Second Harmonic Generation (SHG) 183
The set of equations (3.120) can easily be written for the spectral field envelopes
defined in Eq. (3.123).
The set of coupled equations (3.120) with (3.123) is convenient for a numerical
treatment. No other assumption or approximation has been made, except that the
spectra of the fundamental and second harmonic do not overlap, to be able to
split a single Maxwell’s second-order propagation equation into three coupled
differential equations. The dispersion of the material is contained in the frequency
dependence of the wave vectors k
2
q
() =
2
n
2
()/c
2
. The second derivative of
the envelope with respect to z can generally be neglected, unless the spectral
envelope of the field changes on length scales of the wavelength.
It should be noted that no moving frame of reference has been adopted in this
section. Hence, the fields are propagating at their respective group velocities.
A more convenient representation of the solution uses a frame of reference prop-
agating at one of the group velocities, for instance that of the second harmonic
(see also Section 3.4.1). The temporal (complex) envelopes in this frame of refer-
ence moving at the velocity ν
2
are obtained from the solutions
˜
E
i
(, z) through
the transformation:
˜
E
2
(t, z) =
∞
−∞
˜
E
2
(, z)e
−i[k
2
()−
ν
2
]z
e
−it
d (3.124)
˜
E
o
(t, z) =
∞
−∞
˜
E
o
(, z)e
−i[k
o
()−
ν
2
]z
e
−it
d (3.125)
˜
E
e
(t, z) =
∞
−∞
˜
E
e
(, z)e
−i[k
e
()−
ν
2
]z
e
−it
d. (3.126)
3.4.3. Pulse Shaping in Second Harmonic
Generation (Type II)
In this section we will describe the situation where group velocity mismatch
can be utilized to shape (shorten) ultrashort light pulses as a result of nonlinear
frequency conversion.
Akhmanov et al. [28] analyzed the situation where an SH pulse and a
fundamental pulse are simultaneously incident on a nonlinear crystal with
ν
2
> ν
1
. If a short SH pulse is launched in the trailing edge of a long fundamen-
tal pulse the SH will extract energy from various parts of the fundamental while
moving through the fundamental pulse because of the group velocity mismatch.
High peak powers of the second harmonic and considerable pulse shortening
were predicted.
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