Optics, electromagnetic theory Definition

Second-harmonic generation

Builds on Nonlinear susceptibility Polarization density

Definition

Let a medium have second-order nonlinear susceptibility $\chi^{(2)}$ , and let a monochromatic field $E(t) = \mathcal{E} e^{-i\omega t} + \overline{\mathcal{E}} e^{i\omega t}$ act on it, written for one component. The second-order part of the polarization density is $\varepsilon_0 \chi^{(2)} E(t)^2 = 2\varepsilon_0 \chi^{(2)} \abs{\mathcal{E}}^2 + \varepsilon_0 \chi^{(2)} \left( \mathcal{E}^2 e^{-2i\omega t} + \overline{\mathcal{E}}^2 e^{2i\omega t} \right).$ Second-harmonic generation is the radiation of light at frequency $2\omega$ by the term that oscillates at $2\omega$ .

Remarks

The constant term $2\varepsilon_0 \chi^{(2)} \abs{\mathcal{E}}^2$ is a static polarization, called optical rectification. The generated power grows like the square of the input intensity, as long as the input is not depleted. Efficient conversion needs phase matching, so that the waves radiated at different points of the crystal add in phase.

Example: the crystal potassium titanyl phosphate lacks a centre of inversion, so it has $\chi^{(2)} \ne 0$ and doubles $1064$ nm light to $532$ nm.

Non-example: a glass plate has a centre of inversion in its bulk, so $\chi^{(2)} = 0$ there and it produces essentially no second harmonic.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`PYAEW7M109`
URI	`m2://PYAEW7M109`
Kind	definition
Writing	W0 stub
MSC	78A60
Defines	Second-harmonic generation
Uses	Nonlinear susceptibility, Polarization density
Changed	3 November 2026

Second-harmonic generation

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here