L15 · Population, photons and response

Laser Dynamics & Relaxation Oscillation

Why does a laser overshoot before it settles? Energy sloshes back and forth between the population inversion, which responds slowly, and the light in the cavity, which responds fast. The light overshoots and drains the inversion, then dips until the inversion recovers, in smaller and smaller swings. In class B lasers this shows up as damped relaxation oscillations. Class A lasers have no slow energy store and settle smoothly.

Turn a laser on, nudge its pump or modulate it, and watch the inversion and the photons trade energy. Compare how class A and class B lasers respond to turn-on, to a small pump step and to a measured sinusoidal pump modulation.

Elapsed time $\text{—}$

Threshold-normalized inversion $\text{—}$

Normalized photon density $\text{—}$

Instantaneous output power $\text{—}$

Unmodulated steady output reference $\text{—}$

Largest output in retained experiment $\text{—}$

First half-steady turn-on time $\text{—}$

Damped eigenfrequency $\text{—}$

Pole decay rate $\text{—}$

Population to photon lifetime ratio $\text{—}$

Time-scale regime $\text{—}$

Linearized fractional response $\text{—}$

Linearized output phase $\text{—}$

Measured fractional response $\text{—}$

Measured output phase $\text{—}$

The model is a uniform, four-level, single-mode intensity model. It assumes the polarization relaxes faster than both the population and the photons. Inversion is measured in units of its threshold value without modulation, and photon density is scaled so that its steady value equals the pump minus one. Lifetimes are in microseconds. The pump is an injection rate in units of its threshold value, not an optical or electrical power in watts. The threshold inversion stores an illustrative ten microjoules, and sixty percent of the base photon loss leaves through the output mirror. The run starts from a finite photon seed, but there is no ongoing spontaneous emission and nothing random. The two rate equations are integrated together with an adaptive, step-doubling Runge–Kutta method, using the logarithm of the photon density. In the class A limit the population relaxes much faster than the photons decay; in class B the order is reversed. The eigenvalues keep the damping exactly. A damped frequency of zero means the poles are real, not that the laser does not respond. The small-signal curves are a separate linearization about the unmodulated state above threshold. Measured points come from projecting the output directly onto a sinusoid after transients die out, with a finite modulation depth. Reset or any change to a physical control clears the measurements. The five-percent step starts from steady state, sets the modulation to zero and never takes the pump above its maximum. Complete startup advances at least thirty population lifetimes. The apparatus is schematic, and the marker fractions and logarithmic brightness are only for display. Spatial variation, coherent polarization dynamics, multimode competition, heating and detector noise are left out.

Start with the essentials

Focus question: When does a disturbance in the output die away with oscillations?
One-sentence intuition: In a class B laser, energy swings back and forth between the slowly responding population and the fast-decaying photons. In class A the population relaxes so fast that it no longer acts as a slow store.

Core mathematical model

Normalized rate equations for inversion and photons

\begin{aligned}\dot D&=\frac{r(t)-D-DS}{\tau_L}\\\dot S&=\frac{(D-1)S}{\tau_p}\end{aligned}

The same stimulated-emission term removes population and adds photons. The normalized photon variable is not a power in watts.

Steady state above and below threshold

\begin{aligned}D_s&=1,\quad S_s=r-1\quad(r>1)\\D_s&=r,\quad S_s=0\quad(r\le1)\end{aligned}

This steady state assumes a constant pump and ignores the ongoing seeding by spontaneous emission.

Poles of the linearized equations

\begin{aligned}s_\pm&=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}\\\gamma&=\frac{r}{2\tau_L}\\\omega_0^2&=\frac{r-1}{\tau_L\tau_p}\end{aligned}

Real poles mean the laser relaxes without oscillating; complex poles mean a damped oscillation. Both statements hold above threshold.

Ringing frequency and the two laser classes

\begin{aligned}f_d&=\frac{\sqrt{\max(0,\omega_0^2-\gamma^2)}}{2\pi}\\\text{A: }\tau_L&\ll\tau_p\\\text{B: }\tau_p&\ll\tau_L\end{aligned}

Both limits assume the polarization relaxes much faster still. A large enough pump can overdamp even a class B laser.

Response to a small pump modulation

\begin{aligned}H(s)&=\frac{\delta S/S_s}{\delta r/r}\\&=\frac{r/(\tau_L\tau_p)}{s^2+(r/\tau_L)s+\omega_0^2}\\H(0)&=\frac{r}{r-1}\end{aligned}

The response is the fractional change in output divided by the fractional change in pump. Near threshold its low-frequency value is not one.

Converting to energy and power

\begin{aligned}E_g&=E_0D\\U&=E_0\frac{\tau_p}{\tau_L}S\\P_{\rm out}&=\eta\frac{U}{\tau_p}\\\dot E_g+\dot U&=\frac{E_0}{\tau_L}(r-D-S)\end{aligned}

The illustrative threshold energy sets the scale in watts. The pump input, minus population decay and photon escape, equals the rate at which the stored energy changes.

Common difficulties

The response peak is not a cavity mode

Typical misconceptionThe peak in the response must be one of the cavity's longitudinal optical resonances.

Better mental modelThis frequency comes from energy swapping between population and photons. It lies far below the optical frequency.

Laser class depends on time scales

Typical misconceptionEvery solid-state laser is automatically class B.

Better mental modelThe class comes from the order of the relaxation times in the model, and both classes assume the polarization is already fast.

Measured points need not sit on the curve

Typical misconceptionA measurement with finite modulation depth must land exactly on the linear curve.

Better mental modelThe curve assumes an infinitely small modulation. The measurements come from integrating the full nonlinear equations.

Run the experiment

01
Watch the laser start
Choose Class B storage and press Complete startup. Compare this first turn-on with the later pump-step test.
What to observe: Starting from a small seed, the light waits while the population builds up, then overshoots as stimulated emission drains the inversion.
02
Disturb a steady laser
Press Apply five-percent pump step, then repeat with Class A fast population.
What to observe: The class B output rings. The class A output follows the step with no relaxation resonance.
03
Measure near resonance
Choose Modulate near relaxation and press Measure modulation. Compare the amber point with the curve.
What to observe: A small pump modulation can make the output swing by a much larger fraction.
04
Go beyond the small-signal limit
Raise the modulation depth, choose Near threshold, or choose Modulate too fast.
What to observe: The simulated response can drift away from the linear prediction, and fast modulation is strongly filtered out.