Thermo-Optic Modulator

1. Theory

• Definition: A thermo-optic modulator changes the refractive index of a waveguide by heating it. A resistive heater next to the waveguide, or the doped waveguide itself, raises the temperature by $\Delta T$, and the phase of the light changes by $\Delta \phi =(2\pi L/\lambda )(\partial n_{eff}/\partial T)\Delta T$ (Harris et al.; Liu et al.). Placed in one arm of a Mach-Zehnder interferometer, it makes a switch or an attenuator; placed on a ring, it tunes the resonance.
• Thermo-optic coefficient: Silicon has about $1.86\times 10^{-4}{\mathrm{K}}^{-1}$ near 1550 $\mathrm{nm}$, stoichiometric silicon nitride about $2.4\times 10^{-5}$, silica about $0.95\times 10^{-5}$, and polymer claddings about $-1\times 10^{-4}$ (Harris et al.; Yong et al.; Nejadriahi et al.; Zhong et al.). A $\pi$ shift needs $\Delta T_{\pi }=\lambda /(2L\partial n_{eff}/\partial T)$, which I compute as 68 $\mathrm{K}$ for a 61.6 $\mu \mathrm{m}$ silicon shifter and 14 $\mathrm{K}$ for a 300 $\mu \mathrm{m}$ one.
• Power and speed: The heater power is $P_{\pi }=G\Delta T_{\pi }$, with $G$ the thermal conductance to the surroundings, and the response is a single pole with time constant $\tau =H/G$. Making $G$ small saves power and slows the device, so the product $P_{\pi }\tau =H\Delta T_{\pi }$ is the figure of merit (Harris et al.; Liu et al.).
• Figures of merit: $P_{\pi }$, the rise time, $P_{\pi }\tau$, insertion loss, extinction ratio and thermal crosstalk. Typical values are a few to tens of $\mathrm{mW}$ per $\pi$ and tens of $\mathrm{kHz}$ (Liu et al.).
• Strengths and limits: Heating adds no free carriers, so a silicon thermo-optic shifter has almost no excess loss (0.23 $\mathrm{dB}$ in one device) and an unlimited phase range, but it is slow, dissipates milliwatts, and heats its neighbours.
• Scope: The carrier-based silicon modulators have their own pages, the ring resonance shift is treated on the Silicon Microring Modulator page, and the electro-optic devices are on the Electro-Optic Phase Modulator and Lithium Niobate Mach-Zehnder Modulator pages. This page covers the heated phase shifter and switch.

2. Working Principle

• Heater: The heater is titanium nitride, silicide, tungsten, platinum, aluminium, nickel-chromium or the doped silicon itself (Liu et al.; Zhong et al.; Nejadriahi et al.). Its resistance sets the voltage: $V_{\pi }=\sqrt{P_{\pi }R}$, 4.36 $\mathrm{V}$ across 769 $\Omega$ for 24.8 $\mathrm{mW}$ in Harris et al.
• Heat flow: The heat leaves through the oxide to the silicon substrate. Silicon conducts about 149 $W/(\mathrm{m}\mathrm{K})$ and oxide about two orders less, so oxide channels keep the heat near the waveguide (Harris et al.; Liu et al.).
• Phase: The mean temperature rise of the mode sets the phase, $\Delta \phi \propto L\Delta T$, so a longer heater needs a smaller $\Delta T$ and the same power per $\pi$ if the conductance grows in proportion to the length (background knowledge).
• Switching: In a balanced Mach-Zehnder switch the output is ${\cos }^2(\pi P/2P_{\pi })$. The phase is linear in power and so quadratic in voltage, and half of $P_{\pi }$ gives half of the light.
• Dynamics: The temperature follows the power with a single pole, so the rise time is $2.2\tau =0.35/f_{3dB}$ (background knowledge; the sources define $\tau$ in three different ways, see the questions).
• Efficiency measures: Undercutting the substrate, deep isolation trenches, folded or multi-pass waveguides and serpentine heaters all trade speed or loss for power. Examples are 0.49 $\mathrm{mW}$ with a 144 $\mu \mathrm{s}$ rise (undercut TiN), a three-fold cut in $P_{\pi }$ with trenches in silicon nitride, 1.7 $\mathrm{mW}$ with a 6.5 $\mu \mathrm{s}$ rise and 4.6 $\mathrm{dB}$ loss (7-pass silicon), and a 30 per cent cut from a serpentine heater (Liu et al.; Serkin et al.; Nejadriahi et al.).
• Arrays: Row-column addressing with pulse-amplitude or pulse-width drive, plus a matrix correction for crosstalk, controls 288 shifters on a 6 $\mathrm{m}{\mathrm{m}}^2$ die (Gurses et al.).

3. Applications

• Programmable photonic circuits: Meshes of Mach-Zehnder shifters need hundreds of tuned phases; Gurses et al. built a 288-element array with about 10 $\mathrm{mW}$ per $\pi$ and estimate more than 40,000 on a 30 $\mathrm{mm}$ by 30 $\mathrm{mm}$ reticle.
• Tuning and trimming: Thermo-optic shifters correct fabrication errors in interferometers and ring resonators (Nejadriahi et al.; see the Silicon Microring Modulator page).
• Optical switches and attenuators in silica: A polymer-clad silica Mach-Zehnder switch uses 48 $\mathrm{mW}$ with a 215 $\mu \mathrm{s}$ rise and a 10.1 $\mathrm{dB}$ extinction ratio (Zhong et al.).
• High-extinction filtering: An unbalanced silicon-nitride Mach-Zehnder reached an extinction ratio of about 80 $\mathrm{dB}$ (Serkin et al.).
• Visible-wavelength circuits: Nitride shifters at 445 to 561 $\mathrm{nm}$ need 0.78 to 1.20 $\mathrm{mW}$ per $\pi$ with an undercut oxide bridge (Yong et al.).
• Low-power, slow control: A folded, suspended shifter switches with 50 $\mu \mathrm{W}$ and a response of about a millisecond (Liu et al.).
• Fast heaters: Doped-silicon and silicide shifters reach rise times of 2.2 to 2.8 $\mu \mathrm{s}$ and so about 125 to 160 $\mathrm{kHz}$ (Liu et al., computed bandwidths).

