Abstract

The method of surface impedance generating operator (SIGO) is developed for analyzing complex optical nanostructures. In this method, the main problem is divided into several subproblems. The proposed SIGO method handles these subproblems independently. Therefore, the method is suitable for parallel computing and is numerically efficient for analyzing large-scale optical structures. To formulate the subproblems in integral form, the dyadic Green’s functions need to be derived for all interior and exterior domains. The dyadic Green’s functions of typical exterior problems, e.g., free space, multilayer, periodic, etc., are quite familiar. However, a method based on distribution theory is introduced to obtain the required dyadic Green’s functions of interior problems for scatterers with arbitrary shapes. An important lemma is stated and proved. This lemma preserves the crucial property of Green’s functions, which is the completeness of eigenmodes. The dyadic Green’s functions of the interior problem are specifically derived for the rectangular nanorods. Using the SIGO method and the derived Green’s functions, the current distribution of an optical nano dipole antenna is analyzed. It is shown that, for the same level of accuracy, SIGO can be faster than other conventional formulations and require lower computational resources as well. Therefore, it can be used for successful design and optimization of complex plasmonic circuits.

© 2020 Optical Society of America

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References

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2019 (6)

2018 (2)

R. Janaswamy, “Consistency requirements for integral representations of Green’s functions—part I,” IEEE Trans. Antennas Propag. 66, 4060–4068 (2018).
[Crossref]

C. Han, L. Yang, P. Ye, E. P. J. Parrott, E. Pickwell-Macpherson, and W. Y. Tam, “Three dimensional chiral plasmon rulers based on silver nanorod trimers,” Opt. Express 26, 10315–10325 (2018).
[Crossref]

2017 (3)

2013 (1)

Z. Ma and G. A. E. Vandenbosch, “Systematic full-wave characterization of real-metal nano dipole antennas,” IEEE Trans. Antennas Propag. 61, 4990–4999 (2013).
[Crossref]

2012 (1)

2011 (1)

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

2009 (1)

2004 (2)

A. Borji and S. Safavi-Naeini, “Rapid calculation of the Green’s function in a rectangular enclosure with application to conductor loaded cavity resonators,” IEEE Trans. Microwave Theory Tech. 52, 1724–1731 (2004).
[Crossref]

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

2003 (1)

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003).
[Crossref]

1997 (1)

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 45, 508–519 (1997).
[Crossref]

1982 (1)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

1976 (1)

C.-T. Tai and P. Rozenfeld, “Different representations of dyadic Green’s functions for a rectangular cavity,” IEEE Trans. Microwave Theory Tech. 24, 597–601 (1976).
[Crossref]

1975 (1)

Y. Rahmat-Samii, “On the question of computation of the dyadic Green’s function at the source region in waveguides and cavities (short papers),” IEEE Trans. Microw. Theory Tech. 23, 762–765 (1975).
[Crossref]

1972 (1)

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Ahmed, A.

Alaeian, H.

Atia,

Berini, P.

Borji, A.

A. Borji and S. Safavi-Naeini, “Rapid calculation of the Green’s function in a rectangular enclosure with application to conductor loaded cavity resonators,” IEEE Trans. Microwave Theory Tech. 52, 1724–1731 (2004).
[Crossref]

Chen, Q.

Chen, Y.

Chen, Y. P.

Chew, W. C.

Choy, W. C. H.

Christy, R.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Dajani, K.

de Arquer, G.

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

Ding, L.

Eshrah, I. A.

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

Faraji-Dana, R.

Gholipour, A.

Ghosh, S.

Glisson, A.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Glisson, A. W.

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

Gosciniak, J.

Han, C.

Hanson, G. W.

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics: an Introduction (Springer, 2001).

Heikal, A.

Hu, X.

Janaswamy, R.

R. Janaswamy, “Consistency requirements for integral representations of Green’s functions—part I,” IEEE Trans. Antennas Propag. 66, 4060–4068 (2018).
[Crossref]

Jia, H.

Jiang, L.

Jin, L.

Johnson, P.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Kelley, M.

Khaled, S. R.

Khodami, M.

Kishk, A. A.

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

Lee, A.

Liu, H.

Liu, W.

Liu, Z.

Ma, Z.

Z. Ma and G. A. E. Vandenbosch, “Systematic full-wave characterization of real-metal nano dipole antennas,” IEEE Trans. Antennas Propag. 61, 4990–4999 (2013).
[Crossref]

Michalski, K. A.

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 45, 508–519 (1997).
[Crossref]

Moshchalkov, V. V.

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

Mosig, J. R.

