2.2 The Pockels effect

L1 Phenomenon

bulk; MZM of Li et al. 2026

A field E changes the optical impermeability 1/n² of a non-centrosymmetric crystal linearly. With field and optical polarization both along the polar axis c (= z), the extraordinary index shifts by

Δ n_{e} = - \frac{1}{2} n_{e}^{3} r_{33} E_{z}, Δ φ = \frac{2 π}{λ} Δ n_{e} L

(1)

Sign: a positive r₃₃E_z lowers n_e. A phase shift of π needs E·L = λ / (n_e³ r₃₃) for a uniform field that fills the mode.

Drive voltage V 0.00 V

Phase per arm: ±0.0°
V / V_π: 0.000
Transmission T: 1.000

V_π 2.90 V at 1550 nm reported dashed: engine FEM, V_πL 2.697 V·cm over 0.9 cm engine 9 mm x-cut TFLN on silicon, G-S-G coplanar line, one waveguide per gap (push-pull); T = cos²(πV / 2V_π), zero bias phase light: 1550 nm, false color

Bulk: field and polarization along c

Wavelength λ1.55 µm

Field E_z1.0 V/µm

n_e(1.55 µm)	2.1376		P-db
r₃₃	30.8	pm/V	P-comp
Δn_e	−1.504×10⁻⁴		derived
Δφ / L	−0.6097	rad/mm	derived
E·L for Δφ = π	5.153	kV	derived

r₃₃ is held at its 633 nm clamped value; no 1550 nm EO coefficients exist in the sources held. n_e(λ): Zelmon et al. 1997 Sellmeier, valid 0.4 to 5.0 µm.

MZM, Li et al. 2026: T(V)reported V_π 2.90 V

Drive voltage V0.00 V

T	1.0000
T (dB)	0.00	dB
Δφ = πV / V_π	0.000	π rad	derived
\|Δn_eff\| per arm = λ\|Δφ\| / 4πL	0		derived
\|Δn_eff\| per arm, engine \|dn_A/dV\|·\|V\|	0		engine

9 mm x-cut TFLN on silicon, G-S-G coplanar line, one waveguide per gap: the two arms see opposite fields (push-pull). T = cos²(πV / 2V_π), zero bias phase. Dashed: engine FEM, same cross-section (L3), V_πL 2.697 V·cm over 0.9 cm engine.

L2 The r tensor

point group 3m

Index ellipsoid and its first-order perturbation, contracted index I: 1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz, 6 = xy; j runs over the field components.

\sum_{i, j} {(\frac{1}{n^{2}})}_{i j} x_{i} x_{j} = 1, Δ {(\frac{1}{n^{2}})}_{I} = \sum_{j = 1}^{3} r_{I j} E_{j}, δ n_{k} = - \frac{1}{2} n_{k}^{3} Δ {(\frac{1}{n^{2}})}_{k k}

(2)

Point group 3m, x perpendicular to a mirror plane, z along c: four independent coefficients. The last relation holds for polarization along a principal axis k; off-diagonal terms (I = 4, 5, 6) rotate the principal axes instead.

r_Ij at E = 1 V/µm, λ = 1550 nmKaminow et al. 1980, clamped

Field

Polarization

Set

	E_x	E_y	E_z
1 xx	0	−r₂₂	r₁₃
2 yy	0	r₂₂	r₁₃
3 zz	0	0	r₃₃
4 yz	0	r₅₁	0
5 xz	r₅₁	0	0
6 xy	−r₂₂	0	0

Δ(1/n²)_I = r_IzE
1 xx	7.70×10⁻⁶
2 yy	7.70×10⁻⁶
3 zz	2.88×10⁻⁵	diag k
4 yz	0
5 xz	0
6 xy	0

polarization z; exact principal axes, crystal frame
n_z (n_e, 1.55 µm)	2.1376	P-db
δn_z first order	−1.406×10⁻⁴	derived
δn along (+1.000, +0.000, +0.000)	−4.162×10⁻⁵	along x
δn along (+0.000, +1.000, +0.000)	−4.162×10⁻⁵	along y
δn along (+0.000, +0.000, +1.000)	−1.406×10⁻⁴	along z

off-diagonal terms rotate the principal axes
4 yz	0	no rotation
5 xz	0	no rotation
6 xy	0	no rotation
principal indices	2 equal	uniaxial

