A detailed study of light absorption by silver gratings having two-dimensional periodicity is presented for structures constructed either of channels or of holes with subwavelength dimensions. Rigorous numerical modelling shows a systematic difference between the two structures: hole (cavity) gratings can strongly absorb light provided the cavity is sufficiently deep, when compared to the wavelength, whereas very thin channel gratings can induce total absorption. A detailed analysis is given in the limit when the period tends towards zero, and an explanation of the differences in behavior is presented using the properties of effective optical index of the metamaterial layer that substitutes the periodical structure in the limit when the period tend to zero.
©2012 Optical Society of America
Enhanced light absorption by periodic structures has attracted the attention of scientists and engineers since the first observation of grating anomalies by R. Wood . Resonant excitation of surface plasmon waves, responsible for these anomalies was first proposed by Fano  as the explication of this phenomenon. Later, Hutley and Maystre  have predicted and observed total light absorption (called by them Brewster effect) by relatively shallow gratings, with groove height not exceeding 10% of the wavelength. The resonant excitation of surface plasmons gave explanation to surface-enhanced Raman scattering (SERS)  and was largely used in nonlinear optics .
Since the work of Ebbesen et al.  on the observation of enhanced transmission through metallic hole arrays, surface and volume plasmons became object of so great number of experimental and theoretical studies that gave birth to a new name ‘plasmonics’ of the domain.
Light absorption remains also a separate topic of interest for applications in photovoltaics and microwave isolating. Recently, it has been shown that very shallow periodic structures having one- or two-dimensional periodicity can absorb light totally in relatively large spectral and angular interval [7–9]. The advantage of 2D periodicity is the possibility to enhance absorption in unpolarized light, whereas for 1D gratings this is possible by combining surface plasmon and cavity resonances, which imposes tighter constraints to manufacturing.
There are many yet unsolved problems in the theory of periodic structures. A typical example is the study of the similarities and differences between inductive (with continuous perforated metal layer in the grating region) and capacitive grids (where the metal inclusions are separated from each other inside the grating) . Another problem is the choice of homogetization procedure for structures having very small periods when compared with the wavelength of light, structures known as metamaterials.
Our aim in this paper is to study light absorption by two different types of 2D gratings, constructed of channels or holes. If the substrate is dielectric, and the grating bulk material is metal, these systems are called capacitive and inductive grids, respectively . While channel gratings of just several nm depth can absorb light totally, as already shown in [7, 8], the inverted geometry requires much deeper modulation values to obtain similar performance. This difference persists even when the period is reduced to just several nanometers (1/100 of light wavelength). We propose an explanation from metamaterial point of view by testing two approaches for obtaining the formula for the effective dielectric permittivity when the period of the structure tends to zero. A very good agreement is observed between the rigorous electromagnetic modeling and he effective index approach.
2. Comparative study of channel and hole array gratings
Let us consider the reflection properties of silver-made gratings with 2D periodicity, presented schematically in Fig. 1 . The first system consists of square pillars separated by channels; the second one contains square holes separated by metallic wall. In what follows, the substrate and the grating material is silver, the wavelength is equal to 457 nm, the cladding, channel and hole material is air. The calculations are done using a rigorous electromagnetic home-made code based on the Fourier Modal Method. Li’s factorization rules [11, 12] are applied for square holes and pillars, and Normal Vector Method  is used for circular shapes.
The reflection properties of the first system has been studied in [7, 8] and they are characterized by strong absorption in large angular and spectral intervals, even for quite shallow channels. Figure 2(a) gives the dependence of the reflectivity on the channel depth h and the filling ratio f:
The period d = 250 nm is chosen to avoid non-specular diffraction orders in the visible. As previously explained [7, 8], these regions of high absorption are due to the metamaterial behavior of the grating structure, which has effective permittivity that is characteristic for highly absorbing anisotropic dielectric, rather than metal (see also next section).On the contrary, the holes grating reflectivity remains highs for shallow holes (Fig. 2(b)), and diminishes significantly only when h > 500 nm. An explanation of this difference between hole and channel gratings, obtained from the metamaterial point of view is discussed in the next section.