Thermo-Optic Effect and Phase Shift

Name	Formula	Related Formulae	Variables/Constants
Thermo-optic coefficient	$\frac{dn}{dT}=9.45\times 10^{-5}+3.47\times 10^{-7}T-1.49\times 10^{-10}{T}^2$	$\frac{\partial n_{eff}}{\partial T}={\Gamma }_{core}{\left(\frac{dn}{dT}\right)}_{core}+{\Gamma }_{clad}{\left(\frac{dn}{dT}\right)}_{clad}$	$n:refractiveindex$ $\begin{array}{r}T:temperaturein\mathrm{K},thepolynomialisfor\\ siliconintheCbandfrom300to600\mathrm{K}\\ (Liuetal.)\end{array}$ $n_{eff}:effectiveindexofthewaveguidemode$ $\begin{array}{r}{\Gamma }_{core}and{\Gamma }_{clad}:fractionsofthemodepower\\ inthecoreandinthecladding\\ (Nejadriahietal.2020,Eq.2,written\\ forsilicaandthenitridecore),values\\ in1/\mathrm{K}near1550\mathrm{nm}:silicon1.86\times 10^{-4}\\ (Harris,Nejadriahi)orabout1.8\times 10^{-4}\\ (Liu),stoichiometric\mathrm{S}{\mathrm{i}}_3{\mathrm{N}}_{4}2.45\times 10^{-5}\\ (Yong,Nejadriahi2020),about2\times 10^{-5}\\ (Nejadriahi2021)or2.51\times 10^{-5}(Serkin),\\ silicon-richnitride1.65\times 10^{-4}(index3.1),\\ 5.85\times 10^{-5}(2.25)and1.82\times 10^{-5}(1.92)\\ (Nejadriahi2020),silica0.95\times 10^{-5}(Yong,\\ measuredintheCband)or9\times 10^{-6}\\ (Nejadriahi),polymercladdingabout\\ -1\times 10^{-4}(Zhong)\end{array}$ $\begin{array}{r}(computed:thepolynomialgives1.852\times 10^{-4}\\ at300\mathrm{K},2.095\times 10^{-4}at400\mathrm{K}and2.491\times 10^{-4}\\ at600\mathrm{K};forthevisible-lightnitride\\ shifterofYongetal.\Delta n_{eff}=1.94\times 10^{-4}\\ forariseof8.9\mathrm{K}is\partial n_{eff}/\partial T=\\ 2.18\times 10^{-5}per\mathrm{K},whichwith2.45\times 10^{-5}for\\ thecoreand0.95\times 10^{-5}forsilicaputs0.82\\ ofthemodeinthecoreifthetwo\\ fractionsaddtoone(assumed))\end{array}$
Thermally induced phase shift and pi temperature	$\Delta \phi =\frac{2\pi L}{\lambda }\frac{\partial n_{eff}}{\partial T}\Delta T$ $\Delta T_{\pi }=\frac{\lambda }{2L\partial n_{eff}/\partial T}$	$\Delta \phi \approx 2.4\pi \times 10^{-4}\Delta TL$	$\Delta \phi :phaseshift$ $L:heatedlength$ $\lambda :free-spacewavelength$ $\begin{array}{r}\Delta T:meantemperatureriseofthe\\ waveguide,\Delta T_{\pi }:risethatgivesashift\\ of\pi ,thefirstformisEq.1ofHarriset\\ al.,Yongetal.,Gursesetal.,Serkinet\\ al.andNejadriahietal.,theapproximate\\ formisforsiliconat1550\mathrm{nm}withLin\\ \mu \mathrm{m}androom-temperaturedn/dT(Harriset\\ al.)\end{array}$ $\begin{array}{r}(computedwiththetabulatedcoefficient\\ andthelengthineachpaper:\Delta T_{\pi }=67.6\\ \mathrm{K}forthe61.6\mu \mathrm{m}siliconshifterof\\ Harrisetal.(thedoped-regionoverlapis\\ ignored),13.9\mathrm{K}forthe300\mu \mathrm{m}spiralof\\ Gursesetal.,3.9\mathrm{K}forthe1.2\mathrm{mm}\\ silicon-richnitridespiralofNejadriahi\\ etal.,8.1and8.6\mathrm{K}forthe1.5\mathrm{mm}\\ nitrideshifterofYongetal.at532and\\ 561\mathrm{nm},andatleast154\mathrm{K}fora200\mu \mathrm{m}\\ stoichiometricnitrideheaterat1550\mathrm{nm}\\ (corecoefficient2.51\times 10^{-5},anupperbound\\ ontheefficiency))\end{array}$
Spectral shift of an interferometer or ring	$\frac{\Delta \lambda }{{\lambda }_{null}}=\frac{\Delta n_{eff}}{n_g}\frac{L_{mod}}{\Delta L}$	$\Delta \phi =2\pi \frac{|\Delta \lambda |}{\mathrm{FSR}}$ $\Delta {\lambda }_{res}=\frac{\lambda }{n_g}\frac{\partial n_{eff}}{\partial T}\Delta T$	$\begin{array}{r}\Delta \lambda :shiftofaninterferometernullora\\ resonance\end{array}$ ${\lambda }_{null}:nullwavelength$ $n_g:groupindex$ $L_{mod}:heatedlength$ $\Delta L:armlengthimbalance$ $\begin{array}{r}FSR:freespectralrange,thefirstform\\ isEq.1ofNejadriahietal.2020(heater\\ 480\mu \mathrm{m},imbalance102\mu \mathrm{m}),Serkinetal.\\ printthephaseshiftasthewavelength\\ shiftdividedbytheFSR,withoutthe\\ factor2\pi thatashiftofoneFSR\\ requires,soIhaveaddedit(myreading),\\ theringformisbackgroundknowledge(see\\ theSiliconMicroringModulatorpage)\end{array}$ $\begin{array}{r}(computedfromNejadriahietal.2021:a\\ freespectralrangeofabout0.56\mathrm{nm}means\\ that\pi ishalfofit,0.28\mathrm{nm},reachedat\\ 8\mathrm{mW},so35pm/mW)\end{array}$

