Abstract

Generalized Green’s Function Surface Integral Equation Method (G-GFSIEM) is used to study propagation of surface plasmon polariton waves at interface of two semi-infinite metal-dielectric waveguides. Reflection, transmission, and scattering coefficients for structures with different dielectric constants are calculated by using this method and by using energy conservation law. Conditions where scattering coefficient is maximized or minimized are studied. It is found that by using appropriate materials with specified dielectric constants, structures with required reflection, transmission, and scattering coefficients can be designed.

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References

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  1. H. Raether, Surface plasmons on smooth and rough surfaces and on grating (Springer-Verlag, Berlin, 1988).
  2. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
    [CrossRef]
  3. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
    [CrossRef]
  4. S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, Singapore, 2009).
  5. R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
    [CrossRef]
  6. G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
    [CrossRef]
  7. M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
    [CrossRef]
  8. J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
    [CrossRef] [PubMed]
  9. J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
    [CrossRef]
  10. T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
    [CrossRef]
  11. J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
    [CrossRef]
  12. J. D. Jackson, Classical electrodynamics, (John Wiley & Sons, New York, 1999), p. 479.
  13. F. Paris, and J. Canas, Boundary element method-fundamentals and applications (Oxford University Press, 1997).
  14. E. D. Palik, ed., The Handbook of optical constants of solids (Academic Press, New York, NY 1997)
  15. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
    [CrossRef]

2009 (1)

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[CrossRef]

2008 (3)

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[CrossRef] [PubMed]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[CrossRef]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

2007 (2)

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[CrossRef]

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
[CrossRef]

2006 (2)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[CrossRef]

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

2005 (1)

1981 (1)

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[CrossRef]

Boltasseva, A.

Bozhevolnyi, S. I.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[CrossRef]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
[CrossRef]

Brongersma, M. L.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[CrossRef]

Chandran, A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

Elser, J.

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[CrossRef] [PubMed]

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

Hai, M.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

Hong-Gao, T.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

Jung, J.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[CrossRef]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[CrossRef]

Kjaer, K.

Larsen, M. S.

Leosson, K.

Liu, Y.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[CrossRef]

Maradudin, A. A.

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[CrossRef]

Nikolajsen, T.

Oulton, R. F.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[CrossRef]

Pei, W.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

Pile, D. F.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[CrossRef]

Podolskiy, V. A.

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[CrossRef] [PubMed]

Rahman, T. S.

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[CrossRef]

Schuller, J. A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[CrossRef]

Søndergaard, T.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[CrossRef]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[CrossRef]

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[CrossRef]

Yong, C.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

Zhang, X.

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[CrossRef]

Zhong-Tuan, M.

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

Zia, R.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[CrossRef]

Chin. Phys. Lett. (1)

M. Zhong-Tuan, W. Pei, C. Yong, T. Hong-Gao, and M. Hai, “Pure reflection and refraction of a surface polariton by a matched waveguide structure,” Chin. Phys. Lett. 23(9), 2545–2548 (2006).
[CrossRef]

J. Lightwave Technol. (1)

Mater. Today (1)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip- scale technology,” Mater. Today 9(7-8), 20–27 (2006).
[CrossRef]

Phys. Rev. B (4)

R. F. Oulton, D. F. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: Implication for nanoscale cavities,” Phys. Rev. B 76(3), 035408 (2007).
[CrossRef]

G. I. Stegeman, A. A. Maradudin, and T. S. Rahman, “Refraction of a surface polariton by an interface,” Phys. Rev. B 23(6), 2576–2585 (1981).
[CrossRef]

J. Jung and T. Søndergaard, “Green’s function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77(24), 245310 (2008).
[CrossRef]

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Gap plasmon-polariton nanoresonators: Scattering enhancement and launching of surface plasmon polaritons,” Phys. Rev. B 79(3), 035401 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100(6), 066402 (2008).
[CrossRef] [PubMed]

Phys. Status Solidi (1)

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi 244(10), 3448–3462 (2007) (b).
[CrossRef]

Phys. Today (1)

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface- plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

Other (5)

S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, Singapore, 2009).

