Abstract

The multipolar decomposition of current distributions is used in many branches of physics. Here, we obtain new exact expressions for the dipolar moments of a localized electric current distribution. The typical integrals for the dipole moments of electromagnetically small sources are recovered as the lowest order terms of the new expressions in a series expansion with respect to the size of the source. All the higher order terms can be easily obtained. We also provide exact and approximated expressions for dipoles that radiate a definite polarization handedness (helicity). Formally, the new exact expressions are only marginally more complex than their lowest order approximations.

© 2015 Optical Society of America

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  1. L. D. Landau and E. Lifshitz, The Classical Theory of Fields (Butterworth-Heinemann, 1975).
  2. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  3. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989). Trans. of : Photons et atomes. InterEditions, 1987.
  4. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics: An Introduction to Radiation-molecule Interactions (Academic Press, 1984).
  5. J. D. Walecka, Theoretical Nuclear and Subnuclear Physics (World Scientific, 2004).
    [Crossref]
  6. S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
    [Crossref]
  7. C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
    [Crossref]
  8. P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
    [Crossref]
  9. F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15, 073023 (2013).
    [Crossref]
  10. J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons Inc, 1952).
  11. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [Crossref]
  12. Their eigenvalues can be deduced from the parity transformation properties of a vector field in momentum space, i.e. ΠF(p) = −F(−p), and those of the spherical harmonics, ΠYjm(p̂) = Yjm(−p̂) = (−1)jYjm(p̂).
  13. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill and Kogakusha Book Companies, 1953).
  14. I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, “On the dynamic toroidal multipoles,” arXiv preprint arXiv:1507.00755 (2015).
  15. V. M. Dubovik and A. A. Cheshkov, “Multipole expansion in classical and quantum field theory and radiation,” Sov. J. Part. Nucl. 5, 318–337 (1974).
  16. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
    [Crossref]
  17. T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
    [Crossref] [PubMed]
  18. This can be seen using the complex unitary matrix that transforms spherical into Cartesian components, written on the right of Eq. (63). Calling such matrix C, we start with the dot product in spherical coordinates and transform it to Cartesian coordinates using that (C†C) is the identity: r^†Jω(r)=r^†(C†C)Jω(r)=(Cr^)†(CJω(r))=r^cart†Jω(r)cart=r^cartTJω(r)cart, where the last equality follows because the Cartesian coordinates (x, y, z) are real.
  19. R. Mehrem, J. Londergan, and M. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” J. Phys. A: Math. Gen. 24, 1435 (1991).
    [Crossref]
  20. M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
    [Crossref] [PubMed]
  21. I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, “Dual and Chiral Objects for Optical Activity in General Scattering Directions,” ACS Photonics 2, 376384 (2015).
    [Crossref]
  22. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
    [Crossref]
  23. N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
    [Crossref]
  24. K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
    [Crossref]
  25. I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
    [Crossref]
  26. I. Fernandez-Corbaton, “Helicity and duality symmetry in light matter interactions: Theory and applications,” Ph.D. thesis, Macquarie University (2014). arXiv: 1407.4432.
  27. C. Pfeiffer and A. Grbic, “Metamaterial huygens’ surfaces: Tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
    [Crossref]
  28. M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
    [Crossref]
  29. I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
    [Crossref]
  30. Using that Ylm*=(−1)mYl−m.
  31. G. B. Arfken, Mathematical Methods for Physicists (Academic Press, 1985).
  32. A. Messiah, Quantum Mechanics (Dover, 1999).
  33. Equation (64) is obtained from the relations ê1 × ê−1 = iê0, ê1 × ê0 = iê1 and ê−1 × ê0 = −iê−1, which follow from Eq. (62) and the cross products between the Cartesian basis vectors {x̂, ŷ, ẑ}.

2015 (2)

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, “Dual and Chiral Objects for Optical Activity in General Scattering Directions,” ACS Photonics 2, 376384 (2015).
[Crossref]

2014 (2)

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

2013 (3)

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

C. Pfeiffer and A. Grbic, “Metamaterial huygens’ surfaces: Tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15, 073023 (2013).
[Crossref]

2012 (2)

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

2011 (3)

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
[Crossref]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

2010 (1)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

2002 (1)

E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
[Crossref]

1991 (1)

R. Mehrem, J. Londergan, and M. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” J. Phys. A: Math. Gen. 24, 1435 (1991).
[Crossref]

1974 (2)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

V. M. Dubovik and A. A. Cheshkov, “Multipole expansion in classical and quantum field theory and radiation,” Sov. J. Part. Nucl. 5, 318–337 (1974).

Aizpurua, J.