Heater Power, Time Constant and Crosstalk

Name	Formula	Related Formulae	Variables/Constants
Heater power and thermal conductance	$P=\frac{V^2}{R}=G\Delta T$ $P_{\pi }=G\Delta T_{\pi }$	$V_{\pi }=\sqrt{P_{\pi }R}$ $G=\frac{H}{\tau }$	$P:electricalpowerintheheater$ $V:voltage$ $R:heaterresistance$ $\begin{array}{r}G:thermalconductancefromtheheated\\ waveguidetothesubstrateand\\ surroundingsinasingle-node(lumped)\\ picture\end{array}$ $\Delta T:temperaturerise$ $P_{\pi }:powerforaphaseshiftof\pi$ $V_{\pi }:voltagefor\pi$ $H:heatcapacityoftheheatedregion$ $\begin{array}{r}\tau :thermaltimeconstant,Harrisetal.:\\ P_{\pi }=V_{\pi }^2/R(4.36\mathrm{V}across769\Omega ),Gurses\\ etal.:P_{\pi }=Ak\Delta T_{\pi }withAkthe\\ conductance,thesingle-nodepicture\\ itselfisbackgroundknowledge\end{array}$ $\begin{array}{r}(computed:4.36^2/769=24.72\mathrm{mW}against\\ thereported24.77;G=P_{\pi }/\Delta T_{\pi }=0.37\\ mW/KforHarrisetal.(with67.6\mathrm{K}),0.76\\ mW/Kforthe300\mu \mathrm{m}spiralofGurseset\\ al.(10.6\mathrm{mW}over13.9\mathrm{K},andtheir\\ separatelyextractedGof62.8\mu W/Kgives\\ P_{\pi }/(12G)=14.1\mathrm{K},whichagrees,although\\ thefactor12isnotexplainedinthetext\\ Iread),0.134mW/KforYongetal.(1.09\\ \mathrm{mW}over8.1\mathrm{K},andthesimulated8.9\mathrm{K}for\\ 1.19\mathrm{mW}is7.5K/mW),and2.04mW/Kfor\\ the1.2\mathrm{mm}silicon-richnitridespiral(8\\ \mathrm{mW}over3.9\mathrm{K});Nejadriahietal.:1.5\mathrm{V}\\ and8\mathrm{mW}meanR=281\Omega ;Zhongetal.:48\\ \mathrm{mW}in1.6\mathrm{k\Omega }is8.8\mathrm{V}and8\mathrm{V}gives40\mathrm{mW})\end{array}$
Time constant, bandwidth and rise time	$\tau =\frac{H}{G}$ $f_{3dB}=\frac{1}{2\pi \tau }$	$t_r=2.2\tau =\frac{0.35}{f_{3dB}}$ $\tau =\frac{0.35}{f_{3dB}}\left(\text{as printed}\right)$	$\begin{array}{r}\tau :timeconstantofthesingle-pole\\ thermalresponse\end{array}$ $f_{3dB}:3dBbandwidthofthephaseresponse$ $\begin{array}{r}{t}_r:10to90percentrisetime,thefirst\\ tworelationsand2.2\tau arebackground\\ knowledgeforasinglepole,Harrisetal.\\ andLiuetal.print\tau =0.35/f_{3dB},Gurses\\ etal.useafirst-orderthermalcircuit\\ with\tau =C/(Ak)and{C}_W=(5/2)G\tau ,\\ Harrisetal.:bandwidth130.0\mathrm{kHz}anda\\ reportedtimeconstantof2.69\mu \mathrm{s},Serkin\\ etal.:bandwidth12\mathrm{kHz},rise31.6\mu \mathrm{s}and\\ fall35.4\mu \mathrm{s},Gursesetal.:bandwidth\\ 56.8\mathrm{kHz},C_W=440pJ/KandG=62.8\mu W/K\end{array}$ $\begin{array}{r}(computed:forHarrisetal.1/(2\pi \cdot 130\\ \mathrm{kHz})=1.22\mu \mathrm{s}and2.2\tau =2.69\mu \mathrm{s},sothe\\ 2.69\mu \mathrm{s}istherisetime;forSerkinet\\ al.0.35/12\mathrm{kHz}=29.2\mu \mathrm{s}againstthe\\ measured31.6and\tau =13.3\mu \mathrm{s};forGurses\\ etal.1/(2\pi \cdot 56.8\mathrm{kHz})=2.80\mu \mathrm{s},which\\ equals{C}_W/(2.5G)=2.80\mu \mathrm{s},whereas\\ 0.35/56.8\mathrm{kHz}wouldbe6.16\mu \mathrm{s})\end{array}$