Highlight: diagonal entry for the chosen polarization off-diagonal: axis rotation diagonal entry for another polarization

Field, 1 V/µm along

Optical polarization, 1550 nm

Coefficient set

Index along the polarization: 2.1376 (nₑ)
δn first order: −1.406e−4
Principal indices: uniaxial, 2 equal
Largest principal-axis tilt: 0.0000°
r₁₃ / r₂₂ / r₃₃ / r₅₁ of the set: 7.7 / 3.40 / 28.8 / 18.2 pm/V

Drawn: Δ(1/n²) ×500 before the eigen-decomposition; semi-axis 1 + 10 (n_i/n̄ − 1); wireframe at E = 0. Numbers: exact, unscaled. derived

Coefficients (pm/V)

Set	Condition	r₁₃	r₂₂	r₃₃	r₅₁	Status
Kaminow et al. 1980	clamped (S); preferred by Weis & Gaylord	7.7	3.4	28.8	18.2	P-comp
Turner 1966	clamped (S)	8.6	3.4	30.8	28	P-comp
Zook et al. 1967	free (T)	10.0	6.81	32.2	–	P-comp
Lenzo 1966	free (T)	–	6.7	–	–	P-comp
Bernal 1966	free (T)	–	–	–	32	P-comp

Clamped (S): strain fixed, above the acoustic resonances. Free (T): includes the piezoelectric-photoelastic contribution, below them. No 1550 nm values in the sources held. The L3 engine run uses the Turner 1966 set. Click a row to load it above.

Device mapping

Cut	Propagation	Optical field	Static field	Coefficient	Index change
x-cut	y (in plane)	TE, lateral = z	in-plane, lateral = z	r₃₃	−½ n_e³ r₃₃ E
z-cut	in plane	TM, vertical = z	vertical = z	r₃₃	−½ n_e³ r₃₃ E
z-cut plate	z (along c)	any, in xy	in-plane, x or y	r₂₂	birefringence n_o³ r₂₂ E; axes at 45° for E_x, along x, y for E_y
Li et al. 2026 (x-cut MZM)	y	TE	lateral, 6 µm gap	r₃₃ (99.66% of d n_eff/dV)	L3

The z-cut plate row is the conoscope on the landing page: E ⊥ c makes the uniaxial crystal biaxial.

L3 Evaluating the overlap

FEM: eo-modulator-atlas engine, full-vector solver, geometry of Li et al. 2026, exported 2026-10-25 engine

First-order perturbation of a guided mode normalized to power P, with Δε = −ε Δ(ε⁻¹) ε:

Δ β = \frac{ω ε_{0}}{4 P} \int E^{*} \cdot Δ ε \cdot E d A, P = \frac{1}{2} \int Re (E \times H^{*}) \cdot \hat{z} d A

(3)

x-cut TE, E ≈ E_x along c: Δε_xx = −n_e⁴ r₃₃ E_x^RF, and H_y = n_eff E_x / Z₀ makes P = (n_eff / 2Z₀) ∫|E_x|² dA. With e_x the static field per terminal volt:

\frac{d n_{eff}}{d V} = - \frac{n_{e}^{4} r_{33}}{2 n_{eff}} \frac{\int_{LN} e_{x} {| E_{x} |}^{2} d A}{\int {| E_{x} |}^{2} d A}

(4)

Conventional overlap with gap g, and the push-pull half-wave product it implies (engine: V_πL = λ / 2|dn_A − dn_B|):

Γ = \frac{g}{V} \frac{\int_{LN} E_{x}^{RF} {| E |}^{2} d A}{\int {| E |}^{2} d A}, Δ n = - \frac{1}{2} n_{e}^{3} r_{33} Γ \frac{V}{g}, V_{π} L = \frac{λ g}{2 n_{e}^{3} r_{33} Γ} = \frac{λ}{2 n_{e}^{3} r_{33} \bar{E}}

(5)

Eq. (4) carries n_e⁴/n_eff where eq. (5) carries n_e³; the two agree only for n_eff = n_e. Here n_eff = 1.770 against n_e = 2.138 because only 65% of ∫|E_x|²dA (the TE power density) lies in LN and the rest in SiO₂, so eq. (5) underestimates d n_eff/dV by n_e/n_eff = 1.208 and overestimates V_πL: 3.24 vs 2.70 V cm, measured 2.61 V cm. The engine ratio 1.201 differs from 1.208 through the E_y and E_z terms the vector solver keeps.

