^{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
Correspondence email: benoit.boulanger@grenoble.cnrs.fr
This chapter deals mainly with harmonic generation and parametric interactions in anisotropic crystals. Section 1.7.2
Keywords: ABDP and Kleinmann symmetries; Gaussian beams; Maker fringes; Manley–Rowe relations; Sellmeier equations; acceptance bandwidths; biaxial classes; biaxial crystals; coherence length; contraction; conversion efficiency; dielectric polarization; dielectric susceptibility; dielectric tensor; differencefrequency generation; electric polarization; field tensors; figure of merit; index surface; nonlinear crystals; nonlinear optics; optical parametric oscillation; parametric amplification; phase matching; phase mismatch; polarization; quasi phase matching; second harmonic generation; sumfrequency generation; third harmonic generation; undepleted pump approximation; uniaxial classes; uniaxial crystals; walkoff.
The first nonlinear
The basis of nonlinear optics, including quantummechanical perturbation theory and Maxwell equations
It would take too long here to give a complete historical account of nonlinear optics, because it involves an impressive range of different aspects, from theory to applications, from physics to chemistry, from microscopic to macroscopic aspects, from quantum mechanics of materials to classical and quantum electrodynamics, from gases to solids, from mineral to organic compounds, from bulk to surface, from waveguides to fibres and so on.
Among the main nonlinear optical effects are harmonic generation
This chapter deals mainly with harmonic generation and parametric interactions in anisotropic crystals, which stand out as one of the most important fields in nonlinear optics and certainly one of its oldest and most rigorously treated topics. Indeed, there is a great deal of interest in the development of solidstate laser sources, be they tunable or not, in the ultraviolet, visible and infrared ranges. Spectroscopy, telecommunications, telemetry and optical storage are some of the numerous applications.
The electric field
Here, we do not consider optical interactions at the microscopic level, and we ignore the way in which the atomic or molecular dielectric susceptibility
We shall mainly emphasize propagation aspects, on the basis of Maxwell equations
We analyse in detail second harmonic generation
An overview of the methods of measurement of the nonlinear optical properties is provided, and the chapter concludes with a comparison of the main mineral and organic crystals showing nonlinear optical properties.
The macroscopic electronic polarization of a unit volume of the material system is classically expanded in a Taylor power series of the applied electric field E, according to Bloembergen (1965
A more compact expression for (1.7.2.1)
Tensorial expressions will be formulated within the Cartesian formalism and subsequent multiple lower index notation. The alternative irreducible tensor representation
Let us first consider the firstorder linear response in (1.7.2.1)
The most general expression for P^{(2)}(t) which is quadratic in E(t) isor in Cartesian notationIt can easily be proved by decomposition of T^{(2)} into symmetric and antisymmetric parts and permutation of dummy variables (α, τ_{1}) and (β, τ_{2}), that T^{(2)} can be reduced to its symmetric part, satisfyingFrom time invarianceCausality demands that R^{(2)}(τ_{1}, τ_{2}) cancels for either τ_{1} or τ_{2} negative while R^{(2)} is real. Intrinsic permutation symmetry implies that R_{μαβ}^{(2)}(τ_{1}, τ_{2}) is invariant by interchange of (α, τ_{1}) and (β, τ_{2}) pairs.
The nth order polarization
For similar reasons to those previously stated, it is sufficient to consider the symmetric part of T^{(n)} with respect to the n! permutations of the n pairs (α_{1}, τ_{1}), (α_{2}, τ_{2}) (α_{n}, τ_{n}). The T^{(n) }tensor will then exhibit intrinsic permutation symmetry at the nth order. Timeinvariance considerations will then allow the introduction of the ()thrank real tensor R^{(n)}, which generalizes the previously introduced R operators:R^{(n)} cancels when one of the τ_{i}'s is negative and is invariant under any of the n! permutations of the (α_{i}, τ_{i}) pairs.
Whereas the polarization response has been expressed so far in the time domain, in which causality and time invariance are most naturally expressed, Fourier transformation into the frequency domain permits further simplification of the equations given above and the introduction of the susceptibility tensors
The direct and inverse Fourier transforms of the field are defined aswhere as E(t) is real.
