## Bouncing of a dispersive pulse on an accelerating soliton and stepwise frequency conversion in optical fibers

Optics Express, Vol. 15, Issue 22, pp. 14560-14565 (2007)

http://dx.doi.org/10.1364/OE.15.014560

Acrobat PDF (176 KB)

### Abstract

We demonstrate that a short pulse with spectrum in the range of normal group velocity dispersion can experience periodic reflections on a refractive index maximum created by a co-propagating with it soliton, providing the latter is continuously decelerated by the intrapulse Raman scattering. After each reflection the intensity profile and phase of the pulse are almost perfectly reconstructed, while its frequency is stepwise converted. This phenomenon has direct analogy with the effect of ’quantum bouncing’ known for cold atoms.

© 2007 Optical Society of America

02. A. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercon-tinuum spectra in photonic crystal fibers,” Nature-Photonics **1** (2007); DOI: 10.1038/nphoton.2007.202;
http://xxx.lanl.gov/abs/0706.1187
.

03. A. Gorbach and D.V. Skryabin. “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A (to be published); http://xxx.lanl.gov/abs/0707.1598.

02. A. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercon-tinuum spectra in photonic crystal fibers,” Nature-Photonics **1** (2007); DOI: 10.1038/nphoton.2007.202;
http://xxx.lanl.gov/abs/0706.1187
.

03. A. Gorbach and D.V. Skryabin. “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A (to be published); http://xxx.lanl.gov/abs/0707.1598.

07. J. Gea-Banacloche, “A quantum bouncing ball,” Am. J. Phys. **67**, 776 (1999). [CrossRef]

08. C. V. Saba, P. A. Barton, M. G. Boshier, I. G. Hughes, P. Rosenbusch, B. E. Sauer, and E. A. Hinds, “Reconstruc-tion of a cold atom cloud by magnetic focusing,” Phys. Rev. Lett. **82**, 468–471 (1999). [CrossRef]

09. K. Bongs, S. Burger, G. Birkl, K. Sengstock, W. Ertmer, K. Rzazewski, A. Sanpera, and M. Lewenstein, “Coher-ent evolution of bouncing Bose-Einstein condensates,” Phys. Rev. Lett. **83**, 3577–3580 (1999). [CrossRef]

*Ee*

^{ik0z-iω0t}+

*c.c.*, where

*ω*

_{0}is the zero GVD frequency

*k*

_{0}is the corresponding wavenumber and

*E*=

*A*

_{1}exp[

*ik*

_{1}

*z*-

*iδ*

_{1}

*t*] +

*A*

_{2}exp[

*ik*

_{2}

*z*-

*iδ*

_{2}

*t*]. Here

*A*

_{12}are the slowly varying ampltudes of the two pulses,

*δ*

_{1,2}are their frequency detunings from

*ω*

_{0}and

*k*

_{1,2}=

*k*(

*δ*

_{1,2}) are the corresponding wavenumbers determined via the fiber dispersion. We also assume that

*δ*

_{1,2}are selected in such a way that

*K*́

_{1}=

*k*́

_{2}(

*k*́ =

*∂*) and therefore the initial group velocities of the two pulses are equal. In photonic crystal fibers such group velocity matching is possible across the wide bandwidth [6

_{δ}k06. A.V. Gorbach, D.V. Skryabin, J.M. Stone, and J.C. Knight. “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express **14**, 9854–9863 (2006). [CrossRef] [PubMed]

*A*

_{1,2}are

*x*= (

*t*-

*zk*′

_{1})/√∣

*k*′′

_{1}∣,

*t*is the dimensionless time in the reference frame moving with the light group velocity at

*ω*

_{0}and measured in the units of the input pulse duration τ ∼ 200

*fs*,

*z*is the distance along the fiber measured in the units of any convenient characteristic length

*l*,

*T*= ∣

*k*′′

_{1}∣

^{-1/2}∫

^{∞}

_{0}

*tR*(

*t*)

