Abstract

We study the propagation of chirped (D+1)-dimensional optical pulses in bulk media with periodic dispersion, analytically by using the variational approach and numerically by using a new, to our knowledge, numerical technique relying on the adaptive fast Hankel split-step method using cylindrical and spherical symmetries for two and three dimensions, respectively. Stability criteria for (2+1)- and (3+1)-dimensional solitons are identified, and the long-term dynamics of the solitons are studied with the averaged equations obtained using the Kapitza approach. Also, the slow dynamics of the solitons around the fixed points for the width and the chirp are studied. The importance of this research is in generating dispersion-managed optical solitons in optical communication. Also, this research is applied to the stabilization of the Bose–Einstein condensate in (2+1)- and (3+1)-dimensional optical lattices. We compare results of the new numerical technique with those obtained using the fast Fourier split-step technique.

© 2006 Optical Society of America

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References

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  1. F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
    [CrossRef]
  2. J. N. Kutz, P. Holmes, S. G. Evangelides, Jr., and J. P. Gordon, "Hamiltonian dynamics of dispersion-managed breathers," J. Opt. Soc. Am. B 15, 87-96 (1998).
    [CrossRef]
  3. V. Cautaerts, A. Maruta, and Y. Kodama, "On the dispersion managed soliton," Chaos 10, 515-528 (2000).
    [CrossRef]
  4. F. Kh. Abdullaev, B. B. Baizakov, and M. Salerno, "Stable two-dimensional dispersion-managed soliton," Phys. Rev. E 68, 066605 (2003).
    [CrossRef]
  5. M. Ablowitz and Z. Muslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
    [CrossRef] [PubMed]
  6. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  7. G. Nehmetallah and P. P. Banerjee, "Numerical modeling of (D+1)-dimensional solitons in a sign-alternating nonlinear medium with an adaptive fast Hankel split-step method," J. Opt. Soc. Am. B 22, 2200-2207 (2005).
    [CrossRef]
  8. L. Berge, V. K. Mezentsev, J. Juul Rasmussen, P. L. Christiansen, and Yu. B. Gaididei, "Self-guiding light in layered nonlinear media," Opt. Lett. 25, 1037-1039 (2000).
    [CrossRef]
  9. I. Towers and B. A. Malomed, "Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity," J. Opt. Soc. Am. B 19, 537-543 (2002).
    [CrossRef]
  10. S. K. Adhikari, "Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity," Phys. Rev. E 70, 036608 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams," Phys. Rev. Lett. 76, 4356-4359 (1996).
    [CrossRef] [PubMed]
  13. V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
    [CrossRef]
  14. F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
    [CrossRef]
  15. S. K. Adhikari, "Stabilization of a (3+1)-dimensional soliton in a Kerr medium by a rapidly oscillating dispersion coefficient," Phys. Rev. E 71, 016611 (2005).
    [CrossRef]
  16. M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, "Quasi-discrete Hankel transform," Opt. Lett. 23, 409-411 (1998).
    [CrossRef]
  21. M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
    [CrossRef]
  22. M. Feit and J. Fleck, "Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams," J. Opt. Soc. Am. B 5, 633-640 (1988).
    [CrossRef]
  23. V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, "Character of singularity and stochastic phenomena in self-focusing," Zh. Eksp. Teor. Fiz. Pis'ma Red. 14, 564-568 (1971).
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    [CrossRef]
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    [CrossRef] [PubMed]
  26. J. Talman, "Numerical Fourier and Bessel transforms in logarithmic variables," J. Comput. Phys. 29, 35-48 (1978).
    [CrossRef]

2005 (2)

S. K. Adhikari, "Stabilization of a (3+1)-dimensional soliton in a Kerr medium by a rapidly oscillating dispersion coefficient," Phys. Rev. E 71, 016611 (2005).
[CrossRef]

