Extraordinary optical reflection from sub-wavelength cylinder arrays

doi:10.1364/OE.14.003730

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Extraordinary optical reflection from sub-wavelength cylinder arrays

Raquel Gómez-Medina, Marine Laroche, and Juan J. Sáenz »View Author Affiliations

Optics Express, Vol. 14, Issue 9, pp. 3730-3737 (2006)
http://dx.doi.org/10.1364/OE.14.003730

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▸ Article Outline
– Abstract
1. Introduction
2. Scattering theory for s-polarized waves (Electric field parallel to the cylinder axis)
3. Scattering theory for p-polarized waves (Magnetic field parallel to the cylinder axis)
4. Conclusion
– References and links

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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.

1. Introduction

The study of light scattering from periodic structures has been a topic of interest during the last century. Already in 1902, Wood [1

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).

, 2

R.W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 15, 928 (1935). [CrossRef]

] reported remarkable effects (known as Wood’s anomalies) in the reflectance of one-dimensional (1D) metallic gratings. Two different types of anomalies were definitely identified by Fano [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]

]. One is associated to the discontinuous change of intensity along the spectrum at sharply defined frequencies and was already discussed by Rayleigh [4

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. Roy. Soc. (London) A79, 399 (1907).

]. The other is related to a resonance effect. It occurs when the incoming wave couples with quasi-stationary waves confined in the grating. The nature of the confined waves depends on the details of the periodic structure [5

A. Hessel and A.A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275 (1965) [CrossRef]

] and is usually associated to surface plasmon polaritons in shallow metallic gratings, standing waves in deep grating grooves [5

A. Hessel and A.A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275 (1965) [CrossRef]

] or guided modes in dielectric coated metallic gratings [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) 8, 231 (1977).

Since the observation of enhanced transmission through a metallic film perforated by a 2D array of sub-wavelength holes [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]

], there has been a renewed interest in analyzing and understanding the underlying physics of both reflection and transmission “anomalies” in both 2D hole [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]

] and 1D slit [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]

] arrays. Although the enhanced transmission is commonly associated to the excitation of surface plasmons [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

], dynamical diffraction resonances [10

, 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]

] are also invoked as the origin of the effect. Recent theoretical and experimental works put forward that the excitation of any surface plasmons was not required [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]

]. Consequently, there remains some controversy surrounding the transmission mechanism [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]

]. In this work we discuss the physics behind the (Babinet) complementary problem: the extraordinary reflection from a periodic array of sub-wavelength scatterers.

In absence of resonant surface modes (or surface plasmons for metallic particles) the scattering cross section of subwavelength-sized particles is very small [25

H.C. van de Hulst, Light Scattering by small particles (Dover, New York, 1981).

, 26

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

]. However, in a periodic array of small particles, the coupling of the scattered dipolar field with diffraction modes may induce a geometric resonance close to the onset of new propagating modes (i.e. close to the Rayleigh frequencies). Following a multiple scattering approach [27

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

, 28

K. Ohtaka and H. Numata, “Multiple scattering effects in photon diffraction for an array of cylindrical dielectric,” Phys. Lett. 73A, 411 (1979).

, 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]

], we analytically derive the resonance conditions as a function of the geometry and the polarizabilities of each individual scatterer. As we will show, in absence of absorption, it is possible to have a perfect reflected wave even for vanishingly small scatterers.

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

Let us consider an infinite set of parallel cylinders with their axis along the 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_{0} = k \sin θ u_{x} + k \cos θ u_{y} \equiv Q_{0} u_{x} + q_{0} u_{y},

(1)

and k =ω/c.

Let us first consider an incoming wave with the electric field parallel to the cylinder axis (s-polarized wave, see Fig. 1), E = E(r)u _z = E⁰ e ^{i
Q
₀
x} e ^{i
q
₀
y} u _z. The scattered field from a given cylinder n, can be written as [25

H.C. van de Hulst, Light Scattering by small particles (Dover, New York, 1981).

