## Efficient frequency shifting of dispersive waves at solitons |

Optics Express, Vol. 20, Issue 5, pp. 5538-5546 (2012)

http://dx.doi.org/10.1364/OE.20.005538

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### Abstract

We demonstrate frequency redshifting and blueshifting of dispersive waves at group velocity horizons of solitons in fibers. The tunnelling probability of waves that cannot propagate through the fiber-optical solitons (horizons) is measured and described analytically. For shifts up to two times the soliton spectral width, the waves frequency shift with probability exceeding 90% rather than tunnelling through the soliton in our experiment. We also discuss key features of fiber optical Cherenkov radiation such as high efficiency and large bandwidth within this framework.

© 2012 OSA

## 1. Introduction

1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. **23**, 142–144 (1973). [CrossRef]

2. L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. **45**, 1095–1098 (1980). [CrossRef]

3. K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. **46**, 317–319 (1985). [CrossRef]

9. M. Nazarathy, Z. Zalevsky, A. Rudnitsky, B. Larom, A. Nevet, M. Orenstein, and B. Fischer, “All-optical linear reconfigurable logic with nonlinear phase erasure,” J. Opt. Soc. Am. A **26**, A21–A39 (2009). [CrossRef]

11. M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, “Cross-phase modulation in optical fibers,” Opt. Lett. **12**, 625–627 (1987). [CrossRef] [PubMed]

12. J. P. Gordon, “Dispersive perturbations of solitons of the
nonlinear Schrodinger-equation,” J. Opt. Soc. Am.
B **9**, 91–97
(1992). [CrossRef]

12. J. P. Gordon, “Dispersive perturbations of solitons of the
nonlinear Schrodinger-equation,” J. Opt. Soc. Am.
B **9**, 91–97
(1992). [CrossRef]

15. J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

16. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

17. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. **11**, 464–466 (1986). [CrossRef] [PubMed]

19. L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B **77**, 307–311 (2003). [CrossRef]

20. N. Nishizawa and T. Goto, “Characteristics of pulse trapping by ultrashort soliton pulse in optical fibers across zerodispersion wavelength,” Opt. Express **10**, 1151–1159 (2002). [PubMed]

22. A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. **95**, 213902 (2005). [CrossRef] [PubMed]

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science **319**, 1367–1370 (2008). [CrossRef] [PubMed]

24. S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express **17**13588–13600 (2009). [CrossRef] [PubMed]

25. S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A **81**, 063835 (2010). [CrossRef]

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science **319**, 1367–1370 (2008). [CrossRef] [PubMed]

26. W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. **46**, 1351–1353 (1981). [CrossRef]

23. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science **319**, 1367–1370 (2008). [CrossRef] [PubMed]

**319**, 1367–1370 (2008). [CrossRef] [PubMed]

26. W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. **46**, 1351–1353 (1981). [CrossRef]

28. S. M. Hawking, “Particle creation by black-holes,” Commun. Math. Phys. **43**, 199–220 (1975). [CrossRef]

8. A. Demircan, Sh. Amiranashvili, and G. Steinmeyer, “Controlling light by light with an optical event horizon,” Phys. Rev. Lett. **106**, 163901 (2011). [CrossRef] [PubMed]

22. A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. **95**, 213902 (2005). [CrossRef] [PubMed]

**319**, 1367–1370 (2008). [CrossRef] [PubMed]

## 2. Tunnelling model

*A*

_{1}(

*z*,

*t*) for the probe wave and

*A*

_{2}(

*z*,

*t*) for the soliton. We focus on the principal effects of group velocity dispersion and nonlinearity. We assume a soliton unaffected by higher order effects as well as a weak probe wave with negligible backaction on the soliton. Numerical treatment including these effects can be found in [30

30. 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]

24. S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express **17**13588–13600 (2009). [CrossRef] [PubMed]

31. V. E. Lobanov and A. P. Sukhorukov, “Total reflection, frequency, and velocity tuning in optical pulse collision in nonlinear dispersive media,” Phys. Rev. A , **82**, 033809 (2010). [CrossRef]

32. N. N. Rosanov, N. V. Vysotina, and A. N. Shatsev, “Forward light reflection from a moving inhomogeneity,” JETP Lett. **93**, 308–312 (2011). [CrossRef]

*β*

_{1}and

*β*

_{2}are the first and second derivative of the propagation constant

*β*(

*ω*) at a frequency

*ω*

*of the same group velocity*

_{m}*A*

_{1}representing the probe.

