## Abstract

We investigate the interference of the surface plasmon polariton (SPP) with an incident beam on a metallic slit using the FDTD. We find that the bulk waves radiated at the slit edge by scattering of the SPP leak into the slit and induce accumulated charges within the skin depth, which excite new SPPs on the slit side-walls. The SPP on the top surface of aperture is coupled into the slit and induces the 2D asymmetric field distributions, including the horizontal and vertical Fabry-Perot multi-reflection resonator modes. We show that the addition of these modes with the slit waveguide modes induced by a normally incident beam is the interference between the SPP and the incident beam, which enhances or suppresses the slit transmission, depending on their relative phase.

©2007 Optical Society of America

## 1. Introduction

The extraordinary enhancement of the light transmission through subwavelength aperture arrays in metal films [1], or through a single hole surrounded by a number of grooves [2] excited enormous research interests. Among numerous theoretical explanations, Lezec *et al*. suggested that the enhancement of transmission could be a result of the interference of the composite diffracted evanescent waves with the normally incident light on the apertures. Their theory is within the scalar diffraction theory framework [3]. Schouten *et al*. studied the optical transmission of two subwavelength slits as a function of wavelength and slit separation [4]. They suggested that the interference of the surface plasmon polariton (SPP), which is excited at one slit and is propagated to another slit with the incident beam at the slit, is the source of modulation of the double-slit transmission. We notice that before their arrival to the slit, the impinging SPP and the normally incident beam have orthogonal polarizations and perpendicular wavevectors. It is then important to understand how their interference may occur. Generally speaking, this interference must be mediated by conversion between SPP and free-space bulk waves at the nanoslit. However, the coupling mechanism of the SPP into the slit and the interference with the incident wave has yet to be fully understood.

In this paper we investigate the behavior of the SPP impinging on a metallic nanoslit, using the full vectorial solutions of the Maxwell equations with the finite-difference time-domain method (FDTD). In particular, we are interested in the mechanism of coupling a part of the impinging SPP into the slit and its interference with the normally incident beam. The FDTD has been used to model the optical transmission of subwavelength hole [5,6], array of slits [7 6]. The scattering of SPP by surface defects or by discontinuities of the substrate's dielectric constant is a classical problem investigated by many authors [8,9]. It is well known that the incoming SPP to a surface defect or a material discontinuity, is partially reflected, transmitted, and scattered into the free space above the metal surface. This scattering is relevant to probing of the SPP in near-field optical microscopy. However, to our best knowledge no detailed analysis has been reported on the coupling of the SPP into the slit, and the interference at the slit between the incoming SPP and the incident free-space beam. We find that the bulk waves radiated at the slit edge by scattering of the SPP leak into the slit and induce accumulated charges and new SPPs on the slit side-walls. A sole incident SPP can induce bulk wave multi-reflection resonance cavity modes between the two slit walls, and the SPP slit waveguide modes along the slit axis, with their field distributions depending on the slit's geometry. The addition of those modes with the well-known slit waveguide mode induced by a single normally incident beam [14], is equal to the field in the slit when both the SPP and the plane wave beam are incident on the slit. Therefore, we confirm the interference theory and demonstrate that the slit transmission is enhanced or suppressed, depending on the relative phases of the interfering fields.

## 2. Coupling surface plasmons polaritons into slit

We consider an infinitely long slit perforated in a real metal film, such as silver, along the *z*-axis. The surface plasmon polariton (SPP) is generated by a linearly *p*-polarized beam, with *E _{x}* and

*H*components, of wavelength

_{z}*λ*

_{0}=500

*nm*normally incident on a groove of width 250nm and depth 70

*nm*. As shown on Fig. 1(a), SPPs are generated by diffraction in the groove and propagate away from the groove on the top metal surface in the ±

*x*directions. The internal mechanism of the SPP generation in the groove is not our concern in this paper. Other SPP sources could also be used.

