## General analysis of group velocity effects in collinear optical parametric amplifiers and generators

Optics Express, Vol. 15, Issue 10, pp. 6513-6527 (2007)

http://dx.doi.org/10.1364/OE.15.006513

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### Abstract

Group velocity mismatch (GVM) is a major concern in the design of optical parametric amplifiers (OPAs) and generators (OPGs) for pulses shorter than a few picoseconds. By simplifying the coupled propagation equations and exploiting their scaling properties, the number of free parameters for a collinear OPA is reduced to a level where the parameter space can be studied systematically by simulations. The resulting set of figures show the combinations of material parameters and pulse lengths for which high performance can be achieved, and they can serve as a basis for a design.

© 2007 Optical Society of America

## 1. Introduction

2. G. A. Bukauskas, V. I. Kabelka, A. Piskarskas, and A. Y. Stabinis, “Features of three-photon parametric interaction of ultrashort light packets in the nonlinear amplification regime,” Sov. J. Quantum Electron. **4**, 290–292 (1974). [CrossRef]

3. W. H. Glenn, “Parametric amplification of ultrashort laser pulses,” Appl. Phys. Lett. **11**, 333–335 (1967). [CrossRef]

4. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “H-5 - Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. **4**, 598–605 (1968). [CrossRef]

5. M. F. Becker, C. K. Young, S. R. Gautam, and E. J. Powers, “Three-wave nonlinear optical interactions in dispersive media,” IEEE Journal of Quantum Electronics **18**, 113–123 (1982). [CrossRef]

6. R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, “Traveling-wave parametric generation of widely tunable highly coherent femtosecond light pulses,” J. Opt. Soc. Am. B **10**, 2222–2232 (1993). [CrossRef]

*et al*. [2

2. G. A. Bukauskas, V. I. Kabelka, A. Piskarskas, and A. Y. Stabinis, “Features of three-photon parametric interaction of ultrashort light packets in the nonlinear amplification regime,” Sov. J. Quantum Electron. **4**, 290–292 (1974). [CrossRef]

7. G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. **20**, 1562–1564 (1995). [CrossRef] [PubMed]

8. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. **71**, 1–18 (2003). [CrossRef]

9. V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strizhevskii, “Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. **4**, 1090–1098 (1975). [CrossRef]

10. A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. **22**, 1071–1073 (1997). [CrossRef] [PubMed]

11. A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. **26**, 719–721 (2001). [CrossRef]

6. R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, “Traveling-wave parametric generation of widely tunable highly coherent femtosecond light pulses,” J. Opt. Soc. Am. B **10**, 2222–2232 (1993). [CrossRef]

3. W. H. Glenn, “Parametric amplification of ultrashort laser pulses,” Appl. Phys. Lett. **11**, 333–335 (1967). [CrossRef]

4. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “H-5 - Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. **4**, 598–605 (1968). [CrossRef]

2. G. A. Bukauskas, V. I. Kabelka, A. Piskarskas, and A. Y. Stabinis, “Features of three-photon parametric interaction of ultrashort light packets in the nonlinear amplification regime,” Sov. J. Quantum Electron. **4**, 290–292 (1974). [CrossRef]

5. M. F. Becker, C. K. Young, S. R. Gautam, and E. J. Powers, “Three-wave nonlinear optical interactions in dispersive media,” IEEE Journal of Quantum Electronics **18**, 113–123 (1982). [CrossRef]

6. R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, “Traveling-wave parametric generation of widely tunable highly coherent femtosecond light pulses,” J. Opt. Soc. Am. B **10**, 2222–2232 (1993). [CrossRef]

12. T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO_{4} crystals,” J. Appl. Phys. **77**, 4941–4947 (1995). [CrossRef]

13. T. Nishikawa and N. Uesugi, “Transverse beam profiles on traveling-wave optical parametric generation in KTiOPO_{4} crystals,” J. Appl. Phys. **78**, 6361–6366 (1995). [CrossRef]

