We report beam splitting in a metamaterial composed of a silver-alumina composite covered by a layer of chromium containing one slit. By simulating distributions of energy flow in the metamaterial for H-polarized waves, we find that the beam splitting occurs when the width of the slit is shorter than the wavelength, which is conducive to making a beam splitter in sub-wavelength photonic devices. We also find that the metamaterial possesses deep sub-wavelength resolution capabilities in the far field when there are two slits and the central silver layer is at least 36 nm in thickness, which has potential applications in superresolution imaging.
© 2010 OSA
Metamaterials have attracted much attention in recent years due to the extraordinary electromagnetic properties and their potential applications which include negative refraction [1,2], sub-wavelength imaging , cloaking , absorbers [5–7], filters , couplers [9,10], and polarizers [11,12]. As far as the beam splitters are concerned, photonic crystals are often the choice for them. Some beam splitters simply splits an incident beam into a reflected and transmitted beam [13,14], while others involve two different wavevectors . Rahm et al  also designed a beam splitter with transformation optics. Recently, metal-dielectric composites became more attractive due to their applications in optics. They could be used for superlenses [17,18], hyperlenses [19–22], beam splitting with two different wavevectors , focusing [24–28] and angle compensation  as well as negative refraction .
In this paper, we propose a metamaterial based on a multilayer metal-dielectric composite. Simulation results show that this metamaterial can realize beam splitting for the waves propagating in a sub-wavelength waveguide and deep sub-wavelength resolution in the far field, which has potential applications in nano-optical communications and super-resolution imaging.
Figure 1 shows the cross-sectional diagram of the metamaterial studied. It is made up of silver layers embedded in alumina covered by a chromium layer containing one slit. The structure is investigated using the Radio Frequency (RF) Module of COMSOL Multiphysics 3.5. This RF module has functions for design of electromagnetic structures and simulation of their response to the electromagnetic waves. The permittivities of silver, chromium, and alumina used are extracted from Ref , Ref . and the supporting material of Ref , respectively.
As also shown in Fig. 1, we consider an H-polarized (i.e., H ||y axis) plane wave at normal incidence onto the interface between air and the metamaterial. To satisfy the conditions of our model, we select the two-dimensional TM waves/Harmonic propagation application mode in the RF module. In this mode, we first construct our structure and then set calculation parameters. Other boundaries except the incident boundary are set with perfectly matched layer boundary conditions. The dimension along the x direction in the simulations of this metamaterial is 26a, where a is the period. The maximum mesh sizes for the silver layer and for the slit in the chromium are set to 3 nm, while the maximum mesh size for the other regions is set to 10 nm. GMRES is selected as the solver because of its rapid convergence. The relative error for the last two iterations is less than 10−6. In this structure, the chromium layer with the slit is used to ensure that the incident wave propagates only through the slit, which therefore may be regarded as a waveguide.
3. Simulation results and discussion
With the above structure, we simulated the power flow (or the time-average Poynting vector) distributions for slit widths of 50 nm, 100 nm, 300 nm, and 500 nm, respectively. The corresponding results are shown in Fig. 2(a) to Fig. 2(d), respectively. As seen from the figures, every beam guided from the slit splits into two beams whose k vectors are almost unchanged. Their intensities are found to decay at about 6.9 W/m2/μm in the metamaterial. Unfortunately, the energy flow is not entirely concentrated in the two beams and there is some scattering into other regions. To evaluate the magnitude of the energy flow in different regions, we considered two locations indicated by the short white lines corresponding to h = 1950 nm, as indicated in Fig. 2(a) where their lengths are 500 nm. The calculated results show that the proportion of the energy passing each line to the total energy is about 34.5%. This indicates that the energy lost to other regions is about 31%. However, the proportion lost increases with increasing d. When d is more than 660 nm, the beam splitting disappears. This means that the phenomenon of beam splitting only exists when the slit width is less than the incident wavelength.
It is noted that the position of the slit relative to the silver layers has little effect on the beam splitting, as shown in Fig. 2(a) and Fig. 3 . In these figures, the slit has been shifted from one location to the next in x, as indicated in their captions, but the energy flow distributions do not vary significantly.
4. Analysis of propagation mechanism
4.1 Analysis with negative refraction
In this section, we will try to explain the above beam splitting by first analyzing the wave vector diagram of the metamaterial. For H-polarized waves propagating in the xz plane, the dispersion relation can be described by Fig. 4 . Note that λ = 690 nm is employed as a reference wavelength in order to determine the direction of the group velocity. We also show an equifrequency contour with λ = 660 nm in vacuum (see the circle in Fig. 4). It is known that when electromagnetic waves are incident at the interface between two media, the component of their wave vectors parallel to this interface is conserved. For the case of an incident angle of θ = 20°, λ = 660 nm, and a = 105 nm, the normalized component k // a/2π = sinθk 0 a/2π = 0.054 can be obtained. We then draw the line with k // a/2π = 0.054, the so-called construction line , as displayed in Fig. 4. Clearly, this line intersects the equifrequency contours of λ = 660 nm at points A and B. From the equifrequency contours of λ = 660 nm and λ = 690 nm, we can determine the direction of the associated group velocity vector. Since only the vectors pointing away from the source need to be considered, only the wave vectors at point A are acceptable. Evidently, in this case, negative refraction will occur.
