L09 · Modes and cavity design

Gain Curve Meets Longitudinal Modes

Move the cavity comb through a gain line. Compare shared homogeneous gain, independent spectral classes, standing-wave hole burning and a frequency-selective loss. 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 distinguish transient candidates from surviving modes. Compare the three unfiltered presets at the same pump, then use the filter. Tune through half a mode spacing to record a hop.

Reduced class-A competition model with 31 axial modes, adiabatic inversion and positive modal intensities. Homogeneous gain is Lorentzian; inhomogeneous gain averages Lorentzian classes of 80 MHz half-width over a Gaussian distribution. The bandwidth control is the homogeneous half-width or Gaussian standard deviation, respectively. Exact spectral and whole-cavity spatial overlaps feed a first-order saturation closure; it is a teaching approximation away from threshold. Standing waves increase self-saturation relative to cross-saturation for a uniformly filled cavity. Equal small seeds are classical initial conditions. Intensity and time use saturation-intensity and photon-lifetime units; no watts, beat phase, relaxation oscillations or linewidth are predicted. The separate frequency-pulling readout is a linear single-line threshold estimate, disabled for inhomogeneous, standing-wave or filtered cases. It does not shift the competition spectrum.

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 uses unsaturated gain. Survival depends on self- and cross-saturation. Homogeneous uniform gain favors the mode with the lowest threshold, except at exact degeneracy; spectral or spatial hole burning permits coexistence.

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 sets the spacing. The offset moves the entire comb relative to the atomic line; relative mode orders are retained during tuning.

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}

One class gives a homogeneous Lorentzian. A Gaussian distribution of classes gives a Voigt profile. The normalizer makes the line-center free gain equal to the pump control.

Whole-cavity spatial overlap

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

Unit-mean standing-wave intensities have stronger self-overlap than the overlap between distinct integer axial orders. This assumes uniform gain filling the whole cavity; a short crystal requires another overlap integral.

Reduced 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 overlap is exact within the stated quadrature and geometry. The rational saturation is a positive continuation of the first-order local saturation expansion, not a full spatial Maxwell–Bloch solver.

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 weakest free-gain threshold minimizes loss divided by gain profile. The tiny classical floor allows suppressed modes to recover on tuning. Intensities above one percent of the total are counted as active.

Separate single-line pulling estimate

\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 single-line threshold estimate moves the frequency toward the atomic resonance. A base logarithmic round-trip power loss of 0.08 defines the cold-cavity linewidth. The competition spectrum remains at cold-cavity frequencies, and the estimate is disabled outside its stated assumptions.

Common difficulties

A candidate is not a survivor

Typical misconceptionEvery mode with free gain above loss appears in the final spectrum.

Better mental modelThreshold is evaluated before saturation. The final gain must be calculated with all competing intensities present.

Multimode does not mean mode locked

Typical misconceptionSeveral surviving lines automatically form short pulses.

Better mental modelThis model has no modal phases. It cannot determine beating, phase locking, pulse formation or linewidth; those belong to later Labs.

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 deplete different regions. The travelling-wave reference removes that spatial effect, and exact equal-threshold degeneracies can still share output.

Run the experiment

01
Predict candidates, then watch selection
Choose Homogeneous competition and note the threshold candidate count. Advance three times and compare it with active modes and 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: Different spectral classes reduce cross-saturation. Standing-wave self-overlap also leaves neighboring orders access to gain. These are different mechanisms.
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 pay less loss. A wide gain line does not force broad output if competing thresholds are raised.
04
Record a mode hop
Load Near a mode hop, advance, then tune right and evolve twice. Inspect the axial-order ledger and cold-cavity offset.
What to observe: The winning order switches as the comb passes a half-spacing. Tuning retains existing intensities, so the competition can take time to reverse.