## 1. Introduction

Metal gratings have an interesting property known as resonance absorption in the optics region [Raeter 1982], which causes partial or total absorption of incident light energy. This absorption is associated with the resonant excitation of plasmons on a grating surface; incident light couples with surface plasmons via an evanescent spectral order generated by the grating [Nevièr 1982]. Resonance absorption in metal film gratings has been the subject of many theoretical [Nevièr 1982] and experimental investigations focused on various applications including chemical sensing [DeGrandpre 1990, Zoran 2009], surface enhanced phenomena such as Raman scattering [Nemetz 1994], and photonic bandgaps [Barnes 1995, Tan 1998].

A thin-film metal grating, which is a corrugated thin metal film, also results in absorption similar to that observed for thick gratings [Inagaki, Motosuga 1985, Chen 2008, Bryan-Brown 1991, Davis 2009]. Absorption in thin-film metal gratings, however, is much more complicated than in thick gratings because of the existence of coupled plasmon modes in addition to those observed in thick gratings. If the metal film is sufficiently thick, single-interface surface plasmons (SISPs) alone are excited [Raeter 1977, Okuno 2006, Suyama 2009]. However, if the film is sufficiently thin, simultaneous excitation of surface plasmons occurs on both surfaces; these plasmons interfere with each other and produce two coupled plasmon modes, short-range surface plasmons (SRSPs) and long-range surface plasmons (LRSPs) [Chen 1988, Hibbins 2006].

Most previous studies on resonance absorption have mainly dealt with metal gratings whose surfaces are periodic in one direction. Metal bi-gratings, which are periodic in two directions, also yield plasmon resonance absorption, similar to singly periodic gratings [Glass 1982, Glass 1983, Inagaki, Goudonnet 1985, Harris 1996]. In this work, we therefore investigated coupled plasmon modes excited in multilayered bi-gratings [Matsuda 1993, Matsuda 1996, Suyama 2010]. We anticipated interesting behavior in the resonance phenomenon due to the presence of the double periodicity. Further, in view of the fact that layered gratings are interesting structures for optical device applications, we investigated a multilayered bi-gratings, which is a stack of thin-film bi-gratings made of a dielectric or metal. This paper is structured as follows. After formulating the problem in Section 2, we briefly describe a method for obtaining a solution in Section 3. Focusing our attention on the resonant excitation of plasmon modes, we then show the computational results in Section 4, before presenting the conclusions of this study.

## 2. Formulation of the problem

Here, we formulate the problem of diffraction from multilayered bi-gratings when an electromagnetic plane wave is incident on it. The time-dependent factor, exp(*−iωt*), is suppressed throughout this paper as customary.

### 2.1. Geometry of the gratings

Figure 1 shows the schematic representation of multilayered sinusoidal gratings with double periodicity. The grating, with *L*-1 laminated grating layers, has a period *d* in both the *X*- and *Y*-directions. The semi-infinite regions corresponding to the medium above the

grating and the substrate are denoted by V_{1} and V_{L+1}, respectively. The individual layers in the grating, beginning from the upper layer (light-incidence side), are denoted by

Here,

It should be noted that we can easily generalize the shape of *d* and *X*-and *Y*-directions, by writing them as

### 2.2. Incident wave

The electric and magnetic fields of an incident wave are given by

(2)with

(3)Here, *P* is the position vector for an observation point P(*X, Y, Z*), and

where*k* (=

The amplitude of the incident electric field can be decomposed into TE and TM modes [Chen 1973] (with respect to the *Z*-axis) and is written as

Here, the superscript TE (TM) indicates the absence of a *Z* component of the electric (magnetic) field.

The symbol

### 5.3. Diffracted wave

We denote the diffracted fields as

(C1) Helmholtz equation:

(8)(C2) Radiation conditions:

*Z*-direction.

*Z*-direction.

(C3) Periodicity conditions:

(9) (10)Here, *f* denotes any component of

(C4) Boundary condition (

where

## 3. Mode-matching method

We next explain the mode-matching method [Yasuura 1965, Yasuura 1971, Okuno 1990] for determining the diffracted field produced by the multilayered bi-grating. We introduce vector modal functions in the region

where

where the positive and negative signs match on either side of the equation, and *m,n*)th order diffracted wave given by

Note that the superscripts + and - represent upwardly and downwardly propagating waves in the positive and negative *Z*-direction, respectively.

