L11 · Shape and mode selection

Transverse Laser Modes

Evolve a finite family of Hermite–Gaussian or Laguerre–Gaussian modes. Move the pump and cavity axis, tighten the aperture, and compare predicted output with an ideal field reference.

Evolution time in decay units $\text{—}$

Fundamental waist radius $\text{—}$

Adjacent transverse-order frequency spacing $\text{—}$

Unsaturated candidates above threshold $\text{—}$

Modes carrying at least one percent $\text{—}$

Dominant output mode $\text{—}$

Normalized intracavity power $\text{—}$

Largest modal power share $\text{—}$

Horizontal output beam quality $\text{—}$

Vertical output beam quality $\text{—}$

Selected ideal reference $\text{—}$

Ideal reference pump threshold $\text{—}$

Ideal reference two-pass clipping estimate $\text{—}$

Horizontal reference beam quality $\text{—}$

Vertical reference beam quality $\text{—}$

Dominant mode two-pass clipping estimate $\text{—}$

L11 · Transverse workbench

Make a spatial pattern win

Use circular LG familyOtherwise use rectangular HG modes; families do not compete together.Show ideal reference field phaseHue shows phase on the separate ideal map; output measurements stay unchanged.Reference horizontal or radial order

m\ \text{or}\ p

0

Reference vertical or azimuthal order

n\ \text{or}\ \ell

0

Mirror separation

L

50\,\mathrm{cm}

Both mirror curvature radii

\mathcal R

100\,\mathrm{cm}

Vacuum wavelength

\lambda

1064\,\mathrm{nm}

Waist aperture radius

a/w_0

2.5

Pump radial width

w_P/w_0

0.6

Pump ring radius

r_P/w_0

0

Horizontal pump displacement

x_P/w_0

0

Cavity axis displacement

x_c/w_0

0

Peak small-signal pump gain

p

6

Extra loss per transverse order

\ell_N

0.08

Compare narrow, broad and ring pumps after evolving 300 decay times. Selecting an ideal reference does not seed or force the laser into that mode.

Scalar paraxial modes of an aligned symmetric spherical-mirror cavity in air. The waist is halfway between mirrors; horizontal and vertical coordinates are transverse, and all pump, aperture and axis-offset controls use the fundamental waist radius. Analytic HG and LG fields solve paraxial diffraction and reproduce spherical mirror wavefront curvature. This is a fixed finite modal projection, not a Fox–Li solver for the clipped cavity. Retained families: HG horizontal orders zero through three and vertical orders zero through two; LG radial orders zero through one and nonnegative azimuthal orders zero through three. Only one family competes at a time; a positive LG helicity is selected. Pump gain saturates locally with the incoherent intensity sum and an adiabatic reservoir. Baseline logarithmic round-trip loss is 0.08; estimated two-pass aperture loss and a chosen order penalty add to it. Time is in baseline intensity-decay units, power is normalized, and equal tiny seeds are numerical initial conditions. Axis displacement translates the entire aligned cavity relative to the fixed pump and aperture; it is not a mirror-tilt model. Significant clipping requires diffraction reshaping. Thermal effects, coherent beating, polarization, vector fields and actual measured data are omitted. Maps use peak normalization and square-root display contrast. The profiler always shows predicted output; the separate ideal reference can show phase in hue.

Start with the essentials

Focus question: Can changing only the pump footprint change the winning spatial mode?
One-sentence intuition: Modal thresholds depend on gain overlap and aperture loss. Once modes grow, local saturation changes their competition; an ideal reference picture alone does not prove that a mode oscillates.

Core mathematical model

Cartesian fields

\begin{aligned}u_{mn}&\propto H_m(\sqrt2 X)\\&\quad H_n(\sqrt2 Y)e^{-(X^2+Y^2)}\\X&=x/w,\quad Y=y/w\end{aligned}

Hermite polynomials create rectangular nodes. The actual implementation normalizes each plane to unit integrated power.

Circular fields

\begin{aligned}u_{p\ell}&\propto s^{|\ell|/2}L_p^{|\ell|}(s)\\&\quad e^{-s/2}e^{i\ell\phi}\\s&=2r^2/w^2\end{aligned}

Laguerre polynomials set radial nodes; azimuthal phase winds around the center. The circular family here selects positive helicity.

Mirror boundary and transverse spacing

\begin{aligned}z_R&=\tfrac12\sqrt{L(2\mathcal R-L)}\\w_0^2&=\lambda z_R/\pi\\\Delta\nu_\perp&=\frac{c}{2\pi L}\,2\arctan\frac{L}{2z_R}\end{aligned}

The symmetric cavity fixes the waist and Gouy phase. Each added transverse order shifts the resonance by the displayed spacing; no single longitudinal order is forced.

Local gain competition

\begin{aligned}G_j&=p\int\frac{P\rho_j}{1+\sum_l I_l\rho_l/\rho_0}\,dA\\\dot I_j&=(G_j-\ell_j)I_j\\\rho_0&=2/\pi\end{aligned}

Coordinates are in waist-radius units. The pump has peak one, mode densities integrate to one, and the baseline decay time sets the clock. This adiabatic intensity projection omits coherent interference.

Aperture and order penalty

\begin{aligned}Q_j&=\int_{r<a}\rho_j\,dA\\\ell_j&=1-\frac{2\ln Q_j}{0.08}+\ell_N N_j\end{aligned}

The aperture is traversed twice each round trip. The integral uses the undeformed field; severe clipping changes eigenmodes and exceeds this model.

Ideal-mode beam quality

\begin{aligned}\mathrm{HG}:\quad M_x^2&=2m+1\\M_y^2&=2n+1\\\mathrm{LG}:\quad M_x^2&=M_y^2=2p+|\ell|+1\end{aligned}

These are second-moment quality factors. For an incoherent centered modal sum, each axis quality is the power-weighted modal mean; phase information is excluded.

Common difficulties

Allowed is not oscillating

Typical misconceptionEvery displayed ideal mode must appear in the laser.

Better mental modelA mode must overcome its own losses and compete after local saturation; the reference is a separate unit-power field.

A donut does not imply a unique phase

Typical misconceptionAn intensity image establishes azimuthal phase winding.

Better mental modelDifferent mixtures can produce rings. The phase map belongs to the specified ideal LG field, not an inferred phase measurement.

An aperture integral is not a clipped-cavity solver

Typical misconceptionRemoving power leaves the eigenfield exactly unchanged.

Better mental modelThe fixed basis estimates modal loss. Strong clipping or off-axis apertures require field reshaping and coherent mode coupling beyond this projection.

Run the experiment

01
A narrow pump selects the center
Evolve the centered fundamental preset for 300 decay times. Compare candidate gain with saturated gain.
What to observe: The fundamental survives and clamps its gain to its loss.
02
Broaden the gain footprint
Choose the broad pump and evolve for 300 decay times.
What to observe: More spatial regions retain inversion, allowing several modes to carry power.
03
Create a donut without forcing the field
Choose the ring pump and evolve. Then change the ideal reference orders or phase display.
What to observe: The circular first azimuthal mode wins from gain overlap. Reference changes leave its output state unchanged.
04
Misalign the cavity and restrict the aperture
Move the entire cavity axis relative to the fixed pump and aperture, then choose the clipped preset.
What to observe: Gain overlap and transmission both change. A permitted mode may fail threshold; a reference map remains visible even when output is dark.