Power-speed trade-off and figure of merit	$P_{\pi }\tau =H\Delta T_{\pi }$ $\mathrm{FOM}=P_{\pi }\tau =\frac{\lambda }{2}{C}_p\rho A\frac{dT}{d{n}_{eff}}$	$P_{\pi }=\frac{H\Delta T_{\pi }}{\tau }$ ${\mathrm{FOM}}_2=P_{\pi }\tau \mathrm{IL}$	$FOM:energyfigureofmerit(smallerisbetter)$ $C_p:specificheat$ $\rho :density$ $\begin{array}{r}A:effectiveheatedcross-section\\ (heat-flowarea)\end{array}$ $\begin{array}{r}dT/d{n}_{eff}:inverseofthethermo-optic\\ coefficientofthemode\end{array}$ $\begin{array}{r}IL:insertionlossoftheshifter,Harris\\ etal.:thetrade-off{P}_{\pi }\tau ,Liuetal.:\\ FOMandFO{M}_{2}aswrittenhere,withthe\\ quotedfigures11.1(7-pass)and30.5\\ (dopedsilicon)in\mathrm{mW}\mu \mathrm{s},theforms{P}_{\pi }=\\ H/(\tau \Delta T_{\pi })inonesummaryofHarrisetal.\\ andtheexpressioninLiuetal.and\\ Serkinetal.writtenasapowerare\\ discussedinthequestions\end{array}$ $\begin{array}{r}(computed:Harrisetal.P_{\pi }\tau =24.77\times \\ 2.69=66.6\mathrm{mW}\mu \mathrm{s}(reported66.9),which\\ with{C}_p\rho =1.63\times 10^{6}J/({\mathrm{m}}^3\mathrm{K})(assumedfor\\ silicon)and1.86\times 10^{-4}per\mathrm{K}givesA=9.9\\ \mu {\mathrm{m}}^{2}usingthequoted2.69\mu \mathrm{s}or4.5\mu {\mathrm{m}}^2\\ using1.22\mu \mathrm{s},againsta1\times 1.5\mu {\mathrm{m}}^{2}heat\\ channel;productsofthetabulated{P}_{\pi }and\\ thetimequoted:112.8(TiN,rise5.90\\ \mu \mathrm{s}),56.0(silicide,2.8\mu \mathrm{s}),30.5(12.7\\ \mathrm{mW}\times 2.4\mu \mathrm{s},matchingLiuetal.),11.0\\ (1.7\times 6.5,matchingtheir11.1),29.7\\ (Gursesetal.,2.8\mu \mathrm{s}),70.6and130.3\\ (undercutTiNwith144and266\mu \mathrm{s}),39.0\\ and64.0(suspendedfoldedshifterwith\\ 780and1280\mu \mathrm{s}),187(silicon-rich\\ nitride,0.35/15\mathrm{kHz}),2054(Serkinet\\ al.,31.6\mu \mathrm{s})and621(Yongetal.,570\\ \mu \mathrm{s}),allin\mathrm{mW}\mu \mathrm{s})\end{array}$
Thermal crosstalk and heat spreading	$\Delta \phi =a_0{\rho }_P^m$	$\frac{\Delta {\phi }_{adj}}{\Delta {\phi }_{self}}=\frac{P_{\pi }}{P_{\pi }^{(1)}}$	${\rho }_P:heaterpowerdensity$ $a_0:fitconstant(notreported)$ $\begin{array}{r}m:exponent,0.937plusorminus0.015\\ (Harrisetal.,fromheater-to-waveguide\\ separationsof1,2,4,8,16and32\mu \mathrm{m})\end{array}$ $\begin{array}{r}{P}_{\pi }^{(1)}:powerinaneighbouringheater\\ neededforashiftof\pi inthewaveguide\\ undertest\end{array}$ $\begin{array}{r}{G}_{SS}^{(1)}:substrateconductancebetween\\ neighbours,Gursesetal.:P_{\pi }=10.6\mathrm{mW},\\ P_{\pi }^{(1)}=62.6\mathrm{mW},G_{SS}^{(1)}=4.57\mu W/K(isolated\\ pair)and13.2\mu W/K(tunedinthe32\times 9\\ arraymodel),oxideetchedaroundeach\\ shifter,theratioformismyownreading\\ ofthosetwonumbers(Serkinetal.:\\ isolationtrenchescut{P}_{\pi }from195to65\\ \mathrm{mW}),Harrisetal.:oxideconductsheat\\ abouttwoorderslessthansilicon,Liuet\\ al.:silicon149W/(\mathrm{m}\mathrm{K})\end{array}$ $\begin{array}{r}(computed:62.6/10.6=5.9,soaneighbour\\ atthesamepowergives17percentofthe\\ shift;13.2/4.57=2.9;thetrenchfactor\\ is195/65=3.0;149/1.4=106forsilicon\\ againstsilica,with1.4W/(\mathrm{m}\mathrm{K})for\\ silicaasbackgroundknowledge)\end{array}$

