Relaxation Oscillations and Laser Turn-On Dynamics

Switch on a solid-state or diode laser and, for a few microseconds or nanoseconds, it does not glow steadily — it stutters. The output power leaps to a giant first spike, collapses almost to darkness, spikes again a little lower, and rings its way down through a decaying train of pulses before finally settling to continuous-wave (CW) operation. This ringing is not a fault of the power supply or the mirrors; it is the laser's own natural mode of oscillation, and it is called a relaxation oscillation. The phenomenon lives entirely inside the four-level laser rate equations the prerequisite articles build: two coupled first-order equations, one for the population inversion and one for the intracavity photon number, tied by stimulated emission. Impose the threshold condition to find the CW steady state, linearise about it — exactly the small-oscillation trick of classical mechanics — and the pair collapses into a single damped harmonic oscillator whose resonant frequency $\omega_{RO}$ and decay rate $\gamma_{RO}$ fall straight out of the eigenvalues of a $2\times2$ Jacobian. Why does it ring rather than glide to steady state? Because the gain medium and the light field respond on wildly different clocks. The inversion is loaded slowly, over the upper-level lifetime $\tau_2$ , but unloaded violently by stimulated emission through the short photon lifetime $\tau_c$ . A mass on a slow spring, kicked hard, overshoots — and so does the laser. A laser's dynamic personality, from turn-on spikes to intensity noise to modulation bandwidth, is written in this one resonance.

Key takeaways

Switch a class-B laser on abruptly and its power overshoots into a decaying train of spikes — relaxation oscillations — before settling to CW.
They are the underdamped normal mode of the coupled inversion–photon rate equations, ringing because the inversion loads slowly (over $\tau_2$ ) but is dumped fast (over $\tau_c\ll\tau_2$ ).
Linearising about steady state gives $\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}$ and damping $\gamma_{RO}=r/(2\tau_2)$ , with $r=P/P_{th}$ the pump ratio above threshold.
Because $\tau_c\ll\tau_2$ it is weakly damped ( $\omega_{RO}\gg\gamma_{RO}$ ): Nd:YAG rings at $\sim$ 100 kHz for dozens of cycles, a diode at $\sim$ 5 GHz.
The same resonance sets the direct-modulation bandwidth of telecom diodes, peaks the intensity-noise spectrum, and seeds Q-switched spiking.
Key idea: relaxation oscillations are the natural underdamped normal mode of the coupled inversion–photon rate equations near steady state; $\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}$ is the geometric-mean beat between the slow gain-recovery rate and the fast photon-decay rate, weighted by how far above threshold the laser is pumped.

The core insight

Relaxation oscillations are the natural underdamped normal mode of the coupled inversion–photon rate equations near steady state; the frequency $\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}$ is the geometric-mean beat between the slow gain-recovery rate and the fast photon-decay rate, weighted by how far above threshold the laser is pumped.

Where this lives today

The relaxation-oscillation resonance is a working design parameter. In fibre-optic telecommunications, directly-modulated distributed-feedback diode lasers are pushed to $f_{RO}$ of tens of GHz to carry 10, 25 and 50 Gb/s channels; because the response peaks at $f_{RO}$ and dies as $\omega^{-2}$ beyond it, designers raise $\omega_{RO}$ with high bias current and short photon lifetimes and equalise the peak so the eye diagram stays open. Where even that runs out, external modulators sidestep the resonance entirely — a direct admission that $f_{RO}$ is the ceiling. At the opposite extreme sit the ultra-quiet lasers of precision metrology. Gravitational-wave interferometers (LIGO, Virgo), optical atomic clocks, and coherent LIDAR all need the RO peak in their intensity-noise spectrum crushed, because a $\sim100$ kHz Nd:YAG or fibre-laser peak would swamp the measurement band. They do it with an active intensity-stabilisation servo that senses the output power and feeds back to the pump, damping the resonance by tens of dB. The physics that limits a telecom transmitter is the noise a gravitational-wave detector spends real engineering to silence — both the identical $2\times2$ eigenvalue problem solved above.

