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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 186-190

Section 1.7.3.1. Crystalline linear optical properties

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

1.7.3.1. Crystalline linear optical properties

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We summarize here the main linear optical properties that govern the nonlinear propagation phenomena. The reader may refer to Chapter 1.6[link] for the basic equations.

1.7.3.1.1. Index surface and electric field vectors

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The relations between the different field vectors relative to a propagating electromagnetic wave are obtained from the constitutive relations of the medium and from Maxwell equations.

In the case of a non-magnetic and non-conducting medium, Maxwell equations lead to the following wave propagation equation for the Fourier component at the circular frequency ω defined by (1.7.2.15)[link] and (1.7.2.16)[link] (Butcher & Cotter, 1990[link]):[\nabla{\bf x}\nabla{\bf xE}(\omega)=(\omega^2/c^2){\bf E}(\omega)+\omega^2\mu_0{\bf P}(\omega),\eqno(1.7.3.1)]where [\omega=2\pi c/\lambda], λ is the wavelength and c is the velocity of light in a vacuum; [\mu_0] is the free-space permeability, E(ω) is the electric field vector and P(ω) is the polarization vector.

In the linear regime, [{\bf P}(\omega)=\varepsilon_0\chi^{(1)}(\omega){\bf E}(\omega)], where [epsilon]0 is the free-space permittivity and χ(1)(ω) is the first-order electric susceptibility tensor. Then (1.7.3.1)[link] becomes[\nabla{\bf x}\nabla{\bf xE}(\omega)=(\omega^2/c^2)\varepsilon(\omega){\bf E}(\omega).\eqno(1.7.3.2)][\varepsilon(\omega)=1+\chi^{(1)}(\omega)] is the dielectric tensor. In the general case, [\chi^{(1)}(\omega)] is a complex quantity i.e. [\chi^{(1)}=\chi^{(1)'}+i\chi^{(1)''}]. For the following, we consider a medium for which the losses are small ([\chi^{(1)'}\gg\chi^{(1)''}]); it is one of the necessary characteristics of an efficient nonlinear medium. In this case, the dielectric tensor is real: [\varepsilon = 1 + \chi^{(1)'}].

The plane wave is a solution of equation (1.7.3.2)[link]:[{\bf E}(\omega,X,Y,Z)={\bf e}(\omega){\bf E}(\omega, X, Y, Z)\exp[\pm ik(\omega)Z].\eqno(1.7.3.3)]([X,Y,Z]) is the orthonormal frame linked to the wave, where Z is along the direction of propagation.

We consider a linearly polarized wave so that the unit vector e of the electric field is real ([{\bf e}={\bf e}^*]), contained in the XZ or YZ planes.

[{\bf E}(\omega,X,Y,Z) = A(\omega,X,Y,Z)\exp[i\Phi(\omega,Z)]] is the scalar complex amplitude of the electric field where [\Phi(\omega,Z)] is the phase, and [{\bf E}^*(-\omega,X,Y,Z)= {\bf E}(\omega,X,Y,Z)]. In the linear regime, the amplitude of the electric field varies with Z only if there is absorption.

k is the modulus of the wavevector, real in a lossless medium: [+kZ] corresponds to forward propagation along Z, and [-kZ] to backward propagation. We consider that the plane wave propagates in an anisotropic medium, so there are two possible wavevectors, k+ and k, for a given direction of propagation of unit vector u:[{\bf k}^{\pm}(\omega,\theta,\varphi)=(\omega/c)n^{\pm}(\omega,\theta,\varphi){\bf u}(\theta,\varphi).\eqno(1.7.3.4)]([\theta,\varphi]) are the spherical coordinates of the direction of the unit wavevector u in the optical frame; ([x,y,z]) is the optical frame defined in Section 1.7.2[link].