## 2. Theory

*Z*

_{0}is the vacuum impedance. In order to simplify the equations, GVD is neglected. Since typical nonlinear crystals have GVD coefficients of the order 10

^{-25}s

^{2}, this approximation should be good for pulses as short as about 100 fs and crystals of about 1 cm. Becker

*et al*. [5

5. M. F. Becker, C. K. Young, S. R. Gautam, and E. J. Powers, “Three-wave nonlinear optical interactions in dispersive media,” IEEE Journal of Quantum Electronics **18**, 113–123 (1982). [CrossRef]

*n*is the group index of beam

_{g,j}*j, n*is the refractive index of beam

_{j}*j*, Δ

*k*=

*k*

_{3}-

*k*

_{2}-

*k*

_{1}is the phase-mismatch, and χ

_{eff}is the effective nonlinear susceptibility. The different

*n*allow for GVM. Terms of the form

_{g}*∂*

^{2}

*e*/∂

_{j}*z*

^{2}have been neglected based on the slowly varying envelope approximation, i.e. that the pulse is long compared to the wavelength so that

*∂e*/

_{j}*∂z*≪

*k*. This assumption is consistent with the omission of GVD. The equations can be transformed to the frame moving with velocity

_{j}e_{j}*v*

_{g,1}=

*c*/

*n*

_{g,1}by using the coordinate

*t′*=

*t*-

*z*/

*v*

_{g,1}instead of

*t*. The equations can be further simplified by defining the scaled amplitudes

*k*= 0 for the centre frequencies and obtain

*δn*=

_{g,j}*n*-

_{g, j}*n*. This form of the equations reduces the set of independent parameters to

_{g,1}*δn*

_{g,2},

*δn*

_{g,3}, and the crystal length

*L*, while the wavelengths, the refractive index and χ

_{eff}enter only in the scaling of the amplitudes. The boundary conditions are defined by the input pulses

*U*(

_{j}*z*= 0,

*t′*). The physical significance of

*u*

_{3}is that the steady-state small-signal amplitude gain is cosh(

*u*

_{3}

*L*). The equations are invariant if the GVM parameters

*δn*and the time are scaled by the same factor

_{g,j}*s*, or if

*u*are scaled by a factor

_{j}*s*while time and

*L*are scaled by

*s*

^{-1}. For simplicity of notation, and without loss of generality, I take

*n*

_{g,1}<

*n*

_{g,2}in the rest of the paper. If the equations are written in the frame moving with beam 2, they read

*t′*is now reversed and

*n*

_{g,3}is replaced by

*n*

_{g,1}+

*n*

_{g,2}-

*n*

_{g,3}, the original form of the equations is recovered, except that beams 1 and 2 are swapped. It follows that this replacement of

*n*

_{g,3}is equivalent to reversing the pulses and swapping beams 1 and 2. This symmetry is useful for reducing the set of

*n*

_{g,3}values to simulate.

## 3. Simulations

14. G. Arisholm, “Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators,” J. Opt. Soc. Am. B **16**, 117–127 (1999). [CrossRef]

15. G. Arisholm, J. Biegert, P. Schlup, C. P. Hauri, and U. Keller, “Ultra-broadband chirped-pulse optical parametric amplifier with angularly dispersed beams,” Opt. Express **12**, 518–530 (2004). [CrossRef] [PubMed]

*δn*

_{g,2}and the peak pump amplitude

*u*fixed while varying

_{p}*δn*

_{g,3},

*L*, the shape and duration of the pump pulse, and the intensity, shape and duration of the seed pulse. Scaling

*δn*

_{g,2}by

*s*would be equivalent to scaling

*δn*

_{g,3}and

*t*by

*s*

^{-1}, and scaling

*u*by

_{p}*s*would be equivalent to scaling

*L*and

*t*by s and the seed amplitude

*u*by

_{s}*T*. The duration is measured as the full width at exp(-2) of the peak power, i.e.

