---
title: "Alignment & Mode Matching Interactive Lab | Sandbox Physics"
description: "Why can a centered beam still reflect from a resonant cavity? Centering only fixes where the beam enters and at what angle."
canonical_url: "https://sandboxphysics.com/labs/alignment-mode-matching/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/alignment-mode-matching/"
last_modified: "2026-10-27"
---

L13 · Axis, wavefront and cavity response

# Alignment & Mode Matching

**Why can a centered beam still reflect from a resonant cavity?** Centering only fixes where the beam enters and at what angle. If its width or wavefront curvature differs from the cavity mode's, part of the light goes into higher-order modes that are off resonance, and that part bounces back. Steering mirrors cannot fix this; matching size and curvature takes lenses.

Adjust two steering mirrors and a two-lens telescope to couple a laser beam into a cavity. Tell axis errors apart from a beam with the wrong size or curvature, then scan the cavity frequency to see which transverse orders the beam excites.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

A coherent 1064 nm Gaussian beam enters a passive, lossless cavity in air formed by two identical spherical mirrors. The cavity is 0.5 m long, each mirror has a 0.9 m radius of curvature, and each reflects 98% of the power. Input power is normalized to one. Distances along the beam are measured from the source reference plane: the steering mirrors sit at 0.15 and 0.35 m, the lenses at 0.50 and 0.90 m and the cavity entrance at 1.40 m. The cavity waist is midway between its mirrors. Beam radius means the radius where the intensity falls to one over e squared of its peak. The steering controls tilt each mirror slightly in the vertical plane, and the reflected ray turns by twice that tilt. The ideal fold mirrors are unfolded for the propagation calculation, and their sagittal effects, polarization and mirror aberrations are left out. Positive vertical displacement and angle point upward. The scene exaggerates sideways offsets, and the beam envelopes may be drawn at a different scale. Lenses have infinite aperture and positive focal lengths. The Gaussian field is projected analytically onto the complete set of Hermite modes. The spectrum keeps modes up to total order 32 and treats the remaining power as fully reflected; that remainder is reported, and under strong mismatch the predicted spectrum is only approximate. Each kept order has its own Airy response, shifted by its Gouy phase. An ideal lossless circulator sends the returning light to the reflection meter without changing the input beam parameter. Reflected and transmitted power add up to the incident power. The incident profile and the ideal cavity reference are separate data, drawn on the same scale, normalized to their peaks and shown with square-root contrast. An intensity image alone cannot show phase curvature or tilt. Automatic steering removes the centroid offset and angle but does not fix a wavefront mismatch. The constrained lens search minimizes the centered mismatch and then realigns; it does not guarantee a perfect match for every source. Active gain, thermal distortion, finite apertures, general astigmatism, detector noise and alignment servo dynamics are not modeled.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## Align the axis, then match the wavefront

**Background** A cavity only accepts light that matches one of its spatial modes; a bright spot at the center is not enough. The beam's position, angle, size and wavefront curvature decide how an incoming Gaussian beam splits among those modes.

**Why it matters** Even when the spot looks centered, the power meter can show only a shallow dip in reflection. Each kind of error needs a different control to fix it.

Start with the essentials

- **Focus question**: Can steering mirrors fix a beam that is centered but has the wrong size or curvature?
- **One-sentence intuition**: Two steering mirrors set the beam's position and angle, and two lenses set its complex beam parameter, that is, its size and curvature; only when both are right can the fundamental resonance take in almost all the light.

### Core mathematical model

#### How mirrors, distance and lenses change a ray

$$
\begin{aligned}x^{\prime}&=x+d\theta\\\theta^{\prime}_{\rm mirror}&=\theta+2\delta\\\theta^{\prime}_{\rm lens}&=\theta-x/f\end{aligned}
$$

Over a distance, an angle turns into a position offset, so two mirrors at different planes can set both. A lens also bends a beam whose centroid is off axis.

#### The complex parameter of a Gaussian beam

$$
\begin{aligned}q&=z-z_0+iz_R\\z_R&=\pi w_0^2/\lambda\\q^{\prime}&=(Aq+B)/(Cq+D)\end{aligned}
$$

The real part tells you where the waist is, and the imaginary part sets how quickly the beam spreads by diffraction. Both must match the cavity mode at the entrance.

