Sum/difference frequency generation (two-wave mixing)

Glazer, A. M.; Cox, K. G.

doi:10.1107/97809553602060000633

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.6, p. 151

Section 1.6.2.6. Sum/difference frequency generation (two-wave mixing) $[\varepsilon_o\chi_{ijk}E_j^{\omega_1}E_k^{\omega_2}]$

A. M. Glazer^a ^* and K. G. Cox^b ^‡

^a Department of Physics, University of Oxford, Parks Roads, Oxford OX1 3PU, England, and ^bDepartment of Earth Sciences, University of Oxford, Parks Roads, Oxford OX1 3PR, England
Correspondence e-mail: glazer@physics.ox.ac.uk

1.6.2.6. Sum/difference frequency generation (two-wave mixing) $[\varepsilon_o\chi_{ijk}E_j^{\omega_1}E_k^{\omega_2}]$

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It can happen that when two different light fields of frequencies $[\omega_1]$ and $[\omega_1]$ , such as can be obtained from two lasers, propagate through a crystal, the resulting output frequency can be the sum or difference of the two incident frequencies, through the susceptibility $[\chi_{ijk}(\omega_1\pm\omega_2;\omega_1,\omega_2)]$ . A particular case is of interest: suppose $[\omega_1 = \omega_2 = \omega/2]$ i.e. $[\chi_{ijk}(\omega;\omega/2,\omega/2)]$ , to get $[P_i^\omega = \varepsilon_o\chi_{ijk}E_j^{\omega/2}E_k^{\omega/2}.]$ Here, the light, on passing through the crystal, interacts with itself to produce a doubling of the frequency. This important effect is known as second harmonic generation or SHG and is used for generating different laser frequencies starting from a fundamental frequency. The observation of SHG is also often a good indicator of the lack of a centre of inversion in a crystal. As two optical fields are involved, the incident field inducing the second one, this is a true nonlinear optical effect.

Another effect can be envisaged in which the susceptibility is $[\chi_{ijk}(0;\omega/2,-\omega/2)]$ , i.e. there is a cancellation of the two incident frequencies to produce a polarization in the crystal. This is called optical rectification.

Parametric amplification is an effect caused when an incident beam at frequency $[\omega_1]$ is incident on a nonlinear optical crystal at the same time as an intense pump beam frequency $[\omega_2]$ , where $[\omega_2\,\gt\,\omega_1]$ . The $[\omega_1]$ wave is then amplified accompanied by an `idler' wave of frequency $[\omega_3=\omega_2-\omega_1]$ .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.6, p. 151