1. Theory

Definition: An electro-optic phase modulator changes the optical phase of a beam with a voltage, through the Pockels effect, without changing the optical power; the optical spectrum changes (RP Photonics). A Pockels cell can be seen as a voltage-controlled waveplate (Wikipedia). A phase modulator in one arm of a Mach-Zehnder interferometer is an intensity modulator, the subject of the Lithium Niobate Mach-Zehnder Modulator page of this vault.
Other phase-modulation effects: Liquid crystals, thermally induced index changes and fibre stretching are slower (RP Photonics); the electro-optic type has bandwidths of many gigahertz.
Bulk modulators: A crystal with an aperture of millimetres and a free-space collimated beam, which has very low insertion loss and handles high power, but the electrodes are far apart, so the half-wave voltage is typically hundreds of volts and the bandwidth is limited (RP Photonics). A transverse crystal of lithium niobate with $d / L = 0.1$ needs 519 V at 1550 nm and a longitudinal one 14.96 kV (computed).
Geometries: In the transverse geometry the field is perpendicular to the beam and $V_{π}$ scales with $d / L$ ; in the longitudinal geometry $V_{π} = λ / (n_{o}^{3} r_{13})$ does not depend on the size (Beyersdorf).
Resonant bulk modulator: The Thorlabs EO-PM-R-20-C1 uses MgO-doped lithium niobate with an aperture of 2 mm; $V_{π}$ is 15 V at 633 nm for the resonant and 135 V for the non-resonant version, the operating bandwidth is about 5 % of the centre frequency, and the RF rating is 3 W.
Waveguide modulators: Light is confined in a single-mode channel and the electrodes are close to it, so $V_{π}$ is often below 5 V and the bandwidth tens of GHz, with higher insertion loss and limited power handling (RP Photonics). The iXblue MPZ-LN-10 (1530 to 1625 nm) has $V_{π}$ of 6 V at 10 GHz, an EO bandwidth of 10 to 12 GHz, a loss of 2.5 to $3.5 dB$ and a maximum input of $20 dBm$ ; the Thorlabs LNP6119 (1260 to 1625 nm) has an EO bandwidth of 35 GHz, $V_{π}$ of 7 to 10 V, a loss of 4 to $5.5 dB$ and a maximum input of 100 mW.
Thin film: The phase modulators of Zhu et al. have $V_{π} L$ of 4.7, 1.748 and 3.5 V cm (2016 to 2019), and Hu et al. give about 2 to 2.5 V for a loop-back device; the half-wave voltage of a Mach-Zehnder form is half that of a phase modulator of the same length (Hu et al.).
Spectrum: Sinusoidal drive makes equidistant sidebands whose powers are $J_{n}^{2} (β)$ ; for weak modulation one pair is enough (RP Photonics).

2. Working Principle

Pockels effect: The field changes the index by $Δ n_{e} = - \frac{1}{2} n_{e}^{3} r_{33} E$ , and the phase by $2 π Δ n L / λ$ ; the phase is proportional to the voltage, $Δ φ = π V / V_{π}$ .
Modulation depth: A sine drive of amplitude $V_{p}$ gives $β = π V_{p} / V_{π}$ and the field $E_{0} e^{i (ω t + β s i n Ω t)}$ (Black). The drive power is $P = (β V_{π} / π)^{2} /2 R$ , $15.6 dBm$ for 1 rad at $V_{π} = 6$ V (computed).
Carrier and sidebands: The carrier carries $J_{0}^{2}$ of the power and each sideband $J_{n}^{2}$ ; the carrier vanishes at 2.405 rad, where each first sideband carries 26.95 % (computed).
Frequency view: The instantaneous frequency deviates by $β f_{m}$ at peak (background knowledge), 41.6 GHz for 4.16 rad at 10 GHz (computed).
Bandwidth: A lumped electrode is limited by its capacitance (160 MHz for 20 pF and 50 $Ω$ ) and by the transit time ( $c /2 n L$ ); a traveling-wave electrode with matched velocities removes the second limit, and a resonant circuit multiplies the voltage within 5 % of its centre frequency (Beyersdorf; RP Photonics; Thorlabs).
Power limits: Bulk crystals are limited by the power density (2 to 4 W/mm $^{2}$ for the Thorlabs modulator) and by photorefraction at short wavelengths; fibre-coupled devices by 50 to 300 mW and 2.5 to $5.5 dB$ of loss (Thorlabs; iXblue; RP Photonics).

3. Applications

Laser frequency stabilization: Pound-Drever-Hall locking uses a phase modulator to make sidebands for an error signal; the best depth is 1.08 rad (Black; RP Photonics).
Laser tuning and mode locking: A constant phase change in a single-frequency laser resonator becomes a constant frequency change, and strong sinusoidal drive gives active FM mode locking (RP Photonics).
Suppression of stimulated Brillouin scattering: Strong phase modulation lowers the power per spectral line (RP Photonics); 10 rad gives a strongest line of 10 % (computed).
Interferometers and spectroscopy: Periodic drive of the phase (RP Photonics); the modulator also forms the arms of the Mach-Zehnder modulator.
Frequency combs: Strong modulation gives many sidebands (RP Photonics); Hu et al. report a waveguide comb with 67 lines over 12.6 nm and a cavity comb with an efficiency of 30 % and a span of 132 nm.
Optical communication: Phase-shift keying encodes data in the phase (RP Photonics); Zhu et al. list phase modulators with bandwidths of 40 to 47 GHz.

