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

International Tables for Crystallography (2006). Vol. D, ch. 1.7, pp. 183-212

Section 1.7.3. Propagation phenomena

B. Boulangera* and J. Zyssb

aLaboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:

1.7.3. Propagation phenomena

| top | pdf | 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. 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 ([link] and ([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(]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 ([link] becomes[\nabla{\bf x}\nabla{\bf xE}(\omega)=(\omega^2/c^2)\varepsilon(\omega){\bf E}(\omega).\eqno(][\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 ([link]:[{\bf E}(\omega,X,Y,Z)={\bf e}(\omega){\bf E}(\omega, X, Y, Z)\exp[\pm ik(\omega)Z].\eqno(]([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(]([\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(]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&&(}]nx(ω), ny(ω) and nz(ω) are the principal refractive indices of the index ellipsoid at the circular frequency ω.

Equation ([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 ([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(]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.[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.


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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 &&(}][\| {\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(]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[link] and[link] ]. 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. 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 ([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.[link].


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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(]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(\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(}%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&&(}]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]). 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.[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.


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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(]The propagation along the optic axes leads to the internal conical refraction effect (Schell & Bloembergen, 1978[link]; Fève et al., 1994[link]). 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, &(}]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 ([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(]

  • (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 ([link] and ([link] with [\varphi=\pi/2]. The ordinary walk-off is nil and the extraordinary one is given by ([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 ([link] and ([link] with [\varphi = 0]. The walk-off is relative to the extraordinary wave and is calculated from ([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 ([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 ([link], that is[e_x^o=0\quad e_y^o=-1\quad e_z^o=0.\eqno(]The extraordinary walk-off angle is given by ([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. 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 ([link] and ([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 ([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(][\varepsilon = +1] or [-1] for a positive or a negative optic sign, respectively. Equations of propagation of three-wave and four-wave interactions

| top | pdf | Coupled electric fields amplitudes equations

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The nonlinear crystals considered here are homogeneous, lossless, non-conducting, without optical activity, non-magnetic and are optically anisotropic. The nonlinear regime allows interactions between γ waves with different circular frequencies [\omega_i,i=1,\ldots,\gamma]. The Fourier component of the polarization vector at ωi is [{\bf P}(\omega_i)=\varepsilon_0\chi^{(1)}(\omega_i){\bf E}(\omega_i)+{\bf P}^{NL}(\omega_i)], where [{\bf P}^{NL}(\omega_i)] is the nonlinear polarization corresponding to the orders of the power series greater than 1 defined in Section 1.7.2[link].

Thus the propagation equation of each interacting wave ωi is (Bloembergen, 1965[link])[\nabla x\nabla x{\bf E}(\omega_i) = (\omega_i^2/c^2)\varepsilon (\omega _i){\bf E}(\omega_i) + \omega_i ^2\mu_0 {\bf P}^{NL}(\omega_i).\eqno(]The γ propagation equations are coupled by [{\bf P}^{NL}(\omega_i)]:

  • (1) for a three-wave interaction, γ = 3,[\eqalign{{\bf P}^{NL}(\omega_1)&={\bf P}^{(2)}(\omega_1)=\varepsilon_0\chi^{(2)}(\omega_1=\omega_3-\omega_2)\cdot{\bf E}(\omega_3)\otimes{\bf E}^*(\omega_2),\cr {\bf P}^{NL}(\omega_2)&={\bf P}^{(2)}(\omega_2)=\varepsilon_0\chi^{(2)}(\omega_2=\omega_3-\omega_1)\cdot{\bf E}(\omega_3)\otimes{\bf E}^*(\omega_1),\cr {\bf P}^{NL}(\omega_3)&={\bf P}^{(2)}(\omega_3)=\varepsilon_0\chi^{(2)}(\omega_3=\omega_1+\omega_2)\cdot{\bf E}(\omega_1)\otimes{\bf E}^*(\omega_2)\semi}]

  • (2) for a four-wave interaction[\eqalign{{\bf P}^{NL}(\omega_1)={\bf P}^{(3)}(\omega_1)&=\varepsilon_0\chi^{(3)}(\omega_1=\omega_4-\omega_2-\omega_3)\cr&\quad\cdot{\bf E}(\omega_4)\otimes{\bf E}^*(\omega_2)\otimes{\bf E}^*(\omega_3),\cr {\bf P}^{NL}(\omega_2)={\bf P}^{(3)}(\omega_2)&=\varepsilon_0\chi^{(3)}(\omega_2=\omega_4-\omega_1-\omega_3)\cr&\quad\cdot{\bf E}(\omega_4)\otimes{\bf E}^*(\omega_1)\otimes{\bf E}^*(\omega_3),\cr}][\eqalign{{\bf P}^{NL}(\omega_3)={\bf P}^{(3)}(\omega_3)&=\varepsilon_0\chi^{(3)}(\omega_3=\omega_4-\omega_1-\omega_2)\cr&\quad\cdot{\bf E}(\omega_4)\otimes{\bf E}^*(\omega_1)\otimes{\bf E}^*(\omega_2)\cr {\bf P}^{NL}(\omega_4)={\bf P}^{(3)}(\omega_4)&=\varepsilon_0\chi^{(3)}(\omega_4=\omega_1+\omega_2+\omega_3)\cr&\quad\cdot{\bf E}(\omega_1)\otimes{\bf E}(\omega_2)\otimes{\bf E}(\omega_3).}]

The complex conjugates [{\bf E}^*(\omega_1)] come from the relation [{\bf E}^*(\omega_i)={\bf E}(-\omega_i)].

We consider the plane wave, ([link], as a solution of ([link], and we assume that all the interacting waves propagate in the same direction Z. Each linearly polarized plane wave corresponds to an eigen mode E+ or E defined above. For the usual case of beams with a finite transversal profile and when Z is along a direction where the double-refraction angles can be nonzero, i.e. out of the principal axes of the index surface, it is necessary to specify a frame for each interacting wave in order to calculate the corresponding powers as a function of Z: the coordinates linked to the wave at ωi are written ([X_i,Y_i,Z]), which can be relative to the mode (+) or (−). The systems are then linked by the double-refraction angles ρ: according to Fig.[link], we have [X_j^+=X_i^+ +Z\tan[\rho^+(\omega_j)-\rho^+(\omega_i)], Y_j^+=Y_i^+] for two waves (+) with [\rho^+(\omega_j)>\rho^+(\omega_i)], and [X_j^- =] [X_i^-, Y_j^- =] [Y_i^-] [ +] [Z\tan[\rho^-(\omega_j)-\rho^-(\omega_i)]] for two waves (−) with [\rho^-(\omega_j)>\rho^-(\omega_i)].

The presence of [{\bf P}^{NL}(\omega_i)] in equations ([link] leads to a variation of the γ amplitudes Ei) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λi, which is usually true. Thus we can state[\eqalignno{{1 \over k(\omega_i)}\left| {\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}\right| &\ll \left| E(\omega _i, X_i, Y_i, Z)\right|\hbox{ or}&\cr \left| {\partial ^2 E(\omega _i, X_i, Y_i, Z)\over \partial Z^2 }\right| &\ll k(\omega_i)\left| {\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}\right|.&\cr&&(}]This is called the slowly varying envelope approximation.

Stating ([link], the wave equation ([link] for a forward propagation of a plane wave leads to[\eqalignno{{\partial E(\omega _i, X_i, Y_i, Z)\over \partial Z}&= j\mu_0 {\omega _i^2 \over 2k(\omega_i)\cos^2 \rho (\omega_i)}{\bf e}(\omega_i)\cdot{\bf P}^{NL}(\omega _i, X_i, Y_i, Z)&\cr&\quad\times\exp[- jk(\omega_i)Z].&(}]We choose the optical frame ([x,y,z]) for the calculation of all the scalar products [{\bf e}(\omega_i)\cdot{\bf P}^{NL}(\omega _i)], the electric susceptibility tensors being known in this frame.

For a three-wave interaction, ([link] leads to[\eqalignno{{\partial E_1 (X_1, Y_1, Z)\over \partial Z} &= j\kappa_1 \left[{\bf e}_1 \cdot\varepsilon_0\chi^{(2)}(\omega_1 = \omega_3 -\omega_2)\cdot{\bf e}_3 \otimes {\bf e}_2 \right] &\cr&\quad\times E_3 (X_3, Y_3, Z)E_2^* (X_2, Y_2, Z)\exp(j\Delta kZ)&\cr{\partial E_2 (X_2, Y_2, Z)\over \partial Z}&= j\kappa_2 \left [{\bf e}_2 \cdot \varepsilon_0\chi ^{(2)}(\omega_2 = \omega _3 - \omega_1)\cdot{\bf e}_3 \otimes {\bf e}_1 \right]&\cr&\quad\times E_3 (X_3, Y_3, Z)E_1^* (X_1, Y_1, Z)\exp(j\Delta kZ)&\cr {\partial E _3 (X_3, Y_3, Z)\over \partial Z}&= j\kappa_3 \left [{\bf e}_3 \cdot \varepsilon _0 \chi ^{(2)}(\omega _3 = \omega _1 + \omega _2)\cdot{\bf e}_1 \otimes{\bf e}_2 \right] &\cr&\quad\times E _1 (X_1, Y_1, Z) E _2 (X_2, Y_2, Z)\exp(- j\Delta kZ), &\cr&&(}]with [{\bf e}_i =] [{\bf e}(\omega_i)], [E_i(X_i,Y_i,Z_i) = ] [E(\omega_i,X_i,Y_i,Z)], [\kappa_i =] [ (\mu_o\omega_i^2)/[2k(\omega _i)\cos^2 \rho (\omega_i)]] and [\Delta k=k(\omega_3)-[k(\omega_1)+k(\omega_2)]], called the phase mismatch. We take by convention [\omega_1\,\lt\,\omega_2\,\,(\lt\,\omega_3)].

If ABDP relations, defined in Section[link], are verified, then the three tensorial contractions in equations ([link] are equal to the same quantity, which we write [\varepsilon_0\chi^{(2)}_{\rm eff}], where [\chi^{(2)}_{\rm eff}] is called the effective coefficient:[\eqalignno{\chi _{\rm eff}^{(2)} &= {\bf e}_1 \cdot\chi^{(2)}(\omega_1 = \omega _3 - \omega _2)\cdot{\bf e}_3 \otimes {\bf e}_2 &\cr& = {\bf e}_2 \cdot\chi ^{(2)}(\omega_2 = \omega _3 - \omega _1)\cdot{\bf e}_3 \otimes {\bf e}_1 &\cr& = {\bf e}_3\cdot\chi^{(2)}(\omega _3 = \omega _1 + \omega _2)\cdot{\bf e}_1\otimes {\bf e}_2. &(}]The same considerations lead to the same kind of equations for a four-wave interaction:[\eqalignno{{\partial E_1 (X_1, Y_1, Z)\over \partial Z}&= j\kappa _1 \varepsilon _0 \chi _{\rm eff}^{(3)}E_4 (X_4, Y_4, Z)E_2^* (X_2, Y_2, Z)&\cr&\quad\times E_3^* (X_3, Y_3, Z)\exp(j\Delta kZ)&\cr{\partial E_2 (X_2, Y_2, Z)\over \partial Z}&= j\kappa _2 \varepsilon _0 \chi _{\rm eff}^{(3)}E_4 (X_4, Y_4, Z)E_1^* (X_1, Y_1, Z)&\cr &\quad\times E_3^* (X_3, Y_3, Z)\exp(j\Delta kZ)&\cr{\partial E_3 (X_3, Y_3, Z)\over \partial Z}&= j\kappa _3 \varepsilon _0 \chi _{eff}^{(3)}E_4 (X_4, Y_4, Z)E_1^* (X_1, Y_1, Z)&\cr&\quad\times E_2^* (X_2, Y_2, Z)\exp(j\Delta kZ)&\cr{{\partial E_4 (X_4, Y_4, Z)}\over {\partial Z}}&= j\kappa _4 \varepsilon _0 \chi _{eff}^{(3)}E_1 (X_1, Y_1, Z)E_2 (X_2, Y_2, Z)&\cr&\quad\times E_3 (X_3, Y_3, Z)\exp(- j\Delta kZ).&\cr&&(}]The conventions of notation are the same as previously and the phase mismatch is [\Delta k=k(\omega_4)-[k(\omega_1)+k(\omega_2)+k(\omega_3)]]. The effective coefficient is[\eqalignno{ \chi _{\rm eff}^{(3)} &= {\bf e}_1 \cdot\chi ^{(3)}(\omega _1 = \omega _4 - \omega _2 - \omega _3)\cdot{\bf e}_4 \otimes{\bf e}_2 \otimes {\bf e}_3 &\cr&= {\bf e}_2 \cdot\chi ^{(3)}(\omega _2 = \omega _4 - \omega _1 - \omega _3)\cdot{\bf e}_4 \otimes {\bf e}_1 \otimes {\bf e}_3 &\cr &= {\bf e}_3 \cdot\chi ^{(3)}(\omega _3 = \omega _4 - \omega _1 - \omega _2)\cdot{\bf e}_4 \otimes {\bf e}_1 \otimes {\bf e}_2 &\cr&= {\bf e}_4 \cdot\chi ^{(3)}(\omega _4 = \omega _1 + \omega _2 + \omega _3)\cdot{\bf e}_1 \otimes {\bf e}_2 \otimes {\bf e}_3. & \cr&&(}]Expressions ( for [\chi_{\rm eff}^{(2)}] and ( for [\chi_{\rm eff}^{(3)}] can be condensed by introducing adequate third- and fourth-rank tensors to be contracted, respectively, with [\chi^{(2)}] and [\chi^{(3)}]. For example, [\chi_{\rm eff}^{(2)}=\chi^{(2)}\cdot e_3\otimes e_1\otimes e_2] or [\chi_{\rm eff}^{(3)} =] [\chi^{(3)}\cdot e_4\otimes e_1\otimes e_2\otimes e_3], and similar expressions. By substituting ([link] in ([link], we obtain the derivatives of Manley–Rowe relations ([link] [\partial N(\omega_3,Z)/\partial Z =] [-\partial N(\omega_k,Z)/\partial Z] [(k =] [1,2)] for a three-wave mixing, where [N(\omega_i,Z)] is the Z photon flow. Identically with ([link], we have [\partial N(\omega_4,Z)/\partial Z =] [-\partial N(\omega_k,Z)/\partial Z] [(k = 1,2,3)] for a four-wave mixing.

In the general case, the nonlinear polarization wave and the generated wave travel at different phase velocities, [(\omega_1+\omega_2)/[k(\omega_1)+k(\omega_2)]] and [\omega_3/[k(\omega_3)]], respectively, because of the frequency dispersion of the refractive indices in the crystal. Then the work per unit time Wi), given in ([link], which is done on the generated wave Ei, Z) by the nonlinear polarization PNLi, Z), alternates in sign for each phase shift of π during the Z-propagation, which leads to a reversal of the energy flow (Bloembergen, 1965[link]). The length leading to the phase shift of π is called the coherence length, [L_c=\pi/\Delta k], where Δk is the phase mismatch given by ([link] or ([link]. Phase matching

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The transfer of energy between the waves is maximum for [\Delta k = 0], which defines phase matching: the energy flow does not alternate in sign and the generated field grows continuously. Note that a condition relative to the phases Φ(ωi, Z) also exists: the work of PNLi, Z) on Ei, Z) is maximum if these two waves are π/2 out of phase, that is to say if [\Delta kZ+\Delta\Phi(Z)=\pi/2], where [\Delta\Phi(Z) = \Phi(\omega_3,Z) - [\Phi(\omega_1,Z) + \Phi(\omega_2,Z)]]; thus in the case of phase matching, the phase relation is [\Phi(\omega_3,Z)] [= ] [\Phi(\omega_1,Z) + \Phi(\omega_2,Z) + \pi/2] (Armstrong et al., 1962[link]). The complete initial phase matching is necessarily achieved when at least one wave among all the interacting waves is not incident but is generated inside the nonlinear crystal: in this case, its initial phase is locked on the good one. Phase matching is usually realized by the matching of the refractive indices using birefringence of anisotropic media as it is studied here. From the point of view of the quantum theory of light, the phase matching of the waves corresponds to the total photon-momentum conservation i.e.[\textstyle\sum\limits_{i = 1}^{\gamma - 1}{\hbar k(}\omega _i) = \hbar k(\omega _\gamma)\eqno(]with [\gamma = 3] for a three-photon interaction and [\gamma = 4] for a four-photon interaction.

According to ([link], the phase-matching condition ([link] is expressed as a function of the refractive indices in the direction of propagation considered ([\theta,\varphi]); for an interaction where the γ wavevectors are collinear, it is written[\textstyle\sum\limits_{i = 1}^{\gamma - 1}\omega _i n(\omega _i, \theta, \varphi) = \omega _\gamma n(\omega _\gamma, \theta, \varphi)\eqno(]with[\textstyle\sum\limits_{i = 1}^{\gamma - 1}{\omega _i } = \omega _\gamma.\eqno(]( is the relation of the energy conservation.

The efficiency of a nonlinear crystal directly depends on the existence of phase-matching directions. We shall see by considering in detail the effective coefficient that phase matching is a necessary but insufficient condition for the best expression of the nonlinear optical properties.

In an hypothetical non-dispersive medium [[\partial n(\omega)/\partial \omega=0]], ([link] is always verified for each of the eigen refractive indices n+ or n; then any direction of propagation is a phase-matching direction. In a dispersive medium, phase matching can be achieved only if the direction of propagation has a birefringence which compensates the dispersion. Except for a propagation along the optic axis, there are two possible values, n+ and n given by ([link], for each of the three or four refractive indices involved in the phase-matching relations, that is to say 23 or 24 possible combinations of refractive indices for a three-wave or a four-wave process, respectively.

For a three-wave process, only three combinations among the 23 are compatible with the dispersion in frequency ([link] and with the momentum and energy conservations ([link] and ([link]. Thus the phase matching of a three-wave interaction is allowed for three configurations of polarization given in Table[link].

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Correspondence between the phase-matching relations, the configurations of polarization and the types according to the sum- and difference-frequency generation processes SFG ([\omega_3=\omega_1+\omega_2]), DFG ([\omega_1=\omega_3-\omega_2]) and DFG ([\omega_2=\omega_3-\omega_1])

[{\bf e}^{\pm}] are the unit electric field vectors relative to the refractive indices [n^{\pm}] in the phase-matching direction (Boulanger & Marnier, 1991[link]).

Phase-matching relationsConfigurations of polarizationTypes of interaction
ω3ω1ω2SFG (ω3)DFG (ω1)DFG (ω2)
[\omega_3n_3^-=\omega_1n_1^+=\omega_2n_2^+] e e+ e+ I II III
[\omega_3n_3^-=\omega_1n_1^-=\omega_2n_2^+] e e e+ II III I
[\omega_3n_3^-=\omega_1n_1^+=\omega_2n_2^-] e e+ e III I II

The designation of the type of phase matching, I, II or III, is defined according to the polarization states at the frequencies which are added or subtracted. Type I characterizes interactions for which these two waves are identically polarized; the two corresponding polarizations are different for types II and III. Note that each phase-matching relation corresponds to one sum-frequency generation SFG ([\omega_3=\omega_1+\omega_2]) and two difference-frequency generation processes, DFG ([\omega_1=\omega_3-\omega_2]) and DFG ([\omega_2=\omega_3-\omega_1]). Types II and III are equivalent for SHG because [\omega_1=\omega_2].

For a four-wave process, only seven combinations of refractive indices allow phase matching in the case of normal dispersion; they are given in Table[link] with the corresponding configurations of polarization and types of SFG and DFG.

