Glossary entry

Electro-optic effect (Pockels effect)

A linear change in refractive index produced by an applied electric field in a non-centrosymmetric crystal. The underlying mechanism of fast electro-optic modulators.

In a non-centrosymmetric crystal, an applied DC or low-frequency electric field changes the refractive index linearly with field:

\Delta\!\left(\frac{1}{n^2}\right)_{ij} \;=\; r_{ijk} E_k,

where $r_{ijk}$ is the linear electro-optic (Pockels) tensor with units of m/V. For a one-dimensional approximation along a single principal axis:

\Delta n \;\approx\; -\frac{1}{2} n^3 r_{ij} E.

The effect requires a crystal without inversion symmetry. In centrosymmetric materials (silicon, silica fiber, isotropic glasses), the Pockels effect is forbidden by symmetry. The second-order Kerr effect (quadratic in $E$ ) is permitted but is typically much weaker.

Standard electro-optic materials at 1550 nm:

Material	$r_{33}$ (pm/V)	$V_\pi L$ (V·cm)	Bandgap regime
Lithium niobate (LiNbO $_3$ )	30.8	5 – 15	Mid-infrared transparent
Lithium tantalate (LiTaO $_3$ )	30.5	5 – 12	Telecom-IR
Potassium titanyl phosphate (KTP)	36	4 – 10	Visible / near-IR
Beta-barium borate (BBO)	2.7	$>$ 100	UV transparent
InP (zinc-blende)	1.4	30 – 60	III–V active region
Silicon (forbidden, $\chi^{(2)} = 0$ )	0	n/a	Indirect bandgap
Polymer EO	$>$ 100 (engineered)	$<$ 1	Polymeric, tunable

The figure of merit is $n^3 r$ — higher value gives lower drive voltage. The half-wave voltage of a Mach–Zehnder modulator is set by this product.

Lithium niobate has dominated electro-optic modulator technology for decades. Recent thin-film lithium niobate on insulator (LNOI) combines the EO properties of bulk LiNbO $_3$ with the strong optical confinement of SOI-style waveguides, dropping $V_\pi L$ from 5–15 V·cm (bulk) to 1–3 V·cm (LNOI).

For silicon photonics, since silicon has no Pockels effect, modulators rely instead on:

Free-carrier dispersion — modulating refractive index via injected or depleted carriers (10–100× weaker than Pockels but available in CMOS)
Hybrid integration with EO materials — bonded LiNbO $_3$ or polymer overlays
Strained silicon — broken centrosymmetry from strain induces weak Pockels response

The Kerr effect ( $\chi^{(3)}$ , intensity-dependent index, present in all materials including centrosymmetric ones) is a separate mechanism — distinct from the Pockels effect, although both modulate refractive index. See Kerr effect.