Phase Shift, Half-Wave Voltage and Drive

Name	Formula	Related Formulae	Variables/Constants
Phase shift of a transverse bulk modulator	$Δ φ = \frac{π n _{e}^{3} r _{33}}{λ} \frac{L}{d} V = π \frac{V}{V _{π}}$	$V_{π} = \frac{d}{L} \frac{λ}{n _{e}^{3} r _{33}}$ $V_{π}^{r e t} = \frac{d}{L} \frac{λ}{n _{e}^{3} r _{33} - n _{o}^{3} r _{13}}$	$d : d i s t an ce b e tw ee n t h e e l ec t r o d es (t h e f i e l d i s p er p e n d i c u l a r t o t h e b e am)$ $L : cr y s t a l l e n g t h, l i g h t p o l a r i z e d a l o n g z (ba c k g r o u n d k n o w l e d g e f or t h e s in g l e - p o l a r i z a t i o n f or m)$ $V_{π}^{r e t} : t h e f or m w i t h b o t h coe f f i c i e n t s i s t h e ha l f - w a v e v o l t a g e o f t h e r e t a r d a t i o n b e tw ee n t h e tw o p o l a r i z a t i o n s, a v o l t a g e - co n t r o l l e d w a v e pl a t e (B ey er s d or f; W ik i p e d ia), t h e m o d u l a t i o n d e pt h i s p r o p or t i o na l t o V L / d, f or l i t hi u m ni o ba t e a t 1550 nm n_{e} = 2.13, n_{o} = 2.21, r_{33} = 30.9 an d r_{13} = 9.6 p m / V (Z h u e t a l .), R P P h o t o ni cs g i v es V_{π} o f a b u l k m o d u l a t or a s t y p i c a l l y h u n d r e d s o f v o l t s, W ik i p e d ia a s h u n d r e d s or t h o u s an d s$ $(co m p u t e d, 1550 nm : V_{π} = 519 V f or d / L = 0.1 (1 mm b y 10 mm) an d 51.9 V f or d / L = 0.01; t h e r e t a r d a t i o n f or m g i v es 795 V f or d / L = 0.1, w i t h n_{e}^{3} r_{33} = 2.986 \times 1 0^{- 10} an d n_{o}^{3} r_{13} = 1.036 \times 1 0^{- 10} m / V)$
Longitudinal modulator	$V_{π} = \frac{λ}{n _{o}^{3} r _{13}}$	$Δ φ = \frac{2 π}{λ} n_{o} L - \frac{π}{λ} n_{o}^{3} r_{13} V$	$T h e f i e l d i s a l o n g t h e b e am (z - c u t l i t hi u m ni o ba t e w i t h t r an s p a r e n t e l ec t r o d es), so t h e v o l t a g e d oes n o t d e p e n d o n t h e l e n g t h or o n t h e a p er t u r e (B ey er s d or f)$ $r_{13} : t h e s ma l l coe f f i c i e n t t ha t co u pl es t o a f i e l d a l o n g z f or l i g h t p o l a r i z e d in t h e pl an e, w hi c h mak es t hi s g eo m e t r y t h e l e a s t se n s i t i v e$ $(co m p u t e d, 1550 nm, n_{o} = 2.21 an d r_{13} = 9.6 p m / V : V_{π} = 14.96 k V, 29 t im es t h e 519 V o f t h e t r an s v er se c a se w i t h d / L = 0.1)$
