Abstract

We propose a field-stitching boundary element method (FSBEM) for diffraction analyses for large diffractive optical elements (DOEs). A field stitching (FS) method separately computes the divided sections of a large DOE and provides the field distribution without field mismatching. However, a divided section must be largely overlapped, which is not efficient in terms of computational resources. In the FSBEM, in addition to spatial division, the electromagnetic field is computed for each component—the reflected, difference, and stitching fields—for rapid convergence of the solution. We validate the FSBEM by computing a one-dimensional DOE and evaluating the convergence property. The sample structure is also computed by the conventional FS method, and the advantages of the FSBEM are discussed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. D. Prather, S. Shi, and J. Bergey, “Field stitching algorithm for the analysis of electrically large diffractive optical elements,” Opt. Lett. 24, 273–275 (1999).
    [Crossref]
  27. J. Sugisaka, T. Yasui, and K. Hirayama, “Expansion of the difference-field boundary element method for numerical analyses of various local defects in periodic surface-relief structures,” J. Opt. Soc. Am. A 32, 751–763 (2015).
    [Crossref]
  28. J. Sugisaka, T. Yasui, and K. Hirayama, “Efficient analysis of diffraction grating with 10000 random grooves by difference-field boundary element method,” IEICE Trans. Electron. 100, 27–36 (2017).
    [Crossref]
  29. J. Sugisaka, T. Yasui, and K. Hirayama, “Fast actual-size vectorial simulation of concave diffraction gratings with structural randomness,” J. Opt. Soc. Am. A 34, 2157–2164 (2017).
    [Crossref]
  30. N. Morita, N. Kumagai, and J. R. Mautz, Integral equation methods for electromagnetics, (Artech House, 1990).

2017 (2)

J. Sugisaka, T. Yasui, and K. Hirayama, “Efficient analysis of diffraction grating with 10000 random grooves by difference-field boundary element method,” IEICE Trans. Electron. 100, 27–36 (2017).
[Crossref]

J. Sugisaka, T. Yasui, and K. Hirayama, “Fast actual-size vectorial simulation of concave diffraction gratings with structural randomness,” J. Opt. Soc. Am. A 34, 2157–2164 (2017).
[Crossref]

2015 (2)

J. Sugisaka, T. Yasui, and K. Hirayama, “Expansion of the difference-field boundary element method for numerical analyses of various local defects in periodic surface-relief structures,” J. Opt. Soc. Am. A 32, 751–763 (2015).
[Crossref]

Z. Peng, K. Lim, and J. Lee, “A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities,” J. Comput. Phys. 280, 626–642 (2015).
[Crossref]

2014 (1)

2005 (2)

S. Lee, M. Vouvakis, and J. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys. 203, 1–21 (2005).
[Crossref]

N. Chun-Hui, L. Zhi-Yuan, Y. Jia-Sheng, and G. Ben-Yuan, “Application of an approximate vectorial diffraction model to analysing diffractive micro-optical elements,” Chin. Phys. 14, 1136 (2005).
[Crossref]

2004 (1)

2003 (2)

2002 (1)

2001 (4)

2000 (1)

1999 (1)

1998 (3)

1997 (4)

1996 (2)

1994 (1)

1987 (1)

1983 (1)

Adam, K.

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

Banerjee, P.

Ben-Yuan, G.

N. Chun-Hui, L. Zhi-Yuan, Y. Jia-Sheng, and G. Ben-Yuan, “Application of an approximate vectorial diffraction model to analysing diffractive micro-optical elements,” Chin. Phys. 14, 1136 (2005).
[Crossref]

Bergey, J.

Chen, H.

Chun-Hui, N.

N. Chun-Hui, L. Zhi-Yuan, Y. Jia-Sheng, and G. Ben-Yuan, “Application of an approximate vectorial diffraction model to analysing diffractive micro-optical elements,” Chin. Phys. 14, 1136 (2005).
[Crossref]

Collino, P.

P. Collino, G. Delbue, P. Joly, and A. Piacentini, “A new interface condition in the non-overlapping domain decomposition method for the Maxwell equations,” Comput. Methods Appl. Mech. Eng. 148, 195–207 (1997).
[Crossref]

Collins, J.

Delbue, G.

P. Collino, G. Delbue, P. Joly, and A. Piacentini, “A new interface condition in the non-overlapping domain decomposition method for the Maxwell equations,” Comput. Methods Appl. Mech. Eng. 148, 195–207 (1997).
[Crossref]

Di, F.