We now consider the SPP propagating in the +*x* direction on the film top surface and impinges on the slit from the left ridge. The distance *L* from the groove top-right edge to the slit top-left edge is a parameter associated to the phase of the SPP incident on the slit. By choosing to vary *L* instead to vary the wavelength, we avoid in this paper problems arising from the metal’s dispersion. In order to explicitly monitor the SPP on the metal top surface, the incident plane wave on the groove is artificially narrowed to 290*nm* width, which exceeds by 20*nm* on either side of the groove under illumination. In the FDTD, it is possible to simulate numerically a collimated beam of subwavelength width via the total-field/scattered-field (TF/SF) boundary conditions [10]. The incident narrow beam is collimated via the TF/SF technique until 10*nm* before reaching the top metal surface. Then, the TF/SF boundary conditions are removed to allow the free interaction of the light with the structure. The FDTD is performed with a fine isotropic grid scale of 10*nm* (or λ_{0}/50). The space step Δ*x* is related to the time step Δ*t* via Δ*x*=2c∙Δ*t*, where *c* denotes the speed of light in vacuum. The wavevector of the SPP is computed from the SPP dispersion relation [12]: *k _{spp}*=

*n*

_{spp}*k*

_{0}, where

*n*= √

_{spp}*ε*

_{d}*ε*/(

_{m}*ε*+

_{d}*ε*) with

_{m}*ε*and

_{m}*ε*denoting the dielectric constants of the metal and the adjacent dielectric medium, respectively. The frequency-dependent complex permittivity of metal is described by the Lorentz-Drude model [11]:

_{d}where *ε*
_{r,∞} is the dielectric constant at infinite frequencies, *ω _{p}* the plasma frequency, and

*ω*,

_{k}*f*, and Γ

_{k}*are the resonance frequency, strength and damping frequency, respectively, of*

_{k}*k*th oscillator. The Lorentz-Drude model uses

*K*damped harmonic oscillators to describe the small resonances observed in the metal’s frequency response. The values of the constants in Eq. (1) are taken from [11]. At

*λ*

_{0}=500

*nm*, we obtain the complex permittivity of silver:

*ε*=-7.6321+0.7306

*i*. Perfectly-matched layer boundary conditions were applied at the limits of the computational domain [10]. The slit intensity transmission coefficient

*T*is calculated by integrating the time-averaged modulus of the Poynting vector along a plane “detector” located at the slit bottom exit (0≤

_{s}*x*≤

*a, y*=-10

*nm*), as shown on Fig. 1(a).

We computed the electromagnetic fields and electrical charge distributions induced by the sole incident SPP for a set of slit width a and thickness t with the FDTD. The following is our observations on the results.

## 2.1 Fabry-Perot resonator of SPP between groove and slit

The incident SPP on the metal top surface on the air side is described by *H _{z}* and

*E*, which is parallel and normal to the top surface, respectively. At the slit top-left edge (

_{y}*x*=0,

*y*=

*t*), the abrupt change of permittivity of the material from metal to vacuum partially reflects the incident SPP. The remaining SPP is converted to bulk radiation by scattering. As the groove is present, the reflected SPP is again partially reflected back towards the slit from the top-right edge of the groove (

*x*=-

*L*, y=

*t*). The SPP characteristic propagation length [12]

*L*=1/(2∙Im(

_{spp}*k*))=52

_{spp}*μm*is significantly longer than the propagation distance

*L*in our experiments. Thus, the multiple reflections on the section -

*L*<

*x*<0 of the metal top surface create a Fabry-Perot (F-P) type interference of the SPP. In Fig. 1(b) we show the slit energy transmission

*T*as a function of the cavity length

_{s}*L*, when the sole SPP is incident to the slit from the left ridge. The main features of the curve are its initial rapid decay for 0<

*L*<

*μm*and the oscillation period of

*λ*/2, where the SPP wavelength

_{spp}*λ*=

_{spp}*λ*

_{0}/Re(

*n*)=470

_{spp}*nm*. The resonance peaks of

*T*have low magnitudes corresponding to small SPP reflectivities

_{s}*R*at the slit and groove edges. At the left-end ridge of the metal film the SPP propagating in the −

*x*direction is also partially reflected back. However, the propagation distance of the SPP in the −

*x*direction,

*L*’, is chosen to be 1.5

*λ*such that their interference is destructive and their perturbation to the +

_{spp}*x*propagating SPP on the right-side of the groove is minimized.