G. Nehmetallah and P. P. Banerjee, "Numerical modeling of (D+1)-dimensional solitons in a sign-alternating nonlinear medium with an adaptive fast Hankel split-step method," J. Opt. Soc. Am. B 22, 2200-2207 (2005).
[CrossRef]

2004 (3)

M. Guizar-Sicairos and J. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transform of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

S. K. Adhikari, "Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity," Phys. Rev. E 70, 036608 (2004).
[CrossRef]

2003 (3)

F. Kh. Abdullaev, B. B. Baizakov, and M. Salerno, "Stable two-dimensional dispersion-managed soliton," Phys. Rev. E 68, 066605 (2003).
[CrossRef]

G. Fibich and B. Ilan, "Discretization effects in the nonlinear Schrödinger equation," Appl. Numer. Math. 44, 63-75 (2003).
[CrossRef]

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

2002 (1)

2001 (1)

M. Ablowitz and Z. Muslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

2000 (2)

1998 (3)

1997 (1)

V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

1996 (1)

G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams," Phys. Rev. Lett. 76, 4356-4359 (1996).
[CrossRef] [PubMed]

1993 (1)

1991 (1)

1988 (1)

1981 (1)

M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
[CrossRef]

1978 (1)

J. Talman, "Numerical Fourier and Bessel transforms in logarithmic variables," J. Comput. Phys. 29, 35-48 (1978).
[CrossRef]

1977 (1)

1971 (1)

V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, "Character of singularity and stochastic phenomena in self-focusing," Zh. Eksp. Teor. Fiz. Pis'ma Red. 14, 564-568 (1971).

Abdullaev, F. K.

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, B. B. Baizakov, and M. Salerno, "Stable two-dimensional dispersion-managed soliton," Phys. Rev. E 68, 066605 (2003).
[CrossRef]

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

Ablowitz, M.

M. Ablowitz and Z. Muslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Adhikari, S. K.

S. K. Adhikari, "Stabilization of a (3+1)-dimensional soliton in a Kerr medium by a rapidly oscillating dispersion coefficient," Phys. Rev. E 71, 016611 (2005).
[CrossRef]

S. K. Adhikari, "Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity," Phys. Rev. E 70, 036608 (2004).
[CrossRef]

Agrawal, G. P.

M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Anderson, D.

Assanto, G.

Baizakov, B. B.

F. Kh. Abdullaev, B. B. Baizakov, and M. Salerno, "Stable two-dimensional dispersion-managed soliton," Phys. Rev. E 68, 066605 (2003).
[CrossRef]

Banerjee, P. P.

Batteh, J. H.

M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
[CrossRef]

Berezhiani, V.

V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Berge, L.

Caputo, J. G.

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

Cautaerts, V.

V. Cautaerts, A. Maruta, and Y. Kodama, "On the dispersion managed soliton," Chaos 10, 515-528 (2000).
[CrossRef]

Chen, M.

Chen, W.

Christiansen, P. L.

Desaix, M.

Evangelides, S. G.

Feit, M.

Fibich, G.

G. Fibich and B. Ilan, "Discretization effects in the nonlinear Schrödinger equation," Appl. Numer. Math. 44, 63-75 (2003).
[CrossRef]

G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams," Phys. Rev. Lett. 76, 4356-4359 (1996).
[CrossRef] [PubMed]

Fleck, J.

Gaididei, Yu. B.

Gordon, J. P.

Guizar-Sicairos, M.

Gutiérrez-Vega, J.

Holmes, P.

Huang, M.

Huang, W.

Ilan, B.

G. Fibich and B. Ilan, "Discretization effects in the nonlinear Schrödinger equation," Appl. Numer. Math. 44, 63-75 (2003).
[CrossRef]

Infeld, E.

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Kodama, Y.

V. Cautaerts, A. Maruta, and Y. Kodama, "On the dispersion managed soliton," Chaos 10, 515-528 (2000).
[CrossRef]

Kraenkel, R. A.

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Kutz, J. N.

Lax, M.

