Abstract

A grating coupler is investigated for the case when the incident direction of a guided wave in a planar waveguide is inclined, within the guide plane, to the direction of a grating vector. The coupling and conversion between a TE, or TM, guided mode and the TE and TM radiative and nonradiative modes are considered in a weakly corrugated guide. The amplitude transport equation and the expressions for the leakage parameter, and the phase correction to a wave number of an unperturbed waveguide, are deduced using a quasi-optical technique. The power-conservation relation of a grating coupler for obliquely incident waves is established, and the coupling and conversion efficiencies are examined. It has been found that significant differences exist for excitation, or scattering, of waves at the normal and oblique incidences.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Martelucci, A. N. Chester, eds., Integrated Optics Physics and Applications (Plenum, New York, 1983), pp. 323–333.
  2. M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
    [CrossRef]
  3. A. Jacques, D. B. Ostrowsky, “The grating coupler: comparison of theoretical and experimental results,” Opt. Commun. 13, 74–77 (1975).
    [CrossRef]
  4. C. C. Ghizoni, B. U. Chen, C. L. Tang, “Theory and experiment on grating couplers for thin-film waveguides,” IEEE J. Quantum Electron. QE-12, 69–73 (1976).
    [CrossRef]
  5. D. G. Dalgoutte, C. D. W. Wilkinson, “Thin grating couplers integrated optics: an experimental and theoretical study,” for Appl. Opt. 14, 2983–2998 (1975).
    [CrossRef]
  6. K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
    [CrossRef]
  7. N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
    [CrossRef]
  8. V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron. 4, 872–875 (1975).
    [CrossRef]
  9. J. H. Harris, R. K. Winn, D. G. Dalgoutte, “Theory and design of periodic couplers,” Appl. Opt. 11, 2234–2241 (1972).
    [CrossRef] [PubMed]
  10. M. K. Barnoski, ed., Introduction to Integrated Optics (Plenum, New York, 1976), pp. 315–368.
  11. T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 16, 235–254 (1977).
    [CrossRef]
  12. S. R. Seshadri, “TE–TE mode coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211–220 (1981).
    [CrossRef]
  13. K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering at obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-12, 632–633 (1979).
    [CrossRef]
  14. S. R. Seshadri, M. C. Tsai, “Mode conversion of obliquely incident guided magnetic waves by a grating on a yttrium iron garnet film for the normal magnetization,” J. Appl. Phys. 56, 501–510 (1984).
    [CrossRef]
  15. F. K. Schwering, S. T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microwave Theory Tech. MTT-31, 199–208 (1983).
    [CrossRef]
  16. M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. I: Output coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).
  17. S. R. Seshadri, M. C. Tsai, “Quasi optics of the coupling of guided magnetic waves for the normal magnetization,” J. Appl. Phys. 52, 6401–6410 (1981).
    [CrossRef]
  18. S. R. Seshadri, “Symmetric first-order Bragg interactions in active dielectric waveguides,” Appl. Phys. 15, 337–389 (1978).
    [CrossRef]
  19. M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. II: Input coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).

1984

S. R. Seshadri, M. C. Tsai, “Mode conversion of obliquely incident guided magnetic waves by a grating on a yttrium iron garnet film for the normal magnetization,” J. Appl. Phys. 56, 501–510 (1984).
[CrossRef]

1983

F. K. Schwering, S. T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microwave Theory Tech. MTT-31, 199–208 (1983).
[CrossRef]

1981

S. R. Seshadri, M. C. Tsai, “Quasi optics of the coupling of guided magnetic waves for the normal magnetization,” J. Appl. Phys. 52, 6401–6410 (1981).
[CrossRef]

S. R. Seshadri, “TE–TE mode coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211–220 (1981).
[CrossRef]

1979

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering at obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-12, 632–633 (1979).
[CrossRef]

1978

S. R. Seshadri, “Symmetric first-order Bragg interactions in active dielectric waveguides,” Appl. Phys. 15, 337–389 (1978).
[CrossRef]

1977

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 16, 235–254 (1977).
[CrossRef]

1976

C. C. Ghizoni, B. U. Chen, C. L. Tang, “Theory and experiment on grating couplers for thin-film waveguides,” IEEE J. Quantum Electron. QE-12, 69–73 (1976).
[CrossRef]

1975

D. G. Dalgoutte, C. D. W. Wilkinson, “Thin grating couplers integrated optics: an experimental and theoretical study,” for Appl. Opt. 14, 2983–2998 (1975).
[CrossRef]

A. Jacques, D. B. Ostrowsky, “The grating coupler: comparison of theoretical and experimental results,” Opt. Commun. 13, 74–77 (1975).
[CrossRef]

V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron. 4, 872–875 (1975).
[CrossRef]

1973

K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
[CrossRef]

N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

1972

1970

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Cadilhac, M.

N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Chang, W. S. C.

K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
[CrossRef]

Chen, B. U.

C. C. Ghizoni, B. U. Chen, C. L. Tang, “Theory and experiment on grating couplers for thin-film waveguides,” IEEE J. Quantum Electron. QE-12, 69–73 (1976).
[CrossRef]

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Dalgoutte, D. G.