There exists a physical explanation for relatively large periods that is linked with the existence of cavity modes inside the channels and the holes, modes that can propagate in the vertical direction. In the case of channels, the fundamental mode has no cut-off for 1D gratings and it is equivalent to the TEM mode in metallic plane waveguides. Figure 3 presents the dependence of the real and imaginary part of the normalized propagation constant of the modes for 2D channel and hole gratings of finite conductivity, taking into account the modal interaction through the channel (or hole) walls by the tunneling effect, and the interaction through the channels open in the perpendicular direction.
The modes are determined by the Fourier-modal expansion of the field inside the grating region, an approach widely known as rigorous coupled-wave (RCW) method. Due to the periodicity of the structure in x- and y-direction, and the invariance in z-direction, the electric field vector in monochromatic regime can be represented in a modal form:11–16] that links the electromagnetic field values with its z-derivatives inside the grating layer. In normal incidence:
Because of the invariance of the grating region in z-direction, the mode constants are independent of z. As the filling factor is varied, Fig. 3(a) presents the mode constant having the smallest imaginary part for the channel grating.
In the case of 1D perfectly conducting channels, the TEM mode constant is equal to the refractive index of the channels (equal to one in our case). When f = 0, the pillars in Fig. 1(a) have zero width, so that the fundamental harmonic of the field in Eq. (2) is also equal to one. When the bums are growing in size, the interaction between the modes in the parallel and perpendicular channels leads to increase of the losses (reflected in the increase of the imaginary part of γ).
More interesting is the fact that the real part also grows with f, which can be understood by the fact that the capacitance of the system increases as the gap between the conductive elements decreases. The analysis of the equivalent effective index in the next section for shorter periods shows an increase of the real part of the effective refractive index, which explains the increase of Re(γ).
The existence of this fundamental mode can explain the appearance of consecutive minima and maxima in the reflectivity as a function of h, as well as the decrease of the distance between them as f increases. Indeed, a Fabry-Perot resonance within the grating structure due to the fundamental mode has to be quasiperiodic with respect to h, with the period Dh determined by the real part of the mode propagation constant:Fig. 2(a), which corresponds to γ = 1.53 in Fig. 3(a).
There are higher modes propagating (and evanescent) inside the structure, as observed in Fig. 3(b) for the next three modes having minimum imaginary part. They are characterized by cut-off filling factor of approximately 0.3, 0.75, 0.85. Contrary to the case of hole grating (discussed below), we are not able to identify the role of higher modes in the reflectivity behavior in Fig. 2(a), although they definitely participate in the diffraction process. The modal analysis of the hole array grating has been made in detail after the work of Ebbesen . The hole waveguides do not present a TEM mode that has no cut-off. Figure 4 shows that the first mode becomes propagating for f ~0.3, the second one at f ~0.6, the third one at f ~0.85. Each of them leads to well-distinguished anomalies observed in Fig. 2(b), but even their combined role cannot lead to strong light absorption for shallow gratings, contrary to the channel structure.
3. Metamaterial analysis of channel and hole arrays
Since the works of Maxwell-Garnett  on the homogenization of Maxwell equations for media with inhomogeneous inclusions, there are numerous theoretical, numerical, and experimental works devoted to the determination of an equivalent effective refractive index (or dielectric permittivity and magnetic permeability in case of magnetic properties of the media) that can replace the “alloy” of different substances involved in the structure. It is difficult here to even mention the most important works. One approach is the so called homogenization technique for periodic media, an approach that analyses the optical properties in the quasistatic regime, when the optical frequency tends to zero. While this is interesting and fruitful approach, it is not quite practical in the near-IR, visible and UV domain, where the optical constants depend strongly on the wavelength. To take into account the dispersion, instead of increasing wavelength to infinity, it is possible to diminish the period(s) to zero. If together with that the grating thickness tends to zero, all effects disappear and the limit becomes a reflection from the substrate-superstrate interface. To avoid this, in what follows, we fix the wavelength and the grating thickness h, and decrease the x-y dimensions.
