## Abstract

The authors theoretically demonstrate a plamonic beam deflector based on the particular properties of surface plasmon polaritons in metallic nanoslits. Beam deflection ranging from 0° to 90° can be achieved by designing the deflector with appropriate structural parameters. Numerical illustrations of deflectors for variant deflection angles are presented through finite-difference time-domain simulation, showing good agreement with theoretical analysis. The efficiency and some factors influencing the deflection behavior are also discussed.

©2008 Optical Society of America

## 1. Introduction

The developing trend of miniaturization and integration of optic system poses a great challenge to the traditional optical technology due to the diffraction limit. Recently, there are renewed interests in surface plasmon polariton (SPP), with the development of nano optics including the fabrication of metallic nanostructure and scanning near-field optical microscopy (SNOM), which allows one to measure the surface polariton field directly in nano-scale regions. The SPP is an electromagnetic excitation that propagates in a wave like fashion and localized at the interface between metal and dielectric medium, and its amplitude decays exponentially as it goes farther from this interface [1]. The conversion between SPP and propagating light can be obtained readily by introducing some specific structures, such as metallic grating, slit etc. This provides us an avenue to manipulate light with nanostructures and develop various nano-optic devices [2–7]. For example, a metallic optic lens with nanoslits is designed to collimate, imaging, and split light effectively [4–5]. In particular, some researchers have found that using metallic slits filled with nonlinear medium can result in the beam deflection phenomenon [6], and the deflection angle depends on the intensity of the incident light.

In this paper, a plasmonic beam deflector consisting of metallic nanoslit array is proposed for manipulating beam deflection precisely. Four illustrative design examples with the deflection angles of 30°, 45°, 60°and 80°are given and simulated by means of two-dimensional FDTD method, respectively. In addition, characteristics of resembling “negative refraction” and spatial multiplexing for the deflector are also discussed.

## 2. Principle and design for the plasmonic beam deflector

The deflector’s structure contains a planar thin metal film featured with periodical nanoslit arrays, as shown in Fig. 1. When the normal light with magnetic field polarized in the *y*-direction impinges on the metal surface, SPPs can be excited at the slit entrance. Each nanoslit can be treated as a waveguide constructed by two closely placed parallel metallic plates (inset of Fig. 1). The SPPs propagate inside the slit in the specific waveguide modes, until reaching the slit exit where they are scattered into free space. Since the slit width is assumed to be far less than the wavelength, only the fundamental mode dominates the propagation behavior. The complex propagation constant *k _{spp}* for SPPs inside the slit region can be calculated by

where *k _{0}* is the wave vector of light in free space,

*ε*and

_{m}*ε*are the permittivities for the metal and dielectric medium filled in the slit, and w is the slit width. The real and imaginary parts of

_{d}*k*determine the phase velocity and the propagation loss of SPPs inside the slit, respectively.

_{spp}Given the thickness of metal film (i.e. slit depth) is *d*, we can express the phase retardation of the SPPs transmitted through the slit as the following form [8],

where *Δϕ _{1}* and

*Δϕ*are the accompanied phase shift occurring at the slit entrance and exit, respectively. When the medium on the illuminated and unilluminated side of metal film are the same, such as in our design (where the material is air), the two factors have the equal value but opposite sign, and thus can be cancelled together. The last term

_{2}*Φ*originates from the multiple reflections between the entrance and exit interfaces. Further calculation indicates that when the silt width is larger than 10nm, the retardation contribution from

*Φ*is less than 1 percentage in proportion with the factor (

*k*) in Eq. (2). That is to say, the phase retardation is mainly dominated by the real part of propagation constant, which is approximated as

_{spp}d*Δϕ*=Re(

*k*). Therefore, the phase retardation

_{spp}d*Δϕ*can be tuned by varying the slit width

*w*if the other parameters (including

*k*,

_{0}*d*,

*ε*and

_{m}*ε*) are fixed.

_{d}In order to implement beam deflection at a specific angle *θ*, as shown in Fig. 1, the phase retardation of light transmitted through the slits along the *x* direction should take the form:

where *n* is an integer number. Therefore, the key point of designing a plasmonic beam deflector is to determine the width and position of each nanoslit for the desirable phase retardation calculated from Eq. (3).

Four deflectors (with deflection angles of 30°, 45°, 60°and 80°) are designed to illustrate this method and the wide defection angle range. The deflectors’ thickness and aperture size are fixed as 500 nm and 6µm, respectively. The wavelength of incident light is 650nm, and the corresponding permittivity of used metal silver is *ε _{m}*=-17.36+i0.715 [9]. The medium surrounding the deflector and inside silts are assumed to be air. On the basis of Eqs. (1)–(3), slits with variant widths are designed for different deflections, as plotted in Fig. 2. The designed slits’ width ranges from 10nm to about 50nm, corresponding to phase retardation modulation range of 2π. The metal space between any two adjacent slits is larger than the skin depth, about 24nm for silver at wavelength 650nm [1], to eliminate the plasmonic interaction between neighboring slits which would deteriorate the desired phase retardation.