D. G. Dalgoutte, C. D. W. Wilkinson, “Thin grating couplers integrated optics: an experimental and theoretical study,” for Appl. Opt. 14, 2983–2998 (1975).
[CrossRef]

J. H. Harris, R. K. Winn, D. G. Dalgoutte, “Theory and design of periodic couplers,” Appl. Opt. 11, 2234–2241 (1972).
[CrossRef] [PubMed]

Ghizoni, C. C.

C. C. Ghizoni, B. U. Chen, C. L. Tang, “Theory and experiment on grating couplers for thin-film waveguides,” IEEE J. Quantum Electron. QE-12, 69–73 (1976).
[CrossRef]

Harris, J. H.

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Jacques, A.

A. Jacques, D. B. Ostrowsky, “The grating coupler: comparison of theoretical and experimental results,” Opt. Commun. 13, 74–77 (1975).
[CrossRef]

Kiselev, V. A.

V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron. 4, 872–875 (1975).
[CrossRef]

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Neviere, N.

N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Ogawa, K.

K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
[CrossRef]

Ostrowsky, D. B.

A. Jacques, D. B. Ostrowsky, “The grating coupler: comparison of theoretical and experimental results,” Opt. Commun. 13, 74–77 (1975).
[CrossRef]

Peng, S. T.

F. K. Schwering, S. T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microwave Theory Tech. MTT-31, 199–208 (1983).
[CrossRef]

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 16, 235–254 (1977).
[CrossRef]

Petit, R.

N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Rosenbaum, F. J.

K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
[CrossRef]

Saito, S.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering at obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-12, 632–633 (1979).
[CrossRef]

Sakaki, H.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering at obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-12, 632–633 (1979).
[CrossRef]

Sapori, B. L.

K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
[CrossRef]

Schwering, F. K.

F. K. Schwering, S. T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microwave Theory Tech. MTT-31, 199–208 (1983).
[CrossRef]

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, M. C. Tsai, “Mode conversion of obliquely incident guided magnetic waves by a grating on a yttrium iron garnet film for the normal magnetization,” J. Appl. Phys. 56, 501–510 (1984).
[CrossRef]

S. R. Seshadri, “TE–TE mode coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211–220 (1981).
[CrossRef]

S. R. Seshadri, M. C. Tsai, “Quasi optics of the coupling of guided magnetic waves for the normal magnetization,” J. Appl. Phys. 52, 6401–6410 (1981).
[CrossRef]

S. R. Seshadri, “Symmetric first-order Bragg interactions in active dielectric waveguides,” Appl. Phys. 15, 337–389 (1978).
[CrossRef]

M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. I: Output coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).

M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. II: Input coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).

Tamir, T.

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 16, 235–254 (1977).
[CrossRef]

Tang, C. L.

C. C. Ghizoni, B. U. Chen, C. L. Tang, “Theory and experiment on grating couplers for thin-film waveguides,” IEEE J. Quantum Electron. QE-12, 69–73 (1976).
[CrossRef]

Tsai, M. C.

S. R. Seshadri, M. C. Tsai, “Mode conversion of obliquely incident guided magnetic waves by a grating on a yttrium iron garnet film for the normal magnetization,” J. Appl. Phys. 56, 501–510 (1984).
[CrossRef]

S. R. Seshadri, M. C. Tsai, “Quasi optics of the coupling of guided magnetic waves for the normal magnetization,” J. Appl. Phys. 52, 6401–6410 (1981).
[CrossRef]

Vincent, P.

N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering at obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-12, 632–633 (1979).
[CrossRef]

Wilkinson, C. D. W.

D. G. Dalgoutte, C. D. W. Wilkinson, “Thin grating couplers integrated optics: an experimental and theoretical study,” for Appl. Opt. 14, 2983–2998 (1975).
[CrossRef]

Winn, R. K.

Wlodarczyk, M. T.

M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. I: Output coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).

M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. II: Input coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).

Appl. Opt.

Appl. Phys.

T. Tamir, S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 16, 235–254 (1977).
[CrossRef]

S. R. Seshadri, “TE–TE mode coupling at oblique incidence in a periodic dielectric waveguide,” Appl. Phys. 25, 211–220 (1981).
[CrossRef]

S. R. Seshadri, “Symmetric first-order Bragg interactions in active dielectric waveguides,” Appl. Phys. 15, 337–389 (1978).
[CrossRef]

Appl. Phys. Lett.

M. L. Dakss, L. Kuhn, P. F. Heidrich, B. A. Scott, “Grating couplers for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

for Appl. Opt.

D. G. Dalgoutte, C. D. W. Wilkinson, “Thin grating couplers integrated optics: an experimental and theoretical study,” for Appl. Opt. 14, 2983–2998 (1975).
[CrossRef]

IEEE J. Quantum Electron

K. Ogawa, W. S. C. Chang, B. L. Sapori, F. J. Rosenbaum, “A theoretical analysis of etched grating couplers for integrated optics,” IEEE J. Quantum Electron QE-9, 29–42 (1973).
[CrossRef]

IEEE J. Quantum Electron.

