Study of an optical nanolens with the parallel finite difference time domain technique



[1] In this paper a three-dimensional dispersive finite difference time domain based on the message passing interface architecture is applied for the full wave modeling of a metallic nanolens operating at optical frequencies. The subwavelength imaging potential of the nanodevice is thoroughly studied. The issue of symmetry in the nanolens is exploited, which is shown to have a direct result at the dynamic behavior of the nanolens.

1. Introduction

[2] Metamaterials are artificially constructed structures, which possess extraordinary electromagnetic properties not found in natural materials. One of the most comprehensively studied examples is a material having negative refractive index [Shelby et al., 2001], which can lead to the design of perfect lens with a subwavelength resolution [Pendry, 2000] leading to potential novel imaging systems. To sustain subwavelength imaging, the evanescent electromagnetic fields have to be preserved on their passage from the source to the focal point. In conventional materials, however, these fields decay exponentially. It has been shown [Pendry, 2000] that the evanescent field components are directly involved in the image formation, when metamaterials with negative refraction are used in a lens configuration.

[3] However, metamaterials with negative refraction have inherent limitations in terms of high loss and frequency dispersion [Smith et al., 2004; Podolskiy and Narimanov, 2005] which, combined with the design complexity (especially at optical frequencies), make practical implementation of them limited. A promising alternative, in order to exploit the near fields at optical frequencies in a controlled fashion, can be found in the rapidly emerging research area of plasmonics [Barnes et al., 2003; Maier, 2007]. The plasmons are surface waves confined to the interface between noble metals and surrounding air and occur at the IR and visible frequency regimes, where most metals appear to have negative permittivity. It has been shown that a nanolens, in order to achieve subwavelength resolution, can be constructed with the desired property of lossless plasmonic transfer of the near-field information [Ono et al., 2005; Kawata et al., 2008].

[4] The complexity of simulating such a device leads to a significantly increased numerical simulation time using a conventional dispersive finite difference time domain (FDTD) code; this is due to the large number of cells required to model a three-dimensional (3-D) device with subwavelength features. In this paper, a parallel 3-D dispersive FDTD technique is applied. The convergence and the accuracy of the simulation is improved with an additional spatial averaging scheme applied to the dispersive FDTD algorithm [Zhao et al., 2007]. The combination of dispersion and spatial averaging in a parallel FDTD scheme is unique, leading to a useful numerical tool in the growing research field of plasmonics. The issue of symmetry is explored, which is shown to have a direct result at the time taking the fields to reach their steady state. The numerical convergence in the modeling of symmetric and nonsymmetric nanolenses is thoroughly studied and interesting results are obtained. Finally, note that the parallel FDTD method is proved to be a very useful numerical tool to explore very complicated electromagnetic structures, whose complexity would otherwise lead to prohibitively long simulation times on a single processor computer. Conformal parallel FDTD techniques can also be used to reduce the required spatial resolution. However, late time instabilities and increased complexity of the FDTD algorithm may affect the robustness of the proposed numerical simulation tool.

2. Description of Numerical Method

2.1. A Parallel FDTD Algorithm

[5] The FDTD method is a versatile numerical technique with the major advantages of simplicity in implementation and robust operation. From an engineering point of view, it is advantageous because it can easily compute the transient response and the operational bandwidth of a device. However, for large electromagnetic problems it is computationally intensive, like most numerical simulation methods. Hence, in order to distribute the high requirement for system resources, a parallel version of the FDTD algorithm has to be implemented, which will operate in multiprocessor computers, i.e. computer clusters.

[6] The FDTD technique is inherently parallel in nature. The computational domain is divided into smaller subdomains based on the space decomposition technique [Chew and Fusco, 1995; Gedney, 1995] and every one of them is assigned to one processor. The tangential field components are passed between the adjacent interfaces at each time step with an appropriate synchronization procedure, which is provided by the message passing interface (MPI) library. As a result, few modifications lead to a more efficient algorithm, which can handle the most computationally demanding simulations. The parallel process between two subdomains is depicted in Figure 1. During our simulations, the computational domain will be divided along only one direction (z axis). Future goal will be to implement uniform division of the parallel FDTD domain, simultaneously through x, y and z axis, which is going to reduce the required simulation time even further. Moreover, additional field components are required to pass between the subdomains due to the complexity of the utilized dispersive FDTD code. Note that the material parameters and the geometry of the simulated device are derived for the whole domain, independently from the main parallel procedure, not for each separate subdomain. Therefore, different complex geometries can be modeled with the same parallel FDTD code.

