## Abstract

Finite-difference-time-domain (FDTD) computer simulations reveal interesting features of the transmission of a linearly polarized plane-wave through a periodic array of sub-wavelength slits in a thick metal film (incident *E*-field perpendicular to the slits’ long axis). The results show that slit transmission has a quasi-periodic dependence on both the film thickness and the period of the slits. This indicates that resonant surface waves excited at the top and bottom facets of the metal film as well as resonant guided modes along the depth of the slits play major roles in determining the transmission efficiency of the array. When the slit periodicity is an integer-multiple of the surface-plasmon wavelength, transmission drops to zero regardless of film thickness; in other words, excitation of surface plasmons reduces the transmission efficiency. When the slit periodicity deviates from the aforementioned value, maximum transmission through the slits is achieved by adjusting the film thickness. In the thickness dimension, transmission maxima occur periodically, with a period of half the effective wavelength of the guided mode in each slit waveguide. Optimum transmission is thus achieved by simultaneously adjusting the film thickness and the period of the slits. Computed field profiles clarify the role played by the induced surface charges and currents in enhancing the light’s coupling efficiency into and out of the slits.

©2005 Optical Society of America

## 1. Introduction

Enhanced transmission of electromagnetic waves through sub-wavelength circular apertures (arranged in a periodic array) as well as through single slits in metallic films have been observed from the microwave to the visible range [1,2]. With extensive applications on the horizon (see the summary by Sambles [3]), investigation of the various factors involved in the transmission mechanism, especially the role played by surface plasmons (SP) in periodic structures, has acquired a certain sense of urgency [4–6]. For periodic slit arrays, Porto *et al* [7] disagree with Cao and Lalanne [8] on the effect of SP excitation on the transmission efficiency *η*: Cao and Lalanne use the rigorous coupled-wave method of Lalanne *et al* [9] and arrive at the conclusion that SP’s reduce the efficiency, in contrast to Porto *et al*, who base their arguments on the transfer matrix formalism, and claim that SP excitation enhances the efficiency. For 2D structures, Barnes *et al* [1] argue for a positive role for SP based on the experimental observations of transmission through circular apertures. In addition to the aforementioned methods, transmission through thick metallic gratings has been examined by Kuta *et al* [10] using the method of coupled-wave analysis [11,12]. Although Kuta *et al* anchored their conclusions on a wealth of experimental data, they did not pay explicit attention to SP excitation and its influence on the efficiency *η* of transmission. Treacy [13] uses a rigorous dynamical diffraction model to compute the transmission of light through slit arrays in metallic films. He does not attribute any specific roles to SP excitations; in fact, he refers to all evanescent modes of the structure as surface plasmons, emphasizing the cooperation between evanescent and guided modes in determining the overall transmission efficiency of the slit array. By focusing attention on the properties of a metallic grating of fixed thickness, Treacy also misses the important role played by the film thickness. Overall, the various observations and analyses indicate the possibility of existence of different mechanisms for enhanced as well as attenuated transmission through one-dimensional (1D) and two-dimensional (2D) periodic structures. For slit apertures, in particular, there appears to be no consensus on the specific mechanism(s) responsible for the observed values of *η*.

In this paper we present a study of transmission efficiency through a periodic array of slits for transverse-magnetic (TM) waves at normal incidence on the metallic host surface. To our knowledge, the finite-difference-time-domain (FDTD) method of calculation [14] has not yet been employed to analyze the transmission of light through periodic 1D structures. We use FDTD simulations to investigate the influence of film thickness *t* and slit periodicity *p* on transmission efficiency*η*; all other relevant parameters such as the wavelength of the incident beam, the angle of incidence, the slit-width, and the (complex) permittivity of the host material are kept constant. By choosing to vary the period of the array (as opposed to fixing the period and varying the incident light’s wavelength), we avoid complications arising from the frequency dependence of the host material’s permittivity, which can vary substantially over the frequency range from the infrared to the visible [15], thus shifting the location of the SP excitation in the frequency domain.