C. C. Ghizoni, B. U. Chen, C. L. Tang, “Theory and experiment on grating couplers for thin-film waveguides,” IEEE J. Quantum Electron. QE-12, 69–73 (1976).
[CrossRef]

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering at obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. QE-12, 632–633 (1979).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

F. K. Schwering, S. T. Peng, “Design of dielectric grating antennas for millimeter-wave applications,” IEEE Trans. Microwave Theory Tech. MTT-31, 199–208 (1983).
[CrossRef]

J. Appl. Phys.

S. R. Seshadri, M. C. Tsai, “Mode conversion of obliquely incident guided magnetic waves by a grating on a yttrium iron garnet film for the normal magnetization,” J. Appl. Phys. 56, 501–510 (1984).
[CrossRef]

S. R. Seshadri, M. C. Tsai, “Quasi optics of the coupling of guided magnetic waves for the normal magnetization,” J. Appl. Phys. 52, 6401–6410 (1981).
[CrossRef]

Opt. Commun.

A. Jacques, D. B. Ostrowsky, “The grating coupler: comparison of theoretical and experimental results,” Opt. Commun. 13, 74–77 (1975).
[CrossRef]

N. Neviere, P. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Sov. J. Quantum Electron.

V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron. 4, 872–875 (1975).
[CrossRef]

Other

S. Martelucci, A. N. Chester, eds., Integrated Optics Physics and Applications (Plenum, New York, 1983), pp. 323–333.

M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. I: Output coupler,” (University of Wisconsin-Madison, Madison, Wisc., 1984).

M. K. Barnoski, ed., Introduction to Integrated Optics (Plenum, New York, 1976), pp. 315–368.

M. T. Wlodarczyk, S. R. Seshadri, “Grating coupler for dielectric waveguides. II: Input coupler,” (University of Wisconsin-Madison, Madison, Wisc., 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 (16)

Fig. 1
Fig. 1

Geometry of the planar dielectric waveguide with the periodic structure of shallow grooves of an arbitrary profile formed in (a) upper, (b) lower surface of the guide.

Fig. 2
Fig. 2

Directions n ^ g , n ^ m ( n ), and n ^ p ( n ) of the guided mode exp[g(y sin θg + z cos θg)] and modes exp { i β m ( n ) [ y sin θ m ( n ) + z cos θ m ( n ) ] } , exp [ i β p ( n ) [ y sin θ p ( n ) + z cos θ p ( n ) ] }, with reference to the grating vector z ^K.

Fig. 3
Fig. 3

Zigzag model for the ray directions of a guided wave and the directions of radiation beams incident at the coupler from the cover and substrate regions.

Fig. 4
Fig. 4

Zigzag model for the ray directions of the guided wave propagating in the direction n ^ g, which is inclined at an angle θg to the z axis.

Fig. 5
Fig. 5

Closed surface A used in the power-conservation analysis of the periodic coupler at oblique incidence.

Fig. 6
Fig. 6

Grating profiles investigated in this paper. w/Λ/4 = 0.5 for the trapezoidal and asymmetric triangular gratings. In the case of a TM0 guided mode, the terms rectangular and sawtooth refer to the trapezoidal and asymmetric-triangular gratings, respectively, of w/Λ/4 = 0.01.

Fig. 7
Fig. 7

(a) Normalized leakage parameter Re q0λ and (b) normalized phase correction to the unperturbed wave number Im q0λ as a function of θg; c = 1, f = 3.0, s = 2.3; TE0 guided mode βg = 9.8206/λ, d = 0.1383 λ, K = 12.566/λ. Grating at the upper surface, tg/d = 0.05.

Fig. 8
Fig. 8

(a) Normalized leakage parameter Re q0λ and (b) normalized phase correction to the unperturbed wave number Im q0λ as a function of θg; c = 1, f = 2.3; TM0 guided mode βg = 10.0028/λ, d = 0.5 λ, K = 10.472/λ. Grating at the upper surface, tg/d = 0.05.

Fig. 9
Fig. 9

(a) Normalized leakage parameter Re q0λ and (b) normalized phase correction to the unperturbed wave number Im q0λ as a function of θg; c = 1.0, f = 16.0, ∊s = 10.758, d = 0.4 λ, K = 12.566/λ; TE0 guided mode βg = 24.369 λ. Grating at the upper surface, tg/d = 0.031.

Fig. 10
Fig. 10

(a) Normalized leakage parameter Re q0λ and (b) normalized phase correction to the unperturbed wave number Im q0λ as a function of θg; c = 1.0, f = 16.0, s = 10.758, d = 0.4 λ, K = 12.566/λ; TM0 guided mode βg = 24.16/λ. Grating at the upper surface, tg/d = 0.031.

Fig. 11
Fig. 11

Efficiency of I, mode coupling: TE radiation mode, TE0 guided mode; II, mode conversion: TM radiation mode, TE0 guided mode as a function of θg, for the excitation (a) from the cover region and (b) from the substrate region. Data of Fig. 7 have been used.

Fig. 12
Fig. 12

Efficiency of I, mode coupling: TE radiation mode, TE0 guided mode; II, mode conversion: TM radiation mode, TE0 guided mode as a function of θg; excitation from the substrate region. No radiation exists in the cover region. Data of Fig. 9 have been used.

Fig. 13
Fig. 13

Efficiency of I, mode coupling: TM radiation mode, TM0 guided mode; II, mode conversion: TE radiation mode, TM0 guided mode as a function of θg, for the excitation (a) from the cover region and (b) from the substrate region. Data of Fig. 8 have been used.

Fig. 14
Fig. 14

Efficiency of I, mode coupling: TM radiation mode, TM0 guided mode; II, mode conversion: TE radiation mode, TM0 guided mode as a function of θg; excitation from the substrate region. No radiation exists in the cover region. Data of Fig. 10 have been used.

