## Frequency Generation and Solitonic Decay in ThreeWave Interactions

Optics Express, Vol. 17, Issue 16, pp. 13889-13894 (2009)

http://dx.doi.org/10.1364/OE.17.013889

Acrobat PDF (680 KB)

### Abstract

We consider experimentally three-wave resonant nonlinear interactions of fields propagating in nonlinear media. We investigate the spatial dynamics of two diffractionless beams at frequency *ω*_{1}, *ω*_{2} which mix to generate a field at the sum frequency *ω*_{3}. If the generated field at *ω*_{3} can sustain a soliton, it decays into solitons at *ω*_{1}, *ω*_{2}. We report the experimental evidence of the transition from steady frequency wave generation to solitonic decay in nonlinear optics.

© 2009 Optical Society of America

## 1. Introduction

1. D.J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. **51**, 275–309 (1979). [CrossRef]

2. B. Kim, J. Blake, H. Engan, and H. Shaw, “All-fiber acousto-optic frequency shifter,” Opt. Lett. **11**, 389–391 (1986). [CrossRef] [PubMed]

3. P. Russel, D. Culverhouse, and F. Farahi, “Theory of Forward Stimulated Brillouin Scattering in Dual-Mode Single-Core Fibers,” IEEE J. Quantum Electron. **27**, 836–842 (1991). [CrossRef]

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

5. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express **15**, 12246–12251 (2007). [CrossRef] [PubMed]

6. F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. **44**, 542–546 (2008). [CrossRef]

8. Y. Tsidulko, V. Malkin, and N. Fisch, “Suppression of Superluminous Precursors in High-Power Backward Raman Amplifiers,” Phys. Rev. Lett. **88**, 235004 (2002). [CrossRef] [PubMed]

9. W. Cheng, Y. Avitzour, Y. Ping, S. Suckewer, N. Fisch, M. Hur, and J. Wurtele, “Reaching the Nonlinear Regime of Raman Amplification of Ultrashort Laser Pulses,” Phys. Rev. Lett. **94**, 045003 (2005). [CrossRef] [PubMed]

11. K. Lamb, “Tidally generated near-resonant internal wave triads at shelf break,” Geophys. Res. Lett. **34**, L18607 (2007). [CrossRef]

12. L. Svaasand, “Interaction between elastic surface waves in piezoelectric materials,” Appl. Phys. Lett. **15**, 300–302 (1969). [CrossRef]

14. A. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. **370**, 63–235 (2002). [CrossRef]

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

*ω*

_{1}and

*ω*

_{2}which mix to generate a field at the sum frequency

*ω*

_{3}, in absence of diffraction. Depending on the input intensities, three different regimes exist. Linear regime: the beams at frequency

*ω*

_{1}and

*ω*

_{2}don’t interact. Frequency conversion: the beams at frequencies

*ω*

_{1}and

*ω*

_{2}interact and generate a steady field at the sum frequency

*ω*

_{3}. Solitonic regime: the beams at frequencies

*ω*

_{1}and

*ω*

_{2}interact and generate a field at the sum frequency

*ω*

_{3}; the generated field at

*ω*

_{3}sustains a Zakharov–Manakov soliton which decays into solitons at frequencies

*ω*

_{1}and

*ω*

_{2}[15]. Here we report the experimental evidence of the transition from steady sum frequency generation to solitonic decay in optics. To our knowledge, this is the first experimental observation of diffractionless soliton dynamics.

