Acousto-Optic Modulator

1. Theory

• Definition: An acousto-optic modulator (AOM) diffracts a light beam from a travelling sound wave in a transparent solid. An RF signal on a piezoelectric transducer launches the sound; the strain changes the refractive index (the photoelastic effect), so the crystal acts as a moving grating with period $\Lambda =V/f$ (GH introduction; RP Photonics; Wikipedia; Isomet).
• Bragg diffraction: With the beam at the Bragg angle $\sin {\theta }_B=\lambda f/(2V)$, most of the light goes into one order, deflected by ${\theta }_d\approx \lambda f/V$, twice the Bragg angle (GH introduction; Arizona). The first order is shifted in optical frequency by $\pm f$ (Arizona; Wikipedia).
• Efficiency: $\eta ={\sin }^2[(\pi /\lambda )\sqrt{M_2{P}_{a}L/(2H)}]$ grows with RF power until it saturates; the RF power for the same efficiency scales as ${\lambda }^2$ (Wikipedia). The figure of merit $M_2$ is large for TeO${}_2$, especially in its slow shear mode, and small for fused silica (GH introduction; photonics.com).
• Speed: The rise time is set by the time the sound takes to cross the beam, ${\tau }_r\approx 0.65d/V$, so a smaller beam is faster (Isomet). Typical rise times are 5 to 100 $\mathrm{ns}$ (Wikipedia).
• Three uses of one device: Changing the RF amplitude modulates the intensity, changing the RF frequency steers the beam (a deflector), and the $\pm f$ shift makes a frequency shifter.
• Regimes and sizes: The Klein-Cook parameter $Q$ separates multi-order Raman-Nath diffraction from single-order Bragg diffraction; most modulators work in the Bragg regime (RP Photonics; Isomet). Bulk devices use 27 $\mathrm{MHz}$ to 1 $\mathrm{GHz}$ sound (Wikipedia), and an on-chip surface-acoustic-wave version reaches 5.5 $\mathrm{GHz}$ (Erdil et al.).
• Scope: The electro-optic modulators on the Electro-Optic Phase Modulator and Lithium Niobate Mach-Zehnder Modulator pages use volts; the AOM uses RF watts, roughly 1 to 10 $\mathrm{W}$ (RP Photonics), against drive voltages of 1 to 10 kilovolts for the electro-optic devices (Wikipedia). This page covers the acousto-optic effect, the bulk modulator and deflector, and the integrated version.

2. Working Principle

• Transducer and sound column: The RF drive makes the transducer launch a sound beam of height $H$ and length $L$ through the crystal. The crystal and the acoustic mode set the velocity: 4.2 $mm/\mu \mathrm{s}$ for TeO${}_2$ longitudinal, about 0.62 $mm/\mu \mathrm{s}$ for its slow shear mode, and 5.96 $mm/\mu \mathrm{s}$ for fused silica.
• Alignment: The light enters at the Bragg angle. The diffracted and undiffracted beams leave with a separation of $\lambda f/V$ (about 1 degree typically, RP Photonics; 3.2 degrees computed for the 1550 $\mathrm{nm}$ Isomet device); one is blocked and the other is used.
• Intensity control: The first-order power follows the RF amplitude through the sine-squared law. At the saturation power $P_s$ the efficiency peaks (above 85 per cent for the Isomet 1550 $\mathrm{nm}$ TeO${}_2$ device); at half that power it is about 0.80 of the peak (computed).
• Switching speed: The edge of the optical pulse is the sound front crossing the beam, so a 0.5 $\mathrm{mm}$ beam gives 77 $\mathrm{ns}$ and a 0.25 $\mathrm{mm}$ beam gives 39 $\mathrm{ns}$ in TeO${}_2$. A smaller beam is faster but diverges more.
• Q and aperture: A long transducer raises $Q$ and keeps the light in one order, but it needs a larger acoustic aperture and a narrower band. The transducer bandwidth must be twice the modulation bandwidth (Isomet).
• Deflector: Sweeping $f$ over a band $\Delta F$ steers the beam over $\lambda \Delta F/V$. The number of resolvable spots is the time-bandwidth product $\Delta FD/V$.
• Frequency shifter: The diffracted light is Doppler-shifted by the sound frequency. Two cascaded deflectors rotated by 90 degrees can cancel the shift in a Doppler-free deflector (Kim et al.).
• On-chip version: Interdigital transducers on an AlScN film launch surface waves that phase-modulate a silicon waveguide, producing sidebands at the RF frequency instead of a deflected beam (Erdil et al.).

3. Applications

• Pulse generation in lasers: Q-switching a Nd:YAG laser gives over 10 $\mathrm{kW}$ peak power in 40 to 200 $\mathrm{ns}$ pulses up to 100 $\mathrm{kHz}$; cavity dumping an Ar${}^{+}$ laser gives about 100 $\mathrm{W}$ in 15 $\mathrm{ns}$ pulses up to 1 $\mathrm{MHz}$; mode-locking a Ti:sapphire laser gives pulses under 100 femtoseconds with about 150 $\mathrm{kW}$ peak power (GH introduction).
• Intensity switches and pulse pickers: Over 85 per cent efficiency and a contrast of 1000:1 or better (Isomet M1250; RP Photonics) make the AOM a fast shutter and amplitude control for CW and pulsed lasers.
• Beam deflection and scanning: TeO${}_2$ deflectors resolve over 2000 spots at 20 spots per $\mu \mathrm{s}$, and GaP deflectors about 2000 spots at 1000 spots per $\mu \mathrm{s}$ (photonics.com; see the questions on these figures).
• Atom control: Two-colour, Doppler-free TeO${}_2$ deflectors address arrays of Rb atoms with over 100 resolvable spots for a 4 $\mathrm{mm}$ beam (Kim et al.).
• Frequency shifters and heterodyne systems: The $\pm f$ shift offsets the light by a known radio frequency; the on-chip device was characterised by detecting a beat note of roughly 7 $\mathrm{GHz}$ (Erdil et al.).
• Multi-line laser control: The NEOS PCAOM devices handle an optical bandwidth of 4.0 to 7.5 $\mathrm{nm}$ with one RF setting, at 92 to 93 per cent throughput for 120 to 130 $\mathrm{mW}$ of RF (GH introduction; PCAOM probably stands for polychromatic acousto-optic modulator, my reading).
• Fibre-coupled and infrared devices: Fibre AOMs cover 750 to 1750 $\mathrm{nm}$ with about 3 $\mathrm{dB}$ insertion loss (RP Photonics), and germanium modulators serve the 10.6 $\mu \mathrm{m}$ CO${}_2$ laser at 20 to 30 $\mathrm{W}$ of acoustic power (photonics.com).
• Integrated photonics: An AlScN-on-silicon device in a commercial foundry process works at 5.5 $\mathrm{GHz}$ with 112 to 294 $\mathrm{MHz}$ of bandwidth (Erdil et al.).

