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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 19 — Sep. 17, 2007
  • pp: 12246–12251
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Parametric frequency conversion of short optical pulses controlled by a CW background

Matteo Conforti, Fabio Baronio, Antonio Degasperis, and Stefan Wabnitz  »View Author Affiliations


Optics Express, Vol. 15, Issue 19, pp. 12246-12251 (2007)
http://dx.doi.org/10.1364/OE.15.012246


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Abstract

We predict that parametric sum-frequency generation of an ultra-short pulse may result from the mixing of an ultra-short optical pulse with a quasi-continuous wave control. We analytically show that the intensity, time duration and group velocity of the generated idler pulse may be controlled in a stable manner by adjusting the intensity level of the background pump.

© 2007 Optical Society of America

1. Introduction

Optical parametric amplification in quadratic nonlinear crystals has been studied since the invention of the laser, as it provides a versatile means of achieving widely tunable frequency conversion [1

1. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003). [CrossRef]

]. In parametric processes, the effective interaction length of short optical pulses is limited by temporal walk-off owing to chromatic dispersion, or group velocity mismatch (GVM) [2

2. J.A. Armstrong, S.S. Jha, and N.S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. QE-6, 123–129 (1970) [CrossRef]

, 3

3. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Pulses (AIP, New York, 1992).

]. Compression and amplification of ultra-short laser pulses in second harmonic and sum-frequency (SF) generation in the presence of GVM was theoretically predicted [4

4. Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990) [CrossRef] [PubMed]

, 5

5. A. Stabinis, G. Valiulis, and E.A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991). [CrossRef]

] and observed in several experiments [6

6. Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992). [CrossRef] [PubMed]

, 7

7. C.Y. Chien, G. Korn, J.S. Coe, J. Squier, G. Mourou, and R.S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. 20, 353–355 (1995). [CrossRef] [PubMed]

]. The conversion efficiency of generated SF pulses may be optimised [8

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996). [CrossRef] [PubMed]

, 9

9. E. Ibragimov and A. Struthers, “Three-wave soliton interaction of ultrashort pulses in quadratic media,” J. Opt. Soc. Am. B 14, 1472–1479 (1997). [CrossRef]

, 10

10. E. Ibragimov, A. Struthers, D.J. Kaup, J.D. Khaydarov, and K.D. Singer, “Three-wave interaction solitons in optical parametric amplification,” Phys. Rev. E 59, 6122–6137 (1999). [CrossRef]

, 11

11. E. Ibragimov, A. Struthers, and D.J. Kaup, “Soliton pulse compression in the theory of optical parametric amplification,” Opt. Commun. 152, 101–107 (1998). [CrossRef]

, 12

12. S. Fournier, R.L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, “Solitonlike pulse shortening in a femtosecond parametric amplifier,” Opt. Lett. 23, 627–629 (1998). [CrossRef]

] by operating in the soliton regime [13

13. V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP Lett. , 18, 243–245 (1973).

, 14

14. D. J. Kaup, “The three-wave interaction — a nondispersive phenomenon,” Stud. Appl. Math. 55, 9–44 (1976).

]. In fact, the temporal collision of two short soliton pulses in a quadratic nonlinear crystal may efficiently generate a short, time-compressed SF pulse [8

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996). [CrossRef] [PubMed]

]. However this SF pulse is unstable: its energy decays back into the two incident pulses after a relatively short distance.

In this Paper we consider the parametric SF conversion from the mixing of an ultra-short signal pulse with a quasi-continuous wave (CW) or background pump, in the presence of GVM. Quite surprisingly we find that parametric mixing of these waves may lead to highly efficient generation of stable and ultra-short idler pulses. Indeed, we predict that the interaction of an ultra-short signal with a CW pump may generate a stable three-wave resonant interaction simulton (TWRIS) [15

15. K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Japan 34, 796–800 (1973). [CrossRef]

, 16

16. Y. Ohsawa and K. Nozaki, “Propagation of solitary pulses in interactions of plasma waves. II,” J. Phys. Soc. Japan 36, 591–595 (1974). [CrossRef]

], consisting of a locked bright-bright-dark triplet (signal-idler-pump) that propagates with a single nonlinear velocity [17