J. D. Jackson, Classical electrodynamics, (John Wiley & Sons, New York, 1999), p. 479.

F. Paris, and J. Canas, Boundary element method-fundamentals and applications (Oxford University Press, 1997).

E. D. Palik, ed., The Handbook of optical constants of solids (Academic Press, New York, NY 1997)

H. Raether, Surface plasmons on smooth and rough surfaces and on grating (Springer-Verlag, Berlin, 1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the structure under study. Ω i , i = d1, d2, m1, and m2 present different regions. Solid lines present boundary between different regions. The boundaries B and B are, respectively, located at | x | = and | y | = . Reflection and transmission coefficients are calculated by using points A and B, respectively.

Fig. 2
Fig. 2

a) Magnetic field H z , and b) Electric field E y distributions at the interface for structure with ε d 1 = 2.25 , ε d 2 = 1 , ε m 1 = ε m 2 = 18.3 0.5 i . The wavelength of incident beam is λ = 632.8 n m .

Fig. 3
Fig. 3

Variation of reflection, transmission, and scattering coefficients for a structure with a) ε d 2 = 1 , ε m 1 = ε m 2 = 18.3 0.5 i vs. ε d 1 and b) ε d 1 = 1 , ε m 1 = ε m 2 = 18.3 0.5 i vs. ε d 2 . The wavelength of incident beam is λ = 632.8 n m .

Fig. 4
Fig. 4

Variation of RTS coefficients vs. ε d 1 (a, b) and ε d 2 (c, d). Dielectric constant of A and B are ε A = 20 2 i and ε B = 60 20 i , respectively. The wavelength of the incident beam is λ = 632.8 n m .

Fig. 5
Fig. 5

Variation of d 1 / d 2 vs. dielectric constant ε d for structures investigated in Fig. 4.

Fig. 6
Fig. 6

Variation of RTS coefficients vs. a) | ε m 1 | , b) | ε m 2 | , c) | ε m 1 | , and d) | ε m 2 | . Dielectric constants of A and B are ε A = 20 2 i and ε B = 60 20 i respectively. The wavelength of incident beam is λ = 632.8 n m .

Fig. 7
Fig. 7

Variation of RTS coefficient vs. wavelength of the incident light for two different structures depicted in the inset.

Fig. 8
Fig. 8

Profile of the magnetic field along z-axis, | H z | for wavelength of λ = 563.6 n m and for structures investigated in Fig. 7.

Fig. 9
Fig. 9

Variation of RTS coefficients vs. wavelength for structure consisted of air, Ag and silicon, Al.

Equations (31)