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

Arango, F. B.

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15, 073023 (2013).
[Crossref]

Arfken, G. B.

G. B. Arfken, Mathematical Methods for Physicists (Academic Press, 1985).

Barnett, S. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

Blatt, J. M.

J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons Inc, 1952).

Bliokh, K. Y.

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

Brener, I.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Cameron, R. P.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

Cheshkov, A. A.

V. M. Dubovik and A. A. Cheshkov, “Multipole expansion in classical and quantum field theory and radiation,” Sov. J. Part. Nucl. 5, 318–337 (1974).

Chipouline, A.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989). Trans. of : Photons et atomes. InterEditions, 1987.

Craig, D. P.

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics: An Introduction to Radiation-molecule Interactions (Academic Press, 1984).

Decker, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Dominguez, J.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Dubovik, V. M.

V. M. Dubovik and A. A. Cheshkov, “Multipole expansion in classical and quantum field theory and radiation,” Sov. J. Part. Nucl. 5, 318–337 (1974).

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989). Trans. of : Photons et atomes. InterEditions, 1987.

Etrich, C.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Falkner, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Fedotov, V. A.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

Fernandez-Corbaton, I.

I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, “Dual and Chiral Objects for Optical Activity in General Scattering Directions,” ACS Photonics 2, 376384 (2015).
[Crossref]

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

I. Fernandez-Corbaton, “Helicity and duality symmetry in light matter interactions: Theory and applications,” Ph.D. thesis, Macquarie University (2014). arXiv: 1407.4432.

I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, “On the dynamic toroidal multipoles,” arXiv preprint arXiv:1507.00755 (2015).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill and Kogakusha Book Companies, 1953).

Fruhnert, M.

I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, “Dual and Chiral Objects for Optical Activity in General Scattering Directions,” ACS Photonics 2, 376384 (2015).
[Crossref]

Grahn, P.

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

Grbic, A.

C. Pfeiffer and A. Grbic, “Metamaterial huygens’ surfaces: Tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989). Trans. of : Photons et atomes. InterEditions, 1987.

Helgert, C.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Jackson, J. D.

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

Juan, M. L.

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

Kaelberer, T.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

Kaivola, M.

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

Kivshar, Y. S.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Koenderink, A. F.

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15, 073023 (2013).
[Crossref]

Landau, L. D.

L. D. Landau and E. Lifshitz, The Classical Theory of Fields (Butterworth-Heinemann, 1975).

Lederer, F.

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
[Crossref]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Lifshitz, E.

L. D. Landau and E. Lifshitz, The Classical Theory of Fields (Butterworth-Heinemann, 1975).

Londergan, J.

R. Mehrem, J. Londergan, and M. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” J. Phys. A: Math. Gen. 24, 1435 (1991).
[Crossref]

Macfarlane, M.

R. Mehrem, J. Londergan, and M. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” J. Phys. A: Math. Gen. 24, 1435 (1991).
[Crossref]

Mehrem, R.

R. Mehrem, J. Londergan, and M. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” J. Phys. A: Math. Gen. 24, 1435 (1991).
[Crossref]

Menzel, C.

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
[Crossref]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Messiah, A.

A. Messiah, Quantum Mechanics (Dover, 1999).

Minovich, A.

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

Molina-Terriza, G.

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill and Kogakusha Book Companies, 1953).

Mühlig, S.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
[Crossref]

Nanz, S.

I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, “On the dynamic toroidal multipoles,” arXiv preprint arXiv:1507.00755 (2015).

Neshev, D. N.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Nori, F.

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

Papasimakis, N.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

Pertsch, T.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Petschulat, J.

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Pfeiffer, C.

C. Pfeiffer and A. Grbic, “Metamaterial huygens’ surfaces: Tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

Radescu, E.

E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
[Crossref]

Rockstuhl, C.

I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, “Dual and Chiral Objects for Optical Activity in General Scattering Directions,” ACS Photonics 2, 376384 (2015).
[Crossref]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
[Crossref]

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, “On the dynamic toroidal multipoles,” arXiv preprint arXiv:1507.00755 (2015).

Sáenz, J. J.

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

Schmidt, M. K.

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

Shevchenko, A.

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

Staude, I.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

Thirunamachandran, T.

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics: An Introduction to Radiation-molecule Interactions (Academic Press, 1984).

Tischler, N.

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

Tsai, D. P.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

Vaman, G.

E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
[Crossref]

Vidal, X.

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

Walecka, J. D.

J. D. Walecka, Theoretical Nuclear and Subnuclear Physics (World Scientific, 2004).
[Crossref]

Weisskopf, V. F.

