Abstract

We develop a novel formalism to calculate the optical forces and torques on complex and realistic nanostructures by combining the surface integral equation (SIE) technique with Maxwell’s stress tensor. The optical force is calculated directly on the scatterer surface from the currents obtained from the SIE, which does not require an additional surface to evaluate Maxwell’s stress tensor; this is especially useful for intricate geometries such as plasmonic antennas. SIE enables direct evaluation of forces from the surface currents very efficiently and accurately for complex systems. As a proof of concept, we establish the accuracy of the model by comparing the results with the calculations from the Mie theory. The flexibility of the method is demonstrated by simulating a realistic plasmonic system with intricate geometry.

© 2014 Optical Society of America

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References

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  1. D. G. Grier, Nature 424, 810 (2003).
    [CrossRef]
  2. K. C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004).
    [CrossRef]
  3. J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
    [CrossRef]
  4. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
    [CrossRef]
  5. A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
    [CrossRef]
  6. L. Huang and O. J. F. Martin, Opt. Lett. 33, 3001 (2008).
    [CrossRef]
  7. L. Huang, S. J. Maerkl, and O. J. F. Martin, Opt. Express 17, 6018 (2009).
    [CrossRef]
  8. W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, Nano Lett. 10, 1006 (2010).
    [CrossRef]
  9. M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011).
    [CrossRef]
  10. S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
    [CrossRef]
  11. C. Girard, A. Dereux, and O. J. F. Martin, Phys. Rev. B 49, 13872 (1994).
    [CrossRef]
  12. P. C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 61, 14119 (2000).
    [CrossRef]
  13. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. London A 362, 719 (2004).
    [CrossRef]
  14. M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
    [CrossRef]
  15. A. J. Hallock, P. L. Redmond, and L. E. Brus, Proc. Natl. Acad. Sci. USA 102, 1280 (2005).
    [CrossRef]
  16. A. Salandrino, S. Fardad, and D. N. Christodoulides, J. Opt. Soc. Am. B 29, 855 (2012).
    [CrossRef]
  17. M. Fujii, Opt. Express 18, 27731 (2010).
    [CrossRef]
  18. M. Mansuripur, Nat. Photonics 7, 765 (2013).
  19. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  20. M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
    [CrossRef]
  21. A. Lovera and O. J. F. Martin, Appl. Phys. Lett. 99, 151104 (2011).
    [CrossRef]
  22. F. J. García de Abajo and A. Howie, Phys. Rev. B 65, 115418 (2002).
    [CrossRef]
  23. U. Hohenester and J. Krenn, Phys. Rev. B 72, 195429 (2005).
    [CrossRef]
  24. A. M. Kern and O. J. F. Martin, J. Opt. Soc. Am. A 26, 732 (2009).
    [CrossRef]
  25. B. Gallinet, A. M. Kern, and O. J. F. Martin, J. Opt. Soc. Am. A 27, 2261 (2010).
    [CrossRef]
  26. S. Rao, D. Wilton, and A. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
    [CrossRef]
  27. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  28. Z. Li, M. Käll, and H. Xu, Phys. Rev. B 77, 085412 (2008).
    [CrossRef]
  29. V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
    [CrossRef]
  30. É. Lamothe, G. Lévêque, and O. J. F. Martin, Opt. Express 15, 9631 (2007).
    [CrossRef]

2013 (2)

S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
[CrossRef]

M. Mansuripur, Nat. Photonics 7, 765 (2013).

2012 (2)

M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
[CrossRef]

A. Salandrino, S. Fardad, and D. N. Christodoulides, J. Opt. Soc. Am. B 29, 855 (2012).
[CrossRef]

2011 (2)

M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

A. Lovera and O. J. F. Martin, Appl. Phys. Lett. 99, 151104 (2011).
[CrossRef]

2010 (5)

M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
[CrossRef]

W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, Nano Lett. 10, 1006 (2010).
[CrossRef]

B. Gallinet, A. M. Kern, and O. J. F. Martin, J. Opt. Soc. Am. A 27, 2261 (2010).
[CrossRef]

M. Fujii, Opt. Express 18, 27731 (2010).
[CrossRef]

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

2009 (2)

2008 (4)

Z. Li, M. Käll, and H. Xu, Phys. Rev. B 77, 085412 (2008).
[CrossRef]

L. Huang and O. J. F. Martin, Opt. Lett. 33, 3001 (2008).
[CrossRef]

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
[CrossRef]

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
[CrossRef]

2007 (2)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
[CrossRef]

É. Lamothe, G. Lévêque, and O. J. F. Martin, Opt. Express 15, 9631 (2007).
[CrossRef]

2005 (2)

U. Hohenester and J. Krenn, Phys. Rev. B 72, 195429 (2005).
[CrossRef]

A. J. Hallock, P. L. Redmond, and L. E. Brus, Proc. Natl. Acad. Sci. USA 102, 1280 (2005).
[CrossRef]

2004 (2)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. London A 362, 719 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004).
[CrossRef]

2003 (1)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef]

2002 (1)

F. J. García de Abajo and A. Howie, Phys. Rev. B 65, 115418 (2002).
[CrossRef]

2000 (1)

P. C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 61, 14119 (2000).
[CrossRef]

1994 (1)

C. Girard, A. Dereux, and O. J. F. Martin, Phys. Rev. B 49, 13872 (1994).
[CrossRef]

1982 (1)

S. Rao, D. Wilton, and A. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Block, S. M.

K. C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004).
[CrossRef]

Brus, L. E.

