L13 · Axis, wavefront and cavity response

Alignment & Mode Matching

Adjust two steering mirrors and a two-lens telescope. Separate axis errors from size and curvature mismatch, then use a cavity frequency scan to reveal the excited transverse orders.

Fundamental power overlap $\text{—}$

Centered wavefront mismatch $\text{—}$

Integrated reflected power fraction $\text{—}$

Integrated transmitted power fraction $\text{—}$

Vertical centroid at cavity entrance $\text{—}$

Vertical ray angle at cavity entrance $\text{—}$

Incident radius at entrance $\text{—}$

Cavity reference radius at entrance $\text{—}$

Incident beam parameter real part $\text{—}$

Incident beam parameter imaginary part $\text{—}$

Total first-order power $\text{—}$

Total second-order power $\text{—}$

Unresolved power above order 32 $\text{—}$

L13

Make the beam fit the cavity

Input beam waist radius

w_0

0.21728768082325872\,\mathrm{mm}

Source plane distance since waist

z_s-z_0

-20.432276657060523\,\mathrm{cm}

Source vertical offset

d_s

0\,\mathrm{mm}

Source vertical angle

\theta_s

0\,\mathrm{mrad}

First steering mirror pitch

\delta_1

0\,\mathrm{mrad}

Second steering mirror pitch

\delta_2

0\,\mathrm{mrad}

First matching lens focal length

f_1

60\,\mathrm{cm}

Second matching lens focal length

f_2

50\,\mathrm{cm}

Frequency detuning from fundamental

\Delta\nu

0\,\mathrm{MHz}

First cancel the centroid and angle. Then match the complex wavefront. A dark reflected port alone does not identify which transverse order resonates.

A coherent 1064 nm Gaussian enters a passive, lossless symmetric spherical-mirror cavity in air: length 0.5 m, curvature radius 0.9 m, mirror power reflectivity 98%. Input power is normalized to one. Optical path coordinates run from the source reference plane through steering mirrors at 0.15 and 0.35 m, lenses at 0.50 and 0.90 m, and the cavity entrance at 1.40 m. The cavity waist is midway between its mirrors. Radius denotes the intensity one-over-e-squared radius. Steering controls are small mirror pitch changes in the vertical plane; the reflected ray receives twice that change. Ideal folds are unfolded for propagation; their sagittal effects, polarization and mirror aberrations are omitted. Positive vertical displacement and angle point upward. The transverse scene scale exaggerates offsets; beam envelopes may use a separate scale. Lenses have infinite aperture and positive focal lengths. The Gaussian field is projected analytically into the complete Hermite basis; the spectrum retains total order through 32 and treats the unresolved remainder as fully reflected. Its size is reported, and strong mismatch makes the spectral prediction approximate. Each retained order uses its own Gouy-shifted Airy response. An ideal lossless circulator routes the return port to the reflection meter without changing the input beam parameter. Integrated reflection and transmission sum to the incident power. The incident profile and ideal cavity reference are separate data, shown on the same scale with peak normalization and square root display contrast; an intensity image alone cannot reveal phase curvature or tilt. Automatic steering cancels centroid and angle without repairing wavefront mismatch. The constrained lens search minimizes centered mismatch, then re-aligns; it does not guarantee a perfect solution for every source. No active gain, thermal distortion, finite apertures, general astigmatism, detector noise or alignment servo dynamics.

Start with the essentials

Focus question: Can steering mirrors repair a centered beam with the wrong size or curvature?
One-sentence intuition: Two steering mirrors independently set position and angle. Two lens powers change the complex beam parameter. Both tasks are needed before a fundamental resonance can accept almost all the input.

Core mathematical model

Steering and lens action

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

Distance turns angle into position. Two mirrors at different planes control both; a lens also acts on an off-axis centroid.

A physical Gaussian parameter

\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 locates a waist and the imaginary part determines diffraction. Both must match at the cavity entrance.

Cavity target 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}

The fixed symmetric cavity has its waist halfway between mirrors; the entrance field is converging toward it.

Project 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}

Complex overlap retains curvature and tilt phase. A centered image can still project substantially into even higher orders.

Centered mismatch and 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 is the equal-size common-waist limit, useful for independently checking sensitivity. The general integral also handles simultaneous curvature and axis errors.

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}

Sum the responses weighted by projected power. The two mirrors share power reflectivity; no absorption means total reflection plus transmission is one.

Common difficulties

A centered spot is not a matched field

Typical misconceptionA centered Gaussian intensity image proves good coupling.

Better mental modelTilt and phase curvature are invisible in an intensity image. Use both axis and complex-wavefront diagnostics.

Steering and focusing do different jobs

Typical misconceptionMore mirror adjustment can fix any mismatch.

Better mental modelSteering changes centroid and angle, while lenses change the Gaussian parameter. A centered wavefront error remains after steering.

A reflection dip may belong to another mode

Typical misconceptionEvery dark reflection dip indicates a fundamental resonance.

Better mental modelHigher-order components also resonate at Gouy-shifted frequencies. Check the overlap and the position of the peak.

Run the experiment

01
Recover a displaced axis
Choose displaced source. Adjust the two mirror pitches, or use correct steering mirrors.
What to observe: Entrance position and angle both return to zero, and fundamental overlap recovers.
02
Find a hidden wavefront error
Choose aligned but wrong wavefront. Correct steering and compare the centered mismatch.
What to observe: The axis is already right. Steering does not change the complex beam parameter or the even-order content.
03
Match with lenses
Vary both focal lengths, then search matching lenses for comparison.
What to observe: The size and curvature converge toward the reference; the search also corrects steering after changing optics.
04
Separate frequency from space
Compare matched but off resonance with spatial mismatch. Scan detuning around several peaks.
What to observe: Frequency detuning preserves the modal overlap. Spatial mismatch changes the relative heights of transverse-order peaks.