# Pockels effect

The Pockels effect is the linear electro-optic effect in which the refractive index of a material changes in proportion to an applied electric field.<sup>[1](https://www.rp-photonics.com/pockels_effect.html)</sup> It is one of the two lowest-order electro-optic effects, the other being the [Kerr effect](https://www.edgepedia.org/kerr-effect), in which the index change is proportional to the square of the field.<sup>[2](https://www.sciencedirect.com/topics/chemistry/pockels-effect)</sup> The effect underlies electro-optic modulators, Pockels cells, Q-switches, and electric-field sensors, and it is the basis of electro-optic sampling in ultrafast metrology.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC10058895/)</sup>

| Key fact | Detail |
|---|---|
| Definition | Refractive index change linear in the applied electric field; also called the linear electro-optic (LEO) effect<sup>[1](https://www.rp-photonics.com/pockels_effect.html)</sup> |
| Governing relation | \( n(E) \approx n_0 - \tfrac{1}{2} r n^3 E \), where \( r \) is the Pockels (linear electro-optic) coefficient<sup>[4](https://www.physi.uni-heidelberg.de/Einrichtungen/FP/anleitungen/F85_optics_lab_course-2.pdf)</sup> |
| Typical coefficient size | \( r \) lies between \( 10^{-12} \) and \( 10^{-10} \) m/V for common materials<sup>[4](https://www.physi.uni-heidelberg.de/Einrichtungen/FP/anleitungen/F85_optics_lab_course-2.pdf)</sup> |
| Symmetry requirement | Occurs in non-centrosymmetric materials, including crystals lacking inversion symmetry, such as lithium niobate (LiNbO₃) and gallium arsenide, and in other noncentrosymmetric media such as electric-field poled polymers and glasses<sup>[5](https://www.youtube.com/watch?v=d7YeIxEl9Bo)</sup><sup> • </sup><sup>[26](https://pubs.rsc.org/en/content/articlepdf/2021/nr/d1nr04282c)</sup> |
| Reference material | LiNbO₃, whose largest coefficient \( r_{33} \) is about 30.8 pm/V, has been used for decades in long-haul telecommunications<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC11901258/)</sup><sup> • </sup><sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC10058895/)</sup> |
| Speed | The linear electro-optic response has femtosecond or attosecond response times, supporting modulation bandwidths above 100 GHz<sup>[7](https://pubs.rsc.org/en/content/articlehtml/2023/tc/d3tc01132a)</sup> |
| Figure of merit | Half-wave voltage \( V_\pi \), the voltage producing a π phase shift; thin-film lithium niobate devices have reached voltage-length products of 2.2 to 3.3 V·cm<sup>[8](https://onlinelibrary.wiley.com/doi/full/10.1002/adpr.202200216)</sup> |

## Theory

In an electro-optic medium the refractive index becomes a function of the applied field, \( n = n(E) \). Because the variations are small, \( n(E) \) can be expanded as a [Taylor series](https://www.edgepedia.org/taylor-series) about \( E = 0 \), giving

\[ n(E) = n - \tfrac{1}{2} r n^3 E - \tfrac{1}{2} s n^3 E^2, \]

where \( r \) is the linear (Pockels) coefficient and \( s \) the quadratic (Kerr) coefficient.<sup>[4](https://www.physi.uni-heidelberg.de/Einrichtungen/FP/anleitungen/F85_optics_lab_course-2.pdf)</sup> When \( r \gg s \) the refractive index depends linearly on the field, and the relation reduces to \( n(E) \approx n_0 - \tfrac{1}{2} r n^3 E \); when \( r \ll s \) the quadratic Kerr effect dominates.<sup>[4](https://www.physi.uni-heidelberg.de/Einrichtungen/FP/anleitungen/F85_optics_lab_course-2.pdf)</sup> Equivalently, the electric impermeability \( \eta = \varepsilon_0/\varepsilon \) changes as \( \eta(E) = \eta_0 + rE + sE^2 \).<sup>[9](https://depts.washington.edu/cmdwiki/mediawiki/index.php?title=Second-order_Processes)</sup>