#### The cavity mode at its input mirror

$$
\begin{aligned}z_{R,c}&=\tfrac12\sqrt{L(2R-L)}\\q_c&=-L/2+iz_{R,c}\\w_c^2&=\frac{\lambda|q_c|^2}{\pi z_{R,c}}\end{aligned}
$$

This symmetric cavity has its waist halfway between the mirrors, so at the entrance the mode is still converging toward it.

#### Overlap of fields, not intensities

$$
\begin{aligned}c_{mn}&=\iint u_{mn,c}^{*}u_{\rm in}\,dx\,dy\\\eta_{00}&=|c_{00}|^2\\\sum_{m,n\ge0}|c_{mn}|^2&=1\end{aligned}
$$

The overlap uses complex fields, so it keeps the phase from curvature and tilt. A beam with a centered image can still put a large share of its power into even higher orders.

#### Mismatch of a centered beam, and the alignment limit

$$
\begin{aligned}\eta_{\rm centered}&=\frac{4\,\Im q\,\Im q_c}{|q-q_c^{*}|^2}\\\eta_{\rm waist}&=\exp\!\left[-\frac{d^2}{w^2}-\frac{k^2w^2\theta^2}{4}\right]\end{aligned}
$$

The second line holds when the beam and the cavity mode have the same size and a common waist; it gives an independent check of alignment sensitivity. The general overlap integral also handles curvature and axis errors at the same time.

#### Each transverse order has its own resonance

$$
\begin{aligned}\nu_{\rm FSR}&=c/(2L)\\\Delta\nu_{\perp}&=\frac{\nu_{\rm FSR}}{\pi}\arccos(1-L/R)\\\mathcal T_N&=\frac{(1-\mathcal R)^2}{(1-\mathcal R)^2+4\mathcal R\sin^2(\phi_N/2)}\\\phi_N&=\frac{2\pi(\Delta\nu-N\Delta\nu_{\perp})}{\nu_{\rm FSR}}\end{aligned}
$$

Add up the response of each order, weighted by the power projected into it. Both mirrors have the same power reflectivity, and with no absorption the reflected and transmitted fractions add up to one.

### Common difficulties

#### A centered spot is not a matched field

**Typical misconception** If the intensity image shows a centered Gaussian, the coupling must be good.

**Better mental model** An intensity image cannot show tilt or phase curvature. Check both the axis readings and the complex wavefront.

#### Steering and focusing do different jobs

**Typical misconception** With enough mirror tweaking, any mismatch can be fixed.

**Better mental model** Steering moves the beam's centroid and changes its angle; lenses change its Gaussian beam parameter. A wavefront error in a centered beam is still there after steering.

#### A reflection dip may belong to another mode

**Typical misconception** Whenever the reflection dips, the fundamental mode must be resonating.

**Better mental model** Higher-order components also resonate, at frequencies shifted by the Gouy phase. Check the mode overlap and where the peak sits in frequency.

### Run the experiment

1. 01

   #### Bring a displaced beam back on axis

   Choose Displaced source. Adjust the pitch of both mirrors, or press Correct steering mirrors.

   **What to observe:** The position and angle at the cavity entrance both return to zero, and the overlap with the fundamental mode comes back.

2. 02

   #### Find a hidden wavefront error

   Choose Aligned but wrong wavefront. Press Correct steering mirrors and compare the centered mismatch.

   **What to observe:** The axis is already right. Steering changes neither the complex beam parameter nor the power in even orders.

3. 03

   #### Match with lenses

   Change both focal lengths, then press Search matching lenses and compare.

   **What to observe:** The beam's size and curvature approach the cavity reference. After it changes the lenses, the search also corrects the steering.

4. 04

   #### Tell frequency detuning from spatial mismatch

   Compare Matched but off resonance with a spatially mismatched beam. Scan the detuning across several peaks.

   **What to observe:** Detuning the frequency leaves the mode overlap unchanged. A spatial mismatch changes the relative heights of the peaks from different transverse orders.

Physical conclusion

To find out why coupling is poor, read the axis error, the centered mismatch and a frequency scan together. Higher-order transmission peaks are what tell a spatial mismatch apart from simple frequency detuning.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)