Nonlinear Fiber Optics - 4 ed. Agrawal

500 Chapter 12. Novel Nonlinear Phenomena
Relative Power
Second
Harmonic
Pump
Wavelength (μm)
Figure 12.37: Spectrum at the output a 10-cm-long microstructured fiber when 100-fs pulses at
1064 nm with 250 W peak power are launched into it. (After Ref. [189]; c○2000 OSA.)
having the form [154]
dA p
dz = iγ p(|A p | 2 + 2|A h | 2 )A p + i 2 γ∗ SHA h A ∗ p exp(−iκz), (12.5.4)
dA h
dz = iγ h(|A h | 2 + 2|A p | 2 )A h + iγ SH A 2 p exp(iκz), (12.5.5)
where γ p and γ h are defined similarly to Eq. (2.3.28),
γ SH =(3ω 1 /4n 1 c)ε 2 0 α SH f 112 χ (3) |E p | 2 |E SH |, (12.5.6)
f 112 is an overlap integral (see Section 10.2), κ = Δk p − Δk, and Δk is given by Eq.
(12.5.2) after replacing ω p with ω 1 . The parameter κ is the residual wave-vector mismatch
occurring when ω 1 ≠ ω p . The terms proportional to γ p and γ h are due to SPM
and XPM and must be included in general.
Equations (12.5.4) and (12.5.5) can be solved using the procedure of Section 10.2.
If we assume that the pump remains undepleted (|A h | 2 ≪|A p | 2 ), Eq. (12.5.4) has the
solution A p (z)= √ P p exp(iγ p P p z), where P p is the incident pump power. Introducing
A h = B h exp(2iγ p P p z) in Eq. (12.5.5), we obtain
dB h
dz = iγ SHP p exp(iκz)+2i(γ h − γ p )P p B h . (12.5.7)
Equation (12.5.7) is readily solved to obtain the second-harmonic power as
P h (L)=|B h (L)| 2 = |γ SH P p L| 2 sin2 (κ ′ L/2)
(κ ′ L/2) 2 , (12.5.8)
where κ ′ = κ − 2(γ h − γ p )P p . Physically, SPM and XPM modify κ as they contribute
to the phase-matching condition.
The approximation that the pump remains undepleted begins to break down for
conversion efficiencies >1%. It turns out that Eqs. (12.5.4) and (12.5.5) can be solved