Waveguide phase modulator	$V_{π} = \frac{λ g}{n _{e}^{3} r _{33} Γ L}$	$V_{π} L = \frac{λ g}{n _{e}^{3} r _{33} Γ}$ $V_{π}^{M Z M} = \frac{1}{2} V_{π}^{P M}$	$g : g a p o f t h e e l ec t r o d es a t t h e w a v e g u i d e$ $Γ : o v er l a p f a c t or o f m y m o d e l (see t h e L i t hi u m N i o ba t e M a c h - Z e hn d er M o d u l a t or p a g e o f t hi s v a u l t), R P P h o t o ni cs : V_{π} o f a w a v e g u i d e m o d u l a t or i s o f t e n b e l o w 5 V an d t h e ban d w i d t h t e n s o f G H z, b u t t h e in ser t i o n l oss i s hi g h er an d t h e p o w er han d l in g l imi t e d, Z h u e t a l . l i s t p ha se m o d u l a t or s w i t h V_{π} L o f 4.7 V cm (4.7 V, 10 mm, 40 G H z, 1 d B / c m, 2016), 1.748 V cm (1.9 V, 9.2 mm, 47 G H z es t ima t e d, 7 d B / c m, 2018) an d 3.5 V cm (1.75 V, 20 mm, ab o v e 45 G H z, 0.5 d B / c m, 2019), H u e t a l . g i v e ab o u t 2 t o 2.5 V f or a l oo p - ba c k p ha se m o d u l a t or$ $(co m p u t e d, 1550 nm, g = 5 μ m, Γ = 1 : V_{π} L = 2.595 V cm an d 1.298 V f or 20 mm (0.649 V f or t h e M a c h - Z e hn d er f or m); t h e t h r ee t ab l e v a l u es cor r es p o n d t o g a p s o f 9.05, 3.37 an d 6.74 μ m f or Γ = 1, so t h e i r g a p s or o v er l a p s d i f f er)$
Modulation index for a sinusoidal drive	$β = \frac{π V _{p}}{V _{π}}$	$E (t) = E_{0} e^{i (ω t + β s i n Ω t)}$ $V_{p} = 2 R P$	$V_{p} : am pl i t u d e o f t h e s in e d r i v e$ $P : R F p o w er in t o R = 50 Ω (ba c k g r o u n d k n o w l e d g e)$ $β : m o d u l a t i o n d e pt h in r a d (B l a c k), i X b l u e : V_{π} 4 t o 5 V a t 50 k H z, 6 V a t 10 G H z an d 9 V a t 16 G H z, R F in p u t 28 d B m, T h or l ab s : V_{π} 10 V a t 1 G H z, 7 V a t 10 G H z an d 8.5 t o 9.5 V a t 30 G H z, R F in p u t 24 d B m$ $(co m p u t e d : 28 d B m i s 0.631 W (V_{p} = 7.94 V) an d g i v es β = 4.16 r a d a t 10 G H z an d 2.77 r a d a t 16 G H z; 24 d B m i s 0.251 W (5.01 V) an d g i v es 1.57 r a d a t 1 G H z, 2.25 r a d a t 10 G H z an d 1.65 t o 1.85 r a d a t 30 G H z)$
Radio-frequency power for a given depth	$P = \frac{1}{2 R} (\frac{β V _{π}}{π})^{2}$	$P_{d B m} = 10 lo g_{10} \frac{P}{1 mW}$	$R = 50 Ω (ba c k g r o u n d k n o w l e d g e), t h e r a t e d R F in p u t i s 28 d B m f or t h e i X b l u e d e v i ce, 24 d B m f or t h e T h or l ab s L N P 6119 an d 3 W (35 V p e ak - t o - p e ak) f or t h e T h or l ab s r eso nan t b u l k m o d u l a t or$ $(co m p u t e d : β = 1 r a d n ee d s 15.6 d B m f or V_{π} = 6 V, 19.1 d B m f or 9 V an d 13.1 d B m f or 4.5 V; β = 2.405 r a d (c a r r i er n u l l) n ee d s 23.2, 26.8 an d 20.7 d B m)$

