## Extraordinary optical reflection from sub-wavelength cylinder arrays

Optics Express, Vol. 14, Issue 9, pp. 3730-3737 (2006)

http://dx.doi.org/10.1364/OE.14.003730

Acrobat PDF (324 KB)

### Abstract

A multiple scattering analysis of the reflectance of a periodic array of sub-wavelength cylinders is presented. The optical properties and their dependence on wavelength, geometrical parameters and cylinder dielectric constant are analytically derived for both *s*- and *p*-polarized waves. In absence of Mie resonances and surface (plasmon) modes, and for positive cylinder polarizabilities, the reflectance presents sharp peaks close to the onset of new diffraction modes (Rayleigh frequencies). At the lowest resonance frequency, and in the absence of absorption, the wave is perfectly reflected even for vanishingly small cylinder radii.

© 2006 Optical Society of America

## 1. Introduction

7. T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667 (1998). [CrossRef]

8. E. Popov, M. Neviére, S. Enoch, and R. Reinisch,“Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B **62**, 16100 (2000). [CrossRef]

9. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen,“Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. **86**, 1114 (2001). [CrossRef] [PubMed]

10. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B **67**, 085415 (2003). [CrossRef]

11. F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwave-lenght hole arrays,” Phys. Rev. E , **72**, 016608, (2005). [CrossRef]

12. J. B. Pendry, L. Martín-Moreno, and F.J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847 (2004). [CrossRef] [PubMed]

13. F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface states in structured perfect-conductor surfaces,” Phys. Rev. Lett. **95**, 233901 (2005). [CrossRef]

14. F. J. García de Abajo, J.J. Sáenz, I Campillo, and J.S. Dolado, “Site and Lattice Resonances in Metallic Hole Arrays,” Opt. Express **14**, 7 (2006). [CrossRef] [PubMed]

15. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. **83**, 2845 (1999). [CrossRef]

8. E. Popov, M. Neviére, S. Enoch, and R. Reinisch,“Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B **62**, 16100 (2000). [CrossRef]

16. Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. **86**, 5601 (2001). [CrossRef] [PubMed]

17. F. Yang and J.R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. **89**, 063901 (2002). [CrossRef] [PubMed]

18. F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B **66**, 155412 (2002) [CrossRef]

7. T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667 (1998). [CrossRef]

8. E. Popov, M. Neviére, S. Enoch, and R. Reinisch,“Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B **62**, 16100 (2000). [CrossRef]

9. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen,“Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. **86**, 1114 (2001). [CrossRef] [PubMed]

10. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B **67**, 085415 (2003). [CrossRef]

19. M. M. J. Treacy,“Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B **66**, 195105 (2002). [CrossRef]

20. H. Lezec and T. Thio,“Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express **12**, 3629 (2004). [CrossRef] [PubMed]

11. F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwave-lenght hole arrays,” Phys. Rev. E , **72**, 016608, (2005). [CrossRef]

20. H. Lezec and T. Thio,“Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express **12**, 3629 (2004). [CrossRef] [PubMed]

21. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. **92**, 107401 (2004). [CrossRef] [PubMed]

22. Q. Cao and P. Lalanne, “Negative Role of Surface Plasmons in the Transmission of Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. **88**, 057403(2002). [CrossRef] [PubMed]

23. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B **68**, 125404 (2003). [CrossRef]

20. H. Lezec and T. Thio,“Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express **12**, 3629 (2004). [CrossRef] [PubMed]

24. K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers,“Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory,” Phys. Rev. B **72**, 045421 (2005). [CrossRef]

26. C.F. Bohren and D.R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, New York, 1998). [CrossRef]

*geometric*resonance close to the onset of new propagating modes (i.e. close to the Rayleigh frequencies). Following a multiple scattering approach [27

27. V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am **52**, 145 (1962). [CrossRef] [PubMed]

29. Ch. Kunze and R. Lenk, “A single scatter in a quantum wire: compact reformulation of scattering and transmission,” Sol. State Comm. **84**, 457 (1992). [CrossRef]

