Abstract

The properties of the polariton modes associated with a resonance in the dielectric function of an optical fiber are derived. Particular attention is given to the surface polaritons, whose energy densities and power flows are concentrated close to the interface of the fiber core and its cladding. The results are illustrated with a range of dispersion curves for the two most important cases, when the fiber dielectric function has the reststrahl and plasma forms. General expressions are given and illustrated graphically for the power flow and the power density. It is shown numerically that the energy velocity, or group velocity, can have opposite sign to the phase velocity of the surface polariton.

© 1991 Optical Society of America

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References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  2. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1971).
  3. J. S. Nkoma, R. Loudon, D. R. Tilley, “Elementary properties of surface polaritons,”J. Phys. C 7, 3547–3559 (1974).
    [CrossRef]
  4. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).
  5. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).
  6. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  7. R. Olshansky, D. B. Keck, “Pulse broadening in graded-index optical fibers,” Appl. Opt. 15, 483–491 (1976).
    [CrossRef] [PubMed]
  8. W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single mode fibers,” Microwaves Opt. Acoust. 3, 239–246 (1979).
    [CrossRef]
  9. D. Gloge, E. A. J. Marcatili, D. Marcuse, S. D. Personick, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979).
  10. R. Englman, R. Ruppin, “Optical lattice vibrations in finite ionic crystals,”J. Phys. C 2, 1515–1531 (1968).
    [CrossRef]
  11. C. A. Pfeiffer, E. N. Economou, K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
    [CrossRef]
  12. G. C. Aers, A. D. Boardman, B. V. Paranjape, “Non-radiative surface plasmon-polariton modes in inhomogeneous metal circular cylinder,”J. Phys. F 10, 53–62 (1980).
    [CrossRef]
  13. R. Ruppin, Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982), Chap. 9.
  14. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  15. S. Ushioda, R. Loudon, in Surface Polaritons, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 535–586.
  16. M. G. Cottam, D. R. Tilley, Introduction to Surface and Superlattice Excitations (Cambridge U. Press, Cambridge, 1989).
    [CrossRef]
  17. H. A. Haus, H. Kogelnik, “Electromagnetic momentum and momentum flow in dielectric waveguides,”J. Opt. Soc. Am. 66, 320–327 (1976).
    [CrossRef]
  18. M. Fukui, V. C. Y. So, R. Normandin, “Lifetimes of surface plasmons in thin silver films,” Phys. Status Solidi B 91, K61–K64 (1979).
    [CrossRef]
  19. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
    [CrossRef]

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

1980 (1)

G. C. Aers, A. D. Boardman, B. V. Paranjape, “Non-radiative surface plasmon-polariton modes in inhomogeneous metal circular cylinder,”J. Phys. F 10, 53–62 (1980).
[CrossRef]

1979 (2)

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single mode fibers,” Microwaves Opt. Acoust. 3, 239–246 (1979).
[CrossRef]

M. Fukui, V. C. Y. So, R. Normandin, “Lifetimes of surface plasmons in thin silver films,” Phys. Status Solidi B 91, K61–K64 (1979).
[CrossRef]

1976 (2)

1974 (2)

C. A. Pfeiffer, E. N. Economou, K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[CrossRef]

J. S. Nkoma, R. Loudon, D. R. Tilley, “Elementary properties of surface polaritons,”J. Phys. C 7, 3547–3559 (1974).
[CrossRef]

1968 (1)

R. Englman, R. Ruppin, “Optical lattice vibrations in finite ionic crystals,”J. Phys. C 2, 1515–1531 (1968).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Aers, G. C.

G. C. Aers, A. D. Boardman, B. V. Paranjape, “Non-radiative surface plasmon-polariton modes in inhomogeneous metal circular cylinder,”J. Phys. F 10, 53–62 (1980).
[CrossRef]

Boardman, A. D.

