## The plasmon Talbot effect

Optics Express, Vol. 15, Issue 15, pp. 9692-9700 (2007)

http://dx.doi.org/10.1364/OE.15.009692

Acrobat PDF (2867 KB)

### Abstract

The plasmon analog of the self-imaging Talbot effect is described and theoretically analyzed. Rich plasmon carpets containing hot spots are shown to be produced by a row of periodically-spaced surface features. A row of holes drilled in a metal film and illuminated from the back side is discussed as a realizable implementation of this concept. Self-images of the row are produced, separated from the original one by distances up to several hundreds of wavelengths in the examples under consideration. The size of the image focal spots is close to half a wavelength and the spot positions can be controlled by changing the incidence direction of external illumination, suggesting the possibility of using this effect (and its extension to non-periodic surface features) for far-field patterning and for long-distance plasmon-based interconnects in plasmonic circuits, energy transfer, and related phenomena.

© 2007 Optical Society of America

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**, 189–193 (2006). [CrossRef] [PubMed]

3. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Materials Today **9**, 20–27 (2006). [CrossRef]

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**, 189–193 (2006). [CrossRef] [PubMed]

3. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Materials Today **9**, 20–27 (2006). [CrossRef]

4. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express **15**, 4474–4484 (2007). [CrossRef] [PubMed]

5. C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-Enhanced Raman Scattering from Individual Au Nanoparticles and Nanoparticle Dimer Substrates,” Nano Lett. **5**, 1569–1574 (2005). [CrossRef] [PubMed]

8. K. Patorski, “The self imaging phenomenon and its applications,” Prog. Opt. **27**, 1–108 (1989). [CrossRef]

*revives*(self-images) to its initial configuration after the

*Talbot distance τ*=2

*a*

^{2}/

*λ*, where

*a*is the transverse period and

*λ*is the wavelength. In a simple analytical description, we represent the grating by a periodic function given in Fourier series form,

*x*is the direction of periodicity. The monochromatic wave function emanating from the grating towards the

*y*direction reduces then to

*x*and

*y*in these exponential functions define a vector of magnitude 2

*π*/

*λ*, the light momentum. In the paraxial approximation (λ≪

*a*), the binomial expansion

*m*

^{2}, equivalent to Fresnel diffraction. This yields

*τ*=2

*a*

^{2}/

*λ*is indeed the Talbot distance at which the initial field self-images (except for an overall phase that is washed away when observing intensities), while another image is formed at

*τ*/2, laterally shifted by half a period and leading to an alternate definition of the Talbot distance [9

9. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. **43**, 2139–2164 (1996). [CrossRef]

*y*is a fraction of

*τ*, the field undergoes

*fractional revivals*, which in the ideal case are fractal at irrational values of

*y*/

*τ*[9

9. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. **43**, 2139–2164 (1996). [CrossRef]

10. A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. **2**, 413–415 (1971). [CrossRef]

*a*. In practice, this approximation stands only for a finite number of

*m*’s in (1), but it can be sufficient to render well-defined focal spots, as we shall see below for self-imaging of small features.

9. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. **43**, 2139–2164 (1996). [CrossRef]

10. A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. **2**, 413–415 (1971). [CrossRef]

12. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. **129**, 167–172 (1996). [CrossRef]

13. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express **14**, 3039–3044 (2006). [CrossRef] [PubMed]

14. I. S. Averbukh and N. F. Perelman, “Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A **139**, 449–453 (1989). [CrossRef]

15. M. V. Berry, “Quantum fractals in boxes,” J. Phys. A: Math Gen. **29**, 6617–6629 (1996). [CrossRef]

**43**, 2139–2164 (1996). [CrossRef]

16. E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. **98**, 132–140 (1993). [CrossRef]

17. T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A **21**, 1424–1429 (2004). [CrossRef]

18. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. **57**, 772–778 (1967). [CrossRef]

19. A. W. Lohmann, H. Knuppertz, and J. Jahns, “Fractional Montgomery effect: a self-imaging phenomenon,” J. Opt. Soc. Am. A **22**, 1500–1508 (2005). [CrossRef]

