Abstract

We present a simple analytic model, based on surface plasmon propagation, that explains the depolarization induced by metal hole arrays illuminated with linearly and circularly polarized light of varying numerical apertures. Arrays with square and hexagonal lattices of circular holes are compared. We relate this model to experimental data.

© 2005 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  2. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
    [CrossRef]
  3. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
    [CrossRef]
  4. H. Raether, Surface Plasmons (Springer, 1988).
  5. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
    [CrossRef]
  6. M. Sarrazin and J. P. Vigneron, "Polarization effects in metallic films perforated with a bidimensional array of subwavelength rectangular holes," Opt. Commun. 240, 89-97 (2004).
    [CrossRef]
  7. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
    [CrossRef] [PubMed]
  8. E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization analysis of propagating surface plasmons in a subwavelength hole array," J. Opt. Soc. Am. B 20, 1927-1931 (2003).
    [CrossRef]
  9. E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization tomography of metallic nanohole arrays," Opt. Lett. 30, 90-92 (2005).
    [CrossRef] [PubMed]
  10. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  11. F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
    [CrossRef]
  12. Here, transverse is used to indicate the direction orthogonal to the array surface normal.
  13. C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Optical depolarization induced by arrays of subwavelength metal holes," Phys. Rev. B 71, 033409 (2005).
    [CrossRef]
  14. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
    [CrossRef]
  15. The imaginary part of B vanishes for any integration region that is mirror symmetric in the origin.
  16. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085415 (2003).
    [CrossRef]
  17. C. Genet, M. P. van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Commun. 225, 331-336 (2003).
    [CrossRef]
  18. U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
    [CrossRef]

2005 (2)

E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization tomography of metallic nanohole arrays," Opt. Lett. 30, 90-92 (2005).
[CrossRef] [PubMed]

C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Optical depolarization induced by arrays of subwavelength metal holes," Phys. Rev. B 71, 033409 (2005).
[CrossRef]

2004 (3)

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

M. Sarrazin and J. P. Vigneron, "Polarization effects in metallic films perforated with a bidimensional array of subwavelength rectangular holes," Opt. Commun. 240, 89-97 (2004).
[CrossRef]

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

2003 (3)

E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization analysis of propagating surface plasmons in a subwavelength hole array," J. Opt. Soc. Am. B 20, 1927-1931 (2003).
[CrossRef]

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085415 (2003).
[CrossRef]

C. Genet, M. P. van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Commun. 225, 331-336 (2003).
[CrossRef]

2000 (1)

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

1998 (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

1997 (1)

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

1961 (1)

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

1944 (1)

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Altewischer, E.

Azzam, R. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bethe, H. A.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Brolo, A. G.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Enoch, S.

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Fano, U.

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Genet, C.

C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Optical depolarization induced by arrays of subwavelength metal holes," Phys. Rev. B 71, 033409 (2005).
[CrossRef]

C. Genet, M. P. van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Commun. 225, 331-336 (2003).
[CrossRef]

Ghaemi, H. F.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gordon, R.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

Grupp, D. E.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Kavanagh, K. L.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

Koerkamp, K. J.

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

Kuipers, L.

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

Le Jeune, B.

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Leathem, B.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

McKinnon, A.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

Nevière, M.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons (Springer, 1988).

Rajora, A.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Sarrazin, M.

M. Sarrazin and J. P. Vigneron, "Polarization effects in metallic films perforated with a bidimensional array of subwavelength rectangular holes," Opt. Commun. 240, 89-97 (2004).
[CrossRef]

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Segerink, F. B.

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

Thio, T.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

van Exter, M. P.

C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Optical depolarization induced by arrays of subwavelength metal holes," Phys. Rev. B 71, 033409 (2005).
[CrossRef]

E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization tomography of metallic nanohole arrays," Opt. Lett. 30, 90-92 (2005).
[CrossRef] [PubMed]

E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization analysis of propagating surface plasmons in a subwavelength hole array," J. Opt. Soc. Am. B 20, 1927-1931 (2003).
[CrossRef]

C. Genet, M. P. van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Commun. 225, 331-336 (2003).
[CrossRef]

van Hulst, N. F.

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

Vigneron, J. P.

M. Sarrazin and J. P. Vigneron, "Polarization effects in metallic films perforated with a bidimensional array of subwavelength rectangular holes," Opt. Commun. 240, 89-97 (2004).
[CrossRef]

Vigneron, J.-P.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Vigoureux, J.-M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Woerdman, J. P.