Because electro-optic crystals are generally birefringent, the response depends on the relative orientation of the field, the crystal axes, and the beam path; the scalar \( r \) is in fact one entry of a Pockels coefficient tensor \( r_{ijk} \) whose entries are constrained by the crystal symmetry.<sup>[4](https://www.physi.uni-heidelberg.de/Einrichtungen/FP/anleitungen/F85_optics_lab_course-2.pdf)</sup> For light traversing a waveguide of length \( L \) with field \( E \), the induced phase shift is

\[ \Delta \phi = -\frac{\pi r n^3 E L}{\lambda}, \]

with the field produced by a voltage \( V \) across an electrode spacing \( d \), \( E = V/d \).<sup>[2](https://www.sciencedirect.com/topics/chemistry/pockels-effect)</sup>

At the atomic level, the applied voltage anisotropically distorts the electron density in the material, so the optical beam sees a different polarizability and anisotropy than in the absence of the field; the polarization state, speed, and direction of the light can all change as a result.<sup>[9](https://depts.washington.edu/cmdwiki/mediawiki/index.php?title=Second-order_Processes)</sup>

## Symmetry constraint and relation to the Kerr effect

The Pockels effect is a second-order nonlinear optical effect, and second-order interactions occur only in noncentrosymmetric materials; in centrosymmetric materials the second-order susceptibility \( \chi^{(2)} \) is zero, whereas third-order effects such as the Kerr effect can occur in all materials.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC10058895/)</sup> John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that a DC or low-frequency electric field changes the refractive index of a material; both are nonlinear optical effects that do not require a laser.<sup>[9](https://depts.washington.edu/cmdwiki/mediawiki/index.php?title=Second-order_Processes)</sup> The date of Pockels's description is given differently across sources: some state the linear electro-optic effect was first described by Friedrich Pockels in 1893,<sup>[2](https://www.sciencedirect.com/topics/chemistry/pockels-effect)</sup> while others give 1906.<sup>[1](https://www.rp-photonics.com/pockels_effect.html)</sup><sup> • </sup><sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC10058895/)</sup>

In materials lacking inversion symmetry both Pockels and Kerr effects are present simultaneously, with the linear effect generally dominating the quadratic one.<sup>[10](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.82.013821)</sup> The Pockels effect can also be induced in normally centrosymmetric crystals such as silicon by strain, which breaks the inversion symmetry.<sup>[11](https://arxiv.org/abs/1507.06589)</sup>

## Devices

**Pockels cells.** A Pockels cell is an electro-optic crystal between electrodes, acting as a voltage-controlled waveplate. At the half-wave voltage \( V_\pi \) the relative phase retardation between the two polarizations is \( \pi \), so the cell acts as a [half-wave plate](https://www.edgepedia.org/half-wave-plate) and rotates linear polarization by 90 degrees.<sup>[12](https://synqs.github.io/FP-Optik/f86.html)</sup><sup> • </sup><sup>[13](https://www.coherent.com/web-resources/manual/electro-optic-modulator-user-manual)</sup> Placed between crossed polarizers at 45 degrees to the crystal axes, the cell becomes an intensity modulator with transmittance \( T = \sin^2(\Phi/2) \), a periodic function of the applied voltage.<sup>[12](https://synqs.github.io/FP-Optik/f86.html)</sup> Because the electro-optic changes are small, Pockels cells generally need high drive voltages; for one commercial modulator the half-wave voltage is 9.5 kV at an incident wavelength of 9.36 μm, and \( V_\pi \) is directly proportional to wavelength.<sup>[13](https://www.coherent.com/web-resources/manual/electro-optic-modulator-user-manual)</sup> Rotating polarization in nanoseconds makes the cell an ultra-fast optical shutter between orthogonal polarizing filters.<sup>[13](https://www.coherent.com/web-resources/manual/electro-optic-modulator-user-manual)</sup> Both Pockels and Kerr cells can build optical shutters with response times around \( 10^{-10} \) s, but Pockels cells typically require 5 to 10 times lower voltages than equivalent Kerr cells.<sup>[14](https://labs.phys.utk.edu/mbreinig/phys421/modules/m11/Non-linear%20optics.html)</sup>