## 2. Scattering theory for s-polarized waves (Electric field parallel to the cylinder axis)

*z*-axis, relative dielectric constant ε and radius

*a*much smaller than the wavelength. The cylinders are located at

**r**

_{n}=

*nD*

**u**

_{x}=

*x*

_{n}

**u**

_{x}(with

*n*an integer number). For simplicity, we will assume incoming plane waves with wave vector

**k**

_{0}⊥

**u**

_{z}(i.e. the fields do not depend on the z-coordinate)

*k*=

*ω*/

*c*.

*s*-polarized wave, see Fig. 1),

**E**= E(

**r**)

**u**

_{z}= E

^{0}

*e*

^{iQ0x}

*e*

^{iq0y}

**u**

_{z}. The scattered field from a given cylinder

*n*, can be written as [25, 26

26. C.F. Bohren and D.R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, New York, 1998). [CrossRef]

*G*

_{0}(

**r**,

**r**

_{n}) = (

*i*/4)

*H*

_{0}(

*k*∣

**r**-

**r**

_{n}∣) is the free-space Green function (

*H*

_{0}is the Hankel function), and E

_{in}(

**r**

_{n}) is the incident field on the scatterer. Since for a periodic array E

_{in}(

**r**

_{m}) = E

_{in}(

**r**

_{0})

*e*

^{iQ0xm}, the total scattered field can be written as

*G*(

**r**) is given by:

*K*

_{m}= 2

*πm*/

*D*and

*k*

^{2}=

*Q*

_{0}-

*K*

_{m})

^{2}.

*incident*field on each cylinder [27

27. V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am **52**, 145 (1962). [CrossRef] [PubMed]

29. Ch. Kunze and R. Lenk, “A single scatter in a quantum wire: compact reformulation of scattering and transmission,” Sol. State Comm. **84**, 457 (1992). [CrossRef]

_{in}(

**r**

_{0}) is given by the incoming plane wave plus the scattered fields from other cylinders, i.e.

*G*

_{b}= lim

_{r→r0}[

*G*(

**r**)-

*G*

_{0}(

**r**,

**r**

_{0})]. The calculation of

*G*involves the sum of a (poorly converging) series of Hankel functions. The convergence can be improved by using the well known result ∑

_{b}_{n=1}

*e*

^{-byn}/

*n*= -ln(1-

*e*

^{-by}) [30, 31

31. R.E. Collin and W.H. Eggimann,“Dynamic Interaction Fields in a Two-Dimensional Lattice,” IRE Trans. on Microwave Theory and Techniques , MTT-**9**, 110 (1961). [CrossRef]

*G*

_{b}is found to be given by

_{zz},

*y*∣→∞) where only propagating diffraction orders (or channels) contribute to the scattered power. The total field, for both

*s*- and

*p*-polarizations (see below) can be written in the general form

*πf̂*(

*q*

_{m},

*q*

_{0}) is the scattering amplitude and the sum runs only over modes having ∣

*Q*

_{0}-

*K*

_{m}∣ <

*k*. The transmittance

*T*(reflectance

*R*), defined as the ratio between transmitted (reflected) power and incoming power is shown to be

*s*-polarization (4

*πf̂*(

*q*

_{m},

*q*

_{0}) =

_{zz}

*k*

^{2}) the scattering amplitude is isotropic and

*T*+

*R*= 1), ℑ{1/(

*k*

^{2}α

_{zz}) = - ℑ{

*G*

_{0}(0)}, we obtain

^{2}{

*x*} = (Real{

*x*})

^{2}and ℑ

^{2}{

*x*} = (Imag{

*x*})

^{2}). Figure 1 presents the calculated reflectance in a

*ω*

*vs*.

