Abstract

We prove that spatial Kerr solitons, usually obtained in the frame of a nonlinear Schrödinger equation valid in the paraxial approximation, can be found in a generalized form as exact solutions of Maxwell’s equations. In particular, they are shown to exist, both in the bright and dark version, as TM, linearly polarized, exactly integrable one-dimensional solitons and to reduce to the standard paraxial form in the limit of small intensities. In the two-dimensional case, they are shown to exist as azimuthally polarized, circularly symmetric dark solitons. Both one- and two-dimensional dark solitons exhibit a characteristic signature in that their asymptotic intensity cannot exceed a threshold value in correspondence of which their width reaches a minimum subwavelength value.

© 2005 Optical Society of America

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]
  2. S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).
    [CrossRef]
  3. Y. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  4. M. D. Feit and J. A. Fleck, Jr., "Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams," J. Opt. Soc. Am. B 5, 633-640 (1998).
    [CrossRef]
  5. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, "Does the nonlinear Schrödinger equation correctly describe beam propagation?," Opt. Lett. 15, 411-413 (1993).
    [CrossRef]
  6. G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams," Phys. Rev. Lett. 76, 4356-4359 (1996).
    [CrossRef] [PubMed]
  7. A. P. Sheppard and M. Haelterman, "Nonparaxiality stabilizes three-dimensional soliton beams in Kerr media," Opt. Lett. 23, 1820-1822 (1998).
    [CrossRef]
  8. S. Chi and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995).
    [CrossRef] [PubMed]
  9. G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
    [CrossRef]
  10. A. Ciattoni, C. Conti, E. DelRe, P. Di Porto, B. Crosignani, and A. Yariv, "Polarization and energy dynamics in ultrafocused optical Kerr propagation," Opt. Lett. 27, 734-736 (2002).
    [CrossRef]
  11. P. Kelly, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1508 (1965).
    [CrossRef]
  12. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
    [CrossRef] [PubMed]
  13. G. Fibich and A. L. Gaeta, "Critical power for self-focusing in bulk media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000).
    [CrossRef]
  14. B. Crosignani, A. Cutolo, and P. Di Porto, "Coupled-mode theory of nonlinear propagation in multimode and single-mode fibers: envelope solitons and self-confinement," J. Opt. Soc. Am. 72, 1136-1144 (1982).
    [CrossRef]
  15. Y. Chen and J. Atai, "Maxwell's equations and the vector nonlinear Schrödinger equation," Phys. Rev. E 55, 3652-3657 (1997).
    [CrossRef]
  16. B. Crosignani, A. Yariv, and S. Mookherjea, "Nonparaxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media," Opt. Lett. 29, 1254-1256 (2004).
    [CrossRef] [PubMed]
  17. A. Ciattoni, B. Crosignani, A. Yariv, and S. Mookherjea, "Nonparaxial dark solitons in optical Kerr media," Opt. Lett. 30, 516-518 (2005).
    [CrossRef] [PubMed]
  18. S. Blair, "Nonparaxial one-dimensional spatial solitons," Chaos 10, 570-583 (2000).
    [CrossRef]
  19. A. W. Snyder, D. J. Mitchell, and Y. Chen, "Spatial solitons of Maxwell's equations," Opt. Lett. 19, 524-526 (1994).
    [CrossRef] [PubMed]
  20. E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
    [CrossRef]
  21. E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
    [CrossRef]
  22. E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
    [CrossRef]
  23. B. V. Gisin and B. A. Malomed, "One- and two-dimensional subwavelength solitons in saturable media," J. Opt. Soc. Am. B 18, 1356-1361 (2001).
    [CrossRef]
  24. N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, "'Optical needle' in media with saturating self-focusing nonlinearities," J. Opt. B 3, S96-S98 (2001).
    [CrossRef]

2005 (1)

2004 (1)

2002 (1)

2001 (3)

B. V. Gisin and B. A. Malomed, "One- and two-dimensional subwavelength solitons in saturable media," J. Opt. Soc. Am. B 18, 1356-1361 (2001).
[CrossRef]

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
[CrossRef]

N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, "'Optical needle' in media with saturating self-focusing nonlinearities," J. Opt. B 3, S96-S98 (2001).
[CrossRef]

2000 (3)

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

S. Blair, "Nonparaxial one-dimensional spatial solitons," Chaos 10, 570-583 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

1999 (1)

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

1998 (2)

1997 (2)

Y. Chen and J. Atai, "Maxwell's equations and the vector nonlinear Schrödinger equation," Phys. Rev. E 55, 3652-3657 (1997).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

1994 (1)

1993 (1)

1990 (1)

1982 (1)

1965 (1)

P. Kelly, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1508 (1965).
[CrossRef]

1964 (1)

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

Agrawal, G. P.

