Abstract

We present a novel technique to numerically solve transverse and pulsed optical beam or light bullet propagation in a layered alternating self-focusing and self-defocusing medium based on the scalar nonlinear Schrödinger equation in two and three dimensions with cylindrical and spherical symmetry, respectively. Using fast algorithms for Hankel transform along with adaptive longitudinal stepping and transverse grid management in a symmetrized split-step technique, it is possible to accurately study many nonlinear effects, including the possibility of spatiotemporal collapse, or the collapse-arresting mechanism due to a sign-alternating nonlinearity coefficient. Also, by using the variational approximation technique, we can prove that stable (D+1)-dimensional soliton beams and optical bullets exist in these media.

© 2005 Optical Society of America

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  1. R. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]
  2. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
    [CrossRef] [PubMed]
  3. O. Budneva, V. Zakharov, and V. Synakh, "Certain models of wave collapse," Sov. J. Plasma Phys. 1, 335-338 (1975).
  4. V. E. Zakharov and D. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).
  5. A. Haseqawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 171-172 (1973).
    [CrossRef]
  6. G. Fibich and A. Gaeta, "Critical power for self-focusing in bulk optical media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000).
    [CrossRef]
  7. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  8. 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]
  9. G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams," Phys. Rev. Lett. 76, 4356-4359 (1996).
    [CrossRef] [PubMed]
  10. V. Skarka, V. Berezhiani, and R. Miklaszewski, "Spatiotemporal soliton propagation in saturating nonlinear optical media," Phys. Rev. E 56, 1080-1087 (1997).
    [CrossRef]
  11. 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]
  12. 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]
  13. G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13-15 (1993).
    [CrossRef] [PubMed]
  14. 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).
  15. M. Desaix, D. Anderson, and M. Lisak, "Variational approach to collapse of optical pulses," J. Opt. Soc. Am. B 8, 2082-2086 (1991).
    [CrossRef]
  16. 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]
  17. I. Sneddon, The Use Of Integral Transforms (McGraw-Hill, 1972).
  18. 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]
  19. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, "Quasi-discrete Hankel transform" Opt. Lett. 23, 409-411 (1998).
    [CrossRef]
  20. M. Lax, J. H. Batteh, and G. P. Agrawal, "Channeling of intense electromagnetic beams," J. Appl. Phys. 52, 109-125 (1981).
    [CrossRef]
  21. G. Fibich and B. Ilan, "Discretization effects in the nonlinear Schrödinger equation," Appl. Numer. Math. 44, 63-75 (2003).
    [CrossRef]
  22. A. E. Siegman, "Quasi-fast Hankel transform," Opt. Lett. 1, 13-15 (1977).
    [CrossRef]
  23. J. Talman, "Numerical Fourier and Bessel transform in logarithmic variables," J. Comput. Phys. 29, 35-48 (1978).
    [CrossRef]

2004

2003

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]

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

2002

2000

1998

1997

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

1996

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

1993

1991

1990

1988

1981

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

1978

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

1977

1975

O. Budneva, V. Zakharov, and V. Synakh, "Certain models of wave collapse," Sov. J. Plasma Phys. 1, 335-338 (1975).

1973

A. Haseqawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 171-172 (1973).
[CrossRef]

1972

V. E. Zakharov and D. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

1971

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).

1964

R. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

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]

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.

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.

Budneva, O.

O. Budneva, V. Zakharov, and V. Synakh, "Certain models of wave collapse," Sov. J. Plasma Phys. 1, 335-338 (1975).

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]

Chen, M.

Chen, W.

Chiao, R.

R. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Christiansen, P. L.

Desaix, M.

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 and A. Gaeta, "Critical power for self-focusing in bulk optical media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000).
[CrossRef]

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

Fleck, J.

Gaeta, A.

Gaididei, Yu. B.

Garmire, E.

R. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Guizar-Sicairos, M.

Gutiérrez-Vega, J.

Haseqawa, A.

A. Haseqawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 171-172 (1973).
[CrossRef]

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]

Kivshar, Y. S.

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

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]

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.

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]

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]

Rasmussen, J. Juul

Shabat, D. B.

V. E. Zakharov and D. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Sheik-Bahae, M.

Siegman, A. E.

Silberberg, Y.

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]

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.

O. Budneva, V. Zakharov, and V. Synakh, "Certain models of wave collapse," Sov. J. Plasma Phys. 1, 335-338 (1975).

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 transform in logarithmic variables," J. Comput. Phys. 29, 35-48 (1978).
[CrossRef]

Tappert, F.

A. Haseqawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 171-172 (1973).
[CrossRef]

Towers, I.

Townes, C.

R. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Van Stryland, E. W.

Yu, L.

Zakharov, V.

O. Budneva, V. Zakharov, and V. Synakh, "Certain models of wave collapse," Sov. J. Plasma Phys. 1, 335-338 (1975).

Zakharov, V. E.

V. E. Zakharov and D. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

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.

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

Appl. Phys. Lett.

A. Haseqawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 171-172 (1973).
[CrossRef]

J. Appl. Phys.

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.

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

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. 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]

Phys. Rev. E

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.

R. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

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

Sov. J. Plasma Phys.

O. Budneva, V. Zakharov, and V. Synakh, "Certain models of wave collapse," Sov. J. Plasma Phys. 1, 335-338 (1975).

Sov. Phys. JETP

V. E. Zakharov and D. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Zh. Eksp. Teor. Fiz. Pis/ma Red.

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

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

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

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Figures (9)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Stable 2D soliton generation through the sign-alternating nonlinearity when Δ z = 0.01 , A 0 = 0.2 , w 0 = 10 , γ 0 = 0.5 , γ 1 = 1.5 , and L + = L = 0.1 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 cylindrical samples).

Fig. 3
Fig. 3

Stable 2D soliton generation through the sign-alternating nonlinearity when Δ z = 0.01 , A 0 = 2.1 , w 0 = 1 , γ 0 = 1 , γ 1 = 2 , and L + = L = 0.01 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 cylindrical samples).

Fig. 4
Fig. 4

Decay of 2D beams when Δ z = 0.01 , A 0 = 1 , w 0 = 1 , γ 0 = 1.7 , γ 1 = 2.7 , and L + = L = 0.4 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 cylindrical samples).

Fig. 5
Fig. 5

SF of a 2D optical pulse when Δ z = 0.01 , A 0 = 3 , w 0 = 1 , γ 0 = 1.5 , γ 1 = 0.5 , and L + = L = 0.1 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 radial samples).

Fig. 6
Fig. 6

Variation of the main beam ( D = 2 ) period Λ z with respect to the nonlinearity map period L = [ 0.02 , 0.04 , 0.08 , 0.16 , 0.32 ] for different initial widths ranging from w 0 = [ 1 , 1.5 , 2 , 2.5 ] keeping the same initial beam power I 0 . Superposed on the data are empirical fits of the form Λ z = c 1 + c 2 L η obtained through the fitting of the first, third, and fifth points for each w 0 . FHT, fast Hankel transform; FFT, fast-Fourier transform.

Fig. 7
Fig. 7

Stable 3D soliton generation through the sign-alternating nonlinearity when Δ z = 0.01 , A 0 = 0.18 , w 0 = 10 , γ 0 = 1 , γ 1 = 4 , and L + = L = 0.2 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 radial samples).

Fig. 8
Fig. 8

SF of a 3D optical pulse when Δ z = 0.01 , A 0 = 0.5 , w 0 = 5 , γ 0 = 1 , γ 1 = 1 , and L + = L = 0.2 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 radial samples).

Fig. 9
Fig. 9

Decay of a 3D optical pulse when Δ z = 0.01 , A 0 = 1 , w 0 = 10 , γ 0 = 1 , γ 1 = 4 , and L + = L = 0.2 by use of AFHSS with S = 2 π R 1 R 2 = 2 π × 800 (1600 radial samples).

Equations (46)