J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons Inc, 1952).

Wolf, E.

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[Crossref]

Yao, A. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

Zambrana-Puyalto, X.

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

Zheludev, N. I.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

ACS Photonics (1)

I. Fernandez-Corbaton, M. Fruhnert, and C. Rockstuhl, “Dual and Chiral Objects for Optical Activity in General Scattering Directions,” ACS Photonics 2, 376384 (2015).
[Crossref]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

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R. Mehrem, J. Londergan, and M. Macfarlane, “Analytic expressions for integrals of products of spherical bessel functions,” J. Phys. A: Math. Gen. 24, 1435 (1991).
[Crossref]

Light Sci. Appl. (1)

N. Tischler, I. Fernandez-Corbaton, X. Zambrana-Puyalto, A. Minovich, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Experimental control of optical helicity in nanophotonics,” Light Sci. Appl. 3, e183 (2014).
[Crossref]

Metamaterials (1)

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5, 64–73 (2011).
[Crossref]

New J. Phys. (3)

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15, 073023 (2013).
[Crossref]

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

Phys. Rev. A (2)

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

Phys. Rev. B (1)

C. Rockstuhl, C. Menzel, S. Mühlig, J. Petschulat, C. Helgert, C. Etrich, A. Chipouline, T. Pertsch, and F. Lederer, “Scattering properties of meta-atoms,” Phys. Rev. B 83, 245119 (2011).
[Crossref]

Phys. Rev. E (1)

E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
[Crossref]

Phys. Rev. Lett. (3)

M. K. Schmidt, J. Aizpurua, X. Zambrana-Puyalto, X. Vidal, G. Molina-Terriza, and J. J. Sáenz, “Isotropically polarized speckle patterns,” Phys. Rev. Lett. 114, 113902 (2015).
[Crossref] [PubMed]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111, 060401 (2013).
[Crossref]

C. Pfeiffer and A. Grbic, “Metamaterial huygens’ surfaces: Tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

Science (1)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330, 1510–1512 (2010).
[Crossref] [PubMed]

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Other (16)

Using that Ylm*=(−1)mYl−m.

G. B. Arfken, Mathematical Methods for Physicists (Academic Press, 1985).

A. Messiah, Quantum Mechanics (Dover, 1999).

Equation (64) is obtained from the relations ê1 × ê−1 = iê0, ê1 × ê0 = iê1 and ê−1 × ê0 = −iê−1, which follow from Eq. (62) and the cross products between the Cartesian basis vectors {x̂, ŷ, ẑ}.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric huygens surfaces,” Adv. Opt. Mater. (2015).
[Crossref]

I. Fernandez-Corbaton, “Helicity and duality symmetry in light matter interactions: Theory and applications,” Ph.D. thesis, Macquarie University (2014). arXiv: 1407.4432.

This can be seen using the complex unitary matrix that transforms spherical into Cartesian components, written on the right of Eq. (63). Calling such matrix C, we start with the dot product in spherical coordinates and transform it to Cartesian coordinates using that (C†C) is the identity: r^†Jω(r)=r^†(C†C)Jω(r)=(Cr^)†(CJω(r))=r^cart†Jω(r)cart=r^cartTJω(r)cart, where the last equality follows because the Cartesian coordinates (x, y, z) are real.

Their eigenvalues can be deduced from the parity transformation properties of a vector field in momentum space, i.e. ΠF(p) = −F(−p), and those of the spherical harmonics, ΠYjm(p̂) = Yjm(−p̂) = (−1)jYjm(p̂).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill and Kogakusha Book Companies, 1953).

I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, “On the dynamic toroidal multipoles,” arXiv preprint arXiv:1507.00755 (2015).

J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons Inc, 1952).

L. D. Landau and E. Lifshitz, The Classical Theory of Fields (Butterworth-Heinemann, 1975).

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

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989). Trans. of : Photons et atomes. InterEditions, 1987.

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics: An Introduction to Radiation-molecule Interactions (Academic Press, 1984).

J. D. Walecka, Theoretical Nuclear and Subnuclear Physics (World Scientific, 2004).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

The electromagnetic field radiated by a confined monochromatic current density Jω(r) with Fourier transform Jω(p) only depends on the components of Jω(p) in a spherical shell of radius |p| = ω/c. The relevant part of Jω(p) can hence be expressed as a linear combination of the momentum space vector multipolar functions {Xjm(), Zjm(), Wjm()}, which form an orthonormal basis for functions defined on the shell. The polarization vectors of Xjm() and Zjm() are tangential to the surface of the shell, i.e., orthogonal (transverse) to the momentum vector p. The polarization vector of Wjm() is normal to the surface of the shell, i.e., parallel (longitudinal) to p.

