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Planar lens based on nanoscale slits has been demonstrated theoretically and experimentally. In this paper, we propose a 2D model of a similar planar lens but with side-illumination. The lens consists of a main bus waveguide to transport plasmonic wave and grooves functioning as antennas. The shapes and filling materials of the waveguide and grooves are assumed to be invariant in the third direction. The phase retardation needed for wavefront shaping comes from the transverse propagation of the plasmonic wave in the waveguide and the well-designed groove positions. The concept is applied to the design of planar lenses and axicons. The simulation results demonstrate that such structures can work as good diffractive elements. The side-illumination property of such structure enables the potential integration of lens on chip.
© 2014 Optical Society of America
1. Introduction
In recent years, surface plasmon has been a research focus for its excellent performance in nano optics [1, 2]. The unique property of plasmonic wave makes it possible to develop optical elements to manipulate electromagnetic waves in nanoscale. Many plasmonic elements have been proposed and demonstrated both theoretically and experimentally. Among them, the plasmonic lens based on metal plate with nanoslits has shown potential application and attracted many research interests. Metal planar lenses with varied slit patterns have been proposed and used in imaging [3], beam focusing [4], deflecting [5] and splitting. Design methods based on optimization algorithm [6, 7] and holography [8] are introduced to realize more complex functions, e.g. focusing beam to multiple spots. In all these structures, the incident wave illuminates the lens from the free space, which is similar to that in the conventional imaging system. In this paper, we propose a plasmonic lens with side-illumination and vertical emitting.
The motivation comes from the well-known electromagnetic phased arrays which have been applied in silicon-based integrated-circuits [9]. Such phased arrays are based on a waveguide-antenna system which is used to coupling and decoupling electromagnetic waves. The antenna can be regarded as a point source which is used to generate secondary waves. In plasmonic devices, it is convenient to realize mutual conversion between the localized plasmonic field and the radiation wave in free space by using subwavelength slits and grooves. Motivated by such unique property, we propose a metal-insulator-metal (MIM) waveguide to transport plasmonic energy and metallic grooves as antennas to generate radiation waves. The secondary waves emerging from each groove undergo different phase retardations which are used in the wavefront shaping.
Another background is the planar metallic nanoscale slit lenses for angle compensation [10]. The side-illuminated planar lens can be regarded as a special case where the angle of the incidence is 90°. The grooves are used to compensate the incident angle and meanwhile focus the light. In the following sections, we will first give an introduction about the waveguide-antenna system made of MIM waveguide and grooves. The design methodology of lens is presented in detail and examples are given to demonstrate the proposed idea. The numerical simulations shown in this paper are all carried out using the commercial finite-element method (FEM) software COMSOL Multiphysics.
2. Waveguide-antenna system
A 2D model of the MIM waveguide-antenna system is illustrated schematically in Fig. 1(a). The structure is assumed to be invariant in the third direction. The main bus waveguide is composed of a metal-dielectric-metal three-layer system. The input energy propagates in the main waveguide and then is partly coupled into the side-grooves at particular positions. The MIM waveguide can support single plasmonic mode (TM0) if the core-width w is small enough. The propagation constant β of the TM0 mode can be calculated using the mode equation [11]
with k1 = (β2 − εdk2)1/2 and k2 = (β2 − εmk2)1/2, where k is the wave number in vacuum, εm and εd are relative permittivity of metal and dielectric, respectively.
Fig. 1 (a) Schematic illustration of a MIM waveguide-antenna system. (b) The simulated field distribution (Hy) out of a single groove based on finite-element method. The width of the waveguide is w = 100 nm and the metal barrier thickness is set as t = 50 nm. (c) Dependence of the energy transmission through a single groove upon the barrier thickness t.
The groove functions as an antenna. The energy in the waveguide is partly coupled [12] into the groove and then diffracts into the free space. The coupling efficiency can be varied by changing the thickness of the metal barrier between the main waveguide and the groove. Figure 1(b) shows a FEM simulation of light transmission in MIM waveguide with a single groove. The working wavelength is λ = 637 nm. Gold is used as the metal part and the relative permittivity is εm = −11.04 + 0.78i [13]. In order to reduce the propagation loss in the main waveguide, it is necessary to use dielectric materials with low refractive index [14]. Here the nanoporous silica (PSiO2, n = 1.23) [15] is used as the dielectric core in the waveguide. The width of the waveguide is w = 100 nm. The size of the groove is fixed at 100 nm × 100 nm which is much smaller than the working wavelength. Grooves with such size can be modeled as point sources. The thickness of the metal barrier between the waveguide and the groove is set to be t = 50 nm. The plasmonic wave coupled into the groove decouples into radiation wave at the groove exit and then diffracts into all radial directions similar to a point source (shown in Fig. 1(b)).
