Polymer Electro-Optic Modulator

1. Theory

• Definition: An electro-optic polymer is a polymer that contains push-pull chromophores, dyes with a large first hyperpolarizability, which are aligned by a poling field near the glass transition so that the material has a Pockels coefficient $r_{33}$. A polymer modulator is a waveguide interferometer or resonator in which this material is the active layer (Taghavi et al.).
• Why polymers: Taghavi et al. give $n^3{r}_{33}$ of about 3100 pm/V in device for the best polymers against about 365 pm/V for LiNbO${}_3$ (computed: 10.4 or 8.5 times, depending on the lithium niobate value), and a dielectric constant that is lower than that of lithium niobate, so that the microwave and optical velocities are easier to match (the last point is background knowledge).
• Material families: Guest-host films (a chromophore dispersed in a host polymer, up to 95 pm/V at 25 wt %, Yang et al.), side-chain polymers (150 pm/V at 1310 nm, glass transition not given, Otomo), crosslinkable chromophore glasses (327 to 373 pm/V crosslinked, 386 pm/V uncrosslinked blend, glass transition 184 to 185 ${}^{\circ }\mathrm{C}$, Zeng et al.), and the materials of Table 2 of Taghavi et al. (HLD1/HLD2 460 pm/V with $T_g$ = 170 ${}^{\circ }\mathrm{C}$, BAH13 1100 pm/V, SEO125 125 pm/V with $T_g$ = 145 ${}^{\circ }\mathrm{C}$).
• Oriented-gas model: $r_{33}=2Nf(\omega )\beta ⟨{\cos }^3\theta ⟩/n^4$ with the order parameter $⟨{\cos }^3\theta ⟩=(\mu F/5k_{B}T)[1-L^2(W/k_{B}T)]$ (Yang et al.); the coefficient rises with the density, the dipole moment, $\beta$ and the poling field until intermolecular electrostatic interaction reduces the order (the W term).
• Device types: (a) all-polymer waveguides with coplanar electrodes (Lee et al.: fluorinated polyimide, $T_g$ = 202 ${}^{\circ }\mathrm{C}$, 12 $\mu \mathrm{m}$ gap, 75 V for the Mach-Zehnder device), (b) silicon-organic hybrid (SOH) slot waveguides in which the polymer fills a slot of 75 to 320 nm between two silicon rails, with strip-to-slot converters, (c) slow-light photonic-crystal slots (0.97 V at 300 $\mu \mathrm{m}$), and (d) plasmonic-organic hybrids, which are only quoted as data here (500 GHz and 0.06 $\mathrm{V}\mathrm{mm}$, Taghavi et al.).
• Reported results: SOH Mach-Zehnder modulators with 68 to 70 GHz, 100 to 200 Gbit/s, 0.9 to 1.8 V and 42 to 98 fJ/bit (Table 1 of Taghavi et al.); the silicon-polymer device of Lu et al. runs at 200 Gbit/s up to 110 ${}^{\circ }\mathrm{C}$ with $0.22\mathrm{dB}$/mm; an in-device $n^3{r}_{33}$ of 1021 pm/V.
• Limits: Poling needs fields of 80 to 100 V/$\mu \mathrm{m}$ and leaks by tunnelling above that; the activity relaxes with time and temperature; the RC time constant of the slot rails limits the bandwidth; and the photochemical stability is not covered by the sources read.

2. Working Principle

• Poling: The chromophores are mobile near $T_g$; a field of about 80 to 100 V/$\mu \mathrm{m}$ turns their dipoles, and cooling under the field freezes the order. The result is $r_{33}\propto N\beta \mu E_p/k_{B}T$ at weak order (Yang et al.).
• Index change: The poled film shows $\Delta n=\frac{1}{2}{n}^3{r}_{33}E$. In a slot of width $S_w$ the field is $V/S_w$, 3.1 V/$\mu \mathrm{m}$ for 1 V across 320 nm, and the index change is 20 times that of a 5 $\mu \mathrm{m}$ lithium niobate gap (computed).
• Interferometer: The two arms are poled and driven so that their phases change in opposite directions; with the slots in series the half-wave voltage is $V_{\pi }=\lambda S_w/(n^3{r}_{33}\sigma L)$ (arXiv 1403.0132; the series connection is background knowledge).
• Slow light: A photonic-crystal slot with a group index of about 20 shortens the device (13.4 times, computed) and raises the effective coefficient to 1190 pm/V, at the price of an optical range of 8 nm and $20\mathrm{dB}$ of loss.
• Bandwidth: The microwave and optical velocities limit the product $BW\times L$ to $1.9c/(\pi |\Delta n|)$, and the RC time of the slot limits the bandwidth of SOH devices (Taghavi et al.).
• Stability: The device is run below $T_g$; the relaxation of the order raises $V_{\pi }$ as $r_{33}(0)/r_{33}(t)$.

