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.

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}

Remove the centroid offset and angle first, then match the complex wavefront. A dark reflection port alone does not tell you which transverse order is resonating.

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.

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 misconceptionIf the intensity image shows a centered Gaussian, the coupling must be good.

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

Steering and focusing do different jobs

Typical misconceptionWith enough mirror tweaking, any mismatch can be fixed.

Better mental modelSteering 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 misconceptionWhenever the reflection dips, the fundamental mode must be resonating.

Better mental modelHigher-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

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.
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.
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.
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.