Liquid-Crystal Spatial Light Modulator

1. Theory

• Definition: A liquid-crystal spatial light modulator is a two-dimensional array of nematic liquid-crystal cells, each acting as a voltage-controlled wave plate. A field tilts the director, the effective index for light polarised along it falls from $n_e$ towards $n_o$, and a pixel of gap $d$ adds the retardation $\delta =2\pi d\Delta n_{eff}/\lambda$ (Yang et al.; Zou et al.). In the reflective liquid-crystal-on-silicon (LCoS) form the light crosses the layer twice, so the gap needed is half as large.
• Phase or amplitude: With the input polarisation along the extraordinary axis the panel changes only the phase, and a stroke of $2\pi$ is needed. At 45 degrees to the axis and between polarisers it changes the intensity as ${\sin }^2(\Phi /2)$ or ${\cos }^2(\Phi /2)$ and needs about $\pi$ (Yang et al.; Moreno et al. 2021; the 45 degrees is background knowledge).
• Birefringence and gap: Nematic mixtures for LCoS have $\Delta n$ of 0.2 to 0.215 at 633 $\mathrm{nm}$ (Yang et al.; Zou et al.). The silicon backplane limits the drive to about 5 ${\mathrm{V}}_{\mathrm{rms}}$, at which the director reaches about 84 per cent of the full birefringence (computed from Zou et al.), so the reflective gap for $2\pi$ is 1.9 $\mu \mathrm{m}$ at 633 $\mathrm{nm}$ and 2.9 $\mu \mathrm{m}$ at 905 $\mathrm{nm}$.
• Speed: The response time follows ${\tau }_0={\gamma }_1{d}^2/(K_{11}{\pi }^2)$ and the turn-on and turn-off factors; nematic panels run at a few tens of $\mathrm{Hz}$ to a few hundred $\mathrm{Hz}$, ferroelectric liquid crystals at a few $\mathrm{kHz}$ (Moreno et al. 2021). The best reported total for a reflective cell is 3.14 $\mathrm{ms}$ (Yang et al.) and 2.20 $\mathrm{ms}$ (Zou et al.), and the commercial panels in the sources list 12 to 43 $\mathrm{ms}$.
• Pixelation: The panel is a sampled phase screen with a pitch of 8 to 20 $\mu \mathrm{m}$ (Meadowlark; Holoeye; Hamamatsu), so a blazed grating of $N$ pixels per period diffracts a fraction ${\mathrm{sinc}}^2(1/N)$ into the first order, the fringing field rounds the steps, and the fill factor of 93 to 98 per cent leaves a zero order (Moreno et al.; Varga et al.).
• Figures of merit: Stroke at the wavelength of use, response time, pixel pitch and count, fill factor, diffraction efficiency, phase jitter and polarisation sensitivity. Typical panels have 8-bit phase, 60 $\mathrm{Hz}$ video input, 12 to 16 $\mathrm{ms}$ response in the visible, 78 to 86 per cent efficiency and a phase jitter below 12 to 22 $\mathrm{nm}$ rms (Kuhn and Patapis).
• Scope: This page covers nematic liquid-crystal phase and amplitude modulators and LCoS panels. The voltage-driven crystal modulators are on the Electro-Optic Phase Modulator page, the heated waveguide shifters on the Thermo-Optic Modulator page and the sound-wave deflectors on the Acousto-Optic Modulator page; MEMS and magneto-optic modulators have their own pages.

2. Working Principle

• Cell: A layer of 1.5 to 5 $\mu \mathrm{m}$ of nematic liquid crystal lies between two aligned surfaces; in LCoS one surface is a mirror over a silicon backplane that holds the pixel electrodes and the other is a transparent electrode (Yang et al.; Meadowlark; the backplane details are background knowledge). The aligned director lies in the plane with field off.
• Threshold and tilt: Below $V_{th}=\pi \sqrt{K_{11}/({\epsilon }_0\Delta \epsilon )}$ nothing moves; for 5CB this is 0.70 $\mathrm{V}$. Above it the mid-cell tilt grows as ${\theta }_m=\sqrt{2s/(\kappa +{\rm Y}+1)}$ with $s=(V^2-V_{th}^2)/V_{th}^2$ and the effective index falls from $n_e$ to $n_o$ (Caussarieu et al.; graphene paper). The drive is alternating, at a few $\mathrm{kHz}$ in Caussarieu et al.
• Phase calibration: The retardance is linear in the gray level after calibration: $\pi$ at level 100 and $2\pi$ at level 200 at 625 $\mathrm{nm}$, so a level is 1.8 degrees (Moreno et al. 2021). Panels with 8 to 10 times $\pi$ of range wrap the phase at multiples of $2\pi$ (Moreno et al. 2020).
• Hologram: A blazed phase ramp of period $\Lambda$ pixels steers light by $\sin \theta =\lambda /(\Lambda p_x)$; the steepest ramp, two pixels per period, gives 1.9 degrees at 532 $\mathrm{nm}$ with an 8 $\mu \mathrm{m}$ pitch (computed). Shifting the ramp by $\delta$ pixels adds $2\pi \delta /\Lambda$ to the first order (Varga et al.).
• Speed: The relaxation is limited by the viscoelastic constant ${\gamma }_1/K_{11}$ and falls as $d^2$. Warming to 40 to 50 ${}^{\circ }\mathrm{C}$ speeds up a cell by a factor of 1.5 to 1.7 (computed from the tables of Yang et al.) and nearly doubles the figure of merit (Yang et al.; Zou et al.). Overdriving the voltage steps shortens the turn-on further (background knowledge).
• Crosstalk: The field of a pixel spills into its neighbours, rounds the edges of a binary pattern and cuts the effective phase to 94 per cent at four pixels per period and 89 per cent at two (Moreno et al. 2021).
• Operation: Light must be linearly polarised along the extraordinary axis, off-axis angles are kept below 5 degrees to limit crosstalk, and the exposure should be longer than the 30 $\mathrm{ms}$ refresh jitter (Kuhn and Patapis).

