Abstract

TM modes guided by thin-film waveguides with one or more magnetooptic and anisotropic layers (i.e., substrate, film, and cover layers) have been studied with emphasis on its nonreciprocal characteristics. Our analysis shows that the nonreciprocity of TM modes is the result of nonreciprocal Goos-Haenchen phase shifts at the cover–film and/or film–substrate interfaces. Contributions to the nonreciprocity defined as (N+N) from various regions are identified. Here N± are the effective indices of refraction of two counterpropagating modes. An accurate algebraic approximation for (N+N) has been obtained to provide physical insight into the problem and facilitate numerical calculations. Guidelines for designing nonreciprocal TM mode phase shifters are also listed.

© 1986 Optical Society of America

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References

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  1. M. S. Wang, J. D. Crow, “Studies of the Use of Gyrotropic and Anisotropic Materials for Mode Conversion in Thin-Film Optical Waveguide Applications,” J. Appl. Phys. 43, 1861 (1972).
    [CrossRef]
  2. J. Warner, “Faraday Optical Isolator/Gyrator Design in Planar Dielectric Form,” IEEE Trans. Microwave Theory Tech. MTT-21, 769 (1973).
    [CrossRef]
  3. J. Warner, “Nonreciprocal Magnetooptic Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 70 (1975).
    [CrossRef]
  4. S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
    [CrossRef]
  5. S. Yamamoto, T. Makimoto, “Circuit Theory for a Class of Anisotropic and Gytotropic Thin-Film Optical Waveguides and Design of Nonreciprocal Devices for Integrated Optics,” J. Appl. Phys. 45, 882 (1974).
    [CrossRef]
  6. S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090 (1972).
    [CrossRef]
  7. Y. Okamura, T. Megami, S. Yamamoto, “Integrated Optical Isolator and Circulator Using Nonreciprocal Phase Shifters: a Proposal,” Appl. Opt. 23, 1886 (1984).
    [CrossRef] [PubMed]
  8. Y. Okamura, T. Negami, S. Yamamoto, “A Design for a Nonreciprocal Phase Shifter,” Opt. Quantum Electron. 17, 195 (1985).
    [CrossRef]
  9. F. Auracher, H. H. Witte, “A New Design for an Integrated Optical Isolator,” Opt. Commun. 13, 435 (1975).
    [CrossRef]
  10. P. Hlawiczka, “A Gyrotropic Waveguide with Dielectric Boundaries: the Longitudinally Magnetized Case,” J. Phys. D 11, 1157 (1978).
    [CrossRef]
  11. P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).
  12. T. Mizumoto, Y. Naito, “Nonreciprocal Propagation Characteristics of YIG Thin Film,” IEEE Trans. Microwave Theory Tech. MTT-30, 922 (1982).
    [CrossRef]
  13. T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
    [CrossRef]
  14. P. K. Tien, “Light Waves in Thin Films and Integrated Optics,” Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  15. G. A. Bennett, C. L. Chen, “Wavelength Dispersion of Optical Waveguides,” Appl. Opt. 19, 1990 (1980).
    [CrossRef] [PubMed]
  16. A. E. Turner, R. L. Gunshor, S. Datta, “New Class of Materials for Optical Isolators,” Appl. Opt. 22, 3152 (1983).
    [CrossRef] [PubMed]
  17. L. A. Kolodziejski et al., MBE Growth of Films and Superlattices of Diluted Magnetic Semiconductors,” J. Vac. Sci Technol. B3, 714 (1985).

1986 (1)

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
[CrossRef]

1985 (2)

L. A. Kolodziejski et al., MBE Growth of Films and Superlattices of Diluted Magnetic Semiconductors,” J. Vac. Sci Technol. B3, 714 (1985).