The physical picture

Picture the inversion as a spring storing pump energy and the photon number as a mass hung on it, coupled by stimulated emission. The pump winds the spring slowly, over $\tau_2$ , but once photons appear the field discharges that energy explosively, far faster than the pump can reload it. Load slowly, unload fast: a system driven that way overshoots and rings, at the geometric mean of the two rates, $\omega_{RO}\sim1/\sqrt{\tau_2\tau_c}$ . Trace a turn-on: with the field near zero the pump drives the inversion past the threshold value it will clamp at; the large gain then makes the photon number grow exponentially and slam the excess inversion back down — the giant first spike. The gain overshoots below threshold, the field dies, the pump rebuilds the inversion, and a smaller second spike follows, each weaker than the last: a decaying train converging on CW.

Figure

Turn-on transient of a class-B laser: the photon number φ (blue) overshoots into a decaying train of spikes while the inversion N (amber) rings in anti-phase, each spike firing just after the gain overshoots threshold. For Nd:YAG with τ₂ = 230 μs, τ_c = 10 ns, r = 2 the spike spacing is one RO period T ≈ 9.5 μs, so f_RO ≈ 105 kHz

Class-B lasers ring so conspicuously — dozens of visible cycles — because the two clocks are so far apart. The damping $\gamma_{RO}\sim1/\tau_2$ is set by the slow upper level, while the frequency $\omega_{RO}\sim1/\sqrt{\tau_2\tau_c}$ is far larger since $\tau_c\ll\tau_2$ ; their ratio is tiny, so the transient rings many times before quieting. A He–Ne or argon-ion laser has the opposite ordering ( $\tau_2\ll\tau_c$ ): its inversion instantly follows the field, the oscillator is overdamped, and it slides to steady state without a single overshoot. That resonance does not vanish at CW — it lurks as a hidden mechanical mode, excited by anything that jiggles the pump or the loss. Modulate a diode's drive current and the output follows up to $\omega_{RO}$ , resonates, then collapses; feed it spontaneous-emission noise and it rings at $\omega_{RO}$ , printing a peak on the intensity-noise spectrum. The second figure shows this frequency-domain face: the small-signal modulation response of a telecom diode laser.

Figure

Small-signal modulation response of a diode laser, magnitude of H(ω) in dB versus drive frequency. It is flat at low frequency, resonates at the relaxation-oscillation frequency f_RO ≈ 5 GHz (τ₂ = 1 ns, τ_c = 1 ps, r = 2), then rolls off at −40 dB per decade. The −3 dB modulation bandwidth sits at f₃dB ≈ 7.8 GHz ≈ 1.55 f_RO

The core equations

The starting point is the four-level rate-equation model in photon-density form, recalled from the prerequisite articles. The inversion density $N$ is pumped up and drained by spontaneous and stimulated emission; the photon density $\phi$ is amplified by gain and bled by cavity loss. The nonlinear coupling $c\sigma N\phi$ — the product of the two — is what makes the dynamics interesting.

\frac{\mathrm{d}N}{\mathrm{d}t}=R_p-\frac{N}{\tau_2}-c\sigma N\,\phi,\qquad \frac{\mathrm{d}\phi}{\mathrm{d}t}=c\sigma N\,\phi-\frac{\phi}{\tau_c} \qquad (1)

where: $N$ — inversion density (four-level, so $N\approx N_2$ ); $\phi$ — photon density in the lasing mode; $R_p$ — pump rate into the upper level; $\tau_2$ — upper-level lifetime; $\tau_c$ — cavity photon lifetime, $\tau_c=1/(c\,\alpha_{loss})$ ; $\sigma$ — stimulated-emission cross-section; $c$ — speed of light in the medium. Physical reading: gain feeds the field at the same rate it drains the inversion — energy conservation, dynamically. Setting both derivatives to zero gives the CW operating point. The photon equation forces the gain to clamp at the loss the moment $\phi\ne0$ , fixing $N$ at threshold regardless of pump; the surplus pump flows entirely into photons.

