Abstract

We present a new integral equation method for calculating the electromagnetic modes of photonic crystal fiber (PCF) waveguides. Our formulation can easily handle PCFs with arbitrary hole geometries and irregular hole distributions, enabling optical component manufacturers to optimize hole designs as well as assess the effect of manufacturing defects. The method produces accurate results for both the real and imaginary parts of the propagation constants, which we validated through extensive convergence analysis and by comparison with previously published results.

© 2004 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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  12. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002).
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  13. A.H. Bouk, A. Cucinotta, F. Poli, and S. Selleri, “Dispersion properties of square-lattice photonic crystal fibers,” Opt. Express 12, 941–946 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-941
    [Crossref] [PubMed]
  14. A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
    [Crossref]
  15. X. Wang, J. Lou, C. Lu, C.L. Zhao, and W.T Ang, “Modeling of PCF with multiple reciprocity boundary element method,” Opt. Express 12, 961–966 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-961
    [Crossref] [PubMed]
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  17. T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightwave Technol. 21, 1793–1807 (2003).
    [Crossref]
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    [Crossref]
  19. D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, “Leakage properties of photonic crystal fibers,” Opt. Express 10, 1314–1319 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314
    [Crossref] [PubMed]
  20. H. Cheng, W.Y. Crutchfield, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers II: Acceleration techniques,” in preparation.
  21. A.W. Snyder and J.D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1996).
  22. I. Stakgold, Green’s Functions and Boundary Value Problems, (John Wiley & Sons, New York, 1979).
  23. V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Physics 60, 187–207 (1985).
    [Crossref]
  24. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, (McGraw-Hill, New York, 1953).
  25. R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, (Prentice-Hall, Englewood Cliffs, 1988).
  26. J. Stoer and R. Bulirsch, Introduction to numerical analysis, Second Edition, (Springer-Verlag, New York, 1993).
  27. S. Kapur and V. Rokhlin, “High-order corrected trapezoidal quadrature rules for singular functions,” SIAM J. Numer. Anal. 34, 1331–1356 (1997).
    [Crossref]
  28. J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, 669 (1988).
    [Crossref]
  29. H. Cheng, W.Y. Crutchfield, and L. Greengard, “Sensitivity analysis of photonic crystal fibers,” in preparation.

2004 (2)

2003 (4)

2002 (5)

2001 (2)

1999 (2)

1997 (3)

A. Figotin and Y.A. Godin, “The computation of spetra of some 2D photonic crystals,” J. Comput. Physics 136, 585–598 (1997).
[Crossref]

T.A. Burks, J.C. Knight, and P.S.J. Russell, “Endlessly single-mode photonic crystal fibers,” Opt. Lett. 22, 961–963 (1997).
[Crossref]

S. Kapur and V. Rokhlin, “High-order corrected trapezoidal quadrature rules for singular functions,” SIAM J. Numer. Anal. 34, 1331–1356 (1997).
[Crossref]

1988 (1)

J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, 669 (1988).
[Crossref]

1985 (1)

V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Physics 60, 187–207 (1985).
[Crossref]

Andres, M.V.

Andres, P.

Ang, W.T

Bennett, P.J.

Benson, T.M.

S.V. Boriskina, T.M. Benson, P. Sewell, and A.I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Bjarklev, A.

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic crystal fibers, (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

Bjarklev, A.S.

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic crystal fibers, (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

Boriskina, S.V.

S.V. Boriskina, T.M. Benson, P. Sewell, and A.I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Botten, L. C.

Bouk, A.H.

Broderick, N.G.R.

Broeng, J.

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic crystal fibers, (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

Bulirsch, R.

J. Stoer and R. Bulirsch, Introduction to numerical analysis, Second Edition, (Springer-Verlag, New York, 1993).

Burks, T.A.

Carrier, J.

J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, 669 (1988).
[Crossref]

Cheng, H.

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers II: Acceleration techniques,” in preparation.

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Sensitivity analysis of photonic crystal fibers,” in preparation.

Crutchfield, W.Y.

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Sensitivity analysis of photonic crystal fibers,” in preparation.

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers II: Acceleration techniques,” in preparation.

Cucinotta, A.