Y. Okamura, T. Negami, S. Yamamoto, “A Design for a Nonreciprocal Phase Shifter,” Opt. Quantum Electron. 17, 195 (1985).
[CrossRef]

1984 (1)

1983 (1)

1982 (1)

T. Mizumoto, Y. Naito, “Nonreciprocal Propagation Characteristics of YIG Thin Film,” IEEE Trans. Microwave Theory Tech. MTT-30, 922 (1982).
[CrossRef]

1981 (1)

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

1980 (1)

1978 (1)

P. Hlawiczka, “A Gyrotropic Waveguide with Dielectric Boundaries: the Longitudinally Magnetized Case,” J. Phys. D 11, 1157 (1978).
[CrossRef]

1975 (2)

J. Warner, “Nonreciprocal Magnetooptic Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 70 (1975).
[CrossRef]

F. Auracher, H. H. Witte, “A New Design for an Integrated Optical Isolator,” Opt. Commun. 13, 435 (1975).
[CrossRef]

1974 (1)

S. Yamamoto, T. Makimoto, “Circuit Theory for a Class of Anisotropic and Gytotropic Thin-Film Optical Waveguides and Design of Nonreciprocal Devices for Integrated Optics,” J. Appl. Phys. 45, 882 (1974).
[CrossRef]

1973 (1)

J. Warner, “Faraday Optical Isolator/Gyrator Design in Planar Dielectric Form,” IEEE Trans. Microwave Theory Tech. MTT-21, 769 (1973).
[CrossRef]

1972 (2)

M. S. Wang, J. D. Crow, “Studies of the Use of Gyrotropic and Anisotropic Materials for Mode Conversion in Thin-Film Optical Waveguide Applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

1971 (1)

Auracher, F.

F. Auracher, H. H. Witte, “A New Design for an Integrated Optical Isolator,” Opt. Commun. 13, 435 (1975).
[CrossRef]

Bennett, G. A.

Biolsi, W. A.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

Blank, S. L.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

Chen, C. L.

Crow, J. D.

M. S. Wang, J. D. Crow, “Studies of the Use of Gyrotropic and Anisotropic Materials for Mode Conversion in Thin-Film Optical Waveguide Applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

Datta, S.

Grabbe, P.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

Gunshor, R. L.

Harada, T.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
[CrossRef]

Hlawiczka, P.

P. Hlawiczka, “A Gyrotropic Waveguide with Dielectric Boundaries: the Longitudinally Magnetized Case,” J. Phys. D 11, 1157 (1978).
[CrossRef]

P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).

Kirsch, S. T.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

Kolodziejski, L. A.

L. A. Kolodziejski et al., MBE Growth of Films and Superlattices of Diluted Magnetic Semiconductors,” J. Vac. Sci Technol. B3, 714 (1985).

Koyamada, Y.

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Makimoto, T.

S. Yamamoto, T. Makimoto, “Circuit Theory for a Class of Anisotropic and Gytotropic Thin-Film Optical Waveguides and Design of Nonreciprocal Devices for Integrated Optics,” J. Appl. Phys. 45, 882 (1974).
[CrossRef]

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Martin, R. J.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

Megami, T.

Mizumoto, T.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
[CrossRef]

T. Mizumoto, Y. Naito, “Nonreciprocal Propagation Characteristics of YIG Thin Film,” IEEE Trans. Microwave Theory Tech. MTT-30, 922 (1982).
[CrossRef]

Naito, Y.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
[CrossRef]

T. Mizumoto, Y. Naito, “Nonreciprocal Propagation Characteristics of YIG Thin Film,” IEEE Trans. Microwave Theory Tech. MTT-30, 922 (1982).
[CrossRef]

Negami, T.

Y. Okamura, T. Negami, S. Yamamoto, “A Design for a Nonreciprocal Phase Shifter,” Opt. Quantum Electron. 17, 195 (1985).
[CrossRef]

Okamura, Y.

Oochi, K.

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
[CrossRef]

Tien, P. K.

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

P. K. Tien, “Light Waves in Thin Films and Integrated Optics,” Appl. Opt. 10, 2395 (1971).
[CrossRef] [PubMed]

Turner, A. E.

Wang, M. S.

M. S. Wang, J. D. Crow, “Studies of the Use of Gyrotropic and Anisotropic Materials for Mode Conversion in Thin-Film Optical Waveguide Applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

Warner, J.