Fig. 15
Fig. 15

Optimum coupler length Lopt as a function of θg, calculated for the coupler of Figs. 7 and 8. (a) TE0 guided mode, (b) TM0 guided mode.

Fig. 16
Fig. 16

Optimum coupler length Lopt as a function of θg calculated for the coupler of Figs. 9 and 10. (a) TE0 guide mode, (b) TM0 guided mode.

Equations (141)

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

x = d ( z ) = d ave + δ g ( z ) = d + δ { n = 1 [ η n exp ( - i n K z ) + η n * exp ( i n K z ) ] }
x = 0 ( z ) = δ g ( z ) = δ { n = 1 [ η n exp ( - i n K z ) + η n * exp ( i n K z ) ] }
× E = i ω H ,
× H = - i ω r E ,
E ( x , y , z ) = E 0 ( x , y , z ) + δ E 1 ( x , y , z ) + δ 2 E 2 ( x , y , z ) ,
H ( x , y , z ) = H 0 ( x , y , z ) + δ H 1 ( x , y , z ) + δ 2 H 2 ( x , y , z ) ,
× E i = i ω H i ,             i = 0 , 1 , 2 ,
× H i = i ω r E i ,             i = 0 , 1 , 2.
( 2 x 2 + 2 y 2 + 2 z 2 + ω 2 r ) { E x i H x i } = 0 ,             i = 0 , 1 , 2.
( 2 x 2 + ω 2 r ) E y i = 2 x y E x i + i ω z H x i ,
( 2 x 2 + ω 2 r ) E z i = 2 x z E x i - i ω y H x i ,
( 2 x 2 + ω 2 r ) H y i = - i ω r z E x i + 2 x y H x i ,
( 2 x 2 + ω 2 r ) H z i = i ω r y E x i + 2 x z H x i ,
E tan = E z + d x d z E x is continuous ,
H tan = H z + d x d z H x is continuous ,
E y is continuous ,
H y is continuous .
{ H x 0 c E x 0 c } ( x , y , z ) = B g 0 exp [ - α g ( x - d ) ] × exp [ i β g ( y sin θ g + z cos θ g ) ] ,             x > d ,
{ H x 0 f E x 0 f } ( x , y , z ) = { C g 0 exp [ i k g ( x - d ) ] + D g 0 exp [ - i k g ( x - d ) ] } × exp [ i β g ( y sin θ g + z cos θ g ) ] ,             0 < x < d ,
{ H x 0 s E x 0 s } ( x , y , z ) = F g 0 exp ( γ g x ) exp [ i β g ( y sin θ g + z cos θ g ) ] ,             x < 0 ,
α g 2 = β g 2 - ω 2 c ,             k g 2 = ω 2 f - β g 2 ,             γ g 2 = β g 2 - ω 2 s .
C g 0 / D g 0 = ρ 0 J c ,
ρ 0 J c = exp ( - 2 i Φ g J c ) ,
Φ g H c = tan - 1 ( α g k g ) ,             Φ g E c = tan - 1 ( α g k g p c ) .
D g 0 / C g 0 = ρ 0 J s ,
ρ 0 J s = exp ( - 2 i Φ g J s ) exp ( 2 i k g d ) ,
Φ g H s = tan - 1 ( γ g k g ) ,             Φ g E s = tan - 1 ( γ g k g p s ) ,
p c = f / c ,             p s = f / s .
exp ( - 2 i Φ g J c ) = exp [ 2 i ( Φ g J s - k g d ) ] ,             J = E , H .
{ H x 1 c E x 1 c } = A r j ( 1 ) exp [ - i k c m ( 1 ) ( x - d ) ] exp { i β m ( 1 ) [ y sin θ m ( 1 ) + z cos θ m ( 1 ) ] } + n = 1 B m j ( n ) exp [ i k c m ( n ) ( x - d ) ] × exp { i β m ( n ) [ y sin θ m ( n ) + z cos θ m ( n ) ] } + B p j ( n ) exp [ - α p ( n ) ( x - d ) ] exp { i β p ( n ) × [ y sin θ p ( n ) + z cos θ p ( n ) ] } ,             x > d ,
{ H x 1 f E x 1 f } = n = 1 { C m j ( n ) exp [ - i k f m ( n ) ( x - d ) ] + D m j ( n ) × exp [ i k f m ( n ) ( x - d ) ] } × exp { i β m ( n ) [ y sin θ m ( n ) + z cos θ m ( n ) ] + { C p j ( n ) exp [ - i k f p ( n ) ( x - d ) ] × D p j ( n ) exp [ i k f p ( n ) ( x - d ) ] } exp { i β p ( n ) × [ y sin θ p ( n ) + z cos θ p ( n ) ] } ,             d > x > 0 ,