To verify the negative refraction, we suppose that a Gaussian beam with a radius of 400 nm is obliquely incident onto the metamaterial from air with an incident angle of 20 degrees after the chromium is removed, and then simulate the resulting energy flow. The simulation results are shown in Fig. 5 . Obviously, the negative refraction arises on the two interfaces between the metamaterial and air, in agreement with the result obtained from the wave vector diagram. Utilizing this negative refraction property, we analyze the propagation of the electromagnetic waves guided from the slits into the metamaterial in Fig. 2. The electromagnetic waves are diffracted by the slits, with vectors in x and z, leading to energy flow in all directions and it is difficult to completely explain the advent of the two beams in the metamaterial. We thus also need to resort to include the contribution from surface plasmon polaritons as discussed below.
4.2 Analysis with surface plasmon polaritons
Figure 6 shows the energy flow distributions for different numbers of silver layers. It is seen that when the electromagnetic waves leave the sub-wavelength slit, many waves with high spatial frequencies would arise. After these waves encounter a silver layer, the surface plasmon polaritons (SPPs) on the surface nearer to the slit of the layer are easily excited, as seen in Fig. 6(a). As this layer is so thin, the SPPs would be coupled onto the other surface of this layer. Consequently, surface waves propagate along the z direction accompanying the coupling in the x direction. The S-shaped energy flow seen in Fig. 6(a) is thus formed. When another silver layer is added to the left [see Fig. 6(b)], the same process will occur. Meanwhile, the surface waves on the two silver layers will couple to each other. When more silver layers are added on the two sides of the slit [see Fig. 6(c) to Fig. 6(g)], the surface waves produced in the two layers nearest to the slit are continually coupled sideways, resulting in the beam splitting. Hence the beam splitting results from the waves with high spatial frequencies, while the wave vector diagram analyzed above is only applicable to the propagation of the waves with lower spatial frequencies.
5. Applications to sub-wavelength resolution in the far field
Here, we investigate the sub-wavelength resolution capability of our structure using this beam splitting property. For this purpose, we first assume that the right side of a slit with 50 nm width is aligned with the left side of a certain silver layer. Then we simulate the energy flow for different thicknesses of the silver layer at 26 nm, 36 nm, and 56 nm, respectively. Note that other silver layers are still 16 nm thick. As shown in Fig. 7(a) to Fig. 7(c), it is evident that the right hand side beam is greatly weakened once the thickness is equal to or greater than 36 nm [see Fig. 7(b)], indicating that there is only one beam in this case. We also simulated the energy flow for slit widths of 30 nm and 10 nm. The simulations confirm the same phenomena as seen for the 50 nm slit width.
Subsequently, another slit with a 50 nm width is added and its left side is aligned with the right side of the silver layer. In the two-slit structures, the energy flow for the three corresponding thicknesses of the silver layer is simulated and the results are shown in Fig. 7(d), Fig. 7(e), and Fig. 7(f), respectively. Two beams can be observed in all the three cases. However, according to the analysis of Fig. 7(a) to Fig. 7(c) above, we can infer that each of the two beams in Fig. 7(d) is contributed by the two slits, but any one of the beams in Fig. 7(e) and Fig. 7(f) is primarily the result of only the slit on the same side as the beam. That is to say, the left beam results from the left slit, while the right beam results from the right slit. Therefore, this structure can distinguish the two extremely narrow slits having a spacing of only 36 nm which corresponds to a resolution of 0.055λ. More importantly, this resolution can be realized in the far field.
With the far-field sub-wavelength resolution arising from the metamaterial with two slits, sub-wavelength imaging of a broad luminescent object can be realized by scanning along the x direction over this object. For example, a scanning step length can be set to the summation of the widths of the two slits and of the chromium between them. The information from the two output beams can be recorded by a CCD camera during scanning. This recorded information is then input to a computer for image reconstruction and processing. The output beams recorded in the successive measurements need to be correctly registered in order to form the superresolved image of the object in the scanned direction. The metamaterial can be translated in y or rotated with its rotation axis along z, to implement the same process and obtain the superresolved image in another direction.
In conclusion, we propose a new beam splitter consisting of a silver-alumina composite metamaterial coated by a layer of chromium containing one slit. The simulation results show that beam splitting occurs when the width of the slit is much shorter than the incident wavelength. By adding another slit, the metamaterial structure can also show the deep sub-wavelength resolution capability in the far field when the silver layer under the slits is thicker than a critical value. It is believed that such structures have potential applications in nanophotonics devices and superresolution imaging.
The project is supported by the National Research Foundation, Singapore (NRF-G-CRP 2007-01). Dr. Yan would like to thank Xuzhou Normal University for their permission to join the research team at Nanyang Technological University and Dr Ng Tsu Hau for his suggestion to the draft.
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