G. C. Aers, A. D. Boardman, B. V. Paranjape, “Non-radiative surface plasmon-polariton modes in inhomogeneous metal circular cylinder,”J. Phys. F 10, 53–62 (1980).
[CrossRef]

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Cottam, M. G.

M. G. Cottam, D. R. Tilley, Introduction to Surface and Superlattice Excitations (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Economou, E. N.

C. A. Pfeiffer, E. N. Economou, K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[CrossRef]

Englman, R.

R. Englman, R. Ruppin, “Optical lattice vibrations in finite ionic crystals,”J. Phys. C 2, 1515–1531 (1968).
[CrossRef]

Fukui, M.

M. Fukui, V. C. Y. So, R. Normandin, “Lifetimes of surface plasmons in thin silver films,” Phys. Status Solidi B 91, K61–K64 (1979).
[CrossRef]

Gambling, W. A.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single mode fibers,” Microwaves Opt. Acoust. 3, 239–246 (1979).
[CrossRef]

Gloge, D.

D. Gloge, E. A. J. Marcatili, D. Marcuse, S. D. Personick, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979).

Haus, H. A.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Keck, D. B.

Kogelnik, H.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1971).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1971).

Loudon, R.

J. S. Nkoma, R. Loudon, D. R. Tilley, “Elementary properties of surface polaritons,”J. Phys. C 7, 3547–3559 (1974).
[CrossRef]

S. Ushioda, R. Loudon, in Surface Polaritons, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 535–586.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, D. Marcuse, S. D. Personick, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979).

Marcuse, D.

D. Gloge, E. A. J. Marcatili, D. Marcuse, S. D. Personick, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

Matsumura, H.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single mode fibers,” Microwaves Opt. Acoust. 3, 239–246 (1979).
[CrossRef]

Ngai, K. L.

C. A. Pfeiffer, E. N. Economou, K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[CrossRef]

Nkoma, J. S.

J. S. Nkoma, R. Loudon, D. R. Tilley, “Elementary properties of surface polaritons,”J. Phys. C 7, 3547–3559 (1974).
[CrossRef]

Normandin, R.

M. Fukui, V. C. Y. So, R. Normandin, “Lifetimes of surface plasmons in thin silver films,” Phys. Status Solidi B 91, K61–K64 (1979).
[CrossRef]

Olshansky, R.

Paranjape, B. V.

G. C. Aers, A. D. Boardman, B. V. Paranjape, “Non-radiative surface plasmon-polariton modes in inhomogeneous metal circular cylinder,”J. Phys. F 10, 53–62 (1980).
[CrossRef]

Personick, S. D.

D. Gloge, E. A. J. Marcatili, D. Marcuse, S. D. Personick, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979).

Pfeiffer, C. A.

C. A. Pfeiffer, E. N. Economou, K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[CrossRef]

Ragdale, C. M.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single mode fibers,” Microwaves Opt. Acoust. 3, 239–246 (1979).
[CrossRef]

Ruppin, R.

R. Englman, R. Ruppin, “Optical lattice vibrations in finite ionic crystals,”J. Phys. C 2, 1515–1531 (1968).
[CrossRef]

R. Ruppin, Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982), Chap. 9.

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

So, V. C. Y.

M. Fukui, V. C. Y. So, R. Normandin, “Lifetimes of surface plasmons in thin silver films,” Phys. Status Solidi B 91, K61–K64 (1979).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tilley, D. R.

J. S. Nkoma, R. Loudon, D. R. Tilley, “Elementary properties of surface polaritons,”J. Phys. C 7, 3547–3559 (1974).
[CrossRef]

M. G. Cottam, D. R. Tilley, Introduction to Surface and Superlattice Excitations (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

Ushioda, S.

