Lithium Niobate Mach-Zehnder Modulator

1. Theory

• Definition: A lithium niobate Mach-Zehnder modulator (LN MZM) is an optical intensity modulator: the light in a LiNbO${}_3$ waveguide is split into two arms, a voltage on electrodes along the arms changes their refractive index by the Pockels effect, and the recombined light is bright or dark according to the phase difference. Wikipedia calls the Mach-Zehnder form ‘widely used as intensity modulator’ and says that the most established Pockels-type modulators are based on lithium niobate; a bulk Pockels cell needs hundreds or thousands of volts (Wikipedia).
• Crystal: LiNbO${}_3$ is a trigonal ferroelectric (space group R3c, point group 3m) with a band gap of 3.77 eV, a density of 4.30 $g/c{\mathrm{m}}^3$, a melting point of 1240 ${}^{\circ }\mathrm{C}$ and transparency from 350 to 5200 nm (Wikipedia; Zhu et al. give 350 nm to 5 $\mu \mathrm{m}$ and Hu et al. 0.4 to 5.5 $\mu \mathrm{m}$). The largest Pockels coefficient is $r_{33}\approx 31$ pm/V with the field along the z axis, and the indices at 1550 nm are $n_o=2.21$ and $n_e=2.13$ (Zhu et al.).
• Bulk devices: Titanium in-diffusion (index increase 0.001 to 0.04) or proton exchange makes weakly guiding waveguides with an index contrast of about 0.02, electrodes about 10 $\mu \mathrm{m}$ from the guide and lengths above 5 cm; Wang et al. quote $V_{\pi }\approx 3.5$ V and a bandwidth of about 35 GHz for such a modulator. A commercial 1550 nm device (Optilab IM-1550-40-PM) has $V_{\pi }=6.4$ V at 10 GHz and 8.3 V at 30 GHz, at least 30 GHz of bandwidth, an insertion loss of at most $5\mathrm{dB}$ and an extinction ratio of at least $25\mathrm{dB}$.
• Thin-film devices: In lithium niobate on insulator (LNOI) a film of 300 to 600 nm lies on silica, the index contrast is above 0.7, the ridge is etched (260 to 350 nm) and gold coplanar electrodes sit 5 $\mu \mathrm{m}$ apart; wafers of up to 6 inches are available (Zhu et al.). Wang et al. used a 300 nm slab on 4.7 $\mu \mathrm{m}$ of oxide with an 800 nm wide ridge; the thick-buffer device a 360 nm film, a 260 nm etch and 12 $\mu \mathrm{m}$ of silica; Kharel et al. and Light: AM a quartz handle.
• Electrodes: Coplanar waveguides in push-pull, capacitively loaded traveling-wave electrodes, micro-structured (segmented) electrodes that stop the current in the closest gap region, and superconducting niobium electrodes (Hu et al.; Kharel et al.). Matching the microwave index to the optical group index of about 2.2 is the design goal.
• Results to date: Wang et al. (2018): 1.4 V, 2.3 V and 4.4 V for 20, 10 and 5 mm with bandwidths of more than 45, 80 and 100 GHz and an extinction ratio of $30\mathrm{dB}$. Kharel et al. (2021): 1.3 V for 20 mm with $1.8\mathrm{dB}$ of roll-off at 50 GHz. Thick-buffer device (2024): 1.76 V for 7 mm and above 67 GHz. Light: AM (2025): 2.92 V and above 110 GHz on a 4-inch wafer.
• Figures of merit: $V_{\pi }L$ ($\mathrm{V}\mathrm{cm}$), the electro-optic $3\mathrm{dB}$ bandwidth, the ratio of the two, the static extinction ratio, the insertion loss and the energy per bit. The sources disagree on several of these, which the questions below treat.

2. Working Principle

• Pockels effect: An electric field along z changes the extraordinary index by $\Delta n_e=-\frac{1}{2}{n}_e^3{r}_{33}{E}_z$, and the phase of the light by $2\pi \Delta n_e\Gamma L/\lambda$. One volt over a 5 $\mu \mathrm{m}$ gap changes the index by $3\times 10^{-5}$ (computed).
• Push-pull interference: The two arms see fields of opposite sign, so their phases change by $\pm \Delta \phi$ and the output is $T=\frac{1}{2}[1+\cos (\pi V/V_{\pi }+{\theta }_b)]$. The half-wave voltage of the interferometer is half of that of one arm (Hu et al.), and a single drive on a coplanar line is enough (Kharel et al.).
• Traveling-wave drive: The microwave and the light travel together along the electrode. A velocity mismatch gives a $\sin x/x$ response, the skin-effect loss of the electrode grows as $\sqrt{f}$, and the response of a lossy matched line is $(1-e^{-\alpha L})/(\alpha L)$ (the last two are background knowledge with the loss coefficients of Kharel et al.).
• Trade-off: A narrower gap or a longer electrode lowers $V_{\pi }$ but raises the loss, so the voltage and the bandwidth are traded; the segmented electrode lowers the RF loss to 0.26 from 0.69 $\mathrm{dB}\mathrm{c}{\mathrm{m}}^{-1}\mathrm{GH}{\mathrm{z}}^{-1/2}$ at an unchanged simulated $V_{\pi }L$ of 2.1 $\mathrm{V}\mathrm{cm}$ (Table I of Kharel et al.).
• Bias: The transfer curve is cosine squared. The null point gives the largest extinction for on-off keying and the quadrature point the largest slope for analog links; temperature, photorefraction and material relaxation move the operating point (Agiltron).
• Drive: Binary, 4-level and 8-level signals at 70 to 190 GBd are produced by a drive of 0.06 to 2.5 V, down to the levels of a CMOS driver (Wang et al.; Light: AM).

