Raman Scattering

Particle in Vacuum

Let us consider a charged particle subject to an electromagnetic wave. The equation of motion is

${\frac {d^{2}\mathbf {r} }{dt^{2}}}={\frac {qE}{m}}\left[{\hat {x}}+{\frac {1}{c}}{\frac {d\mathbf {r} }{dt}}\times {\hat {y}}\right]\cos \left(\omega t-z\omega /c\right)$

We define new dimensionless variables $\mathbf {R} =\mathbf {r} /{\frac {qE}{m\omega ^{2}}}$, $\varphi =\omega t$ and $\delta ={\frac {qE}{m\omega c}}$

Let us expand the solution in $\delta$. For simplicity, let us assume that at $t=0$ the particle is at the origin, and that its velocity is zero. If we set $\delta = 0$ then the solution is

${\frac {d^{2}\mathbf {R} _{0}}{d\varphi ^{2}}}={\hat {x}}\cos \varphi \Rightarrow \mathbf {R} _{0}={\hat {x}}\left(1-\cos \varphi \right)$

Expanding to first order in $\delta$

${\frac {d^{2}\mathbf {R} _{1}}{d\varphi ^{2}}}=\delta \sin \varphi \cos \varphi {\hat {z}}\Rightarrow \mathbf {R} _{1}={\frac {\delta }{8}}\left[2\varphi -\sin \left(2\varphi \right)\right]{\hat {z}}$

We get that the particle continuously drifts in the longitudinal direction.

Particle in a Medium

Now suppose that a particle is confined to a medium. We model the effect of the medium as a harmonic oscillator with a frequency $\omega _{m}\ll \omega$. The equation of motion for the particle to first order in $\delta$ becomes

${\frac {d^{2}\mathbf {R} _{1}}{d\varphi ^{2}}}+{\frac {\omega _{m}^{2}}{\omega ^{2}}}\mathbf {R} _{1}=\delta \sin \varphi \cos \varphi {\hat {z}}\Rightarrow \mathbf {R} _{1}={\frac {\delta }{4}}\left[{\frac {\omega }{\omega _{m}}}\sin \left({\frac {\omega _{m}}{\omega }}\varphi \right)-{\frac {1}{2}}\sin \left(2\varphi \right)\right]{\hat {z}}$

We see that in addition to exciting a longitudinal wave with a harmonic of the original frequency, an oscillation with the frequency of the medium is excited. The latter is often called a phonon. Continuing the expansion to second order in $\delta$ yields

${\frac {d^{2}\mathbf {R} _{2}}{d\varphi ^{2}}}={\frac {\delta ^{2}}{4}}\left[{\frac {\omega }{\omega _{m}}}\sin \left({\frac {\omega _{m}}{\omega }}\varphi \right)-{\frac {1}{2}}\sin \left(2\varphi \right)\right]\left(\sin \varphi -\cos \varphi \right)$

The source term on the right hand side contains products of trigonometric functions with argumens $\varphi$ and $\omega _{m}\varphi /\omega$. When multiplied, the resulting modes will have arguments of the form $\varphi \left(1\pm \omega _{m}/\omega \right)$. These modes will have slightly larger or smaller energies than the incident radiation. This is the inelastic Raman scattering. The amplitude of the acceleration of these modes is

$a\approx {\frac {qE}{m}}\left({\frac {qE}{m\omega c}}\right)^{2}{\frac {\omega }{\omega _{m}}}$

and the luminosity of a single particle is given by

$L_{1}\approx {\frac {q^{2}a^{2}}{c^{3}}}\approx {\frac {q^{2}}{c}}{\frac {\omega ^{4}}{\omega _{m}^{2}}}\left({\frac {qE}{m\omega c}}\right)^{6}$

Contrary to elastic scattering (like Compton scattering), the emitted flux is not linear in the incident flux.

Induced Raman Scattering

We recall that the optical depth for the induced Compton scattering is

$\tau _{ic}\approx n_{e}\sigma _{t}x{\frac {kT_{b}}{m_{e}c^{2}}}$

The main difference between Compton and Raman scatterings is that the former is an interaction between a photon and an electron, and the latter is an interaction between a photon and a plasmon. A plasmon is just a quanta of longitudinal Langmuir waves in a plasma. The frequency of a plasmon is just the plasma frequency

$\omega _{p}^{2}\approx {\frac {n_{e}q^{2}}{m_{e}}}$

We assume that $\omega \gg \omega_p$. If a comparable amount of energy is transferred to scattered photons and to plasmons, the ratio between their numbers would be $\omega /\omega _{p}$, and so the optical depth for induced Raman scattering would be boosted by this factor compared with induced Compton scattering

$\tau _{ir}\approx {\frac {\omega }{\omega _{p}}}\tau _{ic}$

By dividing the optical depth by the light crossing time $x/c$, it is possible to obtain the induced Raman growth rate

$\Gamma _{ir}\approx {\frac {c}{x}}\tau _{ir}$

Induced Raman competes with other dissipative processes. The first is Landau damping, which occurs when the wavelength is smaller than the Debye length.

${\frac {c}{\omega }}<\lambda _{d}\approx {\sqrt {\frac {kT}{n_{e}q^{2}}}}$.

Another process that tends to damp out the induced Raman scattering is collisions, where the collision rate is given by

$\Gamma _{c}\approx {\frac {n_{e}q^{4}}{m_{e}^{2}}}\left({\frac {m_{e}}{kT}}\right)^{3/2}$.

Induced Raman scattering can only be effective if its growth rate exceeds the collision rate $\Gamma _{ir}>\Gamma _{c}$.