1. Theory

Definition: A MEMS optical modulator is a light modulator in which a micromachined mechanical element, driven by an electrostatic force, changes the amplitude, phase or direction of the light. The moving element is a suspended waveguide, a micromirror, a ribbon or a mirror of a cavity (Errando-Herranz et al.; Younse; Bloom).
Actuator: A movable plate of area $A$ held by a spring of stiffness $k$ over a gap $g_{0}$ feels the force $F_{e} = ε_{0} A V^{2} / [2 (g_{0} - x)^{2}]$ and balances it with $k x$ . Equilibrium becomes unstable at $x = g_{0} /3$ and at $V_{P I} = 8 k g_{0}^{3} / (27 ε_{0} A)$ ; beyond that the plate collapses (Raghuram et al.; Bifano). The force is attractive only, so the release relies on the spring.
Dynamics: The plate is a mass on a spring with a resonance $f_{0} = k / m / (2 π)$ and a quality factor set by the air film. Photonic devices resonate at roughly 100 $kHz$ to 10 $MHz$ (Errando-Herranz et al.), so MEMS are slower than carrier-based or electro-optic modulators and faster than most heaters of the Thermo-Optic Modulator page.
Three families: Gap-tuned waveguide phase shifters change the effective index by a few nanometres of motion (Sattari et al.; Edinger et al.). Digital micromirrors tilt and land on a stop (Younse; Texas Instruments). Grating light valves deflect ribbons by a quarter wavelength to switch a diffraction grating (Bloom).
Figures of merit: Drive voltage or $V_{π} L_{π}$ , insertion loss, actuation bandwidth, static power (zero for an electrostatic actuator), footprint, contrast and switching time. The silicon phase shifter of Edinger et al. has $V_{π}$ = 10.7 $V$ , $L_{π}$ = 17.2 $μ m$ , 0.33 $dB$ and 1.03 $MHz$ (Edinger et al.).
Scope: This page covers electrostatic MEMS used as optical modulators, switches and phase shifters. The heater-based tuning is on the Thermo-Optic Modulator page, the pixelated liquid-crystal panels on the Liquid-Crystal Spatial Light Modulator page and the crystal modulators on the Electro-Optic Phase Modulator page; magneto-optic modulators have their own page.

2. Working Principle

Electrostatic pull: A voltage between the movable and the fixed electrode charges the capacitor $C = ε_{0} A / (g_{0} - x)$ and stores $C V^{2} /2$ ; the attraction closes the gap until the spring force balances it (Raghuram and Ajoy). Pull-in at $g_{0} /3$ limits the stable analog travel; a step applied suddenly collapses the plate already at the lower dynamic pull-in voltage $V_{D P I} = k g_{0}^{3} / (4 ε_{0} A)$ , 0.919 times $V_{P I}$ (computed).
Hysteresis: Once the plate rests on stoppers the voltage can be lowered until the pull-out voltage $V_{P O}$ before it lets go, so the response is hysteretic. For the actuator of Raghuram et al. the three values are 20.33, 18.67 and 18.0 $V$ (computed).
Phase shifter: A suspended tapered waveguide is pulled towards a fixed bus waveguide in an adiabatic coupler; a few tens of nanometres change the effective index of the supermode and give a phase of $π$ with 19 $V$ (Sattari et al.). The relation $L_{π} Δ n_{e f f} = λ /2$ links the length, the index change and the phase (Errando-Herranz et al.).
Micromirror: A square mirror on a torsion hinge tilts by about 10 to 12 degrees towards one of two landing pads under the bias of an address electrode, and the beams of the two states leave 4 times the tilt apart (Younse; Texas Instruments; the hinge, the landing pads and the beam geometry are background knowledge). A pixel is on or off, and grey is made by time-weighting the binary pattern.
Grating light valve: Alternate ribbons of a reflective grating are pulled down by $λ /4$ , which adds $π$ to their reflected light; the flat mirror then sends the light to the zero order and the deflected state sends 81 per cent into the two first orders (Bloom; the grating algebra is the binary case of the Liquid-Crystal Spatial Light Modulator page).
Cavity: A movable mirror changes the gap of a Fabry-Perot cavity and shifts its resonance in proportion to the gap (background knowledge; no MEMS source was read for this).
Speed and damping: The air film under the plate damps the motion; its quality factor has a strong effect on the release time and less on the switching time (Shekhar et al.). Drive voltages stay below 100 $V$ for phase shifters (Errando-Herranz et al.).

3. Applications

Programmable photonic circuits: Meshes of phase shifters and couplers need hundreds of low-loss, low-power elements, and an electrostatic phase shifter holds its phase with no static power (Errando-Herranz et al.; Edinger et al.).
Optical switches: Out-of-plane directional-coupler switches have reached 50 by 50 and 240 by 240 matrices, and an in-plane tunable coupler an 8 by 8 matrix (Errando-Herranz et al.).
Beam steering: A tunable grating coupler steered by 5.6 degrees at 20 $V$ in an experiment, and the authors estimate up to 90 degrees and 200 $kHz$ in simulation (Errando-Herranz et al.).
Tunable grating couplers: One device shifted its peak by 22.8 $nm$ with 12 $V$ (Errando-Herranz et al.).
Projection and displays: The micromirror array of 1920 by 1080 pixels and the grating light valve write images by spatial light modulation (Texas Instruments; Bloom).
Printing and lithography: Both the micromirror and the grating light valve serve as dynamic masks and print heads (background knowledge for the micromirror; Bloom and Wikipedia for the grating light valve).
Adaptive optics: Deformable-mirror actuators with 8-bit resolution and no hysteresis reach nanometre repeatability (Bifano).
Quantum photonics: Strain tuning of emitters and tunable photonic-crystal cavities are named as uses of MEMS (Errando-Herranz et al.).

