Abstract

The excitation of a whispering gallery resonator by a surface wave guided in a dielectric slab is analyzed with a rigorous volume-integral-equation approach. The analysis is based on the Green’s function concept and the application of the entire-domain Galerkin technique through expansion of the electric field in the resonator in terms of cylindrical wave functions. The algorithm developed yields highly accurate results for the transmission and reflection coefficients in the waveguide. The radiated far field is computed, and the effect of the excitation of a whispering gallery mode on the radiation pattern is studied.

© 2004 Optical Society of America

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References

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  1. G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
    [CrossRef]
  2. D. Cros, P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1673 (1990).
    [CrossRef]
  3. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
    [CrossRef]
  4. K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
    [CrossRef]
  5. S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
    [CrossRef]
  6. S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol. 7, 2033–2038 (1989).
    [CrossRef]
  7. B. E. Little, J.-P. Laine, H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704–715 (1999).
    [CrossRef]
  8. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1983).
  9. A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).
  10. N. K. Uzunoglu, J. G. Fikioris, “Scattering from an inhomogeneity inside a dielectric slab waveguide,” J. Opt. Soc. Am. 72, 628–637 (1982).
    [CrossRef]
  11. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).
  12. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).
  13. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).
  14. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  15. N. K. Uzunoglu, “Scattering from inhomogeneities inside a fiber waveguide,” J. Opt. Soc. Am. 71, 259–273 (1981).
    [CrossRef]
  16. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).
  17. M. J. Jory, E. A. Perkins, J. R. Sambles, “Light emission from whispering-gallery modes in microscopic spheres,” J. Opt. Soc. Am. A 20, 1785–1791 (2003).
    [CrossRef]
  18. F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).
  19. I. D. Chremmos, N. K. Uzunoglu, “Analysis of coupling between two slab waveguides in the presence of ring resonators,” J. Opt. Soc. Am. A 21, 267–279 (2004).
    [CrossRef]

2004 (1)

2003 (2)

M. J. Jory, E. A. Perkins, J. R. Sambles, “Light emission from whispering-gallery modes in microscopic spheres,” J. Opt. Soc. Am. A 20, 1785–1791 (2003).
[CrossRef]

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

1999 (2)

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

B. E. Little, J.-P. Laine, H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704–715 (1999).
[CrossRef]

1997 (2)

G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

1991 (1)

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

1990 (1)

D. Cros, P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1673 (1990).
[CrossRef]

1989 (1)

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

1982 (1)

1981 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Annino, G.

G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
[CrossRef]

Arnold, S.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

Boriskina, S. V.

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

Braun, D.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

Cassetari, M.

G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
[CrossRef]

Chaudhuri, S. K.

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

Chremmos, I. D.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

Cros, D.

D. Cros, P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1673 (1990).
[CrossRef]

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).

Fikioris, J. G.

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Guillon, P.

D. Cros, P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1673 (1990).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1983).

Haus, H. A.

B. E. Little, J.-P. Laine, H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704–715 (1999).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Jory, M. J.

Laine, J.-P.

B. E. Little, J.-P. Laine, H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704–715 (1999).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Libchaber, A.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

Little, B. E.

B. E. Little, J.-P. Laine, H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704–715 (1999).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Longo, I.

G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
[CrossRef]

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).

Martinelli, M.

G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
[CrossRef]

Nosich, A. I.

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

Oda, K.

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

Perkins, E. A.

Sambles, J. R.

Sommerfeld, A. R.

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Teraoka, I.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

Toba, H.

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

Tokato, N.

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

Uzunoglu, N. K.

Vollmer, F.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

Biophys. J. (1)

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. 85, 1974–1979 (2003).