{ H x 1 s E x 1 s } = G r j ( 1 ) exp [ - i k s m ( 1 ) x ] exp { i β m ( 1 ) [ y sin θ m ( 1 ) + z cos θ m ( 1 ) ] } + n = 1 F m j ( n ) exp [ i k s m ( n ) x ] × exp { i β m ( n ) [ y sin θ m ( n ) + z cos θ m ( n ) ] } + F p j ( n ) exp [ γ p ( n ) x ] × exp { i β p ( n ) [ y sin θ p ( n ) + z cos θ p ( n ) ] } ,             x < 0 ,
β m ( n ) sin θ m ( n ) = β g sin θ g ,             β m ( n ) cos θ m ( n ) = β g cos θ g - n K ,
β p ( n ) sin θ p ( n ) = β g sin θ g ,             β p ( n ) cos θ p ( n ) = β g cos θ g + n K ,
k c m ( n ) = { ω 2 c - [ β m ( n ) ] 2 } 1 / 2 ,
k f m ( n ) = { ω 2 f - [ β m ( n ) ] 2 } 1 / 2 ,
k s m ( n ) = { ω 2 s - [ β m ( n ) ] 2 } 1 / 2 ,
α p ( n ) = { [ β p ( n ) ] 2 - ω 2 c } 1 / 2 ,
k f p ( n ) = { ω 2 f - [ β p ( n ) ] 2 } 1 / 2 ,
γ p ( n ) = { [ β p ( n ) ] 2 - ω 2 s } 1 / 2 .
sin θ m ( n ) E y m ( n ) + cos θ m ( n ) [ E z m ( n ) + d x d z E x 0 ] ,
H x m ( n ) is continuous at x = d ( x = 0 ) .
sin θ m ( n ) H y m ( n ) - cos θ m ( n ) [ H z m ( n ) + d x d z H x 0 ] ,
x H x m ( n ) + β m ( n ) β g cos [ θ g - θ m ( n ) ] η n 2 x 2 H x 0             be continuous at x = d ( x = 0 ) .
sin θ m ( n ) E y m ( n ) - cos θ m ( n ) [ E z m ( n ) + d x d z E x 0 ] ,
sin θ m ( n ) H y m ( n ) + cos θ m ( n ) [ H z m ( n ) + d x d z H x 0 ] ,
x E x m ( n ) be continuous at x = d ( x = 0 ) ,
r E x m ( n ) + i η n β m ( n ) β g sin [ θ g - θ m ( n ) ] ω 2 x 2 H x 0             be continuous at x = d ( x = 0 ) .
B m j ( 1 ) = r j A r j ( 1 ) + t j G r j ( 1 ) + η 1 P j J ( 1 ) B g 0 ,             n = 1 ,
B m j ( n ) = η n P j J ( n ) B g 0 ,             n > 1 ,
r j = - R m j ( 1 ) c + R m j ( 1 ) s exp [ 2 i k f m ( 1 ) d ] 1 - R m j ( 1 ) c R m j ( 1 ) s exp [ 2 i k f m ( 1 ) d ] ,
t j = [ 1 + R m j ( 1 ) c ] [ 1 - R m j ( 1 ) s ] 1 - R m j ( 1 ) c R m j ( 1 ) s exp [ 2 i k f m ( 1 ) d ] μ ,
μ = { 1 , j = h p c / p s , j = e ,
R m h ( n ) c = k f m ( n ) - k c m ( n ) k f m ( n ) + k c m ( n ) ,             R m h ( n ) s = k f m ( n ) - k s m ( n ) k f m ( n ) + k s m ( n ) .
R m e ( n ) c = k f m ( n ) - p c k c m ( n ) k f m ( n ) + p c k c m ( n ) ,             R m e ( n ) s = k f m ( n ) s - p s k s m k f m ( n ) + p s k s m ( n ) ,
B p j ( n ) = η n * S j J ( n ) B g 0 .
{ H x 2 c E x 2 c } = B g 2 exp [ - α g ( x - d ) ] exp [ i β g ( y sin θ g + z cos θ g ) ] ,             x > d ,
{ H x 2 f E x 2 f } = { C g 2 exp [ - i k g ( x - d ) ] + D g 2 exp [ i k g ( x - d ) ] } × exp [ i β g ( y sin θ g + z cos θ g ) ] ,             0 < x < d ,
{ H x 2 s E x 2 s } = F g 2 exp ( γ g x ) exp [ i β g ( y sin θ g + z cos θ g ) ] ,             x < 0.
sin θ g E y 2 - cos θ g { E z 2 + d d z × [ n = 1 η n exp ( - i n K z ) + c . c ] E x 1 } ,
sin θ g H y 2 - cos θ g { H z 2 + d d z × [ n = 1 η n exp ( - i n K z ) + c . c ] H x 1 } ,
H x 2 + n = 1 η n 2 2 x 2 H x 0 is continuous ,
x H x 2 + β g ( n = 1 η n * { i ω β m ( n ) sin [ θ g - θ m ( n ) ] c ( p c - 1 ) x E x m ( n ) + 1 β m ( n ) cos [ θ g - θ m ( n ) ] 2 x 2 H x m ( n ) } + η n { i ω β p ( n ) sin [ θ g - θ p ( n ) ] × c ( p c - 1 ) x E x p ( n ) + 1 β p ( n ) cos [ θ g - θ p ( n ) ] 2 x 2 H x p ( n ) } + 1 2 η n 2 1 β g 3 x 3 H x 0 )             is continuous .
sin θ g E y 2 + cos θ g { E z 2 + d d z × [ n = 1 η n exp ( - i n K z ) + c . c . ] E x 1 } ,