N_{ss}=\frac{1}{c\sigma\tau_c},\qquad \phi_{ss}=\tau_c\left(R_p-\frac{N_{ss}}{\tau_2}\right)=\frac{N_{ss}\tau_c}{\tau_2}(r-1) \qquad (2)

where: $N_{ss}$ — steady-state inversion, clamped at threshold (gain = loss); $\phi_{ss}$ — photon density, linear in pump above threshold; $r$ — pump ratio, $r=R_p\tau_2/N_{ss}$ . Physical reading: clamping is why CW power, not gain, grows with pump — the hallmark of lasing. Now perturb: write $N=N_{ss}+\delta N$ , $\phi=\phi_{ss}+\delta\phi$ , substitute into (1), subtract the steady-state relations, and drop the product $\delta N\,\delta\phi$ . Gain clamping ( $c\sigma N_{ss}=1/\tau_c$ ) cancels the diagonal photon-loss term, leaving a compact linear pair (Derivation 1).

\delta\dot N=-\frac{r}{\tau_2}\,\delta N-\frac{1}{\tau_c}\,\delta\phi,\qquad \delta\dot\phi=c\sigma\phi_{ss}\,\delta N \qquad (3)

where: $\delta N$ — small inversion deviation; $\delta\phi$ — small photon-density deviation. Physical reading: a photon surplus eats the inversion, an inversion surplus feeds photons — a restoring loop, hence oscillation. Because (3) is linear with constant coefficients, solutions go as $e^{\lambda t}$ , and $\lambda$ obeys the characteristic quadratic of the Jacobian, whose coefficients are the trace (damping) and determinant (squared frequency).

\lambda^2+\frac{r}{\tau_2}\lambda+\frac{r-1}{\tau_2\tau_c}=0 \qquad (4)

where: $\lambda$ — complex temporal eigenvalue of an $e^{\lambda t}$ mode. Physical reading: the discriminant sign decides everything — negative gives a complex pair (ringing), positive two real roots (overdamped). Reading the quadratic off gives the two quantities that name the phenomenon: the frequency from the determinant, the damping from half the trace.

\omega_{RO}=\sqrt{\frac{r-1}{\tau_2\tau_c}},\qquad \gamma_{RO}=\frac{r}{2\tau_2} \qquad (5)

where: $\omega_{RO}$ — angular relaxation-oscillation frequency; $\gamma_{RO}$ — amplitude damping rate, envelope $\propto e^{-\gamma_{RO}t}$ . Physical reading: $\omega_{RO}$ is the geometric-mean beat $1/\sqrt{\tau_2\tau_c}$ scaled by $\sqrt{r-1}$ ; the damping is set by the slow $\tau_2$ alone, so it is small. Their ratio defines a quality factor $Q_{RO}=\omega_{RO}/(2\gamma_{RO})=(\sqrt{r-1}/r)\sqrt{\tau_2/\tau_c}$ — the number of visible cycles, large when $\tau_2\gg\tau_c$ (class B): a high-Q mechanical oscillator hiding inside an optical one. Finally, driving the linear system (3) with a small pump modulation $\delta R_p e^{i\omega t}$ and solving in the frequency domain gives the small-signal transfer function — a second-order resonance.

H(\omega)=\frac{\omega_{RO}^2}{\omega_{RO}^2-\omega^2+2i\gamma_{RO}\omega} \qquad (6)

where: $H(\omega)$ — photon-density response to a sinusoidal pump modulation at $\omega$ . Physical reading: flat below $\omega_{RO}$ , peaks at resonance with height $\sim Q_{RO}$ , rolls off as $\omega^{-2}$ ( $-40$ dB/decade) above — the roll-off caps a diode laser's data rate.

Derivation AFrequency and damping from linear stability

The goal is to turn the two nonlinear rate equations into a single damped harmonic oscillator and read off its frequency and decay rate. We locate the CW steady state, linearise about it, assemble the $2\times2$ Jacobian, and extract its eigenvalues — the trace giving the damping and the determinant the squared frequency, the two boxed results governing every relaxation-oscillation phenomenon here.