**Q-switches.** In a laser cavity, an electro-optic Q-switch combines a polarizer and a Pockels cell as a polarization rotator. With the voltage applied, the polarization is rotated so light cannot pass the polarizer and the cavity is held at low Q; removing the voltage restores high reflectivity and the laser emits a large pulse. Switching times are typically less than a nanosecond.<sup>[15](https://photonics.creol.ucf.edu/wp-content/uploads/sites/4/2019/06/Course_2_Laser_Systems_and_Applications_2nd_Edition_2016.pdf)</sup>

**Integrated modulators.** Electro-optic modulators encode electrical signals onto optical carriers and are central to high-speed optical transceivers.<sup>[7](https://pubs.rsc.org/en/content/articlehtml/2023/tc/d3tc01132a)</sup> In a Mach-Zehnder modulator, a phase modulator in one arm converts voltage-induced phase changes into intensity modulation through interference.<sup>[16](https://physik.uni-paderborn.de/en/integrated-quantum-optics/quantum-devices)</sup> Silicon's crystal inversion symmetry inhibits the Pockels effect, so silicon modulators rely on weaker carrier-dispersion effects; this has driven interest in Pockels materials such as LiNbO₃, BaTiO₃, PZT, and electro-optic polymers.<sup>[17](https://arxiv.org/abs/2302.13768)</sup><sup> • </sup><sup>[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC11278586/)</sup> [Thin-film lithium niobate](https://www.edgepedia.org/thin-film-lithium-niobate) (TFLN) confines the light in a submicron film, raising the electric-field overlap enough to cut drive voltages below one volt in some designs.<sup>[19](https://technav.ieee.org/topic/electro-optic-effects/)</sup> Reported TFLN devices include a modulator with an extrapolated 3 dB bandwidth of 170 GHz at a voltage-length product of 3.3 V·cm, another with 84 GHz at 2.2 V·cm, and PAM-8 data rates up to 240 Gbit/s.<sup>[8](https://onlinelibrary.wiley.com/doi/full/10.1002/adpr.202200216)</sup>

## Electric-field and charge measurement

Because the sensing element is a dielectric crystal rather than a metallic probe, a Pockels sensor measures an electric field with little disturbance of the field being measured. A 2001 review notes advantages over electrical methods including resistance to electromagnetic interference, high sensitivity, and wide frequency bandwidth.<sup>[20](https://www.cpsjournals.cn/article/doi/10.3321/j.issn%3A0379-4148.2001.02.005)</sup> Pockels devices have been applied to rapidly changing fields in corona discharges with space charge,<sup>[21](https://www.sciencedirect.com/science/article/pii/0304388682900122)</sup> and electro-optic DC field probes have achieved a measurement resolution of 100 V/m with an 80 dB dynamic range and temperature stability of 0.1%/°C near 20 °C.<sup>[22](https://doi.org/10.1016/j.measurement.2006.06.003)</sup> The Pockels technique is considered the only approach so far for measuring two-dimensional dynamic surface charge distributions on solid insulating materials, since the crystal hides under the sample and interferes little with the charge-induced field.<sup>[23](https://ietresearch.onlinelibrary.wiley.com/doi/full/10.1049/hve2.12073)</sup>

## Ultrafast sampling

[Electro-optic sampling](https://www.edgepedia.org/electro-optic-sampling) exploits the Pockels effect as second-order nonlinear mixing of two fields, one of them the quasi-static field of the sampled pulse, which induces birefringence in a crystal and a measurable polarization change in a probe pulse.<sup>[24](https://doi.org/10.1088/1361-6455%2Fac8032)</sup> The technique's bandwidth approaches the terahertz regime, making it a leading method for measuring high-speed devices and integrated circuits.<sup>[25](https://globals.ieice.org/en_transactions/electronics/10.1587/e76-c_1_55/_p)</sup> Inside the crystal, a terahertz field modulates the polarization of a probe laser pulse, which is how THz pulses are measured in the time domain.<sup>[2](https://www.sciencedirect.com/topics/chemistry/pockels-effect)</sup>

