Page 498 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 479

Virtual energy
state

ħω p Ω = ω – ω = ω – ω p
a
s
p
ħω p ħω s ħω a

Vibrational
state
ħΩ

Stokes Raman Anti-Stokes Raman
Scattering Scattering

Figure 10.32 Stokes and anti-Stokes Raman scattering.

independently by Landsberg and Manderlstam [78]) in 1928. The molecules in the medium have several
vibrational states (or phonon modes). When a light wave (photons) interacts with sound waves (phonons), the
frequency of the light wave is shifted up or down. The shift in frequency gives information about the phonon
modes of the molecules. When the scattered photon has a frequency lower than the incident photon, it is
known as Stokes shift. Stokes Raman scattering can be described quantum mechanically as the annihilation
of a pump photon of energy ℏ and the creation of a Stokes photon of lower energy ℏ , and absorption
s
p
of the energy ℏ( − ) by the molecules by making transition from a low-energy vibrational state to a
s
p
high-energy vibrational state (see Fig. 10.32). A lower-energy photon has lower frequency and, therefore,
Stokes Raman scattering leads to a red shift of the incident light wave. If the molecule makes transition from
a high-energy vibrational state to a low-energy vibrational state in the presence of an incident pump of energy
ℏ , the difference in energy is added to the incident photon, leading to a photon of higher energy ℏ (which
p
a
is of higher frequency). This is known as anti-Stokes Raman scattering. Raman scattering is quite useful in
chemistry, since vibrational information is specific to the chemical bonds and symmetry of molecules.
Spontaneous Raman scattering is typically very weak. In 1962, it was found that an intense-pump optical
wave can excite molecular vibrations and, thereby, stimulate molecules to emit photons of reduced energy
(a Stokes wave), to which most of the pump energy is transferred [79]. This is known as stimulated Raman
scattering. The interaction between the pump and the Stokes wave is described by the following coupled
equations under CW conditions [80]:
d
s
= g   −  , (10.406)
s s
R p s
dZ
d
p p
=− g   −  , (10.407)
R p s
p p
dZ
s
where  and  are the optical intensities of the pump and the Stokes waves, respectively, and are the
p s p s
fiber loss coefficients at the pump and the Stokes frequencies, respectively, and g (Ω), Ω= − is the
R p s
Raman gain coefficient. The amplification of the Stokes wave by the pump wave can be understood from
Eqs. (10.406) and (10.407). To simplify the analysis, we assume that the pump intensity is much larger than