Fan, S.

Feng, D.

Fiddy, M.

Gaylord, T.

Glytsis, E.

Grann, E.

Guofan, J.

Hirayama, K.

Hou, Z.

Jarem, J.

Jiang, J.

Jia-Sheng, Y.

N. Chun-Hui, L. Zhi-Yuan, Y. Jia-Sheng, and G. Ben-Yuan, “Application of an approximate vectorial diffraction model to analysing diffractive micro-optical elements,” Chin. Phys. 14, 1136 (2005).
[Crossref]

Jin, G.

Joly, P.

P. Collino, G. Delbue, P. Joly, and A. Piacentini, “A new interface condition in the non-overlapping domain decomposition method for the Maxwell equations,” Comput. Methods Appl. Mech. Eng. 148, 195–207 (1997).
[Crossref]

Kamiya, N.

Kettunen, V.

Kuittinen, M.

Kumagai, N.

N. Morita, N. Kumagai, and J. R. Mautz, Integral equation methods for electromagnetics, (Artech House, 1990).

Layet, B.

Lee, J.

Z. Peng, K. Lim, and J. Lee, “A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities,” J. Comput. Phys. 280, 626–642 (2015).
[Crossref]

S. Lee, M. Vouvakis, and J. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys. 203, 1–21 (2005).
[Crossref]

Lee, S.

S. Lee, M. Vouvakis, and J. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys. 203, 1–21 (2005).
[Crossref]

Lim, K.

Z. Peng, K. Lim, and J. Lee, “A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities,” J. Comput. Phys. 280, 626–642 (2015).
[Crossref]

Liu, H.

Mait, J.

Marom, E.

Mautz, J. R.

N. Morita, N. Kumagai, and J. R. Mautz, Integral equation methods for electromagnetics, (Artech House, 1990).

Mellin, S.

Minxian, W.

Mirotznik, M.

Moharam, M.

Morita, N.

N. Morita, N. Kumagai, and J. R. Mautz, Integral equation methods for electromagnetics, (Artech House, 1990).

Neureuther, A.

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

Nordin, G.

Peng, Z.

Z. Peng, K. Lim, and J. Lee, “A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities,” J. Comput. Phys. 280, 626–642 (2015).
[Crossref]

Piacentini, A.

P. Collino, G. Delbue, P. Joly, and A. Piacentini, “A new interface condition in the non-overlapping domain decomposition method for the Maxwell equations,” Comput. Methods Appl. Mech. Eng. 148, 195–207 (1997).
[Crossref]

Pommet, D.

Prather, D.

Qiaofeng, T.

Shi, S.

Si, J.

Sugisaka, J.

Taghizadeh, M.

Testorf, M.

Turunen, J.

Vallius, T.

Vouvakis, M.

S. Lee, M. Vouvakis, and J. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys. 203, 1–21 (2005).
[Crossref]

Wilson, D.

Yan, Y.

Yasui, T.

Ying, Y.

Yingbai, Y.

Zhang, B.

Zhang, J.

Zhi-Yuan, L.

N. Chun-Hui, L. Zhi-Yuan, Y. Jia-Sheng, and G. Ben-Yuan, “Application of an approximate vectorial diffraction model to analysing diffractive micro-optical elements,” Chin. Phys. 14, 1136 (2005).
[Crossref]

Zuo, H.

Zylberberg, Z.

Appl. Opt. (2)

Chin. Phys. (1)

N. Chun-Hui, L. Zhi-Yuan, Y. Jia-Sheng, and G. Ben-Yuan, “Application of an approximate vectorial diffraction model to analysing diffractive micro-optical elements,” Chin. Phys. 14, 1136 (2005).
[Crossref]

Comput. Methods Appl. Mech. Eng. (1)

P. Collino, G. Delbue, P. Joly, and A. Piacentini, “A new interface condition in the non-overlapping domain decomposition method for the Maxwell equations,” Comput. Methods Appl. Mech. Eng. 148, 195–207 (1997).
[Crossref]

IEICE Trans. Electron. (1)

J. Sugisaka, T. Yasui, and K. Hirayama, “Efficient analysis of diffraction grating with 10000 random grooves by difference-field boundary element method,” IEICE Trans. Electron. 100, 27–36 (2017).
[Crossref]