*dt*is the Raman parameter and

*R*(

*t*) is the Raman response function for silica [1],

*d*

_{1}= 1/2 (anomalous GVD) and

*d*

_{2}= - ∣

*k*′′

_{2}∣/∣2

*k*′′

_{1}∣ < 0 (normal GVD). The fact that GVDs of both components are assumed to be frequency independent is not critical for the effects described below. Choosing

*l*= 50cm and the fiber nonlinear parameter

*γ*= 01/

*W*/

*m*, which is typical for small core PCFs [6

06. A.V. Gorbach, D.V. Skryabin, J.M. Stone, and J.C. Knight. “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express **14**, 9854–9863 (2006). [CrossRef] [PubMed]

*A*

_{1,2}∣

^{2}corresponds to 1/

*γ*/

*l*= 20

*W*in physical units.

*A*

_{2}= 0, one can show that Eq. (1) has an approximate solution in the form of the NLS soliton moving with the constant acceleration

*g*[1, 10

10. K.J. Blow, N.J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am B **5**, 1301–1304 (1988). [CrossRef]

11. L. Gagnon and P.A. Belanger, “Soliton self-frequency shift versus Galilean-like symmetry,” Opt. Lett. **15**, 466–468 (1990). [CrossRef] [PubMed]

*q*> 0 is the soliton parameter. As it has been demonstrated, e.g., in [11

11. L. Gagnon and P.A. Belanger, “Soliton self-frequency shift versus Galilean-like symmetry,” Opt. Lett. **15**, 466–468 (1990). [CrossRef] [PubMed]

*ψ*

_{0}resulting from the above substitution acquires a potential term varying linearly in ξ. It is important to realize that the substitution similar to Eq. (2) can be applied not only to the soliton, but in general for a linear or nonlinear Schrödinger equation with any sign in front of the dispersion term.

*A*

_{2}component added to the soliton. Then Eqs. (1) can be linearized for

*A*

_{2}, and the soliton enters this equation as an external potential moving with acceleration. It is convenient to transform into the accelerating frame of reference, where the soliton is stationary [2

02. A. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercon-tinuum spectra in photonic crystal fibers,” Nature-Photonics **1** (2007); DOI: 10.1038/nphoton.2007.202;
http://xxx.lanl.gov/abs/0706.1187
.

03. A. Gorbach and D.V. Skryabin. “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A (to be published); http://xxx.lanl.gov/abs/0707.1598.

*g*> 0) is accompanied by the red frequency shift in the case of anomalous GVD(

*d*

_{1}> 0) and by the blue shift if GVD is normal (

*d*

_{2}< 0).

*φ*is

*V*(ξ) consists of the localized soliton part and of the linear potential induced by the acceleration:

*V*on the trailing tail of the soliton, see Fig. 1.

*V*localized around the minimum

*ξ*

_{0}and having oscillating tails which decay slowly at

*ξ*→ -∞ [3

*V*can be as high as several dozens [2

**1** (2007); DOI: 10.1038/nphoton.2007.202;
http://xxx.lanl.gov/abs/0706.1187
.

*V*can be well approximated by

*V*(

_{b}*ξ*) = 16

*q*exp(-2√

*q*/

*d*

_{1}

*ξ*) +

*gξ*/(2∣

*d*

_{2}∣), where the soliton part has been replaced by its exponential asymptotic. Then the amplitude

*φ*can be approximated with

*y*(

_{n}*ξ*) are the eigenmodes of

*V*,

_{b}*λ*are the corresponding eigenvalues, -∣

_{n}*d*

_{2}∣∂

^{2}

_{ξ}

*y*+

_{n}*V*

_{b}*y*=

_{n}*λ*, and the summation is truncated when

_{n}y_{n}*n*reaches the number of the last quasi-bound state of

*V*.