The spherical coordinates are related to the Cartesian coordinates ([u_x,u_y,u_z]) by[u_x=\cos\varphi\sin\theta\quad u_y=\sin\varphi\sin\theta\quad u_z=\cos\theta. \eqno(1.7.3.5)]The refractive indices [n^{\pm}(\omega,\theta,\varphi)=[\varepsilon^{\pm}(\omega,\theta,\varphi)]^{1/2}], [(n^+> n^-)], real in the case of a lossless medium, are the two solutions of the Fresnel equation (Yao & Fahlen, 1984[link]):[\eqalignno{n^{\pm}&=\left[2 \over -B \mp (B^2 - 4C)^{1/2}\right]^{1/2}&\cr B&=-u_x^2(b+c)-u_y^2(a+c)-u_z^2(a+b)&\cr C&=u_x^2bc+u_y^2ac+u_z^2ab&\cr a&=n_x^{-2}(\omega),\quad b=n_y^{-2}(\omega), \quad c=n_z^{-2}(\omega).&\cr&&(1.7.3.6)}]nx(ω), ny(ω) and nz(ω) are the principal refractive indices of the index ellipsoid at the circular frequency ω.

Equation (1.7.3.6)[link] describes a double-sheeted three-dimensional surface: for a direction of propagation u the distances from the origin of the optical frame to the sheets (+) and (−) correspond to the roots n+ and n. This surface is called the index surface or the wavevector surface. The quantity ([n^+ -n^-]) or ([n^- -n^+]) is the birefringency. The waves (+) and (−) have the phase velocities [c/n^+] and [c/n^-], respectively.

Equation (1.7.3.6)[link] and its dispersion in frequency are often used in nonlinear optics, in particular for the calculation of the phase-matching directions which will be defined later. In the regions of transparency of the crystal, the frequency law is well described by a Sellmeier equation, which is the case for normal dispersion where the refractive indices increase with frequency (Hadni, 1967[link]):[n^\pm(\omega_i)\,\lt \,n^\pm(\omega_j)\;\hbox{ for }\;\omega_i\,\lt\,\omega_j.\eqno(1.7.3.7)]If ωi or ωj are near an absorption peak, even weak, n±i) can be greater than n±j); this is called abnormal dispersion.

The dielectric displacements [{\bf D}^\pm], the electric fields [{\bf E}^\pm], the energy flux given by the Poynting vector [{\bf S}^\pm = {\bf E}^\pm \times {\bf H}^\pm] and the collinear wavevectors [{\bf k}^\pm] are coplanar and define the orthogonal vibration planes [\Pi^\pm] (Shuvalov, 1981[link]). Because of anisotropy, [{\bf k}^\pm] and [{\bf S}^\pm], and hence [{\bf D}^\pm] and [{\bf E}^\pm], are non-collinear in the general case as shown in Fig. 1.7.3.1[link]: the walk-off angles, also termed double-refraction angles, [\rho^\pm=\arccos({\bf d}^\pm\cdot{\bf e}^\pm) = \arccos({\bf u}\cdot{\bf s}^\pm)] are different in the general case; [{\bf d}^\pm], [{\bf e}^\pm], [{\bf u}] and [{\bf s}^\pm] are the unit vectors associated with [{\bf D}^\pm], [{\bf E}^\pm], [{\bf k}^\pm] and [{\bf S}^\pm], respectively. We shall see later that the efficiency of a nonlinear interaction is strongly conditioned by k, E and ρ, which only depend on χ(1)(ω), that is to say on the linear optical properties.

[Figure 1.7.3.1]

Figure 1.7.3.1 | top | pdf |

Field vectors of a plane wave propagating in an anisotropic medium. ([X,Y,Z]) is the wave frame. Z is along the direction of propagation, X and Y are contained in Π+ and Π respectively, by an arbitrary convention.

The directions S+ and S are the directions normal to the sheets (+) and (−) of the index surface at the points n+ and n.

For a plane wave, the time-average Poynting vector is (Yariv & Yeh, 2002[link])[\eqalignno{\left\| {\bf S}^\pm (\omega)\right\|&= \left\| \textstyle{1 \over 2}Re\left[{\bf E}^\pm (\omega)\times{\bf H}^{\pm *}(\omega)\right]\right\| &\cr&= {\textstyle{1 \over 2}}{\left\| {\bf k}^\pm (\omega)\right\| \over \mu_0 \omega}\left\|{\bf E}^ \pm(\omega)\right\|^2 \cos^2 \rho ^ \pm (\omega).&\cr &&(1.7.3.8)}][\| {\bf S ^ \pm }\|] is the energy flow [I = \hbar \omega N^ \pm], which is a power per unit area i.e. the intensity, where [\hbar\omega] is the energy of the photon and [N^ \pm] are the photons flows. ρ±(ω) is the angle between S± and u; it is detailed later on.