3. Applications

• Optical communication: Wang et al. sent 70 Gbit/s with a 200 mV CMOS-level drive and 210 Gbit/s with 8-ASK; Light: AM modulated 190 GBd PAM-4 (380 Gbit/s) and expects 400 Gbit/s per lane; Hu et al. quote 1.96 Tbit/s for a dual-polarization IQ modulator, and Zhu et al. list monolithic IQ modulators with 48 and 67 GHz.
• Coherent transmitters: Two nested interferometers make the IQ modulator of Hu et al. and Zhu et al. (the nesting is background knowledge); the 64-QAM projection of Wang et al. is 1.2 Tbit/s per polarization at 200 Gbaud (computed).
• Analog and microwave photonic links: Biased at quadrature with a small-signal modulation index of 0.1 to 0.2 $V_{\pi }$ for low distortion (Agiltron); the third harmonic is 35.5 to $47.7\mathrm{dB}$ below the fundamental (computed).
• Pulse picking and gating: Agiltron describes pulse-picker drivers with amplitudes up to 5 V.
• Combs, frequency shifters and transducers: Hu et al. report electro-optic combs of 67 lines over 12.6 nm, frequency shifters of more than 99 % efficiency and a cavity transducer of 1.02 % efficiency.
• Volume manufacturing: A 4-inch wafer of Light: AM holds 12 chips with ten modulators each, with 6 of 12 meeting more than 110 GHz and $V_{\pi }<3$ V; the yield definition is ambiguous (question 5).
• Related page: The traveling-wave electrode and the velocity matching are the same ideas as in the traveling-wave detector of the Waveguide Photodiode page of this vault.

Pockels Effect and Half-Wave Voltage

Name	Formula	Related Formulae	Variables/Constants
Pockels effect in lithium niobate	$\Delta n_e=-\frac{1}{2}{n}_e^3{r}_{33}{E}_z$	$\Delta {\left(\frac{1}{n^2}\right)}_i={\sum }_{j=1}^3{r}_{ij}{E}_j$ $E_z=\frac{V}{g}$	$\begin{array}{r}{n}_e:extraordinaryindex,2.13at1550nm\\ (ordinary2.21)inTable1ofZhuetal.,\\ whileWikipediagives{n}_e=2.2116and{n}_o=\\ 2.3007atthesodiumDline\end{array}$ $\begin{array}{r}{r}_{33}:largestPockelscoefficient,30.9pm/V\\ inTable1ofZhuetal.(about31pm/Vin\\ theirtext,31pm/VinHuetal.,{}^{\prime }ofthe\\ orderof30pm/V^{\prime }inRPPhotonics),the\\ otherelementsare{r}_{13}=9.6,r_{22}=6.8and\\ r_{51}=32.6pm/V(Zhuetal.)\end{array}$ $E_z:fieldalongthecrystalzaxis$ $\begin{array}{r}g:electrodegap,theclampedrelative\\ permittivityis28alongzand44alongx\\ andy(Zhuetal.),theindex-ellipsoid\\ formisthestandardone(RPPhotonics)\end{array}$ $\begin{array}{r}(computed:n_e^3{r}_{33}=2.986\times 10^{-10}m/Vfor\\ 2.13and30.9pm/Vand3.343\times 10^{-10}m/V\\ (+11.9\%)fortheD-lineindex2.2116;1\\ Vacrossagapof5\mu \mathrm{m}(E_z=2\times 10^{5}V/m)\\ gives\Delta n_e=2.99\times 10^{-5};afieldof5.19\times 10^5\\ V/mshiftsthephaseof1cmofguideby\\ \pi )\end{array}$
Phase shift and half-wave voltage of a phase modulator	$\Delta \phi =\frac{2\pi }{\lambda }\Delta n_e\Gamma L=\pi \frac{V}{V_{\pi }}$	$V_{\pi }^{PM}=\frac{\lambda g}{n_e^3{r}_{33}\Gamma L}$ $E(t)=E_0{e}^{i{\omega }_{c}t}{e}^{i\beta \sin {\omega }_{m}t},\beta =\frac{\pi V_p}{V_{\pi }}$	$\lambda :vacuumwavelength$ $L:electrodelength$ $\begin{array}{r}\Gamma :overlapoftheopticalmodewiththe\\ appliedfield(backgroundknowledge,a\\ factorofmymodelthatthesourcesdonot\\ give)\end{array}$ $\begin{array}{r}{V}_{\pi }^{PM}:voltageforaphasechangeof\pi in\\ thephasemodulator\end{array}$ $\begin{array}{r}\beta :modulationindexofthephase-modulator\\ fieldofZhuetal.\end{array}$ $\begin{array}{r}{V}_p:driveamplitude,asmallergaplowers\\ V_{\pi }butmayaddabsorptioninthemetaland\\ alongerelectrodelowers{V}_{\pi }butlowers\\ thebandwidth(Huetal.)\end{array}$ $\begin{array}{r}(computed,\lambda =1550nm,g=5\mu \mathrm{m},\Gamma =1,\\ 2.13and30.9pm/V:V_{\pi }L=2.595\mathrm{V}\mathrm{cm}for\\ thephasemodulatorand2.319\mathrm{V}\mathrm{cm}with\\ theD-lineindex2.2116)\end{array}$
Half-wave voltage of the push-pull Mach-Zehnder modulator	$V_{\pi }^{MZM}=\frac{\lambda g}{2n_e^3{r}_{33}\Gamma L}=\frac{1}{2}{V}_{\pi }^{PM}$	$V_{\pi }L=\frac{\lambda g}{2n_e^3{r}_{33}\Gamma }$ ${\Gamma }_{eff}=\frac{(V_{\pi }L{)}_{ideal}}{(V_{\pi }L{)}_{meas}}$	$\begin{array}{r}Thetwoarmslieinthetwogapsofa\\ coplanarwaveguideandseefieldsof\\ oppositesign,sothephasedifferenceis\\ twicethatofonearmand{V}_{\pi }ishalfof\\ thatofaphasemodulatorofthesame\\ length(Huetal.;Agiltron)\end{array}$ $\begin{array}{r}{V}_{\pi }:voltagethatswitchestheoutputfrom\\ amaximumtothenearestminimum(Kharel\\ etal.)\end{array}$ $\begin{array}{r}g:gapbetweenthesignalandground\\ electrodes,5\mu \mathrm{m}inKhareletal.andin\\ thethick-bufferdeviceandnotstatedby\\ Wangetal.,measured{V}_{\pi }L:2.2,2.3and\\ 2.8\mathrm{V}\mathrm{cm}for5,10and20mm(Wangetal.;\\ the2.8is1.4Vtimes2cm,computed),\\ 2.1(simulated)and2.3\mathrm{V}\mathrm{cm}(Kharelet\\ al.),1.25\mathrm{V}\mathrm{cm}(thickbuffer,simulated\\ 1.2),2.92\mathrm{V}\mathrm{cm}(Light:AM)\end{array}$ $\begin{array}{r}(computed,\Gamma =1,g=5\mu \mathrm{m},1550nm:V_{\pi }L\\ =1.298\mathrm{V}\mathrm{cm}for2.13and30.9pm/Vand\\ 1.159\mathrm{V}\mathrm{cm}for2.2116;implied{\Gamma }_{eff}=0.618\\ (2.1),0.564(2.3),0.499(1.3V,20mm,\\ 2.6)and0.481(1.35V,2.7)forKharelet\\ al.and1.038(1.25)forthethick-buffer\\ device,whichwouldneed{r}_{33}=32.1pm/V\\ for\Gamma =1)\end{array}$
Wavelength scaling of the half-wave voltage	$V_{\pi }L\propto \frac{\lambda g}{\Gamma }$	$\frac{{\left(V_{\pi }L\right)}_{\lambda }}{{\left(V_{\pi }L\right)}_{1550}}\approx \frac{\lambda }{1550\mathrm{nm}}$	$\begin{array}{r}Theindexand{r}_{33}alsochangewith\\ wavelength(backgroundknowledge)andare\\ takenasconstanthere,Huetal.report\\ V_{\pi }Laslowas0.17\mathrm{V}\mathrm{cm}inthevisible\\ withabandwidthofabout35GHz(thetext\\ says450nmandthecaptionofFig.3says\\ 400nm),Agiltronsaysthatthecrystalis\\ unstablebelowabout750nmbecauseof\\ photo-inducedcharge\end{array}$ $\begin{array}{r}(computed:1310nm0.845,780nm0.503and\\ 450nm0.290ofthevalueat1550nm;2.1\\ \mathrm{V}\mathrm{cm}at1550nmwouldbecome0.61\mathrm{V}\mathrm{cm}at\\ 450nm,3.6timesmorethanthereported\\ 0.17\mathrm{V}\mathrm{cm},sothevisibledevicemustalso\\ haveasmallergaporalargeroverlap(my\\ reading))\end{array}$

