1. Theory

Definition: A magneto-optic modulator changes the polarisation of light with a magnetised medium and turns the change into an intensity change with a polariser pair. The effect is the rotation of the plane of polarisation, in transmission the Faraday effect and in reflection the Kerr effect (RP Photonics; Wikipedia).
Faraday effect: Left and right circular polarisations see different indices, $n_{L} - n_{R} = (λ / π) V B$ , and the plane turns by $β = V B L$ (RP Photonics). The rotation depends on the field direction and not on the propagation direction, so it doubles on a round trip, which is the basis of the optical isolator and its 45 degrees (RP Photonics). The sign and size of $V$ change with the material and strongly with the wavelength.
Two kinds of material: Paramagnetic crystals and glasses such as TGG have $V$ = 134 $rad/ (T m)$ at 632 $nm$ and 40 $rad/ (T m)$ at 1064 $nm$ , so 45 degrees takes 1.96 $cm$ at 1 $T$ (RP Photonics; MT-Berlin). Magnetic garnets rotate by their own magnetisation, $θ_{F} = F (M / M_{s}) l$ (Tsuzuki et al.), with 2500 to 6200 $^{\circ} / cm$ at 1550 $nm$ in Bi-substituted films (US 7133579; MIT thesis), so 45 degrees takes tens to a few hundred micrometres.
Kerr effect: In reflection the polarisation change is the Kerr rotation and ellipticity, measured in the polar, longitudinal and transverse geometries; the transverse one changes the intensity and the other two the polarisation (Wikipedia). The rotation of metallic films is only milliradians (Kharratian et al.).
Figures of merit: Rotation per unit length, loss, their ratio in degrees per decibel, switching field or current, switching energy and speed, pixel pitch, bias field and bandwidth. A record film has 770 $^{\circ} / dB$ at 1550 $nm$ (MIT thesis), against most dielectrics below 1 $^{\circ} / dB$ (Kharratian et al.).
Speed: A garnet modulator driven near its ferromagnetic resonance worked from 1 $MHz$ to 1.8 $GHz$ with a measured rise time of 343 to 435 $ps$ (US 7133579); a magnetic pixel switches in tens of nanoseconds or less (Kato et al.).
Scope: This page covers Faraday and Kerr modulators, garnet waveguide modulators, magneto-optic spatial light modulators and the magnetostatic-wave Bragg cell. The sound-wave deflectors are on the Acousto-Optic Modulator page, the MEMS micromirrors on the MEMS Optical Modulator page and the pixelated liquid-crystal panels on the Liquid-Crystal Spatial Light Modulator page.

2. Working Principle

Rotation to intensity: Light passes a polariser, the magnetised layer turns the plane by $θ$ and an analyser at angle $ψ$ transmits $I_{0} cos^{2} (ψ - θ)$ . At $ψ$ = 45 degrees the response is linear around half intensity, $m \approx 2 θ$ (background knowledge; the patent states that an analyser converts the polarisation modulation into intensity modulation).
Waveguide modulator: A 1.0 $μ m$ Bi-YIG film on a gadolinium gallium garnet substrate sits in a static in-plane bias field. A current in a microstrip line adds a small field of 0.3 $mT$ peak to peak that tips the magnetisation along the light and converts polarisation (US 7133579). Below the resonance the response is flat to about 370 $MHz$ , and near it the precession peaks at 1.5 $GHz$ for 30 $mT$ and 842 $MHz$ for 15 $mT$ , a tuning slope of 42 $GHz/T$ (US 7133579).
Limits of that speed: Pulses of 300 $ps$ or longer are reproduced with little distortion, shorter ones are not, and the simulated conversion falls from 42 per cent for 1 $ns$ pulses to 1 per cent for 50 $ps$ ones (US 7133579).
Binary pixel: A pixel magnetised up or down turns the light by plus or minus $θ$ . With the analyser set to extinguish one state the other transmits $I_{0} sin^{2} 2 θ$ , and full amplitude modulation needs a swing of 90 degrees (Kharratian et al.; the formula is my arithmetic). A back reflector doubles the rotation in reflection (Kharratian et al.).
Writing a pixel: A field from crossing drive lines (LIGHT-MOD), a laser pulse that heats the film (thermomagnetic), a spin-polarised current (spin-transfer or spin-orbit torque) or a strain from a piezoelectric layer (multiferroic) reverses the magnetisation (Kharratian et al.; Kato et al.; Aoshima et al.). The current-driven energy of 1 to 10 $mJ/c m^{2}$ is far above the 1 to 500 $μ J/c m^{2}$ of the strain route (Kharratian et al.).
Multi-tone pixels: Tilting the magnetisation into the plane of the layer rotates the light in proportion to $M / M_{s}$ , which gives grey levels (Tsuzuki et al.).
Magnetostatic-wave Bragg cell: A microstrip transducer excites a magnetostatic wave in a YIG-GGG waveguide, which diffracts the light as an acoustic wave does in the Acousto-Optic Modulator page, with a carrier of 0.5 to 40 $GHz$ tuned by the bias field (Laser Focus World).