Fig. 2
Fig. 2

Relative error in the magnetic dipole moment of an infinitesimally thin circular current loop of radius a (shown in the inset) due to the small 2πa/λ0 approximation. Solid red line: Error due to taking only the first term in the small argument expansion of the spherical Bessel function in Eq. (20). Such first order gives the typical integral for the magnetic dipole moment of electromagnetically small sources [see Eq. (27)]. Dashed black line: Error due to taking the first two terms in the expansion, i.e. Eq. (27) plus Eq. (32).

Equations (93)

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J ( r , t ) = [ 0 + d ω 2 π exp ( i ω t ) J ω ( r ) ] = [ 0 + d ω 2 π exp ( i ω t ) d 3 p ( 2 π ) 3 J ω ( p ) exp ( i p r ) ] ,
X j m ( p ^ ) = 1 j ( j + 1 ) L Y j m ( p ^ ) , Z j m ( p ^ ) = i p ^ × X j m ( p ^ ) , W j m ( p ^ ) = p ^ Y j m ( p ^ ) .
J 2 Q j m ( p ^ ) = j ( j + 1 ) Q j m ( p ^ ) , J Z Q j m ( p ^ ) = m Q j m ( p ^ ) ,
Π X j m ( p ^ ) = X j m ( p ^ ) = ( 1 ) j + 1 X j m ( p ^ ) , Π Z j m ( p ^ ) = Z j m ( p ^ ) = ( 1 ) j Z j m ( p ^ ) , Π W j m ( p ^ ) = W j m ( p ^ ) = ( 1 ) j W j m ( p ^ ) .
A | B = d p ^ A ( p ^ ) B ( p ^ ) ,
J ˚ ω ( p ^ ) = j m a j m ω Z j m ( p ^ ) + b j m ω X j m ( p ^ ) + c m ω W j m ( p ^ ) ,
q j m ω = Q j m | J ˚ ω = d p ^ Q j m ( p ^ ) J ˚ ω ( p ^ ) .
a ˜ j m ω = 1 k d 3 r ( × j j ( k r ) X j m ( r ^ ) ) J ω ( r ) , b ˜ j m ω = d 3 r j j ( k r ) X j m ( r ^ ) J ω ( r ) ,
a ^ j m ω = i k j ( j + 1 ) d 3 r j j ( k r ) Y j m * ( r ^ ) L ( × J ω ( r ) ) , b ^ j m ω = k 2 j ( j + 1 ) d 3 r j j ( k r ) Y j m * ( r ^ ) L J ω ( r ) ,
[ a 11 ω a 10 ω a 1 1 ω ] d 3 r J ω ( r ) , [ b 11 ω b 10 ω b 1 1 ω ] d 3 r r × J ω ( r ) ,
J ˚ ω ( p ^ ) = 1 ( 2 π ) 3 d 3 r J ω ( r ) exp ( i ω c p ^ r ) .
q j m ω = 1 ( 2 π ) 3 d p ^ Q j m ( p ^ ) d 3 r J ω ( r ) exp ( i ω c p ^ r ) .
exp ( i ω c p ^ r ) = ( 4 π ) l ¯ , m ¯ ( i ) l ¯ Y l ¯ m ¯ * ( r ^ ) Y l ¯ m ¯ ( p ^ ) j l ¯ ( k | r | ) ,
( 2 π ) 3 4 π q j m ω = l ¯ m ¯ ( i ) l ¯ d p ^ Q j m ( p ^ ) Y l ¯ m ¯ ( p ^ ) d 3 r J ω ( r ) Y l ¯ m ¯ * ( r ^ ) j l ¯ ( k r ) .
i ( 2 π ) 3 4 π b 1 m ω = m ¯ = 1 m ¯ = 1 d p ^ X 1 m ( p ^ ) Y 1 m ¯ ( p ^ ) d 3 r J ω ( r ) Y 1 m ¯ * ( r ^ ) j 1 ( k r ) .