Sidebands and Modulation Depth

Name	Formula	Related Formulae	Variables/Constants
Sideband spectrum	$E (t) = E_{0} \sum_{n = - \infty}^{\infty} J_{n} (β) e^{i (ω + n Ω) t}$	$P_{n} = J_{n}^{2} (β) P_{0}$ $\sum_{n} J_{n}^{2} (β) = 1$	$T h e s i d e ban d s a r e e q u i d i s t an t, s p a ce d b y t h e m o d u l a t i o n f r e q u e n cy, w i t h p o w er s g i v e n b y B esse l f u n c t i o n s (R P P h o t o ni cs; t h e T h or l ab s man u a l : t h e n t h p ai r l i es a t f_{0} \pm n f_{m} w i t h p o w er J_{n}^{2}), W ik i p e d ia : p ha se m o d u l a t i o n g i v es t h e c a r r i er an d in t h eor y an in f ini t e n u mb er o f s i d e ban d s$ $B l a c k : P_{c} = J_{0}^{2} P_{0} an d P_{s} = J_{1}^{2} P_{0}$ $(co m p u t e d, J_{0}^{2} / J_{1}^{2} / s t r o n g es t l in e / l in es ab o v e 1 % : β = 0.5 0.881 / 0.059 / 0.881 / 3; β = 1 0.586 / 0.194 / 0.586 / 5; β = 2 0.050 / 0.333 / 0.333 / 7; β = 4.16 0.146 / 0.016 / 0.189 / 11; β = 10 0.061 / 0.002 / 0.101 / 17)$
Small-depth limit	$E \approx E_{0} [1 + i β sin Ω t] e^{iω t}$	$J_{0} \approx 1, J_{1} \approx \frac{β}{2}, P_{s} \approx \frac{β ^{2}}{4} P_{0}$	$V a l i d f or a p ha se e x c u r s i o n w e l l b e l o w 1 r a d, w h er e o n e p ai r o f s i d e ban d s s u f f i ces an d t h e s i d e ban d am pl i t u d e i s p r o p or t i o na l t o β (R P P h o t o ni cs; B l a c k), t h e er r or o f J_{1} \approx β /2 i s t h e q u an t i t y in t h e n e x t co l u mn$ $(co m p u t e d : J_{1} / (β /2) = 0.969 a t β = 0.5 an d 0.880 a t β = 1)$
Carrier null	$J_{0} (β) = 0 : β = 2.405, 5.520, 8.654$	$V_{p} = 0.7655 V_{π} at β = 2.405$	$T h e T h or l ab s man u a l s a y s t ha t a t a d e pt h o f ab o u t 2.4 r a d a l m os t a l l o f t h e e n er g y i s t r an s f er r e d f r o m t h e c a r r i er t o t h e s i d e ban d s, t h e z er os o f J_{0} a r e t h e s t an d a r d v a l u es (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : a t 2.405 r a d t h e f i r s t s i d e ban d s c a r r y 0.2695 e a c h, t h e seco n d 0.1864 an d t h e t hi r d 0.0396 o f t h e p o w er; V_{p} = 2.405/ π V_{π})$
Optimum depth for Pound-Drever-Hall locking	$\frac{d}{d β} [J_{0} (β) J_{1} (β)] = 0$	$P_{c} P_{s} = P_{0} J_{0} J_{1}$ $\frac{P _{s}}{P _{c}} = (\frac{J _{1}}{J _{0}})^{2}$	$B l a c k : t h e er r or s i g na l i s p r o p or t i o na l t o 2 P_{c} P_{s}, t h e s im pl e o pt im u m i s P_{s} / P_{c} = 1/2 (P_{c} / P_{0} = 1/2) an d t h e r e f in e d o pt im u m w i t h t h e f u l l B esse l f u n c t i o n s i s β = 1.08 r a d w i t h P_{s} / P_{c} \approx 0.42, t h e d i scr iminan t i s D = 2 8 P_{c} P_{s} / δ ν w i t h δ ν t h e l in e w i d t h o f t h e c a v i t y (B l a c k; t h e e x t r a c t w a s g a r b l e d an d t h e f a c t or s h o u l d b e c h ec k e d)$ $(co m p u t e d : t h e ma x im u m o f J_{0} J_{1} i s a t β = 1.082, J_{0} = 0.7281, J_{1} = 0.4656, J_{0} J_{1} = 0.339, P_{s} / P_{c} = 0.409 an d P_{c} / P_{0} = 0.530; P_{s} / P_{c} = 1/2 w i t h t h e e x a c t f u n c t i o n s i s a t β = 1.161; t h e s ma l l - d e pt h f or m (β /2)^{2} w o u l d g i v e 0.292 a t 1.08)$
Instantaneous frequency and deviation	$f_{in s t} (t) = f_{0} + β f_{m} cos (2 π f_{m} t)$	$Δ f_{p k} = β f_{m}$	$T h e f r e q u e n cy i s t h e t im e d er i v a t i v e o f t h e p ha se d i v i d e d b y 2 π (ba c k g r o u n d k n o w l e d g e), R P P h o t o ni cs : a co n s t an t p ha se c han g e in a l a ser r eso na t or b eco m es a co n s t an t f r e q u e n cy c han g e an d s t r o n g s in u so i d a l d r i v e i s u se d f or F M m o d e l oc k in g, H u e t a l . r e p or t a w a v e g u i d e co mb o f 67 l in es o v er 12.6 nm w i t h 3.8 W o f mi cr o w a v e p o w er$ $(co m p u t e d : β = 4.16 a t 10 G H z g i v es a p e ak d e v ia t i o n o f 41.6 G H z, β = 2.77 a t 16 G H z 44.3 G H z an d β = 1 a t 1 G H z 1 G H z; 12.6 nm a t 1550 nm i s 1.57 T H z, so 67 l in es a r e ab o u t 23.8 G H z a p a r t i f t h e s p ec t r u m i s f l a t)$
Spreading of power over many lines	$P_{ma x} = max_{n} J_{n}^{2} (β)$	$Δ_{d B} = - 10 lo g_{10} P_{ma x}$	$R P P h o t o ni cs nam es t h e s u pp r ess i o n o f s t im u l a t e d B r i l l o u in sc a tt er in g b y s t r o n g p ha se m o d u l a t i o n, w hi c h l o w er s t h e p o w er p er s p ec t r a l l in e, a s an a ppl i c a t i o n, t h e n u mb er o f l in es an d t h e r e d u c t i o n a r e t h e q u an t i t i es o f t hi s r o w$ $(co m p u t e d : β = 1 2.32 d B (5 l in es ab o v e 1 %), 2.405 5.69 d B (6), 4 7.33 d B (9), 5 8.15 d B (11), 10 9.96 d B (17); β = 10 n ee d s V_{p} = 3.18 V_{π}, t ha t i s 19.1 V or 35.6 d B m f or 6 V, ab o v e t h e 28 d B m r a t in g o f t h e i X b l u e d e v i ce)$