Mach-Zehnder Transfer Function, Extinction Ratio and Bias

Name	Formula	Related Formulae	Variables/Constants
Intensity transfer function	$T=\frac{P_{out}}{P_{in}}=\frac{1}{2}\left[1+\cos \left(\frac{\pi V}{V_{\pi }}+{\theta }_b\right)\right]$	${\theta }_b=-\frac{\pi }{2}:T_Q=\frac{1}{2}\left[1+\sin \frac{\pi V}{V_{\pi }}\right]$ ${\frac{dT}{dV}|}_Q=\frac{\pi }{2V_{\pi }}$	$\begin{array}{r}T:powertransmissionofanideal\\ loss-freeinterferometerwith50:50\\ splitters(backgroundknowledge)\end{array}$ $\begin{array}{r}{\theta }_b:biasphasesetbytheDCvoltage,\\ Agiltrondescribesthetransfercurveas\\ cosinesquared,thenullpointasthe\\ pointofmaximumextinctionforon-off\\ keyingandthequadraturepoint(3dB)as\\ thepointofgreatestslopeforanalogand\\ advancedformats,the{V}_{\pi }ofthebiasport\\ oftheOptilabdeviceis5Vorlessat1\\ kHz\end{array}$ $\begin{array}{r}(computed:theslopeatquadrature\pi /(2V_{\pi })\\ is0.683{\mathrm{V}}^{-1}for2.3Vand1.122{\mathrm{V}}^{-1}for1.4\\ V)\end{array}$
Static extinction ratio and arm imbalance	$ER=10{\log }_{10}\frac{P_{max}}{P_{min}}=20{\log }_{10}\frac{1+\rho }{1-\rho }$	$\rho =\frac{10^{ER/20}-1}{10^{ER/20}+1}$ $\frac{P_2}{P_1}={\rho }^2$	$\begin{array}{r}\rho :ratioofthefieldamplitudesofthe\\ twoarmsattheoutput(background\\ knowledge)\end{array}$ $\begin{array}{r}{P}_2/P_1:powerimbalance,measuredstatic\\ extinctionratiosareabout30dB(Wanget\\ al.;thethick-bufferdevice),about20dB\\ (Khareletal.;Zhuetal.list10to40\\ dBforthedevicesoftheirtable),at\\ least25dB(atleast30dBfortheHE\\ version)fortheOptilabdevice\end{array}$ $\begin{array}{r}(computed:40dBneeds\rho =0.9802(0.174\\ dBimbalance),30dB\rho =0.9387(0.550\\ dB),25dB0.8935(0.978dB),20dB0.8182\\ (1.743dB)and10dB0.5195(5.688dB))\end{array}$
Dynamic extinction ratio from the drive swing	$E{R}_{dyn}=10{\log }_{10}\frac{1+\sin {\phi }_0}{1-\sin {\phi }_0},{\phi }_0=\frac{\pi V_p}{V_{\pi }}$	$V_p=\frac{V_{\pi }}{\pi }\arcsin \frac{10^{E{R}_{dyn}/10}-1}{10^{E{R}_{dyn}/10}+1}$	$Biasatquadrature$ $\begin{array}{r}{V}_p:peakdriveamplitude(background\\ knowledge),thethick-bufferdevicegives\\ adynamicextinctionratioof7.14dB(80\\ Gb/son-offkeying),4.42dB(112Gb/s),\\ 5.05dB(PAM-4at40GBd)and5.09dB(50\\ GBd)againstastatic30dB,andthe\\ Light:AMdevice3.212dBat190GBd\\ PAM-4,thedriveamplitudesarenotgiven\\ intheextractsread\end{array}$ $\begin{array}{r}(computed,implieddrivewith{V}_{\pi }=1.76V\\ (thickbuffer)and2.92V(Light:AM):\\ 7.14dB{V}_p=0.416V(peak-to-peak0.832\\ V,0.47V_{\pi }),4.42dB0.274V,5.05dB\\ 0.309V,5.09dB0.311Vand3.212dB\\ 0.336V(0.672Vpeak-to-peak,0.23V_{\pi }))\end{array}$
Harmonic distortion at quadrature	$\frac{P_{3f}}{P_f}={\left[\frac{J_3({\phi }_0)}{J_1({\phi }_0)}\right]}^2$	$T_Q=\frac{1}{2}+{\sum }_{k\mathrm{odd}}{J}_k({\phi }_0)\sin \left(k\omega t\right)$ ${\phi }_0=\frac{\pi V_p}{V_{\pi }}$	$\begin{array}{r}{P}_{3f}/P_f:ratiooftheelectricalpowersof\\ thethirdharmonicandofthefundamental\\ afterdetection(backgroundknowledge)\end{array}$ $\begin{array}{r}{J}_k:Besselfunctionofthefirstkind,the\\ secondharmoniciszeroatexact\\ quadrature,Agiltronadvisesa\\ small-signalmodulationindexof0.1to\\ 0.2V_{\pi }(myreading:V_p/V_{\pi })andadrive\\ wellbelow{V}_{\pi }\end{array}$ $\begin{array}{r}(computed,20{\log }_{10}(J_3/J_1):V_p/V_{\pi }=0.05-59.7\\ dB,0.10-47.7dB,0.15-40.6dB,0.20\\ -35.5dB,0.30-28.1dBand0.50-18.3dB;\\ at{V}_p=V_{\pi }thethirdharmonicislarger\\ thanthefundamental)\end{array}$
Bias error and loss of extinction ratio	$E{R}_{bias}=-10{\log }_{10}\left[{\sin }^2\left(\frac{\pi \Delta V}{2V_{\pi }}\right)\right]$	$\frac{\Delta V}{V_{\pi }}=\frac{2}{\pi }\arcsin \sqrt{10^{-ER/10}}$	$\begin{array}{r}\Delta V:erroroftheDCbiasatthenull\\ point(backgroundknowledge,ideal50:50\\ interferometer),Agiltronnames\\ temperature,photorefractiveeffectsand\\ materialrelaxationasthecausesofthe\\ driftoftheoperatingpoint,commercial\\ modulatorsuseamonitorphotodiodeanda\\ heaterwithaditherordither-free\\ feedbackloop,nodriftrateisgivenin\\ thesourcesread\end{array}$ $\begin{array}{r}(computed:\Delta V/V_{\pi }=0.005,0.01,0.02,\\ 0.05and0.10limittheextinctionratio\\ to42.1,36.1,30.1,22.1and16.1dB;30\\ dBneeds\Delta Vof2.01\%of{V}_{\pi },thatis\\ 0.101Vfor5V(Optilabbiasport),0.059\\ Vfor2.92Vand0.028Vfor1.4V;25dB\\ needs3.58\%and20dB6.38\%)\end{array}$