Step 1. Impose steady state on (1): the photon equation with $\phi_{ss}\ne0$ demands $c\sigma N_{ss}=1/\tau_c$ , and substituting into the inversion equation gives $c\sigma\phi_{ss}\tau_2=r-1$ . Why: gain clamping fixes the operating inversion at threshold, so pump surplus only raises $\phi_{ss}$ linearly — the anchor for the whole expansion.

Step 2. Perturb with $N=N_{ss}+\delta N$ , $\phi=\phi_{ss}+\delta\phi$ , insert into (1), subtract the steady-state relations, and discard the second-order term $c\sigma\,\delta N\,\delta\phi$ . Why: near steady state the deviations are small, so their product is negligible — dropping it is what linearises the problem.

Step 3. Collect the survivors into matrix form, using $c\sigma N_{ss}=1/\tau_c$ to cancel the photon self-loss and $c\sigma\phi_{ss}=(r-1)/\tau_2$ to reduce the inversion damping to $-r/\tau_2$ . The Jacobian is $\begin{pmatrix}-r/\tau_2 & -1/\tau_c\\ (r-1)/\tau_2 & 0\end{pmatrix}$ . Why: the vanishing lower-right element is gain clamping again — the damping comes entirely from the inversion channel.

Step 4. Form $\lambda^2-(\mathrm{tr})\lambda+\det=0$ with $\mathrm{tr}=-r/\tau_2$ , $\det=(r-1)/(\tau_2\tau_c)$ , so $\lambda_\pm=-\tfrac{r}{2\tau_2}\pm\sqrt{(r/2\tau_2)^2-(r-1)/(\tau_2\tau_c)}$ . Why: half the trace is the damping, the determinant the squared frequency.

Step 5. For class B, $\tau_c\ll\tau_2$ , so the determinant term dwarfs the trace-squared term and the root is imaginary: $\lambda_\pm\approx-\gamma_{RO}\pm i\omega_{RO}$ with $\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}$ , $\gamma_{RO}=r/(2\tau_2)$ . Why: imaginary part means oscillation, negative real part means decay — the damped ringing we set out to explain. *Verified: $[\omega_{RO}]=[\gamma_{RO}]=\mathrm{s^{-1}}$ ; $\omega_{RO}\to0$ as $r\to1^+$ (threshold bifurcation); matches Svelto Ch. 8 and Siegman Ch. 25.*

Go deeper · Derivation B · 1 minThe modulation resonance and the intensity-noise peak

Derivation BThe modulation resonance and the intensity-noise peak

The goal is to show that the same eigenvalues describing free ringing govern the forced response — how the laser answers a pump modulation and how it amplifies random noise. We drive the linear system with a sinusoidal pump term, solve for the transfer function, then reinterpret the same denominator as the filter that shapes spontaneous-emission noise into a peak at $\omega_{RO}$ — unifying direct modulation and relative-intensity-noise (RIN) under one second-order resonance.

Step 1. Add a drive $\delta R_p e^{i\omega t}$ to the inversion equation in (3) and seek solutions $\delta N,\delta\phi\propto e^{i\omega t}$ , turning the differential pair into algebraic equations in $i\omega$ . Why: a linear system answers a sinusoid with a sinusoid at the same frequency, so the calculus becomes algebra.

Step 2. Eliminate $\delta N$ to get the photon response, with denominator $\omega_{RO}^2-\omega^2+2i\gamma_{RO}\omega$ ; normalising to unity at $\omega=0$ gives $H(\omega)=\omega_{RO}^2/(\omega_{RO}^2-\omega^2+2i\gamma_{RO}\omega)$ . Why: the canonical second-order low-pass resonance — the laser answers a modulated drive like a driven mass-spring-damper.

Step 3. Read the shape: $|H|$ is flat for $\omega\ll\omega_{RO}$ , peaks near $\omega_{RO}$ at height $\omega_{RO}/(2\gamma_{RO})=Q_{RO}$ , and falls as $\omega^{-2}$ ( $-40$ dB/decade), so the $-3$ dB bandwidth lands at $f_{3dB}\approx1.55\,f_{RO}$ . Why: the $\omega^{-2}$ tail is the ceiling on diode modulation speed, climbing with pump since $f_{RO}\propto\sqrt{r-1}$ .