Bragg Diffraction and Frequency Shift

Name	Formula	Related Formulae	Variables/Constants
Acoustic wavelength and Bragg condition	$\Lambda =\frac{V}{f}$ $\sin {\theta }_B=\frac{\lambda f}{2V}$	$2\Lambda \sin {\theta }_{in}=\frac{\lambda }{n}$ $\sin {\theta }_B=n\sin {\theta }_{in}$	$\Lambda :acousticwavelength$ $V:acousticvelocity$ $f:acoustic(RF)frequency$ $\begin{array}{r}{\theta }_B:Bragganglebetweentheincidentbeam\\ andthesoundwavefronts,outsidethe\\ crystal\end{array}$ $\lambda :opticalwavelengthinvacuumorair$ $n:refractiveindex$ $\begin{array}{r}{\theta }_{in}:thesameangleinsidethecrystal,GH\\ introduction:\sin {\theta }_b=\lambda f_a/(2V_a)with\lambda in\\ air,Isometcrib:{\theta }_{Bragg}=\lambda f/(2V),Arizona\\ Eq.1:\sin \theta =m\lambda /(2\Lambda ),therefractionlink(n\\ cancelsoutsidethecrystal)isbackground\\ knowledge\end{array}$ $\begin{array}{r}(computedfor632.8\mathrm{nm},40\mathrm{MHz}and3800\\ m/s(Arizonanumbers):\Lambda =95\mu \mathrm{m}and{\theta }_B=\\ 3.33\mathrm{mrad}(0.191degrees);theArizona\\ textprints9.5\mu \mathrm{m},seethequestions)\end{array}$
Deflection angle and optical frequency shift	${\theta }_d=2{\theta }_B\approx \frac{\lambda f}{V}$ ${\nu }_1={\nu }_0\pm f$	$\sin {\theta }_m=\frac{m\lambda }{2\Lambda }$ ${\nu }_m={\nu }_0+mf$	${\theta }_d:anglebetweenthezeroandfirstorders$ ${\nu }_0:opticalfrequencyoftheincidentlight$ ${\nu }_1:frequencyofthefirstorder$ $\begin{array}{r}m:diffractionorder,thesignisplus\\ whentheacousticwavefrontsmovetoward\\ theincomingbeamandminuswhentheymove\\ away(Arizona),withastandingwavethe\\ shiftsarewholemultiplesoff\\ (Wikipedia),GHintroduction:the\\ deflectionangleistwicetheBraggangle,\\ Wikipedia:adjacentordersareseparated\\ bytwicetheBraggangle,ArizonaEq.3:\theta \\ \approx \lambda f/Vforthefirstorder,GH35210:17\\ \mathrm{mrad}at488\mathrm{nm}and210\mathrm{MHz}infused\\ silica,IsometM1250:\mathrm{Te}{\mathrm{O}}_2,150\mathrm{MHz},\\ acousticvelocity4.2mm/\mu \mathrm{s},RPPhotonics:\\ deflectionanglenormallyabout1degree\end{array}$ $\begin{array}{r}(computed:fortheGH35210with5960m/s\\ (assumed,thefused-silicalongitudinal\\ velocityoftheGHintroductiontable)\lambda \\ f/V=17.2\mathrm{mrad}againstthestated17;for\\ theIsometM1250with4.2mm/\mu \mathrm{s}(assumed,\\ longitudinalmode,myreadingoftheLin\\ thepartnumber)55.4\mathrm{mrad}(3.17degrees)\\ at1550\mathrm{nm}and150\mathrm{MHz};theshiftof150\\ \mathrm{MHz}on193.4\mathrm{THz}isafractionof7.8\times 10^{-7})\end{array}$
Raman-Nath and Bragg regimes: the Q parameter	$Q=\frac{2\pi \lambda L{f}^2}{n{V}^2}$	$Q=\frac{2\pi \lambda L}{n{\Lambda }^2}$ ${\eta }_m=J_m^2\left(\Delta \varphi \right)\left(Q\ll 1\right)$	$Q:Klein-Cookparameter$ $L:interactionlength(transducerlength)$ $n:refractiveindex$ $\begin{array}{r}{J}_m:Besselfunctionofthefirstkindand\\ orderm\end{array}$ $\begin{array}{r}\Delta \varphi :peakphaseexcursionofthelight,\\ Isometapplicationnote:Qabove10gives\\ strongBraggdiffraction,thedesign\\ targetis9to15andbelow5significant\\ lightgoesintohigherorders(theformula\\ wasstrippedfromthenote,sothe\\ expressionisthestandardone,background\\ knowledge),RPPhotonics:Raman-Nathfor\\ veryshortinteractionlengthswith\\ severalorders,Braggformostmodulators\end{array}$ $\begin{array}{r}(computedfor\mathrm{Te}{\mathrm{O}}_{2}at1550\mathrm{nm}and150\mathrm{MHz}\\ withn=2.18andV=4.2mm/\mu \mathrm{s}(nfrom\\ theIsomet\mathrm{Te}{\mathrm{O}}_{2}sheet,Vassumed\\ longitudinal):Q=5.70per\mathrm{mm}oflength,\\ soQ=10needs1.75\mathrm{mm}andQ=4\pi needs\\ 2.2\mathrm{mm};fusedsilicaat488\mathrm{nm}and210\mathrm{MHz}\\ withn=1.46and5960m/s:Q=2.61per\\ \mathrm{mm};Qvariesas{f}^2)\end{array}$