## 2.2 Bulk waves Fabry-Perot resonator modes between slit walls

Of the bulk waves converted from the incident SPP at the slit top-left edge, a major part is scattered into the half-space above the metal film, see Fig. 1(a). Our FDTD computation shows that a small part of the scattered divergent bulk waves leak into the slit. The bulk waves, mainly consisting of the *H _{z}* and

*E*components, as for the incident SPP, hit and are reflected from the right-side-wall of the slit, and are then reflected back by the left-side-wall of the slit. The multiple reflections between the two slit walls form F-P resonator modes with slightly inclined bulk wave input from the top of the cavity. When the slit width is close to integer multiples of

_{y}*λ*

_{0}/2, the cavity resonance creates strong standing wave fringes of

*E*and

_{y}*H*with a fringe period of

_{z}*λ*

_{0}/2 as shown in Fig. 2(a) and 2(c). The fringes of

*H*are interlaced with the fringes of

_{z}*E*. The maxima of

_{y}*H*are at the slit walls, while the maxima of

_{z}*E*are shifted by

_{y}*λ*

_{0}/4 from the slit walls [5]. These fringes represent the energy stored in the slit cavity. At the resonance, the slit’s stored energy and transmission

*T*are maximal. Out of the resonance, both

_{s}*E*and

_{y}*H*in the slit decrease to low intensities and

_{z}*T*is minimal, as pictured in Fig. 2(b).

_{s}As shown in Fig. 2, the multiple reflection interference patterns of the scattered waves inside the slit overflow the top and bottom openings of the slit due to the weak confinement in this open cavity. The FDTD computation shows the new SPPs excited on both the top and the bottom surfaces of the metal film on the right-side of the slit (*x*>*a*, *y*=0 and *y*=*t*). On the top surface, the SPP on the right-side of the slit is referred to as the transmitted SPP over the slit. This result is consistent with previous studies [8,9,13]. Indeed, when the slit is filled with another material of dielectric constant *ε*
_{2}, or when the slit is replaced by a surface defect, such as a groove or bump of any shape, one can write down the analytic expressions for *H _{z}* in the free space above the top surface (

*y*≥

*t*), which should satisfy the boundary conditions at the surface

*y*=

*t*[8,9,13]. When the slit of substrate

*ε*

_{2}has a width

*a*>>

*λ*the structure is referred to as a SPP Fabry-Perot interferometer. Its SPP energy transmission coefficient

_{spp}*T*oscillates with period Δ

*a*=

*λ*/2 [13]. In the case of surface defects, Sanchez-Gil

_{spp}*et al*. found that the SPP intensity transmission coefficient

*T*across the defect exhibits a cavity-like oscillation with a period Δ

*a*∼1.4(

*λ*/2) [8]. They also predicted that incident SPP energy is partially stored in the groove [9]. Using the FDTD we are able to show the field in the slit below the interface as well as the energy stored inside the slit through F-P resonator modes.

_{spp}Sanchez-Gil *et al*. showed in the case of a simple surface defect (groove or bump) that the scattering efficiency *S* of the waves into the free space above the top surface depends on the SPP transmission *T* (Fig. 1). When the surface defect is replaced by a slit, we observed that the scattering *S* and its direction of maximum radiation both depend on the F-P resonator modes in the slit, which also determine the SPP transmission *T* on the top surface and the slit transmission Ts coefficients. The latter is the energy leaked from the slit’s bottom exit in the form of the transmitted bulk waves and of the SPPs excited on the bottom plane of the metal film (*x*<0 and *x*>*a*, *y*=0).

