Resonant Excitation of Plasmons in Bi-Gratings

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 [Nevier 1982]. Resonance absorption in metal film gratings has been the subject of many theoretical [Nevier 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].


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, 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, singleinterface 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, 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.

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. 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

Geometry of the gratings
The profile of S  is sinusoidal and is given by Here, p  represents the phase of S  in each direction, and e  denotes the average distance between S  and 1 l S + . The value h  is regarded as the groove depth of the boundary S  .
It should be noted that we can easily generalize the shape of S  by distinguishing between d and p  in the X-and Y-directions, by writing them as x d , y d , x p  and y p  , respectively. A singly periodic grating is a special case where y d → ∞ in the doubly periodic case. In the present paper, we concentrate our attention on describing the analysis only for the doubly periodic case, since it reduces to the singly periodic case through a simple procedure.

Incident wave
The electric and magnetic fields of an incident wave are given by Here, P is the position vector for an observation point P(X, Y, Z), and i k is the wave vector of the incident wave defined by The symbol k (= 2π / λ ) is the wave number in vacuum and λ is the wavelength of the incident wave. We define θ and ϕ as polar and azimuth angles, respectively, as shown in Fig. 1(a), and the superscript "T" denotes a transposition.
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 T TE sin cos 0 , , T TM cos cos cos sin sin , , Here, the superscript TE (TM) indicates the absence of a Z component of the electric (magnetic) field.

Diffracted wave
We denote the diffracted fields as Here, f denotes any component of ( ) (C4) Boundary condition ( 0 X d < < ; 0 Y d < < ; 1,2,..., L =  ): Resonant Excitation of Plasmons in Bi-Gratings 317 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 V  : Here, are therefore determined such that the solutions approximately satisfy the boundary conditions (C4). In the mode-matching method [Yasuura 1996, Yasuura 1971, Okuno 1990, the least-squares method is employed to fit the solution to the boundary conditions [Hugonin 1981]. That is, we find coefficients that minimize the weighted mean-square error by Here, ' S  is a one-period cell of the interface S  , and Γ  is the intrinsic impedance of the medium in V  .
To solve the least-squares problem on a computer, we first discretize the weighted meansquare error N I by applying a two-dimensional trapezoidal rule where the number of divisions in the x-and y-direction is chosen as 2(2N + 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].

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.

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 with sufficiently large 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 1 V and L 1 V + are given by The coefficient defined above is the power carried away by propagating diffraction orders normalized by the incident power. We calculate the total diffraction efficiency  denotes a summation over the propagating orders.
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  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  in which the resonance angle θd, 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 φ = 45°. The resonance angle calculated from the present metod for these parameters is θd = 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 bigrating 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 n2 = 0.07+i4.20 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 n2 = 0.07+i4.20 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.

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 V1 and V2, respectively. V1 is vacuum (V) with a relative refractive index n1=1 and V2 consists of a lossy metal characterized by a complex refractive index n2. Figure 4 shows the total diffraction efficiency of a sinusoidal silver bi-grating as functions of a polar angle θ when the azimuthal angle φ = 30 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.

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 A − TM 1( 1,0) at the angles of incidence θ = 9.5 º , i.e., dip A, and A − TM 1(0, 1) at the angles of incidence 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)storder evanescent mode, respectively, with surface plasmons, although we do not include any numerical example here.

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)storder evanescent mode couples with surface plasmons. We calculate the electric field of the / .
x y z x y z h y 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.

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] that the polarization conversion , 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

Prediction of resonance angles
We seek to determine a complex incidence angle θc for which total or partial absorption occurs, i.e., 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 SP θ . The data demonstrate that the estimated resonance angle SP θ agrees with d θ which is the absorption peak in Fig. 4. Figure 8 shows the estimated resonance angle SP θ as a function of the azimuthal angle ϕ for the sinusoidal bi-grating considered in Fig. 4. Note that points A, B, C, and D in Fig. 8 are results obtained from the absorption peak of total-efficiency curves in Fig. 4. From this figure, we can find the resonance angle for each azimuthal angle.