M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
[CrossRef]

Lisak, M.

Malomed, B. A.

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

I. Towers and B. A. Malomed, "Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity," J. Opt. Soc. Am. B 19, 537-543 (2002).
[CrossRef]

Maruta, A.

V. Cautaerts, A. Maruta, and Y. Kodama, "On the dispersion managed soliton," Chaos 10, 515-528 (2000).
[CrossRef]

Matuszewski, M.

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Mezentsev, V. K.

Miklaszewski, R.

V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Muslimani, Z.

M. Ablowitz and Z. Muslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Nehmetallah, G.

Rasmussen, J. Juul

Salerno, M.

F. Kh. Abdullaev, B. B. Baizakov, and M. Salerno, "Stable two-dimensional dispersion-managed soliton," Phys. Rev. E 68, 066605 (2003).
[CrossRef]

Sheik-Bahae, M.

Siegman, A. E.

Skarka, V.

V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Skorupski, A. A.

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Sneddon, I.

I. Sneddon, The Use Of Integral Transforms (McGraw-Hill, 1972).

Sobolev, V. V.

V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, "Character of singularity and stochastic phenomena in self-focusing," Zh. Eksp. Teor. Fiz. Pis'ma Red. 14, 564-568 (1971).

Stegeman, G. I.

Synakh, V. S.

V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, "Character of singularity and stochastic phenomena in self-focusing," Zh. Eksp. Teor. Fiz. Pis'ma Red. 14, 564-568 (1971).

Talman, J.

J. Talman, "Numerical Fourier and Bessel transforms in logarithmic variables," J. Comput. Phys. 29, 35-48 (1978).
[CrossRef]

Towers, I.

Trippenbach, M.

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Van Stryland, E. W.

Yu, L.

Zakharov, V. E.

V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, "Character of singularity and stochastic phenomena in self-focusing," Zh. Eksp. Teor. Fiz. Pis'ma Red. 14, 564-568 (1971).

Zhu, Z.

Appl. Numer. Math. (1)

G. Fibich and B. Ilan, "Discretization effects in the nonlinear Schrödinger equation," Appl. Numer. Math. 44, 63-75 (2003).
[CrossRef]

Chaos (1)

V. Cautaerts, A. Maruta, and Y. Kodama, "On the dispersion managed soliton," Chaos 10, 515-528 (2000).
[CrossRef]

J. Appl. Phys. (1)

M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
[CrossRef]

J. Comput. Phys. (1)

J. Talman, "Numerical Fourier and Bessel transforms in logarithmic variables," J. Comput. Phys. 29, 35-48 (1978).
[CrossRef]

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

J. Opt. Soc. Am. B (5)

Opt. Lett. (4)

Phys. Rev. A (1)

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Phys. Rev. E (6)

S. K. Adhikari, "Stabilization of a (3+1)-dimensional soliton in a Kerr medium by a rapidly oscillating dispersion coefficient," Phys. Rev. E 71, 016611 (2005).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and A. A. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

S. K. Adhikari, "Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity," Phys. Rev. E 70, 036608 (2004).
[CrossRef]

F. Kh. Abdullaev, B. B. Baizakov, and M. Salerno, "Stable two-dimensional dispersion-managed soliton," Phys. Rev. E 68, 066605 (2003).
[CrossRef]

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams," Phys. Rev. Lett. 76, 4356-4359 (1996).
[CrossRef] [PubMed]

M. Ablowitz and Z. Muslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. Pis'ma Red. (1)

V. E. Zakharov, V. V. Sobolev, and V. S. Synakh, "Character of singularity and stochastic phenomena in self-focusing," Zh. Eksp. Teor. Fiz. Pis'ma Red. 14, 564-568 (1971).

Other (2)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

I. Sneddon, The Use Of Integral Transforms (McGraw-Hill, 1972).

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Figures (9)

Fig. 1
Fig. 1

AFHSS algorithm, a symmetrized version of the split-step fast Fourier transform using the cylindrical or spherical Fourier–Bessel transform instead and using adaptive longitudinal stepping and transverse grid management.