, 26

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

E_{n}^{scatt} (r) = α_{zz} E_{in} (r_{n}) k^{2} G_{0} (r, r_{n})

(2)

where

α_{zz} \approx π a^{2} (ε - 1) {[1 - i \frac{π}{4} {(ka)}^{2} (ε - 1)]}^{- 1},

(3)

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 ^{i
Q
₀
x_m}, the total scattered field can be written as

E^{scatt} (r) = (α_{zz} E_{in} (r_{0})) k^{2} G (r)

(4)

Fig. 1. (s-polarization) Calculated reflectance R in a frequency ω versus in-plane wave number Q ₀ = (ω/c)sin(θ). The reflectance along the vertical lines is shown in the inset.

where the total Green function G(r) is given by:

G (r) \equiv \sum_{n = - \infty}^{+ \infty} e^{i Q_{0} x_{n}} G_{0} (r, r_{n}) = \frac{1}{D} \sum_{m = - \infty}^{\infty} e^{i (Q_{0} - K_{m}) x} (\frac{i}{2 q_{m}} e^{i q_{m} ∣ y ∣})

(5)

with K_m = 2πm/D and k ² =

q_{m}^{2}

+(Q ₀ -K_m )².

Multiple scattering effects manifest themselves in the actual incident field on each cylinder [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]

]: E_in(r ₀) is given by the incoming plane wave plus the scattered fields from other cylinders, i.e.

E_{in} (r_{0}) = E^{0} + α_{zz} k^{2} \sum_{m \neq 0} E_{in} (r_{m}) G_{0} (r_{0}, r_{m}) = E^{0} + α_{zz} k^{2} E_{in} (r_{0}) G_{b}

= {(1 - α_{zz} k^{2} G_{b})}^{- 1} E^{0}

(6)

where G_b = lim_{r→r
₀} [G(r)-G ₀(r,r ₀)]. The calculation of G_b involves the sum of a (poorly converging) series of Hankel functions. The convergence can be improved by using the well known result ∑_n=1 e ^-byn/n = -ln(1-e ^-by) [30

P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7.

, 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]

]. Finally, G_b is found to be given by

G_{b} = i {\frac{1}{2 D q_{0}} - \frac{1}{4}} + \frac{1}{2 D} \sum_{m = 1}^{\infty} (\frac{i}{q_{m}} + \frac{i}{q_{- m}} - \frac{2}{K_{m}}) + \frac{1}{2 π} (\ln {\frac{kD}{4 π}} + γ_{E}) .

(7)

The total scattered field (eq. 4) can then be rewritten as

E^{scatt} (r) = {\hat{α}}_{zz} E^{0} k^{2} G (r)

(8)

where all multiple scattering effects are included in the renormalized polarizability,

\hat{α}

_zz,

k^{2} {\hat{α}}_{zz} = {(\frac{1}{k^{2} α_{zz}} - G_{b})}^{- 1} .

(9)

Fig. 2. (s-polarization) (a) Plot of ℜ{G_b } and ℜ{1/(k ²α_zz} versus frequency along the constant Q ₀ line in Fig. 1 (Q ₀ = 0.8π/D)). The crossing points (open circles) correspond to different resonant frequencies ω _0m. (b) Calculated reflectance R versus ω. The inset shows a zoom-out of the ω ₀₁ resonance. Dashed lines corresponds to the approximate expression given in eq. 15 with no fitting parameters.

In order to calculate the transmittance/reflectance we consider the far-field limit (∣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

ψ (r) = ψ^{0} e^{i Q_{0} x} e^{i q_{0} y} + ψ^{0} \sum_{m}^{Prop} i \frac{\hat{4 πf} (q_{m}, q_{0})}{2 D q_{m}} e^{i (Q_{0} - K_{m}) x} e^{i q_{m} ∣ y ∣}

(10)

where 4πf̂(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

T = 1 - 2 \frac{ℑ {\hat{4 πf} (q_{0}, q_{0})}}{2 D q_{0}} + \frac{1}{D^{2}} \sum_{n}^{Prop} (\frac{{∣ \hat{4 πf} (q_{0}, q_{0}) ∣}^{2}}{4 q_{n} q_{0}})

(11)

R = \frac{1}{D^{2}} \sum_{n}^{Prop} (\frac{{∣ \hat{4 πf} (- q_{n}, q_{0}) ∣}^{2}}{4 q_{n} q_{0}}) .