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Correspondence between the phase-matching relations, the configurations of polarization and the types according to SFG ([\omega_4=\omega_1+\omega_2+\omega_3]), DFG ([\omega_1=\omega_4-\omega_2-\omega_3]), DFG ([\omega_2=\omega_4-\omega_1-\omega_3]) and DFG ([\omega_3=\omega_4-\omega_1-\omega_2]) (Boulanger et al., 1993[link])

Phase-matching relationsConfigurations of polarizationTypes of interaction
ω4ω1ω2ω3SFG (ω4)DFG (ω1)DFG (ω2)DFG (ω3)
[\omega_4 n_4^-=\omega_1 n_1^+ +\omega_2 n_2^+ +\omega_3 n_3^+] e e+ e+ e+ I II III IV
[\omega_4n_4^-=\omega_1n_1^-+\omega_2n_2^-+\omega_3n_3^+] e e e e+ II III IV I
[\omega_4n_4^-=\omega_1n_1^-+\omega_2n_2^++\omega_3n_3^-] e e e+ e III IV I II
[\omega_4n_4^-=\omega_1n_1^++\omega_2n_2^-+\omega_3n_3^-] e e+ e e IV I II IV
[\omega_4n_4^-=\omega_1n_1^-+\omega_2n_2^++\omega_3n_3^+] e e e+ e+ V4 V1 V2 V3
[\omega_4n_4^-=\omega_1n_1^++\omega_2n_2^-+\omega_3n_3^+] e e+ e e+ VI4 VI1 VI2 VI3
[\omega_4n_4^-=\omega_1n_1^++\omega_2n_2^++\omega_3n_3^-] e e+ e+ e VII4 VII1 VII2 VII3

The convention of designation of the types is the same as for three-wave interactions for the situations where one polarization state is different from the three others, leading to the types I,  II, III and IV. The criterion corresponding to type I cannot be applied to the three other phase-matching relations where two waves have the same polarization state, different from the two others. In this case, it is convenient to refer to each phase-matching relation by the same roman numeral, but with a different index: Vi, VIi and VIIi, with the index [i = 1,2,3,4] corresponding to the index of the frequency generated by the SFG or DFG. For THG ([\omega_1=\omega_2=\omega_3]), types II, III and IV are equivalent, and so are types V4, VI4 and VII4.

The index surface allows the geometrical determination of the phase-matching directions, which depend on the relative ellipticity of the internal (−) and external (+) sheets divided by the corresponding wavelengths: according to Tables and[link] the directions are given by the intersection of the internal sheet of the lowest wavelength [[n^ - (\lambda _\gamma, \theta, \varphi)] /(\lambda _\gamma)] with a linear combination of the internal and external sheets at the other frequencies [\textstyle\sum_{i = 1}^{\gamma - 1}[n^ \pm (\lambda _i, \theta, \varphi)]/(\lambda _i)]. The existence and loci of these intersections depend on specific inequalities between the principal refractive indices at the different wavelengths. Note that independently of phase-matching considerations, normal dispersion and energy conservation impose [\textstyle\sum_{i = 1}^{\gamma - 1}[n_a (\lambda _i)]/(\lambda _i)] [\lt\, [n_a (\lambda _\gamma)]/(\lambda _\gamma)] with [a = x, y, z]. Cubic crystals

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There is no possibility of collinear phase matching in a dispersive cubic crystal because of the absence of birefringence. In a hypothetical non-dispersive anaxial crystal, the 23 three-wave and 24 four-wave phase-matching configurations would be allowed in any direction of propagation. Uniaxial crystals

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The configurations of polarization in terms of ordinary and extraordinary waves depend on the optic sign of the phase-matching direction with the convention given in Section[link]: Tables[link] and[link] must be read by substituting (+, −) by (e, o) for a positive crystal and by (o, e) for a negative one.

Because of the symmetry of the index surface, all the phase-matching directions for a given type describe a cone with the optic axis as a revolution axis. Note that the previous comment on the anaxial class is valid for a propagation along the optic axis ([n_o=n_e]).

Fig.[link] shows the example of negative uniaxial crystals ([n_o>n_e]) like β-BaB2O4 (BBO) and KH2PO4 (KDP).


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Index surface sections in a plane containing the optic axis z of a negative uniaxial crystal allowing collinear type-I phase matching for SFG ([\omega_3=\omega_1+\omega_2]), [\gamma = 3], or for SFG ([\omega_4=\omega_1+\omega_2+\omega_3]), [\gamma = 4]. [{\bf u}^{\rm I}_{\rm PM}] is the corresponding phase-matching direction.

From Fig.[link], it clearly appears that the intersection of the sheets is possible only if [(n_{e_\gamma })/(\lambda _\gamma)\,\lt\, \textstyle\sum_{i = 1}^{\gamma - 1}(n_{o_i })/(\lambda _i) ] [[\lt\,(n_{o_\gamma })/(\lambda _\gamma)]] with [\gamma = 3] for a three-wave process and [\gamma = 4] for a four-wave one. The same considerations can be made for the positive sign and for all the other types of phase matching. There are different situations of inequalities allowing zero, one or several types: Table[link] gives the five possible situations for the three-wave interactions and Table[link] the 19 situations for the four-wave processes.

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Classes of refractive-index inequalities for collinear phase matching of three-wave interactions in positive and negative uniaxial crystals

Types I, II and III refer to SFG; the types of the corresponding DFG are given in Table[link] (Fève et al., 1993[link]).

Positive sign ([n_e> n_o])Negative sign ([n_o> n_e])Types of SFG
[{n_{o3}\over \lambda_3}\,\lt\,{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2};{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}] [{n_{o1}\over\lambda_1}+{n_{e2}\over\lambda_2},{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{e3}\over\lambda_3}] I, II, III
[{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{o3}\over\lambda_3}\,\lt\,{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}] [{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}\,\lt\,{n_{e3}\over\lambda_3}\,\lt\,{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}] I, II
[{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}\,\lt\,{n_{o3}\over \lambda_3}\,\lt\,{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}] [{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{e3}\over\lambda_3}\,\lt\,{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}] I, III
[{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}, {n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\, {n_{o3}\over \lambda_3}\,\lt\, {n_{e_1}\over\lambda_1}+{n_{e2}\over\lambda_2}] [{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}, {n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\, {n_{e3}\over \lambda_3}\,\lt\, {n_{o_1}\over\lambda_1}+{n_{o2}\over\lambda_2}] I
[{n_{e_1}\over\lambda_1}+{n_{e2}\over\lambda_2}\,\lt\,{n_{o3}\over \lambda_3}] [{n_{o_1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{e3}\over \lambda_3}] None

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Classes of refractive-index inequalities for collinear phase matching of four-wave interactions in positive ([n_a=n_e, n_b=n_o]) and negative ([n_a=n_o, n_b=n_e]) uniaxial crystals with [(n_{b4}/\lambda_4)\,\lt\,(n_{a1}/\lambda_1)+(n_{a2}/\lambda_2)+(n_{a3}/\lambda_3)]

If this inequality is not verified, no phase matching is allowed. The types of phase matching refer to SFG; the types of the corresponding DFG are given in Table[link] (Fève, 1994[link]).

Positive sign ([n_e> n_o])Negative sign ([n_o> n_e])Types of SFG
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}]   I
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}, {n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}]   I, V4
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}, {n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}]   I, VI4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}} \,\lt\, {n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}]   I, VII4
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] [{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}} \,\lt\, {n_{b4}\over\lambda_{4}}] I, V4, VI4
[{n_{b4}\over\lambda_{4}} \,\lt\, {n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, V4, VI4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] [{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}} \,\lt\, {n_{b4}\over\lambda_{4}}] I, V4, VII4
[{n_{b4}\over\lambda_{4}} \,\lt\, {n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, III, V4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] [{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}} \,\lt\, {n_{b4}\over\lambda_{4}}] I, VI4, VII4
  [{n_{b4}\over\lambda_{4}} \,\lt\, {n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, IV, VI4, VII4
[{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}}+{n_{a2}\over\lambda_{2}}+{n_{b3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}}+{n_{b2}\over\lambda_{2}}+{n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] [{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}] I, V4, VI4, VII4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, V4, VI4, VII4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, III, V4, VI4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, IV, V4, VI4, VII4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, III, V4, VI4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, IV, V4, VI4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, III, IV, V4, VI4, VII4
[{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] All Biaxial crystals

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The situation of biaxial crystals is more complicated, because the two sheets that must intersect are both elliptical in several cases. For a given interaction, all the phase-matching directions generate a complicated cone which joins two directions in the principal planes; the possible loci a, b, c, d are shown on the stereographic projection given in Fig.[link].


Figure | top | pdf |

Stereographic projection on the optical frame of the possible loci of phase-matching directions in the principal planes of a biaxial crystal.

The basic inequalities of normal dispersion ([link] forbid collinear phase matching for all the directions of propagation located between two optic axes at the two frequencies concerned.

Tables[link] and[link] give, respectively, the inequalities that determine collinear phase matching in the principal planes for the three types of three-wave SFG and for the seven types of four-wave SFG.

Table| top | pdf |
Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for three-wave SFG

a, b, c, d refer to the areas given in Fig.[link]. The types corresponding to the different DFGs are given in Table[link] (Fève et al., 1993[link]).

Types of SFGPhase-matching loci in the principal planesInequalities determining three-wave collinear phase matching in biaxial crystals
Positive biaxial crystalNegative biaxial crystal
[n_x(\omega_i) \,\lt\, n_y(\omega_i)\,\lt\, n_z(\omega_i)][n_x(\omega_i)> n_y(\omega_i)> n_z(\omega_i)]
Type I a [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{z3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
b [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{x3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
c [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
d [{n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
Type II a [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{z3}\over\lambda_{3}}] [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
b [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{x3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
c [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}};{n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}}] [{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
c* [{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}};{n_{y3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
d [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
d* [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}};{n_{z3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}]
Type III a [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{z3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
b [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}] [{n_{x3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
c [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
c* [{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}};{n_{y3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}] [{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
d [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}};{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
d* [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}] [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}};{n_{z3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
Conditions c, d are applied if [{n_{y1}\over\lambda_{1}} - {n_{x1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{x2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} - {n_{x3}\over\lambda_{3}}] [{n_{y1}\over\lambda_{1}} - {n_{z1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{z2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} - {n_{z3}\over\lambda_{3}}]
Conditions c*, d* are applied if [{n_{y3}\over\lambda_{3}} - {n_{x3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} - {n_{x1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{x2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}} - {n_{z3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} - {n_{z1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{z2}\over\lambda_{2}}]

Table| top | pdf |
Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for four-wave SFG

The types corresponding to the different DFGs are given in Table[link] (Boulanger et al., 1993[link]).

(a) SFG type I.

Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z4}\over\lambda_{4}}] [{n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}}]
b [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{x4}\over\lambda_{4}}]
c [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}}] [{n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}}]
d [{n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}}]

(b) SFG type II ([i=1,j=2,k=3]), SFG type III ([i=3,j=1,k=2]), SFG type IV ([i=2,j=3,k=1]).

Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}] [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
b [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}]
c [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}] [{n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
c* [{n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} ] [{n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
d [{n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}]
d* [{n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
SFG type II [ (i,j) = (1,2)]; SFG type III [(i,j) = (1,3)]; SFG type IV [(i,j) = (2,3)]
Conditions c, d are applied if [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{xi}\over\lambda_{i}} - {n_{xj}\over\lambda_{j}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}}] [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{zi}\over\lambda_{i}} - {n_{zj}\over\lambda_{j}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}}]
Conditions c*, d* are applied if [{n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{xi}\over\lambda_{i}} - {n_{xj}\over\lambda_{j}}] [{n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{zi}\over\lambda_{i}} - {n_{zj}\over\lambda_{j}}]

(c) SFG type V4 ([i=1,j=2,k=3]), SFG type VI4 ([i=2,j=3,k=1]), SFG type VII4 ([i=3,j=1,k=2]).

Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}] [{n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
b [{n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}]
c [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}] [{n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
c** [{n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
d [{n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}]
d** [{n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
SFG type V4, ([i = 1]); SFG type VI4 ([i = 2]) ; SFG type VII4 ([i = 3])
Conditions c′, d′ are applied if [{n_{yi}\over\lambda_{i}} - {n_{xi}\over\lambda_{i}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}} ] [{n_{yi}\over\lambda_{i}} - {n_{zi}\over\lambda_{i}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}}]
Conditions c**, d** are applied if [{n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} - {n_{xi}\over\lambda_{i}}] [{n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} - {n_{zi}\over\lambda_{i}}]

The inequalities in Table[link] show that a phase-matching cone which would join the directions a and d is not possible for any type of interaction, because the corresponding inequalities have an opposite sense. It is the same for a hypothetical cone joining b and c.

The existence of type-II or type-III SFG phase matching imposes the existence of type I, because the inequalities relative to type I are always satisfied whenever type II or type III exists. However, type I can exist even if type II or type III is not allowed. A type-I phase-matched SFG in area c forbids phase-matching directions in area b for type-II and type-III SFG. The exclusion is the same between d and a. The consideration of all the possible combinations of the inequalities of Table[link] leads to 84 possible classes of phase-matching cones for both positive and negative biaxial crystals (Fève et al., 1993[link]; Fève, 1994[link]). There are 14 classes for second harmonic generation (SHG) which correspond to the degenerated case ([\omega_1=\omega_2]) (Hobden, 1967[link]).

The coexistence of the different types of four-wave phase matching is limited as for the three-wave case: a cone joining a and d or b and c is impossible for type-I SFG. Type I in area d forbids the six other types in a. The same restriction exists between c and b. Types II, III, IV, V4, VI4 and VII4 cannot exist without type I; other restrictions concern the relations between types II, III, IV and types V4, VI4, VII4 (Fève, 1994[link]). The counting of the classes of four-wave phase-matching cones obtained from all the possible combinations of the inequalities of Table[link] is complex and it has not yet been done.

For reasons explained later, it can be interesting to consider a non-collinear interaction. In this case, the projection of the vectorial phase-matching relation ([link] on the wavevector [{\bf k}(\omega_\gamma,\theta_\gamma,\varphi_\gamma)] of highest frequency [\omega_\gamma] leads to[\textstyle\sum\limits_{i = 1}^{\gamma - 1}{\omega _i }n(\omega _i, \theta _i, \varphi _i)\cos \alpha _{i\gamma } = \omega _\gamma n(\omega _\gamma, \theta _\gamma, \varphi _\gamma),\eqno(]where [\alpha_{i\gamma}] is the angle between [{\bf k}(\omega_i,\theta_i,\varphi_i)] and [{\bf k}(\omega_\gamma,\theta_\gamma,\varphi_\gamma)], with [\gamma = 3] for a three-wave interaction and [\gamma = 4] for a four-wave interaction. The phase-matching angles ([\theta_\gamma,\varphi_\gamma]) can be expressed as a function of the different ([\theta_i,\varphi_i]) by the projection of ([link] on the three principal axes of the optical frame.

The configurations of polarization allowing non-collinear phase matching are the same as for collinear phase matching. Furthermore, non-collinear phase matching exists only if collinear phase matching is allowed; the converse is not true (Fève, 1994[link]). Note that collinear or non-collinear phase-matching conditions are rarely satisfied over the entire transparency range of the crystal. Quasi phase matching

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When index matching is not allowed, it is possible to increase the energy of the generated wave continuously during the propagation by introducing a periodic change in the sign of the nonlinear electric susceptibility, which leads to a periodic reset of π between the waves (Armstrong et al., 1962[link]). This method is called quasi phase matching (QPM). The transfer of energy between the nonlinear polarization and the generated electric field never alternates if the reset is made at each coherence length. In this case and for a three-wave SFG, the nonlinear polarization sequence is the following:

  • (i) from 0 to Lc, [{\bf P}^{NL}(\omega_3) =] [\varepsilon_0\chi^{(2)}(\omega_3){\bf e}_1{\bf e}_2 E_1E_2\exp\{i[k(\omega_1)] [+] [k(\omega_2)]Z\}];

  • (ii) from Lc to 2Lc, [{\bf P}^{NL}(\omega_3) =] [-\varepsilon_0\chi^{(2)}(\omega_3){\bf e}_1{\bf e}_2 E_1E_2\exp\{i[k(\omega_1)] [+] [k(\omega_2)]Z\}], which is equivalent to [{\bf P}^{NL}(\omega_3) =] [\varepsilon_0\chi^{(2)}(\omega_3){\bf e}_1{\bf e}_2 E_1E_2\exp(i\{[k(\omega_1)] [+] [k(\omega_2)]Z-\pi\})].

QPM devices are a recent development and are increasingly being considered for applications (Fejer et al., 1992[link]). The nonlinear medium can be formed by the bonding of thin wafers alternately rotated by π; this has been done for GaAs (Gordon et al., 1993[link]). For ferroelectric crystals, it is possible to form periodic reversing of the spontaneous polarization in the same sample by proton- or ion-exchange techniques, or by applying an electric field, which leads to periodically poled (pp) materials like ppLiNbO3 or ppKTiOPO4 (Myers et al., 1995[link]; Karlsson & Laurell, 1997[link]; Rosenman et al., 1998[link]).

Quasi phase matching offers three main advantages when compared with phase matching: it may be used for any configuration of polarization of the interacting waves, which allows us to use the largest coefficient of the [\chi^{(2)}] tensor, as explained in the following section; QPM can be achieved over the entire transparency range of the crystal, since the periodicity can be adjusted; and, finally, double refraction and its harmful effect on the nonlinear efficiency can be avoided because QPM can be realized in the principal plane of a uniaxial crystal or in the principal axes of biaxial crystals. Nevertheless, there are limitations due to the difficulty in fabricating the corresponding materials: diffusion-bonded GaAs has strong reflection losses and periodic patterns of ppKTP or ppLN can only be written over a thickness that does not exceed 3 mm, which limits the input energy. Effective coefficient and field tensor

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The refractive indices and their dispersion in frequency determine the existence and loci of the phase-matching directions, and so impose the direction of the unit electric field vectors of the interacting waves according to ([link]. The effective coefficient, given by ([link] and ([link], depends in part on the linear optical properties via the field tensor, which is the tensor product of the interacting unit electric field vectors (Boulanger, 1989[link]; Boulanger & Marnier, 1991[link]; Boulanger et al., 1993[link]; Zyss, 1993[link]). Indeed, the effective coefficient is the contraction between the field tensor and the electric susceptibility tensor of corresponding order:

  • (i) For three-wave mixing,[\eqalignno{ \chi _{\rm eff}^{(2)}(\omega _a, \omega _b, \omega _c, \theta, \varphi) &= \textstyle\sum\limits_{ijk}{\chi _{ijk}}(\omega _a)F_{ijk}(\omega _a, \omega _b, \omega _c, \theta, \varphi) &\cr&= \chi ^{(2)}(\omega _a)\cdot F^{(2)}(\omega _a, \omega _b, \omega _c, \theta, \varphi), &\cr&&(}]with [F^{(2)}(\omega _a, \omega _b, \omega _c, \theta, \varphi) = {\bf e}(\omega _a, \theta, \varphi) \otimes {\bf e}(\omega _b, \theta, \varphi) \otimes {\bf e}(\omega _c, \theta, \varphi), \eqno(]where [\omega_a, \omega_b, \omega_c] correspond to [\omega_3, \omega_1, \omega_2] for SFG ([\omega_3 =] [ \omega_1+ \omega_2]); to [\omega_1, \omega_3, \omega_2] for DFG ([\omega_1= \omega_3- \omega_2]); and to [\omega_2, \omega_3, \omega_1] for DFG ([\omega_2= \omega_3- \omega_1]).