After describing the details of the simulation in Section 2, we discuss in Section 3 several important features of *η* observed in computer simulations. By comparing the SP wavelength *λ*
_{SP} to the slit periods *p*
_{max} and *p*
_{min} that exhibit high and low transmission, respectively, we conclude that the excitation of SP in fact reduces the transmission efficiency. Our results thus agree with those reached by Cao and Lalanne [8], despite our different methods of calculation (they used a large number of diffraction orders to match the boundary conditions). To understand the interaction of the incident light with the slits and with the host material, we present in Section 4 the electromagnetic field distributions and explore the relationship between transmission efficiency and the *E*- and *H*-field profiles. The strength of the *E*-field component *E _{z}* (i.e., the component parallel to the propagation direction

*z*) near the top and bottom surfaces of the host metallic film is evidence of a significant accumulation of electrical charge at and around the slit corners; this indicates, in the present case of periodic structures, the relevance of the edge-dipole explanation invoked for single slits in our previous paper [16]. The existence of SP is further confirmed in our Poynting vector plots of Section 4. Generally speaking, large transmission appears to be associated with the presence of strong surface charges and currents in the vicinity of the slits, which help to launch the incident optical power into the guided mode of these sub-wavelength waveguides. A summary of our results appears in Section 5.

## 2. Simulation setup

A monochromatic plane-wave (vacuum wavelength *λ*
_{0}=1.0 µm), propagating along the -z direction, illuminates (at normal incidence) a periodic array of slits in a metal film surrounded by free-space; see Fig. 1. The thickness *t* and period *p* of the slits are variables of the simulation, while the slit-width is constant at *W*=*λ*
_{0}/10=0.1µm. The host material is silver, featuring a large, negative *ε*
_{real} and a small (but finite) *ε*
_{imag} at the incidence frequency. The smallness of *ε*
_{imag} and the large index contrast between the host material and the surrounding environment (free-space) ensures that absorption is weak, and that the observed behavior is primarily dominated by reflection and transmission. Since the slits are aligned with the *x*-axis, the simulation domain is the *Y Z*-plane. The fields, which are now independent of *x*, are decoupled into *E*
_{‖} (i.e., incident *E*-field parallel to the slit direction *x*; also referred to as transverse electric or TE) and *E*
_{⊥} (transverse magnetic or TM); see the lower part of Fig. 1. The focus of the present paper is solely on *E*
_{⊥} modes, as the modes associated with *E*
_{‖} rapidly decay through sub-half-wavelength slits [16].

In our simulations the computation domain in the *Y Z*-plane is (*L _{y}*,

*L*)=(

_{z}*p*, 6.0 µm). Since the incidence is normal, periodic boundary conditions can be applied in the

*y*-direction in order to simulate an infinite array of slits, repeated regularly at intervals of

*L*=

_{y}*p*along the

*y*-axis. The perfectly-matched-layer (PML) boundary condition is applied in the

*z*-direction to absorb the back-reflected as well as transmitted radiation at the top and bottom boundaries of the mesh. The origin of the coordinate system is at the center of the slit aperture, while the light source is placed two grid points below the top PML. Along the y-axis, the grid is uniform with a cell size of 3.0 nm; the grid along the

*z*-axis is non-uniform (cell size =3.0 nm at the center, increasing to 8.0 nm near the top and bottom boundaries). The time duration

*T*over which the Maxwell equations are integrated is proportional to the period

*p*of the slits; specifically,

*T*in units of femtoseconds is chosen to be equal to

*p*in nanometers. Such long integration times are needed if the strong resonances appearing on the top and bottom facets of the metallic film are to be properly taken into account in FDTD simulations.

## 3. Map of transmission efficiency

The slit transmission efficiency *η* is defined as the ratio of two quantities: The numerator is the total energy flux along the propagation direction (-*z*), integrated over a period *p* in the region immediately below a single aperture. The denominator is the total incident power across the width *W* of the slit, i.e., the magnitude of the incident plane-wave’s Poynting vector component *S _{z}* multiplied by the slit-width

*W*. This definition is appropriate for our purposes, as the large absorption coefficient of the fairly thick metal film eliminates the possibility of any incident light leaking through the film. (The same definition for

*η*is used extensively in the literature of sub-wavelength apertures.) For each setup having fixed values of

*t*and

*p*(range:

*t*=0.1-0.8 µm,

*p*=0.2-2.4 µm), we calculated

*η*from the

*E*and

*H*-fields obtained in FDTD simulations. Subsequently, we interpolated (linearly) between the 2664 simulated data points thus obtained, and plotted the data as the color map of Fig. 2. Since a main focus of the present paper is the dependence of

*η*on slit periodicity

*p*, we also display, at the bottom of Fig. 2, plots of the minimum and maximum of

*η*versus

*p*, obtained by searching over the entire range of simulated film thickness

*t*.

