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. 211215
Section 1.7.3.3.5. Differencefrequency generation (DFG)^{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 
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.
References
Baumgartner, R. A. & Byer, R. L. (1979). Optical parametric amplification. IEEE J. Quantum Electron. QE15, 432–444.Boulanger, B., Fève, J. P., Ménaert, B. & Marnier, G. (1999). PCT/FR98/02563 Patent No. WO99/28785.
Breitenbach, G., Schiller, S. & Mlynek, J. (1995). 81% conversion efficiency in frequencystable continuous wave parametric oscillator. J. Opt. Soc. Am. B, 12(11), 2095–2101.
Brosnan, S. J. & Byer, R. L. (1979). Optical parametric oscillator threshold and linewidth studies. IEEE J. Quantum Electron. QE15(6), 415–431.
Byer, R. L. (1973). Treatise in quantum electronics, edited by H. Rabin & C. L. Tang. New York: Academic Press.
Chung, J. & Siegman, E. (1993). Singly resonant continuouswave modelocked KTiOPO_{4} optical parametric oscillator pumped by a Nd:YAG laser. J. Opt. Soc. Am. B, 10(9), 2201–2210.
Debuisschert, T., Sizmann, A., Giacobino, E. & Fabre, C. (1993). TypeII continuouswave optical parametric oscillator: oscillation and frequency tuning characteristics. J. Opt. Soc. Am. B, 10(9), 1668–1690.
Ebrahimzadeh, M. & Dunn, M. H. (2000). Optical parametric oscillators. In Handbook of optics, Vol. IV, pp. 2201–2272. New York: McGrawHill.
Ebrahimzadeh, M., Henderson, A. J. & Dunn, M. H. (1990). An excimerpumped βBaB_{2}O_{4} optical parametric oscillator tunable from 354 nm to 2.370 µm. IEEE J. Quantum Electron. QE26(7), 1241–1252.
Ebrahimzadeh, M., Turnbull, G. A., Edwards, T. J., Stothard, D. J. M., Lindsay, I. D. & Dunn, M. H. (1999). Intracavity continuouswave singly resonant optical parametric oscillators. J. Opt. Soc. Am. B, 16, 1499–1511.
Fève, J. P., Pacaud, O., Boulanger, B., Ménaert, B., Hellström, J., Pasiskeviscius, V. & Laurell, F. (2001). Widely and continuously tuneable optical parametric oscillator using a cylindrical periodically poled KTiOPO_{4} crystal. Opt. Lett. 26, 1882–1884.
Fève, J. P., Pacaud, O., Boulanger, B., Ménaert, B. & Renard, M. (2002). Tunable phasematched optical parametric oscillators based on a cylindrical crystal. J. Opt. Soc. Am. B, 19, 222–233.
Harris, S. E. (1969). Tunable optical parametric oscillators. Proc. IEEE, 57(12), 2096–2113.
Kato, K. (1986). Secondharmonic generation to 2048 Å in βBaB_{2}O_{4}. IEEE J. Quantum Electron. QE22, 1013–1014.
Kato, K. (1991). Parametric oscillation at 3.3 µm in KTP pumped at 1.064 µm. IEEE J. Quantum Electron. QE27, 1137–1140.
Khodja, S., Josse, D. & Zyss, J. (1995a). First demonstration of an efficient nearinfrared optical parametric oscillator with an organomineral crystal. Proc. CThC2, CLEO'95 (Baltimore), pp. 267–268.
Ledoux, I., Lepers, C., Perigaud, A., Badan, J. & Zyss, J. (1990). Linear and nonlinear optical properties of N4nitrophenylLprolinol single crystals. Optics Comm. 80, 149–154.
Louisell, W. H., Yariv, A. & Siegman, A. E. (1961). Quantum fluctuations and noise in parametric processes. I. Phys. Rev. 124, 1646.
Marshall, L. R. & Kaz, A. (1993). Eyesafe output from noncritically phasematched parametric oscillators. J. Opt. Soc. Am. B, 10(9), 1730–1736.
Myers, L. E., Eckardt, R. C., Fejer, M. M., Byer, R. L. & Bosenberg, W. R. (1996). Multigrating quasiphasematched optical parametric oscillator in periodically poled LiNbO_{3}. Opt. Lett. 21(8), 591–593.
Pacaud, O., Fève, J. P., Boulanger, B. & Ménaert, B. (2000). Cylindrical KTiOPO_{4} crystal for enhanced angular tunability of phasematched optical parametric oscillators. Opt. Lett. 25, 737–739.
Powers, P. E., Kulp, T. J. & Bisson, S. E. (1998). Continuous tuning of a continuouswave periodically poled lithium niobate optical parametric oscillator by use of a fanout grating design. Opt. Lett. 23, 159–161.
Reid, D. T., Kennedy, G. T., Miller, A., Sibbett, W. & Ebrahimzadeh, M. (1998). Widely tunable near to midinfrared femtosecond and picosecond optical parametric oscillators using periodically poled LiNbO_{3} and RbTiOAsO_{4}. IEEE J. Sel. Top. Quantum Electron. 4, 238–248.
Ruffing, B., Nebel, A. & Wallenstein, R. (1998). Allsolidstate CW modelocked picosecond KTiOAsO_{4} (KTA) optical parametric oscillator. Appl. Phys. B67, 537–544.
Scheidt, M., Beier, B., Knappe, R., Bolle, K. J. & Wallenstein, R. (1995). Diodelaserpumped continuous wave KTP optical parametric oscillator. J. Opt. Soc. Am. B, 12(11), 2087–2094.
Unschel, R., Fix, A., Wallenstein, R., Rytz, D. & Zysset, B. (1995). Generation of tunable narrowband midinfrared radiation in a type I potassium niobate optical parametric oscillator. J. Opt. Soc. Am. B, 12, 726–730.
Yang, S. T., Eckardt, R. C. & Byer, R. L. (1993). Power and spectral characteristics of continuouswave parametric oscillators: the doubly to singly resonant transition. J. Opt. Soc. Am. B, 10(9), 1684–1695.