*m*| for

*m*

^{2}in Eq. (3), and obviously maintaining the property (4), but not (5). Recent work on a metal film perforated by quasiperiodic hole arrays has also revealed concentration of transmitted light intensity in hot spots at large distances from the film [20

20. F. M. Huang, N. Zheludev, Y. Chen, and F. J. García de Abajo, “Focusing of light by a nano-hole array,” Appl. Phys. Lett. **90**, 091,119 (2007). [CrossRef]

## 2. Self-focusing of plasmon carpets on metals: the plasmon Talbot effect

*a*. Light is partly transmitted into plasmons on the exit side of the film, thus deploying a complex carpet pattern. The field from each of the nanoholes is modeled as a dipole, oscillating with a frequency corresponding to the incident wavelength

*λ*

_{0}. This oscillation sets up surface plasmons, propagating into the plasmonic far-field with wavelength

*ε*of the metal. We shall concentrate our description on the situation most likely to find practical application, with small attenuation and |

*ε*|≫1, implying that λ

_{SP}≈λ

_{0}. We shall also concentrate on values of the periodicity

*a*of similar lengthscale to the plasmon wavelength

*λ*

_{SP}. In our graphical illustrations, we model a silver surface with incident wavelength

*λ*

_{0}=1.55

*µ*m, for which

*ε*=-130.83+i3.32 [21

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

*λ*

_{SP}=1.544

*µ*m.

*y*direction at position

**R**

_{0}infinitesimally close to the metal surface, incorporating direct propagation and reflection. The electric field, made dimensionless through multiplication by

*λ*

^{3}

_{0}, reads [22

22. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. **113**, 195–287 (1984). [CrossRef]

*k*=2

*π*/

*λ*

_{0}is the free-space light momentum,

**Q**=(

*k*,

_{x}*k*) is the projection of the wavevector

_{y}**k**into the plane of the metal,

**k**̂,

**ê**

*,*

_{p}**ê**

*} is the natural orthonormal basis for*

_{s}**k**, defined as

**ê**

*=*

_{s}**ẑ**×

**k**̂/|

**ẑ**×

**k**̂| and

**ê**

*=*

_{p}**ê**

*×*

_{s}**k**̂, and

*r*=(

_{p}*εk*-

_{z}*k*′

*)/(*

_{z}*εk*+

_{z}*k*′

_{z}) and

*r*=(

_{s}*k*-

_{z}*k*′

*)/(*

_{z}*k*+

_{z}*k*′

*) are the appropriate Fresnel reflection coefficients for TM (*

_{z}*p*) and TE (

*s*) polarization, with

**E**

_{single}is

*E*, and this is strongest on the metal plane for

_{z}**r**-

**R**

_{0}in the direction of the dipole (the

*y*direction). Therefore, to maximize the observable effect, we choose to make the periodic dipole array in the

*x*direction, with the plasmons propagating in

*y*.

**R**

*=(*

_{n}*na*,0,0). Using the Poisson sum formula [24], Σ

*exp(i*

_{n}*k*)=(2

_{x}na*π*/

*a*)Σ

*mδ*(

*k*-2

_{x}*πm*/

*a*), the infinite sum can be rewritten as a Rayleigh expansion,

**Q**

*=(2*

_{m}*πm*/

*a*,

*k*), and

_{y}**F**(

**Q**) is defined in Eq. (7). In the second line, 2

*π*/

*a*times the integral has been written as the

*y*- and

*z*-dependent Fourier coefficient

**F**

*(*

_{m}*y*,

*z*). Numerical evaluation of this field, for the values of the parameters above and various choices of a are shown in Fig. 2(a–c).