3. Applications

• Datacentre transceivers: Lu et al. report 200 Gbit/s on one lane with signal fidelity up to 110 ${}^{\circ }\mathrm{C}$, and Schwarzenberger et al. 200 Gbit/s PAM-4 and 100 Gbit/s on-off keying after 2300 h at 120 ${}^{\circ }\mathrm{C}$.
• Low-voltage drive: SOH devices reach 0.9 V (0.99 $\mathrm{V}\mathrm{mm}$) and the slot photonic-crystal device 0.97 V (0.292 $\mathrm{V}\mathrm{mm}$), against 1.4 V (28 $\mathrm{V}\mathrm{mm}$) for the best thin-film lithium niobate device of Wang et al.
• Compact silicon-photonic circuits: SOH devices with lengths of 0.5, 1.1 and 8 mm (computed from the loss and voltage-length checks above, Table 1 of Taghavi et al.).
• Energy-efficient links: 42 fJ/bit for the 200 Gbit/s device (Table 1 of Taghavi et al.), dynamic energy only.
• Harsh environments: High-temperature operation is the aim of the hybrid device of Lu et al. (they name harsh-environment uses in the title) and of the crosslinked polymers of Zeng et al.
• Large-area and low-cost waveguides: All-polymer modulators with coplanar electrodes can be made by spin-coating and etching on oxidized silicon (Lee et al.), at the price of 75 V half-wave voltage.

Material, Poling and Temperature

Name	Formula	Related Formulae	Variables/Constants
Oriented-gas model of the electro-optic coefficient	$r_{33}=\frac{2Nf(\omega )\beta ⟨{\cos }^3\theta ⟩}{n^4}$	$⟨{\cos }^3\theta ⟩=\frac{\mu F}{5k_{B}T}\left[1-L^2\left(\frac{W}{k_{B}T}\right)\right]$ $F=f(0)E_p$	$N:numberdensityofalignedchromophores$ $f(\omega ):local-fieldfactor$ $\beta :firsthyperpolarizability$ $⟨{\cos }^3\theta ⟩:acentricorderparameter$ $\mu :dipolemoment$ $E_p:polingfield$ $L:Langevinfunction$ $\begin{array}{r}W:intermolecularelectrostaticenergy,\\ withoutintermolecularinteractions{r}_{33}\\ shouldincreaselinearlywithdensity,\\ dipolemoment,\beta andpolingfield(Yanget\\ al.),whoreportupto95pm/Vfor\\ Y-shapedchromophoresat25wt\%in\\ guest-hostfilms(coronapoled10Kabove\\ theglasstransition),Zengetal.givea\\ numberdensityof4.74to4.97\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3}for\\ theirneatfilms,assumingamassdensity\\ of1g/c{m}^3\end{array}$ $k_B:Boltzmannconstant$ $\begin{array}{r}(computed:r_{33}/(E_p{r}_N)forthefirstfilm\\ ofZengetal.is3.03/4.97=0.61to\\ 3.03/4.74=0.64,andthepapergives0.51\\ to0.81foritsfilms)\end{array}$