  • (ii) For four-wave mixing,[\eqalignno{ \chi _{\rm eff}^{(3)}(\omega _a, \omega _b, \omega _c, \omega _d, \theta, \varphi) &= \textstyle\sum\limits_{ijkl}{\chi _{ijkl}}(\omega _a)F_{ijkl}(\omega _a, \omega _b, \omega _c, \omega _d, \theta, \varphi) & \cr &= \chi ^{(3)}(\omega _a)\cdot F^{(3)}(\omega _a, \omega _b, \omega _c, \omega _d, \theta, \varphi), &\cr&&(}]with[\displaylines{F^{(3)}(\omega _a, \omega _b, \omega _c, \omega _d \theta, \varphi) \hfill\cr\quad= {\bf e}(\omega _a, \theta, \varphi) \otimes {\bf e}(\omega _b, \theta, \varphi) \otimes {\bf e}(\omega _c, \theta, \varphi) \otimes {\bf e}(\omega _d, \theta, \varphi),\hfill\cr\hfill(}]where [\omega_a, \omega_b, \omega_c, \omega_d] correspond to [\omega_4, \omega_1, \omega_2,\omega_3] for SFG ([\omega_4 =] [ \omega_1+ \omega_2 + \omega_3]); to [\omega_1, \omega_4, \omega_2,\omega_3] for DFG ([\omega_1 =] [ \omega_4 - \omega_2 - \omega_3]); to [\omega_2, \omega_4, \omega_1, \omega_3] for DFG ([\omega_2 =] [ \omega_4 - \omega_1-\omega_3]); and to [\omega_3,] [ \omega_4, ] [\omega_1,] [\omega_2] for DFG ([\omega_3 =] [ \omega_4 - \omega_1 - \omega_2]).

Each [{\bf e}(\omega_i,\theta,\varphi)] corresponds to a given eigen electric field vector.

The components of the field tensor are trigonometric functions of the direction of propagation.

Particular relations exist between field-tensor components of SFG and DFG which are valid for any direction of propagation. Indeed, from ([link] and ([link], it is obvious that the field-tensor components remain unchanged by concomitant permutations of the electric field vectors at the different frequencies and the corresponding Cartesian indices (Boulanger & Marnier, 1991[link]; Boulanger et al., 1993[link]):[\eqalignno{F_{ijk}^{{\bf e}_3{\bf e}_1{\bf e}_2}(\omega_3=\omega_1+\omega_2) &=F_{jik}^{{\bf e}_1{\bf e}_3{\bf e}_2}(\omega_1=\omega_3-\omega_2)&\cr &=F_{kij}^{{\bf e}_2{\bf e}_3{\bf e}_1}(\omega_2=\omega_3-\omega_1)&(}]and[\eqalignno{&F_{ijkl}^{{\bf e}_4{\bf e}_1{\bf e}_2{\bf e}_3}(\omega_4=\omega_1+\omega_2+\omega_3)&\cr&=F_{jikl}^{{\bf e}_1{\bf e}_4{\bf e}_2{\bf e}_3}(\omega_1=\omega_4-\omega_2-\omega_3)&\cr&=F_{kijl}^{{\bf e}_2{\bf e}_4{\bf e}_1{\bf e}_3}(\omega_2=\omega_4-\omega_1-\omega_3)&\cr&=F_{lijk}^{{\bf e}_3{\bf e}_4{\bf e}_1{\bf e}_2}(\omega_3=\omega_4-\omega_1-\omega_2),&\cr&&(}]where ei is the unit electric field vector at ωi.

For a given interaction, the symmetry of the field tensor is governed by the vectorial properties of the electric fields, detailed in Section[link]. This symmetry is then characteristic of both the optical class and the direction of propagation. These properties lead to four kinds of relations between the field-tensor components described later (Boulanger & Marnier, 1991[link]; Boulanger et al., 1993[link]). Because of their interest for phase matching, we consider only the uniaxial and biaxial classes.

(a) The number of zero components varies with the direction of propagation according to the existence of nil electric field vector components. The only case where all the components are nonzero concerns any direction of propagation out of the principal planes in biaxial crystals.

(b) The orthogonality relation ([link] between any ordinary and extraordinary waves propagating in the same direction leads to specific relations independent of the direction of propagation. For example, the field tensor of an (eooo) configuration of polarization (one extraordinary wave relative to the first Cartesian index and three ordinary waves relative to the three other indices) verifies [F_{xxij} + F_{yyij}] [(+ \,\,F_{zzij} = 0)=] [ F_{xixj} + F_{yiyj}] [ (+\,\, F_{zizj} = 0)=] [F_{xijx} + F_{yijy}] [ (+ \,\,F_{zijz} = 0)=] [], with i and j equal to x or y; the combination of these three relations leads to [F_{xxxx}=] [-F_{yyxx}=] [ -F_{yxyx}=] [-F_{yxxy}], [F_{yyyy}=] [-F_{xxyy}=] [-F_{xyxy}=] [-F_{xyyx}] and [F_{yxyy}=] [F_{yyxy}=] [F_{yyyx}=] [ -F_{xyxx}=] [F_{xxyx} =] [-F_{xxxy}]. In a biaxial crystal, this kind of relation does not exist out of the principal planes.

(c) The fact that the direction of the ordinary electric field vectors in uniaxial crystals does not depend on the frequency, ([link], leads to symmetry in the Cartesian indices relative to the ordinary waves. These relations can be redundant in comparison with certain orthogonality relations and are valid for any direction of propagation in uniaxial crystals. It is also the case for biaxial crystals, but only in the principal planes xz and yz. In the xy plane of biaxial crystals, the ordinary wave, ([link], has a walk-off angle which depends on the frequency, and the extraordinary wave, ([link], has no walk-off angle: then the field tensor is symmetric in the Cartesian indices relative to the extraordinary waves. The walk-off angles of ordinary and extraordinary waves are nil along the principal axes of the index surface of biaxial and uniaxial crystals and so everywhere in the xy plane of uniaxial crystals. Thus, any field tensor associated with these directions of propagation is symmetric in the Cartesian indices relative to both the ordinary and extraordinary waves.

(d) Equalities between frequencies can create new symmetries: the field tensors of the uniaxial class for any direction of propagation and of the biaxial class in only the principal planes xz and yz become symmetric in the Cartesian indices relative to the extraordinary waves at the same frequency; in the xy plane of a biaxial crystal, this symmetry concerns the indices relative to the ordinary waves. Equalities between frequencies are the only situations for which the field tensors are partly symmetric out of the principal planes of a biaxial crystal: the symmetry concerns the indices relative to the waves (+) with identical frequencies; it is the same for the waves (−): for example, [F_{ijk}^{-++}(2\omega=] [\omega+\omega)=] [F_{ikj}^{-++}(2\omega=] [\omega+\omega)], [F_{ijkl}^{-++-}(\omega_4=] [\omega+\omega+\omega_3)=] [F_{ikjl}^{-++-}(\omega_4=] [\omega+\omega+\omega_3)], [F_{ijkl}^{---+}(\omega_4=] [\omega+\omega+\omega_3)=] [F_{ikjl}^{---+}(\omega_4=] [\omega+\omega+\omega_3)] and so on. Uniaxial class

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The field-tensor components are calculated from ([link] and ([link]. The phase-matching case is the only one considered here: according to Tables[link] and[link], the allowed configurations of polarization of three-wave and four-wave interactions, respectively, are the 2o.e (two ordinary and one extraordinary waves), the 2e.o and the 3o.e, 3e.o, 2o.2e.

Tables[link] and[link] give, respectively, the matrix representations of the three-wave interactions (eoo), (oee) and of the four-wave (oeee), (eooo), (ooee) interactions for any direction of propagation in the general case where all the frequencies are different. In this situation, the number of independent components of the field tensors are: 7  for 2o.e, 12 for 2e.o, 9 for 3o.e, 28 for 3e.o and 16 for 2o.2e. Note that the increase of the number of ordinary waves leads to an enhancement of symmetry of the field tensors.

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Matrix representations of the (oee) and (eoo) field tensors of the uniaxial class and of the biaxial class in the principal planes xz and yz, with [\omega_1\ne \omega_2] (Boulanger & Marnier, 1991[link])

[Scheme scheme1]

InteractionsThree-rank [F_{ijk}(\theta,\varphi)] field tensors
Type eoo [Scheme scheme2]
SFG (ω3) type I < 0
DFG (ω1) type I > 0
DFG (ω2) type I > 0
Type oee [Scheme scheme3]
SFG (ω3) type I > 0
DFG (ω1) type I < 0
DFG (ω2) type I < 0

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Matrix representations of the (oeee), (eooo) and (ooee) field tensors of the uniaxial class and of the biaxial class in the principal planes xz and yz, with [\omega_1\ne\omega_2\ne\omega_3] (Boulanger et al., 1993[link])

[Scheme scheme4]

InteractionsFour-rank [F_{ijkl}(\theta,\varphi)] field tensors
Type oeee [Scheme scheme5]
SFG(ω4) type I > 0
DFG (ω1) type I < 0
DFG (ω2) type I < 0
DFG (ω3) type I < 0
Type eooo [Scheme scheme6]
SFG (ω4) type I < 0
DFG (ω1) type I > 0
DFG (ω2) type I > 0
DFG (ω3) type I > 0
Type ooee [Scheme scheme7]
SFG (ω4) type V4 > 0
DFG (ω1) type V1 > 0
DFG (ω2) type V2 > 0
DFG (ω3) type V3 > 0

If there are equalities between frequencies, the field tensors oee, oeee and ooee become totally symmetric in the Cartesian indices relative to the extraordinary waves and the tensors eoo and eooo remain unchanged.

Table[link] gives the field-tensor components specifically nil in the principal planes of uniaxial and biaxial crystals. The nil components for the other configurations of polarization are obtained by permutation of the Cartesian indices and the corresponding polarizations.

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Field-tensor components specifically nil in the principal planes of uniaxial and biaxial crystals for three-wave and four-wave interactions

[(i,j,k) = x, y \hbox{ or } z].

Configurations of polarizationNil field-tensor components
(xy) plane(xz) plane(yz) plane
[e o o] [F_{xjk}= 0; F_{yjk}= 0] [F_{ixk}= F_{ijx}= 0 ] [F_{iyk}= F_{ijy} = 0 ]
    [F_{yjk}= 0] [F_{xjk}= 0]
[o e e] [F_{ixk}= F_{ijx}= 0] [F_{iyk}= F_{ijy}= 0] [F_{ixk}= F_{ijx}= 0]
  [F_{iyk}= F_{ijy}= 0] [F_{xik}= 0] [F_{yjk}= 0]
[e o o o] [F_{xjkl}= 0; F_{yjkl}= 0] [F_{ixkl}= F_{ijxl}= F_{ijkx}= 0] [F_{iykl}= F_{ijyl}= F_{ijky}= 0]
    [F_{yjkl}= 0] [F_{xjkl}= 0]
[o e e e] [F_{ixkl}= F_{ijxl}= F_{ijkx}= 0] [F_{iykl}= F_{ijyl}= F_{ijky}= 0] [F_{ixkl}= F_{ijxl}= F_{ijkx}= 0]
  [F_{iykl}= F_{ijyl}= F_{ijky}= 0] [F_{xjkl}= 0 ] [F_{yjkl}= 0]
[o o e e] [F_{ijxl}= F_{ijkx}= 0] [F_{xjkl}= F_{ixkl}= 0] [F_{yjkl}= F_{iykl}= 0]
  [F_{ijyl}= F_{ijky}= 0] [F_{ijyl}= F_{ijky}= 0] [F_{ijxl}= F_{ijkx}= 0]

From Tables[link] and[link], it is possible to deduce all the other 2e.o interactions (eeo), (eoe), the 2o.e interactions (ooe), (oeo), the 3o.e interactions (oooe), (oeoo), (ooeo), the 3e.o interactions (eoee), (eeoe), (eeeo) and the 2o.2e inter­actions (oeoe), (eoeo), (eeoo), (oeeo), (eooe). The corresponding interactions and types are given in Tables[link] and[link]. According to ([link] and ([link], the magnitudes of two permutated components are equal if the permutation of polarizations are associated with the corresponding frequencies. For example, according to Table[link], two permutated field-tensor components have the same magnitude for permutation between the following 3o.e interactions:

  • (i) (eooo) SFG (ω4) type I < 0 and the three (oeoo) interactions, DFG (ω1) type II < 0, DFG (ω2) type III < 0, DFG (ω3) type IV < 0;

  • (ii) the three (oooe) interactions, SFG (ω4) type II > 0, DFG (ω1) type III > 0, DFG (ω2) type IV > 0 and (eooo) DFG (ω3) type I > 0;

  • (iii) the two (ooeo) interactions SFG (ω4) type III > 0, DFG (ω1) type IV > 0, (eooo) DFG (ω2) type I > 0, and (oooe) DFG (ω3) type II > 0;

  • (iv) (oeoo) SFG (ω4) type IV > 0, (eooo) DFG (ω1) type I > 0, and the two interactions (ooeo) DFG (ω2) type II > 0, DFG (ω3) type III > 0.

The contraction of the field tensor and the uniaxial dielectric susceptibility tensor of corresponding order, given in Tables[link] to[link][link][link], is nil for the following uniaxial crystal classes and configurations of polarization: D4 and D6 for 2o.e, C4v and C6v for 2e.o, D6, D6h, D3h and C6v for 3o.e and 3e.o. Thus, even if phase-matching directions exist, the effective coefficient in these situations is nil, which forbids the interactions considered (Boulanger & Marnier, 1991[link]; Boulanger et al., 1993[link]). The number of forbidden crystal classes is greater under the Kleinman approximation. The forbidden crystal classes have been determined for the particular case of third harmonic generation assuming Kleinman conjecture and without consideration of the field tensor (Midwinter & Warner, 1965[link]). Biaxial class

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The symmetry of the biaxial field tensors is the same as for the uniaxial class, though only for a propagation in the principal planes xz and yz; the associated matrix representations are given in Tables[link] and[link], and the nil components are listed in Table[link]. Because of the change of optic sign from either side of the optic axis, the field tensors of the interactions for which the phase-matching cone joins areas b and a or a and c, given in Fig.[link], change from one area to another: for example, the field tensor (eoee) becomes an (oeoo) and so the solicited components of the electric susceptibility tensor are not the same.

The nonzero field-tensor components for a propagation in the xy plane of a biaxial crystal are: [F_{zxx}], [F_{zyy}], [F_{zxy}\ne F_{zyx}] for (eoo); [F_{xzz}], [F_{yzz}] for (oee); [F_{zxxx}], [F_{zyyy}], [F_{zxyy}\ne F_{zyxy}\ne F_{zyyx}], [F_{zxxy}\ne F_{zxyx} \ne F_{zyxx}] for (eooo); [F_{xzzz}], [F_{yzzz}] for (oeee); [F_{xyzz}\ne F_{yxzz}], [F_{xxzz}], [F_{yyzz}] for (ooee). The nonzero components for the other configurations of polarization are obtained by the associated permutations of the Cartesian indices and the corresponding polarizations.

The field tensors are not symmetric for a propagation out of the principal planes in the general case where all the frequencies are different: in this case there are 27 independent components for the three-wave interactions and 81 for the four-wave interactions, and so all the electric susceptibility tensor components are solicited.

As phase matching imposes the directions of the electric fields of the interacting waves, it also determines the field tensor and hence the effective coefficient. Thus there is no possibility of choice of the [\chi^{(2)}] coefficients, since a given type of phase matching is considered. In general, the largest coefficients of polar crystals, i.e. [\chi_{zzz}], are implicated at a very low level when phase matching is achieved, because the corresponding field tensor, i.e. [F_{zzz}], is often weak (Boulanger et al., 1997[link]). In contrast, QPM authorizes the coupling between three waves polarized along the z axis, which leads to an effective coefficient which is purely [\chi_{zzz}], i.e. [\chi_{\rm eff}=(2/\pi)\chi_{zzz}], where the numerical factor comes from the periodic character of the rectangular function of modulation (Fejer et al., 1992[link]). Integration of the propagation equations

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The resolution of the coupled equations ([link] or ([link] over the crystal length L leads to the electric field amplitude [E_i(X,Y,L)] of each interacting wave. The general solutions are Jacobian elliptic functions (Armstrong et al., 1962[link]; Fève, Boulanger & Douady, 2002[link]). The integration of the systems is simplified for cases where one or several beams are held constant, which is called the undepleted pump approximation. We consider mainly this kind of situation here. The power of each interacting wave is calculated by integrating the intensity over the cross section of each beam according to ([link]. For our main purpose, we consider the simple case of plane-wave beams with two kinds of transverse profile:[\eqalignno{{\bf E}(X,Y,Z)&={\bf e}E_o(Z)\quad\hbox{for }(X,Y)\in[-w_o,+w_o]&\cr{\bf E}(X,Y,Z)&=0\phantom{E_o(Z)\quad}\hbox{elsewhere}&(}]for a flat distribution over a radius wo;[{\bf E}(X,Y,Z)={\bf e}E_o(Z)\exp[-(X^2+Y^2)/w_o^2]\eqno(]for a Gaussian distribution, where wo is the radius at ([1/e]) of the electric field and so at ([1/e^2]) of the intensity.

The associated powers are calculated according to ([link], which leads to[P(L)=m(n/2)(\varepsilon_o/\mu_o)^{1/2}|E_o|^2\pi w_o^2\eqno(]where [m=1] for a flat distribution and [m = 1/2] for a Gaussian profile.

The nonlinear interaction is characterized by the conversion efficiency, which is defined as the ratio of the generated power to the power of one or several incident beams, according to the different kinds of interactions.

For pulsed beams, it is necessary to consider the temporal shape, usually Gaussian:[P(t)=P_c\exp(-2t^2/\tau^2)\eqno(]where Pc is the peak power and τ the half ([1/e^2]) width.

For a repetition rate f (s−1), the average power [\tilde P] is then given by[{\tilde P}=P_c\tau f(\pi/2)^{1/2}={\tilde E}f\eqno(]where [\tilde E] is the energy per Gaussian pulse.

When the pulse shape is not well defined, it is suitable to consider the energies per pulse of the incident and generated waves for the definition of the conversion efficiency.

The interactions studied here are sum-frequency generation (SFG), including second harmonic generation (SHG: [\omega+\omega=2\omega]), cascading third harmonic generation (THG: [\omega+2\omega=3\omega]) and direct third harmonic generation (THG: [\omega+\omega+\omega=3\omega]). The difference-frequency generation (DFG) is also considered, including optical parametric amplification (OPA) and oscillation (OPO).

We choose to analyse in detail the different parameters relative to conversion efficiency (figure of merit, acceptance bandwidths, walk-off effect etc.) for SHG, which is the prototypical second-order nonlinear interaction. This discussion will be valid for the other nonlinear processes of frequency generation which will be considered later. Second harmonic generation (SHG)

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According to Table[link], there are two types of phase matching for SHG: type I and type II (equivalent to type III).

The fundamental waves at ω define the pump. Two situations are classically distinguished: the undepleted pump approximation, when the power conversion efficiency is sufficiently low to consider the fundamental power to be undepleted, and the depleted case for higher efficiency. There are different ways to realize SHG, as shown in Fig.[link]: the simplest one is non-resonant SHG, outside the laser cavity; other ways are external or internal resonant cavity SHG, which allow an enhancement of the single-pass efficiency conversion.


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Schematic configurations for second harmonic generation. (a) Non-resonant SHG; (b) external resonant SHG: the resonant wave may either be the fundamental or the harmonic one; (c) internal resonant SHG. [P^{\omega,2\omega}] are the fundamental and harmonic powers; [{\rm HT}^\omega] and [{\rm HR}^{\omega,2\omega}] are the high-transmission and high-reflection mirrors at ω or 2ω and [T^{\omega,2\omega}] are the transmission coefficients of the output mirror at ω or 2ω. NLC is the nonlinear crystal with a nonzero χ(2). Non-resonant SHG with undepleted pump in the parallel-beam limit with a Gaussian transverse profile

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We first consider the case where the crystal length is short enough to be located in the near-field region of the laser beam where the parallel-beam limit is a good approximation. We make another simplification by considering a propagation along a principal axis of the index surface: then the walk-off angle of each interacting wave is nil so that the three waves have the same coordinate system ([X,Y,Z]).