## 3. Simulation and discussion

A finite-difference time-domain (FDTD) method is employed with a simulation size of 6µm×µm, surrounded with a boundary of perfect matched layer (PML). TM polarized plane wave (at a wavelength of 650nm) is incident from the left side. The deflector structure is positioned at the region between Z=0.2µm and Z=0.7µm. The simulated phase distributions of light field for the four deflectors, which are perpendicular to the propagation direction of the deflected light, are presented in Fig. 3. Theoretically, the deflection angle can be determined by *θ*=arcsin(*Nλ/D*), where *N* is the number of phase fringes which are close to the deflector’s exit side and confined in the simulation region, and *D* is aperture size of the deflector. With *N* for four deflectors approximately equal to 4.5, 6.5, 8 and 9 from Fig. 3, the calculated values show good agreement with design result.

A detailed evaluation of the deflection effect can be made with the angular spectrum in the far field, as presented in Fig. 4. As expected, the maximum angular spectrum peak is localized at the angle of 30.12°,45.26°,60.40° and 81.55° for the four deflectors. The slight angular deviation probably arises from the coupling effect of SPPs at both slits’ entrance and exit. Moreover, due to the diffraction effect, it can be seen that lager deflection angle delivers greater divergence. The inset of Fig. 4 depicts the angular spectrum of deflection behavior for the case of *θ*=45°, with the deflector’s aperture ranging from 0.35µm (5 slits) to 6µm (60 slits). It is clearly shown that the better deflection behavior occurs with the wider aperture and more slits employed in the deflector. However, even when the aperture size of the deflector is close to half of incident wavelength (cyan line), there still exists obvious deflection phenomenon and the deflection angle agrees with the designed one, except for the broadened angular spectra width due to the diffraction effect.

Efficiency, another important parameter for the beam deflector evaluation, is defined as the percentage of incoming power distributed in the central lobe of angular spectrum confined by the angles of two closely adjacent minimum [10]. The calculated four deflectors’ efficiencies are 47.04%, 52.72%, 51.74% and 53.38%, corresponding to the *θ* of 30°, 45°, 60° and 80°. The efficiencies are not too high, but still acceptable for practical applications (like optical interconnects and switching etc). The energy loss mainly arises from the great reflection at the illuminating side. The propagation loss in the nano slits, however, can be omitted due to the short propagation distance (i.e. slit depth of 0.5µm) and small imaginary part of effective index in slit region determined by Eq. (1). One interesting point is that the plasmonic interaction effect occurred at the entrance and exit of slits [11] help to increase deflector’s efficiency with the enhancement factor about 2.5 (slits’ total width only accounts for about 20% that of the deflector’s aperture). It is believed that higher efficiency can be obtained by optimizing structure parameters.

If a tile angle is introduced for the incident beam, deflection phenomenon resembling “negative refraction” can be delivered by redesigning the deflector. Fig. 5 gives an instance of such case, in which the incident angle is set to 30° and the refractive angle is designed as - 30°. The deflector is designed with the aperture of 4µm and thickness of 800nm. From the phase distribution and far field angular spectrum (Fig. 5(a) and Fig. 5(b)), it is clearly shown that the deflection angle agrees well with the designed one, and the emitted beam resembles “negative refraction” at the interface of deflector and external medium.

The deflector also works well with incident light other than that at the designed wavelength. Figure 6 presents the calculated deflection angles for light ranging from 550nm to 750nm with a step of 10nm, while the designed working wavelength and deflection angle are 650nm and 45°, respectively. There is a nearly linear deflection variance with incident wavelength, approximately 0.9°/10nm. This implies that the plasmonic deflector can also be used for spatial and spectral multiplexing, in addition to manipulating beam deflection.

Although above simulations are performed with precisely designed parameters, however, owing to fabrication imprecision, the final structure maybe deviates from the designed one. Therefore it is of great interest to better understand the sensitivity of slit’s effective refractive index *N _{eff}* to the fabrication error. In Fig. 7,

*N*and phase retardation inside the slit are calculated as a function of slit width. It is shown that only slight variance of

_{eff}*N*and phase retardation (less than 10%) can be observed with the slit width error of +/-3nm. Moreover, the variance even becomes much smaller with widening the slit. This indicates the deflector possesses acceptable tolerance for fabrication error.

_{eff}## 4. Conclusion

In conclusion, a method to design metallic beam deflector is presented based on the phase delay properties of SPPs within the nanostructures. The numerical simulation results show that manipulation of arbitrary deflection angle from 0° to 90° is possible with efficiency of about 50%. The plasmonic beam deflector, characterized with small dimension in wavelength scale and binary structure, is believed to display potential applications in integrated and near-field optics, such as plasmonic based antenna and near-field scanning.

## Acknowledgments

The work was supported by 973 Program of China (No.2006-CB302900) and the Chinese Nature Science Grant (60507014). Authors would like to thank Yongqi Fu and Haofei Shi for their kind contribution for the work.

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