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 45, 508–519 (1997).
[Crossref]

Mozumdar, M.

Obayya, S. S. A.

Parrott, E. P. J.

Pelayo, F.

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

Pickwell-Macpherson, E.

Rahman, B. M. A.

Rahmat-Samii, Y.

Y. Rahmat-Samii, “On the question of computation of the dyadic Green’s function at the source region in waveguides and cavities (short papers),” IEEE Trans. Microw. Theory Tech. 23, 762–765 (1975).
[Crossref]

Rao, S.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Rasras, M.

Rozenfeld, P.

C.-T. Tai and P. Rozenfeld, “Different representations of dyadic Green’s functions for a rectangular cavity,” IEEE Trans. Microwave Theory Tech. 24, 597–601 (1976).
[Crossref]

Safavi-Naeini, S.

A. Borji and S. Safavi-Naeini, “Rapid calculation of the Green’s function in a rectangular enclosure with application to conductor loaded cavity resonators,” IEEE Trans. Microwave Theory Tech. 52, 1724–1731 (2004).
[Crossref]

Said, A.

Su, Q.

Tai, C.-T.

C.-T. Tai and P. Rozenfeld, “Different representations of dyadic Green’s functions for a rectangular cavity,” IEEE Trans. Microwave Theory Tech. 24, 597–601 (1976).
[Crossref]

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).

Tam, W. Y.

Taskinen, M.

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003).
[Crossref]

Vandenbosch, G. A. E.

A. Gholipour, R. Faraji-Dana, and G. A. E. Vandenbosch, “High performance analysis of layered nanolithography masks by a surface impedance generating operator,” J. Opt. Soc. Am. A 34, 464–471 (2017).
[Crossref]

Z. Ma and G. A. E. Vandenbosch, “Systematic full-wave characterization of real-metal nano dipole antennas,” IEEE Trans. Antennas Propag. 61, 4990–4999 (2013).
[Crossref]

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

Verellen, N.

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

Volski, V.

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

Wan, P.

Wei, E. I.

Wen, L.

Wilton, D.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Xu, W.

Yakovlev, A. B.

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics: an Introduction (Springer, 2001).

Yang, C.

Yang, L.

Ye, P.

Yla-Oijala, P.

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003).
[Crossref]

Zhong, Y.

Zhu, J.

Zhu, T.

Appl. Opt. (1)

IEEE Antennas Propag. Mag. (1)

I. A. Eshrah, A. B. Yakovlev, A. A. Kishk, A. W. Glisson, and G. W. Hanson, “The TE00 waveguide mode- the complete story,” IEEE Antennas Propag. Mag. 46, 33–41 (2004).
[Crossref]

IEEE Trans. Antennas Propag. (6)

F. Pelayo, G. de Arquer, V. Volski, N. Verellen, G. A. E. Vandenbosch, and V. V. Moshchalkov, “Engineering the input impedance of optical nano dipole antennas: materials, geometry and excitation effect,” IEEE Trans. Antennas Propag. 59, 3144–3153 (2011).
[Crossref]

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 45, 508–519 (1997).
[Crossref]

R. Janaswamy, “Consistency requirements for integral representations of Green’s functions—part I,” IEEE Trans. Antennas Propag. 66, 4060–4068 (2018).
[Crossref]

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Z. Ma and G. A. E. Vandenbosch, “Systematic full-wave characterization of real-metal nano dipole antennas,” IEEE Trans. Antennas Propag. 61, 4990–4999 (2013).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

Y. Rahmat-Samii, “On the question of computation of the dyadic Green’s function at the source region in waveguides and cavities (short papers),” IEEE Trans. Microw. Theory Tech. 23, 762–765 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

A. Borji and S. Safavi-Naeini, “Rapid calculation of the Green’s function in a rectangular enclosure with application to conductor loaded cavity resonators,” IEEE Trans. Microwave Theory Tech. 52, 1724–1731 (2004).
[Crossref]

C.-T. Tai and P. Rozenfeld, “Different representations of dyadic Green’s functions for a rectangular cavity,” IEEE Trans. Microwave Theory Tech. 24, 597–601 (1976).
[Crossref]

J. Lightwave Technol. (3)

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

J. Opt. Soc. Am. B (4)

Opt. Express (3)

Phys. Rev. B (1)

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Other (2)

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics: an Introduction (Springer, 2001).

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).

