L21 · Distributed gain and feedback

Fiber Laser & FBG Cavity

Does more fiber always give more useful output? No. The pump is absorbed as it travels along the fiber. Past the point where it is used up, extra fiber adds loss instead of gain. The gratings at the two ends must also reflect the same wavelength; shift the output grating's wavelength and the feedback can fail completely.

Build a fiber laser between two fiber Bragg gratings, check the pump and signal power at any point along the fiber, and record output versus pump. Shift the output grating to take away the feedback at the fixed laser wavelength.

Front signal output $\text{—}$

Rear signal escape $\text{—}$

Absorbed pump $\text{—}$

Residual pump $\text{—}$

Zero-field round-trip log margin $\text{—}$

Relative mirror-boundary error $\text{—}$

Output FBG reflection at held wavelength $\text{—}$

Bare-fiber longitudinal spacing $\text{—}$

Position probe $\text{—}$

Local excited fraction $\text{—}$

Local pump power $\text{—}$

Local forward signal $\text{—}$

Local backward signal $\text{—}$

Local net intensity gain $\text{—}$

Quantum-defect minimum channel $\text{—}$

Forward nonlinear phase diagnostic $\text{—}$

Captured pump settings $\text{—}$

Captured final-step slope $\text{—}$

Held-mode branch $\text{—}$

This steady-state teaching model works at a single signal wavelength with ytterbium-like parameters. At each point a balance between two energy manifolds is coupled to a forward pump wave and to signal waves travelling both ways. The pump is at 976 nanometres and the signal is held at 1064 nanometres. Dopant density, cross-sections and lifetime are fixed illustrative values, not fitted to a real fiber product. The core is uniformly doped. The pump fills a fixed 125-micrometre cladding, and the signal overlap is 0.8. Stimulated emission at the pump wavelength and reabsorption of the signal are included. A fourth-order integration along the fiber, with shooting on the signal amplitude, meets the power conditions at both gratings. Below the zero-field threshold the coherent signal is zero; fluorescence and amplified spontaneous emission (ASE) are left out. The rear grating has a fixed strength of four, and the output grating is adjustable. Both are lossless, uniform, one centimetre long and have an index of 1.45. The signal stays at the rear grating's Bragg wavelength. Shifting the front grating changes its reflection at that wavelength, but the model does not solve for wavelength selection or competition between longitudinal modes. The probe reads the same steady profile. The 21 points of a sweep are independent solutions of the boundary problem, not measured hardware data or startup dynamics. The slope is taken over the last one-watt step. The quantum-defect readout is only the minimum power set by the energy difference between absorbed pump photons and signal photons; it is not the total heat or a temperature. The forward nonlinear phase is a diagnostic integral using an illustrative nonlinear index for silica. It does not feed back into the linear cavity, does not predict a damage threshold and does not simulate Raman, Brillouin, thermal or modal instabilities. The drawing straightens the fiber, enlarges the core and the grating periods, and uses visible colors for infrared light.

Start with the essentials

Focus question: Does more fiber always give more useful output?
One-sentence intuition: The laser runs only when the gain along the fiber and the reflections at both gratings together bring the signal back to the same power after a round trip.

Core mathematical model

Population balance at each point

\begin{aligned}f&=\frac{W_{pa}+W_{sa}}{W_{pa}+W_{pe}+W_{sa}+W_{se}+\tau^{-1}}\\ W_{jk}&=\frac{\Gamma_j\sigma_{jk}P_j}{h\nu_j A}\end{aligned}

Absorption fills the upper manifold; stimulated and spontaneous emission empty it. The signal power here is the sum of both directions.

One population sets pump absorption and signal gain

\begin{aligned}\frac{dP_p}{dz}&=-\Gamma_pN[\sigma_{pa}(1-f)-\sigma_{pe}f]P_p\\ g_s&=\Gamma_sN[\sigma_{se}f-\sigma_{sa}(1-f)]-\alpha_s\end{aligned}

The pump overlap is the core area divided by the cladding area. A larger core therefore changes the pump overlap as well as the signal intensity.

Signal waves and the two grating conditions

\begin{aligned}\frac{dP_+}{dz}&=g_sP_+\\\frac{dP_-}{dz}&=-g_sP_-\\ P_+(0)&=R_1P_-(0)\\ P_-(L)&=R_2P_+(L)\end{aligned}

In this model the product of the forward and backward powers stays constant along the fiber. The solver adjusts the launched forward signal until the condition at the output grating is met.

Threshold with no signal present

\mathcal M_0=\ln(R_1R_2)+2\int_0^L g_{s,0}(z)\,dz

If this zero-field round-trip margin is positive, a saturated lasing solution can exist. Reaching transparency at one point is a weaker condition than reaching threshold for the whole cavity.

Useful power leaves through the front grating

P_{\rm out}=(1-R_2)P_+(L)

Making the output grating reflect almost everything can raise the power inside the cavity and still lower the useful output.

Two diagnostics that do not change the solution

\begin{aligned}Q_{\rm qd}&=\left(1-\frac{\lambda_p}{\lambda_s}\right)P_{p,\rm abs}\\ B_+&=\int_0^L\frac{2\pi n_2P_+(z)}{\lambda_s A}\,dz\end{aligned}

The first is the minimum power set by the photon-energy difference between absorbed pump and signal. The second is the nonlinear phase of the forward signal. Neither one changes the steady cavity solution, and neither is a high-power solver.

Common difficulties

More fiber is always better

Typical misconceptionA longer fiber absorbs more pump, so the output has to go up.

Better mental modelA tail that gets almost no pump, together with background loss, can outweigh that extra absorption.

Mismatched gratings mean no laser

Typical misconceptionIf the two gratings are offset, the laser cannot start at any pump power.

Better mental modelThis fixed-wavelength model only checks whether gain beats loss, and enough gain can make up for a weak reflection. It does not solve for other wavelengths.

Run the experiment

01
Break and restore feedback
Choose Gratings mismatched, then set Output Bragg offset to zero.
What to observe: The only thing that changes is the output grating's reflection at the fixed signal wavelength.
02
Follow the powers along the fiber
In Working cavity and in Long fiber, move the probe along the fiber with Probe position fraction.
What to observe: The leftover pump, the local excitation and the backward signal all come from the same solution.
03
Measure the useful output
Press Acquire pump sweep for Working cavity and for Nearly closed output.
What to observe: The signal stored in the cavity and the useful output do not have to rise together.