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[ ε ( r , ω ) . ε ( r , ω ) + 2 + k 0 2 ε ( r , ω ) ] H Z ( r , ω ) = 0.
[ ε ( r , ω ) . ε ( r , ω ) + 2 + k 0 2 ε ( r , ω ) ] g ( r , r , ω ) = δ ( r , r ) .
1 2 H Z , 1 ( r ) = B 1 , 2 ( g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y B 1 ( g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y ,
1 2 H Z , 2 ( r ) = B 2 , 1 ( g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ) d y + B 2 ( g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ) d y .
1 2 H Z , 1 ( r ) = B 1 , 2 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B 1 [ g ( r , r ) x H i ( r ) H i ( r ) x g ( r , r ) ] d y r B 1 [ g ( r , r ) x H r ( r ) H r ( r ) x g ( r , r ) ] d y .
1 2 H Z , 2 ( r ) = B 2 , 1 [ g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ] d y + t B 2 [ g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ] d y .
H i = N i exp ( i k s p p x ) exp ( k s p p y ε d 1 / ε m 1 ) H r = N r exp ( i k s p p x ) exp ( k s p p y ε d 1 / ε m 1 ) H t = N t exp ( i k s p p x ) exp ( k s p p y ε d 2 / ε m 2 ) ,
H i = N i exp ( i k s p p x ) exp ( k s p p y ε m 1 / ε d 1 ) H r = N r exp ( i k s p p x ) exp ( k s p p y ε m 1 / ε d 1 ) H t = N t exp ( i k s p p x ) exp ( k s p p y ε m 2 / ε d 2 ) .
H Z , 1 ( r ) = B 1 , 2 ( g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y B 1 ( g ( r , r ) x H i ( r ) H i ( r ) x g ( r , r ) ) d y r B 1 ( g ( r , r ) x H r ( r ) H r ( r ) x g ( r , r ) ) d y .
H Z , 2 ( r ) = B d 2 , 1 ( ε d 2 ε d 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y B m 2 , 1 ( ε m 2 ε m 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ) d y + t B 2 ( g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ) d y .
d d = Re ( [ ( ε d + ε m ) ] 1 / 2 / k 0 ε d ) ,
d m = Re ( [ ( ε d + ε m ) ] 1 / 2 / k 0 ε m ) .
δ ( r r ) H Z ( r ) d a = B i [ g ( r , r ) n . H Z ( r ) H Z ( r ) n . g ( r , r ) ] d l ,
H Z , d 1 ( r ) = B d 1 , 2 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B d 1 , m 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d x + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d x ,
0 = B m 1 , 2 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B m 1 , d 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d x + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d x .
g ( x , y , x , y = 0 + ) = ε m 1 ε d 1 g ( x , y , x , y = 0 ) ,
n g ( x , y , x , y = 0 + ) = n g ( x , y , x , y = 0 ) .
B m 1 , d 1 [ g ( r , r ) n m 1 . H m 1 ( r ) H m 1 ( r ) n m 1 . g ( r , r ) ] d x = B d 1 , m 1 [ g ( r , r ) n m 1 . H d 1 ( r ) H d 1 ( r ) n m 1 . g ( r , r ) ] d x = B d 1 , m 1 [ g ( r , r ) n d 1 . H d 1 ( r ) H d 1 ( r ) n d 1 . g ( r , r ) ] d x ,
H Z , 1 ( r ) = B d 1 , 2 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B m 1 , 2 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d y + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d y + B m 1 [ g ( r , r ) n m 1 . H Z , m 1 ( r ) H Z , m 1 ( r ) n m 1 . g ( r , r ) ] d x + B d 1 [ g ( r , r ) n d 1 . H Z , d 1 ( r ) H Z , d 1 ( r ) n d 1 . g ( r , r ) ] d x .
H Z , 1 ( r ) = B 1 , 2 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B 1 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y .
H 1 = H i + r H r + H S ,
H 2 = t H t + H S .
0 = B 1 , 2 [ g ( r A , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r A , r ) ] d y B 1 [ g ( r A , r ) x ( H i ( r ) + r H r ( r ) ) ( H i ( r ) + r H r ( r ) ) x g ( r A , r ) ] d y .
r = f i ( r A ) + B 12 [ g ( r A , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r A , r ) ] d y f r ( r A ) ,
f i ( r A ) = B 1 [ g ( r A , r ) x H i ( r ) H i ( r ) x g ( r A , r ) ] d y ,
f r ( r A ) = B 1 [ g ( r A , r ) x H r ( r ) H r ( r ) x g ( r A , r ) ] d y .
t = B 21 [ g ( r B , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r B , r ) ] d y f t ( r B ) ,
f t ( r B ) = B 2 [ g ( r B , r ) x H t ( r ) H t ( r ) x g ( r B , r ) ] d y .
1 2 H Z , 1 ( r ) = B 12 [ g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B 1 [ g ( r , r ) x H i ( r ) H i ( r ) x g ( r , r ) ] d y r B 1 [ g ( r , r ) x H r ( r ) H r ( r ) x g ( r , r ) ] d y .
1 2 H Z , 2 ( r ) = B 21 [ g ( r , r ) x H Z , 2 ( r ) H Z , 2 ( r ) x g ( r , r ) ] d y + t B 2 [ g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ] d y .
1 2 H Z , 1 ( r ) = B d 2 , 1 [ ε d 2 ε d 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y B m 2 , 1 [ ε m 2 ε m 1 g ( r , r ) x H Z , 1 ( r ) H Z , 1 ( r ) x g ( r , r ) ] d y + t B 2 [ g ( r , r ) x H t ( r ) H t ( r ) x g ( r , r ) ] d y .

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