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, 2] 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]. One is associated to the discontinuous change of intensity along the spectrum at sharply defined frequencies and was already discussed by Rayleigh [4]. 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] and is usually associated to surface plasmon polaritons in shallow metallic gratings, standing waves in deep grating grooves [5] or guided modes in dielectric coated metallic gratings [6].

Since the observation of enhanced transmission through a metallic film perforated by a 2D array of sub-wavelength holes [7], there has been a renewed interest in analyzing and understanding the underlying physics of both reflection and transmission “anomalies” in both 2D hole [8, 9, 10, 11, 12, 13, 14] and 1D slit [15, 8, 16, 17, 18] arrays. Although the enhanced transmission is commonly associated to the excitation of surface plasmons [7, 8, 9], dynamical diffraction resonances [10, 19, 20] 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, 20]. Consequently, there remains some controversy surrounding the transmission mechanism [21, 22, 23, 20, 24]. 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, 26]. 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, 28, 29], 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)

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 ^iQ₀x e ^iq₀y u _z. The scattered field from a given cylinder n, can be written as [25, 26]:

where

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 ^iQ₀x_m, the total scattered field can be written as

 

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.

Download Full Size | PDF

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

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

where G_b = lim_r→r0 [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, 31]. Finally, G_b is found to be given by

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

where all multiple scattering effects are included in the renormalized polarizability, $\widehat{\alpha }$ _zz,

 

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.

Download Full Size | PDF

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

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

For s-polarization (4πf̂(q_m ,q ₀) = $\widehat{\alpha }$ _zz k ²) the scattering amplitude is isotropic and

since $\Im \left\{G\left(0\right)\right\}={\sum }_n^{\mathrm{Prop}}\frac{1}{2D{q}_n}.$

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

(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 → ${\omega }_m^{(-)}$ = 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

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(${\omega }_m^2$ - ${\omega }_{0m}^2$)/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, 8, 16,17, 18] 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 ^iQ₀x e ^iq₀y u _z. The scattered (magnetic) field from a given cylinder [25, 26] can be cast in the form:

where

and H_in(r _n) = H_in(r ₀)e ^iQ₀x_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=rn:

 

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.

Download Full Size | PDF

where ${\mathit{\partial }}_x^2$ = lim_r→r0 ${\mathit{\partial }}_x^2$[G(r) - G ₀(r, r ₀)]. The resulting series can be written as [31]:

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

where all the multiple scattering effects have been included in a renormalized polarizability tensor $\hat{\overleftrightarrow{\alpha }}$:

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

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{${\mathit{\partial }}_x^2$ G_b } ≈ - (Q ₀ - K_m )²(2D q_m )^-1 at the Rayleigh frequencies (in contrast ℜ{${\mathit{\partial }}_y^2$ 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, 32, 33]: Geometric resonances had been discussed in the context of electronic transport in waveguides [34], ultracold atomic collisions [37] and light-atom interactions in confined geometries [35, 36, 38]. 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.