*u*

_{3}(0,

*t*) =

*u*exp(- (2

_{p}*t*/

*T*)

^{2}). In the high-gain regime, the signal pulse is shaped by the gain, so provided the seed pulse is long enough to overlap the pump, its exact shape is not critical. Furthermore, the simulations show that it makes very little difference whether beam 1 or beam 2 is seeded. For these reasons, the input idler is taken to be zero and only the signal is seeded, and the seed pulse is taken to have the same shape and duration as the pump pulse. With these assumptions for the input pulses, the parameters that must be varied are

*δn*

_{g,3},

*L, T*, and seed intensity.

*n*

_{g,3}by

*n*

_{g,1}+

*n*

_{g,2}-

*n*

_{g,3}is equivalent to swapping beams 1 and 2 and reversing the pulses. Since the input pulses are symmetric and it makes little difference which input beam is seeded, it is only necessary to perform simulations for

*n*

_{g,3}≥ (

*n*

_{g,1}+

*n*

_{g,2})/2. Results for a smaller

*n*

_{g,3}can be obtained by swapping and reversing the output pulses from the corresponding simulation with

*n*

_{g,1}+

*n*

_{g,2}-

*n*

_{g,3}. Pulse quality and energy are of course unaffected by time reversal.

**4**, 290–292 (1974). [CrossRef]

3. W. H. Glenn, “Parametric amplification of ultrashort laser pulses,” Appl. Phys. Lett. **11**, 333–335 (1967). [CrossRef]

4. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “H-5 - Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. **4**, 598–605 (1968). [CrossRef]

**18**, 113–123 (1982). [CrossRef]

**10**, 2222–2232 (1993). [CrossRef]

*T*and

*n*

_{g,3}. In Fig. 1(a), with

*n*

_{g,3}= (

*n*

_{g,1}+

*n*

_{g,2})/2 = 1.63 and a short (200 fs) pump pulse, the peak of the pump pulse has been depleted at

*z*= 4.8 mm. Backconversion regenerates a pump pulse at

*z*= 5.8mm, but because the signal and idler pulses walk off in opposite directions the backcon-version stops before the signal and idler pulses are severely distorted. At position

*z*= 7.2 mm, the regenerated pump has generated weak side pulses for the signal and idler, so the crystal should be kept shorter than this if clean pulses are desired. In Fig. 1(b), with the same

*n*

_{g,3}and a long (4 ps) pump pulse, the middle part of the pump pulse has been depleted at

*z*= 4 mm and backconversion has just started. Because the temporal walk-off is now small compared to the pulse length, it cannot suppress backconversion, and at

*z*= 5 mm the middle part of the pulse has been completely backconverted. At

*z*= 5.8 mm a second round of conversion and backconversion has taken place, and the pulses have broken up completely. Figure 1(c), with

*n*

_{g,3}= 1.72 and long pump pulse, shows similar behaviour, but the pulses become asymmetric because of the temporal walk-off of the pump. The case with

*n*

_{g,3}>

*n*

_{g,2}and a short pump pulse is not shown because the temporal walk-off leads to low gain [1] and very small conversion.

**10**, 2222–2232 (1993). [CrossRef]

*n*

_{g,3}inside and outside the interval [

*n*

_{g,1},

*n*

_{g,2}]. In Fig. 2(a), with

*n*

_{g,3}= 1.63, all pulses grow rapidly to saturation, and only the shortest pulses experience a somewhat reduced gain coefficient. Pulses longer than about 0.3 ps are in the long-pulse regime, i.e. they have a gain coefficient nearly as high as for continuous beams. The high gain, in spite of temporal walk-off can be understood by considering the signal and idler pulses in the frame moving with the pump pulse: Since they walk in opposite directions, the signal light being left behind by the trailing end of the pump pulse has already generated idler light that walks in the other direction. Conversely, when idler light leaves the leading end of the pump pulse, it has generated signal light, and this maintains positive feedback between signal and idler within the pump pulse.

*n*

_{g,3}=

*n*

_{g,2}= 1.66, the gain coefficients for the short pulses are lower, but even the shortest pulses continue growing through the whole crystal [1] and eventually reach pump depletion, so there is not a minimum pulse length for efficient interaction.