3. Applications

• Beam shaping and holography: Phase-only panels write computer-generated holograms for beam shaping, optical tweezers and structured light (background knowledge); Yang et al. name adaptive optics and beam shaping for lithography.
• Wavelength-selective switching: Telecom switches steer each wavelength channel with a phase grating on an LCoS panel (Yang et al.).
• LiDAR beam steering: Fast LCoS phase modulators for 905 $\mathrm{nm}$ are the subject of Zou et al., who give a reflective gap of 2.91 $\mu \mathrm{m}$ and a response of 5.9 $\mathrm{ms}$.
• Coronagraphy: A phase mask on an LCoS panel nulls a star; Kuhn and Patapis reach an on-axis null of about 1.4 to 2.2 times $10^{-2}$ at 640 $\mathrm{nm}$ with 12 per cent bandwidth.
• Quantum optics: Pure-phase panels generate spatial qudits, with the amplitude encoded in the grating depth and the phase in its displacement (Varga et al.).
• Pulse and wavefront shaping: Linear panels shape ultrashort pulses in a spectral plane and correct aberrations (background knowledge).
• Displays and projectors: The same cells with polarisers give amplitude light valves (background knowledge); Yang et al. state that a 240 $\mathrm{Hz}$ frame rate needs a response below 4 $\mathrm{ms}$.

Liquid-Crystal Phase Retardation

Name	Formula	Related Formulae	Variables/Constants
Phase retardation of a liquid-crystal layer	$\delta =\frac{2\pi d\Delta n_{eff}}{\lambda }$ ${\delta }_{refl}=2\cdot \frac{2\pi d\Delta n_{eff}}{\lambda }$	$d_{2\pi }=\frac{\lambda }{\Delta n_{eff}}$ $d_{2\pi ,refl}=\frac{\lambda }{2\Delta n_{eff}}$ $\Delta n=n_e-n_o$	$\delta :phaseretardationin\mathrm{rad}$ $d:liquid-crystalcellgap$ $\begin{array}{r}\lambda :free-spacewavelength,\Delta n=n_e-n_o:\\ birefringenceofthenematic\end{array}$ $\begin{array}{r}\Delta n_{eff}:changeoftheeffectiveindex\\ betweenthelowestandthehighestdrive\\ voltage(atmost\Delta n),thefactor2inthe\\ reflectiveformisthedoublepassofa\\ liquid-crystal-on-siliconcell(Zouet\\ al.),amplitudemodulationneedsa\\ retardationofabout\pi andphase-only\\ operationneeds2\pi (Yangetal.),\\ birefringencevalues:TL-2160.205and\\ UCF-2160.215at633\mathrm{nm}and22{}^{\circ }C(Yanget\\ al.),A49070.200at633\mathrm{nm}and0.185at\\ 905\mathrm{nm}at40{}^{\circ }C(Zouetal.)\end{array}$ $\begin{array}{r}(computedfromthegapsquotedfor2\pi at5\\ {\mathrm{V}}_{\mathrm{rms}}:\lambda /(2d)gives\Delta n_{eff}=0.1684forthe\\ 1.88\mu \mathrm{m}reflectivegapat633\mathrm{nm}and\\ 0.1555forthe2.91\mu \mathrm{m}gapat905\mathrm{nm},\\ whichare0.842and0.841oftheA4907\\ values0.200and0.185,sothedrive\\ reachesabout84percentofthefull\\ birefringence;thetransmissivegapsof\\ Yangetal.at50{}^{\circ }C,4.14\mu \mathrm{m}forUCF-216\\ and4.93\mu \mathrm{m}forTL-216,correspondto\\ \Delta n_{eff}=0.153and0.128;the1.5\mu \mathrm{m}\\ reflectivegapthatYangetal.pairwith\\ \Delta n=0.211isexactly\lambda /(2\Delta n)=1.50\mu \mathrm{m})\end{array}$