Eggleton, B.J.

Ferrando, A.

Ferrarini, D.

Feshbach, H.

P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, (McGraw-Hill, New York, 1953).

Figotin, A.

A. Figotin and Y.A. Godin, “The computation of spetra of some 2D photonic crystals,” J. Comput. Physics 136, 585–598 (1997).
[Crossref]

Godin, Y.A.

A. Figotin and Y.A. Godin, “The computation of spetra of some 2D photonic crystals,” J. Comput. Physics 136, 585–598 (1997).
[Crossref]

Greengard, L.

J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, 669 (1988).
[Crossref]

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers II: Acceleration techniques,” in preparation.

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Sensitivity analysis of photonic crystal fibers,” in preparation.

Guan, N.

Guenther, R. B.

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, (Prentice-Hall, Englewood Cliffs, 1988).

Habu, S.

Hale, A.

Himeno, K.

Joannopoulos, J.D.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, New Jersey, 1995).

Kapur, S.

S. Kapur and V. Rokhlin, “High-order corrected trapezoidal quadrature rules for singular functions,” SIAM J. Numer. Anal. 34, 1331–1356 (1997).
[Crossref]

Kerbage, C.

Knight, J.C.

Kominsky, D.

Kuhlmey, B. T.

Lee, J. W.

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, (Prentice-Hall, Englewood Cliffs, 1988).

Lou, J.

Love, J.D.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1996).

Lu, C.

Lu, T.

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Meade, R.D.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, New Jersey, 1995).

Miret, J.J.

Monro, T.M.

Morse, P.M.

P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, (McGraw-Hill, New York, 1953).

Nosich, A.I.

S.V. Boriskina, T.M. Benson, P. Sewell, and A.I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Osgood, R.M.

Pickrell, G.

Poli, F.

Renversez, G.

Richardson, D.J.

Rokhlin, V.

S. Kapur and V. Rokhlin, “High-order corrected trapezoidal quadrature rules for singular functions,” SIAM J. Numer. Anal. 34, 1331–1356 (1997).
[Crossref]

J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, 669 (1988).
[Crossref]

V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Physics 60, 187–207 (1985).
[Crossref]

Russell, P.

P. Russell “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Russell, P.S.J.

Selleri, S.

Sewell, P.

S.V. Boriskina, T.M. Benson, P. Sewell, and A.I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Silvestre, E.

Snyder, A.W.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1996).

Stakgold, I.

I. Stakgold, Green’s Functions and Boundary Value Problems, (John Wiley & Sons, New York, 1979).

Steel, M.J.

Stoer, J.

J. Stoer and R. Bulirsch, Introduction to numerical analysis, Second Edition, (Springer-Verlag, New York, 1993).

Stolen, R.

Takenaga, K.

Vincent, L.

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

Vincetti, L.

Wada, A.

Wang, X.

Westbrook, P.S.

White, T. P.

Windeler, R.S.

Winn, J.N.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, New Jersey, 1995).

Yevick, D.

Zhao, C.L.

Zoboli, M.

D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, “Leakage properties of photonic crystal fibers,” Opt. Express 10, 1314–1319 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314
[Crossref] [PubMed]

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

IEEE J. Selected Topics in Quantum Electron. (1)

S.V. Boriskina, T.M. Benson, P. Sewell, and A.I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

J. Comput. Physics (2)

A. Figotin and Y.A. Godin, “The computation of spetra of some 2D photonic crystals,” J. Comput. Physics 136, 585–598 (1997).
[Crossref]

V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Physics 60, 187–207 (1985).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (2)

Opt. Express (4)

Opt. Lett. (4)

Science (1)

P. Russell “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

SIAM J. Numer. Anal. (1)

S. Kapur and V. Rokhlin, “High-order corrected trapezoidal quadrature rules for singular functions,” SIAM J. Numer. Anal. 34, 1331–1356 (1997).
[Crossref]

SIAM J. Sci. Statist. Comput. (1)

J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm for particle simulations,” SIAM J. Sci. Statist. Comput. 9, 669 (1988).
[Crossref]

Other (9)

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Sensitivity analysis of photonic crystal fibers,” in preparation.

P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, (McGraw-Hill, New York, 1953).