*n*

_{g,3}>

*n*

_{g,2}, as shown in Fig. 2(c) and (d), the temporal walk-off does not only reduce the gain coefficient for short pulses, but it also clamps their total gain at a level below the limit set by pump depletion. Intuitively, any part of the signal or idler pulse can only grow as long as it overlaps the pump pulse, so the maximum effective gain length for a pulse of length

*T*equals the the distance in which the pump pulse separates from the signal and idler pulses by

*T*:

*K*is a constant of order 1 and

*c*is the speed of light.

*δn*is taken to correspond to the temporal walk-off between the pump and the generated pulse with the best overlap, i.e.

_{g,p}*δn*= min(|

_{g,p}*n*

_{g,3}-

*n*

_{g,1}|,|

*n*

_{g,3}-

*n*

_{g,2}|). The total gain is limited either by this gain length or by pump depletion, whichever occurs first. Corresponding to

*L*is a minimum pulse length

_{e}*T*

_{min}required to obtain an amplitude gain of

*G*= cosh(

*g*) (in the absence of pump depletion). This is determined by

*u*(

_{p}L_{e}*T*

_{min}) =

*g*, and assuming Eq. (12) for

*L*,

_{e}*T*

_{min}is at least consistent with the scaling properties of Eqs. (6–8) in that

*T*

_{min}∝

*u*

^{-1}

_{p}. If

*δn*

_{g,2}and

*δn*

_{g,3}vary in proportion,

*T*

_{min}is ∝

*δn*

_{g,3}. If

*δn*

_{g,2}is small compared to

*δn*

_{g,3}, it can be neglected as far as gain is concerned, although it does determine the signal gain bandwidth. In this case,

*T*

_{min}∝

*δn*

_{g,3}, and the parameter space for simulations can be simplified by keeping

*δn*

_{g,3}fixed and varying only

*T, L*, and the seed intensity.

16. G. Rousseau, N. McCarthy, and M. Piche, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett. **27**, 1649–1651 (2002). [CrossRef]

*n*

_{g,3}= 1.63 = (

*n*

_{g,1}+

*n*

_{g,2})/2. The signal and idler have similar pulse quality because they walk off symmetrically with respect to the pump, so only the idler is shown. The same data are shown as images for intuitive interpretation (a, b) and as graphs for quantitative analysis (c, d). The first feature to note is that for sufficiently long pulses, a specific value of

*L*(4 mm in this case), combines moderately high conversion and good pulse quality. The approximate independence of

*T*means that these pulses are in the long-pulse limit, where the effect of temporal walk-off is small. For comparison, the optimal crystal length for a continuous-wave pump with amplitude

*u*would be 3.8 mm. The limited conversion efficiency can be explained by the time-dependent intensity of the pump pulses - no crystal length is optimal for the whole range of pump intensities that occur within the pulse. Longer crystals lead to back conversion, and although conversion can be higher for some crystal lengths, the pulse quality suffers. Short pulses grow more slowly, but for crystals longer than 4 mm they can reach even higher conversion than the long pulses, and the pulse quality remains high for long crystals. This is a beneficial effect of temporal walk-off, as seen in Fig. 1(a): Backconversion is suppressed because the signal and idler pulses walk off in opposite directions. This mechanism appears to work well for pulses up to about 0.2 ps, which have a signal-idler walk-off length

_{p}*L*=

_{w}*Tc*/(

*n*

_{g,2}-

*n*

_{g,1}) = 1 mm. For comparison, the inverse of the peak gain coefficient is 1/

*u*= 0.26 mm.