Effective index of a tilted director	$n_{eff}(\theta )=\frac{n_o{n}_e}{\sqrt{n_e^2{\sin }^2\theta +n_o^2{\cos }^2\theta }}$	$\phi \approx \frac{{\phi }_0\nu }{4}{\theta }_m^2$ ${\phi }_0=\frac{2\pi L{n}_e}{\lambda }$ $\nu =\frac{n_e^2-n_o^2}{n_e^2}$	$\begin{array}{r}{n}_{eff}:indexseenbynormallyincidentlight\\ polarisedintheplaneofthedirector\end{array}$ $\begin{array}{r}\theta :tiltofthedirectorfromthesubstrate\\ plane(zerowiththefieldoff,90degrees\\ fullyswitched)\end{array}$ $\begin{array}{r}{n}_{o}and{n}_e:ordinaryandextraordinary\\ indices,thefirstformisthe\\ index-ellipsoidresult(background\\ knowledge)\end{array}$ ${\theta }_m:tiltatthemiddleofthecell$ $L:cellgap$ $\begin{array}{r}\phi :magnitudeofthephasechangejust\\ abovethresholdforasinusoidaltilt\\ profile,{\phi }_{0}and\nu aswritteninCaussarieu\\ etal.(Eq.8andthetext;5CBwith\nu =\\ 0.2457,632.8\mathrm{nm},Labout15\mu \mathrm{m})\end{array}$ $\begin{array}{r}(computed:withtheterahertzindicesof\\ 5CB,n_o=1.58and{n}_e=1.77(graphene\\ paper),atiltof45degreesgives{n}_{eff}=\\ 1.667,betweenthetwo;thesameindices\\ give\nu =0.203,notthe0.2457ofthe\\ visible-lightmeasurement,sincethe\\ indicesdifferbetweenthetwofrequencies\\ (assumed))\end{array}$
Phase against gray level	$\Phi (g)={\Phi }_{max}\frac{g}{g_{max}}$ $\Delta {\Phi }_{step}=\frac{2\pi }{g_{2\pi }}$	${\Phi }_{max}=2\pi \frac{g_{max}}{g_{2\pi }}$	$\Phi :retardance$ $g:graylevel(0to255for8bits)$ $g_{max}:highestgraylevel$ $\begin{array}{r}{g}_{2\pi }:numberofgraylevelsper2\pi of\\ retardance,therelationislinearinthe\\ calibrationofMorenoetal.2021(\Phi =\pi \\ atg=100and2\pi atg=200at625\mathrm{nm}for\\ theHamamatsu792by600panel),Morenoet\\ al.2020giveabout64levelsper2\pi at\\ 520\mathrm{nm}and50at458\mathrm{nm}witharangeofup\\ to8\pi and10\pi \end{array}$ $\begin{array}{r}(computed:at625\mathrm{nm}onelevelis0.0314\\ \mathrm{rad},1.8degrees,and256levelsspan\\ 1.28\times 2\pi ;256/64=4gives8\pi at520\mathrm{nm}and\\ 256/50=5.12gives10.2\pi at458\mathrm{nm},which\\ agreewiththequoted8\pi and10\pi to\\ rounding)\end{array}$
Amplitude modulation between polarisers	$I_{\perp }={\sin }^2\frac{\Phi }{2}$ $I_{\parallel }={\cos }^2\frac{\Phi }{2}$	$\Phi =\frac{2\pi d\Delta n_{eff}}{\lambda }$ $I_{unmod}={\sin }^2\psi$	$\begin{array}{r}{I}_{\perp }and{I}_{\parallel }:transmittedfractionbetween\\ crossedandparallelpolarisers\end{array}$ $\begin{array}{r}\Phi :retardanceofthecell,thetwoforms\\ arefromMorenoetal.2021andholdfor\\ inputlightat45degreestothedirector\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}{I}_{unmod}:fractionoftheinputintensitythat\\ isnotphase-modulatedwhentheinput\\ polarisationmakesanangle\psi withthe\\ extraordinaryaxisinaphase-onlypanel\\ (backgroundknowledge),KuhnandPatapis\\ rotatedtheMeadowlarkpanelby90degrees\\ relativetotheHoloeyeonebecausethe\\ twomakersplacetheaxisdifferently\end{array}$ $\begin{array}{r}(computed:fulltransmissionbetween\\ crossedpolarisersat\Phi =\pi ,halfofa2\pi \\ stroke,and0.5at\Phi =\pi /2;amisalignment\\ of5degreesleaves0.76percentofthe\\ intensityunmodulated)\end{array}$

