Abstract

We derive a general expression of the electric dyadic Green function in a time-reversal cavity, based on vector diffraction theory in the frequency domain. Our theory gives a rigorous framework to time-reversal experiments using electromagnetic waves and suggests a methodology to design structures generating subwavelength focusing after time reversal.

© 2007 Optical Society of America

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References

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  1. M. Fink, Phys. Today 20(12), 34 (1997).
    [CrossRef]
  2. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
    [CrossRef] [PubMed]
  3. C. Draeger and M. Fink, Phys. Rev. Lett. 79, 407 (1997).
    [CrossRef]
  4. J. de Rosny and M. Fink, Phys. Rev. Lett. 89, 124301 (2002).
    [CrossRef] [PubMed]
  5. R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
    [CrossRef]
  6. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
    [CrossRef] [PubMed]
  7. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Science, 2006).
  8. D. Cassereau and M. Fink, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 579 (1992).
    [CrossRef] [PubMed]
  9. The vector form of Sommerfeld's radiation condition reads limr′-->∞[∇r′×G⃡(r′,r,ω)−iku×G⃡(r′,r,ω)]=0, where k=ω/c, r′=∣r′∣, and u=r′/r′. This condition states that G⃡ behaves as an outgoing (retarded) wave.
  10. For two well-behaved vector fields A(r) and B(r), the second Green identity reads as ∫V[A∙∇×∇×B−B∙∇×∇×A]d3r=∫S[B×∇×A−A×∇×B]∙nd2r, where V is a volume enclosed by surface S and n is the outward normal. See Refs. .
  11. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  12. For any dyadic D⃡ and any vectors U and V, one has U∙D⃡V=V∙D⃡TU.
  13. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 89.
  14. R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998).
    [CrossRef]
  15. J.-J. Greffet and R. Carminati, Prog. Surf. Sci. 56, 133 (1997).
    [CrossRef]
  16. E. R. Méndez, J.-J. Greffet, and R. Carminati, Opt. Commun. 142, 7 (1997).
    [CrossRef]
  17. R. L. Weaver and O. I. Lobkis, Phys. Rev. Lett. 87, 134301 (2001).
    [CrossRef] [PubMed]
  18. K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
    [CrossRef]

2007 (1)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

2006 (1)

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Science, 2006).

2004 (1)

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

2003 (1)

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
[CrossRef]

2002 (1)

J. de Rosny and M. Fink, Phys. Rev. Lett. 89, 124301 (2002).
[CrossRef] [PubMed]

2001 (1)

R. L. Weaver and O. I. Lobkis, Phys. Rev. Lett. 87, 134301 (2001).
[CrossRef] [PubMed]

2000 (1)

R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
[CrossRef]

1998 (1)

1997 (4)

M. Fink, Phys. Today 20(12), 34 (1997).
[CrossRef]

J.-J. Greffet and R. Carminati, Prog. Surf. Sci. 56, 133 (1997).
[CrossRef]

E. R. Méndez, J.-J. Greffet, and R. Carminati, Opt. Commun. 142, 7 (1997).
[CrossRef]

C. Draeger and M. Fink, Phys. Rev. Lett. 79, 407 (1997).
[CrossRef]

1992 (1)

D. Cassereau and M. Fink, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 579 (1992).
[CrossRef] [PubMed]

1984 (1)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 89.

1953 (1)

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

Carminati, R.

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
[CrossRef]

R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
[CrossRef]

R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998).
[CrossRef]

J.-J. Greffet and R. Carminati, Prog. Surf. Sci. 56, 133 (1997).
[CrossRef]

E. R. Méndez, J.-J. Greffet, and R. Carminati, Opt. Commun. 142, 7 (1997).
[CrossRef]

Cassereau, D.

D. Cassereau and M. Fink, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 579 (1992).
[CrossRef] [PubMed]

de Rosny, J.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

J. de Rosny and M. Fink, Phys. Rev. Lett. 89, 124301 (2002).
[CrossRef] [PubMed]

Derode, A.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

Draeger, C.

C. Draeger and M. Fink, Phys. Rev. Lett. 79, 407 (1997).
[CrossRef]

Feshbach, H.

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

Fink, M.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

J. de Rosny and M. Fink, Phys. Rev. Lett. 89, 124301 (2002).
[CrossRef] [PubMed]

M. Fink, Phys. Today 20(12), 34 (1997).
[CrossRef]

C. Draeger and M. Fink, Phys. Rev. Lett. 79, 407 (1997).
[CrossRef]

D. Cassereau and M. Fink, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 579 (1992).
[CrossRef] [PubMed]

Greffet, J.-J.

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
[CrossRef]

R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
[CrossRef]

R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998).
[CrossRef]

J.-J. Greffet and R. Carminati, Prog. Surf. Sci. 56, 133 (1997).
[CrossRef]

E. R. Méndez, J.-J. Greffet, and R. Carminati, Opt. Commun. 142, 7 (1997).
[CrossRef]

Joulain, K.