The unit electric field vectors e+ and eare calculated from the propagation equation projected on the three axes of the optical frame. We obtain, for each wave, three equations which relate the three components ([e_x,e_y,e_z]) to the unit wavevector components ([u_x,u_y,u_z]) (Shuvalov, 1981[link]):[(n^\pm)^2(e^\pm_p-u_p[u_xe^\pm_x+u_ye^\pm_y+u_ze^\pm_z])=(n_p)^2e_p^\pm\quad (p=x, y\hbox{ and }z)\eqno(1.7.3.9)]with [(e_x^\pm)^2+(e_y^\pm)^2+(e_z^\pm)^2=1.]

The vibration planes [\Pi^\pm] relative to the eigen polarization modes [{\bf e}^\pm] are called the neutral vibration planes associated with u: an incident linearly polarized wave with a vibration plane parallel to [\Pi^+] or [\Pi^-] is refracted inside the crystal without depolarization, that is to say in a linearly polarized wave, e+ or e, respectively. For any other incident polarization the wave is refracted in the two waves e+ and e, which propagate with the difference of phase [(\omega/c)(n^ + - n^-)Z].

The existence of equalities between the principal refractive indices determines the three optical classes: isotropic for the cubic system; uniaxial for the tetragonal, hexagonal and trigonal systems; and generally biaxial for the orthorhombic, monoclinic and triclinic systems [Nye (1957[link]) and Sections 1.1.4.1[link] and 1.6.3.2[link] ].

1.7.3.1.2. Isotropic class

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The isotropic class corresponds to the equality of the three principal indices: the index surface is a one-sheeted sphere, so [n^+=n^-], [\rho^+=\rho^-=0] for all directions of propagation, and any electric field vector direction is allowed as in an amorphous material.

1.7.3.1.3. Uniaxial class

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The uniaxial class is characterized by the equality of two principal indices, called ordinary indices ([n_x=n_y=n_o]); the other index is called the extraordinary index ([n_z=n_e]). Then, according to (1.7.3.6)[link], the index surface has one umbilicus along the z axis, [n^+(\theta=0)=n^-(\theta=0)], called the optic axis, which is along the fold rotation axis of greatest order of the crystal. The two other principal axes are related to the symmetry elements of the orientation class according to the standard conventions (Nye, 1957[link]). The ordinary sheet is spherical i.e. [n_o(\theta,\varphi)=n_o], so an ordinary wave has no walk-off for any direction of propagation in a uniaxial crystal; the extraordinary sheet is ellipsoidal i.e. [n_e (\theta, \varphi) = [(\cos^2\theta)/(n_o^2)+(\sin^2\theta)/(n_e^2)]^{-1/2}]. The sign of the uniaxial class is defined by the sign of the birefringence [n_e-n_o]. Thus, according to these definitions, ([n_e,n_o]) corresponds to ([n^+,n^-]) for the positive class ([n_e>n_o]) and to ([n^-,n^+]) for the negative class ([n_e\,\lt\,n_o]), as shown in Fig. 1.7.3.2[link].

[Figure 1.7.3.2]

Figure 1.7.3.2 | top | pdf |

Index surfaces of the negative and positive uniaxial classes. [{\bf E}_{o,e}^ \pm ] are the ordinary (o) and extraordinary (e) electric field vectors relative to the external (+) or internal (−) sheets. OA is the optic axis.

The ordinary electric field vector is orthogonal to the optic axis ([e_z^o=0]), and also to the extraordinary electric field vector, leading to[{\bf e}^o(\omega_i,\theta,\varphi)\cdot{\bf e}^e(\omega_j,\theta,\varphi)=0.\eqno(1.7.3.10)]This relation is satisfied when ωi and ωj are equal or different and for any direction of propagation ([\theta,\varphi]).