Poling efficiency and the field it implies	${\eta }_p=\frac{r_{33}}{E_p}$	$E_p=\frac{r_{33}}{{\eta }_p}$	$\begin{array}{r}{\eta }_{p}in\mathrm{n}{\mathrm{m}}^2/{\mathrm{V}}^2(pm/VperV/\mu \mathrm{m}),typical\\ polingfieldsare80to100V/\mu \mathrm{m}(Zenget\\ al.;Leeetal.;theslotdeviceofarXiv\\ 1403.0132)andleakageabove100V/\mu \mathrm{m}is\\ dominatedbyFowler-Nordheimtunnelling\\ (Taghavietal.),Table2ofTaghaviet\\ al.(asextracted):HLD1/HLD2r_{33}=460\\ pm/V,fieldprintedas1300V/\mu \mathrm{m},{\eta }_p=\\ 2.3,BAH131100pm/V,about85V/\mu \mathrm{m},11.6,\\ YLD124/PSLD41285bulkand230indevice,\\ 250V/\mu \mathrm{m},0.92,YLD124(ref.30)230and\\ 30,150V/\mu \mathrm{m},0.23,PSLD4190and98,312\\ V/\mu \mathrm{m},0.26,DLD164133and150,280V/\mu \mathrm{m},\\ 0.54,Zengetal.give{\eta }_p=2.43to3.89\\ andfieldsofabout80(uncrosslinked)and\\ about100V/\mu \mathrm{m}(crosslinked)\end{array}$ $\begin{array}{r}(computed:theimpliedfieldsare200V/\mu \mathrm{m}\\ forHLD(printed1300),94.8forBAH13(85\\ gives12.9),250forYLD124/PSLD41only\\ with230(1.14with285),280forDLD164\\ onlywith150;forZengetal.QLD1to\\ QLD4give80.5to83.5V/\mu \mathrm{m},theblend\\ QLD1:QLD299.2andthecrosslinkedfilms\\ 121to128V/\mu \mathrm{m})\end{array}$
Material figure of merit	$Fo{M}_{mat}=n^3{r}_{33}$	$V_{\pi }L\propto \frac{1}{n^3{r}_{33}}$	$\begin{array}{r}Taghavietal.giveabout3100pm/Vin\\ devicefortheHLDmaterials(r_{33}=460\\ pm/V,n=1.89)againstabout365pm/Vfor\\ LiNb{O}_3(Section2.2),Table2alsolists\\ BAH13with{n}^3{r}_{33}=4700(bulk)and1317(in\\ device),AJ/CKL1390(bulk)and3183(in\\ device),Luetal.reportanin-device\\ n^3{r}_{33}of1021pm/Vforapolymerwitha\\ glasstransitionat172{}^{\circ }C\end{array}$ $\begin{array}{r}(computed:n_e^3{r}_{33}ofLiNb{O}_3=298.6pm/V\\ for2.13and30.9pm/V(334.3for2.2116),\\ so3100is10.4timesthat(8.5times\\ 365);1.89^3\times 460=3106and1.63^3\times 735=\\ 3183agreewiththetable,but1.85^3\times 1100\\ =6965andnot4700,and4700wouldmean\\ r_{33}=742pm/V;1021pm/Vwithanassumedn\\ =1.7means{r}_{33}=208pm/V)\end{array}$
Poling and operating temperature	$T_{pole}\approx T_g+(5\mathrm{to}10)\mathrm{K}$	$T_{op}<T_g$ $\Delta T_{margin}=T_g-T_{op}$	$\begin{array}{r}Thefilmsarepoledafewkelvinabovethe\\ glasstransitionandcooledunderthe\\ field,Taghavietal.describeheatingat\\ about10K/minto95{}^{\circ }C,holding10min\\ per10Kstepuptotheglasstransition\\ andcoolingrapidlywiththefieldon,\\ Zengetal.poleduncrosslinkedfilms5to\\ 10Kabove{T}_g(chromophores74to122{}^{\circ }C),\\ step-poledcrosslinkedfilmsat130,140\\ and150{}^{\circ }Candcrosslinkedat150{}^{\circ }Cfor\\ 30mintoreach184to185{}^{\circ }C,ariseof\\ about90K,Leeetal.poledat200{}^{\circ }Cfor\\ T_g=202{}^{\circ }C,Luetal.use{T}_g=172{}^{\circ }C,\\ beyond{T}_{g}thepolymerslosetheir\\ electro-opticproperties(Taghavietal.)\end{array}$ $\begin{array}{r}(computed:marginsof62K(172and110\\ {}^{\circ }C),52K(172and120{}^{\circ }C),87K(172and\\ 85{}^{\circ }C),100K(185and85{}^{\circ }C)and52K\\ (202and150{}^{\circ }C))\end{array}$