Bandwidth and Resonant Drive

Name	Formula	Related Formulae	Variables/Constants
RC limit of a lumped electrode	$f_{R C} = \frac{1}{2 π R _{s} C}$	$C_{E O M} \approx 14 pF$	$R_{s} : so u r ce im p e d an ce$ $C : c a p a c i t an ce o f t h e e l ec t r o d es, B ey er s d or f : C = 20 pF an d R_{s} = 50 Ω g i v e ab o u t 160 M H z, T h or l ab s : an in p u t c a p a c i t an ce o f 14 pF (t y p i c a l, n o n - r eso nan t v er s i o n) an d a s t a t e d r an g e o f D C t o 100 M H z$ $(co m p u t e d : 159.2 M H z f or 20 pF an d 227.4 M H z f or 14 pF)$
Transit-time limit of a lumped electrode	$m (f) = \frac{s i n x}{x}, x = \frac{π f n L}{c}$	$f_{c} = \frac{c}{2 n L} (x = π /2, m = 0.64)$ $f_{3 d B} = \frac{0.443 c}{n L}$	$n : r e f r a c t i v e in d e x o f t h e l i g h t$ $L : e l ec t r o d e l e n g t h, B ey er s d or f : f_{c} = c / (2 n L) i s w h er e t h e m o d u l a t i o n f a l l s t o 0.64 o f t h e v a l u e w i t h o u t a v er a g in g an d t h e l imi t d i s a pp e a r s f or a t r a v e l in g - w a v e e l ec t r o d e w i t h e q u a l v e l oc i t i es (R P P h o t o ni cs : ma t c h e d p ha se v e l oc i t i es a l l o w e l ec t r o d es o f se v er a l mi cr o w a v e w a v e l e n g t h s), f_{3 d B} i s t h e p o in t o f 1/ 2 (20 l o g co n v e n t i o n o f t h e L i t hi u m N i o ba t e M a c h - Z e hn d er M o d u l a t or p a g e, x = 1.392)$ $(co m p u t e d : n = 2 an d L = 3 c m g i v e f_{c} = 2.50 G H z (r es p o n se 0.637, - 3.92 d B) an d f_{3 d B} = 2.21 G H z; n = 2.2 an d L = 2 c m g i v e 3.41 an d 3.02 G H z; t h e R C l imi t o f 159 M H z i s 15.7 t im es l o w er t han f_{c})$
Series-resonant drive	$f_{0} = \frac{1}{2 π L _{l o a d} C}, \frac{∣ V _{E O M} ∣}{V _{m}} = \frac{1}{ω _{0} ( R _{s} + R _{l o a d} ) C}$	$Q = \frac{1}{ω _{0} ( R _{s} + R _{l o a d} ) C}$ $Δ f = \frac{f _{0}}{Q}$	$B ey er s d or f g i v es t h e e x am pl e R_{s} = R_{l o a d} = 50 Ω, C = 20 pF an d f_{0} = 10 M H z an d a r a t i o o f ab o u t 80 (t h e p o w er f or m u l a in t h e s am e e x t r a c t i s g a r b l e d), T h or l ab s : t h e r eso nan t m o d u l a t or s ha v e a t y p i c a l o p er a t in g ban d w i d t h o f 5 % o f t h e ce n t r e f r e q u e n cy, an in p u t im p e d an ce o f 50 Ω an d a ma x im u m R F p o w er o f 3 W (35 V p e ak - t o - p e ak), t h e o p er a t in g f r e q u e n cy i s f i x e d a t p u r c ha se$ $(co m p u t e d : t h e e x am pl e g i v es 7.96, n o t 80, w i t h L_{l o a d} = 12.67 μ H an d a ban d w i d t h o f 1.26 M H z; 5 % ban d w i d t h m e an s Q = 20 an d 1.5 M H z a t 30 M H z; 35 V p e ak - t o - p e ak i s V_{p} = 17.5 V an d 3.06 W in t o 50 Ω)$
Resonant against non-resonant half-wave voltage	$G_{V} = \frac{V _{π, n r}}{V _{π, r}}$	$β = \frac{π V _{p}}{V _{π}}$	$T h e T h or l ab s man u a l g i v es V_{π} = 15 V a t 633 nm f or t h e r eso nan t v er s i o n an d 135 V f or t h e n o n - r eso nan t o n e (F i g . 1 pl o t s V_{π} a g ain s t w a v e l e n g t h f r o m 400 t o 1600 nm), t h e cr y s t a l i s M g O - d o p e d l i t hi u m ni o ba t e w i t h a c l e a r a p er t u r e o f 2 mm an d t h e cr y s t a l l e n g t h i s n o t g i v e n, a s an e x am pl e an 8 V p e ak s in e d r i v e a t 30 M H z g a v e a d e pt h o f 2.1 r a d$ $(co m p u t e d : G_{V} = 9.0; 8 V p e ak an d 15 V g i v e 1.68 r a d, w h er e a s 2.1 r a d n ee d s V_{π} = 11.97 V (0.80 o f 15 V), w hi c h w o u l d b e 505 nm i f V_{π} w er e p r o p or t i o na l t o t h e w a v e l e n g t h (ba c k g r o u n d k n o w l e d g e); 17.5 V p e ak an d 15 V g i v e 3.67 r a d)$
Half-wave voltage against frequency	$V_{π} (f) = V_{π} (0) 1 0^{R_{E O} (f) /20}$	$R_{E O} = 20 lo g_{10} \frac{V _{π} ( f )}{V _{π} ( 0 )}$	$T h e s am e 20 l o g co n v e n t i o n a s o n t h e L i t hi u m N i o ba t e M a c h - Z e hn d er M o d u l a t or p a g e, i X b l u e : E O ban d w i d t h o f 10 t o 12 G H z an d u s ab l e ban d w i d t h o f 16 G H z, V_{π} 4 t o 5 V a t 50 k H z, 6 V a t 10 G H z, 9 V a t 16 G H z an d 4 t o 7 V a t D C (bia s p or t), T h or l ab s : E O ban d w i d t h 35 G H z, V_{π} 7 t o 10 V o v er a l l, 10 V a t 1 G H z, 7 V a t 10 G H z an d 8.5 t o 9.5 V a t 30 G H z$ $(co m p u t e d : i X b l u e 9/4.5 = 2.0 (6.02 d B) an d 6/4.5 = 1.33 (2.5 d B), 4.5 \times 1.414 = 6.4 V; T h or l ab s 7 V a t 1 G H z an d 9 V a t 30 G H z w o u l d b e 2.2 d B, w h er e a s 10 V a t 1 G H z t o 7 V a t 10 G H z f a l l s b y 3.1 d B)$