*Q*

_{0}map (frequency versus in-plane wave number

*Q*

_{0}=

*k*sin

*θ*). For simplicity, we have considered a real dielectric constant (ε > 1) independent of the frequency (the results correspond to (2

*πa*/

*D*)

^{2}(ε - 1) ≈ 4/9). The physics of the reflectance/transmittance can be understood from a simple argument (see Fig. 2): For small cylinders and ε > 1, ℜ{1/(

*k*

^{2}α

_{zz})} > 0 is large and dominates the renormalized polarizability. However, approaching the threshold of a new propagating channel (i.e.

*ω*→

*c*∣

*Q*

_{0}-

*K*

_{m}∣)

*q*

_{m}goes to zero. Then, the contribution of the lowest evanescent mode to ℜ{

*G*

_{b}} outweighs all the others and ℜ{

*G*

_{b}} ≈ (2

*D*

*q*

_{m})

^{-1}diverging at the threshold. The precise compensation of these two large terms at

*ω*=

*ω*

_{0m}(see Fig. 2(a)) gives rise to a geometric resonance. Very close to each resonance, the reflectance along the vertical lines in Fig. 1 (i.e. for a given

*Q*

_{0}) can then be approximated as

*R*

_{max}=(2

*D*

*q*

_{0}ℑ{

*G*(0)})

^{-1}and γ= ℑ{

*G*(0)}/ℜ{1/(

*k*

^{2}α

_{zz})}. As shown in Fig. 2(b), the reflection resonances present typical asymmetric Fano line shapes: The reflectance presents sharp maxima

*R*=

*R*

_{max}at frequencies

*ω*=

*ω*

_{0m}⪅

*ω*

_{m}. Just at the onset of a new diffraction channel, i.e. at the Rayleigh frequencies

*ω*=

*ω*

_{m}the reflectance goes to zero. For

*ω*=

*ω*

_{01}, i.e. at the lowest resonance frequency, there is a perfect reflection (

*R*= 1) even for vanishingly small cylinders (although, in these extreme cases, the resonance width Γ ≈ γ(

*ω*

_{m}goes to zero). Notice that for metallic cylinders or strips (with α < 0) there will be no sharp resonances. This is consistent with the Babinet complementary system of a periodic array of slits [15

15. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. **83**, 2845 (1999). [CrossRef]

**62**, 16100 (2000). [CrossRef]

16. Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. **86**, 5601 (2001). [CrossRef] [PubMed]

17. F. Yang and J.R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. **89**, 063901 (2002). [CrossRef] [PubMed]

18. F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B **66**, 155412 (2002) [CrossRef]

*p*-polarized waves, sharp transmittance peaks only appear for deep enough gratings (i.e. when the phase shift inside the slit changes the sign of the effective polarizability).

## 3. Scattering theory for p-polarized waves (Magnetic field parallel to the cylinder axis)

*p*-polarized wave),

**H**= H(

**r**)

**u**

_{z}= H

^{0}

*e*

^{iQ0x}

*e*

^{iq0y}

**u**

_{z}. The scattered (magnetic) field from a given cylinder [25, 26

26. C.F. Bohren and D.R. Huffman, *Absorption and Scattering of Light by Small Particles* (John Wiley & Sons, New York, 1998). [CrossRef]

_{in}(

**r**

_{n}) = H

_{in}(

**r**

_{0})

*e*

^{iQ0xn}is the incident field on the scatterer. For

*p*-polarization, multiple scattering effects manifest themselves in the actual incident magnetic field

*gradient*on each cylinder, ∇H

_{in}(

**r**)∣

_{r=rn}:

_{r→r0}

*G*(

**r**) -

*G*

_{0}(

**r**,

**r**

_{0})]. The resulting series can be written as [31

31. R.E. Collin and W.H. Eggimann,“Dynamic Interaction Fields in a Two-Dimensional Lattice,” IRE Trans. on Microwave Theory and Techniques , MTT-**9**, 110 (1961). [CrossRef]