3. Applications

Optical isolators and circulators: The non-reciprocal rotation of 45 degrees in TGG or in garnet films on silicon is the main use of the effect (RP Photonics; Kharratian et al.; MIT thesis).
Holography and three-dimensional displays: Magneto-optic spatial light modulators with pixel pitches of 5 $μ m$ and below one micrometre aim at viewing zones above 30 degrees (Kato et al.; Aoshima et al.).
Augmented and virtual reality and head-up displays: Named by Kharratian et al., who also name LiDAR, beam steering and optical communication.
High-speed modulation of light: The garnet waveguide modulator offers polarisation or intensity modulation up to the gigahertz range (US 7133579).
Radio-frequency signal processing and switching: The magnetostatic-wave Bragg cell is proposed as a high-speed multiple-port optical switch with a carrier of up to 40 $GHz$ (Laser Focus World).
Magneto-optic data storage: Discs introduced in 1985 are written by heating above the Curie temperature in a field and read through the Kerr rotation (Wikipedia).
Imaging of magnetisation: The Kerr microscope converts polarisation changes into visible intensity changes (Wikipedia).

Faraday Rotation and Magnetic Garnet Films

Name	Formula	Related Formulae	Variables/Constants
Faraday rotation and circular birefringence	$β = V B L$ $n_{L} - n_{R} = \frac{λ}{π} V B$	$L_{45} = \frac{π}{4 V B}$ $V \propto λ^{- 2} (roughly)$	$β : r o t a t i o n o f t h e pl an e o f p o l a r i s a t i o n in r a d ian s$ $V : V er d e t co n s t an t$ $B : ma g n e t i c f l ux d e n s i t y a l o n g t h e p r o p a g a t i o n d i r ec t i o n$ $L : l e n g t h o f t h e m e d i u m, n_{L} - n_{R} : in d e x d i f f er e n ce b e tw ee n t h e l e f t an d t h e r i g h t c i r c u l a r p o l a r i s a t i o n$ $λ : f r ee - s p a ce w a v e l e n g t h$ $L_{45} : l e n g t h f or a r o t a t i o n o f π /4 (R P P h o t o ni cs), a p os i t i v e V r o t a t es t o t h e l e f t f or l i g h t t r a v e l l in g a l o n g t h e f i e l d, p a r ama g n e t i c ma t er ia l s ha v e l a r g er v a l u es t han d iama g n e t i c o n es, t h e r o t a t i o n d e p e n d s o n t h e f i e l d d i r ec t i o n an d n o t o n t h e p r o p a g a t i o n d i r ec t i o n, so i t d o u b l es o n a r o u n d t r i p (R P P h o t o ni cs), V er d e t co n s t an t s (R P P h o t o ni cs; M T - B er l in) : T GG 134 rad/ (T m) a t 632 nm (R P P h o t o ni cs p r in t s 623.8 nm f or t h e H e N e l in e, m y r e a d in g i s 632.8) an d 40 rad/ (T m) a t 1064 nm, M O S - 4 g l a ss 73 a t 633 nm an d 21 a t 1060 nm, M O S - 10 g l a ss 87 an d 26, t h e T GG w in d o w i s 400 t o 1100 nm w i t h o u t 470 t o 500 nm, t h e M O S g l a sses l ose 0.001 c m^{- 1} a t 633 an d 1060 nm, a r ecor d t er bi u m - d o p e d f ib r e ha s 32 rad/ (T m), an d Y I G i s u se d ab o v e 1100 nm (R P P h o t o ni cs)$ $(co m p u t e d : L_{45} = 1.963 cm f or T GG a t 1064 nm an d 1 T, e q u a l t o t h e 1.96 cm q u o t e d, an d 5.86 mm a t 633 nm; n_{L} - n_{R} = 2.70 \times 1 0^{- 5} f or T GG a t 633 nm an d 1 T; V (633) / V (1060 or 1064) = 3.35 f or T GG, 3.48 f or M O S - 4 an d 3.35 f or M O S - 10, a g ain s t (1064/633)^{2} = 2.83, w hi c h g i v es l oc a l e x p o n e n t s o f 2.33, 2.40 an d 2.33 (2.42 an d 2.34 f or t h e M O S g l a sses w i t h 1060 nm))$
Garnet films: specific rotation, loss and figure of merit	$θ_{F} = F \frac{M}{M _{s}} l$ $FoM = \frac{Θ [ ^{\circ} / cm ]}{α [ dB/cm ]}$	$l_{45} = \frac{4 5 ^{\circ}}{Θ}$ $θ_{r e f l} = 2 θ_{F}$	$θ_{F} : F a r a d a y r o t a t i o n o f a l a y er$ $F : s p ec i f i c r o t a t i o n o f t h e s a t u r a t e d f i l m (T s u z u k i e t a l . c a l l i t t h e V er d e t co n s t an t)$ $M / M_{s} : ma g n e t i s a t i o n r e l a t i v e t o i t s s a t u r a t i o n v a l u e$ $l : l a y er t hi c k n ess$ $Θ : r o t a t i o n p er u ni t l e n g t h$ $α : o pt i c a l l oss$ $F o M : f i g u r e o f m er i t in d e g r ees p er d ec ib e l (E q . 3 o f K ha r r a t ian e t a l .)$ $l_{45} : t hi c k n ess f or 45 d e g r ees, a ba c k r e f l ec t or d o u b l es t h e r o t a t i o n see n in r e f l ec t i o n (K ha r r a t ian e t a l .), t h e B i - s u b s t i t u t e d Y I G f i l m o f t h e p a t e n t (1.0 μ m l i q u i d - p ha se e p i t a x y o n a g a d o l ini u m g a l l i u m g a r n e t s u b s t r a t e) ha s 11000^{\circ} / cm a t 630 nm, ab o u t 5400^{\circ} / cm a t 800 nm an d ab o u t 2500^{\circ} / cm a t 1550 nm, t h e s i g n b e in g n e g a t i v e (U S 7133579), p o l y cr y s t a l l in e B i 0.03 T b I G ha s 6200 pl u s or min u s 300^{\circ} / cm a t 1550 nm, an d t o p - d o w n cr y s t a l l i se d B iY I G / Y I G o n s i l i co n r e a c h es a f i g u r e o f m er i t u p t o 770^{\circ} / dB a t 1550 nm (M I T t h es i s), K ha r r a t ian e t a l . : m os t r e p or t e d d i e l ec t r i c f i g u r es o f m er i t a r e b e l o w 1^{\circ} / dB an d B i s u b s t i t u t i o n im p r o v es i t s i g ni f i c an tl y, t h e B i : Y I G ma g n e t o p h o t o ni c cr y s t a l o f t h e i r o w n e a r l i er w or k r o t a t es 20 d e g r ees a t 494 nm, 55 a t 541 nm an d 30 a t 630 nm, an d K er r r o t a t i o n o f m e t a l f i l m s i s g e n er a l l y o n l y mi l l i r a d ian s (K ha r r a t ian e t a l .)$ $(co m p u t e d : l_{45} = 40.9 μ m a t 630 nm (11000^{\circ} / cm), 83.3 μ m a t 800 nm, 180 μ m a t 1550 nm w i t h 2500^{\circ} / cm an d 72.6 μ m w i t h 6200^{\circ} / cm; t h e 1.0 μ m f i l m i t se l f r o t a t es 1.10, 0.54, 0.25 an d 0.62 d e g r ees; a t 770^{\circ} / dB a 45 d e g r ee r o t a t i o n cos t s 0.058 dB; t h e T GG l e n g t h o f 1.963 cm a t 1 T i s 109 an d 270 t im es t h ose 180 an d 72.6 μ m)$