In terms of the linear combinations of the vector modal functions, we form approximate solutions for the diffracted electric and magnetic fields in

(20) |

with

(21)Here,

The approximate solutions

(22) |

Here,

To solve the least-squares problem on a computer, we first discretize the weighted mean-square error *x*-and *y*-direction is chosen as 2(2*N* + 1) [Yasuura 1965, Yasuura 1971, Okuno 1990]. We then solve the discretized least-squares problem by the QR decomposition method. Computational implementation of the least-squares problem is detailed in the literature [Lawson 1974, Matsuda 1966, Suyama 2008].

## 4. Numerical results

Here we show some numerical results obtained by the method described in the preceding section. After making necessary preparation, we show the results for three the bi-grating, thin-film bi-grating and multilayered thin-film bi-gratings cases.

### 4.1. Preparation

It is known that the solutions obtained by mode-matching method [Yasuura 1965, Yasuura 1971, Okuno 1990] have proof of convergence. We, therefore, can employ the coefficient *N* for which the coefficients are stable in evaluating diffracted fields.

The power reflection and transmission coefficient of the (*m*, *n*) order propagating mode in

The coefficient defined above is the power carried away by propagating diffraction orders normalized by the incident power. We calculate the total diffraction efficiency

Although it is known that the solutions obtained by mode-matching modal expansion method [Yasuura 1965, Yasuura 1971, Okuno 1990] have proof of convergence for problems of diffraction by gratings, we compare our results with other existing theoretical [Glass 1983] and experimental results [Inagaki, Goudonnet 1985] on plasmon resonance absorption in bi-gratings to show the validity of the present method. Figure 3 shows the reflectivity curves calculated by the present method and those from Rayleigh's method [Glass 1983] for three sinusoidal silver bi-gratings with different corrugation amplitudes. As confirmed in Fig. 3, the reflectivity curves from the present method are coincident with those of the Rayleigh’s method [Glass 1983]. Next, we make comparison with the experimental results [Inagaki, Goudonnet 1985] in which the resonance angle *θ*_{d,} i.e., the polar angle at which the dip of reflectivity occurs, is observed near 55^{°} for a sinusoidal silver bi-grating with *h* = 0.048 µm, *d* = 2.186 µm, *λ* = 0.633 µm, and ^{°}. The resonance angle calculated from the present metod for these parameters is *θ*_{d} = 54.1^{°}, which is close to the experimental data. These examples show that the present method gives reliable results for the analysis of plasmon-resonance absorption in metal bi-gratings.

In the numerical examples presented here, we deal with a shallow sinusoidal silver bi-grating with height *h* = 0.030 µm and period *d* = 0.556 µm. The wavelength of an incident light is chosen as *λ* = 0.650 µm. We take *n*_{2} = 0.07+*i*4.20 as the refractive index of silver at this wavelength [Hass 1963].

It should be noted, however, that the index of a metal film depends not only on the wavelength but also on the thickness of the film, in particular when the film is extremely thin it may take unusual values if circumstances require. When dealing with a thin metal structure, hence, we should be careful in using the index value given in the literature. As for the value taken in our computation, we assume that *n*_{2} = 0.07+*i*4.20 is available even for the case of *e*/*d* = 0.02. This is because a similar assumption was supported by experimental data in a problem of diffraction by an aluminum grating with a thin gold over-coating.

### 4.2. A bi-grating case

Using the numerical algorithm stated in the previous section, we first investigate the absorption in a metal bi-gratings by *L* = 1 as shown in Fig. 1(c). The semi-infinite regions corresponding to the medium above the grating and the substrate are denoted by V_{1} and V_{2}, respectively. V_{1} is vacuum (V) with a relative refractive index *n*_{1}=1 and V_{2} consists of a lossy metal characterized by a complex refractive index *n*_{2}.

#### 4.2.1. Diffraction efficiency

Figure 4 shows the total diffraction efficiency of a sinusoidal silver bi-grating as functions of a polar angle *θ* when the azimuthal angle ^{º} is fixed. In the efficiency curves we observe four dips which occur at the same angles of incidence for both TE and TM polarized incident light. In this subsection, we demonstrate that the dips are associated with absorption that is caused by the coupling of surface plasmons with an evanescent mode diffracted by a sinusoidal bi-grating. For convenience, the four dips in Fig. 4 are labeled as A, B, C, and D.

