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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 215

Section Phase-matching directions and associated acceptance bandwidths

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: Phase-matching directions and associated acceptance bandwidths

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The very early stage of crystal growth of a new material usually provides a powder with particle sizes less than 100 µm. It is then impossible to measure the phase-matching loci. Nevertheless, careful SHG experiments performed on high-quality crystalline material may indicate whether the SHG is phase-matched or not by considering the dependence of the SHG intensity on the following parameters: the angle between the detector and the direction of the incident fundamental beam, the powder layer thickness, the average particle size and the laser beam diameter (Kurtz & Perry, 1968[link]). However, powder measurements are essentially used for the detection in a simple and quick way of noncentrosymmetry of crystals, this criterion being necessary to have a nonzero χ(2) tensor (Kurtz & Dougherty, 1978[link]). They also allow, for example, the measurement of the temperature of a possible centrosymmetric/noncentrosymmetric transition (Marnier et al., 1989[link]).

For crystal sizes greater than few hundred µm, it is possible to perform direct measurements of phase-matching directions. The methods developed at present are based on the use of a single crystal ground into an ellipsoidal (Velsko, 1989[link]) or spherical shape (Marnier & Boulanger, 1989[link]; Boulanger, 1989[link]; Boulanger et al., 1998[link]); a sphere is difficult to obtain for sample diameters less than 2 mm, but it is the best geometry for large numbers and accurate measurements because of normal refraction for every chosen direction of propagation. The sample is oriented using X-rays, placed at the centre of an Euler circle and illuminated with fixed and appropriately focused laser beams. The experiments are usually performed with SHG of different fundamental wavelengths. The sample is rotated in order to propagate the fundamental beam in different directions: a phase-matching direction is then detected when the SHG conversion efficiency is a maximum. It is then possible to describe the whole phase-matching cone with an accuracy of 1°. A spherical crystal also allows easy measurement of the walk-off angle of each of the waves (Boulanger et al., 1998[link]). It is also possible to perform a precise observation and study of the internal conical refraction in biaxial crystals, which leads to the determination of the optic axis angle V(ω), given by relation ([link], for different frequencies (Fève et al., 1994[link]).

Phase-matching relations are often poorly calculated when using refractive indices determined by the prism method or by measurement of the critical angle of total reflection. Indeed, all the refractive indices concerned have to be measured with an accuracy of 10−4 in order to calculate the phase-matching angles with a precision of about 1°. Such accuracies can be reached in the visible spectrum, but it is more difficult for infrared wavelengths. Furthermore, it is difficult to cut a prism of few mm size with plane faces.

If the refractive indices are known with the required accuracy at several wavelengths well distributed across the transparency region, it is possible to fit the data with a Sellmeier equation of the following type, for example:[n_i^2(\lambda)=A_i+{B_i\lambda^2\over\lambda^2-C_i}+D_i\lambda^2.\eqno(]ni is the principal refractive index, where [i = o] (ordinary) and e (extraordinary) for uniaxial crystals and [i = x, y] and z for biaxial crystals.

It is then easy to calculate the phase-matching angles (θPM, [\varphi]PM) from ([link] using equations ([link] or ([link] where the angular variation of the refractive indices is given by equation ([link].

The measurement of the variation of intensity of the generated beam as a function of the angle of incidence can be performed on a sphere or slab, leading, respectively, to internal and external angular acceptances. The thermal acceptance is usually measured on a slab which is heated or cooled during the frequency conversion process. The spectral acceptance is not often measured, but essentially calculated from Sellmeier equations ([link] and the expansion of Δk in the Taylor series ([link] with [\xi=\lambda].


Boulanger, B. (1989). Synthèse en flux et étude des propriétés optiques cristallines linéaires et non linéaires par la méthode de la sphère de KTiOPO4 et des nouveaux composés isotypes et solutions solides de formule générale (K,Rb,Cs)TiO(P,As)O4. PhD Dissertation, Université de Nancy I, France.
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.
Fève, J. P., Boulanger, B. & Marnier, G. (1994). Experimental study of internal and external conical refractions in KTP. Optics Comm. 105, 243–252.
Kurtz, S. K. & Dougherty, J. P. (1978). Systematic materials analysis. Vol. IV, edited by J. H. Richardson. New York: Academic Press.
Kurtz, S. K. & Perry, T. T. (1968). A powder technique for the evaluation of nonlinear optical materials. J. Appl. Phys. 39(8), 3978–3813.
Marnier, G. & Boulanger, B. (1989). The sphere method: a new technique in linear and non linear crystalline optical studies. Optics Comm. 72(3–4), 139–143.
Marnier, G., Boulanger, B. & Ménaert, B. (1989). Melting and ferroelectric transition temperature of new compounds: CsTiOAsO4 and CsxM1−xTiOAs1−yO4 with M = K or Rb. J. Phys. Condens. Matter, 1, 5509–5513.
Velsko, S. P. (1989). Direct measurements of phase matching properties in small single crystals of new nonlinear materials. Soc. Photo-Opt. Instrum. Eng. Conf. Laser Nonlinear Opt. Eng. 28, 76–84.

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