Traveling-Wave Electrode, Bandwidth and Loss

Name	Formula	Related Formulae	Variables/Constants
Velocity mismatch and bandwidth	$m(f)=\left|\frac{\sin x}{x}\right|,x=\frac{\pi fL|n_{RF}-n_g|}{c}$	$f_{3dB}=\frac{1.392c}{\pi L|n_{RF}-n_g|}$	$n_{RF}:microwaveindexoftheelectrode$ $\begin{array}{r}{n}_g:opticalgroupindex,theloss-free\\ form(backgroundknowledge),m(f)isthe\\ voltage-likeresponsethatfallsby3dB\\ (20logconvention)atx=1.392,the\\ groupindicesofthethin-filmdesignare\\ bothabout2.2andtheoxidethicknessof\\ 4.7\mu \mathrm{m}waschosenforthematch(Wanget\\ al.),themeasuredRFindexis2.23for\\ thesegmentedelectrodeandabout1.8for\\ theregularelectrodeonquartz(Kharelet\\ al.),therelativepermittivityofbulk\\ LiNb{O}_{3}is28atmicrowaveand5atoptical\\ frequencies(Wangetal.)\end{array}$ $\begin{array}{r}(computed:for{n}_g=2.2and{n}_{RF}=1.8,f_{3dB}\\ =16.6GHzfor20mmand33.2GHzfor10\\ mm;for{n}_{RF}=2.23(\Delta n=0.03)and20mm\\ 221GHz;TableIofKhareletal.gives13\\ GHzfortheregularquartzelectrode,with\\ themismatchassumed)\end{array}$
Half-wave voltage against frequency	$V_{\pi }(f)=V_{\pi }(0)10^{R_{EO}(f)/20}$	$V_{\pi ,3dB}\approx 1.4V_{\pi ,1GHz}$ $V_{\pi ,6dB}\approx 2V_{\pi ,1GHz}$	$\begin{array}{r}{R}_{EO}:electro-opticroll-offindBrelative\\ tolowfrequency(20logconvention,as\\ thefactorsaboveshow),Khareletal.\\ give{V}_{\pi }=1.35V(DC),1.3V(1GHz)and\\ 1.6V(50GHz,1.8dBroll-off)for20mm,\\ 2.3V(1GHz)and0.8dBat50GHzfor10\\ mm,thethick-bufferdevice1.3dBat67\\ GHz,theOptilabdevice{V}_{\pi }=6.4Vat10\\ GHzand8.3Vat30GHz\end{array}$ $\begin{array}{r}(computed:1.8dBmeansafactor1.230and\\ 1.6/1.3=1.231(1.185for1.35V);1.3dB\\ 1.161,so1.76Vbecomes2.04Vat67GHz;\\ 3dB1.413and6dB1.995,asinthetwo\\ rulesabove;theOptilabvoltagesriseby\\ 1.297(2.26dB)from10to30GHz)\end{array}$
Radio-frequency loss of the electrode	$A_{RF}(f)={\alpha }_0\sqrt{f}L$	$A_{RF}[\mathrm{dB}]=8.686\alpha L$	$\begin{array}{r}{\alpha }_0:skin-effectlosscoefficientin\\ \mathrm{dB}\mathrm{c}{\mathrm{m}}^{-1}\mathrm{GH}{\mathrm{z}}^{-1/2}(A_{RF}indBfortheRF\\ amplitude,\alpha innepersperlength),\\ measured0.69fortheregularelectrode\\ and0.26forthesegmentedelectrode\\ (Khareletal.),simulated(TableI)0.75,\\ 0.64and0.57forregularelectrodeson\\ siliconwithgapsof5,7and10\mu \mathrm{m},0.37\\ forregularonquartzand0.21for\\ segmentedonquartz,thelinearabsorption\\ onLN-on-quartzisatmost0.007dBc{m}^{-1}\\ GH{z}^{-1}(simulated0.003)\end{array}$ $\begin{array}{r}(computed,2cm:at50GHz9.76dB\\ (regular,0.69)and3.68dB(segmented,\\ 0.26),at100GHz13.8dBand5.20dB;the\\ RFlossreaches3dBat4.7GHzforthe\\ regularand33.3GHzforthesegmented\\ electrode)\end{array}$
Loss-limited electro-optic response	$m_{loss}(f)=\frac{1-e^{-\alpha (f)L}}{\alpha (f)L}$	$\alpha (f)L=\frac{A_{RF}(f)}{8.686}$ $f_{3dB}={\left(\frac{6.41}{{\alpha }_{0}L}\right)}^2$	$\begin{array}{r}Matchedvelocitiesandimpedance\\ (backgroundknowledge),f_{3dB}inGHzfor{\alpha }_0\\ Lintheunitsoftherowabove,6.41dB\\ isthelossatwhich{m}_{loss}=1/\sqrt{2}(\alpha L=\\ 0.738nepers),Khareletal.give1.8dB\\ roll-offat50GHzand0.8dBforthe10\\ mmdevice,anextrapolated\\ conductor-loss-limited3dBbandwidthof\\ 180GHzfor20mm(Zhuetal.list175\\ GHz)againstabout40GHzforaregular\\ thin-filmelectrode,andasimulated228\\ GHzinTableI\end{array}$ $\begin{array}{r}(computed:0.26and2cmgive1.77dBat\\ 50GHz(reported1.8),2.47dBat100GHz\\ and{f}_{3dB}=152GHz;0.21with2.0and2.1\\ cmgives233and212GHz(TableI228\\ GHz);regularsilicon0.75with2.1cm\\ gives16.6GHz(TableI27GHz)and\\ regularquartz0.37gives68GHz(TableI\\ 13GHz,mismatch);0.69and2cmgive4.43\\ dBat50GHz)\end{array}$
Impedance mismatch of the electrode	${\Gamma }_Z=\frac{Z_0-Z_s}{Z_0+Z_s}$	$\frac{V_{\pi }^{\prime }}{V_{\pi }}=\frac{Z_0+Z_s}{2Z_0}$ $S_{11}=20{\log }_{10}|{\Gamma }_Z|$	$\begin{array}{r}{Z}_0:impedanceoftheelectrode,Z_s=50\Omega \\ sourceandload,anelectrodeof40\Omega \\ raises{V}_{\pi }by12.5\%(Zhuetal.),the\\ lineimpedanceofthethin-filmdesignis\\ near50\Omega (Wangetal.)and{S}_{11}isbelow\\ -15dBforthe20mmand10mmdevicesof\\ Khareletal.(regularelectrodes39to48\\ \Omega intheirTableI)\end{array}$ $\begin{array}{r}(computed:40\Omega gives{\Gamma }_Z=-0.111,S_{11}=\\ -19.1dBandafactor1.125;45\Omega 1.056\\ (-25.6dB);48\Omega 1.021(-33.8dB);S_{11}=\\ -15dBcorrespondsto|{\Gamma }_Z|=0.178,that\\ is{Z}_{0}between34.9and71.6\Omega )\end{array}$
Bandwidth-voltage figure of merit	$FoM=\frac{f_{3dB}}{V_{\pi }}$	$f_{3dB}=FoM\cdot V_{\pi }$	$f_{3dB}:electro-optic3dBbandwidthinGHz$ $\begin{array}{r}{V}_{\pi }:half-wavevoltageinV,Wangetal.\\ quoteabout30GHz/Vfortheirdevicesand\\ about35GHzwith{V}_{\pi }of3.5Vfora\\ conventionalmodulator,thebandwidthof\\ thethick-bufferdevice(above67GHz)is\\ limitedbythenetworkanalyserand\\ extrapolatedtoabove120GHz,thatof\\ Light:AMisabove110GHzatchiplevel\\ and95GHzpackaged\end{array}$ $\begin{array}{r}(computed:Wang5,10and20mm22.7,34.8\\ and32.1GHz/V,conventionalLiNb{O}_{3}10.0;\\ thickbuffer38.1(67GHz)and68.2(120\\ GHz);Khareletal.133(180GHz)and130\\ (175GHz);Light:AM37.7(110GHz)and\\ 32.5(95GHz);Optilab4.7(30GHz,6.4V)\\ and3.6(30GHz,8.3V))\end{array}$