Garnet Waveguide Modulator

Name	Formula	Related Formulae	Variables/Constants
Polarisation analysis and small-signal intensity modulation	$I = I_{0} cos^{2} (ψ - θ)$ $I_{45} = \frac{I _{0}}{2} (1 + sin 2 θ)$	$m \approx 2 θ (θ ≪ 1)$ $θ (t) = θ_{F} \frac{M _{∥} ( t )}{M _{s}}$	$I : in t e n s i t y b e hin d t h e ana l y ser$ $I_{0} : in t e n s i t y b e hin d t h e f i r s t p o l a r i ser$ $ψ : an g l e b e tw ee n t h e ana l y ser an d t h e in p u t p o l a r i s a t i o n$ $θ : F a r a d a y r o t a t i o n$ $I_{45} : t h e c a se ψ = 45 d e g r ees, in w hi c h t h e r es p o n se i s l in e a r a r o u n d t h e ha l f - in t e n s i t y p o in t$ $m : m o d u l a t i o n d e pt h$ $M_{∥} : co m p o n e n t o f t h e ma g n e t i s a t i o n a l o n g t h e p r o p a g a t i o n d i r ec t i o n$ $M_{s} : s a t u r a t i o n ma g n e t i s a t i o n (t h e p a t e n t s a y s o n l y t ha t a p o l a r i s a t i o n ana l y ser co n v er t s t h e p o l a r i s a t i o n - m o d u l a t e d s i g na l t o in t e n s i t y m o d u l a t i o n; M a l u s^{'} l a w an d t h e 45 d e g r ee bia s a r e ba c k g r o u n d k n o w l e d g e, an d a w a v e g u i d e a dd s i t s o w n bi r e f r in g e n ce, so t hi s i s an i d e a l i s a t i o n o f t h e b u l k f i l m), in t h e d e v i ce o f t h e p a t e n t a s t a t i c in - pl an e f i e l d se t s t h e ma g n e t i s a t i o n an d a c u r r e n t in a mi cr os t r i p l in e t i p s i t a l o n g t h e l i g h t (U S 7133579)$ $(co m p u t e d w i t h t h e r o t a t i o n s o f t h e p r e v i o u s t ab l e an d M_{∥} = M_{s}, w hi c h i s an u pp er b o u n d : m = s in 2 θ = 3.8 p er ce n t f or t h e 1.0 μ m f i l m a t 630 nm (1.10 d e g r ees), 1.9 p er ce n t a t 800 nm, 0.87 p er ce n t a t 1550 nm w i t h 2500^{\circ} / cm an d 2.2 p er ce n t w i t h 6200^{\circ} / cm)$
Ferromagnetic resonance and field tuning	$f_{F M R} \approx f_{0} + S_{B} B_{bia s}$ $f_{K} = \frac{γ}{2 π} B (B + μ_{0} M_{s})$	$\frac{γ}{2 π} \approx 28 GHz/T$	$f_{F M R} : f r e q u e n cy o f t h e f er r o ma g n e t i c r eso nan ce$ $S_{B} : t u nin g s l o p e w i t h t h e bia s f i e l d B_{bia s}$ $f_{0} : o f f se t o f t h e s t r ai g h t - l in e f i t (m y d e f ini t i o n)$ $f_{K} : K i tt e l f r e q u e n cy f or an in - pl an e ma g n e t i se d f i l m (ba c k g r o u n d k n o w l e d g e)$ $γ /2 π : g y r o ma g n e t i c r a t i o o f t h e f r ee e l ec t r o n, μ_{0} M_{s} : s a t u r a t i o n ma g n e t i s a t i o n in t es l a, t h e p a t e n t (U S 7133579) : S_{B} = 42 GHz/T, F M R p e ak a t 1.5 GHz f or 30 mT an d a t 842 MHz f or 15 mT, s a t u r a t i o n ma g n e t i s a t i o n o f 16 mT in o n e e x am pl e an d 9 mT in t h e l a t er d e v i ce e x am pl es, bia s f i e l d s f r o m 6 t o 45 mT, r es p o n se f l a t u p t o ab o u t 370 MHz in t h e 15 mT c a se, o p er a t i o n b e tw ee n 1 MHz an d 1.8 GHz$ $(co m p u t e d : t h e tw o m e a s u r e d p e ak s g i v e a s l o p e o f 43.9 GHz/T an d an o f f se t o f 0.18 GHz; 42 GHz/T t h r o ug h t h e or i g in w o u l d g i v e 1.26 GHz a t 30 mT an d 0.63 GHz a t 15 mT; t h e K i tt e l f or m u l a g i v es 1.04 an d 0.60 GHz w i t h 16 mT an d 0.96 an d 0.53 GHz w i t h 9 mT)$
Rise time, bandwidth and drive of the microstrip modulator	$t_{r} \approx \frac{0.35}{f _{3 d B}}$ $t_{d e v} = t_{r}^{2} - t_{in}^{2}$	$B_{z} = \frac{μ _{0} I}{2 π r}$ $P = I_{r m s}^{2} R$	$t_{r} : m e a s u r e d r i se t im e$ $f_{3 d B} : ban d w i d t h o f a s in g l e p o l e$ $t_{in} : r i se t im e o f t h e e l ec t r i c a l in p u t$ $t_{d e v} : r i se t im e o f t h e d e v i ce a l o n e i f t h e t im es a dd in q u a d r a t u r e (m y a ss u m pt i o n)$ $B_{z} : f i e l d o f a t hin s t r ai g h t w i r e a t t h e d i s t an ce r$ $I_{r m s} : r m s c u r r e n t$ $R : l in e im p e d an ce, p a t e n t (U S 7133579) : m e a s u r e d r i se t im e 435 ps f or an e l ec t r i c a l in p u t o f 320 ps, f a l l in g t o 357 an d 343 ps a t hi g h er bia s f i e l d s, p u l ses o f 300 ps or l o n g er ma p w i t h l i ttl e d i s t or t i o n an d p u l ses o f 100 ps or s h or t er d o n o t, d r i v e 74 m A_{pp} g i v in g a p e ak - t o - p e ak f i e l d o f ab o u t 0.3 mT w i t h 34 mW in a 50 Ω l in e, s im u l a t e d co n v er s i o n e f f i c i e n c i es o f 42, 11, 4 an d 1 p er ce n t f or 500 mA p u l ses o f 1 ns, 200 ps, 100 ps an d 50 ps w i t h a f i e l d o f 50 mT a t 2 μ m f r o m t h e l in e, s t a t i c co n v er s i o n o f 80 p er ce n t a t 300 μ m in s im u l a t i o n, t h e p a t e n t g i v es n o in ser t i o n l oss an d t h e hi g h es t f r e q u e n cy t es t e d i s 1.8 GHz$ $(co m p u t e d : 0.35/435 ps = 0.80 GHz, 0.98 an d 1.02 GHz f or 357 an d 343 ps; t_{d e v} = 295 ps, e q u a l t o 1.19 GHz; 0.3 mT p er 74 mA i s 4.05 mT/A an d t h e w i r e f or m u l a g i v es 100 mT/A a t 2 μ m, w hi c h i s 24.7 t im es m or e an d g i v es 50 mT f or 500 mA a s q u o t e d; 4.05 mT/A cor r es p o n d s t o r = 49 μ m in t h e w i r e f or m u l a; a s in u so i d o f 74 m A_{pp} ha s I_{r m s} = 26.2 mA an d 34.2 mW in 50 Ω, a s q u o t e d)$