According to these results, the coplanarity of the field vectors imposes the condition that the double-refraction angle of the extraordinary wave is in a plane containing the optic axis. Thus, the components of the ordinary and extraordinary unit electric field vectors eo and ee at the circular frequency ω are[\displaylines{\hfill e_x^o=-\sin\varphi\quad e_y^o=+\cos\varphi\quad e_z^o=0\hfill(1.7.3.11)\cr e_x^e=-\cos[\theta\pm\rho^\mp(\theta,\omega)]\cdot\cos\varphi\cr e^e_y=-\cos[\theta\pm\rho^\mp(\theta,\omega)]\cdot\sin\varphi\cr \hfill e_z^e=\sin[\theta\pm\rho^\mp(\theta,\omega)]\hfill(1.7.3.12)}%fd1.7.3.12]with [-\rho^+(\theta,\omega)] for the positive class and [+\rho^-(\theta,\omega)] for the negative class. [\rho^\pm(\theta,\omega)] is given by[\eqalignno{\rho^\pm(\theta,\omega)&=\arccos({\bf d}^\pm\cdot{\bf e}^\pm)=\arccos({\bf u}^\pm\cdot{\bf s}^\pm)&\cr &=\arccos\left\{\left[{\cos^2\theta \over n_o^2(\omega)} + {\sin^2\theta \over n_e^2(\omega)}\right]\left[{\cos^2\theta \over n_o^4(\omega)} + {\sin^2\theta \over n_e^4(\omega)}\right]^{-1/2}\right\}.&\cr&&(1.7.3.13)}]Note that the extraordinary walk-off angle is nil for a propagation along the optic axis ([\theta=0]) and everywhere in the xy plane ([\theta=\pi/2]).

1.7.3.1.4. Biaxial class

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In a biaxial crystal, the three principal refractive indices are all different. The graphical representations of the index surfaces are given in Fig. 1.7.3.3[link] for the positive biaxial class ([n_x\,\lt\,n_y\,\lt\, n_z]) and for the negative one ([n_x>n_y>n_z]), both with the usual conventional orientation of the optical frame. If this is not the case, the appropriate permutation of the principal refractive indices is required.

[Figure 1.7.3.3]

Figure 1.7.3.3 | top | pdf |

Index surfaces of the negative and positive biaxial classes. [{\bf E}_{o.e}^{\pm}] are the ordinary (o) and extraordinary (e) electric field vectors relative to the external (+) or internal (−) sheets for a propagation in the principal planes. OA is the optic axis.

In the orthorhombic system, the three principal axes are fixed by the symmetry; one is fixed in the monoclinic system; and none are fixed in the triclinic system. The index surface of the biaxial class has two umbilici contained in the xz plane, making an angle V with the z axis:[\sin^2V(\omega)={n^{-2}_y(\omega)-n^{-2}_x(\omega)\over n^{-2}_z(\omega)-n_x^{-2}(\omega)}.\eqno(1.7.3.14)]The propagation along the optic axes leads to the internal conical refraction effect (Schell & Bloembergen, 1978[link]; Fève et al., 1994[link]).

1.7.3.1.4.1. Propagation in the principal planes

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It is possible to define ordinary and extraordinary waves, but only in the principal planes of the biaxial crystal: the ordinary electric field vector is perpendicular to the z axis and to the extraordinary one. The walk-off properties of the waves are not the same in the [xy] plane as in the [xz] and [yz] planes.

  • (1) In the xy plane, the extraordinary wave has no walk-off, in contrast to the ordinary wave. The components of the electric field vectors can be established easily with the same considerations as for the uniaxial class:[\eqalignno{e_x^o&=-\sin[\varphi\pm\rho^\mp(\varphi,\omega)]&\cr e_y^o&=\cos[\varphi\pm\rho^\mp(\varphi,\omega)]&\cr e_z^o&=0, &(1.7.3.15)}]with [+\rho^-(\varphi,\omega)] for the positive class and [-\rho^+(\varphi,\omega)] for the negative class. [\rho^\pm(\varphi,\omega)] is the walk-off angle given by (1.7.3.13)[link], where [\theta] is replaced by [\varphi], no by ny and ne by nx:[e_x^e=0\quad e_y^e=0\quad e_z^e=1.\eqno(1.7.3.16)]

  • (2) The yz plane of a biaxial crystal has exactly the same characteristics as any plane containing the optic axis of a uniaxial crystal. The electric field vector components are given by (1.7.3.11)[link] and (1.7.3.12)[link] with [\varphi=\pi/2]. The ordinary walk-off is nil and the extraordinary one is given by (1.7.3.13)[link] with [n_o=n_y] and [n_e=n_z].