*γ*is the fiber nonlinearity and

*r*is a factor accounting for the reduction in cross-phase modulation due to conditions such as the relative polarization orientation or mode size mismatch. We change to the coordinates

*τ*=

*t*–

*z*/

*u*and

*ζ*=

*z*/

*u*of the moving frame of the soliton with velocity

*u*. Hence, we neglect self-phase modulation of the weak probe and assume a soliton, unaffected by the probe, of the form |

*A*

_{2}(

*τ*)|

^{2}=

*P*

_{0}sech

^{2}(

*τ*/

*T*

_{0}), where

*P*

_{0}and

*T*

_{0}are the soliton amplitude and length. We arrive at: We introduce the frequency of a wave in the reference frame moving with the group velocity of the soliton,

*ω*′ =

*ω*–

*β*

*u*=

*ω*{1 –

*n*(

*ω*,

*I*(

*τ*))/(

*β*

_{1}

*c*)}. The refractive index

*n*has a nonlinear contribution from the soliton intensity

*I*(

*τ*) due to the Kerr effect. Finally, we use the ansatz

*A*

_{1}= 𝒜

*exp*(

*iϕ*(

*ζ*,

*τ*)+

*i*

*ω*′

_{m}*ζ*), where

*ϕ*is the optical phase and

*ω*′

*is the moving frame frequency at the group velocity matched lab frequency*

_{m}*ω*

*. The nonlinear Schrödinger equation becomes: where we define Ω′ =*

_{m}*ω*′ –

*ω*′

*. Equation (3) is formally identical to the Schrödinger equation in quantum mechanics. This analogy allows us to investigate quantum mechanical potential problems with classical nonlinear fiber optics. As*

_{m}*β*

_{2}can either be positive or negative, we can realize attractive or repulsive potentials. Input and converted modes are connected: the waves follow contours of constant

*ω*′ (moving frame frequency) as this is a conserved quantity in Eq. (3) [23

**319**, 1367–1370 (2008). [CrossRef] [PubMed]

*ω*′ is conserved even in the presence of higher order dispersion acting on the probe. Therefore, the waves are shifted in frequency in the laboratory frame by an amount that depends on the soliton velocity and the refractive index profile of the fiber mode only. In addition to the modes let us consider the efficiency of the process, i.e. the probability of conversion from the input mode to the output mode. The scattering at solitons represents a spatially constant one-dimensional potential, for which the transmission and reflection coefficients,

*T*and

*R*, are known [33]. In the variables of fiber optics these coefficients are: where Ω =

*ω*–

*ω*

*is the detuning of the probe wave from the group velocity matched frequency. The transmission through the barrier therefore is determined by the two parameters Ω*

_{m}*T*

_{0}, the ratio of detuning to soliton bandwidth, and

*B*> 0 (

*B*< 0) implies a potential barrier (pit). For

*B*= 1 the reflectivity has the sech

^{2}-form of the pulse and the interaction decreases exponentially with detuning. At large

*B*, the effective wave number becomes complex (cf. Eq. (3)) and the dispersive wave can only pass the soliton by tunnelling. Accordingly, tunneling takes place if the effective nonlinearity (i.e.