A. J. Hallock, P. L. Redmond, and L. E. Brus, Proc. Natl. Acad. Sci. USA 102, 1280 (2005).
[CrossRef]

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
[CrossRef]

Chaumet, P. C.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. London A 362, 719 (2004).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 61, 14119 (2000).
[CrossRef]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
[CrossRef]

Christodoulides, D. N.

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Crozier, K. B.

S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
[CrossRef]

Dereux, A.

C. Girard, A. Dereux, and O. J. F. Martin, Phys. Rev. B 49, 13872 (1994).
[CrossRef]

Dholakia, K.

M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
[CrossRef]

M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
[CrossRef]

Dickinson, M.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
[CrossRef]

Fardad, S.

Fujii, M.

Gallinet, B.

García de Abajo, F. J.

F. J. García de Abajo and A. Howie, Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Girard, C.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
[CrossRef]

C. Girard, A. Dereux, and O. J. F. Martin, Phys. Rev. B 49, 13872 (1994).
[CrossRef]

Glisson, A.

S. Rao, D. Wilton, and A. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

Grier, D. G.

D. G. Grier, Nature 424, 810 (2003).
[CrossRef]

Grigorenko, A.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
[CrossRef]

Hallock, A. J.

A. J. Hallock, P. L. Redmond, and L. E. Brus, Proc. Natl. Acad. Sci. USA 102, 1280 (2005).
[CrossRef]

Hohenester, U.

U. Hohenester and J. Krenn, Phys. Rev. B 72, 195429 (2005).
[CrossRef]

Howie, A.

F. J. García de Abajo and A. Howie, Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Huang, L.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Jin, Y.

S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
[CrossRef]

Johansson, P.

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Juan, M. L.

M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

Käll, M.

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

Z. Li, M. Käll, and H. Xu, Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Kern, A. M.

Krauss, T. F.

M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
[CrossRef]

Krenn, J.

U. Hohenester and J. Krenn, Phys. Rev. B 72, 195429 (2005).
[CrossRef]

Lamothe, É.

Lévêque, G.

Li, Z.

Z. Li, M. Käll, and H. Xu, Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Lin, S.

S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
[CrossRef]

Lovera, A.

A. Lovera and O. J. F. Martin, Appl. Phys. Lett. 99, 151104 (2011).
[CrossRef]

Maerkl, S. J.

Mansuripur, M.

M. Mansuripur, Nat. Photonics 7, 765 (2013).

Martin, O. J. F.

Mazilu, M.

M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
[CrossRef]

M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
[CrossRef]

Miljkovic, V. D.

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. London A 362, 719 (2004).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 61, 14119 (2000).
[CrossRef]

Pakizeh, T.

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

Ploschner, M.

M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
[CrossRef]

Quidant, R.

M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
[CrossRef]

Rahmani, A.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. London A 362, 719 (2004).
[CrossRef]

Rao, S.

S. Rao, D. Wilton, and A. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

Redmond, P. L.

A. J. Hallock, P. L. Redmond, and L. E. Brus, Proc. Natl. Acad. Sci. USA 102, 1280 (2005).
[CrossRef]

Righini, M.

M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
[CrossRef]

Roberts, N.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
[CrossRef]

Rudhall, A.

M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
[CrossRef]

Salandrino, A.

Santschi, C.

W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, Nano Lett. 10, 1006 (2010).
[CrossRef]

Sepulveda, B.

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
[CrossRef]

Wilton, D.

S. Rao, D. Wilton, and A. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

Wright, E. M.

M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
[CrossRef]

Xu, H.

Z. Li, M. Käll, and H. Xu, Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Zelenina, A. S.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
[CrossRef]

Zhang, W.

W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, Nano Lett. 10, 1006 (2010).
[CrossRef]

Zhang, Y.

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
[CrossRef]

Zhu, W.