_{p}*n*

_{g,3}in the range 1.645–2. Note that the range of the T-axis is not the same in all the graphs. In these examples, with

*n*

_{g,3}> (

*n*

_{g,1}+

*n*

_{g,2}) /2, the pulse quality is often better for the signal than for the idler. Simulations with

*n*

_{g,3}< (

*n*

_{g,1}+

*n*

_{g,2})/2 give better pulse quality for the idler, whereas seeding beam 1 instead of beam 2 makes little difference, as expected in the high-gain regime. Thus, most of the difference in pulse quality can be ascribed to the group velocites – the pulse with group velocity closer to pump has better quality. As the GVM

*δn*

_{g,3}increases, high conversion is no longer possible for short pulses, and the minimum pulse lengths from the figures are in fair agreement with Eq. (13) if

*K*≈ 0.5. For long pulses,

*L*= 4 mm remains the optimal crystal length. The sensitivity to

*δn*

_{g,3}for short pulses is striking, and this is illustrated in detail in Fig. 5. For short pulses,

*n*

_{g,3}= (

*n*

_{g,1}+

*n*

_{g,2})/2 is the optimal value, and the efficiency drops sharply when

*n*

_{g,3}approaches the ends of the interval [

*n*

_{g,1},

*n*

_{g,2}]. The curves are very nearly symmetric about

*n*

_{g,3}= (

*n*

_{g,1}+

*n*

_{g,2})/2, and the slight deviation from symmetry occurs because the signal is seeded and the idler is not.

*n*and the maximum permissible pump amplitude

_{g,j}*u*

_{p,max}are given by the material and the desired wavelengths,

*T*is given by the available pump laser, and the parameters that can be adjusted in the design are

*L*and

*u*≤

_{p}*u*

_{p,max}. From the scaling arguments in the beginning of this section, reducing the pump amplitude is equivalent to moving towards the upper left corner in the performance maps of Figs. 3–4. In the case with

*n*

_{g,3}∈ [

*n*

_{g,1},

*n*

_{g,2}], it can be desirable to scale up down (and maybe increase

*L*to make up for the reduced gain) to move into the short-pulse regime where temporal walk-off can reduce back-conversion. For

*n*

_{g,3}∉[

*n*

_{g,1},

*n*

_{g,2}], scaling down

*u*is not advantageous. If the scaled

_{p}*T*exceeds

*T*

_{min}, high performance can be restored by increasing

*L*, but if the scaled

*T*becomes too short, the reduced pump amplitude cannot be compensated by crystal length. If the pump amplitude is scaled up from a level that works well, high performance can be restored by reducing

*L*, but the figures indicate that there is not much performance to gain by increasing the pump above such a level. Hence, for

*n*

_{g,3}∉ [

*n*

_{g,1},

*n*

_{g,2}], operating with

*u*=

_{p}*u*

_{p,max}can give nearly optimal performance in most cases.

*n*

_{g,3}= 1.63 and

*n*

_{g,3}= 2. As expected, the maps of conversion and pulse quality are qualitatively similar to those in Figs. 3 and 4, but they are shifted to shorter crystal lengths. If

*n*

_{g,3}∉ [

*n*

_{g,1},

*n*

_{g,2}], the gain for short pulses saturates after a short propagation distance, and a minimum seed intensity is required in order to reach a significant conversion efficiency. Another way to see this is that

*T*

_{min}in Eq. (13) depends on

*g*, which is of course lower for a stronger seed. This is consistent with Fig. 6(c), where

*T*

_{min}is slightly smaller than in Fig. 4(i). Simulations with a seed pulse 3 times longer than the pump pulse yielded results nearly identical to those with equal seed and pump pulses.

## 4. Application to real crystals

*s*

_{1}=

*u*/

_{p}*u′*and

_{p}*S*

_{2}=

*δn*

_{g,2}/

*δn′*

_{g,2}. From Section 3,

*u′*= 3840m

_{p}^{-1}and

*δn′*

_{g,2}= 0.06.

*δn*

_{g,2}can be found from the material data, and from Eqs. (1–5)

*n*

_{g,3}∈ [

*n*

_{g,1},

*n*

_{g,2}], and an OPA should work well with a wide range of pulse lengths.