3.1. Numerical results for short periods
The intuitive assumption that the effective permittivity would be the average of the permitivities of the participants fails to be true. The first contra-argument is that it is not clear when to take arithmetic average, when geometric, when some other. The second argument is that the average values for two media are symmetric when the permittivities are mutually permuted, simultaneously with the filling factor.
Numerical results for very small periods, when compared to the wavelength show that the peculiarities in the properties of the gratings, presented in Fig. 3(a) and 3(b) persist, as can be observed in Fig. 5 that shows the dependence of the reflectivity on the filling factor f and grating depth h in the regions of high absorption, obtained by the rigorous numerical method for the two structures of Fig. 1 with period d = 3 nm. In particular, very shallow channel gratings can totally absorb the incident light, which is not the case of hole arrays as seen in Fig. 2(b). Figure 6 shows the dependence on the filling factor of the reflectivity of the grating of Fig. 2(a) for square pillars with period 10 and 3nm, together with a circular pillar reflectivity having period of 3 nm. The grating thickness h is equal to 15 nm (a) and to 415 nm (b). Results obtained for d = 1 nm practically coincide with the results for d = 3 nm. We see that for h = 15 nm the fill-factor dependence of the reflectivity goes down to less than 0.1%, whereas it remains greater than 94.5% for the gratings consisting of circular or square holes. The blue and cyan lines in the figure describe the effective-index results of the effective-index approach, presented further on.
Figure 7 presents similar results of the reflectivity of a square hole arrays for two different hole depths as a function of the filling factor. Similarly to Fig. 6(b), there is no difference between the gratings with periods 3 and 10 nm.
Two conclusions could be drawn from the results:
- 1) The behavior of very small pitch gratings shows similar peculiarities, independent on the pitch size; the position of the anomalies tends toward a limit position as the period tends to zero. Channel gratings properties differ significantly from the hole arrays whatever small the period is.
3.2. Equivalent effective-index analysis
In order to obtain a simple model that enables us to better understand the behavior of these short-pitch structures, we use an effective-index approach based on the Maxwell-Garnett approach  and Green tensor singularity analysis .
Let us consider at first the Green’s function approach to light scattering by an object having permittivity ε2 embedded in a medium with permittivity ε1. The incident field will create a total field , which is a sum of the incident and the scattered field:18]:17. In particular, for a sphere or a cube:Equations (8) and (10) permit immediately to obtain the well-known electric response of a spherical inclusion (or cavity):Eqs. (8) and (11), the link between the scattered and the incident field is given as:Eqs. (13) and (15) is denoted as Q. In what follows, numerical study will distinguish the choice between Eqs. (13) and (15).
The original approach proposed by Maxwell-Garnett  enables us to find the effective permittivity εeff of the inhomogeneous medium, based on a simple hypothesis that it gives the link between the mean values of electric and displacement fields:Eqs. (17)-(19) into Eq. (16) results in the following relation:Eq. (21) is simplified into:Eqs. (11) and (13), the effective permittivity represent anisotropic uniaxial medium with axis along the cylinder axis. The ordinary part of the permittivity (in the x-y plane) is given by:Eqs. (14) and (15), the resulting tensor of effective permittivity is also uniaxial, but it contains the mean arithmethic value of the two permittivities in the x-y plane:
There are two arguments to choose the first approach (the singularity of infinitely long cylinder. First, when the unit cell cross-section tends to zero with the reduction of the periods and keeping the thickness h fixed, the cross-section diameter can become infinitely smaller than h. Second, Eqs. (25) and (26) are symmetrical when permuting simultaneously with , i.e., channel and hole gratings would be equivalent by interchanging f to 1-f, which is not the case.
If we go back to Fig. 6, the results using the two approaches are presented in blue (for infinitely long cylinders) and cyan (infinitely thin disks). As can be expected, the first approach, Eqs. (23) and (24), gives much better results, which are quite close to the numerical values for circular pillars.
Similar correspondence between the results of the effective-index approach and the numerical method are observed in Fig. 7 for cavity resonances.