(12)

For s-polarization (4πf̂(q_m ,q ₀) =

\hat{α}

_zz k ²) the scattering amplitude is isotropic and

R = \frac{ℑ {G (0)}}{2 D q_{0}} {∣ {\hat{α}}_{zz} k^{2} ∣}^{2} .

(13)

since

ℑ {G (0)} = \sum_{n}^{Prop} \frac{1}{2 D q_{n}} .

In absence of absorption (T +R = 1), ℑ{1/(k ²α_zz) = - ℑ{G ₀(0)}, we obtain

R = \frac{ℑ G (0)}}{2 D q_{0}} {(ℜ^{2} {\frac{1}{k^{2} α_{zz}} - G_{b}} + ℑ^{2} G (0)})}^{- 1}

(14)

(where ℜ²{x} = (Real{x})² and ℑ²{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 = ksin θ). For simplicity, we have considered a real dielectric constant (ε > 1) independent of the frequency (the results correspond to (2πa/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 ²α_zz)} > 0 is large and dominates the renormalized polarizability. However, approaching the threshold of a new propagating channel (i.e. ω →

ω_{m}^{(-)}

= 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 } ≈ (2D 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 (ω ≲ ω_{m}) \approx R_{max} {(\frac{1}{γ^{2}} {(1 - \sqrt{\frac{ω_{m}^{2} - ω_{m}^{2}}{ω_{m}^{2} - ω^{2}}})}^{2} + 1)}^{- 1}

(15)

where R _max=(2D q ₀ℑ{G(0)})^-1 and γ= ℑ{G(0)}/ℜ{1/(k ²α_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}^{2}

ω_{0 m}^{2}

)/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

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]

] where, for 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)

Let us now consider an incoming wave with the magnetic field parallel to the cylinder axis (p-polarized wave), H = H(r)u _z = H⁰ e ^{i
Q
₀
x} e ^{i
q
₀
y} u _z. The scattered (magnetic) field from a given cylinder [25

H.C. van de Hulst, Light Scattering by small particles (Dover, New York, 1981).

, 26

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

] can be cast in the form:

H_{n}^{scatt} (r) = - {α_{yy} \partial_{x} H_{in} (r)}_{r = r_{n}} \partial_{x} G_{0} (r, r_{n}) - {α_{xx} \partial_{y} H_{in} (r)}_{r = r_{n}} \partial_{y} G_{0} (r, r_{n})

(16)

where

α_{xx} = α_{yy} \approx 2 π a^{2} \frac{ε - 1}{ε + 1} {[1 - i \frac{π}{4} {(ka)}^{2} \frac{ε - 1}{ε + 1}]}^{- 1}

(17)

α_{xy} = α_{yx} = 0

(18)

and H_in(r _n) = H_in(r ₀)e ^{i
Q
₀
x_n} 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=r
_n}:

Fig. 3. (p-polarization) Calculated reflectance R in a frequency ω versus in-plane wave number Q ₀ = (ω/c)sin(θ). The reflectance along the vertical lines is shown in the inset.

lim_{r \to r_{0}} \partial_{x} H_{in} (r) = i Q_{0} H^{0} - α_{yy} \partial_{x} H_{in} (r ∣_{r = r_{0}} \partial_{x}^{2} G_{b} = i Q_{0} H^{0} {(1 + α_{yy} \partial_{x}^{2} G_{b})}^{- 1}

(19)

where

\partial_{x}^{2}

= lim_{r→r
₀}

\partial_{x}^{2}

[G(r) - G ₀(r, r ₀)]. The resulting series can be written as [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]

\partial_{x}^{2} G_{b} = - \frac{1}{2 D} \sum_{m = 1}^{\infty} {\frac{i {(K_{m} - Q_{0})}^{2}}{q_{m}} + \frac{i {(K_{m} + Q_{0})}^{2}}{q - m} - 2 K_{m} - \frac{k^{2}}{K_{m}}}

- \frac{k^{2}}{4 π} (\ln {\frac{kD}{4 π}} + γ_{E} - \frac{1}{2}) + \frac{1}{6} \frac{π}{D^{2}} + i (\frac{k^{2}}{8} - \frac{Q_{0}^{2}}{2 D q_{0}})

(20)