S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
[CrossRef]

Annu. Rev. Biochem. (1)

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, Annu. Rev. Biochem. 77, 205 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

A. Lovera and O. J. F. Martin, Appl. Phys. Lett. 99, 151104 (2011).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Rao, D. Wilton, and A. Glisson, IEEE Trans. Antennas Propag. 30, 409 (1982).
[CrossRef]

J. Nanophoton. (1)

M. Ploschner, M. Mazilu, T. F. Krauss, and K. Dholakia, J. Nanophoton. 4, 041570 (2010).
[CrossRef]

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

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

J. Phys. Chem. C (1)

V. D. Miljković, T. Pakizeh, B. Sepulveda, P. Johansson, and M. Käll, J. Phys. Chem. C 114, 7472 (2010).
[CrossRef]

J. Phys. Condens. Matter (1)

M. Mazilu, A. Rudhall, E. M. Wright, and K. Dholakia, J. Phys. Condens. Matter 24, 464117 (2012).
[CrossRef]

Nano Lett. (2)

W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, Nano Lett. 10, 1006 (2010).
[CrossRef]

S. Lin, W. Zhu, Y. Jin, and K. B. Crozier, Nano Lett. 13, 559 (2013).
[CrossRef]

Nat. Photonics (3)

M. L. Juan, M. Righini, and R. Quidant, Nat. Photonics 5, 349 (2011).
[CrossRef]

M. Mansuripur, Nat. Photonics 7, 765 (2013).

A. Grigorenko, N. Roberts, M. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
[CrossRef]

Nat. Phys. (1)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, Nat. Phys. 3, 477 (2007).
[CrossRef]

Nature (1)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phil. Trans. R. Soc. London A (1)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. London A 362, 719 (2004).
[CrossRef]

Phys. Rev. B (6)

F. J. García de Abajo and A. Howie, Phys. Rev. B 65, 115418 (2002).
[CrossRef]

U. Hohenester and J. Krenn, Phys. Rev. B 72, 195429 (2005).
[CrossRef]

C. Girard, A. Dereux, and O. J. F. Martin, Phys. Rev. B 49, 13872 (1994).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 61, 14119 (2000).
[CrossRef]

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Z. Li, M. Käll, and H. Xu, Phys. Rev. B 77, 085412 (2008).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

A. J. Hallock, P. L. Redmond, and L. E. Brus, Proc. Natl. Acad. Sci. USA 102, 1280 (2005).
[CrossRef]

Rev. Sci. Instrum. (1)

K. C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004).
[CrossRef]

Other (1)

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Traditional approach used to calculate forces on nanostructures requires integrating Maxwell’s stress tensor as a function of electric (E) and magnetic (H) fields over a fictitious surface (e.g., a sphere) surrounding the structure; (b) in this Letter, we show that Maxwell’s stress tensor can be obtained as a function of surface electric (J) and magnetic (M) currents directly over the surface mesh elements.

Fig. 2.
Fig. 2.

Relative error in force (F) and torque (T) as a function of the number of triangles the surface is discretized into for a sphere of 30 nm radius illuminated at λ=390nm.

Fig. 3.
Fig. 3.

Wavelength dependence of (a) z component, and (b) x component of optical force on one sphere of a two-sphere system composed of two silver spheres of 30 nm radius separated by a center to center distance of d. The system is illuminated by a plane wave incident along z and polarized along x, as illustrated in the inset of panel (a), and the force is computed for the sphere on the left.

Fig. 4.
Fig. 4.

Distance dependence of (a) z component, and (b) x component of optical force on each sphere of a two-sphere system composed of two silver spheres of 30 nm radius separated by a center to center distance of d. The illumination geometry is the same as that in the inset of Fig. 3(a), and the wavelength of the incident light is λ=370nm. Particle 1 is the sphere on the left.

Fig. 5.
Fig. 5.

Dependence of the y-component of torque (Ty) on the angle of incidence (θ) for the realistic structure shown in the inset for TM-polarized plane wave illumination at λ=500nm.

Equations (16)

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

(Pmech+Pfield)it=SjσijnjdS,
σij=ϵ0ϵEiEj+μ0μHiHj12(ϵ0ϵEkEk+μ0μHkHk)δij,
σij=12R[ϵ0ϵEiEj*+μ0μHiHj*12(ϵ0ϵEkEk*+μ0μHkHk*)δij].
Fi=SσijnjdS.
Fi=TTσijnjdST,
J=n^×H,
M=n^×E.
H=J×n^,
E=n^×M.
E=iωϵ(·J)n^,
H=iωμ(·M)n^.
E=iωϵ(·J)n^+n^×M,
H=iωμ(·M)n^+J×n^.
F=TTdST{(·M)(·M*)2ω2μ*n^+(·J)(·J*)2ω2ϵ*n^+iω[μμ*(J×n^)(·M*)+ϵϵ*(n^×M)(·J*)]12(ϵM·M*+μJ·J*)n^}.
J=γ1(rr1)+γ2(rr2)+γ3(rr3),
M=δ1(rr1)+δ2(rr2)+δ3(rr3),

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