Magneto-Optic Spatial Light Modulators

Name	Formula	Related Formulae	Variables/Constants
Magneto-optic pixel and amplitude contrast	$I_{o n} = I_{0} sin^{2} (2 θ)$ $I_{o f f} = 0 (ψ = 9 0^{\circ} - θ)$	$θ = θ_{F} \frac{M}{M _{s}}$ $Δ θ_{z o n e} = 2 arcsin \frac{λ}{2 p}$	$I_{o n}, I_{o f f} : in t e n s i t i es o f a p i x e l ma g n e t i se d u p an d d o w n f or an i d e a l ana l y ser se t t o e x t in g u i s h t h e d o w n s t a t e$ $θ : r o t a t i o n o f o n e s t a t e (t h e p i x e l g i v es pl u s an d min u s θ), θ_{F} M / M_{s} : m u l t i - t o n e r o t a t i o n o f a p i x e l ma g n e t i se d p a r tl y a l o n g t h e l i g h t (T s u z u k i e t a l .)$ $Δ θ_{z o n e} : f u l l d i f f r a c t i o n (v i e w in g) an g l e o f a p i x e l p i t c h p a t w a v e l e n g t h λ (ba c k g r o u n d k n o w l e d g e), f u l l am pl i t u d e m o d u l a t i o n n ee d s a p o l a r i s a t i o n s w in g o f 90 d e g r ees, w hi c h i s θ = 45 d e g r ees (K ha r r a t ian e t a l .; m y a r i t hm e t i c), T s u z u k i e t a l . : p i x e l s o f 16 μ m w i t h a 2 μ m g a p an d 4 μ m e l ec t r o d e w i d t h, d r i v e c u r r e n t f r o m 0 t o 28 mA (t h e e l ec t r o d e b r o k e ab o v e, w hi c h t h e a u t h or s a tt r ib u t e t o h e a t in g), p r in t s t h e t r an s mi tt e d in t e n s i t y a s I = cos θ (I g i v e t h e s q u a r e d f or m), n o a r r a y s i z e, s p ee d or w a v e l e n g t h g i v e n, K a t o e t a l . : p i t c h 5 μ m, 2048 b y 2048 p a tt er n, G d - F e l i g h t - m o d u l a t i o n l a y er w i t h a K er r r o t a t i o n o f ab o u t 0.14 d e g r ees a t 780 nm in a g ian t - ma g n e t or es i s t an ce s t a c k (T b - F e - C o / C o - F e p inn e d l a y er, A g s p a cer), h o l o g r am s w i t c h e d b y an e x t er na l f i e l d o f pl u s or min u s 200 Oe, ima g e r eco n s t r u c t e d a t 632.8 nm, A os hima e t a l . (2023) : s p in - p o l a r i se d c u r r e n t inj ec t i o n, a v i e w in g z o n e ab o v e 30 d e g r ees an d p o t e n t ia l f or p i x e l s b e l o w o n e mi cr o m e t r e, K ha r r a t ian e t a l . : p i t c h b e l o w 1 μ m i s n ee d e d in t h e v i s ib l e f or a g oo d v i e w in g an g l e, s ma l l es t p i t c h es in p r o d u c t s a r e 3.74 μ m (l i q u i d cr y s t a l o n s i l i co n) an d 5.4 μ m (mi cr o mi r r or d e v i ce)$ $(co m p u t e d : 0.14 d e g r ees = 2.44 mrad, so I_{o n} / I_{0} = s i n^{2} (0.28 d e g r ees) = 2.4 \times 1 0^{- 5}; ha l f in t e n s i t y n ee d s θ = 22.5 d e g r ees; f u l l v i e w in g an g l es a r e 30.9 d e g r ees f or p = 1 μ m a t 532 nm, 64.3 d e g r ees f or 0.5 μ m, an d 7.3 d e g r ees f or 5 μ m a t 632.8 nm)$
