L04 · Gain and feedback

Single-Pass Optical Amplifier

Send a seed through a pumped medium. Compare its amplified power with a forward ASE channel, follow local inversion depletion, and make a bright output without any feedback mirror.

Input seed power $\text{—}$

Output seed-channel power $\text{—}$

Selected forward ASE power $\text{—}$

Seed power gain $\text{—}$

Small-signal reference gain $\text{—}$

Saturation power $\text{—}$

Exit local gain $\text{—}$

Exit inversion fraction remaining $\text{—}$

Total forward photon flux $\text{—}$

Steady, homogeneous, ideal four-level forward-channel model at 1064 nm. Fixed mode area 0.5 mm² and recovery lifetime 0.5 ms; continuous density, 40 rounded apparatus markers. ASE is one selected forward spectral/angular channel with collection fraction 0.0001, not total emitted power. Backward ASE, pump propagation, diffraction and thermal effects are omitted. Field size and brightness are schematic.

Start with the essentials

Focus question: Can output remain when the seed is removed, and does that establish oscillation?
One-sentence intuition: The seed and selected ASE share the same depleted reservoir. Increasing either optical channel reduces the inversion available to both.

Core mathematical model

Frequency-dependent cross section

\sigma(\Delta)=\frac{\sigma_0}{1+(\Delta/\gamma)^2},\qquad g_0=\sigma N_0

Detuning is measured in gain half-widths. Cross section is per emitter; density is per cubic metre and gain is per metre.

Local steady depletion

N(z)=\frac{N_0}{1+[P_s(z)+P_a(z)]/P_{\mathrm{sat}}},\quad P_{\mathrm{sat}}=\frac{h\nu A}{\sigma\tau}

A fixed recovery reservoir drives upper density toward its unsaturated value. The lower state is empty. Mode area is 0.5 square millimetres and recovery time is 0.5 milliseconds.

Two forward channels, one reservoir

\frac{dP_s}{dz}=\sigma NP_s,\qquad\frac{dP_a}{dz}=\sigma NP_a+\beta\frac{h\nu AN}{\tau}

The selected spontaneous collection fraction is 0.0001 when enabled. The source term and seed amplification consume the same upper-state energy. No backward wave is included.

Independent limits

\ln\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}+\frac{P_{\mathrm{out}}-P_{\mathrm{in}}}{P_{\mathrm{sat}}}=g_0L\quad(\beta=0)

With ASE disabled this implicit relation checks the numerical propagation. A sufficiently weak seed recovers exponential gain, while added optical power is bounded by the replenishing pump reservoir.

Common difficulties

Forward ASE is a selected channel

Typical misconceptionThe displayed ASE power equals all spontaneous output.

Better mental modelOnly a specified forward angular and spectral channel is collected. Backward and uncollected emission are omitted, so this readout is not total fluorescence.

Saturation is a population effect

Typical misconceptionThe detector or display caps the power.

Better mental modelThe numerical density is depleted by the total modeled optical power. The linear power readout remains independent of schematic scene brightness.

Run the experiment

01
Find the weak-seed limit
Choose Weak seed. Disable the spontaneous channel and reduce the logarithmic seed setting. Compare seed gain with the small-signal reference.
What to observe: At low photon flux the inversion remains nearly uniform and gain approaches the reference.
02
Exhaust local inversion
Choose Saturation and increase medium length. Compare the signal trace, exit inversion and local gain.
What to observe: The upper density falls along the propagation direction; signal gain grows less than the unsaturated exponential prediction.
03
Make output without a seed
Choose ASE without seed. Disable and re-enable the spontaneous channel.
What to observe: Zero seed and zero spontaneous source remain dark. A nonzero source is amplified during one pass, without any feedback loop.
04
Miss the gain line
Choose Weak seed, then increase detuning while retaining all other settings.
What to observe: Both stimulated cross section and local saturation power change consistently. Stronger intensity is not a substitute for frequency overlap.