Figure 1.

The field components in two different subdomains in parallel FDTD simulations. The red arrows are the transferred field components from the neighboring subdomain during the data communication process, which are used to update the field components on the boundary of the current subdomain.

[7] The computer cluster in the Antenna Group at Queen Mary, University of London consists of one head node to monitor the processors and 15 compute nodes to perform calculations. Every node has Dual Intel Xeon E5405 (Quad Core 2.0 GHz) central processing units (CPUs) and there are 128 cores and 512 GB memory in total. The GNU C compiler (GCC) is used to compile the parallel dispersive FDTD algorithm. Finally, a free version of MPI, MPI Chameleon (MPICH), developed by Argonne National Laboratory, is utilized to handle the intercore data communications.

2.2. Spatial Averaging at the Material Interface in FDTD Modeling of Nanolens

[8] The metallic nanorods are composed of silver, which has an approximate permittivity equation image = −9.121 + equation image0.304 at the excitation frequency used throughout the paper f = 614.75 THz [Johnson and Christy, 1972]. In order to model silver materials with the FDTD method at optical frequencies, dispersion models have to be introduced to the conventional FDTD code. Here, the permittivity is mapped with the well-known and widely used Drude dispersive material model:

equation image

where ɛ0 is the free space permittivity, ωp is the plasma frequency and γ is the collision frequency, which characterizes the losses of the dispersive material. The plasma frequency ωp is varying in order to simulate the material properties of silver.

[9] The FDTD method is based on the temporal and spatial discretization of Faraday's and Ampere's laws. Harmonic time dependence exp(equation imageωt) of the field components is assumed throughout the paper. Faraday's and Ampere's Laws are discretized with the common procedure [Taflove and Hagness, 2005], resulting to the conventional updating FDTD equations:

equation image
equation image

where Δt is the temporal discretization, equation image is the discrete curl operator and n the number of the current time step.

[10] For the dispersive FDTD method used throughout this paper, the electric field constitutive equation has to be also discretized, which is given by the following equation:

equation image

where the permittivity equation image can have scalar or tensor form. The auxiliary differential equation (ADE) method [Gandhi et al., 1993] is used, based on the Drude model, to produce the updating FDTD equations. Equation (4) is discretized, as in [Zhao et al., 2007], and the final updating FDTD equation is:

equation image

The temporal discretization is always chosen according to the Courant stability condition [Taflove and Hagness, 2005] and is given by Δt = Δx/equation imagec, where c is the speed of light in free space and Δx is the uniform spatial resolution.

[11] A spatial averaging technique is used to improve the accuracy of the simulation and to increase its efficiency [Zhao et al., 2007]. At the interfaces between the dispersive material (described by equation (1)) and the surrounding free space, the permittivity is calculated from the arithmetic mean of the two materials:

equation image

where ɛ0 is the permittivity of free space and ɛ is the dispersive permittivity from equation (1). The above technique is used to avoid artificial numerical resonances of the surface plasmon wave confined at the interface of the dispersive device and the surrounding free space. Finally, note that the averaging technique is only applied to the field components, which are tangential to the material interfaces. As a result, it applied only at the planar y-z surfaces of the nanorod and not at the curved surfaces of the cylindrical structure.

3. Numerical Results of the Metallic Nanolens

[12] The nanolens is composed of a hexagonal arrangement of silver nanorods, operating at optical frequencies. The excitation signal has frequency f = 615 THz, placed at the middle of the optical spectrum. The rod diameter is d = 20 nm and the optimum value of the pitch is a = 40 nm, chosen according to the study in [Ono et al., 2005]. The length of the nanorod (L) is varying from 50 nm to 90 nm, in order to study the subwavelength imaging potential of the device. The front view and the profile of the lens' geometry can be seen in Figures 2a and 2b, respectively.

Figure 2.

Geometry of the (a) front view and the (b) profile of the hexagonal arrangement of silver nanorods. The green shaded rods are later removed in order to simulate the asymmetric structure.