In the bus waveguide, the field amplitude of the fundamental mode decreases exponentially, normal to the metal-dielectric interface [1]. The value of the skin depth at which the field falls to 1/e is only tens of nanometers at visible light. The coupling efficiency between the main waveguide and the groove can be varied by changing the barrier thickness t. Using FEM simulations, we calculate the dependence of the energy transmission at the groove upon the barrier thickness t. To get the transmission, the simulation is performed twice. First, the groove is removed and the incident energy flux Pin in the waveguide is measured 200 nm before the position of groove. Then, the groove is added and the energy flux Pout is measured at the outlet of the groove. The normalized transmission can be calculated as Pout/Pin. The parametric sweep is used to calculate all transmission results at different values of t. The results are shown in Fig. 1(c). As the barrier becomes thinner, the transmission increases gradually. When t = 0, the transmittance is about 34% (not shown in Fig. 1(c)). When t = 50 nm, the transmittance drops to only 1%. In such waveguide-antenna system, a low coupling efficiency between the main waveguide and the groove is essential. We want to make sure that each groove obtains as same amount of power as possible. The influence of the barrier thickness t to the focusing performance of the lens will be discussed in detail in the following sections.
3. Design of diffractive elements
3.1. Lens
Figure 2 illustrates the schematic of a side-illuminated plasmonic lens. The sizes of all grooves are identical. The phase retardation needed for wavefront shaping comes from the transverse propagation of waves in the bus waveguide. To converge the light to a single point, the waves emerging from the grooves should interfere constructively at the focus. According to the equal optical path principle, the phase of waves emerging from the middle and the mth groove take the form
where f is the preset focal length and xm is the position of the mth groove. ϕ0 and ϕm are the phase retardation when plasmonic wave passes through the middle and the mth groove, respectively. Since the sizes of all grooves are identical, ϕ0 and ϕm are assumed to have same values. After a simple derivation, the groove positions xm can be obtained. The design formula (2) is similar to that of the binary Fresnel Zone Plate (FZP) except for the extra phase retardation of βxm. As shown in Eq. (2), the extra phase will increase the order m, which means the lens can include more Fresnel Zones in a fixed aperture than the traditional FZP. For a normal incident FZP with radius of R, the number of Fresnel Zone is .Fig. 2 Schematic illustration of a side-illuminated plasmonic diffractive element. The incident energy in the main waveguide is coupled partly to the side-grooves and then diffracts into free space.
To demonstrate the idea, a lens with a fixed focal length is designed and simulated. The half-size of the lens aperture is set to be a = 4 μm and the focal length is set at f = 4 μm. The parameters of the bus waveguide and the grooves are kept the same as those used in Fig. 1(b). The relative propagation constant (β/k) of the TM0 mode is calculated according to Eq. (1): β/k = 1.621 + 0.016i. Since only the real part of β contributes to the phase retardation, the imaginary part is ignored in the design process. Using Eq. (2), all grooves positions xm (m = −7 to m = 12) in the range of the preset aperture are obtained. For a normal incident FZP with the same size (R = a), the order of Fresnel Zone covers only from m = −2 to m = 2. We validate the design with numerical simulation based on FEM. The calculated intensity and phase distribution are shown in Figs. 3(a) and 3(b), respectively. From Fig. 3(a), we can see a clear focal spot locating at z = 3.9 μm which is very close to the preset focal length: f = 4 μm. As shown in Fig. 3(b), a hyperbolic phase profile lies near the aperture of the lens. The intensity distribution on the axis and focal plane are extracted and depicted in Figs. 3(c) and 3(d), respectively. Despite of the asymmetric distribution of the grooves, the center of the focal spot is still located at the axis. The full-width at half-maximum (FWHM) of the focus along the axis is 350 nm.
Fig. 3 (a) Magnetic field intensity (|Hy|2) of the designed side-illuminated lens obtained by finite-element method. (b) A transient phase distribution of Hy. (c) The extracted intensity distribution along the axis (x = 0). (d) The intensity distribution at the real focal plane (z = 3.9 μm).
As shown in Fig. 3(a), the distribution of the grooves is asymmetric with respect to the axis. There are more grooves at the right half part of the lens, which means the energy flux at the right part will be greater than that at the left half part. But meanwhile, the incidence of energy is also asymmetric. The energy will be coupled to the left grooves first and the remaining energy will attenuate along the metal waveguide. These two kinds of asymmetry have opposing effect on the energy flux at the two half sides of the lens. The asymmetric energy flux of the lens results in an asymmetric diffraction pattern. A proper choice for the barrier thickness t should be found to keep the balance. Figures 4(a)–4(f) show the simulation results of field distribution of lens with t = 0 nm, 20 nm and 80 nm, respectively. Since the transmittance of the three structures differ greatly, we normalize the field values to ranges which are described by color bars aside the figures. As shown in Fig. 4(a), when the metal barrier between the bus waveguide and the groove is removed, most of the incident energy flood into the first few grooves. The focal spot obviously tilts with respect to the axis of the lens. Because of the low transmittance, the lens with t = 80 nm (shown in Figs. 4(c) and 4(f)) has a better symmetry in diffraction field and more regular phase profile than others.