The main disadvantage of this simplified approach is that the results depend only on the filling factor, while not taking into account the form of the cross-section of the pillars or holes, while numerical results given in Fig. 6 show that there is a slight difference in the position of the reflectivity dip for circular and square cross-sections. In order to take into account the cross-section form, it is necessary to use more sophisticated approaches that have been developed during the last 120 years. An interested reader can find a detailed work on homogenization theory in a recent review by G. W. Milton . The specific form can be taken into account by the concept of polarisability, and the necessity to use the average values not only of the fields but of the modulus square of the electric field (see ch.16 of  and ref .). However, we are not able to directly apply this approach, because it is developed in the quasistatic limit as the frequency tends to zero, while in our case the frequency is finite, and the periods in the grating plane tend to zero.
4. Single-mode model and effective index behavior
This quite good correspondence between the rigorous numerical results and the metamaterial approximation enables us to obtain a deeper insight in the origin of the strong light absorption, as well as of the difference between channel and hole structures. Similar to Fig. 2, well-designed regions of low reflectivity are observed, pointing toward some type of modal structure for d = 3 nm in Fig. 5. This can be observed in detail in Fig. 8 that presents the results of the metamaterial behavior of the two structures in Fig. 1 by using Eqs. (23) and (24) to model the grating region, i.e., the structures are represented as a homogeneous anisotropic layer deposited on a metallic substrate. The square regions bordered by black lines in the figure correspond to the scale limits of Figs. 5(a) and 5(b).
In normal incidence, the normalized propagation constant of the waves propagating in direction to +/− z is equal to the effective refractive index in direction x for the waves polarized in y. The values of neff,xx are given in Fig. 9 . For the channel gratings, as f increases, the real part of the effective index (and thus the real part of the normalized propagation constant of the vertical mode) increases, which explains why the distance in h of the resonances in Fig. 8(a) decreases with h (see Eq. (4)). In fact, for f < 0.7, the metamaterial behaves like a lossy dielectric with large optical index, having real part that varies from 1 to 6. The same behavior was observed for d = 250 nm in sec.1. As f approaches 0.7, both real and imaginary part grow significantly, thus a very thin optical layer (several nanometers thick) can totally absorb the incident light.
When the channel width decreases (f > 0.7), the effective medium is transferred into a lossy metal, with optical proparties tending towards the properties of the metal of the pillar.
Let us consider now the case of hole gratings. When the holes cover completely the grating region (f = 1), the modulated layer is air. Reducing the hole size, the real part of the effective index decreases, in contrast with Fig. 9(a). The other important fact is the small increase of its imaginary part, as compared with Fig. 9(a). Close to the cut-off of the mode near f = 0.7, both the real and the imaginary parts remain small, which fact explains why for small values of h there is no strong absorption, as seen in Fig. 8(b).
As already observed in [7–9], the reflection properties of subwavelength two-dimensional grating depend significantly on the form of the structure. When it consists of pillars separated by channels, there exists a fundamental mode that propagates in vertical direction due to the field interaction through the channels, even for narrow channels, with width less than 5% of the light wavelength. Due to the existence of this mode, the grating structure behaves like a lossy high-index anisotropic dielectric, with characteristic Fabry-Perot resonances observed as a function of its thickness. For channels narrower than 5%, this fundamental mode becomes evanescent and the structure behaves like lossy metal. Close to the cut-off, due to the very high losses and large real part of the effective index, one can observe total light absorption by the layer having thickness of the order of 10 nm.
The corresponding structure that consists of hole instead of pillars is characterized by a vertical mode that has a cut-off (fact well known for single apertures and periodical hole arrays). When propagating, its real part is smaller than unity; this is why strong light absorption and Fabry-Perot resonances appear for thicknesses comparable to the wavelength of light.
These conclusions are also valid in the limit as the structure periods in both x- and y-directions tend toward zero, as shown numerically. A simple Maxwell-Garnett homogenization approach gives very good coincidence with the rigorous numerical results. An open question remains to explain the small difference between the numerical results for circular and square-form pillars, difference that cannot be explained using effective-index approximation that depends only on the filling factor.
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