The integration of equations ([link] over the crystal length Z in the undepleted pump approximation, i.e. [\partial E_1^\omega(X,Y,Z)/\partial Z=] [\partial E_2^\omega(X,Y,Z)/\partial Z=] [], with [E_3^{2\omega}(X,Y,0)=] [], leads to[\eqalignno{|E_3^{2\omega}(X,Y,L)|^2&=\{K_3^{2\omega}[\varepsilon_o\chi_{\rm eff}^{(2)}]\}^2|E_1^\omega(X,Y,0)E_2^\omega(X,Y,0)|^2&\cr&\quad\times L^2\sin c^2[(\Delta k\cdot L)/2].&(}]([link] implies a Gaussian transversal profile for [|E_3^{2\omega}(X,Y,L)|] if [|E_1^{\omega}(X,Y,0)|] and [|E_2^{\omega}(X,Y,0)|] are Gaussian. The three beam radii are related by [(1/w_{o3}^2)=(1/w_{o1}^2)+(1/w_{o2}^2)], so if we assume that the two fundamental beams have the same radius [w_o^\omega], which is not an approximation for type I, then [w_o^{2\omega}=[w_o^\omega/(2^{1/2})]]. Two incident beams with a flat distribution of radius [w_o^\omega] lead to the generation of a flat harmonic beam with the same radius [w_o^{2\omega}=w_o^\omega].

The integration of ([link] according to ([link]–([link][link] for a Gaussian profile gives in the SI system[\eqalignno{P^{2\omega}(L)&=BP_1^\omega(0)P_2^\omega(0){L^2\over w_o^2}\sin c^2\left({\Delta k\cdot L\over 2}\right)&\cr B&={32\pi \over \varepsilon_o c}{2N-1 \over N}{d_{\rm eff}^2 \over \lambda_\omega^2}{T^{2\omega}_3T^\omega_1T^\omega_2 \over n^{2\omega}_3n^\omega_1n^\omega_2},\quad ({\rm W}^{-1})&\cr&&(}]where [c = 3\times 10^8] m s−1, [\varepsilon_o= 8.854\times 10^{-12}] A s V−1 m−1 and so [(32\pi/\varepsilon_oc)=37.85\times 10^3] V A−1. L (m) is the crystal length in the direction of propagation. [\Delta k=k_3^{2\omega}-k_1^\omega-k_2^\omega] is the phase mismatch. [n_3^{2\omega}], [n_1^\omega] and [n_2^\omega] are the refractive indices at the harmonic and fundamental wavelengths λ and λω (µm): for the phase-matching case, [\Delta k=0], [n_3^{2\omega}=n^-(2\omega)], [n_1^\omega=n_2^\omega=n^+(\omega)] for type I (the two incident fundamental beams have the same polarization contained in Π+, with the harmonic polarization contained in Π) and [n_1^\omega=n^+(\omega)\ne n_2^\omega=n^-(\omega)] for type II (the two solicited eigen modes at the fundamental wavelength are in Π+ and Π, with the harmonic polarization contained in Π). [T_3^{2\omega}], [T_1^{\omega}] and [T_2^{\omega}] are the transmission coefficients given by [T_i=4n_i/(n_i+1)^2]. deff (pm V−1) [= (1/2)\chi_{\rm eff}=(1/2)[F^{(2)}\cdot \chi^{(2)}]] is the effective coefficient given by ([link] and ([link]. [P_1^\omega(0)] and [P_2^\omega(0)] are the two incident fundamental powers, which are not necessarily equal for type II; for type I we have obviously [P_1^\omega(0) = P_2^\omega(0)= (P_{\rm tot}^\omega/2)]. N is the number of independently oscillating modes at the fundamental wavelength: every longitudinal mode at the harmonic pulsation can be generated by many combinations of two fundamental modes; the [(2N-1)/N] factor takes into account the fluctuations between these longitudinal modes (Bloembergen, 1963[link]).

The powers in ([link] are instantaneous powers P(t).

The second harmonic (SH) conversion efficiency, ηSHG, is usually defined as the ratio of peak powers [P_c^{2\omega}(L)/P_{c, {\rm tot}}^\omega(0)], or as the ratio of the pulse total energy [{\tilde E}^{2\omega}(L)/{\tilde E}_{\rm tot}^\omega(0)]. For Gaussian temporal profiles, the SH [(1/e^2)] pulse duration [\tau_{2\omega}] is equal to [\tau_\omega/(2^{1/2})], because [P_{2\omega}] is proportional to [P_{\omega}^2], and so, according to ([link], the pulse average energy conversion efficiency is [1/(2^{1/2})] smaller than the peak power conversion efficiency given by ([link]. Note that the pulse total energy conversion efficiency is equivalent to the average power conversion efficiency [{\tilde P}^{2\omega}(L)/{\tilde P}_{\rm tot}^\omega(0)], with [{\tilde P}={\tilde E}\cdot f] where f is the repetition rate.

Formula ([link] shows the importance of the contribution of the linear optical properties to the nonlinear process. Indeed, the field tensor F(2), the transmission coefficients Ti and the phase mismatch [\Delta k] only depend on the refractive indices in the direction of propagation considered.

  • (i) Figure of merit.

    The contribution of F(2) was discussed previously, where it was shown that the field tensor is nil in particular directions of propagation or everywhere for particular crystal classes and configurations of polarization (even if the nonlinearity χ(2) is high).

    The field tensor F(2) of SHG can be written with the contracted notation of d(2); according to Table[link] and to the contraction conventions given in Section[link], the contracted field-tensor components for the phase-matched SHG are[\eqalign{F_{i1}&={\bf e}_i^-(2\omega)[{\bf e}_x^+(\omega)]^2\cr F_{i2}&={\bf e}_i^-(2\omega)[{\bf e}_y^+(\omega)]^2\cr F_{i3}&={\bf e}_i^-(2\omega)[{\bf e}_z^+(\omega)]^2\cr F_{i4}&=2{\bf e}_i^-(2\omega){\bf e}_y^+(\omega){\bf e}_z^+(\omega)\cr F_{i5}&=2{\bf e}_i^-(2\omega){\bf e}_x^+(\omega){\bf e}_z^+(\omega)\cr F_{i6}&=2{\bf e}_i^-(2\omega){\bf e}_x^+(\omega){\bf e}_y^+(\omega)}] for type I and[\eqalign{F_{i1}&={\bf e}_i^-(2\omega){\bf e}_x^+(\omega){\bf e}_x^-(\omega)\cr F_{i2}&={\bf e}_i^-(2\omega){\bf e}_y^+(\omega){\bf e}_y^-(\omega)\cr F_{i3}&={\bf e}_i^-(2\omega){\bf e}_z^+(\omega){\bf e}_z^-(\omega)\cr F_{i4}&={\bf e}_i^-(2\omega)[{\bf e}_y^+(\omega){\bf e}_z^-(\omega)+{\bf e}_y^-(\omega){\bf e}_z^+(\omega)]\cr F_{i5}&={\bf e}_i^-(2\omega)[{\bf e}_x^+(\omega){\bf e}_z^-(\omega)+{\bf e}_x^-(\omega){\bf e}_z^+(\omega)]\cr F_{i6}&={\bf e}_i^-(2\omega)[{\bf e}_x^+(\omega){\bf e}_y^-(\omega)+{\bf e}_x^-(\omega){\bf e}_y^+(\omega)]}]for type II, with [i=(1,2,3)] for Fij, corresponding to [i=(x,y,z)] for [{\bf e}_i^-(2\omega)].

    The ratio [d_{\rm eff}^2/n_3^{2\omega}n_1^\omega n_2^\omega] in formula ([link] is called the figure of merit of the direction considered. The effective coefficient is given in Section 1.7.5[link] for the main nonlinear crystals and for chosen SHG wavelengths.

  • (ii) Effect of the phase mismatch.

    The interference function [\sin c^2(\Delta kL/2)] is a maximum and equal to unity only for [\Delta k = 0], which defines the phase-matching condition. Fig.[link] shows the effect of the phase mismatch on the growth of second harmonic conversion efficiency, ηSHG, with interaction distance Z.


    Figure | top | pdf |

    Spatial growth evolution of second harmonic conversion efficiency, ηSHG, for non phase matching (NPM), [\Delta k\ne 0], and phase matching (PM), [\Delta k=0], in a `continuous' crystal, and for quasi phase matching (QPM) in a periodic structure. The dashed curve corresponds to (4/π2PM(Z) where ηPM is the conversion efficiency of the phase-matched SHG. [l_c=\pi/\Delta k] is the coherence length.

    The conversion efficiency has a Z2 dependence in the case of phase matching. The harmonic power oscillates around Z2 for quasi phase matching, but is reduced by a factor of 4/π2 compared with that of phase-matched interaction (Fejer et al., 1992[link]).

    An SHG phase-matching direction ([\theta_{\rm PM}, \varphi_{\rm PM}]) for given fundamental wavelength (λPM) and type of interaction, I or II, is defined at a given temperature (TPM). It is important to consider the effect of deviation of Δk from 0 due to variations of angles ([\theta_{\rm PM}\pm{\rm d}\theta,\varphi_{\rm PM}\pm{\rm d}\varphi]), of temperature ([T_{\rm PM}\pm{\rm d}T]) and of wavelength ([\lambda_{\rm PM}\pm{\rm d}\lambda]) on the conversion efficiency. The quantities that characterize these effects are the acceptance bandwidths δξ ([\xi = \theta, \varphi, T, \lambda]), usually defined as the deviation from the phase-matching value ξPM leading to a phase-mismatch variation Δk from 0 to 2π/L, where L is the crystal length. Then δξ is also the full width of the peak efficiency curve plotted as a function of ξ at 0.405 of the maximum, as shown in Fig.[link].


    Figure | top | pdf |

    Conversion efficiency evolution as a function of ξ for a given crystal length. ξ denotes the angle (θ or [\varphi]), the temperature (T) or the wavelength (λ). ξPM represents the parameter allowing phase matching.

    Thus Lδξ is a characteristic of the phase-matching direction. Small angular, thermal and spectral dispersion of the refractive indices lead to high acceptance bandwidths. The higher Lδξ, the lower is the decrease of the conversion efficiency corresponding to a given angular shift, to the heating of the crystal due to absorption or external heating, or to the spectral bandwidth of the fundamental beam.

    The knowledge of the angular, thermal and spectral dispersion of the refractive indices allows an estimation of δξ by expanding Δk in a Taylor series about ξPM:[{2\pi \over L}=\Delta k=\left.{\partial(\Delta k)\over\partial \xi}\right|_{\xi_{\rm PM}}\delta\xi+{1 \over 2}\left.{\partial^2(\Delta k)\over \partial\xi^2}\right|_{\xi_{\rm PM}}(\delta\xi)^2+\ldots.\eqno(]When the second- and higher-order differential terms in ([link] are negligible, the phase matching is called critical (CPM), because [L\delta\xi\simeq|2\pi/[\partial(\Delta k)/\partial\xi|_{\xi_{\rm PM}}]|] is small. For the particular cases where [\partial(\Delta k)/\partial\xi|_{\xi_{\rm PM}}=0], [L\delta\xi =] [\{|4\pi L/[\partial^2(\Delta k)/\partial\xi^2|_{\xi_{\rm PM}}]|\}^{1/2}] is larger than the CPM acceptance and the phase matching is called non-critical (NCPM) for the parameter ξ considered.

    We first consider the case of angular acceptances. In uniaxial crystals, the refractive indices do not vary in [\varphi], leading to an infinite [\varphi] angular acceptance bandwidth. δθ is then the only one to consider. For directions of propagation out of the principal plane ([\theta_{\rm PM}\ne \pi/2]), the phase matching is critical. According to the expressions of no and ne(θ) given in Section[link], we have

    • (1) for type I in positive crystals, [n_e(\theta,\omega)=n_o(2\omega)] and[L\delta\theta\simeq 2\pi/\{-(\omega/c)n_o^3(2\omega)[n_e^{-2}(\omega)-n_o^{-2}(\omega)]\sin 2\theta_{\rm PM}\};\eqno(]

    • (2) for type II in positive crystals, [2n_o(2\omega)=n_e(\theta,\omega)+n_o(\omega)] and[\eqalignno{L\delta\theta&\simeq 2\pi/\{-(\omega/2c)[2n_o(2\omega)-n_o(\omega)]^3&\cr&\quad\times [n_e^{-2}(\omega)-n_o^{-2}(\omega)]\sin 2\theta_{\rm PM}\}\semi&(}]

    • (3) for type I in negative crystals, [n_e(\theta,2\omega)=n_o(\omega)] and[L\delta\theta\simeq 2\pi/\{-(\omega/c)n_o^3(\omega)[n_o^{-2}(2\omega)-n_e^{-2}(2\omega)]\sin 2\theta_{\rm PM}\};\eqno(]

    • (4) for type II in negative crystals, [2n_e(\theta,2\omega)=] [n_e(\theta,\omega)] [+] [n_o(\omega)] and[\eqalignno{L\delta\theta&\simeq \left|2\pi/\{-(\omega/c)n_e^3(\theta,2\omega)[n_e^{-2}(2\omega)-n_o^{-2}(2\omega)]\sin2\theta_{\rm PM}\right.\cr&\left.\quad+\,(\omega/2c)n_e^3(\theta,\omega)[n_e^{-2}(\omega)-n_o^{-2}(\omega)]\sin 2\theta_{\rm PM}\}\right|.&\cr&&(}]

    CPM acceptance bandwidths are small, typically about one mrad cm, as shown in Section 1.7.5[link] for the classical nonlinear crystals.

    When [\theta_{\rm PM}=\pi/2], [\partial\Delta k/\partial\theta=0] and the phase matching is non-critical:

    • (1) for type I in positive crystals, [n_e(\omega)=n_o(2\omega)] and[L\delta\theta\simeq\left(2\pi L/\{-(\omega/c)n_o^3(2\omega)[n_e^{-2}(\omega)-n_o^{-2}(\omega)]\}\right)^{1/2};\eqno(]

    • (2) for type II in positive crystals, [2n_o(2\omega)=n_e(\omega) + n_o(\omega)] and[L\delta\theta\simeq\left(2\pi L/\{-(\omega/2c)n_e^3(\omega)[n_e^{-2}(\omega)-n_o^{-2}(\omega)]\}\right)^{1/2};\eqno(]

    • (3) for type I in negative crystals, [n_o(\omega)=n_e(2\omega)] and[L\delta\theta\simeq\left(2\pi L/\{(\omega/c)n_o^3(\omega)[n_e^{-2}(2\omega)-n_o^{-2}(2\omega)]\}\right)^{1/2};\eqno(]

    • (4) for type II in negative crystals, [2n_e(2\omega)=n_e(\omega)+n_o(\omega)] and[\eqalignno{L\delta\theta &\simeq\big(|2\pi L/\{-(\omega/c)n_e^3(2\omega)[n_e^{-2}(2\omega)-n_o^{-2}(2\omega)]&\cr &\quad +(\omega/2c)n_e^3(\omega)[n_e^{-2}(\omega)-n_o^{-2}(\omega)]\}|\big)^{1/2}.&\cr&&(}]

    Values of NCPM acceptance bandwidths are given in Section 1.7.5[link] for the usual crystals. From the previous expressions for CPM and NCPM angular acceptances, it appears that the angular bandwidth is all the smaller since the birefringence is high.

    The situation is obviously more complex in the case of biaxial crystals. The [\varphi] acceptance bandwidth is not infinite, leading to a smaller anisotropy of the angular acceptance in comparison with uniaxial crystals. The expressions of the θ and [\varphi] acceptance bandwidths have the same form as for the uniaxial class only in the principal planes. The phase matching is critical (CPM) for all directions of propagation out of the principal axes x, y and z: in this case, the mismatch Δk is a linear function of small angular deviations from the phase-matching direction as for uniaxial crystals. There exist six possibilities of NCPM for SHG, types I and II along the three principal axes, corresponding to twelve different index conditions (Hobden, 1967[link]):

    • (1) for positive biaxial crystals[\matrix{\hbox{Type I }(x)\hfill &n_{2\omega}^y=n_\omega^z\hfill\cr \hbox{Type I }(y)\hfill &n_{2\omega}^x=n_\omega^z\hfill\cr \hbox{Type I }(z)\hfill &n_{2\omega}^x=n_\omega^y\hfill\cr \hbox{Type II }(x)\hfill &n_{2\omega}^y={\textstyle{1\over 2}}(n_\omega^y+n_\omega^z)\hfill\cr \hbox{Type II }(y)\hfill &n_{2\omega}^x={\textstyle{1\over 2}}(n_\omega^x+n_\omega^z)\hfill\cr \hbox{Type II }(z)\hfill &n_{2\omega}^x={\textstyle{1\over 2}}(n_\omega^x+n_\omega^y)\semi\hfill\cr}\eqno(]

    • (2) for negative biaxial crystals[\matrix{\hbox{Type I }(x)\hfill &n_{2\omega}^z=n_\omega^y\hfill\cr \hbox{Type I }(y)\hfill &n_{2\omega}^z=n_\omega^x\hfill\cr \hbox{Type I }(z)\hfill &n_{2\omega}^y=n_\omega^x\hfill\cr \hbox{Type II }(x)\hfill &n_{2\omega}^z={\textstyle{1\over 2}}(n_\omega^y+n_\omega^z)\hfill\cr \hbox{Type II }(y)\hfill &n_{2\omega}^z={\textstyle{1\over 2}}(n_\omega^x+n_\omega^z)\hfill\cr \hbox{Type II }(z)\hfill &n_{2\omega}^y={\textstyle{1\over 2}}(n_\omega^x+n_\omega^y).\hfill\cr}]

    The NCPM angular acceptances along the three principal axes of biaxial crystals can be deduced from the expressions relative to the uniaxial class by the following substitutions:

    Along the x axis:[\eqalign{& L\delta\varphi\hbox{ (type I}>0)=(\hbox{ with }n_o(\omega)\rightarrow n_z(\omega),\cr&\quad n_e(2\omega)\rightarrow n_y(2\omega)\hbox{ and }n_o(2\omega)\rightarrow n_x(2\omega)\cr &L\delta\theta\hbox{ (type I}>0)=(\hbox{ with }n_o(2\omega)\rightarrow n_y(2\omega),\cr&\quad n_e(\omega)\rightarrow n_z(\omega)\hbox{ and }n_o(\omega)\rightarrow n_x(\omega)\cr &L\delta\varphi\hbox{ (type II}>0)=(\hbox{ with }n_e\rightarrow n_y\hbox{ and }n_o\rightarrow n_x\cr &L\delta\theta\hbox{ (type II}>0)=(\hbox{ with }n_e(\omega)\rightarrow n_z(\omega)\cr&\quad\hbox{ and }n_o(\omega)\rightarrow n_x(\omega)\cr &L\delta\varphi\hbox{ (type I}\,\lt\,0)=(\hbox{ with }n_o(2\omega)\rightarrow n_z(2\omega),\cr&\quad n_e(\omega)\rightarrow n_x(\omega)\hbox{ and }n_o(\omega)\rightarrow n_y(\omega)\cr &L\delta\theta\hbox{ (type I}\,\lt\,0)=(\hbox{ with }n_o(\omega)\rightarrow n_y(\omega),\cr&\quad n_e(2\omega)\rightarrow n_z(2\omega)\hbox{ and }n_o(2\omega)\rightarrow n_x(2\omega)\cr &L\delta\varphi\hbox{ (type II}\,\lt\,0)=(\hbox{ with }n_e(\omega)\rightarrow n_x(\omega)\cr&\quad\hbox{ and }n_o(\omega)\rightarrow n_y(\omega)\cr &L\delta\theta\hbox{ (type II}\,\lt\,0)=(\hbox{ with }n_e\rightarrow n_z\hbox{ and }n_o\rightarrow n_x.\cr}]

