Abstract

It seems to be self-evident that stable optical pulses cannot be considerably shorter than a single oscillation of the carrier field. From the mathematical point of view the solitary solutions of pulse propagation equations should loose stability or demonstrate some kind of singular behavior. Typically, an unphysical cusp develops at the soliton top, preventing the soliton from being too short. Consequently, the power spectrum of the limiting solution has a special behavior: the standard exponential decay is replaced by an algebraic one. We derive the shortest soliton and explicitly calculate its spectrum for the so-called short pulse equation. The latter applies to ultra-short solitons in transparent materials like fused silica that are relevant for optical fibers.

© 2014 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics (Academic, New York, 2008), 3rd ed.
  2. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [Crossref]
  3. G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express 15, 5382–5387 (2007).
    [Crossref] [PubMed]
  4. T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
    [Crossref]
  5. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
    [Crossref]
  6. S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
    [Crossref]
  7. A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
    [Crossref]
  8. A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
    [Crossref]
  9. S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
    [Crossref]
  10. S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A 87, 013805 (2013).
    [Crossref]
  11. R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
    [Crossref] [PubMed]
  12. S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
    [Crossref]
  13. S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
    [Crossref]
  14. D. Gabor, “Theory of communication,” Journal of the Institute of Electrical Engineers 93, 429–457 (1946).

2013 (1)

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A 87, 013805 (2013).
[Crossref]

2011 (1)

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[Crossref]

2008 (1)

S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[Crossref]

2007 (2)

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express 15, 5382–5387 (2007).
[Crossref] [PubMed]

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

2006 (1)

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[Crossref]

2005 (1)

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

2004 (1)

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

1997 (3)

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[Crossref]

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
[Crossref]

1993 (1)

R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

1946 (1)

D. Gabor, “Theory of communication,” Journal of the Institute of Electrical Engineers 93, 429–457 (1946).

Akhmediev, N.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A 87, 013805 (2013).
[Crossref]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[Crossref]

Amiranashvili, S.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A 87, 013805 (2013).
[Crossref]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[Crossref]

S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[Crossref]

Bandelow, U.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A 87, 013805 (2013).
[Crossref]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[Crossref]

S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, New York, 2008), 3rd ed.

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[Crossref]

Camassa, R.

R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

Dudley, J. M.

Gabor, D.

D. Gabor, “Theory of communication,” Journal of the Institute of Electrical Engineers 93, 429–457 (1946).

Genty, G.

Holm, D.

R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

Kartashov, D. V.

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

Kibler, B.

Kim, A. V.

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

Kinsler, P.

Kozlov, S. A.

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
[Crossref]

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[Crossref]

Oughstun, K. E.

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[Crossref]

Sakovich, A.

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[Crossref]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

Sakovich, S.

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[Crossref]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

Sazonov, S. V.

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
[Crossref]

Schäfer, T.

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

Skobelev, S. A.

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

Vladimirov, A. G.

S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[Crossref]

Wayne, C. E.

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

Xiao, H.

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[Crossref]

J. Phys. A (1)

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[Crossref]

J. Phys. Soc. Jpn. (1)

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

JETP (1)

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
[Crossref]

Journal of the Institute of Electrical Engineers (1)

D. Gabor, “Theory of communication,” Journal of the Institute of Electrical Engineers 93, 429–457 (1946).

Opt. Express (1)

Phys. Rev. A (3)

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[Crossref]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A 87, 013805 (2013).
[Crossref]

S. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[Crossref]

Phys. Rev. Lett. (4)

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642–645 (1997).
[Crossref]

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[Crossref]

Physica D (1)

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, New York, 2008), 3rd ed.

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Figures (2)

Fig. 1
Fig. 1 (a) An exemplary potential U(a) from Eq. (13) for γ = 9/8 − δ with δ = 10−3. The red point labels the upper value of a(ξ) from the inequality (14). As δ → 0, an infinite wall is formed at a = 1 / 6, resulting in cusp formation at the top of the soliton. (b) Shape of the shortest soliton calculated from Eq. (15).
Fig. 2
Fig. 2 (a) Phase of the limiting soliton and (b) its power spectrum. Blue line: numerical solution, red line: expression (22). The two auxiliary thin lines show the spectrum of the fundamental soliton (dashed), and the Ω−4 power law (solid).

