International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 186215
Section 1.7.3. Propagation phenomena^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France 
We summarize here the main linear optical properties that govern the nonlinear propagation phenomena. The reader may refer to Chapter 1.6 for the basic equations.
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 nonmagnetic and nonconducting medium, Maxwell equations lead to the following wave propagation equation for the Fourier component at the circular frequency ω defined by (1.7.2.15) and (1.7.2.16) (Butcher & Cotter, 1990):where , λ is the wavelength and c is the velocity of light in a vacuum; is the freespace permeability, E(ω) is the electric field vector and P(ω) is the polarization vector.
In the linear regime, , where _{0} is the freespace permittivity and χ^{(1)}(ω) is the firstorder electric susceptibility tensor. Then (1.7.3.1) becomes is the dielectric tensor. In the general case, is a complex quantity i.e. . For the following, we consider a medium for which the losses are small (); it is one of the necessary characteristics of an efficient nonlinear medium. In this case, the dielectric tensor is real: .
The plane wave is a solution of equation (1.7.3.2):() 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 (), contained in the XZ or YZ planes.
is the scalar complex amplitude of the electric field where is the phase, and . 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: corresponds to forward propagation along Z, and 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:() are the spherical coordinates of the direction of the unit wavevector u in the optical frame; () is the optical frame defined in Section 1.7.2.
The spherical coordinates are related to the Cartesian coordinates () byThe refractive indices , , real in the case of a lossless medium, are the two solutions of the Fresnel equation (Yao & Fahlen, 1984):n_{x}(ω), n_{y}(ω) and n_{z}(ω) are the principal refractive indices of the index ellipsoid at the circular frequency ω.
Equation (1.7.3.6) describes a doublesheeted threedimensional 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 () or () is the birefringency. The waves (+) and (−) have the phase velocities and , respectively.
Equation (1.7.3.6) and its dispersion in frequency are often used in nonlinear optics, in particular for the calculation of the phasematching 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):If ω_{i}_{ }or ω_{j} are near an absorption peak, even weak, n^{±}(ω_{i}) can be greater than n^{±}(ω_{j}); this is called abnormal dispersion.
The dielectric displacements , the electric fields , the energy flux given by the Poynting vector and the collinear wavevectors are coplanar and define the orthogonal vibration planes (Shuvalov, 1981). Because of anisotropy, and , and hence and , are noncollinear in the general case as shown in Fig. 1.7.3.1: the walkoff angles, also termed doublerefraction angles, are different in the general case; , , and are the unit vectors associated with , , and , 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.
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 timeaverage Poynting vector is (Yariv & Yeh, 2002) is the energy flow , which is a power per unit area i.e. the intensity, where is the energy of the photon and 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 e^{−}are 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 () to the unit wavevector components () (Shuvalov, 1981):with
The vibration planes relative to the eigen polarization modes are called the neutral vibration planes associated with u: an incident linearly polarized wave with a vibration plane parallel to or 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 .
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) and Sections 1.1.4.1 and 1.6.3.2 ].
The isotropic class corresponds to the equality of the three principal indices: the index surface is a onesheeted sphere, so , for all directions of propagation, and any electric field vector direction is allowed as in an amorphous material.
The uniaxial class is characterized by the equality of two principal indices, called ordinary indices (); the other index is called the extraordinary index (). Then, according to (1.7.3.6), the index surface has one umbilicus along the z axis, , 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). The ordinary sheet is spherical i.e. , so an ordinary wave has no walkoff for any direction of propagation in a uniaxial crystal; the extraordinary sheet is ellipsoidal i.e. . The sign of the uniaxial class is defined by the sign of the birefringence . Thus, according to these definitions, () corresponds to () for the positive class () and to () for the negative class (), as shown in Fig. 1.7.3.2.
The ordinary electric field vector is orthogonal to the optic axis (), and also to the extraordinary electric field vector, leading toThis relation is satisfied when ω_{i} and ω_{j} are equal or different and for any direction of propagation ().
According to these results, the coplanarity of the field vectors imposes the condition that the doublerefraction 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 e^{o} and e^{e} at the circular frequency ω arewith for the positive class and for the negative class. is given byNote that the extraordinary walkoff angle is nil for a propagation along the optic axis () and everywhere in the xy plane ().
In a biaxial crystal, the three principal refractive indices are all different. The graphical representations of the index surfaces are given in Fig. 1.7.3.3 for the positive biaxial class () and for the negative one (), 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.
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:The propagation along the optic axes leads to the internal conical refraction effect (Schell & Bloembergen, 1978; Fève et al., 1994).
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 walkoff properties of the waves are not the same in the plane as in the and planes.
It is impossible to define ordinary and extraordinary waves out of the principal planes of a biaxial crystal: according to (1.7.3.6) and (1.7.3.9), e^{+} and e^{−} have a nonzero projection on the z axis. According to these relations, it appears that e^{+} and e^{−} are not perpendicular, so relation (1.7.3.10) is never verified. The walkoff angles ρ^{+} and ρ^{−} are nonzero, different, and can be calculated from the electric field vectors: or for a positive or a negative optic sign, respectively.
The nonlinear crystals considered here are homogeneous, lossless, nonconducting, without optical activity, nonmagnetic and are optically anisotropic. The nonlinear regime allows interactions between γ waves with different circular frequencies . The Fourier component of the polarization vector at ω_{i} is , where is the nonlinear polarization corresponding to the orders of the power series greater than 1 defined in Section 1.7.2.
Thus the propagation equation of each interacting wave ω_{i} is (Bloembergen, 1965)The γ propagation equations are coupled by :
The complex conjugates come from the relation .
We consider the plane wave, (1.7.3.3), as a solution of (1.7.3.19), 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 doublerefraction 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 ω_{i} are written (), which can be relative to the mode (+) or (−). The systems are then linked by the doublerefraction angles ρ: according to Fig. 1.7.3.1, we have for two waves (+) with , and for two waves (−) with .
The presence of in equations (1.7.3.19) leads to a variation of the γ amplitudes E(ω_{i}) with Z. In order to establish the equations of evolution of the wave amplitudes, we assume that their variations are small over one wavelength λ_{i}, which is usually true. Thus we can stateThis is called the slowly varying envelope approximation.
Stating (1.7.3.20), the wave equation (1.7.3.19) for a forward propagation of a plane wave leads toWe choose the optical frame () for the calculation of all the scalar products , the electric susceptibility tensors being known in this frame.
For a threewave interaction, (1.7.3.21) leads towith , , and , called the phase mismatch. We take by convention .
If ABDP relations, defined in Section 1.7.2.2.1, are verified, then the three tensorial contractions in equations (1.7.3.22) are equal to the same quantity, which we write , where is called the effective coefficient:The same considerations lead to the same kind of equations for a fourwave interaction:The conventions of notation are the same as previously and the phase mismatch is . The effective coefficient isExpressions (1.7.3.23) for and (1.7.3.25) for can be condensed by introducing adequate third and fourthrank tensors to be contracted, respectively, with and . For example, or , and similar expressions. By substituting (1.7.3.8) in (1.7.3.22), we obtain the derivatives of Manley–Rowe relations (1.7.2.40) for a threewave mixing, where is the Z photon flow. Identically with (1.7.3.24), we have for a fourwave mixing.
In the general case, the nonlinear polarization wave and the generated wave travel at different phase velocities, and , respectively, because of the frequency dispersion of the refractive indices in the crystal. Then the work per unit time W(ω_{i}), given in (1.7.2.39), which is done on the generated wave E(ω_{i}, Z) by the nonlinear polarization P^{NL}(ω_{i}, Z), alternates in sign for each phase shift of π during the Zpropagation, which leads to a reversal of the energy flow (Bloembergen, 1965). The length leading to the phase shift of π is called the coherence length, , where Δk is the phase mismatch given by (1.7.3.22) or (1.7.3.24).
The transfer of energy between the waves is maximum for , 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(ω_{i}, Z) also exists: the work of P^{NL}(ω_{i}, Z) on E(ω_{i}, Z) is maximum if these two waves are π/2 out of phase, that is to say if , where ; thus in the case of phase matching, the phase relation is (Armstrong et al., 1962). 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 photonmomentum conservation i.e.with for a threephoton interaction and for a fourphoton interaction.
According to (1.7.3.4), the phasematching condition (1.7.3.26) is expressed as a function of the refractive indices in the direction of propagation considered (); for an interaction where the γ wavevectors are collinear, it is writtenwith(1.7.3.28) is the relation of the energy conservation.
The efficiency of a nonlinear crystal directly depends on the existence of phasematching 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 nondispersive medium [], (1.7.3.27) is always verified for each of the eigen refractive indices n^{+} or n^{−}; then any direction of propagation is a phasematching 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 (1.7.3.6), for each of the three or four refractive indices involved in the phasematching relations, that is to say 2^{3} or 2^{4 }possible combinations of refractive indices for a threewave or a fourwave process, respectively.
For a threewave process, only three combinations among the 2^{3} are compatible with the dispersion in frequency (1.7.3.7) and with the momentum and energy conservations (1.7.3.27) and (1.7.3.28). Thus the phase matching of a threewave interaction is allowed for three configurations of polarization given in Table 1.7.3.1.