Reorientation, Threshold and Switching Time

Name	Formula	Related Formulae	Variables/Constants
Freedericksz threshold voltage	$V_{th}=\pi \sqrt{\frac{K_{11}}{{\epsilon }_0\Delta \epsilon }}$	$\Delta \epsilon ={\epsilon }_{\parallel }-{\epsilon }_{\perp }$ $s=\frac{V^2-V_{th}^2}{V_{th}^2}$ ${\theta }_m=\sqrt{\frac{2s}{\kappa +{\rm Y}+1}}$	$\begin{array}{r}{V}_{th}:thresholdofthesplay(Freedericksz)\\ reorientationofaplanarcell\end{array}$ $K_{11}:splayelasticconstant$ ${\epsilon }_0:vacuumpermittivity8.854\times 10^{-12}F/m$ $\begin{array}{r}\Delta \epsilon :dielectricanisotropy,thethreshold\\ isindependentofthecellgap\end{array}$ $s:reducedvoltagejustabovethreshold$ $\begin{array}{r}{\theta }_m:mid-celltilt,{\rm Y}=\Delta \epsilon /{\epsilon }_{\perp }and\kappa :\\ elasticanisotropyparameter(Caussarieu\\ etal.,Eq.4),5CBat25{}^{\circ }C:K_{11}=\\ 5.24\times 10^{-12}\mathrm{N}and\Delta \epsilon =11.92(graphene\\ paper)\end{array}$ $\begin{array}{r}(computed:theformulagives0.7000\mathrm{V}for\\ 5CB,inagreementwiththe0.7\mathrm{V}ofthe\\ graphenepaper;Caussarieuetal.measure\\ 0.711\mathrm{V}fromthefluctuationsand0.705\mathrm{V}\\ fromanumericalfit,1.6and0.7percent\\ higher)\end{array}$
Relaxation, turn-on and turn-off times	${\tau }_0=\frac{{\gamma }_1{d}^2}{K_{11}{\pi }^2}$ ${\tau }_{on}=\frac{{\tau }_0}{(V_2/V_{th}{)}^2-1}$ ${\tau }_{off}=\frac{{\tau }_0}{|(V_1/V_{th}{)}^2-1|}$	${\tau }_{total}={\tau }_{on}+{\tau }_{off}$ $f_{max}\approx \frac{1}{{\tau }_{total}}$ ${\tau }_{refl}\approx \frac{{\tau }_{trans}}{4}$	${\tau }_0:freerelaxationtime$ ${\gamma }_1:rotationalviscosity$ $K_{11}:splayelasticconstant$ $d:cellgap$ $\begin{array}{r}{V}_{1}and{V}_2:voltagesbeforeandafterthe\\ switchingstep\end{array}$ $V_{th}:thresholdvoltage$ $\begin{array}{r}{\tau }_{on}and{\tau }_{off}:turn-onandturn-off\times (Yanget\\ al.,Eqs.3to5,andZouetal.,Eqs.4\\ to6)\end{array}$ $\begin{array}{r}{f}_{max}:frameratethattheresponseallows,\\ thelastformismyreadingofthetable\\ ofYangetal.,inwhichthereflective\\ cellhashalfthegapofthetransmissive\\ cellforthesame2\pi \end{array}$ $\begin{array}{r}(computed:thereflectiveentriesofYang\\ etal.are0.249to0.250ofthe\\ transmissiveones,ratios4.02(48.2to\\ 12.0\mathrm{ms}),4.00(17.32to4.33),4.00\\ (40.22to10.06)and4.01(27.7to6.9);\\ 1/3.14\mathrm{ms}=318\mathrm{Hz},and240\mathrm{Hz}allows4.17\\ \mathrm{ms}inall;for{V}_1=0theturn-offtime\\ equals{\tau }_0)\end{array}$
Temperature dependence and figure of merit	$\Delta n=\Delta n_0{\left(1-\frac{T}{T_c}\right)}^{\beta }$ $\frac{{\gamma }_1}{K_{11}}=\frac{A\exp \left(\frac{E_a}{k_{B}T}\right)}{{\left(1-\frac{T}{T_c}\right)}^{\beta }}$ $\mathrm{FoM}=\frac{\Delta n^2}{{\gamma }_1/K_{11}}$	$\Delta n(\lambda )=\frac{G{\lambda }^2{\lambda }^{\ast 2}}{{\lambda }^2-{\lambda }^{\ast 2}}$	$\Delta n_0:birefringenceatzerotemperature$ $T:temperaturein\mathrm{K}$ $T_c:clearingtemperature$ $\beta :exponent$ $A:constant$ $E_a:activationenergy$ $k_B:Boltzmannconstant1.380649\times 10^{-23}J/K$ $\begin{array}{r}FoM:figureofmeritofYangetal.and\\ Zouetal.\end{array}$ $\begin{array}{r}G:proportionalityconstantofthe\\ single-banddispersion\end{array}$ $\begin{array}{r}{\lambda }^{\ast }:resonancewavelength(thedispersion\\ formisEq.2ofZouetal.;theconstants\\ oftheirTables1and2werenotinthe\\ textread)\end{array}$ $\begin{array}{r}(fromZouetal.:operatingA4907at40{}^{\circ }C\\ nearlydoublesthefigureofmerit\\ comparedwith22{}^{\circ }Candthefigureof\\ meritpeaksabout20{}^{\circ }Cbelow{T}_c;computed\\ fromYangetal.:UCF-216witha5.01\mu \mathrm{m}\\ gapspeedsupfrom40.22to25.94\mathrm{ms}\\ between40and50{}^{\circ }C,afactor1.55)\end{array}$
Gap and speed against wavelength	$d_{2\pi }(\lambda )=\frac{\lambda }{2\Delta n_{eff}(\lambda )}$ $\tau \propto d_{2\pi }^2$	$\frac{d_{2\pi }({\lambda }_2)}{d_{2\pi }({\lambda }_1)}=\frac{{\lambda }_2}{{\lambda }_1}\frac{\Delta n_{eff}({\lambda }_1)}{\Delta n_{eff}({\lambda }_2)}$	$\begin{array}{r}{d}_{2\pi }:reflectivegapfora2\pi stroke,\Delta \\ n_{eff}(\lambda ):usablechangeoftheeffective\\ indexatthewavelength\lambda \end{array}$ $\begin{array}{r}\tau :responsetimeatafixedliquidcrystal\\ anddrive,thesecondformisthe{d}^2\\ dependenceof{\tau }_{0}above,quoted\\ standard-speedresponsetimesofthe\\ MeadowlarkE-series:atmost13.4\mathrm{ms}at\\ 405\mathrm{nm},14.0at532,14.5at635,20.5at\\ 785,25.0at1064and43.0at1550\mathrm{nm}\\ (datasheet;ambiguoustablelayout)\end{array}$ $\begin{array}{r}(computed:theA4907gapsofZouetal.\\ growby2.91/1.88=1.548from633to905\\ \mathrm{nm},andthewavelengthratio1.430\\ multipliedbythebirefringenceratio\\ 0.200/0.185gives1.546;theMeadowlark\\ timegrowsby3.21from405to1550\mathrm{nm},\\ againstawavelengthratioof3.83anda\\ squaredratioof14.6,anexponentof\\ 0.87)\end{array}$