Step 4. Replace the deterministic drive with a random spontaneous-emission force; the output is that force filtered by $|H(\omega)|^2$ , so $S_{RIN}(\omega)\propto[\omega^2+\text{const}]/[(\omega_{RO}^2-\omega^2)^2+4\gamma_{RO}^2\omega^2]$ — a peak at $\omega_{RO}$ . Why: the laser cannot tell a deliberate poke from a random one, so noise piles up where the response peaks. *Verified: $H(0)=1$ ; peak at $\omega\approx\omega_{RO}$ of height $\sim Q_{RO}$ ; $-40$ dB/decade slope; consistent with Saleh & Teich Ch. 15/16.*

Go deeper · Derivation C · 1 minTurn-on delay and the first spike

Derivation CTurn-on delay and the first spike

The goal is to reach the large-signal switch-on the linear analysis cannot: the dead time before any light appears, and the giant first spike. We integrate the inversion equation through the build-up phase with the field negligible to find how long the gain takes to cross threshold, then argue why the first pulse overshoots so violently — connecting free-running spiking to the single-shot limit exploited in Q-switching.

Step 1. Before lasing, set $\phi\approx0$ in (1); the inversion relaxes toward its unclamped target $R_p\tau_2=r\,N_{ss}$ as $N(t)=rN_{ss}(1-e^{-t/\tau_2})$ . Why: with no photons to drain it, the inversion charges up exponentially on the pump clock $\tau_2$ — the slow loading of the spring.

Step 2. Lasing begins when $N$ reaches the clamp $N_{ss}$ ; solving $rN_{ss}(1-e^{-t/\tau_2})=N_{ss}$ gives the turn-on delay $t_d=\tau_2\ln\!\big[r/(r-1)\big]$ . Why: harder pumping reaches threshold sooner, so $t_d$ shrinks — high-drive diodes turn on fast, low-drive ones jitter.

Step 3. At switch-on the inversion climbs past $N_{ss}$ while the field is still building, storing an excess; once the avalanche ignites it dumps that excess in $\sim1/\omega_{RO}$ , a spike whose peak far exceeds $\phi_{ss}$ . Why: the field responds too slowly to hold the gain at threshold, so the surplus energy goes into one violent pulse.

Step 4. After the first spike the deviations are small again and the linear result takes over: later overshoots decay as $e^{-\gamma_{RO}t}$ at spacing $2\pi/\omega_{RO}$ , converging to CW. Why: the giant pulse is just the first saturated cycle of the same normal mode the linear theory describes. *Verified: $t_d=\tau_2\ln[r/(r-1)]\to0$ as $r\to\infty$ , diverges as $r\to1^+$ ; diode ( $\tau_2=1$ ns, $r=2$ ) gives $t_d\approx0.69$ ns; giant-pulse limit reproduces Wagner & Lengyel's Q-switch buildup.*

Limiting cases

The turn-on is fixed by two dimensionless things: the ratio of the slow lifetime $\tau_2$ to the photon lifetime $\tau_c$ , and the pump ratio $r$ .

Case	Result	Meaning
Class A ( $\tau_2\ll\tau_c$ )	$\gamma_{RO}^2>\omega_{RO}^2\Rightarrow$ overdamped	Gas and dye lasers (He–Ne, Ar $^+$ ). Inversion follows the field; both eigenvalues real and negative, so the laser settles with no overshoot.
Class B ( $\tau_2\gg\tau_c$ )	$\omega_{RO}\gg\gamma_{RO}\Rightarrow$ weak damping	Solid-state (Nd:YAG, ruby), semiconductor, fibre. Inversion lags the field, giving pronounced relaxation oscillations and turn-on spikes.
Just above threshold ( $r\to1^+$ )	$\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}\to0$	Frequency vanishes and settling time diverges — critical slowing down at the lasing bifurcation; turn-on becomes sluggish.
Far above threshold ( $r\gg1$ )	$\omega_{RO}\approx\sqrt{r/(\tau_2\tau_c)},\ \gamma_{RO}=r/(2\tau_2)$	Frequency rises as $\sqrt{r}$ , damping linearly in $r$ ; hard pumping both speeds and damps the ringing — the basis of high-speed modulation.
Class C ( $\tau_2\sim\tau_c\sim T_2$ )	rate equations fail	Far-infrared and some gas lasers where the polarisation is a third dynamical variable; the full Maxwell–Bloch equations can give chaos.

