Nonlinear Fiber Optics - 4 ed. Agrawal

12.5. Harmonic Generation 501
analytically even when the pump is allowed to deplete. The solution in the form of
elliptic functions was first obtained in 1962 [173]. The periodic nature of the elliptic
functions implies that the power flows back to the pump after the SHG power attains
its maximum value. The analysis also predicts the existence of a parametric mixing
instability induced by SPM and XPM [171]. The instability manifests as doubling of
the spatial period associated with the frequency-conversion process. As pump depletion
is negligible in most experimental situations, such effects are difficult to observe
in optical fibers.
The derivation of Eq. (12.5.8) assumes that the χ (2) grating is created coherently
throughout the fiber. This would be the case if the pump used during the preparation
process were a CW beam of narrow spectral width. In practice, mode-locked pulses of
duration ∼100 ps are used. The use of such short pulses affects grating formation in
two ways. First, a group-velocity mismatch between the pump and second-harmonic
pulses leads to their separation within a few walk-off lengths L W .IfweuseT 0 ≈ 80 ps
and |d 12 |≈80 ps/m in Eq. (1.2.14), the values appropriate for 1.06-μm experiments,
L W ≈ 1 m. Thus, the χ (2) grating stops to form within a distance ∼1 m for pump pulses
of 100 ps duration. Second, SPM-induced spectral broadening reduces the coherence
length L coh over which the χ (2) grating can generate the second harmonic coherently.
It turns out that L coh sets the ultimate limit because L coh < L W under typical experimental
conditions. This can be seen by noting that each pump frequency creates its
own grating with a slightly different period 2π/Δk p , where Δk p is given by Eq. (12.5.2).
Mathematically, Eq. (12.5.1) for P dc should be integrated over the pump-spectral range
to include the contribution of each grating. Assuming Gaussian spectra for both pump
and second-harmonic waves, the effective dc polarization becomes [161]
P eff
dc = P dc exp[−(z/L coh ) 2 ], L coh = 2/|d 12 δω p |, (12.5.9)
where d 12 is defined in Eq. (1.2.13) and δω p is the spectral half-width (at 1/e point).
For |d 12 | = 80 ps/m and 10-GHz spectral width (FWHM), L coh ≈ 60 cm.
In most experiments performed with 1.06-μm pump pulses, the spectral width at
the fiber input is ∼10 GHz. However, SPM broadens the pump spectrum as the pump
pulse travels down the fiber [see Eqs. (4.1.6) and (4.1.11)]. This broadening reduces the
coherence length considerably, and L coh ∼ 10 cm is expected. Equation (12.5.8) should
be modified if L coh < L. In a simple approximation [154], L is replaced by L coh in Eq.
(12.5.8). This amounts to assuming that Pdc eff = P dc for z ≤ L coh and zero for z > L coh .
One can improve over this approximation using Eq. (12.5.9). Its use requires that γ SH
in Eq. (12.5.7) be multiplied by the exponential factor exp[−(z/L coh ) 2 ].IfL coh ≪ L,
Eq. (12.5.7) can be integrated with the result [161]
P 2 (κ)=(π/4)|γ SH P p L coh | 2 exp(− 1 2 κ2 Lcoh 2 ). (12.5.10)
This expression is also approximate as it is based on Eqs. (12.5.4) and (12.5.5) that are
valid only under quasi-CW conditions. For short pump pulses, the GVD effects can be
included in Eqs. (12.5.4) and (12.5.5) by replacing the spatial derivatives by a sum of
partial derivatives as indicated in Eq. (10.2.23).