## 2. Frequency conversion and Solitonic Decay: Theoretical Concepts

*k*

_{1},

*k*

_{2},

*k*

_{3}and frequencies

*ω*

_{1},

*ω*

_{2},

*ω*

_{3}, which propagate in a dispersive nonlinear medium with quadratic nonlinearity, interact efficiently with each other and exchange energy if the resonance conditions

*k*

_{1}+

*k*

_{2}=

*k*

_{3},

*ω*

_{1}+

*ω*

_{2}=

*ω*

_{3}, are satisfied. The evolution equations for the waves in a two dimensional medium, after a multiscale expansion of Maxwell equations, turn out to be:

*ϕ*are complex wave-packet amplitudes at frequencies

_{n}*ω*,

_{n}*V*are characteristic velocities,

_{n}*η*are signs, and

_{n}*K*is a complex coupling coefficient. The variables

*x*and

*z*stand for transverse and longitudinal coordinate, respectively. Here, we consider the three-wave exchange type interaction [1

1. D.J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. **51**, 275–309 (1979). [CrossRef]

*η*

_{1},

*η*

_{2},

*η*

_{3}) = (+, +, -) and

*V*

_{1}<

*V*

_{3}<

*V*

_{2}. It should be pointed out that the physical meaning of the waves

*ϕ*, and of the variable

_{n}*x*and

*z*depends on the particular physical domain that one has in mind (plasma physics, nonlinear optics, fluid dynamics).

*x*and the other one in

*z*(the Zakharov–Manakov (ZM) eigenvalue problem [18]). This fact gives a way to set up a nonlinear generalization of the Fourier analysis of solutions of the associated initial value problem, namely the Inverse Scattering Technique (IST). In particular, this generalization leads to decompose a given solution

*ϕ*

_{1}(

*x*,

*z*),

*ϕ*

_{2}(

*x*,

*z*),

*ϕ*

_{3}(

*x*,

*z*) as functions of

*x*at a given fixed

*z*in its continuum spectrum component (radiation) and in discrete spectrum component (solitons).

*ω*

_{1}and

*ω*

_{2}which mix to generate a field at the sum frequency

*ω*

_{3}. Depending on the input intensities, three different regimes exist, which we define as linear, frequency conversion, and solitonic regime (Fig. 1). Figure 1 shows numerical simulations, with

*V*

_{1}= -

*V*

_{2}and

*V*

_{3}= 0, of the typical wave dynamics in the

*x*–

*z*plane, corresponding to the initial data at

*z*= 0:

*ϕ*

_{1}(

*x*,0) ≠ 0,

*ϕ*

_{2}(

*x*,0) ≠ 0,

*ϕ*

_{3}(

*x*,0) = 0. In particular, figure 1.a shows the low-intensity linear regime: the beams at frequency

*ω*

_{1}and

*ω*

_{2}do not interact and propagate with their own characteristic velocities

*V*

_{1}and

*V*

_{2}. Note that, from a theoretical IST perspective, the decomposition of the initial data

*ϕ*

_{1}(

*x*,

*z*= 0) ,

*ϕ*

_{2}(

*x*,

*z*= 0) ,

*ϕ*

_{3}(

*x*,

*z*= 0) in a continuum spectrum component (radiation) and a discrete spectrum component (solitons) is irrelevant in this case since the wave–wave interaction is quite negligible. Figure 1.b shows that, at moderate input intensity, the beams at frequency

*ω*

_{1}and

*ω*

_{2}interact and generate a steady field at the sum frequency

*ω*

_{3}. When the faster wave overtakes the slower one, we observe energy conversion to the sum frequency wave

*ϕ*

_{3}; next the three waves propagate alone with their own characteristic velocities. In this case, the initial data

*ϕ*

_{1}(

*x*,

*z*= 0) ,

*ϕ*

_{2}(

*x*,

*z*= 0) ,

*ϕ*

_{3}(

*x*,

*z*= 0) are composed of a continuum spectrum component (radiation) and no discrete spectrum component (solitons). Figure 1.c shows that, at relatively high input intensity, the beams at frequency

*ω*

_{1}and

*ω*

_{2}interact and generate a field at the sum frequency

*ω*

_{3}. The sum frequency wave

*ϕ*

_{3}has enough energy to sustain at least one soliton. After the non–collinear nonlinear interaction the radiation parts of the waves at

