Abstract

Electromagnetic wave theory is used to predict the radiation forces exerted upon a micrometer-sized spherical particle illuminated by evanescent waves penetrating across a dielectric interface. These forces are quantified for two incident beam polarizations (p and s polarization) and for different refractive-index media. The electromagnetic formalism that we use is based on theoretical studies of Barton et al. [ J. Appl. Phys. 64, 1632 ( 1988); J. Appl. Phys. 66, 4594 ( 1989)]. The novelty of the present research is to apply this general formalism to the calculation of forces when the evanescent field is identified with the incident field. Our theoretical results for the horizontal and vertical force components are shown graphically in nondimensional form as functions of the size parameter of the sphere. Moreover, our results are found to be in reasonable agreement with recent experimental findings of Kawata and Sugiura [ Opt. Lett. 17, 772 ( 1992)].

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288 (1986).
    [CrossRef] [PubMed]
  2. A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517 (1987); "Internal cell manipulation using infrared laser traps," Proc. Natl. Acad. Sci. (USA) 86, 7914 (1989).
    [CrossRef] [PubMed]
  3. A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
    [CrossRef]
  4. W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
    [CrossRef]
  5. R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am. B 9, 1922 (1992).
    [CrossRef]
  6. W. H. Wright, G. J. Sonek, and M. W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33, 1735 (1994).
    [CrossRef] [PubMed]
  7. S. Kawata and T. Sugiura, "Movement of micrometer-sized particles in the evanescent field of a laser beam," Opt. Lett. 17, 772 (1992).
    [CrossRef] [PubMed]
  8. G. Roosen, "La lévitation optique de sphères," Can. J. Phys. 57, 1260 (1979).
    [CrossRef]
  9. G. Mie, "Beitrage zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. (Leipzig) 25, 377 (1908).
  10. P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57 (1909).
  11. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  12. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  13. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991).
  14. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).
  15. I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133 (1979).
    [CrossRef]
  16. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632 (1988).
    [CrossRef]
  17. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594 (1989).
    [CrossRef]
  18. H. Chew, D. S. Wang, and M. Kerker, "Elastic scattering of evanescent electromagnetic waves," Appl. Opt. 18, 2679 (1979).
    [CrossRef] [PubMed]
  19. D. C. Prieve and J. Y. Walz, "Scattering of an evanescent surface wave by a microscopic dielectric sphere," Appl. Opt. 32, 1629 (1993).
    [CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (2)

1990 (2)

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
[CrossRef]

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632 (1988).
[CrossRef]

1987 (1)

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517 (1987); "Internal cell manipulation using infrared laser traps," Proc. Natl. Acad. Sci. (USA) 86, 7914 (1989).
[CrossRef] [PubMed]

1986 (1)

1979 (3)

H. Chew, D. S. Wang, and M. Kerker, "Elastic scattering of evanescent electromagnetic waves," Appl. Opt. 18, 2679 (1979).
[CrossRef] [PubMed]

G. Roosen, "La lévitation optique de sphères," Can. J. Phys. 57, 1260 (1979).
[CrossRef]

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133 (1979).
[CrossRef]

1909 (1)

P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57 (1909).

1908 (1)

G. Mie, "Beitrage zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. (Leipzig) 25, 377 (1908).

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632 (1988).
[CrossRef]

Ashkin, A.

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517 (1987); "Internal cell manipulation using infrared laser traps," Proc. Natl. Acad. Sci. (USA) 86, 7914 (1989).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288 (1986).
[CrossRef] [PubMed]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632 (1988).
[CrossRef]

Berns, M. W.

W. H. Wright, G. J. Sonek, and M. W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33, 1735 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
[CrossRef]

Bjorkholm, J. E.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991).

Brevik, I.

R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am. B 9, 1922 (1992).
[CrossRef]

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133 (1979).
[CrossRef]

Chew, H.

Chu, S.

Debye, P.