*a*=

*λ*

_{SP}, the Talbot effect is not yet developed, although an interesting periodic pattern appears that could be employed to imprint hight-quality 2D arrays. When we move to larger spacing [

*a*=5

*λ*

_{SP}in Fig. 2(b)], clear evidence of self-imaging is observed, which is particularly intense at half the Talbot distance. With even larger spacing [

*a*=20

*λ*

_{SP}in Fig. 2(c)] a fine Talbot carpet is deployed, showing structures reminiscent of cusp caustics at

*τ*and

*τ*/2 [25]. The focal-spot intensities decrease with distance from the hole array due to plasmon attenuation (≈1.26 mm for silver at

*λ*

_{0}=1.55

*µ*m), to which image contrast is however insensitive at these low-absorption levels.

*λ*

_{SP}, whereas its extension along

*y*is considerably larger. This type of behavior is also observed for other values of the period and for spots at integer Talbot distances. The width along

*x*varies from case to case, but it is always close to half a wavelength.

## 3. Analytical approach

*k*, particularly in the plasmon far-field, comes from the pole of the

_{y}*r*reflection coefficient, in the

_{p}*Q*upper-half complex plane. After all, the plasmon dispersion relation derives from that pole (i.e.,

*εk*+

_{z}*k*′

*=0), so that the plasmon itself posses*

_{z}*p*symmetry. This contribution may be approximated by the Cauchy integral theorem using the

*r*plasmon pole of wavenumber

_{p}*λ*

_{SP}=2

*π*/

*ℜ*{

*Q*

_{SP}} [22

22. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. **113**, 195–287 (1984). [CrossRef]

*z*component of the wavevector is

*m*|≤

*N*, where

*N*≈

*a*/

*λ*

_{SP}. For

*m*in this range, the approximation gives

*Q*

_{SP}is small enough to be neglected), apart from the extreme non-paraxiality of the regime under consideration. It should be noted that the

*m*dependence of

**F**

*is only in the vector and in the exponent of*

_{m}*y*, and therefore, the Talbot carpet is independent of

*z*in this plasmon-pole approximation, except for a global exponential decay away from the surface.

*a*≤20

*λ*

_{SP}in Fig. 2 are in the non-paraxial regime. In Ref. [9

**43**, 2139–2164 (1996). [CrossRef]

*m*

^{4}. The inclusion of this and later terms implies that the field is no longer perfectly periodic, and that the distance in y at which the (imperfect) self-imaging occurs is less than

*τ*(as in Fig. 2). However, as our simulations and analytic approximation demonstrate, good, if not perfect, Talbot focusing of plasmons should nevertheless be possible in practice (similar effects have been noticed in free-space propagation [16

16. E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. **98**, 132–140 (1993). [CrossRef]

17. T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A **21**, 1424–1429 (2004). [CrossRef]

*x*,

*y*)=(

*a*/2,

*τ*/2), calculated from Eq. (8), is illustrated in Fig. 4, which shows a complex evolution of the spot positions, generally below

*y*=

*τ*/2. An interesting consequence of these results is that the position of the focal spot can be controlled through small changes in wavelength.

*ε*|, electric dipoles parallel to the metal surface are quenched by their image charges. Then, the transmission through the holes depicted in Fig. 1 relies on parallel magnetic dipoles (provided such dipoles can be induced, for instance under the condition that the metal skin depth is small compared to the hole size [26]). Magnetic dipoles couple best to plasmons propagating in the

*y*direction when they are oriented along

*x*. The above analysis remains valid in that case, and in particular Eq. (9) is only corrected by a factor

**ẑ**) are also relevant under these conditions, induced by

*p*-polarized light under oblique incidence. Again, Eq. (9) can be still applied, amended by a factor √

*ε*/ζ

*.*

_{m}## 4. Discussion

*x*: the self-image is displaced along

*y*from the Talbot distance and it is also laterally shifted along

*x*, as shown both theoretically and experimentally in Ref. [12

12. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. **129**, 167–172 (1996). [CrossRef]

*k*, enters Eq. (9) through an uninteresting overall phase factor, but also through the coefficient of the exponential in

^{i}_{x}*y*,

*Q*

_{SP}ζ

*, which becomes*

_{m}*τ*×(

*k*

*,*

_{y}_{SP}/

*Q*

_{SP})

^{3}, where

*x*a distance

*y*

*k*/

^{i}_{x}*k*

_{y,SP}that increases with separation from the array. Thus, the position of the focal spots can be controlled through obliquity of the external illumination in a setup as in Fig. 1. One should therefore be able to raster the plasmon focus with nanometer accuracy for potential applications in nanolithography and biosensing.

27. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. García de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive sub-wavelength control of nano-optical fields,” Nature **446**, 301–304 (2007). [CrossRef] [PubMed]

28. M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A: Math Gen. **39**, 6965–6977 (2006). [CrossRef]

20. F. M. Huang, N. Zheludev, Y. Chen, and F. J. García de Abajo, “Focusing of light by a nano-hole array,” Appl. Phys. Lett. **90**, 091,119 (2007). [CrossRef]

29. F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range coupled surface exciton polaritons,” Phys. Rev. Lett. **64**, 559–562 (1990). [CrossRef] [PubMed]

30. R. Ulrich and M. Tacke, “Submillimeter waveguiding on periodic metal structure,” Appl. Phys. Lett. **22**, 251–253 (1972). [CrossRef]

31. A. Mugarza, A. Mascaraque, V. Pérez-Dieste, V. Repain, S. Rousset, F. J. García de Abajo, and J. E. Ortega, “Electron confinement in surface states on a stepped gold surface revealed by angle-resolved photoemission,” Phys. Rev. Lett. **87**, 107,601 (2001). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

2. | E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science |

3. | R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Materials Today |

4. | J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express |

5. | C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, “Surface-Enhanced Raman Scattering from Individual Au Nanoparticles and Nanoparticle Dimer Substrates,” Nano Lett. |

6. | H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. |

7. | Lord Rayleigh, “On copying diffraction-grating and on some phenomena connected with therewith,” Philos.Mag. |

8. | K. Patorski, “The self imaging phenomenon and its applications,” Prog. Opt. |

9. | M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. |

10. | A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. |

11. | A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik |

12. | M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. |

13. | K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express |

14. | I. S. Averbukh and N. F. Perelman, “Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A |

15. | M. V. Berry, “Quantum fractals in boxes,” J. Phys. A: Math Gen. |

16. | E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. |

17. | T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, “Exact self-imaging of transversely periodic fields,” J. Opt. Soc. Am. A |

18. | W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. |

19. | A. W. Lohmann, H. Knuppertz, and J. Jahns, “Fractional Montgomery effect: a self-imaging phenomenon,” J. Opt. Soc. Am. A |

20. | F. M. Huang, N. Zheludev, Y. Chen, and F. J. García de Abajo, “Focusing of light by a nano-hole array,” Appl. Phys. Lett. |

21. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

22. | G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. |

23. | J. D. Jackson, |

24. | W. Rudin, |

25. | M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. |

26. | F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. (in press). |

27. | M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. García de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive sub-wavelength control of nano-optical fields,” Nature |

28. | M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A: Math Gen. |

29. | F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range coupled surface exciton polaritons,” Phys. Rev. Lett. |

30. | R. Ulrich and M. Tacke, “Submillimeter waveguiding on periodic metal structure,” Appl. Phys. Lett. |

31. | A. Mugarza, A. Mascaraque, V. Pérez-Dieste, V. Repain, S. Rousset, F. J. García de Abajo, and J. E. Ortega, “Electron confinement in surface states on a stepped gold surface revealed by angle-resolved photoemission,” Phys. Rev. Lett. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: June 11, 2007

Revised Manuscript: July 13, 2007

Manuscript Accepted: July 13, 2007

Published: July 19, 2007

**Citation**

Mark R. Dennis, Nikolay I. Zheludev, and F. Javier García de Abajo, "The plasmon Talbot effect," Opt. Express **15**, 9692-9700 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9692