17. F. Calogero and A. Degasperis, “Novel solution of the system describing the resonant interaction of three waves,” Physica D 200, 242–256 (2005). [CrossRef]

, 18

18. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. 97, 093901 (2006). [CrossRef] [PubMed]

]. We analytically show that the intensity, time duration and group velocity of the generated idler pulse may be controlled in a stable manner [19

19. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E 74, 065602(R) (2006). [CrossRef]

] by means of simply adjusting the intensity level of the CW background. Although we shall restrict our attention in this work to a travelling-wave interaction geometry, we may anticipate that our results will have important ramifications in the optimization of the efficiency of ultrashort pulse optical parametric oscillators [20

20. A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. 23, 1808–1810 (1998). [CrossRef]

, 21

21. 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]

].

2. Three-wave-interaction equations

The equations describing the quadratic resonant interaction of three waves in a nonlinear medium read as:

A1ξ+δ1A1τ=iA2*A3,
A2ξ+δ2A2τ=iA1*A3,
A3ξ+δ3A3τ=iA1A2,
(1)

with

Aj=πχ(2)njω1ω2ω3n1n2n3ωjEj.
(2)

Here τ=t/t0, t0 is an arbitrary time parameter; ξ=z/z0, z0 is an unit space-propagation parameter. Ej are the slowly varying electric field envelopes of the waves at frequencies ωj, nj are the refractive indexes, χ (2) is the quadratic nonlinear susceptibility, δj=z0/(vjt0) with vj the linear group velocities, and j=1,2,3. We assume that the group velocity v3 of the wave with the highest frequency (ω312) lies between the group velocities of the other waves, i.e. v1>v3>v2. With no loss of generality, we shall write the Eqs. (1) in a coordinate system such that δ 1=0, which implies 0<δ 3<δ 2. Eqs. (1) exhibit the conserved quantities

U13=U1+U3=12+(A12+A32)dτ,
(3)
U23=U2+U3=12+(A22+A32)dτ,
(4)
U=U1+U2+2U3=12+(A12+A22+2A32)dτ.
(5)

where U 1, U 2 and 2U 3 represent the energies at the frequenciesω1, ω2 and ω3.

3. Soliton-based parametric sum-frequency conversion

Figure 1 illustrates a typical example of the efficient SF parametric interaction of two short optical pulses in the soliton regime [8

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996). [CrossRef] [PubMed]

].

Fig. 1. Sum-frequency parametric interaction of two short optical signals atω1 and ω2. The characteristic delays are δ1=0,δ2=2,δ3=1.

At the crystal input, two isolated pulses A 1 and A 2 with frequencies ω1 and ω2 propagate with speeds v1 and v2. Whenever the faster pulse overtakes the slower one, an idler pulse A 3 at the SF ω12 is generated and propagates with the linear speed v3. Depending on the time widths and intensities of the input pulses, the duration of the SF pulse is reduced with respect to the input pulse widths. Correspondingly, the SF pulse peak intensity grows larger than the input pulse intensities. Figure 1 shows that, eventually, the SF idler pulse decays back into the two original isolated pulses at frequencies ω1 and ω2. Note that the shapes, intensities and widths of the input pulses are left unchanged in spite of their interaction. As shown in Ref. [8

8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996). [CrossRef] [PubMed]

], the above discussed SF pulse generation process may be analytically described in terms of soliton solutions of Eqs. (1) [13

13. V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP Lett. , 18, 243–245 (1973).

, 14

14. D. J. Kaup, “The three-wave interaction — a nondispersive phenomenon,” Stud. Appl. Math. 55, 9–44 (1976).

]. The decay of the SF pulse which is shown in Fig. 1 may be a significant drawback in practical applications, since it implies that a given nonlinear crystal length yields efficient conversion for a limited range of input pulse intensities and time widths only.

Fig. 2. Sum-frequency parametric interaction of a short pulse atω2 and a quasi-CW control at ω1. The characteristic delays are δ1=0,δ2=2,δ3=1.

Here we demonstrate that the parametric sum-frequency conversion of an ultra-short signal and a quasi-CW background pump-control may be exploited as a means to reduce or even eliminate the decay of the generated idler wave. In the presence of GVM, the parametric SF conversion of an ultra-short optical signal and a quasi-CW pump typically leads to the generation of a low-intensity and relatively long idler pulse, whose duration is associated with the interaction distance in the crystal. This scenario changes dramatically in the soliton regime. Figure 2 illustrates the efficient generation of a stable, ultra-short SF idler pulse from the parametric SF conversion of a properly prepared ultra-short signal and an arbitrary intensity level CW background control.