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 0.01d above the one-unit cell of the grating surface are plotted as the vector ( ) S ,S X Y . We further observe in Fig. 10 that the simultaneous excitation of the surface plasmons waves causes the strong absorption for both TE-and TM-polarized incident light.
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.

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.

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 00 ρ in 1 V (Vacuum) (a) and the transmission coefficient 00 τ in 3 V (Vacuum) (b) as functions of the incident angle θ for two values of d e / when the azimuth angle 0 φ =  is fixed; 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 8.0 curve, and a SRSP and LRSP corresponding to the dips in the / 0.08 e d = curve at 6.54 θ =  and 8.8  , respectively. When the grating is thick ( 0.4 / e d = ), the power can be seen in 1 V alone and no transmitted power exists in 3 V in Fig. 11. Increasing θ from 0  , we first observe the dip at 8.0 θ =  corresponding to absorption in Fig. 11(a). If the grating is relatively thin ( / 0.08 e d = ), the power exists in both 1 V and 3 V . Although the power in 3 V is generally small, it becomes large at the incidence angles for which absorption was observed in Fig. 11. This suggests that coupled oscillations occur on the upper and lower surfaces of the grating.

Expansion coefficients
We examined the same phenomena observing the modal expansion coefficients in V ( 1,3) l l = .
Figures 12 and 13 illustrate the (-1,0)th-order coefficients A + TM 1(-1,0) for / 0.4 e d = and 0.08, respectively. We observe, in Fig. 12(a)  A − = remains unchanged, as seen in Fig. 12(b). This means that the incident wave illuminating the grating at this angle causes coupling between the (-1,0)th-order evanescent mode and some oscillation excited on the upper surface of the grating. The oscillation exists locally in the vicinity of the illuminated surface and hence does not have any influence on the field in 3 V .

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].     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 λ . The parallel broken lines represent the grating surfaces. corresponds to the single dip in Fig. 11(a). Figures 15 and 16 show the same thing for the / 0.08 e d = case. Figure 15 illustrates the results at 6.54 θ =  , where the left dip is observed in Fig. 11(a). On the other hand, Fig. 16 depicts the results at 8.8 θ =  , corresponding to the right dip in Fig. 11(a). In Figs. 14 to 16, we observe strong enhancement of X E and Z E (note that the magnitude of the incident radiation is 1), which is observed at the incidence angles where absorption occurs.
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 Z E in Fig. 14(a). The state of affairs is nearly the same in the metal region except for the rapid decay. Because the grating is thick, the oscillation near the upper surface does not reach the lower surface and, hence, the field below the grating is zero. Figure 14(b) illustrates the energy flow S, which is magnified by 15 in the metal region. We see that the energy flow is almost in the 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 X E and Z E on the upper and lower surface of the silver-film grating, respectively. The rate of enhancement in Fig. 15(a) is not as large as that in Fig. 16(a). Z E is strongly enhanced at both the upper and lower surfaces of the grating and exponentially decays away from each surface. We thus observe the simultaneous excitation of surface plasmons at the surfaces. We can understand the difference of the field distributions assuming that the former and the latter refer to

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 V/Ag/SiO /Ag/V ) that consists of a stack of silver and SiO2 films pairs. As listed in the figure, the values of the parameters are the same as those in Fig. 11  , the extinction power is more than 50% for all angles of incidence θ ranging from 0  to 12  .

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)thorder diffracted electric field We observe in the figure that the field distributions of E total inside the SiO2 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 3(0 ,0) TM E . Hence, we conclude that the wide absorption observed in the multilayered grating V/Ag/SiO2/Ag/V is associated with resonance of the (0, 0)th-order diffracted wave 3( 0 ,0) TM E in the SiO2 film sandwiched by a sinusoidal silver film grating.

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.