Two characteristic length-scales feature prominently in the efficiency map of Fig. 2. In the *y*-direction, the horizontally propagating surface plasmon wavevector ${k}_{\mathrm{sp}}=\sqrt{{\epsilon}_{1}{\epsilon}_{2}\u2044\left({\epsilon}_{1}+{\epsilon}_{2}\right)}{k}_{o}=\left(1.0104+0.0007i\right)(2\pi \u2044{\lambda}_{o})$, yields the corresponding SP wavelength λ_{SP}~0.99µm [17]. (Although *λ*
_{SP} is very close to *λ*
_{o}, the minima of *η* at *p*=*λ*
_{SP} and 2*λ*
_{SP} in Fig. 2 should not be confused with the Rayleigh-Wood anomalies that occur when *p* is an integer multiple of λ_{o}; our simulations have sufficient resolution to distinguish these phenomena from each other.) In the *z*-direction, constructive/destructive interference between forward- and backward-propagating modes of the slit waveguide plays a crucial role in determining the transmission efficiency [8,9]. Although the boundary conditions at the (vertical) walls of each slit in the presence of adjacent slits differ from those of an isolated slit, it turns out that, so long as the adjacent slits are more than a few skin-depths apart, the effective index neff of the waveguide does not deviate perceptibly from that of an isolated slit. The effective wavelength for propagation up and down the individual slits of the array is thus *λ _{eff}*≈0.83µm, which was obtained in [16] for an isolated slit under similar conditions.

The most striking feature of the efficiency map of Fig. 2 is the quasi-periodic dependence of *η* on both the thickness *t* of the film and the period *p* of the slits. The maximum *η* is larger for shorter periods, when *p*<*λ*
_{sp}, compared to cases when *p*>*λ*
_{SP}, indicating that the strength of the interaction between adjacent slits is related to their absolute separation. Since, at *λ*
_{o}=1 µm, the absorption by silver in the walls of the slit waveguides is rather small, we observe, for fixed *p*, only an insignificant drop in *η*
_{max} when the thickness *t* increases by ½*λ*
_{eff}. Note that *t* and *p* can be traded off against each other (i.e., adjusted jointly) over a reasonable range to achieve a given value of *η*. Adjusting *p*, however, cannot be of much help in improving the transmission efficiency if *t* is confined to an inappropriate range.

When *p* is at (or very close to) an integer-multiple of λ_{SP}, the efficiency map of Fig. 2 shows that *η* vanishes for all values of *t*; both *η*
_{min} and *η*
_{max} go to zero at such locations (see the bottom plots in Fig. 2). We thus concur with the conclusion of Cao and Lalanne [8] on the negative role of surface plasmons. The reported high efficiencies that several authors have attributed to the excitation of SP, however, may be understood in light of the fact that the peak efficiencies occur when *p* is only slightly below λ_{SP} (or an integer-multiple thereof). For example, *η* in Fig. 2 reaches ~500% when *p*=0.96 µm and *t*=0.16µm. Compared to the single (i.e., isolated), 100 nm-wide slits studied in [16], the maximum transmission efficiency per slit through a periodic array of such slits is more than doubled. At the same time, the minimum *η* for a periodic array can reach well below that of isolated slits of the same width. Periodic structures thus have the ability to substantially magnify or attenuate *η* when operated in the vicinity of the period *p* at which surface plasmons are excited.

The plot of *η*
_{max} versus *p* in Fig. 2 shows *η*
_{max} growing (almost linearly) from ~200% at *p*=0.2 µm to nearly 700% at *p*=0.85 µm. Considering that the simulated slit-width in Fig. 2 is *W*=0.1 µm, the fraction of total incident optical power that passes through the array at *p*=0.85 µm, *t*=0.25 µm is ~82%, not far from the so-called perfect transmission [7].

## 4. Electromagnetic field profiles

To gain a better appreciation for the results of the preceding section, we present in Fig. 3 plots of the |*E _{z}*| and |

*H*| field amplitudes as well as the corresponding distribution of the Poynting vector

_{x}*S*in three cases that exhibit either high or low transmission. The strength of

*E*at the top and bottom facets of the film is a measure of the accumulated surface charge at these free-space-to-metal junctions. The profile of |

_{z}*E*| can thus be used to estimate the extent of SP excitation and the strength of the induced electric dipoles at the sharp edges of the apertures (the so-called edge-dipoles). The magnitude of

_{z}*H*at the metallic surfaces is directly proportional to the strength of the local surface current, which flows back and forth between nearby reservoirs of surface-charge. The Poynting vector

_{x}*is useful for visualizing the flow of energy in and around the slits.*

**S**In Fig. 3, the left-hand column corresponds to a slit array in a relatively thin (but opaque) film whose period *p* is just below λ_{SP}; this array exhibits a transmission efficiency of ~500%. The middle column shows the field profiles for a highly transmissive (*η*~400%) array of slits through a thick film with a fairly long period *p*. The low-efficiency array (*η*<1%) depicted in the right-hand column has period *p*=2*λ*
_{SP}.