Fig. 2
Fig. 2

(a)–(c) On-axis-amplitude stable 2D soliton evolution through the sign-alternating dispersion when D = 2 with the parameters f = [ 1 , 2 ] , Δ z = 0.001 , M = 4000 , A 0 = 1.5 , and w 0 = 1 , for different values of the dispersion period ranging from L = 0.004 to L = 0.2 , using AFHSS with S = 2 π R 1 R 2 = 2 π ( 200 ) (400 cylindrical samples).

Fig. 3
Fig. 3

Stable 2D soliton generation through the sign-alternating dispersion. (a) The pulse evolution with the parameters f = [ 1 , 2 ] , Δ z = 0.001 , M = 8000 , A 0 = 1.5 , w 0 = 1 , and L = 0.04 . (b) The on-axis amplitude for both the AFFSS and the AFHSS methods for the same parameters, using AFHSS with S = 2 π R 1 R 2 = 2 π ( 200 ) (400 cylindrical samples).

Fig. 4
Fig. 4

Stable 2D soliton generation through the sign-alternating dispersion. (a) The pulse evolution with the parameters f = [ 2 , 4 ] , Δ z = 0.001 , M = 8000 , A 0 = 1 , w 0 = 2 , and L = 0.04 . (b) The on-axis amplitude for the AFHSS methods for the same parameters, using AFHSS with S = 2 π R 1 R 2 = 2 π ( 200 ) (400 cylindrical samples).

Fig. 5
Fig. 5

(a)–(c) Comparison of the on-axis amplitude between the AFFSS and the AFHSS for the parameters shown in each inset, using AFHSS with S = 2 π R 1 R 2 = 2 π ( 200 ) (400 cylindrical samples) and N 2 = ( 2 8 ) 2 for the AFFSS 2D mesh.

Fig. 6
Fig. 6

Stable 3D soliton generation through the sign-alternating dispersion when f = [ 2 , 4 ] , Δ z = 0.005 , M = 4000 , A 0 = 0.2 , w 0 = 15 , and L = 4 , using AFHSS with S = 2 π R 1 R 2 = 2 π ( 800 ) (1600 radial samples). (a) Pulse evolution and (b) on-axis amplitude.

Fig. 7
Fig. 7

Stable 3D soliton generation through the sign-alternating dispersion when f = [ 2 , 4 ] , Δ z = 0.005 , M = 7200 , A 0 = 1 , w 0 = 5 , and L = 0.015 using AFHSS with S = 2 π R 1 R 2 = 2 π ( 800 ) (1600 radial samples). (a) Pulse evolution and (b) on-axis amplitude.

Fig. 8
Fig. 8

Decay of a 3D optical pulse when f = [ 2 , 4 ] , Δ z = 0.005 , M = 1200 , A 0 = 2 , w 0 = 5 , and L = 0.01 using AFHSS with S = 2 π R 1 R 2 = 2 π ( 800 ) (1600 radial samples). (a) Pulse evolution and (b) on-axis amplitude.

Fig. 9
Fig. 9

Collapse of a 3D optical pulse when f = [ 2 , 5 ] , Δ z = 0.005 , M = 1800 , A 0 = 1 , w 0 = 10 , and L = 0.015 using AFHSS with S = 2 π R 1 R 2 = 2 π ( 800 ) (1600 radial samples). (a) Pulse evolution and (b) on-axis amplitude.