Electrostatic Actuation of a MEMS Element

Name	Formula	Related Formulae	Variables/Constants
Electrostatic force and static equilibrium of a parallel-plate actuator	$F_{e} = \frac{ε _{0} A V ^{2}}{2 ( g _{0} - x ) ^{2}}$ $k x = F_{e}$	$C (x) = \frac{ε _{0} A}{g _{0} - x}$ $U = \frac{1}{2} C V^{2}$	$F_{e} : e l ec t r os t a t i c f or ce$ $ε_{0} : v a c uu m p er mi tt i v i t y 8.854 \times 1 0^{- 12} F/m$ $A : e l ec t r o d e a r e a$ $V : a ppl i e d v o l t a g e$ $g_{0} : ini t ia l g a p$ $x : d i s pl a ce m e n t o f t h e m o v ab l e pl a t e$ $k : s p r in g co n s t an t o f t h e s u s p e n s i o n$ $C : c a p a c i t an ce$ $U : s t or e d e n er g y (B i f an o w i t h L w f or A an d ε_{r} = 1 f or ai r, an d E q . 1 o f R a g h u r am an d A j oy), t h e e q u i l ib r i u m co n d i t i o n i s a c u bi c in x an d g i v es t h e d i s pl a ce m e n t a s a f u n c t i o n o f V^{2} (B i f an o)$ $(co m p u t e d w i t h t h e a c t u a t or o f R a g h u r am e t a l ., A = 960 \times 1 0^{- 12} m^{2}, g_{0} = 3 μ m, k = 0.439 N/m : C (0) = 2.83 fF, an d 4.25 fF a t t h e s t a t i c p u l l - in d i s pl a ce m e n t g_{0} /3; B i f an o q u o t es ab o u t 100 fF f or a d e f or mab l e - mi r r or a c t u a t or, w hi c h a t 150 an d 200 V s t or es 1.1 an d 2.0 nJ)$
Pull-in, dynamic pull-in and pull-out voltages	$V_{P I} = \frac{8 k g _{0}^{3}}{27 ε _{0} A}, x_{P I} = \frac{g _{0}}{3}$	$V_{D P I} = \frac{k g _{0}^{3}}{4 ε _{0} A}, x_{D P I} = \frac{g _{0}}{2}$ $V_{P O} = \frac{2 k h _{s}^{2} ( g _{0} - h _{s} )}{ε _{0} A}$ $V_{c} \approx \frac{5 E t ^{3} g _{0}^{3}}{ε _{0} L ^{4}}$	$V_{P I} : s t a t i c p u l l - in v o l t a g e, ab o v e w hi c h t h e pl a t e co l l a p ses$ $x_{P I} : d i s pl a ce m e n t a t p u l l - in, o n e t hi r d o f t h e g a p an d in d e p e n d e n t o f t h e s t i f f n ess$ $V_{D P I} an d x_{D P I} : d y nami c p u l l - in v o l t a g e an d d i s pl a ce m e n t f or a s u dd e n l y a ppl i e d s t e p w i t h o u t d am p in g$ $V_{P O} : p u l l - o u t (r e l e a se) v o l t a g e w h e n t h e pl a t e r es t s o n s t o pp er s o f h e i g h t h_{s}, t h e f i r s t t h r ee a r e T ab l e I I I o f R a g h u r am e t a l . an d E q s . 2 an d 3 o f R a g h u r am an d A j oy$ $V_{c} : p u l l - in o f a f i x e d - f i x e d b e am o f l e n g t h L, t hi c k n ess t an d Y o u n g m o d u l u s E, w i t h a s t i f f n ess o f ab o u t 192 E I / L^{3} (B i f an o; t h e p r in t e d seco n d f or m i s g a r b l e d), t h e b e am r e a c h es ab o u t 0.4 g_{0} b e f or e co l l a p se (B i f an o)$ $(co m p u t e d f or A = 960 \times 1 0^{- 12} m^{2}, g_{0} = 3 μ m, k = 0.439 N/m, h_{s} = 1.4 μ m : V_{P I} = 20.33 V, V_{D P I} = 18.67 V, V_{P O} = 18.00 V, e q u a l t o t h e q u o t e d 20.33, 18.67 an d 18 V; V_{D P I} / V_{P I} = 27/32 = 0.919; B i f an o^{'} s e x am pl e E = 170 GPa, t = 3 μ m, g_{0} = 5 μ m, L = 300 μ m g i v es 200.0 V w i t h t h e f a c t or 5, an d 194.7 V w i t h 128/27 = 4.74, w hi c h f o l l o w s f r o m k = 16 E w t^{3} / L^{3} an d an e l ec t r o d e a r e a L w (m y d er i v a t i o n))$
Resonance, quality factor and settling of the moving element	$f_{0} = \frac{1}{2 π} \frac{k}{m}$ $m \overset{x}{¨} + c \overset{x}{˙} + k x = F_{e}$	$Q = \frac{k m}{c}$ $c = 2 m ω_{0} ζ$	$f_{0} : u n d am p e d r eso nan ce f r e q u e n cy$ $m : m o v in g ma ss$ $k : s p r in g co n s t an t$ $c : d am p in g coe f f i c i e n t$ $Q : q u a l i t y f a c t or, ω_{0} = 2 π f_{0}$ $ζ : d am p in g r a t i o (t h e e q u a t i o n o f m o t i o n i s E q . 1 o f R a g h u r am an d A j oy an d t h e d am p in g f or m i s f r o m R a g h u r am e t a l .; Q = k m / c i s ba c k g r o u n d k n o w l e d g e), t h e s q u eez e f i l m o f ai r u n d er t h e pl a t e i s t h e u s u a l so u r ce o f c in ai r (ba c k g r o u n d k n o w l e d g e), S h e k ha r e t a l . s t a t e t ha t t h e q u a l i t y f a c t or ha s an a pp r ec iab l e e f f ec t o n t h e r e l e a se t im e b u t l ess o n t h e s w i t c hin g t im e f or S O I p a r a l l e l - b e am s w i t c h es a t 10 t o 30 V in ai r, E r r an d o - H er r an z e t a l . : p h o t o ni c M E M S r eso na t e a t r o ug h l y 100 kHz t o 10 MHz, o u t - o f - pl an e e l ec t r os t a t i c a c t u a t or s e x cee d 2 MHz an d in - pl an e o n es a r e n e a r 100 kHz, an d s u b - 10 V d e v i ces t e n d t o ha v e s t i f f n ess b e l o w 0.1 N/m an d so a r e m or e se n s i t i v e t o n o i se$ $(co m p u t e d f or t h e a c t u a t or o f R a g h u r am e t a l ., k = 0.439 N/m an d m = 4.4736 \times 1 0^{- 12} kg : f_{0} = 49.9 kHz, a p er i o d o f 20.1 μ s; n o d am p in g w a s u se d in t ha t w or k)$