Efficiency, Figure of Merit and RF Power

Name	Formula	Related Formulae	Variables/Constants
Diffraction efficiency	$\eta ={\sin }^2\left[\frac{\pi }{\lambda }\sqrt{\frac{M_2{P}_{a}L}{2H}}\right]$	$\eta \approx \frac{{\pi }^2{M}_2{P}_{a}L}{2{\lambda }^{2}H}\left(\text{low power}\right)$ $\eta \left(P\right)={\eta }_{max}{\sin }^2\left(\frac{\pi }{2}\sqrt{\frac{P}{P_s}}\right)$	$\eta :first-orderdiffractionefficiency$ $M_2:acousto-opticfigureofmerit$ $P_a:acousticpower$ $L:interactionlength$ $\begin{array}{r}H:transducerheight(thewidthofthe\\ soundcolumn)\end{array}$ $P:appliedRFpower$ $P_s:optimumRFpower$ $\begin{array}{r}{\eta }_{max}:efficiencyat{P}_s,GHintroductioneq.\\ 2(garbledinthesource,reconstructedby\\ thesummaryinthisstandardform),\\ WikipediaandRPPhotonics:proportional\\ topoweratlowdriveandthensaturating,\\ thelastformwith{\eta }_{max}ismyownwayof\\ fittingdatasheetsthatquotelessthan\\ 100percentatfulldrive(myreading),\\ IsometM1250:morethan83percent(0.25\\ \mathrm{mm}beam)andmorethan85percent(0.5\mathrm{mm}\\ beam)at100percentpeakdrive,more\\ than62and65percentat50percent\\ drive\end{array}$ $\begin{array}{r}(computed:athalfthesaturationpower\\ {\sin }^2(\pi /\sqrt{8})=0.803,so0.83and0.85give\\ 66.6and68.2percent,whichmeetsthe62\\ and65percentofthedatasheet;\\ low-powerlimitfor\mathrm{Te}{\mathrm{O}}_{2}at1550\mathrm{nm}with\\ M_2=34.5\times 10^{-15}{\mathrm{s}}^3/kg,1\mathrm{W},L=2.17\mathrm{mm}and\\ H=0.5\mathrm{mm}(bothassumed):\eta =0.31)\end{array}$
Optimum RF power and wavelength scaling	$P_s=\frac{k{\lambda }^{2}H}{2M_{2}L}$	$P_s\propto {\lambda }^2$ $\frac{P_s\left(1550\right)}{P_s\left(633\right)}={\left(\frac{1550}{633}\right)}^2$	$\begin{array}{r}{P}_s:RFpoweratwhichthesine-squaredlaw\\ reachesitsfirstmaximum\end{array}$ $\begin{array}{r}k:transducerconversionloss(1.12\\ typical,Isometapplicationnote)\end{array}$ $\begin{array}{r}H:activeapertureheight,myderivation\\ setstheargumentofthesineto\pi /2,the\\ IsometcribprintsHsquaredinthis\\ expression(readthroughasummary),which\\ doesnothavetheunitofpower(seethe\\ questions),Wikipedia:theRFpowerfor\\ thesameefficiencyscaleswiththesquare\\ ofthewavelength,IsometM1250:peakRF\\ drive9.0\mathrm{W}(mean1.3\mathrm{W},2.0\mathrm{W}\\ duty-cycled),GH35210:6\mathrm{W}rated,more\\ than70percentat488\mathrm{nm},acoustic\\ aperture0.13\mathrm{mm}inair\end{array}$ $\begin{array}{r}(computedwithk=1.12andH=0.5\mathrm{mm}\\ (assumedtoequaltheactiveaperture):\\ 19.5\mathrm{W}per\mathrm{mm}oflengthfor\mathrm{Te}{\mathrm{O}}_2\\ longitudinalat1550\mathrm{nm},0.85\mathrm{W}per\mathrm{mm}for\\ slowshear(793\times 10^{-15}{\mathrm{s}}^3/kg)and448\mathrm{W}per\\ \mathrm{mm}forfusedsilica;9.0\mathrm{W}impliesL=\\ 2.17\mathrm{mm}andQ=12.3,whichisinsidethe\\ Isometdesignrangeof9to15andisa\\ consistencycheck,notareportedlength;\\ theGH35210at6\mathrm{W}withH=0.13\mathrm{mm}\\ (assumed)impliesL=1.9\mathrm{mm}andQ=5.0;\\ theratiobetween1550and633\mathrm{nm}is6.0)\end{array}$
Acousto-optic figure of merit	$M_2=\frac{n^6{p}^2}{\rho V^3}$	$p_{eff}=\sqrt{\frac{M_2\rho V^3}{n^6}}$	$M_2:figureofmerit(in{\mathrm{s}}^3/kg)$ $n:refractiveindex$ $p:effectivephotoelasticcoefficient$ $\rho :density$ $\begin{array}{r}V:acousticvelocityofthechosenmode,\\ thisexpressionisbackgroundknowledge\\ andisnotinthesourcesIread,values:\\ fusedsilicalongitudinal1.5,\mathrm{Te}{\mathrm{O}}_2\\ longitudinal34to34.5,\mathrm{Te}{\mathrm{O}}_{2}slowshear\\ 750(GHintroduction),793(photonics.com)\\ or1200(MT-Berlin),alltimes1\times 10^{-15}\\ {\mathrm{s}}^3/kg\end{array}$ $\begin{array}{r}(computed:fusedsilicawithn=1.46,p=\\ 0.27and\rho =2200kg/{\mathrm{m}}^3(pand\rho assumed)\\ gives1.52\times 10^{-15}{\mathrm{s}}^3/kgagainstthe\\ tabulated1.5;for\mathrm{Te}{\mathrm{O}}_{2}withn=2.25and\\ \rho =6000kg/{\mathrm{m}}^3(assumed)theeffective\\ coefficientis0.35forthelongitudinal\\ modeand0.091,0.093and0.114forthe\\ threeshearvalues,intheorder750,793\\ and1200)\end{array}$
Throughput, insertion loss and contrast	$T=\mathrm{DE}\left(1-\mathrm{IL}\right)$ $\mathrm{MT}=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$	$\mathrm{CR}=\frac{I_{max}}{I_{min}}$ ${\mathrm{CR}}_{dB}=10{\log }_{10}\mathrm{CR}$	$T:throughput$ $\begin{array}{r}DE:diffractionefficiency,definedasthe\\ ratioofthediffractedpowertothe\\ zero-orderpowerwiththeRFoff(Isomet\\ crib)\end{array}$ $\begin{array}{r}IL:staticinsertionloss,1minusthe\\ zero-orderpoweroverthelaserpowerwith\\ theRFoff(Isometcrib),typically3to7\\ percent(Isometapplicationnote)\end{array}$ $MT:modulationcontrast$ $CR:contrastratioofthediffractedbeam$ $\begin{array}{r}{I}_{max}and{I}_{min}:highestandlowest\\ first-orderintensities,GHintroduction\\ printsthecontrastratioas{I}_{min}/I_{max},\\ whichistheinverseoftheusagehere,\\ IsometM1250:contrastmorethan1000:1\\ minimumandmorethan2000:1typical,\\ staticinsertionlossunder3percent,RP\\ Photonics:contrastabout1000and\\ fibre-coupledlossabout3\mathrm{dB}\end{array}$ $\begin{array}{r}(computed:DE85percentwithILof3,5\\ and7percentgives82.5,80.8and79.1\\ percent;acontrastof1000:1is30\mathrm{dB}\\ withMT=0.998and2000:1is33\mathrm{dB})\end{array}$

