Manley–Rowe relations

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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. 1.7, p. 182

Section 1.7.2.2.1.2. Manley–Rowe relations

B. Boulanger^a ^* and J. Zyss^b

^a Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and ^bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: benoitb@satie-bourgogne.fr

1.7.2.2.1.2. Manley–Rowe relations

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An important consequence of overall permutation symmetry is the Manley–Rowe power relations, which account for energy exchange between electromagnetic waves in a purely reactive (e.g. non-dissipative) medium. Calling W_i the power input at frequency ω_i into a unit volume of a dielectric polarizable medium, $[W_i=\left\langle {\bf E}(t)\cdot{{\rm d}{\bf P} \over {\rm d}t}(t)\right\rangle,\eqno(1.7.2.37)]$ where the averaging is performed over a cycle and $[\eqalignno{{\bf E}(t)&=Re[E_{\omega_i}\exp(-j\omega_i t)]&\cr {\bf P}(t)&=Re[P_{\omega_i}\exp(-j\omega_i t)].&(1.7.2.38)}]$ The following expressions can be derived straightforwardly: $[W_i=\textstyle{1 \over 2}\omega_i \,Re(iE_{\omega_i}\cdot P_{\omega_i})=\textstyle{1 \over 2}\omega_i \,Im(E_{\omega_i}^* \cdot P_{\omega_i}).\eqno(1.7.2.39)]$ Introducing the quadratic induced polarization P⁽²⁾, Manley–Rowe relations for sum-frequency generation state $[{W_1 \over \omega_1}={W_2 \over \omega_2}=-{W_3 \over \omega_3}.\eqno(1.7.2.40)]$ Since $[\omega_1+\omega_2=\omega_3]$ , (1.7.2.40) leads to an energy conservation condition, namely [W_3+W_1+W_2=0] , which expresses that the power generated at ω₃ is equal to the sum of the powers lost at ω₁ and ω₂.

A quantum mechanical interpretation of these expressions in terms of photon fusion or splitting can be given, remembering that $[W_i/\hbar\omega_i]$ is precisely the number of photons generated or annihilated per unit volume in unit time in the course of the nonlinear interactions.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 182