Switch Transfer and Reported Devices

Name	Formula	Related Formulae	Variables/Constants
Mach-Zehnder switch transfer	$T={\cos }^2\left(\frac{\Delta \phi }{2}\right),\Delta \phi =\pi \frac{P}{P_{\pi }}=\pi \frac{V^2}{V_{\pi }^2}$	$T\left(P\right)={\cos }^2\left(\frac{\pi P}{2P_{\pi }}\right)$ $\mathrm{ER}=\frac{T_{max}}{T_{min}},T_{min}={\left(\frac{1-x}{1+x}\right)}^2$	$\begin{array}{r}T:powertransmissionofoneoutputofa\\ balancedMach-Zehnderswitchwithequal\\ armsandidealcouplersatzerodrive\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}\Delta \phi :phasedifference,linearintheheater\\ powerandsoquadraticinthevoltage\end{array}$ $ER:extinctionratio$ $\begin{array}{r}x:ratioofthetwoarmamplitudesatthe\\ output,Serkinetal.:ERabout80\mathrm{dB}\\ (unbalancedMZI),Gursesetal.:-22.8\mathrm{dB},\\ Nejadriahietal.2021:morethan15\mathrm{dB},\\ Zhongetal.:10.12\mathrm{dB}\end{array}$ $\begin{array}{r}(computed:T=0.5atP=P_{\pi }/2,sohalf\\ thepowerisnothalfthelight;for80\mathrm{dB}\\ thearmamplitudesmustmatchtox=\\ 0.9998,animbalanceof0.02percent)\end{array}$
Doped-silicon and metal heaters on silicon	${\mathrm{FOM}}_2=P_{\pi }\tau \mathrm{IL}$	$f_{3dB}\approx \frac{0.35}{t_r}$	$IL:insertionloss$ $\begin{array}{r}{t}_r:10to90percentrisetime,Harriset\\ al.:220\mathrm{nm}siliconwitha90\mathrm{nm}slab,\\ dopedwaveguideasitsownheater,length\\ 61.6\mu \mathrm{m},P_{\pi }=24.77plusorminus0.43\mathrm{mW},\\ 769\Omega ,130\mathrm{kHz},loss0.23plusorminus\\ 0.13\mathrm{dB}over21devices,Liuetal.(P_{\pi }in\\ \mathrm{mW}andtimesin\mu \mathrm{s}):TiN19.12,rise5.90,\\ fall8.97,silicide20,2.8and2.2,\\ directdopedsilicon12.7,2.2and2.4\\ withabout0.5\mathrm{dB}loss,undercutTiN0.49,\\ 144and122,foldedandsuspended0.05,\\ 780and500,folded2.56,34.8and34.4,\\ 7-pass1.7,6.4and6.6with4.6\mathrm{dB}loss,\\ Gursesetal.:300\mu \mathrm{m}spiral,10.6\mathrm{mW},\\ 56.8\mathrm{kHz},Liuetal.:modulationbandwidth\\ belowabout100\mathrm{kHz}\end{array}$ $\begin{array}{r}(computed:bandwidthsfromtherisetimes\\ are0.35/2.8=125\mathrm{kHz},0.35/2.4=146\mathrm{kHz}\\ and0.35/2.2=159\mathrm{kHz}forthethree\\ fastestshifters,and2.4\mathrm{kHz}forthe\\ undercutTiNwith144\mu \mathrm{s};Harrisetal.at\\ 130\mathrm{kHz}isthereforebeyondthe100\mathrm{kHz}of\\ Liuetal.)\end{array}$
Silicon nitride and silicon-rich nitride shifters	$\Delta T_{\pi }\ge \frac{\lambda }{2L{\left(dn/dT\right)}_{core}}$	$\Delta T_{\pi }=\frac{P_{\pi }}{G}$	$\begin{array}{r}theinequalityholdsbecausetheeffective\\ indexalsoseessilica,whosecoefficient\\ issmaller(backgroundknowledgefor\\ positivecoefficients),Serkinetal.:220\\ \mathrm{nm}\mathrm{S}{\mathrm{i}}_3{\mathrm{N}}_{4}corein2.5\mu \mathrm{m}ofoxide,titanium\\ heater100\mathrm{nm}thickand2.0\mu \mathrm{m}wide,at\\ least200\mu \mathrm{m}long,trenchesatleast4\mu \mathrm{m}\\ deepat0.5\mu \mathrm{m}fromtheheater,P_{\pi }=65\mathrm{mW}\\ withand195\mathrm{mW}withouttrenches\\ (simulated42\mathrm{mW}atthebesttrench\\ positionand64\mathrm{mW}forthefinaldesign),\\ ERabout80\mathrm{dB},12\mathrm{kHz},rise31.6\mu \mathrm{s}and\\ fall35.4\mu \mathrm{s},simulatedcorenear508\mathrm{K}\\ andheaternear710\mathrm{K}withthesubstrate\\ at293\mathrm{K},Yongetal.:150\mathrm{nm}nitride,TiN\\ heater500\mu \mathrm{m}long(1190\Omega ),1.5\mathrm{mm}of\\ waveguideinthreepassesoveranundercut\\ oxidebridge,P_{\pi }=0.78,0.93,1.09and\\ 1.20\mathrm{mW}at445,488,532and561\mathrm{nm},rise\\ 570\mu \mathrm{s}andfall590\mu \mathrm{s},Nejadriahietal.\\ 2021:silicon-richnitride(index3.1),\\ 1.2\mathrm{mm}spiralin65\times 65\mu {\mathrm{m}}^2,Ni:Crheater\\ 300\mathrm{nm}thick,P_{\pi }=8plusorminus0.25\mathrm{mW}\\ (serpentine)or11.5(rectangular),15\\ \mathrm{kHz},lossestimatedat1.2\mathrm{dB}ormore\end{array}$ $\begin{array}{r}(computed:theSerkinetal.coreriseof\\ 215\mathrm{K}(508minus293)isabovethe154\mathrm{K}\\ minimumfora200\mu \mathrm{m}heater,whichfitsif\\ thesimulationisatabout\pi ,butthetext\\ readdoesnotsayso(myreading);P_{\pi }/\lambda \\ forYongetal.risesfrom1.75to2.14\mu \mathrm{W}\\ per\mathrm{nm}overthefourwavelengths,22per\\ cent,whichsuggeststhatmoreofthemode\\ sitsinsilicaatthelongerwavelengths\\ (myreading);theserpentineheatercuts\\ P_{\pi }by30percent;1.2\mathrm{mm}ofsilicon-rich\\ nitrideat7\mathrm{dB}per\mathrm{cm}is0.84\mathrm{dB},andtwo\\ Y-branchesat0.159\mathrm{dB}make1.16\mathrm{dB},close\\ tothe1.2\mathrm{dB}estimate)\end{array}$
Polymer-clad silica switch	${\eta }_{red}=1-\frac{P_{sw}}{{\overline{P}}_{ref}}$	$\frac{{\left(dn/dT\right)}_{polymer}}{{\left(dn/dT\right)}_{silica}}\approx -10$	${\eta }_{red}:reductionofswitchingpower$ $P_{sw}:switchingpowerofthepolymer-claddevice$ $\begin{array}{r}{\overline{P}}_{ref}:meanpoweroftheall-silica\\ switchesinthecomparison,Zhongetal.:\\ Mach-ZehnderswitchwithaGe-dopedsilica\\ coreofabout3\times 3\mu \mathrm{m}anda6.5\mu \mathrm{m}\\ UV-curedepoxyuppercladding(index1.46,\\ thermalconductivity0.2W/(\mathrm{m}\mathrm{K})),\\ aluminiumheater100\mathrm{nm}\times 5\mu \mathrm{m}\times 7\mathrm{mm}with\\ 1.6\mathrm{k\Omega },switchingpower48\mathrm{mW}(simulated\\ 37.9),rise215\mu \mathrm{s}andfall271\mu \mathrm{s}\\ (simulated181and176),ER10.12\mathrm{dB},\\ insertionloss8.53\mathrm{dB}ofwhichabout2\mathrm{dB}\\ perfacetiscleaving,thecomparison\\ devicesuse433.24,300(404.82inthe\\ text),246.6and300\mathrm{mW}with650,509.7,\\ 900and2000\mu \mathrm{s},P_{\pi }isnotreported\end{array}$ $\begin{array}{r}(computed:themeanofthefourpowersis\\ 320\mathrm{mW}(346with404.82),so48\mathrm{mW}isan\\ 85percentreduction(86percent),not\\ the75percentclaimed,whichwouldneed\\ ameanof192\mathrm{mW};theratioof-1\times 10^{-4}to\\ thesilicacoefficient0.95\times 10^{-5}is-10.5)\end{array}$

Important Questions

Ques: Is the thermal time constant $0.35/f_{3dB}$ or $1/(2\pi f_{3dB})$?