Equations (22)

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ε ( ω ) = n s 2 ( 1 ω p 2 / ω 2 ) ,
t 2 ( ε ^ E + χ ( 3 ) E 3 ) c 2 z 2 E = 0 , where ε ^ [ ω E ω e i ω t ] = ω ε ( ω ) E ω e i ω t .
n s 2 ( t 2 E + ω p 2 E ) c 2 z 2 E + χ ( 3 ) t 2 ( E 3 ) = 0 ,
ω 0 t β 0 z = ω 0 [ t n s c z ( 1 μ 2 ) 1 / 2 ] = ω 0 ( t n s c z ) + μ 2 n s ω 0 2 c z + O ( μ 4 ) .
τ = ω 0 ( t n s c z ) , ζ = μ 2 n s ω 0 c z , ( 3 χ ( 3 ) 8 ) 1 / 2 E ( z , t ) = μ n s F ( ζ , τ ) .
2 ζ τ F + F + 8 3 τ 2 ( F 3 ) = μ 2 ζ 2 F .
ζ τ + 1 2 + 1 3 τ 2 ( 3 | | 2 + 3 ) = 0 .
ζ τ + 1 2 + τ 2 ( | | 2 ) = 0 .
V phase nonlinear = γ V ph = 2 γ ν 2 and V group nonlinear = γ V gr = 2 γ ν 2 ,
( ζ , τ ) = f ( ν τ ζ 2 γ ν ) exp [ i ( ν τ + ζ 2 γ ν ) ] ,
( f 2 γ ν 2 | f | 2 f ) ( γ 1 ) f + 4 i γ ν 2 ( | f | 2 f ) + 2 γ ν 2 | f | 2 f = 0 ,
ϕ = ( 3 4 a 2 ) a 2 ( 1 2 a 2 ) 2 .
a 2 + U ( a ) = 0 , U ( a ) = ( γ 1 ) ( 1 3 a 2 ) a 2 ( 1 6 a 2 ) 2 + ( 1 7 a 2 + 12 a 4 ) a 4 ( 1 2 a 2 ) 2 ( 1 6 a 2 ) 2 .
0 a [ 4 γ 3 ( 9 8 γ ) 1 / 2 8 γ ] 1 / 2 ,
a 2 = a 2 ( 1 3 a 2 ) 8 ( 1 2 a 2 ) 2 with a ( 0 ) = 1 6 a e B ( a ) = Λ e | ξ | 2 2 ,
B ( a ) [ 2 3 ( 1 3 a 2 ) 1 / 2 ln [ 1 + ( 1 3 a 2 ) 1 / 2 ] ] 0 a , Λ e B ( 1 6 ) 6 0.3935 .
ϕ = 0 ϕ ( ξ ) d ξ = 0 ( 3 4 a 2 ) a 2 ( 1 2 a 2 ) 2 ( d ξ d a d a ) = 4 ( 2 1 ) 3 arcsin 1 3 ,
X = a e B ( a ) = a a 3 4 + a 5 8 + 49 a 7 192 + a = X + X 3 4 + X 5 16 61 X 7 192 +
ϕ = ϕ + 3 2 X 2 + 19 X 4 2 2 + 457 X 6 24 2 + ,
a e i ϕ = e i ϕ ( X + 1 + 12 i 2 4 X 3 + 143 + 88 i 2 16 X 5 + ) ,
( a e i ϕ ) Ω = a ( ξ ) e i ϕ ( ξ ) e i Ω ξ d ξ = 0 a ( ξ ) e i ϕ ( ξ ) e i Ω ξ d ξ + c . c . ,
( a e i ϕ ) Ω = Λ R 1 + S 1 Ω 1 + 8 Ω 2 + Λ 3 R 3 + S 3 Ω 9 + 8 Ω 2 + Λ 5 R 5 + S 5 Ω 25 + 8 Ω 2 + ,

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