J. Comput. Phys. (2)

S. Lee, M. Vouvakis, and J. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys. 203, 1–21 (2005).
[Crossref]

Z. Peng, K. Lim, and J. Lee, “A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities,” J. Comput. Phys. 280, 626–642 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

E. Glytsis and T. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
[Crossref]

D. Pommet, M. Moharam, and E. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[Crossref]

K. Hirayama, E. Glytsis, and T. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[Crossref]

K. Hirayama, D. Wilson, E. Glytsis, and T. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[Crossref]

D. Prather, M. Mirotznik, and J. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[Crossref]

D. Prather, J. Mait, M. Mirotznik, and J. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[Crossref]

J. Sugisaka, T. Yasui, and K. Hirayama, “Fast actual-size vectorial simulation of concave diffraction gratings with structural randomness,” J. Opt. Soc. Am. A 34, 2157–2164 (2017).
[Crossref]

V. Kettunen, M. Kuittinen, and J. Turunen, “Effects of abrupt surface-profile transitions in nonparaxial diffractive optics,” J. Opt. Soc. Am. A 18, 1257–1260 (2001).
[Crossref]

F. Di, Y. Yingbai, J. Guofan, T. Qiaofeng, and H. Liu, “Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm,” J. Opt. Soc. Am. A 20, 1739–1746 (2003).
[Crossref]

M. Testorf and M. Fiddy, “Efficient optimization of diffractive optical elements based on rigorous diffraction models,” J. Opt. Soc. Am. A 18, 2908–2914 (2001).
[Crossref]

B. Layet and M. Taghizadeh, “Electromagnetic analysis of fan-out gratings and diffractive cylindrical lens arrays by field stitching,” J. Opt. Soc. Am. A 14, 1554–1561 (1997).
[Crossref]

T. Vallius, M. Kuittinen, J. Turunen, and V. Kettunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129–1135 (2002).
[Crossref]

J. Sugisaka, T. Yasui, and K. Hirayama, “Expansion of the difference-field boundary element method for numerical analyses of various local defects in periodic surface-relief structures,” J. Opt. Soc. Am. A 32, 751–763 (2015).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (1)

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

Other (1)

N. Morita, N. Kumagai, and J. R. Mautz, Integral equation methods for electromagnetics, (Artech House, 1990).

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Figures (9)

Fig. 1
Fig. 1 Cross section of large-sized DOEs. (a) surface-relief structure on the dielectric substrate, (b) metallic features on the dielectric substrate, (c) holographic DOE. Regions S1 and S2 are semi-infinite, and the incident wave propagates from the S2 side.
Fig. 2
Fig. 2 Geometries of the boundaries of (a) the large-sized DOE, PA+B; (b) the left-side section, PA, and (c) the right-side section, PB, of PA+B; and (d) the flat substrate with no features P0.
Fig. 3
Fig. 3 Field expressions in the regions (left column) and those on the boundaries (right column) for (a) the large-sized DOE, PA+B; (b) the left-side section, PA, and (c) the right-side section, PB, of PA+B; and (d) the flat substrate with no features P0.
Fig. 4
Fig. 4 Isolated scatterer that consists of a uniform medium. The scattered fields are given by Eqs. (7) and (8).
Fig. 5
Fig. 5 Total fields of (a) PA and (b) PB, (c) stitching field of PA+B, (d) total field of PA+B computed by the FSBEM, and (e) total field of PA+B directly computed by the DFBEM.
Fig. 6
Fig. 6 Error distributions of (a) the total fields computed by the FSBEM and (b) the total field without the stitching-field component.
Fig. 7
Fig. 7 Convergence properties of the total field computed by the FSBEM with respect to M, which corresponds to the extent of the analysis region for the computation of the stitching field.
Fig. 8
Fig. 8 Computation of the field distribution over the features (indicated by the red dashed lines) by the conventional FS algorithm. After computing the fields for sections (a) PA, (b) PB, and (c) PC, the field distribution of (d) the original structure is configured.
Fig. 9
Fig. 9 Convergence properties of the total field computed by the conventional FS algorithm with respect to M.