Speed and Bandwidth

Name	Formula	Related Formulae	Variables/Constants
Rise time and modulation bandwidth	${\tau }_r=0.65\frac{d}{V}$ ${\mathrm{BW}}_{mod}=\frac{0.35}{{\tau }_r}$	${\tau }_r\approx 0.66\frac{d}{V}$ ${\tau }_r=1.5\frac{d}{V}$ $\mathrm{BW}=\frac{0.5}{{\tau }_r}$	${\tau }_r:opticalrisetime$ $\begin{array}{r}d:beamdiameteratthe1/esquared\\ intensitypointsinsidethecrystal\end{array}$ $\begin{array}{r}V:acousticvelocity,{\mathrm{BW}}_{mod}:modulation\\ bandwidth,theconventionsdisagree:0.65\\ and0.35intheIsometcrib,0.66inRP\\ Photonics(about100\mathrm{ns}per\mathrm{mm}in\\ practice),1.5inGHintroductioneq.3,\\ andabandwidthof0.5overtherisetime\\ intheArizonanumbers(seethe\\ questions),RPPhotonicsandGH\\ introduction:theswitchingtimeis\\ limitedbythetransittimeofthesound\\ acrossthebeam,IsometM1250:risetime\\ 39\mathrm{ns}(0.25\mathrm{mm}beam)and77\mathrm{ns}(0.5\mathrm{mm}\\ beam),GH35210:under15\mathrm{ns}fora0.1\mathrm{mm}\\ waist,Arizona:170\mathrm{ns}(1.0\mathrm{mm})and110\mathrm{ns}\\ (0.65\mathrm{mm})with2.9and4.5\mathrm{MHz}\end{array}$ $\begin{array}{r}(computedwith0.65:38.7and77.4\mathrm{ns}for\\ theIsometbeams(4.2mm/\mu \mathrm{s}),10.9\mathrm{ns}for\\ theGH35210(5.96mm/\mu \mathrm{s})and171and111\\ \mathrm{ns}fortheArizonabeams(3.8mm/\mu \mathrm{s});the\\ factor1.5wouldgive89,179,25,395and\\ 257\mathrm{ns};per\mathrm{mm}ofbeam:155ns/mmfor\mathrm{Te}{\mathrm{O}}_2\\ longitudinal,109ns/mmforfusedsilica\\ and1053ns/mmforslowshear)\end{array}$
Modulation transfer function	$\mathrm{MTF}\left(f_m\right)=\exp \left[-{\left(\frac{f_m}{1.2f_o}\right)}^2\right]$	$f_o=\frac{0.35}{{\tau }_r}$ $\mathrm{MTF}\left(f_o\right)=\exp \left[-{\left(\frac{1}{1.2}\right)}^2\right]=0.499$	$\begin{array}{r}\mathrm{MTF}:modulationtransferfunction,the\\ fractionofthemodulationdepththat\\ survivesatthemodulationfrequency{f}_m\end{array}$ $\begin{array}{r}{f}_o:characteristicfrequency,GH\\ introductioneq.4(reconstructedbythe\\ summary),thefactor1.2makestheMTF\\ equaltoonehalfat{f}_o(computed),Isomet\\ M1250:modulationbandwidthatanMTFof\\ 0.5is8.7\mathrm{MHz}(0.25\mathrm{mm}beam)and5\mathrm{MHz}\\ (0.5\mathrm{mm}beam)\end{array}$ $\begin{array}{r}(computed:f_o=9.05\mathrm{MHz}and4.52\mathrm{MHz}for\\ thetwoIsometbeamsagainst8.7and5.0\\ \mathrm{MHz}onthedatasheet,sothedata-sheet\\ ratiois1.74wherethebeamsizeswould\\ give2;withtheGHintroductionrisetime\\ of1.5d/Vthe0.25\mathrm{mm}beamwouldhave{f}_o\\ =3.9\mathrm{MHz}andanMTFof0.03at8.7\mathrm{MHz})\end{array}$
Transducer bandwidth and centre frequency	$B_T=2{\mathrm{BW}}_{mod}$ $f_c\ge 2B_T$	$\Delta {\theta }_{sep}=\frac{\lambda \Delta f}{V}$	$B_T:requiredtransducer(RF)bandwidth$ $f_c:centrefrequency$ $\Delta f:RFbandwidth$ $\begin{array}{r}\Delta {\theta }_{sep}:separationangleacrossthatband\\ (Isometcrib),Isometapplicationnote:\\ thetransducerbandwidthmustbetwicethe\\ modulationbandwidthandtheminimum\\ centrefrequencyistwicethebandwidth\\ (thenotedoesnotsaywhichbandwidth,\\ andIreaditasthetransducer\\ bandwidth),IsometM1250:centrefrequency\\ 150\mathrm{MHz},RFbandwidth100\mathrm{MHz}minimum,\\ mid-banddegeneracyfrequenciesofon-axis\\ \mathrm{Te}{\mathrm{O}}_{2}fromtheapplicationnote:83,54,\\ 39,25and16\mathrm{MHz}at442,532,633,800\\ and1064\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:forthe8.7\mathrm{MHz}modulation\\ bandwidth{B}_T=17.4\mathrm{MHz}and{f}_{c}isatleast\\ 34.8\mathrm{MHz},whichthe150\mathrm{MHz}centre\\ frequencyand100\mathrm{MHz}bandsatisfy;the\\ productsofthefivedegeneracy\\ frequenciesandthesquareofthe\\ wavelengthare16.2,15.3,15.6,16.0and\\ 18.1times1\times 10^6\mathrm{MHz}\mathrm{n}{\mathrm{m}}^2,sothefrequency\\ fallsaboutas1/{\lambda }^2(myreadingoffive\\ points);the100\mathrm{MHz}bandat1550\mathrm{nm}and\\ 4.2mm/\mu \mathrm{s}spans36.9\mathrm{mrad})\end{array}$

Deflectors and Resolution

Name	Formula	Related Formulae	Variables/Constants
Deflection range and resolvable spots	$\Delta \theta =\frac{\lambda \Delta F}{V}$ $N=\frac{\Delta \theta }{\Delta \Theta }=\frac{\Delta FD}{V}$	$\Delta \Theta =\frac{\lambda }{D}$ $N=\Delta F\Delta T$	$\Delta \theta :totalangularsweep$ $\Delta F:acousticbandwidthofthedeflector$ $\Delta \Theta :opticalresolution(onespot)$ $D:opticalaperture$ $\begin{array}{r}N:numberofresolvablespots,\Delta T=D/V:\\ aperturetime,GHintroductioneqs.9and\\ 10(thefirstwithitsdefinitionofthe\\ secondasthetime-bandwidthproduct),\\ Arizona:asweepfrom30to50\mathrm{MHz}gives\\ 3.3\mathrm{mrad}inthetextand3.2\mathrm{mrad}inits\\ table,time-bandwidthproduct100fora20\\ \mathrm{MHz}deflectionbandwidth\end{array}$ $\begin{array}{r}(computed:3.33\mathrm{mrad}fortheArizonasweep\\ (632.8\mathrm{nm},3800m/s);atime-bandwidth\\ productof100at20\mathrm{MHz}meansanaperture\\ timeof5\mu \mathrm{s}andanapertureof19\mathrm{mm}at\\ 3.8mm/\mu \mathrm{s},whichismyinferenceasthe\\ apertureisnotstated;theIsometM1250\\ bandof100\mathrm{MHz}is36.9\mathrm{mrad}anda0.5\mathrm{mm}\\ beamresolves3.1\mathrm{mrad},so11.9spotsif\\ thatmodulatorwereusedasadeflector,\\ whichitisnotdesignedfor)\end{array}$
Access time and time-bandwidth product	$T_a=\frac{D}{V}$ $\mathrm{TBP}=\Delta F{T}_a$	$\dot{N}=\frac{N}{T_a}=\Delta F$	$\begin{array}{r}{T}_a:accesstime,thetimeforthesoundto\\ crossthebeamortheaperture,\mathrm{TBP}:\\ time-bandwidthproduct(equaltoN)\end{array}$ $\begin{array}{r}\dot{N}:scanspeedinspotsperunittime,the\\ lastrelationismyownreadingofthe\\ photonics.comtable,inwhichthescan\\ speedinspotsper\mu \mathrm{s}equalsthebandwidth\\ in\mathrm{MHz}(\mathrm{Te}{\mathrm{O}}_{2}20,GaP1000),Kimetal.:\\ slow-shear\mathrm{Te}{\mathrm{O}}_2,crystalabout1\mathrm{cm}long,\\ 0.617mm/\mu \mathrm{s}assumedfromthephotonics.com\\ table,centrefrequencies108and191\mathrm{MHz},\\ designbandwidths21and34\mathrm{MHz},\\ time-bandwidthproducts336and560,\\ accesstime16\mu \mathrm{s}forthefullaperture,\\ morethan100spotsfora4\mathrm{mm}beam,\\ walk-offangle32degrees,measured\\ efficiency50to60percent\end{array}$ $\begin{array}{r}(computed:a10\mathrm{mm}aperturegives{T}_a=\\ 16.2\mu \mathrm{s}and21and34\mathrm{MHz}give340and\\ 551,againstthe336and560reported;a4\\ \mathrm{mm}beamgives6.5\mu \mathrm{s}and136spotsat21\\ \mathrm{MHz};thecascadeddesignwitha1.2\mathrm{mm}\\ beamgives1.9\mu \mathrm{s}againstitsquoted2\mu \mathrm{s})\end{array}$
Acoustic lens from a chirped drive	$F_L=\frac{V^2}{\lambda \frac{d{F}_a}{dt}}$	$\frac{d{F}_a}{dt}=\frac{\Delta F}{T_a}$	$F_L:focallengthoftheacousticlens$ $\begin{array}{r}\frac{d{F}_a}{dt}:chirprateoftheacoustic\\ frequency,GHintroductioneq.13(given\\ inthisstandardformbythesummary),\\ Photonics.com:achirped\mathrm{Te}{\mathrm{O}}_{2}deflector\\ holdsabout2000spotswithascanspeed\\ over20spotsper\mu \mathrm{s}\end{array}$ $\begin{array}{r}(computedforanassumedsweepof20\mathrm{MHz}\\ in5\mu \mathrm{s}at632.8\mathrm{nm}and3800m/s:5.7\mathrm{m})\end{array}$

