Optics 103 · Mode competition

Laser Startup from Noise

How does a laser grow from random noise into one dominant mode? Spontaneous emission seeds every cavity mode with a little noise. Above threshold, gain beats cavity loss, and the mode with the most net gain grows exponentially, faster than the rest. As it grows it saturates the gain, pulling it down until gain just equals loss, and that mode ends up dominating.

The page opens on a recorded start-up from pump turn-on. Every longitudinal mode starts from spontaneous noise; watch them grow together on the camera until one is left.

Time to the first spike $\text{—}$

Side-mode suppression at the end $\text{—}$

Modes above 1 % of the strongest $\text{—}$

Output power at the end $\text{—}$

Recorded time after pump turn-on $\text{—}$

Inversion reaches threshold (analytic) $\text{—}$

Inversion over line-centre threshold $\text{—}$

Threshold of the best mode $\text{—}$

Strongest mode share of the light $\text{—}$

Intracavity photons at the end $\text{—}$

Clamped photon number (analytic) $\text{—}$

Steady side-mode suppression (analytic) $\text{—}$

Small-signal ringing frequency (record) $\text{—}$

Relaxation-oscillation frequency (analytic) $\text{—}$

Ringing decay time (analytic) $\text{—}$

Peak output power $\text{—}$

Mode spacing $\text{—}$

Modes inside the gain FWHM $\text{—}$

Photon lifetime $\text{—}$

Ion and photon bookkeeping residual $\text{—}$

Optics 103

How does a laser grow from random noise into one dominant mode?

Pump over line-centre threshold

r

3

Gain linewidth, full width at half maximum

\Delta\nu_g

8\,\mathrm{GHz}

Mirror spacing

L

45\,\mathrm{cm}

Output coupler transmission

T

5\,\mathrm{\%}

Nearest mode offset from line centre, in mode spacings

\delta

0.15

Upper-level lifetime

\tau_L

100\,\mathrm{\mu s}

Other round-trip loss

\alpha_{\rm i}

1\,\mathrm{\%}

Spontaneous fraction into the line-centre mode, power of ten

\log_{10}\beta

-9

Every preset records the start-up from pump turn-on. The record is computed once; the replay never adds photons or events.

Mean-field rate equations for a homogeneously broadened four-level gain medium in a linear two-mirror cavity. All longitudinal modes share one inversion, and the gain and the spontaneous emission into each mode follow one Lorentzian line. Each mode gains photons by stimulated emission in proportion to the photons already in it plus one; the extra one is the spontaneous seed. Photons leave with the photon lifetime set by the round-trip time, the output coupler and the other round-trip loss. Inversion is given in units of the threshold inversion of a mode at line centre, and the pump in units of that mode's threshold pump. The spontaneous fraction sets how many ions that threshold needs, so it also sets the photon numbers and output power; the power uses a 1064 nanometre photon energy. Each record starts with no inversion and no photons when the pump turns on, and runs for six upper-level lifetimes. Spontaneous emission enters as its mean value: the model has no individual random events, optical phase or linewidth. The single-mode laser start-up Lab shows discrete spontaneous events. Spatial hole burning is left out. In a real standing-wave cavity it lets a few more modes survive; a unidirectional ring cavity removes it. The offset of the nearest mode from line centre is a sub-wavelength mirror shift and stays fixed while the mirror spacing changes. There are no thermal effects, transverse modes or polarization. Analytic values are single-mode limits for the best mode, shown separately from the record. The ringing frequency from the record uses only peaks within twenty percent of the clamped photon number. The scene uses one display scale for mirror spacing; components, the grating dispersion and the camera size are schematic, and brightness is logarithmic on a fixed scale.

Start with the essentials

Focus question: Why does one mode win, and how long does it take?
One-sentence intuition: All modes draw on one shared inversion. Once the best mode clamps it at that mode's threshold, every other mode has less gain than loss, so it decays back to its spontaneous glow.

Core mathematical model

One inversion, one equation per mode

\begin{aligned}\frac{dx}{dt}&=\frac{r-x}{\tau_L}-\frac{x}{\tau_p N_{\rm th}}\sum_k \ell_k n_k\\\frac{dn_k}{dt}&=\frac{x\,\ell_k\,(n_k+1)-n_k}{\tau_p}\end{aligned}

Here x is the inversion over the threshold of a line-centre mode, r the pump over that mode's threshold pump, and the plus one is spontaneous emission into the mode.