Pixelated Phase Gratings and Diffraction Efficiency

Name	Formula	Related Formulae	Variables/Constants
Blazed-grating efficiency against phase depth	${\eta }_1={\mathrm{sinc}}^2\left(1-\frac{{\phi }_0}{2\pi }\right)$	${\eta }_a(M)={\mathrm{sinc}}^2(M-a)$ $\mathrm{sinc}(u)=\frac{\sin (\pi u)}{\pi u}$	$\begin{array}{r}{\eta }_1:first-orderefficiencyofanideal,\\ continuousblazedprofile\end{array}$ $\begin{array}{r}{\phi }_0:phasedepthofthegrating,100per\\ centat{\phi }_0=2\pi (Vargaetal.,Eq.4)\end{array}$ $a:diffractionorder$ $\begin{array}{r}M:depthinunitsof2\pi (Morenoetal.\\ 2020,Eq.4withNtendingtoinfinity)\end{array}$ $\begin{array}{r}(computed:{\phi }_0=0.5\times 2\pi gives40.5per\\ cent,0.75\times 2\pi gives81.1,0.9\times 2\pi gives\\ 96.8and0.95\times 2\pi gives99.2percent,soa\\ panelthatfalls10percentshortof2\pi \\ losesabout3percentofthelight)\end{array}$
Efficiency with N phase levels per period	${\eta }_1={\mathrm{sinc}}^2\left(\frac{1}{N}\right)={\left[\frac{\sin (\pi /N)}{\pi /N}\right]}^2$	${\phi }_0=\frac{N-1}{N}2\pi$ ${\varphi }_{shift}=\frac{2\pi \delta }{\Lambda }$	$\begin{array}{r}N:numberofdiscretephasestepsinone\\ gratingperiod,equaltotheperiodin\\ pixelsforaone-pixelstep(thefirst\\ formisEq.6ofMorenoetal.2020,the\\ classicalrelation)\end{array}$ $\begin{array}{r}{\phi }_0:maximumphasedepthwithNlevels\\ (Vargaetal.)\end{array}$ $\delta :lateralshiftofthegratinginpixels$ $\Lambda :gratingperiodinpixels$ $\begin{array}{r}{\varphi }_{shift}:phasethatalateralshiftaddsto\\ thefirstorder,minimumstep2\pi /\Lambda ,and\\ Vargaetal.used\Lambda =4,8and16and\\ chose16\end{array}$ $\begin{array}{r}(computed:N=2gives40.5percent,3\\ gives68.4,4gives81.1,5gives87.5,8\\ gives95.0,16gives98.7and200gives\\ 99.99percent;Morenoetal.quote40.5\\ and68.4percentforN=2and3;ashift\\ ofonepixelina16-pixelperiodis22.5\\ degrees)\end{array}$
Binary grating and the sinusoidal limit	$i_0={\cos }^2\frac{\phi }{2}$ $i_1=\frac{4}{{\pi }^2}{\sin }^2\frac{\phi }{2}$	$i_m=J_m^2({\phi }_{max})$	$\begin{array}{r}{i}_{0}and{i}_1:zero-andfirst-order\\ intensitiesofabinaryphasegratingof\\ depth\phi (Morenoetal.2021)\end{array}$ $\begin{array}{r}{i}_m:ordermofasinusoidalphasegrating\\ ofpeakphase{\phi }_{max}\end{array}$ $\begin{array}{r}{J}_m:Besselfunctionofthefirstkind(the\\ fullyblurredlimitofthefringing\\ field),maximaof40.5,33.9and23.6per\\ centforthefirstorder(binary,\\ sinusoidal)andthesecondorder\\ (sinusoidal)arequotedbyMorenoetal.\\ 2021\end{array}$ $\begin{array}{r}(computed:4/{\pi }^2=40.53percentat\phi =\pi ;\\ themaximumof{J}_1^{2}is33.86percentat\\ {\phi }_{max}=1.841andthatof{J}_2^{2}is23.67per\\ centat3.054,whichmatchthequoted33.9\\ and23.6percent)\end{array}$
Fringing-field smoothing of a binary phase pattern	$\phi (x)=M\pi \frac{1}{C}\arctan \left[a\cos (\gamma x)\right]$ $\gamma =\frac{2\pi }{\Lambda }$	$H(p)=\exp \left[-{\left(\frac{|p|}{\sigma }\right)}^m\right]$	$\begin{array}{r}\phi (x):smoothedphaseofthepatternalong\\ thegratingdirection\end{array}$ $a:roundnessparameter(largeforsharpedges)$ $\begin{array}{r}C:normalisationofthearctan,takenas\arctan \\ a(myreading)\end{array}$ $M:depthfactor$ $\begin{array}{r}\Lambda :gratingperiodinpixels,H(p):\\ broadeningkernelinFourierspace\end{array}$ $p:spatialfrequency$ $\begin{array}{r}\sigma andm:fitparameters,fromMorenoet\\ al.2021(HamamatsuX10468-01,20\mu \mathrm{m}\\ pitch,625\mathrm{nm}):a=20.2,12.2,8.8and\\ 3.4forperiodsof10,6,4and2pixels,\\ aneffectivephaseof94percentat4\\ pixelsand89percentat2pixels,kernel\\ fitm=1.21,\sigma =0.74pixels\end{array}$ $\begin{array}{r}(myreconstructionoftheprofilewitha\\ swingof\pi giveseffectivephasesof90\\ and83percent(zeroorder)or92and85\\ percent(firstorder)at4and2pixels,\\ not94and89,soIcouldnotreproduce\\ thedefinitionoftheeffectivephaseand\\ quotethepape{r}^{\prime }svaluesasread)\end{array}$
Deflection angle and steering limit	$\sin {\theta }_1=\frac{\lambda }{\Lambda p_x}$	$\sin {\theta }_{max}=\frac{\lambda }{2p_x}$	$\begin{array}{r}{\theta }_1:first-orderdeflectionofablazed\\ gratingofperiod\Lambda pixels\end{array}$ $p_x:pixelpitch$ $\lambda :wavelength$ $\begin{array}{r}{\theta }_{max}:limitattheNyquistperiodoftwo\\ pixels,wherefringingisstrongest(the\\ gratingequationisbackgroundknowledge)\end{array}$ $\begin{array}{r}(computed:for8\mu \mathrm{m}pitchat532\mathrm{nm}the\\ limitis1.91degrees,and0.95degreesat\\ fourpixels(81percent)and0.48degrees\\ ateight(95percent);at1550\mathrm{nm}the\\ samepitchgives5.56degrees;9.2\mu \mathrm{m}at\\ 633\mathrm{nm}gives1.97,8\mu \mathrm{m}at633\mathrm{nm}gives\\ 2.27and20\mu \mathrm{m}at625\mathrm{nm}gives0.90\\ degrees)\end{array}$

