Optics, electromagnetic theory Definition

Nonlinear optics

Builds on Polarization density Electric field Constitutive relations of electromagnetism Power series

Definition

Consider a medium in which an electric field $\mathbf{E}$ induces a polarization density $\mathbf{P}$ , as part of the constitutive relations. The medium responds linearly when $\mathbf{P}$ is a linear function of $\mathbf{E}$ , and nonlinearly otherwise. Nonlinear optics is the study of the propagation of light in media that respond nonlinearly. In the usual setting, the response is a power series in the field: $P_i = \varepsilon_0 \Big( \sum_j \chi^{(1)}_{ij} E_j + \sum_{j,k} \chi^{(2)}_{ijk} E_j E_k + \sum_{j,k,l} \chi^{(3)}_{ijkl} E_j E_k E_l + \cdots \Big).$ Here $\varepsilon_0$ is the vacuum permittivity, and the terms of order at least $2$ are the nonlinear part.

Remarks

The power series is valid when the field is weak compared with the fields that bind electrons in atoms, so that each term is much smaller than the one before. This is why nonlinear effects need intense light.

Example: if $\chi^{(2)} \ne 0$ and $\mathbf{E}$ oscillates at frequency $\omega$ , the term $\chi^{(2)} E^2$ oscillates at $0$ and at $2\omega$ . The medium then radiates light at $2\omega$ , which is second-harmonic generation.

Non-example: in vacuum $\mathbf{P} = 0$ for every field in classical electrodynamics, so Maxwell's equations are linear and there is no nonlinear optics.

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Constitutive relations of electromagnetism concept
Electric field concept
Polarization density concept
Power series concept
Nonlinear optics definition

Tag	`T0R3ZXZHZC`
URI	`m2://T0R3ZXZHZC`
Kind	definition
Writing	W0 stub
MSC	78A60
Defines	Nonlinear optics
Uses	Polarization density, Electric field, Constitutive relations of electromagnetism, Power series
Changed	3 November 2026

Nonlinear optics

Definition

Remarks

Reading path

Further reading

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