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 phasematching relation corresponds to one sumfrequency generation SFG () and two differencefrequency generation processes, DFG () and DFG (). Types II and III are equivalent for SHG because .
For a fourwave process, only seven combinations of refractive indices allow phase matching in the case of normal dispersion; they are given in Table 1.7.3.2 with the corresponding configurations of polarization and types of SFG and DFG.

The convention of designation of the types is the same as for threewave 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 phasematching relations where two waves have the same polarization state, different from the two others. In this case, it is convenient to refer to each phasematching relation by the same roman numeral, but with a different index: V^{i}, VI^{i} and VII^{i}, with the index corresponding to the index of the frequency generated by the SFG or DFG. For THG (), types II, III and IV are equivalent, and so are types V^{4}, VI^{4} and VII^{4}.
The index surface allows the geometrical determination of the phasematching directions, which depend on the relative ellipticity of the internal (−) and external (+) sheets divided by the corresponding wavelengths: according to Tables 1.7.3.1 and 1.7.3.2 the directions are given by the intersection of the internal sheet of the lowest wavelength with a linear combination of the internal and external sheets at the other frequencies . The existence and loci of these intersections depend on specific inequalities between the principal refractive indices at the different wavelengths. Note that independently of phasematching considerations, normal dispersion and energy conservation impose with .
There is no possibility of collinear phase matching in a dispersive cubic crystal because of the absence of birefringence. In a hypothetical nondispersive anaxial crystal, the 2^{3} threewave and 2^{4} fourwave phasematching configurations would be allowed in any direction of propagation.
The configurations of polarization in terms of ordinary and extraordinary waves depend on the optic sign of the phasematching direction with the convention given in Section 1.7.3.1: Tables 1.7.3.1 and 1.7.3.2 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 phasematching 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 ().
Fig. 1.7.3.4 shows the example of negative uniaxial crystals () like βBaB_{2}O_{4 }(BBO) and KH_{2}PO_{4} (KDP).