Switching current, energy and power	$E_{p i x} = e_{A} A_{p}$ $j = \frac{I}{A _{c}}$	$B = \frac{μ _{0} I}{2 π r}$	$E_{p i x} : e n er g y f or o n e s w i t c h o f a p i x e l$ $e_{A} : e n er g y p er a r e a$ $A_{p} : p i x e l a r e a$ $j : c u r r e n t d e n s i t y in t h e s w i t c hin g co n d u c t or$ $I : c u r r e n t$ $A_{c} : i t s cr oss - sec t i o n$ $B : f i e l d o f a t hin w i r e a t t h e d i s t an ce r, K ha r r a t ian e t a l . : e n er g y p er s w i t c h o f 1 t o 10 mJ/c m^{2} f or c u r r e n t - d r i v e n s w i t c hin g, 1 t o 500 μ J/c m^{2} f or m u l t i f er r o i c (s t r ain) s w i t c hin g, an d v o l t a g e - d r i v e n s w i t c hin g ab o u t tw o or d er s o f ma g ni t u d e b e l o w c u r r e n t - d r i v e n, c u r r e n t d e n s i t i es o f 1 \times 1 0^{5} t o 1 \times 1 0^{7} A/c m^{2} w i t h ab o u t 1 \times 1 0^{7} A/c m^{2} f or s p in - t r an s f er t or q u e, t h e f i e l d - d r i v e n L I G H T - M O D (O er s t e d, 1983) c u t i t s d r i v e c u r r e n t f r o m o v er 100 mA t o b e l o w 10 mA, 10 ns p u l ses o f a 532 nm l a ser o n am or p h o u s T b F e (t h er m o ma g n e t i c), t h e f i r s t p h o t oco n d u c t or d e v i ce s w i t c h e d a t 100 Oe w i t h a 1 μ W H e N e l a ser, p r o d u c t s u se 1 t o 50 W an d l i q u i d - cr y s t a l o n es o v er 10 W a t 60 Hz ab o v e 1024 b y 1024 p i x e l s, ma g n e t o - o pt i c p r o t o t y p es b e l o w 1 W w i t h o u t d r i v er s an d o pt i cs, K a t o e t a l . : s p in - t r an s f er s w i t c hin g^{'} a s f a s t a s t e n s o f nan oseco n d s or l es s^{'}$ $(co m p u t e d f or t h e 5 μ m p i x e l o f K a t o e t a l ., A_{p} = 2.5 \times 1 0^{- 7} c m^{2} : 0.25 t o 2.5 nJ p er s w i t c h a t 1 t o 10 mJ/c m^{2} an d 0.25 t o 125 pJ a t 1 t o 500 μ J/c m^{2})$
Pixel speed and frame rate	$f_{f r am e} = \frac{1}{N _{p i x} t _{s w}} (serial)$	$\frac{t _{D M D}}{t _{s w}}$	$f_{f r am e} : f r am e r a t e i f t h e p i x e l s a r e w r i tt e n o n e a f t er t h e o t h er (m y a ss u m pt i o n, a s t h e so u r ces d o n o t d escr ib e t h e a dd r ess in g)$ $N_{p} i x : n u mb er o f p i x e l s$ $t_{s} w : s w i t c hin g t im e o f o n e p i x e l, K ha r r a t ian e t a l . : ma g n e t i s a t i o n s w i t c hin g in nan oseco n d s d o w n t o f e m t oseco n d s, f r am e r a t es ab o v e 1 kHz in e a r l y v i s ib l e - r an g e p r o t o t y p es, l i q u i d - cr y s t a l - o n - s i l i co n r es p o n se o f mi l l i seco n d s, mi cr o mi r r or r es p o n se ab o u t 10 μ s, a s in g l e 40 fs c i r c u l a r l y p o l a r i se d p u l se o n G d F e C o, a 200 fs p u l se o n D y F e O 3 a c t in g l ik e a f i e l d u p t o 5 T, an d a 50 fs p u l se in C o - s u b s t i t u t e d Y I G w i t h t h e s w i t c h e d d o main a pp e a r in g a t ab o u t 20 ps an d s t ab l e a f t er ab o u t 60 ps (t h e a l l - o pt i c a l r es u l t s a r e n o t p i x e l s p ee d s)$ $(co m p u t e d : 2048 b y 2048 = 4.19 Mpixel; a t 10 ns p er p i x e l in ser i es (a ss u m e d) o n e f r am e t ak es 41.9 ms, 23.8 frames/s; 1 kHz n ee d s 0.24 ns p er p i x e l in ser i es, so r o w s or b l oc k s m u s t b e w r i tt e n in p a r a l l e l; 10 μ s /10 ns = 1000)$