\partial_{y}^{2} G_{b} = - \frac{1}{2 D} \sum_{m = 1}^{\infty} {i q_{m} + i q_{- m} + 2 K_{m} - \frac{k^{2}}{K_{m}}}

- \frac{k^{2}}{4 π} (\ln {\frac{kD}{4 π}} + γ_{E} + \frac{1}{2}) + \frac{1}{6} \frac{π}{D^{2}} + i (\frac{k^{2}}{8} - \frac{q_{0}^{2}}{2 D q_{0}})

(21)

(notice that ∇² G_b +k ² G_b = 0). The total scattered field can now be written as

H_{n}^{scatt} (r) = - i H^{0} {\hat{α}}_{yy} Q_{0} \partial_{x} G (r) - i H^{0} {\hat{α}}_{xx} q_{0} \partial_{y} G (r)

(22)

where all the multiple scattering effects have been included in a renormalized polarizability tensor

\hat{\overset{\leftrightarrow}{α}}

{\hat{α}}_{xx} = {(1 + α_{xx} \partial_{y}^{2} G_{b})}^{- 1} α_{xx}

(23)

{\hat{α}}_{yy} = {(1 + α_{yy} \partial_{x}^{2} G_{b})}^{- 1} α_{yy} .

(24)

The transmittance/reflectance are given by the general equations 11 and 12 with a non-isotropic scattering amplitude

\hat{4 πf} (q_{m}, q_{0}) = {\hat{α}}_{yy} Q_{0} (Q_{0} - K_{m}) + {\hat{α}}_{xx} q_{0} q_{m} .

(25)

Figure 3 presents the calculated reflectance in a ω - Q ₀ map. (The results correspond to non-absorbing cylinders with (2πa/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{

\partial_{x}^{2}

G_b } ≈ - (Q ₀ - K_m )²(2D q_m )^-1 at the Rayleigh frequencies (in contrast ℜ{

\partial_{y}^{2}

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

Similar resonances to the ones discussed above appear in very different contexts under the label of Fano or Feshbach geometric resonances [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]

, 32

H. Feshbach,“Unified theory of nuclear reactions, I”, Ann. Phys. (N.Y.) 5, 357 (1958); “A unified theory of nuclear reactions, II,” Ann. Phys. (N.Y.) 19, 287 (1962).

, 33

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

]: Geometric resonances had been discussed in the context of electronic transport in waveguides [34

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

], ultracold atomic collisions [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]

] and light-atom interactions in confined geometries [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]

]. In general, Fano-Feshbach resonances occur when the energy (frequency) of the incoming wave is tuned to the energy (frequency) of a quasi-bound-state (QBS). These QBS may have a (multiple scattering -dynamical diffraction-) 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.

To summarize, in this work we have discussed the optical properties of an array of sub-wavelength Rayleigh cylinders. In absence of particle resonant modes, the cross-section of a single non-resonant cylinder can be extremely small and resonant effects are associated to 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

We thank S. Albaladejo, R. Carminati, J. García de Abajo, J.J. Greffet, L. Froufe-Pérez and M. Thomas for interesting discussions. This work has been supported by the Spanish MCyT (Ref No. BFM2003-01167) and the EU Integrated Project “Molecular Imaging” (EU contract LSHG-CT-2003-503259).

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 18, 269 (1902).
2.	R.W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 15, 928 (1935). [CrossRef]
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]
4.	Lord Rayleigh, “On the dynamical theory of gratings,” Proc. Roy. Soc. (London) A79, 399 (1907).
5.	A. Hessel and A.A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275 (1965) [CrossRef]
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) 8, 231 (1977).
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]
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]
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]
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]
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]
25.	H.C. van de Hulst, Light Scattering by small particles (Dover, New York, 1981).
26.	C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998). [CrossRef]
27.	V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am 52, 145 (1962). [CrossRef] [PubMed]
28.	K. Ohtaka and H. Numata, “Multiple scattering effects in photon diffraction for an array of cylindrical dielectric,” Phys. Lett. 73A, 411 (1979).
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]
30.	P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7.
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]
32.	H. Feshbach,“Unified theory of nuclear reactions, I”, Ann. Phys. (N.Y.) 5, 357 (1958); “A unified theory of nuclear reactions, II,” Ann. Phys. (N.Y.) 19, 287 (1962).
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]
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]
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]
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]