    Along the y axis:[\eqalign{&L\delta\varphi\hbox{ is the same as along the }x\hbox{ axis for all interactions}\hfill\cr&L\delta\theta\hbox{ (type I}>0)=(\hbox{ with }n_o(2\omega)\rightarrow n_x(2\omega),\cr&\quad n_e(\omega)\rightarrow n_z(\omega)\hbox{ and }n_o(\omega)\rightarrow n_y(\omega)\cr &L\delta\theta\hbox{ (type II}>0)=(\hbox{ with }n_e(\omega)\rightarrow n_z(\omega)\cr&\quad\hbox{ and }n_o(\omega)\rightarrow n_y(\omega)\cr &L\delta\theta\hbox{ (type I}\,\lt\,0)=(\hbox{ with }n_o(\omega)\rightarrow n_x(\omega),\cr&\quad n_e(2\omega)\rightarrow n_z(2\omega)\hbox{ and }n_o(2\omega)\rightarrow n_y(2\omega)\cr &L\delta\theta\hbox{ (type II}\,\lt\,0)=(\hbox{ with }n_e\rightarrow n_z\hbox{ and }n_o\rightarrow n_y.\cr}]

    Along the z axis:[\eqalign{&L\delta\theta_{xz}\hbox{ (type I}>0)=(\hbox{ with }n_o(2\omega)\rightarrow n_y(2\omega),\cr&\quad n_e(\omega)\rightarrow n_x(\omega)\hbox{ and }n_o(\omega)\rightarrow n_z(\omega)\cr &L\delta\theta_{yz}\hbox{ (type I}>0)=(\hbox{ with }n_o(2\omega)\rightarrow n_x(2\omega),\cr&\quad n_e(\omega)\rightarrow n_y(\omega)\hbox{ and }n_o(\omega)\rightarrow n_z(\omega)\cr &L\delta\theta_{xz}\hbox{ (type II}>0)=(\hbox{ with }n_e(\omega)\rightarrow n_x(\omega)\cr&\quad\hbox{ and }n_o(\omega)\rightarrow n_z(\omega)\cr &L\delta\theta_{yz}\hbox{ (type II}>0)=(\hbox{ with }n_e(\omega)\rightarrow n_y(\omega)\cr&\quad\hbox{ and }n_o(\omega)\rightarrow n_z(\omega)\cr &L\delta\theta_{xz}\hbox{ (type I}\,\lt\,0)=(\hbox{ with }n_o(\omega)\rightarrow n_y(\omega),\cr&\quad n_e(2\omega)\rightarrow n_z(2\omega)\hbox{ and }n_o(2\omega)\rightarrow n_x(2\omega)\cr &L\delta\theta_{yz}\hbox{ (type I}\,\lt\,0)=(\hbox{ with }n_o(\omega)\rightarrow n_x(\omega),\cr&\quad n_e(2\omega)\rightarrow n_z(2\omega)\hbox{ and }n_o(2\omega)\rightarrow n_y(2\omega)\cr }][\eqalign{&L\delta\theta_{xz}\hbox{ (type II}\,\lt\,0)=(\hbox{ with }n_e\rightarrow n_x\hbox{ and }n_o\rightarrow n_z\cr &L\delta\theta_{yz}\hbox{ (type II}\,\lt\,0)=(\hbox{ with }n_e\rightarrow n_y\hbox{ and }n_o\rightarrow n_z.\cr}]

    The above formulae are relative to the internal angular acceptance bandwidths. The external acceptance angles are enlarged by a factor of approximately n(ω) for type I or [[n_1(\omega)+n_2(\omega)]/2] for type II, due to refraction at the input plane face of the crystal. The angular acceptance is an important issue connected with the accuracy of cutting of the crystal.

    Temperature tuning is a possible alternative for achieving NCPM in a few materials. The corresponding temperatures for different interactions are given in Section 1.7.5[link].

    Another alternative is to use a special non-collinear configuration known as one-beam non-critical non-collinear phase matching (OBNC): it is non-critical with respect to the phase-matching angle of one of the input beams (referred to as the non-critical beam). It has been demonstrated that the angular acceptance bandwidth for the non-critical beam is exceptionally large, for example about 50 times that for the critical beam for type-I SHG at 1.338 µm in 3-methyl-4-nitropyridine-N-oxide (POM) (Dou et al., 1992[link]).

    The typical values of thermal acceptance bandwidth, given in Section 1.7.5[link], are of the order of 0.5 to 50 K cm. The thermal acceptance is an important issue for the stability of the harmonic power when the absorption at the wavelengths concerned is high or when temperature tuning is used for the achievement of angular NCPM. Typical spectral acceptance bandwidths for SHG are given in Section 1.7.5[link]. The values are of the order of 1 nm cm, which is much larger than the linewidth of a single-frequency laser, except for some diode or for sub-picosecond lasers with a large spectral bandwidth.

    Note that a degeneracy of the first-order temperature or spectral derivatives ([\partial\Delta k/\partial T|_{T_{\rm PM}}=0] or [\partial\Delta k/\partial\lambda|_{\lambda_{\rm PM}}=0]) can occur and lead to thermal or spectral NCPM.

    Consideration of the phase-matching function [\lambda_{\rm PM}=f(\xi_{\rm PM})], where [\chi_{\rm PM}=T_{\rm PM}], [\theta_{\rm PM}], [\varphi_{\rm PM}] or all other dispersion parameters of the refractive indices, is useful for a direct comparison of the situation of non-criticality of the phase matching relative to [\lambda_{\rm PM}] and to the other parameters [\xi_{\rm PM}]: a nil derivative of [\lambda_{\rm PM}] with respect to [\xi_{\rm PM}], i.e. [{\rm d}\lambda_{\rm PM}/{\rm d}\xi_{\rm PM}=0] at the point ([\lambda_{\rm PM}^o,\xi_{\rm PM}^o]), means that the phase matching is non-critical with respect to [\xi_{\rm PM}] and so strongly critical with respect to [\lambda_{\rm PM}], i.e. [{\rm d}\xi_{\rm PM}/{\rm d}\lambda_{\rm PM}=\infty] at this point. Then, for example, an angular NCPM direction is a spectral CPM direction and the reverse is also so.

  • (iii) Effect of spatial walk-off.

    The interest of the NCPM directions is increased by the fact that the walk-off angle of any wave is nil: the beam overlap is complete inside the nonlinear crystal. Under CPM, the interacting waves propagate with different walk-off angles: the conversion efficiency is then attenuated because the different Poynting vectors are not collinear and the beams do not overlap. Type I and type II are not equivalent in terms of walk-off angles. For type I, the two fundamental waves have the same polarization E+ and the same walk-off angle ρ+, which is different from the harmonic one; thus the coordinate systems that are involved in equations ([link] are [(X_1,Y_1,Z)=(X_2,Y_2,Z)=] [(X_\omega^+, Y_\omega^+,Z)] and [(X_3,Y_3,Z)=(X_{2\omega}^-,Y_{2\omega}^-,Z)]. For type II, the two fundamental waves have necessarily different walk-off angles ρ+ and ρ, which forbids the nonlinear interaction beyond the plane where the two fundamental beams are completely separated. In this case we have three different coordinate systems: [(X_1,Y_1,Z)] [=(X_\omega^+,Y_\omega^+,Z)], [(X_2,Y_2,Z)=(X_\omega^-,Y_\omega^-,Z)] and [(X_3,Y_3,Z)=] [ (X_{2\omega}^-,Y_{2\omega}^-,Z)].

    The three coordinate systems are linked by the refraction angles ρ of the three waves as explained in Section[link]. We consider Gaussian transverse profiles: the electric field amplitude is then given by ([link]. In these conditions, the integration of ([link] over ([X,Y,Z]) by assuming [\tan\rho=\rho], the non-depletion of the pump and, in the case of phase matching, [ \Delta k=0] leads to the efficiency ηSHG(L) given by formula ([link] with [\sin c^2(\Delta kL/2)=1] and multiplied by the factor [[G(L,w_o,\rho)]/[\cos^2\rho(2\omega)]] where [\rho(2\omega)] is the harmonic walk-off angle and [G(L,w_o,\rho)] is the walk-off attenuation function.

    For type I, the walk-off attenuation is given by (Boyd et al., 1965[link])[G_I(t)=(\pi^{1/2}/t)\,{\rm erf}(t)-(1/t^2)[1-\exp(-t^2)]]with [t=(\rho L/w_o)\eqno(]and[{\rm erf}(x)=(2/\pi^{1/2})\textstyle \int \limits_{0}^{x}\exp(-t^2)\;{\rm d}t.]

    For uniaxial crystals, [\rho=\rho^e(2\omega)] for a 2oe interaction and [\rho=\rho^e(\omega)] for a 2eo interaction. For the biaxial class, [\rho=\rho^e(2\omega)] for a 2oe interaction and [\rho=\rho^e(\omega)] for a 2eo interaction in the xz and yz planes, [\rho=\rho^o(\omega)] for a 2oe interaction and [\rho=\rho^o(2\omega)] for a 2eo interaction in the xy plane. For any direction of propagation not contained in the principal planes of a biaxial crystal, the fundamental and harmonic waves have nonzero walk-off angles, respectively ρ+(ω) and ρ(2ω). In this case, ([link] can be used with [\rho=|\rho^+(\omega)-\rho^-(2\omega)|].

    • (a) For small t ([t\ll1]), [G_I(t)\simeq 1] and [P^{2\omega}(L)\equiv L^2],

    • (b) For large  t ([t\gg 1]), [G_I(t)\simeq(\pi^{1/2}/t)] and so [P^{2\omega}(L)\equiv L/\rho] according to ([link] with [\Delta k=0].

    For type II, we have (Mehendale & Gupta, 1988[link])[G_{II}(t)=(2/\pi^{1/2})\textstyle \int \limits_{-\infty}^{+\infty}F^2(a,t)\;{\rm d}a]with[F(a,t)=(1/t)\exp(-a^2)\textstyle \int \limits_{0}^{t}\exp[-(a+\tau)^2]\;{\rm d}\tau\eqno(]and[a={r \over w_o}\quad\tau={\rho u\over w_o}\quad t={\rho L\over w_o}.]r and u are the Cartesian coordinates in the walk-off plane where u is collinear with the three wavevectors, i.e. the phase-matching direction.

    [\rho=\rho^e(\omega)] for (oeo) in uniaxial crystals and in the xz and yz planes of biaxial crystals. [\rho=\rho^o(\omega)] in the xy plane of biaxial crystals for an (eoe) interaction.

    For the interactions where ρ(2ω) and ρ(ω) are nonzero, we assume that they are close and contained in the same plane, which is generally the case. Then we classically take ρ to be the maximum value between [|\rho^-(2\omega)-\rho^+(\omega)|] and [|\rho^-(\omega)-\rho^+(\omega)|]. This approximation concerns the (eoe) configuration of polarization in uniaxial crystals and for biaxial crystals in the xz and yz planes, in the xy plane for (oeo) and out of the principal planes for all the configurations of polarization.

    The exact calculation of G, which takes into account the three walk-off angles, ρ(ω), ρ+(ω) and ρ(2ω), was performed in the case where these three angles were coplanar (Asaumi, 1992[link]). The exact calculation in the case of KTiOPO4 (KTP) for type-II SHG at 1.064 µm gives the same result for [L/z_R\,\lt\,1] as for one angle defined as previously (Fève et al., 1995[link]), which includes the parallel-beam limit [L/z_R\,\lt] 0.3–0.4: [z_R=[k(\omega)w_o^2]/2] is the Rayleigh length of the fundamental beam inside the crystal.

    • (a) For [t\ll 1], [G_{II}(t) \simeq 1], leading to the L2 dependence of [P^{2\omega}(L)].

    • (b) For [t\gg 1], [G_{II}(t) \simeq (t_a^2 / t^2)] with [t_a=[(2)^{1/2}\arctan(2^{1/2})]^{1/2}], corresponding to a saturation of [P^{2\omega}(L)] because of the walk-off between the two fundamental beams as shown in Fig.[link].


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      Beam separation in the particular case of type-II (oeo) SHG out of the xy plane of a positive uniaxial crystal or in the xz and yz planes of a positive biaxial crystal. [{\bf S}^{\omega, o}], [{\bf S}^{\omega, e}] and [{\bf S}^{2\omega, o}] are the fundamental and harmonic Poynting vectors; [{\bf k}^\omega] and [{\bf k}^{2\omega}] are the associated wavevectors collinear to the CPM direction. wo is the fundamental beam radius and ρ is the walk-off angle. Lsat is the saturation length.

    The saturation length, Lsat, is defined as [2.3 t_a w_o / \rho], which corresponds to the length beyond which the SHG conversion efficiency varies less than 1% from its saturation value [BP^\omega(0)t_a^2/\rho^2].

    The complete splitting of the two fundamental beams does not occur for type I, making it more suitable than type II for strong focusing. The fundamental beam splitting for type II also leads to a saturation of the acceptance bandwidths δξ ([\xi = \theta, \varphi, T, \lambda]), which is not the case for type I (Fève et al., 1995[link]). The walk-off angles also modify the transversal distribution of the generated harmonic beam (Boyd et al., 1965[link]; Mehendale & Gupta, 1988[link]): the profile is larger than that of the fundamental beam for type I, contrary to type II.

    The walk-off can be compensated by the use of two crystals placed one behind the other, with the same length and cut in the same CPM direction (Akhmanov et al., 1975[link]): the arrangement of the second crystal is obtained from that of the first one by a π rotation around the direction of propagation or around the direction orthogonal to the direction of propagation and contained in the walk-off plane as shown in Fig.[link] for the particular case of type II (oeo) in a positive uniaxial crystal out of the xy plane.


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    Twin-crystal device allowing walk-off compensation for a direction of propagation θPM in the yz plane of a positive uniaxial crystal. ([X,Y,Z]) is the wave frame and ([x,y,z]) is the optical frame. The index surface is given in the yz plane. [{\bf k}^\omega] is the incident fundamental wavevector. The refracted wavevectors [{\bf k}^{\omega,o}], [{\bf k}^{\omega,e}] and [{\bf k}^{2\omega,o}] are collinear and along [{\bf k}^\omega]. [{\bf S}^{\omega,o}], [{\bf S}^{\omega,e}] and [{\bf S}^{2\omega,o}] are the Poynting vectors of the fundamental and harmonic waves. [{\bf E}^{\omega,o}], [{\bf E}^{\omega,e}] and [{\bf E}^{2\omega,o}] are the electric field vectors. ρ is the walk-off angle.

    The twin-crystal device is potentially valid for both types I and II. The relative sign of the effective coefficients of the twin crystals depends on the configuration of polarization, on the relative arrangement of the two crystals and on the crystal class. The interference between the waves generated in the two crystals is destructive and so cancels the SHG conversion efficiency if the two effective coefficients have opposite signs: it is always the case for certain crystal classes and configurations of polarization (Moore & Koch, 1996[link]).

    Such a tandem crystal was used, for example, with KTiOPO4 (KTP) for type-II SHG at [\lambda_\omega =1.3] µm ([\rho=2.47^\circ]) and [\lambda_\omega =2.532] µm ([\rho=2.51^\circ]): the conversion efficiency was about 3.3 times the efficiency in a single crystal of length 2L, where L is the length of each crystal of the twin device (Zondy et al., 1994[link]). The two crystals have to be antireflection coated or contacted in order to avoid Fresnel reflection losses.

    Non-collinear phase matching is another method allowing a reduction of the walk-off, but only in the case of type II (Dou et al., 1992[link]). Fig.[link] illustrates the particular case of (oeo) type-II SHG for a propagation out of the xy plane of a uniaxial crystal, or in the xz or yz plane of a biaxial crystal.


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    Comparison between (a) collinear and (b) special non-collinear phase matching for (oeo) type-II SHG. [{\bf k}^{\omega,o}], [{\bf k}^{\omega,e}] and [{\bf k}^{2\omega,o}] are the wavevectors, [{\bf S}^{\omega,o}], [{\bf S}^{\omega,e}] and [{\bf S}^{2\omega,o}] are the Poynting vectors of the fundamental and harmonic waves, and [{\bf E}^{\omega,o}], [{\bf E}^{\omega,e}] and [{\bf E}^{2\omega,o}] are the electric field vectors; ρ is the walk-off angle in the collinear case and the angle between [{\bf k}^{\omega,o}] and [{\bf k}^{\omega,e}] inside the crystal for the non-collinear interaction.

    In the configuration of special non-collinear phase matching, the angle between the fundamental beams inside the crystal is chosen to be equal to the walk-off angle ρ. Then the associated Poynting vectors [{\bf S}^{\omega,o}] and [{\bf S}^{\omega,e}] are along the same direction, while that of the generated wave deviates from them only by approximately ρ/2. The calculation performed in the case of special non-collinear phase matching indicates that it is possible to increase type-II SHG conversion efficiency by 17% for near-field undepleted Gaussian beams (Dou et al., 1992[link]). Another advantage of such geometry is to turn type II into a pseudo type I with respect to the walk-off, because the saturation phenomenon of type-II CPM is avoided.

  • (iv) Effect of temporal walk-off.

    Even if the SHG is phase matched, the fundamental and harmonic group velocities, [v_g(\omega)=\partial\omega/\partial k], are generally mismatched. This has no effect with continuous wave (c.w.) lasers. For pulsed beams, the temporal separation of the different beams during the propagation can lead to a decrease of the temporal overlap of the pulses. Indeed, this walk-off in the time domain affects the conversion efficiency when the pulse separations are close to the pulse durations. Then after a certain distance, Lτ, the pulses are completely separated, which entails a saturation of the conversion efficiency, for both types I and II (Tomov et al., 1982[link]). Three group velocities must be considered for type II. Type I is simpler, because the two fundamental waves have the same velocity, so [L_\tau=\tau/[v_g^{-1}(\omega)-v_g^{-1}(2\omega)]], which defines the optimum crystal length, where τ is the pulse duration. For type-I SHG of 532 nm in KH2PO4 (KDP), vg(266 nm) [=1.84\times 10^8] m s−1 and vg(532 nm) [=1.94\times 10^8] m s−1, so Lτ [=3.5] mm for 1 ps. For the usual nonlinear crystals, the temporal walk-off must be taken into account for pico- and femtosecond pulses. Non-resonant SHG with undepleted pump and transverse and longitudinal Gaussian beams

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We now consider the general situation where the crystal length can be larger than the Rayleigh length.

The Gaussian electric field amplitudes of the two eigen electric field vectors inside the nonlinear crystal are given by[\eqalignno{E^ \pm(X,Y,Z) &= E_o^ \pm {{w_o }\over {w(Z)}}\exp \Bigg[{- {{({X + \rho ^ + Z})^2 + ({Y + \rho ^ - Z})^2 }\over {w^2 (Z)}}}\Bigg] &\cr&\quad \times \exp \Bigg(i\Bigg\{ k^ \pm Z - \arctan(Z /z_R) &\cr&\quad+ {{k^ \pm \left [(X + \rho ^ + Z)^2 + (Y + \rho ^ - Z)^2 \right]}\over {2Z\left[1 + (z_R^2/Z^2)\right]}}\Bigg\}\Bigg)&\cr&&(}]with [\rho^-=0] for E+ and [\rho^+=0] for E.

([X,Y,Z]) is the wave frame defined in Fig.[link]. [E_o^\pm] is the scalar complex amplitude at [(X,Y,Z) = (0,0,0)] in the vibration planes [\Pi^\pm].

We consider the refracted waves E+ and E to have the same longitudinal profile inside the crystal. Then the [(1/e^2)] beam radius is given by [w(Z)^2 = w_o^2 [1 + ({Z^2 }/{z_R^2 })]], where wo is the minimum beam radius located at [Z = 0] and [z_R=kw_o^2/2], with [k=(k^++k^-)/2]; zR is the Rayleigh length, the length over which the beam radius remains essentially collimated; [k^\pm] are the wavevectors at the wavelength λ in the direction of propagation Z. The far-field half divergence angle is [\Delta\alpha=2/kw_o].

The coordinate systems of ([link] are identical to those of the parallel-beam limit defined in (iii[link]).