Silicon-Photonic MEMS Phase Shifters

Name	Formula	Related Formulae	Variables/Constants
Phase shift of a MEMS-tuned waveguide	$Δ φ = \frac{2 π L}{λ} Δ n_{e f f} (g)$ $L_{π} Δ n_{e f f} = \frac{λ}{2}$	$φ = \frac{2 π n _{e f f} L}{λ}$ $S_{g} = \frac{Δ φ}{Δ g}$	$Δ φ : p ha se c han g e$ $L : in t er a c t i o n l e n g t h$ $λ : f r ee - s p a ce w a v e l e n g t h, Δ n_{e f f} (g) : c han g e o f t h e e f f ec t i v e in d e x o f t h e s u p er m o d e a s t h e g a p g b e tw ee n t h e s u s p e n d e d an d t h e f i x e d w a v e g u i d e c han g es$ $L_{π} : l e n g t h f or a p ha se o f π$ $V_{π} : v o l t a g e f or a p ha se o f π$ $S_{g} : p ha se p er u ni t g a p c han g e (m y d e f ini t i o n), t h e p ha se i s c han g e d t h r o ug h t h e e f f ec t i v e in d e x or t h e l e n g t h (E r r an d o - H er r an z e t a l ., E q . 2), S a tt a r i e t a l . : an a d iaba t i c co u pl er p u l l s a s u s p e n d e d t a p er e d w a v e g u i d e t o a f i x e d b u s w a v e g u i d e an d t u n es t h e v er t i c a l g a p, π a t 19 V w i t h a d i s pl a ce m e n t o f 19 nm, o v er 1.5 t o 1.6 μ m, w i t h an a v er a g e in ser t i o n l oss b e l o w 0.3 dB (2019), t h e 2017 co n f er e n ce p a p er g i v es 20 nm, 19 V, π an d an e x cess l oss b e l o w 0.1 dB, a f oo tp r in t o f 60 b y 44 μ m an d a g a p r an g e o f 150 t o 185 nm in t h e pl o tt e d s im u l a t i o n (C O M S O L p a g e), E d in g er e t a l . (im ec i S i P P 50 G f o u n d r y pl a t f or m, 1550 nm) : p ha se 2.9 π pl u s or min u s π, V_{π} = 10.7 V (+ 2.2, - 1.4), L_{π} = 17.2 μ m (+ 8.8, - 4.3), in ser t i o n l oss 0.33 dB, a c t u a t i o n ban d w i d t h 1.03 MHz in ai r$ $(co m p u t e d : w i t h L_{π} = 17.2 μ m a t 1550 nm (a ss u m e d), Δ n_{e f f} = λ / (2 L_{π}) = 0.0451; t h e p ha se p er g a p c han g e o f S a tt a r i e t a l . i s π /19 nm = 0.165 rad p er nm, 9.5 d e g r ees p er nm (9.0 w i t h 20 nm); V_{π} L_{π} = 184 V μ m f or E d in g er e t a l .)$
Tunable coupler, ring shift and insertion loss	$\frac{P _{2}}{P _{1}} = sin^{2} (\frac{π L Δ n _{e}}{λ})$	$ΔΦ = 2 π \frac{Δ λ}{FSR}$ $IL [dB] = α [dB/cm] C$	$P_{2} / P_{1} : p o w er r a t i o o f t h e tw o o u tp u t s o f a d i r ec t i o na l co u pl er$ $L : co u pl in g l e n g t h$ $Δ n_{e} : in d e x d i f f er e n ce b e tw ee n t h e tw o s u p er m o d es (E q . 3 o f E r r an d o - H er r an z e t a l .)$ $Δ λ : r eso nan ce s hi f t o f a r in g$ $F S R : f r ee s p ec t r a l r an g e$ $ΔΦ : p ha se s hi f t, t h e p a p er w r i t es ΔΦ = Δ λ / FSR an d I a dd e d t h e f a c t or 2 π, a s f or t h e T h er m o - O pt i c M o d u l a t or p a g e (m y r e a d in g)$ $I L : in ser t i o n l oss$ $α : p r o p a g a t i o n l oss$ $C : r o u n d - t r i p l e n g t h, E r r an d o - H er r an z e t a l . : s t an d a r d s t r i p w a v e g u i d es l ose 1 t o 2 dB/cm, t r an s i t i o n s o f f u l l y s u s p e n d e d s in g l e - e t c h d e v i ces 0.1 t o 1 dB, t a p er e d an c h or s 0.05 dB an d c l a dd in g c han g es 0.1 dB in t h e a ss u m e d b u d g e t, w i t h a t a r g e t b e l o w 0.1 dB$ $(co m p u t e d : a f u l l t r an s f er b e tw ee n t h e o u tp u t s n ee d s Δ n_{e} L = λ /2, t h e s am e co n d i t i o n a s t h e L_{π} ab o v e; a M E M S p ha se s hi f t er w i t h t h e 0.33 dB o f E d in g er e t a l . l oses 7.3 p er ce n t o f t h e p o w er)$
Drive voltage, energy and speed of MEMS compared with heating	$E_{s w} = \frac{1}{2} C V^{2}$ $f_{3 d B} \approx f_{0} (Q \leq 1)$	$P_{s t a t i c} \approx 0$	$E_{s w} : e n er g y t o c ha r g e t h e a c t u a t or o n ce$ $C an d V : c a p a c i t an ce an d d r i v e v o l t a g e$ $f_{3 d B} : a c t u a t i o n ban d w i d t h, t ak e n a s t h e r eso nan ce f or a d am p e d sy s t e m (m y r e a d in g)$ $P_{s t a t i c} : e l ec t r os t a t i c a c t u a t or s d r a w n o s t a t i c c u r r e n t, w h er e a s e l ec t r o t h er ma l o n es d r a w s i g ni f i c an t p o w er (E r r an d o - H er r an z e t a l .), t y p i c a l d r i v e : b e l o w 100 V f or p ha se s hi f t er s, ab o u t 5 V f or c a v i t y - ba se d co u pl er s an d s w i t c h es an d 30 t o 50 V f or l a r g e w a v e g u i d e s t r u c t u r es (E r r an d o - H er r an z e t a l .), E d in g er e t a l . : 1.03 MHz, E r r an d o - H er r an z e t a l . : M E M S a r e n o t e x p ec t e d t o r e pl a ce hi g h - s p ee d m o d u l a t i o n an d a r e p r o p ose d a s l o w - p o w er r eco n f i g u r a t i o n b es i d e f o u n d r y m o d u l a t or s, K han e t a l . (C L E O 2022) : n o n - v o l a t i l e M E M S a c t u a t or s o n a f o u n d r y pl a t f or m (n o f i g u r es in t h e ab s t r a c t), t h er m o - o pt i c s hi f t er s o f t h e T h er m o - O pt i c M o d u l a t or p a g e : 125 t o 160 kHz f or t h e f a s t h e a t er s an d 2.4 kHz f or t h e u n d er c u t s hi f t er$ $(co m p u t e d : a 1.03 MHz ban d w i d t h cor r es p o n d s t o a t im e co n s t an t o f 155 ns an d a r i se t im e o f 0.34 μ s f or a s in g l e p o l e (a ss u m e d); i t i s 6.4 t o 8.2 t im es t h e f a s t h e a t er s an d ab o u t 430 t im es t h e 2.4 kHz u n d er c u t t h er m o - o pt i c s hi f t er; t h e T h er m o - O pt i c M o d u l a t or p a g e l i s t s a f e w t o t e n s o f mW p er π f or t y p i c a l s hi f t er s an d 0.49 mW f or t h e u n d er c u t o n e, w h er e a s t h e e l ec t r os t a t i c M E M S h o l d s i t s p ha se w i t h n o s t a t i c p o w er)$