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]
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]
F. J. García de Abajo, R. Gómez-Medina and J. J. Sáenz, "Full transmission through perfect-conductor subwavelenght hole arrays," Phys. Rev. E, 72, 016608, (2005). [CrossRef]
J. B. Pendry, L. Martín-Moreno, F.J. García-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004). [CrossRef] [PubMed]
F. J. García de Abajo, and J. J. Sáenz, "Electromagnetic surface states in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005). [CrossRef]
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]
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]
Y. Takakura, "Optical Resonance in a Narrow Slit in a Thick Metallic Screen," Phys. Rev. Lett. 86, 5601 (2001). [CrossRef] [PubMed]
F. Yang and J.R. Sambles, "Resonant transmission of microwaves through a narrow metallic slit," Phys. Rev. Lett. 89, 063901 (2002). [CrossRef] [PubMed]
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]
M. M. J. Treacy,"Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002). [CrossRef]
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]
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]
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]
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]
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]
H.C. van de Hulst, Light Scattering by small particles (Dover, New York, 1981).
C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (JohnWiley & Sons, New York, 1998). [CrossRef]
V. Twersky, "Multiple scattering of waves and optical phenomena," J. Opt. Soc. Am 52, 145 (1962). [CrossRef] [PubMed]
K. Ohtaka and H. Numata, "Multiple scattering effects in photon diffraction for an array of cylindrical dielectric," Phys. Lett. 73A, 411 (1979).
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]
P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7.
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]
H. Feshbach,"Unified theory of nuclear reactions, I", Ann. Phys. (N.Y.) 5, 357 (1958); "A unified theory of nuclear reactions, II," Ann. Phys. (N.Y.) 19, 287 (1962).
U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866 (1961). [CrossRef]
J.U. Nöckel and A.D. Stone, "Resonance line shapes in quasi-one-dimensional scattering," Phys. Rev. B 50, 17415 (1994). [CrossRef]
R. G Gómez-Medina, P. San Jose, A. García- Martín, M. Lester, M. Nieto-Vesperinas, and J.J. Sáenz, "Resonant radiation pressure on neutral particles in a waveguide," Phys. Rev. Lett. 86, 4275 (2001). [CrossRef] [PubMed]
R. Gómez-Medina, and J.J. Sáenz, "Unusually strong optical interactions between particles in quasi-onedimensional geometries," Phys. Rev. Lett. 93, 243602 (2004). [CrossRef]
M. Olshanii, "Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons," Phys. Rev. Lett. 81, 938 (1998). [CrossRef]
P. Horak, P. Domokos and H. Ritsch, "Giant Lamb shift of atoms near lossy multimod optical micro-waveguides," Europhys. Lett. 61, 459 (2003). [CrossRef]

Barnes, W. L.
Campillo, I
Cao, Q.
Chavel, P.
Collin, R.E.
Devaux, E.
Dintinger, J.
Dolado, J.S.
Domokos, P.
Ebbesen, T. W.
Ebbesen, T.W.
Eggimann, W.H.
Enoch, S.
Fano, U.
García, F. J.
García, F.J.
García de Abajo, F. J.
García-Vidal, F. J.
Ghaemi, H.F.
Gómez-Medina, R.
Gómez-Medina, R. G
Hessel, A.
Horak, P.
Hugonin, J. P.
Klein Koerkamp, K. J.
Kuipers, L.
Kunze, Ch.
Lalanne, P.
Lenk, R.
Lezec, H.
Lezec, H.J.
Martín-Moreno, L.
Maystre, D.
Murray, W. A.
Nevière, M.
Nöckel, J.U.
Numata, H.
Ohtaka, K.
Oliner, A.A.
Olshanii, M.
Pendry, J. B.
Popov, E.
Porto, J. A.
Rayleigh, Lord
Reinisch, R.
Ritsch, H.
Rodier, J. C.
Sáenz, J. J.
Sáenz, J.J.
Sambles, J.R.
San Jose, P.
Sarrazin, M.
Sauvan, C.
Segerink, F. B.
Stone, A.D.
Takakura, Y.
Thio, T.
Treacy, M. M. J.
Twersky, V.
van der Molen, K. L.
van Hulst, N. F.
Vigneron, J.-P.
Vigoureux, J.-M.
Vincent, P.
Wolf, P.A.
Wood, R.W.
Yang, F.

