L21 · Distributed gain and feedback

Fiber Laser & FBG Cavity

Build a distributed fiber oscillator, inspect local pump and signal powers, and acquire a pump–output sweep. Shift the output grating to break the held-mode feedback.

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 $\text{—}$

A stationary single-wavelength Yb-like teaching model couples a local two-manifold balance to forward pump and two signal waves. The pump is 976 nanometres and the held signal is 1064 nanometres. Density, cross-sections and lifetime are illustrative fixed parameters, not a fitted fiber product. The core is uniformly doped; the pump fills a fixed 125-micrometre cladding, while the signal overlap is 0.8. Pump stimulated emission and signal reabsorption are included. Fourth-order spatial integration and signal-amplitude shooting enforce both FBG power boundaries. Below the zero-field threshold the selected coherent field is zero; fluorescence and ASE are omitted. The rear grating has fixed strength four; the output grating is variable. Both are lossless uniform one-centimetre gratings with an index of 1.45. The signal stays at the rear Bragg wavelength: shifting the front grating changes its reflection there, but does not solve wavelength selection or longitudinal-mode competition. A probe reads the same stationary profile. The 21 sweep records are independent boundary solves; they are not measured hardware data or startup dynamics. The slope is the last one-watt increment. The quantum-defect channel is only the minimum energy difference associated with absorbed pump photons, not total heat or temperature. The forward nonlinear phase is a diagnostic integral using an illustrative silica nonlinear index; it does not feed back into the linear cavity, predict a damage threshold, or simulate Raman, Brillouin, thermal or modal instabilities. The drawing straightens the fiber, enlarges the core and grating periods, and uses visible colors for infrared.

Start with the essentials

Focus question: Does more fiber always give more useful output?
One-sentence intuition: Gain and feedback must close the same physical boundary problem.

Core mathematical model

Local two-manifold balance

\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 populates the upper manifold; stimulated and spontaneous emission drain it. The signal power is the sum of both directions.

One population drives all powers

\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 core area divided by cladding area. Increasing the core changes pump filling as well as signal intensity.

Close both signal boundaries

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

Their product is constant inside this model. The launched forward signal is shot until the output boundary closes.

Threshold is an unsaturated limit

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

Positive zero-field margin allows a saturated branch. Local transparency is weaker than whole-cavity threshold.

Useful power escapes the front grating

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

Almost closing the output can increase intracavity power while reducing useful escape.

Diagnostics are not high-power solvers

\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 channel is a minimum photon-energy difference. The second is a forward nonlinear phase; neither modifies this stationary cavity solution.

Common difficulties

More fiber is always better

Typical misconceptionLonger length increases absorption, so output must rise.

Better mental modelUnpumped tail and background loss can defeat that increase.

A mismatched grating guarantees darkness

Typical misconceptionTwo offset gratings cannot lase at any power.

Better mental modelThis fixed-wavelength model checks a gain–loss condition; enough gain can overcome weak reflection. Other wavelengths are outside the solve.

Run the experiment

01
Break and restore feedback
Choose gratings mismatched, then set the output Bragg offset to zero.
What to observe: Only the reflection at the held signal wavelength changes.
02
Follow the spatial balance
Move the position probe along the working and long-fiber presets.
What to observe: Residual pump, local excitation and backward signal all come from the same solution.
03
Measure useful escape
Acquire a sweep for working cavity and nearly closed output.
What to observe: Stored signal and useful output need not move together.