Abstract

We demonstrate that the reflecting properties of a single particle nanoantenna can be extremely sensitive to its distance from a quantum emitter at frequencies lower than the plasmon resonance. The phenomenon is shown to arise from rapid phase variations of the emitter field at short distances associated with a phase of the antenna particle polarizability lower than π/4.

© 2011 Optical Society of America

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References

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  1. S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
    [CrossRef] [PubMed]
  2. A. Devilez, B. Stout, and N. Bonod, Phys. Rev. B 81, 245128 (2010).
    [CrossRef]
  3. L. Novotny and N. van Hulst, Nat. Photon. 5, 83 (2011).
    [CrossRef]
  4. J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
    [CrossRef] [PubMed]
  5. P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
    [CrossRef] [PubMed]
  6. S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
    [CrossRef] [PubMed]
  7. H. Mertens, A. F. Koenderink, and A. Polman, Phys. Rev. B 76, 115123 (2007).
    [CrossRef]
  8. L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007).
    [CrossRef] [PubMed]
  9. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
    [CrossRef]
  10. H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
    [CrossRef] [PubMed]
  11. J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007).
    [CrossRef]
  12. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
    [CrossRef]
  13. A. F. Koenderink, Nano Lett. 9, 4228 (2009).
    [CrossRef] [PubMed]
  14. A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010).
    [CrossRef] [PubMed]
  15. A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
    [CrossRef] [PubMed]
  16. N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
    [CrossRef]
  17. R. Esteban, T. V. Teperik, and J. J. Greffet, Phys. Rev. Lett. 104, 026802 (2010).
    [CrossRef] [PubMed]
  18. E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).
  19. B. Stout, A. Devilez, B. Rolly, and N. Bonod, J. Opt. Soc. Am. B 28, 1213 (2011).
    [CrossRef]
  20. L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge Univ. Press, 2006), pp. 250–303.
  21. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

2011 (3)

L. Novotny and N. van Hulst, Nat. Photon. 5, 83 (2011).
[CrossRef]

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

B. Stout, A. Devilez, B. Rolly, and N. Bonod, J. Opt. Soc. Am. B 28, 1213 (2011).
[CrossRef]

2010 (5)

A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010).
[CrossRef] [PubMed]

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

R. Esteban, T. V. Teperik, and J. J. Greffet, Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

A. Devilez, B. Stout, and N. Bonod, Phys. Rev. B 81, 245128 (2010).
[CrossRef]

2009 (3)

S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
[CrossRef] [PubMed]

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

A. F. Koenderink, Nano Lett. 9, 4228 (2009).
[CrossRef] [PubMed]

2008 (1)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
[CrossRef]

2007 (3)

J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007).
[CrossRef]

H. Mertens, A. F. Koenderink, and A. Polman, Phys. Rev. B 76, 115123 (2007).
[CrossRef]

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007).
[CrossRef] [PubMed]

2006 (2)

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

2005 (1)

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
[CrossRef] [PubMed]

Acimovic, S. S.

S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
[CrossRef] [PubMed]

Agio, M.

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Aouani, H.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Avlasevich, Y.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Bidault, S.

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bonod, N.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, J. Opt. Soc. Am. B 28, 1213 (2011).
[CrossRef]

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

A. Devilez, B. Stout, and N. Bonod, Phys. Rev. B 81, 245128 (2010).
[CrossRef]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010).
[CrossRef] [PubMed]

Curto, A.

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

Devaux, E.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Devilez, A.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, J. Opt. Soc. Am. B 28, 1213 (2011).
[CrossRef]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010).
[CrossRef] [PubMed]

A. Devilez, B. Stout, and N. Bonod, Phys. Rev. B 81, 245128 (2010).
[CrossRef]

Ebbesen, T. W.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Eisler, H.-J.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
[CrossRef] [PubMed]

Engheta, N.

J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007).
[CrossRef]

Esteban, R.

R. Esteban, T. V. Teperik, and J. J. Greffet, Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

Fan, S.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

Farahani, J. N.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
[CrossRef] [PubMed]

Ghosh, G.

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).

Gonzalez, M. U.

S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
[CrossRef] [PubMed]

Greffet, J. J.

R. Esteban, T. V. Teperik, and J. J. Greffet, Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

Hakanson, U.

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Hecht, B.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
[CrossRef] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge Univ. Press, 2006), pp. 250–303.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Kaminski, F.

Kinkhabwala, A.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

Koenderink, A. F.

A. F. Koenderink, Nano Lett. 9, 4228 (2009).
[CrossRef] [PubMed]

H. Mertens, A. F. Koenderink, and A. Polman, Phys. Rev. B 76, 115123 (2007).
[CrossRef]

Kreuzer, M.

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

Kreuzer, M. P.

S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
[CrossRef] [PubMed]

Kuhn, S.

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Li, J.

J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007).
[CrossRef]

Mahboub, O.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Mertens, H.

H. Mertens, A. F. Koenderink, and A. Polman, Phys. Rev. B 76, 115123 (2007).
[CrossRef]

Moerner, W. E.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

Mullen, K.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

Novotny, L.

L. Novotny and N. van Hulst, Nat. Photon. 5, 83 (2011).
[CrossRef]

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge Univ. Press, 2006), pp. 250–303.