P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57 (1909).

Dziedzic, J. M.

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517 (1987); "Internal cell manipulation using infrared laser traps," Proc. Natl. Acad. Sci. (USA) 86, 7914 (1989).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288 (1986).
[CrossRef] [PubMed]

Euteneuer, U.

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

Gussgard, R.

Kawata, S.

Kerker, M.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Lindmo, T.

Mie, G.

G. Mie, "Beitrage zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. (Leipzig) 25, 377 (1908).

Prieve, D. C.

Roosen, G.

G. Roosen, "La lévitation optique de sphères," Can. J. Phys. 57, 1260 (1979).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632 (1988).
[CrossRef]

Schliwa, M.

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

Schüutze, K.

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

Sonek, G. J.

W. H. Wright, G. J. Sonek, and M. W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33, 1735 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
[CrossRef]

Stratton, J. A.

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

Sugiura, T.

Tadir, Y.

W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
[CrossRef]

Walz, J. Y.

Wang, D. S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991).

Wright, W. H.

W. H. Wright, G. J. Sonek, and M. W. Berns, "Parametric study of the forces on microspheres held by optical tweezers," Appl. Opt. 33, 1735 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
[CrossRef]

Ann. Phys. (2)

G. Mie, "Beitrage zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. (Leipzig) 25, 377 (1908).

P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57 (1909).

Appl. Opt. (3)

Can. J. Phys. (1)

G. Roosen, "La lévitation optique de sphères," Can. J. Phys. 57, 1260 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. H. Wright, G. J. Sonek, Y. Tadir, and M. W. Berns, "Laser trapping in cell biology," IEEE J. Quantum Electron. 26, 2148 (1990).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594 (1989).
[CrossRef]

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

Nature (1)

A. Ashkin, K. Schüutze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, "Force generation of organelle transport measured in vivo by an infrared laser trap," Nature (London) 348, 346 (1990).
[CrossRef]

Opt. Lett. (2)

Phys. Rep. (1)

I. Brevik, "Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor," Phys. Rep. 52, 133 (1979).
[CrossRef]

Science (1)

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517 (1987); "Internal cell manipulation using infrared laser traps," Proc. Natl. Acad. Sci. (USA) 86, 7914 (1989).
[CrossRef] [PubMed]

Other (4)

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

M. Kerker, The Scattering of Light (Academic, New York, 1969).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Particle of radius a situated in the evanescent field region x′ > 0. A laser beam is incident from below at an angle of incidence θ1 > θcrit in the substrate.

Fig. 2
Fig. 2

Definition sketch of polar angle θ and azimuthal angle ϕ. The origin of the coordinates is at the center of the sphere.

Fig. 3
Fig. 3

Definition sketch of the polarization directions. The normal components E ( 1 ) and E ( 2 ) are directed out of the page.

Fig. 4
Fig. 4

Nondimensional vertical radiation force Q x = F x / ( 0 E 0 2 a 2 ) versus particle size parameter α = k2a = 2πa/λ2 for the set of refractive indices shown. In the case of p polarization, E 0 = E ( 1 ); whereas, in the case of s polarization, E 0 = E ( 1 ). Here, and in all subsequent figures, θ1 = 51°. Similarly here, and in all subsequent figures, h = a.

Fig. 5
Fig. 5

Nondimensional horizontal radiation force Q z = F z / ( 0 E 0 2 a 2 ) versus α for the same refractive indices as in Fig. 4. The meaning of E0 for each state of polarization is the same as above.

Fig. 6
Fig. 6

Vertical force: same as for Fig. 4 but with a higher value of the refractive index (n3 = 1.60) in the sphere.

Fig. 7
Fig. 7

Horizontal force: same as for Fig. 5 but with n3 = 1.60.

Fig. 8
Fig. 8

Vertical force: same as for Fig. 4 but with vacuum (n2 = 1) surrounding the sphere.