Magnetostatic-Wave Bragg Cell

Name	Formula	Related Formulae	Variables/Constants
Magnetostatic-wave Bragg cell	$2 Λ sin θ_{B} = \frac{λ}{n}$ $Λ = \frac{v _{m s}}{f _{R F}}$	$f_{R F} \propto B_{bia s}$	$Λ : w a v e l e n g t h o f t h e ma g n e t os t a t i c w a v e in t h e Y I G f i l m$ $θ_{B} : B r a g g an g l e$ $λ : f r ee - s p a ce w a v e l e n g t h$ $n : in d e x o f t h e w a v e g u i d e$ $v_{m s} : w a v e v e l oc i t y$ $f_{R F} : r a d i o - f r e q u e n cy c a r r i er$ $B_{bia s} : bia s f i e l d (t h e B r a g g co n d i t i o n i s t h e o n e o f t h e A co u s t o - O pt i c M o d u l a t or p a g e, an d t h e p r o p or t i o na l i t y o f t h e c a r r i er t o t h e bia s f i e l d i s ba c k g r o u n d k n o w l e d g e f or t hi s g eo m e t r y; t h e so u r ce s a y s o n l y t ha t t h e c a r r i er i s t u n e d b y t h e f i e l d), L a ser F oc u s W or l d (1997) : t a p er e d Y I G - GGG w a v e g u i d e w i t h c u r v e d h y b r i d l e n ses an d a s in g l e mi cr os t r i p t r an s d u cer o f 60 μ m l in e w i d t h, c a r r i er 0.5 t o 40 GHz, ab o u t t e n t im es hi g h er t han a co u s t o - o pt i c d e v i ces, e l ec t r o ni c a l l y t u nab l e w i t h t h e bia s f i e l d, s w i t c hin g s ai d t o b e 10 t o 1000 t im es f a s t er t han w i t h a co u s t i c w a v es, a p r o p os a l f or a hi g h - s p ee d m u l t i pl e - p or t s w i t c h$ $(co m p u t e d : 40/0.5 = a s p an o f 80 in t h e c a r r i er; n o w a v e v e l oc i t y w a s f o u n d, so t h e B r a g g an g l e i s n o t co m p u t e d; t h e 1997 a r t i c l e g i v es n o e f f i c i e n cy, ban d w i d t h or l oss)$

Important Questions

Ques: Does the Verdet constant fall as the inverse square of the wavelength?