J. Warner, “Nonreciprocal Magnetooptic Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 70 (1975).
[CrossRef]

J. Warner, “Faraday Optical Isolator/Gyrator Design in Planar Dielectric Form,” IEEE Trans. Microwave Theory Tech. MTT-21, 769 (1973).
[CrossRef]

Witte, H. H.

F. Auracher, H. H. Witte, “A New Design for an Integrated Optical Isolator,” Opt. Commun. 13, 435 (1975).
[CrossRef]

Yamamoto, S.

Y. Okamura, T. Negami, S. Yamamoto, “A Design for a Nonreciprocal Phase Shifter,” Opt. Quantum Electron. 17, 195 (1985).
[CrossRef]

Y. Okamura, T. Megami, S. Yamamoto, “Integrated Optical Isolator and Circulator Using Nonreciprocal Phase Shifters: a Proposal,” Appl. Opt. 23, 1886 (1984).
[CrossRef] [PubMed]

S. Yamamoto, T. Makimoto, “Circuit Theory for a Class of Anisotropic and Gytotropic Thin-Film Optical Waveguides and Design of Nonreciprocal Devices for Integrated Optics,” J. Appl. Phys. 45, 882 (1974).
[CrossRef]

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Microwave Theory Tech. (3)

T. Mizumoto, Y. Naito, “Nonreciprocal Propagation Characteristics of YIG Thin Film,” IEEE Trans. Microwave Theory Tech. MTT-30, 922 (1982).
[CrossRef]

J. Warner, “Faraday Optical Isolator/Gyrator Design in Planar Dielectric Form,” IEEE Trans. Microwave Theory Tech. MTT-21, 769 (1973).
[CrossRef]

J. Warner, “Nonreciprocal Magnetooptic Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 70 (1975).
[CrossRef]

IEEE/OSA Lightwave Technol. (1)

T. Mizumoto, K. Oochi, T. Harada, Y. Naito, “Measurement of Optical Nonreciprocal Phase Shift in a Bi-substituted Gd3Fe5O12 Film and Application to Waveguide-type Optical Circulator,” IEEE/OSA Lightwave Technol. LT-4, 347 (1986).
[CrossRef]

J. Appl. Phys. (4)

S. T. Kirsch, W. A. Biolsi, S. L. Blank, P. K. Tien, R. J. Martin, P. Grabbe, “Semileaky Thin Film Optical Isolator,” J. Appl. Phys. 52, 3190 (1981).
[CrossRef]

S. Yamamoto, T. Makimoto, “Circuit Theory for a Class of Anisotropic and Gytotropic Thin-Film Optical Waveguides and Design of Nonreciprocal Devices for Integrated Optics,” J. Appl. Phys. 45, 882 (1974).
[CrossRef]

S. Yamamoto, Y. Koyamada, T. Makimoto, “Normal-Mode Analysis of Anisotropic and Gyrotropic Thin-Film Waveguides for Integrated Optics,” J. Appl. Phys. 43, 5090 (1972).
[CrossRef]

M. S. Wang, J. D. Crow, “Studies of the Use of Gyrotropic and Anisotropic Materials for Mode Conversion in Thin-Film Optical Waveguide Applications,” J. Appl. Phys. 43, 1861 (1972).
[CrossRef]

J. Phys. D (1)

P. Hlawiczka, “A Gyrotropic Waveguide with Dielectric Boundaries: the Longitudinally Magnetized Case,” J. Phys. D 11, 1157 (1978).
[CrossRef]

J. Vac. Sci Technol. (1)

L. A. Kolodziejski et al., MBE Growth of Films and Superlattices of Diluted Magnetic Semiconductors,” J. Vac. Sci Technol. B3, 714 (1985).

Opt. Commun. (1)

F. Auracher, H. H. Witte, “A New Design for an Integrated Optical Isolator,” Opt. Commun. 13, 435 (1975).
[CrossRef]

Opt. Quantum Electron. (1)

Y. Okamura, T. Negami, S. Yamamoto, “A Design for a Nonreciprocal Phase Shifter,” Opt. Quantum Electron. 17, 195 (1985).
[CrossRef]

Other (1)

P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).