sin θ g H y 2 - cos θ g ( H z 2 + d x d z H x 1 ) ,
x E x 2 + n = 1 η n * β g β m ( n ) [ ( cos [ θ g - θ m ( n ) ] × { 2 x 2 - [ β m ( n ) ] 2 } + β m ( n ) β g ) E x m ( n ) + i sin [ θ g - θ m ( n ) ] ω H x m ( n ) x ] + n = 1 η n β g β p ( n ) [ ( cos [ θ g - θ p ( n ) ] { 2 x 2 - [ β p ( n ) ] 2 } + β p ( n ) β g ) E x p ( n ) + i sin [ θ g - θ p ( n ) ] ω H x p ( n ) x ] + n = 1 η n 2 2 x 3 E x 0             is continuous at x = d ( x = 0 ) ,
r E x 2 + n = 1 η n * β g β m ( n ) ( cos [ θ g - θ m ( n ) ] x [ r E x m ( n ) ] - i ω sin [ θ g - θ m ( n ) ] { 2 x 2 - [ β m ( n ) ] 2 } H x m ( n ) ) + n = 1 η n β g β p ( n ) ( cos [ θ g - θ p ( n ) ] x [ r E x p ( n ) ] - i ω sin [ θ g - θ p ( n ) ] { 2 x 2 - [ β p ( n ) ] 2 } E x p ( n ) ) + n = 1 η n 2 2 x 2 ( r E x 0 )             is continuous at x = d ( x = 0 ) .
C g 2 = D g 2 ρ 0 J c + τ J e c A r e ( 1 ) + τ J h c A r h ( 1 ) + τ J e s G r e ( 1 ) + τ J h s G r h ( 1 ) + Q 0 J B g 0 ,
τ H h c = η 1 * ( k g - i α g ) β g β r cos ( θ g - θ r ) ( 1 + r h ) ,
τ H e c = η 1 * ( k g - i α g ) β g β r sin ( θ g - θ r ) ( 1 - r e ) ,
τ H h s = η 1 * ( k g - i α g ) β g β r cos ( θ g - θ r ) t h ,
τ H e s = - η 1 * ( k g - i α g ) β g β r sin ( θ g - θ r ) t e ,
Q 0 H = - ( k g - i α g ) n = 1 η n 2 ( 2 i α g + i β g β m ( n ) { k c m ( n ) ω × sin [ θ g - θ m ( n ) ] P e H ( n ) - cos [ θ g - θ m ( n ) ] P h H ( n ) } + i β g β p ( n ) { i α p ( n ) ω sin [ θ g - θ p ( n ) ] × S e H ( n ) - cos [ θ g - θ p ( n ) ] S h H ( n ) } ) .
τ E e c = η 1 * p c - 1 p c β g β r 1 k g + i α g p c [ i β g β r ( 1 + r e ) - α g k c p c ( 1 - r e ) cos ( θ g - θ r ) ] ,
τ E h c = η 1 * p c - 1 p c β g β r sin ( θ g - θ r ) α g ω p c k g + i α g p c ( 1 + r h ) ,
τ E e s = η 1 * p c - 1 p c β g β r t e k g + i α g p c × [ i β g β r + α g k c p c cos ( θ g - θ r ) ] ,
τ E h s = η 1 * p c - 1 p c β g β r sin ( θ g - θ r ) α g ω p c k g + i g p c t h ,
Q 0 E = p c - 1 p c ( k g + i α g p c ) n = 1 η n 2 ( β g β m ( n ) × { cos [ θ g - θ m ( n ) ] α g k c m ( n ) p c + i β g β m ( n ) } P e E ( n ) + β g β p ( n ) i { cos [ θ g - θ p ( n ) ] α g α p ( n ) p c - β g β p ( n ) } S e E ( n ) + β g β m ( n ) sin [ θ g - θ m ( n ) ] α g ω p c P h E ( n ) + β g β p ( n ) sin [ θ g - θ p ( n ) ] α g ω p c S h E ( n ) - 2 i α g ( α g 2 p c + β g 2 ) ) .
ρ 2 J s D g 2 C g 2 = exp ( 2 i k g d ) exp ( - 2 i Φ g J s ) = ρ 0 J s ,             J = E , H .
C g = D g ρ 0 J + τ J e c A r e ( 1 ) + τ J h c A r h ( 1 ) + τ J e s G r e ( 1 ) + τ J h s G r h ( 1 ) + Q 0 J B g 0 ,             J = E , H ,
C g = C g 0 + δ 2 C g 2 ,             D g = D g 0 + δ 2 D g 2 .
D g ( z ) exp [ i β g ( sin θ g s b J + cos θ g s b J ) ] ,
s b J = 2 tan θ 0 d + s J c + s J s ,
s J c = 2 Φ g J c β g ,             s J s = 2 Φ g J s β g ,             J = E , H .
C g ( z - s b J cos θ g ) exp [ i β g ( sin θ g s b J + cos θ g s b J ) ] exp ( - 2 i Φ g J s + 2 i k g d ) .
D g ( z ) = C g ( z - s b J cos θ g ) exp [ - 2 i Φ g J s + 2 i k g d ] .
z B g 0 = q J e c A r e ( 1 ) + q J h c A r h ( 1 ) + q J e s G r e ( 1 ) + q J h s G r h ( 1 ) - q 0 J B g 0 ,