Micromirror and Grating Light Valve

Name	Formula	Related Formulae	Variables/Constants
Digital micromirror pixel and array	$θ_{b e am} = 2 θ_{m}$ $Δ θ_{o n - o f f} = 4 θ_{m}$	$W = N_{x} p$ $η_{D M D} = η_{f i l l} η_{d i f f} η_{r e f l} η_{w in}$	$θ_{m} : m ec hani c a l t i l t o f a mi r r or ab o u t i t s t or s i o n hin g e$ $θ_{b e am} : t i l t o f t h e r e f l ec t e d b e am f r o m t h e f l a t - s t a t e d i r ec t i o n$ $Δ θ_{o n - o f f} : an g l e b e tw ee n t h e b e am s o f t h e tw o l an d e d s t a t es (b o t h f r o m r e f l ec t i o n g eo m e t r y, ba c k g r o u n d k n o w l e d g e)$ $W : a r r a y w i d t h$ $N_{x} : mi r r or s a l o n g t h e w i d t h$ $p : p i t c h$ $η : e f f i c i e n cy f a c t or s o f f i l l, d i f f r a c t i o n, r e f l ec t i v i t y an d w in d o w, Y o u n se (1997) : s q u a r e mi r r or s 16 b y 16 μ m o n a 17 μ m p i t c h, t i l t pl u s or min u s 10 d e g r ees, f i l l f a c t or 74 p er ce n t, r es p o n se b e l o w 20 μ s w i t h a g o a l o f 10 μ s, 15 V a cr oss o n e e l ec t r o d e an d 10 V a cr oss t h e o t h er, co n t r a s t g o a l s ab o v e 50 : 1 an d ab o v e 150 : 1, m e a s u r e d b e tt er t han 140 : 1 an d 200 : 1 in tw o - d im e n s i o na l a r r a y s, D L P 9500 (J u l y 2023) : 1920 b y 1080 mi r r or s, 10.8 μ m p i t c h, pl u s or min u s 12 d e g r ees, s w i t c hin g 12.5 μ s t y p i c a l (cr osso v er 3 μ s), a r r a y s w i t c hin g 56 μ s w i t h g l o ba l r ese t, p a tt er n r a t e 23, 148 Hz bina r y an d 2, 893 Hz f or 8 - bi t g r a y, w in d o w t r an s mi ss i o n 96 p er ce n t, f i l l 94 p er ce n t, e f f i c i e n cy 68 p er ce n t (94, 87, 89 an d 96 p er ce n t), 400 t o 700 nm$ $(co m p u t e d : 1920 t im es 10.8 μ m = 20.736 mm an d 1080 t im es 10.8 μ m = 11.664 mm, a s q u o t e d; t h e b e am s o f t h e tw o s t a t es a r e 48 d e g r ees a p a r t f or 12 d e g r ees o f t i l t; 1/23, 148 Hz = 43.2 μ s an d t h e q u o t e d a r r a y s w i t c hin g t im e o f 56 μ s cor r es p o n d s t o 17, 857 Hz; 23, 148 p a tt er n s p er seco n d o f 2.07 mi l l i o n p i x e l s i s 48.0 Gpixel/s; t h e e f f i c i e n cy f a c t or s m u l t i pl y t o 69.9 p er ce n t, n o t 68; t h e 16 b y 16 μ m mi r r or o n a 17 μ m p i t c h ha s an a r e a f i l l o f 88.6 p er ce n t, a g ain s t t h e 74 p er ce n t q u o t e d, w hi c h in c l u d es o t h er s t r u c t u r e)$
