## Abstract

The launching of surface plasmons by micro-gratings of subwavelength apertures milled in a thick metal film is important for the development of surface plasmon based circuits. By comparing the near-field optical images of such surface plasmon sources with the results of a Huygens-Fresnel principle based scattering model, we show that the properties of the locally launched SP beams such as divergence or uniformity can be tuned by adjusting the shape of the micro-gratings. This allows us to propose an optimized source array well adapted for providing a narrow, collimated and uniform beam.

©2007 Optical Society of America

## 1. Introduction

In the context of the development of highly integrated photonic circuits using surface plasmons (SPs), various components must be developed to launch, control and process the SPs in the circuit. Previous experiments have shown that subwavelength hole arrays perforated on thick metal film can be used as source arrays for launching SPs along a flat metal surface and controlling the propagation direction [1,2]. Other experiments and finite-difference-time-domain (FDTD) simulations investigated to what extent SP mode intensity and propagation direction could be tuned by varying the parameters of the excitation light impinging on the micro-grating [3]. These studies have shown that varying the polarization, incidence angle and wavelength of the incident field are key features for controlling the two-dimensional launching of SPs.

On another hand, considerable work is being accomplished to develop SP-based components analogous to classical optical devices such as mirrors or lenses, the next step being the realization of active devices as circuit elements [4,5]. In this context, source arrays turn out to be useful devices to supply the necessary SP beam. Indeed, the use of apertures in optically thick films, such as those studied here, prevents strong radiation damping of the SPs into the substrate supporting the metal film and allows the coupling of light to SPs through back side illumination, avoiding noise and blinding due to the excitation. However, SP beams locally launched by finite size gratings of nano-holes exhibit usually a complex shape and/or transverse structure resulting from the interference of the SPP scattered by each nano-holes of the grating. In this work, we show that the shape of the micro-grating itself can be engineered to improve the properties of the SPP beams such as the divergence or uniformity. These features could be of practical interest for applications where mode profile matching is needed.

Modelling the scattering of an incident light field by a finite-size grating is necessary to optimize the properties of the locally launched SPP beams. The most basic description of SP launching by a finite-size bi-gratings with periods *d _{x}* and

*d*along the

_{y}*x*and

*y*directions relies on momentum conservation that can be written:

where ${\overrightarrow{k}}_{SP}$
and ${\overrightarrow{k}}_{in}^{\left|\right|}$
denote the wave-vector components in the (*x*, *y*) plane of respectively the
scattered SP and the incident light and where *i* and *j* are integers [6]. This reciprocal space
argument determines the SPP beam propagation directions, however it does not offer any information on the beams structure. One approach to obtain the SPP beam field distributions is to use FDTD simulations [3]. They show good agreement with experimental observations, however, such fully numerical calculations do not lead to simple physical interpretations of the SP field intensities.

In this paper, we suggest a different approach based on a real-space analysis. The scattering dynamics of SP waves will be described following a Huygens-Fresnel model. This simple model provides simulations that are in good agreement with experimental results. At the same time, it offers a clear physical description of the phenomenon of SP beaming from which it is possible to design an efficient source array that delivers a well collimated and uniform SP beam.

## 2. Modelling

Our approach is based on a point scattering description of the source array where each sub-wavelength hole of the array is treated as a discrete secondary dipolar source [7]. Under coherent, polarized illumination, each point sources behaves as a coherent dipole which radiates SP waves along the surface of the metal film. The global SP field is eventually the coherent summation of all these elementary fields.

The incident field ${\overrightarrow{E}}_{in}$
is taken as a plane wave with wave vector ${\overrightarrow{k}}_{in}$
. For a given hole, this incoming field is converted into two-dimensional surface waves that propagate away from the hole as a spherical (Huygens) wave exp(*ik _{sp}r_{n}*)/$\sqrt{{r}_{n}}$
. The distance between the

*n*

^{th}source and the point of observation is given by

*r*and

_{n}*k*is the complex amplitude of the SP wave vector, defined by the standard dispersion relation ${k}_{\mathit{SP}}=\frac{2\pi}{\lambda}\sqrt{\left(\frac{{\epsilon}_{m}{\epsilon}_{d}}{{\epsilon}_{m}+{\epsilon}_{d}}\right)}$ on a smooth metal-dielectric interface with respective dielectric functions

_{sp}*ϵ*and

_{m}*ϵ*[8]. This wave vector amplitude is complex and its imaginary part determines the damping of the surface wave as it propagates along the interface. In our simulations, we took

_{d}*ϵ*=

_{d}*1*since we measure the SP field intensity on the metal-air interface and used tabulated experimental values for