m = 1 d p ^ [ Y 10 ( p ^ ) 2 Y 11 ( p ^ ) 2 0 ] Y 1 m ¯ ( p ^ ) , m = 0 d p ^ [ Y 1 1 ( p ^ ) 2 0 Y 11 ( p ^ ) 2 ] Y 1 m ¯ ( p ^ ) , m = 1 d p ^ [ 0 Y 1 1 ( p ^ ) 2 Y 10 ( p ^ ) 2 ] Y 1 m ¯ ( p ^ ) ,
m = 1 1 2 ( 0 1 0 ) ( 1 0 0 ) , ( 0 0 0 ) m = 0 1 2 ( 0 0 1 ) ( 0 0 0 ) , ( 1 0 0 ) m = 1 1 2 ( 0 0 0 ) ( 0 0 1 ) . ( 0 1 0 )
r ^ = r | r | = 2 π 3 [ Y 11 * ( r ^ ) Y 10 * ( r ^ ) Y 1 1 * ( r ^ ) ] ,
b 11 ω = 3 2 π i d 3 r ( J 0 ω r ^ 1 J 1 ω r ^ 0 ) j 1 ( k r ) , b 10 ω = 3 2 π i d 3 r ( J 1 ω r ^ 1 J 1 ω r ^ 1 ) j 1 ( k r ) , b 1 1 ω = 3 2 π i d 3 r ( J 1 ω r ^ 0 J 0 ω r ^ 1 ) j 1 ( k r ) .
[ b 11 ω b 10 ω b 1 1 ω ] = 3 2 π d 3 r r ^ × J ω ( r ) j 1 ( k r ) .
[ a 11 ω a 10 ω a 1 1 ω ] = 1 π 3 d 3 r J ω ( r ) j 0 ( k r ) l ¯ = 0 1 2 π 3 d 3 r { 3 [ r ^ J ω ( r ) ] r ^ J ω ( r ) } j 2 ( k r ) l ¯ = 2 ,
[ c 11 ω c 10 ω c 1 1 ω ] = 1 π 6 d 3 r J ω ( r ) j 0 ( k r ) l ¯ = 0 1 π 6 d 3 r { 3 [ r ^ J ω ( r ) ] r ^ J ω ( r ) } j 2 ( k r ) l ¯ = 2 ,
d 3 r J ω ( r ) Y l m * ( r ^ ) j l ( k r ) .
l ¯ , m ¯ 4 π i l ¯ ( 2 π ) 3 d 3 p J ω ( p ) Y l ¯ m ¯ * ( p ^ ) d 3 r Y l ¯ m ¯ ( r ^ ) Y l m * ( r ^ ) j l ¯ ( | p | r ) j l ( k r ) .
d r r 2 j l ( | p | r ) j l ( k r ) = π 2 k 2 δ ( | p | k ) ,
4 π i l ( 2 π ) 3 d 3 p J ω ( p ) Y l m * ( p ^ ) π 2 k 2 δ ( | p | k ) = 4 π i l ( 2 π ) 3 d p ^ Y l m * ( p ^ ) 0 d p p 2 J ω ( p ) π 2 k 2 δ ( | p | k ) = 1 k 2 π 2 i l d p ^ J ˚ ω ( p ^ ) Y l m * ( p ^ ) .
[ b 11 ω b 10 ω b 1 1 ω ] 1 2 π 3 k d 3 r r × J ω ( r ) ,
[ a 11 ω a 10 ω a 1 1 ω ] 1 π 3 d 3 r J ω ( r ) l ¯ = 0
1 π 3 k 2 d 3 r 1 10 { [ r J ω ( r ) ] r 2 r 2 J ω ( r ) } l ¯ = 0 , l ¯ = 2 ,
[ c 11 ω c 10 ω c 1 1 ω ] 1 π 6 d 3 r J ω ( r ) l ¯ = 0
1 π 2 3 k 2 d 3 r 1 10 { 2 [ r J ω ( r ) ] r + r 2 J ω ( r ) } l ¯ = 0 , l ¯ = 2 .
3 k 3 60 π d 3 r [ r × J ω ( r ) ] r 2 ,
k 4 140 π 3 d 3 r { [ r J ω ( r ) ] r 3 2 J ω ( r ) r 2 } r 2 ,
k 4 70 π 6 d 3 r { [ r J ω ( r ) ] r + 1 4 J ω ( r ) r 2 } r 2 .
J ω ( r ) = ϕ ^ I 0 δ ( r a ) 1 r δ ( θ π 2 ) ,
m = z ^ 3 I 0 a j 1 ( k a ) .
m k a 1 ( 1 ) = z ^ 3 I 0 k a 2 3 , m k a 1 ( 2 ) = z ^ 3 I 0 k a 2 3 [ 1 ( k a ) 2 / 10 ] .