RP Photonics says a strong wavelength dependence is typical and that roughly $λ^{- 2}$ is often seen. The MT-Berlin table gives TGG as 134 $rad/ (T m)$ at 632 $nm$ and 40 at 1064 $nm$ , MOS-4 as 73 and 21, and MOS-10 as 87 and 26. The ratios are 3.35, 3.48 and 3.35, while $(1064/633)^{2}$ is 2.83 (computed), so the local exponents are 2.33, 2.40 and 2.33. The two statements are compatible only if ‘roughly’ includes an exponent above two. I regard the table as the better guide here, because it is measured on these materials, and the exponent as 2.3 to 2.4 between the two lines (my reading). RP Photonics also prints 623.8 $nm$ for the helium-neon line; I take that to be a slip for 632.8 $nm$ , since both pages give 134. Reference: RP Photonics Encyclopedia, ‘Faraday Effect’ (fetch-tool summary; no figures for modulators); MT-Berlin, ‘Faraday Crystals TGG and Glasses MOS-4 and MOS-10’ (fetch-tool summary; temperature dependence not given).

Ques: Is the figure of merit of a magneto-optic film below 1 or up to 770 degrees per decibel?

Kharratian et al. write that most reported dielectric figures of merit are smaller than 1 $^{\circ} / dB$ , and that Bi substitution in YIG improves them significantly. The MIT thesis reports up to 770 $^{\circ} / dB$ at 1550 $nm$ for crystallised BiYIG/YIG films on silicon, with 6200 plus or minus 300 $^{\circ} / cm$ for Bi0.03TbIG. At 770 $^{\circ} / dB$ a 45 degree rotation costs 0.058 $dB$ ; at 1 $^{\circ} / dB$ it would cost 45 $dB$ (computed), a ratio of 770. The review dates from 2019 or earlier by its references, while the thesis is from 2022, and the review’s statement concerns dielectrics in general and not garnets (my reading). I regard 770 $^{\circ} / dB$ as the figure for the best garnet films and 1 $^{\circ} / dB$ as the typical value for other materials. Reference: Kharratian, Urey and Onbasli (Koc University), ‘Advanced Materials and Device Architectures for Magnetooptical Spatial Light Modulators’, arXiv:1909.07494 (year not stated, latest references 2019) (fetch-tool summary; Table 3 not in the text read); Fakhrul (MIT doctoral thesis, advisor C. A. Ross), ‘Iron Garnet Thin Films for Integrated Photonics and Spintronics’ (February 2022) (abstract page only).

Ques: Is the ferromagnetic resonance of the garnet modulator consistent with 42 gigahertz per tesla?

The patent states a tuning slope of 42 $GHz/T$ , and gives peaks of 1.5 $GHz$ at 30 $mT$ and 842 $MHz$ at 15 $mT$ . Through the origin, 42 $GHz/T$ gives 1.26 and 0.63 $GHz$ (computed), too low. The two measured points give 43.9 $GHz/T$ with an offset of 0.18 $GHz$ (computed), so the data fit the stated slope only with an offset. The Kittel formula for an in-plane film, with 28 $GHz/T$ and the 16 $mT$ magnetisation of one example, gives 1.04 and 0.60 $GHz$ , and with 9 $mT$ it gives 0.96 and 0.53 $GHz$ (computed, background knowledge). Both are well below the peaks. The patent also gives conflicting resonance values in places. I regard the measured peaks as more likely than either the stated slope or my Kittel estimate. Reference: US patent 7133579 B2, ‘High-speed magneto-optic modulator’ (Google Patents text read through a fetch-tool summary of the first 100,000 of 111,571 characters; inventors and date not read).

Ques: Is the garnet modulator a multi-gigahertz device?

The patent speaks of high-speed and multi-gigahertz operation, but the measured evidence is smaller. Operation was tested between 1 $MHz$ and 1.8 $GHz$ , the response is flat up to about 370 $MHz$ in the 15 $mT$ case, and the rise time of 435 $ps$ for a 320 $ps$ input corresponds to 0.80 $GHz$ , or 1.19 $GHz$ for the device alone if the times add in quadrature (computed, assumed). The simulated conversion drops to 4 per cent for 100 $ps$ pulses and 1 per cent for 50 $ps$ . I regard about 1 $GHz$ as the demonstrated bandwidth and the multi-gigahertz claim as a projection that rests partly on simulation. The patent gives no insertion loss, and its drive of 34 $mW$ was for a sinusoid of 74 $m A_{pp}$ . Reference: US patent 7133579 B2, ‘High-speed magneto-optic modulator’ (Google Patents text read through a fetch-tool summary of the first 100,000 of 111,571 characters; inventors and date not read).

Ques: Why does the field per current in the patent differ by a factor of 25 between experiment and simulation?