Index surface sections in a plane containing the optic axis z of a negative uniaxial crystal allowing collinear typeI phase matching for SFG (), , or for SFG (), . is the corresponding phasematching direction. 
From Fig. 1.7.3.4, it clearly appears that the intersection of the sheets is possible only if with for a threewave process and for a fourwave 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 1.7.3.3 gives the five possible situations for the threewave interactions and Table 1.7.3.4 the 19 situations for the fourwave processes.


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 phasematching 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. 1.7.3.5.

Stereographic projection on the optical frame of the possible loci of phasematching directions in the principal planes of a biaxial crystal. 
The basic inequalities of normal dispersion (1.7.3.7) forbid collinear phase matching for all the directions of propagation located between two optic axes at the two frequencies concerned.
Tables 1.7.3.5 and 1.7.3.6 give, respectively, the inequalities that determine collinear phase matching in the principal planes for the three types of threewave SFG and for the seven types of fourwave SFG.


The inequalities in Table 1.7.3.5 show that a phasematching 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 typeII or typeIII 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 typeI phasematched SFG in area c forbids phasematching directions in area b for typeII and typeIII SFG. The exclusion is the same between d and a. The consideration of all the possible combinations of the inequalities of Table 1.7.3.5 leads to 84 possible classes of phasematching cones for both positive and negative biaxial crystals (Fève et al., 1993; Fève, 1994). There are 14 classes for second harmonic generation (SHG) which correspond to the degenerated case () (Hobden, 1967).
The coexistence of the different types of fourwave phase matching is limited as for the threewave case: a cone joining a and d or b and c is impossible for typeI 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, V^{4}, VI^{4} and VII^{4} cannot exist without type I; other restrictions concern the relations between types II, III, IV and types V^{4}, VI^{4}, VII^{4} (Fève, 1994). The counting of the classes of fourwave phasematching cones obtained from all the possible combinations of the inequalities of Table 1.7.3.6 is complex and it has not yet been done.
For reasons explained later, it can be interesting to consider a noncollinear interaction. In this case, the projection of the vectorial phasematching relation (1.7.3.26) on the wavevector of highest frequency leads towhere is the angle between and , with for a threewave interaction and for a fourwave interaction. The phasematching angles () can be expressed as a function of the different () by the projection of (1.7.3.26) on the three principal axes of the optical frame.
The configurations of polarization allowing noncollinear phase matching are the same as for collinear phase matching. Furthermore, noncollinear phase matching exists only if collinear phase matching is allowed; the converse is not true (Fève, 1994). Note that collinear or noncollinear phasematching conditions are rarely satisfied over the entire transparency range of the crystal.
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). 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 threewave SFG, the nonlinear polarization sequence is the following:
QPM devices are a recent development and are increasingly being considered for applications (Fejer et al., 1992). 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). For ferroelectric crystals, it is possible to form periodic reversing of the spontaneous polarization in the same sample by proton or ionexchange techniques, or by applying an electric field, which leads to periodically poled (pp) materials like ppLiNbO_{3} or ppKTiOPO_{4} (Myers et al., 1995; Karlsson & Laurell, 1997; Rosenman et al., 1998).
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 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: diffusionbonded 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.
The refractive indices and their dispersion in frequency determine the existence and loci of the phasematching directions, and so impose the direction of the unit electric field vectors of the interacting waves according to (1.7.3.9). The effective coefficient, given by (1.7.3.23) and (1.7.3.25), 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; Boulanger & Marnier, 1991; Boulanger et al., 1993; Zyss, 1993). Indeed, the effective coefficient is the contraction between the field tensor and the electric susceptibility tensor of corresponding order:
Each 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 fieldtensor components of SFG and DFG which are valid for any direction of propagation. Indeed, from (1.7.3.31) and (1.7.3.33), it is obvious that the fieldtensor components remain unchanged by concomitant permutations of the electric field vectors at the different frequencies and the corresponding Cartesian indices (Boulanger & Marnier, 1991; Boulanger et al., 1993):andwhere e_{i} is the unit electric field vector at ω_{i}.
For a given interaction, the symmetry of the field tensor is governed by the vectorial properties of the electric fields, detailed in Section 1.7.3.1. 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 fieldtensor components described later (Boulanger & Marnier, 1991; Boulanger et al., 1993). 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 (1.7.3.10) 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 , with i and j equal to x or y; the combination of these three relations leads to , and . 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, (1.7.3.11), 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, (1.7.3.15), has a walkoff angle which depends on the frequency, and the extraordinary wave, (1.7.3.16), has no walkoff angle: then the field tensor is symmetric in the Cartesian indices relative to the extraordinary waves. The walkoff 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, , , and so on.
The fieldtensor components are calculated from (1.7.3.11) and (1.7.3.12). The phasematching case is the only one considered here: according to Tables 1.7.3.1 and 1.7.3.2, the allowed configurations of polarization of threewave and fourwave interactions, respectively, are the 2o.e (two ordinary and one extraordinary waves), the 2e.o and the 3o.e, 3e.o, 2o.2e.
Tables 1.7.3.7 and 1.7.3.8 give, respectively, the matrix representations of the threewave interactions (eoo), (oee) and of the fourwave (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.