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, (Prentice-Hall, Englewood Cliffs, 1988).

J. Stoer and R. Bulirsch, Introduction to numerical analysis, Second Edition, (Springer-Verlag, New York, 1993).

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic crystal fibers, (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press, Princeton, New Jersey, 1995).

H. Cheng, W.Y. Crutchfield, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers II: Acceleration techniques,” in preparation.

A.W. Snyder and J.D. Love, Optical Waveguide Theory, (Chapman & Hall, London, 1996).

I. Stakgold, Green’s Functions and Boundary Value Problems, (John Wiley & Sons, New York, 1979).

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Figures (10)

Fig. 1.
Fig. 1.

Cross-section of a model PCF, whose longitudinal axis runs in the z-direction. V 0 denotes the glass matrix with index n 0. In this case, there are only four “holes” in the fiber V 1,…,V 4 made of materials with refractive indices n 1,…,n 4. At each point on a material interface, ν denotes the unit normal vector and τ denotes the unit tangent vector.

Fig. 2.
Fig. 2.

Six circular holes.

Fig. 3.
Fig. 3.

Convergence study for the second mode

Fig. 4.
Fig. 4.

Six cookie-shaped holes.

Fig. 5.
Fig. 5.

Convergence study of the PCF depicted in Fig.4 (h=6%)

Fig. 6.
Fig. 6.

A PCF with circular holes.

Fig. 7.
Fig. 7.

A PCF with irregular shaped holes.

Fig. 8.
Fig. 8.

Convergence study of the PCF depicted in Fig.7

Fig. 9.
Fig. 9.

Dispersion curve (real part).

Fig. 10.
Fig. 10.

Dispersion curve (imaginary part).

Tables (3)

Tables Icon

Table 1. Effective index of the PCF depicted in Fig. 2

Tables Icon

Table 2. Effective index of the PCF depicted in Fig. 4

Tables Icon

Table 3. Effective index of the fundamental mode of PCFs with one to three layers of irregular holes and irregular locations (Fig.7). The results in last row are for the PCF with three layers of regular holes (Fig.6).

Equations (53)