In these conditions and by assuming the undepleted pump approximation, the integration of ([link] over ([X,Y,Z]) leads to the following expression of the power conversion efficiency (Zondy, 1991[link]):[\eta_{\rm SHG}(L)={P^{2\omega}(L)\over P^\omega(0)}=CLP^\omega(0){h(L,w_o,\rho,f,\Delta k)\over \cos^2\rho_{2\omega}}]with[C=5.95\times 10^{-2}{2N-1\over N}{d_{\rm eff}^2 \over \lambda_\omega^3}{n_1^\omega+n_2^\omega\over 2}{T^{2\omega}_3T_1^\omega T_2^\omega\over n_3^{2\omega}n_1^\omega n_2^\omega}\quad({\rm W}^{-1}\;{\rm m}^{-1})\eqno(]in the same units as equation ([link].

For type I, [n_1^\omega=n_2^\omega], [T_1^\omega=T_2^\omega], and for type II [n_1^\omega\neq n_2^\omega], [T_1^\omega\neq T_2^\omega].

The attenuation coefficient is written[h(L,w_o,\rho,f,\Delta k)=[2z_R(\pi)^{1/2}/L]\textstyle \int \limits_{-\infty}^{+\infty}|H(a)|^2\exp(-4a^2)\;{\rm d}a]with[\displaylines{H(a)={1\over (2\pi)^{1/2}}\displaystyle \int \limits_{-fL/z_R}^{L(1-f)/z_R}{{\rm d}\tau\over 1+i\tau}\exp\left[-\gamma^2\left(\tau+{fL \over z_R}\right)^2-i\sigma\tau\right]\cr\hbox{for type I: }\gamma=0\hbox{ and }\sigma=\Delta kz_R+4{\rho z_R\over w_o}a\cr \hbox{for type II: }\gamma={\rho z_R\over w_o(2)^{1/2}}\hbox{ and }\sigma=\Delta kz_R+2{\rho z_R\over w_o}a,\cr\hfill(}]where f gives the position of the beam waist inside the crystal: [f=0] at the entrance and [f=1] at the exit surface. The definition and approximations relative to ρ are the same as those discussed for the parallel-beam limit. Δk is the mismatch parameter, which takes into account first a possible shift of the pump beam direction from the collinear phase-matching direction and secondly the distribution of mismatch, including collinear and non-collinear interactions, due to the divergence of the beam, even if the beam axis is phase-matched.

The computation of [h(L,w_o,\rho,f,\Delta k)] allows an optimization of the SHG conversion efficiency which takes into account [L/z_R], the waist location f inside the crystal and the phase mismatch Δk.

Fig.[link] shows the calculated waist location which allows an optimal SHG conversion efficiency for types I and II with optimum phase matching. From Fig.[link], it appears that the optimum waist location for type I, which leads to an optimum conversion efficiency, is exactly at the centre of the crystal, [f_{\rm opt} = 0.5]. For type II, the focusing ([L/z_R]) is stronger and the walk-off angle is larger, and the optimum waist location is nearer the entrance of the crystal. These facts can be physically understood: for type I, there is no walk-off for the fundamental beam, so the whole crystal length is efficient and the symmetrical configuration is obviously the best one; for type II, the two fundamental rays can be completely separated in the waist area, which has the strongest intensity, when the waist location is far from the entrance face; for a waist location nearer the entrance, the waist area can be selected and the enlargement of the beams from this area allows a spatial overlap up to the exit face, which leads to a higher conversion efficiency.


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Position fopt of the beam waist for different values of walk-off angles and [L/z_R], leading to an optimum SHG conversion efficiency. The value [f_{\rm opt}=0.5] corresponds to the middle of the crystal and [f_{\rm opt}=0] corresponds to the entrance surface (Fève & Zondy, 1996[link]).

The divergence of the pump beam imposes non-collinear interactions such that it could be necessary to shift the direction of propagation of the beam from the collinear phase-matching direction in order to optimize the conversion efficiency. This leads to the definition of an optimum phase-mismatch parameter [\Delta k_{\rm opt}] ([\neq 0]) for a given [L/z_R] and a fixed position of the beam waist f inside the crystal.

The function [h(L,w_o,\rho,f_{\rm opt},\Delta k_{\rm opt})], written [h_m(B,L)], is plotted in Fig.[link] as a function of [L/z_R] for different values of the walk-off parameter, defined as B = [(1/2)\rho\{[(k_o^\omega] [+] [k_e^\omega)/2]L\}^{1/2}], at the optimal waist location and phase mismatch.


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Optimum walk-off function [h_m(B,L)] as a function of [L/z_R] for various values of [B=(1/2)\rho\{[(k_o^\omega+k_e^\omega)/2]L\}^{1/2}]. The curve at [B=0] is the same for both type-I and type-II phase matching. The full lines at [B\ne 0] are for type II and the dashed line at [B=16] is for type I. (From Zondy, 1990[link]).

Consider first the case of angular NCPM ([B=0]) where type-I and -II conversion efficiencies obviously have the same [L/z_R] evolutions. An optimum focusing at [L/z_R=5.68] exists which defines the optimum focusing [z_{R_{\rm opt}}] for a given crystal length or the optimal length [L_{\rm opt}] for a given focusing. The conversion efficiency decreases for [L/z_R>5.68] because the increase of the `average' beam radius over the crystal length due to the strong focusing becomes more significant than the increased peak power in the waist area.

In the case of angular CPM ([B\ne 0]), the [L/z_R] variation of type-I conversion efficiency is different from that of type II. For type I, as B increases, the efficiency curves keep the same shape, with their maxima abscissa shifting from [L/z_R=5.68] ([B=0]) to 2.98 ([B=16]) as the corresponding amplitudes decrease. For type II, an optimum focusing becomes less and less appearent, while [(L/z_R)_{\rm opt}] shifts to much smaller values than for type I for the same variation of B; the decrease of the maximum amplitude is stronger in the case of type II. The calculation of the conversion efficiency as a function of the crystal length L at a fixed [z_R] shows a saturation for type II, in contrast to type I. The saturation occurs at [B\simeq 3] with a corresponding focusing parameter [L/z_R\simeq 0.4], which is the limit of validity of the parallel-beam approximation. These results show that weak focusing is suitable for type II, whereas type I allows higher focusing.

The curves of Fig.[link] give a clear illustration of the walk-off effect in several usual situations of crystal length, walk-off angle and Gaussian laser beam. The SHG conversion efficiency is calculated from formula ([link] and from the function ([link] at fopt and Δkopt.


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Type-I and -II conversion efficiencies calculated as a function of [L/z_R] for different typical walk-off angles ρ: (a) and (c) correspond to a fixed focusing condition (wo = 30 µm); the curves (b) and (d) are plotted for a constant crystal length (L = 5 mm); all the calculations are performed with the same effective coefficient (deff = 1 pm V−1), refractive indices ([n_3^{2\omega}n_1^\omega n_2^\omega= 5.83]) and fundamental power [Pω(0 = 1 W]. B is the walk-off parameter defined in the text (Fève & Zondy, 1996[link]). Non-resonant SHG with depleted pump in the parallel-beam limit

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The analytical integration of the three coupled equations ([link] with depletion of the pump and phase mismatch has only been done in the parallel-beam limit and by neglecting the walk-off effect (Armstrong et al., 1962[link]; Eckardt & Reintjes, 1984[link]; Eimerl, 1987[link]; Milton, 1992[link]). In this case, the three coordinate systems of equations ([link] are identical, ([X,Y,Z]), and the general solution may be written in terms of the Jacobian elliptic function [{\rm sn}(m,\alpha)].

For the simple case of type I, i.e. [E_1^\omega(X,Y,Z)=] [ E_2^\omega(X,Y,Z) =] [ E^\omega(X,Y,Z)= ] [ E_{\rm tot}^\omega(X,Y,Z)/(2^{1/2})], the exit second harmonic intensity generated over a length L is given by (Eckardt & Reintjes, 1984[link])[I^{2\omega}(X,Y,L)=I_{\rm tot}^\omega(X,Y,0)T^{2\omega}T^{\omega}v_b^2{\rm sn}^2\left[{\Gamma(X,Y)L\over v_b}, v_b^4\right].\eqno(][I_{\rm tot}^\omega(X,Y,0) = 2 I^\omega(X,Y,0)] is the total initial fundamental intensity, [T^{2\omega}] and [T^\omega] are the transmission coefficients, [{1 \over v_b}={\Delta s\over 4}+\left[1+\left({\Delta s\over 4}\right)^2\right]^{1/2}]with[\Delta s=(k^{2\omega}-k^{\omega})/\Gamma]and[\Gamma(X,Y)={\omega d_{\rm eff} \over cn^{2\omega}}(T^\omega)^{1/2}|E_{\rm tot}^\omega(X,Y,0)|.\eqno(]For the case of phase matching ([k^\omega = k^{2\omega}], [T^\omega = T^{2\omega}]), we have [ \Delta s=0] and [v_b=1], and the Jacobian elliptic function [{\rm sn}(m,1)] is equal to [\tanh(m)]. Then formula ([link] becomes[I^{2\omega}(X,Y,L)=I_{\rm tot}^\omega(X,Y,0)(T^\omega)^2\tanh^2[\Gamma(X,Y)L], \eqno(]where [\Gamma(X,Y)] is given by ([link].

The exit fundamental intensity [I^\omega(X,Y,L)] can be established easily from the harmonic intensity ([link] according to the Manley–Rowe relations ([link], i.e.[I^{\omega}(X,Y,L)=I_{\rm tot}^\omega(X,Y,0)(T^\omega)^2{\rm sech}^2[\Gamma(X,Y)L].\eqno(]For small [\Gamma L], the functions [\tanh^2(\Gamma L) \simeq \Gamma^2L^2] and [{\rm sn}^2[(\Gamma L/v_b),v_b^4]\simeq\sin^2(\Gamma L/v_b)] with [v_b\simeq 2/\Delta s].

The first consequence of formulae ([link]–([link] is that the various acceptance bandwidths decrease with increasing ΓL. This fact is important in relation to all the acceptances but in particular for the thermal and angular ones. Indeed, high efficiencies are often reached with high power, which can lead to an important heating due to absorption. Furthermore, the divergence of the beams, even small, creates a significant dephasing: in this case, and even for a propagation along a phase-matching direction, formula ([link] is not valid and may be replaced by ([link] where [k(2\omega) - k(\omega)] is considered as the `average' mismatch of a parallel beam.

In fact, there always exists a residual mismatch due to the divergence of real beams, even if not focused, which forbids asymptotically reaching a 100% conversion efficiency: [I^{2\omega}(L)] increases as a function of ΓL until a maximum value has been reached and then decreases; [I^{2\omega}(L)] will continue to rise and fall as ΓL is increased because of the periodic nature of the Jacobian elliptic sine function. Thus the maximum of the conversion efficiency is reached for a particular value (ΓL)opt. The determination of (ΓL)opt by numerical computation allows us to define the optimum incident fundamental intensity [I_{\rm opt}^\omega] for a given phase-matching direction, characterized by K, and a given crystal length L.

The crystal length must be optimized in order to work with an incident intensity [I_{\rm opt}^\omega] smaller than the damage threshold intensity [I_{\rm dam}^\omega] of the nonlinear crystal, given in Section 1.7.5[link] for the main materials.

Formula ([link] is established for type I. For type II, the second harmonic intensity is also an sn2 function where the intensities of the two fundamental beams [I_1^\omega(X,Y,0)] and [I_2^\omega(X,Y,0)], which are not necessarily equal, are taken into account (Eimerl, 1987[link]): the tanh2 function is valid only if perfect phase matching is achieved and if [I_1^\omega(X,Y,0)=I_2^\omega(X,Y,0)], these conditions being never satisfied in real cases.

The situations described above are summarized in Fig.[link].


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Schematic SHG conversion efficiency for different situations of pump depletion and dephasing. (a) No depletion, no dephasing, [\eta = \Gamma^2L^2]; (b) no depletion with constant dephasing δ, [\eta = \Gamma^2L^2\sin c^2\delta]; (c) depletion without dephasing, [\eta = \tanh^2(\Gamma L)]; (d) depletion and dephasing, [\eta] [=] [\eta_m{\rm sn}^2(\Gamma L/v_b,v_b^4)].

We give the example of type-II SHG experiments performed with a 10 Hz injection-seeded single-longitudinal-mode ([N=1]) 1064 nm Nd:YAG (Spectra-Physics DCR-2A-10) laser equipped with super Gaussian mirrors; the pulse is 10 ns in duration and is near a Gaussian single-transverse mode, the beam radius is 4 mm, non-focused and polarized at π/4 to the principal axes of a 10 mm long KTP crystal (Lδθ = 15 mrad cm, Lδϕ = 100 mrad cm). The fundamental energy increases from 78 mJ (62 MW cm−2) to 590 mJ (470 MW cm−2), which correponds to the damage of the exit surface of the crystal; for each experiment, the crystal was rotated in order to obtain the maximum conversion efficiency. The peak power SHG conversion efficiency is estimated from the measured energy conversion efficiency multiplied by the ratio between the fundamental and harmonic pulse duration ([\tau_\omega/\tau_{2\omega}=2^{1/2}]). It increases from 50% at 63 MW cm−2 to a maximum value of 85% at 200 MW cm−2 and decreases for higher intensities, reaching 50% at 470 MW cm−2 (Boulanger, Fejer et al., 1994[link]).

The integration of the intensity profiles ([link] and ([link] is obvious in the case of incident fundamental beams with a flat energy distribution ([link]. In this case, the fundamental and harmonic beams inside the crystal have the same profile and radius as the incident beam. Thus the powers are obtained from formulae ([link] and ([link] by expressing the intensity and electric field modulus as a function of the power, which is given by ([link] with [m=1].

For a Gaussian incident fundamental beam, ([link], the fundamental and harmonic beams are not Gaussian (Eckardt & Reintjes, 1984[link]; Pliszka & Banerjee, 1993[link]).

All the previous intensities are the peak values in the case of pulsed beams. The relation between average and peak powers, and then SHG efficiencies, is much more complicated than the ratio [\tau^{2\omega}/\tau^\omega] of the undepleted case. Resonant SHG

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When the single-pass conversion efficiency SHG is too low, with c.w. lasers for example, it is possible to put the nonlinear crystal in a Fabry–Perot cavity external to the pump laser or directly inside the pump laser cavity, as shown in Figs.[link](b) and (c). The second solution, described later, is generally used because the available internal pump intensity is much larger.

We first recall some basic and simplified results of laser cavity theory without a nonlinear medium. We consider a laser in which one mirror is 100% reflecting and the second has a transmission T at the laser pulsation ω. The power within the cavity, Pin(ω), is evaluated at the steady state by setting the round-trip saturated gain of the laser equal to the sum of all the losses. The output laser cavity, Pout(ω), is given by (Siegman, 1986[link])[P_{\rm out}(\omega)=TP_{\rm in}(\omega)]with[P_{\rm in}(\omega)={2g_oL'-(\gamma+T)\over 2S(T+\gamma)}.\eqno(][L'] is the laser medium length, [g_o=\sigma N_o] is the small-signal gain coefficient per unit length of laser medium, σ is the stimulated-emission cross section, No is the population inversion without oscillation, S is a saturation parameter characteristic of the nonlinearity of the laser transition, and [\gamma=\gamma_L=2\alpha _LL'+\beta] is the loss coefficient where αL is the laser material absorption coefficient per unit length and β is another loss coefficient including absorption in the mirrors and scattering in both the laser medium and mirrors. For given go, S, αL, β and [L'], the output power reaches a maximum value for an optimal transmission coefficient Topt defined by [[\partial P_{\rm out}(\omega)/\partial T]_{T_{\rm opt}}=0], which gives[T_{\rm opt}=(2g_oL'\gamma)^{1/2}-\gamma.\eqno(]The maximum output power is then given by[P_{\rm out}^{\rm max}(\omega)=(1/2S)[(2g_oL')^{1/2}-\gamma^{1/2}]^2.\eqno(]

In an intracavity SHG device, the two cavity mirrors are 100% reflecting at ω but one mirror is perfectly transmitting at 2ω. The presence of the nonlinear medium inside the cavity then leads to losses at ω equal to the round-trip-generated second harmonic (SH) power: half of the SH produced flows in the forward direction and half in the backward direction. Hence the highly transmitting mirror at 2ω is equivalent to a nonlinear transmission coefficient at ω which is equal to twice the single-pass SHG conversion efficiency ηSHG.

The fundamental power inside the cavity Pin(ω) is given at the steady state by setting, for a round trip, the saturated gain equal to the sum of the linear and nonlinear losses. Pin(ω) is then given by ([link], where T and γ are (Geusic et al., 1968[link]; Smith, 1970[link])[T=2\eta_{\rm SHG}=[P_{\rm out}(2\omega)/P_{\rm in}(\omega)]\eqno(]and[\gamma=\gamma_L+\gamma_{NL}.\eqno(]ηSHG is the single-pass conversion efficiency. γL and γNL are the loss coefficients at ω of the laser medium and of the nonlinear crystal, respectively. L is the nonlinear medium length. The two faces of the nonlinear crystal are assumed to be antireflection-coated at ω.

In the undepleted pump approximation, the backward and forward power generated outside the nonlinear crystal at 2ω is[P_{\rm out}(2\omega)=2KP_{\rm in}^2(\omega)\eqno(]with[K=B(L^2/w_o^2)\sin c^2(\Delta kL/2), ]where[B={32\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_\omega^2}{T^{2\omega}_3T_1^\omega T_2^\omega\over n^{2\omega}_3n_1^\omega n_2^\omega}\quad({\rm W}^{-1}).]

The intracavity SHG conversion efficiency is usually defined as the ratio of the SH output power to the maximum output power that would be obtained from the laser without the nonlinear crystal by optimal linear output coupling.

Maximizing ([link] with respect to K according to ([link], ([link] and ([link] gives (Perkins & Fahlen, 1987[link])[K_{\rm opt}=(\gamma_L+\gamma_{NL})S\eqno(]and[P_{\rm out}^{\rm max}(2\omega)=(1/2S)[(2g_oL')^{1/2}-(\gamma_L+\gamma_{NL})^{1/2}]^2.\eqno(]([link] shows that for the case where [\gamma_{\rm NL}\ll\gamma_L] ([\gamma\simeq\gamma_L]), the maximum SH power is identically equal to the maximum fundamental power, ([link], available from the same laser for the same value of loss, which, according to the previous definition of the intracavity efficiency, corresponds to an SHG conversion efficiency of 100%. [P_{\rm out}^{\rm max}(2\omega)] strongly decreases as the losses ([\gamma_L + \gamma_{\rm NL}]) increase . Thus an efficient intracavity device requires the reduction of all losses at ω and 2ω to an absolute minimum.

([link] indicates that Kopt is independent of the operating power level of the laser, in contrast to the optimum transmitting mirror where Topt, given by ([link], depends on the laser gain. Kopt depends only on the total losses and saturation parameter. For given losses, the knowledge of Kopt allows us to define the optimal parameters of the nonlinear crystal, in particular the figure of merit, [d_{\rm eff}^2/n_3^{2\omega}n_1^\omega n_2^\omega] and the ratio (L/wo)2, in which the walk-off effect and the damage threshold must also be taken into account.

Some examples: a power of 1.1 W at 0.532 µm was generated in a TEMoo c.w. SHG intracavity device using a 3.4 mm Ba2NaNb5O15 crystal within a 1.064 µm Nd:YAG laser cavity (Geusic et al., 1968[link]). A power of 9.0 W has been generated at 0.532 µm using a more complicated geometry based on an Nd:YAG intracavity-lens folded-arm cavity configuration using KTP (Perkins & Fahlen, 1987[link]). High-average-power SHG has also been demonstrated with output powers greater than 100 W at 0.532 µm in a KTP crystal inside the cavity of a diode side-pumped Nd:YAG laser (LeGarrec et al., 1996[link]).