S. Ushioda, R. Loudon, in Surface Polaritons, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 535–586.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Phys. C (2)

J. S. Nkoma, R. Loudon, D. R. Tilley, “Elementary properties of surface polaritons,”J. Phys. C 7, 3547–3559 (1974).
[CrossRef]

R. Englman, R. Ruppin, “Optical lattice vibrations in finite ionic crystals,”J. Phys. C 2, 1515–1531 (1968).
[CrossRef]

J. Phys. F (1)

G. C. Aers, A. D. Boardman, B. V. Paranjape, “Non-radiative surface plasmon-polariton modes in inhomogeneous metal circular cylinder,”J. Phys. F 10, 53–62 (1980).
[CrossRef]

Microwaves Opt. Acoust. (1)

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single mode fibers,” Microwaves Opt. Acoust. 3, 239–246 (1979).
[CrossRef]

Phys. Rev. B (1)

C. A. Pfeiffer, E. N. Economou, K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[CrossRef]

Phys. Rev. Lett. (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[CrossRef]

Phys. Status Solidi B (1)

M. Fukui, V. C. Y. So, R. Normandin, “Lifetimes of surface plasmons in thin silver films,” Phys. Status Solidi B 91, K61–K64 (1979).
[CrossRef]

Other (10)

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

D. Gloge, E. A. J. Marcatili, D. Marcuse, S. D. Personick, in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979).

R. Ruppin, Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, Chichester, UK, 1982), Chap. 9.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

S. Ushioda, R. Loudon, in Surface Polaritons, V. M. Agranovich, D. L. Mills, eds. (North-Holland, Amsterdam, 1982), pp. 535–586.

M. G. Cottam, D. R. Tilley, Introduction to Surface and Superlattice Excitations (Cambridge U. Press, Cambridge, 1989).
[CrossRef]

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1971).

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Figures (18)

Fig. 1
Fig. 1

Illustration of guided-mode and surface-mode windows in the ωq plane for the case when 2 = constant and 1 has the reststrahl form [Eq. (2)], with > 2. Guided-mode windows are regions 1 and 3; the surface-mode window is region 2.

Fig. 2
Fig. 2

Illustration of guided-mode and surface-mode windows in the ωq plane for the case when 2 = constant and 1 has the plasma form [Eq. (3)], with > 2. The guided-mode window is region 3; the surface-mode window is region 2.

Fig. 3
Fig. 3

Dispersion curves for m = 0 in the upper guided-mode window with 2 = 1 and 1 in the reststrahl form [Eq. (2)]. Parameters are = 10.9, ωL/ωT = 1.08, and fiber radius a = 1.36c/ωT.

Fig. 4
Fig. 4

Dispersion curves for m = 1 for the parameters of Fig. 3.

Fig. 5
Fig. 5

Dispersion curves for m = 0 in the lower guided-mode window with 2 = 1 and 1 in the reststrahl form [Eq. (2)]. Parameters are = 10.9, ωL/ωT = 1.08, and fiber radius a = 1.36c/ωT. The hatched region near ωT corresponds to the accumulation of modes described in the text.

Fig. 6
Fig. 6

Surface-mode dispersion curves for m = 0 with 2 = 1 and 1 in the reststrahl form [Eq. (2)]. Parameters are = 10.9, ωL/ωT = 1.08. Curves are marked with values of the dimensionless radius T/c, and the flat-surface curve, given by [Eq. (20)], is shown dashed. The three points marked for ω/ωT = 1.05 correspond to the power and energy curves drawn in Figs. 10 and 11 below.

Fig. 7
Fig. 7

Surface-mode dispersion curves for m = 1 with 2 = 1 and 1 in the reststrahl form [Eq. (2)] and parameters and conventions as in Fig. 6. The three q values marked for T/c = 0.289 and ω/ωT = 1.0708 correspond to the power and energy curves drawn in Figs. 14 and 15 below.

Fig. 8
Fig. 8

Surface-mode dispersion curves for m = 0 with 2 = 1 and 1 in the plasma form [Eq. (2)], with = 2. Curves are marked with values of the dimensionless fiber radius p/c; the flat-surface curve, given by Eq. (20), is shown dashed.