  • (3) In the xz plane, the optic axes create a discontinuity of the shape of the internal and external sheets of the index surface leading to a discontinuity of the optic sign and of the electric field vector. The birefringence, [n_e-n_o], is nil along the optic axis, and its sign changes on either side. Then the yz plane, xy plane and xz plane from the x axis to the optic axis have the same optic sign, the opposite of the optic sign from the optic axis to the z axis. Thus a positive biaxial crystal is negative from the optic axis to the z axis. The situation is inverted for a negative biaxial crystal. It implies the following configuration of polarization:

    • (i) From the x axis to the optic axis, eo and ee are given by (1.7.3.11)[link] and (1.7.3.12)[link] with [\varphi = 0]. The walk-off is relative to the extraordinary wave and is calculated from (1.7.3.13)[link] with [n_o=n_x] and [n_e = n_z].

    • (ii) From the optic axis to the z axis, the vibration plane of the ordinary and extraordinary waves corresponds respectively to a rotation of π/2 of the vibration plane of the extraordinary and ordinary waves for a propagation in the areas of the principal planes of opposite sign; the extraordinary electric field vector is given by (1.7.3.12)[link] with [\varphi = 0], [-\rho^-(\varphi,\omega)] for the positive class and [+\rho^+(\varphi,\omega)] for the negative class, and the ordinary electric field vector is out of phase by π in relation to (1.7.3.11)[link], that is[e_x^o=0\quad e_y^o=-1\quad e_z^o=0.\eqno(1.7.3.17)]The extraordinary walk-off angle is given by (1.7.3.13)[link] with [n_o = n_x] and [n_e = n_z].

    The π/2 rotation on either side of the optic axes is well observed during internal conical refraction (Fève et al., 1994[link]).

    Note that for a biaxial crystal, the walk-off angles are all nil only for a propagation along the principal axes.

1.7.3.1.4.2. Propagation out of the principal planes

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It is impossible to define ordinary and extraordinary waves out of the principal planes of a biaxial crystal: according to (1.7.3.6)[link] and (1.7.3.9)[link], e+ and e have a nonzero projection on the z axis. According to these relations, it appears that e+ and e are not perpendicular, so relation (1.7.3.10)[link] is never verified. The walk-off angles ρ+ and ρ are nonzero, different, and can be calculated from the electric field vectors:[\rho^\pm(\theta,\varphi,\omega)=\varepsilon\arccos[{\bf e}^\pm(\theta,\varphi,\omega)\cdot{\bf u}(\theta,\varphi,\omega)]-\varepsilon\pi/2.\eqno(1.7.3.18)][\varepsilon = +1] or [-1] for a positive or a negative optic sign, respectively.

References

Butcher, P. N. & Cotter, D. (1990). The elements of nonlinear optics. Cambridge series in modern optics. Cambridge University Press.
Fève, J. P., Boulanger, B. & Marnier, G. (1994). Experimental study of internal and external conical refractions in KTP. Optics Comm. 105, 243–252.
Hadni, A. (1967). Essentials of modern physics applied to the study of the infrared. Oxford: Pergamon Press.
Nye, J. F. (1957). Physical properties of crystals. Oxford: Clarendon Press.
Schell, A. J. & Bloembergen, N. (1978). Laser studies of internal conical refraction. I. Quantitative comparison of experimental and theoretical conical intensity distribution in aragonite. J. Opt. Soc. Am. 68, 1093–1106.
Shuvalov, L. A. (1981). Modern crystallography IV – Physical properties of crystals. Springer Series in solid-state sciences No. 37. Heidelberg: Springer Verlag.
Yao, J. Q. & Fahlen, T. S. (1984). Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4. J. Appl. Phys. 55, 65–68.
Yariv, A. & Yeh, P. (2002). Optical waves in crystals. New York: Wiley.








































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