L14 · Saturation, loss and useful power

Output Coupler Optimization

Which output mirror gives a laser the most output power? One that lets out a moderate fraction of the light. With too little transmission the light stays trapped inside and internal losses eat it; with too much, the gain cannot keep up and the laser stops. Where the best value lies depends on the gain and on the internal loss.

A more transmissive output mirror lets more light out but weakens the light circulating inside. Record power-meter readings as you change the mirror, pick the best setting you measured, and compare it with the optimum of the continuous model.

Steady oscillator state $\text{—}$

Recorded mirror settings $\text{—}$

Circulating power before gain $\text{—}$

Power incident on output mirror $\text{—}$

Useful output power $\text{—}$

Internal dissipated power $\text{—}$

Stimulated power added per round trip $\text{—}$

Saturated intensity log gain $\text{—}$

Total intensity log loss $\text{—}$

Largest transmission at threshold $\text{—}$

Useful share of stimulated power $\text{—}$

Power-balance residual $\text{—}$

Low-loss optimum estimate $\text{—}$

Continuous reference optimum transmission $\text{—}$

Continuous reference peak output $\text{—}$

Best recorded transmission $\text{—}$

Best recorded output power $\text{—}$

The model is a single-mode, one-way ring laser running continuously at steady state. Ideal fold mirrors keep the mode unchanged. The scene is a schematic, with no cavity-stability or diffraction calculation. A single lumped gain element holds the whole round-trip logarithmic intensity gain; it is followed by a lumped parasitic power loss and then the output coupler. All other mirrors are lossless. Circulating power is referenced just before the gain element, and the saturation power is defined at the same plane. The saturated logarithmic gain follows a phenomenological reciprocal saturation law. The threshold uses exact power survival factors, and every displayed power follows the same sequence: gain, then loss, then coupler, then back to the gain. The output mirror transmission is a power fraction, not an amplitude coefficient. Below threshold there is no steady stimulated power; spontaneous fluorescence and ASE are left out. You set the gain directly; it is not calibrated to pump power in watts. The extraction share is the useful output divided by the total stimulated power added. It is not a pump efficiency or an electrical efficiency. This is not the distributed Rigrod solution with counter-propagating beams. Gain depletion along the medium, spatial hole burning, startup, relaxation, heating, multimode competition, finite apertures and output-mirror absorption are all left out. Each record is a noiseless simulated power-meter reading. Changing the mirror transmission keeps the records; changing the gain, saturation power or internal loss clears them. The automatic scan records 71 settings in whole-percent steps from zero to 70%. The best recorded setting is picked from those records alone, separately from the optimum found numerically on the continuous model. The square-root estimate for low loss is only an asymptotic comparison. Brightness in the scene is normalized schematic contrast, not a power scale.

Start with the essentials

Focus question: How does the best output mirror change when gain or internal loss changes?
One-sentence intuition: The best mirror lies between trapping the light inside and letting out so much that lasing stops. To find it, fix the plane where circulating power is measured, follow every survival factor around the ring, and compare recorded outputs with the medium held fixed.

Core mathematical model

Round-trip survival and the lasing threshold

\begin{aligned}S&=(1-L_i)(1-T)\\\ell&=-\ln S\\G_0&>\ell\quad\text{for lasing}\end{aligned}

The gain is the logarithm of the intensity growth over one full round trip. Internal loss and transmission are power fractions, so the fractions that survive each of them multiply.

Gain saturation in steady state

\begin{aligned}G(P)&=\frac{G_0}{1+P/P_s}\\P&=P_s\max\!\left(0,\frac{G_0}{\ell}-1\right)\\e^{G(P)}S&=1\quad(P>0)\end{aligned}

The circulating power is taken just before the gain element. Above threshold, the saturated gain is pinned at the total logarithmic loss.

Output uses the power arriving at the mirror

\begin{aligned}P_{\rm OC}&=P/(1-T)\\P_{\rm out}&=TP_{\rm OC}\\P_{\rm gain,out}&=P_{\rm OC}/(1-L_i)\\P_{\rm loss}&=L_iP_{\rm gain,out}\end{aligned}

In the ring sequence, the reference circulating power comes after the coupler. Multiplying it directly by the transmission would use the power at the wrong plane, and the error matters once the coupling is appreciable.

Power balance around the ring

\begin{aligned}\Delta P_{\rm stim}&=P_{\rm gain,out}-P\\\Delta P_{\rm stim}&=P_{\rm out}+P_{\rm loss}\\\eta_{\rm extraction}&=\frac{P_{\rm out}}{\Delta P_{\rm stim}}\end{aligned}

This is a steady-state balance of optical power, not an electrical or pump conversion efficiency. No calibrated pump power enters the model.

Threshold and a low-loss estimate

\begin{aligned}T_{\rm th}&=1-\frac{e^{-G_0}}{1-L_i}\\P_{\rm out}&\simeq P_sT\!\left(\frac{G_0}{L_i+T}-1\right)\\T_{\rm opt,low}&\simeq\sqrt{G_0L_i}-L_i\end{aligned}

If the threshold transmission comes out negative, no mirror can keep the laser oscillating. The square-root estimate drops higher-order corrections to the survival factors, so compare it only when losses are small.

Best sample versus continuous optimum

\begin{aligned}T_{\rm recorded}&=\underset{T_j}{\operatorname{argmax}}\,P_{{\rm out},j}\\T_{\rm reference}&=\underset{0\le T\le0.7}{\operatorname{argmax}}\,P_{\rm out}(T)\end{aligned}

The first line picks from recorded values only. The second searches the model over a continuous range; a scan in whole-percent steps can miss a true maximum that falls between samples.

Common difficulties

A bright cavity does not mean a bright output

Typical misconceptionThe more reflective the output mirror, the more power comes out.

Better mental modelA fully closed mirror lets nothing out. Optimize the useful output, not the power circulating inside.

Extraction is not pump efficiency

Typical misconceptionA high useful fraction means the pump or power supply is efficient.

Better mental modelThe fraction only compares the light extracted with the stimulated power added. This model does not calibrate the pump.

The best sample is not the exact optimum

Typical misconceptionThe best whole-percent setting must be the exact optimum.

Better mental modelThe continuous peak can fall between two samples. Keep the measured choice and the ideal reference apart.

Run the experiment

01
Close the mirror
Choose Closed output mirror and compare the circulating power with the useful output.
What to observe: A saturated field can still persist inside, but all the power the gain adds is dissipated internally, and the useful output is zero.
02
Extract too much
Choose Too much extraction and lower the transmission until lasing starts again.
What to observe: Lasing needs the small-signal gain to exceed the total logarithmic loss. Below threshold there is no steady stimulated output.
03
Choose from a measured scan
Press Scan 71 settings, then Use best recorded mirror, and compare the result with the continuous reference.
What to observe: The best recorded output comes close to the model's peak, but the chosen transmission is still one of the sampled settings.
04
Change the loss and the gain
Compare High parasitic loss with Stronger available gain, and repeat the acquisition each time.
What to observe: The best transmission moves. Changing the saturation power scales every power in watts but leaves the best transmission unchanged.