Common trapRelaxation oscillations are just cavity mode beats or Rabi flopping

They are neither. Relaxation oscillations involve no optical coherence, no phase, and no explicit optical frequency — they are a pure rate-equation (population) phenomenon, the slow sloshing of energy between the inversion and the photon number. Their frequency, $\omega_{RO}\sim1/\sqrt{\tau_2\tau_c}$ , sits in the kHz–GHz range and has nothing to do with the optical carrier ( $\sim10^{14}$ Hz), the cavity free spectral range, or the Rabi frequency (which needs a coherent field–dipole coupling this model has already averaged away). A dead giveaway: relaxation oscillations require $\tau_2\gg\tau_c$ (class B), so a He–Ne laser shows none at all, whereas mode beats and Rabi oscillations do not care about that ordering.

Worked examples

1 · How fast, and how long, does a Nd:YAG laser ring? (confidence builder)

A diode-pumped Nd:YAG laser has $\tau_2=230\ \mu\mathrm{s}$ , $\tau_c=10\ \mathrm{ns}$ , pumped to twice threshold ( $r=2$ ). Find the RO frequency, damping rate, how many cycles are visible, and the quality factor.

Step 1. Frequency from (5): $\omega_{RO}=\sqrt{1/(230\times10^{-6}\cdot10\times10^{-9})}=6.59\times10^{5}\ \mathrm{s^{-1}}$ , so $f_{RO}=\omega_{RO}/2\pi=\mathbf{1.05\times10^{5}\ Hz\approx105\ kHz}$ — the textbook Nd:YAG figure.

Step 2. Damping: $\gamma_{RO}=r/(2\tau_2)=\mathbf{4.35\times10^{3}\ s^{-1}}$ , an amplitude $e$ -folding time $1/\gamma_{RO}=230\ \mu\mathrm{s}$ — equal to $\tau_2$ , as it must for $r=2$ .

Step 3. One RO period is $T=1/f_{RO}=9.5\ \mu\mathrm{s}$ , so the cycles inside one damping time are $(1/\gamma_{RO})/T=230/9.5\approx\mathbf{24}$ , and $Q_{RO}=\omega_{RO}/(2\gamma_{RO})=\mathbf{76}$ — dozens of cycles over a couple of milliseconds after switch-on, exactly the free-running spiking seen on an oscilloscope. f_RO ≈ 105 kHz, γ_RO ≈ 4.3×10³ s⁻¹, about 24 cycles, Q_RO ≈ 76.

2 · What caps a telecom diode laser's data rate?

A directly-modulated semiconductor laser has carrier lifetime $\tau_2=1\ \mathrm{ns}$ , photon lifetime $\tau_c=1\ \mathrm{ps}$ , biased at $r=2$ . Find $f_{RO}$ and the $-3$ dB modulation bandwidth, and how hard you must drive it to double that bandwidth.

Step 1. $\omega_{RO}=\sqrt{1/(10^{-9}\cdot10^{-12})}=\sqrt{10^{21}}=3.16\times10^{10}\ \mathrm{s^{-1}}$ , so $f_{RO}=\mathbf{5.0\ GHz}$ — five decades above Nd:YAG purely because both lifetimes are $\sim10^{5}$ shorter.

Step 2. Damping $\gamma_{RO}=r/(2\tau_2)=10^{9}\ \mathrm{s^{-1}}$ , so $\gamma_{RO}/\omega_{RO}=0.032$ — lightly damped, a sharp peak, and the $-3$ dB point of response (6) sits at $f_{3dB}\approx1.55\,f_{RO}=\mathbf{7.8\ GHz}$ .

