Nonlinear Fiber Optics - 4 ed. Agrawal

88 Chapter 4. Self-Phase Modulation
Figure 4.7: SPM-induced spectral broadening of a partially coherent CW beam for several values
of Z. The curve marked Z = 0 shows the input Gaussian spectrum.
When the coherence time becomes shorter than the pulse width, effects of partial
coherence must be included [14]–[21]. In the case of a CW beam, SPM can lead to
spectral broadening during its propagation inside an optical fiber. The physical reason
behind such broadening can be understood by noting that partially coherent light exhibits
both intensity and phase fluctuations. The SPM converts intensity fluctuations
into additional phase fluctuations [see Eq. (4.1.5)] and broadens the optical spectrum.
Alternatively, SPM reduces the coherence time T c as the CW beam propagates inside
the fiber, making it less and less coherent.
The optical spectrum of partially coherent light at the fiber output is obtained using
the Wiener–Khintchine theorem [22]:
∫ ∞
S(ω)= Γ(z,τ)exp(iωτ)dτ, (4.1.18)
−∞
where the coherence function Γ(z,τ) is defined as
Γ(z,τ)=〈U ∗ (z,T )U(z,T + τ)〉. (4.1.19)
The optical field U(z,T ) inside the fiber at a distance z is known from Eq. (4.1.4). The
angle brackets denote an ensemble average over fluctuations in the input field U(0,T ).
The statistical properties of U(0,T ) depend on the optical source and are generally
quite different for laser and nonlaser sources.
The average in Eq. (4.1.19) can be performed analytically for thermal sources because
both the real and imaginary parts of U(0,T ) follow a Gaussian distribution for
such a source. Even though the laser light used commonly in nonlinear-optics experiments
is far from being thermal, it is instructive to consider the case of a thermal field.