#### 4.2.2. Expansion coefficients

In Fig. 5 we plot the expansion coefficients of the (0, -1)st-order and (-1, 0)th-order TM vector modal function, which are two evanescent modes, as a functions of *θ* under the same parameters as in Fig. 4. Solid curves in Fig. 5 represent the real part of the expansion coefficient, and dashed curves for the imaginary part. We observe a resonance curve of the expansion coefficient *θ* = 9.5^{ º}, i.e., dip A, and *θ* = 23.3^{ º}, 41.5^{ º} and 49.5^{ º}, i.e., dips B, C, and D in Fig. 4 for both TE and TM incidence.

This implies that the TM component of the (0, -1)st-order and (-1, 0)th-order evanescent mode couples with surface plasmons at dips B and D. We can similarly confirm that dips A and C are associated with the coupling of the TM component of the (-1, 0)th- and (-1, -1)st-order evanescent mode, respectively, with surface plasmons, although we do not include any numerical example here.

#### 4.2.3. Field distributions and energy flows

In order to investigate the resonant excitation of surface plasmons, we study field distributions and energy flows in the vicinity of the grating surface when the absorption occurs. Here we consider the case of the dip B at which the TM component of the (0, -1)st-order evanescent mode couples with surface plasmons. We calculate the electric field of the TM component of the (0, -1)st-order evanescent mode *E*^{ t}. The magnitude of these fields along the Z-axis where

Next we show the energy flows S that are the real part of Poynting's vectors for the total field. The X and Y components of the energy flows S are plotted as the vector (S_{X}*,* S_{Y}*)* in Fig. 6(b). The energy flows are calculated over the region close to the grating surface:

The energy flow at a point *P* is given by Re[*S*(*P*)], where *S*(*P*)=(1/ 2) *E*^{t}(*P*)× *H*^{t} (*P*) stands for Poynting’s vector, *E*^{ t} and *H*^{ t} denote total fields, and the over-bar means complex conjugate. We calculate the energy flow at each point located densely near the grating surface and show the results in Fig. 6(b).

Figure 6(b) shows that the energy of electromagnetic fields in the vicinity of the grating surface flows uniformly in the direction that the (0, -1)st-order evanescent mode travels in the *XY* plane. We thus confirm that surface plasmons are excited on the grating surface through the coupling of the TM component of an evanescent mode.

#### 4.2.4. Polarization conversion through plasmon resonance absorption

Diffracted fields from a sinusoidal metal bi-grating have both TE and TM component for an arbitrary polarized incident light. We therefore observe polarization conversion that a TM (or TE) component of the incident light is converted into a TE (or TM) component of the reflected light. It has been pointed out [Chen 1973, Inagaki, Goudonnet 1985] that the polarization conversion [Elston 1991, Matsuda 1999, Suyama 2007] is strongly enhanced when the plasmon-resonance absorption occurs in a sinusoidal metal bi-grating. Our study confirms the enhancement of polarization conversion through plasmon-resonance absorption. In Fig. 7, the TE and TM component of the diffraction efficiency *h*.

#### 4.2.5. Prediction of resonance angles

We seek to determine a complex incidence angle *θ*_{c} for which total or partial absorption occurs, i.e., *X* and *Y* components of the surface plasmon wave vector normalized by the wave number *k*_{1}:

In reality, we cannot realize a complex incidence angle

If the wavevector of the surface plasmon (*m*, *n*)th-order evanescent mode:

We solve the homogenous problem [Nevièr 1982] for a sinusoidal metal bi-grating by present method and then obtain the surface-plasmon wave vector. Table 1 shows the propagation constants of the surface plasmon and the resonance angles

#### 4.2.6. Simultaneous resonance absorption

From Fig. 8, it is predicted the angle of incidence at which the simultaneous resonance absorption occurs from the position of the intersection of the (-1,0) and (0,-1) curve. At the intersection E, the (0,-1)st- and (-1,0)th-order evanescent modes couple simultaneously with two surface-plasmon waves at the same angle of incidence. Thus, two surface-plasmon waves are excited simultaneously in directions symmetric with respect to the plane of incidence and interact with each other. The interference of the surface-plasmon waves causes the standing wave in the vicinity of the grating surface. This is confirmed from Figs. 9, the strong fields along the Z-axis where Y=X, and where the *X* and *Y* components of Poynting's vectors S on a surface 0.01*d* above the one-unit cell of the grating surface are plotted as the vector

It has been reported [Barnes 1995, Ritchie 1968] that surface-plasmon band gaps exist at the angles of incidence at which simultaneous excitation of plasmon waves occurs, and that the appearance of the band gaps depends strongly on the surface profile. Hence, there is a possibility that a band gap will be observed at the point *E* in Figs. 9 and 10 provided that the grating profile is appropriately chosen, because two plasmon modes are excited at that point.