By substitution of (1.7.2.15)
In these equations, to satisfy the energy conservation condition that will be generalized in the following. In order to ensure convergence of χ^{(1)}, ω has to be taken in the upper half plane of the complex plane. The reality of R^{(1)} implies that .
Substitution of (1.7.2.15)
Substitution of (1.7.2.15)
All frequencies must lie in the upper half complex plane and reality of χ^{(n)} imposesIntrinsic permutation symmetry implies that is invariant with respect to the n! permutations of the (α_{i}, ω_{i}) pairs.
Optical fields are often superpositions of monochromatic waves which, due to spectral discretization, will introduce considerable simplifications in previous expressions such as (1.7.2.20)
The Fourier transform of the induced polarization
In practical cases where the applied field is a superposition of monochromatic waveswith . By Fourier transformation of (1.7.2.26)
Insertion of (1.7.2.26)
The K factor allows the avoidance of discontinuous jumps in magnitude of the elements when some frequencies are equal or tend to zero, which is not the case for the other conventions (Shen, 1984
The induced nonlinear polarization
The K convention described above is often used, but may lead to errors in cases where two of the interacting waves have the same frequency but different polarization states
Because of the possible nondegeneracy with respect to the direction of polarization of the electric fields at the same frequency, it is suitable to consider a harmonic generation process, second harmonic generation (SHG)
According to (1.7.2.33)
According to (1.7.2.33)
Intrinsic permutation symmetry, as already discussed, imposes the condition that the nth order susceptibility
Let us consider as an application the quantum expression of the quadratic susceptibility
An important consequence of overall permutation symmetry is the Manley–Rowe power relations, which account for energy exchange between electromagnetic waves in a purely reactive (e.g. nondissipative) medium. Calling W_{i} the power input at frequency ω_{i} into a unit volume of a dielectric polarizable medium,where the averaging is performed over a cycle andThe following expressions can be derived straightforwardly:Introducing the quadratic induced polarization
A quantum mechanical interpretation of these expressions in terms of photon fusion or splitting can be given, remembering that is precisely the number of photons generated or annihilated per unit volume in unit time in the course of the nonlinear interactions.
The tensors or are invariant with respect to (α, β) permutation as a consequence of the intrinsic permutation symmetry. Independently, it is not possible to distinguish the coefficients and by SHG
Therefore, these thirdrank tensors can be represented in contracted form as matrices and , where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction
For example, (1.7.2.35)
Centrosymmetry is the most detrimental crystalline symmetry constraint that will fully cancel all oddrank tensors such as the d^{(2)} [or χ^{(2)}] susceptibilities. Intermediate situations, corresponding to noncentrosymmetric crystalline point groups, will reduce the number of nonzero coefficients without fully depleting the tensors.
Tables 1.7.2.2
We summarize here the main linear optical properties that govern the nonlinear propagation phenomena. The reader may refer to Chapter
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)
In the linear regime, , where ɛ_{0} is the freespace permittivity
The plane wave is a solution of equation (1.7.3.2)
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
Equation (1.7.3.6)
Equation (1.7.3.6)
The dielectric displacements
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
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
The vibration planes relative to the eigen
The existence of equalities between the principal refractive indices determines the three optical classes: isotropic for the cubic system; uniaxial
The isotropic class corresponds to the equality of the three principal indices: the index surface
The uniaxial class
The ordinary electric field vector is orthogonal to the optic axis
According to these results, the coplanarity of the field vectors imposes the condition that the doublerefraction
In a biaxial crystal
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
It is possible to define ordinary and extraordinary waves, but only in the principal planes of the biaxial crystal
It is impossible to define ordinary and extraordinary waves out of the principal planes of a biaxial crystal
The nonlinear crystals
Thus the propagation equation of each interacting wave ω_{i} is (Bloembergen, 1965
The complex conjugates come from the relation .