Fig. 9
Fig. 9

Horizontal force: same as for Fig. 5 but with n2 = 1.

Tables (1)

Tables Icon

Table 1 Specification of the Symbols used in Eqs. (11)(16)a

Equations (51)

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

f = - ½ E 2 ,
¯ 3 = 3 + i ( σ 3 / ω ) ,
k 3 = n ¯ 3 ( ω / c ) ,             k 2 = n 2 ( ω / c ) .
Y l m ( θ , ϕ ) = [ 2 l + 1 4 π ( l - m ) ! ( l + m ) ! ] 1 / 2 P l m ( cos θ ) exp ( i m ϕ ) ,
ψ l ( x ) = x j l ( x ) ( π x 2 ) 1 / 2 J ν ( x ) ,
E r ( i ) = E 0 r ˜ 2 l = 1 m = - l l l ( l + 1 ) A l m ψ l ( α r ˜ ) Y l m ( θ , ϕ ) , H r ( i ) = H 0 r ˜ 2 l = 1 m = - l l l ( l + 1 ) B l m ψ l ( α r ˜ ) Y l m ( θ , ϕ ) ,
α = k 2 a ,             r ˜ = r / a ,             H 0 = 0 / μ 0 E 0 .
A l m = ( b / a ) 2 E 0 l ( l + 1 ) ψ l ( k 2 b ) Ω E r ( i ) ( b , θ , ϕ ) Y l m * ( θ , ϕ ) d Ω ,
B l m = ( b / a ) 2 H 0 l ( l + 1 ) ψ l ( k 2 b ) Ω H r ( i ) ( b , θ , ϕ ) Y l m * ( θ , ϕ ) d Ω ,
ξ l ( 1 ) ( x ) = x h l ( 1 ) ( x ) = ( π x 2 ) 1 / 2 H ν ( 1 ) ( x )
E r ( p ) = E 0 r ˜ 2 l = 1 m = - l l l ( l + 1 ) K l m z l ( n ¯ α r ˜ ) Y l m ,
E θ ( p ) = α E 0 r ˜ l = 1 m = - l l [ n ¯ K l m z l ( n ¯ α r ˜ ) Y l m θ - m n 2 K ˜ l m z l ( n ¯ α r ˜ ) Y l m sin θ ] ,
E ϕ ( p ) = α E 0 r ˜ l = l m = - l l [ i m n ¯ K l m z l ( n ¯ α r ˜ ) Y l m sin θ - i n 2 K ˜ l m z l ( n ¯ α r ˜ ) Y l m θ ] ,
H r ( p ) = H 0 r ˜ 2 l = 1 m = - l l l ( l + 1 ) K ˜ l m z l ( n ¯ α r ˜ ) Y l m ,
H θ ( p ) = α H 0 r ˜ l = 1 m = - l l [ n ¯ K ˜ l m z l ( n ¯ α r ˜ ) Y l m θ + m n 2 n ¯ 2 K l m z l ( n ¯ α r ˜ ) Y l m sin θ ] ,
H ϕ ( p ) = α r ˜ H 0 l = 1 m = - l l [ i m n ¯ K ˜ l m z l ( n ¯ α r ˜ ) Y l m sin θ + i n 2 n ¯ 2 K l m z l ( n ¯ α r ˜ ) Y l m θ ] .
a l m = ψ l ( α ) ψ l ( n ¯ 32 α ) - n ¯ 32 ψ l ( n ¯ 32 α ) ψ l ( α ) n ¯ 32 ψ l ( n ¯ 32 α ) ξ l ( 1 ) ( α ) - ξ l ( 1 ) ( α ) ψ l ( n ¯ 32 α ) A l m ,
b l m = n ¯ 32 ψ l ( α ) ψ l ( n ¯ 32 α ) - ψ l ( n ¯ 32 α ) ψ l ( α ) ψ l ( n ¯ 32 α ) ξ l ( 1 ) ( α ) - n ¯ 32 ξ l ( 1 ) ( α ) ψ l ( n ¯ 32 α ) B l m ,
