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. 184

Section 1.7.2.1.4.2. Convention used in this chapter

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

1.7.2.1.4.2. Convention used in this chapter

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The K convention described above is often used, but may lead to errors in cases where two of the interacting waves have the same frequency but different polarization states. Indeed, as demonstrated in Chapter 1.6[link] and recalled in Section 1.7.3[link], a direction of propagation in an anisotropic crystal allows in the general case two different directions of polarization of the electric field vector, written E+ and E. Then any nonlinear coupling in this medium occurs necessarily between these eigen modes at the frequencies concerned.

Because of the possible non-degeneracy with respect to the direction of polarization of the electric fields at the same frequency, it is suitable to consider a harmonic generation process, second harmonic generation (SHG) or third harmonic generation (THG) for example, like any other non-degenerated interaction. We do so for the rest of this chapter. Then all terms derived from the permutation of the fields with the same frequency are taken into account in the expression of the induced nonlinear polarization and the K factor in equation (1.7.2.29)[link] disappears: hence, in the general case, the induced nonlinear polarization is written[\eqalignno{P_\mu^{(n)}(\omega_\sigma) &=\varepsilon_o\textstyle \sum \limits_{\alpha_1,\ldots,\alpha_n}\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-\omega_\sigma\semi\omega_1,\ldots,\omega_n)&\cr&\quad\times E_{\alpha_1}^{\pm}(\omega_1)\ldots E_{\alpha_n}^{\pm}(\omega_n), &(1.7.2.33)}]where [+] and − refer to the eigen polarization modes.

According to (1.7.2.33)[link], the nth harmonic generation induced polarization is expressed as[\eqalignno{P_\mu^{(n)}(n\omega) &=\varepsilon_o\textstyle \sum \limits_{\alpha_1,\ldots,\alpha_n}\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-n\omega\semi\omega,\ldots,\omega)&\cr&\quad\times E_{\alpha_1}^{\pm}(\omega_1)\ldots E_{\alpha_n}^{\pm}(\omega_n).&(1.7.2.34)}]For example, in the particular case of SHG where the two waves at ω have different directions of polarization E+(ω) and E(ω) and where the only nonzero [\chi^{(2)}_{yij}] coefficients are [\chi_{yxz}] and [\chi_{yzx}], (1.7.2.34)[link] gives[\eqalignno{P_y^{(2)}(2\omega) &=\varepsilon_o[\chi_{yxz}(-2\omega\semi\omega,\omega)E_x^+(\omega)E_z^-(\omega) &\cr &\quad +\chi_{yzx}(-2\omega\semi\omega,\omega)E_z^+(\omega)E_x^-(\omega)].&\cr&&(1.7.2.35)}]The two field component products are equal only if the two eigen modes are the same, i.e. [+] or −.

According to (1.7.2.33)[link] and (1.7.2.34)[link], we note that [\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\ldots,\omega_n)] changes smoothly to [\chi^{(n)}_{\mu\alpha_1\ldots\alpha_n}(-n\omega;] [\omega,\ldots,\omega)] when all the [\omega_1,\ldots\omega_n] approach continuously the same value ω.








































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