Materials and On-Chip Devices

Name	Formula	Related Formulae	Variables/Constants
Material trade-off between power and speed	$\frac{P_{s,A}}{P_{s,B}}=\frac{M_{2,B}}{M_{2,A}}$ $\frac{{\tau }_{r,A}}{{\tau }_{r,B}}=\frac{V_B}{V_A}$	${\alpha }_a\left(f\right)={\alpha }_0{\left(\frac{f}{f_0}\right)}^2$	$\begin{array}{r}AandB:twomaterialsatthesame\\ wavelength,aperture,lengthandbeamsize\end{array}$ ${\alpha }_a:acousticattenuation$ $\begin{array}{r}{f}_0:frequencyatwhichitisgiven,thef\\ squaredlawisbackgroundknowledge,\\ tabulatedvalues(velocityinmm/\mu \mathrm{s}and{M}_2\\ in1\times 10^{-15}{\mathrm{s}}^3/kg):\mathrm{Te}{\mathrm{O}}_{2}longitudinal4.2to\\ 4.26and34to34.5,\mathrm{Te}{\mathrm{O}}_{2}slowshear0.617\\ to0.62and750to1200,fusedsilica\\ longitudinal5.96and1.5,fusedsilica\\ shear3.76and0.46,\mathrm{PbMo}{\mathrm{O}}_{4}3.63and36.3,\\ GaPlongitudinal6.32to6.65and29to\\ 30,GaAslongitudinal5.15and104,\\ germanium5.5and180,lithiumniobate6.6\\ and7(GHintroduction,photonics.com,\\ MT-Berlin),MT-Berlin:\mathrm{Te}{\mathrm{O}}_{2}acoustic\\ attenuation2.3\mathrm{dB}per\mathrm{cm}at500\mathrm{MHz},\\ refractiveindices2.26and2.41at633\\ \mathrm{nm},density6.00g/c{\mathrm{m}}^3,transparentfrom\\ 0.35to5.0\mu \mathrm{m},Isomet:nof2.180and\\ 2.320at1550\mathrm{nm},density5.99g/c{\mathrm{m}}^3,\\ transmission0.34to4.5\mu \mathrm{m}\end{array}$ $\begin{array}{r}(computed:slow-shear\mathrm{Te}{\mathrm{O}}_2(793\times 10^{-15})\\ needs23timeslessRFpowerthan\\ longitudinal\mathrm{Te}{\mathrm{O}}_2(34.5\times 10^{-15})butis6.8\\ timesslower,andfusedsilicaneeds23\\ timesmorepowerthanlongitudinal\mathrm{Te}{\mathrm{O}}_2;\\ withthefsquaredlawthe\mathrm{Te}{\mathrm{O}}_2\\ attenuationof2.3\mathrm{dB}per\mathrm{cm}at500\mathrm{MHz}\\ becomes0.21\mathrm{dB}per\mathrm{cm}at150\mathrm{MHz}(assumed\\ law,andthesourcedoesnotsaywhich\\ modethe2.3refersto))\end{array}$
On-chip surface-acoustic-wave modulator	${\eta }_{AOM}=20{\log }_{10}\left(\frac{\alpha \left(L\right)}{2}\right)$ $\alpha \left(L\right)=-\frac{2\pi \Delta n_{eff}L}{{\lambda }_0}$	$V_{\pi }=\frac{\pi }{\alpha \left(L\right)}{V}_{dev}$ $\mathrm{FOM}\mathrm{II}=\frac{\mathrm{BW}}{V_{\pi }L}$	$\begin{array}{r}{\eta }_{AOM}:powerofasidebandrelativetothe\\ carrier(in\mathrm{dB})\end{array}$ $\alpha (L):phaseexcursionofthelight$ $\begin{array}{r}\Delta n_{eff}:acousticallyinducedchangeof\\ effectiveindex\end{array}$ $L:transducerlength$ ${\lambda }_0:free-spacewavelength$ $V_{dev}:drivevoltageamplitude$ $V_{\pi }:half-wavevoltage$ $\begin{array}{r}{V}_{\pi }L:half-wavevoltagetimeslength,\mathrm{BW}:\\ bandwidth,Erdiletal.:340\mathrm{nm}AlScNfilm\\ with30percentscandiumonasilicon\\ waveguideinacommercialfoundryprocess,\\ interdigitaltransducerpitch700\mathrm{nm},\\ transducer150\mu \mathrm{m}long,resonanceabout\\ 5.5\mathrm{GHz},V_{\pi }Lof1.12\mathrm{V}\mathrm{cm}(20finger\\ pairs)and3.25\mathrm{V}\mathrm{cm}(5pairs),bandwidth\\ 112and294\mathrm{MHz},efficiency-18.3and\\ -19.4\mathrm{dB}(loweranduppersideband,20\\ pairs)and-24.2and-25.7\mathrm{dB}(5pairs)at\\ anRFpowerof23.2\mathrm{dBm}\end{array}$ $\begin{array}{r}(computed:-18.3\mathrm{dB}is\alpha =0.243\mathrm{rad}and\\ 1.48percentofthecarrierpower;FOMII\\ is100MHz/(\mathrm{V}\mathrm{cm})for20pairsand90\\ MHz/(\mathrm{V}\mathrm{cm})for5pairsfromthenumbers\\ above,andthepape{r}^{\prime }sownfigureswere\\ notread;the700\mathrm{nm}pitchand5.5\mathrm{GHz}\\ giveanacousticvelocityof3.85km/sif\\ thepitchistheacousticwavelengthand\\ 7.7km/sifthewavelengthistwicethe\\ pitch)\end{array}$

Important Questions

Ques: Is the rise time 0.65, 0.66 or 1.5 times $d/V$, and is the bandwidth $0.35/{\tau }_r$ or $0.5/{\tau }_r$?

