(i) Figure of merit.
The contribution of F^{(2)} was discussed previously, where it was shown that the field tensor is nil in particular directions of propagation or everywhere for particular crystal classes and configurations of polarization (even if the nonlinearity χ^{(2)} is high).
The field tensor F^{(2)} of SHG can be written with the contracted notation of d^{(2)}; according to Table 1.7.3.1 and to the contraction conventions given in Section 1.7.2.2, the contracted fieldtensor components for the phasematched SHG arefor type I andfor type II, with for F_{ij}, corresponding to for .
The ratio in formula (1.7.3.42) is called the figure of merit of the direction considered. The effective coefficient is given in Section 1.7.5 for the main nonlinear crystals and for chosen SHG wavelengths.
(ii) Effect of the phase mismatch.
The interference function is a maximum and equal to unity only for , which defines the phasematching condition. Fig. 1.7.3.7 shows the effect of the phase mismatch on the growth of second harmonic conversion efficiency, η_{SHG}, with interaction distance Z.
The conversion efficiency has a Z^{2} dependence in the case of phase matching. The harmonic power oscillates around Z^{2} for quasi phase matching, but is reduced by a factor of 4/π^{2} compared with that of phasematched interaction (Fejer et al., 1992).
An SHG phasematching direction () for given fundamental wavelength (λ_{PM}) and type of interaction, I or II, is defined at a given temperature (T_{PM}). It is important to consider the effect of deviation of Δk from 0 due to variations of angles (), of temperature () and of wavelength () on the conversion efficiency. The quantities that characterize these effects are the acceptance bandwidths δξ (), usually defined as the deviation from the phasematching value ξ_{PM} leading to a phasemismatch variation Δk from 0 to 2π/L, where L is the crystal length. Then δξ is also the full width of the peak efficiency curve plotted as a function of ξ at 0.405 of the maximum, as shown in Fig. 1.7.3.8.

Figure 1.7.3.8
 top  pdf  Conversion efficiency evolution as a function of ξ for a given crystal length. ξ denotes the angle (θ or ), the temperature (T) or the wavelength (λ). ξ_{PM} represents the parameter allowing phase matching.

Thus Lδξ is a characteristic of the phasematching direction. Small angular, thermal and spectral dispersion of the refractive indices lead to high acceptance bandwidths. The higher Lδξ, the lower is the decrease of the conversion efficiency corresponding to a given angular shift, to the heating of the crystal due to absorption or external heating, or to the spectral bandwidth of the fundamental beam.
The knowledge of the angular, thermal and spectral dispersion of the refractive indices allows an estimation of δξ by expanding Δk in a Taylor series about ξ_{PM}:When the second and higherorder differential terms in (1.7.3.43) are negligible, the phase matching is called critical (CPM), because is small. For the particular cases where , is larger than the CPM acceptance and the phase matching is called noncritical (NCPM) for the parameter ξ considered.
We first consider the case of angular acceptances. In uniaxial crystals, the refractive indices do not vary in , leading to an infinite angular acceptance bandwidth. δθ is then the only one to consider. For directions of propagation out of the principal plane (), the phase matching is critical. According to the expressions of n_{o} and n_{e}(θ) given in Section 1.7.3.1, we have
CPM acceptance bandwidths are small, typically about one mrad cm, as shown in Section 1.7.5 for the classical nonlinear crystals.
When , and the phase matching is noncritical:
Values of NCPM acceptance bandwidths are given in Section 1.7.5 for the usual crystals. From the previous expressions for CPM and NCPM angular acceptances, it appears that the angular bandwidth is all the smaller since the birefringence is high.
The situation is obviously more complex in the case of biaxial crystals. The acceptance bandwidth is not infinite, leading to a smaller anisotropy of the angular acceptance in comparison with uniaxial crystals. The expressions of the θ and acceptance bandwidths have the same form as for the uniaxial class only in the principal planes. The phase matching is critical (CPM) for all directions of propagation out of the principal axes x, y and z: in this case, the mismatch Δk is a linear function of small angular deviations from the phasematching direction as for uniaxial crystals. There exist six possibilities of NCPM for SHG, types I and II along the three principal axes, corresponding to twelve different index conditions (Hobden, 1967):
The NCPM angular acceptances along the three principal axes of biaxial crystals can be deduced from the expressions relative to the uniaxial class by the following substitutions:
Along the x axis:
Along the y axis:
Along the z axis:
The above formulae are relative to the internal angular acceptance bandwidths. The external acceptance angles are enlarged by a factor of approximately n(ω) for type I or for type II, due to refraction at the input plane face of the crystal. The angular acceptance is an important issue connected with the accuracy of cutting of the crystal.