Insertion Loss and Optical Power

Name	Formula	Related Formulae	Variables/Constants
Insertion loss and transmitted power	$T = 1 0^{- I L /10}$	$P_{o u t} = T P_{in}$	$I L : in ser t i o n l oss in d B, i X b l u e 2.5 t o 3.5 d B, T h or l ab s 4 t o 5.5 d B, R P P h o t o ni cs a r o u n d 4 d B f or f ib r e - co u pl e d d e v i ces an d a p o w er l imi t o f f or e x am pl e 50 mW, ma x im u m o pt i c a l in p u t p o w er 20 d B m f or i X b l u e an d 100 mW f or T h or l ab s, R P P h o t o ni cs nam es a 30 G H z d e v i ce w i t h 300 mW$ $(co m p u t e d : 2.5 t o 3.5 d B t r an s mi t 0.56 t o 0.45, 4 d B 0.40 an d 4 t o 5.5 d B 0.40 t o 0.28; 20 d B m i s 100 mW, 50 mW i s 17.0 d B m an d 300 mW i s 24.8 d B m)$
Optical power density in a bulk crystal	$I = \frac{P}{A}$	$A = π (\frac{D}{2})^{2}$	$T h e T h or l ab s man u a l g i v es a d ama g e l imi t o f 2 W/m m^{2} a t 532 nm an d 4 W/m m^{2} a t 1064 nm f or a c l e a r a p er t u r e o f 2 mm, l i t hi u m ni o ba t e w i t h s t an d s p u l se d p o w er d e n s i t i es o f u p t o 1 M W / c m^{2} b u t s h or t w a v e l e n g t h s a r e l imi t e d b y p h o t or e f r a c t i v e e f f ec t s (T h or l ab s), R P P h o t o ni cs : l i t hi u m ni o ba t e ha s a l o w er d ama g e t h r es h o l d t han B B O or K D * P an d b u l k d e v i ces han d l e hi g h p o w er w i t h v er y l o w l oss$ $(co m p u t e d : t h e a p er t u r e i s 3.14 m m^{2}, so 2 an d 4 W/m m^{2} w o u l d b e 6.3 an d 12.6 W f or a u ni f or m b e am o v er t h e w h o l e a p er t u r e; a G a u ss ian b e am o f s ma l l er r a d i u s r e a c h es t h e l imi t a t l o w er p o w er)$

Important Questions

Ques: Why does a bulk lithium niobate modulator need hundreds or thousands of volts?

The half-wave voltage of a transverse crystal is $V_{π} = (d / L) λ / (n_{e}^{3} r_{33})$ . With $n_{e} = 2.13$ and $r_{33} = 30.9$ pm/V at 1550 nm, a crystal 1 mm thick and 10 mm long needs 519 V, and 51.9 V if $d / L$ is 0.01 (computed). The retardation form, with the $n_{o}^{3} r_{13}$ term of a waveplate, gives 795 V for $d / L = 0.1$ (computed). In the longitudinal geometry $V_{π} = λ / (n_{o}^{3} r_{13})$ is 14.96 kV whatever the size, because $r_{13}$ is small (computed). RP Photonics says hundreds of volts for bulk modulators and Wikipedia hundreds or thousands, and both fit the transverse and longitudinal values. Resonant circuits bring the figure down: 15 V against 135 V for the Thorlabs modulator, a factor of 9 (computed). Reference: Beyersdorf, ‘Chapter 8: Electrooptic devices’, San Jose State University lecture notes (Physics 208); RP Photonics, ‘Phase modulators’; Wikipedia, ‘Electro-optic modulator’; Thorlabs, EO-PM-R-20-C1 resonant phase modulator manual, Rev. B (copy at wiki.physics.wisc.edu); Zhu et al., ‘Integrated photonics on thin-film lithium niobate’ (arXiv 2102.11956; Tables 1 and 3).

Ques: Is the voltage gain of the series-resonant circuit 80 or 8?

The lecture notes of Beyersdorf give the example $R_{s} = R_{l o a d} = 50 Ω$ , $C = 20$ pF and $f_{0} = 10$ MHz, with $V_{E O M} / V_{m} \approx 80$ . The stated formula $1/ (ω_{0} (R_{s} + R_{l o a d}) C)$ gives 7.96 with these values, not 80 (computed), and the load inductance needed is 12.67 $μ$ H and the bandwidth $f_{0} / Q = 1.26$ MHz (computed). A gain of 80 would need a total resistance of 10 $Ω$ or a capacitance of 200 pF. The tool extract of the notes also had a garbled power formula, so the 80 may be a transcription fault or a typing error (my reading). The Thorlabs resonant modulators have a bandwidth of 5 % of the centre frequency, which means $Q = 20$ (computed), and so a gain near 8 is more plausible than 80. Reference: Beyersdorf, ‘Chapter 8: Electrooptic devices’, San Jose State University lecture notes (Physics 208); Thorlabs, EO-PM-R-20-C1 resonant phase modulator manual, Rev. B (copy at wiki.physics.wisc.edu).

Ques: Do 15 V and 2.1 rad at 8 V peak agree for the resonant modulator?

The Thorlabs manual gives $V_{π} = 15$ V at 633 nm for the resonant version and, as an example, an 8 V peak sine drive that gave 2.1 rad at 30 MHz. The formula $β = π V_{p} / V_{π}$ gives 1.68 rad for 15 V (computed), and 2.1 rad needs $V_{π} = 11.97$ V, 0.80 of the stated value. If $V_{π}$ is proportional to the wavelength (background knowledge), that is 505 nm, and the manual’s Fig. 1 plots $V_{π}$ for 400 to 1600 nm, so the example was probably taken at a shorter wavelength, not stated in the extract (my reading). The rating of 3 W at 35 V peak-to-peak agrees with 50 $Ω$ , since 17.5 V peak is 3.06 W (computed), and at 15 V it would give 3.67 rad. Reference: Thorlabs, EO-PM-R-20-C1 resonant phase modulator manual, Rev. B (copy at wiki.physics.wisc.edu).