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Figures (8)

Fig. 1.
Fig. 1. Object with an arbitrary shape in a stratified media. ${ J_s}$ and ${ M_s}$ on ${ S}$ are the equivalent surface electric and magnetic currents, respectively.
Fig. 2.
Fig. 2. Separation of interior and exterior subproblems. The subproblems can be handled independently. The results are then combined to find the solution of the main problem.
Fig. 3.
Fig. 3. Typical periodic structure. The unit cell is divided into subproblems; the interior problem is similar to the interior problem of Fig. 2, and the exterior problem is a homogenous domain with periodic boundary conditions.
Fig. 4.
Fig. 4. Geometry of a rectangular monomer.
Fig. 5.
Fig. 5. (a) Typical triangulation of a rectangular nanorod. (b) Typical RWG basis function that is defined on a sample edge.
Fig. 6.
Fig. 6. Geometry of a 3D chiral nanorod trimer [23].
Fig. 7.
Fig. 7. Optical nano dipole antenna. Two excitation models are considered: plane wave excitation and gap excitation. Inset shows the gap excitation model, which is used to compute the input impedance.
Fig. 8.
Fig. 8. Current distribution along the length of the dipole for $G=5$ and $30\;\text{nm}$.

Equations (62)

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

E int ( r ) = j ω μ S d s G ¯ e int ( r , r ) J S ( r ) , r V , r S ,
× × G ¯ e int ( r , r ) ω 2 μ ε G ¯ e int ( r , r ) = I ¯ δ ( r , r ) .
n ^ × [ × G ¯ e int ( r , r ) ] | S = 0 .
M S ( r ) = n ^ × E int ( r ) , r S .
n ^ × M S ( r ) = j ω μ S d s G ¯ e int ( r , r ) J S ( r ) , r , r S .
Z ( ) = j ω μ S G ¯ e int ( r , r ) ( ) d s , r , r S .
n ^ × M S ( r ) = Z { J S ( r ) } .
× × G ¯ m int ( r , r ) k 2 G ¯ m int ( r , r ) = × I ¯ δ ( r , r ) .
k 2 G ¯ e int ( r , r ) = × G ¯ m int ( r , r ) I ¯ δ ( r , r ) .
G ¯ m int ( r , r ) = × G ¯ e int ( r , r ) .
G ¯ e int ( r , r ) = G ¯ e 0 ( r , r ) 1 k 2 I ¯ δ ( r , r ) ,
G ¯ e 0 ( r , r ) = 1 k 2 × G ¯ m int ( r , r ) .
n ^ × G ¯ m int ( r , r ) = 0 .
2 G ¯ A int ( r , r ) + k 2 G ¯ A int ( r , r ) = I ¯ δ ( r r ) .
G ¯ e int ( r , r ) = ( I ¯ + 1 k 2 ) G ¯ A int ( r , r ) .
2 G ¯ m int ( r , r ) + k 2 G ¯ m int ( r , r ) = × I ¯ δ ( r r ) .
2 g ¯ m ( r , r ) + k 2 g ¯ m ( r , r ) = I ¯ δ ( r r ) .
n ^ × g ¯ m ( r , r ) = 0 .
G ¯ m int ( r , r ) = d v g ¯ m ( r , r ) × I ¯ δ ( r r ) .
( 2 + k 2 ) g m ii = δ ( r r ) ,
g m ii = X ii ( x ) Y ii ( y ) Z ii ( z ) .
n [ n ^ G ¯ m int ( r , r ) ] | S = 0 .
P j = 0.
n ^ × P j = 0.
P j = P tj t ^ + P nj n ^ ,
P j = t P tj + P nj n = 0.
P nj n = n [ n ^ P j ] = 0.
n [ n ^ P j ] = 0 j = 1 , 2 , 3.
n [ n ^ g ¯ m ( r , r ) ] | S = 0 .
r V E ( r ) r V 0 } = E exc ( r ) + L E ( r , r ) J S ( r ) + K E ( r , r ) M S ( r ) ,
L E ( r , r ) = j ω d r G ¯ e out ( r , r ) μ ( r ) ,
K E ( r , r ) = ε 1 ( r ) d r × G ¯ m out ( r , r ) ε ( r ) ,
E exc ( r ) = L E ( r , r ) J S ( r ) + K E ( r , r ) M S ( r ) .