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

Akhmediev, N.

Ankiewicz, A.

Atai, J.

Y. Chen and J. Atai, "Maxwell's equations and the vector nonlinear Schrödinger equation," Phys. Rev. E 55, 3652-3657 (1997).
[CrossRef]

Blair, S.

S. Blair, "Nonparaxial one-dimensional spatial solitons," Chaos 10, 570-583 (2000).
[CrossRef]

Chen, Y.

Y. Chen and J. Atai, "Maxwell's equations and the vector nonlinear Schrödinger equation," Phys. Rev. E 55, 3652-3657 (1997).
[CrossRef]

A. W. Snyder, D. J. Mitchell, and Y. Chen, "Spatial solitons of Maxwell's equations," Opt. Lett. 19, 524-526 (1994).
[CrossRef] [PubMed]

Chi, S.

Chiao, R. Y.

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

Ciattoni, A.

Conti, C.

Crosignani, B.

Cutolo, A.

DelRe, E.

Di Porto, P.

Feit, M. D.

Fibich, G.

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
[CrossRef]

G. Fibich and A. L. Gaeta, "Critical power for self-focusing in bulk 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. A.

Gaeta, A. L.

Garmire, E.

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

Gisin, B. V.

Granot, E.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

Guo, Q.

Haelterman, M.

Ilan, B.

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
[CrossRef]

Isbi, Y.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

Kelly, P.

P. Kelly, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1508 (1965).
[CrossRef]

Kivshar, Y.

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

Lewis, A.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

Malomed, B.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

Malomed, B. A.

Mitchell, D. J.

Mookherjea, S.

Rosanov, N. N.

N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, "'Optical needle' in media with saturating self-focusing nonlinearities," J. Opt. B 3, S96-S98 (2001).
[CrossRef]

Semenov, V. E.

N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, "'Optical needle' in media with saturating self-focusing nonlinearities," J. Opt. B 3, S96-S98 (2001).
[CrossRef]

Sheppard, A. P.

Silberberg, Y.

Snyder, A. W.

Soto-Crespo, J. M.

Sternklar, S.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

Torruellas, W.

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).
[CrossRef]

Townes, C. H.

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

Trillo, S.

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).
[CrossRef]

Vyssotina, N. V.

N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, "'Optical needle' in media with saturating self-focusing nonlinearities," J. Opt. B 3, S96-S98 (2001).
[CrossRef]

Yariv, A.

Chaos (1)

S. Blair, "Nonparaxial one-dimensional spatial solitons," Chaos 10, 570-583 (2000).
[CrossRef]

J. Opt. B (1)

N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, "'Optical needle' in media with saturating self-focusing nonlinearities," J. Opt. B 3, S96-S98 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Sub-wavelength non-local spatial solitons," Opt. Commun. 166, 121-126 (1999).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "On the existence of subwavelength spatial solitons," Opt. Commun. 178, 431-435 (2000).
[CrossRef]

Opt. Lett. (10)

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, "Does the nonlinear Schrödinger equation correctly describe beam propagation?," Opt. Lett. 15, 411-413 (1993).
[CrossRef]

Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
[CrossRef] [PubMed]

A. W. Snyder, D. J. Mitchell, and Y. Chen, "Spatial solitons of Maxwell's equations," Opt. Lett. 19, 524-526 (1994).
[CrossRef] [PubMed]

S. Chi and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995).
[CrossRef] [PubMed]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, "Subwavelength spatial solitons," Opt. Lett. 22, 1290-1292 (1997).
[CrossRef]

A. P. Sheppard and M. Haelterman, "Nonparaxiality stabilizes three-dimensional soliton beams in Kerr media," Opt. Lett. 23, 1820-1822 (1998).
[CrossRef]