We consider the plane wave, (1.7.3.3)
The presence of in equations (1.7.3.19)
Stating (1.7.3.20)
For a threewave interaction, (1.7.3.21)
If ABDP relations
In the general case, the nonlinear polarization
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
According to (1.7.3.4)
The efficiency of a nonlinear crystal
In an hypothetical nondispersive medium [], (1.7.3.27)
For a threewave process, only three combinations among the 2^{3} are compatible with the dispersion in frequency (1.7.3.7)
The designation of the type of phase matching, I, II or III, is defined according to the polarization states
For a fourwave process, only seven combinations of refractive indices allow phase matching in the case of normal dispersion
The convention of designation of the types is the same as for threewave interactions for the situations where one polarization state
The index surface
There is no possibility of collinear phase matching in a dispersive cubic crystal because of the absence of birefringence
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
Because of the symmetry of the index surface
Fig. 1.7.3.4
From Fig. 1.7.3.4
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
The basic inequalities of normal dispersion (1.7.3.7)
Tables 1.7.3.5
The inequalities in Table 1.7.3.5
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
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
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)
The configurations of polarization
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
QPM devices are a recent development and are increasingly being considered for applications (Fejer et al., 1992
Quasi phase matching offers three main advantages when compared with phase matching
The refractive indices and their dispersion in frequency determine the existence and loci of the phasematching
Each corresponds to a given eigen electric field
The components of the field tensor
Particular relations exist between fieldtensor components of SFG and DFG which are valid for any direction of propagation. Indeed, from (1.7.3.31)
For a given interaction, the symmetry of the field tensor
(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
(c) The fact that the direction of the ordinary electric field vectors in uniaxial crystals
(d) Equalities between frequencies can create new symmetries: the field tensors
The fieldtensor components are calculated from (1.7.3.11)
Tables 1.7.3.7
If there are equalities between frequencies, the field tensors
Table 1.7.3.9
From Tables 1.7.3.7

The contraction
The symmetry of the biaxial field tensors
The nonzero fieldtensor components for a propagation in the xy plane of a biaxial crystal
The field tensors
As phase matching
The resolution of the coupled equations (1.7.3.22)
The associated powers are calculated according to (1.7.3.8)
The nonlinear interaction is characterized by the conversion efficiency
For pulsed beams, it is necessary to consider the temporal shape, usually Gaussian
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)
We choose to analyse in detail the different parameters relative to conversion efficiency
According to Table 1.7.3.1
The fundamental waves at ω define the pump. Two situations are classically distinguished: the undepleted pump approximation
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
The integration of equations (1.7.3.22)
The integration of (1.7.3.41)
The powers in (1.7.3.42)
The second harmonic (SH) conversion efficiency
Formula (1.7.3.42)

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
() is the wave frame defined in Fig. 1.7.3.1
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 coordinate systems of (1.7.3.22)
In these conditions and by assuming the undepleted pump approximation
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
The computation of allows an optimization of the SHG conversion efficiency
Fig. 1.7.3.12
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
The function , written , is plotted in Fig. 1.7.3.13
Consider first the case of angular NCPM () where typeI and II conversion efficiencies
In the case of angular CPM (), the variation of typeI conversion efficiency
The curves of Fig. 1.7.3.14
The analytical integration of the three coupled equations (1.7.3.22)
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
The exit fundamental intensity can be established easily from the harmonic intensity (1.7.3.60)
The first consequence of formulae (1.7.3.58)
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
The crystal length must be optimized in order to work with an incident intensity smaller than the damage threshold intensity of the nonlinear crystal
Formula (1.7.3.57)
The situations described above are summarized in Fig. 1.7.3.15
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
The integration of the intensity profiles (1.7.3.58)
For a Gaussian
All the previous intensities are the peak values in the case of pulsed beams. The relation between average and peak powers, and then SHG
When the singlepass conversion efficiency
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
In an intracavity 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)
In the undepleted pump approximation
The intracavity SHG conversion efficiency
Maximizing (1.7.3.67)
(1.7.3.68)
Some examples: a power of 1.1 W at 0.532 µm was generated in a TEM_{oo} c.w. SHG
For typeII phase matching
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
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
Fig. 1.7.3.16
We consider the case of the situation in which the SHG is phasematched
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)
For typeII SHG
The cascading conversion efficiency
(n^{ω}, T^{ω}) are relative to the phasematched
In the undepleted pump approximation
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
(1.7.3.73
For a nonzero SFG phase mismatch
Therefore (1.7.3.75)
As for the cascading process, we consider a flat plane wave which propagates in a direction without walkoff
According to (1.7.3.36)
The different types of phase matching
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
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)
The resolution of system (1.7.3.22)
or .