Equations (72)

Equations on this page are rendered with MathJax. Learn more.

i u z + f ( z ) 2 1 r D 1 r ( r D 1 u r ) + g ( z ) u 2 u = 0 ,
L = i 2 ( u * u z u u * z ) r D 1 + f ( z ) 2 u r 2 r D 1 1 2 r D 1 g ( z ) u 4 ,
z L ( u z ) + r L ( u r ) L u = 0 .
u = u ( r , z ) = A ( z ) exp [ r 2 2 w 2 ( z ) + i b ( z ) r 2 ] ,
u ( r , z ) = A ( z ) sech [ r w ( z ) ] exp [ i b ( z ) r 2 ] ,
L = 0 L d r = i 2 ( A d A * d z A * d A d z ) w D α D 1 + A 2 w D + 2 ( d b d z + 2 f b 2 ) α D + 1 + f 2 A 2 w D 2 γ D 1 1 2 w D β D 1 g ( z ) A 4 ,
α D = 0 r D exp ( r 2 ) d r , β D = 0 r D exp ( 2 r 2 ) d r ,
γ D 1 = α D + 1
α D = 0 r D sech 2 ( r ) d r , β D = 0 r D sech 4 ( r ) d r ,
γ D = α D β D
A 2 w D = A 0 2 w 0 D = N 0 ,
b = 1 2 f w d w d z ,
d ϕ d z = [ γ D 1 α D 1 f ( z ) w 2 + ( 1 + D 4 ) g ( z ) β D 1 α D 1 A 2 ] ,
d 2 w d z 2 = γ D 1 α D + 1 f 2 ( z ) w 3 N 0 D β D 1 2 α D + 1 f ( z ) g ( z ) w D + 1 + 1 f ( z ) d w d z d f d z ,
f ( z ) = { f + 0 < z < L + f L + < z < L + + L } .
d 2 w d z 2 = { F + 4 w 3 0 < z < L + F 4 w 3 L + < z < L + + L } ,
( d v d z ) 2 = F ± + 8 T 0 ± v ,
v j r = [ v 0 r + ( L 2 v 0 r ) ( v 0 r 2 + F r ) ] 2 + F r ( v 0 r 2 + F r ) v 0 r ,
v j r = v 0 r + ( L 2 v 0 r ) ( v 0 r 2 + F r ) ,
v 0 r = v ¯ 0 r = [ v j r + ( 1 2 v j r ) ( v j r 2 + 1 ) ] 2 + 1 ( v j r 2 + 1 ) v j r ,
v 0 r = v ¯ 0 r = v j r + ( 1 2 v j r ) ( v j r 2 + 1 ) .
f ± = c 2 2 c 1 [ 1 ± ( 1 + F ± c 1 c 2 2 ) 1 2 ] ,
f ¯ = [ ( L + 1 ) + L ( 1 + c 1 F + c 2 2 ) 1 2 ( 1 + c 1 F c 2 2 ) 1 2 ] [ 2 c 1 c 2 ( L + 1 ) ] > 0 .
d b d z = c 1 f c 2 2 w 4 2 f b 2 ,
d w d z = 2 f b w .
d w ̃ d z = 2 f 0 w ̃ b ̃ + 2 w ̃ δ w f 1 sin Ω z + 2 b ̃ δ w f 1 sin Ω z + 2 f 0 δ w δ b ,
d δ w d z = 2 w ̃ b ̃ f 1 sin Ω z + 2 f 0 w ̃ δ b + 2 f 0 b ̃ δ w ,
d b ̃ d z = 1 w ̃ 4 [ c 1 f 0 c 2 2 ( 1 + 10 δ w 2 w ̃ 2 ) 2 c 1 f 1 δ w sin Ω z w ̃ ] 2 f 0 b ̃ 2 2 f 0 δ b 2 4 b ̃ f 1 δ b sin Ω z ,