Fig. 9
Fig. 9

Surface-mode dispersion curves for m = 1 with 2 = 1 and 1 in the plasma form [Eq. (3)]. Parameters and conventions as for Fig. 8.

Fig. 10
Fig. 10

Power-flow density rSz as a function of radius for m = 0 surface polaritons with 2 = 1 and 1 in the reststrahl form [Eq. (2)]. Frequency ω/ωT = 1.05. Curves are shown for the three values of dimensionless radius T/c, used for the dispersion curves (Fig. 6) and are normalized so that the area under each curve is unity. The q values used are marked in Fig. 6.

Fig. 11
Fig. 11

Energy density rU corresponding to Fig. 10; curves are normalized to unit area.

Fig. 12
Fig. 12

Integrated power Pz(q, ω) [Eqs. (43)(45)] for m = 0 and T/c = 1.87 (solid curve). ω and q are related by the dispersion curve in Fig. 6. Also shown is the integrated energy per unit length Y(q, ω) (dashed curve) given by Eqs. (51)(54). Normalizing factors are A = (πa2c∊0E0z2)−1 and B = (πa20E0z2)−1.

Fig. 13
Fig. 13

Energy velocity vE(q, ω) = Pz(q, ω)/Y(q, ω) for m = 0 and T/c = 1.87. This velocity equals the gradient of the corresponding dispersion curve in Fig. 6.

Fig. 14
Fig. 14

Power-flow density rSz as a function of radius for m = 1 surface polaritons with 2 = 1 and 1 in the reststrahl form [Eq. (2)]. The three curves shown are all for radius T/c = 0.289 and frequency ω/ωT = 1.0708; they correspond to the three possible q values for this frequency, as marked on Fig. 7.

Fig. 15
Fig. 15

Energy-density curves rU corresponding to Fig. 14.

Fig. 16
Fig. 16

Integrated power flow Pz(q, ω) for m = 1 and T/c = 0.289. ω and q are related by the dispersion curve in Fig. 7. The normalizing factor is A = (πa2c∊0E0z2)−1.

Fig. 17
Fig. 17

Integrated energy Y(q, ω) corresponding to Fig. 16. The normalizing factor is B = (πa20E0z2)−1.

Fig. 18
Fig. 18

Energy velocity vE(q, ω) for m = 1 and T/c = 0.289 found from Figs. 16 and 17. It is equal to the slope of the corresponding dispersion curve in Fig. 7.

Equations (61)