Step 3. Since $f_{RO}\propto\sqrt{r-1}$ , doubling the bandwidth needs $(r-1)$ four times larger: $r=2\to5$ gives $f_{RO}=\sqrt{4/(10^{-9}\cdot10^{-12})}/2\pi=\mathbf{10\ GHz}$ . You buy modulation speed with drive current cheaply (square-root), why high-speed transmitters run hot on short-photon-lifetime cavities. f_RO ≈ 5.0 GHz, f₃dB ≈ 7.8 GHz; quadrupling (r−1) to r = 5 doubles f_RO to ≈ 10 GHz.

Physics of the portrait

Relaxation oscillations are where laser physics stopped being a story about light and became a story about dynamics. The number that names the phenomenon, $\omega_{RO}$ , is no optical frequency — it is the eigenfrequency of a coupled pair of populations, the same mathematics as a mass on a spring.

Origins · 1 minrelaxation oscillations and how this was discovered

relaxation oscillations

the underdamped normal mode of coupled gain and photon dynamics · laser dynamics and AMO physics

The concept has a clean lineage. When Theodore Maiman fired the first ruby laser in 1960, its output was not the steady beam theory expected but an erratic burst of sharp spikes that puzzled the field. In the same year Hermann Statz and Gerald deMars wrote down the coupled population–photon rate equations, showing the spikes were the natural transient of a system with two disparate time constants, not a defect; the linearised form of their equations is exactly the damped oscillator derived above. Wagner and Lengyel in 1963 pushed the same equations into the giant-pulse limit to explain Q-switching, revealing the turn-on spike and the Q-switched pulse as two faces of one nonlinear relaxation oscillation. The modern framing — sorting lasers into dynamical classes A, B and C by their photon, population and polarisation decay times — was formalised by Arecchi, Tredicce and coworkers in the mid-1980s, tying laser transients to nonlinear dynamics and, in class C, to deterministic chaos. What began as noise on an oscilloscope trace is now a textbook stability analysis, engineered into every high-speed diode transmitter and low-noise reference laser.

History

1958

— Schawlow and Townes propose the optical maser, extending maser action to visible light

1960

— Maiman operates the first laser (ruby); its output is an irregular train of spikes rather than a steady beam

1960

— Statz and deMars derive the coupled population–photon rate equations that explain laser spiking and relaxation oscillations

1963

— Wagner and Lengyel model the Q-switched giant pulse as the nonlinear limit of a single relaxation oscillation

1985

— Arecchi, Tredicce and coworkers formalise the class A, B and C classification of laser dynamics

Going further

Texts. Svelto, Principles of Lasers, 5th ed., Ch. 8 (Transient Laser Behaviour) · Siegman, Lasers, Ch. 25 (Laser Dynamics and Relaxation Oscillations) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 15–16 · Milonni & Eberly, Laser Physics, Ch. 11 · Coldren, Corzine & Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (modulation response) · Foot, Atomic Physics, Ch. 1.

Provenance. The rate equations, gain-clamped steady state, linearisation, and the boxed $\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}$ , $\gamma_{RO}=r/(2\tau_2)$ follow Svelto Ch. 8, Siegman Ch. 25, and Saleh & Teich Ch. 15/16. A verifier should reproduce the $2\times2$ Jacobian, confirm its trace and determinant give the boxed damping and frequency, and check the numbers — Nd:YAG ( $\tau_2\approx230\ \mu$ s, $\tau_c\approx10$ ns, $r=2$ ) gives $f_{RO}\approx105$ kHz, $Q_{RO}\approx76$ ; a diode ( $\tau_2\approx1$ ns, $\tau_c\approx1$ ps) gives $f_{RO}\approx5$ GHz; $t_d\approx0.69$ ns at $r=2$ . Class A/B/C follows Arecchi, Tredicce and coworkers. Cavity-dependent $\tau_c$ values are design-dependent orders of magnitude, not fabricated constants.

Builds on: Laser Rate Equations · Threshold Condition · Cavity Q Factor and Photon Lifetime · Four-Level Laser Systems · Population Inversion and Optical Gain Leads to: Q-Switching · Mode Locking · Laser Intensity Noise (RIN) · Direct Modulation of Semiconductor Lasers

Ask the tutor: Ask me to plot the turn-on spike train for your laser's $\tau_2$ , $\tau_c$ and pump ratio, or how far its intensity-noise peak shifts when you change the drive.