d δ b d z = [ 2 ( c 1 f 0 c 2 ) w ̃ 5 ] δ w + ( c 1 2 w ̃ 4 2 b ̃ 2 ) f 1 sin Ω z 4 f 0 b ̃ δ b .
d w ̃ d z = 2 f 0 w ̃ b ̃ + 2 w ̃ δ w f 1 sin Ω z + 2 b ̃ δ w f 1 sin Ω z + 2 f 0 δ w δ b ,
d b ̃ d z = 1 w ̃ 4 [ c 1 f 0 c 2 2 ( 1 + 10 δ w 2 w ̃ 2 ) 2 c 1 f 1 δ w sin Ω z w ̃ ] 2 f 0 b ̃ 2 2 f 0 δ b 2 4 b ̃ f 1 δ b sin Ω z .
d w d z = 2 f 0 w b + 2 w δ w f 1 sin Ω z + 2 b δ w f 1 sin Ω z + 2 f 0 δ w δ b ,
d b d z = 1 w 4 [ c 1 f 0 c 2 2 ( 1 + 10 δ w 2 w 2 ) 2 c 1 f 1 δ w sin Ω z w ] 2 f 0 b 2 2 f 0 δ b 2 4 b f 1 δ b sin Ω z ,
d δ w d z = 2 w b f 1 sin Ω z + 2 f 0 w δ b + 2 f 0 b δ w ,
d δ b d z = [ 2 ( c 1 f 0 c 2 ) w 5 ] δ w + ( c 1 2 w 4 2 b 2 ) f 1 sin Ω z 4 f 0 b δ b .
[ 0 2 w f 0 Ω 2 b f 0 2 w f 0 0 2 b f 0 Ω Ω 4 b f 0 0 κ 4 b f 0 Ω κ 0 ] ( S 1 S 2 S 3 S 4 ) = ( 0 2 w b f 1 0 η ) .
S 1 = 2 κ b w f 1 ( 2 w f 0 κ + 8 b 2 f 0 2 + Ω 2 ) M D + 4 b f 0 ( w f 0 κ + Ω 2 + 4 b 2 f 0 2 ) η M D ,
S 2 = 4 Ω b 2 w f 1 f 0 κ Ω ( 2 w f 0 κ + Ω 2 + 4 b 2 f 0 2 ) η M D ,
S 3 = 4 b 2 f 0 w f 1 ( 4 w f 0 κ + 16 b 2 f 0 2 + Ω 2 ) M D 2 w f 0 ( 2 w f 0 κ + Ω 2 + 8 b 2 f 0 2 ) η M D ,
S 4 = 2 b w f 1 Ω ( 2 w f 0 κ + Ω 2 + 16 b 2 f 0 2 ) 4 b f 0 2 w Ω η M D ,
δ w = f 0 f 1 c 1 w 3 Ω 2 sin Ω z 2 w b f 1 Ω Ω 2 cos Ω z ,
δ b = 2 b f 1 ( c 1 f 0 c 2 ) w 4 Ω 2 sin Ω z c 1 f 1 2 w 4 Ω cos Ω z .
d w d z = 2 w b ( f 0 + f 1 2 Ω 2 σ w 4 ) ,
d b d z = σ 2 w 4 + 4 b 2 f 1 2 σ w 4 Ω 2 + c 1 2 f 1 2 f 0 w 8 Ω 2 ,
d 2 w d z 2 = f 0 σ w 3 + 2 c 1 2 f 1 2 f 0 2 w 7 Ω 2 + ( d w d z ) 2 ( 1 + 2 f 1 2 σ f 0 w 5 Ω 2 ) ,
d v d z = c 1 f w 3 c 2 w 4 ,
d w d z = 2 f v ,
d w d z = 2 f 0 v + 2 f 1 δ v sin Ω z + 2 f 1 δ v sin Ω z ,
d v d z = ( c 1 f 0 w c 2 w 4 ) + ( 6 c 1 f 0 w 10 c 2 w 6 ) δ w 2 3 c 1 f 1 w 4 δ w sin Ω z ,
d δ w d z = 2 v f 1 sin Ω z + 2 f 0 δ v ,
d δ v d z = ( 3 c 1 f 0 w + 4 c 2 w 5 ) δ w + c 1 f 1 w 3 sin Ω z .