Optical Loss, Data Rate and Drive Power

Name	Formula	Related Formulae	Variables/Constants
Optical loss budget	$I{L}_{fibre}=2{\alpha }_c+{\alpha }_{p}L+I{L}_{exc}$	$I{L}_{chip}={\alpha }_{p}L+I{L}_{exc}$	${\alpha }_c:couplinglossperfacet$ ${\alpha }_p:waveguidepropagationloss$ $\begin{array}{r}I{L}_{exc}:excessloss,Wangetal.give5dB\\ perfacet,0.2dB/cm(Zhuetal.list0.3\\ dB/cm),anon-chiplossbelow0.5dBand\\ metalabsorptionbelow0.03dB/cm,Kharel\\ etal.give6dBperfacetwithgrating\\ couplers,below1dBonchipand13dB\\ fibretofibre(about4dBwithedge\\ couplersorbackreflectors)\end{array}$ $\begin{array}{r}Light:AMgives2dBperfacet,below0.4\\ dB/cm(0.2dB/cminoneblock)and6.5dB\\ forthepackaged10mmdevice,theOptilab\\ devicehasaninsertionlossof5dBor\\ less\end{array}$ $\begin{array}{r}(computed:Wang20mm0.2\times 2=0.4dB\\ (0.6dBwith0.3dB/cm,whichwouldexceed\\ 0.5dB)andabout10.5dBfibretofibre;\\ Kharel12+1=13dB;Light:AM2\times 2+0.4\times 1=\\ 4.4dB,so2.1dBofthe6.5dBisexcess\\ lossorpackaging)\end{array}$
Bit rate of a multilevel signal	$R_b=B{\log }_{2}M$	$R_{net}=\frac{R_b}{1+OH}$	$B:symbolrateinbaud$ $M:numberoflevels$ $\begin{array}{r}OH:overheadoftheforwarderror\\ correction,Wangetal.report70Gbit/s\\ on-offkeyingwithabiterrorratiobelow\\ 10^{-6}at200mVpeak-to-peak,100Gbaud\\ binary(biterrorratio7.8\times 10^{-5}),140\\ Gbit/s(4-ASK,70Gbaud,2.1\times 10^{-5})and210\\ Gbit/s(8-ASK,1.5\times 10^{-2},threshold1.9\times 10^{-2}\\ with20\%overhead)andprojectmorethan\\ 1Tbit/sperpolarizationwith64-QAMat\\ 200Gbaud\end{array}$ $\begin{array}{r}Light:AMreaches190GBdPAM-4and\\ expects400Gbit/sperlaneat200GBd,Hu\\ etal.quote1.96Tbit/sfora\\ dual-polarizationIQmodulator\end{array}$ $\begin{array}{r}(computed:70Gbaud4-ASK140and8-ASK\\ 210Gbit/s;190GBdPAM-4380Gbit/sand\\ 200GBd400Gbit/s;64-QAMat200Gbaud\\ 1.2Tbit/s;210Gbit/swith20\%overhead\\ is175Gbit/snet)\end{array}$
Electrical energy per bit	$E_{bit}=\frac{V_{rms}^2}{R_d{R}_b}$	$E_{bit}\propto V_{rms}^2$	$V_{rms}:root-mean-squaredrivevoltage$ $R_d:driverimpedance$ $\begin{array}{r}{R}_b:bitrate,Wangetal.give0.37fJ/bit\\ at70Gbit/swith{V}_{rms}=36mV(200mV\\ peak-to-peak),37aJ/bitforthe60mV\\ driveand14fJ/bitat210Gbit/swith{V}_{rms}\\ =360mV,R_{d}isintheirMethodsandis\\ notintheextractread\end{array}$ $\begin{array}{r}(computedwith{R}_d=50\Omega :0.370fJfor36\\ mVand70Gbit/s;33.3aJfor10.8mV(V_{rms}\\ scaledwiththe60mVdrive);12.3fJfor\\ 360mVand210Gbit/s;14fJwouldneed\\ 44.1\Omega or{V}_{rms}=383mV)\end{array}$
Drive power from the radio-frequency rating	$P=\frac{V_p^2}{2R}$	$P_{dBm}=10{\log }_{10}\frac{P}{1\mathrm{mW}}$	$R=50\Omega$ $\begin{array}{r}{V}_p:peakvoltageonasinewave,the\\ OptilabdeviceisratedatamaximumRF\\ inputpowerof27dBmandamaximum\\ opticalinputpowerof100mW,with{V}_{\pi }=\\ 6.4Vat10GHzand8.3Vat30GHz,its\\ RFporthasanimpedanceof50\Omega ,Agiltron\\ warnsagainstelectrodeheatingand\\ permanentdamageandgivesnolimit\end{array}$ $\begin{array}{r}(computed:27dBmis0.501W,V_p=7.08V\\ and14.2Vpeak-to-peak,2.21times6.4V\\ and1.70times8.3V;aswingof{V}_{\pi }\\ peak-to-peakneeds20.1dBmat10GHzand\\ 22.4dBmat30GHz)\end{array}$

Important Questions

Ques: Which constants belong in the half-wave voltage: $n_e=2.13$ or $2.2116$, and $r_{33}$ = 30, 30.9 or 31 pm/V?