q J j c = τ J j c s b J cos θ g 2 k g k g - i α g p ,
q J j s = τ J j s s b J cos θ g 2 k g k g - i α g p ,
q 0 j = Q 0 J s b J cos θ g 2 k g k g - i α g p ,
s b J = 2 β g k g t eff J ,             J = E , H ,
t eff H = d + 1 α g + 1 γ g ,
t eff E = d + p c α g α g 2 + k g 2 α g 2 p c 2 + k g 2 + p s γ g γ g 2 + k g 2 γ g 2 p s 2 + k g 2 .
d d z B g 0 = - q 0 j B g 0 .
A S ( x , z ) · n ^ d a = 0.
z P g z J ( z ) = - [ S r x ( z , x = ) - S r x ( z , x = - ) ] ,
P g z J ( z ) = N J B g 0 ( z ) 2 ,             J = E , H ,
N E = 1 4 k g 2 + α g 2 p c 2 k g 2 c 2 f ω β g cos θ g t eff E ,
N H = 1 4 k g 2 + α g 2 k g 2 ω β g cos θ g t eff H .
S r x ( z ) = k c 2 ω ( ω β r ) 2 [ - A r h ( 1 ) ( z ) 2 + B m h ( 1 ) ( z ) 2 + c ( - A r e ( 1 ) ( z ) 2 + B m e ( 1 ) ( z ) 2 ) ] .
S r x ( z ) = k s 2 ω ( ω β r ) 2 [ G r h ( 1 ) ( z ) 2 - F m h ( 1 ) ( z ) 2 + s ( G r e ( 1 ) ( z ) 2 - F m e ( 1 ) ( z ) 2 ) ] .
d d z B g 0 ( z ) 2 = { B g 0 * ( z ) [ q J e c A r e ( 1 ) ( z ) + q J h c A r h ( 1 ) ( z ) + q J e s G r e ( 1 ) ( z ) + q J h s G r h ( 1 ) ( z ) ] + c . c . } - 2 Re q 0 J B g 0 ( z ) 2 ,             J = E , H .
d d z P g z J ( z ) = N J { B g 0 * ( z ) [ q J e c A r e ( 1 ) ( z ) + q J h c A r h ( 1 ) ( z ) + q J e s G r e ( 1 ) ( z ) + q J h s G r h ( 1 ) ( z ) ] + c . c . } - 2 Re q 0 J P g z J ( z ) .
B g 0 ( z ) = B g 0 ( 0 ) exp ( - q 0 z ) .
κ j J out ( k ) ( L ) = P r x j ( k ) / P g z J ( L ) .
κ j J out ( k ) ( L ) = N j l ( k ) η k 2 P j J ( k ) 2 N J 2 Re q 0 J [ 1 - exp ( - 2 Re q 0 J L ) ] ,
N h c ( k ) = k c ( k ) 2 ω [ ω β m ( k ) ] 2 ,
N e c ( k ) = c N h c ( k ) ,
N h s ( k ) = k s ( k ) 2 ω [ ω β m ( k ) ] 2 ,
N e s ( k ) = s N h s ( k ) .
2 N H Re q 0 H = ( k c 2 ω ) ( ω β r ) 2 η 1 2 [ cos 2 ( θ g - θ r ) ] P ˜ h H ( 1 ) 2 + c sin 2 ( θ g - θ r ) 2 P ˜ e H ( 1 ) 2 ] ,
P h H ( 1 ) = P ˜ h H ( 1 ) cos [ θ g - θ m ( 1 ) ] ,             P e H ( 1 ) = P ˜ e H ( 1 ) cos [ θ g - θ m ( 1 ) ] .
κ h H out ( 1 ) = 1 1 + c tan 2 ( θ g - θ r ) | P ˜ e H ( 1 ) P ˜ h H ( 1 ) | 2 × [ 1 - exp ( - 2 Re q 0 H L ) ]
κ e H out ( 1 ) = c c + cot 2 ( θ g - θ r ) | P ˜ h H ( 1 ) P ˜ e H ( 1 ) | 2 × [ 1 - exp ( - 2 Re q 0 H L )
θ g C c ( n ) = cos - 1 ( n 2 K 2 + α g 2 2 n K β g )
θ g C s ( n ) = cos - 1 ( n 2 K 2 + γ g 2 2 n K β g )
B g 0 ( z ) = q J j q 0 J + i β 2 exp ( - q 0 J Z ) × 0 z V j ( z ) exp [ ( i β 2 + q 0 J ) z ] d z .
κ j J in ( L ) P g z J ( L ) / P r x j ,
κ j J in ( L ) = N J q J j 2 | 0 L V j ( z ) exp ( Re q 0 J z ) d z | 2 Re 2 q 0 J N j exp ( 2 Re q 0 J L ) 0 L V j ( z ) 2 d z ,
κ h H in = 2 1 1 + c tan 2 ( θ g - θ r ) | P e H ( 1 ) P h H ( 1 ) | 2 F ( Re q 0 H L )
κ e H in = 2 c c + cot 2 ( θ g - θ r ) | P h H ( 1 ) P e H ( 1 ) | 2 F ( Re q 0 H L )
F ( Re q 0 J L ) = [ 1 - exp ( - Re q 0 J L ) ] 2 Re q O J L ,             J = E , H .
δ 0 : E z 0 , H z 0 , E y 0 , H y 0 are continuous ,
δ 1 : E z 1 + [ n = 1 η n exp ( - i n K z ) + c . c . ] x E z 0 + d d z [ n = 1 η n exp ( - i n K z ) + c . c . ] E x 0             is continuous ,
H z 1 + [ n = 1 η n exp ( - i n K z ) + c . c . ] x H z 0 + d d z [ n = 1 η n exp ( - i n K z ) + c . c . ] H x 0             is continuous ,