The undepleted wave at ω_{p}, the pump, is mixed in the nonlinear crystal
For a small conversion efficiency
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
DFG is defined by with or with . The DFG phasematching
or .
The resolution of system (1.7.3.22)
or .
The resolution of system (1.7.3.22)
or .
The initial conditions are the same as in Section 1.7.3.3.5.2
Equations (1.7.3.90)
The gain of OPA can be defined as (Harris, 1969
According to (1.7.3.91)
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
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
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
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
The intensity threshold is calculated by assuming that the pump beam is undepleted. For a phasematched
(1.7.3.95)
For increasing pump powers above the oscillation threshold
The total signal or idler conversion efficiency
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
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
We consider here two of the most frequently encountered methods at present: for birefringence

We review here the different methods that are used for the study of nonlinear crystals
The very early stage of crystal growth of a new material usually provides a powder with particle sizes less than 100 µm. It is then impossible to measure the phasematching loci. Nevertheless, careful SHG
For crystal sizes greater than few hundred µm, it is possible to perform direct measurements of phasematching directions. The methods developed at present are based on the use of a single crystal ground into an ellipsoidal (Velsko, 1989
Phasematching relations are often poorly calculated when using refractive indices determined by the prism method or by measurement of the critical angle of total reflection. Indeed, all the refractive indices concerned have to be measured with an accuracy of 10^{−4} in order to calculate the phasematching angles with a precision of about 1°. Such accuracies can be reached in the visible spectrum, but it is more difficult for infrared wavelengths. Furthermore, it is difficult to cut a prism of few mm size with plane faces.
If the refractive indices are known with the required accuracy at several wavelengths well distributed across the transparency region, it is possible to fit the data with a Sellmeier equation
It is then easy to calculate the phasematching angles (θ_{PM}, _{PM}) from (1.7.4.1)
The measurement of the variation of intensity of the generated beam as a function of the angle of incidence can be performed on a sphere or slab, leading, respectively, to internal and external angular acceptances
The knowledge of the absolute magnitude and of the relative sign of the independent elements of the tensors χ^{(2)} and χ^{(3)} is of prime importance not only for the qualification of a new crystal, but also for the fundamental engineering of nonlinear optical materials
However, disparities in the published values of the nonlinear coefficients of the same crystal exist, even if it is a well known material that has been used for a long time in efficient devices (Eckardt & Byer, 1991
Accurate measurements require mmsize crystals with high optical quality of both surface and bulk.
The main techniques used are based on nonphasematched SHG
The conversion efficiency
When the crystal is rotated, the harmonic and fundamental waves are refracted with different angles, which leads to a variation of the coherence length
A continuous variation of the phase mismatch
It is necessary to take into account a multiple reflection factor in the expression of .
The Makerfringes
The use of phasematched interactions is suitable for absolute and accurate measurements (Eckardt & Byer, 1991
Recent experiments have been performed in a KTP
For both nonphasematched and phasematched techniques, it is important to know the refractive indices and to characterize the spatial, temporal and spectral properties of the pump beam carefully. The considerations developed in Section 1.7.3
Tables 1.7.5.1
A complete review of mineral crystals is given in Bordui & Fejer (1993
A new generation of materials has been developed since 1995 for the design of new compact allsolidstate laser sources. These optical materials are multifunction crystals, such as LiNbO_{3}:Nd^{3+}















