L14 · Saturation, loss and useful power

Output Coupler Optimization

Trade circulating power against extraction. Record output meter readings while changing the mirror, find the best sampled setting, and compare it with the continuous model optimum.

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 $\text{—}$

A single-mode unidirectional ring is modeled at continuous-wave steady state. Ideal folds preserve the mode; the scene is schematic and contains no stability or diffraction solver. One lumped gain element represents the complete round-trip intensity logarithmic gain, followed by a lumped parasitic power loss and an output coupler. All other mirrors are lossless. The reference circulating power is just before the gain element; saturation power is defined at that same plane. Saturated logarithmic gain follows a phenomenological reciprocal saturation law. Threshold uses exact power survival factors, and each displayed power comes from the same gain → loss → coupler → return sequence. The output mirror transmission is a power fraction, not an amplitude coefficient. Below threshold, steady stimulated power is zero; spontaneous fluorescence and ASE are omitted. Gain is controlled directly and is not calibrated to pump watts. Extraction share is useful output divided by total stimulated power added; it is neither pump efficiency nor electrical efficiency. The model is not the distributed counter-propagating Rigrod solution: gain depletion along a material, spatial hole burning, startup, relaxation, heating, multimode competition, finite apertures and output-mirror absorption are omitted. Records are noiseless simulated power-meter readings. Changing mirror transmission keeps records; gain, saturation or internal-loss changes clear them. The automatic scan samples 71 integer-percent settings from zero to 70%; the best recorded setting is chosen from those records, independently of a continuous numerical reference optimum. A low-loss square-root estimate is only an asymptotic comparison. Display brightness 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 optimum is between trapped light and loss of oscillation. Define a circulating reference plane, follow all survival factors, then compare acquired output readings at fixed medium conditions.

Core mathematical model

Exact survival and logarithmic threshold

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

Gain is an intensity logarithm over a complete circuit. Internal loss and transmission are power fractions, so their survival factors multiply.

Steady reciprocal gain saturation

\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}

Circulating power is defined before the lumped gain. Above threshold the saturated gain clamps to total logarithmic loss.

Output must use the incident mirror power

\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}

The reference circulating power is after the coupler in the ring sequence. Multiplying it directly by transmission would use the wrong plane at appreciable coupling.

One conserved power budget

\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 optical balance, not electrical or pump conversion efficiency. No calibrated pump power enters this 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}

Negative threshold means no mirror can sustain oscillation. The square-root estimate neglects higher-order survival corrections; compare it only in the low-loss regime.

Sampled choice and continuous reference

\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 selection uses acquired values only. The second uses the model continuously; an integer-percent scan can miss the true maximum between samples.

Common difficulties

Intracavity brightness is not delivery

Typical misconceptionThe most reflective output mirror always yields the most output.

Better mental modelA fully closed mirror extracts nothing. Optimize useful output rather than circulating power.

Extraction is not pump efficiency

Typical misconceptionA high useful fraction proves an efficient pump or power supply.

Better mental modelThe fraction only compares optical extraction with stimulated power added. The pump is not calibrated in this model.

A sampled maximum is not an analytic optimum

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

Better mental modelThe continuous peak can lie between samples. Keep measured choice and ideal reference separate.

Run the experiment

01
Close the mirror
Choose closed output mirror and compare circulating and useful output powers.
What to observe: A saturated field can persist while all stimulated addition is dissipated internally; useful output is zero.
02
Extract too much
Choose too much extraction and decrease transmission until lasing resumes.
What to observe: The small-signal gain must exceed total log loss. Below threshold there is no steady stimulated output.
03
Choose from a measured scan
Scan 71 settings, use the best recorded mirror, then compare against the continuous reference.
What to observe: Recorded output approaches the model peak but the selected transmission remains a sampled setting.
04
Change the loss budget
Compare high parasitic loss and stronger gain. Repeat the acquisition each time.
What to observe: The optimum moves; saturation power scales watts without changing the optimal transmission.