(a) |E_x|² / max, optical, P = 1 Wpeak |E_x| 2.339×10⁷ V/m

–|E_x|² / |E_x|²_max–

(b) e_x, static field per terminal voltmin 0, max 0.6254 (V/µm)/V

–(V/µm)/V–

–1/(V µm²)–

x – µmy – µm–|E_x|²/max –|E_x| – V/me_x –e_y – (V/µm)/Vintegrand –r₃₃ – 1/(V µm²)

(d) ∫ integrand dx vs ybars: total; line: r₃₃

y – µm total – r₃₃ – 1/(V µm)

∑ rows × Δy = −1.4424×10⁻⁵ 1/V (nodal grid sum). Hover or tap a bar.

Quantity	Value	Unit	Source
d n_eff/dV, arm A
engine, FEM quadrature	−1.4367×10⁻⁵	1/V	engine
grid, nodal sum ∑ f Δx² +0.39%	−1.4424×10⁻⁵	1/V	derived
grid, column trapezoid in LN +0.36%	−1.4419×10⁻⁵	1/V	derived
r₃₃ only: engine	−1.4319×10⁻⁵	1/V	engine
r₃₃ only: grid nodal +0.40%	−1.4375×10⁻⁵	1/V	derived
by region (grid, column trapezoid)
rib, y 0.12 to 0.3 µm 61.4%	−8.856×10⁻⁶	1/V	derived
slab under the rib, \|x\| < 1.854 µm 38.3%	−5.523×10⁻⁶	1/V	derived
slab outside the rib 0.3%	−4.063×10⁻⁸	1/V	derived
by tensor entry (engine surface integrals)
r₃₃ 99.66%	−1.432×10⁻⁵	1/V	engine
r₁₃ 0.31%	−4.515×10⁻⁸	1/V	engine
r₅₁ 0.02%	−3.304×10⁻⁹	1/V	engine
r₂₂ 0.00%	0	1/V	engine
push-pull
d n_eff/dV, arm B	1.4366×10⁻⁵	1/V	engine
balance d n_B / d n_A	-0.9999		engine
overlap
Γ_A, vector \|E\|²	0.4771		engine
Γ_x, grid \|E_x\|², grid window	0.4760		derived
Ē_A = ΓV/g	0.079510	(V/µm)/V	engine
gap g	6.0	µm	engine
indices and power
n_e, engine material	2.138		engine
n_eff, TE₀	1.7696		engine
n_g	2.135		engine
n_e / n_eff	1.2082		derived
∫_LN\|E_x\|² / ∫\|E_x\|², grid	65.0%		derived
∫\|E_x\|²dA, grid vs 2Z₀P/n_eff = 425.8	421.5	V²	derived
d n/dV and V_πL
eq. (4) scalar TE, grid fields	−1.4425×10⁻⁵	1/V	derived
eq. (5) −½n_e³r₃₃Ē_A	−1.1966×10⁻⁵	1/V	derived
V_πL engine check λ/2\|dn_A−dn_B\| = 2.697	2.697	V cm	engine
V_πL conventional, eq. (5)	3.238	V cm	engine
V_πL measured	2.61	V cm	reported
ratio conventional / engine	1.2006		derived
V_π at 9 mm: engine / measured	2.997 / 2.90	V	engine reported

Grid 241 × 81 nodes, Δx = 0.03 µm; optical field and integrand are undefined outside the engine's arm window |x| ≤ 2.9 µm (blank). Interface nodes take whichever side the engine's point locator returns: 67 of 117 nodes on the rib top y = 0.3 µm return LN values. The column trapezoid replaces interface values by linear extrapolation from the two nearest interior nodes and changes the sum by −0.03%; both grid estimates exceed the engine quadrature by about 0.4%, a residual not isolated here (the engine integrates on its own triangles; its accuracy note quotes 1.4e-3 to 3.5e-3 relative against independent references on its test cases). Γ_x and eq. (4) use |E_x|² integrated over the grid window y −0.90 to 1.5 µm; the engine window extends to y −1.5 to 2.0 µm.