Abstract

A method to determine the field scattered by a two-dimensional rectangular grating of finite length is developed. Both periodic or aperiodic and lossy or lossless structures can be treated. The grating is built up as a finite sequence of two alternating types of waveguide sections connected by means of step discontinuities. The mode-matching method is employed to determine the field scattered by a single step. In combining the scattering properties of the separate steps and waveguide sections to describe the entire grating, the scattering matrix formalism was employed. With the proposed method we have examined the 10- and the 20-period configurations described by Shigesawa and Tsuji [ IEEE Trans. Microwave Theory Tech. 37, 3– 14 ( 1989)] and by Liu and Chew [ IEEE Trans. Microwave Theory Tech. 39, 422– 430 ( 1991)], which show good agreement. Furthermore, we have successfully extended these configurations to gratings with 100 periods.

© 1995 Optical Society of America

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References

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  1. H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
    [CrossRef]
  2. Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
    [CrossRef]
  3. M. Tomita, “Thin-film waveguide with a periodic groove structure of finite extent,” J. Opt. Soc. Am. A 6, 1455–1464 (1989).
    [CrossRef]
  4. T. Rozzi, “A rigorous analysis of DFB lasers with large and aperiodic corrugations,” IEEE J. Quantum Electron. 27, 212–223 (1991).
    [CrossRef]
  5. C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 12, 1666–1698 (1976).
    [CrossRef]
  6. C. Vassallo, “Radiating normal modes of lossy planar waveguides,” J. Opt. Soc. Am. A 69, 311–316 (1979).
    [CrossRef]
  7. C. Vassallo, Théorie des Guides D’Ondes Electromagnétiques (Eyrolles, Paris, 1985), Chap. 3, pp. 151–248.
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 1, pp. 1–30.
  9. S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
    [CrossRef]
  10. W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
    [CrossRef]
  11. W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2: Numerical Analysis of TE-polarized fields,” Opt. Quantum Electron. 18,229–238 (1986).
    [CrossRef]
  12. H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
    [CrossRef]
  13. M. Abramovitz, I. A. Stegun, Handbook of Mathematics (Dover, New York, 1965), Chap. 22, pp. 778–779.
  14. P-P. Borsboom, “Field analysis of two-dimensional inte-grated-optical gratings,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).
  15. R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, 2nd ed., E. F. Beckenbach, ed., University of California Engineering Extension Series (McGraw-Hill, New York, 1961), pp. 282–337.
  16. P-P. Borsboom, H. J. Frankena, “Field analysis of two-dimensional focusing grating couplers,” J. Opt. Soc. Am. A 12, 1142–1146 (1995).
    [CrossRef]

1995 (1)

1991 (2)

T. Rozzi, “A rigorous analysis of DFB lasers with large and aperiodic corrugations,” IEEE J. Quantum Electron. 27, 212–223 (1991).
[CrossRef]

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

1989 (2)

M. Tomita, “Thin-film waveguide with a periodic groove structure of finite extent,” J. Opt. Soc. Am. A 6, 1455–1464 (1989).
[CrossRef]

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

1986 (3)

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2: Numerical Analysis of TE-polarized fields,” Opt. Quantum Electron. 18,229–238 (1986).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

1979 (1)

C. Vassallo, “Radiating normal modes of lossy planar waveguides,” J. Opt. Soc. Am. A 69, 311–316 (1979).
[CrossRef]

1976 (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 12, 1666–1698 (1976).
[CrossRef]

1975 (1)

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematics (Dover, New York, 1965), Chap. 22, pp. 778–779.

Beal, J. C.

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Biehlig, W.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2: Numerical Analysis of TE-polarized fields,” Opt. Quantum Electron. 18,229–238 (1986).
[CrossRef]

Borsboom, P-P.

P-P. Borsboom, H. J. Frankena, “Field analysis of two-dimensional focusing grating couplers,” J. Opt. Soc. Am. A 12, 1142–1146 (1995).
[CrossRef]

P-P. Borsboom, “Field analysis of two-dimensional inte-grated-optical gratings,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).

Chew, W. C.

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 12, 1666–1698 (1976).
[CrossRef]

Frankena, H. J.

Hehl, K.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Langbein, U.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Lederer, F.

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

Liu, Q.-H.

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Mahmoud, S. F.

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 1, pp. 1–30.

Redheffer, R.

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, 2nd ed., E. F. Beckenbach, ed., University of California Engineering Extension Series (McGraw-Hill, New York, 1961), pp. 282–337.

Rozzi, T.

T. Rozzi, “A rigorous analysis of DFB lasers with large and aperiodic corrugations,” IEEE J. Quantum Electron. 27, 212–223 (1991).
[CrossRef]

Shigesawa, H.