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 1.7.3.9 gives the fieldtensor 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.

From Tables 1.7.3.7 and 1.7.3.8, 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 1.7.3.1 and 1.7.3.2. According to (1.7.3.31) and (1.7.3.33), the magnitudes of two permutated components are equal if the permutation of polarizations are associated with the corresponding frequencies. For example, according to Table 1.7.3.2, two permutated fieldtensor components have the same magnitude for permutation between the following 3o.e interactions:

The contraction of the field tensor and the uniaxial dielectric susceptibility tensor of corresponding order, given in Tables 1.7.2.2 to 1.7.2.5, is nil for the following uniaxial crystal classes and configurations of polarization: D_{4} and D_{6} for 2o.e, C_{4v} and C_{6v} for 2e.o, D_{6}, D_{6h}, D_{3h} and C_{6v} for 3o.e and 3e.o. Thus, even if phasematching directions exist, the effective coefficient in these situations is nil, which forbids the interactions considered (Boulanger & Marnier, 1991; Boulanger et al., 1993). 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).
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 1.7.3.7 and 1.7.3.8, and the nil components are listed in Table 1.7.3.9. Because of the change of optic sign from either side of the optic axis, the field tensors of the interactions for which the phasematching cone joins areas b and a or a and c, given in Fig. 1.7.3.5, 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 fieldtensor components for a propagation in the xy plane of a biaxial crystal are: , , for (eoo); , for (oee); , , , for (eooo); , for (oeee); , , 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 threewave interactions and 81 for the fourwave 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 coefficients, since a given type of phase matching is considered. In general, the largest coefficients of polar crystals, i.e. , are implicated at a very low level when phase matching is achieved, because the corresponding field tensor, i.e. , is often weak (Boulanger et al., 1997). In contrast, QPM authorizes the coupling between three waves polarized along the z axis, which leads to an effective coefficient which is purely , i.e. , where the numerical factor comes from the periodic character of the rectangular function of modulation (Fejer et al., 1992).
The resolution of the coupled equations (1.7.3.22) or (1.7.3.24) over the crystal length L leads to the electric field amplitude of each interacting wave. The general solutions are Jacobian elliptic functions (Armstrong et al., 1962; Fève, Boulanger & Douady, 2002). 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 (1.7.3.8). For our main purpose, we consider the simple case of planewave beams with two kinds of transverse profile:for a flat distribution over a radius w_{o};for a Gaussian distribution, where w_{o} is the radius at () of the electric field and so at () of the intensity.
The associated powers are calculated according to (1.7.3.8), which leads towhere for a flat distribution and 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:where P_{c} is the peak power and τ the half () width.
For a repetition rate f (s^{−1}), the average power is then given bywhere 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 sumfrequency generation (SFG), including second harmonic generation (SHG: ), cascading third harmonic generation (THG: ) and direct third harmonic generation (THG: ). The differencefrequency 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, walkoff effect etc.) for SHG, which is the prototypical secondorder nonlinear interaction. This discussion will be valid for the other nonlinear processes of frequency generation which will be considered later.
According to Table 1.7.3.1, 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. 1.7.3.6: the simplest one is nonresonant SHG, outside the laser cavity; other ways are external or internal resonant cavity SHG, which allow an enhancement of the singlepass efficiency conversion.

Schematic configurations for second harmonic generation. (a) Nonresonant SHG; (b) external resonant SHG: the resonant wave may either be the fundamental or the harmonic one; (c) internal resonant SHG. are the fundamental and harmonic powers; and are the hightransmission and highreflection mirrors at ω or 2ω and are the transmission coefficients of the output mirror at ω or 2ω. NLC is the nonlinear crystal with a nonzero χ^{(2)}. 
1.7.3.3.2.1. Nonresonant SHG with undepleted pump in the parallelbeam limit with a Gaussian transverse profile
We first consider the case where the crystal length is short enough to be located in the nearfield region of the laser beam where the parallelbeam limit is a good approximation. We make another simplification by considering a propagation along a principal axis of the index surface: then the walkoff angle of each interacting wave is nil so that the three waves have the same coordinate system ().
The integration of equations (1.7.3.22) over the crystal length Z in the undepleted pump approximation, i.e. = , with , leads to(1.7.3.41) implies a Gaussian transversal profile for if and are Gaussian. The three beam radii are related by , so if we assume that the two fundamental beams have the same radius , which is not an approximation for type I, then . Two incident beams with a flat distribution of radius lead to the generation of a flat harmonic beam with the same radius .
The integration of (1.7.3.41) according to (1.7.3.36)–(1.7.3.38) for a Gaussian profile gives in the SI systemwhere m s^{−1}, A s V^{−1} m^{−1} and so V A^{−1}. L (m) is the crystal length in the direction of propagation. is the phase mismatch. , and are the refractive indices at the harmonic and fundamental wavelengths λ_{2ω} and λ_{ω} (µm): for the phasematching case, , , for type I (the two incident fundamental beams have the same polarization contained in Π^{+}, with the harmonic polarization contained in Π^{−}) and for type II (the two solicited eigen modes at the fundamental wavelength are in Π^{+} and Π^{−}, with the harmonic polarization contained in Π^{−}). , and are the transmission coefficients given by . d_{eff} (pm V^{−1}) is the effective coefficient given by (1.7.3.30) and (1.7.3.31). and are the two incident fundamental powers, which are not necessarily equal for type II; for type I we have obviously . 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 factor takes into account the fluctuations between these longitudinal modes (Bloembergen, 1963).
The powers in (1.7.3.42) are instantaneous powers P(t).
The second harmonic (SH) conversion efficiency, η_{SHG}, is usually defined as the ratio of peak powers , or as the ratio of the pulse total energy . For Gaussian temporal profiles, the SH pulse duration is equal to , because is proportional to , and so, according to (1.7.3.40), the pulse average energy conversion efficiency is smaller than the peak power conversion efficiency given by (1.7.3.42). Note that the pulse total energy conversion efficiency is equivalent to the average power conversion efficiency , with where f is the repetition rate.
Formula (1.7.3.42) shows the importance of the contribution of the linear optical properties to the nonlinear process. Indeed, the field tensor F^{(2)}, the transmission coefficients T_{i} and the phase mismatch only depend on the refractive indices in the direction of propagation considered.