Modulator Efficiency

Name	Formula	Related Formulae	Variables/Constants
Modulation sensitivity and voltage-length product of a phase shifter	$S_p=\frac{\partial n_{eff}}{\partial V}=\frac{1}{2}\frac{n^3{r}_{33}\Gamma }{d}$	$V_{\pi }L=\frac{\lambda }{2S_p}=\frac{\lambda d}{n^3{r}_{33}\Gamma }$	$d:electrode(slot)gap$ $\begin{array}{r}\Gamma :overlapofthemodewiththepoled\\ polymer,thelossefficiencyis\alpha V_{\pi }L\\ (Taghavietal.,Eqs.inSection2.2),the\\ formisthesameasforlithiumniobateon\\ theElectro-OpticPhaseModulatorpageof\\ thisvault\end{array}$ $\begin{array}{r}(computed,n=1.63,r_{33}=89pm/V,\Gamma =\\ 0.33,d=320nmand1550nm:S_p=\\ 1.99\times 10^{-4}{\mathrm{V}}^{-1}and{V}_{\pi }L=3.90\mathrm{V}\mathrm{mm})\end{array}$
Slot-waveguide Mach-Zehnder modulator with series push-pull	$V_{\pi }=\frac{\lambda S_w}{n^3{r}_{33}\sigma L}$	$\frac{\Delta n}{V}=\frac{n^3{r}_{33}}{2S_w}$ $E=\frac{V}{S_w}$	$S_w:slotwidth$ $\begin{array}{r}\sigma :fractionofthemodeintheslot\\ (simulated0.33),thetwoslotsarein\\ seriesandeachseesV/2(background\\ knowledge),whichiswhythefactor2ofa\\ coplanarpush-pullmodulatorisabsent,\\ arXiv1403.0132:S_w=320nm,L=300\mu \mathrm{m},\\ n=1.63,V_{\pi }=0.97\pm 0.02V(0.973V),V_{\pi }\\ L=0.291\pm 0.006\mathrm{V}\mathrm{mm}(0.292inthetext)\\ forpush-pulland0.582\mathrm{V}\mathrm{mm}forasingle\\ arm,Taghavietal.:slotsasnarrowas75\\ nm,\Gamma typically10timesthatofa\\ non-slottedwaveguide,strip-to-slot\\ converter0.02\pm 0.02dB\end{array}$ $\begin{array}{r}(computed:\Delta n/Vwithout\sigma =6.02\times 10^{-4}for\\ 320nmand2.57\times 10^{-3}for75nm(r_{33}=89\\ pm/V),against2.99\times 10^{-5}fora5\mu \mathrm{m}gapof\\ lithiumniobate,thatis20and86times,\\ or6.7timeswith\sigma =0.33;1Vacross320\\ nmis3.1V/\mu \mathrm{m}andacross75nm13.3V/\mu \mathrm{m})\end{array}$
Slow-light enhancement of the effective coefficient	$r_{33,eff}=\frac{\lambda S_w}{n^3{V}_{\pi }\sigma L}$	$L=\frac{\lambda }{2\sigma n_g}\frac{n}{\Delta n},\Delta n=\frac{n^3{r}_{33}{V}_{\pi }}{2S_w}$	$\begin{array}{r}{n}_g:groupindex,about20.4(\pm 10\%)over\\ 8.2nmforthephotonic-crystalsection,\\ r_{33,eff}=1190pm/V,thein-devicevalue\\ withoutslowlightis89pm/V(51\%above\\ the59pm/Voftheearlierwork)andthe\\ bulkfilmvalue125pm/Vat1310nm,the\\ opticalrangeofstable{V}_{\pi }is8nm,the\\ insertionlossis20dBincluding6.5dB\\ perfacet,V_{\pi }wasmeasuredat100kHzand\\ theelectricalbandwidthisnotreported\\ (arXiv1403.0132)\end{array}$ $\begin{array}{r}(computed:Eq.1gives1189pm/Vand\\ 1189/89=13.4;Eq.2givesL=320\mu \mathrm{m},\\ 6.7\%above300\mu \mathrm{m};withoutslowlight{V}_{\pi }\\ L=3.90\mathrm{V}\mathrm{mm}and{V}_{\pi }=13.0Vfor300\mu \mathrm{m},\\ 13.4timesthemeasured0.973V;89/125=\\ 0.71;20-13=7dBoflossonchip)\end{array}$
Loss efficiency	$\alpha V_{\pi }L[\mathrm{V}\mathrm{dB}]$	$IL\approx \alpha L$	$\begin{array}{r}\alpha :propagationlossindB/mm,V_{\pi }Lin\\ \mathrm{V}\mathrm{mm},Table1ofTaghavietal.(unitsas\\ extracted):SOH3.16(0.22dB/mm,14.4\\ \mathrm{V}\mathrm{mm}),7.2(7.2dB/mm,0.99\mathrm{V}\mathrm{mm})and\\ 46.53(4.23dB/mm,11\mathrm{V}\mathrm{mm}),POH72,30\\ and49,LiNb{O}_{3}0.7(0.025dB/mm,28\mathrm{V}\mathrm{mm}),\\ 102(1.5dB/mm,67\mathrm{V}\mathrm{mm})and18.3(0.83\\ dB/mm,22\mathrm{V}\mathrm{mm}),thefirstSOHrowhas68\\ GHz,200Gbit/sand42fJ/bit,thesame\\ lossandrateasthedeviceofLuetal.\end{array}$ $\begin{array}{r}(computed:3.17,7.13and46.53forSOH,\\ 0.70,100.5(printed102)and18.26for\\ LiNb{O}_3;IL=\alpha Lgives1.76dBfor0.22\times 8\\ mm,7.92dBfor7.2\times 1.1mm(printed8)and\\ 2.1dBfor4.23\times 0.5mm(printed2))\end{array}$
Voltage-length ratio against lithium niobate	$R_V=\frac{{\left(V_{\pi }L\right)}_{LN}}{{\left(V_{\pi }L\right)}_{pol}}$	$R_V>1\Rightarrow \mathrm{polymer}\mathrm{more}\mathrm{efficient}$	$\begin{array}{r}Taghavietal.listlithiumniobatewith\\ 22to67\mathrm{V}\mathrm{mm}andpolymerhybridswith\\ 0.06to14.4\mathrm{V}\mathrm{mm},thethin-filmlithium\\ niobatedeviceofWangetal.has28\mathrm{V}\mathrm{mm}\\ (2.8\mathrm{V}\mathrm{cm},asontheLithiumNiobate\\ Mach-ZehnderModulatorpageofthis\\ vault),theall-polymerdeviceofLeeet\\ al.has75Vanda5mmelectrodeandno\\ stated{V}_{\pi }L,andTaghavietal.saythat\\ SOHhasabout10timesbettermodulation\\ efficiencythanfree-carrierdevices\end{array}$ $\begin{array}{r}(computedwith28\mathrm{V}\mathrm{mm}:SOH14.4\mathrm{V}\mathrm{mm}1.9\\ and0.99\mathrm{V}\mathrm{mm}28.3;slow-lightdevice\\ 0.292\mathrm{V}\mathrm{mm}95.9;POH0.192,0.07and0.06\\ \mathrm{V}\mathrm{mm}146,400and467;theall-polymer\\ deviceofLeeetal.,75Vtimes5mm=\\ 375\mathrm{V}\mathrm{mm},0.075,thatis13timesworse)\end{array}$