*r*

*γ*

*P*

_{0}) or the dispersion length (

*r*= 2, copolarized). For

*B*< 0, the probe wave experiences anomalous dispersion and reflection is limited to detunings of much less than the spectral width of the soliton. In the case of

*B*> 0, however, effective reflection can be achieved for large detunings. The contour for 90% reflection is given by (

*B*≫ 1,Ω

*T*

_{0}> 1): This is a remarkable result. In the photon picture, the mode conversion may be thought of as an annihilation of a photon in the input mode and creation of a photon in the output mode, which has a different laboratory frequency. The energy difference has to come from the pulse. In the simplest way, a photon of one of the modes of the pulse is annihilated and another photon is generated with the corresponding energy difference. This is a simple four-wave mixing process well known in third order nonlinear materials. Hence, we would expect the spectral width of the pulse to severely limit the frequency shift as it is the maximum energy difference between the modes that can be compensated for. For most pulses, e.g. solitons, the spectrum exponentially decreases away from the carrier and an exponential decrease in frequency conversion is expected. Equation (5), however, states that efficient conversion is possible even for very large detunings, provided

*B*is sufficiently large. The required height of the barrier

*B*increases quadratically rather than exponentially. In principle, condition 5 can always be fulfilled in a medium with very small magnitude group velocity dispersion

*β*

_{2}at the probe frequency. Therefore, the mode conversion is a collective effect of the modes of the soliton and the probe rather than a phasematched mixing of only four modes.

## 3. Tunnelling experiment

*D*(

*D*

_{s}) and

*λ*(

*λ*

_{s}) are the dispersion (in ps/nm/km) and wavelength of the probe (soliton). To achieve large

*B*we use a highly nonlinear photonic crystal fiber that exhibits anomalous dispersion in the infrared for soliton creation. We choose the probe wavelength shorter than the soliton wavelength beyond the zero dispersion point of the fiber, where there is a group-velocity matched wavelength

*λ*

*. As is indicated in the inset in Fig. 4, the integral over the dispersion curve, which is the group delay, vanishes between*

_{m}*λ*

*and*

_{m}*λ*

*. Hence, for the photonic crystal fiber (pcf) we used (NL-1.5–670, NKT Photonics, Inc.), we obtain*

_{s}*B*= 22.4 with dispersions of −256ps/(nm km) and 144ps/(nm km) for probe (

*λ*

*= 532nm) and soliton (*

_{p}*λ*

*∼ 840nm), respectively. The group index of the fiber was determined approximately using a simple silica strand model which reproduced the group velocity matching condition as observed [34].*

_{s}*η*

_{int}of the probe light interacts with the pulse in the finite length of fiber

*L*. Thus the probe is weak and the soliton recoil is negligible. Hence the conversion efficiency

*R*(Eq. (4)) is reduced by

*η*

_{int}to the total efficiency

*η*: where

*ν*is the pulse repetition rate and

*δ*

*n*

*is the difference in group indices of pulse and probe wave.*

_{g}*R*(blue) and

*η*

_{int}(green) to the total efficiency

*η*(red) are displayed according to the model, in which the experimental parameters were used (identical to Fig. 5). The larger the detuning of the probe from the group velocity of the soliton, the more light collides with the soliton (green). For small detunings there is negligible tunnelling and the probe is nearly perfectly reflected. At a detuning of approximately 12 THz, tunnelling sets in and rapidly increases until the probe light is no longer reflected at about 25 THz (blue). According to Fig. 5, Ω

*T*

_{0}≈ 2, i.e. frequency shifting is expected up to a detuning of twice the soliton bandwidth. The resulting efficiency is displayed in red. The curves are slightly asymmetric because of the higher order dispersion in the fiber.

25. S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A **81**, 063835 (2010). [CrossRef]

*ω*′, including higher order dispersion, as a red solid line. The figure shows that the center of the probe spectrum shifts as expected according to the fiber dispersion. Thus the soliton propagates approximately with constant velocity, unaffected by higher order effects. The efficiency of reflection and frequency shifting depends on the interaction efficiency

*η*

_{int}and the soliton reflectivity

*R*. The former is a dispersive property, while the latter depends on the refractive index barrier. To find the total observed efficiency, we integrated the shifted spectra to find the shifted power and normalized to the power at the initial probe wavelength of 532nm, measured separately. For the spectra of Fig. 3,

*η*= 1.210

^{−4}(

*η*= 1.110

^{−4}), which results in an observed reflectivity

*R*= 96.3% (

*R*= 58.3%) for this particular blue (red) shifting.

*R*, however, is significant (>10%) for frequency shifts up to ±20 THz (±19 nm).