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematics (Dover, New York, 1965), Chap. 22, pp. 778–779.

Tomita, M.

Tsuji, M.

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

Vassallo, C.

C. Vassallo, “Radiating normal modes of lossy planar waveguides,” J. Opt. Soc. Am. A 69, 311–316 (1979).
[CrossRef]

C. Vassallo, Théorie des Guides D’Ondes Electromagnétiques (Eyrolles, Paris, 1985), Chap. 3, pp. 151–248.

IEEE J. Quantum Electron. (1)

T. Rozzi, “A rigorous analysis of DFB lasers with large and aperiodic corrugations,” IEEE J. Quantum Electron. 27, 212–223 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

H. Shigesawa, M. Tsuji, “Mode propagation through a step discontinuity planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-34, 205–212 (1986).
[CrossRef]

H. Shigesawa, M. Tsuji, “A new equivalent network method for analyzing discontinuity properties of open dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 3–14 (1989).
[CrossRef]

Q.-H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

S. F. Mahmoud, J. C. Beal, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-23, 193–198 (1975).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Quantum Electron. (2)

W. Biehlig, K. Hehl, U. Langbein, F. Lederer, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part I: operator formalism,” Opt. Quantum Electron. 18, 219–228 (1986).
[CrossRef]

W. Biehlig, “Light propagation in a planar dielectric slab waveguide with step discontinuities. Part 2: Numerical Analysis of TE-polarized fields,” Opt. Quantum Electron. 18,229–238 (1986).
[CrossRef]

Proc. IEEE (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 12, 1666–1698 (1976).
[CrossRef]

Other (5)

C. Vassallo, Théorie des Guides D’Ondes Electromagnétiques (Eyrolles, Paris, 1985), Chap. 3, pp. 151–248.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 1, pp. 1–30.

M. Abramovitz, I. A. Stegun, Handbook of Mathematics (Dover, New York, 1965), Chap. 22, pp. 778–779.

P-P. Borsboom, “Field analysis of two-dimensional inte-grated-optical gratings,” Ph.D. dissertation (Delft University of Technology, Delft, The Netherlands, 1994).

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, 2nd ed., E. F. Beckenbach, ed., University of California Engineering Extension Series (McGraw-Hill, New York, 1961), pp. 282–337.

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Figures (12)

Fig. 1
Fig. 1

Relief-type grating consisting of a cascade of junctions between two types of waveguide.

Fig. 2
Fig. 2

N-layer planar waveguide.

Fig. 3
Fig. 3

Two waveguides connected at a step discontinuity at z = 0. The x coordinates x = Xj,p denote the boundary between the domains Dj,p and Dj,p+1, where p = 1, …, N − 1 and j = 1, 2 denote the waveguides for z < 0 and z > 0, respectively.

Fig. 4
Fig. 4

TE0 mode incident upon a symmetrical grating consisting of a cascade of junctions between two types of waveguide that are 0.32 and 0.48 μm thick, respectively. The wavelength of the incident guided wave is λ0= 1.508 μm. The grating length is assumed to be L μm.

Fig. 5
Fig. 5

Reflection coefficient R0 of the TE0 mode as a function of the grating period (in wavelengths) for gratings consisting of 10 (solid curve) and 20 (dotted–dashed curve) periods.

Fig. 6
Fig. 6

Transmission coefficient T0 of the TE0 mode as a function of the grating period (in wavelengths) for gratings consisting of 10 (solid curve) and 20 (dashed curve) periods.

Fig. 7
Fig. 7

Reflected and transmitted radiation power as a function of the grating period (in wavelengths) for gratings consisting of 10 (solid curves) and 20 (dashed curves) periods.

Fig. 8
Fig. 8

Net power propagating in the positive z direction in the region where z < −L divided by the net power propagating in the same direction in the region where z > 0 for gratings of 10 (solid curve) and 20 (dashed curve) periods.

Fig. 9
Fig. 9

Reflection coefficient of the TE0 mode as a function of the grating period for a grating consisting of 100 periods.

Fig. 10
Fig. 10

Transmission coefficient of the TE0 mode as a function of the grating period for a grating consisting of 100 periods.

Fig. 11
Fig. 11

Reflected (solid curve) and transmitted (dashed curve) radiation power as a function of the grating period (in wavelengths) for a grating consisting of 100 periods.

Fig. 12
Fig. 12

Net power propagating to the right in front of the waveguide divided by the net power propagating to the right behind the grating of 100 periods.