A. Ciattoni, C. Conti, E. DelRe, P. Di Porto, B. Crosignani, and A. Yariv, "Polarization and energy dynamics in ultrafocused optical Kerr propagation," Opt. Lett. 27, 734-736 (2002).
[CrossRef]

B. Crosignani, A. Yariv, and S. Mookherjea, "Nonparaxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media," Opt. Lett. 29, 1254-1256 (2004).
[CrossRef] [PubMed]

A. Ciattoni, B. Crosignani, A. Yariv, and S. Mookherjea, "Nonparaxial dark solitons in optical Kerr media," Opt. Lett. 30, 516-518 (2005).
[CrossRef] [PubMed]

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

Phys. Rev. E (1)

Y. Chen and J. Atai, "Maxwell's equations and the vector nonlinear Schrödinger equation," Phys. Rev. E 55, 3652-3657 (1997).
[CrossRef]

Phys. Rev. Lett. (3)

P. Kelly, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1508 (1965).
[CrossRef]

R. Y. Chiao, E. Garmire, and C. H. 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]

Physica D (1)

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
[CrossRef]

Other (2)

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).
[CrossRef]

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

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Figures (12)

Fig. 1
Fig. 1

Plot of phase portrait of Eqs. (6) associated to bright solitons for u x 0 = 1 , 2 , 3 , 4 . Each bow-tie-shaped curve is obtained by plotting the level set defined in Eq. (14) with β given by Eq. (16). Any piece of curve starting from and ending at the origin (left or right loop of each bow tie) is associated with a single bright soliton.

Fig. 2
Fig. 2

Plot of (a) the transverse component u x ( ξ ) and (b) longitudinal component u z ( ξ ) of bright solitons for u x 0 = 1 , 2 , 3 , 4 (same cases as in Fig. 1) and β > 0 .

Fig. 3
Fig. 3

Bright soliton existence curve (solid curve), relating the FWHM, Δ bright , of the amplitude u x ( ξ ) to u x 0 . For very small and very large u x 0 , the FWHM diverges and vanishes, respectively. The dashed curve represents the FWHM, Δ ̃ bright , of paraxial bright solitons. Note the complete overlapping of the two curves for u x 0 < 0.2 .

Fig. 4
Fig. 4

Plot of phase portrait of Eqs. (6) associated with dark solitons for u x = 0.1 , 0.2 , 0.3 , 0.4 . Each loop is obtained by plotting the level set defined in Eq. (10) with β and F 0 given in Eqs. (19). Any piece of curve joining the points ( u x , 0 ) and ( u x , 0 ) is associated with a single dark soliton.

Fig. 5
Fig. 5

Plot of (a) the transverse component u x ( ξ ) and (b) longitudinal component u z ( ξ ) of dark solitons for u x = 0.1 , 0.2 , 0.3 , 0.4 (same cases as in Fig. 4) and β > 0 .

Fig. 6
Fig. 6

Dark soliton existence curve (solid curve), relating the FWHM, Δ dark of the amplitude u x ( ξ ) to u x . For very small u x the FWHM diverges, whereas at the threshold value u x = 1 6 it attains its minimum value 3 . The dashed curve represents the FWHM, Δ ̃ dark , of paraxial dark solitons. Note the complete overlap for most of the values of u x .

Fig. 7
Fig. 7

Normalized optical intensity S ( ξ ) I 0 of (a) bright and (b) dark solitons evaluated from Eq. (21) for the same soliton conditions as in Fig. 1 (for bright solitons) and Fig. 5 (for dark solitons).

Fig. 8
Fig. 8

Values of f x = ( n 2 n 0 ) 1 2 V x and Q x (black region), where no modulational instability is present in defocusing ( γ = 1 ) media.

Fig. 9
Fig. 9

Two-dimensional dark soliton profile u ( ρ ) for various values of u .

Fig. 10
Fig. 10

Normalized asymptotic optical intensity I I 0 as a function of the asymptotic dimensionless field amplitude u . Note that two solitons exist for any allowed asymptotic optical intensity.

Fig. 11
Fig. 11

Existence curve relating the normalized soliton optical intensity HWHM to u .

Fig. 12
Fig. 12

Plot of the ratio R ( ρ ) = u tanh ( u ρ 2 ) u ( ρ ) for different values of u .

