L09 · Modes and cavity design

Gain Curve Meets Longitudinal Modes

Which cavity modes survive after they saturate the gain? If all modes draw on the same gain, usually only the one with the lowest threshold survives; it pulls the gain below what the others need. Hole burning breaks that sharing. When modes feed on different groups of atoms in the spectrum, or on different spots along the cavity, several can lase at once. A filter that adds loss at the rivals' frequencies raises their thresholds and leaves one.

Slide the cavity's mode comb across a gain line and see which modes survive. Compare shared homogeneous gain, separate spectral classes of atoms, standing-wave hole burning and a frequency-selective loss, then record a mode hop.

Elapsed photon lifetimes $\text{—}$

Cold-cavity mode spacing $\text{—}$

Unsaturated threshold candidates $\text{—}$

Modes above one-percent share $\text{—}$

Lowest unsaturated pump threshold $\text{—}$

Total normalized intensity $\text{—}$

Dominant mode share $\text{—}$

Dominant relative mode order $\text{—}$

Dominant cold-cavity frequency offset $\text{—}$

Dominant saturated net gain $\text{—}$

Side-mode suppression $\text{—}$

Single-line threshold pulling estimate $\text{—}$

L09 · Mode selection

Crossing threshold is only the beginning

Inhomogeneous spectral classesUse a Gaussian distribution of transition frequencies instead of one shared line.Standing-wave spatial hole burningWhole-cavity standing-wave overlaps; off means a uniform travelling-wave reference.Optical cavity length

L

60\,\mathrm{cm}

Gain width parameter

b

0.65\,\mathrm{GHz}

Peak free gain relative to loss

p

1.8

Comb displacement in mode spacings

\delta

0.12

Quadratic filter loss strength

f

0

Filter center offset

\nu_f-\nu_a

0\,\mathrm{GHz}

Advance several times to tell short-lived candidates from modes that survive. Compare the three unfiltered presets at the same pump, then add the filter. Tune through half a mode spacing to record a hop.

This is a simplified class-A competition model with 31 axial modes. The inversion is eliminated adiabatically, and mode intensities stay positive. Homogeneous gain has a Lorentzian line. Inhomogeneous gain averages Lorentzian classes of 80 MHz half-width over a Gaussian distribution. The gain width control sets the homogeneous half-width in the first case and the Gaussian standard deviation in the second. The spectral overlaps and whole-cavity spatial overlaps are exact. They feed a first-order saturation closure, which is a teaching approximation away from threshold. In a uniformly filled cavity, standing waves make self-saturation stronger than cross-saturation. All modes start from equal small seeds as classical initial conditions. Intensity is measured in units of the saturation intensity and time in photon lifetimes, so the model predicts no watts, beat phases, relaxation oscillations or linewidths. The frequency-pulling readout is a separate linear estimate for a single line at threshold. It is switched off for inhomogeneous, standing-wave or filtered cases, and it does not shift the competition spectrum.

Physics tutorial

Threshold candidates compete for a finite gain

BackgroundA cavity allows only a set of discrete frequencies. The gain line and any frequency-dependent loss decide which of these modes can grow from a weak start. Once the gain saturates, the outcome can change.

Why it mattersSeveral modes can start above threshold, yet the steady state need not keep them all. What happens depends on whether the modes draw on the same atoms, on groups of atoms with different transition frequencies, or on different regions of a standing wave.

Start with the essentials

Focus question: Why do identical cavity modes behave differently when the gain broadening or spatial overlap changes?
One-sentence intuition: Threshold is judged with unsaturated gain, but survival depends on how each mode saturates its own gain and that of the others. Homogeneous, spatially uniform gain favors the lowest-threshold mode unless two thresholds are exactly equal; spectral or spatial hole burning lets several modes coexist.

Core mathematical model

Cold-cavity comb

\begin{aligned}\mathrm{FSR}&=\frac{c}{2L}\\\nu_j-\nu_a&=(j+\delta)\mathrm{FSR}\end{aligned}

The optical length of the cavity sets the mode spacing. The offset slides the whole comb relative to the atomic line, and each mode keeps its relative order number while you tune.