*ω*

_{1}and

*ω*

_{2}and the wave at

*ω*

_{3}propagate alone with their own characteristic velocities; furthermore, the component at

*ω*

_{3}decays into two waves at

*ω*

_{1}and

*ω*

_{2}. This decay process may be described in terms of the analytical solutions of equations (1) which were discovered in the 70

*s*[15] (the ZM TWRI solitons). From a theoretical IST perspective, the initial data

*ϕ*

_{1}(

*x*,

*z*= 0) ,

*ϕ*

_{2}(

*x*,

*z*= 0) ,

*ϕ*

_{3}(

*x*,

*z*= 0) are composed of both a continuum spectrum component (radiation) and a discrete spectrum component (solitons). Indeed the input waves

*ϕ*

_{1}and

*ϕ*

_{2}contain one soliton each plus radiation. The solitons in the waves

*ϕ*

_{1}and

*ϕ*

_{2}interact and generate a soliton in wave

*ϕ*

_{3}which, due to its finite lifetime, eventually decays into solitons in the waves

*ϕ*

_{1}and

*ϕ*

_{2}.

## 3. Frequency conversion and Solitonic Decay: Experiments

*ω*

_{1}=

*ω*

_{2}=ω,

*E*and

^{e}_{ω}*E*, were injected into the nonlinear birefringent medium to cross and overlap at the input face of the crystal (see Fig. 2). Each field is linearly polarized and aligned with a polarization eigenstate of the crystal. In the present scheme the two input waves required for the parametric processes to occur, and for the subsequent observation of solitonic decay, had tilted wave–fronts. In the overlapping area the harmonic

^{o}_{ω}*E*

^{e}_{2ω}at

*ω*

_{3}= 2

*ω*is generated along a direction which maximized the conversion efficiency as it was fixed by the noncollinear phase–matching conditions. The sum–frequency harmonic propagation direction was found in between the directions of the two input waves.

*ps*pulses at

*λ*= 1064nm. Then laser light passed through a spatial telescope made from two lenses,

*L*

_{1}and

*L*

_{2}. We introduced a Glan polarizer to obtain, after passage of the light through

*L*

_{2}, two independent beams with perpendicular linear polarization states. A half-wave plate placed before the prism served to adjust the intensity

*I*

_{1}and

*I*

_{2}of the two beams at an equal level

*I*. By means of two highly reflecting mirrors, a beam splitter and a third lens

*L*

_{3}, both beams were focused and spatially superimposed in the plane of their beam waist with a circular shape of 120

*μm*width (FWHMI). A 3

*cm*long type II KTP crystal cut for second harmonic generation was positioned such that its input face corresponds to the plane of superposition of the two input beams. The crystal was oriented for perfect phase matching. The directions of the linear polarization state of the two beams were adjusted to coincide with the ordinary and the extraordinary axes, respectively, of the KTP crystal. The wave vectors of the input fields were tilted at angles of

*θ*

_{1}= 0.7

^{o}and

*θ*

_{2}= -0.7

^{o}(in the crystal) with respect to the direction of perfect collinear phase matching for the extraordinary and the ordinary components, respectively. These parameters corresponded inside the crystal to a tilt between the input beams equal to 3.7 times the natural walk–off angle but introduced along the ordinary noncritical plane. The sum frequency direction lies in between the input waves directions. With these values of parameters, spatial diffraction and temporal dispersion were negligible. Therefore, the dynamics of the waves could be well described by the TWRI (1+1)D spatial model equations (1) in the ordinary plane. The spatial waves’ patterns at the output of the crystal were imaged with magnification onto a CCD camera and analyzed. We used alternately different filters and polarizers to select either the IR or the green output. As the intensities of the input fields were varied in a suitable range, we observed the TWRI linear, frequency conversion, and solitonic regimes in the ordinary KTP plane (Fig. 3). For each regime, Fig. 3 shows i) the output profile of the waves at frequency