*δn*′

_{g,3}= 0.016 corresponds approximately (by symmetry) to

*n*

_{g,3}= 1.645, or Fig. 4(a,b). The parameters shown in the table,

*T*= 0.75ps and

*L*= 40 mm correspond to a short pulse for which temporal walk-off suppresses backconversion Example 2 is in the regime with

_{c}*n*

_{g,3}much greater than

*n*

_{g,1}and

*n*

_{g,2}, so

*T*= 6.5ps is the minimum pulse length for efficient conversion. In example 3,

*n*

_{g,3}is just slightly smaller than

*n*

_{g,2}. Figure 4(c,d) can be used for a first estimate of the performance, but because of the strong sensitivity when

*n*

_{g,3}≈

*n*

_{g,2}as seen in Fig. 5, more accurate calculations should be carried out for a detailed design. Example 4 has

*n*

_{g,3}near the optimal value (

*n*

_{g,1}+

*n*

_{g,2})/2, and Fig. 3 or 4(a,b) should be applied.

*T*

_{min}must be taken into account.

## 5. Optical parametric generators

*n*

_{g,3}= 1.63) is similar to 3(a), except that the crystal must be about 1 mm longer to give sufficient gain for the OPG. Fig. 7(b) differs noticeably from 3(b) in that the pulse quality is always poor for long pulses, as explained above. For shorter pulses, there is a range of parameters in Fig. 7(a,b) that combine high conversion and high pulse quality.

*n*

_{g,3}= 1.66, in Fig. 7(c,d), there is high performance for a parameter range near

*L*= 5mm and

*T*= 1.4 ps, but the range of pulse lengths that combine moderately high conversion and good pulse quality is narrow. With

*n*

_{g,3}= 1.80 (Fig. 7(e,f)), it is no longer possible to combine these features, as the minimum pulse length for high conversion is already too long for high pulse quality. In the OPG, the pulse quality of the signal and idler differ less than in Fig. 4. This is because the spectra of both beams are now mainly determined by the gain bandwidth.

## 6. Transverse effects

*is*important. Spatio-temporal simulations of a specific KTP-based short-pulse OPA have been reported [12

12. T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO_{4} crystals,” J. Appl. Phys. **77**, 4941–4947 (1995). [CrossRef]

13. T. Nishikawa and N. Uesugi, “Transverse beam profiles on traveling-wave optical parametric generation in KTiOPO_{4} crystals,” J. Appl. Phys. **78**, 6361–6366 (1995). [CrossRef]

20. G. Arisholm, R. Paschotta, and T. Südmeyer, “Limits to the power scalability of high-gain optical parametric amplifiers,” J. Opt. Soc. Am. B **21**, 578–590 (2004). [CrossRef]

20. G. Arisholm, R. Paschotta, and T. Südmeyer, “Limits to the power scalability of high-gain optical parametric amplifiers,” J. Opt. Soc. Am. B **21**, 578–590 (2004). [CrossRef]

*n*

_{g,3}≈ (

*n*

_{g,1}+

*n*

_{g,2})/2, where Figs. 3 and 4(a,b) show that the tolerance for crystal length, and hence pump intensity, is considerable. This may allow high conversion, beam- and pulse-quality even for relatively wide beams.

*n*

_{g,3}= 1.63 or 1.80 and four different combinations of pulse length and pump beam radius. The pulse- and beam-quality of the signal are not shown because they are similar to those of the idler. The pump- and seed-beams are Gaussian with equal waist radii

*w*

_{0}= 30 or 200

*μ*m. The corresponding Rayleigh length in the nonlinear crystal is about 4 mm or 20 cm, respectively, so these correspond to cases with strong and weak diffraction. In order to avoid unnecessary complications, I have assumed noncritical phase matching (i.e. no transverse walk-off) and cylindrical symmetry.

*n*

_{g,3}= 1.63,

*T*= 3ps,

*w*

_{0}=200

*μ*m(as in Fig. 8(d)), and

*L*= 7 mm. The conversion efficiency in this case is 0.36, but backconversion breaks up the pulse and gives rise to intensity oscillations in time and along the radial position. Integration over the pulse or beam masks the oscillations, so the poor quality is not so clearly seen in the totals in Fig. 9(b) and (c).

