Electro-optic effect (Pockels effect)

The linear electro-optic (Pockels) effect is a change in refractive index proportional to an applied DC or radio-frequency electric field. It occurs only in materials without inversion symmetry, such as non-centrosymmetric crystals and poled polymers. The index changes are small: in lithium niobate, with $r_{33}=30.8$ pm/V and extraordinary index $n_e=2.138$ at 1550 nm, a field of 1 V/μm gives $\mathrm{\Delta }n\approx 1.5\times {10}^{-4}$. Because the response is electronic and ionic, it follows the field to well beyond 100 GHz, and modulator bandwidth is limited by electrode design.

The general form uses the index ellipsoid (impermeability tensor):

$$\mathrm{\Delta }\text{\hspace{0.17em}⁣}{\left(\frac{1}{n^2}\right)}_{ij}=r_{ijk}{E}_k$$

where $r_{ijk}$ is the linear electro-optic tensor, in m/V (usually quoted in pm/V). For light polarized along a principal axis and a field along a single axis, this reduces to

$$\mathrm{\Delta }n\approx -\frac{1}{2}{n}^{3}rE\mathrm{.}$$

Symmetry. In centrosymmetric materials (silicon, silica fiber, isotropic glasses), the Pockels effect is forbidden by symmetry. The quadratic electro-optic (DC Kerr) effect, a ${\chi }^{(3)}$ process, is permitted but is typically much weaker. The tensor also makes the effect strongly dependent on orientation: lithium niobate has $r_{13}\approx 8.6$ pm/V, and combined with the ordinary index $n_o=2.211$ it gives an index change about 3.2 times smaller than the $r_{33}$ configuration for the same field. Lithium niobate modulators therefore align both the optical polarization and the applied field with the crystal $z$ axis (TE light in x-cut waveguides).

Typical values

Standard electro-optic materials (coefficient near 1550 nm unless noted):

The material figure of merit is $n^{3}r$; a higher value gives a lower drive voltage. For lithium niobate $n_e^3{r}_{33}\approx 300$ pm/V. The half-wave voltage of a Mach–Zehnder modulator is set by this product, the electrode gap, and the overlap between optical mode and applied field.

Half-wave voltage

For a phase shifter of length $L$ with electrode gap $d$ and field–mode overlap $\mathrm{\Gamma }$, a phase of $\pi$ requires

$$V_{\pi }L=\frac{\lambda d}{n^{3}r\mathrm{\Gamma }}\mathrm{.}$$

Worked example. Take thin-film lithium niobate at 1550 nm with $d=5$ μm and $\mathrm{\Gamma }=0.5$. A single arm gives $V_{\pi }L\approx 5.1$ V·cm; a push-pull Mach–Zehnder, where the two arms shift in opposite directions, halves this to about 2.6 V·cm. Bulk titanium-diffused waveguides have weakly confined modes several micrometers across and need wider gaps, hence 5–15 V·cm.

Free-space Pockels cells. In a longitudinal cell the field is applied along the beam, and the half-wave voltage is independent of crystal length: $V_{\pi }=\lambda \mathrm{/}(2n_o^3{r}_{63})$. For KD*P with $r_{63}\approx 24$ pm/V and $n_o\approx 1.49$, this gives about 6.7 kV at 1064 nm, and Pockels cells used for Q-switching are accordingly driven at kilovolt levels (the quarter-wave voltage is half this).

Measurement

The most direct measurement is the transfer curve of an intensity modulator: sweep the bias and read $V_{\pi }$ as the voltage between a transmission maximum and the adjacent minimum. Bulk crystals are measured through the field-induced birefringence between crossed polarizers. At high frequency, $V_{\pi }$ is inferred from the ratio of carrier to sideband power on an optical spectrum analyzer, which avoids the need for a fast photodiode. Values measured at low frequency (unclamped crystal) and at microwave frequencies (clamped crystal, where acoustic contributions cannot follow) typically differ by 5–15 percent.

Where it is used

Lithium niobate has dominated electro-optic modulator technology for decades. 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 1.5–3 V·cm. Other uses include phase modulators for sideband generation in Pound–Drever–Hall locking, Pockels cells for Q-switching and pulse picking, and barium titanate, which has a much larger Pockels coefficient and can be grown on silicon.

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

• Free-carrier dispersion (plasma dispersion effect): modulating refractive index via injected or depleted carriers (depletion devices reach a $V_{\pi }L$ of 1–3 V·cm, similar to thin-film lithium niobate, at the cost of optical loss and a phase that is nonlinear in voltage; available in CMOS)
• Hybrid integration with EO materials: bonded LiNbO${}_3$ or polymer overlays
• Strained silicon: broken centrosymmetry from strain induces a weak Pockels response

Pitfalls

Bias drift. Lithium niobate modulators show slow drift of the operating point, caused by charge migration in the buffer layers and the pyroelectric effect, so most need a bias controller.

Photorefractive damage. At visible and near-UV wavelengths, light-induced index changes in undoped lithium niobate distort the waveguide; MgO doping reduces the effect.

Tensor element. Using $r_{33}$ for a configuration that actually uses $r_{13}$ overestimates the response by roughly the factor of 3.2 given above.

Common questions

What is the difference between the Pockels and Kerr effects?

The Pockels effect is linear in the applied field and requires a non-centrosymmetric crystal. The Kerr effect is quadratic in field (or linear in optical intensity), arises from ${\chi }^{(3)}$, and is present in all materials including centrosymmetric ones. Both modulate refractive index, but with very different drive characteristics and applications.

Why does silicon have no electro-optic effect?

Crystalline silicon has a center of inversion, so every element of the second-order tensor, including $r_{ijk}$, vanishes. Silicon modulators therefore use carrier effects or heterogeneous materials.

References: Yariv & Yeh, Photonics 6th ed. 2007; Saleh & Teich, Fundamentals of Photonics 3rd ed. 2019; Boyd, Nonlinear Optics 4th ed. 2020.