The experiment gives a peak-to-peak field of about 0.3 $mT$ for 74 $m A_{pp}$ , which is 4.05 $mT/A$ (computed). The simulation uses 500 $mA$ pulses and a field of 50 $mT$ at 2 $μ m$ from the current, which is 100 $mT/A$ and matches the thin-wire formula $μ_{0} I / (2 π r)$ (computed). The wire formula would match the experiment only at 49 $μ m$ . The first is a field at the film from a real microstrip line and the second a point close to an idealised wire, so I regard 4 $mT/A$ as the value to use for the device as built and 100 $mT/A$ as a limit for a thin line placed 2 $μ m$ from the film (my reading). The ground plane of a microstrip line also reduces the field, which is background knowledge, so the factor is not a contradiction. Reference: US patent 7133579 B2, ‘High-speed magneto-optic modulator’ (Google Patents text read through a fetch-tool summary of the first 100,000 of 111,571 characters; inventors and date not read).

Ques: Are magneto-optic spatial light modulators faster than micromirror devices?

Kato et al. say that spin-transfer switching takes tens of nanoseconds or less, which they state generally and did not measure. Kharratian et al. give 10 $ns$ thermomagnetic pulses, a micromirror response of about 10 $μ s$ , and frame rates above 1 $kHz$ for early visible prototypes. A pixel at 10 $ns$ would be 1000 times faster than the micromirror (computed), yet a frame of 2048 by 2048 pixels written one after the other would take 41.9 $ms$ , or 23.8 $frames/s$ (computed, assumed), and 1 $kHz$ would need rows or blocks written in parallel. I regard the pixel as faster and the frame rate as limited by addressing, not by the magnetisation, and no source gives a measured frame rate for a large array. The femtosecond all-optical results are not pixel speeds. Reference: Kato, Aoshima, Machida, Emoto, Kinjo, Kuga, Ono, Ishibashi, Kikuchi and Shimidzu, ‘Holographic images reconstructed from GMR-based fringe pattern’, EPJ Web of Conferences (JEMS 2012), 16006 (fetch-tool summary); Kharratian, Urey and Onbasli (Koc University), ‘Advanced Materials and Device Architectures for Magnetooptical Spatial Light Modulators’, arXiv:1909.07494 (year not stated, latest references 2019) (fetch-tool summary; Table 3 not in the text read).

Ques: How large is the intensity change of a Kerr rotation of 0.14 degrees, and is the transmitted intensity cos theta or its square?

Kato et al. measured a Kerr rotation of about 0.14 degrees in their Gd-Fe layer at 780 $nm$ , which is 2.44 $mrad$ (computed). With an analyser set to extinguish one state, the other transmits $sin^{2} (2 θ)$ = $2.4 \times 1 0^{- 5}$ of the incident light (computed), so a half-intensity swing would need 22.5 degrees. Kharratian et al. write that Kerr rotation is generally only milliradians and not large enough for practical devices, which agrees. Tsuzuki et al. print the transmitted intensity as $I = cos θ$ . Malus’ law gives $cos^{2}$ , which is what I use (my reading, background knowledge), and the summary may also have dropped the square. This is why Kato et al. form a hologram, where the diffraction pattern, not the transmitted amplitude, carries the information. Reference: Kato, Aoshima, Machida, Emoto, Kinjo, Kuga, Ono, Ishibashi, Kikuchi and Shimidzu, ‘Holographic images reconstructed from GMR-based fringe pattern’, EPJ Web of Conferences (JEMS 2012), 16006 (fetch-tool summary); Kharratian, Urey and Onbasli (Koc University), ‘Advanced Materials and Device Architectures for Magnetooptical Spatial Light Modulators’, arXiv:1909.07494 (year not stated, latest references 2019) (fetch-tool summary; Table 3 not in the text read); Tsuzuki, Uchida, Takagi, Lim and Inoue, ‘Formation and Properties of Multiple-Tone Spatial Light Modulator using Garnet Film with In-Plane Magnetization’, Journal of Magnetics 11(3):143 (2006) (fetch-tool summary).

Ques: Why does a garnet rotator need micrometres where terbium gallium garnet needs centimetres?

A paramagnetic crystal such as TGG turns the light in proportion to an applied field: 45 degrees needs 1.96 $cm$ at 1 $T$ and 1064 $nm$ , equal to the 1.96 $cm$ quoted (computed). A garnet film has its own magnetisation, and its rotation is $F (M / M_{s}) l$ , so it needs only a small field to saturate. With 2500 $^{\circ} / cm$ the length is 180 $μ m$ , and with 6200 $^{\circ} / cm$ it is 72.6 $μ m$ , which are 109 and 270 times shorter (computed). The cost is the absorption of the iron, so what matters is degrees per decibel, as in the 770 $^{\circ} / dB$ of the best film. The garnet length is for the free film, not for a waveguide, where birefringence limits the conversion (background knowledge). Reference: RP Photonics Encyclopedia, ‘Faraday Effect’ (fetch-tool summary; no figures for modulators); US patent 7133579 B2, ‘High-speed magneto-optic modulator’ (Google Patents text read through a fetch-tool summary of the first 100,000 of 111,571 characters; inventors and date not read); Fakhrul (MIT doctoral thesis, advisor C. A. Ross), ‘Iron Garnet Thin Films for Integrated Photonics and Spintronics’ (February 2022) (abstract page only); MT-Berlin, ‘Faraday Crystals TGG and Glasses MOS-4 and MOS-10’ (fetch-tool summary; temperature dependence not given).

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Magneto-Optic Modulator