Grating light valve diffraction	$I_{0} = cos^{2} (\frac{2 π d}{λ})$ $I_{\pm 1} = \frac{8}{π ^{2}} sin^{2} (\frac{2 π d}{λ})$	$d_{b r i g h t} = \frac{λ}{4}$ $I_{\pm 1, ma x} = \frac{8}{π ^{2}}$	$I_{0} : in t e n s i t y in t h e z er o or d er (t h e mi r r or s t a t e)$ $I_{\pm 1} : in t e n s i t y in t h e tw o f i r s t or d er s t o g e t h er$ $d : p u l l - d o w n o f t h e m o v ab l e r ibb o n s r e l a t i v e t o t h e f i x e d o n es$ $λ : w a v e l e n g t h, t h e r o u n d t r i p d o u b l es t h e p a t h, so d = λ /4 g i v es a p ha se o f π an d t h e e q u a l - w i d t h r ibb o n g r a t in g i s a bina r y p ha se g r a t in g (ba c k g r o u n d k n o w l e d g e; t h e s am e g r a t in g e x p r ess i o n s a s o n t h e L i q u i d - C r y s t a l S p a t ia l L i g h t M o d u l a t or p a g e, i_{0} = cos^{2} (φ /2) an d i_{1} = (4/ π^{2}) sin^{2} (φ /2) p er or d er)$ $B l oo m : r ibb o n s 20 μ m l o n g o n a 5 μ m p i t c h, f o u r r ibb o n s (tw o f i x e d, tw o m o v ab l e) p er 20 μ m s q u a r e p i x e l, p u l l - d o w n ab o u t 1300 an g s t r o m, ab o u t a q u a r t er o f a g r ee n w a v e l e n g t h, t h e v o l t a g e w in d o w l i es b e tw ee n V_{1} an d V_{2} w i t h V_{2} / V_{1} ab o u t 2, 81 p er ce n t o f t h e l i g h t co l l ec t e d in t h e b r i g h t s t a t e, f i l l r a t i o 67.5 p er ce n t w i t h 1.25 μ m d es i g n r u l es, co n t r a s t b e tt er t han 200 : 1, s w i t c hin g in 20 ns, 8 - bi t g r a y sc a l e an d 1000 : 1 d y nami c r an g e$ $(co m p u t e d : 4 t im es 1300 an g s t r o m = 520 nm; 4 t im es 5 μ m = 20 μ m; 8/ π^{2} = 81.06 p er ce n t, e q u a l t o t h e 81 p er ce n t co l l ec t e d; t h e f i r s t - or d er s ha r e i s 7.7, 28.0, 53.1, 73.3 an d 81.1 p er ce n t f or d / λ = 0.05, 0.10, 0.15, 0.20 an d 0.25, an d t h e z er o or d er f a l l s t o z er o a t 0.25)$
Speed and contrast of MEMS light modulators	$τ_{s w} \approx \frac{0.35}{f _{3 d B}}$	$\frac{τ _{D M D}}{τ _{G L V}}$	$τ_{s w} : s w i t c hin g t im e o f a s in g l e - p o l e r es p o n se (ba c k g r o u n d k n o w l e d g e)$ $f_{3 d B} : ban d w i d t h, G L V (B l oo m; W ik i p e d ia) : s w i t c hin g f r o m u n d e f l ec t e d t o ma x im u m d e f l ec t i o n in 20 ns, r ibb o n e x c u r s i o n o f a f e w h u n d r e d nan o m e t r es, n o co n t a c t b e tw ee n m o v in g p a r t s an d so m or e t han 210 bi l l i o n cy c l es$ $D M D : b e l o w 20 μ s (Y o u n se 1997) an d 12.5 μ s (D L P 9500), mi r r or s l an d o n p a d s a t t h e mi r r or p o t e n t ia l, s i l i co n p h o t o ni c p ha se s hi f t er s : 1.03 MHz (E d in g er e t a l .), d e f or mab l e mi r r or s : r e p e a t abi l i t y 13 pm a t J P L an d 2 nm a t B os t o n U ni v er s i t y, 8 - bi t v o l t a g e r eso l u t i o n, n o h y s t er es i s (B i f an o), W ik i p e d ia s t a t es t ha t t h e G L V i s ab o u t 1000 t im es f a s t er t han a D M D an d ab o u t a mi l l i o n t im es f a s t er t han l i q u i d - cr y s t a l d e v i ces$ $(co m p u t e d : 12.5 μ s /20 ns = 625, 20 μ s /20 ns = 1000 an d 10 μ s /20 ns = 500; 0.35/20 ns g i v es a ban d w i d t h o f 17.5 MHz f or t h e G L V i f i t s r es p o n se i s a s in g l e p o l e (a ss u m e d))$