^{2}

*G*

_{b}+

*k*

^{2}

*G*

_{b}= 0). The total scattered field can now be written as

*ω*-

*Q*

_{0}map. (The results correspond to non-absorbing cylinders with (2

*πa*/

*D*)

^{2}(ε - 1)/(ε + 1) ≈ 8/9). The physics behind the reflectance presents significant differences with respect to

*s*-polarization Sharp peaks in the reflectance (which now appear for ε > 1 or ε < - 1) are associated to the resonant coupling of electric dipoles pointing along the

*y*-axis which lead to the divergence of ℜ{

*G*

_{b}} ≈ - (

*Q*

_{0}-

*K*

_{m})

^{2}(2

*D*

**q**

_{m})

^{-1}at the Rayleigh frequencies (in contrast ℜ{

*G*

_{b}} remains finite). In absence of absorption, the reflectance at the lowest resonant frequency can be very large but, in contrast with

*s*-waves, strictly less than 1.

## 4. Conclusion

3. U. Fano, “The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves),” J. Opt. Soc. Am. **31**, 213 (1941). [CrossRef]

33. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866 (1961). [CrossRef]

34. J.U. Nöckel and A.D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B **50**, 17415 (1994). [CrossRef]

37. M. Olshanii, “Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons,” Phys. Rev. Lett. **81**, 938 (1998). [CrossRef]

35. R. Gómez-Medina, P. San José, A. García-Martín, M. Lester, M. Nieto-Vesperinas, and J.J. Saénz, “Resonant radiation pressure on neutral particles in a waveguide,” Phys. Rev. Lett. **86**, 4275 (2001). [CrossRef] [PubMed]

36. R. Gómez-Medina and J.J. Sáenz, “Unusually strong optical interactions between particles in quasi-one-dimensional geometries,” Phys. Rev. Lett. **93**, 243602 (2004). [CrossRef]

38. P. Horak, P. Domokos, and H. Ritsch, “Giant Lamb shift of atoms near lossy multimod optical micro-waveguides,” Europhys. Lett. **61**, 459 (2003). [CrossRef]

*geometrical*origin or can be an internal property of each scatterer. From the discussion above, reflectance resonances, for both

*s*and

*p*polarized waves, have a geometrical origin for dielectric cylinders. The existence of particle surface modes or plasmons would reflect itself in a resonant behavior of the bare polarizabilities (for p-polarized fields) and ℜ{1/α

_{ii}} would present sharp maxima and minima around each internal resonant frequency. Surface modes would then induce new peaks in the reflectance or, when the surface resonance frequency is close to a Rayleigh frequency, they would mix with geometrical resonances leading to more complex reflectance patterns.

*geometrical resonances*. We have shown that, for non-absorbing scatterers, it is possible to have a perfect reflected wave even for vanishingly small cylindrical radii. We believe that our study of reflectance resonances provide a new physical insight into the general mechanisms of light interactions with periodic structures of sub-wavelength objects.

## Acknowledgements

## References and links

1. | R.W. Wood, “On the remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. R. Soc. London A |

2. | R.W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. |

3. | U. Fano, “The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves),” J. Opt. Soc. Am. |

4. | Lord Rayleigh, “On the dynamical theory of gratings,” Proc. Roy. Soc. (London) |

5. | A. Hessel and A.A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. |

6. | M. Neviére, D. Maystre, and P. Vincent,“Application du calcul des modes de propagation a letude theorique des anomalies des reseaux recouverts de dielectrique,” J. Opt. (Paris) |

7. | T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolf, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

8. | E. Popov, M. Neviére, S. Enoch, and R. Reinisch,“Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B |

9. | L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen,“Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. |

10. | M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B |

11. | F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwave-lenght hole arrays,” Phys. Rev. E , |

12. | J. B. Pendry, L. Martín-Moreno, and F.J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