Equations (36)

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E y ( x , z ) = m = 0 M 1 [ A m exp ( j κ m z ) + B m exp ( j κ m z ) ] ψ m ( x ) 0 k x 1 k z 1 [ a 1 ( k x 1 ) exp ( j k z 1 z ) + b 1 ( k x 1 ) exp ( j k z 1 z ) ] ψ A ( k x 1 , x ) d k x 1 0 k x N k z N [ a N ( k x N ) exp ( j k z N z ) + b N ( k x N ) exp ( j k z N z ) ] ψ B ( k x N , x ) d k x N .
ψ A ( x ) = { T N + ( k x 1 ) exp [ j k x N ( x X N 1 ) ] for x > X N 1 T p + ( k x 1 ) exp [ j k x p ( x X p 1 ) ] + R p + ( k x 1 ) × exp [ j k x p ( x X p 1 ) ] for X p 1 < exp [ j k x 1 ( x X 1 ) ] + R 1 + ( k x 1 ) × exp [ j k x 1 ( x X 1 ) ] for x < X 1 x < X p ,
ψ B ( x ) = { exp [ j k x N ( x X N 1 ) ] + R N ( k x N ) × exp [ j k x N ( x X N 1 ) ] for x > X N 1 T p ( k x N ) exp [ j k x p ( x X p 1 ) ] + R N ( k x N ) × exp [ j k x p ( x X p 1 ) ] for X p 1 < x < X p T 1 ( k x N ) exp [ j k x 1 ( x X 1 ) ] for x < X 1 .
ψ m ( x ) = { A m , N exp [ j k x N ( x X N 1 ) ] for x > X N 1 B m , p exp [ j k x p ( x X p ) ] + C m , p exp [ j k x p ( x X p ) ] for X p 1 < x < X p D m , 1 exp [ j k x 1 ( x X 1 ) ] for x < X 1 .
μ 1 ( x ) ψ m ( x ) ψ n ( x ) d x = δ m , n ,
μ 1 ( x ) ψ A , B ( x , k z ) ψ A , B ( x , k z ) d x = N k z a , b δ ( k z k z ) ,
ψ E = ψ A + ψ B ,
ψ O = ψ A ψ B ,
A m ( 1 ) + B m ( 1 ) = n = 0 M ( 2 ) 1 [ A n ( 2 ) + B n ( 2 ) ] F m n 0 k x 1 ( 2 ) k z ( 2 ) { a 1 ( 2 ) [ k x 1 ( 2 ) ] + b 1 ( 2 ) [ k x 1 ( 2 ) ] } F m A [ k x 1 ( 2 ) ] d k x 1 ( 2 ) 0 k x N ( 2 ) k z ( 2 ) { a N ( 2 ) [ k x N ( 2 ) ] + b N ( 2 ) [ k x N ( 2 ) ] } F m B [ k x N ( 2 ) ] d k x N ( 2 ) ,
2 π k x N ( 1 ) k z ( 1 ) R 1 1 , + [ k x 1 ( 1 ) ] { a 1 ( 1 ) [ k x 1 ( 1 ) ] + b 1 ( 1 ) [ k x 1 ( 1 ) ] } = n = 0 M ( 2 ) 1 [ A n ( 2 ) + B n ( 2 ) ] F A n [ k x 1 ( 1 ) ] 0 k x 1 ( 2 ) k z ( 2 ) { a 1 ( 2 ) [ k x 1 ( 2 ) ] + b 1 ( 2 ) [ k x 1 ( 2 ) ] } F A A [ k x 1 ( 1 ) , k x 1 ( 2 ) ] d k x 1 ( 2 ) 0 k x N ( 2 ) k z ( 2 ) { a N ( 2 ) [ k x N ( 2 ) ] + b N ( 2 ) [ k x N ( 2 ) ] } F A B [ k x 1 ( 1 ) , k x N ( 2 ) ] d k x N ( 2 ) k x 1 ( 1 ) k z ( 1 ) { a 1 ( 2 ) [ k x 1 ( 1 ) ] + b 1 ( 2 ) [ k x 1 ( 1 ) ] } F A A δ [ k x 1 ( 1 ) ] ,