Tunable Fabry-Perot Cavity

Name	Formula	Related Formulae	Variables/Constants
Tunable Fabry-Perot cavity with a movable mirror	$T (λ) = \frac{1}{1 + F s i n ^{2} ( \frac{2 π n g}{λ} )}$ $λ_{m} = \frac{2 n g}{m}$	$\frac{Δ λ _{m}}{λ _{m}} = \frac{Δ g}{g}$ $F = \frac{4 R}{( 1 - R ) ^{2}}$	$T : t r an s mi ss i o n o f an i d e a l l oss l ess c a v i t y a t n or ma l in c i d e n ce$ $F : coe f f i c i e n t o f f in esse$ $R : p o w er r e f l ec t an ce o f e a c h mi r r or$ $n : in d e x in t h e g a p (1 f or ai r)$ $g : g a p$ $m : or d er$ $λ_{m} : r eso nan ce w a v e l e n g t h$ $Δ g : g a p c han g e se t b y t h e e l ec t r os t a t i c a c t u a t or (a l l ba c k g r o u n d k n o w l e d g e; n o M E M S F ab r y - P er o t p a p er w a s r e a d, see t h e a u d i t), t h e g a p r an g e o f an a c t u a t or i s l imi t e d t o ab o u t g_{0} /3 b y p u l l - in$ $(co m p u t e d : a g a p o f 1.55 μ m in ai r ha s t h e r eso nan ce m = 2 a t 1550 nm, an d t h e p u l l - in t r a v e l o f o n e t hi r d o f t h e g a p, 0.52 μ m, w o u l d m o v e i t b y 33 p er ce n t o f t h e w a v e l e n g t h i f t h e e l ec t r o d e g a p i s t h e c a v i t y g a p (a ss u m e d))$

Important Questions

Ques: Does the silicon MEMS phase shifter of Sattari et al. need 19 or 20 nanometres, and is its loss below 0.3 or 0.1 decibel?