The gain line and the mode comb

\begin{aligned}\ell_k&=\frac{1}{1+\left[2(k+\delta)\,\Delta\nu_{\rm FSR}/\Delta\nu_g\right]^2}\\\Delta\nu_{\rm FSR}&=\frac{c}{2L}\end{aligned}

Mode k sits k spacings from the mode nearest line centre, which is offset by delta spacings. A longer cavity packs more modes under the same line.

Photon lifetime

\tau_p=\frac{2L/c}{-\ln\left[(1-T)(1-\alpha_{\rm i})\right]}

The round-trip time divided by the logarithmic round-trip loss of the output coupler and the other losses.

Threshold and the first crossing

\begin{aligned}x_{\rm th}&=\frac{1}{\ell_0}\\t_{\rm th}&=\tau_L\ln\frac{r\ell_0}{r\ell_0-1}\end{aligned}

Before the light is strong the inversion rises as r times one minus the exponential decay. The best mode reaches threshold at this time, and the first spike comes a little later.

Clamped steady state

\begin{aligned}x\,\ell_0&\to1\\n_0&=\frac{r-1/\ell_0}{\beta}\\n_{k\neq0}&=\frac{\ell_k}{\ell_0-\ell_k}\end{aligned}

Extra pump only adds photons to the winner. Each losing mode keeps a spontaneous level set by how close its gain is to the winner's.

Relaxation oscillation

\begin{aligned}\omega_R^2&=\frac{r\ell_0-1}{\tau_L\tau_p}-\gamma_R^2\\\gamma_R&=\frac{r\ell_0}{2\tau_L}\end{aligned}

After the first spike, the photons and the inversion trade energy and ring down at this rate. The record's late peaks are compared with this small-signal result.

Why the losers fade

\frac{d\ln n_k}{dt}\approx-\frac{1-\ell_k/\ell_0}{\tau_p}

Once the inversion is clamped, a mode whose gain is almost as high as the winner's loses almost nothing per round trip and fades slowly.

Common difficulties

Clamping does not mean the pump stops mattering

Typical misconceptionOnce the gain is clamped, pumping harder changes nothing.

Better mental modelThe inversion stays at threshold, but the extra pump goes into photons in the winning mode, so the output power keeps rising.

Homogeneous gain alone is not the whole story

Typical misconceptionEvery laser with a homogeneous gain line runs on one mode.

Better mental modelThis model shares one inversion among all modes. In a real standing-wave cavity, spatial hole burning leaves unused inversion between the standing-wave peaks, so extra modes can survive.

The first spike is not the steady power

Typical misconceptionThe height of the first spike is what the laser will deliver.

Better mental modelThe inversion overshoots threshold before the light catches up, so the first spike is far above the steady power and is followed by ringing.

Run the experiment

01
Watch the record start from noise
Keep the default preset, Noise grows into one mode, and watch the line camera through one replay.
What to observe: The gain medium brightens first. Then many faint spots on the camera grow at once, a spike appears, and soon one spot is left.
02
Turn the pump down below threshold
Load Below threshold: only glow.
What to observe: The inversion settles at the pump level without ever reaching threshold. Every mode keeps only a weak glow, shaped by the gain line.
03
Broaden the gain line and lengthen the cavity
Load Broad gain line: modes linger, or drag the output coupler outward.
What to observe: More modes fit under the line and their gains are closer together, so several modes are still lasing at the end of the record.
04
Put line centre between two modes
Load Two modes almost tied and compare the two strongest modes in the growth plot.
What to observe: Two modes grow almost equally. The slightly better one wins, but the suppression at the end is far below the steady value.

Laser Startup from Noise

Watch noise turn into one laser mode

Core mathematical model

One inversion, one equation per mode

The gain line and the mode comb

Photon lifetime

Threshold and the first crossing

Clamped steady state

Relaxation oscillation

Why the losers fade

Common difficulties

Clamping does not mean the pump stops mattering

Homogeneous gain alone is not the whole story

The first spike is not the steady power

Run the experiment

Watch the record start from noise

Turn the pump down below threshold

Broaden the gain line and lengthen the cavity

Put line centre between two modes

Related experiments and articles

From a seed to laser light

Controlled light and precision measurement