IEEE Trans. Microwave Theory Tech. (3)

G. Annino, M. Cassetari, I. Longo, M. Martinelli, “Whispering gallery modes in dielectric resonators: characterization at the millimeter wavelengths,” IEEE Trans. Microwave Theory Tech. 45, 2025–2034 (1997).
[CrossRef]

D. Cros, P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1673 (1990).
[CrossRef]

S. V. Boriskina, A. I. Nosich, “Radiation and absorption losses of the WGM dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
[CrossRef]

J. Lightwave Technol. (4)

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

K. Oda, N. Tokato, H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems,” J. Lightwave Technol. 9, 728–736 (1991).
[CrossRef]

B. E. Little, J.-P. Laine, H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704–715 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Other (7)

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1983).

A. R. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).

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Figures (10)

Fig. 1
Fig. 1

Dielectric slab waveguide coupled to a ring WG resonator.

Fig. 2
Fig. 2

Integration contour C and Re ( γ 0 ) = 0 branch cut on the complex λ plane.

Fig. 3
Fig. 3

Application of Cauchy’s residue theorem for z > 0 .

Fig. 4
Fig. 4

Application of the steepest-descent method on the complex g plane. The initial integration contour C is deformed to the steepest-descent path passing through the saddle point π - θ . The strips P i ( I i ) , i = 1 , 2, 3, 4, are the images of the ith quadrant of the proper (improper) Riemann sheet on the g plane.

Fig. 5
Fig. 5

Transmitted (solid curve), reflected (dashed curve), and radiated (dotted curve) power profiles versus w = d ( k 1 2 - k 0 2 ) 1 / 2 for n 1 = 1.5 , n 2 = 3 , a = d , b = 2 d , c = 3 d .

Fig. 6
Fig. 6

Transmission profiles for a = 0.2 d (solid curve), a = 0.6 d (dashed curve), a = d (dotted curve), a = 1.4 d (dashed-dotted curve), a = 1.8 d (heavy solid curve) and n 1 = 1.5 , n 2 = 3 , b = 2 d , c = 3 d .

Fig. 7
Fig. 7

Normalized radiation patterns for n 1 = 1.5 , n 2 = 3 , a = d , b = 2 d , c = 3 d , and for the off-resonance cases w = 1.1 (solid curve), w = 1.3 (dotted curve), w = 1.5 (dashed curve). The horizontal axis is the scattering angle θ = tan - 1 ( x / z ) .

Fig. 8
Fig. 8

Normalized radiation pattern for n 1 = 1.5 , n 2 = 3 , a = d , b = 2 d , c = 3 d , and w = 1.3875 (solid curve), w = 1.335 (dotted curve), w = 1.4 (dashed curve).

Fig. 9
Fig. 9

Geometrical optics model for the field emission from an excited WG mode in the region θ > 90 ° .

Fig. 10
Fig. 10

Comparison of the backscattered radiation field between Eq. (25) (dotted curve) and that computed with the integral-equation approach (solid curve) on the resonance w = 1.3875 , for n 1 = 1.5 , n 2 = 3 , a = d , b = 2 d , c = 3 d .

Tables (1)

Tables Icon

Table 1 Convergence Pattern of Forward-Scattering and Backscattering Coefficients for w = 1.5 , n 1 = 1.5 , n 2 = 3 , a = d , b = 2 d , c = 3 d

Equations (55)