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 89.

Lerosey, G.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 89.

Lobkis, O. I.

R. L. Weaver and O. I. Lobkis, Phys. Rev. Lett. 87, 134301 (2001).
[CrossRef] [PubMed]

Méndez, E. R.

E. R. Méndez, J.-J. Greffet, and R. Carminati, Opt. Commun. 142, 7 (1997).
[CrossRef]

Montaldo, G.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

Morse, P. M.

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

Mulet, J.-P.

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Science, 2006).

R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
[CrossRef]

R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, J. Opt. Soc. Am. A 15, 706 (1998).
[CrossRef]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 89.

Sáenz, J. J.

R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
[CrossRef]

Tourin, A.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

Weaver, R. L.

R. L. Weaver and O. I. Lobkis, Phys. Rev. Lett. 87, 134301 (2001).
[CrossRef] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

D. Cassereau and M. Fink, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 579 (1992).
[CrossRef] [PubMed]

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

Opt. Commun. (1)

E. R. Méndez, J.-J. Greffet, and R. Carminati, Opt. Commun. 142, 7 (1997).
[CrossRef]

Phys. Rev. A (1)

R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Phys. Rev. A 62, 012712 (2000).
[CrossRef]

Phys. Rev. B (1)

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, Phys. Rev. B 68, 245405 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

R. L. Weaver and O. I. Lobkis, Phys. Rev. Lett. 87, 134301 (2001).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Phys. Rev. Lett. 92, 193904 (2004).
[CrossRef] [PubMed]

C. Draeger and M. Fink, Phys. Rev. Lett. 79, 407 (1997).
[CrossRef]

J. de Rosny and M. Fink, Phys. Rev. Lett. 89, 124301 (2002).
[CrossRef] [PubMed]

Phys. Today (1)

M. Fink, Phys. Today 20(12), 34 (1997).
[CrossRef]

Prog. Surf. Sci. (1)

J.-J. Greffet and R. Carminati, Prog. Surf. Sci. 56, 133 (1997).
[CrossRef]

Science (1)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

Other (6)

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Science, 2006).

The vector form of Sommerfeld's radiation condition reads limr′-->∞[∇r′×G⃡(r′,r,ω)−iku×G⃡(r′,r,ω)]=0, where k=ω/c, r′=∣r′∣, and u=r′/r′. This condition states that G⃡ behaves as an outgoing (retarded) wave.

For two well-behaved vector fields A(r) and B(r), the second Green identity reads as ∫V[A∙∇×∇×B−B∙∇×∇×A]d3r=∫S[B×∇×A−A×∇×B]∙nd2r, where V is a volume enclosed by surface S and n is the outward normal. See Refs. .

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

For any dyadic D⃡ and any vectors U and V, one has U∙D⃡V=V∙D⃡TU.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 89.

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Figures (1)

Fig. 1
Fig. 1

Scheme of a time-reversal cavity. (a) Direct situation. (b) Time-reversed situation.

Equations (8)

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r × r × E ( r , ω ) ω 2 c 2 ϵ ( r , ω ) E ( r , ω ) = μ 0 ω 2 δ ( r r s ) p ,
r × r × G ( r , r , ω ) ω 2 c 2 ϵ ( r , ω ) G ( r , r , ω ) = δ ( r r ) I ,
V [ E ( r , ω ) r × r × G ( r , r , ω ) C G ( r , r , ω ) C r × r × E ( r , ω ) ] d 3 r = S [ G ( r , r , ω ) C × r × E ( r , ω ) E ( r , ω ) × r × G ( r , r , ω ) C ] n d 2 r .
E ( r ) r × r × G ( r , r ) C G ( r , r ) C r × r × E ( r ) = ω 2 c 2 [ ϵ ( r ) G ( r , r ) C E ( r ) ϵ ( r ) E ( r ) G ( r , r ) C ] E ( r ) C δ ( r r ) + μ 0 ω 2 G ( r , r ) C p δ ( r r s ) .
E ( r , ω ) C = μ 0 ω 2 G ( r , r s , ω ) p C S [ G ( r , r , ω ) C × r × E ( r , ω ) E ( r , ω ) × r × G ( r , r , ω ) C ] n d 2 r .
E TRC ( r , ω ) C = S [ G ( r , r , ω ) C × r × E * ( r , ω ) E * ( r , ω ) × r × G ( r , r , ω ) C ] n d 2 r .
E * ( r , ω ) C = μ 0 ω 2 G ( r , r s , ω ) p * C S [ G ( r , r , ω ) C × r × E * ( r , ω ) E * ( r , ω ) × r × G ( r , r , ω ) C ] n d 2 r .
E TRC ( r , ω ) = 2 i μ 0 ω 2 Im [ G ( r , r s , ω ) ] p * .

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