Equations (64)

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× E = i ω B ,
× B = i ω c 2 n 0 2 E i ω μ 0 P nl ,
P nl = 4 3 ϵ 0 n 0 n 2 [ E 2 E + 1 2 ( E E ) E * ] ,
× × E = k 2 E + k 2 4 3 n 2 n 0 [ E 2 E + 1 2 ( E E ) E * ] ,
E ( x , y , z ) = exp ( i α z ) [ U x ( x ) e ̂ x + i U z ( x ) e ̂ z ] ,
α d U z d x = [ ( α 2 k 2 ) 2 k 2 n 2 n 0 ( U x 2 + 1 3 U z 2 ) ] U x ,
d 2 U z d x 2 α d U x d x = k 2 [ 1 + 2 n 2 n 0 ( 1 3 U x 2 + U z 2 ) ] U z ,
d u x d ξ = [ β 2 ( 1 2 3 γ u x 2 + 2 γ u z 2 ) + 4 3 ( γ + 2 u x 2 + 2 3 u z 2 ) u x 2 ] β [ 1 + γ ( 6 u x 2 + 2 3 u z 2 ) ] u z Q x ( u x , u z β ) ,
d u z d ξ = 1 β [ ( β 2 1 ) 2 γ ( u x 2 + 1 3 u z 2 ) ] u x Q z ( u x , u z β ) ,
1 + γ ( 6 u x 2 + 2 3 u z 2 ) 0
0 = d F d ξ F u x d u x d ξ + F u z d u z d ξ = F u x Q x + F u z Q z .
F ( u x , u z β ) = 2 u x 6 + 4 3 u x 4 u z 2 + 2 9 u x 2 u z 4 1 2 γ ( 3 β 2 4 ) u x 4 + 1 3 γ ( 2 β 2 ) u x 2 u z 2 + 1 2 γ β 2 u z 4 1 2 ( β 2 1 ) u x 2 + 1 2 β 2 u z 2
F ( u x , u z β ) = F 0 .
( u x ( 0 ) u z ( 0 ) ) = ( u x 0 u z 0 ) u 0 , ( u x ( + ) u z ( + ) ) = ( u x u z ) u .
F ( u x 0 , u z 0 β ) = F 0 ,
F ( u x , u z β ) = F 0 .
Q x ( u x , u z β ) = 0 ,
Q z ( u x , u z β ) = 0 .
F ( u x 0 , u z 0 β ) = 0 .
β 2 = ( 1 + 2 γ u x 0 2 ) 2 1 + 3 γ u x 0 2 .
β = ± 1 + 2 u x 0 2 ( 1 + 3 u x 0 2 ) 1 2 ,
β 2 = 1 + 2 γ u x 2 .
F 0 = γ 2 ( 1 + 2 γ u x 2 ) u x 4 ,
u x 4 = γ u z 0 2 u z 0 4 .
β = ± ( 1 2 u x 2 ) 1 2 ,
u z 0 = ± { 1 2 [ 1 ( 1 4 u x 2 ) 1 2 ] } 1 2 ,
F 0 = 1 2 ( 1 2 u x 2 ) u x 4 ,
B ( x , y , z ) = k ω ( n 0 n 2 ) 1 2 exp ( i β k z ) ( β u x d u z d ξ ) ξ = k x e ̂ y .
S = I 0 β [ 1 + 2 γ ( u x 2 + 1 3 u z 2 ) ] u x 2 e ̂ z β β I e ̂ z ,
I bright = I 0 ( 1 + 3 u x 0 2 ) 1 2 u x 0 2 ,
I dark = I 0 ( 1 2 u x 2 ) 1 2 u x 2 .
u x 1 ,
u z u x .
d 2 u x d ξ 2 = ( β 2 1 ) u x 2 γ u x 3 .
β ̃ u x + 1 2 d 2 u x d ξ 2 = γ u x 3 ,
E x ( x , z ) = V x exp [ i k Γ ( V z V x x + z ) ] ,
E z ( x , z ) = V z exp [ i k Γ ( V z V x x + z ) ] ,