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Figures (6)

Fig. 1
Fig. 1

Three configurations of magnetooptic thin-film waveguides.

Fig. 2
Fig. 2

Reflection of plane waves from a magnetooptic–isotropic interface.

Fig. 3
Fig. 3

Normalized b-V diagram of TM0 modes guided by magnetooptic thin-film waveguides (a = 7.00, c = 0.905, and d = 0.238).

Fig. 4
Fig. 4

(b+b) of TM0 modes guided by various magnetooptic waveguides (a = 7.00, c = 0.905, and d = 0.238).

Fig. 5
Fig. 5

Distribution of electric fields of TM0 mode (a = 7.00, c = 0.905, d = 0.238, δf = δs, = 0.1, and V = 2.004).

Fig. 6
Fig. 6

Contributions to the nonreciprocity of TM0 modes from the cover, film, and substrate layers (a = 26.45, c = 0.967, and d = 0.107).

Equations (45)

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( n 1 2 0 j δ 0 0 n 2 2 0 - j δ 0 n 3 2 ) .
N ( θ ) = n 1 2 n 3 2 - δ 2 n 3 2 sin 2 θ - n 1 2 cos 2 θ .
Γ ( θ ) = n a 2 n 1 2 N ( θ ) cos θ + ( n 1 2 n 3 - δ 2 ) n a 2 - N 2 ( θ ) sin 2 θ + j δ N ( θ ) sin θ n a 2 n 1 2 N ( θ ) cos θ - ( n 1 2 n 3 2 - δ 2 ) n a 2 - N 2 ( θ ) sin 2 θ - j δ N ( θ ) sin θ .
Γ ( θ ) = n a 2 n 1 2 N ( θ ) cos θ + j [ n a 2 δ N ( θ ) sin θ - ( n 1 2 n 3 2 - δ 2 ) N 2 ( θ ) sin 2 θ - n a 2 ] n a 2 n 1 2 N ( θ ) cos θ - j [ n a 2 δ N ( θ ) sin θ - ( n 1 2 n 3 2 - δ 2 ) N 2 ( θ ) sin 2 θ - n a 2 ] .
Φ ( θ ) = 2 tan - 1 [ n a 2 δ N ( θ ) sin θ - ( n 1 2 n 3 2 - δ 2 ) N 2 ( θ ) sin 2 θ - n a 2 n a 2 n 1 2 N ( θ ) cos θ ]
n 1 = n 3 ( 1 + Δ ) .
n 1 I 2 + j δ 1 E z = η 0 N H y ;
- j δ 1 E x + n 3 I 2 E z = 1 ω 0 d H y d x ;
j k 0 N E x + d E z d x = j ω μ 0 H z ;
E x = η 0 ξ I ( n 3 I 2 N H y - δ I k 0 d H y d x ) ;
E z = j η 0 ξ I ( δ I H y - n 1 I 2 k 0 d H y d x ) ;
d 2 H d x 2 + k 0 2 ( ξ I n 1 I 2 - n 3 I 2 n 1 I 2 N 2 ) H y = 0.
H y ( x ) = D exp ( - γ c x ) ,             for x h ,
H y ( x ) = B cos K f x + C sin K f x ,             for 0 x h ,
H y ( x ) = A exp ( γ s x ) ,             for - < x 0 ,
K f = k 0 ξ f n 1 f 2 - n 3 f 2 n 1 f 2 N 2 ,
γ I = k o n 3 I 2 n 1 I 2 N 2 - ξ I n 1 I 2 ,
E x = η 0 ξ f [ ( ( N n 3 f 2 B - δ f K f k 0 C ) cos K f x + ( N n 3 f 2 C + δ f K f k 0 B ) sin K f x ] ,