## 2.3 Excitation of new SPPs on slit walls

As noted previously, the SPP on the air side of the top surface contains no *E _{x}* component. The scattered bulk waves entering the slit contains

*E*which is responsible for the downward component of the bulk wave vector

_{x}*k*⃗

_{0}. In addition, new

*E*fields are excited on the slit walls. Indeed, the

_{x}*H*of the incident SPP induces surface currents parallel to the top metal surface (-

_{z}*L*≤

*x*≤0,

*y*=

*t*), while the bulk

*H*leaked into the slit induces surface currents parallel to the slit left-side wall (

_{z}*x*=0, 0≤

*y*≤

*t*). The surface currents associated with the incoming SPP must turn around the top-left corner (

*x*=0,

*y*=

*t*) as shown in Fig. 3. There is a corresponding sharp discontinuity in the amplitude of

*H*on the top surface SPP and that of the leaked bulk

_{z}*H*, in the vicinity of the slit top-left edge, as shown in Fig. 2. According to the charge conservation equation, ∁ ∙

_{z}*J*⃑ = -

*∂ρ*/

*∂t*, the corresponding current discontinuity creates the accumulation of electric charges at the slit corner as an oscillating monopole, which can be observed by a high concentration of

*E*on the top surface (

_{y}*x*≤0,

*y*=

*t*) and of

*E*on the slit left-side wall (

_{x}*x*=0, 0≤

*y*≤

*t*), as shown in Fig. 2. The oscillating monopole radiates waves which excites the SPP on the slit left-side wall (

*x*=0, 0≤

*y*≤

*t*).

On the right-side wall of the slit (*x*=*a*, 0≤*y*≤*t*), there is no significant *E _{x}* when the slit width

*a*>>

*λ*

_{0}. However, the scattered bulk

*H*induces surface currents and accumulated charges on the slit right-side wall within the skin depth. The FDTD calculation shows that the phase of

_{z}*H*is shifted by

_{z}*π*from the left-side wall to the right-side wall of the slit, as shown in Fig. 2(d), so that the surface currents are in the same direction along both walls. As a consequence, the accumulated charges on the slit right-side wall are of the opposite sign to that at the slit top-left edge, and the two poles are in-phase in their oscillation with time resulting in an oblique dipole across the slit as shown on Fig. 3. This dipole is the source of excitation of new SPPs on both slit walls, although the two poles can have different amount of charges, and the rightside pole is not necessarily located at the slit bottom-right corner but near the middle of the right-side wall. The distributions of field components

*E*and

_{x}*H*and of the accumulated charges on the two walls vary with the slit geometry and are asymmetric with respect to the slit axis, see Fig. 2(c).

_{z}The excited SPPs are sustained by the *E _{x}* and

*H*on the slit side walls. In contrast with the

_{z}*E*, whose maximum is shifted from the reflecting wall surfaces by

_{y}*λ*

_{0}/4, the

*E*is stuck on the walls. Apparently, the multiple reflection resonances of the bulk waves between the two slit walls have little effect to the

_{x}*E*on the walls. When the bulk wave is out of resonance,

_{x}*E*is minimum inside the slit while the

_{y}*E*and

_{x}*H*components are non null on the slit walls, as shown in Fig. 2(b), where they sustain the SPPs on the slit side walls and contribute to coupling the incoming SPP on the top surface into the slit cavity.

_{z}## 2.4 Coupling the incident SPP into the slit

Figure 3 shows the Poynting vectors *S*⃑ associated with a single SPP incident on the slit. Since the magnetic field has only *H _{z}* component, the direction of

*S*⃑ is determined by the local ratio of

*E*and

_{x}*E*. On the top surface, the

_{y}*S*⃑ of the incident SPP is in +

*x*direction. The

*S*⃑ then enters inside the slit by turning around the positively charged pole at the slit top-left edge due to the strong presence of accumulated charges at the edge. The

*S*⃑ then flows downward supported by the excited SPP (

*E*) on the slit left-side wall. After propagating through a small depth in the slit, the magnitude of Ex diminishes, and

_{x}*S*⃑ changes the direction right conducted by the rising of

*E*in the middle of the slit, as shown in Fig. 2(c), to meet the right-side wall of the slit where the negatively charged pole and the excited SPP, sustained by

_{y}*E*and

_{x}*H*, leads the

_{z}*S*⃑ flux towards the bottom-right edge and to the slit bottom exit. In addition, new SPPs are excited on the bottom-right metal plane (

*x*≥

*a*,

*t*=0).