Sort: Year | Journal | Reset

### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- E. Ozbay, "Plasmonics: merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006). [CrossRef] [PubMed]
- R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, "Plasmonics: the next chip-scale technology," Materials Today 9, 20-27 (2006). [CrossRef]
- J. A. Conway, S. Sahni, and T. Szkopek, "Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs," Opt. Express 15, 4474-4484 (2007). [CrossRef] [PubMed]
- C. E. Talley, J. B. Jackson, C. Oubre, N. K. Grady, C. W. Hollars, S. M. Lane, T. R. Huser, P. Nordlander, and N. J. Halas, "Surface-Enhanced Raman Scattering from Individual Au Nanoparticles and Nanoparticle Dimer Substrates," Nano Lett. 5, 1569-1574 (2005). [CrossRef] [PubMed]
- H. F. Talbot, "Facts relating to optical science, No. IV," Philos. Mag. 9, 401-407 (1836).
- Lord Rayleigh, "On copying diffraction-grating and on some phenomena connected with therewith," Philos.Mag. 11, 196-205 (1881).
- K. Patorski, "The self imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989). [CrossRef]
- M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996). [CrossRef]
- A. W. Lohmann and D. E. Silva, "An interferometer based on the Talbot effect," Opt. Commun. 2, 413-415 (1971). [CrossRef]
- A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).
- M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, "Talbot effect for oblique angle of light propagation," Opt. Commun. 129, 167-172 (1996). [CrossRef]
- K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 14, 3039-3044 (2006). [CrossRef] [PubMed]
- I. S. Averbukh and N. F. Perelman, "Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics," Phys. Lett. A 139, 449-453 (1989). [CrossRef]
- M. V. Berry, "Quantum fractals in boxes," J. Phys. A: Math Gen. 29, 6617-6629 (1996). [CrossRef]
- E. Noponen and J. Turunen, "Electromagnetic theory of Talbot imaging," Opt. Commun. 98, 132-140 (1993). [CrossRef]
- T. Saastamoinen, J. Tervo, P. Vahimaa, and J. Turunen, "Exact self-imaging of transversely periodic fields," J. Opt. Soc. Am. A 21, 1424-1429 (2004). [CrossRef]
- W. D. Montgomery, "Self-imaging objects of infinite aperture," J. Opt. Soc. Am. 57, 772-778 (1967). [CrossRef]
- A. W. Lohmann, H. Knuppertz, and J. Jahns, "Fractional Montgomery effect: a self-imaging phenomenon," J. Opt. Soc. Am. A 22, 1500-1508 (2005). [CrossRef]
- F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garc’ıa de Abajo, "Focusing of light by a nano-hole array," Appl. Phys. Lett. 90, 091,119 (2007). [CrossRef]
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- G. W. Ford and W. H. Weber, "Electromagnetic interactions of molecules with metal surfaces," Phys. Rep. 113, 195-287 (1984). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
- W. Rudin, Real and Complex Analysis (McGraw-Hill, London, 1941).
- M. V. Berry and E. Bodenschatz, "Caustics, multiply reconstructed by Talbot interference," J. Mod. Opt. 46, 349-365 (1999).
- F. J. Garc’ıa de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. (in press).
- M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. Garc’ıa de Abajo,W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive sub-wavelength control of nano-optical fields," Nature 446, 301-304 (2007). [CrossRef] [PubMed]
- M. V. Berry and S. Popescu, "Evolution of quantum superoscillations and optical superresolution without evanescent waves," J. Phys. A: Math Gen. 39, 6965-6977 (2006). [CrossRef]
- F. Yang, J. R. Sambles, and G. W. Bradberry, "Long-range coupled surface exciton polaritons," Phys. Rev. Lett. 64, 559-562 (1990). [CrossRef] [PubMed]
- R. Ulrich andM. Tacke, "Submillimeter waveguiding on periodic metal structure," Appl. Phys. Lett. 22, 251-253 (1972). [CrossRef]
- A. Mugarza, A. Mascaraque, V. P’erez-Dieste, V. Repain, S. Rousset, F. J. Garc’ıa de Abajo, and J. E. Ortega, "Electron confinement in surface states on a stepped gold surface revealed by angle-resolved photoemission," Phys. Rev. Lett. 87, 107,601 (2001). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.