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. 199215
Section 1.7.3.3. Integration of the propagation equations^{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 
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.
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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