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E = y ˆ Ψ ( x ,   z ) ,
[ 2 / x 2 + 2 / z 2 + k 2 n ( r ) 2 ] G ( r | r ) = - δ ( r - r ) ,
G ( r | r ) = - j 4   H 0 ( 2 ) ( k 0 | r - r | ) + - + d λ f ( λ ) × exp [ j λ ( z - z ) - γ 0 ( x + x ) ] , x > d ,
f ( λ ) = ( k 1 2 - k 0 2 ) exp ( 2 γ 0 d ) sinh ( γ 1 d ) cosh ( γ 1 d ) / ( 4 π γ 0 P e P o ) ,
P e = γ 0 cosh ( γ 1 d ) + γ 1 sinh ( γ 1 d ) ,
P o = γ 0 sinh ( γ 1 d ) + γ 1 cosh ( γ 1 d ) ,
γ 0 = ( λ 2 - k 0 2 ) 1 / 2 , γ 1 = ( λ 2 - k 1 2 ) 1 / 2 ,
Ψ s ( x ,   z ) = K s exp ( - j β s z ) × cos ( a s d ) sin ( a s d ) exp [ - γ s ( x - d ) ] , x > d cos ( a s x ) sin ( a s x ) , | x | < d cos ( a s d ) - sin ( a s d ) exp [ γ s ( x + d ) ] , x < - d ,
a s = ( k 1 2 - β s 2 ) 1 / 2 , γ s = ( β s 2 - k 0 2 ) 1 / 2 ,
P e ( β s ) = 0 , even TE mode ,
P o ( β s ) = 0 , odd TE mode ,
Ψ ( x ,   z ) = Ψ inc ( x ,   z ) + ( k 2 2 - k 0 2 ) R G ( x ,   z | x ,   z ) × Ψ ( x ,   z ) d x d z ,
Ψ ( x ,   z ) = n = - + α n s [ J n ( k 2 ρ ) + Λ ( n ) Y n ( k 2 ρ ) ] exp ( jn ϕ ) ,
ϕ = 0 2 π d ϕ   exp [ jz   cos ( ϕ - q ) - jm ϕ ] = 2 π j m J m ( z ) exp ( - jmq )
0 2 π d ϕ   exp ( - j ν ϕ ) Ψ inc ( ρ ,   ϕ )
= 2 π j - ν Z s - ν A s e , o J ν ( k 0 ρ ) ,
Z s = β s / k 0 + ( β s 2 / k 0 2 - 1 ) 1 / 2 ,
A s e , o = K s cos ( a s d ) sin ( a s d ) exp [ - γ s ( c - d ) ] ,
H 0 ( 2 ) ( k 0 | ρ - ρ | ) = m = - + J m ( k 0 ρ < ) H m ( 2 ) ( k 0 ρ > ) exp [ jm ( ϕ - ϕ ) ] ,
( k 2 2 - k 0 2 ) ϕ = 0 2 π d ϕ   exp ( - j ν ϕ ) R ρ d ρ d ϕ
× - j 4   H 0 ( 2 ) ( k 0 | ρ - ρ | ) Ψ ( ρ ,   ϕ )
= 2 π α ν s [ J ν ( k 2 ρ ) + Λ ( ν ) Y ν ( k 2 ρ ) ] + π α ν s J ν ( k 0 ρ ) K ν ,
( k 2 2 - k 0 2 ) ϕ = 0 2 π d ϕ   exp ( - j ν ϕ ) R ρ d ρ d ϕ
× - + d λ f ( λ ) exp [ j λ ( z - z ) - γ 0 ( x + x ) ] Ψ ( ρ ,   ϕ )
= π J ν ( k 0 ρ ) k = - + α k s L k K ν k ,
k = - + α k s ( K ν δ ν k + L k K ν k ) = - 2 j - ν Z s - ν A s e , o ,
α s = [ α - N s ,   α - N + 1 s , ,   α N s ] ,
G ( r | r ) | | x | < d = - + d λ   exp [ j λ ( z - z ) + γ 0 ( d - x ) ] × cosh ( γ 1 x ) 4 π P e + sinh ( γ 1 x ) 4 π P o ,
Ψ ( x ,   z ) = Ψ inc ( x ,   z ) + ( 1 / 2 ) n = - + α n s j - n L n Z n ( x ,   z ) ,
Z n ( x ,   z ) = - + d λ   exp [ j λ z + γ 0 ( d - c ) + jn ϕ 0 ] × cosh ( γ 1 x ) P e + sinh ( γ 1 x ) P o .
Ψ scat s i = 1 N e τ i , s e K i cos ( a i x ) exp ( - j β i e z ) + i = 1 N o τ i , s o K i sin ( a i x ) exp ( - j β i o z ) , z + i = 1 N e ρ i , s e K i cos ( a i x ) exp ( j β i e z ) + i = 1 N o ρ i , s o K i sin ( a i x ) exp ( j β i o z ) , z - ,
Ψ scat = ( 1 / 2 ) n = - + α n s L n
× [ - π jH n ( 2 ) ( k 0 ρ ) exp ( jn ϕ ) + j - n Φ n ( x ,   z ) ] , x > d j - n Ξ n ( x ,   z ) , x < - d   ,
 