*ϵ*[9]. In relation with SPs excited on smooth interfaces, the surface waves are elliptically polarized with the longest axis being of course normal to the surface. However, to take into account the coupling efficency between the polarization of the incoming light and the excited SP wave, the in-plane polarization component of the surface waves, taken along their direction of propagation with a unitary polarization vector ${\overrightarrow{e}}_{SP}={\overrightarrow{r}}_{n}/{r}_{n}$ , is considered. Given the incident field polarization ${\overrightarrow{E}}_{in}$ , it is natural to introduce this coupling efficiency by ${\overrightarrow{e}}_{SP}\cdot {\overrightarrow{E}}_{in}$ , which leads to a dipolar emission pattern by each individual hole as experimentally observed [10]. The scattering process is coherent and each elementary surface wave emitted at a given hole

_{m}*n*has a fixed phase relation ${e}^{i{\overrightarrow{k}}_{in}\cdot {\overrightarrow{\delta}}_{n}}$ with the incoming light, where

*δ*is the distance between the

_{n}*n*

^{th}source and center of the array. Therefore, the initial phase differences that appear between SP waves emitted at different holes under oblique illumination (${\overrightarrow{k}}_{in}\cdot {\overrightarrow{\delta}}_{n}\ne 0$ ) are accounted for. The global SP field is defined as the sum of all the elementary SP fields, and the SP intensity

*I*can then be written as:

_{sp}Interferences naturally occur where the different coherent SP fields overlap. The SP beam which is measured by near-field optical (NFO) imaging is the direct expression of the SP intensity interference pattern produced by a given set of holes and illumination conditions. Eq. (2) provides a straightforward evaluation of this interference pattern.

It should be stressed that the direct transmission of the incoming field through the holes is neglected in our model and, as a consequence, Fano-type interferences between surface waves and propagating waves is not included in the model [11]. Finally, even though multiple scattering is not taken into account, as we shall see it is sufficient to obtain remarkable agreement with the experimental results.

## 3. Comparison between experiments and simulations

The source arrays were prepared with a FEI Dual Beam Strata 235 Focused Ion Beam in 160nm-thick gold films deposited by electron beam evaporation on ITO coated BK7 substrates. The NFO measurements were carried out with a setup that has been described in details in Ref. [12]. We first studied source arrays consisting of square lattices of 11×11 holes with period 760nm and hole diameter 220nm. In the NFO measurements, the illumination was done at 800nm at a 47° incidence angle in order to match the (-2,0) mode on the air side as explained further down [6]. The laser beam was only slightly focused (±8°) to produce a 10μm diameter spot on the sample incident on the glass-metal interface of the array, a condition fairly compatible with our model assumption of plane wave illumination. Comparisons between the NFO images and the simulations are presented in Fig. 1, showing a good agreement between experimental and computed field distributions. In particular, the computed and experimental transverse cross-sections of the beam show that close to the edge of the source array, the beam exhibits two transverse maxima that merge into a single transverse peak, giving the SP beam a self-focusing shape. Note that, as discussed earlier, the model does not take into account the direct illumination whereas it is collected by the fiber tip, resulting in higher intensities over the hole array in the NFO image as compared to the simulated data.

Experimental results and the corresponding simulations are also presented for an array with triangular lattice (Fig. 2). The previous period was adapted to this new lattice, and the experiment was carried out under the same illumination conditions. A triple beam output is observed, a slight misalignment between the illumination and the array weakening one of the beams by comparison with the simulation in the ideal case.

## 4. Angle of incidence

A source array produces a number of beams depending on the lattice of the array, as we have just seen, as well as on the optical characteristics of the excitation: past experiments performed on square arrays displayed 2 symmetrical beams under normal incidence, by coupling simultaneously into the (±1,0) modes [2]. Such multiple beams configurations can be useful in the context of two-dimensional optical devices when addressing simultaneously different elements. However, in many cases it is critical to generate a single beam propagating though the circuit. For periodic arrays, such single beam can only be obtained by playing with the angle of incidence to break the symmetry.

With this in mind, oblique illumination together with the choice of an appropriate period of the array were studied since it should allow the enhancement of a single SP beam at expense of the other(s). Simulations were carried out at different incidence angles for a square array and the intensities of the SP beams were integrated on each side of the source, as presented in Fig. 3. Resonances build up every time a Bragg-type condition is met. These situations correspond to those predicted by the classical wave-vector argument and can be related to the usual (*i,j*) surface modes indexing. For a period of 760nm, two distinct resonances appear at 43° and 47° on the air-metal interface, corresponding respectively to the excitation of the (0,0) mode, propagating along the same direction as the illumination, and to the excitation of the (-2,0) mode, counter propagative by comparison to the illumination direction. Illuminating this array at 47° with a weakly focused beam enables to excite the (-2,0) mode only, giving thus rise to a single (counter propagating) SP beam, as illustrated in Fig. 3.

