L04 · Gain and feedback

Single-Pass Optical Amplifier

Why does an inverted medium stop giving exponential gain? Because the light it amplifies spends the inversion. Each new photon from stimulated emission moves an atom out of the upper level, and the seed and the spontaneous emission all draw on the same inversion. The more the light grows, the lower the gain, until it saturates. Bright output from a medium with no mirrors is amplified spontaneous emission, not a laser.

Send a seed beam through a pumped medium. Compare its amplified power with a forward ASE channel, follow how the inversion is used up along the medium, and get a bright output with no 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 $\text{—}$

L04 · Experiment

Why does an inverted medium stop giving exponential gain?

Seed onRemove the input; the medium stays pumped.Selected spontaneous channelCompare exact darkness with ASE at zero seed.Seed power, logarithmic

\log_{10}(P_s/\mathrm{mW})

0

Unsaturated upper density

\frac{N_0}{10^{23}\,\mathrm{m}^{-3}}

2

Line-center cross section

\frac{\sigma_0}{10^{-23}\,\mathrm{m}^{2}}

2

Medium length

L

50\,\mathrm{cm}

Detuning in gain half-widths

\Delta/\gamma

0

Raise the seed power until the inversion at the exit drops below half its unsaturated value. Then remove the seed and compare the forward output, still with no feedback.

The model is a steady, homogeneous, ideal four-level medium at 1064 nm, with light traveling forward only. The mode area is fixed at 0.5 square millimetres and the recovery lifetime at 0.5 ms. The density is continuous; the 40 markers in the apparatus are rounded for display. ASE is one forward channel with a chosen spectral and angular range and a collection fraction of 0.0001, not the total emitted power. Backward ASE, pump propagation, diffraction and thermal effects are left out. The drawn field size and brightness are schematic.

Start with the essentials

Focus question: Does light still come out when the seed is removed, and would that prove the medium is oscillating?
One-sentence intuition: The seed and the collected ASE draw on the same stored inversion, so more power in either one leaves less inversion for 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.

Steady depletion at each point

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}

The pump keeps restoring the upper-level density toward its unsaturated value with a fixed recovery time. The lower level stays empty. The mode area is 0.5 square millimetres and the recovery time is 0.5 milliseconds.

Seed and ASE share one inversion

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

When the spontaneous channel is on, its collection fraction is 0.0001. That source term and the seed amplification use up the same upper-level energy. No backward wave is included.

An independent check without ASE

\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 switched off, this implicit relation checks the numerical propagation. A weak enough seed gets plain exponential gain, and the optical power added is limited by the pump that replenishes the inversion.

Common difficulties

The ASE readout is one forward channel

Typical misconceptionThe ASE power shown is all the spontaneous light the medium gives off.

Better mental modelOnly one forward channel, with a set range of angles and frequencies, is collected. Backward and uncollected emission are left out, so this readout is not the total fluorescence.

Saturation comes from the populations

Typical misconceptionThe detector or the display is what caps the power.

Better mental modelThe total optical power in the model depletes the computed upper-level density. The power readout is linear and does not depend on how bright the schematic scene looks.

Run the experiment

01
Find the weak-seed limit
Choose Weak seed. Turn off Selected spontaneous channel and lower the logarithmic seed power. Compare the seed gain with the small-signal reference.
What to observe: At low photon flux, the inversion stays nearly uniform along the medium and the gain approaches the reference value.
02
Use up the inversion along the medium
Choose Saturation and increase the medium length. Compare the signal trace, the inversion at the exit and the local gain.
What to observe: The upper-level density drops along the beam path, and the signal gain stays below the unsaturated exponential prediction.
03
Get output without a seed
Choose ASE without seed. Turn Selected spontaneous channel off, then on again.
What to observe: With no seed and no spontaneous source, the output stays dark. Any nonzero source is amplified in a single pass, with no feedback loop at all.
04
Tune off the gain line
Choose Weak seed, then increase the detuning and leave every other setting as it is.
What to observe: The stimulated cross section and the local saturation power change together, consistent with each other. More intensity cannot make up for missing frequency overlap.