In Fig. 2 we injected in the quadratic nonlinear crystal the short signal at frequency ω2, along with a delayed and relatively long pump-control pulse at frequency ω1. Initially, the two pulses propagate uncoupled; as soon as the faster pulse starts to overlap in time with the slower quasi-CW control, their nonlinear mixing generates a short SF idler pulse. The sum-frequency process displayed in Fig. 2 can be analytically explained and explored in terms of stable TWRIS solutions [18

18. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. 97, 093901 (2006). [CrossRef] [PubMed]

]. In the notation of Eqs. (1), the TWRIS solution reads as

A1={1+2pb*b2+a2[1tanh[B(τ+δξ)]]}iag1exp(iq1τ1)g(δ2δ3)
A2=2pab2+a2g2g(δ2δ3)exp[i(q2τ2+χτ+ωξ)]cosh[B(τ+δξ)],
A3=2pb*b2+a2g3g(δ2δ3)exp[i(q3τ3χτωξ)]cosh[B(τ+δξ)],
(6)

where

b=(Q1)(p+ikQ),r=p2k2a2,
Q=1p12[r+r2+4k2p2],
B=p[δ2+δ3Q(δ2δ3)](δ2δ3),
δ=2δ2δ3[δ2+δ3Q(δ2δ3)],
χ=k[δ2+δ3(δ2δ3)Q](δ2δ3),
ω=2kδ2δ3(δ2δ3),τn=τ+δnξ
qn=q(δn+1δn+2),gn=(δnδn+1)(δnδn+2)12
g=g1g2g3,n=1,2,3mod(3).
(7)

At the input, the properly prepared short pulse at frequency ω2 and with a speed v2 is a stable single component TWRIS (6) with parameters p>0,k,q,a=0. The background control in the interaction region can be modeled with A 1(τ)=Ce-iγτ. When this faster pulse, pre-delayed with respect to the slower quasi-CW pump at frequency ω1, overtakes the background (at τ=0, in Fig. 2), their collision leads to the generation of a short idler pulse at the SF ω3. Additionally, a dip appears in the quasi CW-control; whereas the intensity, duration and propagation speed of the input wave at frequency ω2 are modified. Indeed, the signal-pump interaction generates a new stable TWRIS (6), with parameters p̄,k̄,q̄,ā, moving with the locked nonlinear velocity v̄=z0/(t0δ¯), where δ¯ is given in (7). It is remarkable that we may analytically predict the parameters p̄,k̄,q̄,ā of the generated TWRIS from the corresponding parameters of the input single wave TWRIS and the complex amplitude of the pump control. This can be achieved by supposing that the input TWRIS adiabatically (i.e., without emission of radiation) reshapes into a new TWRI simulton after its collision with the quasi-CW pump at a given point in time (say, at τ=0). Under this basic hypothesis, the conservative nature of the three-wave interaction permits us to suppose that: i) the energy U 23 (4) of the input TWRI soliton is conserved in the generated TWRI simulton; ii) the phase of the ω2 frequency components of the input TWRI soliton and of the generated TWRI simulton is continuous across their time interface (i.e., at τ=0); iii) the amplitude and phase of the control pump coincide with the corresponding values of the asymptotic plateau of the generated TWRI simulton component at frequency ω1. By imposing the above three conditions, after some straightforward calculations we obtain the following relations that relate the parameters of the incident and of the transmitted TWRIS

p¯=p,a¯=Cδ2δ3,q¯=γ,k¯=k+(q¯q)(δ2δ3)2.
(8)

As an example, in Fig. 2 the input TWRI soliton at frequency ω2 is described by Eqs. (6) with p=1.3,k=0,q=0,a=0, and the background control amplitude with C=1.7,γ=0. After the collision with the CW background, the above equations predict that the generated TWRIS is again described by Eqs. (6), with p̄=1.3,k̄=0,q̄=0, and ā=1.2. The accuracy of this prediction is well confirmed by its comparison with the numerical solutions of the TWRI Eqs. (1). Indeed, Fig. 3 compares the numerical with the analytical evolutions (along the crystal length ξ) of the energy, the pulse duration and the velocity of the idler and signal pulses which correspond to the case shown in Fig. 2. We performed further extensive numerical simulations, which confirmed the general validity of the above described adiabatic transition model for TWRIS generation upon collision with a CW background.