Γ = [ 1 + γ 2 n 2 n 0 ( V x 2 + V z 2 ) 1 + V z 2 V x 2 ] 1 2 .
E x ( x , z ) = [ V x + v x ( x , z ) ] exp [ i k Γ ( V z V x x + z ) ] ,
E z ( x , z ) = [ V z + v z ( x , z ) ] exp [ i k Γ ( V z V x x + z ) ] ,
3 Γ 2 ( 2 3 Γ 2 ) Q x 4 + Γ 2 ( 2 + Γ 2 ) Q z 4 + 2 ( 3 Γ 4 + 2 Γ 2 2 ) Q x 2 Q z 2 + 2 Γ 2 ( 1 Γ 2 ) ( 2 + Γ 2 ) Q x 2 2 Γ 2 ( 3 Γ 4 + 5 Γ 2 2 ) Q z 2 = 0
E ( r , φ , z ) = E φ ( r , z ) e ̂ φ + E z ( r , z ) e ̂ z ,
2 E z r z = 0 ,
2 E φ z 2 + z ( E φ r + E φ r ) = k 2 E φ k 2 4 3 n 2 n 0 [ E 2 E φ + 1 2 ( E E ) E φ * ] ,
2 E z r 2 + 1 r E z r = k 2 E z k 2 4 3 n 2 n 0 [ E 2 E z + 1 2 ( E E ) E z * ] .
2 E φ z 2 + r ( E φ r + E φ r ) = k 2 E φ 2 k 2 n 2 n 0 E φ 2 E φ .
2 U ζ 2 + 2 ρ ( U ρ + U ρ ) = U 2 γ U 2 U ,
U ( ρ , ζ ) = exp ( i α ζ ) u ( ρ ) ,
d d ρ ( d u d ρ + u ρ ) = 1 2 ( α 2 1 ) u γ u 3 .
lim ρ u ( ρ ) = u ,
d d ρ ( d u d ρ + u ρ ) = 1 2 ( α 2 1 ) u + u 3 ,
α = ± ( 1 2 u 2 ) 1 2 .
u < 1 2 ,
d d ρ ( d u d ρ + u ρ ) = ( u 2 u 2 ) u .
E = ( n 0 n 2 ) 1 2 exp ( i α k z ) u ( 2 k r ) e ̂ φ ,
B = ( n 0 n 2 ) 1 2 exp ( i α k z ) k ω [ α u e ̂ r + i 2 ( d u d ρ + u ρ ) e ̂ z ] ρ = 2 k r .
S = 1 2 μ 0 Re ( E × B * )
S ( r ) = α k 2 ω μ 0 n 0 n 2 u 2 ( 2 k r ) e ̂ z = α k 2 ω μ 0 E 2 e ̂ z .
I ( u ) = I 0 u 2 ( 1 2 u 2 ) 1 2 ,
d 2 u d ρ 2 = ( u 2 u 2 ) u ,
ρ 2 { ρ 4 cos 2 ϕ [ 2 cos 2 ϕ + 4 3 cos 2 ϕ sin 2 ϕ + 2 9 sin 4 ϕ ] + γ ρ 2 [ 1 2 ( 4 3 β 2 ) cos 4 ϕ + 1 3 ( 2 β 2 ) cos 2 ϕ sin 2 ϕ + 1 2 β 2 sin 4 ϕ ] + 1 2 [ ( 1 β 2 ) cos 2 ϕ + β 2 sin 2 ϕ ] } = 0 .
tan 2 ϕ 0 = 1 1 β 2 u 0 x 2 4 u 0 x 2 + γ ( 1 + 2 γ u x 0 2 ) 2 ,
u z 2 = 8 3 u x 4 2 3 ( 1 + 2 u x 2 ) u x 2 + ( 1 2 u x 2 ) 8 9 u x 2 2 ( 1 u x 2 ) + { [ 8 3 u x 4 2 3 ( 1 + 2 u x 2 ) u x 2 + ( 1 2 u x 2 ) ] 2 [ 16 9 u x 2 4 ( 1 2 u x 2 ) ] [ 4 u x 6 ( 1 + 6 u x 2 ) u x 4 + 2 u x 2 u x 2 ( 1 2 u x 2 ) u x 4 ] } 1 2 [ 8 9 u x 2 2 ( 1 2 u x 2 ) ] ,
4 u x 6 ( 1 + 6 u x 2 ) u x 4 + 2 u x 2 u x 2 ( 1 2 u x 2 ) u x 4 < 0 .

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