Panel Parameters and Limits

Name	Formula	Related Formulae	Variables/Constants
Fill factor, pixel gap and zero order	$F=\frac{a_p}{p_x}$ $g_p=p_x(1-F)$	$I_0\approx F^2$	$F:fillfactor$ $a_p:activepixelsize$ $p_x:pixelpitch$ $g_p:gapbetweenpixelsifFisalinearratio$ $\begin{array}{r}{I}_0:intensityoftheundiffractedorder\\ causedbythepixelstructure,Morenoet\\ al.2021definethefillfactorasthe\\ ratioofpixelsizetopitchandquote{F}^2\\ =96percentforafillfactorabove98\\ percent,quotedfillfactors:Meadowlark\\ E-series95.6percentat8.0\mu \mathrm{m},Holoeye\\ PLUTO93percentat8\mu \mathrm{m},Meadowlark\\ P192095.7percentat9.2\mu \mathrm{m}(Kuhnand\\ Patapis)\end{array}$ $\begin{array}{r}(computed:0.98^2=96.0percent;the\\ E-seriesgapis0.35\mu \mathrm{m}ifFislinear,\\ and{F}^2=91.4percent;theHoloeyepanel\\ gives86.5percentagainstitsquoted\\ diffractionefficiencyofabout86per\\ cent,andtheP1920gives91.6percent\\ againstabout78percent)\end{array}$
Active area and pixel count	$A_x=N_x{p}_x$ $A_y=N_y{p}_y$	$N_{pix}=N_x{N}_y$	$N_{x}and{N}_y:pixelsalongthetwoaxes$ $p_{x}and{p}_y:pitches$ $A_{x}and{A}_y:active-areawidthandheight$ $\begin{array}{r}{N}_{pix}:numberofindependentphasepixels\\ (quotedpanels:MeadowlarkE-series1920\\ by1200at8.0\mu \mathrm{m}with15.36by9.60\mathrm{mm},\\ HoloeyePLUTO1920by1080at8\mu \mathrm{m},\\ MeadowlarkP1920at9.2\mu \mathrm{m}listedas1920\\ by1152inatableand1920by1080inthe\\ textofKuhnandPatapis,HamamatsuX10468\\ 792by600at20\mu \mathrm{m})\end{array}$ $\begin{array}{r}(computed:1920\times 8.0\mu \mathrm{m}=15.36\mathrm{mm}and\\ 1200\times 8.0\mu \mathrm{m}=9.60\mathrm{mm},asquoted;Holoeye\\ 15.36by8.64\mathrm{mm};P192017.66by10.60\mathrm{mm}\\ for1152rowsor9.94\mathrm{mm}for1080;\\ Hamamatsu15.84by12.00\mathrm{mm};theE-series\\ has2.304millionpixels)\end{array}$
Chromatic phase error	$\Delta \phi \approx {\phi }_0\frac{\Delta \lambda }{{\lambda }_0}$	$\phi (\lambda )=2\pi \frac{{\lambda }_0}{\lambda }(\Delta n\mathrm{constant})$	$\begin{array}{r}\Delta \phi :phaseerroratawavelength\Delta \lambda away\\ fromthedesignwavelength{\lambda }_0\end{array}$ $\begin{array}{r}{\phi }_0:strokeatthedesignwavelength,the\\ dispersionof\Delta nisneglected(background\\ knowledge),thephasedelayofanLCoS\\ panelischromaticbecauseitisascalar\\ delayandnotageometricphase(Kuhnand\\ Patapis),whoestimateachromaticnull\\ leakageof2.8\times 10^{-3}fora12percentband\\ at640\mathrm{nm}\end{array}$ $\begin{array}{r}(computed:a2\pi maskusedoverplusor\\ minus6percentofthewavelengthhasa\\ phaseerrorofplusorminus0.377\mathrm{rad},\\ 21.6degrees,atthebandedges)\end{array}$
Phase noise of the panel	$\delta {\phi }_{rms}=\frac{2\pi \delta {\ell }_{rms}}{\lambda }$	$\delta {\ell }_{rms}=\frac{\lambda }{2\pi }\delta {\phi }_{rms}$	$\delta {\phi }_{rms}:rmsphasejitter$ $\begin{array}{r}\delta {\ell }_{rms}:thesameexpressedasanoptical\\ pathlength\end{array}$ $\begin{array}{r}\lambda :wavelengthatwhichthejitterwas\\ specified,quotedphasejitterbelow22\mathrm{nm}\\ rmsfortheHoloeyePLUTOandbelow12\mathrm{nm}\\ rmsfortheMeadowlarkP1920(Kuhnand\\ Patapis),phaserippleoftheE-series\\ typically0.10to0.30percent,1to2\\ percentinsomeconfigurations(data\\ sheet)\end{array}$ $\begin{array}{r}(computedwith\lambda =633\mathrm{nm},thecalibration\\ wavelengthofbothpanels(assumed):22\mathrm{nm}\\ is0.218\mathrm{rad}and12\mathrm{nm}is0.119\mathrm{rad})\end{array}$

Important Questions

Ques: Is the reflective gap for $2\pi$ equal to $\lambda /(2\Delta n)$, and why are the quoted gaps larger?

A reflective LCoS cell is crossed twice, so $d_{2\pi }=\lambda /(2\Delta n_{eff})$ (Zou et al.). With the full A4907 birefringence of 0.200 at 633 $\mathrm{nm}$ this gives 1.58 $\mu \mathrm{m}$ (computed), yet the quoted gap is 1.88 $\mu \mathrm{m}$, 19 per cent larger. At 905 $\mathrm{nm}$ the same happens: 2.91 $\mu \mathrm{m}$ against 2.45 $\mu \mathrm{m}$ for 0.185 (computed). Both gaps correspond to $\Delta n_{eff}$ at 84 per cent of $\Delta n$ (computed), which I read as the director not reaching full tilt at the 5 ${\mathrm{V}}_{\mathrm{rms}}$ limit of the backplane (my reading). Yang et al. pair a 1.5 $\mu \mathrm{m}$ gap with $\Delta n$ = 0.211, which is exactly $\lambda /(2\Delta n)$ (computed), so that example ignores the incomplete tilt. I regard $\lambda /(2\Delta n_{eff})$ with $\Delta n_{eff}$ near 0.84 of $\Delta n$ as the practical rule. Reference: Zou et al., ‘Fast-Response Liquid Crystal for Spatial Light Modulator and LiDAR Applications’, Crystals 11(2):93 (2021) (fetch-tool summary; the authors after the first were not in the text read, and the tables with the fitted birefringence constants were missing); Yang, Zou, Li and Wu, ‘Fast-Response Liquid Crystal Phase Modulators with an Excellent Photostability’, Crystals 10(9):765 (2020), DOI 10.3390/cryst10090765 (fetch-tool summary).

Ques: Is the response time of the best LCoS cell 3 or 3.14 milliseconds, and does it scale as the square of the gap?