Tables (1)

Tables Icon

Table 1 Center Positions xi and Widths wi of pai and pbi on the Sample Structure

Equations (26)

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p , q Γ f Γ { G p ( ρ ; ρ ) f ( ρ ) n f ( ρ ) G p ( ρ ; ρ ) n } d l ,
p , q Γ f Γ { G p ( ρ ; ρ ) η p η q f ( ρ ) n f ( ρ ) G p ( ρ ; ρ ) n } d l .
G p ( ρ ; ρ ) = j 4 H 0 ( 2 ) ( k p | ρ ρ | ) ,
k p = ω n p c
1 2 f 1 ( ρ ) = 1 , 2 C f 1 ( ρ C )
1 2 f 1 ( ρ ) = f ( inc ) ( ρ ) 2 , 1 C f 1 ( ρ C )
f 1 ( ρ ) = 1 , 2 C f 1 ( ρ S 1 )
f 2 ( ρ ) = f ( inc ) ( ρ ) 2 , 1 C f 1 ( ρ S 2 ) .
{ A 1 , A 2 , A 3 , A 4 } + B = C ( ρ { G 1 , G 2 , G 3 , G 4 } ) .
A 1 ( ρ ) + B ( ρ ) = C ( ρ ) ( ρ G 1 ) ,
A 2 ( ρ ) + B ( ρ ) = C ( ρ ) ( ρ G 2 ) ,
A 3 ( ρ ) + B ( ρ ) = C ( ρ ) ( ρ G 3 ) ,
A 4 ( ρ ) + B ( ρ ) = C ( ρ ) ( ρ G 4 ) ,
{ 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( A ) + 1 2 Δ f 1 , 1 2 f a m , 1 2 f b m , 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( B ) + 1 2 Δ f 1 } = 1 , 2 C a 0 ( f 1 0 + Δ f 1 ( A ) + Δ f 1 ) + 1 , 2 C b 0 ( f 1 0 + Δ f 1 ( B ) + Δ f 1 ) + n = 1 N a 1 , a n C a 1 n f a n ( A + B ) + n = 1 N b 1 , b n C b 1 n f b n ( A + B ) ( ρ { C a 0 , C a 1 m , C b 1 m , C b 0 } ) .
{ 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( A ) , 1 2 f a m ( A ) , 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( A ) , 1 2 Δ f 1 ( 0 ) + 1 2 Δ f 1 ( A ) } = 1 , 2 C a 0 ( f 1 ( 0 ) + Δ f 1 ( A ) ) + 1 , 2 C b 0 ( f 1 ( 0 ) + Δ f 1 ( A ) ) + n = 1 N a 1 , a n C a 1 n f a n ( A ) + n = 1 N b 1 , 2 C b 1 n ( f 1 ( 0 ) + Δ f 1 ( A ) ) ( ρ { C a 0 , C a 1 m , C b 1 m , C b 0 } ) ,
{ 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( B ) , 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( B ) , 1 2 f b m ( B ) , 1 2 f 1 ( 0 ) + 1 2 Δ f 1 ( B ) } = 1 , 2 C a 0 ( f 1 0 + Δ f 1 ( B ) ) + 1 , 2 C b 0 ( f 1 0 + Δ f 1 ( B ) ) + n = 1 N a 1 , 2 C a 1 n ( f 1 ( 0 ) + Δ f 1 ( B ) ) + n = 1 N b 1 , b n C b 1 n f b n ( B ) ( ρ { C a 0 , C a 1 m , C b 1 m , C b 0 } ) ,
{ 1 2 f 1 ( 0 ) , 1 2 f 1 ( 0 ) , 1 2 f 1 ( 0 ) , 1 2 f 1 ( 0 ) } = 1 , 2 C a 0 f 1 0 + 1 , 2 C b 0 f 1 0 + n = 1 N a 1 , 2 C a 1 n f 1 ( 0 ) + n = 1 N b 1 , 2 C b 1 n f 1 ( 0 ) ( ρ { C a 0 , C a 1 m , C b 1 m , C b 0 } ) .