13. T. Nishikawa and N. Uesugi, “Transverse beam profiles on traveling-wave optical parametric generation in KTiOPO_{4} crystals,” J. Appl. Phys. **78**, 6361–6366 (1995). [CrossRef]

20. G. Arisholm, R. Paschotta, and T. Südmeyer, “Limits to the power scalability of high-gain optical parametric amplifiers,” J. Opt. Soc. Am. B **21**, 578–590 (2004). [CrossRef]

## 7. Conclusion

*δn*

_{g,3}=

*n*

_{g,3}-

*n*

_{g,1}. The performance of a wide range of OPAs can be estimated by computing the scaled parameters and reading these maps. As reported before [1, 6

**10**, 2222–2232 (1993). [CrossRef]

*n*

_{g,3}∈ [

*n*

_{g,1},

*n*

_{g,2}], high conversion and pulse quality can be obtained for a wide range of pulse lengths, and short pulses even benefit from temporal walk-off for suppressing backconversion. In the second regime, with

*n*

_{g,3}∉ [

*n*

_{g,1},

*n*

_{g,2}], only pulses longer than a minimum duration

*T*

_{min}∝

*δn*

_{g,3}/

*u*are converted efficiently, where

_{p}*u*is the amplitude gain coefficient.

_{p}*n*

_{g,3}∈ [

*n*

_{g,1},

*n*

_{g,2}], and a pump pulse short enough to take advantage of temporal walk-off for suppressing back-conversion.

## Acknowledgments

## References and links

1. | A. P. Sukhorukov and A. K. Shchednova, “Parametric amplification of light in the field of a modulated laser wave,” Sov. Phys. JETP |

2. | G. A. Bukauskas, V. I. Kabelka, A. Piskarskas, and A. Y. Stabinis, “Features of three-photon parametric interaction of ultrashort light packets in the nonlinear amplification regime,” Sov. J. Quantum Electron. |

3. | W. H. Glenn, “Parametric amplification of ultrashort laser pulses,” Appl. Phys. Lett. |

4. | S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “H-5 - Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. |

5. | M. F. Becker, C. K. Young, S. R. Gautam, and E. J. Powers, “Three-wave nonlinear optical interactions in dispersive media,” IEEE Journal of Quantum Electronics |

6. | R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, “Traveling-wave parametric generation of widely tunable highly coherent femtosecond light pulses,” J. Opt. Soc. Am. B |

7. | G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. |

8. | G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. |

9. | V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strizhevskii, “Method for compensating the phase-matching dispersion in nonlinear optics,” Sov. J. Quantum Electron. |

10. | A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. |

11. | A. V. Smith, “Group-velocity-matched three-wave mixing in birefringent crystals,” Opt. Lett. |

12. | T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO |

13. | T. Nishikawa and N. Uesugi, “Transverse beam profiles on traveling-wave optical parametric generation in KTiOPO |

14. | G. Arisholm, “Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators,” J. Opt. Soc. Am. B |

15. | G. Arisholm, J. Biegert, P. Schlup, C. P. Hauri, and U. Keller, “Ultra-broadband chirped-pulse optical parametric amplifier with angularly dispersed beams,” Opt. Express |

16. | G. Rousseau, N. McCarthy, and M. Piche, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett. |

17. | H. Vanherzeele, J. D. Bierlein, and F. C. Zumsteg, “Index of refraction measurements and parametric generation in hydrothermally grown KTiOPO |

18. | D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, 22, 1553–1555 (1997). [CrossRef] |

19. | D. E. Zelmon, E. A. Hanning, and P. G. Schunemann, “Refractive-index measurements and Sellmeier coefficients for zinc-germanium phosphide from 2 to 9 |

20. | G. Arisholm, R. Paschotta, and T. Südmeyer, “Limits to the power scalability of high-gain optical parametric amplifiers,” J. Opt. Soc. Am. B |