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( ω ) = + S ω T 2 ω T 2 - ω 2 - i ω Γ ,
( ω ) = ω L 2 - ω 2 ω T 2 - ω 2 ,
( ω ) = [ 1 - ω p 2 ω ( ω + i Γ ) ] .
ω p 2 = n e 2 0 m * ,
0 < ω < ω T
ω 2 < ω ,
ω 1 < ω < ω T ,
ω T < ω < ω S ,
0 < ω < ω S
exp ( i q z + i m θ - i ω t ) .
W i 2 = - U i 2 = q 2 - 1 ω 2 / c 2 ,             i = 1 , 2.
ω 2 c 2 ( η + γ ) ( 1 η + 2 γ ) = m 2 q 2 a 2 ( 1 U 1 2 + 1 W 2 2 ) 2
ω 2 c 2 ( γ - η 1 ) ( 2 γ - 1 η 1 ) = m 2 q 2 a 2 ( 1 W 2 2 - 1 W 1 2 ) 2
γ = K m ( W 2 a ) W 2 K m ( W 2 a ) ,
η = J m ( U 1 a ) U 1 J m ( U 1 a ) ,
η 1 = I m ( W 1 a ) W 1 I m ( W 1 a ) ,
γ - 1 / W 2 ,
η 1 1 / W 1 ,             as a
1 W 1 + 1 W 2 = 0
1 W 1 + 2 W 2 = 0.
( γ - η 1 ) ( 2 γ - 1 η 1 ) = m 2 q 2 a 2 ω 2 c 2 ( 2 - 1 ) 2 W 1 4 W 2 4 .
1 + 2 = m 2 a 2 ω 2 c 2 ( 2 - 1 ) 2 q 2 .
1 ( ω S ) + 2 = 0 ,
K m ( W 2 a ) ~ ½ Γ ( m ) ( ½ W 2 a ) - m ,
K m ( W 2 a ) ~ - ¼ Γ ( m ) ( ½ W 2 a ) - m - 1 .
ω 2 c 2 ( γ - η 1 ) ( 2 γ - 1 η 1 ) ~ 2 γ 2 ω 2 c 2 ~ 2 ω 2 c 2 m 2 a 2 W 2 4 ,
m 2 q 2 a 2 ( 1 W 2 2 - 1 W 1 2 ) 2 ~ q 2 m 2 a 2 W 2 4 .
2 ω 2 / c 2 < q 2 < 1 ω 2 / c 2 .
q 2 > 1 ω 2 / c 2
q 2 > 2 ω 2 / c 2 ,
1 ( ω ) < - 2 .
S = E × H .
S ( r ) = Re [ E ( r , t ) ] × Re [ H ( r , t ) ] = ½ Re E ( r , t ) × H * ( r , t ) ,
P ( q , ω ) = 0 r d r 0 2 π d θ S ( r ) .
S z = ω q 2 U 1 4 ( 1 0 E 0 z 2 cos 2 m θ + μ 0 H 0 z 2 sin 2 m θ ) [ U 1 J m ( U 1 r ) 2 J m ( U 1 a ) ] 2 + ω q 2 r 2 U 1 4 ( 1 0 E 0 z 2 sin 2 m θ + μ 0 H 0 z 2 cos 2 m θ ) [ J m ( U 1 r ) J m ( U 1 a ) ] 2 + i m 2 r U 1 4 [ ( 1 0 μ 0 ω 2 + q 2 ) E 0 z H 0 z ] [ J m ( U 1 r ) J m ( U 1 a ) ] [ U 1 J m ( U 1 r ) J m ( U 1 a ) ] ( guided modes , r < a ) ,
S z = formula ( 35 ) , with trasformations { U 1 to W 1 J m to I m J m to I m }             ( surface modes , r < a ) ,
S z = formula ( 35 ) , with transformations { U 1 to W 2 ; 1 to 2 J m to K m J m to K m }             ( guided and surface modes , r > a ) ,
S z ~ ω q 2 W 1 2 0 1 E 0 z 2 cos 2 m θ [ I m ( W 1 r ) I m ( W 1 a ) ] 2 ,             r < a ,
S z ~ ω q 2 W 2 2 0 2 E 0 z 2 cos 2 m θ [ K m ( W 2 r ) K m ( W 2 a ) ] 2 ,             r > a .
r = a + x ,
S z ~ 0 1 q ω 2 W 1 2 E 0 z 2 cos 2 m θ exp ( 2 W 1 x ) ,             r < a and x < 0 ,