E z = j η 0 ξ f [ ( B δ f N - n 1 f 2 K f k 0 C ) cos K f x + ( C δ f N + n 1 f 2 K f k 0 B ) sin k f x ] ,
E x = η 0 ξ c ( n 3 c 2 N + δ c γ c k 0 ) D exp ( - γ c x ) ,
E z = j η 0 ξ c ( N δ c + n 1 c 2 γ c k 0 ) D exp ( - γ c x ) ,
E x = η 0 ξ s ( n 3 s 2 N - δ s γ s k 0 ) B exp ( γ s x ) ,
E z = j η 0 ξ s ( N δ s - n 1 s 2 γ s k 0 ) B exp ( γ s x ) .
tan K f h = P + Q 1 - P Q ,
P = k 0 ξ f n 1 f 2 K f ( n 1 s 2 γ s k 0 - N δ s ξ s + N δ f ξ f ) ,
Q = k 0 ξ f n 1 2 K f ( n 1 c 2 γ c k 0 + N δ c ξ c - N δ f ξ f ) .
a = ( n 3 s 2 - n 3 c 2 ) / ( n 3 f 2 - n 3 s 2 ) ;
b = ( N 2 - n 3 s 2 ) / ( n 3 f 2 - n 3 s 2 ) ;
c = n 3 s 2 / n 3 f 2 ;
d = n 3 c 2 / n 3 f 2 ;
V = k 0 h n 3 f 2 - n 3 s 2 .
b 0 = ( N 0 2 - n 3 s 2 ) / ( n 3 f 2 - n 3 s 2 ) .
P 0 = 1 c b 0 1 - b 0 , Q 0 = 1 d a + b 0 1 - b 0 , k f 0 = k 0 n 3 f 2 - n 0 2 , γ I 0 = k 0 N 0 2 - n I 2 ,             I = c or s ,
tan V 1 - b 0 = P 0 + Q 0 1 - P 0 Q 0 ,
P = P 0 + P 1 , Q = Q 0 + Q 1 , N = N 0 + N 1 ,
P 1 = P 0 [ - K f 1 K f 0 + γ s 1 γ s 0 - k 0 N 0 γ s 0 ( δ s n 3 s 2 - δ f n 4 s 2 n 3 f 4 ) ] ,
Q 1 = Q 0 [ - K f 1 K f 0 + γ c 1 γ c 0 + k 0 N 0 γ c 0 ( δ c n 3 c 2 - n 3 c 2 δ f n 3 f 4 ) ] ,
K f 1 = k 0 N n 3 f 2 - N 0 2 ( Δ f N 0 - N 1 ) ,
γ I 1 = k 0 N 0 N 0 2 - n 3 I 2 ( N 1 - Δ I N 0 )             I = c or s .
N 1 N 0 Δ f W [ V 1 - b 0 + 1 1 - b 0 ( P 0 1 + P 0 2 + Q 0 1 + Q 0 2 ) ] + Δ s W b 0 P 0 1 + P 0 2 + Δ c W ( a + b 0 ) Q 0 1 + Q 0 2 + 1 W b 0 b 0 + c 1 - c ( δ s n 3 s 2 - c δ f n 3 f 2 ) P 0 1 + P 0 2 + 1 W a + b 0 b 0 + c 1 - c ( d δ f n 3 f 2 - δ c n 3 c 2 ) Q 0 1 + Q 0 2 ,
W = V 1 - b 0 + 1 b 0 ( 1 - b 0 ) P 0 1 + P 0 2 + 1 + a ( a + b 0 ) ( 1 - b 0 ) Q 0 1 + Q 0 2 .
N + - N - = ( N + - N - ) c + ( N + - N - ) f + ( N + - N - ) s ,
( N + - N - ) c = - 2 δ c ( 1 - c ) ( 1 - b 0 ) W [ d 2 ( 1 - b 0 ) + a + b 0 ] ,
( N + - N - ) s = + 2 δ s ( 1 - c ) ( 1 - b 0 ) W [ c 2 ( 1 - b 0 ) + b 0 ] ,
( N + - N - ) f = - 2 δ f { ( 1 - c ) ( 1 - b 0 ) W [ c 2 c 2 ( 1 - b 0 ) + b 0 - d 2 d 2 ( 1 - b 0 ) + a + b 0 ] } .

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