The Isomet crib gives ${\tau }_r=0.65d/V$, RP Photonics 0.66, and the GH introduction 1.5 (eq. 3). The data sheets favour 0.65: the Isomet M1250 states 39 and 77 $\mathrm{ns}$ for 0.25 and 0.5 $\mathrm{mm}$ beams, and 0.65 $d/V$ with 4.2 $mm/\mu \mathrm{s}$ gives 38.7 and 77.4 $\mathrm{ns}$ (computed); the Arizona 170 and 110 $\mathrm{ns}$ for 1.0 and 0.65 $\mathrm{mm}$ compare with 171 and 111 $\mathrm{ns}$ at its 3800 $m/s$ (computed); the GH 35210 states under 15 $\mathrm{ns}$ for 0.1 $\mathrm{mm}$, where 0.65 gives 10.9 and 1.5 gives 25 $\mathrm{ns}$ (computed). I regard 0.65 as more likely for a 10 to 90 per cent rise time. The bandwidths also differ: the Arizona 2.9 and 4.5 $\mathrm{MHz}$ equal $0.5/{\tau }_r$ (computed), about 1.4 times the $0.35/{\tau }_r$ that Isomet uses, and Isomet’s own 8.7 and 5.0 $\mathrm{MHz}$ differ by 1.74 where the beam sizes would give 2 (computed). Reference: Isomet, acousto-optic modulator crib (isomet.com manual, file AN2600; fetch-tool summary, which reconstructed garbled formulas from the variable key); RP Photonics Encyclopedia, ‘Acousto-optic modulators’ (rp-photonics.com; fetch-tool summary); Gooch and Housego (GH) introduction to acousto-optics (IntroductionAO_GH.pdf at wiki.physics.wisc.edu; the title is my reading of the file name; the text is partly garbled and the summary reconstructed some equations; it carries NEOS device values); Isomet M1250-T150L-0.5 TeO2 modulator data sheet for 1550 $\mathrm{nm}$ (isomet.com PDF; fetch-tool summary); Gooch and Housego (GH) 35210 fused-silica modulator data sheet (G&H_AOM_35210-BR.pdf at wiki.physics.wisc.edu; fetch-tool summary); acousto-optic modulator notes from the Milster group, University of Arizona (wp.optics.arizona.edu PDF; fetch-tool summary; Greek symbols were lost in the extraction).

Ques: Is the figure of merit of slow-shear TeO${}_2$ 793, 1200 or 750 (times $1\times 10^{-15}$ ${\mathrm{s}}^3/kg$)?

The photonics.com table gives $793\times 10^{-15}$ ${\mathrm{s}}^3/kg$ at 0.617 $mm/\mu \mathrm{s}$, the GH introduction $750\times 10^{-15}$ at 0.62 $mm/\mu \mathrm{s}$, and MT-Berlin $1200\times 10^{-18}$ ${\mathrm{s}}^3/g$. Since one ${\mathrm{s}}^3/g$ is 1000 ${\mathrm{s}}^3/kg$, the MT-Berlin value is $1200\times 10^{-15}$ ${\mathrm{s}}^3/kg$ (computed). Its longitudinal value of $34.5\times 10^{-18}$ ${\mathrm{s}}^3/g$ equals the $34.5\times 10^{-15}$ ${\mathrm{s}}^3/kg$ of photonics.com after the same conversion, so the units do not explain the gap (my reading). With an assumed density of 6000 $kg/{\mathrm{m}}^3$ and n = 2.25, the three values imply effective photoelastic coefficients of 0.093 (793), 0.091 (750) and 0.114 (1200), against 0.35 for the longitudinal mode (computed). Two sources agree within 6 per cent and 1200 is 51 per cent above 793, so I regard 750 to 800 as more likely; a different crystal cut or polarisation could explain 1200, but none of the pages says so. Reference: photonics.com article ‘Acousto-Optic Devices’ (photonics.com/Articles/Acousto-Optic_Devices/a25112; fetch-tool summary; it reproduces tables whose column alignment was garbled, which the summary flagged); Gooch and Housego (GH) introduction to acousto-optics (IntroductionAO_GH.pdf at wiki.physics.wisc.edu; the title is my reading of the file name; the text is partly garbled and the summary reconstructed some equations; it carries NEOS device values); MT-Berlin TeO${}_2$ description page (mt-berlin.com; fetch-tool summary).

Ques: Is the deflection angle of an AOM the Bragg angle or twice the Bragg angle?

The Bragg condition $\sin {\theta }_B=\lambda f/(2V)$ (GH introduction; Isomet) gives the angle between the incident beam and the sound wavefronts, while the first order leaves at twice that angle, ${\theta }_d\approx \lambda f/V$ (GH introduction and Wikipedia: twice the Bragg angle; Arizona Eq. 3). The GH 35210 states 17 $\mathrm{mrad}$ at 488 $\mathrm{nm}$ and 210 $\mathrm{MHz}$ in fused silica. With 5960 $m/s$, $\lambda f/V$ is 17.2 $\mathrm{mrad}$ and the Bragg angle only 8.6 $\mathrm{mrad}$ (computed), so the stated value is the deflection. The Arizona sweep of 30 to 50 $\mathrm{MHz}$ changes the deflection by 3.33 $\mathrm{mrad}$ (computed), which coincides numerically with the Bragg angle at 40 $\mathrm{MHz}$ (3.33 $\mathrm{mrad}$, computed), so the two are easy to confuse. I regard the deflection reading as more likely for both. Reference: Gooch and Housego (GH) introduction to acousto-optics (IntroductionAO_GH.pdf at wiki.physics.wisc.edu; the title is my reading of the file name; the text is partly garbled and the summary reconstructed some equations; it carries NEOS device values); Wikipedia, ‘Acousto-optic modulator’ (en.wikipedia.org; fetch-tool summary; the equations were stripped from the text); Gooch and Housego (GH) 35210 fused-silica modulator data sheet (G&H_AOM_35210-BR.pdf at wiki.physics.wisc.edu; fetch-tool summary); acousto-optic modulator notes from the Milster group, University of Arizona (wp.optics.arizona.edu PDF; fetch-tool summary; Greek symbols were lost in the extraction); Isomet, acousto-optic modulator crib (isomet.com manual, file AN2600; fetch-tool summary, which reconstructed garbled formulas from the variable key).

