## Abstract

It is demonstrated that multiple directional beaming effect can be realized by a metallic subwavelength slit surrounded by finite number of grooves based on mode expansion method. Each of the directional beaming is formed by superimposing two diffraction orders of spoof surface plasmon excited on the two corrugated sides of the slit. This delivers high contrast and considerably uniform energy distribution for the beaming directions.

© 2008 Optical Society of America

## 1. Introduction

Since H. J. Lezec, *et al.*, reported experimentally the beaming light from a subwavelength aperture surrounded by surface corrugations on a thin metal film in 2002 [1], the inherent mechanism and the potential applications of the beaming effect have triggered enormous interests [2–12]. Until now, the diffraction of the excited surface plasmon wave on the corrugated metal surface is generally believed to play an important role in governing the unexpected beaming behavior. By assuming the perfect conductor condition, L. Martin-Moreno, *et al.*, proposed the mode expansion method to describe the connection of the excited surface mode in the grooves and the distribution of electromagnetic field in the free space [2]. On the basis of the model, C. Wang, *et al.*, proposed a revised quasi perfect conductor model which can deal with the light diffraction behavior of slit-grooves structures for true metals [5]. By combining the applications of low profile microwave antenna and high density optical data storage, the beaming and focusing properties as a function of parameters of the subwavelength metallic slit-grooves structures has been systematically investigated [6–11]. So far, most investigation is focused on the beaming effect in one direction normal to the corrugated surface, with a few works attributed to the beaming in two directions [12]. However, no investigation has been made for the beaming behavior for slit-grooves structures in three or even more directions, which would inevitably improve the flexibility of light manipulation and have potential applications in complex subwavelength photonic devices.

In this paper, we investigated the multiple directional beaming effect of a subwavelength slit patterned with periodically positioned metallic grooves at exit plane. The investigation was carried out on the basis of the mode expansion method and structures for beaming in multiple directions ranging from 1 to 6 were designed and simulated. Also performed is the discussion of the beaming geometrical parameters, the divergence of beaming light and diffraction efficiency with grating diffraction equation and spoof surface plasmon (spoof SP) theory.

## 2. Beaming structure and simulation results

The schematic of the metallic structure investigated in this paper is shown in Fig. 1. A subwavelength slit is symmetrically surrounded by 2*N* grooves on the exit side of a metallic film. The width of the grooves and slit is *a*, the period is *d*, and the groove depth is *h*. The normal incident light form left is TM polarized with magnetic field *H _{y}* for the coordinate depicted in Fig. 1.

As pointed in many works [1–6], wavelength (*λ*) and period (*d*) of grooves mainly determine the beaming direction or the peak position in the angular spectra. This can be clearly seen from the angular spectra of diffracted light as a function of *λ*/*d* as plotted in Fig. 2. The geometrical parameters of slit-grooves structure are *a*=0.110*λ*, *h*=0.145*λ*, *N*=10. The normalized angular spectrum is defined as *I*(*θ*)=|*H _{slit-grooves}*|

^{2}/|

*H*

_{no-grooves}|

^{2}, where

*H*(

*θ*) denotes the diffracted angular spectra for the structure depicted in Fig. 1. On the basis of mode expansion method [2], the calculations were performed by scanning period

*d*with fixed

*λ*.

Clearly, there are multiple bright lines observed in Fig. 2, each of which represents a diffraction order of surface wave in the periodic grooves. Note they are from the two sides of the central slit and appear alternatively in the spectra. The spectra from right to left corresponds to increased diffraction order. For some specific *λ*/*d*, the intersection of the two bright lines forms highlight spots. The spots indicate that the radiation directions of diffraction orders from both sides coincide to each other. The highlight spots have large intensity for the constructive combination of energy from both diffraction orders, while they also display small divergence angle for the doubled radiation size. For instance, the highlight spot located in the vicinity of *λ*/*d*=1, represents the well known one directional beaming at 0°, which is superimposed by the +1 and -1 order from the two sides of the central slit. Similarly, multiple highlight spots can be observed in the intersections of the two diffraction orders, indicating the beaming effect in multiple directions.