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 bywith for E^{+} and for E^{−}.
() is the wave frame defined in Fig. 1.7.3.1. is the scalar complex amplitude at in the vibration planes .
We consider the refracted waves E^{+} and E^{–} to have the same longitudinal profile inside the crystal. Then the beam radius is given by , where w_{o} is the minimum beam radius located at and , with ; z_{R} is the Rayleigh length, the length over which the beam radius remains essentially collimated; are the wavevectors at the wavelength λ in the direction of propagation Z. The farfield half divergence angle is .
The coordinate systems of (1.7.3.22) are identical to those of the parallelbeam limit defined in (iii).
In these conditions and by assuming the undepleted pump approximation, the integration of (1.7.3.22) over () leads to the following expression of the power conversion efficiency (Zondy, 1991):within the same units as equation (1.7.3.42).
For type I, , , and for type II , .
The attenuation coefficient is writtenwithwhere f gives the position of the beam waist inside the crystal: at the entrance and at the exit surface. The definition and approximations relative to ρ are the same as those discussed for the parallelbeam limit. Δk is the mismatch parameter, which takes into account first a possible shift of the pump beam direction from the collinear phasematching direction and secondly the distribution of mismatch, including collinear and noncollinear interactions, due to the divergence of the beam, even if the beam axis is phasematched.
The computation of allows an optimization of the SHG conversion efficiency which takes into account , the waist location f inside the crystal and the phase mismatch Δk.
Fig. 1.7.3.12 shows the calculated waist location which allows an optimal SHG conversion efficiency for types I and II with optimum phase matching. From Fig. 1.7.3.12, 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, . For type II, the focusing () is stronger and the walkoff 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 walkoff 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.

Position f_{opt} of the beam waist for different values of walkoff angles and , leading to an optimum SHG conversion efficiency. The value corresponds to the middle of the crystal and corresponds to the entrance surface (Fève & Zondy, 1996). 
The divergence of the pump beam imposes noncollinear interactions such that it could be necessary to shift the direction of propagation of the beam from the collinear phasematching direction in order to optimize the conversion efficiency. This leads to the definition of an optimum phasemismatch parameter () for a given and a fixed position of the beam waist f inside the crystal.
The function , written , is plotted in Fig. 1.7.3.13 as a function of for different values of the walkoff parameter, defined as B = , at the optimal waist location and phase mismatch.

Optimum walkoff function as a function of for various values of . The curve at is the same for both typeI and typeII phase matching. The full lines at are for type II and the dashed line at is for type I. (From Zondy, 1990). 
Consider first the case of angular NCPM () where typeI and II conversion efficiencies obviously have the same evolutions. An optimum focusing at exists which defines the optimum focusing for a given crystal length or the optimal length for a given focusing. The conversion efficiency decreases for 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 (), the variation of typeI 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 () to 2.98 () as the corresponding amplitudes decrease. For type II, an optimum focusing becomes less and less appearent, while 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 shows a saturation for type II, in contrast to type I. The saturation occurs at with a corresponding focusing parameter , which is the limit of validity of the parallelbeam approximation. These results show that weak focusing is suitable for type II, whereas type I allows higher focusing.
The curves of Fig. 1.7.3.14 give a clear illustration of the walkoff effect in several usual situations of crystal length, walkoff angle and Gaussian laser beam. The SHG conversion efficiency is calculated from formula (1.7.3.56) and from the function (1.7.3.57) at f_{opt} and Δk_{opt}.

TypeI and II conversion efficiencies calculated as a function of for different typical walkoff angles ρ: (a) and (c) correspond to a fixed focusing condition (w_{o} = 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 (d_{eff} = 1 pm V^{−1}), refractive indices () and fundamental power [P_{ω}(0 = 1 W]. B is the walkoff parameter defined in the text (Fève & Zondy, 1996). 
The analytical integration of the three coupled equations (1.7.3.22) with depletion of the pump and phase mismatch has only been done in the parallelbeam limit and by neglecting the walkoff effect (Armstrong et al., 1962; Eckardt & Reintjes, 1984; Eimerl, 1987; Milton, 1992). In this case, the three coordinate systems of equations (1.7.3.22) are identical, (), and the general solution may be written in terms of the Jacobian elliptic function .
For the simple case of type I, i.e. , the exit second harmonic intensity generated over a length L is given by (Eckardt & Reintjes, 1984) is the total initial fundamental intensity, and are the transmission coefficients, withandFor the case of phase matching (, ), we have and , and the Jacobian elliptic function is equal to . Then formula (1.7.3.58) becomeswhere is given by (1.7.3.59).
The exit fundamental intensity can be established easily from the harmonic intensity (1.7.3.60) according to the Manley–Rowe relations (1.7.2.40), i.e.For small , the functions and with .
The first consequence of formulae (1.7.3.58)–(1.7.3.59) 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 phasematching direction, formula (1.7.3.60) is not valid and may be replaced by (1.7.3.58) where 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: increases as a function of ΓL until a maximum value has been reached and then decreases; 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 for a given phasematching direction, characterized by K, and a given crystal length L.
The crystal length must be optimized in order to work with an incident intensity smaller than the damage threshold intensity of the nonlinear crystal, given in Section 1.7.5 for the main materials.
Formula (1.7.3.57) is established for type I. For type II, the second harmonic intensity is also an sn^{2} function where the intensities of the two fundamental beams and , which are not necessarily equal, are taken into account (Eimerl, 1987): the tanh^{2} function is valid only if perfect phase matching is achieved and if , these conditions being never satisfied in real cases.
The situations described above are summarized in Fig. 1.7.3.15.