Ques: Is the acoustic wavelength 9.5 or 95 microns for 3800 m/s at 40 MHz?

The Arizona notes print $\Lambda$ = 9.5 $\mu \mathrm{m}$ for sound at 3800 $m/s$ and 40 $\mathrm{MHz}$, but $\Lambda =V/f$ gives 95 $\mu \mathrm{m}$ (computed). The same notes give a deflection of about 6.7 $\mathrm{mrad}$ at 632.8 $\mathrm{nm}$, and $\lambda /\Lambda$ gives 6.66 $\mathrm{mrad}$ for 95 $\mu \mathrm{m}$ against 66.6 $\mathrm{mrad}$ (3.8 degrees) for 9.5 $\mu \mathrm{m}$ (computed). The printed 9.5 $\mu \mathrm{m}$ therefore contradicts the notes’ own deflection, and I regard it as a decimal slip (my reading). For comparison, longitudinal TeO${}_2$ at 4.2 $mm/\mu \mathrm{s}$ and 100 $\mathrm{MHz}$ has $\Lambda$ = 42 $\mu \mathrm{m}$ (computed), and RP Photonics quotes typical acoustic wavelengths of 10 to 100 $\mu \mathrm{m}$, a range that the corrected 95 $\mu \mathrm{m}$ still fits. The first-order Bragg angle at 95 $\mu \mathrm{m}$ is 3.3 $\mathrm{mrad}$, not the 33 $\mathrm{mrad}$ that a 9.5 $\mu \mathrm{m}$ spacing would give (computed). Reference: acousto-optic modulator notes from the Milster group, University of Arizona (wp.optics.arizona.edu PDF; fetch-tool summary; Greek symbols were lost in the extraction); RP Photonics Encyclopedia, ‘Acousto-optic modulators’ (rp-photonics.com; fetch-tool summary); Isomet M1250-T150L-0.5 TeO2 modulator data sheet for 1550 $\mathrm{nm}$ (isomet.com PDF; fetch-tool summary).

Ques: Is the optimum RF power proportional to H or to H squared?

The Isomet crib, read through a summary, prints $P_s=k{\lambda }^2{H}^2/(2L{M}_2)$. With $M_2$ in ${\mathrm{s}}^3/kg$, ${\lambda }^2{H}^2/(L{M}_2)$ has the unit watt metre, not watt (computed by dimensions). With $H$ to the first power the unit is watt, and setting the argument of the sine-squared law to $\pi /2$ gives $P_s={\lambda }^{2}H/(2M_{2}L)$ before the loss factor $k$ (my derivation). I therefore use $H$ once. As a check, the M1250 at 9.0 $\mathrm{W}$, 1550 $\mathrm{nm}$, longitudinal TeO${}_2$, $k$ = 1.12 and an assumed $H$ = 0.5 $\mathrm{mm}$ gives $L$ = 2.17 $\mathrm{mm}$ and $Q$ = 12.3, inside the 9 to 15 design range (computed). The squared form would give $L$ = 1.1 $\mu \mathrm{m}$ (computed), which is not a transducer. I regard the first-power form as more likely; a slip in the summary is also possible. Reference: Isomet, acousto-optic modulator crib (isomet.com manual, file AN2600; fetch-tool summary, which reconstructed garbled formulas from the variable key); Isomet application note on acousto-optic modulation (AO_Modulation_isomet.pdf at wiki.physics.wisc.edu; fetch-tool summary; the equations were stripped from the text, so only definitions and numbers are used); Isomet M1250-T150L-0.5 TeO2 modulator data sheet for 1550 $\mathrm{nm}$ (isomet.com PDF; fetch-tool summary); Wikipedia, ‘Acousto-optic modulator’ (en.wikipedia.org; fetch-tool summary; the equations were stripped from the text).

Ques: Can an AOM switch in a few nanoseconds?

RP Photonics reports a vendor claim of switching down to a few nanoseconds for fibre AOMs and a user answer calling nanosecond switching impossible, and the photonics.com table lists 4 to 6 $\mathrm{ns}$ for TeO${}_2$ modulators. With ${\tau }_r=0.65d/V$ and 4.2 $mm/\mu \mathrm{s}$, 4 to 6 $\mathrm{ns}$ needs a beam of 26 to 39 $\mu \mathrm{m}$ (computed), whereas 100 $\mu \mathrm{m}$ gives 15.5 $\mathrm{ns}$ and the Isomet 0.5 $\mathrm{mm}$ beam gives 77 $\mathrm{ns}$. The same table’s bandwidth-rise-time products range from 0.25 (glass) to 1.05 (GaP), so it mixes conventions (computed). A few nanoseconds is therefore possible only for a tightly focused beam, and I regard 10 to 20 $\mathrm{ns}$ (the GH 35210 starts at about 11 $\mathrm{ns}$) as the practical limit; the claim is not contradicted by transit-time physics, only by usable beam size (my reading). Reference: RP Photonics Encyclopedia, ‘Acousto-optic modulators’ (rp-photonics.com; fetch-tool summary); photonics.com article ‘Acousto-Optic Devices’ (photonics.com/Articles/Acousto-Optic_Devices/a25112; fetch-tool summary; it reproduces tables whose column alignment was garbled, which the summary flagged); Isomet M1250-T150L-0.5 TeO2 modulator data sheet for 1550 $\mathrm{nm}$ (isomet.com PDF; fetch-tool summary); Gooch and Housego (GH) 35210 fused-silica modulator data sheet (G&H_AOM_35210-BR.pdf at wiki.physics.wisc.edu; fetch-tool summary).

Ques: Which value of $Q$ marks the Bragg regime: 5, 10 or $4\pi$?

The Isomet application note says $Q$ above 10 gives strong Bragg diffraction, sets 9 to 15 as the design target, and warns that below 5 significant light goes into higher orders; the equation was stripped, so I use the standard $Q=2\pi \lambda L{f}^2/(n{V}^2)$ (background knowledge). From memory the Klein-Cook criterion is Bragg for $Q$ of $4\pi$ (12.6) or more, which I could not check. For TeO${}_2$ at 1550 $\mathrm{nm}$, 150 $\mathrm{MHz}$, n = 2.18 and 4.2 $mm/\mu \mathrm{s}$, $Q$ is 5.70 per $\mathrm{mm}$ (computed), so 10 needs 1.75 $\mathrm{mm}$ and $4\pi$ needs 2.2 $\mathrm{mm}$; halving the frequency divides $Q$ by four. Fused silica at 488 $\mathrm{nm}$ and 210 $\mathrm{MHz}$ gives 2.61 per $\mathrm{mm}$, so a cell about 2 $\mathrm{mm}$ long (assumed) has $Q$ near 5, which may explain the modest 70 per cent of the GH 35210. I regard 10 as the practical threshold. Reference: Isomet application note on acousto-optic modulation (AO_Modulation_isomet.pdf at wiki.physics.wisc.edu; fetch-tool summary; the equations were stripped from the text, so only definitions and numbers are used); Gooch and Housego (GH) 35210 fused-silica modulator data sheet (G&H_AOM_35210-BR.pdf at wiki.physics.wisc.edu; fetch-tool summary); RP Photonics Encyclopedia, ‘Acousto-optic modulators’ (rp-photonics.com; fetch-tool summary).