[ Ω 0 0 κ 0 Ω κ 0 0 2 f 0 Ω 0 2 f 0 0 0 Ω ] ( S 1 S 2 S 3 S 4 ) = ( 0 η 0 2 f 1 v ) .
S 1 = 2 κ v f 1 Ω 2 + 2 f 0 κ ,
S 2 = η Ω Ω 2 + 2 f 0 κ ,
S 3 = 2 η f 0 Ω 2 + 2 f 0 κ ,
S 4 = 2 v f 1 Ω Ω 2 + 2 f 0 κ .
δ w = 2 c 1 f 1 f 0 w 3 [ Ω 2 + 2 f 0 ( 4 c 2 3 c 1 f 0 w ) w 5 ] sin Ω z 2 v Ω f 1 [ Ω 2 + 2 f 0 ( 4 c 2 3 c 1 f 0 w ) w 5 ] cos Ω z ,
δ v = 2 f 1 v ( 4 c 2 3 c 1 f 0 w ) w 5 [ Ω 2 + 2 f 0 ( 4 c 2 3 c 1 f 0 w ) w 5 ] sin Ω z c 1 f 1 Ω w 3 [ Ω 2 + 2 f 0 ( 4 c 2 3 c 1 f 0 w ) w 5 ] cos Ω z .
d w d z = 2 v ( f 0 f 1 2 Ω 2 σ w 5 ) ,
d v d z = c 1 f 0 w 3 c 2 w 4 + 3 c 1 2 f 1 2 f 0 w 7 Ω 2 + 12 c 1 f 1 2 f 0 w 5 Ω 2 v 2 20 f 1 2 c 2 w 6 Ω 2 v 2 .
H = ( 3 c 1 f 0 w 2 c 2 6 w 3 + c 1 2 f 1 2 f 0 6 Ω 2 w 6 ) + v 2 ( f 0 + f 1 2 σ Ω 2 w 5 ) ,
d 2 w d z 2 = 2 f 0 2 c 1 w 3 2 f 0 c 2 w 4 + 6 c 1 2 f 1 2 f 0 2 w 7 Ω 2 + ( d w d z ) 2 ( 6 c 1 f 1 2 w 5 Ω 2 10 f 1 2 c 2 f 0 w 6 Ω 2 ) ,
T eff = c 1 f 0 2 w 2 2 f 0 c 2 3 w 3 + c 1 2 f 1 2 f 0 2 w 6 Ω 2 .
H [ u ( r , z ) ] = u H ( ρ m , z ) = b a r u ( r , z ) [ J l ( r ρ m ) Y l ( b ρ m ) J l ( b ρ m ) Y l ( r ρ m ) ] d r ,
u ( r , z ) = π 2 2 m ρ m 2 J l 2 ( ρ m a ) u H ( ρ m , z ) J l 2 ( ρ m b ) J l 2 ( ρ m a ) [ J l ( ρ m r ) Y l ( b ρ m ) J l ( ρ m b ) Y l ( r ρ m ) ] .
U H ( m ) = n = 1 N C m n U ( n ) , U ( n ) = m = 1 N C n m U H ( m ) ,
C m n = 2 S J l ( κ l n κ l m S ) J l + 1 1 ( κ l n ) J l + 1 1 ( κ l m ) ,
U ( n ) = u ( κ l n 2 π R 2 , z ) J l + 1 1 ( κ l n ) R 1 ,
U H ( m ) = u H ( κ l m 2 π R 1 , z ) J l + 1 1 ( κ l m ) R 2 .
( 2 r 2 + D 1 r r ) u = 0 .
u ( r , z ) = 0 u H ( ρ , z ) j l 1 2 ( ρ r ) ρ l + 3 2 d ρ = π 2 r l 0 u H ( ρ , z ) J l ( ρ r ) d ρ ,
u H ( ρ , z ) = 2 π 0 u ( r , z ) j l 1 2 ( ρ r ) r l + 3 2 d r = 2 π ρ l 0 r l + 1 u ( r , z ) J l ( ρ r ) d r ,

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