Glossary entry

Pockels effect (linear electro-optic effect)

The change in refractive index linearly proportional to an applied electric field, occurring only in crystals without inversion symmetry. The basis for the fastest commercial optical modulators.

The Pockels effect is the linear electro-optic response: an applied electric field $E_\text{DC}$ produces a refractive index change

\Delta n \;=\; -\tfrac{1}{2} n_0^3 \, r \, E_\text{DC},

where $n_0$ is the zero-field index and $r$ is the relevant Pockels coefficient (a tensor element with units of m/V or pm/V). The linear dependence on field distinguishes the Pockels effect from the quadratic Kerr effect, which dominates in materials without Pockels response.

The effect occurs only in crystals lacking inversion symmetry (non-centrosymmetric). Centrosymmetric materials — silicon, fused silica, all amorphous glasses — have zero Pockels coefficient and rely instead on the Kerr effect (or other mechanisms such as plasma dispersion in silicon) for electro-optic modulation.

Common Pockels-active materials and key tensor elements:

Material	Coefficient	Value (pm/V)	Notes
Lithium niobate (LiNbO $_3$ )	$r_{33}$	30.8	Workhorse telecom modulator material; field along c-axis
Lithium niobate	$r_{13}$	8.6	Transverse geometry
Lithium tantalate (LiTaO $_3$ )	$r_{33}$	30.5	Similar to LiNbO $_3$
Potassium dihydrogen phosphate (KDP)	$r_{63}$	10.5	Pockels cells for Q-switching
KD*P (deuterated)	$r_{63}$	24	Higher coefficient than KDP
Beta barium borate (BBO)	$r_{22}$	2.7	Pockels cells, frequency conversion
Gallium arsenide (GaAs)	$r_{41}$	1.5	III-V semiconductor EO modulators
Indium phosphide (InP)	$r_{41}$	1.6	III-V EO modulators
Barium titanate (BaTiO $_3$ ) thin film	effective	$>$ 100 (engineered)	Emerging silicon photonic integration

$V_\pi$ — the modulator drive parameter. For a Mach-Zehnder modulator of arm length $L$ and electrode gap $g$ :

V_\pi \;=\; \frac{\lambda_0 \, g}{n_0^3 \, r \, L \, \Gamma},

where $\Gamma \leq 1$ is the optical-electrical field overlap factor. The $V_\pi \cdot L$ product is the conventional figure of merit for electro-optic platforms:

Platform	$V_\pi \cdot L$ at 1550 nm
LiNbO $_3$ (bulk)	18 – 22 V·cm
Thin-film LiNbO $_3$ on Si	2 – 5 V·cm
InP MQW (band-edge)	1 – 3 V·cm
GaAs/AlGaAs	12 – 20 V·cm
Silicon (plasma dispersion, not Pockels)	1 – 3 V·cm
Polymer EO (engineered chromophores)	0.5 – 2 V·cm

The Pockels effect is intrinsically extremely fast — limited only by the lattice phonon response, typically allowing modulation bandwidths exceeding 100 GHz in well-designed structures. Material quality, electrode design, and microwave–optical velocity matching are usually the practical bandwidth bottlenecks rather than the EO response itself.

Recent silicon-photonic platforms have developed thin-film LiNbO $_3$ heterogeneously integrated on SOI, combining the strong Pockels effect of LiNbO $_3$ with the dense passive circuitry of silicon photonics. These have demonstrated $V_\pi < 1$ V and bandwidths $> 100$ GHz simultaneously.