E y 1 + [ n = 1 η n exp ( - i n K z ) + c . c . ] E y 0 x             is continuous ,
H y 1 + [ n = 1 η n exp ( - i n K z ) + c . c . ] H y 0 x             is continuous ,
δ 2 : E z 2 + [ n = 1 η n exp ( - i n K z ) + c . c . ] E z 1 x + 1 2 [ n = 1 η n exp ( - i n K z ) + c . c . ] 2 2 E z 0 x 2 + d d z [ n = 1 η n exp ( - i n K z ) + c . c . ] × { E x 1 + [ n = 1 η n exp ( - i n K z ) + c . c . ] E x 0 x }             is continuous ,
H z 2 + [ n = 1 η n exp ( - i n K z ) + c . c . ] H z 1 x + 1 2 [ n = 1 η n exp ( - i n K z ) + c . c . ] 2 H z 0 x 2 + d d z [ n = 1 η n exp ( - i n K z ) + c . c . ] × { H x 1 + [ n = 1 η n exp ( - i n K z ) + c . c . ] H x 0 x }             is continuous ,
E y 2 + [ n = 1 η n exp ( - i n K z ) + c . c . ] E y 1 x + 1 2 [ n = 1 η n exp ( - i n K z ) + c . c . ] 2 E y 0 x 2             is continuous ,
H y 2 + [ n = 1 η n exp ( - i n K z ) + c . c . ] H y 1 x + 1 2 [ n = 1 η n exp ( - i n K z ) + c . c . ] 2 H y 0 x 2             is continuous .
P h H ( n ) = i β m ( n ) β g [ k f m ( n ) - k c m ( n ) ] cos [ θ g - θ m ( n ) ] 1 + R m h ( n ) s exp [ 2 i k f m ( n ) d ] 1 - R m h ( n ) c R m h ( n ) s exp [ 2 i k f m ( n ) d ] ,
P e H ( n ) = - i β m ( n ) β g sin [ θ g - θ m ( n ) ] ω k f m ( n ) ( p c - 1 ) { 1 - R m h ( n ) s exp [ 2 i k f m ( n ) d ] } [ k f m ( n ) + k c m ( m ) p c ] { 1 - R m e ( n ) c R m e ( n ) s exp [ 2 i k f m ( n ) d ] } ,
P h E ( n ) = - β m ( n ) β g sin [ θ g - θ m ( n ) ] α g ω [ k f m ( n ) - k c m ( n ) ] 1 + R m h ( n ) s exp [ 2 i k f m ( n ) d ] 1 - R m h ( n ) c R m h ( n ) s exp [ 2 i k f m ( n ) d ] ,
P e E ( n ) = β m ( n ) β g ( p c - 1 ) { i β g β m ( n ) + α g k f m ( n ) cos [ θ g - θ m ( n ) ] } R m e ( n ) s exp [ 2 i k f m ( n ) d ] + { i β g β m ( n ) - α g k f m ( n ) cos [ θ g - θ m ( n ) ] } [ k f m ( n ) + k c m ( m ) p c ] { 1 - R m e ( n ) c R m e ( n ) s exp [ 2 i k f m ( n ) d ] } ,
S h H ( n ) = i β p ( n ) β g cos [ θ g - θ p ( n ) ] [ k f p ( n ) - i α p ( n ) ] ( 1 + exp { 2 i [ k f p ( n ) d - Φ p h ( n ) s ] } ) 1 - exp { 2 i [ k f p ( n ) d - Φ p h ( n ) c - Φ p h ( n ) s ] } ,
S e H ( n ) = - i β p ( n ) β g sin [ θ g - θ p ( n ) ] k f p ( n ) ( α g 2 + k g 2 ) ( 1 - exp { 2 i [ k f p ( n ) d - Φ p e ( n ) s ] } ) ω c [ k f p ( n ) + i α p ( n ) p c ] ( 1 - exp { 2 i [ k f p ( n ) d - Φ p e ( n ) c - Φ p e ( n ) s ] } ) ,
Φ p j ( n ) c = tan - 1 [ α p ( n ) k f p ( n ) p 1 j ] ,             Φ p j ( n ) s = tan - 1 [ γ p ( n ) k f p ( n ) p 2 j ] ,
p 1 h = p 2 h = 1 ,             p 1 e = p c ,             p 2 e = p s ,
S h H ( n ) = - β p ( n ) β g sin [ θ g - θ p ( n ) ] α g ω [ k f p ( n ) - i α p ( n ) ] 1 + exp { 2 i [ k f p ( n ) d - Φ p h ( n ) s ] } 1 - exp { 2 i [ k f p ( n ) d - Φ p h ( n ) c - Φ p h ( n ) s ] } ,
S e E ( n ) = β p ( n ) β g ( p c - 1 ) { i β g β p ( n ) + α g k f p ( n ) cos [ θ g - θ p ( n ) ] } exp { 2 i [ k f p ( n ) d - Φ p e ( n ) s ] } + { i β g β p ( n ) - α g k f p ( n ) cos [ θ g - θ p ( n ) ] } [ k f p ( n ) + i α p ( n ) p c ] ( 1 - exp { 2 i [ k f p ( n ) d - Φ p e ( n ) c - Φ p e ( n ) s ] } ) .

Metrics