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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 185

Section 1.7.2.2.1.2. Manley–Rowe relations

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:  benoit.boulanger@grenoble.cnrs.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 Wi the power input at frequency ωi 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)[link] 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.








































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