2 π k x N ( 1 ) k z ( 1 ) R N 1 , [ k x N ( 1 ) ] { a N ( 1 ) [ k x N ( 1 ) ] + b N ( 1 ) [ k x N ( 1 ) ] } = n = 0 M ( 2 ) 1 [ A n ( 2 ) + B n 2 ] F B n [ k x N ( 1 ) ] 0 k x 1 ( 2 ) k z ( 2 ) { a 1 ( 2 ) [ k x 1 ( 2 ) ] + b 1 ( 2 ) [ k x 1 ( 2 ) ] } F B A [ k x N ( 1 ) , k x 1 ( 2 ) ] d k x 1 ( 2 ) 0 k x N ( 2 ) k z ( 2 ) { a N ( 2 ) [ k x N ( 2 ) ] + b N ( 2 ) [ k x N ( 2 ) ] } F B B [ k x N ( 1 ) , k x N ( 2 ) ] d k x N ( 2 ) k x N ( 1 ) k z ( 1 ) { a N ( 2 ) [ k x N ( 1 ) ] + b N ( 2 ) [ k x N ( 1 ) ] } F B B δ [ k x N ( 1 ) ] . ( 11 )
F m n = μ ( 1 ) ( x ) 1 ψ m ( 1 ) ( x ) ψ n ( 2 ) ( x ) d x ,
F m A = μ ( 1 ) ( x ) 1 ψ m ( 1 ) ( x ) ψ A ( 2 ) [ x , k x 1 ( 2 ) ] d x ,
F A B = μ ( 1 ) ( x ) 1 ψ A ( 1 ) [ x , k x 1 ( 1 ) ] ψ B ( 2 ) [ x , k x N ( 2 ) ] d x .
μ ( 1 ) ( x ) 1 ψ A ( 1 ) [ x , k x 1 ( 1 ) ] ψ A ( 2 ) [ x , k x 1 ( 2 ) ] d x = F A A [ k x 1 ( 1 ) , k x 1 ( 2 ) ] + F A A δ [ k x 1 ( 1 ) ] δ [ k x 1 ( 1 ) k x 1 ( 2 ) ] ,
μ ( 1 ) ( x ) 1 ψ B ( 1 ) [ x , k x N ( 1 ) ] ψ B ( 2 ) [ x , k x N ( 2 ) ] d x = F B B [ k x N ( 1 ) , k x N ( 2 ) ] + F B B δ [ k x N ( 1 ) ] δ [ k x N ( 1 ) k x N ( 2 ) ] .
F A A δ = π R 1 1 , + exp { j k x 1 ( 2 ) [ X 1 ( 1 ) X 1 ( 2 ) ] } + π R 1 2 , + exp { j k x 1 ( 2 ) [ x 1 ( 1 ) X 1 ( 2 ) ] } ,
T q ( u 1 , N ) ( 1 u 1 , N 2 ) 1 / 2 ( q = 0 , . . . , O A , B 1 ) ,
( W 2 + W 1 ) = ( S 11 S 12 S 21 S 22 ) ( W 2 + W 1 ) ,
( W k + 1 + W k ) = ( P 0 0 P ) ( W k + W k + 1 ) ,
P = ( P guid 0 0 P rad ) ,
P guid ( m ) = exp [ j κ m ( Z k + 1 Z k ) ] ,
P rad ( p , q ) = 1 1 T p ( u 1 ) T q ( u 1 ) exp [ j k z p ( z k + 1 z k ) ] d u 1 1 u 1 2 .
( W 2 + W 1 ) = ( S 11 ( 1 ) S 12 ( 1 ) S 21 ( 1 ) S 22 ( 1 ) ) ( W 1 + W 2 ) ,
( W 3 + W 2 ) = ( S 11 ( 2 ) S 12 ( 2 ) S 21 ( 2 ) S 22 ( 2 ) ) ( W 2 + W 3 ) .
( W 3 + W 1 ) = ( S 11 ( 3 ) S 12 ( 3 ) S 21 ( 3 ) S 22 ( 3 ) ) ( W 1 + W 3 ) .
S 11 ( 3 ) = S 11 ( 2 ) [ I S 12 ( 1 ) S 21 ( 2 ) ] 1 S 11 ( 1 ) ,
S 12 ( 3 ) = S 11 ( 2 ) [ I S 12 ( 1 ) S 21 ( 2 ) ] 1 S 11 ( 1 ) S 22 ( 2 ) + S 12 ( 2 ) ,
S 21 ( 3 ) = S 22 ( 1 ) [ I S 21 ( 2 ) S 12 ( 1 ) ] 1 S 21 ( 2 ) S 11 ( 1 ) + S 21 ( 1 ) ,
S 22 ( 3 ) = S 22 ( 1 ) [ I S 21 ( 2 ) S 12 ( 1 ) ] 1 S 22 ( 2 ) .
S 3 = S 1 * S 2 .
S 10 = S per * . . . * S per 10 .
S 10 = S per , 1 * . . . * S per , 10 , 10
R 0 ( P λ 0 ) = 10 10 log [ | B 0 ( 1 ) | 2 | A 0 ( 1 ) | 2 ] ( dB ) ,
T 0 ( P λ 0 ) = 10 10 log [ | A 0 ( 2 ) | 2 | A 0 ( 1 ) | 2 ] ( dB ) ,
ϒ = P z > 0 + P z > 0 P z < L + P z < 0 L ,

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