Palik, E.

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).

Pohl, D. W.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
[CrossRef] [PubMed]

Polman, A.

H. Mertens, A. F. Koenderink, and A. Polman, Phys. Rev. B 76, 115123 (2007).
[CrossRef]

Popov, E.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Quidant, R.

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
[CrossRef] [PubMed]

Rigneault, H.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Rogobete, L.

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007).
[CrossRef] [PubMed]

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Rolly, B.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, J. Opt. Soc. Am. B 28, 1213 (2011).
[CrossRef]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

Salandrino, A.

J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007).
[CrossRef]

Sandoghdar, V.

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007).
[CrossRef] [PubMed]

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Segerink, F. B.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
[CrossRef]

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
[CrossRef]

Stout, B.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, J. Opt. Soc. Am. B 28, 1213 (2011).
[CrossRef]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010).
[CrossRef] [PubMed]

A. Devilez, B. Stout, and N. Bonod, Phys. Rev. B 81, 245128 (2010).
[CrossRef]

Taminiau, T.

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

Taminiau, T. H.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
[CrossRef]

Teperik, T. V.

R. Esteban, T. V. Teperik, and J. J. Greffet, Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

van Hulst, N.

L. Novotny and N. van Hulst, Nat. Photon. 5, 83 (2011).
[CrossRef]

van Hulst, N. F.

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
[CrossRef]

Volpe, G.

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

Wenger, J.

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Yu, Z.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

ACS Nano (2)

S. S. Acimovic, M. P. Kreuzer, M. U. Gonzalez, and R. Quidant, ACS Nano 3, 1231 (2009).
[CrossRef] [PubMed]

A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010).
[CrossRef] [PubMed]

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

Nano Lett. (2)

A. F. Koenderink, Nano Lett. 9, 4228 (2009).
[CrossRef] [PubMed]

H. Aouani, O. Mahboub, N. Bonod, E. Devaux, E. Popov, H. Rigneault, T. W. Ebbesen, and J. Wenger, Nano Lett. 11, 637 (2011).
[CrossRef] [PubMed]

Nat. Photon. (3)

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, Nat. Photon. 3, 654 (2009).
[CrossRef]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nat. Photon. 2, 234 (2008).
[CrossRef]

L. Novotny and N. van Hulst, Nat. Photon. 5, 83 (2011).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (4)

H. Mertens, A. F. Koenderink, and A. Polman, Phys. Rev. B 76, 115123 (2007).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, Phys. Rev. B 81, 245128 (2010).
[CrossRef]

J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007).
[CrossRef]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, Phys. Rev. B 82, 115429 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

R. Esteban, T. V. Teperik, and J. J. Greffet, Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, Phys. Rev. Lett. 95, 017402 (2005).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Science (1)

A. Curto, G. Volpe, T. Taminiau, M. Kreuzer, R. Quidant, and N. F. van Hulst, Science 329, 930 (2010).
[CrossRef] [PubMed]

Other (3)

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1998).

L. Novotny and B. Hecht, Principles of Nano-Optics(Cambridge Univ. Press, 2006), pp. 250–303.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (2)

Fig. 1
Fig. 1

(a) An electric dipole emitter oriented along the z axis is coupled to a sphere whose center is at a distance d along the x axis. (b) Reflected power efficiency P ref (left scale, top full lines) and total decay rates Γ tot (right scale, logarithmic scale, bottom full lines) with respect to the separation d: crosses, dipolar approximation; squares, multipolar calculations. (c), (d) Radiation patterns with separation distances, equal to (c)  d = 59 nm and (d)  d = 49 nm .

Fig. 2
Fig. 2

Triangles, ϕ d for an emitter transversely coupled to a spherical scatterer as a function of k d ; dashed line, dipolar field contribution; circles, far-field term contribution. Vertical line marks the k d = 0.93 abscissa. Inset (circles, left scale), polarizability phase, Arg ( α ) , and (squares, right scale) scattering efficiency, Q scat = σ s / π a 2 , of a 80 nm diameter silver sphere, both as a function of the vacuum wavelength.

Equations (6)

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

E ( d ) = e i k d 4 π ε m ϵ 0 d 3 [ k 2 d 2 ( x ^ × p em ) × x ^ + ( 1 i k d ) ( 3 ( x ^ · p em ) x ^ p em ) ] = e i k d 4 π ε m ϵ 0 d 3 ( 1 i k d k 2 d 2 ) z ^ ,
p in = α e i k d 4 π d 3 ( 1 i k d k 2 d 2 ) z ^ .
Δ P ( r , θ , φ ) = P ( r , θ , φ ) + P ( r , π θ , π + φ ) = ω 3 k | p 1 | | p 2 | 8 π 2 ϵ 0 c 2 r 2 { sin ϕ sin [ k d sin θ cos φ ] } sin 2 θ e r ,
Δ P ( x , d ) = ω 3 k | p 1 | | p 2 | 8 π 2 ϵ 0 c 2 x 2 sin ϕ sin ( k d ) x ^ .
ϕ ( k d ) arg ( p in · z ^ p em · z ^ ) = arg [ α e i k d ( 1 i k d k 2 d 2 ) ] .
ϕ d ϕ arg ( α ) = arg [ e i k d ( 1 i k d k 2 d 2 ) ] .

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