Zhu et al. give $n_e=2.13$ and $n_o=2.21$ at 1550 nm in their Table 1, while Wikipedia gives $n_e=2.2116$ and $n_o=2.3007$ at the sodium D line. For a modulator at 1550 nm the first pair is the right one, because the Pockels change scales with $n_e^3$ and the index falls with wavelength (background knowledge). The D-line index would give $n_e^3=10.82$ instead of 9.66 and a half-wave voltage 10.7 % too low (computed). For $r_{33}$ Zhu et al. list 30.9 pm/V in the table and about 31 in the text, Hu et al. give 31, and RP Photonics ‘of the order of 30’. The 3 % spread changes $V_{\pi }$ by 3 % (computed), which is smaller than the uncertainty of the overlap, so 30.9 pm/V with 2.13 is the pair to use. Reference: Zhu et al., ‘Integrated photonics on thin-film lithium niobate’, Adv. Opt. Photon. 13, 242-352 (2021) (arXiv 2102.11956; Table 1 material data and Table 3 of reported modulators); Wikipedia, ‘Lithium niobate’; Hu et al., ‘Integrated electro-optics on thin-film lithium niobate’ (arXiv 2404.06398, 2024; review); RP Photonics, ‘Pockels effect’.

Ques: Why does a Pockels cell need hundreds of volts and an integrated modulator about one volt?

Wikipedia says that a Pockels cell usually needs hundreds or even thousands of volts. The formula $V_{\pi }=\lambda g/(n_e^3{r}_{33}\Gamma L)$ shows why: for phase modulation of one polarization in a crystal 1 mm thick and 10 mm long at 1550 nm it gives 519 V for $\Gamma =1$ (computed). An integrated push-pull modulator has a gap of $g=5\mu \mathrm{m}$, 200 times smaller, and a length of 20 mm, which together with the factor 2 of the push-pull layout gives 0.65 V for $\Gamma =1$ (computed). The measured values of 1.3 to 1.4 V (Kharel et al.; Wang et al.) are about twice this, so the overlap is near 0.5 (computed, my reading). The saving comes from the gap and the guided light, not from a better material. Reference: Wikipedia, ‘Electro-optic modulator’; Zhu et al., ‘Integrated photonics on thin-film lithium niobate’, Adv. Opt. Photon. 13, 242-352 (2021) (arXiv 2102.11956; Table 1 material data and Table 3 of reported modulators); Kharel et al., ‘Breaking voltage-bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes’, Optica 8, 357-363 (2021) (arXiv 2011.13422); Wang, Zhang, Chen, Bertrand, Shams-Ansari, Chandrasekhar, Winzer and Loncar, ‘Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages’, Nature 562, 101-104 (2018) (author PDF, read as a fetch-tool extract).

Ques: Two modulators have a 5 $\mu$m gap: why are their $V_{\pi }L$ values 1.25 and 2.1 to 2.7 V cm?

Kharel et al. use a 600 nm film, a 350 nm etch and 1 $\mu$m of silica cladding and give 2.1 V cm (simulated), 2.3 V cm (their Fig. 4b) and about 2.6 V cm from 1.3 V and 20 mm (computed). The thick-buffer device uses a 360 nm film, a 260 nm etch and 100 nm of cladding and gives 1.25 V cm. With $n_e=2.13$ and 30.9 pm/V the ideal value for a 5 $\mu$m gap is 1.298 V cm, so the overlap is 0.48 to 0.62 for the first and 1.04 for the second (computed). A value above 1 cannot be right for these constants, unless $r_{33}$ is 32.1 pm/V or the effective gap is smaller than 5 $\mu$m (my reading). The extracts do not explain the difference. Reference: Kharel et al., ‘Breaking voltage-bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes’, Optica 8, 357-363 (2021) (arXiv 2011.13422); ‘High-performance thin-film lithium niobate Mach-Zehnder modulator on thick silica buffering layer’ (arXiv 2412.12556, 2024); Zhu et al., ‘Integrated photonics on thin-film lithium niobate’, Adv. Opt. Photon. 13, 242-352 (2021) (arXiv 2102.11956; Table 1 material data and Table 3 of reported modulators).