**OCIS Codes**

(140.4480) Lasers and laser optics : Optical amplifiers

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 26, 2007

Revised Manuscript: May 9, 2007

Manuscript Accepted: May 10, 2007

Published: May 11, 2007

**Citation**

Gunnar Arisholm, "General analysis of group velocity effects in collinear optical parametric amplifiers and generators," Opt. Express **15**, 6513-6527 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6513

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### References

- A. P. Sukhorukov and A. K. Shchednova, "Parametric amplification of light in the field of a modulated laser wave," Sov. Phys. JETP 33, 677-682 (1971).
- G. A. Bukauskas, V. I. Kabelka, A. Piskarskas, and A. Y. Stabinis, "Features of three-photon parametric interaction of ultrashort light packets in the nonlinear amplification regime," Sov. J. Quantum Electron. 4, 290-292 (1974). [CrossRef]
- W. H. Glenn, "Parametric amplification of ultrashort laser pulses," Appl. Phys. Lett. 11, 333-335 (1967). [CrossRef]
- S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "H-5 - Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. 4, 598-605 (1968). [CrossRef]
- M. F. Becker, C. K. Young, S. R. Gautam, and E. J. Powers, "Three-wave nonlinear optical interactions in dispersive media," IEEE Journal of Quantum Electronics 18, 113-123 (1982). [CrossRef]
- R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable highly coherent femtosecond light pulses," J. Opt. Soc. Am. B 10, 2222-2232 (1993). [CrossRef]
- G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, "Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator," Opt. Lett. 20, 1562-1564 (1995). [CrossRef] [PubMed]
- G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 71, 1-18 (2003). [CrossRef]
- V. D. Volosov, S. G. Karpenko, N. E. Kornienko, and V. L. Strizhevskii, "Method for compensating the phasematching dispersion in nonlinear optics," Sov. J. Quantum Electron. 4, 1090-1098 (1975). [CrossRef]
- A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, "Nonlinear second-harmonic pulse compression with tilted pulses," Opt. Lett. 22, 1071-1073 (1997). [CrossRef] [PubMed]
- A. V. Smith, "Group-velocity-matched three-wave mixing in birefringent crystals," Opt. Lett. 26, 719-721 (2001). [CrossRef]
- T. Nishikawa and N. Uesugi, "Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals," J. Appl. Phys. 77, 4941-4947 (1995). [CrossRef]
- T. Nishikawa and N. Uesugi, "Transverse beam profiles on traveling-wave optical parametric generation in KTiOPO4 crystals," J. Appl. Phys. 78, 6361-6366 (1995). [CrossRef]
- G. Arisholm, "Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators," J. Opt. Soc. Am. B 16, 117-127 (1999). [CrossRef]
- G. Arisholm, J. Biegert, P. Schlup, C. P. Hauri, and U. Keller, "Ultra-broadband chirped-pulse optical parametric amplifier with angularly dispersed beams," Opt. Express 12, 518-530 (2004). [CrossRef] [PubMed]
- G. Rousseau, N. McCarthy, and M. Piche, "Description of pulse propagation in a dispersive medium by use of a pulse quality factor," Opt. Lett. 27, 1649-1651 (2002). [CrossRef]
- H. Vanherzeele, J. D. Bierlein, and F. C. Zumsteg, "Index of refraction measurements and parametric generation in hydrothermally grown KTiOPO4," Appl. Opt. 27, 3314-3316 (1988). [CrossRef] [PubMed]
- D. H. Jundt, "Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate," Opt. Lett. 22, 1553-1555 (1997). [CrossRef]
- D. E. Zelmon, E. A. Hanning, and P. G. Schunemann, "Refractive-index measurements and Sellmeier coefficients for zinc-germanium phosphide from 2 to 9 m with implications for phase matching in optical frequencyconversion devices," J. Opt. Soc. Am. B 18, 1307-1310 (2001). [CrossRef]
- G. Arisholm, R. Paschotta, and T. S¨udmeyer, "Limits to the power scalability of high-gain optical parametric amplifiers," J. Opt. Soc. Am. B 21, 578-590 (2004). [CrossRef]

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