## 5. Optimizing the SP beam

Keeping the angle of incidence of the illumination at 47°, the next step of this study was to design an optimized source that can generate a uniform and collimated SP beam. Design of diffractive components, such as a mirror or a lens, have been proposed by geometrically positioning individual surface defects in such a way that constructive interference in the focal point of such devices occur [13]. Following this idea, we investigated to what extent the SP beam can be modified when adding or removing holes with respect to a chosen lattice (a square lattice in this context) with all other experimental conditions fixed. Figure 4 shows examples of convex and concave shaped source arrays. The concave shape results in a slightly more uniform beam with a wider transverse size, however in both cases the beam remains yet spoilt by interferences and self-focusing. Note that, experimentally, these structures have more holes than the initial 11×11 square array and thus cannot be considered as uniformly covered by the laser spot. This had to be taken into account in our simulations by giving, in the plane of the holes, a finite-size gaussian profile to the incident field that corresponds to the experimental size of the laser spot on the sample. Figure 4(c) and 4(d) clearly illustrate this point, with fig. 4(c) being closer to the experiment than fig. 4(d).

Although the convex and concave shapes did not lead to a diffractive behaviour similar to the one of Ref. [13], these results allow understanding more deeply the relation between the number of holes, their position, and the optical properties of the generated SP beam. It is clear that when each elementary source has an equal contribution to the SP beam, the beam interference pattern will present a maximum contrast. This can be seen on Fig. 4(d) where the holes are uniformly illuminated. Therefore, adding more holes allows coupling more light to SP, but holes located away from the optical axis defined by the SP beam also induce a more complicated interference pattern on the beam and the self-focusing phenomenon. On the contrary, with a finite illumination spot, a lower contrast of the fringes is observed. This is due to the progressive reduction of the contribution of those off-axis holes (Fig. 4(c)).

Following this idea, we looked for reducing such contributions: starting from the square source, we removed holes as one goes away from the optical axis, leading to a diamond shaped array. Our simulations showed that the best collimated uniform beam was indeed achieved with this diamond shaped source array with SP launched along the diagonal. Once again, simulations and experiments are in very good agreement as can be seen in Fig. 5.

Intensity profiles integrated over the width of the beam along the direction of propagation as well as cross-sections are also shown. A noticeable difference between measured and simulated intensity profiles can be observed close to the edge of the source array. As discussed earlier, the contributions of multiple scattering and of interferences between surface waves and propagating modes are not taken into account in our single scattering approach, and simulating the field in the immediate vicinity of the array might not be accurate enough using a spherical wavefront description as Eq. (2) relies on the condition *k _{sp}r_{n}* ≫1. Moreover, the fiber tip is very sensitive to the direct illumination through the holes, which indeed results in higher collection in the vicinity of the array: on the contrary, as one moves away from the array, this difference decreases until the simulated curve fits the experimental one. The SP intensity decay length measured far from the edge of the sources is 42 ± 2 μm, which is in good agreement with the calculated intensity decay length 1/2Im(

*k*)≈45μm based on the dielectric constant for gold at 800nm. This means that losses due to surface roughness that would lead to conversion of SP back to freely propagating light does not seem to be an important contribution in these samples.

_{SP}## 6. Conclusion

We have presented a real space approach that allows a simple physical understanding of the SP beaming and emission pattern from source arrays. It has to be stressed that SP beams naturally appear in the simulation method when, for a given set of parameters such as lattice, period of the array and incidence angle of the illumination, positive interference builds in certain directions and for certain frequencies, giving rise to a SP resonance that forms the beam. Although no consideration of reciprocal space is used in the model, resonances appear for the same set of parameters every time a Bragg-type condition is met.

Our model can still be extended to account for multiple scattering contributions and for interferences between surface waves and propagation modes (as direct transmission through the array). However, this simple model already enables one to foresee easily and quickly at arbitrary experimental conditions the SP field intensity that will be measured in the near-field. It is clear from this work that the illumination conditions and the parameters of the source array must be fully taken into account in order to optimize beam shape, and that under the right conditions high quality SP beams can be launched.

## Acknowledgments

The authors acknowledge the support from the European Network of Excellence, PLASMO-NANO-DEVICES (FP6-2002-IST-1-507879) and from the STREP SPP (FP6-NMP4-CT-2003-505699).

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