Yang et al. write ‘about 3 $\mathrm{ms}$’ at 50 ${}^{\circ }\mathrm{C}$ in the abstract and 3.14 $\mathrm{ms}$ in the body for the 2.07 $\mu \mathrm{m}$ reflective UCF-216 cell, with ${\tau }_{total}={\tau }_{on}+{\tau }_{off}$. Since 1/3.14 $\mathrm{ms}$ is 318 $\mathrm{Hz}$, matching their ‘roughly 300 $\mathrm{Hz}$’ (computed), I regard 3.14 as the value and 3 as its rounding. Their reflective entries are one quarter of the transmissive ones (ratios 4.02, 4.00, 4.00, 4.01, computed), the $d^2$ law of ${\tau }_0$ for half the gap. The law fails across drive voltages: gaps of 5.01 and 4.00 $\mu \mathrm{m}$ differ by 1.57 in $d^2$ but 2.32 in time (computed), because the turn-on factor $(V/V_{th}{)}^2-1$ differs between 2.93 and 5.0 ${\mathrm{V}}_{\mathrm{rms}}$. Zou et al. report 2.20 $\mathrm{ms}$ at 50 ${}^{\circ }\mathrm{C}$. Reference: Yang, Zou, Li and Wu, ‘Fast-Response Liquid Crystal Phase Modulators with an Excellent Photostability’, Crystals 10(9):765 (2020), DOI 10.3390/cryst10090765 (fetch-tool summary); Zou et al., ‘Fast-Response Liquid Crystal for Spatial Light Modulator and LiDAR Applications’, Crystals 11(2):93 (2021) (fetch-tool summary; the authors after the first were not in the text read, and the tables with the fitted birefringence constants were missing).

Ques: Does the response time of a commercial LCoS panel grow with the wavelength or with its square?

The Meadowlark E-series sheet lists standard-speed response times of at most 13.4 $\mathrm{ms}$ at 405 $\mathrm{nm}$, 14.0 at 532, 14.5 at 635, 20.5 at 785, 25.0 at 1064 and 43.0 at 1550 $\mathrm{nm}$. From 405 to 1550 $\mathrm{nm}$ the time grows by 3.21, the wavelength by 3.83 and its square by 14.6 (computed), an exponent of 0.87. With one liquid crystal and a gap proportional to $\lambda$ the time would follow ${\lambda }^2$ through ${\tau }_0\propto d^2$, so the vendor must change the mixture, the drive or the overdrive across the range (my reading). The summary of the sheet also warned that some rows list two times and that their assignment to coating ranges is unclear, so I would check the vendor before relying on single numbers. I regard growth close to linear in wavelength as the sheet’s message. Reference: Meadowlark Optics E-Series 1920 by 1200 analog spatial light modulator data sheet (SLM-E19x12, Rev. 0822), as hosted by Laser 2000 (fetch-tool summary; the model number was incomplete and the response-time table layout was ambiguous).

Ques: Is the fill factor a ceiling for the diffraction efficiency?

Moreno et al. define the fill factor as the ratio of pixel size to pitch and give $F^2$ = 96 per cent for a panel above 98 per cent (computed: $0.98^2$ = 96.0 per cent). For the Holoeye PLUTO, 93 per cent gives 86.5 per cent against a quoted efficiency of about 86 per cent, but the Meadowlark P1920 gives 91.6 per cent against about 78 per cent (computed), so other losses matter there. The E-series sheet lists a zero-order efficiency of 76 to 91 per cent, whose top equals $0.956^2$ = 91.4 per cent (computed), but the dielectric-mirror versions list 92 to 98 per cent, above that value. I read the sheet’s figure as normalised in some other way, so I regard $F^2$ as a rough guide and not a bound (my reading). Reference: Moreno, Sanchez-Lopez, Davis and Cottrell, ‘Simple method to evaluate the pixel crosstalk caused by fringing field effect in liquid-crystal spatial light modulators’, Journal of the European Optical Society-Rapid Publications (2021), DOI 10.1186/s41476-021-00174-7 (fetch-tool summary); Kuhn and Patapis, ‘Active focal-plane coronagraphy with liquid-crystal spatial-light modulators: Broadband contrast performance in the visible’, arXiv:2210.14000 (fetch-tool summary); Meadowlark Optics E-Series 1920 by 1200 analog spatial light modulator data sheet (SLM-E19x12, Rev. 0822), as hosted by Laser 2000 (fetch-tool summary; the model number was incomplete and the response-time table layout was ambiguous).

Ques: How much does the fringing field cost for a two-pixel grating, 94 or 89 per cent of the phase?

Moreno et al. fit a roundness parameter $a$ of 20.2, 12.2, 8.8 and 3.4 for periods of 10, 6, 4 and 2 pixels and quote an effective phase of 94 per cent at four pixels and 89 per cent at two. My reconstruction of their profile with a swing of $\pi$ gives 90 and 83 per cent from the zero order, or 92 and 85 per cent from the first order (computed), so I could not reproduce either figure; the summary does not give the normalisation constant or the definition of the effective phase (my reading). Their other numbers are reproduced: the sinusoidal limits are 33.86 and 23.67 per cent from the Bessel functions (computed), against 33.9 and 23.6 quoted. I regard 94 and 89 per cent as the paper’s values and my reconstruction as incomplete. Reference: Moreno, Sanchez-Lopez, Davis and Cottrell, ‘Simple method to evaluate the pixel crosstalk caused by fringing field effect in liquid-crystal spatial light modulators’, Journal of the European Optical Society-Rapid Publications (2021), DOI 10.1186/s41476-021-00174-7 (fetch-tool summary).

Ques: Why does an 8-bit panel give only 81 per cent in a four-pixel blazed grating?