The 2019 journal abstract gives a $π$ shift at 19 $V$ with a displacement of 19 $nm$ and an average insertion loss below 0.3 $dB$ over 1.5 to 1.6 $μ m$ . The 2017 conference abstract gives 20 $nm$ at 19 $V$ and an excess loss below 0.1 $dB$ . The phase per displacement is 0.165 $rad$ per $nm$ for 19 $nm$ and 0.157 for 20 $nm$ (computed), a difference of 5 per cent. I regard the journal values as the more likely, since they are the later and refereed form (my reading). The two losses may count different things, an average over a band against an excess loss at one wavelength, so I would not call either paper wrong. The imec devices of Edinger et al. lose 0.33 $dB$ , nearer the 0.3 figure. Reference: Sattari, Graziosi, Kiss, Seok, Han, Wu and Quack, ‘Silicon Photonic MEMS Phase-Shifter’, Optics Express 27(13):18959 (2019), DOI 10.1364/OE.27.018959 (abstract page only); Sattari et al., ‘Analog silicon photonic MEMS phase-shifter with double-step electrostatic actuation’, 22nd International Conference on Optical MEMS and Nanophotonics (OMN 2017) (abstract page only); Edinger, Takabayashi, Errando-Herranz, Khan, Sattari, Verheyen, Bogaerts, Quack and Gylfason, ‘Silicon photonic microelectromechanical phase shifters for scalable programmable photonics’, Optics Letters 46(22):5671 (2021) (abstract page only).

Ques: Is the pull-in voltage of a fixed-fixed beam found with a factor of 5 or 128/27?

Bifano gives $V_{c} \approx 5 E t^{3} g_{0}^{3} / (ε_{0} L^{4})$ and an example of 170 $GPa$ , 3 $μ m$ thickness, 5 $μ m$ gap and 300 $μ m$ length. The factor 5 gives 200.0 $V$ (computed). Taking the beam stiffness $k = 16 E w t^{3} / L^{3}$ (the centre-load value $192 E I / L^{3}$ , my derivation, since the printed second form was garbled) and an electrode area $L w$ in the parallel-plate $V_{P I}$ gives a factor of 128/27 = 4.74 and 194.7 $V$ (computed), 2.7 per cent lower. I regard 4.74 as the self-consistent value and 5 as the rounded value of the slides. A beam bends unevenly and Bifano quotes a travel of about 0.4 of the gap before collapse, against one third for a rigid plate, so neither number is exact for a real beam (background knowledge). Reference: Bifano, ‘Parallel Plate Electrostatic Actuation for High-Resolution Deformable Mirrors’, Boston University, CfAO TMT MEMS deformable-mirror workshop, 19 August 2004 (fetch-tool summary; a second form of the beam stiffness was garbled).

Ques: Why is the phase shifter analog while the micromirror is digital?

A phase shifter is held in equilibrium before pull-in. Sattari et al. move the waveguide by 19 $nm$ in a simulated gap of 150 to 185 $nm$ (COMSOL page), which is 10 to 13 per cent of the gap (computed), well inside the one-third limit of pull-in. The phase then follows the voltage smoothly, and the title of the 2017 paper calls the shifter analog. The micromirror is run the other way: Younse applies 15 $V$ and 10 $V$ across its two electrodes, and the mirror tilts to a landing stop at plus or minus 10 degrees and is held there. It is bistable, with hysteresis, and grey is made by time-weighting binary frames at 23,148 $Hz$ for the DLP9500 (Texas Instruments). I regard the stop as the difference: collapse is avoided in one design and used in the other. Reference: Sattari, Graziosi, Kiss, Seok, Han, Wu and Quack, ‘Silicon Photonic MEMS Phase-Shifter’, Optics Express 27(13):18959 (2019), DOI 10.1364/OE.27.018959 (abstract page only); Sattari et al., ‘Analog silicon photonic MEMS phase-shifter with double-step electrostatic actuation’, 22nd International Conference on Optical MEMS and Nanophotonics (OMN 2017) (abstract page only); COMSOL user story (December 2021) on the EPFL silicon photonic MEMS phase shifter of Sattari et al. (fetch-tool summary; values were only in figures); Younse (Texas Instruments), ‘High Definition Display Based on the Digital Micromirror Device (DMD) Spatial Light Modulator’, Wright Laboratory report WL-TR-97-1003 (January 1997) (fetch-tool summary); Texas Instruments, DLP9500 DLP 0.95 1080p 2x LVDS Type A DMD data sheet, document DLPS025F, revision F (July 2023) (fetch-tool summary).

Ques: Is the efficiency of a digital micromirror device 68 or 69.9 per cent?

The DLP9500 sheet lists an efficiency of 68 per cent, from a fill factor of 94 per cent, a diffraction efficiency of 87 per cent, a mirror reflectivity of 89 per cent and a window transmission of 96 per cent. These four multiply to 69.9 per cent (computed), 1.9 points above the quoted value, a ratio of 0.973. The sheet may apply a further loss or round the factors; rounding explains most of it, since 94, 87, 89 and 96 per cent each carry half a point, which moves the product by up to about 1.5 points (computed), but not all of the 1.9. I regard 68 per cent as the figure to use in a design, as it is the number given for the device, and the product as an upper estimate (my reading). It applies to the 400 to 700 $nm$ band. Reference: Texas Instruments, DLP9500 DLP 0.95 1080p 2x LVDS Type A DMD data sheet, document DLPS025F, revision F (July 2023) (fetch-tool summary).