Fig. 4 The intensity and phase distribution of the designed side-illuminated lens obtained by finite-element method. The barrier thickness is t = 0 nm ((a), (d)), 20 nm ((b), (e)) and 80 nm ((c), (f)), respectively. Other parameters remain unchanged as those in Fig. 3(a).
Other examples with different preset focal lengths f are designed and simulated to check the performance of the proposed structure. We study the real focal position (position of the peak irradiance zm) for lenses with different values of f. The results are shown in Fig. 5 using red circles. The black dotted line represents the ideal focal position which is equal to the preset focal length f. As shown in the figure, when the preset focal length f increases, the position of the peak irradiance zm shifts gradually from the ideal focal position toward the aperture of the lens. Such focal-shift effect originates from the diffraction effect at the edges of lens aperture and is highly related to the size of the lens aperture [16, 17]. We can obtain the upper limit of the focal length using the following method. In the design of lenses using Eq. (2), when f is increased, the interval of the adjacent grooves gradually becomes a constant: 2π/β. Such structure will produce an uniform phase profile. The intensity distribution of lens with both uniform phase and intensity profile can be estimated using equation [17]
where A is a constant, C and S are the Fresnel cosine integral and Fresnel sine integral, . Equation (3) gives an intensity maximum at z = 34.3 μm for 637 nm incident light passing through an aperture with half-size a = 4 μm. The real focal position zm will not exceed this limit. The result is depicted in Fig. 5 using red dashed line.Fig. 5 Real focal position zm as a function of the preset focal length f. The red circles represents the simulation results of the peak position of the field intensity along the axis. The black dotted line represents the ideal focal position which is equal to the preset focal length. The red dashed line shows the upper limit (34.3 μm) of the focal length for a planar lens with aperture size 2a = 8 μm.
3.2. Axicon
Axicons [18] are conical shaped lenses that can be used to generate non-diffracting Bessel beams. They have many applications in telescopes, microscopes, eyes surgery, and atom traps. Here, we design a side-illuminated cylindrical axicon lens. The structure is also assumed to be uniform in the third direction. The phase equation (2) is modified as
where α is the base angle of the axicon. An example with α = 10° is given. The material parameters and the incident wavelength are kept unchanged as those used in previous section. The geometric parameters of the main waveguide and the grooves also remain the same. Figures 6(a) and 6(b) show the intensity and phase distribution of the designed axicon obtained by FEM. The preset depth of focus (DOF) [18, 19] is a/tanα = 22.7 μm (a is the half-size of the axicon) which is plotted in Fig. 6(a). The energy distribution along the axis and transverse distribution at the position of the peak irradiance (z = 11.18 μm) are extracted and presented in Figs. 6(c) and 6(d), respectively. To show the performance of the designed structure in long-depth focusing, the transverse distribution at other two positions (z = 8.38 μm and z = 14.13 μm) where the intensity is 80.0% of the maximal intensity are also depicted in Fig. 6(d) using dashed and dotted lines, respectively. From the figure, we can see that the full width of the main lobe of the three curves is very close to each other. The full-width at half-maximum (FWHM) at the peak irradiance and the other two positions is 0.82 μm, 0.89 μm and 0.95 μm, respectively.Fig. 6 (a) Magnetic field intensity (|Hy|2) of the designed side-illuminated axicon obtained by finite-element method. (b) A transient phase distribution of Hy. (c) The extracted intensity distribution along the axis (x = 0). (d) The intensity distribution at the peak irradiance (z = 11.18μm) and other two positions (z = 8.38μm and z = 14.13μm) where the intensity is 80.0% of the maximal intensity. The FWHM at these positions is 0.82 μm, 0.89 μm and 0.95 μm, respectively.
4. Conclusion
In this paper, a plasmonic waveguide-antennas system composed of metal-insulator-metal waveguide and grooves is proposed to construct side-illuminated plasmonic lens. The transverse phase retardation of waves in the waveguide is used for the wavefront shaping. The number of Fresnel Zones is bigger than that of the traditional Fresnel Zone Plate due to the introduction of extra phase retardation. To produce the preset diffraction pattern and maintain the symmetry, a low coupling efficiency between the main waveguide and the grooves is essential. Lenses with single focus and axicon are designed as examples to demonstrate the idea. The focal length of such lens is also limited by the size of the aperture. We believe such lenses can be integrated on chips due to the side-illumination property. It is worth mentioning that the energy transmission at each groove element can be controlled precisely by adjusting the thickness of the metal barrier between the main waveguide and the grooves. The shapes and sizes of the grooves can also be custom-made to introduce extra phase retardations at different grooves.
Acknowledgments
This work was supported by the National Basic Research Program of China (Grant No. 2013CBA01702), the National Nature Science Foundation of China (Grant No. 61275117), the Heilongjiang Province Science Foundation (Grant No. F201112) and the Open Fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province, (Heilongjiang University), P. R. China.
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