13. | F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface states in structured perfect-conductor surfaces,” Phys. Rev. Lett. |

14. | F. J. García de Abajo, J.J. Sáenz, I Campillo, and J.S. Dolado, “Site and Lattice Resonances in Metallic Hole Arrays,” Opt. Express |

15. | J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. |

16. | Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. |

17. | F. Yang and J.R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. |

18. | F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B |

19. | M. M. J. Treacy,“Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B |

20. | H. Lezec and T. Thio,“Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express |

21. | W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. |

22. | Q. Cao and P. Lalanne, “Negative Role of Surface Plasmons in the Transmission of Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. |

23. | P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B |

24. | K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers,“Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory,” Phys. Rev. B |

25. | H.C. van de Hulst, |

26. | C.F. Bohren and D.R. Huffman, |

27. | V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am |

28. | K. Ohtaka and H. Numata, “Multiple scattering effects in photon diffraction for an array of cylindrical dielectric,” Phys. Lett. |

29. | Ch. Kunze and R. Lenk, “A single scatter in a quantum wire: compact reformulation of scattering and transmission,” Sol. State Comm. |

30. | P.M. Morse and H. Feshbach, |

31. | R.E. Collin and W.H. Eggimann,“Dynamic Interaction Fields in a Two-Dimensional Lattice,” IRE Trans. on Microwave Theory and Techniques , MTT- |

32. | H. Feshbach,“Unified theory of nuclear reactions, I”, Ann. Phys. (N.Y.) |

33. | U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. |

34. | J.U. Nöckel and A.D. Stone, “Resonance line shapes in quasi-one-dimensional scattering,” Phys. Rev. B |

35. | R. Gómez-Medina, P. San José, A. García-Martín, M. Lester, M. Nieto-Vesperinas, and J.J. Saénz, “Resonant radiation pressure on neutral particles in a waveguide,” Phys. Rev. Lett. |

36. | R. Gómez-Medina and J.J. Sáenz, “Unusually strong optical interactions between particles in quasi-one-dimensional geometries,” Phys. Rev. Lett. |

37. | M. Olshanii, “Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons,” Phys. Rev. Lett. |

38. | P. Horak, P. Domokos, and H. Ritsch, “Giant Lamb shift of atoms near lossy multimod optical micro-waveguides,” Europhys. Lett. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 14, 2006

Revised Manuscript: April 7, 2006

Manuscript Accepted: April 13, 2006

Published: May 1, 2006

**Citation**

Raquel Gómez-Medina, Marine Laroche, and Juan J. Sáenz, "Extraordinary optical reflection from sub-wavelength cylinder arrays," Opt. Express **14**, 3730-3737 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3730

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### References

- R.W. Wood, "On the remarkable case of uneven distribution of light in a diffraction grating spectrum," Proc. R. Soc. London A 18, 269 (1902).
- R.W. Wood, "Anomalous Diffraction Gratings," Phys. Rev. 15, 928 (1935). [CrossRef]
- U. Fano, "The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves)," J. Opt. Soc. Am. 31, 213 (1941). [CrossRef]
- Lord Rayleigh, "On the dynamical theory of gratings," Proc. Roy. Soc. (London) A79, 399 (1907).
- A. Hessel and A.A. Oliner, "A new theory of Wood’s anomalies on optical gratings," Appl. Opt. 4, 1275 (1965) [CrossRef]
- M. Nevière, D. Maystre, P. Vincent,"Application du calcul des modes de propagation a letude theorique des anomalies des reseaux recouverts de dielectrique," J. Opt. (Paris) 8, 231 (1977).
- T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolf, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667 (1998). [CrossRef]
- E. Popov, M. Nevière, S. Enoch and R. Reinisch,"Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100 (2000). [CrossRef]
- L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen," Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays," Phys. Rev. Lett. 86, 1114 (2001). [CrossRef] [PubMed]
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