g j m + ω = b j m ω + a j m ω 2 G j m + ( p ^ ) = X j m ( p ^ ) + Z j m ( p ^ ) 2 = 1 + i p ^ × 2 L Y j m j ( j + 1 ) , g j m ω = b j m ω a j m ω 2 G j m ( p ^ ) = X j m ( p ^ ) Z j m ( p ^ ) 2 = 1 i p ^ × 2 L Y j m j ( j + 1 ) , g j m 0 ω = c j m ω G j m 0 ( p ^ ) = W j m ( p ^ ) = p ^ Y j m ,
Λ = J P | P | i p ^ × ,
2 π 6 [ g 1 λ ω g 0 λ ω g 1 λ ω ] = d 3 r { 3 j 1 ( k r ) r ^ + λ [ 2 j 0 ( k r ) 1 + ( 3 r ^ r ^ 1 ) j 2 ( k r ) ] } J ω ( r ) .
2 π 6 [ g 1 λ ω g 0 λ ω g 1 λ ω ] = d 3 r { k r × + λ [ 2 1 + k 2 5 ( r r 2 r 2 1 ) ] } J ω ( r ) .
ρ ( r , t ) = [ 0 + d ω 2 π exp ( i ω t ) ρ ω ( r ) ] = [ 0 + d ω 2 π exp ( i ω t ) d 3 p ( 2 π ) 3 ρ ω ( p ) exp ( i p r ) ] , J ( r , t ) = [ 0 + d ω 2 π exp ( i ω t ) J ω ( r ) ] = [ 0 + d ω 2 π exp ( i ω t ) d 3 p ( 2 π ) 3 J ω ( p ) exp ( i p r ) ] .
ϕ ω ( r ) = 1 ε d 3 r ρ ω ( r ) exp ( i k | r r | ) 4 π | r r | A ω ( r ) = μ d 3 r J ω ( r ) exp ( i k | r r | ) 4 π | r r | ,
exp ( i k | r r | ) 4 π | r r | = i k l = 0 h l ( 1 ) ( k r ) j l ( k r ) m = l m = l Y l m ( r ^ ) Y l m * ( r ^ )
A ω ( r ) = i μ k l , m h l ( 1 ) ( k r ) Y l m ( r ^ ) d 3 r J ω ( r ) j l ( k r ) Y l m * ( r ^ ) Γ l m ,
J ω ( r ) = d 3 p ( 2 π ) 3 J ω ( p ) exp ( i p r ) ,
exp ( i p r ) = ( 4 π ) l ¯ = 0 m ¯ = l ¯ m ¯ = l ¯ i l ¯ Y l ¯ m ¯ ( r ^ ) Y l ¯ m ¯ * ( p ^ ) j l ¯ ( | p | | r | ) ,
Γ l m 4 π = l ¯ m ¯ i l ¯ d 3 r d 3 p ( 2 π ) 3 J ω ( p ) j l ¯ ( | p | r ) j l ( k r ) Y l ¯ m ¯ * ( p ^ ) Y l ¯ m ¯ ( r ^ ) Y l m * ( r ^ ) .
Γ l m 4 π = i l d 3 p ( 2 π ) 3 J ω ( p ) Y l m * ( p ^ ) d r ( r ) 2 j l ( | p r | ) j l ( k r ) .
d r ( r ) 2 j l ( | p | r ) j l ( k r ) = π 2 k 2 δ ( | p | k ) .
Γ l m = 4 π ( 2 π ) 3 i l d p ^ Y l m * ( p ^ ) d p p 2 J ω ( p ) π 2 k 2 δ ( p k ) = i l 2 π d p ^ J ω ( p , | p | = k ) Y l m * ( p ^ ) .
ρ ω ( r ) = d 3 p ( 2 π ) 3 ρ ω ( p ) exp ( i p r ) ,
E ω ( r ) = i ω A ω ( r ) ϕ ω ( r ) , B ω ( r ) = × A ω ( r ) .
d p ^ W j m ( p ^ ) Y l ¯ m ¯ ( p ^ ) = d p ^ [ p ^ Y j m ( p ^ ) ] Y l ¯ m ¯ ( p ^ ) , d p ^ Z j m ( p ^ ) Y l ¯ m ¯ ( p ^ ) = d p ^ [ i p ^ × X j m ( p ^ ) ] Y l ¯ m ¯ ( p ^ ) ,
d Ω Y l 1 m 1 ( Ω ) Y l 2 m 2 ( Ω ) Y l 3 m 3 ( Ω ) = ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ( l 1 l 2 l 3 0 0 0 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) ,