Schematic SHG conversion efficiency for different situations of pump depletion and dephasing. (a) No depletion, no dephasing, ; (b) no depletion with constant dephasing δ, ; (c) depletion without dephasing, ; (d) depletion and dephasing, . 
We give the example of typeII SHG experiments performed with a 10 Hz injectionseeded singlelongitudinalmode () 1064 nm Nd:YAG (SpectraPhysics DCR2A10) laser equipped with super Gaussian mirrors; the pulse is 10 ns in duration and is near a Gaussian singletransverse mode, the beam radius is 4 mm, nonfocused 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 (). 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).
The integration of the intensity profiles (1.7.3.58) and (1.7.3.60) is obvious in the case of incident fundamental beams with a flat energy distribution (1.7.3.36). 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 (1.7.3.58) and (1.7.3.60) by expressing the intensity and electric field modulus as a function of the power, which is given by (1.7.3.38) with .
For a Gaussian incident fundamental beam, (1.7.3.37), the fundamental and harmonic beams are not Gaussian (Eckardt & Reintjes, 1984; Pliszka & Banerjee, 1993).
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 of the undepleted case.
When the singlepass 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. 1.7.3.6(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, P_{in}(ω), is evaluated at the steady state by setting the roundtrip saturated gain of the laser equal to the sum of all the losses. The output laser cavity, P_{out}(ω), is given by (Siegman, 1986)with is the laser medium length, is the smallsignal gain coefficient per unit length of laser medium, σ is the stimulatedemission cross section, N_{o} is the population inversion without oscillation, S is a saturation parameter characteristic of the nonlinearity of the laser transition, and is the loss coefficient where α_{L} 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 g_{o}, S, α_{L}, β and , the output power reaches a maximum value for an optimal transmission coefficient T_{opt} defined by , which givesThe maximum output power is then given by
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 roundtripgenerated 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 singlepass SHG conversion efficiency η_{SHG}.
The fundamental power inside the cavity P_{in}(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. P_{in}(ω) is then given by (1.7.3.62), where T and γ are (Geusic et al., 1968; Smith, 1970)andη_{SHG} is the singlepass conversion efficiency. γ_{L} and γ_{NL} 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 antireflectioncoated at ω.
In the undepleted pump approximation, the backward and forward power generated outside the nonlinear crystal at 2ω iswithwhere
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 (1.7.3.67) with respect to K according to (1.7.3.62), (1.7.3.65) and (1.7.3.66) gives (Perkins & Fahlen, 1987)and(1.7.3.69) shows that for the case where (), the maximum SH power is identically equal to the maximum fundamental power, (1.7.3.64), 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%. strongly decreases as the losses () increase . Thus an efficient intracavity device requires the reduction of all losses at ω and 2ω to an absolute minimum.
(1.7.3.68) indicates that K_{opt} is independent of the operating power level of the laser, in contrast to the optimum transmitting mirror where T_{opt}, given by (1.7.3.63), depends on the laser gain. K_{opt} depends only on the total losses and saturation parameter. For given losses, the knowledge of K_{opt} allows us to define the optimal parameters of the nonlinear crystal, in particular the figure of merit, and the ratio (L/w_{o})^{2}, in which the walkoff 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 TEM_{oo} c.w. SHG intracavity device using a 3.4 mm Ba_{2}NaNb_{5}O_{15} crystal within a 1.064 µm Nd:YAG laser cavity (Geusic et al., 1968). A power of 9.0 W has been generated at 0.532 µm using a more complicated geometry based on an Nd:YAG intracavitylens foldedarm cavity configuration using KTP (Perkins & Fahlen, 1987). Highaveragepower 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 sidepumped Nd:YAG laser (LeGarrec et al., 1996).
For typeII 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).
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). Furthermore, such geometry allows us to avoid losses of the backward SH beam in the laser medium and in other optical components behind.
Externalcavity 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). For example, a bowtie configuration allowed the generation of 6.5 W of TEM_{oo} c.w. 0.532 µm radiation in a 6 mm LiB_{3}O_{5} (LBO) crystal; the Nd:YAG laser was an 18 W c.w. laser with an injectionlocked single frequency (Yang et al., 1991).
Fig. 1.7.3.16 shows the three possible ways of achieving THG: a cascading interaction involving two χ^{(2)} processes, i.e. and , in two crystals or in the same crystal, and direct THG, which involves χ^{(3)}, i.e. .