*ω*, ii) the profile of the wave at the sum frequency, iii) the profile of the extraordinary polarized component and iv) the profile of the ordinary polarized component at frequency

*ω*. In the low–intensity (

*I*= 1

*MW*/

*cm*

^{2}) linear regime, the input waves did not interact and propagated without diffraction in the KTP crystal following their own characteristic directions (Fig. 3.a). The waves

*E*and

^{e}_{ω}*E*, superimposed at the crystal entrance, were spatially separated at the crystal output. At moderate input intensity (

^{o}_{ω}*I*= 0.1

*GW*/

*cm*

^{2}), parts of the beams

*E*and

^{e}_{ω}*E*interacted and generated a steady field at the sum frequency. At the crystal output we observed a component of beams

^{o}_{ω}*E*and

^{e}_{ω}*E*which did not interact and behave linearly and the generated wave at the sum frequency

^{o}_{ω}*E*whose propagation direction lies in between the input waves directions in the ordinary plane (Fig. 3.b). This regime corresponds to the well known optical non–collinear sum frequency wave generation. At higher input intensity (

^{e}_{2ω}*I*=2.5

*GW*/

*cm*

^{2}), we observed again the amount of the wave

*E*and

^{e}_{ω}*E*which did not interact and behave as linear waves (Fig. 3.c).Moreover, we observed the birth of two well defined ordinary and extraordinary waves

^{o}_{ω}*E*and

^{e}_{ω}*E*, which turned out to be spatially shifted with respect to the linear components. The observation of the two new spatially shifted waves provided a clear evidence that a TWRI soliton was generated at the sum frequency within the crystal: such soliton had subsequently decayed into two solitons in the fundamentals right before the end face of the crystal. In fact, the wave at the sum frequency disappeared. Indeed, by removing the attenuation filters used to protect the camera, we may observe a faint component at the sum frequency 2

^{o}_{ω}*ω*which is generated in part through type II interaction and mainly through type I SHG by the extraordinary wave at

*ω*(see inset of Fig. 3.c). Numerical simulations of the optical interactions, described by the equations (1), well reproduce the experimental results (see Fig. 4). The above discussed results represent the experimental evidence of the transition from a sum–frequency wave generation to a TWRI solitonic decay.

## 4. Conclusions

*ω*

_{1},

*ω*

_{2}which mix to generate a field at the sum frequency

*ω*

_{3}. Depending on the intensity, when the generated field at

*ω*

_{3}can sustain a TWRI soliton, it decays into solitons at

*ω*

_{1}and

*ω*

_{2}. Our experimental findings demonstrate the possibility of reaching soliton regimes in non-diffractive TWRI systems; these nonlinear regimes could pave the way to the construction of novel systems for storing, retrieving and processing information in the optical and plasma domains.

## Acknowledgements

## References and links

1. | D.J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium,” Rev. Mod. Phys. |

2. | B. Kim, J. Blake, H. Engan, and H. Shaw, “All-fiber acousto-optic frequency shifter,” Opt. Lett. |

3. | P. Russel, D. Culverhouse, and F. Farahi, “Theory of Forward Stimulated Brillouin Scattering in Dual-Mode Single-Core Fibers,” IEEE J. Quantum Electron. |

4. | E. Ibragimov and A. Struthers, “Second harmonic pulse compression in the soliton regime,” Opt. Lett. |

5. | M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express |

6. | F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. |

7. | A. Hasegawa, |

8. | Y. Tsidulko, V. Malkin, and N. Fisch, “Suppression of Superluminous Precursors in High-Power Backward Raman Amplifiers,” Phys. Rev. Lett. |

9. | W. Cheng, Y. Avitzour, Y. Ping, S. Suckewer, N. Fisch, M. Hur, and J. Wurtele, “Reaching the Nonlinear Regime of Raman Amplification of Ultrashort Laser Pulses,” Phys. Rev. Lett. |