C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Optical depolarization induced by arrays of subwavelength metal holes," Phys. Rev. B 71, 033409 (2005).
[CrossRef]

E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization tomography of metallic nanohole arrays," Opt. Lett. 30, 90-92 (2005).
[CrossRef] [PubMed]

E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Polarization analysis of propagating surface plasmons in a subwavelength hole array," J. Opt. Soc. Am. B 20, 1927-1931 (2003).
[CrossRef]

C. Genet, M. P. van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Commun. 225, 331-336 (2003).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

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

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Opt. Commun. (2)

M. Sarrazin and J. P. Vigneron, "Polarization effects in metallic films perforated with a bidimensional array of subwavelength rectangular holes," Opt. Commun. 240, 89-97 (2004).
[CrossRef]

C. Genet, M. P. van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Commun. 225, 331-336 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (2)

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Phys. Rev. B (4)

C. Genet, E. Altewischer, M. P. van Exter, and J. P. Woerdman, "Optical depolarization induced by arrays of subwavelength metal holes," Phys. Rev. B 71, 033409 (2005).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes," Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef] [PubMed]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of sub-wavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

Prog. Quantum Electron. (1)

F. Le Roy-Brehonnet and B. Le Jeune, "Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties," Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Other (4)

Here, transverse is used to indicate the direction orthogonal to the array surface normal.

The imaginary part of B vanishes for any integration region that is mirror symmetric in the origin.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

H. Raether, Surface Plasmons (Springer, 1988).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup used to measure the Mueller matrix of the hole arrays. The incident polarization state is set by a quarter-wave plate (QWP) or half-wave plate (HWP); the output polarization state is measured with a combination of a QWP, polarizer (POL), and CCD. Lenses L form a symmetric telescope, with the hole array placed in its focus; an aperture ( A ) sets the maximum NA (opening angle θ A ) of the illumination. The two figures at the top contain sketches of the field profiles and local polarization (arrows) in transverse planes just in front of and behind a square hole array in which diagonal SP modes are excited. The incident circular field profile is distorted by the SP propagation to a shape with lobes in the propagation directions. The incident uniform vertical polarization is thus changed by the array to a nonuniform polarization distribution.

Fig. 2
Fig. 2

Theoretical DOP Π i behind the (a) square and (b) hexagonal hole array as a function of the dimensionless quantity ϴ (lower scale) and the NA of the incident coherent light (upper scale), calculated with the Lorentz model; the multiplication factors relating the two scales are Δ θ = 1 30 (square) and Δ θ = 1 90 (hexagonal). Triangles are used for Π 1 , squares for Π 2 , and circles for Π 3 .

Fig. 3
Fig. 3

Measured far-field transmission of the hexagonal hole array for illumination with (a) horizontally oriented linear-polarized light (as indicated by the arrow in the inset) and (b) circularly polarized light. The pictures span 23 ° × 17 ° each. The inset shows the orientation of the array (lattice spacing 886 nm). The (predominant) polarization of the output lobes is indicated by the arrows around the figure edges.

Fig. 4
Fig. 4

Measured DOP Π i behind the (a) square and (b) hexagonal hole array as a function of the NA of the incident coherent light. Triangles are used for Π 1 , squares for Π 2 , and circles for Π 3 . Note that, for the square array, Π 1 corresponds to a polarization along the array diagonals, i.e., the SP propagation directions; in Ref. [9] we used a different convention.

Fig. 5
Fig. 5

Measured transmission of the hexagonal hole array for illumination with an approximately plane wave at normal incidence (black curve). Also shown is a theoretical fit based on a Fano-type model (gray curve).

Fig. 6
Fig. 6

Measured DOP of the hexagonal hole array (points) compared with the best fit obtained from the Lorentz model (dashed curve) and two theoretical fits based on the Fano-type model: the dark solid curve belongs to values in agreement with the spectral fit of Fig. 5 with α = 6.1 , δ ω = 2.50 , and Δ θ = 0.012 ; the light solid curve corresponds to α = 0.98 , δ ω = 1.02 , and Δ θ = 0.14 .

Equations (11)

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

t ( θ ) = k f k ( θ ) e k e k T ,
A s , h f k ( θ ) 2 d θ x d θ y ,
B s , h f k ( θ ) f * ( θ ) d θ x d θ y .
ω res , k ( θ ) = ω 0 [ 1 sin ( θ k ) n eff ] ω 0 ( 1 + θ k n eff ) .
t ( ω ) 1 ( ω ω 0 + i Δ ω ) ,
t ( θ , ω ) k 1 ω ω res , k ( θ ) + i Δ ω = pairs f k ( θ ) ,
f k ( θ ) = C [ 1 ( θ k Δ θ + i ) Δ ω + 1 ( θ k Δ θ + i ) Δ ω ] = C Δ θ 2 θ k 2 + Δ θ 2 ,
A s = A h = π ϴ 2 1 + ϴ 2 ,
B s = 2 π arctan ( ϴ 2 2 1 + ϴ 2 ) ,
B h = ( 4 π 3 ) arctan ( 3 ϴ 2 4 1 + ϴ 2 )
f k ( θ , ω ) = C { 1 + α + i β 2 [ 1 δ ω ( θ k Δ θ ) + i + 1 δ ω + ( θ k Δ θ ) + i ] } .

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