c l m = i n ¯ 32 2 ψ l ( n ¯ 32 α ) ξ l ( 1 ) ( α ) - n ¯ 32 ξ l ( 1 ) ( α ) ψ l ( n ¯ 32 α ) A l m ,
d l m = i ψ l ( n ¯ 32 α ) ξ l ( 1 ) ( α ) - n ¯ 32 ξ l ( 1 ) ( α ) ψ l ( n ¯ 32 α ) B l m .
F x + i F y 0 E 0 2 a 2 = i α 2 4 l = 1 m = - l l { [ ( l + m + 2 ) ( l + m + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) ] 1 / 2 × l ( l + 2 ) ( 2 n 2 2 a l m a l + 1 , m + 1 * + n 2 2 a l m A l + 1 , m + 1 * ) + n 2 2 A l m a l + 1 , m + 1 * + 2 b l m b l + 1 , m + 1 * + b l m B l + 1 , m + 1 * + B l m b l + 1 , m + 1 * ) + [ ( l - m + 1 ) ( l - m + 2 ) ( 2 l + 1 ) ( 2 l + 3 ) ] 1 / 2 × l ( l + 2 ) ( 2 n 2 2 a l + 1 , m - 1 a l m * + n 2 2 a l + 1 , m - 1 A l m * + n 2 2 A l + 1 , m - 1 a l m * + 2 b l + 1. m - 1 b l m * + b l + 1 , m - 1 B l m * + B l + 1 , m - 1 b l m * ) - [ ( l + m + 1 ) ( l - m ) ] 1 / 2 × n 2 ( - 2 a l m b l , m + 1 * + 2 b l m a l , m + 1 * - a l m B l , m + 1 * + b l m A l , m + 1 * + B l m a l , m + 1 * - A l m b l , m + 1 * ) } ,
F z 0 E 0 2 a 2 = - α 2 2 l = 1 m = - l l [ ( l - m + 1 ) ( l + m + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) ] 1 / 2 × l ( l + 2 ) Im [ 2 n 2 2 a l + 1 , m a l m * + n 2 2 a l + 1 , m A l m * + n 2 2 A l + 1 , m a l m * + 2 b l + 1 , m b l m * + b l + 1 , m B l m * + B l + 1 , m b l m * + n 2 m ( 2 a l m b l m * + a l m B l m * + A l m b l m * ) ] .
E ( 2 ) = x ^ E ( 2 ) sin θ 2 + y ^ E ( 2 ) - z ^ E ( 2 ) cos θ 2 .
T = E ( 2 ) E ( 1 ) | x = 0 ,             T = E ( 2 ) E ( 1 ) | x = 0 .
cos θ 2 = i n 21 ( sin 2 θ 1 - n 21 2 ) 1 / 2 ,             sin θ 2 = 1 n 21 sin θ 1 .
T = 2 n 21 cos θ 1 n 21 2 cos θ 1 + i ( sin 2 θ 1 - n 21 2 ) 1 / 2 ,
T = 2 cos θ 1 cos θ 1 + i ( sin 2 θ 1 - n 21 2 ) 1 / 2 .
β = n 1 ω c ( sin 2 θ 1 - n 21 2 ) 1 / 2 ,
γ = n 1 ω c sin θ 1 ,
E ( 2 ) = { 1 n 21 T E ( 1 ) [ x ^ sin θ 1 - i z ^ ( sin 2 θ 1 - n 21 2 ) 1 / 2 ] + y ^ T E ( 1 ) } × exp ( - β x + i γ z ) .
x = x + h ,             y = y ,             z = z ,
E r ( 2 ) = E x ( 2 ) sin θ cos ϕ + E y ( 2 ) sin θ sin ϕ + E z ( 2 ) cos θ .