## References

1. [Pockels effect, RP Photonics Encyclopedia](https://www.rp-photonics.com/pockels_effect.html)
2. [Pockels Effect - an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/chemistry/pockels-effect)
3. [An Introduction to Nonlinear Integrated Photonics Devices, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC10058895/)
4. [Electro-Optic Effect, Optics Lab Course F85, University of Heidelberg](https://www.physi.uni-heidelberg.de/Einrichtungen/FP/anleitungen/F85_optics_lab_course-2.pdf)
5. [Pockels effect (video description), Audiopedia](https://www.youtube.com/watch?v=d7YeIxEl9Bo)
6. [Advanced Crystallization Methods for Thin-Film Lithium Niobate, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC11901258/)
7. [Perspectives of thin-film lithium niobate and electro-optic polymers for high-performance electro-optic modulation, J. Mater. Chem. C](https://pubs.rsc.org/en/content/articlehtml/2023/tc/d3tc01132a)
8. [Thin-Film Lithium Niobate Optical Modulators with an Extrapolated Bandwidth of 170 GHz, Advanced Photonics Research](https://onlinelibrary.wiley.com/doi/full/10.1002/adpr.202200216)
9. [Second-order Processes, CleanEnergyWIKI, University of Washington](https://depts.washington.edu/cmdwiki/mediawiki/index.php?title=Second-order_Processes)
10. [Direct Kerr electro-optic effect in noncentrosymmetric materials, Phys. Rev. A 82, 013821 (2010)](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.82.013821)
11. [Modeling of strain-induced Pockels effect in Silicon, arXiv](https://arxiv.org/abs/1507.06589)
12. [Part 2: Electro-optic effect, TP Optics in Heidelberg](https://synqs.github.io/FP-Optik/f86.html)
13. [Coherent Electro-Optic Modulator User Manual](https://www.coherent.com/web-resources/manual/electro-optic-modulator-user-manual)
14. [Non-linear optics, University of Tennessee physics course notes](https://labs.phys.utk.edu/mbreinig/phys421/modules/m11/Non-linear%20optics.html)
15. [Electro-Optic Q-Switches, Laser Systems and Applications course, CREOL/UCF](https://photonics.creol.ucf.edu/wp-content/uploads/sites/4/2019/06/Course_2_Laser_Systems_and_Applications_2nd_Edition_2016.pdf)
16. [Quantum Devices - Integrated Quantum Optics, Paderborn University](https://physik.uni-paderborn.de/en/integrated-quantum-optics/quantum-devices)
17. [High speed silicon photonic electro-optic Kerr modulation, arXiv](https://arxiv.org/abs/2302.13768)
18. [Towards High-Performance Pockels Effect-Based Modulators: Review and Projections, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC11278586/)
19. [Electro-optic effects, IEEE Technology Navigator](https://technav.ieee.org/topic/electro-optic-effects/)
20. [Application of the Pockels effect to electricity measurement, Physics (China) 30(2): 86-89 (2001)](https://www.cpsjournals.cn/article/doi/10.3321/j.issn%3A0379-4148.2001.02.005)
21. [A method for measuring electric field in space charge by means of Pockels device, Journal of Electrostatics 11(3): 195-211 (1982)](https://www.sciencedirect.com/science/article/pii/0304388682900122)
22. [Validation of electro-optic sensors for measurement of DC fields in the presence of space charge, Measurement](https://doi.org/10.1016/j.measurement.2006.06.003)
23. [Review of surface transient charge measurement on solid insulating materials via the Pockels technique, High Voltage](https://ietresearch.onlinelibrary.wiley.com/doi/full/10.1049/hve2.12073)
24. [Recent advances in petahertz electric field sampling, Journal of Physics B](https://doi.org/10.1088/1361-6455%2Fac8032)
25. [Measurement of High-Speed Devices and Integrated Circuits Using Electro-Optic Sampling Technique, IEICE Transactions on Electronics](https://globals.ieice.org/en_transactions/electronics/10.1587/e76-c_1_55/_p)
26. [pubs.rsc.org](https://pubs.rsc.org/en/content/articlepdf/2021/nr/d1nr04282c)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Quantum optics and photonics › Laser physics*

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