Abstract

We provide a full theoretical understanding of the recent observations of excitation of Raman two-peak states in solid-core photonic crystal fibers. Based on a “gravity-like” potential approach we derive simple equations for the “magic” peak power ratio and the temporal separation between pulses forming these two-peak states. We develop a model to calculate the magic input power of the input pulse around which the phenomenon can be observed. We also predict the existence of exotic multipeak states that strongly violate the perturbative pulse splitting law, and we study their stability and excitation conditions.

© 2010 Optical Society of America

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  1. K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–395 (1988).
    [CrossRef] [PubMed]
  2. Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
    [CrossRef]
  3. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  4. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  5. A. Podlipensky, P. Szarniak, N. Y. Joly, C. G. Poulton, and P. St. J. Russell, “Bound soliton pairs in photonic crystal fibers,” Opt. Express 15, 1653–1662 (2007).
    [CrossRef] [PubMed]
  6. A. Podlipensky, P. Szarniak, N. Y. Joly, and P. St. J. Russell, “Anomalous pulse breakup in small-core photonic crystal fibers,” J. Opt. Soc. Am. B 25, 2049–2056 (2008).
    [CrossRef]
  7. P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [CrossRef] [PubMed]
  8. N. Akhmediev, W. Królikovski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
    [CrossRef]
  9. L. Gagnon and P. A. Bélanger, “Soliton self-frequency shift versus Galilean-like symmetry,” Opt. Lett. 15, 466–468 (1990).
    [CrossRef] [PubMed]
  10. A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
    [CrossRef]
  11. A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibers,” Nature Photon. 1, 653–657 (2007).
    [CrossRef]
  12. Equation has to be interpreted as an eigen problem for f(ξ) and q, with b=32τRq2/15 imposed by the physics of RSFS. The boundary conditions that one has to impose to find the soliton solutions of Eq. by using the shooting method are as follows: (i) the function values at the leading edge boundary ξ1 and the trailing edge boundary ξ2 are vanishing: f(ξ1)=f(ξ2)=0; (ii) the first-order derivative at the trailing edge boundary ξ2 is vanishing: fξ(ξ2)=0. The numerical solution may shift the position of the maximum slightly with respect to ξ=0 in order to satisfy the above conditions, due to the presence of the oscillating and slowly decaying Airy tail on the leading edge of the soliton.
  13. D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
    [CrossRef] [PubMed]
  14. V. I. Karpman and V. V. Solovev, “A perturbational approach to the two-solution systems,” Physica D 3, 487–502 (1981).
    [CrossRef]
  15. A. Hause, Tr. X. Tran, F. Biancalana, A. Podlipensky, P. S. Russell, and F. Mitschke, “Understanding Raman-shifting multipeak states in photonic crystal fibers: two convergent approaches,” Opt. Lett. 35, 2167–2169 (2010).
    [CrossRef] [PubMed]
  16. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  17. F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
    [CrossRef]

2010 (1)

2008 (1)

2007 (4)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

A. Podlipensky, P. Szarniak, N. Y. Joly, C. G. Poulton, and P. St. J. Russell, “Bound soliton pairs in photonic crystal fibers,” Opt. Express 15, 1653–1662 (2007).
[CrossRef] [PubMed]

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibers,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

2004 (1)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[CrossRef]

2003 (1)

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

1996 (1)

N. Akhmediev, W. Królikovski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[CrossRef]

1990 (1)

1988 (1)

1987 (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

1986 (2)

1985 (1)

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

1981 (1)

V. I. Karpman and V. V. Solovev, “A perturbational approach to the two-solution systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Akhmediev, N.

N. Akhmediev, W. Królikovski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[CrossRef]

Anderson, D.

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

Bekki, N.

Bélanger, P. A.

Biancalana, F.

A. Hause, Tr. X. Tran, F. Biancalana, A. Podlipensky, P. S. Russell, and F. Mitschke, “Understanding Raman-shifting multipeak states in photonic crystal fibers: two convergent approaches,” Opt. Lett. 35, 2167–2169 (2010).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[CrossRef]

Gagnon, L.

Gorbach, A. V.

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibers,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

Gordon, J. P.

Hasegawa, A.

K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–395 (1988).
[CrossRef] [PubMed]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Hause, A.

Joly, N. Y.

Karpman, V. I.

V. I. Karpman and V. V. Solovev, “A perturbational approach to the two-solution systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Królikovski, W.

N. Akhmediev, W. Królikovski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[CrossRef]

Lisak, M.

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

Lowery, A. J.