E r ( i ) = { 1 n 21 T E ( 1 ) [ sin θ 1 sin θ cos ϕ - i ( sin 2 θ 1 - n 21 2 ) 1 / 2 cos θ ] + T E ( 1 ) sin θ sin ϕ } × exp [ - β ( x + h ) + i γ z ] .
H ( 2 ) H ( 1 ) | x = 0 = n 21 T ,             H ( 2 ) H ( 1 ) | x = 0 = n 21 T ,
H r ( i ) = { T H ( 1 ) [ - sin θ 1 sin θ cos ϕ + i ( sin 2 θ 1 - n 21 2 ) 1 / 2 cos θ ] + n 21 T H ( 1 ) sin θ sin ϕ } × exp [ - β ( x + h ) + i γ z ] .
A l m ( p - pol . ) = ( b / a ) 2 T n 21 l ( l + 1 ) ψ l ( k 2 b ) Ω [ sin θ 1 sin θ cos ϕ - i ( sin 2 θ 1 - n 21 2 ) 1 / 2 cos θ ] Y l m * × exp [ - β ( x + h ) + i γ z ] d Ω ,
x = b sin θ cos ϕ ,             z = b cos θ .
B l m ( p - pol . ) = ( b / a ) 2 n 2 T l ( l + 1 ) ψ l ( k 2 b ) Ω sin θ sin ϕ Y l m * × exp [ - β ( x + h ) + i γ z ] d Ω .
A l m ( s - pol . ) = T n 2 T B l m ( p - pol . ) ,
B l m ( s - pol . ) = - n 2 T T A l m ( p - pol . ) .
A l m ( p - pol . ) = α 1 ( l , m ) n 21 T exp ( - β h ) [ sin θ 1 Q 1 ( l , m ) - i ( sin 2 θ 1 - n 21 2 ) 1 / 2 Q 2 ( l , m ) ] ,
α 1 ( l , m ) = [ 2 l + 1 4 π ( l - m ) ! ( l + m ! ) ! ] 1 / 2 ( b / a ) 2 l ( l + 1 ) ψ l ( k 2 b ) ,
Q 1 ( l , m ) = 0 π d θ sin 2 θ P l m ( cos θ ) exp ( i γ b cos θ ) × 0 2 π d ϕ cos ϕ exp ( - β b sin θ cos ϕ - i m ϕ ) ,
Q 2 ( l , m ) = 0 π d θ sin θ cos θ P l m ( cos θ ) exp ( i γ b cos θ ) × 0 2 π d ϕ exp ( - β b sin θ cos ϕ - i m ϕ ) .
Q 1 ( l , m ) = 2 π ( - 1 ) m - 1 0 π / 2 d θ sin 2 θ { cos i sin } ( γ b cos θ ) × P l m ( cos θ ) [ I m - 1 ( β b sin θ ) + I m + 1 ( β b sin θ ) ] ,
Q 2 ( l , m ) = 2 π ( - 1 ) m 0 π / 2 d θ sin θ cos θ { i sin cos } × ( γ b cos θ ) P l m ( cos θ ) I m ( β b sin θ ) .
I ν - 1 ( z ) - I ν + 1 ( z ) = 2 ν z I ν ( z ) ,
B l m ( p - pol . ) = n 2 α 1 ( l , m ) T exp ( - β h ) Q 3 ( l , m ) ,
Q 3 ( l , m ) = 0 π d θ sin 2 θ P l m ( cos θ ) exp ( i γ b cos θ ) × 0 2 π d ϕ sin ϕ exp ( - β b sin θ cos ϕ - i m ϕ ) = 2 π i ( - 1 ) m m β b 0 π / 2 d θ sin θ { cos i sin } ( γ b cos θ ) × P l m ( cos θ ) I m ( β b sin θ )
| f l k = 1 l f k < 10 - 5 | .
Q x = F x 0 E 0 2 a 2 ,             Q z = F z 0 E 0 2 a 2 .

Metrics