Bandwidth, Loss, Energy and Stability

Name	Formula	Related Formulae	Variables/Constants
Velocity-mismatch limit of the bandwidth	${\left(BW\times L\right)}_{max}\approx \frac{1.9c}{\pi |n_{RF}-n_o|}$	$B{W}_{max}=\frac{1.9c}{\pi L|n_{RF}-n_o|}$	$n_{RF}:microwaveindex$ $\begin{array}{r}{n}_o:optical(group)index,Taghavietal.\\ (Section2.3),thefactor1.9isthepoint\\ wheretheresponsefallstoonehalf(x=\\ 1.8955ofthe\sin x/xresponse),whichis\\ the3dBpointoftheopticalpowerand6\\ dBinthe20logconvention,whereasthe\\ LithiumNiobateMach-ZehnderModulator\\ pageusesx=1.392for3dB(20log),\\ Taghavietal.saythattheRCtime\\ constantlimitstheSOHdevicesandnamea\\ BaTi{O}_{3}slotlinewithcapacitivecoupling\\ asawayroundit,andthatpolymer\\ modulatorsreach{}^{\prime }exceeding100GH{z}^{\prime }\end{array}$ $\begin{array}{r}(computed,GHzmmfor|\Delta n|=0.1,0.2,\\ 0.5and1.0:1813,907,363and181with\\ 1.9,and1328,664,266and133withx=\\ 1.392)\end{array}$
Optical loss budget	$I{L}_{fibre}=2{\alpha }_c+{\alpha }_{p}L+I{L}_{exc}$	$I{L}_{chip}={\alpha }_{p}L+I{L}_{exc}$	$\begin{array}{r}Leeetal.(all-polymer,3.7\mu \mathrm{m}core,rib\\ 1.2\mu \mathrm{m}):coupling3.3dBperfacet(4\mu \mathrm{m}\\ core)and2.9dB(6\mu \mathrm{m}),propagation1.9\\ (TE)and2.2(TM)dB/cmfor4\mu \mathrm{m}and1.4\\ and1.6dB/cmfor6\mu \mathrm{m},polingaddsabout\\ 0.5dB/cm,Luetal.andTable1of\\ Taghavietal.:0.22dB/mmforthe\\ silicon-polymerhybrid,arXiv1403.0132:\\ 20dBintotalwith6.5dBperfacet\end{array}$ $\begin{array}{r}(computed:a20mmall-polymerdevicewith\\ a6\mu \mathrm{m}core,TE,2\times 2.9+1.4\times 2+0.5\times 2=9.6\\ dB,witha4\mu \mathrm{m}core,TM,12.0dB;a10mm\\ device7.7dB;thehybrid0.22\times 8mm=1.76\\ dBonchip)\end{array}$
Dynamic energy and power per bit	$E_{bit}\propto C{V}_{pp}^2$	$P_{dyn}=E_{bit}{R}_b$	$C:capacitanceofthemodulator$ $V_{pp}:driveswing$ $\begin{array}{r}{R}_b:bitrate,Taghavietal.listdynamic\\ energiesonly:42fJ/bit(SOH,200\\ Gbit/s),98fJ/bit(SOH,100Gbit/s),110\\ fJ/bit(POH,72Gbit/s)and14fJ/bitfor\\ thelithiumniobatedeviceofWangetal.\\ (210Gbit/s),laserwall-plugefficiency\\ isbelow10\%(Taghavietal.)\end{array}$ $\begin{array}{r}(computed:42fJ\times 200Gbit/s=8.4mW,98\\ fJ\times 100Gbit/s=9.8mW,110fJ\times 72\\ Gbit/s=7.9mWand14fJ\times 210Gbit/s=\\ 2.9mW)\end{array}$
Retention of electro-optic activity	$\frac{r_{33}(t)}{r_{33}(0)}=e^{-t/\tau }$	$\frac{V_{\pi }(t)}{V_{\pi }(0)}=\frac{r_{33}(0)}{r_{33}(t)}$ $\tau =-\frac{t}{\ln \left(r_{33}(t)/r_{33}(0)\right)}$	$\begin{array}{r}First-orderdecayisasimplificationof\\ mine(backgroundknowledge),reported\\ tests:Zengetal.morethan99.63\%\\ after500hat85{}^{\circ }C(crosslinkedfilms),\\ Otomomorethan78\%after500hat85{}^{\circ }C\\ (side-chainpolymer),Leeetal.morethan\\ 90\%after500hat150{}^{\circ }C(T_g=202{}^{\circ }C),\\ Taghavietal.nodegradationof{r}_{33}upto\\ 2000hat85{}^{\circ }Candamodelextrapolation\\ to25yearsat85{}^{\circ }C(Fig.5b),GR-468\\ passed,thepreprint:after300hof\\ burn-in{V}_{\pi }rosebylessthan15\%in2400\\ hat85{}^{\circ }Candtheactivitydecaysand\\ thenconvergestoaconstantlevel(T_g=\\ 172{}^{\circ }C),Schwarzenbergeretal.:2300hat\\ 120{}^{\circ }Candstill200Gbit/sPAM-4and100\\ Gbit/son-offkeying\end{array}$ $\begin{array}{r}(computed:\tau =134900h(Zeng),2012h\\ (Otomo),4746h(Lee,at150{}^{\circ }C)andat\\ least17170h(preprint,1/1.15);the\\ retentionafter25years(219000h)would\\ be0.197,5\times 10^{-48},9\times 10^{-21}and3\times 10^{-6},which\\ showsthatfirst-orderdecayisnota\\ forecast;25yearsis91times2400h\\ (0.27years))\end{array}$
In-device against film coefficient and the field in the core	$\kappa =\frac{r_{33,dev}}{r_{33,film}}$	$E_{core}={\kappa }_E\frac{V}{g}$	$\begin{array}{r}Leeetal.:film27.7pm/V(1310nm),8.42\\ pm/V(polarizationmodulator,12\mu \mathrm{m}gap,\\ 128V,5mm)and7.22pm/V(Mach-Zehnder,\\ 75V)obtainedfrom{V}_{\pi }with{r}_{33}=3r_{13}\\ assumed,polingvoltageupto1.8kV,core\\ fieldsof100,77and63V/\mu \mathrm{m}forgapsof\\ 12,16and20\mu \mathrm{m},andafinite-element\\ fieldof36V/\mu \mathrm{m}against50V/\mu \mathrm{m}fora\\ uniform1kVover20\mu \mathrm{m},biasdriftof7\\ \%inonehourat167V,Table2of\\ Taghavietal.listsbulkandin-device\\ values,e.g.YLD124(ref.30)230and30\\ pm/V,AJ/CKL190and735pm/V,BAH131100\\ and208pm/V\end{array}$ $\begin{array}{r}(computed:\kappa =0.304and0.261forLeeet\\ al.;{\kappa }_E=100/150=0.667,77/112.5=\\ 0.684,63/90=0.700and36/50=0.72;\\ Taghavietal.:YLD1240.13and0.81,\\ PSLD411.09,AJ/CKL18.2,BAH130.19,\\ SEO1259.5(withslowlight)or0.71\\ (without);theidealMach-Zehndervoltage\\ ofLeeetal.for27.7pm/V,g=12\mu \mathrm{m},5\\ mmand\Gamma =1is15.6Vagainst75V\\ measured,sotheoverlaptimesthepoling\\ factoris0.21;11.7Vofbiasdrift)\end{array}$

