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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 215-217

Section Non-phase-matched interaction method

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: Non-phase-matched interaction method

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The main techniques used are based on non-phase-matched SHG and THG performed in several samples cut in different directions. The classical method, termed the Maker-fringes technique (Jerphagnon & Kurtz, 1970[link]; Herman & Hayden, 1995[link]), consists of the measurement of the harmonic power as a function of the angle between the fundamental laser beam and the rotated slab sample, as shown in Fig.[link](a).


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(a) The Maker-fringes technique; (b) the wedge-fringes technique.

The conversion efficiency is weak because the interaction is non-phase-matched. In normal incidence, the waves are collinear and so formulae ([link] for SHG and ([link] for THG are valid. These can be written in a more convenient form where the coherence length appears:[\eqalignno{P^{n\omega}(L)&=A^{n\omega}[P^{\omega}(0)]^n(d_{\rm eff}^{n\omega}\cdot l^{n\omega}_c)^2\sin^2(\pi L/2l^{n\omega}_c) &\cr l^{2\omega}_c&=(\pi c/\omega)(2n_3^{2\omega}-n_1^{\omega}-n_2^\omega)^{-1}&\cr l^{3\omega}_c&=(\pi c/\omega)(3n_4^{3\omega}-n_1^{\omega}-n_2^\omega-n_3^\omega)^{-1}.&(}]The coefficient [A^{n\omega}] depends on the refractive indices in the direction of propagation and on the fundamental beam geometry: [A^{2\omega}] and [A^{3\omega}] can be easily expressed by identifying ([link] with ([link] and ([link], respectively.

When the crystal is rotated, the harmonic and fundamental waves are refracted with different angles, which leads to a variation of the coherence length and consequently to an oscillation of the harmonic power as a function of the angle of incidence, α, of the fundamental beam. Note that the oscillation exists even if the refractive indices do not vary with the direction of propagation, which would be the case for an interaction involving only ordinary waves during the rotation. The most general expression of the generated harmonic power, i.e. [P^{n\omega}(\alpha)=j(\alpha)\sin^2\Psi(\alpha)], must take into account the angular dependence of all the refractive indices, in particular for the calculation of the coherence length and transmission coefficients (Herman & Hayden, 1995[link]). The effective coefficient is then deduced from the angular spacing of the Maker fringes and from the conversion efficiency at the maxima of oscillation.

A continuous variation of the phase mismatch can also be performed by translating a wedged sample as shown in Fig.[link](b) (Perry, 1991[link]). The harmonic power oscillates as a function of the displacement x. In this case, the interacting waves stay collinear and the oscillation is only caused by the variation of the crystal length. Relation ([link] is then valid, by considering a variable crystal length [L(x)=x\tan\beta]; [A^{n\omega}] and [l_c^{n\omega}] are constant. The space between two maxima of the wedge fringes is [\Delta x_c=2l_c/\tan\beta], which allows the determination of lc. Then the measurement of the harmonic power, [P_{\rm max}^{n\omega}], generated at a maximum leads to the absolute value of the effective coefficient:[\eqalignno{|d_{\rm eff}^{n\omega}|&=\left\{{P_{\rm max}^{n\omega}\over A^{n\omega}[P^{\omega}(0)]^2l_c^2}\right\}^{1/2}&\cr l_c&=(\Delta x_c\tan\beta/2).&(}]

It is necessary to take into account a multiple reflection factor in the expression of [A^{n\omega}].

The Maker-fringes and wedge-fringes techniques are essentially used for relative measurements referenced to a standard, usually KH2PO4 (KDP) or quartz (α-SiO2).


Herman, W. N. & Hayden, L. M. (1995). Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials. J. Opt. Soc. Am. B, 12, 416–427.
Jerphagnon, J. & Kurtz, S. K. (1970). Optical nonlinear susceptibilities: accurate relative values for quartz, ammonium dihydrogen phosphate, and potassium dihydrogen phosphate. Phys. Rev. B, 1(4), 1739–1744.
Perry, J. W. (1991). Nonlinear optical properties of molecules and materials. In Materials for nonlinear optics, chemical perspectives, edited by S. R. Marder, J. E. Sohn & G. D. Stucky, pp. 67–88. ACS Symp. Ser. No. 455. Washington: American Chemical Society.

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