Spectral overlap

\begin{aligned}H_{jk}&=\frac{1}{1+[(\nu_j-\nu_k)/\gamma_h]^2}\\h_j&=\frac{\sum_kw_kH_{jk}}{\mathcal N}\end{aligned}

A single class of atoms gives a homogeneous Lorentzian line. A Gaussian spread of classes gives a Voigt profile. The normalization makes the unsaturated gain at line center equal to the pump setting.

Whole-cavity spatial overlap

\langle f_jf_l\rangle=\begin{cases}3/2&j=l\\1&j\ne l\end{cases}

With intensities normalized to a mean of one, a standing wave overlaps more with itself than with the standing wave of a different axial order. This assumes gain spread evenly through the whole cavity; a short crystal needs a different overlap integral.

Simplified saturation and competition

\begin{aligned}C_{jl}&=\frac{\sum_kw_kH_{jk}H_{lk}}{\mathcal N}\langle f_jf_l\rangle\\G_j&=\frac{ph_j}{1+\sum_lC_{jl}I_l/h_j}\end{aligned}

The overlaps are exact for the chosen numerical integration and geometry. The saturation formula extends the first-order expansion of local saturation in a form that stays positive; it is not a full spatial Maxwell–Bloch solution.

Mode growth and frequency-dependent loss

\begin{aligned}u&=t/\tau_p\\\frac{dI_j}{du}&=(G_j-\ell_j)I_j+\epsilon\\\ell_j&=1+f\left(\frac{\nu_j-\nu_f}{\mathrm{FSR}}\right)^2\end{aligned}

The mode with the smallest ratio of loss to gain profile has the lowest unsaturated threshold. A tiny classical floor lets suppressed modes grow back after tuning. A mode with more than one percent of the total intensity counts as active.

Frequency pulling, estimated separately

\begin{aligned}\nu_L&=\frac{\gamma_h\nu_c+\gamma_c\nu_a}{\gamma_h+\gamma_c}\\\gamma_c&=\frac{\kappa}{4\pi}\end{aligned}

This linear estimate for a single line at threshold pulls the lasing frequency toward the atomic resonance. The cold-cavity linewidth comes from a base logarithmic round-trip power loss of 0.08. The competition spectrum stays at the cold-cavity frequencies, and the estimate is switched off when its assumptions do not hold.

Common difficulties

Above threshold does not mean surviving

Typical misconceptionEvery mode whose unsaturated gain exceeds its loss shows up in the final spectrum.

Better mental modelThreshold is checked before the gain saturates. The final gain has to be worked out with all the competing modes present.

Multimode does not mean mode locked

Typical misconceptionSeveral surviving lines automatically form short pulses.

Better mental modelThis model tracks no mode phases, so it cannot predict beating, phase locking, pulse formation or linewidth. Later Labs cover those.

Uniform gain does not mean spatially uniform saturation

Typical misconceptionA homogeneously broadened standing-wave laser must have one mode.

Better mental modelDifferent standing-wave patterns can use up the gain in different places. The travelling-wave reference removes this spatial effect, and even then modes with exactly equal thresholds can share the output.

Run the experiment

01
Predict candidates, then watch selection
Choose Homogeneous competition and note how many modes are threshold candidates. Advance three times, then compare that number with the active modes and the side-mode suppression.
What to observe: Several modes initially qualify, but the mode with the smallest threshold depletes the shared gain and suppresses its neighbors.
02
Separate two kinds of hole burning
Compare Spectral hole burning and Standing-wave hole burning at the same pump and cavity length. Advance each three times.
What to observe: In spectral hole burning, the modes draw on different classes of atoms, so they saturate each other less. In standing-wave hole burning, each mode overlaps most with itself, which leaves gain for neighboring orders. The two mechanisms are different.
03
Force one mode to win
Load Filtered inhomogeneous gain, then move the filter center. Inspect both loss and saturated gain before and after advancing.
What to observe: Modes near the filter center see less loss. A wide gain line need not give broad output once the competing thresholds are raised.
04
Record a mode hop
Load Near a mode hop and advance, then use Tune right and evolve twice. Watch the dominant mode order and the cold-cavity offset.
What to observe: The winning mode changes when the comb moves past half a mode spacing. Tuning keeps the existing intensities, so the takeover can take time.