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. 200209
Section 1.7.3.3.2. Second harmonic generation (SHG)^{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 
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.
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).
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