For type-II phase matching, a rotated quarter waveplate is useful in order to reinstate the initial polarization of the fundamental waves after a round trip through the nonlinear crystal, the retardation plate and the mirror (Perkins & Driscoll, 1987[link]).

If the nonlinear crystal surface on the laser medium side has a 100% reflecting coating at 2ω and if the other surface is 100% transmitting at 2ω, it is possible to extract the full SH power in one direction (Smith, 1970[link]). Furthermore, such geometry allows us to avoid losses of the backward SH beam in the laser medium and in other optical components behind.

External-cavity SHG also leads to good results. The resonated wave may be the fundamental or the harmonic one. The corresponding theoretical background is detailed in Ashkin et al. (1966[link]). For example, a bow-tie configuration allowed the generation of 6.5 W of TEMoo c.w. 0.532 µm radiation in a 6 mm LiB3O5 (LBO) crystal; the Nd:YAG laser was an 18 W c.w. laser with an injection-locked single frequency (Yang et al., 1991[link]). Third harmonic generation (THG)

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Fig.[link] shows the three possible ways of achieving THG: a cascading interaction involving two χ(2) processes, i.e. [\omega+\omega=2\omega] and [\omega+2\omega=3\omega], in two crystals or in the same crystal, and direct THG, which involves χ(3), i.e. [\omega+\omega+\omega=3\omega].


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Configurations for third harmonic generation. (a) Cascading process SHG ([\omega+\omega=2\omega]): SFG ([\omega+2\omega=3\omega]) in two crystals NLC1 and NLC2 and (b) in a single nonlinear crystal NLC; (c) direct process THG ([\omega+\omega+\omega=3\omega]) in a single nonlinear crystal NLC. SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) in different crystals

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We consider the case of the situation in which the SHG is phase-matched with or without pump depletion and in which the sum-frequency generation (SFG) process ([\omega+2\omega=3\omega]), phase-matched or not, is without pump depletion at [\omega] and [2\omega]. All the waves are assumed to have a flat distribution given by ([link] and the walk-off angles are nil, in order to simplify the calculations.

This configuration is the most frequently occurring case because it is unusual to get simultaneous phase matching of the two processes in a single crystal. The integration of equations ([link] over Z for the SFG in the undepleted pump approximation with [E_1^\omega(Z_{\rm SFG}=0)=] [E_1^\omega(L_{\rm SHG})], [E_2^{2\omega}(Z_{\rm SFG}=0)=] [E_2^{2\omega}(L_{\rm SHG})] and [E_3^{3\omega}(Z_{\rm SFG}=0)=] [], followed by the integration over the cross section leads to[\displaylines{P^{3\omega}(L_{\rm SFG})\hfill\cr\quad=B_{\rm SFG}[aP^\omega(L_{\rm SHG})]P^{2\omega}(L_{\rm SHG}){L^2_{\rm SFG}\over w_o^2}\sin c^2{\Delta k_{\rm SFG}L_{\rm SFG}\over 2}\quad({\rm W})\hfill}]with[\displaylines{B_{\rm SFG}={72\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_\omega^2}{T^{3\omega}_3T_1^\omega T_2^{2\omega}\over n^{3\omega}_3n_1^\omega n_2^{2\omega}}\quad({\rm W}^{-1})\cr a=1\hbox{ for type-I SHG,}\quad a={\textstyle{1 \over 2}}\hbox{ for type-II SHG}.\cr\hfill(}]Pω(LSHG) and P2ω(LSHG) are the fundamental and harmonic powers, respectively, at the exit of the first crystal. LSHG and LSFG are the lengths of the first and the second crystal, respectively. [\Delta k_{\rm SFG} = k^{3\omega} - (k^\omega + k^{2\omega})] is the SFG phase mismatch. λω is the fundamental wavelength. The units and other parameters are as defined in ([link].

For type-II SHG, the fundamental waves are polarized in two orthogonal vibration planes, so only half of the fundamental power can be used for type-I, -II or -III SFG ([a=1/2]), in contrast to type-I SHG ([a=1]). In the latter case, and for type-I SFG, it is necessary to set the fundamental and second harmonic polarizations parallel.

The cascading conversion efficiency is calculated according to ([link] and ([link]; the case of type-I SHG gives, for example,[\eqalignno{\eta_{\rm THG}(L_{\rm SHG},L_{\rm SFG})&={P^{3\omega}(L_{\rm SFG})\over P_{\rm tot}^\omega(0)}&\cr &=B_{\rm SFG}(T^\omega)^4P_{\rm tot}^\omega(0)\tanh^2(\Gamma L_{\rm SHG})&\cr&\quad\times{\rm sech}^2(\Gamma L_{\rm SHG}){L^2_{\rm SFG}\over w_o^2}\sin c^2\left({\Delta k_{\rm SFG}L_{\rm SFG}\over 2}\right),&\cr&&(}]where Γ is as in ([link].

(nω, Tω) are relative to the phase-matched SHG crystal and ([n_1^\omega,n_2^{2\omega},n_3^{3\omega}, T_1^\omega,T_2^{2\omega},T_3^{3\omega}]) correspond to the SFG crystal.

In the undepleted pump approximation for SHG, ([link] becomes (Qiu & Penzkofer, 1988[link])[\displaylines{\eta_{\rm THG}(L_{\rm SHG},L_{\rm SFG})\hfill\cr\quad=BT^\omega\left[{P^\omega(0)\over w_o^2}\right]^2L^2_{\rm SHG}L^2_{\rm SFG}\sin c^2\left({\Delta k_{\rm SFG}L_{\rm SFG}\over 2}\right)\hfill\cr\hfill(}]with[\eqalign{B&=B_{\rm SHG}\cdot B_{\rm SFG}\cr&={576\pi^2\over\varepsilon_o^2c^2}\left({2N-1\over N}\right)^2{d_{{\rm eff}_{\rm SHG}}^2d_{{\rm eff}_{\rm SFG}}^2\over \lambda_\omega^4}\left({T_{\rm SHG}^3\over n_{\rm SHG}^3}\right)\left({T_{\rm SFG}^3\over n_{\rm SFG}^3}\right)}] in W−2, where[{T_{\rm SHG}^3\over n_{\rm SHG}^3}={(T^\omega)^3\over(n^\omega)^3}\;\hbox{ and }\;{T_{\rm SFG}^3\over n_{\rm SFG}^3}={T_3^{3\omega}T_1^\omega T_2^{2\omega}\over n_3^{3\omega}n_1^\omega n_2^{2\omega}}.]The units are the same as in ([link].

A more general case of SFG, where one of the two pump beams is depleted, is given in Section[link]. SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) in the same crystal

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When the SFG conversion efficiency is sufficiently low in comparison with that of the SHG, it is possible to integrate the equations relative to SHG and those relative to SFG separately (Boulanger, Fejer et al., 1994[link]). In order to compare this situation with the example taken for the previous case, we consider a type-I configuration of polarization for SHG. By assuming a perfect phase matching for SHG, the amplitude of the third harmonic field inside the crystal is (Boulanger, 1994[link])[\eqalignno{E^{3\omega}(X,Y,Z)&=jK^{3\omega}(\varepsilon_o\chi_{{\rm eff}_{\rm SFG}})&\cr&\quad\times\textstyle \int \limits_{0}^{L}E_{\rm tot}^\omega(X,Y,Z)E^{2\omega}(X,Y,Z)\exp(j\Delta k_{\rm SFG}Z)\,\,{\rm d}Z&\cr&&(}]with[\eqalignno{E^{2\omega}(X,Y,Z)&=(T^\omega)^{1/2}|E_{\rm tot}^\omega(0)|\tanh(\Gamma Z)&\cr\hbox{and }\,\,E_{\rm tot}^\omega(X,Y,Z)&=(T^\omega)^{1/2}|E_{\rm tot}^\omega(0)|\,{\rm sech}(\Gamma Z).&\cr&&(}]Γ is as in ([link].

([link]) can be analytically integrated for undepleted pump SHG; [{\rm sech}(m)\rightarrow 1], [\tanh(m)\rightarrow m], and so we have[\eta_{\rm THG}(L)=P^{3\omega}(L)/P_{\rm tot}^{\omega}(0)\eqno(]with[\displaylines{P^{3\omega}(L)\hfill\cr\quad={576\pi^2\over\varepsilon_o^2c^2}\left({2N-1\over N}\right)^2T^{3\omega}{d_{{\rm eff}_{\rm SHG}}^2d_{{\rm eff}_{\rm SFG}}^2\over n^{3\omega}(n^\omega)^3(n^{2\omega})^2}{[T^\omega P^\omega_{\rm tot}(0)]^3\over w_o^4\lambda_\omega^4}J(L),\hfill}]where the integral J(L) is[J(L)=\left|\textstyle \int \limits_{0}^{L}Z\exp(i\Delta k_{\rm SFG}Z)\;{\rm d}Z\right|^2.\eqno(]

For a nonzero SFG phase mismatch, [\Delta k_{\rm SFG}\ne 0],[J(L)\simeq L^2/(\Delta k_{\rm SFG})^2.\eqno(]

For phase-matched SFG, [\Delta k_{\rm SFG}=0],[J(L)=L^4/4.\eqno(]

Therefore ([link] according to ([link] is equal to ([link] with [L_{\rm SHG}=L_{\rm SFG}=L/2], [\Delta k_{\rm SFG}=0] and 100% transmission coefficients at ω and 2ω between the two crystals. Direct THG ([\omega+\omega+\omega=3\omega])

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As for the cascading process, we consider a flat plane wave which propagates in a direction without walk-off. The integration of equations ([link] over the crystal length L, with [E_4^{3\omega}(X,Y,0)=0] and in the undepleted pump approximation, leads to[\eqalignno{E_4^{3\omega}(X,Y,L)&=jK^{3\omega}_4[\varepsilon_o\chi^{(3)}_{\rm eff}]E_1^{\omega}(X,Y,0)E_2^{\omega}(X,Y,0)E_3^{\omega}(X,Y,0)&\cr&\quad\times L\sin c[(\Delta k\cdot L)/2]\exp(-j\Delta kL/2).&\cr&&(}]

According to ([link] and ([link], the integration of ([link] over the cross section, which is the same for the four beams, leads to[\eta_{\rm THG}(L)={P^{3\omega}(L)\over P^\omega(0)}=B_{\rm THG}[P^\omega(0)]^2{L^2\over w_o^4}\sin c^2[(\Delta k\cdot L)/2]]with[B_{\rm THG}={576\over \varepsilon_o^2c^2}{d_{\rm eff}^2\over\lambda_\omega^2}{T_4^{3\omega}(T_1^\omega)^2T_2^\omega\over n_4^{3\omega}(n_1^\omega)^2n_2^\omega}\quad({\rm m}^{2}\;{\rm W}^{-2}),\eqno(]where [d_{\rm eff}=(1/4)\chi_{\rm eff}^{(3)}] is in m2 V−2 and λω is in m. The statistical factor is assumed to be equal to 1, which corresponds to a longitudinal single-mode laser.

The different types of phase matching and the associated relations and configurations of polarization are given in Table[link] by considering the SFG case with [\omega_1=\omega_2=\omega_3=\omega_4/3]. Sum-frequency generation (SFG)

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SHG ([\omega+\omega=2\omega]) and SFG ([\omega+2\omega=3\omega]) are particular cases of three-wave SFG. We consider here the general situation where the two incident beams at ω1 and ω2, with [\omega_1\,\lt\,\omega_2], interact with the generated beam at ω3, with [\omega_3=\omega_1+\omega_2], as shown in Fig.[link]. The phase-matching configurations are given in Table[link].


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Frequency up-conversion process [\omega_1+\omega_2=\omega_3]. The beam at ω1 is mixed with the beam at ω2 in the nonlinear crystal NLC in order to generate a beam at ω3. [P^{\omega_1,\omega_2,\omega_3}] are the different powers.

From the general point of view, SFG is a frequency up-conversion parametric process which is used for the conversion of laser beams at low circular frequency: for example, conversion of infrared to visible radiation.

The resolution of system ([link] leads to Jacobian elliptic functions if the waves at ω1 and ω2 are both depleted. The calculation is simplified in two particular situations which are often encountered: on the one hand undepletion for the waves at ω1 and ω2, and on the other hand depletion of only one wave at ω1 or ω2. For the following, we consider plane waves which propagate in a direction without walk-off so we consider a single wave frame; the energy distribution is assumed to be flat, so the three beams have the same radius wo. SFG ([\omega_1+\omega_2=\omega_3]) with undepletion at [\omega_1] and [\omega_2]

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The resolution of system ([link] with [E_1(X,Y,0)\ne 0], [E_2(X,Y,0)\ne 0], [\partial E_1(X,Y,Z)/\partial Z=\partial E_2(X,Y,Z)/\partial Z=0] and [E_3(X,Y,0)= 0], followed by integration over [(X,Y)], leads to[\eqalignno{P^{\omega_1}(L)&=(T^{\omega_1})^2P^{\omega_1}(0)&(\cr P^{\omega_2}(L)&=(T^{\omega_2})^2P^{\omega_2}(0)&(\cr P^{\omega_3}(L)&=BP^{\omega_1}(0)P^{\omega_2}(0){L^2\over w_o^2}\sin c^2{\Delta k\cdot L\over 2}&(}%fd1.7.3.83]with[B_{\rm SFG}={72\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_\omega^2}{T^{\omega_3}T^{\omega_1}T^{\omega_2}\over n^{\omega_3}n^{\omega_1}n^{\omega_2}}\quad({\rm W}^{-1})]in the same units as equation ([link]. SFG ([\omega_s+\omega_p=\omega_i]) with undepletion at [\omega_p]

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[(\omega_s,\omega_p,\omega_i) = (\omega_1,\omega_2,\omega_3)] or [(\omega_2,\omega_1,\omega_3)].

The undepleted wave at ωp, the pump, is mixed in the nonlinear crystal with the depleted wave at ωs, the signal, in order to generate the idler wave at [\omega_i=\omega_s+\omega_p]. The integrations of the coupled amplitude equations over ([X,Y,Z]) with [E_s(X,Y,0)\ne 0], [E_p(X,Y,0)\ne 0], [\partial E_p(X,Y,Z)/\partial Z=0] and [E_i(X,Y,0)= 0] give[\eqalignno{P_p(L)&=T_p^2P_p(0)&(\cr P_i(L)&={\omega_i\over \omega_s}P_s(0)\Gamma^2L^2{\sin^2\{\Gamma^2L^2+[(\Delta k\cdot L)/2]^2\}^{1/2}\over \Gamma^2L^2+[(\Delta k\cdot L)/2]^2}&\cr&&(\cr P_s(L)&=P_s(0)\left[1-{\omega_s\over\omega_i}{P_i(L)\over P_s(0)}\right],&(}%fd1.7.3.86]with [\Delta k=k_i-(k_s+k_p)] and [\Gamma^2=[B_sP_p(0)]/w_o^2], where[B_s={8\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over \lambda_s\lambda_i}{T_sT_pT_i\over n_sn_pn_i}.]Thus, even if the up-conversion process is phase-matched ([\Delta k=0]), the power transfers are periodic: the photon transfer efficiency is then 100% for [\Gamma L=(2m+1)(\pi/2)], where m is an integer, which allows a maximum power gain [\omega_i/\omega_s] for the idler. A nonlinear crystal with length [L = (\pi/2\Gamma)] is sufficient for an optimized device.

For a small conversion efficiency, i.e. ΓL weak, ([link] and ([link] become[P_i(L)\simeq P_s(0){\omega_i\over \omega_s}\Gamma^2L^2\sin c^2{\Delta k\cdot L\over2}\eqno(]and [P_s(L)\simeq P_s(0).\eqno(]The expression for Pi(L) with [\Delta k=0] is then equivalent to ([link] with [\omega_p = \omega_1] or [\omega_2], [\omega_i=\omega_3] and [\omega_s = \omega_2] or [\omega_1].

For example, the frequency up-conversion interaction can be of great interest for the detection of a signal, ωs, comprising IR radiation with a strong divergence and a wide spectral bandwidth. In this case, the achievement of a good conversion efficiency, Pi(L)/Ps(0), requires both wide spectral and angular acceptance bandwidths with respect to the signal. The double non-criticality in frequency and angle (DNPM) can then be used with one-beam non-critical non-collinear phase matching (OBNC) associated with vectorial group phase matching (VGPM) (Dolinchuk et al., 1994[link]): this corresponds to the equality of the absolute magnitudes and directions of the signal and idler group velocity vectors i.e. [{\rm d}\omega_i/{\rm d}{\bf k}_i={\rm d}\omega_s/{\rm d}{\bf k}_s]. Difference-frequency generation (DFG)

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DFG is defined by [\omega_3-\omega_1=\omega_2] with [E_2(X,Y,0)=0] or [\omega_3-\omega_2=\omega_1] with [E_1(X,Y,0)=0]. The DFG phase-matching configurations are given in Table[link]. As for SFG, the solutions of system ([link] are Jacobian elliptic functions when the incident waves are both depleted. We consider here the simplified situations of undepletion of the two incident waves and depletion of only one incident wave. In the latter, the solutions differ according to whether the circular frequency of the undepleted wave is the highest one, i.e. ω3, or not. We consider the case of plane waves that propagate in a direction without walk-off and we assume a flat energy distribution for the three beams. DFG ([\omega_p-\omega_s=\omega_i]) with undepletion at [\omega_p] and [\omega_s]

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[(\omega_s,\omega_i,\omega_p) = (\omega_1,\omega_2,\omega_3)] or [(\omega_2,\omega_1,\omega_3)].

The resolution of system ([link] with [E_s(X,Y,0)\ne 0], [E_p(X,Y,0)\ne 0], [\partial E_p(X,Y,Z)/\partial Z=\partial E_s(X,Y,Z)/\partial Z = 0] and [E_i(X,Y,0)=0], followed by integration over ([X,Y]), leads to the same solutions as for SFG with undepletion at ω1 and ω2, i.e. formulae ([link], ([link] and ([link], by replacing ω1 by ωs, ω2 by ωp and ω3 by ωi. A schematic device is given in Fig.[link] by replacing (ω1, ω2, ω3) by (ω1, ω3, ω2) or (ω2, ω3, ω1). DFG ([\omega_s-\omega_p=\omega_i]) with undepletion at [\omega_p]

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[(\omega_s,\omega_i,\omega_p) = (\omega_3,\omega_1,\omega_2)] or [(\omega_3,\omega_2,\omega_1)].

The resolution of system ([link] with [E_s(X,Y,0)\ne 0], [E_p(X,Y,0)\ne 0], [\partial E_p(X,Y,Z)/\partial Z = 0] and [E_i(X,Y,0)=0], followed by the integration over ([X,Y]), leads to the same solutions as for SFG with undepletion at ω1 or ω2: formulae ([link], ([link] and ([link]. DFG ([\omega_p-\omega_s=\omega_i]) with undepletion at [\omega_p] – optical parametric amplification (OPA), optical parametric oscillation (OPO)

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[(\omega_s,\omega_i,\omega_p) = (\omega_1,\omega_2,\omega_3)] or [(\omega_2,\omega_1,\omega_3)].

The initial conditions are the same as in Section[link], except that the undepleted wave has the highest circular frequency. In this case, the integrations of the coupled amplitude equations over ([X,Y,Z]) lead to[P_p(L)=T_p^2P_p(0),\eqno(][P_i(L)=P_s(0){\omega_i\over\omega_s}\Gamma^2L^2{{\rm sinh}^2\{\Gamma^2L^2-[(\Delta k\cdot L)/2]^2\}^{1/2}\over \Gamma^2L^2-[(\Delta k\cdot L)/2]^2}\eqno(]and[\eqalignno{P_s(L)&=P_s(0)\left[1 + {\omega_s\over\omega_i}{P_i(L)\over P_s(0)}\right]&\cr &=P_s(0)\left(1+\Gamma^2L^2{{\rm sinh}^2\{\Gamma^2L^2-[(\Delta k \cdot L)/2]^2\}^{1/2}\over \Gamma^2L^2-[(\Delta k \cdot L)/2]^2}\right)&\cr&&(}]with [\Delta k = k_p - (k_i+k_s)] and [\Gamma^2=[B_iP_p(0)]/w_o^2], where wo is the beam radius of the three beams and [B_i={8\pi\over\varepsilon_oc}{2N-1\over N}{d_{\rm eff}^2\over\lambda_s\lambda_i}{T_sT_pT_i\over n_sn_pn_i}.]The units are the same as in equation ([link].

