and an
ordinary (o) and extraordinary (e) fundamental. Group velocity mismatch of these
three waves does not always lead to pulse broadening. In the case of SHG type II,
it is possible to achieve significant pulse compression at either the fundamental
or the up-converted frequency.
To describe type II frequency conversion we extend the system of
equations (3.102) and (3.103). We choose a retarded time frame of reference trav-
eling with the second harmonic signal at its group velocity ν
2
. The fundamental
pulse has a component
˜
E
o
(t) exp[i(ω
1
t − k
o
z)] propagating as an ordinary wave
(subscript o) at the group velocity ν
o
, and a component
˜
E
e
(t) exp[i(ω
1
t − k
e
z)]
propagating as an extraordinary wave (subscript e) at the group velocity ν
e
.
The system of equations describing the evolution of the fundamental pulses
˜
E
o
and
˜
E
e
, and the generation of the SH wave
˜
E
2
is:
∂
∂z
+
1
ν
o
−
1
ν
2
∂
∂t
˜
E
o
+ D
o
=−iχ
(2)
ω
2
1
4c
2
k
o
˜
E
∗
e
˜
E
2
e
ikz
(3.112)
∂
∂z
+
1
ν
e
−
1
ν
2
∂
∂t
˜
E
e
+ D
e
=−iχ
(2)
ω
2
1
4c
2
k
e
˜
E
∗
o
˜
E
2
e
ikz
(3.113)
∂
∂z
˜
E
2
+ D
2
=−iχ
(2)
ω
2
1
c
2
k
2
˜
E
e
˜
E
0
e
−ikz
, (3.114)
where k = k
o
+ k
e
− k
2
is the wave vector mismatch calculated at the pulse
carrier frequency. The phase matching condition k
0
= k
o
(ω
1
) + k
e
(ω
1
) −
k
2
(2ω
1
) = 0 implies that the phase velocities are matched. The fact that the
waves at ω
1
and ω
2
remain in phase does not necessarily imply that pulses reach
simultaneously the end of the crystal. The three wave packets propagate at group
velocities ν
o
, ν
e
, and ν
2
that, in general, are different. The expression Eq. (3.106)
found for type-I SH generation without pump depletion can be regarded a special
solution of Eqs. (3.112–3.114).
Second Harmonic Generation (SHG) 181
SHG for Short Pulses—Treatment in the Frequency Domain
When dealing with the conversion of short pulses, it is not sufficient to include
dispersion only up to first order, that is D = 0. For D = 0, however, the system of
equations (3.112)–(3.114) contains higher-order time derivatives whose treatment
is numerically difficult. The problem can be stated more clearly in the frequency
domain, using the complete functional dependence of the k vectors (or the indices
of refraction), rather than power series. We start from the wave equation for the
electric field
∂
2
∂z
2
−
1
c
2
∂
2
∂t
2
˜
E(t, z ) = µ
0
∂
2
∂t
2
˜
P
L
(t, z) +
˜
P
NL
(t, z)
(3.115)
where the electric field is the sum of the three participating waves
˜
E(t, z ) =
ˆ
e
o
˜
E
o
(t, z) +
ˆ
e
e
˜
E
e
(t, z) +
ˆ
e
e
˜
E
2
(t, z) (3.116)
and the nonlinear polarization
˜
P
NL
(t, z) =
0
χ
(2)
6
˜
E
o
(t, z)
˜
E
e
(t, z)
ˆ
e
e
+
˜
E
2
(t, z)
˜
E
∗
o
(t, z)
ˆ
e
e
+
˜
E
2
(t, z)
˜
E
∗
e
(t, z)
ˆ
e
o
7
.
(3.117)
Without loss of generality, we have assumed that the SH field
˜
E
2
propagates
as an extraordinary wave with polarization vector
ˆ
e
e
. The nonlinear polarization
terms are responsible for the evolution of the SH, the fundamental e-wave and
the fundamental o-wave, respectively.
Following the same procedure as in Section 1.2, we take the Fourier transform
of Eq. (3.115):
∂
2
∂z
2
+ µ
0
2
()
˜
E(, z) =−µ
0
2
˜
P
NL
(, z) (3.118)
where we used the expressions (1.70) for the linear polarization and Eq. (1.73)
for the dielectric constant. The nonlinear polarization in the frequency domain is
a sum of three convolution integrals; the first member of the sum, for example,
is
0
χ
(2)
ˆe
e
E
o
(
, z )E
e
( −
, z )d
.
For the electric field components we make the ansatz
˜
E
q
(, z) =
1
2
˜a
q
(, z)e
−ik
q
()z
. (3.119)
where the subscript q stands for o, e, or 2. The amplitudes ˜a
q
(, z) peak at the
central frequencies of the corresponding pulse. The ansatz is a solution of the
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