Ques: Is the half-wave voltage of the Thorlabs LNP6119 10 V at 1 GHz and 7 V at 10 GHz?

The listing gives 7 to 10 V overall, 10 V at 1 GHz, 7 V at 10 GHz and 8.5 to 9.5 V at 30 GHz, with an EO bandwidth of 35 GHz. A roll-off should raise the voltage with frequency, and 10 V at 1 GHz falling to 7 V at 10 GHz is a decrease of $3.1 dB$ (computed), which a lumped or traveling-wave electrode does not do. The reverse reading, 7 V at 1 GHz and 9 V at 30 GHz, is a roll-off of $2.2 dB$ (computed), consistent with a $3 dB$ bandwidth of 35 GHz. The first two values are therefore probably interchanged or refer to different conditions (my reading). The same listing gives ‘DC to 40 MHz’, probably 40 GHz for a 35 GHz device. Reference: Thorlabs LNP6119 phase modulator (gophotonics.com listing).

Ques: Is the bandwidth of the iXblue modulator 10 to 12 GHz or 16 GHz?

The listing gives an EO bandwidth of 10 to 12 GHz and a usable bandwidth of 16 GHz, with $V_{π}$ of 4 to 5 V at 50 kHz, 6 V at 10 GHz and 9 V at 16 GHz. With 4.5 V at low frequency, 9 V at 16 GHz is a factor 2.0, $6.02 dB$ in the 20 log convention, and 4.5 times 1.414 is 6.4 V, close to the 6 V at 10 GHz (computed). The 10 to 12 GHz is therefore the $3 dB$ point and 16 GHz the $6 dB$ point, which is how the usable bandwidth appears to be defined (my reading). The figures agree, and the difference is the definition. The device cannot be driven to the full depth over this range: $28 dBm$ gives 4.16 rad at 10 GHz and 2.77 rad at 16 GHz (computed). Reference: iXblue Photonics (Exail), MPZ-LN-10-00-P-P-FA-FA phase modulator (gophotonics.com listing).

Ques: Is the power limit of a fibre-coupled modulator 50, 100 or 300 mW?

RP Photonics says that fibre-coupled devices have around $4 dB$ of insertion loss and ‘can handle only limited power levels, e.g. 50 mW’. The iXblue listing gives $20 dBm$ (100 mW) with 2.5 to $3.5 dB$ , the Thorlabs listing 100 mW with 4 to $5.5 dB$ , and RP Photonics names a 30 GHz device of 300 mW ( $24.8 dBm$ , computed). The 50 mW is an example and not a rating, so the datasheet values are the ones to use (my reading). The loss of $4 dB$ transmits 40 % (computed), and the difference between 2.5 and $5.5 dB$ is a factor of 2 in transmitted power (0.56 against 0.28, computed). The 300 mW device operates in the near infrared, and the extract does not give its loss or its half-wave voltage. Reference: RP Photonics, ‘Electro-optic modulators’; RP Photonics, ‘Phase modulators’; iXblue Photonics (Exail), MPZ-LN-10-00-P-P-FA-FA phase modulator (gophotonics.com listing); Thorlabs LNP6119 phase modulator (gophotonics.com listing).

Ques: What is the best modulation depth for Pound-Drever-Hall locking: 1.08 rad, or $P_{s} / P_{c} = 1/2$ ?

Black gives a simple optimum at $P_{s} / P_{c} = 1/2$ , that is $P_{c} / P_{0} = 1/2$ , and a refined one at $β = 1.08$ rad with $P_{s} / P_{c} \approx 0.42$ . The error signal is proportional to $J_{0} J_{1}$ , and its maximum is at $β = 1.082$ with $P_{s} / P_{c} = 0.409$ and $P_{c} / P_{0} = 0.530$ (computed). The exact Bessel functions give $P_{s} / P_{c} = 1/2$ at $β = 1.161$ (computed), 7 % higher in depth. The simple rule comes from the small-depth form $(β /2)^{2}$ , which gives 0.292 at 1.08 (computed), so it is not an exact statement. The 0.42 of the text is 3 % above my 0.409, probably a rounding. The depth of 1.08 rad and a ratio of about 0.41 are the more likely values. Reference: Black, ‘An introduction to Pound-Drever-Hall laser frequency stabilization’, Am. J. Phys. 69, 79 (2001).

Ques: Which depth removes the carrier, and does it spread the power evenly?

The Thorlabs manual says that at about 2.4 rad almost all the energy goes from the carrier to the sidebands, and the first zero of $J_{0}$ is 2.405 (background knowledge). At this depth each first sideband carries 26.95 % of the power, each second sideband 18.64 % and each third 3.96 % (computed), so the strongest line is still $5.7 dB$ below the total. RP Photonics names the suppression of stimulated Brillouin scattering as an application, which needs many lines: $β = 10$ gives a strongest line of 10 % ( $9.96 dB$ , 17 lines above 1 %, computed). This needs $V_{p} = 3.18 V_{π}$ , 19.1 V or $35.6 dBm$ for a 6 V device, above the $28 dBm$ rating of the iXblue modulator (computed). A single device cannot do both, so the strong-modulation uses need low- $V_{π}$ or resonant structures. Reference: Thorlabs, EO-PM-R-20-C1 resonant phase modulator manual, Rev. B (copy at wiki.physics.wisc.edu); RP Photonics, ‘Phase modulators’; iXblue Photonics (Exail), MPZ-LN-10-00-P-P-FA-FA phase modulator (gophotonics.com listing).