Equations ([link] and ([link] show that both idler and signal powers grow exponentially. So, firstly, the generation of the idler is not detrimental to the signal power, in contrast to DFG ([\omega_s-\omega_p=\omega_i]) and SFG ([\omega_s+\omega_p=\omega_i]), and, secondly, the signal power is amplified. Thus DFG ([\omega_p-\omega_s=\omega_i]) combines two interesting functions: generation at [\omega_i] and amplification at [\omega_s]. The last function is called optical parametric amplification (OPA).

The gain of OPA can be defined as (Harris, 1969[link])[G(L)=\left|{P_s(L)\over P_s(0)}-1\right|.\eqno(]For example, Baumgartner & Byer (1979[link]) obtained a gain of about 10 for the amplification of a beam at 0.355 µm by a pump at 1.064 µm in a 5 cm long KH2PO4 crystal, with a pump intensity of 28 MW cm−2.

According to ([link], for [\Delta k^2L^2/4\gg\Gamma^2L^2], [{\rm sinh}^2(im)\rightarrow] [-\sin^2(m)] and so the gain is given by[G_{{\rm small\,\, gain}}\simeq \Gamma^2L^2\sin c^2\left({\Delta k\cdot L\over 2}\right).\eqno(]Formula ([link] shows that frequencies can be generated around ωs. The full gain linewidth of the signal, Δωs, is defined as the linewidth leading to a maximum phase mismatch [\Delta k=2\pi/L]. If we assume that the pump wave linewidth is negligible, i.e. [\Delta\omega_p=0], it follows, by expanding Δk in a Taylor series around ωi and ωs, and by only considering the first order, that [\left|\Delta\omega_s^{{\rm small\,\,gain}}\right|=\left|\Delta\omega_i^{{\rm small \,\, gain}}\right|\simeq(2\pi/Lb)\eqno(]with [b=[1/v_g(\omega_i)]-[1/v_g(\omega_s)]], where [v_g(\omega)=\partial \omega/\partial k] is the group velocity.

This linewidth can be termed intrinsic because it exists even if the pump beam is parallel and has a narrow spectral spread.

For type I, the spectral linewidth of the signal and idler waves is largest at the degeneracy: [b=0] because the idler and signal waves have the same polarization and so the same group velocity at degeneracy, i.e. [\omega_i= \omega_s= \omega_p/2]. In this case, it is necessary to consider the dispersion of the group velocity [\partial^2\omega/\partial^2k] for the calculation of Δωs and Δωi. Note that an increase in the crystal length allows a reduction in the linewidth.

For type II, b is never nil, even at degeneracy.

A parametric amplifier placed inside a resonant cavity constitutes an optical parametric oscillator (OPO) (Harris, 1969[link]; Byer, 1973[link]; Brosnan & Byer, 1979[link]; Yang et al., 1993[link]). In this case, it is not necessary to have an incident signal wave because both signal and idler photons can be generated by spontaneous parametric emission, also called parametric noise or parametric scattering (Louisell et al., 1961[link]): when a laser beam at ωp propagates in a χ(2) medium, it is possible for pump photons to spontaneously break down into pairs of lower-energy photons of circular frequencies ωs and ωi with the total photon energy conserved for each pair, i.e [\omega_s+\omega_i=\omega_p]. The pairs of generated waves for which the phase-matching condition is satisfied are the only ones to be efficiently coupled by the nonlinear medium. The OPO can be singly resonant (SROPO) at ωs or ωi (Yang et al., 1993[link]; Chung & Siegman, 1993[link]), doubly resonant (DROPO) at both ωs and ωi (Yang et al., 1993[link]; Breitenbach et al., 1995[link]) or triply resonant (TROPO) (Debuisschert et al., 1993[link]; Scheidt et al., 1995[link]). Two main techniques for the pump injection exist: the pump can propagate through the cavity mirrors, which allows the smallest cavity length; for continuous waves or pulsed waves with a pulsed duration greater than 1 ns, it is possible to increase the cavity length in order to put two 45° mirrors in the cavity for the pump, as shown in Fig.[link]. This second technique allows us to use simpler mirror coatings because they are not illuminated by the strong pump beam.


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Schematic OPO configurations. [P^{\omega_p}] is the pump power. (a) can be a SROPO, DROPO or TROPO and (b) can be a SROPO or DROPO, according to the reflectivity of the cavity mirrors (M1, M2).

The only requirement for making an oscillator is that the parametric gain exceeds the losses of the resonator. The minimum intensity above which the OPO has to be pumped for an oscillation is termed the threshold oscillation intensity Ith. The oscillation threshold decreases when the number of resonant frequencies increases: [I_{\rm th}^{\omega_p}({\rm SROPO})] [>] [I_{\rm th}^{\omega_p}({\rm DROPO})] [>] [I_{\rm th}^{\omega_p}({\rm TROPO})]; on the other hand the instability increases because the condition of simultaneous resonance is critical.

The oscillation threshold of a SROPO or DROPO can be decreased by reflecting the pump from the output coupling mirror M2 in configuration (a) of Fig.[link] (Marshall & Kaz, 1993[link]). It is necessary to pump an OPO by a beam with a smooth optical profile because hot spots could damage all the optical components in the OPO, including mirrors and nonlinear crystals. A very high beam quality is required with regard to other parameters such as the spectral bandwidth, the pointing stability, the divergence and the pulse duration.

The intensity threshold is calculated by assuming that the pump beam is undepleted. For a phase-matched SROPO, resonant at ωs or ωi, and for nanosecond pulsed beams with intensities that are assumed to be constant over one single pass, [I_{\rm th}^{\omega_p}] is given by[I_{\rm th}^{\omega_p}={1.8\over KL^2(1+\gamma)^2}\left\{{25L\over c\tau}+2\alpha L+Ln\left[{1\over (1-T)^{1/2}}\right]+Ln(2)\right\}^2.\eqno(][K=(\omega_s\omega_i\chi_{\rm eff}^2)/[2n(\omega_s)n(\omega_i)n(\omega_p)\varepsilon_oc^3]]; L is the crystal length; γ is the ratio of the backward to the forward pump intensity; τ is the 1/e2 half width duration of the pump beam pulse; and 2α and T are the linear absorption and transmission coefficients at the circular frequency of the resonant wave ωs or ωi. In the nanosecond regime, typical values of [I_{\rm th}^{\omega_p}] are in the range 10–100 MW cm−2.

([link] shows that a small threshold is achieved for long crystal lengths, high effective coefficient and for weak linear losses at the resonant frequency. The pump intensity threshold must be less than the optical damage threshold of the nonlinear crystal, including surface and bulk, and of the dielectric coating of any optical component of the OPO. For example, a SROPO using an 8 mm long KNbO3 crystal ([d_{\rm eff}\simeq 10] pm V−1) as a nonlinear crystal was performed with a pump threshold intensity of 65 MW cm−2 (Unschel et al., 1995[link]): the 3 mm-diameter pump beam was a 10 Hz injection-seeded single-longitudinal-mode Nd:YAG laser at 1.064 µm with a 9 ns pulse of 100 mJ; the SROPO was pumped as in Fig.[link](a) with a cavity length of 12 mm, a mirror M1 reflecting 100% at the signal, from 1.4 to 2 µm, and a coupling mirror M2 reflecting 90% at the signal and transmitting 100% at the idler, from 2 to 4 µm.

For increasing pump powers above the oscillation threshold, the idler and signal powers grow with a possible depletion of the pump.

The total signal or idler conversion efficiency from the pump depends on the device design and pump source. The greatest values are obtained with pulsed beams. As an example, 70% peak power conversion efficiency and 65% energy conversion of the pump to both signal (λs = 1.61 µm) and idler (λi = 3.14 µm) outputs were obtained in a SROPO using a 20 mm long KTP crystal (deff = 2.7 pm V−1) pumped by an Nd:YAG laser (λp=1.064 µm) for eye-safe source applications (Marshall & Kaz, 1993[link]): the configuration is the same as in Fig.[link](a) where M1 has high reflection at 1.61 µm and high transmission at 1.064 µm, and M2 has high reflection at 1.064 µm and a 10% transmission coefficient at 1.61 µm; the Q-switched pump laser produces a 15 ns pulse duration (full width at half maximum), giving a focal intensity around 8 MW cm−2 per mJ of pulse energy; the energy conversion efficiency from the pump relative to the signal alone was estimated to be 44%.

OPOs can operate in the continuous-wave (cw) or pulsed regimes. Because the threshold intensity is generally high for the usual nonlinear materials, the cw regime requires the use of DROPO or TROPO configurations. However, cw-SROPO can run when the OPO is placed within the pump-laser cavity (Ebrahimzadeh et al., 1999[link]). The SROPO in the classical external pumping configuration, which leads to the most practical devices, runs very well with a pulsed pump beam, i.e. Q-switched laser running in the nanosecond regime and mode-locked laser emitting picosecond or femtosecond pulses. For nanosecond operation, the optical parametric oscillation is ensured by the same pulse, because several cavity round trips of the pump are allowed during the pulse duration. It is not possible in the ultrafast regimes (picosecond or femtosecond). In these cases, it is necessary to use synchronous pumping: the round-trip transit time in the OPO cavity is taken to be equal to the repetition period of the pump pulse train, so that the resonating wave pulse is amplified by successive pump pulses [see for example Ruffing et al. (1998[link]) and Reid et al. (1998[link])].

OPOs are used for the generation of a fixed wavelength, idler or signal, but have potential for continuous wavelength tuning over a broad range, from the near UV to the mid-IR. The tuning is based on the dispersion of the refractive indices with the wavelength, the direction of propagation, the temperature or any other variable of dispersion. More particularly, the crystal must be phase-matched for DFG over the widest spectral range for a reasonable variation of the dispersion parameter to be used. Several methods are used: variation of the pump wavelength at a fixed direction, fixed temperature etc.; rotation of the crystal at a fixed pump wavelength, fixed temperature etc.; or variation of the crystal temperature at a fixed pump wavelength, fixed direction etc.

We consider here two of the most frequently encountered methods at present: for birefringence phase matching, angle tuning and pump-wavelength tuning; and the case of quasi phase matching.

  • (i) OPO with angle tuning.

    The function of a tunable OPO is to generate the signal and idler waves over a broad range, Δωs and Δωi, respectively, from a fixed pump wave at ωp. The spectral shifts [\Delta\omega_s=\omega_s^+-\omega_s^-] and [\Delta\omega_i=\omega_i^+-\omega_i^-] are obtained by rotating the nonlinear crystal by an angle [\Delta\alpha=\alpha^+-\alpha^-] in order to achieve phase matching over the spectral range considered: [\omega_pn(\omega_p,\alpha)=] [\omega_in(\omega_i,\alpha)] [+] [\omega_sn(\omega_s,\alpha)] with [\omega_p=\omega_i+\omega_s] from [(\omega_s^+,\omega_i^-,\alpha^\pm)] to [(\omega_s^-,\omega_i^+,\alpha^\mp)], where (−) and (+), respectively, denote the minimum and maximum values of the data considered. Note that [\Delta\omega_s=-\Delta\omega_i] and so [(\Delta\lambda_i/\lambda_i^+\lambda_i^-)=] [ -(\Delta\lambda_s/\lambda_s^+\lambda_s^-)] if the spectral bandwidth of the pump, δωp, is zero.

    In the case of parallelepipedal nonlinear crystals, the tuning rate [\Delta\omega_{i,s}/\Delta\alpha] has to be high because Δα cannot exceed about 30° of arc, i.e. 15° on either side of the direction normal to the plane surface of the nonlinear crystal: in fact, the refraction can lead to an attenuation of the efficiency of the parametric interaction for larger angles. For this reason, a broad-band OPO necessarily requires angular critical phase matching (CPM) directions over a broad spectral range. However, the angular criticality is detrimental to the spectral stability of the signal and idler waves with regard to the pointing fluctuations of the pump beam: a pointing instability of the order of 100 µrad is considered to be acceptable for OPOs based on KTP or BBO crystals. Fig.[link] shows the phase-matching tuning curves λi(α) and λs(α) for (a) BBO pumped at λp = 355 nm and (b) KTP pumped at λp = 1064 nm, where [\alpha=\theta] or [\varphi] is an internal angle: the calculations were carried out using the refractive indices given in Kato (1986[link]) for BBO and in Kato (1991[link]) for KTP.


    Figure | top | pdf |

    Calculated angular tuning curves. θ and [\varphi] are the spherical coordinates of the phase-matching directions. θd is the phase-matching angle of the degeneracy process ([\lambda_i^d=\lambda_s^d=2\lambda_p]). [\lambda_i^o] and [\lambda_s^o] are the idler and signal wavelengths, respectively, generated at θo. Ordinary and extraordinary refer to the polarization.

    The divergence of the pump beam may increase the spectral bandwidths δωs and δωi: the higher the derivatives [\partial\lambda_{i,s}/\partial\alpha] are, the higher the spectral bandwidths for a given pump divergence are. Furthermore, [\partial\lambda_{i,s}/\partial\alpha] vary as a function of the phase-matching angle α. The derivative is a maximum at the degeneracy [\lambda_i=\lambda_s=2\lambda_p], when the idler and signal waves are identically polarized: this is the case for BBO as shown in Fig.[link](a). We give another example of a type-I BBO OPO pumped at 308 nm by a narrow-band injection-seeded ultraviolet XeCl excimer laser (Ebrahimzadeh et al., 1990[link]): the spectral bandwidth, expressed in cm−1 [(\partial\lambda_{i,s}/\lambda_{i,s}^2=\partial\omega_{i,s}/2\pi c)], varies from ~78 cm−1 to ~500 cm−1 for a crystal length of 1.2 cm, corresponding to a signal bandwidth δλs ≃ 1.8 nm at 480 nm and δλs ≃ 18 nm at 600 nm, respectively. The degeneracy is not a particular situation with respect to the derivative of the phase-matching curve when the idler and signal waves are orthogonally polarized as shown in Fig.[link](b) with the example of KTP.

    The way currently used for substantial reduction of the spectral bandwidth is to introduce bandwidth-limiting elements in the OPO cavity, such as a grazing grating associated with a tuning mirror reflecting either the signal or the idler according to the chosen resonant wavelength. The rotations of the nonlinear crystal and of the restricting element have to be synchronized in order to be active over all the wavelength range generated. Narrow bandwidths of about 0.1 cm−1 can be obtained in this way, but the gain of such a device is low. High energy and narrow spectral bandwidth can be obtained at the same time by the association of two OPOs: an OPO pumped at ωp and without a restricting element inside the cavity is seeded by the idler or signal beam emitted by a narrow spectral bandwidth OPO also pumped at ωp.

    The disadvantages of parallelepipedal crystals can be circumvented by using a nonlinear crystal cut as a cylindrical plate, with the cylinder axis orthogonal to the OPO cavity axis and to the plane of the useful phase-matching directions (Boulanger et al., 1999[link]; Pacaud et al., 2000[link]; Fève, Pacaud et al., 2002[link]). Such a geometry allows us to consider any phase-matching range by rotation of the cylinder around its revolution axis. It is then possible to use interactions with a weak angular tuning rate to reduce the spectral bandwidth and increase the stability of the generated beams. Moreover, the propagation of the beams is at normal incidence for any direction, so collinear phase matching can be maintained, leading to better spatial and spectral transverse profiles. Because of the cylindrical geometry of the nonlinear crystal, it is necessary to focus the pump beam and to collect the signal and idler beams with cylindrical lenses. The cavity mirrors, plane or cylindrical, are then placed between the nonlinear crystal and the lenses. The diameter of the crystal being about a few tenths of a millimetre, the associated focal distance is short, i.e. a few millimetres, which leads to a strong spatial filtering effect, preventing the oscillation of beams with a quality factor [M^2] bigger than about 1.5.

  • (ii) OPO with a tuning pump.

    The nonlinear crystal is fixed and the pump frequency can vary over Δωp, leading to a variation of the signal and idler frequencies such that [\Delta\omega_i+\Delta\omega_s=\Delta\omega_p].

    In Fig.[link], the example of N-(4-nitrophenyl)-L-propinol (NPP) pumped between 610 and 621 nm is shown (Ledoux et al., 1990[link]; Khodja et al., 1995a[link]). The phase-matching curve λi,sp) is calculated from the Sellmeier equations of Ledoux et al. (1990[link]) for the case of identical polarizations for the signal and idler waves. The tuning rate is a maximum at the degeneracy, as for angular tuning with identical polarizations.


    Figure | top | pdf |

    Calculated pump wavelength tuning curve. [\lambda_p^d] is the pump wavelength leading to degeneracy for the direction considered ([\theta=12.5], [\varphi=0^\circ]). Ordinary and extraordinary refer to the polarization.

    For any configuration of polarization, the most favourable direction of propagation of an OPO with a tuning pump is a principal axis of the index surface, because the phase matching is angular non-critical and so wavelength critical. In this optimal situation, the OPO has a low sensitivity to the divergence and pointing stability of the pump beam; furthermore, the walk-off angle is nil, which provides a higher conversion efficiency.

  • (iii) Quasi-phase-matched OPO with a tunable periodicity.

    In a QPM device, the interacting frequencies are fixed by the frequency dispersion of the birefringence of the nonlinear material and by the periodicity of the grating. A first possibility is to fabricate a series of gratings with different periodicities in the same nonlinear crystal; the translation of this crystal with respect to the fixed pump beam allows us to address the different gratings and thus to generate different couples (ωs, ωi). Because the tuning is obtained in discrete steps, it is necessary to combine temperature or angle tuning with the translation of the sample in order to interpolate smoothly between the steps. For example, a device based on a periodically poled LiNbO3 (ppLN) crystal with a thickness of 0.5 mm and a length along the periodicity vector of 1 cm has been developed (Myers et al., 1996[link]). A total of 25 gratings with periods between 26 and 32 µm were realized in 0.25 µm increments. The OPO was pumped at 1.064 µm and generated a signal between 1.35 and 1.98 µm, with the corresponding idler between 4.83 and 2.30 µm.

    Fan-shaped gratings have been demonstrated as an alternative approach for continuous tuning (Powers et al., 1998[link]). However, such a structure has the disadvantage of introducing large spectral heterogeneity to the generated beams, because the grating period is not constant over the pump beam diameter.

    Finally, the most satisfactory alternative for continuous tuning is the use of a cylindrical crystal with one single grating (Fève et al., 2001[link]). The variations of the signal and idler wavelengths are then obtained by rotation of the cylinder around its revolution axis, which is orthogonal to the OPO cavity axis and to the plane containing the frame vector [\Lambda]. For a direction of propagation making an angle [\alpha] with [\Lambda], the effective period of the grating as seen by the collinear interacting wavevectors is [\Lambda_{\alpha}=(\Lambda/\cos\alpha)], leading to a continuous spectral tuning. For example, a rotation over an [\alpha] range of 26° of a ppKTP cylinder pumped at 1064 nm leads to a signal tuning range of 520 nm, between 1515 and 2040 nm, while the corresponding idler is tuned over 1340 nm, between 2220 and 3560 nm.

    For an overview of OPO and OPA, the reader may refer to the following special issues of the Journal of the Optical Society of America B: (1993), 10(9), 1656–1794; (1993), 10(11), 2148–2239 and (1995), 12(11), 2084–2310; and to the Handbook of Optics devoted to OPO (Ebrahimzadeh & Dunn, 2000[link]).


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