g m xx x | x = 0 , W = 0 ; g m xx | y = 0 , L = 0 ; g m xx | z = 0 , T = 0 ;
g m yy | x = 0 , W = 0 ; g m yy y | y = 0 , L = 0 ; g m yy | z = 0 , T = 0 ;
g m zz | x = 0 , W = 0 ; g m zz | y = 0 , L = 0 ; g m zz z | z = 0 , T = 0 ,
g ¯ m ( r , r ) = p = 0 q = 0 i = 0 ϵ p ϵ q ϵ i T W L Γ p q i 2 [ C x p C x p S y q S y q S z i S z i x ^ x ^ + S x p S x p C y q C y q S z i S z i y ^ y ^ + S x p S x p S y q S y q C z i C z i z ^ z ^ ] ,
Γ p q i 2 = k 2 ( p π W ) 2 ( q π L ) 2 ( i π T ) 2
ϵ n = { 1 n = 0 2 n 0
S x p = sin p π W x , C x p = cos p π W x ,
S y q = sin q π L y , C y q = cos q π L y ,
S z i = sin i π T z , C z i = cos i π T z .
g ¯ m ( r , r ) × I ¯ δ ( r r ) = [ 0 g m xx δ ( r r ) z g m xx δ ( r r ) y g m yy δ ( r r ) z 0 g m yy δ ( r r ) x g m zz δ ( r r ) y g m zz δ ( r r ) x 0 ] .
G ¯ m i n t ( r , r ) = p = 0 q = 0 i = 0 ϵ p ϵ q ϵ i T W L Γ p q i 2 { i π T C x p C x p S y q S y q S z i C z i x ^ y ^ q π L C x p C x p S y q C y q S z i S z i x ^ z ^ i π T S x p S x p C y q C y q S z i C z i y ^ x ^ + p π W S x p C x p C y q C y q S z i S z i y ^ z ^ + q π L S x p S x p S y q C y q C z i C z i z ^ x ^ p π W S x p C x p S y q S y q C z i C z i z ^ y ^ } .
[ z δ ( z z ) ] f d v = f z | z = z .
G ¯ e 0 ( r , r ) = 1 k 2 p = 0 q = 0 i = 0 ϵ p ϵ q ϵ i T W L Γ p q i 2 { [ ( q π L ) 2 + ( i π T ) 2 ] S x p S x p C y q C y q C z i C z i x ^ x ^ + [ ( p π W ) 2 + ( i π T ) 2 ] C x p C x p S y q S y q C z i C z i y ^ y ^ + [ ( p π W ) 2 + ( q π L ) 2 ] C x p C x p C y q C y q S z i S z i z ^ z ^ q π L p π W C x p S x p S y q C y q C z i C z i y ^ x ^ i π T p π W C x p S x p C y q C y q S z i C z i z ^ x ^ p π W q π L S x p C x p C y q S y q C z i C z i x ^ y ^ i π T q π L C x p C x p C y q S y q S z i C z i z ^ y ^ p π W i π T S x p C x p C y q C y q C z i S z i x ^ z ^ q π L i π T C x p C x p S y q C y q C z i S z i y ^ z ^ } .
G ¯ e int ( r , r ) = G ¯ e 0 ( r , r ) 1 k 2 ( x ^ x ^ + y ^ y ^ + z ^ z ^ ) δ ( r r )
G ¯ e int ( r , r ) = [ G ¯ e int ( r , r ) ] T .
J S ( r ) = n = 1 N α n f n ( r ) ,
M S ( r ) = n = 1 N β n f n ( r ) ,
f n ( r ) = { ± L n 2 A n ± ( r p n ± ) r T n ± 0 otherwise ,
n = 1 N β n n ^ ( r ) × f n ( r ) = j ω μ n = 1 N α n T n ± d s G ¯ e int ( r , r ) f n ( r ) .
[ C m n ] [ β n ] = [ A m n int ] [ α n ] ,
A m n int = j ω μ S m d s f m ( r ) S n d s G ¯ e int ( r , r ) f n ( r )
C m n = S m d s f m ( r ) [ n ^ ( r ) × f n ( r ) ] ,
[ β n ] = [ C m n ] 1 [ A m n int ] [ α n ] = [ Z m n int ] [ α n ] .
[ V m ] = [ L m n ] [ α n ] + [ K m n ] [ β n ] ,
L m n = f m ( r ) , L E ( r , r ) , f n ( r ) ,
K m n = f m ( r ) , K E ( r , r ) , f n ( r ) ,
V m = f m ( r ) , E exc ( r ) .
[ V m ] = ( [ L m n ] + [ K m n ] [ Z m n int ] ) [ α n ] = [ Z m n tot ] [ α n ] .
[ [ β n 1 ] 1 [ β n b ] N b ] = [ [ Z m n int ] 1 0 0 0 [ Z m n int ] 2 0 0 0 0 0 [ Z m n int ] N b ] [ [ α n 1 ] 1 [ α n b ] N b ] .

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