Important Questions

Ques: What is the best electro-optic coefficient: 460, 1100, 386, 327 or 150 pm/V?

The values come from different materials, conditions and definitions. Taghavi et al. give 460 pm/V for HLD1/HLD2 and 1100 pm/V for BAH13 (bulk), Zeng et al. 386 pm/V for an uncrosslinked blend and 327 to 373 pm/V for crosslinked films, Otomo 150 pm/V for a side-chain polymer and Yang et al. 95 pm/V for a guest-host film; the measurement methods differ between the papers and were not all read. The crosslinked films are lower than the blend because crosslinking hinders poling (Zeng et al.). In devices the number changes again, from 30 to 1190 pm/V. A coefficient is therefore meaningless without the material, the wavelength, the poling field and whether the value is film or in device. For a record I would quote the 1100 pm/V as unverified, because the table that gives it is not self-consistent (my reading). Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract); Zeng et al., ‘Electro-optic crosslinkable chromophores with ultrahigh electro-optic coefficients and long-term stability’, Chem. Sci. 13, 13393-13402 (2022); NICT abstract page (Otomo101111) of a 2010 J. Polym. Sci. Part A paper on a benzyloxy-modified FTC side-chain polymer; Yang et al., Y-shaped push-pull chromophores, J. Mater. Chem. C (accepted manuscript, RSC c5tc02764k; title not recorded).

Ques: Does BAH13 have 1100 pm/V or 742 pm/V?

The table of Taghavi et al. lists BAH13 with a bulk $r_{33}$ of 1100 pm/V, a refractive index of 1.85 and $n^3{r}_{33}=4700$ pm/V, and the text says ‘as high as 1100 pm/V’ with a poling efficiency of 11.6 $\mathrm{n}{\mathrm{m}}^2/{\mathrm{V}}^2$. The product $1.85^3\times 1100$ is 6965 pm/V, not 4700, and 4700 would mean 742 pm/V (computed). The in-device pair, 208 pm/V and 1317 pm/V, is consistent (computed). Either the index, the product or the coefficient is misprinted. The poling efficiency favours 1100: 11.6 with a field of 94.8 V/$\mu$m (computed), close to the 85 V/$\mu$m printed, which gives 12.9. The 742 pm/V would give 7.8 at 95 V/$\mu$m (computed). I take 1100 pm/V as the more likely, with the product 4700 being the misprint (my reading). The original paper was not read. Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract).

Ques: Is the poling field of HLD 1300 or 200 V/$\mu$m?

Table 2 of Taghavi et al. (as extracted) gives a poling field of 1300 V/$\mu$m for HLD1/HLD2 together with $r_{33}=460$ pm/V and a poling efficiency of 2.3 $\mathrm{n}{\mathrm{m}}^2/{\mathrm{V}}^2$. The efficiency is $r_{33}/E_p$, so 2.3 means a field of 200 V/$\mu$m (computed), and 1300 V/$\mu$m would give 0.35 (computed). A field of 1300 V/$\mu$m is also far above the 80 to 100 V/$\mu$m of the other sources and above the level where tunnelling leakage dominates (Taghavi et al.). The 1300 is probably a misprint or an extraction fault (my reading). The basis of the efficiency is also mixed: it agrees with the in-device value for YLD124/PSLD41 (230 over 250 gives 0.92) and DLD164 (150 over 280 gives 0.54), but with the bulk value for HLD and BAH13 (computed). Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract); Zeng et al., ‘Electro-optic crosslinkable chromophores with ultrahigh electro-optic coefficients and long-term stability’, Chem. Sci. 13, 13393-13402 (2022).