*y*

_{0}and shifted by the distance

*L*from the minimum of the potential:

*L*≫

*w*, but

*L*<

*L*(see Fig. 1), the expansion of the gaussian into

_{tr}*y*requires taking into account large values of

_{n}*n*, see Fig 2(a). Then, the expected evolution of the wavepacket should be close to the evolution of a classical particle (correspondence principle). Namely, the pulse initially should roll down the linear potential, reflect from the soliton created potential wall, move upwards, fall down again and so on. This bouncing dynamics has been found in the direct numerical integration of the coupled Eqs. (1), see Fig. 2(b). The same bouncing is more clearly seen in the integration of the linearized Eq. (5), see Figs. 2(c), (d). A notable feature of the bouncing effect is almost perfect reconstruction of the initial pulse after the bouncing period

*Z*∼ 2√2

_{b}*L*/

*g*. The latter estimate for the period is readily derived using analogy with the Newtonian particle which travels the distance

*L*with acceleration

*g*and zero initial velocity and then bounces back returning into its initial position. For the two cases of different initial deviations

*L*shown in Fig. 2 it gives

*Z*≈ 0.56 and

_{b}*Z*, 0.89, respectively.

_{b}*D*(

*z*) ≡ ∫

*dξ*∣

*φ*(

*ξ*,

*z*)-

*φ*(

*ξ*,

*z*)∣

^{2}, which measures the deviation of the pulse from its initial profile with propagation distance. Using (7) we derive a rather simple expression for

*D*(

*z*) [12

12. I. S. Averbukh and N. F. Perelman, “Dynamics of wave-packets from highly excited atomic and molecular-states,” Usp. Fiz. Nauk **161**, 41–81 (1991). [CrossRef]

*D*(

*z*) ≈ 2∑

*∣*

_{n}*c*∣

_{n}^{2}[1 -cos(

*λ*)]. One can see, Fig. 2(e) and (f), that

_{n}z*D*(

*z*) evolves almost periodically and comes fairly close to zero after each bouncing period

*Z*. This indicates that not only the intensity profile is restored (as seen in Figs. 2(a)–(c)), but also is the phase.

_{b}*δ*plays the role of the particle momentum. Thus, the frequency of

*φ*evolves in two sequentially repeated stages: linear decrease (particle roles down the potential) and stepwise jump (elastic reflection from the soliton induced barrier), see Fig. 3(a). The linear change of frequency seen in Fig. 3(a) between the jumps is the result of transformation (4), while the jumps are intrinsic. In an inertial frame of reference the frequency of the bouncing pulse remains fixed between the jumps, see Fig. 3(b),(c). The straightforward estimate for frequency change during the jump is Δ =

*gZ*/∣

_{b}*d*

_{2}∣. One jump in Figs. 3(b,c) corresponds to the frequency change order of 0.1THz in physical units. Thus, after each period

*Z*, the pulse is almost completely restored in time domain, but its frequency is discretely shifted by Δ. The overall process is schematically shown in Fig. 1. Note, that the reflections of the radiation from the soliton are not perfect, so that a portion of radiation escapes after each collsion. Naturally, this becomes more pronounced for larger values of

_{b}*L*, see Fig. 3(c).

13. D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E **72**, 016619 (2005). [CrossRef]

13. D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E **72**, 016619 (2005). [CrossRef]

13. D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E **72**, 016619 (2005). [CrossRef]

**72**, 016619 (2005). [CrossRef]

**1** (2007); DOI: 10.1038/nphoton.2007.202;
http://xxx.lanl.gov/abs/0706.1187
.

## References and links

01. | G. P. Agrawal, |

02. | A. Gorbach and D.V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercon-tinuum spectra in photonic crystal fibers,” Nature-Photonics |

03. | A. Gorbach and D.V. Skryabin. “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A (to be published); http://xxx.lanl.gov/abs/0707.1598. |

04. | P. Beaud, W. Hodel, B. Zysset, and H.P. Weber. “Ultrashort pulse-propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. |

05. | N. Nishizawa and T. Goto. “Experimental analysis of ultrashort pulse propagation in optical fibers around zero-dispersion region using cross-correlation frequency resolved optical gating,” Opt. Express |