Fig. 3. Numerical evolution (lines) and theoretical predictions (circles) of energy, pulse duration and velocity of idler (red) and signal (blue) waves reported in Fig.2.

Indeed, by increasing or decreasing the CW background amplitude |C| in the range [0, p δ2δ3], we observed that stable TWRISs with different velocity, duration and energy distributions may be adiabatically shaped. The important consequence of this result is that, by means of Eqs. (6)(8), we may analytically predict and control the characteristics of the generated idler pulse (namely, its velocity, time duration and energy) simply as a function of the intensity level of the CW pump. Moreover, we would like to emphasize that the stability of the whole conversion process is ensured by the underlying stability of the generated TWRIS [19

19. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E 74, 065602(R) (2006). [CrossRef]

].

We would like to point out that the observation of the above described sum–frequency short pulse generation phenomena appears to be readily achievable in nonlinear optical experiments. For instance, let us consider the eee interaction of three-waves with carrier wavelengths of λ 1=1.55µm, λ2=3.4µm, λ 3=1.064µm in a 2cm (8cm) long periodically poled bulk Lithium Niobate crystal with 28µm periodicity. In this case, the parametric mixing of a 100 fs (1ps) incident pulse with a quasi–CW (say, with a 3ps (30ps) time duration) control pulse leads to the generation of an ultrashort sum-frequency pulse of approximately the same time width of the incident short pulse whenever the field intensities of the two input pulses are of the order of a few hundreds of MW/cm2 (or a few MW/cm2, respectively).

4. Conclusions

In conclusion, we demonstrated the parametric SF conversion of an ultra-short pulse from the mixing of an ultra-short optical pulse with a quasi-continuous wave control in quadratic nonlinear crystals in the presence of GVM. We analytically showed that the intensity, time duration and group velocity of the generated pulses may be controlled in a stable manner by simply adjusting the intensity level of the background pump.

Acknowledgements

The present research of A.D. and S.W. is within the French CNRS project GDR 3073.

References and links

1.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003). [CrossRef]

2.

J.A. Armstrong, S.S. Jha, and N.S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. QE-6, 123–129 (1970) [CrossRef]

3.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Pulses (AIP, New York, 1992).

4.

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990) [CrossRef] [PubMed]

5.

A. Stabinis, G. Valiulis, and E.A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991). [CrossRef]

6.

Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992). [CrossRef] [PubMed]

7.

C.Y. Chien, G. Korn, J.S. Coe, J. Squier, G. Mourou, and R.S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. 20, 353–355 (1995). [CrossRef] [PubMed]

8.

E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996). [CrossRef] [PubMed]

9.

E. Ibragimov and A. Struthers, “Three-wave soliton interaction of ultrashort pulses in quadratic media,” J. Opt. Soc. Am. B 14, 1472–1479 (1997). [CrossRef]

10.

E. Ibragimov, A. Struthers, D.J. Kaup, J.D. Khaydarov, and K.D. Singer, “Three-wave interaction solitons in optical parametric amplification,” Phys. Rev. E 59, 6122–6137 (1999). [CrossRef]

11.

E. Ibragimov, A. Struthers, and D.J. Kaup, “Soliton pulse compression in the theory of optical parametric amplification,” Opt. Commun. 152, 101–107 (1998). [CrossRef]

12.

S. Fournier, R.L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, “Solitonlike pulse shortening in a femtosecond parametric amplifier,” Opt. Lett. 23, 627–629 (1998). [CrossRef]

13.

V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP Lett. , 18, 243–245 (1973).

14.

D. J. Kaup, “The three-wave interaction — a nondispersive phenomenon,” Stud. Appl. Math. 55, 9–44 (1976).

15.

K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Japan 34, 796–800 (1973). [CrossRef]

16.

Y. Ohsawa and K. Nozaki, “Propagation of solitary pulses in interactions of plasma waves. II,” J. Phys. Soc. Japan 36, 591–595 (1974). [CrossRef]

17.