Temperature tuning is a possible alternative for achieving NCPM in a few materials. The corresponding temperatures for different interactions are given in Section 1.7.5.
Another alternative is to use a special noncollinear configuration known as onebeam noncritical noncollinear phase matching (OBNC): it is noncritical with respect to the phasematching angle of one of the input beams (referred to as the noncritical beam). It has been demonstrated that the angular acceptance bandwidth for the noncritical beam is exceptionally large, for example about 50 times that for the critical beam for typeI SHG at 1.338 µm in 3methyl4nitropyridineNoxide (POM) (Dou et al., 1992).
The typical values of thermal acceptance bandwidth, given in Section 1.7.5, are of the order of 0.5 to 50 K cm. The thermal acceptance is an important issue for the stability of the harmonic power when the absorption at the wavelengths concerned is high or when temperature tuning is used for the achievement of angular NCPM. Typical spectral acceptance bandwidths for SHG are given in Section 1.7.5. The values are of the order of 1 nm cm, which is much larger than the linewidth of a singlefrequency laser, except for some diode or for subpicosecond lasers with a large spectral bandwidth.
Note that a degeneracy of the firstorder temperature or spectral derivatives ( or ) can occur and lead to thermal or spectral NCPM.
Consideration of the phasematching function , where , , or all other dispersion parameters of the refractive indices, is useful for a direct comparison of the situation of noncriticality of the phase matching relative to and to the other parameters : a nil derivative of with respect to , i.e. at the point (), means that the phase matching is noncritical with respect to and so strongly critical with respect to , i.e. at this point. Then, for example, an angular NCPM direction is a spectral CPM direction and the reverse is also so.
(iii) Effect of spatial walkoff.
The interest of the NCPM directions is increased by the fact that the walkoff angle of any wave is nil: the beam overlap is complete inside the nonlinear crystal. Under CPM, the interacting waves propagate with different walkoff angles: the conversion efficiency is then attenuated because the different Poynting vectors are not collinear and the beams do not overlap. Type I and type II are not equivalent in terms of walkoff angles. For type I, the two fundamental waves have the same polarization E^{+} and the same walkoff angle ρ^{+}, which is different from the harmonic one; thus the coordinate systems that are involved in equations (1.7.3.22) are and . For type II, the two fundamental waves have necessarily different walkoff angles ρ^{+} and ρ^{−}, which forbids the nonlinear interaction beyond the plane where the two fundamental beams are completely separated. In this case we have three different coordinate systems: , and .
The three coordinate systems are linked by the refraction angles ρ of the three waves as explained in Section 1.7.3.2.1. We consider Gaussian transverse profiles: the electric field amplitude is then given by (1.7.3.37). In these conditions, the integration of (1.7.3.22) over () by assuming , the nondepletion of the pump and, in the case of phase matching, leads to the efficiency η_{SHG}(L) given by formula (1.7.3.42) with and multiplied by the factor where is the harmonic walkoff angle and is the walkoff attenuation function.
For type I, the walkoff attenuation is given by (Boyd et al., 1965)with and
For uniaxial crystals, for a 2oe interaction and for a 2eo interaction. For the biaxial class, for a 2oe interaction and for a 2eo interaction in the xz and yz planes, for a 2oe interaction and for a 2eo interaction in the xy plane. For any direction of propagation not contained in the principal planes of a biaxial crystal, the fundamental and harmonic waves have nonzero walkoff angles, respectively ρ^{+}(ω) and ρ^{−}(2ω). In this case, (1.7.3.53) can be used with .
For type II, we have (Mehendale & Gupta, 1988)withandr and u are the Cartesian coordinates in the walkoff plane where u is collinear with the three wavevectors, i.e. the phasematching direction.
for (oeo) in uniaxial crystals and in the xz and yz planes of biaxial crystals. in the xy plane of biaxial crystals for an (eoe) interaction.
For the interactions where ρ^{−}(2ω) and ρ^{−}(ω) are nonzero, we assume that they are close and contained in the same plane, which is generally the case. Then we classically take ρ to be the maximum value between and . This approximation concerns the (eoe) configuration of polarization in uniaxial crystals and for biaxial crystals in the xz and yz planes, in the xy plane for (oeo) and out of the principal planes for all the configurations of polarization.