*η*. The only adjustable parameter in these curves is

*r*= 1.7, the effective cross-phase modulation strength. Note that

*r*also includes effects such as coupling of light to higher order fiber modes and deviations from the amplitude required for a perfect

*N*= 1 soliton. The agreement with the experimental data is very good and shows that the reflectivity of the soliton can be described by our tunnelling model. We also inserted a copy of a typical soliton input spectrum into the figure for comparison. According to Eq. (5), 90% reflectivity occurs at a detuning Ω

*T*

_{0}= 2.0. The frequency shift thus can exceed the spectral width of the soliton considerably before eventually decreasing. In our experiment, the reflectivity does not decrease for a frequency shift about two times the soliton spectral width, beyond which it rapidly decreases. For higher values of the barrier

*B*an even wider range of frequency shifts is expected. Note that in [23

**319**, 1367–1370 (2008). [CrossRef] [PubMed]

*B*= 0.85, a regime described by the other branch in Eq. (4). Our experiment is thus performed in a novel regime with a 25 times stronger barrier.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. |

2. | L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. |

3. | K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. |

4. | N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. |

5. | S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. |

6. | M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, and H. A. Haus, “Ultrafast switching with a single-fiber interferometer,” Opt. Lett. |

7. | D. A. B. Miller, “Are optical transistors the logical next step?” Nat. Photonics |

8. | A. Demircan, Sh. Amiranashvili, and G. Steinmeyer, “Controlling light by light with an optical event horizon,” Phys. Rev. Lett. |

9. | M. Nazarathy, Z. Zalevsky, A. Rudnitsky, B. Larom, A. Nevet, M. Orenstein, and B. Fischer, “All-optical linear reconfigurable logic with nonlinear phase erasure,” J. Opt. Soc. Am. A |

10. | S. Akhmanov, A. Sukhorukov, and A. Chirkin, “Nonstationary phenomena and spacetime analogy in nonlinear optics,” Sov. Phys. JETP |

11. | M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, “Cross-phase modulation in optical fibers,” Opt. Lett. |

12. | J. P. Gordon, “Dispersive perturbations of solitons of the
nonlinear Schrodinger-equation,” J. Opt. Soc. Am.
B |

13. | V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP |

14. | J. R. Taylor, |

15. | J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

16. | P. Russell, “Photonic crystal fibers,” Science |

17. | P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. |

18. | N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A |

19. | L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B |

20. | N. Nishizawa and T. Goto, “Characteristics of pulse trapping by ultrashort soliton pulse in optical fibers across zerodispersion wavelength,” Opt. Express |

21. | A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics |

22. | A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. |

23. | T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science |

24. | S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express |

25. | S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A |

26. | W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett. |

27. | S. M. Hawking, “Black-hole explosions,” Nature |

28. | S. M. Hawking, “Particle creation by black-holes,” Commun. Math. Phys. |

29. | G. P. Agrawal, |

30. | 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 |

31. | V. E. Lobanov and A. P. Sukhorukov, “Total reflection, frequency, and velocity tuning in optical pulse collision in nonlinear dispersive media,” Phys. Rev. A , |

32. | N. N. Rosanov, N. V. Vysotina, and A. N. Shatsev, “Forward light reflection from a moving inhomogeneity,” JETP Lett. |

33. | L. D. Landau and E. M. Lifshitz, |

34. | Details of this technique will be published elsewhere. |

35. | H. Tu and S. A. Boppart, “Optical frequency up-conversion by supercontinuum-free widely-tunable fiber-optic Cherenkov radiation,” Opt. Express |

36. | H. Tu and S. A. Boppart, “Ultraviolet-visible non-supercontinuum ultrafast source enabled by switching single silicon strand-like photonic crystal fibers,” Opt. Express |

37. | G. Q. Chang, L. J. Chen, and F. X. Kärtner, “Highly efficient Cherenkov radiation in photonic crystal fibers for broadband visible wavelength generation,” Opt.Lett. |

38. | G. Q. Chang, L. J. Chen, and F. X. Kärtner, “Fiber-optic Cherenkov radiation in the few-cycle regime,” Opt. Express |

**OCIS Codes**

(060.7140) Fiber optics and optical communications : Ultrafast processes in fibers