Ques: Which field produced 386 and 327 pm/V in Zeng et al.: 80, 100 or 99 to 128 V/$\mu$m?

Zeng et al. state field limits of about 80 V/$\mu$m for uncrosslinked and about 100 V/$\mu$m for crosslinked films. Their poling efficiency is $r_{33}/E_p$, and the four single chromophores give 80.5 to 83.5 V/$\mu$m, in agreement (computed). The uncrosslinked blend, 386 pm/V with 3.89 $\mathrm{n}{\mathrm{m}}^2/{\mathrm{V}}^2$, implies 99.2 V/$\mu$m, and the crosslinked films, 327, 311 and 373 pm/V with 2.68, 2.43 and 3.08, imply 122, 128 and 121 V/$\mu$m (computed), above the stated limit. The table may use the field at the end of step poling, or the stated limits may refer to a different stage, and the extract does not say which (my reading). The records of 327 and 373 pm/V are therefore tied to fields of 120 V/$\mu$m or more, and the claim that they are high ‘among crosslinked systems’ should be read with that in mind. Reference: Zeng et al., ‘Electro-optic crosslinkable chromophores with ultrahigh electro-optic coefficients and long-term stability’, Chem. Sci. 13, 13393-13402 (2022).

Ques: Is $n^3{r}_{33}$ of lithium niobate 365 or 299 pm/V, and is the polymer 8.5 or 10 times better?

Taghavi et al. state about 365 pm/V for lithium niobate and about 3100 pm/V for the polymer, a ratio of 8.5 (computed). With the 1550 nm values of Zhu et al., $n_e=2.13$ and $r_{33}=30.9$ pm/V, I get 298.6 pm/V, and 334.3 pm/V with the sodium-line index 2.2116 (computed), so the ratio is 10.4 or 9.3 (computed). The 365 would need an index of 2.28 for 30.9 pm/V (computed), a value of the ordinary index, not of the extraordinary one, or a higher coefficient. The 1550 nm index and the coefficient of 30.9 pm/V are the ones that match the modulators (my reading), so the polymer advantage is nearer 10 than 8.5, but this is only the material figure: the devices differ by far more or less, as the voltage-length ratios show. Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract); Zhu et al. (as read on the Lithium Niobate Mach-Zehnder Modulator page of this vault).

Ques: Why is the in-device coefficient of Lee et al. 7 to 8 pm/V when the film has 27.7 pm/V?

Lee et al. measure 27.7 pm/V on a thin film at 1310 nm and derive 8.42 pm/V (polarization modulator) and 7.22 pm/V (Mach-Zehnder) from the half-wave voltages, assuming $r_{33}=3r_{13}$. The ratios are 0.30 and 0.26 (computed). The field in the core is lower than $V/g$: their finite-element result is 36 V/$\mu$m against 50 V/$\mu$m for a uniform field, a ratio of 0.72, and the core fields of 100, 77 and 63 V/$\mu$m are 0.67 to 0.70 of the nominal values (computed). With 27.7 pm/V and full overlap the Mach-Zehnder voltage would be 15.6 V, against 75 V measured, so the product of overlap and poling factor is 0.21 (computed). The poling field and the overlap together explain most of it, but the extract does not separate the two (my reading). Reference: Lee et al., ‘Coplanar Electrode Polymer Modulators Incorporating Fluorinated Polyimide Backbone Electro-Optic Polymer’, Photonics 7(4), 100 (2020); Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract).

Ques: Is the effective coefficient of 1190 pm/V a material property?

No. The paper on the slot photonic-crystal device defines $r_{33,eff}=\lambda S_w/(n^3{V}_{\pi }\sigma L)$ with $V_{\pi }=0.973$ V, $S_w=320$ nm, $\sigma =0.33$ and $L=300\mu$m, which gives 1189 pm/V (computed). It contains the slow-light factor: the same device without slow light has 89 pm/V, 13.4 times less, and the film value of SEO125 is 125 pm/V, so the poled polymer in the slot reaches 0.71 of its film value (computed). Without slow light $V_{\pi }$ would be 13.0 V for this length (computed). The gain costs optical range (8 nm) and loss ($20\mathrm{dB}$ in total, including $6.5\mathrm{dB}$ per facet), and the electrical bandwidth was not reported, so 1190 pm/V is a device figure and not comparable to the 460 or 1100 pm/V of bulk materials (my reading). Reference: ‘Wide optical spectrum range, sub-volt, compact modulator …’ (arXiv 1403.0132; title truncated in the search result), electro-optic polymer SEO125 in a silicon slot photonic-crystal waveguide; Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract).

Ques: Do polymer modulators exceed 100 GHz: 68 to 70, 100 or 500 GHz?