The exact calculation of G, which takes into account the three walkoff angles, ρ^{−}(ω), ρ^{+}(ω) and ρ^{−}(2ω), was performed in the case where these three angles were coplanar (Asaumi, 1992). The exact calculation in the case of KTiOPO_{4} (KTP) for typeII SHG at 1.064 µm gives the same result for as for one angle defined as previously (Fève et al., 1995), which includes the parallelbeam limit 0.3–0.4: is the Rayleigh length of the fundamental beam inside the crystal.
The saturation length, L_{sat}, is defined as , which corresponds to the length beyond which the SHG conversion efficiency varies less than 1% from its saturation value .
The complete splitting of the two fundamental beams does not occur for type I, making it more suitable than type II for strong focusing. The fundamental beam splitting for type II also leads to a saturation of the acceptance bandwidths δξ (), which is not the case for type I (Fève et al., 1995). The walkoff angles also modify the transversal distribution of the generated harmonic beam (Boyd et al., 1965; Mehendale & Gupta, 1988): the profile is larger than that of the fundamental beam for type I, contrary to type II.
The walkoff can be compensated by the use of two crystals placed one behind the other, with the same length and cut in the same CPM direction (Akhmanov et al., 1975): the arrangement of the second crystal is obtained from that of the first one by a π rotation around the direction of propagation or around the direction orthogonal to the direction of propagation and contained in the walkoff plane as shown in Fig. 1.7.3.10 for the particular case of type II (oeo) in a positive uniaxial crystal out of the xy plane.
The twincrystal device is potentially valid for both types I and II. The relative sign of the effective coefficients of the twin crystals depends on the configuration of polarization, on the relative arrangement of the two crystals and on the crystal class. The interference between the waves generated in the two crystals is destructive and so cancels the SHG conversion efficiency if the two effective coefficients have opposite signs: it is always the case for certain crystal classes and configurations of polarization (Moore & Koch, 1996).
Such a tandem crystal was used, for example, with KTiOPO_{4} (KTP) for typeII SHG at µm () and µm (): the conversion efficiency was about 3.3 times the efficiency in a single crystal of length 2L, where L is the length of each crystal of the twin device (Zondy et al., 1994). The two crystals have to be antireflection coated or contacted in order to avoid Fresnel reflection losses.
Noncollinear phase matching is another method allowing a reduction of the walkoff, but only in the case of type II (Dou et al., 1992). Fig. 1.7.3.11 illustrates the particular case of (oeo) typeII SHG for a propagation out of the xy plane of a uniaxial crystal, or in the xz or yz plane of a biaxial crystal.
In the configuration of special noncollinear phase matching, the angle between the fundamental beams inside the crystal is chosen to be equal to the walkoff angle ρ. Then the associated Poynting vectors and are along the same direction, while that of the generated wave deviates from them only by approximately ρ/2. The calculation performed in the case of special noncollinear phase matching indicates that it is possible to increase typeII SHG conversion efficiency by 17% for nearfield undepleted Gaussian beams (Dou et al., 1992). Another advantage of such geometry is to turn type II into a pseudo type I with respect to the walkoff, because the saturation phenomenon of typeII CPM is avoided.
(iv) Effect of temporal walkoff.
Even if the SHG is phase matched, the fundamental and harmonic group velocities, , are generally mismatched. This has no effect with continuous wave (c.w.) lasers. For pulsed beams, the temporal separation of the different beams during the propagation can lead to a decrease of the temporal overlap of the pulses. Indeed, this walkoff in the time domain affects the conversion efficiency when the pulse separations are close to the pulse durations. Then after a certain distance, L_{τ}, the pulses are completely separated, which entails a saturation of the conversion efficiency, for both types I and II (Tomov et al., 1982). Three group velocities must be considered for type II. Type I is simpler, because the two fundamental waves have the same velocity, so , which defines the optimum crystal length, where τ is the pulse duration. For typeI SHG of 532 nm in KH_{2}PO_{4} (KDP), v_{g}(266 nm) m s^{−1} and v_{g}(532 nm) m s^{−1}, so L_{τ} mm for 1 ps. For the usual nonlinear crystals, the temporal walkoff must be taken into account for pico and femtosecond pulses.