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 26, 2011

Revised Manuscript: December 30, 2011

Manuscript Accepted: February 5, 2012

Published: February 22, 2012

**Citation**

Amol Choudhary and Friedrich König, "Efficient frequency shifting of dispersive waves at solitons," Opt. Express **20**, 5538-5546 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5538

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### References

- A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett.23, 142–144 (1973). [CrossRef]
- L. F. Mollenauer, R. H. Stolen, and J. G. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett.45, 1095–1098 (1980). [CrossRef]
- K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett.46, 317–319 (1985). [CrossRef]
- N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett.13, 56–58 (1988). [CrossRef] [PubMed]
- S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett.13, 672–674 (1988). [CrossRef] [PubMed]
- M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, and H. A. Haus, “Ultrafast switching with a single-fiber interferometer,” Opt. Lett.14311–313 (1989). [CrossRef] [PubMed]
- D. A. B. Miller, “Are optical transistors the logical next step?” Nat. Photonics4, 3–5 (2010). [CrossRef]
- A. Demircan, Sh. Amiranashvili, and G. Steinmeyer, “Controlling light by light with an optical event horizon,” Phys. Rev. Lett.106, 163901 (2011). [CrossRef] [PubMed]
- M. Nazarathy, Z. Zalevsky, A. Rudnitsky, B. Larom, A. Nevet, M. Orenstein, and B. Fischer, “All-optical linear reconfigurable logic with nonlinear phase erasure,” J. Opt. Soc. Am. A26, A21–A39 (2009). [CrossRef]
- S. Akhmanov, A. Sukhorukov, and A. Chirkin, “Nonstationary phenomena and spacetime analogy in nonlinear optics,” Sov. Phys. JETP28, 748–757 (1969).
- M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, “Cross-phase modulation in optical fibers,” Opt. Lett.12, 625–627 (1987). [CrossRef] [PubMed]
- J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger-equation,” J. Opt. Soc. Am. B9, 91–97 (1992). [CrossRef]
- V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–69 (1972).
- J. R. Taylor, Optical Solitons Theory and Experiment (Cambridge Press, 2005).
- J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett.21, 1547–1549 (1996). [CrossRef] [PubMed]
- P. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003). [CrossRef] [PubMed]
- P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett.11, 464–466 (1986). [CrossRef] [PubMed]
- N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995). [CrossRef] [PubMed]
- L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77, 307–311 (2003). [CrossRef]
- N. Nishizawa and T. Goto, “Characteristics of pulse trapping by ultrashort soliton pulse in optical fibers across zerodispersion wavelength,” Opt. Express10, 1151–1159 (2002). [PubMed]
- A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics1, 653–656 (2007). [CrossRef]
- A. Efimov, A. Yulin, D. Skryabin, J. C. Knight, N. Joly, F. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett.95, 213902 (2005). [CrossRef] [PubMed]
- T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science319, 1367–1370 (2008). [CrossRef] [PubMed]
- S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express1713588–13600 (2009). [CrossRef] [PubMed]
- S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: the effect of Raman deceleration,” Phys. Rev. A81, 063835 (2010). [CrossRef]
- W. G. Unruh, “Experimental black-hole evaporation,” Phys. Rev. Lett.46, 1351–1353 (1981). [CrossRef]
- S. M. Hawking, “Black-hole explosions,” Nature248, 30–31 (1974). [CrossRef]
- S. M. Hawking, “Particle creation by black-holes,” Commun. Math. Phys.43, 199–220 (1975). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006).
- 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. E72, 016619 (2005). [CrossRef]
- V. E. Lobanov and A. P. Sukhorukov, “Total reflection, frequency, and velocity tuning in optical pulse collision in nonlinear dispersive media,” Phys. Rev. A, 82, 033809 (2010). [CrossRef]
- N. N. Rosanov, N. V. Vysotina, and A. N. Shatsev, “Forward light reflection from a moving inhomogeneity,” JETP Lett.93, 308–312 (2011). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics3, (Butterworth-Heinemann, 1981).
- Details of this technique will be published elsewhere.
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