### 4.3. A Thin-film Bi-grating case

As a numerical example, we consider a sinusoidal silver (Ag) film bi-grating having a common period as shown in Fig. 1(b). The values of the parameters are the same as those in Fig. 4 except for the thickness of the silver film. Using the present algorithm, we calculated the diffraction efficiencies and field distributions to clarify the properties of the coupled surface plasmon modes.

#### 4.3.1. Diffraction efficiency

First, we consider a sinusoidal silver-film bi-grating. The bi-grating is denoted by *L* = 2 (V/Ag/V). Figure 11 shows the (0,0)th order power reflection *e* is the thickness of the silver film. We observe partial absorption of the incident light as dips in the efficiency curves in Fig. 11(a), in addition to the constant absorption corresponding to the reflectivity of silver. We assume that the dips are caused by resonant excitation of surface plasmons. If this is the case, each of the dips can be related to one of the three types of plasmon modes: a SISP that is observed as a single dip at

When the grating is thick (

#### 4.3.2. Expansion coefficients

We examined the same phenomena observing the modal expansion coefficients in

In Fig. 13 (

#### 4.3.3. Field distributions and energy flows

We consider the same phenomena observing the field distributions and energy flows near the grating surfaces. In the former we find that the total field is enhanced. In the latter we observe the symmetric (even) and anti-symmetric (odd) nature of the oscillations, which correspond to the LRSP and SRSP [Raeter 1977].

Figures 14, 15, and 16 show the field distributions of the *X*- and *Z*-components of the total electric fields (a) and energy flows (b) in the vicinity of the silver-film grating at the incidence angles at which absorption was observed in Fig. 11. The abscissa and ordinate show the magnitude and distance in the *Z* direction normalized by the wavelength

Figure 14 shows *S*_{X} and *S*_{Z} (b) for the case of

We observed that the total field above the grating surface decays exponentially in the *Z* direction and the magnitude of the total field is almost *X* direction and that it goes in opposite directions in vacuum and in metal. This is commonly observed when a SISP is excited.

In Figs. 15(a) and 16(a), we again see the enhancement of

### 4.4. Multilayered thin-film bi-gratings case

Next, we consider multilayered thin-film bi-gratings as shown in Fig. 1(a) indicated by *L* = 4 (_{2} films pairs. As listed in the figure, the values of the parameters are the same as those in Fig. 11 except for *L* = 4,

#### 4.4.1. Diffraction efficiency

Figure 17 shows the (0,0)th order power reflection _{5} (Vacuum) (b) as functions of the incident angle

This is related to the resonant excitation of a guided wave supported by the SiO_{2} film. This absorption is characterized by its occurrence over a wider range of

#### 4.4.2. Field distributions

In order to examine the properties of the wide absorption found in Fig. 17, we investigated the field distributions of the total electric field *E*^{total} and the TM component of the (0, 0)th-order diffracted electric field _{2} film. The magnitude of *E*^{total} and *l* =1,2,...5) along the *Z*-axis are plotted in Fig. 18 where *E*^{total} inside the SiO_{2} film indicates a standing wave pattern corresponding to the normal mode of a one dimensional cavity resonator, and that the distribution is almost close to that of_{2}/Ag/V is associated with resonance of the (0, 0)th-order diffracted wave _{2} film sandwiched by a sinusoidal silver film grating.

## 5. Conclusions

We have investigated the resonance absorption associated with the resonant excitation of surface plasmons in bi-gratings. Calculating diffraction efficiency, expansion coefficients, field profiles, and energy flows, we examined the characteristics of the resonant excitation of surface plasmons in detail. Interesting phenomena were revealed, including the conversion of a TM (or TE) component of the incident light into a TE (or TM) component at several different incidence angles, strong field enhancement on the grating surface where surface plasmons are excited, and simultaneous resonance absorption that does not occur in the case of a singly periodic grating in general. The results presented here facilitate a clear understanding of the coupled plasmon modes, SISP, SRSP and LRSP, excited in a thin film doubly periodic metal grating.

### Acknowledgement

This work was supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (Grant number 23560404). The authors thank Shi Bai and Qi Zhao for help in numerical computation and in preparation of the manuscript.