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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 183

Section Superposition of monochromatic waves

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: Superposition of monochromatic waves

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Optical fields are often superpositions of monochromatic waves which, due to spectral discretization, will introduce considerable simplifications in previous expressions such as ([link] relating the induced polarization to a continuous spectral distribution of polarizing field amplitudes.

The Fourier transform of the induced polarization is given by[{\bf P}^{(n)}(\omega)=(1/2\pi)\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}t\;{\bf P}^{(n)}(t)\exp(i\omega t).\eqno(]Replacing P(n)(t) by its expression as from ([link] and applying the well known identity[(1/2\pi)\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}t\;\exp[i(\omega-\omega_\sigma)t]=\delta(\omega-\omega_\sigma)\eqno(]leads to[\eqalignno{{\bf P}^{(n)}(\omega)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_n\,\,\chi^{(n)}(-\omega_\sigma\semi\omega_1,\omega_2,\ldots\omega_n)&\cr&\quad\times {\bf E}(\omega_1){\bf E}(\omega_2)\ldots {\bf E}(\omega_n)\delta(\omega-\omega_\sigma).&(}]

In practical cases where the applied field is a superposition of monochromatic waves[{\bf E}(t)=(1/2)\textstyle \sum \limits_{\omega'}[E_{\omega'}\exp(-i\omega't)+E_{-\omega'}\exp(i\omega't)]\eqno(]with [E_{-\omega'}=E_{\omega'}^*]. By Fourier transformation of ([link][{\bf E}(\omega)=(1/2)\textstyle \sum \limits_{\omega'}[E_{\omega'}\delta(\omega-\omega')+E_{-\omega'}\delta(\omega+\omega')].\eqno(]The optical intensity for a wave at frequency [\omega'] is related to the squared field amplitude by[I_{\omega'}=\varepsilon_o c n(\omega')\langle {\bf E}^2(t)\rangle_t=\textstyle{1\over 2}\varepsilon_ocn(\omega')|E_{\omega'}|^2.\eqno(]The averaging as represented above by brackets is performed over a time cycle and [n(\omega')] is the index of refraction at frequency [\omega'].

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