Harris et al. and Liu et al. print $\tau =0.35/f_{3dB}$, which gives 2.69 $\mu \mathrm{s}$ for 130 $\mathrm{kHz}$. Gurses et al. write $C_W=(5/2)G\tau$ with $C_W$ = 440 $pJ/K$ and $G$ = 62.8 $\mu W/K$, and these give $\tau$ = 2.80 $\mu \mathrm{s}$, which is exactly $1/(2\pi \cdot 56.8\mathrm{kHz})$, not the 6.16 $\mu \mathrm{s}$ of $0.35/f$ (computed). For a single pole the time constant is $1/(2\pi f)$, and $0.35/f$ is the 10 to 90 per cent rise time, 2.2 times larger (background knowledge). Serkin et al. support this: $0.35/12$ $\mathrm{kHz}$ is 29.2 $\mu \mathrm{s}$ against a measured rise of 31.6 $\mu \mathrm{s}$ (computed). Harris et al.’s 2.69 $\mu \mathrm{s}$ is therefore a rise time, the time constant is 1.22 $\mu \mathrm{s}$, and their $P_{\pi }\tau$ of 66.9 $\mathrm{mW}\mu \mathrm{s}$ becomes 30.3 (computed). I regard $1/(2\pi f)$ as the time constant and the $0.35/f$ values as rise times. Reference: Harris, Ma, Mower, Baehr-Jones, Englund, Hochberg and Galland, ‘Efficient, compact and low loss thermo-optic phase shifter in silicon’, arXiv:1410.3616 (2014) (fetch-tool summary; two summaries of the same paper gave different forms of the power-speed relation); Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo, ‘Thermo-optic phase shifters based on silicon-on-insulator platform: state-of-the-art and a review’, Frontiers of Optoelectronics 15(1):9 (2022), DOI 10.1007/s12200-022-00012-9 (PMC9756266; fetch-tool summary); Gurses, Fatemi, Khachaturian and Hajimiri (Caltech), ‘Large-scale crosstalk-corrected thermo-optic phase shifter arrays in silicon photonics’, arXiv:2206.04525 (fetch-tool summary; the wavelength and the thermo-optic coefficient are not given in the text read); Serkin, Buzaverov, Baburin, Sergeev, Avdeev, Lotkov, Bukatin, Stepanov, Kramarenko, Amiraslanov, Ryzhikov and Rodionov, ‘Thermo-optic modulator with ultra-high extinction ratio for low-loss silicon nitride integrated photonics’, arXiv:2601.19732 (2026) (fetch-tool summary).

Ques: Is the time constant of the undercut shifter 144 or 266 microseconds?

Harris et al. list the undercut shifter with $P_{\pi }$ = 0.49 $\mathrm{mW}$ and $\tau$ = 144 $\mu \mathrm{s}$. Liu et al. list the same $P_{\pi }$ with a rise of 144 $\mu \mathrm{s}$ and a fall of 122 $\mu \mathrm{s}$, and give $\tau$ as about 266 $\mu \mathrm{s}$; for the suspended folded Michelson they list 780 and 500 $\mu \mathrm{s}$ and $\tau$ = 1.28 $\mathrm{ms}$. In both cases their $\tau$ is the sum of rise and fall (144 + 122 = 266 and 780 + 500 = 1280, computed), a third definition next to $0.35/f$ and $1/(2\pi f)$. The bandwidth from the rise time is 2.4 $\mathrm{kHz}$, consistent with the review’s statement of under 2.6 $\mathrm{kHz}$, whereas 266 $\mu \mathrm{s}$ would give 1.3 $\mathrm{kHz}$ (computed). I regard 144 $\mu \mathrm{s}$ as the rise time and 266 $\mu \mathrm{s}$ as a sum that should not be used as $\tau$. Reference: Harris, Ma, Mower, Baehr-Jones, Englund, Hochberg and Galland, ‘Efficient, compact and low loss thermo-optic phase shifter in silicon’, arXiv:1410.3616 (2014) (fetch-tool summary; two summaries of the same paper gave different forms of the power-speed relation); Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo, ‘Thermo-optic phase shifters based on silicon-on-insulator platform: state-of-the-art and a review’, Frontiers of Optoelectronics 15(1):9 (2022), DOI 10.1007/s12200-022-00012-9 (PMC9756266; fetch-tool summary).

Ques: Is the power-speed relation $P_{\pi }=H\Delta T_{\pi }/\tau$ or $H/(\tau \Delta T_{\pi })$, and is $(\lambda /2)(dT/dn)(C_p\rho S/\eta )$ a power?

Two summaries of Harris et al. give $P_{\pi }=H\Delta T_{\pi }/\tau$ and $P_{\pi }=H/(\tau \Delta T_{\pi })$. Only the first has the unit of power, joule per kelvin times kelvin per second (computed by dimensions), and it is the lumped result $P=G\Delta T$ with $G=H/\tau$. Serkin et al. (Eq. 3) and Liu et al. write $P_{\pi }=(\lambda /2)(dT/d{n}_{eff})(C_p\rho S/\eta )$. With $\lambda$ in metres, $C_p\rho S$ in joule per metre kelvin and $dT/dn$ in kelvin, this has the unit joule (computed), and Liu et al. use the same expression as the figure of merit $P_{\pi }\tau$. It equals $H\Delta T_{\pi }$ with $H=C_p\rho SL$, and the length cancels. I regard $H\Delta T_{\pi }/\tau$ as the power and the second expression as $P_{\pi }\tau$; the definition of $\eta$ could hide a time, which the summaries do not show (my reading). Reference: Harris, Ma, Mower, Baehr-Jones, Englund, Hochberg and Galland, ‘Efficient, compact and low loss thermo-optic phase shifter in silicon’, arXiv:1410.3616 (2014) (fetch-tool summary; two summaries of the same paper gave different forms of the power-speed relation); Serkin, Buzaverov, Baburin, Sergeev, Avdeev, Lotkov, Bukatin, Stepanov, Kramarenko, Amiraslanov, Ryzhikov and Rodionov, ‘Thermo-optic modulator with ultra-high extinction ratio for low-loss silicon nitride integrated photonics’, arXiv:2601.19732 (2026) (fetch-tool summary); Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo, ‘Thermo-optic phase shifters based on silicon-on-insulator platform: state-of-the-art and a review’, Frontiers of Optoelectronics 15(1):9 (2022), DOI 10.1007/s12200-022-00012-9 (PMC9756266; fetch-tool summary).