In fact, the number of beaming directions of the subwavelength metallic slit-grooves structure is completely controlled by *d* for fixed *λ*. The multiple directional beaming effect with coinciding diffraction orders occurs with some specific *λ*/*d* positions, which are listed in table 1 with beaming directions ranging from one to six. Large period yields much more beaming directions, as can be well understood from diffraction behaviors of gratings. Figure 3 and Fig. 4 present the angular spectra and space distribution of light behind the exit plane for each parameter listed in table 1. Obvious beaming phenomena in multiple directions can be observed both in the angular and space distributions of light. The beaming peaks display high contrast with the peak intensity that is about 5 to 20 times higher than that of radiation of single slit. The beaming light’s divergence is also very small (the full width at half maximum (FWHM) is usually less than 3°) due to the increased emission size. Around the diffraction peaks is the radiation background with the averaged normalized intensity of about 1 and some slight oscillations resulting from the interference effect, which is obviously caused by the directly emitted light from the central slit. Another interesting phenomenon is that the intensity of each beaming direction is nearly uniform, no matter how many beaming directions are obtained. This characteristic makes it convenient for beam shaping system.

## 3. Discussions

#### 3.1 Determination of parameters for multiple beaming

The incident light was converted into spoof SP by the periodically patterned metallic grooves at the exit of the central slit. The spoof SP can be decomposed into a series of Fourier plane wave components with discrete transversal wave vectors expressed as *k _{xn}*=

*k*+2

_{sp}*nπ*/

*d*, here

*k*is the wave vector of the spoof SP. In fact,

_{sp}*k*is always slightly larger than

_{sp}*k*. To make the analysis convenient,

_{0}*k*is assumed to be equal to

_{sp}*k*. So we can see that When

_{0}*λ*/

*d*>2, i.e. |

*k*|>|

_{xn}*k*|, the spoof SP displays evanescent properties in the direction normal to the grating surface and the corresponding EM field is confined to the grating surface. Therefore, we can not observe any beaming effect in this case. Otherwise, some plane wave components become propagating state and light can be radiated into the free space in the specified direction

_{0}*k*sin

_{0}*θ*=

*k*-2

_{sp}*nπ*/

*d*. For example, if 1<

*λ*/

*d*<2, only one

*k*is localized in the region [-

_{xn}*k*,

_{0}*k*]. But it can be noted that light is diffracted in two symmetrical directions due to the surface plasmon diffraction occurred at the two sides of the central slit. For

_{0}*λ*/

*d*<1, there exist more than one

*k*for light diffraction in the free space, implying that the spoof SP wave could be coupled with the plane waves in the multiple directions.

_{xn}The optimum cases for strong beaming effect occur with the light spots in Fig. 2. The parameters for multiple directional beaming listed in Table 1 are the results of calculation by scanning *λ*/*d* as shown in Fig. 2. But we can give an explicit approximation of these parameters from the fact that multiple directional beaming arises from the superimposing of the diffraction orders. Based on grating equation, the diffraction of spoof SP on the two sides of the central slit can be expressed as

where *n* and *m* are the diffraction order of the upper sides and the lower sides of the slits respectively. According to Eq. (1), when

the strong beaming effect can be observed. Considering *k _{sp}*≈

*k*, multiple directional beaming occurs with

_{0}*λ*/

*d*=1, 2/3, 2/4, 2/5, 2/6, 2/7…. Also we can obtain the approximated beaming directions and the corresponding diffraction orders for different sets of parameters.

Figure 5 presents the beaming conditions and directions calculated by mode expansion method and the above analysis. Clearly, Eq. (2) gives a good approximation to the numerical calculation with mode expansion method. This approximation usually displays a negative error due to the fact that *k _{sp}*>

*k*. Consequently, the radiation angular is slightly larger than the prediction of the grating equation. It can also be seen that the approximation works well for small

_{0}*λ*/

*d*and the error increases for larger

*d*. We believe that this point can be explained qualitatively in terms of the dispersion relation of spoof SP on 1D grooves corrugated metallic surface, which can be expressed as $\frac{\sqrt{{k}_{\mathrm{sp}}^{2}-{k}_{0}^{2}}}{{k}_{0}}=\left(\frac{a}{d}\right)\mathrm{tan}\left({k}_{0}h\right)$ in the condition

*d*≪

*λ*[13]. Considering

*k*≈

_{sp}*k*, we get

_{0}The difference curve is plotted in the inset of Fig. 5, together with the error between Eq. (2) and those for multiple beaming effect listed in Table 1. They are not coinciding to each other, but display same changing feature. The great deviation, we think, arises from the condition *d*≪*λ* required for Eq. (3) is not fully satisfied in our simulations.