N. Akhmediev, W. Królikovski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[CrossRef]

Mitschke, F.

Mitschke, F. M.

Mollenauer, L. F.

Podlipensky, A.

Poulton, C. G.

Russell, P. S.

Russell, P. St. J.

Skryabin, D. V.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibers,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[CrossRef]

Solovev, V. V.

V. I. Karpman and V. V. Solovev, “A perturbational approach to the two-solution systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Szarniak, P.

Tai, K.

Tran, Tr. X.

Yulin, A. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature Photon. (1)

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibers,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

Opt. Commun. (1)

N. Akhmediev, W. Królikovski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. A (2)

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

Phys. Rev. E (1)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[CrossRef]

Physica D (1)

V. I. Karpman and V. V. Solovev, “A perturbational approach to the two-solution systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Science (1)

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

Other (2)

Equation has to be interpreted as an eigen problem for f(ξ) and q, with b=32τRq2/15 imposed by the physics of RSFS. The boundary conditions that one has to impose to find the soliton solutions of Eq. by using the shooting method are as follows: (i) the function values at the leading edge boundary ξ1 and the trailing edge boundary ξ2 are vanishing: f(ξ1)=f(ξ2)=0; (ii) the first-order derivative at the trailing edge boundary ξ2 is vanishing: fξ(ξ2)=0. The numerical solution may shift the position of the maximum slightly with respect to ξ=0 in order to satisfy the above conditions, due to the presence of the oscillating and slowly decaying Airy tail on the leading edge of the soliton.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

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Figures (8)

Fig. 1
Fig. 1

Profiles of multipeak soliton solutions. Blue dashed lines show the gravity-like potential U ( ξ ) created by the most intense soliton in its leading edge into which all other solitons with decreasing powers can be progressively fitted in an organ-pipe fashion.

Fig. 2
Fig. 2

Temporal evolution of multipeak solitons.

Fig. 3
Fig. 3

(a) Plot of the peak power ratio r and (b) the initial temporal separation ξ 0 as functions of q. Blue dots and red solid lines indicate the numerical and analytical results, respectively. Parameter is τ R = 0.1 .

Fig. 4
Fig. 4

Peak power ratio r versus initial temporal separation ξ 0 . (a),(b),(c),(d) Results for q = 0.369 , 0.3164, 0.1151, and 0.0616, respectively. Red dotted lines are calculated based on Eq. (5), blue solid curves are obtained from Eq. (7), and crossing points of these curves correspond to the solution of Eq. (8). Single big black dots are magic points. Green curves with square markers are obtained by numerically modeling Eq. (3) for a propagation length z = 160 with two hyperbolic secant solitons initially having the same frequency ( α = 0 ) as initial input values. The curve with triangular markers in (b) shows the case α = 0.2 .

Fig. 5
Fig. 5

Spectral evolution with increasing soliton number N. The propagation length is z = 0.45 .

Fig. 6
Fig. 6

Formation of two-peak localized states. (a),(b),(c),(d) Temporal evolution of pulses with input soliton orders N = 9.63 , 12.075, 12.65, and 14.65, respectively.

Fig. 7
Fig. 7

Formation of a three-peak Raman state. (a) Temporal evolution of a pulse with input soliton order N = 13.81 . (b) Intensity profile of this three-peak Raman state at z = 0.8 .

Fig. 8
Fig. 8

Power dependence of the spectra obtained experimentally at the end of a highly nonlinear PCF (18 cm long).

Equations (9)

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P j = P 0 ( 2 N 2 j + 1 ) 2 / N 2 ,
T j = T 0 / ( 2 N 2 j + 1 ) ,
i z ψ + 1 2 t 2 ψ + | ψ | 2 ψ τ R ψ t | ψ | 2 = 0 ,
1 2 f ξ ξ ( q + b ξ ) f + | f | 2 f τ R f ( | f | 2 ) ξ = 0 ,
ξ 0 ξ 1 ξ 2 q ( 1 r ) / b .
L p π 2 ( 2 ν ) 2 exp [ y 0 4 ] ,
π 2 16 15 τ R 1 r 2 ( 1 + r ) 4 exp [ 1 + r 2 ξ 0 q ] = ξ 0 .
( π 16 15 τ R ) 2 2 ( 1 + r ) q ( 1 + r ) 4 exp [ 15 ( 1 + r ) ( 1 r ) 32 τ R 2 q ] = 1.
i z ψ + 1 2 t 2 ψ + ψ ( z , t ) t R ( t ) | ψ ( z , t t ) | 2 d t = 0 ,

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