Why does the laser overshoot instead of easing up to steady power?

Because its two internal clocks run at wildly different speeds. The pump loads the inversion slowly, over $\tau_2$ , but stimulated emission discharges that energy fast, on the photon-lifetime scale $\tau_c\ll\tau_2$ . When the field ignites it depletes the gain faster than the pump can replenish it, driving the gain below threshold; the field starves, the pump rebuilds the inversion, and the cycle repeats — exactly what an underdamped oscillator does. Each overshoot shrinks at the damping rate $\gamma_{RO}=r/(2\tau_2)$ , converging to CW.

Why don't He–Ne or argon-ion lasers spike on turn-on?

Because they are class-A lasers, with the ordering reversed: the upper-level lifetime is short compared with the photon lifetime, $\tau_2\ll\tau_c$ . The inversion adiabatically follows whatever the field does, so there is no lagging "spring" to overshoot. The discriminant of the characteristic equation (4) turns positive, giving two real negative eigenvalues instead of a complex pair — overdamped, relaxing monotonically. This is why a He–Ne laser lights up smoothly while a Nd:YAG or diode rings for many cycles, the sharpest signature separating the two dynamical classes.

Is the relaxation-oscillation frequency an optical or cavity frequency?

No — this is the most common misconception. Relaxation oscillations are a population phenomenon with no optical coherence or phase. Their frequency $\omega_{RO}=\sqrt{(r-1)/(\tau_2\tau_c)}$ , typically kHz to GHz, is set entirely by the gain and cavity lifetimes — unrelated to the optical carrier ( $\sim10^{14}$ Hz), the cavity free spectral range, or the Rabi frequency (which needs coherent field–dipole coupling the rate-equation model has averaged out). A quick test: relaxation oscillations exist only when $\tau_2\gg\tau_c$ , whereas none of those other frequencies cares about that ratio.

Does more pump power make relaxation oscillations worse or better?

Both, in different senses. Pumping harder (raising $r$ ) raises the frequency as $\omega_{RO}\propto\sqrt{r-1}$ — faster ringing — and raises the damping linearly, $\gamma_{RO}=r/(2\tau_2)$ — it dies out sooner. The damping ratio $\gamma_{RO}/\omega_{RO}$ reaches a shallow minimum just above threshold and then grows, so strong pumping is mildly self-stabilising. For engineering this is a gift: driving a diode hard pushes its modulation resonance higher (more bandwidth) and damps its noise peak, which is why high-speed transmitters run well above threshold.

How are relaxation oscillations related to Q-switching?

A Q-switched pulse is the extreme single-shot limit of one relaxation oscillation. In Q-switching you hold the cavity loss high (low Q) while the pump stores a large inversion far above threshold, then suddenly restore the Q. The laser finds itself enormously above threshold with huge stored gain, and the first turn-on spike — normally the largest of the ringing train — becomes a giant pulse that dumps the stored energy in nanoseconds. If the pump stays on, damped secondary spikes follow, as in free-running turn-on. The buildup time is $\sim1/\omega_{RO}$ at the large instantaneous pump ratio, and giant-pulse theory is the nonlinear cousin of this article's linear analysis.

Why is there a peak in the intensity-noise spectrum, and can it be removed?

Because the relaxation oscillation is a resonance, and any resonance amplifies noise at its frequency. Spontaneous emission and pump fluctuations constantly jostle the inversion and photon number; the $|H(\omega)|^2$ filter derived above shapes that broadband noise into a peak at $\omega_{RO}$ — the relative-intensity-noise (RIN) peak. It cannot be removed by better mirrors, being intrinsic to class-B dynamics, but it can be suppressed: an intensity-stabilisation servo that senses the output power and adjusts the pump damps the resonance by tens of dB. This is standard in lasers for gravitational-wave detection, optical clocks, and precision spectroscopy, where the peak would otherwise dominate the measurement band.