#### 3.2 Beaming intensity and efficiency analysis

One of the interesting features of the multiple directional beaming effect is the distribution of light among multiple beaming directions. As can be clearly seen in Fig. 3, the intensity of each beaming radiation is nearly identical, which provides a potential application in beam shaping optics. This phenomenon seems rather confusing since light diffraction for gratings usually display irregular diffraction efficiency for different orders except that the grating structures are carefully optimized. This point can be well understood from the strong diffraction of a subwavelength groove (with width of about 1/10 wavelength in our simulations) illuminated by surface waves, which shows almost uniform angular spectra distribution in the far field. The multiple directional beaming effect can be regarded as the interference pattern of the scattered for a number of subwavelength grooves. So the diffraction for each direction gets almost uniform intensity. In addition, the overlapping of diffraction orders from grooves at the both sides of slits also attribute to the nearly identical beaming intensity. Take the four directional beaming as an example, the beaming happens at the four directions as *θ*=+/-36.87° and +/-11.54°. The former is generated by the +(-)1st diffraction order from the upper (lower) side and the -(+)4th diffraction order from the lower (upper) side. While the latter originates from the +(-)2nd diffraction order of the upper (lower) side and the -(+)3rd diffraction order of the lower (upper) side. It can be found that the sum of the diffraction orders |*n*|+|*m*| for each beaming direction is fixed. So the variation of beaming intensity at different directions can be compensated by combining higher and lower diffraction orders.

Another thing should be noted in Fig. 3 is that the divergence of beams is smaller for large *θ*. Taking the four beaming example as well, the FWHM of beaming divergence is 1.8° and 1.44° for the beams in *θ*=±36.87° and *θ*=±11.54° respectively. This can be easily understood from the relationship between *k _{x}* and

*θ*, which yields Δ

*θ*=

*λ*(2

*Nd*cos

*θ*) illustrating the FWHM increases with large

*θ*or

*λ*/

*d*. But the FWHM for beams of four beaming example derived from the equation is 1.46° and 1.20°, slightly smaller than the simulated results. This is because the derivation does not take into account that the propagation of the spoof SP decreases with the number of beams presented, thus reducing the effective

*N*values in the equation. This can also be further confirmed by the FWHM of beaming at the same value of

*θ*but with different number of beaming directions. For instance, the simulated FWHM of beaming divergence at

*θ*=0 for one, three and five directions is 3.82°, 1.78° and 1.22° respectively, and the values derived from the equation are 3.09°, 1.48° and 0.97°.

Plotted in Fig. 6 are the total diffraction efficiency and the beam peak intensity as a function of number of beaming directions. Here the diffraction efficiency is defined by the ratio of the total energy confined in the angular vicinity of beaming peaks defined by the divergence width and all the light energy radiated into the free space. With the increase of beaming direction number, the diffraction efficiency and the maximum intensity drop rapidly. The diffraction efficiency of single directional beaming reaches 0.43, while this value falls to about 0.22 for six directional beaming. This great decrease can also be observed for the beaming intensity. It seems that the presence of multiple beaming directions implying more radiation channels for the spoof SP, leading to a less resonant behavior.

## 4. Conclusion

In conclusion, multiple directional beaming of subwavelength slit-grooves structures was investigated with mode expansion method computational numerical analysis. The origin of this effect is the superimposing of different diffraction orders formed by exciting and diffraction of surface mode in periodic grooves corrugated at the double sides of the metal film. The grating equation is employed to predict the structure parameters and directions for the multiple directional beaming effect. The multiple directional beaming displays advantages of high contrast and nearly uniform intensity for each beaming directions, which make it a good candidate for the application of subwavelength beam manipulation devices.

## Acknowledgments

This work was supported by 973 Program of China (No.2006CB302900) and National Natural Science Foundation of China (No.60507014, No.60528003 and No.60778018).

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