RP Photonics
Encyclopedia
… combined with a great Buyer's Guide!
 part of theVirtualLibrary

# Nonlinear Polarization

Buyer's Guide

Use the RP Photonics Buyer's Guide to find suppliers for photonics products! You will hardly find a more convenient resource.

Ask RP Photonics how calculations and simulations of nonlinear processes may benefit your product development.

Definition: the part of the light-induced electric polarization which depends nonlinearly on the electric field of the light

German: nichtlineare Polarisation

When light propagates in a transparent medium, its electric field causes some amount of electric polarization in the medium, i.e. some density of electric dipole moment. (This must not be confused with the polarization of the light field, which is the direction of its electric field.) That polarization propagates together with the electromagnetic field in the form of a polarization wave. Whereas at low light intensities the electric polarization is proportional to the electric field strength, nonlinear contributions become important at high optical intensities, as they can e.g. be produced with lasers.

## Second-order Nonlinear Polarization

The second (lowest) order of nonlinear polarization can arise from a χ(2) nonlinearity which can occur only in crystal materials with a non-centrosymmetric crystal structure. (Nonlinear effects at crystal surfaces are an exception.) The nonlinear polarization then has a component which depends quadratically on the electric field of an incident light wave. More precisely, the tensor nature of the nonlinear susceptibility needs to be considered:

where Pi is the i-th Cartesian coordinate of the polarization, χ(2) is the nonlinear susceptibility, and E(t) is the optical electric field. More commonly, this is written as

with the nonlinear tensor d. Many tensor components can actually be zero for symmetry reasons, depending on the crystal class.

The nonlinear polarization contains frequency components which are not present in the exciting beam(s). Light with such frequencies can then be generated in the medium (→ nonlinear frequency conversion). For example, if the input field is monochromatic, the nonlinear polarization also has a component with twice the input frequency (→ frequency doubling). As the polarization has the form of a nonlinear polarization wave, the frequency-doubled light is also radiated in the direction of the input beam. Other examples are sum and difference frequency generation, optical rectification, parametric amplification and oscillation.

## Third-order Nonlinear Polarization

The next higher order of nonlinear polarization can arise from a χ(3) nonlinearity, as it occurs in basically all media. This can give rise to various phenomena:

## Phase Matching

In many cases, the nonlinear mixing products can be efficiently accumulated over a greater length of crystal only if phase matching is achieved. Otherwise, the field amplitudes at the exit face, generated at different locations in the crystal, essentially cancel each other, and the apparent nonlinearity is weak. Some nonlinear effects, however, are either automatically phase-matched (e.g. self-phase modulation) or do not need phase matching (e.g. Raman scattering).

### Bibliography

 [1] D. A. Kleinman, “Nonlinear dielectric polarization in optical media”, Phys. Rev. 126 (6), 1977 (1962)

(Suggest additional literature!)

## How do you rate this article?

 Your general impression: don't know poor satisfactory good excellent Technical quality: don't know poor satisfactory good excellent Usefulness: don't know poor satisfactory good excellent Readability: don't know poor satisfactory good excellent Comments: Found any errors? Suggestions for improvements? Do you know a better web page on this topic? Spam protection: (enter the value of 5 + 8 in this field!)

If you want a response, you may leave your e-mail address in the comments field, or directly send an e-mail.

If you like our website, you may also want to get our newsletters!

If you like this article, share it with your friends and colleagues, e.g. via social media:

# A Quiz Question

taken from the Photonics Quiz:

Assume that a periodic picosecond pulse train is generated with a gain-switched laser diode. Does the spectrum of the pulse train have the shape of a frequency comb, consisting of sharp lines?

(a) Yes, it does, because periodic pulse trains always have such a spectrum.

(b) No, it doesn't, because the pulses are not mutually coherent.

(c) This depends on the quality of the electronic driver of the laser diode.

After selecting your answer(s) and pressing this button, find the explanations on the left side.

– Show all banners –

– Get your own banner! –