( 1 j l ¯ 0 0 0 )
| j 1 j 2 | j 3 j 1 + j 2 | 1 j | l ¯ j + 1
l ¯ = j 1 or j + 1 if j > 0 , l ¯ = 1 if j > 0 .
d p ^ X j m ( p ^ ) Y l ¯ m ¯ ( p ^ )
| 0 j | l ¯ j + 0 l ¯ = j .
a = a 1 e ^ 1 + a 0 e ^ 0 + a 1 e ^ 1 ,
e ^ 1 = x ^ + i y ^ 2 e ^ 0 = z ^ e ^ 1 = x ^ i y ^ 2 .
[ a 1 a 0 a 1 ] = [ 1 2 i 2 0 0 0 1 1 2 i 2 0 ] [ a x a y a z ] , [ a x a y a z ] = [ 1 2 0 1 2 i 2 0 i 2 0 1 0 ] [ a 1 a 0 a 1 ] .
a × b = i [ a 1 b 0 a 0 b 1 a 1 b 1 a 1 b 1 a 0 b 1 a 1 b 0 ] .
p ^ = p | p | = [ p ^ 1 p ^ 0 p ^ 1 ] = 2 π 3 [ Y 1 1 Y 10 Y 11 ] = 2 π 3 [ Y 11 * Y 10 * Y 1 1 * ] ,
X j m ( p ^ ) = 1 j ( j + 1 ) [ j ( j + 1 ) m ( m 1 ) 2 Y j ( m 1 ) ( p ^ ) m Y j m ( p ^ ) j ( j + 1 ) m ( m + 1 ) 2 Y j ( m + 1 ) ( p ^ ) ] .
L = [ L x + i L y 2 L z L x + i L y 2 ] = [ L down 2 L 0 L up 2 ] ,
L up Y j m = { j ( j + 1 ) m ( m + 1 ) Y j ( m + 1 ) if | m + 1 | j 0 else , L down Y j m = { j ( j + 1 ) m ( m 1 ) Y j ( m 1 ) if | m 1 | j 0 else .
a 1 m ω = a 1 m ω l ¯ = 0 + a 1 m ω l ¯ = 2 .
a 1 m l ¯ = 0 = 1 ( 2 π ) 3 d p ^ Z 1 m ( p ^ ) d 3 r J ω ( r ) j 0 ( k r ) .
m = 1 i p ^ × X 11 ( p ^ ) = 2 π 3 [ Y 10 2 Y 11 Y 1 1 Y 11 Y 10 Y 11 2 ] , m = 0 i p ^ × X 10 ( p ^ ) = 2 π 3 [ Y 10 Y 1 1 2 Y 11 Y 1 1 Y 10 Y 11 ] , m = 1 i p ^ × X 1 1 ( p ^ ) = 2 π 3 [ Y 1 1 2 Y 10 Y 1 1 Y 10 2 Y 11 Y 1 1 ] .
[ a 11 ω a 10 ω a 1 1 ω ] l ¯ = 0 = 1 π 3 d 3 r J ω ( r ) j 0 ( k r ) .
( 2 π ) 3 4 π a 1 m l ¯ = 2 = m ¯ = 2 m ¯ = 2 d p ^ Z 1 m ( p ^ ) Y 2 m ¯ d 3 r J ω ( r ) Y 2 m ¯ * j 2 ( k r ) ,
d p ^ Y l 1 m 1 ( p ^ ) Y l 2 m 2 ( p ^ ) Y l 3 m 3 ( p ^ ) = ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ( l 1 l 2 l 3 0 0 0 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) ,
m = 1 1 30 ( 0 0 6 ) ( 0 3 0 ) ( 1 0 0 ) ( 0 0 0 ) ( 0 0 0 ) m = 0 1 30 ( 0 0 0 ) ( 0 0 3 ) ( 0 2 0 ) ( 3 0 0 ) ( 0 0 0 ) m = 1 1 30 ( 0 0 0 ) ( 0 0 0 ) ( 0 0 1 ) ( 0 3 0 ) ( 6 0 0 )
a 11 ω l ¯ = 2 = 1 ( 2 π ) 3 4 π 30 d 3 r ( 6 J 1 ω Y 2 2 + 3 J 0 ω Y 2 1 + J 1 ω Y 20 ) j 2 ( k r ) , a 10 ω l ¯ = 2 = 1 ( 2 π ) 3 4 π 30 d 3 r ( 3 J 1 ω Y 2 1 2 J 0 ω Y 20 3 J 1 ω Y 21 ) j 2 ( k r ) , a 1 1 ω l ¯ = 2 = 1 ( 2 π ) 3 4 π 30 d 3 r ( J 1 ω Y 20 + 3 J 0 ω Y 21 + 6 J 1 ω Y 22 ) j 2 ( k r ) .