The efficiency ${\mathrm{sinc}}^2(1/N)$ depends on the number $N$ of phase steps in one period, which for a one-pixel step is the period in pixels, and not on the bit depth. With 200 usable levels per $2\pi$ at 625 $\mathrm{nm}$ (Moreno et al. 2021) the continuous limit would be 99.99 per cent, but a four-pixel period shows four steps and so 81.1 per cent; eight pixels give 95.0 and sixteen give 98.7 per cent (computed). The price of finer steps is a smaller angle: for an 8 $\mu \mathrm{m}$ pitch at 532 $\mathrm{nm}$ the deflection is 1.91 degrees at two pixels, 0.95 at four and 0.48 at eight (computed). Efficiency and steering range trade against each other, and the Meadowlark sheet shows the first-order efficiency only as a plot against the ramp period (fetch-tool summary). Reference: Moreno, Gutierrez, Sanchez-Lopez, Davis, Khanal and Cottrell, ‘Diffraction efficiency of stepped gratings using high phase-modulation spatial light modulator’ (2020), DOI 10.1016/j.optlaseng.2019.105910 (fetch-tool summary; the journal, Optics and Lasers in Engineering, is inferred from the DOI and ISSN, my reading, and Eq. 7 of the summary is garbled); Moreno, Sanchez-Lopez, Davis and Cottrell, ‘Simple method to evaluate the pixel crosstalk caused by fringing field effect in liquid-crystal spatial light modulators’, Journal of the European Optical Society-Rapid Publications (2021), DOI 10.1186/s41476-021-00174-7 (fetch-tool summary); Meadowlark Optics E-Series 1920 by 1200 analog spatial light modulator data sheet (SLM-E19x12, Rev. 0822), as hosted by Laser 2000 (fetch-tool summary; the model number was incomplete and the response-time table layout was ambiguous).

Ques: Is the Freedericksz threshold of 5CB 0.70 or 0.71 volt?

The terahertz graphene paper gives $V_{th}=\pi \sqrt{K_{11}/({\epsilon }_0\Delta \epsilon )}$ = 0.7 $\mathrm{V}$ for $K_{11}$ = $5.24\times 10^{-12}$ $\mathrm{N}$ and $\Delta \epsilon$ = 11.92, and I get 0.7000 $\mathrm{V}$ (computed). Caussarieu et al. measure $0.711\pm 0.001$ $\mathrm{V}$ from the director fluctuations and $0.705\pm 0.003$ $\mathrm{V}$ from a numerical fit, 1.6 and 0.7 per cent higher (computed). Their threshold is written $U_c^2={\pi }^2{k}_1/{ϵ}_a$ without ${\epsilon }_0$, which is dimensionally consistent only if ${ϵ}_a$ is an absolute permittivity difference (my reading). Their cell has a pretilt of about 0.05 $\mathrm{rad}$ and finite anchoring, which can raise a measured threshold (my reading), and the elastic constant of a particular sample can differ. I regard the formula as good to about 2 per cent and independent of the gap. Reference: Terahertz liquid-crystal phase shifter with graphene electrodes, arXiv:1309.1595 (fetch-tool summary; the authors and the title were not read, and equations 1, 3 and 4 were garbled); Caussarieu, Petrosyan and Ciliberto, measurement of the Freedericksz threshold of a planar 5CB cell from director fluctuations, arXiv:1308.4702 (fetch-tool summary; the title was not read).

Ques: How can the same panel act as a phase or an amplitude modulator?

The retardance $\Phi =2\pi d\Delta n_{eff}/\lambda$ is the same in both cases; what differs is the input polarisation. Along the extraordinary axis the light stays linear and only its phase changes, which needs a stroke of $2\pi$ (Yang et al.). At 45 degrees to the axis and between crossed polarisers the intensity is ${\sin }^2(\Phi /2)$ (Moreno et al. 2021), so full transmission needs only $\Phi =\pi$, half the stroke (computed). Alignment matters: a polarisation 5 degrees off the axis leaves ${\sin }^2{5}^{\circ }$ = 0.76 per cent of the intensity unmodulated (computed, background knowledge), a zero-order leak. The axis also differs between makers, so Kuhn and Patapis turned the Meadowlark panel by 90 degrees relative to the Holoeye one. Reference: Yang, Zou, Li and Wu, ‘Fast-Response Liquid Crystal Phase Modulators with an Excellent Photostability’, Crystals 10(9):765 (2020), DOI 10.3390/cryst10090765 (fetch-tool summary); Moreno, Sanchez-Lopez, Davis and Cottrell, ‘Simple method to evaluate the pixel crosstalk caused by fringing field effect in liquid-crystal spatial light modulators’, Journal of the European Optical Society-Rapid Publications (2021), DOI 10.1186/s41476-021-00174-7 (fetch-tool summary); Kuhn and Patapis, ‘Active focal-plane coronagraphy with liquid-crystal spatial-light modulators: Broadband contrast performance in the visible’, arXiv:2210.14000 (fetch-tool summary).

Ques: Is the Meadowlark P1920 panel 1920 by 1152 or 1920 by 1080 pixels?

Kuhn and Patapis list 1920 by 1152 pixels at 9.2 $\mu \mathrm{m}$ pitch for the Meadowlark P1920-0635-HDMI in their Table 1, but write 1920 by 1080 in the text, as they do for the Holoeye PLUTO. The active heights are 10.60 $\mathrm{mm}$ for 1152 rows and 9.94 $\mathrm{mm}$ for 1080 (computed), and the width is 17.66 $\mathrm{mm}$. The two agree on the width but not on the number of independent phase pixels, 2.21 million against 2.07 million (computed). I regard the table as more likely correct because it is the dedicated specification and the text appears to reuse the Holoeye format, but the summary does not show the vendor data, so this is a reading and not a check (my reading). For the E-series sheet, 1920 by 1200 at 8.0 $\mu \mathrm{m}$ gives 15.36 by 9.60 $\mathrm{mm}$ (computed), as quoted. Reference: Kuhn and Patapis, ‘Active focal-plane coronagraphy with liquid-crystal spatial-light modulators: Broadband contrast performance in the visible’, arXiv:2210.14000 (fetch-tool summary); Meadowlark Optics E-Series 1920 by 1200 analog spatial light modulator data sheet (SLM-E19x12, Rev. 0822), as hosted by Laser 2000 (fetch-tool summary; the model number was incomplete and the response-time table layout was ambiguous).

Related pages in this vault: Acousto-Optic Modulator, Magneto-Optic Modulator, Electro-Optic Phase Modulator, Thermo-Optic Modulator, MEMS Optical Modulator.