10. | A. Craik, |

11. | K. Lamb, “Tidally generated near-resonant internal wave triads at shelf break,” Geophys. Res. Lett. |

12. | L. Svaasand, “Interaction between elastic surface waves in piezoelectric materials,” Appl. Phys. Lett. |

13. | Y. N. Karamzin and A. P. Sukhorukov “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JEPT Lett. |

14. | A. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. |

15. | V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” Jept. Lett. |

16. | K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Jpn. |

17. | V. E. Zakharov, |

18. | V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitajevski, |

**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

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 12, 2009

Manuscript Accepted: June 18, 2009

Published: July 27, 2009

**Citation**

Fabio Baronio, Matteo Conforti, Marco Andreana, Vincent Couderc, Costantino De Angelis, Stefan Wabnitz, Alain Barthélémy, and Antonio Degasperis, "Frequency Generation and Solitonic Decay in ThreeWave Interactions," Opt. Express **17**, 13889-13894 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13889

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### References

- D. J. Kaup, A. Reiman, and A. Bers, "Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium," Rev. Mod. Phys. 51, 275-309 (1979). [CrossRef]
- B. Kim, J. Blake, H. Engan, and H. Shaw, "All-fiber acousto-optic frequency shifter," Opt. Lett. 11, 389-391 (1986). [CrossRef] [PubMed]
- P. Russel, D. Culverhouse, and F. Farahi, "Theory of Forward Stimulated Brillouin Scattering in Dual-Mode Single-Core Fibers," IEEE J. Quantum Electron. 27, 836-842 (1991). [CrossRef]
- E. Ibragimov, and A. Struthers, "Second harmonic pulse compression in the soliton regime," Opt. Lett. 21, 1582-1584 (1996). [CrossRef] [PubMed]
- M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, "Parametric frequency conversion of short optical pulses controlled by a CW background," Opt. Express 15, 12246-12251 (2007). [CrossRef] [PubMed]
- F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, "Three-wave trapponic solitons for tunable highrepetition rate pulse train generation, " IEEE J. Quantum Electron. 44, 542-546 (2008). [CrossRef]
- A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, Berlin, 2001).
- Y. Tsidulko, V. Malkin, and N. Fisch, "Suppression of Superluminous Precursors in High-Power Backward Raman Amplifiers," Phys. Rev. Lett. 88, 235004 (2002). [CrossRef] [PubMed]
- W. Cheng, Y. Avitzour, Y. Ping, S. Suckewer, N. Fisch, M. Hur, and J. Wurtele, "Reaching the Nonlinear Regime of Raman Amplification of Ultrashort Laser Pulses," Phys. Rev. Lett. 94, 045003 (2005). [CrossRef] [PubMed]
- A. Craik, Wave Interactions and Fluid Flows (Cambridge Univ. Press, Cambridge, 1985).
- K. Lamb, "Tidally generated near-resonant internal wave triads at shelf break," Geophys. Res. Lett. 34, L18607 (2007). [CrossRef]
- L. Svaasand, "Interaction between elastic surface waves in piezoelectric materials," Appl. Phys. Lett. 15, 300-302 (1969). [CrossRef]
- Y. N. Karamzin, and A. P. Sukhorukov, "Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters," JEPT Lett. 20, 339-344 (1974).
- A. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, " Phys. Rep. 370,63-235 (2002). [CrossRef]
- V. E. Zakharov, and S. V. Manakov, "Resonant interaction of wave packets in nonlinear media," Jept. Lett. 18, 243-245 (1973).
- K. Nozaki, and T. Taniuti, "Propagation of solitary pulses in interactions of plasma waves," J. Phys. Soc. Jpn. 34, 796-800 (1973). [CrossRef]
- V. E. Zakharov, What is integrability? (Springer Verlag, Berlin, 1991).
- V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitajevski, The Theory of Solitons: The Inverse Problem Method (Nauka, Moskow, 1980).

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