## 2.5 Fabry-Perot resonator modes along the slit axis

It is well known that when a plane wave beam is normally incident on a nanoslit from free-space, the impedance mismatch between the region in the slit and the free-space region outside the slit is sufficiently important to allow strong F-P resonance modes along the slit axis [14]. Our FDTD computation shows that the incident SPP on the top surface may alone excite the slit waveguide modes for sufficiently thick films. When the slit width *a*≤*λ*
_{0}/4, the *E _{x}* distribution is oblique only at the top of the slit inside the first period, (

*t-λ*)/2<

_{spp}*y*<

*t*, of the standing-wave fringes. As shown in Fig. 4(b), the standing-wave fringes of

*E*become uniform across the slit width for the remaining slit length, 0<

_{x}*y*<(

*t-λ*/2), as in the conventional slit F-P waveguide modes for the slit illuminated by a normal incident beam.

_{spp}## 2.6 Field distributions inside the slit

The pure bulk wave resonator modes between the two slit walls (horizontal F-P), as shown in Fig. 4(c), and the pure SPP resonator modes along the slit axis (vertical F-P), as shown in Fig. 4(b), represent the two extreme cases for wide and narrow slit, characterized by *a*≥*λ*
_{0} and *a*≥*λ*0/4 respectively. In general, a single incident SPP on the top surface induces 2D field distributions in the slit, which are asymmetric with respect to the slit axis.

Figure 4(a) shows the slit energy transmission *T _{s}* normalized by the slit width

*a*as a function of slit width

*a*and thickness

*t*, when a single SPP on the top surface impinges on the slit. Peak transmission occurs when both the vertical and horizontal F-P resonator modes are at resonance. Our computations show that the values of

*a*and

*t*for the resonances are shifted from those for the classical cavity resonances,

*a*=

*n*(

*λ*

_{0}/2) and

*t*=

*m*(

*λ*/2), where

_{eff}*m*and

*n*are integers and

*λ*=

_{eff}*λ*/

_{spp}*n*is the effective wavelength with slit waveguide mode effective index

_{eff}*n*[14]. For slit widths

_{eff}*a*≤

*λ*

_{0}/4, the F-P resonances along the slit thickness

*t*are clearly visible in Fig. 4(a), which occur when

*t*is

*m*(

*λ*/2). The peak regions of

_{eff}*T*have an inclined shape because

_{s}*λ*and

_{eff}*n*are functions of the slit width

_{eff}*a*. With increasing of the slit thickness

*t*,

*T*decreases due to attenuation losses of the SPPs’ propagation along the slit walls. When the slit width

_{s}*a*is around

*λ*

_{0}/2, the F-P resonances along the slit thickness

*t*are clearly visible in Fig. 4(a), which occur when

*t*is

*m*(

*λ*/2). The peak regions of

_{eff}*T*have an inclined shape because

_{s}*λ*and

_{eff}*n*are functions of the slit width

_{eff}*a*. With increasing of the slit thickness

*t*,

*T*decreases due to attenuation losses of the SPPs’ propagation along the slit walls. When the slit width a is around

_{s}*λ*

_{0}/2, the F-P resonances as a function of

*t*are still present, but with different shapes and locations of the

*T*peaks in the

_{s}*a*-

*t*map due to the onset of bulk wave horizontal F-P modes. When the slit width

*a*is around

*λ*

_{0}and 3

*λ*

_{0}/4, the peaks of

*T*with

_{s}*t*dependence can still be seen in Fig. 4(a). However, the appearance of the resonances with variation of

*t*are not as regular as for small widths,

*a*≤

*λ*

_{0}/4, because of the 2D asymmetric nature of the field distributions in the slit region and the weak confinement ability of the slit cavities for large widths

*a*and small thicknesses

*t*.