06. | A.V. Gorbach, D.V. Skryabin, J.M. Stone, and J.C. Knight. “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express |

07. | J. Gea-Banacloche, “A quantum bouncing ball,” Am. J. Phys. |

08. | C. V. Saba, P. A. Barton, M. G. Boshier, I. G. Hughes, P. Rosenbusch, B. E. Sauer, and E. A. Hinds, “Reconstruc-tion of a cold atom cloud by magnetic focusing,” Phys. Rev. Lett. |

09. | K. Bongs, S. Burger, G. Birkl, K. Sengstock, W. Ertmer, K. Rzazewski, A. Sanpera, and M. Lewenstein, “Coher-ent evolution of bouncing Bose-Einstein condensates,” Phys. Rev. Lett. |

10. | K.J. Blow, N.J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am B |

11. | L. Gagnon and P.A. Belanger, “Soliton self-frequency shift versus Galilean-like symmetry,” Opt. Lett. |

12. | I. S. Averbukh and N. F. Perelman, “Dynamics of wave-packets from highly excited atomic and molecular-states,” Usp. Fiz. Nauk |

13. | D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.5650) Nonlinear optics : Raman effect

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 28, 2007

Revised Manuscript: October 3, 2007

Manuscript Accepted: October 3, 2007

Published: October 19, 2007

**Citation**

A. V. Gorbach and D. V. Skryabin, "Bouncing of a dispersive pulse on an accelerating soliton and stepwise frequency conversion in optical fibers," Opt. Express **15**, 14560-14565 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14560

Sort: Year | Journal | Reset

### References

- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001), 3rd ed.
- A. Gorbach and D.V. Skryabin, "Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic crystal fibers," Nature-Photonics 1 (2007); http://xxx.lanl.gov/abs/0706.1187>.
- A. Gorbach and D. V. Skryabin. "Theory of radiation trapping by the accelerating solitons in optical fibers," Phys. Rev. A (to be published); http://xxx.lanl.gov/abs/0707.1598>.
- P. Beaud, W. Hodel, B. Zysset, and H. P. Weber. "Ultrashort pulse-propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber," IEEE J. Quantum Electron. 23, 1938-1946 (1987). [CrossRef]
- N. Nishizawa and T. Goto. "Experimental analysis of ultrashort pulse propagation in optical fibers around zerodispersion region using cross-correlation frequency resolved optical gating," Opt. Express 8, 328-334 (2001). [CrossRef] [PubMed]
- A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight. "Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum," Opt. Express 14, 9854- 9863 (2006). [CrossRef] [PubMed]
- J. Gea-Banacloche, "A quantum bouncing ball," Am. J. Phys. 67, 776 (1999). [CrossRef]
- C. V. Saba, P. A. Barton, M. G. Boshier, I. G. Hughes, P. Rosenbusch, B. E. Sauer, and E. A. Hinds, "Reconstruction of a cold atom cloud by magnetic focusing," Phys. Rev. Lett. 82, 468-471 (1999). [CrossRef]
- K. Bongs, S. Burger, G. Birkl, K. Sengstock, W. Ertmer, K. Rzazewski, A. Sanpera, and M. Lewenstein, "Coherent evolution of bouncing Bose-Einstein condensates," Phys. Rev. Lett. 83, 3577-3580 (1999). [CrossRef]
- K. J. Blow, N. J. Doran, and D. Wood, "Suppression of the soliton self-frequency shift by bandwidth-limited amplification," J. Opt. Soc. Am B 5, 1301-1304 (1988). [CrossRef]
- L. Gagnon and P. A. Belanger, "Soliton self-frequency shift versus Galilean-like symmetry," Opt. Lett. 15, 466- 468 (1990). [CrossRef] [PubMed]
- I. S. Averbukh and N. F. Perelman, "Dynamics of wave-packets from highly excited atomic and molecular-states," Usp. Fiz. Nauk 161, 41-81 (1991). [CrossRef]
- D. V. Skryabin and A. V. Yulin, "Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers," Phys. Rev. E 72, 016619 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.