Appl. Opt.

A. Hessel and A.A. Oliner, "A new theory of Wood’s anomalies on optical gratings," Appl. Opt. 4, 1275 (1965) [CrossRef]

Europhys. Lett.

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

J. Opt. (Paris)

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).

J. Opt. Soc. Am

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

J. Opt. Soc. Am.

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]

Nature

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]

On Microwave Theory and Techniques, MTT

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]

Opt. Express

Phys. Lett.

K. Ohtaka and H. Numata, "Multiple scattering effects in photon diffraction for an array of cylindrical dielectric," Phys. Lett. 73A, 411 (1979).

Phys. Rev.

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866 (1961). [CrossRef]
R.W. Wood, "Anomalous Diffraction Gratings," Phys. Rev. 15, 928 (1935). [CrossRef]

Phys. Rev. B

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]
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]
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]
M. M. J. Treacy,"Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002). [CrossRef]
J.U. Nöckel and A.D. Stone, "Resonance line shapes in quasi-one-dimensional scattering," Phys. Rev. B 50, 17415 (1994). [CrossRef]
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]
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]

Phys. Rev. E

F. J. García de Abajo, R. Gómez-Medina and J. J. Sáenz, "Full transmission through perfect-conductor subwavelenght hole arrays," Phys. Rev. E, 72, 016608, (2005). [CrossRef]

Phys. Rev. Lett.

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]
Y. Takakura, "Optical Resonance in a Narrow Slit in a Thick Metallic Screen," Phys. Rev. Lett. 86, 5601 (2001). [CrossRef] [PubMed]
F. Yang and J.R. Sambles, "Resonant transmission of microwaves through a narrow metallic slit," Phys. Rev. Lett. 89, 063901 (2002). [CrossRef] [PubMed]
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]
R. G Gómez-Medina, P. San Jose, A. García- Martín, M. Lester, M. Nieto-Vesperinas, and J.J. Sáenz, "Resonant radiation pressure on neutral particles in a waveguide," Phys. Rev. Lett. 86, 4275 (2001). [CrossRef] [PubMed]
R. Gómez-Medina, and J.J. Sáenz, "Unusually strong optical interactions between particles in quasi-onedimensional geometries," Phys. Rev. Lett. 93, 243602 (2004). [CrossRef]
M. Olshanii, "Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons," Phys. Rev. Lett. 81, 938 (1998). [CrossRef]
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]
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]
F. J. García de Abajo, and J. J. Sáenz, "Electromagnetic surface states in structured perfect-conductor surfaces," Phys. Rev. Lett. 95, 233901 (2005). [CrossRef]

Proc. R. Soc. London A

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).

Proc. Roy. Soc. (London)

Lord Rayleigh, "On the dynamical theory of gratings," Proc. Roy. Soc. (London) A79, 399 (1907).

Science

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

Sol. State Comm.

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]

Other

P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 7.
H.C. van de Hulst, Light Scattering by small particles (Dover, New York, 1981).
C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (JohnWiley & Sons, New York, 1998). [CrossRef]
H. Feshbach,"Unified theory of nuclear reactions, I", Ann. Phys. (N.Y.) 5, 357 (1958); "A unified theory of nuclear reactions, II," Ann. Phys. (N.Y.) 19, 287 (1962).

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Extraordinary optical reflection from sub-wavelength cylinder arrays

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Abstract

1. Introduction

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

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

4. Conclusion

Acknowledgements

References and links

References

Appl. Opt.

Europhys. Lett.

J. Opt. (Paris)

J. Opt. Soc. Am

J. Opt. Soc. Am.

Nature

On Microwave Theory and Techniques, MTT

Opt. Express

Phys. Lett.

Phys. Rev.

Phys. Rev. B

Phys. Rev. E

Phys. Rev. Lett.

Proc. R. Soc. London A

Proc. Roy. Soc. (London)

Science

Sol. State Comm.

Other

Cited By

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