For a given slit thickness *t*∼*λ _{spp}*/2=235

*nm*, the peaks of

*T*for the slit widths

_{s}*a*around

*λ*

_{0}/2,

*λ*

_{0}and 3

*λ*

_{0}/2, are due to the horizontal bulk wave F-P resonances. The peaks at

*a*∼

*λ*

_{0}/2 and ⃑∼λ

*λ*

_{0}are spaced with an interval slightly larger than

*λ*

_{0}/2.

## 3. Enhancement and suppression of transmission through field interference

We first computed the fields of *E _{x}*,

*E*,

_{y}*H*and electrical charge distributions when a single

_{z}*p*-polarized narrow beam is normally incident on the groove. As described in Section 2, this beam generates the SPP propagating on the top metal surface and impinging on the slit from the left side.

Then, we computed the field distributions when a single *p*-polarized narrow beam is normally incident on the slit. For this latter configuration, the FDTD results have been well described in [5,6]. The incident beam induces surface currents and accumulated electrical charges in the vicinity of the slit, which form two electric dipoles of nearly the same phase on the top and the bottom ridges of the aperture and parallel to the film surfaces. Fig. 5 shows the computed Poynting vectors, surface currents and induced electric dipoles in the vicinity of the slit for this case. The waves traveling along the slit axis form the F-P slit waveguide modes. In addition, the SPPs are excited and propagate away from the nanoslit in the ±*x* directions on both the top and bottom metal film surfaces.

We added the two individual field distributions by the instantaneous amplitudes of each component in the slit neighborhood, and then we computed the field distributions produced when the two beams are normally incident on the groove and the slit respectively and simultaneously. We found that the addition of the two individual field and charge distributions resulting from the two respective incident beams is identical to that resulting from the two simultaneously incident beams. Thus, we confirm that the interference between the normally incident beam and the incident SPP does occur around and inside the slit. This is natural since the structure responds linearly to the two individual incident beams when nonlinear effects are not taken into account.

The 2D field interference pattern on the top and bottom metal planes and inside the slit varies with the slit width *a*, thickness *t* and the incident SPP propagation distance *L*, which controls the relative phase of the SPP-induced distribution with respect that induced by the normally incident beam. The slit transmission is enhanced by the incident SPP, when the two distributions are superposed constructively. It is reduced when the two distributions are superposed destructively.

Figure 6 shows the slit intensity transmission coefficient *T _{s}* with the two incident beams, on the groove and on the slit, normalized by that of a single beam normally incident on the slit, as a function of the SPP propagation distance

*L*. For constructive interference, the bare slit transmittance can be enhanced by more than 1.3 by the incident SPP on the slit. When the two distributions are superposed destructively, the transmittance can drop as low as 0.72 compared to the bare slit transmittance.

## 4. Conclusion

We have investigated the interference of the surface plasmon polariton (SPP) with an incident beam on a metallic nanoslit using the FDTD. The incident SPP on the film top surface is coupled into the slit by inducing oscillating electric charges at the slit edges which can emulate an oblique dipole that reradiates bulk waves inside and out of the slit, and excites new SPPs on the slit walls. The scattered bulk waves leaked into the slit form Fabry-Perot (F-P) resonator modes between the two slit walls. The excited new SPPs on the slit walls create the F-P resonator modes along the slit axis. The values of the slit width and thickness for the two types of cavity resonances are shifted from that of the standard F-P resonance conditions because of the asymmetric nature of the field distributions in the slit. We have demonstrated the interference between the impinging SPP and the normally incident beam by addition of their respectively induced modes in the slit through SPP/bulk-wave conversion. The slit transmission is enhanced or suppressed by the interference, depending on the relative phase between the incident SPP and the incident beam.

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