International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 190-191
Section 1.7.3.2.1. Coupled electric fields amplitudes equations^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France |
The nonlinear crystals considered here are homogeneous, lossless, non-conducting, without optical activity, non-magnetic and are optically anisotropic. The nonlinear regime allows interactions between γ waves with different circular frequencies . The Fourier component of the polarization vector at ω_{i} is , where is the nonlinear polarization corresponding to the orders of the power series greater than 1 defined in Section 1.7.2.
Thus the propagation equation of each interacting wave ω_{i} is (Bloembergen, 1965)The γ propagation equations are coupled by :
The complex conjugates come from the relation .
We consider the plane wave, (1.7.3.3), as a solution of (1.7.3.19), and we assume that all the interacting waves propagate in the same direction Z. Each linearly polarized plane wave corresponds to an eigen mode E^{+} or E^{−} defined above. For the usual case of beams with a finite transversal profile and when Z is along a direction where the double-refraction angles can be nonzero, i.e. out of the principal axes of the index surface, it is necessary to specify a frame for each interacting wave in order to calculate the corresponding powers as a function of Z: the coordinates linked to the wave at ω_{i} are written (), which can be relative to the mode (+) or (−). The systems are then linked by the double-refraction angles ρ: according to Fig. 1.7.3.1, we have for two waves (+) with , and for two waves (−) with .
The presence of in equations (1.7.3.19) leads to a variation of the γ amplitudes E(ω_{i}) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λ_{i}, which is usually true. Thus we can stateThis is called the slowly varying envelope approximation.
Stating (1.7.3.20), the wave equation (1.7.3.19) for a forward propagation of a plane wave leads toWe choose the optical frame () for the calculation of all the scalar products , the electric susceptibility tensors being known in this frame.
For a three-wave interaction, (1.7.3.21) leads towith , , and , called the phase mismatch. We take by convention .
If ABDP relations, defined in Section 1.7.2.2.1, are verified, then the three tensorial contractions in equations (1.7.3.22) are equal to the same quantity, which we write , where is called the effective coefficient:The same considerations lead to the same kind of equations for a four-wave interaction:The conventions of notation are the same as previously and the phase mismatch is . The effective coefficient isExpressions (1.7.3.23) for and (1.7.3.25) for can be condensed by introducing adequate third- and fourth-rank tensors to be contracted, respectively, with and . For example, or , and similar expressions. By substituting (1.7.3.8) in (1.7.3.22), we obtain the derivatives of Manley–Rowe relations (1.7.2.40) for a three-wave mixing, where is the Z photon flow. Identically with (1.7.3.24), we have for a four-wave mixing.
In the general case, the nonlinear polarization wave and the generated wave travel at different phase velocities, and , respectively, because of the frequency dispersion of the refractive indices in the crystal. Then the work per unit time W(ω_{i}), given in (1.7.2.39), which is done on the generated wave E(ω_{i}, Z) by the nonlinear polarization P^{NL}(ω_{i}, Z), alternates in sign for each phase shift of π during the Z-propagation, which leads to a reversal of the energy flow (Bloembergen, 1965). The length leading to the phase shift of π is called the coherence length, , where Δk is the phase mismatch given by (1.7.3.22) or (1.7.3.24).
References
Bloembergen, N. (1965). Nonlinear optics. New York: Benjamin.