Configurations for third harmonic generation. (a) Cascading process SHG (): SFG () in two crystals NLC1 and NLC2 and (b) in a single nonlinear crystal NLC; (c) direct process THG () in a single nonlinear crystal NLC. 
We consider the case of the situation in which the SHG is phasematched with or without pump depletion and in which the sumfrequency generation (SFG) process (), phasematched or not, is without pump depletion at and . All the waves are assumed to have a flat distribution given by (1.7.3.36) and the walkoff 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 (1.7.3.22) over Z for the SFG in the undepleted pump approximation with , and , followed by the integration over the cross section leads towithP^{ω}(L_{SHG}) and P^{2}^{ω}(L_{SHG}) are the fundamental and harmonic powers, respectively, at the exit of the first crystal. L_{SHG} and L_{SFG} are the lengths of the first and the second crystal, respectively. is the SFG phase mismatch. λ_{ω} is the fundamental wavelength. The units and other parameters are as defined in (1.7.3.42).
For typeII SHG, the fundamental waves are polarized in two orthogonal vibration planes, so only half of the fundamental power can be used for typeI, II or III SFG (), in contrast to typeI SHG (). In the latter case, and for typeI SFG, it is necessary to set the fundamental and second harmonic polarizations parallel.
The cascading conversion efficiency is calculated according to (1.7.3.61) and (1.7.3.70); the case of typeI SHG gives, for example,where Γ is as in (1.7.3.59).
(n^{ω}, T^{ω}) are relative to the phasematched SHG crystal and () correspond to the SFG crystal.
In the undepleted pump approximation for SHG, (1.7.3.71) becomes (Qiu & Penzkofer, 1988)within W^{−2}, whereThe units are the same as in (1.7.3.42).
A more general case of SFG, where one of the two pump beams is depleted, is given in Section 1.7.3.3.4.
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). In order to compare this situation with the example taken for the previous case, we consider a typeI 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)withΓ is as in (1.7.3.59).
(1.7.3.73) can be analytically integrated for undepleted pump SHG; , , and so we havewithwhere the integral J(L) is
For a nonzero SFG phase mismatch, ,
Therefore (1.7.3.75) according to (1.7.3.78) is equal to (1.7.3.72) with , and 100% transmission coefficients at ω and 2ω between the two crystals.
As for the cascading process, we consider a flat plane wave which propagates in a direction without walkoff. The integration of equations (1.7.3.24) over the crystal length L, with and in the undepleted pump approximation, leads to
According to (1.7.3.36) and (1.7.3.38), the integration of (1.7.3.79) over the cross section, which is the same for the four beams, leads towithwhere is in m^{2} V^{−2} and λ_{ω} is in m. The statistical factor is assumed to be equal to 1, which corresponds to a longitudinal singlemode laser.
The different types of phase matching and the associated relations and configurations of polarization are given in Table 1.7.3.2 by considering the SFG case with .
SHG () and SFG () are particular cases of threewave SFG. We consider here the general situation where the two incident beams at ω_{1} and ω_{2}, with , interact with the generated beam at ω_{3}, with , as shown in Fig. 1.7.3.17. The phasematching configurations are given in Table 1.7.3.1.