F. Calogero and A. Degasperis, “Novel solution of the system describing the resonant interaction of three waves,” Physica D 200, 242–256 (2005). [CrossRef]

18.

A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. 97, 093901 (2006). [CrossRef] [PubMed]

19.

M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E 74, 065602(R) (2006). [CrossRef]

20.

A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. 23, 1808–1810 (1998). [CrossRef]

21.

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]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.7110) Nonlinear optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 19, 2007
Revised Manuscript: July 16, 2007
Manuscript Accepted: July 22, 2007
Published: September 11, 2007

Citation
Matteo Conforti, Fabio Baronio, Antonio Degasperis, and Stefan Wabnitz, "Parametric frequency conversion of short optical pulses controlled by a CW background," Opt. Express 15, 12246-12251 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12246


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References

  1. G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003). [CrossRef]
  2. J. A. Armstrong, S. S. Jha, and N. S. Shiren, "Some effects of group-velocity dispersion on parametric interactions," IEEE J. Quantum Electron. QE-6, 123-129 (1970). [CrossRef]
  3. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Pulses (AIP, New York, 1992).
  4. Y. Wang and R. Dragila, "Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion," Phys. Rev. A 41, 5645-5649 (1990) [CrossRef] [PubMed]
  5. A. Stabinis, G. Valiulis and E. A. Ibragimov, "Effective sum frequency pulse compression in nonlinear crystals," Opt. Commun. 86, 301-306 (1991). [CrossRef]
  6. Y. Wang and B. Luther-Davies, "Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses," Opt. Lett. 17, 1459-1461 (1992). [CrossRef] [PubMed]
  7. C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou and R. S. Craxton, "Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses," Opt. Lett. 20, 353-355 (1995). [CrossRef] [PubMed]
  8. E. Ibragimov and A. Struthers, "Second-harmonic pulse compression in the soliton regime," Opt. Lett. 21, 1582-1584 (1996). [CrossRef] [PubMed]
  9. E. Ibragimov and A. Struthers, "Three-wave soliton interaction of ultrashort pulses in quadratic media," J. Opt. Soc. Am. B 14, 1472-1479 (1997). [CrossRef]
  10. E. Ibragimov, A. Struthers, D. J. Kaup, J. D. Khaydarov, and K. D. Singer, "Three-wave interaction solitons in optical parametric amplification," Phys. Rev. E 59, 6122-6137 (1999). [CrossRef]
  11. E. Ibragimov, A. Struthers, and D. J. Kaup, "Soliton pulse compression in the theory of optical parametric amplification," Opt. Commun. 152, 101-107 (1998). [CrossRef]
  12. S. Fournier, R. L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, "Solitonlike pulse shortening in a femtosecond parametric amplifier," Opt. Lett. 23, 627-629 (1998). [CrossRef]
  13. V. E. Zakharov and S. V. Manakov, "Resonant interaction of wave packets in nonlinear media," Sov. Phys. JETP Lett. 18, 243-245 (1973).
  14. D. J. Kaup, "The three-wave interaction - a nondispersive phenomenon," Stud. Appl. Math. 55, 9-44 (1976).
  15. K. Nozaki and T. Taniuti, "Propagation of solitary pulses in interactions of plasma waves," J. Phys. Soc. Japan 34, 796-800 (1973). [CrossRef]
  16. Y. Ohsawa and K. Nozaki, "Propagation of solitary pulses in interactions of plasma waves. II," J. Phys. Soc. Japan 36, 591-595 (1974). [CrossRef]
  17. F. Calogero and A. Degasperis, "Novel solution of the system describing the resonant interaction of three waves," Physica D 200, 242-256 (2005). [CrossRef]
  18. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, "Stable control of pulse speed in parametric three-wave solitons," Phys. Rev. Lett. 97, 093901 (2006). [CrossRef] [PubMed]
  19. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, "Inelastic scattering and interactions of three-wave parametric solitons," Phys. Rev. E 74, 065602(R) (2006). [CrossRef]
  20. A. Picozzi and M. Haelterman, "Spontaneous formation of symbiotic solitary waves in a backward quasi-phasematched parametric oscillator," Opt. Lett. 23, 1808-1810 (1998). [CrossRef]
  21. 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]

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