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 Nonlinear coefficients

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: Nonlinear coefficients

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The knowledge of the absolute magnitude and of the relative sign of the independent elements of the tensors χ(2) and χ(3) is of prime importance not only for the qualification of a new crystal, but also for the fundamental engineering of nonlinear optical materials in connection with microscopic aspects.

However, disparities in the published values of the nonlinear coefficients of the same crystal exist, even if it is a well known material that has been used for a long time in efficient devices (Eckardt & Byer, 1991[link]; Boulanger, Fève et al., 1994[link]). The disagreement between the different absolute magnitudes is sometimes a result of variation in the quality of the crystals, but mainly arises from differences in the measurement techniques. Furthermore, a considerable amount of confusion exists as a consequence of the difference between the conventions taken for the relation between the induced nonlinear polarization and the nonlinear susceptibility, as explained in Section[link].

Accurate measurements require mm-size crystals with high optical quality of both surface and bulk. 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). Phase-matched interaction method

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The use of phase-matched interactions is suitable for absolute and accurate measurements (Eckardt & Byer, 1991[link]; Boulanger, Fève et al., 1994[link]). The sample studied is usually a slab cut in a phase-matching direction. The effective coefficient is determined from the measurement of the conversion efficiency using the theoretical expressions given by ([link] and ([link] for SHG, and by ([link] for THG, according to the validity of the corresponding approximations. Because of phase matching, the generated harmonic power is not weak and it is measurable with very good accuracy, even with a c.w. conversion efficiency.

Recent experiments have been performed in a KTP crystal cut as a sphere (Boulanger et al., 1997[link], 1998[link]): the absolute magnitudes of the quadratic effective coefficients are measured with an accuracy of 10%, which is comparable with typical experiments on a slab.

For both non-phase-matched and phase-matched techniques, it is important to know the refractive indices and to characterize the spatial, temporal and spectral properties of the pump beam carefully. The considerations developed in Section 1.7.3[link] about effective coefficients and field tensors allow judicious choices of configurations of polarization and directions of propagation for the determination of the absolute value and relative sign of the independent coefficients of tensors χ(2) and χ(3), given in Tables[link] to[link][link][link] for the different crystal point groups.


Boulanger, B., Fève, J. P., Marnier, G., Bonnin, C., Villeval, P. & Zondy, J. J. (1997). Absolute measurement of quadratic nonlinearities from phase-matched second-harmonic-generation in a single crystal cut as a sphere. J. Opt. Soc. Am. B, 14, 1380–1386.
Boulanger, B., Fève, J. P., Marnier, G. & Ménaert, B. (1998). Methodology for nonlinear optical studies: application to the isomorph family KTiOPO4, KTiOAsO4, RbTiOAsO4 and CsTiOAsO4. Pure Appl. Opt. 7, 239–256.
Boulanger, B., Fève, J. P., Marnier, G., Ménaert, B., Cabirol, X., Villeval, P. & Bonnin, C. (1994). Relative sign and absolute magnitude of d(2) nonlinear coefficients of KTP from second-harmonic-generation measurements. J. Opt. Soc. Am. B, 11(5), 750–757.
Eckardt, R. C. & Byer, R. L. (1991). Measurement of nonlinear optical coefficients by phase-matched harmonic generation. SPIE. Inorganic crystals for optics, electro-optics and frequency conversion, 1561, 119–127.
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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