Ques: Is the access time of the Doppler-free deflector 4, 6.5 or 16 microseconds?

Kim et al. give 16 $\mu \mathrm{s}$ for the full aperture, and their abstract claims 4 $\mu \mathrm{s}$ or less for a 4 $\mathrm{mm}$ beam. At the slow-shear velocity of 0.617 $mm/\mu \mathrm{s}$ (assumed from the photonics.com table), a 4 $\mathrm{mm}$ beam takes 6.5 $\mu \mathrm{s}$ to cross, and 4 $\mu \mathrm{s}$ corresponds to 2.5 $\mathrm{mm}$ (computed). A 10 $\mathrm{mm}$ aperture gives 16.2 $\mu \mathrm{s}$ (the crystal is about 1 $\mathrm{cm}$ long), and 21 and 34 $\mathrm{MHz}$ then give time-bandwidth products of 340 and 551 against the reported 336 and 560 (computed). The cascaded design with a 1.2 $\mathrm{mm}$ beam gives 1.9 $\mu \mathrm{s}$ against its quoted 2 $\mu \mathrm{s}$ (computed). Simple transit time therefore reproduces 16 and 2 $\mu \mathrm{s}$ but not 4 $\mu \mathrm{s}$ for 4 $\mathrm{mm}$, and I regard 6.5 $\mu \mathrm{s}$ as the transit time and read the 4 $\mu \mathrm{s}$ as a different definition (my reading). Reference: Kim, McLeod, Saffman and Wagner, ‘Doppler-free, multi-wavelength acousto-optic deflector for two-photon addressing arrays of Rb atoms in a quantum information processor’, arXiv:0711.3244 (fetch-tool summary); photonics.com article ‘Acousto-Optic Devices’ (photonics.com/Articles/Acousto-Optic_Devices/a25112; fetch-tool summary; it reproduces tables whose column alignment was garbled, which the summary flagged).

Ques: Can a TeO${}_2$ deflector really resolve more than 2000 spots at 20 spots per microsecond?

The photonics.com deflector table lists TeO${}_2$ with a resolution over 2000 and a scan speed of 20 spots per $\mu \mathrm{s}$. Because spots per time equal the bandwidth (computed from $N=\Delta F{T}_a$), 20 spots per $\mu \mathrm{s}$ is 20 $\mathrm{MHz}$, and 2000 spots then need an access time of 100 $\mu \mathrm{s}$, an aperture of 420 $\mathrm{mm}$ in the longitudinal mode or 62 $\mathrm{mm}$ in slow shear (computed). The 1 $\mathrm{cm}$ slow-shear crystal of Kim et al. gives 16 $\mu \mathrm{s}$ and 340 to 560 spots. Reaching 2000 in 16 $\mu \mathrm{s}$ would need 123 $\mathrm{MHz}$ of bandwidth, more than the 108 $\mathrm{MHz}$ centre frequency (computed). The table lists the chirped type separately at about 2000, so I read the plain figure as a long-path or differently defined device (my reading) and regard 300 to 600 spots as the realistic value for a 1 $\mathrm{cm}$ crystal. Reference: photonics.com article ‘Acousto-Optic Devices’ (photonics.com/Articles/Acousto-Optic_Devices/a25112; fetch-tool summary; it reproduces tables whose column alignment was garbled, which the summary flagged); Kim, McLeod, Saffman and Wagner, ‘Doppler-free, multi-wavelength acousto-optic deflector for two-photon addressing arrays of Rb atoms in a quantum information processor’, arXiv:0711.3244 (fetch-tool summary).

Ques: How much does efficiency fall at half the RF drive?

For a sine-squared law that peaks at the saturation power, half the power gives ${\sin }^2(\pi /\sqrt{8})=0.803$ of the maximum (computed). The Isomet M1250 data sheet lists more than 83 per cent (0.25 $\mathrm{mm}$ beam) and more than 85 per cent (0.5 $\mathrm{mm}$ beam) at 100 per cent peak drive, and more than 62 and 65 per cent at 50 per cent. Multiplying gives 66.6 and 68.2 per cent (computed), which meets both, so I read 50 per cent drive as half the power. If it meant half the amplitude, a quarter of the power would give 0.50 of the maximum, 41 to 43 per cent (computed), which would fail the data sheet. Wikipedia and RP Photonics describe only a rise proportional to power that then saturates, so I regard the power reading as more likely. Reference: Isomet M1250-T150L-0.5 TeO2 modulator data sheet for 1550 $\mathrm{nm}$ (isomet.com PDF; fetch-tool summary); Wikipedia, ‘Acousto-optic modulator’ (en.wikipedia.org; fetch-tool summary; the equations were stripped from the text); RP Photonics Encyclopedia, ‘Acousto-optic modulators’ (rp-photonics.com; fetch-tool summary).

Ques: Does the on-chip AlScN modulator compete with bulk AOMs?

Not on efficiency. The sideband of the Penn device is -18.3 $\mathrm{dB}$ at 23.2 $\mathrm{dBm}$ of RF, which is 1.48 per cent of the carrier power (computed), against 85 per cent diffraction for the bulk Isomet device at 9.0 $\mathrm{W}$ peak, a factor of 57 (computed). Its strengths are frequency and integration: 5.5 $\mathrm{GHz}$ against 150 $\mathrm{MHz}$, 112 $\mathrm{MHz}$ of bandwidth (294 $\mathrm{MHz}$ with 5 finger pairs), and a 150 $\mu \mathrm{m}$ transducer in a commercial silicon-photonics process. The figure of merit BW over $V_{\pi }L$ is 100 $MHz/(\mathrm{V}\mathrm{cm})$ for 20 pairs and 90 for 5 pairs (computed; the paper’s own values were not read). The pitch of 700 $\mathrm{nm}$ and 5.5 $\mathrm{GHz}$ give 3.85 $km/s$ if the pitch is the acoustic wavelength and 7.7 $km/s$ if it is half of it (computed); I regard 3.85 as more plausible for a film on oxide (my reading). Reference: Erdil, Izhar, Deng, Tang, Idjadi, Ashtiani, Aflatouni and Olsson III (University of Pennsylvania), ‘Wideband, efficient AlScN-Si acousto-optic modulator in a commercially available silicon photonics process’, arXiv:2402.01127 (fetch-tool summary); Isomet M1250-T150L-0.5 TeO2 modulator data sheet for 1550 $\mathrm{nm}$ (isomet.com PDF; fetch-tool summary).

Related pages in this vault: Electro-Optic Phase Modulator, Lithium Niobate Mach-Zehnder Modulator, Magneto-Optic Modulator, Liquid-Crystal Spatial Light Modulator, MEMS Optical Modulator.