Nonlinear Fiber Optics - 4 ed. Agrawal

504 Chapter 12. Novel Nonlinear Phenomena
Figure 12.39: Measured third-harmonic spectrum when 60-fs pulses at 1240 nm with 0.5 nJ
energy were launched into a 8-cm-long fiber. (After Ref. [206]; c○2006 APS.)
condition Δβ = 0 is satisfied when third harmonic is shifted from 3ω p by an amount
Ω = −(Δβ 0 /Δβ 1 ). This shift depends on the group-velocity mismatch and can be
positive or negative depending on the fiber mode for which phase matching occurs.
Equation (12.5.12) does not include the contribution of SPM and XPM. This nonlinear
contribution to the phase mismatch also affects the exact value of frequency shift Ω.
It is possible for the THG spectrum to exhibit several distinct peaks, if the phasematching
condition in Eq. (12.5.12) is satisfied for several different values of Ω such
that each peak corresponds to THG in a different fiber mode [205]. This feature was
observed in a 2006 experiment in which 60-fs pulses at a wavelength of 1240 nm were
launched into a 8-cm-long microstructured fiber with 4-μm core diameter [206]. Figure
12.39 shows the spectrum of third harmonic at the fiber output for pulses with 0.5 nJ
energy. The peak located near 420 nm is shifted from the third harmonic of 1240 nm
at 413.3 nm by more than 6 nm. Moreover, several other peaks occur that are shifted
by as much as 60 nm from this wavelength. A theoretical model predicts that three
higher-order modes, EH 14 ,TE 04 , and EH 52 , provide phase matching for spectral peaks
located near 370, 420, and 440 nm, respectively.
The simple theory developed earlier for SHG can be extended for the case of THG,
with only minor changes, when pump pulses are relatively broad. However, one must
include the dispersive effects in the case of short pump pulses by replacing dA/dz as
indicated in Eq. (10.2.23). The resulting set of equations can be written as
∂A p
∂z + 1 ∂A p
+ iβ 2p ∂ 2 A p
v gp ∂t 2 ∂t 2 = iγ p (|A p | 2 + 2|A h | 2 )A p + i 3 γ∗ THA h A ∗ p, (12.5.14)
∂A h
∂z + 1 ∂A h
+ iβ 2h ∂ 2 A h
v gh ∂t 2 ∂t 2 = iγ h (|A h | 2 + 2|A p | 2 )A h + iγ TH A 3 pe iΔβ0z ,(12.5.15)
where γ j is the nonlinear parameter, as defined in Eq. (2.3.29), β 2 j is the GVD parameter,
and j = p or h for the pump and its third harmonic, respectively. The THG growth
is governed by γ TH .