Y 22 = 10 π 3 Y 11 2 , Y 21 = 20 π 3 Y 10 Y 11 Y 20 = 5 π ( Y 10 2 1 4 π ) Y 2 2 = 10 π 3 Y 1 1 2 , Y 2 1 = 20 π 3 Y 10 Y 1 1 ,
a 11 ω l ¯ = 2 = 2 3 d 3 r [ Y 1 1 ( J 1 ω Y 1 1 + J 0 ω Y 10 ) + J 1 ω 2 ( Y 10 2 1 4 π ) ] j 2 ( k r ) , a 10 ω l ¯ = 2 = 2 3 d 3 r [ Y 10 ( J 1 ω Y 1 1 + J 1 ω Y 11 ) + J 0 ω ( Y 10 2 1 4 π ) ] j 2 ( k r ) , a 1 1 ω l ¯ = 2 = 2 3 d 3 r [ Y 11 ( J 0 ω Y 10 + J 1 ω Y 11 ) + J 1 ω 2 ( Y 10 2 1 4 π ) ] j 2 ( k r ) .
3 4 π = | Y 10 | 2 + | Y 11 | 2 + | Y 1 1 | 2 Y 10 2 1 4 π = 1 2 π + 2 Y 11 Y 1 1 .
Y 11 J 1 ω + Y 10 J 0 ω + Y 1 1 J 1 ω = 1 2 3 π [ r ^ J ω ( r ) ] ,
[ a 11 ω a 10 ω a 1 1 ω ] l ¯ = 2 = 1 2 π 3 d 3 r { 3 [ r ^ J ω ( r ) ] r ^ J ω ( r ) } j 2 ( k r ) .
2 l + 1 x j l ( x ) = j l 1 ( x ) + j l + 1 ( x ) , ( 2 l + 1 ) d d x j l ( x ) = l j l 1 ( x ) ( l + 1 ) j l + 1 ( x ) ,
[ a 11 ω a 10 ω a 1 1 ω ] = 1 2 π 3 d 3 r J ω r ( r ) 6 k r j 1 ( k r ) 1 2 π 3 d 3 r 3 J ω t ( r ) ( 1 k r + d d ( k r ) ) j 1 ( k r ) .
i π 2 c 00 = m ¯ = 1 m ¯ = 1 d p ^ W 00 ( p ^ ) d 3 r J ω ( r ) Y 1 m ¯ * j 1 ( k r ) = m ¯ = 1 m ¯ = 1 d p ^ ( p ^ ) Y 1 m ¯ ( p ^ ) d 3 r J ω ( r ) Y 1 m ¯ * j 1 ( k r ) .
i π 2 c 00 = d 3 r 2 π 3 ( J 1 ω Y 11 * + J 0 ω Y 10 * J 1 ω Y 1 1 * ) j 1 ( k r ) = d 3 r [ r ^ J ω ( r ) ] j 1 ( k r ) .
c 00 i 2 π k 3 d 3 r [ r J ω ( r ) ] .
c 1 m ω = c 1 m ω l ¯ = 0 + c 1 m ω l ¯ = 2 .
c 1 m l ¯ = 0 = 4 π ( 2 π ) 3 d p ^ W 1 m ( p ^ ) 1 4 π d 3 r J ω ( r ) 1 4 π j 0 ( k r ) , = 1 ( 2 π ) 3 d p ^ [ p ^ Y 1 m ( p ^ ) ] d 3 r J ω ( r ) j 0 ( k r ) .
m = 1 : 2 π 3 [ 1 0 0 ] , m = 0 : 2 π 3 [ 0 1 0 ] , m = 1 : 2 π 3 [ 0 0 1 ] .
[ c 11 ω c 10 ω c 1 1 ω ] l ¯ = 0 = 1 π 6 d 3 r J ω ( r ) j 0 ( k r ) .
c 1 m l ¯ = 2 = 4 π ( 2 π ) 3 m ¯ = 2 m ¯ = 2 d p ^ [ p ^ Y 1 m ( p ^ ) ] Y 2 m ¯ d 3 r J ω ( r ) Y 2 m ¯ * j 2 ( k r ) ,
m = 1 1 15 ( 0 0 6 ) ( 0 3 0 ) ( 1 0 0 ) ( 0 0 0 ) ( 0 0 0 ) m = 0 1 15 ( 0 0 0 ) ( 0 0 3 ) ( 0 2 0 ) ( 3 0 0 ) ( 0 0 0 ) m 1 1 15 ( 0 0 0 ) ( 0 0 0 ) ( 0 0 1 ) ( 0 3 0 ) ( 6 0 0 )
[ c 11 ω c 10 ω c 1 1 ω ] = 1 π 6 d 3 r J ω ( r ) j 0 ( k r ) l ¯ = 0 1 π 6 d 3 r { 3 [ r ^ J ω ( r ) ] r ^ J ω ( r ) } j 2 ( k r ) . l ¯ = 2

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