In the left-hand column of Fig. 3, strong electric dipoles appear at the sharp edges of the input/output apertures. Intense surface currents feed these dipoles in the horizontal direction and connect them to each other in the vertical direction; see the |*H _{x}*| plot. On the horizontal facets, the surface currents connect the charges of one slit to those of its neighbors. The current along the (vertical) slit walls supports the guided mode, which transports the optical energy through the slit. The strong accumulation of electrical charge at the sharp edges of the aperture should be contrasted with the charges appearing elsewhere on the top and bottom surfaces; ultimately, of course, all these charges are induced by the evanescent (i.e., non-propagating) modes of the structure that reside near the top and bottom facets of the film [13].

The middle column in Fig. 3 shows the field profiles for a case with *λ*
_{SP}<*p*<2*λ*
_{SP} and large transmission efficiency (*η*~400%). The log |* S*| plot shows strong coupling into and out of the aperture, indicating that the boundary conditions at the top and bottom of the slit favor the launching of a strong guided mode into the slit waveguide.

The right-hand column in Fig. 3 corresponds to the case of *p*=2*λ*
_{SP}, where strong SP modes are excited in the top facet of the film, inhibiting transmission of optical energy through the slits [8]. The |*E _{z}*| plot shows how SP-related charges accumulate at a distance away from the edges of the aperture. By the same token, the |

*H*| plot shows strong surface currents flowing in the regions between adjacent apertures. As a result, the edge dipoles, which helped the transmission so much in the two previous examples, are absent in the present case. It is thus clear that, with strong SP modes excited on the top surface, the coupling efficiency into the slit waveguide suffers; radiation into the aperture in this case is nearly three orders of magnitude weaker than that in the previous two examples.

_{x}## 5. Summary and conclusions

Our computer simulations show quasi-periodic behavior for the transmission of light through periodic arrays of slits in thick metallic films surrounded by free space. One can qualitatively treat such slit arrays in terms of two scale parameters, λ_{SP} and λ_{eff}, which depend on the incident wavelength λ_{o}, the dielectric constant ε of the host medium, and the width *W* of the desired slit apertures. Given these two characteristic lengths, one can design (coarsely) an array that will exhibit a large transmission efficiency *η*.

Excitation of SP modes weakens the transmission substantially, as the incident light is coupled into surface waves bounded primarily on the top air-metal interface. (As an aside we note that, when SP modes dominate, the simulations take much longer to reach the steady-state.) The periodic structure of the array enhances the variations of *η* compared to the case of isolated slits, making the maximum *η* larger and the minimum *η* smaller. The adjacent slits in an array exchange electrical charge via surface currents, and the interactions appear to weaken when the period *p* is increased by integer-multiples of λ_{SP}.

By fixing the wavelength *λ*
_{o} of the incident light, we have removed from the discussion the variations with *λ*
_{o} of the real and imaginary parts of the dielectric constant of the host material (silver). We have thus confined our attention to two variables: the grating period *p* and the film thickness *t*. Figure 2, a 3D plot of transmission versus *p* and *t*, is a comprehensive result that has taken several days of computation on a large cluster of computers; this result, which has never before appeared in the literature, clearly shows the negative role of SP in transmission through the grating (see the vertical dark lines at those values of *p* that excite the surface plasmons).

The interaction between adjacent slits in a periodic array helps to create strong electrical dipoles at the edges of the slits, which dipoles subsequently radiate into the slit aperture. The process is governed by the slit periodicity *p*, while coupling between the top and the bottom of each slit is a strong function of the film thickness *t*. These two mechanisms are responsible for the observed quasi-periodic behavior of slit arrays in thick metallic films.

## Acknowledgments

This work has been supported by the AFOSR contracts F496-2003-10194, FA9550-04-1-0213, and F496-2002-10380 awarded by the Joint Technology Office, by the NSF contract DMS 0335101, and by the National Science Foundation STC Program, under Agreement No. DMR-0120967.

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