Frequency upconversion process . The beam at ω_{1} is mixed with the beam at ω_{2} in the nonlinear crystal NLC in order to generate a beam at ω_{3}. are the different powers. 
From the general point of view, SFG is a frequency upconversion 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 (1.7.3.22) leads to Jacobian elliptic functions if the waves at ω_{1} and ω_{2} are both depleted. The calculation is simplified in two particular situations which are often encountered: on the one hand undepletion for the waves at ω_{1} and ω_{2}, and on the other hand depletion of only one wave at ω_{1} or ω_{2}. For the following, we consider plane waves which propagate in a direction without walkoff so we consider a single wave frame; the energy distribution is assumed to be flat, so the three beams have the same radius w_{o}.
The resolution of system (1.7.3.22) with , , and , followed by integration over , leads towithin the same units as equation (1.7.3.70).
or .
The undepleted wave at ω_{p}, the pump, is mixed in the nonlinear crystal with the depleted wave at ω_{s}, the signal, in order to generate the idler wave at . The integrations of the coupled amplitude equations over () with , , and givewith and , whereThus, even if the upconversion process is phasematched (), the power transfers are periodic: the photon transfer efficiency is then 100% for , where m is an integer, which allows a maximum power gain for the idler. A nonlinear crystal with length is sufficient for an optimized device.
For a small conversion efficiency, i.e. ΓL weak, (1.7.3.85) and (1.7.3.86) becomeand The expression for P_{i}(L) with is then equivalent to (1.7.3.83) with or , and or .
For example, the frequency upconversion interaction can be of great interest for the detection of a signal, ω_{s}, comprising IR radiation with a strong divergence and a wide spectral bandwidth. In this case, the achievement of a good conversion efficiency, P_{i}(L)/P_{s}(0), requires both wide spectral and angular acceptance bandwidths with respect to the signal. The double noncriticality in frequency and angle (DNPM) can then be used with onebeam noncritical noncollinear phase matching (OBNC) associated with vectorial group phase matching (VGPM) (Dolinchuk et al., 1994): this corresponds to the equality of the absolute magnitudes and directions of the signal and idler group velocity vectors i.e. .
DFG is defined by with or with . The DFG phasematching configurations are given in Table 1.7.3.1. As for SFG, the solutions of system (1.7.3.22) 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. ω_{3}, or not. We consider the case of plane waves that propagate in a direction without walkoff and we assume a flat energy distribution for the three beams.
or .
The resolution of system (1.7.3.22) with , , and , followed by integration over (), leads to the same solutions as for SFG with undepletion at ω_{1} and ω_{2}, i.e. formulae (1.7.3.81), (1.7.3.82) and (1.7.3.83), by replacing ω_{1} by ω_{s}, ω_{2} by ω_{p} and ω_{3} by ω_{i}. A schematic device is given in Fig. 1.7.3.17 by replacing (ω_{1}, ω_{2}, ω_{3}) by (ω_{1}, ω_{3}, ω_{2}) or (ω_{2}, ω_{3}, ω_{1}).
or .
The resolution of system (1.7.3.22) with , , and , followed by the integration over (), leads to the same solutions as for SFG with undepletion at ω_{1} or ω_{2}: formulae (1.7.3.84), (1.7.3.85) and (1.7.3.86).
1.7.3.3.5.3. DFG () with undepletion at – optical parametric amplification (OPA), optical parametric oscillation (OPO)
or .
The initial conditions are the same as in Section 1.7.3.3.5.2, except that the undepleted wave has the highest circular frequency. In this case, the integrations of the coupled amplitude equations over () lead toandwith and , where w_{o} is the beam radius of the three beams and The units are the same as in equation (1.7.3.42).
Equations (1.7.3.90) and (1.7.3.91) 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 () and SFG (), and, secondly, the signal power is amplified. Thus DFG () combines two interesting functions: generation at and amplification at . The last function is called optical parametric amplification (OPA).
The gain of OPA can be defined as (Harris, 1969)For example, Baumgartner & Byer (1979) 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 KH_{2}PO_{4} crystal, with a pump intensity of 28 MW cm^{−2}.
According to (1.7.3.91), for , and so the gain is given byFormula (1.7.3.93) shows that frequencies can be generated around ω_{s}. The full gain linewidth of the signal, Δω_{s}, is defined as the linewidth leading to a maximum phase mismatch . If we assume that the pump wave linewidth is negligible, i.e. , it follows, by expanding Δk in a Taylor series around ω_{i} and ω_{s}, and by only considering the first order, that with , where 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: because the idler and signal waves have the same polarization and so the same group velocity at degeneracy, i.e. . In this case, it is necessary to consider the dispersion of the group velocity for the calculation of Δω_{s} and Δω_{i}. 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; Byer, 1973; Brosnan & Byer, 1979; Yang et al., 1993). 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): when a laser beam at ω_{p} propagates in a χ^{(2)} medium, it is possible for pump photons to spontaneously break down into pairs of lowerenergy photons of circular frequencies ω_{s} and ω_{i} with the total photon energy conserved for each pair, i.e . The pairs of generated waves for which the phasematching condition is satisfied are the only ones to be efficiently coupled by the nonlinear medium. The OPO can be singly resonant (SROPO) at ω_{s} or ω_{i} (Yang et al., 1993; Chung & Siegman, 1993), doubly resonant (DROPO) at both ω_{s} and ω_{i} (Yang et al., 1993; Breitenbach et al., 1995) or triply resonant (TROPO) (Debuisschert et al., 1993; Scheidt et al., 1995). 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. 1.7.3.18. This second technique allows us to use simpler mirror coatings because they are not illuminated by the strong pump beam.

Schematic OPO configurations. 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 (M_{1}, M_{2}). 
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 I_{th}. The oscillation threshold decreases when the number of resonant frequencies increases: ; 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 M_{2} in configuration (a) of Fig. 1.7.3.18 (Marshall & Kaz, 1993). 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 phasematched SROPO, resonant at ω_{s} or ω_{i}, and for nanosecond pulsed beams with intensities that are assumed to be constant over one single pass, is given by; L is the crystal length; γ is the ratio of the backward to the forward pump intensity; τ is the 1/e^{2} 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 ω_{s} or ω_{i}. In the nanosecond regime, typical values of are in the range 10–100 MW cm^{−2}.
(1.7.3.95) 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 KNbO_{3} crystal ( pm V^{−1}) as a nonlinear crystal was performed with a pump threshold intensity of 65 MW cm^{−2 }(Unschel et al., 1995): the 3 mmdiameter pump beam was a 10 Hz injectionseeded singlelongitudinalmode Nd:YAG laser at 1.064 µm with a 9 ns pulse of 100 mJ; the SROPO was pumped as in Fig. 1.7.3.18(a) with a cavity length of 12 mm, a mirror M_{1} reflecting 100% at the signal, from 1.4 to 2 µm, and a coupling mirror M_{2} 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 (λ_{s} = 1.61 µm) and idler (λ_{i} = 3.14 µm) outputs were obtained in a SROPO using a 20 mm long KTP crystal (d_{eff} = 2.7 pm V^{−1}) pumped by an Nd:YAG laser (λ_{p} = 1.064 µm) for eyesafe source applications (Marshall & Kaz, 1993): the configuration is the same as in Fig. 1.7.3.18(a) where M_{1} has high reflection at 1.61 µm and high transmission at 1.064 µm, and M_{2} has high reflection at 1.064 µm and a 10% transmission coefficient at 1.61 µm; the Qswitched 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 continuouswave (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, cwSROPO can run when the OPO is placed within the pumplaser cavity (Ebrahimzadeh et al., 1999). 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. Qswitched laser running in the nanosecond regime and modelocked 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 roundtrip 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) and Reid et al. (1998)].
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 midIR. 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 phasematched 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 pumpwavelength tuning; and the case of quasi phase matching.

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