L11 · Shape and mode selection

Transverse Laser Modes

Which beam pattern wins when a laser's transverse modes compete for gain? The pattern that overlaps the pumped region best and loses least at the aperture reaches threshold first. Once modes start to grow, each one eats into the gain where it is bright, and that reshuffles the competition. Moving the pump spot alone can change which pattern wins.

Let a small set of Hermite–Gaussian or Laguerre–Gaussian modes compete for gain. Move the pump and the cavity axis, close down the aperture, and compare the predicted output with an ideal reference field.

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

Each setting automatically evolves for 300 decay times. Compare the narrow, broad and ring pumps. Choosing an ideal reference does not seed the laser or force it into that mode.

The model uses scalar paraxial modes of an aligned, symmetric spherical-mirror cavity in air. The waist sits halfway between the mirrors. Horizontal and vertical are both transverse coordinates, and the pump, aperture and axis-offset controls are all measured in fundamental waist radii. The analytic HG and LG fields solve paraxial diffraction and match the wavefront curvature of the spherical mirrors. The Lab projects onto a fixed, finite set of modes; it is not a Fox–Li solver for the clipped cavity. The HG family keeps horizontal orders zero through three and vertical orders zero through two. The LG family keeps radial orders zero and one and nonnegative azimuthal orders zero through three. Only one family competes at a time, and the LG modes have positive helicity. Pump gain saturates locally under the incoherent sum of mode intensities, with an adiabatic reservoir. The baseline logarithmic round-trip loss is 0.08; the estimated two-pass aperture loss and a chosen order penalty add to it. Time is counted in baseline intensity-decay times, and power is normalized. All modes start from equal tiny seeds, which are only a numerical initial condition. Axis displacement moves the whole aligned cavity relative to the fixed pump and aperture; it does not model mirror tilt. Significant clipping would reshape the field by diffraction, which this projection does not include. Thermal effects, coherent beating, polarization, vector fields and real measured data are left out. Maps are peak-normalized and shown with square-root contrast. An ideal upright, unit-magnification relay images the waist onto the profiler; its internal lenses are enclosed and perfect imaging is assumed. The camera has a fixed physical field of view and can clip the image; the data map separately shows a range of four waist radii. Mirror separation, transverse dimensions and mirror curvature use separate schematic scales. The profiler always shows the predicted output; the separate ideal reference can show phase as hue.

Start with the essentials

Focus question: Can changing only the pump footprint change the winning spatial mode?
One-sentence intuition: A mode's threshold depends on how well it overlaps the gain and how much light the aperture removes, and once modes grow, local saturation changes the competition. An ideal reference picture by itself does not show that a mode oscillates.

Core mathematical model

Hermite–Gaussian modes in Cartesian coordinates

\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 put the nodes on a rectangular grid. The Lab normalizes each mode so that its power integrated over the plane is one.

Laguerre–Gaussian modes in polar coordinates

\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}

The azimuthal factor makes nonzero-order modes vanish on the axis: that is why their intensity can form a ring. Laguerre polynomials add radial nodes, and the phase winds around the center. This Lab uses only positive helicity for the circular family.

The mirrors set the waist and the transverse mode 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 the Gouy phase. Each step up in transverse order shifts the resonance by this spacing; the model does not force a single longitudinal order.

Modes compete for local gain

\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}

Lengths are measured in waist radii. The pump peaks at one, each mode density integrates to one, and time is counted in baseline decay times. The model projects intensities adiabatically and leaves out coherent interference.

Aperture loss 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}

Light passes the aperture twice on each round trip. The integral uses the undeformed mode; severe clipping changes the eigenmodes themselves and goes beyond 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. When centered modes add incoherently, the quality on each axis is the power-weighted mean of the modal values; phase plays no part.

Common difficulties

An allowed mode is not necessarily lasing

Typical misconceptionEvery ideal mode shown on screen must appear in the laser.

Better mental modelA mode has to overcome its own losses and then win the competition after local saturation. The reference is a separate field drawn at unit power.

A donut does not fix the phase

Typical misconceptionA ring in the intensity image proves that the phase winds around the center.

Better mental modelDifferent mode mixtures can also produce rings. The phase map shows the chosen ideal LG field; it is not an inferred phase measurement.

An aperture integral cannot solve a clipped cavity

Typical misconceptionAn aperture removes power but leaves the eigenfield exactly the same.

Better mental modelThe fixed set of modes only estimates each mode's loss. Strong clipping or an off-axis aperture reshapes the field and couples modes coherently, which goes beyond this projection.

Run the experiment

01
A narrow pump picks the center
Choose Centered fundamental. The model has already evolved for 300 decay times. Open Gain competition and independent ideal mode to compare candidate and saturated gains.
What to observe: The fundamental mode survives, and its gain is clamped to its loss.
02
Widen the pumped area
Predict whether a wider pumped area can support more than one pattern, then choose Broad pump, many modes. Compare the camera image and modal power shares.
What to observe: Inversion survives in more places across the beam, so several modes carry power.
03
Get a donut without forcing the field
Choose Ring pump selects a donut. In Advanced controls, change the reference orders or its phase display, then compare with the output camera.
What to observe: The circular mode with azimuthal order one wins through its overlap with the gain. Changing the reference leaves the output unchanged.
04
Shift the cavity and close the aperture
Move the whole cavity axis relative to the fixed pump and aperture, then choose Aperture extinguishes output.
What to observe: Both the gain overlap and the aperture transmission change. An allowed mode can fall below threshold, and its reference map stays visible even when the output is dark.

Transverse Laser Modes

Diffraction sets the family; pump and loss select the winner

Core mathematical model

Hermite–Gaussian modes in Cartesian coordinates

Laguerre–Gaussian modes in polar coordinates

The mirrors set the waist and the transverse mode spacing

Modes compete for local gain

Aperture loss and order penalty

Ideal-mode beam quality

Common difficulties

An allowed mode is not necessarily lasing

A donut does not fix the phase

An aperture integral cannot solve a clipped cavity

Run the experiment

A narrow pump picks the center

Widen the pumped area

Get a donut without forcing the field

Shift the cavity and close the aperture

Related experiments and articles

Confine and select a mode