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

E ( x , y , z , t ) = E β ( x , y ) e i ( β z ω t ) = e i ( β z ω t ) ( E 1 β ( x , y ) , E 2 β ( x , y ) , E 3 β ( x , y ) ) , H ( x , y , z , t ) = H β ( x , y ) e i ( β z ω t ) = e i ( β z ω t ) ( H 1 β ( x , y ) , H 2 β ( x , y ) , H 3 β ( x , y ) ) .
[ ν × E β ] = 0 , [ ν × H β ] = 0 ,
2 E + ( k 2 β 2 ) E = 0 ,
2 H + ( k 2 β 2 ) H = 0 ,
[ E ] = 0 ,
[ H ] = 0 ,
[ β k 2 β 2 E τ ] = [ k vac k 2 β 2 H n ] ,
[ β k 2 β 2 H τ ] = [ k 2 k vac k 2 β 2 E n ] .
E j ( P ) = Γ j ( s E ( 2 , j ) G j ( P , Q ) σ j E ( Q ) + d E ( 2 , j ) G j ( P , Q ) n Q μ j E ( Q ) ) d s Q ,
H j ( P ) = Γ j ( s H ( 2 , j ) G j ( P , Q ) σ j H ( Q ) + d H ( 2 , j ) G j ( P , Q ) n Q μ j H ( Q ) ) d s Q .
E 0 ( P ) = j = 1 M Γ j ( s E ( 1 , j ) G 0 ( P , Q ) σ j E ( Q ) + d E ( 1 , j ) G 0 ( P , Q ) n Q μ j E ( Q ) ) d s Q ,
H 0 ( P ) = j = 1 M Γ j ( s H ( 1 , j ) G 0 ( P , Q ) σ j H ( Q ) + d H ( 1 , j ) G 0 ( P , Q ) n Q μ j H ( Q ) ) d s Q .
lim P P 0 P 0 Γ j P glass E 0 ( P ) = 1 2 d E ( 1 , j ) μ j E ( P 0 )
+ j = 1 N Γ j ( s E ( 1 , j ) G 0 ( P 0 , Q ) σ j E ( Q ) + d E ( 1 , j ) G 0 ( P 0 , Q ) n Q μ j E ( Q ) ) d s Q .
lim P P 0 P 0 Γ j P glass E 0 ( P ) τ = 1 2 d E ( 1 , j ) μ j E ( P 0 ) τ
+ j = 1 N Γ j ( s E ( 1 , j ) G 0 ( P 0 , Q ) τ σ j E ( Q ) + d E ( 1 , j ) 2 G 0 ( P 0 , Q ) τ n Q μ j E ( Q ) ) d s Q ,
lim P P 0 P 0 Γ j P glass E 0 ( P ) n P = 1 2 s E ( 1 , j ) σ j E ( P 0 )
+ j = 1 N Γ j ( s E ( 1 , j ) G 0 ( P 0 , Q ) n P σ j E ( Q ) + d E ( 1 , j ) 2 G 0 ( P 0 , Q ) n P n Q μ j E ( Q ) ) d s Q ,
lim P P 0 P 0 Γ j P V j E j ( P ) = 1 2 d E ( 2 , j ) μ j E ( P 0 )
+ Γ j ( s E ( 2 , j ) G j ( P 0 , Q ) σ j E ( Q ) + d E ( 2 , j ) G j ( P 0 , Q ) n Q μ j E ( Q ) ) d s Q .
lim P P 0 P 0 Γ j P V j E j ( P ) τ = 1 2 d E ( 2 , j ) μ j E ( P 0 ) τ
+ Γ j ( s E ( 2 , j ) G j ( P 0 , Q ) τ σ j E ( Q ) + d E ( 2 , j ) 2 G j ( P 0 , Q ) τ n Q μ j E ( Q ) ) d s Q ,
lim P P 0 P 0 Γ j P V j E j ( P ) n P = 1 2 s E ( 2 , j ) σ j E ( P 0 )
+ Γ j ( s E ( 2 , j ) G j ( P 0 , Q ) n P σ j E ( Q ) + d E ( 2 , j ) 2 G j ( P 0 , Q ) n P n Q μ j E ( Q ) ) d s Q ,
0 = 1 2 ( d E ( 1 , j ) + d E ( 2 , j ) ) μ j E ( P 0 )
+ j ' = 1 , j ' j N Γ j ' ( s E ( 1 , j ' ) G 0 ( P 0 , Q ) σ j ' E ( Q ) + d E ( 1 , j ' ) G 0 ( P 0 , Q ) n Q μ j ' E ( Q ) ) d s Q
+ Γ j ( [ s E ( 1 , j ) G 0 ( P 0 , Q ) s E ( 2 , j ) G j ( P 0 , Q ) ] · σ j E ( Q ) )
+ [ d E ( 1 , j ) G 0 ( P 0 , Q ) n Q d E ( 2 , j ) G j ( P 0 , Q ) n Q ] · μ j E ( Q ) ) d s Q ,
0 = 1 2 ( d H ( 1 , j ) + d H ( 2 , j ) ) μ j H ( P 0 )
+ j ' = 1 , j ' j N Γ j ' ( s H ( 1 , j ' ) G 0 ( P 0 , Q ) σ j ' H ( Q ) + d H ( 1 , j ' ) G 0 ( P 0 , Q ) n Q μ j ' H ( Q ) ) d s Q