Ques: Is the thermo-optic coefficient of stoichiometric silicon nitride 2.45, 2 or 2.51 times $1\times 10^{-5}$ per kelvin?

Yong et al. give $2.45\times 10^{-5}$ per $\mathrm{K}$, and Nejadriahi et al. (2020) quote the same value from Arbabi and Goddard. Nejadriahi et al. (2021) write about $2\times 10^{-5}$, and Serkin et al. state $2.51\times 10^{-5}$. I read the $2\times 10^{-5}$ as a rounded figure (my reading). The silicon-rich series shows how strongly the coefficient depends on composition: $1.82\times 10^{-5}$ at index 1.92, $5.85\times 10^{-5}$ at 2.25 and $1.65\times 10^{-4}$ at 3.1 (Nejadriahi et al. 2020), so a film near index 2.0 sits at the low end, consistent with $2.4\times 10^{-5}$ given the quoted uncertainty of $0.8\times 10^{-5}$ on the 1.92 sample. Silica at $0.95\times 10^{-5}$ (Yong et al.) and $9\times 10^{-6}$ (Nejadriahi et al.) differ by 5 per cent (computed). For a waveguide in oxide the cladding share matters more than 2.45 against 2.51. I regard 2.4 to $2.5\times 10^{-5}$ as more likely. Reference: Yong, Chen, Luo, Govdeli, Chua, Azadeh, Stalmashonak, Lo, Poon and Sacher, ‘Power-efficient silicon nitride thermo-optic phase shifters for visible light’, arXiv:2111.07890 (fetch-tool summary of a truncated text); Nejadriahi, Friedman, Sharma, Pappert, Fainman and Yu, ‘Thermo-optic properties of silicon-rich silicon nitride for on-chip applications’, arXiv:2005.13348 (fetch-tool summary); Nejadriahi, Pappert, Fainman and Yu, ‘Efficient and compact thermo-optic phase shifter in silicon-rich silicon nitride’, arXiv:2106.09476 (fetch-tool summary; its platform comparison table was not in the text read); Serkin, Buzaverov, Baburin, Sergeev, Avdeev, Lotkov, Bukatin, Stepanov, Kramarenko, Amiraslanov, Ryzhikov and Rodionov, ‘Thermo-optic modulator with ultra-high extinction ratio for low-loss silicon nitride integrated photonics’, arXiv:2601.19732 (2026) (fetch-tool summary).

Ques: Which thermo-optic coefficient of silicon should be used when the waveguide is heated by 70 K?

Harris et al. and Nejadriahi et al. use $1.86\times 10^{-4}$ per $\mathrm{K}$, Liu et al. say about $1.8\times 10^{-4}$ and give a polynomial for 300 to 600 $\mathrm{K}$ that evaluates to $1.852\times 10^{-4}$ at 300 $\mathrm{K}$ (computed), so the three agree at room temperature. The polynomial rises to $2.095\times 10^{-4}$ at 400 $\mathrm{K}$ and $2.491\times 10^{-4}$ at 600 $\mathrm{K}$ (computed), 13 and 35 per cent higher. For the 61.6 $\mu \mathrm{m}$ shifter of Harris et al., integrating the polynomial from 300 $\mathrm{K}$ gives a $\pi$ shift at a rise of 65.1 $\mathrm{K}$ instead of 67.6 $\mathrm{K}$ (computed), a mean coefficient of $1.93\times 10^{-4}$, 4 per cent above the room-temperature value. The correction is small compared with the unknown overlap of the heated region and the mode, so I regard $1.85\times 10^{-4}$ as the room-temperature value and the temperature dependence as a modest correction. Reference: Harris, Ma, Mower, Baehr-Jones, Englund, Hochberg and Galland, ‘Efficient, compact and low loss thermo-optic phase shifter in silicon’, arXiv:1410.3616 (2014) (fetch-tool summary; two summaries of the same paper gave different forms of the power-speed relation); Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo, ‘Thermo-optic phase shifters based on silicon-on-insulator platform: state-of-the-art and a review’, Frontiers of Optoelectronics 15(1):9 (2022), DOI 10.1007/s12200-022-00012-9 (PMC9756266; fetch-tool summary); Nejadriahi, Pappert, Fainman and Yu, ‘Efficient and compact thermo-optic phase shifter in silicon-rich silicon nitride’, arXiv:2106.09476 (fetch-tool summary; its platform comparison table was not in the text read).

Ques: Does a polymer cladding cut the switching power by 75 or 85 per cent, and is the polymer coefficient ten times or one to two orders above silica?

Zhong et al. report 48 $\mathrm{mW}$ for their polymer-clad silica switch and claim a 75 per cent reduction. The four all-silica switches they compare use 433.24, 300, 246.6 and 300 $\mathrm{mW}$ (404.82 for the second in the text), a mean of 320 $\mathrm{mW}$ (346), so 48 $\mathrm{mW}$ is an 85 per cent reduction (86 per cent), and 75 per cent would need a mean of 192 $\mathrm{mW}$ (computed). The introduction calls the polymer coefficient both ten times and one to two orders of magnitude larger than that of silica; $-1\times 10^{-4}$ against $0.95\times 10^{-5}$ is a factor of 10.5 (computed), so ten times fits. Their 48 $\mathrm{mW}$ is a switching power, not $P_{\pi }$, and the material is named 2003T7 in the text and 2002T7 in the table. I regard 85 per cent and ten times as more likely. Reference: Zhong, Qin, Liu, Xie, An, Qin and Yi, ‘Low power consumption silica thermo-optic switch based on polymer cladding’, Polymers 17(23):3214 (2025), DOI 10.3390/polym17233214 (PMC12694417; fetch-tool summary); Yong, Chen, Luo, Govdeli, Chua, Azadeh, Stalmashonak, Lo, Poon and Sacher, ‘Power-efficient silicon nitride thermo-optic phase shifters for visible light’, arXiv:2111.07890 (fetch-tool summary of a truncated text).

Ques: Is the thermal conductivity of air 0.31 or 0.026 W/(m K)?