Taghavi et al. say that polymer modulators reach ‘exceeding 100 GHz’, but their Table 1 lists 68 and 70 GHz for two SOH devices and 100 GHz for a third (a short, high-voltage one), and 500 GHz and 70 GHz for plasmonic-organic hybrids. The 100 GHz claim therefore rests on one SOH row and on the plasmonic ones (my reading). Their formula $BW\times L\le 1.9c/(\pi |\Delta n|)$ gives 363 GHz mm for a mismatch of 0.5 (computed), but it uses the 1.9 factor of the optical $3\mathrm{dB}$ point, not the 20 log electrical convention with 1.392 that gives 266 GHz mm (computed). Bandwidths from different papers can differ by this convention alone, a factor of 1.36 (computed), and the RC time constant is the more usual limit. Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract); Lithium Niobate Mach-Zehnder Modulator page of this vault.

Ques: Which voltage-length product is typical for a polymer modulator: 375, 14.4, 0.99, 0.292 or 0.06 V mm?

There is no typical value; it depends on the architecture. The all-polymer device of Lee et al. has 75 V and a 5 mm electrode, 375 V mm (computed from their values), the SOH rows of Taghavi et al. 14.4 and 0.99 V mm and the slot photonic-crystal device 0.292 V mm, while plasmonic-organic hybrids have 0.06 to 0.192 V mm. Relative to the 28 V mm of the thin-film lithium niobate device of Wang et al., these are 0.075, 1.9, 28, 96 and 146 to 467 times (computed). Taghavi et al. give 0.07 V mm in the text and 0.06 in the table for the same plasmonic device, and the claim of ‘about 10 times’ better efficiency than free-carrier devices cannot be checked from overlapping table ranges. The 0.99 V mm is the usual SOH figure (my reading). Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract); Lee et al., ‘Coplanar Electrode Polymer Modulators Incorporating Fluorinated Polyimide Backbone Electro-Optic Polymer’, Photonics 7(4), 100 (2020); ‘Wide optical spectrum range, sub-volt, compact modulator …’ (arXiv 1403.0132; title truncated in the search result), electro-optic polymer SEO125 in a silicon slot photonic-crystal waveguide; Wang et al., Nature 562, 101 (2018) (as read on the Lithium Niobate Mach-Zehnder Modulator page).

Ques: Will a polymer modulator last 25 years at 85 ${}^{\circ }$C?

Taghavi et al. cite a model extrapolation to 25 years of reliable operation at 85 ${}^{\circ }$C (their Fig. 5b), and the measured data are much shorter: no degradation of $r_{33}$ up to 2000 h, and in the preprint a rise of $V_{\pi }$ by less than 15 % over 2400 h after a burn-in of 300 h. 2400 h is 0.27 years, and 25 years is 91 times that (computed). A first-order decay fitted to other tests gives retentions after 25 years between 0.197 (Zeng et al., 99.63 % in 500 h) and $10^{-48}$ (Otomo, 78 % in 500 h), so the chemistry decides (computed). The preprint says that the activity decays and then converges to a constant level, which a first-order law does not describe. A 25-year lifetime is a model result for a particular crosslinked material and is not demonstrated (my reading). Reference: Taghavi et al., ‘Polymer modulators in silicon photonics: review and projections’, Nanophotonics (2022) (PMC11502111; Tables 1 and 2 read in a fetch-tool extract); ‘Demonstration of long-term thermally stable Silicon-Organic Hybrid Modulators at 85 deg C’ (Optica Open preprint, abstract page); Zeng et al., ‘Electro-optic crosslinkable chromophores with ultrahigh electro-optic coefficients and long-term stability’, Chem. Sci. 13, 13393-13402 (2022); NICT abstract page (Otomo101111) of a 2010 J. Polym. Sci. Part A paper on a benzyloxy-modified FTC side-chain polymer; Schwarzenberger et al., ECOC 2023 (We.A.4.2), ‘First Demonstration of a Silicon-Organic Hybrid (SOH) Modulator Based on a Long-Term-Stable Crosslinked Electro-Optic Material’ (abstract page of the KIT repository).

Ques: Is the series connection the reason that the slot-modulator voltage has no factor 2?

For a coplanar push-pull lithium niobate modulator the half-wave voltage is $\lambda g/(2n^3{r}_{33}\Gamma L)$, with the factor 2 from the two arms. The slot-modulator paper uses $V_{\pi }=\lambda S_w/(n^3{r}_{33}\sigma L)$ and states ‘push-pull’. The formula reproduces its own result: 1189 pm/V against 1190 pm/V printed (computed). A factor 2 would give 595 pm/V (computed). The two slots are connected in series, so each has $V/2$ and opposite field directions, which gives the same phase difference as a coplanar push-pull with $V$ across each gap (my reading, background knowledge). The consequence is that slot widths and lithium niobate gaps can be compared only after the topology is fixed: with the series connection the voltage is $V_{\pi }=\lambda S_w/(n^3{r}_{33}\sigma L)$ for the whole interferometer. Reference: ‘Wide optical spectrum range, sub-volt, compact modulator …’ (arXiv 1403.0132; title truncated in the search result), electro-optic polymer SEO125 in a silicon slot photonic-crystal waveguide; Lithium Niobate Mach-Zehnder Modulator page of this vault.

Related pages in this vault: Lithium Niobate Mach-Zehnder Modulator, Silicon Carrier-Depletion Mach-Zehnder Modulator, Electro-Optic Phase Modulator, Silicon MOS-Capacitor Modulator, Thermo-Optic Modulator, Lithium Niobate Electro-Optic Modulators.