{ 1 2 Δ f 1 1 2 Δ f 1 ( B ) , 1 2 f a m 1 2 Δ f 1 ( B ) 1 2 Δ f a m ( A ) , 1 2 f b m 1 2 f 1 ( B ) 1 2 Δ f b m ( A ) , 1 2 Δ f 1 1 2 Δ f 1 ( A ) } = 1 , 2 C a 0 ( Δ f 1 Δ f 1 ( B ) ) + n = 1 N a 1 , a n C a 1 n ( f a n ( A + B ) f a n ( A ) ) n = 1 N a 1 , 2 C a 1 n Δ f 1 ( B ) + 1 , 2 C b 0 ( Δ f 1 Δ f 1 ( A ) ) + n = 1 N b 1 , b n C b 1 n ( f b n ( A + B ) f b n ( B ) ) n = 1 N b 1 , 2 C b 1 n Δ f 1 ( A ) ( ρ { C a 0 , C a 1 m , C b 1 m , C b 0 } ) .
{ 1 2 Δ f 1 1 2 Δ f 1 ( B ) , 1 2 f a m 1 2 Δ f 1 ( B ) 1 2 Δ f a m ( A ) , 1 2 f b m 1 2 f 1 ( B ) 1 2 Δ f b m ( A ) , 1 2 Δ f 1 1 2 Δ f 1 ( A ) } = 1 , 2 C a 0 ( Δ f 1 Δ f 1 ( B ) ) + n = 1 N a 1 , a n C a 1 n ( f a n ( A + B ) f a n ( A ) ) n = 1 N a 1 , 2 C a 1 n Δ f 1 ( B ) + 1 , 2 C b 0 ( Δ f 1 Δ f 1 ( A ) ) + n = 1 N b 1 , b n C b 1 n ( f b n ( A + B ) f b n ( B ) ) n = 1 N b 1 , 2 C b 1 n Δ f 1 ( A ) ( ρ { C a 0 , C a 1 m , C b 1 m , C b 0 } ) .
{ 1 2 Δ f 1 1 2 Δ f 1 ( B ) , 1 2 f a m 1 2 Δ f 2 ( B ) 1 2 f a m ( A ) , 1 2 f b m Δ f 2 ( A ) 1 2 f b m ( B ) , 1 2 Δ f 1 1 2 Δ f 1 ( A ) } = 2 , 1 C a 0 ( Δ f 1 Δ f 1 ( B ) ) + n = 1 N a 2 , a n C a 1 n ( f a n ( A + B ) f a n ( A ) ) + n = 1 N a 2 , 1 C a 1 n Δ f 1 ( B ) 2 , 1 C b 0 ( Δ f 1 Δ f 1 ( A ) ) n = 1 N b 2 , b n C b 2 n ( f b n ( A + B ) f b n ( B ) ) + n = 1 N b 2 , 1 C b 1 n Δ f 1 ( A ) ( ρ { C a 0 , C a 2 m , C b 2 m , C b 0 } ) .
1 2 f a m = a m , 1 C a 1 m f a m + a m , 2 C a 2 m f a m ( ρ C a 1 m , C a 2 m ) .
1 2 f b m = b m , 1 C b 1 m f b m + b m , 2 C b 2 m f b m ( ρ C b 1 m , C b 2 m ) .
Δ f 1 ( ρ ) = 1 , 2 C a 0 ( Δ f 1 + Δ f 1 ( B ) ) + 1 , 2 C b 0 ( Δ f 1 Δ f 1 ( A ) ) + n = 1 N a 1 , a n C a 1 n f a n ( A + B ) n = 1 N a 1 , 2 C a 1 n Δ f 1 ( B ) n = 1 N a 1 , a n C a 1 n f a n ( A ) + n = 1 N b 1 , b n C b 1 n f b n ( A + B ) n = 1 N b 1 , 2 C b 1 n Δ f 1 ( A ) n = 1 N b 1 , b n C b 1 n f b n ( B ) ( ρ S 1 ) ,
Δ f 2 ( ρ ) = 2 , 1 C a 0 ( Δ f 1 Δ f 1 ( B ) ) + 2 , 1 C b 0 ( Δ f 1 Δ f 1 ( A ) ) n = 1 N a 2 , a n C a 2 n f a n ( A + B ) + n = 1 N a 2 , a n C a 2 n f a n ( A ) + n = 1 N a 2 , 1 C a 1 n Δ f 1 ( B ) + n = 1 N b 2 , b n C b 2 n f b n ( A + B ) + n = 1 N b 2 , b n C b 2 n f b n ( B ) + n = 1 N b 2 , 1 C b 1 n Δ f 1 ( A ) ( ρ S 2 ) ,
f a n ( A + B ) ( ρ ) = a n , 1 C a 1 n f a n ( A + B ) + a n , 2 C a 2 n f a n ( A + B ) ( ρ S a n ) ,
f b n ( A + B ) ( ρ ) = b n , 1 C b 1 n f b n ( A + B ) + b n , 2 C b 2 n f b n ( A + B ) ( ρ S b n ) .

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