L27 · Scatter, oscillate, cascade

Raman & Brillouin Lasers

Pump a silica fiber, cross a Stokes threshold and watch the next order clamp the previous one. Capture a transient and calculate a steady pump sweep.

Scattering process $\text{—}$

Elapsed round trips $\text{—}$

Physical elapsed time $\text{—}$

First Stokes frequency shift $\text{—}$

First Stokes wavelength $\text{—}$

Effective gain bandwidth $\text{—}$

Feedback free spectral range $\text{—}$

Gain overlap fraction $\text{—}$

First oscillation threshold $\text{—}$

Live pump in the fiber $\text{—}$

First-order extracted power $\text{—}$

Second-order extracted power $\text{—}$

Third-order extracted power $\text{—}$

Live phonon heat power $\text{—}$

Live passive loss power $\text{—}$

Live total Stokes output $\text{—}$

Separate steady output reference $\text{—}$

Integrated energy residual $\text{—}$

Captured transient samples $\text{—}$

Computed equilibrium samples $\text{—}$

Frozen captured Stokes output $\text{—}$

Stokes propagation $\text{—}$

First-threshold regime $\text{—}$

L27

Can transparent glass become a laser gain medium?

Raman / Brillouin

m

0

Optical pump power

P_{\rm in}

3\,\mathrm{W}

Pump wavelength

\lambda_0

1064\,\mathrm{nm}

Silica gain fiber length

L

100\,\mathrm{m}

Effective mode area

A_{\rm eff}

60\,\mathrm{\mu m^2}

Stokes output coupling

T

10\,\mathrm{\%}

Additional round-trip loss

\ell

2\,\mathrm{\%}

Allowed Stokes orders

M

3

Spacing error in gain FWHM units

\delta/\Gamma

0

Prescribed Brillouin pump linewidth

\Delta\nu_p

1\,\mathrm{MHz}

Detail: fiber / backward / forward

j

1

Capture a cascade after evolving the cavity, then compare the steady pump sweep. Broadening the Brillouin pump raises the first threshold even at the same total pump power.

A reduced, spatially uniform cavity power model with one replenished optical pump and up to three Stokes modes. The pump supplies power to a nonresonant transit reservoir; it is not a coherent resonantly driven pump cavity. Raman modes co-propagate; backward bulk Brillouin scattering alternates direction at each order in an ideal bidirectional feedback loop. Mode frequencies are prescribed near the gain peak; the model assumes cavity modes exist there and omits their discrete resonance alignment and dispersion. Assumed silica parameters: refractive index 1.45, sound speed 5960 metres per second, Raman shift 13.2 terahertz, a six-terahertz Gaussian Raman gain envelope, thirty-megahertz Lorentzian Brillouin gain and illustrative peak gain coefficients. A Lorentzian pump spectrum reduces and broadens Brillouin gain by convolution; the same prescribed pump linewidth is used at every cascade step. This is not an emitted linewidth calculation. Fast material responses are eliminated; polarization, acoustic transients, thermal/quantum noise, anti-Stokes processes, Kerr effects and spatial pump profiles are absent. Fixed one-nanowatt initial Stokes seeds trigger deterministic growth; they do not model spontaneous emission. Power equations retain photon-energy ratios: missing optical energy becomes phonon heat. Propagation loss uses the effective interaction length and uniform cavity attenuation. The logarithm of coupler survival defines a continuous mean-field extraction rate; extracted power is that rate times circulating power, not an exact discrete coupler calculation. Feedback path length is the fiber length plus two metres. Captured transients copy actual evolution; the separately labelled 151-point sweep solves model equilibria and does not represent acquired hardware or a time sweep. Coil shape, route arrows and colors are schematic, including infrared light. The energy residual includes initial seeds, stored fields, escaped pump, extracted light, passive loss and phonon heat.

Start with the essentials

Focus question: Can transparent glass become a laser gain medium?
One-sentence intuition: The next Stokes order can drain and clamp its parent instead of simply adding more output.

Core mathematical model

One photon becomes light and vibration

\begin{aligned}\nu_{j-1}&=\nu_j+\Omega_j\\h\nu_{j-1}&=h\nu_j+h\Omega_j\end{aligned}

Photon number transfers between neighboring orders, while their optical energies differ.

Average gain over the fiber

\begin{aligned}C_j&=g_jL_{\rm eff}/A_{\rm eff}\\L_{\rm eff}&=(1-e^{-\alpha L})/\alpha\end{aligned}

Gain overlap includes the prescribed spacing error. Pump spectral broadening reduces Brillouin gain.

Stokes growth and depletion

\begin{aligned}\dot P_j&=G_j-D_j\\G_j&=C_jP_{j-1}P_j-\Lambda P_j\\D_j&=r_jC_{j+1}P_jP_{j+1}\end{aligned}

The dot uses time in round trips. The frequency ratio in depletion pays for the emitted phonon.

First oscillation threshold

\begin{aligned}P_{0,\rm th}&=\Lambda/C_1\\P_{\rm in,th}&=a\Lambda/C_1\end{aligned}

The pump transit reservoir has decay coefficient a. Additional Stokes orders raise the demand on their parent.

Count the phonon heat

\begin{aligned}Q_{\rm ph}&=\sum_j(r_{j-1}-1)S_j\\S_j&=C_jP_{j-1}P_j\\r_{j-1}&=\nu_{j-1}/\nu_j\end{aligned}

The fractional optical energy deficit is small for Brillouin scattering and much larger for Raman scattering.

A steady parent is clamped

\begin{aligned}C_jP_{j-1}&=\Lambda+r_jC_{j+1}P_{j+1}\\P_j&>0\end{aligned}

This relation solves each occupied equilibrium branch. The last order has no further depletion channel.

Common difficulties

Gain width is output linewidth

Typical misconceptionThe displayed gain bandwidth predicts an emitted laser linewidth.

Better mental modelThis model solves power dynamics only. Phase noise, mode selection and intrinsic linewidth require another model.

More pump always grows the first order

Typical misconceptionEvery Stokes order increases monotonically with pump.

Better mental modelOnce the next order oscillates, it drains the parent and creates a clamped branch.

Run the experiment

01
Cross a real threshold
Evolve the below-threshold preset, then the Raman preset. Capture each transient.
What to observe: The fixed seed decays below threshold and grows above it. Pump depletion limits the growth.
02
Follow the energy into another order
Evolve the cascaded preset and calculate its steady pump sweep. Compare one allowed order.
What to observe: As a new order grows, its parent becomes clamped. The separately computed sweep exposes successive thresholds.
03
Broaden the pump
Compare narrow and broad Brillouin pumping at the same power. Inspect the backward and forward ports.
What to observe: Less overlap raises threshold. Odd and even Brillouin orders leave opposite ports.