Ques: Does the grating light valve have four ribbons or six per pixel?

Bloom describes ribbons 20 $μ m$ long on a 5 $μ m$ pitch and four ribbons, two fixed and two movable, per pixel; four times 5 $μ m$ gives a square pixel of 20 $μ m$ (computed). The Wikipedia page for the device describes six ribbons per pixel. Six ribbons at the 5 $μ m$ pitch would be 30 $μ m$ wide, or a 3.33 $μ m$ pitch in a 20 $μ m$ pixel (computed), and neither is stated by Bloom. I regard four as the more likely, as it comes from the inventor’s own description and matches the geometry. The count may differ between generations of the product, since the sources do not give a date for the six-ribbon form, and a grating of more ribbons per pixel would change the sharpness of the diffraction orders but not the 81 per cent first-order limit of the binary case. Reference: Bloom (Silicon Light Machines), ‘The Grating Light Valve: revolutionizing display technology’ (date not stated) (fetch-tool summary); Wikipedia, ‘Grating light valve’ (fetch-tool summary; the page names no sources for the numbers read).

Ques: Is the grating light valve a thousand times faster than the micromirror?

The Wikipedia page for the device states that it is about 1000 times faster than a micromirror device. Bloom gives a switching time of 20 $ns$ . Against the 20 $μ s$ of Younse (1997) the ratio is 1000, and against the 10 $μ s$ goal it is 500. The DLP9500 sheet gives 12.5 $μ s$ as the typical mirror switching time, which makes the ratio 625 (computed). I regard 625 as the right figure for a current device, and 1000 as the value against the early micromirror. These are the speeds of single elements; the sheet also lists 56 $μ s$ for the whole array, including the reset of all mirrors, and the 20 $ns$ of Bloom is for the light valve with no time given for the full pixel array or the electronics. Reference: Wikipedia, ‘Grating light valve’ (fetch-tool summary; the page names no sources for the numbers read); Bloom (Silicon Light Machines), ‘The Grating Light Valve: revolutionizing display technology’ (date not stated) (fetch-tool summary); Younse (Texas Instruments), ‘High Definition Display Based on the Digital Micromirror Device (DMD) Spatial Light Modulator’, Wright Laboratory report WL-TR-97-1003 (January 1997) (fetch-tool summary); Texas Instruments, DLP9500 DLP 0.95 1080p 2x LVDS Type A DMD data sheet, document DLPS025F, revision F (July 2023) (fetch-tool summary).

Ques: Is the fill factor of a micromirror 74 per cent or about 89 per cent?

Younse gives mirrors of 16 by 16 $μ m$ on a 17 $μ m$ pitch and a fill factor of 74 per cent. The area of the mirror over the area of the cell is $(16/17)^{2}$ = 88.6 per cent (computed), 15 points more. The DLP9500 sheet lists 94 per cent for a 10.8 $μ m$ pitch. I read Younse’s 74 per cent as the fraction of the cell that reflects usefully in the 1997 design, with the hinge, the support post and the tilted edges taking the rest (my reading); the sources do not say which areas are lost. The better value of the modern device may come from the smaller gap between mirrors in relation to the pitch (my reading). I regard the geometric ratio as an upper bound on the optical fill and 74 per cent as the measured early value. Reference: Younse (Texas Instruments), ‘High Definition Display Based on the Digital Micromirror Device (DMD) Spatial Light Modulator’, Wright Laboratory report WL-TR-97-1003 (January 1997) (fetch-tool summary); Texas Instruments, DLP9500 DLP 0.95 1080p 2x LVDS Type A DMD data sheet, document DLPS025F, revision F (July 2023) (fetch-tool summary).

Ques: Why do the sources not give the damping of a MEMS plate, and what can be said about it?

The squeeze-film formulas were not available to me: one full text could not be read, and the abstract of Shekhar et al. gives only the result, that the quality factor has an appreciable effect on the release time and less on the switching time for parallel-beam silicon-on-insulator switches at 10 to 30 $V$ in air. Raghuram et al. work without damping and obtain a resonance of 49.9 $kHz$ (computed), a period of 20.1 $μ s$ . So the table here has no formula for the damping coefficient; the form $Q = k m / c$ is background knowledge. The faster 1.03 $MHz$ phase shifter of Edinger et al. works in air, and I regard its bandwidth as the actuation bandwidth the authors measured, not a number derived from a damping model (my reading). Reference: Shekhar, Vinoy and Ananthasuresh, ‘Switching and Release Dynamics of an Electrostatically Actuated MEMS Switch under the influence of Squeeze-film Damping’, TechConnect Briefs 2:625 (2012) (abstract page only); Raghuram, Ajoy and Padmanabhan (IIT Palakkad), ‘Analysis of Electrostatic MEMS Using Energy-Charge Landscape’, arXiv:2002.12430 (year not stated) (fetch-tool summary); Edinger, Takabayashi, Errando-Herranz, Khan, Sattari, Verheyen, Bogaerts, Quack and Gylfason, ‘Silicon photonic microelectromechanical phase shifters for scalable programmable photonics’, Optics Letters 46(22):5671 (2021) (abstract page only).

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