+ Γ j ( [ s H ( 1 , j ) G 0 ( P 0 , Q ) s H ( 2 , j ) G j ( P 0 , Q ) ] · σ j H ( Q ) )
+ [ d H ( 1 , j ) G 0 ( P 0 , Q ) n Q d H ( 2 , j ) G j ( P 0 , Q ) n Q ] · μ j H ( Q ) ) d s Q ,
0 = 1 2 ( b t E ( 1 , j ) d E ( 1 , j ) + b t E ( 2 , j ) d E ( 2 , j ) ) μ j E ( P 0 ) τ
+ b t E ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s E ( 1 , j ' ) G 0 ( P 0 , Q ) τ σ j ' E ( Q ) + d E ( 1 , j ' ) 2 G 0 ( P 0 , Q ) τ n Q μ j ' E ( Q ) ) d s Q
+ Γ j ( [ b t E ( 1 , j ) s E ( 1 , j ) G 0 ( P 0 , Q ) τ b t E ( 2 , j ) s E ( 2 , j ) G j ( P 0 , Q ) τ ] · σ j E ( Q )
+ [ b t E ( 1 , j ) d E ( 1 , j ) 2 G 0 ( P 0 , Q ) τ n Q b t E ( 2 , j ) d E ( 2 , j ) 2 G j ( P 0 , Q ) τ n Q ] · μ j E ( Q ) ) d s Q
1 2 ( b t H ( 1 , j ) s H ( 1 , j ) + b n H ( 2 , j ) s H ( 2 , j ) ) σ j H ( P 0 )
+ b n H ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s H ( 1 , j ' ) G 0 ( P 0 , Q ) n P σ j ' E ( Q ) + d H ( 1 , j ' ) 2 G 0 ( P 0 , Q ) n P n Q μ j ' H ( Q ) ) d s Q
+ Γ j ( [ b n H ( 1 , j ) s H ( 1 , j ) G 0 ( P 0 , Q ) n P b n H ( 2 , j ) s H ( 2 , j ) G j ( P 0 , Q ) n P ] · σ j H ( Q )
+ [ b n H ( 1 , j ) d H ( 1 , j ) 2 G 0 ( P 0 , Q ) n P n Q b n H ( 2 , j ) d H ( 2 , j ) 2 G j ( P 0 , Q ) n P n Q ] · μ j H ( Q ) ) d s Q ,
0 = 1 2 ( b t H ( 1 , j ) d H ( 1 , j ) + b t H ( 2 , j ) d H ( 2 , j ) ) μ j H ( P 0 ) τ
+ b t H ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s H ( 1 , j ' ) G 0 ( P 0 , Q ) τ σ j ' H ( Q ) + d H ( 1 , j ' ) 2 G 0 ( P 0 , Q ) τ n Q μ j ' H ( Q ) ) d s Q
+ Γ j ( [ b t H ( 1 , j ) s H ( 1 , j ) G 0 ( P 0 , Q ) τ b t H ( 2 , j ) s H ( 2 , j ) G j ( P 0 , Q ) τ ] · σ j H ( Q )
+ [ b t H ( 1 , j ) d H ( 1 , j ) 2 G 0 ( P 0 , Q ) τ n Q b t H ( 2 , j ) d H ( 2 , j ) 2 G j ( P 0 , Q ) τ n Q ] · μ j H ( Q ) ) d s Q
1 2 ( b t E ( 1 , j ) s E ( 1 , j ) + b n E ( 2 , j ) s E ( 2 , j ) ) σ j E ( P 0 )
+ b n E ( 1 , j ) j ' = 1 , j ' j N Γ j ' ( s E ( 1 , j ' ) G 0 ( P 0 , Q ) n P σ j ' E ( Q ) + d E ( 1 , j ' ) 2 G 0 ( P 0 , Q ) n P n Q μ j ' E ( Q ) ) d s Q
+ Γ j ( [ b n E ( 1 , j ) s E ( 1 , j ) G 0 ( P 0 , Q ) n P b n E ( 2 , j ) s E ( 2 , j ) G j ( P 0 , Q ) n P ] · σ j E ( Q )
+ [ b n E ( 1 , j ) d E ( 1 , j ) 2 G 0 ( P 0 , Q ) n P n Q b n E ( 2 , j ) d E ( 2 , j ) 2 G j ( P 0 , Q ) n P n Q ] · μ j E ( Q ) ) d s Q ,
b n E ( 1 , j ) d E ( 1 , j ) = b n E ( 2 , j ) , d E ( 2 , j ) b n H ( 1 , j ) d H ( 1 , j ) = b n H ( 2 , j ) d H ( 2 , j ) ,
b n E ( 1 , j ) s E ( 1 , j ) + b n E ( 2 , j ) s E ( 2 , j ) 0 , b n H ( 1 , j ) s H ( 1 , j ) + b n H ( 2 , j ) s H ( 2 , j ) 0 .
A ( β ) · x = 0 .
L = 20 ln ( 10 ) · 2 π λ · ( n eff ) · 10 9 ,
r ( θ ) c i r

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