Ques: Can one modulator have $V_{\pi }$ of 1 V and a bandwidth of 100 GHz?

Hu et al. describe thin-film Mach-Zehnder modulators with $V_{\pi }\approx 1$ V, an extinction ratio above $20\mathrm{dB}$ and a bandwidth above 100 GHz. The table of Zhu et al. does not show this in one device: 1.4 V goes with more than 45 GHz (20 mm) and 100 GHz with 4.4 V (5 mm) in Wang et al. The ratio of bandwidth to voltage is 22.7 to 34.8 GHz/V for Wang et al. and 38 GHz/V for the thick-buffer device, whose 67 GHz is limited by the analyser (computed). Only the extrapolated 175 to 180 GHz with 1.35 V of Kharel et al. gives 130 GHz/V (computed), and their sub-volt designs (0.65 V, above 200 GHz) are simulations. The summary is a statement about the platform, not about a measured device (my reading). Reference: Hu et al., ‘Integrated electro-optics on thin-film lithium niobate’ (arXiv 2404.06398, 2024; review); Zhu et al., ‘Integrated photonics on thin-film lithium niobate’, Adv. Opt. Photon. 13, 242-352 (2021) (arXiv 2102.11956; Table 1 material data and Table 3 of reported modulators); Wang, Zhang, Chen, Bertrand, Shams-Ansari, Chandrasekhar, Winzer and Loncar, ‘Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages’, Nature 562, 101-104 (2018) (author PDF, read as a fetch-tool extract); ‘High-performance thin-film lithium niobate Mach-Zehnder modulator on thick silica buffering layer’ (arXiv 2412.12556, 2024); Kharel et al., ‘Breaking voltage-bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes’, Optica 8, 357-363 (2021) (arXiv 2011.13422).

Ques: Is the Light: AM half-wave voltage of 2.92 V measured at 500 kHz or 1 GHz, and is the yield 50 %?

The text of Liu et al. says that $V_{\pi }$ was measured with a 500 kHz triangular sweep, while the label of their Fig. 4c says 1 GHz. Kharel et al. found 1.35 V at DC and 1.3 V at 1 GHz, a difference of 3.7 % (computed), so on either basis the value is between 2.8 and 2.9 V and the statement ‘below 3 V’ holds. The yield of 50 % is 6 of 12, but a chip holds ten modulators, so the count could be per chip or per device, and a per-device yield is not given. The criteria are more than 110 GHz and $V_{\pi }$ below 3 V. The packaged bandwidth of 95 GHz is below the criterion, so the 50 % applies to chip-level tests (my reading). Reference: Liu et al., ‘Volume manufacturing of thin-film lithium niobate modulators with bandwidth > 110 GHz based on 4-inch wafer with a quartz handle’, Light: Advanced Manufacturing 6, 47 (2025); Kharel et al., ‘Breaking voltage-bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes’, Optica 8, 357-363 (2021) (arXiv 2011.13422).

Ques: Which conventional lithium niobate numbers are right: 3.5 V and 35 GHz, 4.5 V, or 6.4 and 8.3 V?

Wang et al. quote $V_{\pi }\approx 3.5$ V and a bandwidth of about 35 GHz for conventional lithium niobate. The Optilab IM-1550-40-PM datasheet gives 6.4 V at 10 GHz, 8.3 V at 30 GHz and at least 30 GHz, but its feature list says ‘low drive voltage of 4.5 volt’, and the bias port is rated at 5 V or less at 1 kHz. The 4.5 V is most likely a low-frequency (DC) value, 0.70 of the 6.4 V (computed, my reading); the RF values are the ones that set the drive amplitude. The 3.5 V and 35 GHz belong to a good bulk device, not to a typical one, giving 10 GHz/V against 4.7 and 3.6 GHz/V for Optilab (computed). Reference: Wang, Zhang, Chen, Bertrand, Shams-Ansari, Chandrasekhar, Winzer and Loncar, ‘Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages’, Nature 562, 101-104 (2018) (author PDF, read as a fetch-tool extract); Optilab, datasheet IM-1550-40-PM (copy at sphotonics.ru).

Ques: Is the waveguide loss 0.2 or $0.3\mathrm{dB}$/cm, and is the record $0.027\mathrm{dB}$/cm, $0.03\mathrm{dB}$/cm or $1\mathrm{dB}$/m?

Wang et al. give a propagation loss of about $0.2\mathrm{dB}$/cm and an on-chip loss below $0.5\mathrm{dB}$, whereas Zhu et al. list $0.3\mathrm{dB}$/cm for the same row of their table. For the 20 mm device $0.2\mathrm{dB}$/cm gives $0.4\mathrm{dB}$, while $0.3\mathrm{dB}$/cm gives $0.6\mathrm{dB}$, above the stated bound (computed), so 0.2 is the more likely. Zhu et al. give a record of $0.027\mathrm{dB}$/cm, and Hu et al. give $0.03\mathrm{dB}$/cm in a figure caption but ‘about $1\mathrm{dB}$/m’ in the text. A resonator with $Q=10^7$ and $n_g=2.2$ has a loss of $3.9\mathrm{dB}$/m (computed, background knowledge), which supports $0.03\mathrm{dB}$/cm and suggests that the $1\mathrm{dB}$/m belongs to another quantity or reference (my reading). Reference: Wang, Zhang, Chen, Bertrand, Shams-Ansari, Chandrasekhar, Winzer and Loncar, ‘Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages’, Nature 562, 101-104 (2018) (author PDF, read as a fetch-tool extract); Zhu et al., ‘Integrated photonics on thin-film lithium niobate’, Adv. Opt. Photon. 13, 242-352 (2021) (arXiv 2102.11956; Table 1 material data and Table 3 of reported modulators); Hu et al., ‘Integrated electro-optics on thin-film lithium niobate’ (arXiv 2404.06398, 2024; review).

Ques: Does the energy per bit of 0.37 fJ, 37 aJ and 14 fJ follow from one formula?

