L15 · Population, photons and response

Laser Dynamics & Relaxation Oscillation

Drive the coupled inversion and photon reservoirs. Compare turn-on, a small pump step, and measured sinusoidal response in class A and class B regimes.

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 $\text{—}$

A uniform four-level single-mode intensity model assumes polarization relaxes faster than both population and photons. Inversion is normalized to the unmodulated threshold; photon density is normalized so steady density equals pump minus one. Lifetimes are in microseconds; pump is a threshold-normalized injection rate, not optical or electrical watts. The threshold inversion energy is an illustrative ten microjoules, and sixty percent of base photon loss exits through the output mirror. There is a finite initial photon seed but no continuing spontaneous source or stochastic events. The two rate equations are integrated together with adaptive step-doubled Runge–Kutta in logarithmic photon density. The class A limit requires population relaxation much faster than photon decay; the class B regime reverses this hierarchy. The eigenvalues retain damping exactly; a zero damped frequency means real poles, not zero response. Small-signal curves are a separate linearization about an above-threshold unmodulated state. Measured points come from direct sinusoidal projection after transient settling, with finite modulation. Reset and physical control changes clear measurements. The five-percent step starts at steady state and sets modulation to zero; it is capped at the maximum pump. The full-startup action advances at least thirty population lifetimes. The apparatus is schematic; marker fractions and logarithmic brightness are presentation only. Spatial variation, coherent polarization dynamics, multimode competition, thermal effects and detector noise are omitted.

Start with the essentials

Focus question: When does an output disturbance decay with oscillation?
One-sentence intuition: A class B laser exchanges energy between a slow population and a fast photon reservoir. Class A removes the slow storage variable.

Core mathematical model

Coupled normalized rates

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

One stimulated term depletes population and amplifies photons. The normalized photon variable is not watts.

The stationary reference

\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}

The reference assumes a constant pump and neglects continuing spontaneous seeding.

Exact linearized poles

\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 give nonoscillatory relaxation; complex poles give damped oscillation. These statements apply above threshold.

Damped frequency and class limits

\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 a much faster polarization relaxation. A large pump can overdamp even a class B reservoir pair.

Fractional pump transfer

\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 fractional output divided by fractional pump. Its low-frequency value is not one near threshold.

One energy interpretation

\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 fixes the watt scale. Population decay and photon escape close the energy budget.

Common difficulties

A resonance is not a new cavity mode

Typical misconceptionThe response peak must be an optical longitudinal resonance.

Better mental modelThis frequency comes from population–photon energy exchange, far below the optical carrier.

Class labels need time scales

Typical misconceptionEvery solid-state laser is automatically class B.

Better mental modelClassify the modeled relaxation hierarchy; polarization must already be fast.

Measurement and reference differ

Typical misconceptionA finite-depth measurement must lie exactly on the linear curve.

Better mental modelThe curve is infinitesimal; measurements integrate the full nonlinear equations.

Run the experiment

01
Watch startup
Choose class B and complete startup. Compare the first turn-on time with the pump step experiment.
What to observe: A finite seed, population buildup and stimulated depletion produce a delay and overshoot.
02
Disturb a steady laser
Apply a five-percent pump step, then repeat in class A.
What to observe: The class B output rings; the class A output follows without a relaxation resonance.
03
Measure near resonance
Choose modulation near relaxation and measure. Compare the amber point with the curve.
What to observe: Small modulation can cause a much larger fractional output oscillation.
04
Leave the linear regime
Raise modulation depth, approach threshold, or modulate too fast.
What to observe: The integrated response can depart from linearization; fast input is strongly filtered.