Ques: Is a phase modulator twice as high in voltage as a Mach-Zehnder modulator of the same length?

Hu et al. say that the Mach-Zehnder form has half the $V_{π}$ of a phase modulator of the same length, because the arms have opposite phase shifts. The phase modulators in Table 3 of Zhu et al. do not show a clean factor of two against the Mach-Zehnder devices. They give 4.7, 1.748 and 3.5 V cm, which for $Γ = 1$ need gaps of 9.05, 3.37 and 6.74 $μ$ m (computed), while the 20 mm Mach-Zehnder device of Wang et al. has 2.8 V cm (computed). The ideal value of a phase modulator with a 5 $μ$ m gap is 2.595 V cm, twice the 1.298 V cm of the Mach-Zehnder form (computed). The devices differ in gap and overlap, so the rule holds for equal geometry and not for the table (my reading). Reference: Hu et al., ‘Integrated electro-optics on thin-film lithium niobate’ (arXiv 2404.06398); Zhu et al., ‘Integrated photonics on thin-film lithium niobate’ (arXiv 2102.11956; Tables 1 and 3).

Ques: Where is the bandwidth limit of a lumped modulator: 160 MHz or 2.5 GHz?

Beyersdorf gives two limits. The RC limit, $1/ (2 π R_{s} C)$ , is 160 MHz for 20 pF and 50 $Ω$ (159.2 MHz, computed), and the transit-time limit $c / (2 n L)$ is 2.5 GHz for 3 cm and $n = 2$ . The first applies to the electrode as a lumped capacitor and is 15.7 times lower than the second (computed), so a lumped modulator of this size is limited by the RC time. The transit-time value is where the response is $sin x / x = 0.64$ , or $- 3.92$ dB in the 20 log convention, while the $3 dB$ point is at 2.21 GHz, 0.886 of it (computed). With $6 dB$ ( $3 dB$ optical) the point is higher, 3.02 GHz for the same size (computed). A traveling-wave electrode removes the second limit when the velocities match. Reference: Beyersdorf, ‘Chapter 8: Electrooptic devices’, San Jose State University lecture notes (Physics 208); RP Photonics, ‘Electro-optic modulators’; Lithium Niobate Mach-Zehnder Modulator page of this vault.

Quartz 5

Explorer

Electro-Optic Phase Modulator

1. Theory

2. Working Principle

3. Applications

Phase Shift, Half-Wave Voltage and Drive

Sidebands and Modulation Depth

Bandwidth and Resonant Drive

Insertion Loss and Optical Power

Important Questions

Ques: Why does a bulk lithium niobate modulator need hundreds or thousands of volts?

Ques: Is the voltage gain of the series-resonant circuit 80 or 8?

Ques: Do 15 V and 2.1 rad at 8 V peak agree for the resonant modulator?

Ques: Is the half-wave voltage of the Thorlabs LNP6119 10 V at 1 GHz and 7 V at 10 GHz?

Ques: Is the bandwidth of the iXblue modulator 10 to 12 GHz or 16 GHz?

Ques: Is the power limit of a fibre-coupled modulator 50, 100 or 300 mW?

Ques: What is the best modulation depth for Pound-Drever-Hall locking: 1.08 rad, or $P_{s} / P_{c} = 1/2$ ?

Ques: Which depth removes the carrier, and does it spread the power evenly?

Ques: Is a phase modulator twice as high in voltage as a Mach-Zehnder modulator of the same length?

Ques: Where is the bandwidth limit of a lumped modulator: 160 MHz or 2.5 GHz?

Graph View

Table of Contents

Backlinks

Quartz 5

Explorer

Electro-Optic Phase Modulator

1. Theory

2. Working Principle

3. Applications

Phase Shift, Half-Wave Voltage and Drive

Sidebands and Modulation Depth

Bandwidth and Resonant Drive

Insertion Loss and Optical Power

Important Questions

Ques: Why does a bulk lithium niobate modulator need hundreds or thousands of volts?

Ques: Is the voltage gain of the series-resonant circuit 80 or 8?

Ques: Do 15 V and 2.1 rad at 8 V peak agree for the resonant modulator?

Ques: Is the half-wave voltage of the Thorlabs LNP6119 10 V at 1 GHz and 7 V at 10 GHz?

Ques: Is the bandwidth of the iXblue modulator 10 to 12 GHz or 16 GHz?

Ques: Is the power limit of a fibre-coupled modulator 50, 100 or 300 mW?

Ques: What is the best modulation depth for Pound-Drever-Hall locking: 1.08 rad, or Ps​/Pc​=1/2?

Ques: Which depth removes the carrier, and does it spread the power evenly?

Ques: Is a phase modulator twice as high in voltage as a Mach-Zehnder modulator of the same length?

Ques: Where is the bandwidth limit of a lumped modulator: 160 MHz or 2.5 GHz?

Graph View

Table of Contents

Backlinks

Ques: What is the best modulation depth for Pound-Drever-Hall locking: 1.08 rad, or $P_{s} / P_{c} = 1/2$ ?