Wang et al. estimate the energy as $V_{rms}^2/(R_d{R}_b)$. With $R_d=50\Omega$ the first case, 36 mV at 70 Gbit/s, gives exactly 0.370 fJ (computed). The other two cases, if the root-mean-square voltage scales with the drive, give 33.3 aJ for the 60 mV drive (10.8 mV) and 12.3 fJ for 360 mV at 210 Gbit/s (computed), against 37 aJ and 14 fJ printed, which are 11 to 14 % higher. An impedance of 44 $\Omega$ would fit the last two but not the first, and 14 fJ at 50 $\Omega$ needs 383 mV. The driver impedance is in the Methods and was not in the extract, so I cannot decide (my reading: rounding of the voltages). In all cases the energy is below 15 fJ/bit. Reference: Wang, Zhang, Chen, Bertrand, Shams-Ansari, Chandrasekhar, Winzer and Loncar, ‘Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages’, Nature 562, 101-104 (2018) (author PDF, read as a fetch-tool extract).

Ques: Why is the dynamic extinction ratio 3 to $7\mathrm{dB}$ when the static one is $30\mathrm{dB}$?

The static value is measured by sweeping the bias over the full transfer curve, while a data signal swings only a fraction of it around quadrature. The relation $E{R}_{dyn}=10{\log }_{10}[(1+\sin {\phi }_0)/(1-\sin {\phi }_0)]$ turns a measured ratio into a drive. For $7.14\mathrm{dB}$ and $V_{\pi }=1.76$ V it gives a peak voltage of 0.416 V, 0.47 $V_{\pi }$ peak-to-peak, and for $3.212\mathrm{dB}$ at 190 GBd with 2.92 V it gives 0.336 V, 0.23 $V_{\pi }$ (computed, drive implied, since the amplitudes are not in the extracts). Low swings are the choice of PAM-4 links and CMOS drivers, and a bandwidth roll-off at 112 Gb/s lowers the ratio further ($4.42\mathrm{dB}$). The low dynamic values therefore do not contradict the $30\mathrm{dB}$ static ratio (my reading). Reference: ‘High-performance thin-film lithium niobate Mach-Zehnder modulator on thick silica buffering layer’ (arXiv 2412.12556, 2024); Liu et al., ‘Volume manufacturing of thin-film lithium niobate modulators with bandwidth > 110 GHz based on 4-inch wafer with a quartz handle’, Light: Advanced Manufacturing 6, 47 (2025).

Ques: Is the extrapolated bandwidth 175, 180 or 228 GHz?

Zhu et al. list 175 GHz for the segmented 20 mm device, extrapolated from $1.8\mathrm{dB}$ of roll-off at 50 GHz. Kharel et al. say 180 GHz in the text and give a simulated 228 GHz in Table I for $V_{\pi }$ of 1 V. My loss-limited model, $(1-e^{-\alpha L})/(\alpha L)$ with the skin-effect loss of the electrode, gives $1.77\mathrm{dB}$ at 50 GHz for 0.26 $\mathrm{dB}\mathrm{c}{\mathrm{m}}^{-1}\mathrm{GH}{\mathrm{z}}^{-1/2}$ and 2 cm, which agrees with the $1.8\mathrm{dB}$ reported (computed). It gives 152 GHz for the measured coefficient and 212 to 233 GHz for the simulated 0.21 (computed). The 175 to 180 GHz lie between these and are the likely extrapolation for the fabricated device, while 228 GHz is the simulation. Nothing was measured above 50 GHz. Reference: Zhu et al., ‘Integrated photonics on thin-film lithium niobate’, Adv. Opt. Photon. 13, 242-352 (2021) (arXiv 2102.11956; Table 1 material data and Table 3 of reported modulators); Kharel et al., ‘Breaking voltage-bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes’, Optica 8, 357-363 (2021) (arXiv 2011.13422).

Ques: How accurately must the bias be held for an extinction ratio of $30\mathrm{dB}$?

A bias error $\Delta V$ at the null point leaves a transmission of ${\sin }^2(\pi \Delta V/2V_{\pi })$, so the extinction ratio is $-10{\log }_{10}$ of it (background knowledge). For $30\mathrm{dB}$ the error must stay within 2.01 % of $V_{\pi }$, which is 0.101 V for the 5 V bias port of the Optilab device, 0.059 V for 2.92 V and 0.028 V for 1.4 V; for $25\mathrm{dB}$ it is 3.58 % and for $20\mathrm{dB}$ 6.38 % (computed). Agiltron names temperature, photorefractive effects and material relaxation as causes of drift, and says that commercial devices use a monitor photodiode and a heater in a feedback loop. No drift rate is given in the sources read, so the interval between corrections cannot be estimated. Reference: Agiltron, ‘Instructions for LiNbO3 Modulator use’ (4 November 2025); Optilab, datasheet IM-1550-40-PM (copy at sphotonics.ru).

Ques: What does a modulation index of 0.1 to 0.2 $V_{\pi }$ mean for the distortion of an analog link?

Agiltron advises a small-signal modulation index of 0.1 to 0.2 $V_{\pi }$ and a drive well below $V_{\pi }$. If this means $V_p/V_{\pi }$, then at quadrature the second harmonic is zero and the third is $47.7\mathrm{dB}$ below the fundamental at 0.1 and $35.5\mathrm{dB}$ at 0.2 (computed, Bessel functions). If it meant a phase amplitude ${\phi }_0$ of 0.1 to 0.2 rad, the values would be 67.6 and $55.6\mathrm{dB}$ below, from the small-argument form $J_3/J_1\approx ({\phi }_0/2{)}^2/6$ (computed). The first reading is the more likely, because the phrase uses $V_{\pi }$ as the unit (my reading). At $V_p/V_{\pi }=0.5$ the third harmonic is only $18.3\mathrm{dB}$ down, so large-signal drive suits digital formats only. Reference: Agiltron, ‘Instructions for LiNbO3 Modulator use’ (4 November 2025).

Related pages in this vault: Electro-Optic Phase Modulator, Polymer Electro-Optic Modulator, Silicon Carrier-Depletion Mach-Zehnder Modulator, Indium Phosphide Quantum-Well Electroabsorption Modulator, Thermo-Optic Modulator, Waveguide Photodiode, Lithium Niobate Electro-Optic Modulators, Uni-Traveling-Carrier Photodiode (UTC-PD).