Liu et al. list 149 $W/(\mathrm{m}\mathrm{K})$ for silicon and 0.31 $W/(\mathrm{m}\mathrm{K})$ for air. The silicon value agrees with the usual 148 (background knowledge). Air at room temperature is about 0.026 $W/(\mathrm{m}\mathrm{K})$ (background knowledge), twelve times below 0.31 (computed). The polymer cladding of Zhong et al. has 0.2 $W/(\mathrm{m}\mathrm{K})$, and air should conduct much less than a polymer, so 0.31 would put air above it (computed comparison). The value matters for undercut and suspended shifters, where the air gap is a main thermal path and the lower conductivity is what makes them efficient: 0.49 $\mathrm{mW}$ per $\pi$ for the undercut design against 19 to 25 $\mathrm{mW}$ for oxide-clad ones (Liu et al.; Harris et al.). I regard 0.026 as correct and the printed 0.31 as an error whose origin I cannot identify. Reference: Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo, ‘Thermo-optic phase shifters based on silicon-on-insulator platform: state-of-the-art and a review’, Frontiers of Optoelectronics 15(1):9 (2022), DOI 10.1007/s12200-022-00012-9 (PMC9756266; fetch-tool summary); Zhong, Qin, Liu, Xie, An, Qin and Yi, ‘Low power consumption silica thermo-optic switch based on polymer cladding’, Polymers 17(23):3214 (2025), DOI 10.3390/polym17233214 (PMC12694417; fetch-tool summary).

Ques: Are thermo-optic shifters limited to 100 kHz?

Liu et al. state that the bandwidth is below about 100 $\mathrm{kHz}$, but Harris et al. measure 130 $\mathrm{kHz}$, and the review’s own table lists silicide and doped-silicon shifters with rise times of 2.2 to 2.8 $\mu \mathrm{s}$, which give 125 to 159 $\mathrm{kHz}$ from $0.35/t_r$ (computed). Gurses et al. simulate 367 $\mathrm{kHz}$ for a crosstalk-corrected array driven by pulse-amplitude modulation, which I read as a gain from pre-emphasis of the drive and not from a faster heater (my reading). Efficient designs are far slower: 15 $\mathrm{kHz}$ for silicon-rich nitride, 12 $\mathrm{kHz}$ for nitride with trenches and 2.4 $\mathrm{kHz}$ for the undercut shifter. The speed cannot be raised without raising $P_{\pi }$, because $P_{\pi }\tau$ is fixed by the heated volume. I regard 100 $\mathrm{kHz}$ as typical for efficient devices and about 150 $\mathrm{kHz}$ as the demonstrated limit of a plain heater. Reference: Liu, Feng, Tian, Zhao, Jin, Ouyang, Zhu and Guo, ‘Thermo-optic phase shifters based on silicon-on-insulator platform: state-of-the-art and a review’, Frontiers of Optoelectronics 15(1):9 (2022), DOI 10.1007/s12200-022-00012-9 (PMC9756266; fetch-tool summary); Harris, Ma, Mower, Baehr-Jones, Englund, Hochberg and Galland, ‘Efficient, compact and low loss thermo-optic phase shifter in silicon’, arXiv:1410.3616 (2014) (fetch-tool summary; two summaries of the same paper gave different forms of the power-speed relation); Gurses, Fatemi, Khachaturian and Hajimiri (Caltech), ‘Large-scale crosstalk-corrected thermo-optic phase shifter arrays in silicon photonics’, arXiv:2206.04525 (fetch-tool summary; the wavelength and the thermo-optic coefficient are not given in the text read); Nejadriahi, Pappert, Fainman and Yu, ‘Efficient and compact thermo-optic phase shifter in silicon-rich silicon nitride’, arXiv:2106.09476 (fetch-tool summary; its platform comparison table was not in the text read); Serkin, Buzaverov, Baburin, Sergeev, Avdeev, Lotkov, Bukatin, Stepanov, Kramarenko, Amiraslanov, Ryzhikov and Rodionov, ‘Thermo-optic modulator with ultra-high extinction ratio for low-loss silicon nitride integrated photonics’, arXiv:2601.19732 (2026) (fetch-tool summary).

Ques: Is the neighbour coupling of the array 4.57 or 13.2 microwatts per kelvin?

Gurses et al. extract $G_{SS}^{(1)}$ = 4.57 $\mu W/K$ from a measurement in which a neighbouring heater needs 62.6 $\mathrm{mW}$ for a $\pi$ shift, but their array model uses 13.2 $\mu W/K$, 2.9 times more (computed). The summary I read does not explain the difference. Two reasons are plausible (my reading): in the array each shifter has several neighbours that also leak heat through the substrate, so the fitted coupling is larger than for an isolated pair; or the single pair has more etched oxide between the heaters. Relative to the self conductance of 62.8 $\mu W/K$ the two values are 7 and 21 per cent (computed). For a correction matrix in a real array the larger value is the one that reproduces the measured crosstalk, so I regard 13.2 $\mu W/K$ as the better number for an array and 4.57 $\mu W/K$ as the pair value. Reference: Gurses, Fatemi, Khachaturian and Hajimiri (Caltech), ‘Large-scale crosstalk-corrected thermo-optic phase shifter arrays in silicon photonics’, arXiv:2206.04525 (fetch-tool summary; the wavelength and the thermo-optic coefficient are not given in the text read).

Ques: Is the phosphorus doping of the Harris test resistor $5\times 10^{-3}$ per cubic centimetre?

Harris et al. list the n-doped test resistor, 503.52 $\Omega$, with a phosphorus doping that the summary prints as $5\times 10^{-3}$ $\mathrm{c}{\mathrm{m}}^{-3}$. That is not a doping level, since it is far less than one dopant atom per cubic centimetre. The same paper gives boron levels of $1.7\times 10^{20}$ and $7\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ for the shifter, so the intended value is probably in the range $1\times 10^{17}$ to $1\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (my reading), and the exponent cannot be recovered from the text. The heater power in the decay experiment is set by the measured resistance and the applied bias, so the fitted exponent of 0.937 for the power-law decay with heater power density is not affected. I regard the printed exponent as a sign or digit error, either in the paper or in its summary, and I do not use the number. Reference: Harris, Ma, Mower, Baehr-Jones, Englund, Hochberg and Galland, ‘Efficient, compact and low loss thermo-optic phase shifter in silicon’, arXiv:1410.3616 (2014) (fetch-tool summary; two summaries of the same paper gave different forms of the power-speed relation).

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