Φ n ( x ,   z ) = ( k 1 2 - k 0 2 ) - + d λ   exp [ j λ z + γ 0 ( 2 d - c - x ) + jn ϕ 0 ]   sinh ( γ 1 d ) cosh ( γ 1 d ) γ 0 P e P o ,
Ξ n ( x ,   z ) = - + d λ   exp [ j λ z + γ 0 ( 2 d - c + x ) + jn ϕ 0 ]   γ 1 P e P o .
Φ n ( r ,   θ ) = C d gR n ( g ) exp [ jk 0 r   cos ( g + θ ) ] ,
Ψ ( r ,   θ ) | r + - ( π / 2 k 0 r ) 1 / 2 exp [ - j ( k 0 r - π / 4 ) ] × n = - + α n s L n j n { j   exp [ j ( k 0 c   sin   θ + n θ ) ] + ( - 1 ) n R n ( π - θ ) } , 0 < θ < π j - n Q n ( π + θ ) , -   π < θ < 0 ,
R n ( g ) Q n ( g ) = j   exp [ jk 0 ( 2 d - c ) sin   g + jng ] P e P o
× ( k 1 2 - k 0 2 ) sinh ( γ 1 d ) cosh ( γ 1 d ) γ 0 γ 1.
U ( θ ) = | F ( θ ) | max { | F ( θ ) | }   ,
F ( θ ) = 1 + R ( θ ) exp [ - jk Δ ( θ ) ] ,
R ( θ ) = ( k 1 2 - k 0 2 ) sinh ( γ 1 d ) cosh ( γ 1 d ) / ( P e P o ) ,
γ 0 = jk 0 sin   θ ,
γ 1 = j ( k 1 2 - k 0 2 cos 2   θ ) 1 / 2 ,
Δ = 2 n 2 b ( π - θ ) + 2 n 0 ( c - d ) sin   θ ,
Λ ( n )
= n 0 J n + 1 ( k 0 a ) J n ( k 2 a ) - n 2 J n ( k 0 a ) J n + 1 ( k 2 a ) n 2 Y n + 1 ( k 2 a ) J n ( k 0 a ) - n 0 Y n ( k 2 a ) J n + 1 ( k 0 a ) .
( π jkb ) - 1 K n
= n 0 H n + 1 ( 2 ) ( k 0 b ) [ J n ( k 2 b ) + Λ ( n ) Y n ( k 2 b ) ] - n 2 H n ( 2 ) ( k 0 b ) [ J n + 1 ( k 2 b ) + Λ ( n ) Y n + 1 ( k 2 b ) ] ,
( kb ) - 1 L n
= n 2 J n ( k 0 b ) [ J n + 1 ( k 2 b ) + Λ ( n ) Y n + 1 ( k 2 b ) ] - n 0 J n + 1 ( k 0 b ) [ J n ( k 2 b ) + Λ ( n ) Y n ( k 2 b ) ] ,
K ν k = 4 π j ν - k - + d λ f ( λ ) exp [ - 2 γ 0 c + j ( ν + k ) ϕ 0 ] ,
λ = k 0 cos   ϕ 0 γ 0 = jk 0 sin   ϕ 0     ϕ 0 = tan - 1 ( - j γ 0 / λ ) ,
τ i , s e , o ρ i , s e , o = - π j   exp [ 2 γ i ( d - c ) ] γ i a i 2 β i ( 1 + γ i d ) ( k 1 2 - k 0 2 ) A i e , o n = - + α n s L n j ± n z ± n ( β i ) .

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