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

i E Z + 1 2 k 0 ( 2 E ) k 2 2 2 E t 2 + γ ( Z ) E 2 E = 0 ,
i u z + 1 2 ( 2 u s d 2 u τ 2 ) + γ ( z ) u 2 u = 0 ,
i u z + 1 2 1 r D 1 r ( r D 1 u r ) + γ ( z ) u 2 u = 0 ,
N = u 2 d D r ,
H = [ u 2 F ( u 2 ) ] d D r ,
M = i ( u * u u u * ) d D r ,
d 2 σ 2 ( z ) d z 2 = 1 N ( 8 H + 4 R 4 M 2 N ) ,
u ( r , z ) 1 ( z c z ) s Q [ r ( z c z ) s ] ,
L = i 2 ( u * u z u u * z ) r D 1 + 1 2 u r 2 r D 1 1 2 r D 1 γ ( 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 ) sec h [ 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 b 2 ) α D + 1 + 1 2 A 2 w D 2 σ D 1 1 2 w D β D 1 γ ( z ) A 4 ,
α D = 0 r D exp ( r 2 ) d r , β D = 0 r D exp ( 2 r 2 ) d r , σ D 1 = α D + 1
α D = 0 r D sec h 2 ( r ) d r , β D = 0 r D sec h 4 ( r ) d r , σ D = α D β D
A 2 w D = A 0 2 W 0 D = I 0 ,
b = 1 2 w d w d z ,
d ϕ d z = [ σ D 1 α D 1 1 w 2 + ( 1 + d 4 ) β D 1 α D 1 A 2 ] ,
d 2 w d z 2 = σ D 1 α D + 1 1 w 3 I 0 D β D 1 2 α D + 1 γ ( z ) w D + 1 ,
d 2 w d z 2 = σ 1 α 3 1 w 3 I 0 β 1 α 3 γ ( Z ) w 3 = Γ ( z ) 4 w 3 ,
d 2 w d z 2 = ε 0 + ε sin ω z w 3 ,
d 2 δ w d z 2 = w ¯ ε sin ω z + 3 w ¯ 4 ε 0 δ w ,
d 2 w ¯ d z 2 = w ¯ 3 ε 0 6 w ¯ 5 ε 0 δ w 2 3 w ¯ 4 ε δ w sin ω z ,
δ w ( z ) = ε sin ω z w ¯ 3 ( ω 2 + 3 w ¯ 4 ε 0 ) ,
d 2 w ¯ d z 2 = w ¯ 3 [ ε 0 3 ε 0 ε 2 ( ω 2 w ¯ 4 + 3 ε 0 ) 2 + 3 2 ε 2 ( ω 2 w ¯ 4 + 3 ε 0 ) ] .
w ¯ 4 = 1 ω 2 [ 3 4 ε 2 ε 0 2 ( 1 + 1 16 3 ε 2 ε 0 2 ) 3 ε 0 ] .
d 2 w d z 2 = T w = σ 2 α 4 1 w 3 3 I 0 β 2 2 α 4 γ ( Z ) w 4 .
d 2 w d z 2 = η w 3 + ε 0 + ε sin ω z w 4 ,
d 2 δ w d z 2 = ( 4 ε 0 w ¯ 1 3 η ) w ¯ 4 δ ω + w ¯ 4 ε sin ω z ,
d 2 w ¯ d z 2 = η w ¯ 3 ε 0 w ¯ 4 + ( 6 η w ¯ 5 10 ε 0 w ¯ ) δ w 2 ε 4 w ¯ 5 δ w sin ω z .
δ w ( z ) = w ¯ ε sin ω z w ¯ 5 ω 2 3 η w ¯ + 4 ε 0 .
d 2 w ¯ d z 2 = η w ¯ 3 ε 0 w ¯ 4 + 2 ε 2 w ¯ 4 w ¯ 5 ω 2 3 η w ¯ + 4 ε 0 + w ¯ 4 ε 2 ( 3 η w ¯ 5 ε 0 ) ( w ¯ 5 ω 2 3 η w ¯ + 4 ε 0 ) 2 .
η ω 4 w ¯ 11 ω 4 ε 0 w ¯ 10 + 6 η 2 ω 2 w ¯ 7 14 η ω 2 ε 0 w ¯ 6 + 2 ω 2 w ¯ 5 ( 4 ε 0 2 + ε 2 ) + 9 η 3 w ¯ 3 33 η ε 0 w ¯ 2 + η ( 40 ε 0 2 3 ε 2 ) w ¯ + 3 ε 0 ε 2 16 ε 0 3 = 0 .
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 ( r , z ) = m c l m J l [ κ l m ( r a ) ] , 0 r a ,
c l m = [ 1 a 2 J l + 1 2 ( κ l m ) ] 0 a r u ( r , z ) J l [ κ l m ( r a ) ] d r
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 ln κ l m S ) J 1 1 ( κ ln ) J 1 1 ( κ l m ) ,
U ( n ) = u ( κ ln 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 .
( 2 r 2 + 1 r r l 2 r 2 ) v = 0 AFHSS 4 π 2 ρ 2 v H ( ρ , z ) ,
v H ( ρ , z ) = H l [ v ( r , z ) ] = H l r l u ( r , z ) u ( r , z ) = r l H l [ v H ( ρ , z ) ] ,
u ( r , z ) = 0 u H ( ρ , z ) j l 1 2 ( ρ r ) ρ l + 3 2 d ρ = π 2 r 1 0 ρ l + 1 μ 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 ,
j l ( x ) = ( x ) l ( 1 x d d x ) sin ( x ) x = π 2 x ) J l + 1 2 ( x )

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