S z ~ 0 2 q ω 2 W 2 2 E 0 z 2 cos 2 m θ exp ( - 2 W 2 x ) ,             r > a and x > 0.
P z ( q , ω ) = P 1 ( q , ω ) + P 2 ( q , ω ) ,
P 1 ( q , ω ) = - π a 2 E 0 z 2 0 1 q ω 2 W 1 2 { [ I 1 ( W 1 a ) I 0 ( W 1 a ) ] 2 - ( 1 + 1 W 1 2 a 2 ) [ I 1 ( W 1 a ) I 0 ( W 1 a ) ] 2 }
P 2 ( q , ω ) = π a 2 E 0 z 2 0 2 q ω 2 W 2 2 { [ K 1 ( W 2 a ) K 0 ( W 2 a ) ] 2 - ( 1 + 1 W 2 2 a 2 ) [ K 1 ( W 2 a ) K 0 ( W 2 a ) ] 2 } .
U = 1 4 [ 0 ( + ω d d ω ) E · E * + μ 0 H · H * ] .
E · E * = [ m 2 r 2 W 1 4 ( ω 2 μ 0 2 H 0 z 2 cos 2 m θ + q 2 E 0 z 2 sin 2 m θ ) + E 0 z 2 cos 2 m θ ] [ I m ( W 1 r ) I m ( W 1 a ) ] 2 + [ 1 W 1 2 ( ω 2 μ 0 2 H 0 z 2 sin 2 m θ + q 2 E 0 z 2 cos 2 m θ ) ] [ I m ( W 1 r ) I m ( W 1 a ) ] 2 + [ 2 i m r W 1 3 q ω μ 0 E 0 z H 0 z ] [ I m ( W 1 r ) I m ( W 1 a ) ] [ I m ( W 1 r ) I m ( W 1 a ) ] ,
H · H * = [ m 2 r 2 W 1 4 ( ω 2 1 2 0 2 E 0 z 2 sin 2 m θ + q 2 H 0 z 2 cos 2 m θ ) + H 0 z 2 sin 2 m θ ] [ I m ( W 1 r ) I m ( W 1 a ) ] 2 + [ 1 W 1 2 ( ω 2 1 2 0 2 E 0 z 3 cos 2 m θ + q 2 H 0 z 2 sin 2 m θ ) ] [ I m ( W 1 r ) I m ( W 1 a ) ] 2 + [ 2 i m r W 1 3 ω 1 0 q E 0 z H 0 z ] [ I m ( W 1 r ) I m ( W 1 a ) ] [ I m ( W 1 r ) I m ( W 1 a ) ] ,
E · E * = formula ( 47 ) with transformations × { W 1 to W 2 ; 1 to 2 I m to K m I m to K m } ,
H · H * = formula ( 48 ) with transformations ( 49 ) .
Y ( q , ω ) = 0 r d r 0 2 π d θ U .
Y ( q , ω ) = Y 1 ( q , ω ) + Y 2 ( q , ω )
Y 1 ( q , ω ) = π 0 E 0 z 2 2 I 0 2 ( W 1 a ) { ( 1 + ω d 1 d ω ) [ 0 a r I 0 2 ( W 1 r ) d r + q 2 W 1 2 0 a r I 1 2 ( W 1 r ) d r ] + μ 0 0 ω 2 1 2 W 1 2 0 a r I 1 2 ( W 1 r ) d r } ,
Y 2 ( q , ω ) = π 0 2 E 0 z 2 2 K 0 2 ( W 2 a ) [ a r K 0 2 ( W 2 r ) d r + q 2 W 2 2 a r K 1 2 ( W 2 r ) d r + μ 0 0 ω 2 1 W 2 2 a r K 1 2 ( W 2 r ) d r ] .
v E ( q , ω ) = P z ( q , ω ) Y ( q , ω ) .
v g ( q , ω ) = d ω / d q .
S i z ( x ) = ( 0 i q ω 0 2 / 2 W i 2 ) exp ( - 2 W i x ) ,
P ( q , ω ) = 0 q 0 2 4 ( 1 W 1 3 + 2 W 2 3 ) .
P ( q , ω ) = 0 2 q 0 2 4 W 2 3 ( 1 - 2 2 1 2 ) ,
E i · E i * = ( 1 + q 2 / W i 2 ) 0 2 exp ( - 2 W i x ) ,             i = 1 , 2 ,
H i · H i * = ( 0 2 i 2 ω 2 / W i 2 ) 0 2 exp ( - 2 W i x ) ,             i = 1 , 2.

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