[13] The computational domain of the parallel FDTD is obtained in Figure 3. The simulation domain is terminated with modified material-independent perfectly matched layers (PML) [Zhao et al., 1998] due to the proximity of the dispersive device with the edges of the FDTD domain. A uniform spatial discretization is used, with an uniform FDTD cell size of Δx = Δy = Δz = λ/488, where λ = 488 nm is the wavelength of the source signal. Very fine spatial resolution is employed to avoid staircasing approximations caused from the nanorods' curved structure. Six infinitesimal small coherent electric dipoles (Ex) are placed in a hexagonal formation and are depicted as red spots in Figure 3, placed 10 nm in front of the lens. The distance between the sources is a = 40 nm, which leads to a subwavelength resolution of λ/12, in case of perfect image formation of the coherent source from the other side of the nanolens. It is going to be shown that perfect image formation happens only at a particular length of the nanorod equal to L = 80 nm.

Figure 3.

Perspective view of the nanolens structure. The red spots in hexagonal formation indicate the six infinitesimal small electric dipoles that act as the source image. The green shaded nanorods are removed in order to simulate a nonsymmetric structure of the nanolens.

[14] First, the symmetrical structure of Figure 3 is simulated with the parallel FDTD method for different nanorod lengths. The structure is symmetrical because three or two nanorods exist at both sides (z axis) of the hexagonal source formation, different from the nonsymmetric structure, where the nanorods emphasized with green (see Figures 2a and 3) are removed. The nonsymmetric device will be studied later at this paper. Note that the computational domain of the symmetric nanolens could be reduced by a factor of 4 due to the symmetry of the device. The dimensions of the computational domain are 150 × 350 × 350 cells and the parallel code uses 35 processors, each of them composed of a subdomain of 150 × 350 × 10 cells. There is an almost linear relationship between the number of the utilized processors and the speedup in the computation time of the FDTD simulation. The fields are monitored at the other side of the hexagonal source to exploit the imaging capability of the device, 10 nm away of the lens. The total memory required for the parallel FDTD simulation is 52 GB of RAM. Convergence is reached after approximately 60,000 time steps, when the total energy of the electromagnetic fields is less than 0.1% different compared to the energy calculated during the previous time step. An estimated average of 0.514 seconds per time step per processor is needed to compute the fields. From a physics perspective, the convergence is reached after approximately 0.1 ps, or 71 wave periods at the current excitation frequency.

[15] Different lengths of nanorods are chosen (L = 50–90 nm) and the fields can be seen in Figures 4a–4e. The hexagonal source is depicted in Figure 4f, where the unwanted reflection of the device have been removed for better picture clarity. The nanorod lenses with lengths of L = 50 nm and L = 90 nm tend to focus the energy of the source at the central nanorod (Figures 4a and 4e) and they cannot be used as an imaging device. When the length of the lens is L = 60 nm and, especially, L = 70 nm the image starts to appear, similar to the source surface in Figure 4f. However, the best subwavelength imaging performance is achieved when the length of the nanorod is L = 80 nm. This effect is due to constructive interference of the optical coherent sources, valid only close to the Fabry-Pérot resonance condition, when the length of the nanorod is equal to: Lλg/2 ≃ λ/(2∣ɛ∣1/2) ≃ 80 nm, as was numerically predicted.

Figure 4.

(a–e) The amplitude of the electric field component (Ex) for lenses with different lengths ranging from L = 50 nm to L = 90 nm. The fields are monitored 10 nm away behind the lens. (f) The hexagonal source formation monitored 10 nm away, in front of the device.

[16] Next, the nonsymmetric nanolens is simulated, i.e. without including the marked nanorods shown in Figure 3, using exactly the same FDTD computational scenario. It is obtained that the simulation time to achieve steady state increases with a factor of three for the nonsymmetric structure. Hence, convergence is reached now, after approximately 180,000 time steps, a major disadvantage in the efficient modeling of the nanolens. This is due to more intense oscillations at the nonsymmetric structure's edges leading to higher ringing of the fields. However, when the lens simulation is converged, the results are similar to the previously mentioned symmetric device. To exploit this effect, the nonsymmetric device is simulated with two different nanorod lengths (L = 50 nm and L = 0 nm). The image formation of the nanolens with length L = 50 nm can be seen in Figures 5a and 5b for 60,000 and 150,000 time steps, respectively. The nonsymmetric image pattern is obvious in Figure 5a, differently from the Figure 5b, where the image starts to formate. The same experiment is repeated for the nanolens with length L = 80 nm and the results are observed in Figures 5c and 5d for 60,000 and 150,000 time steps, respectively. For this particular case, there is no image formation at 60,000 time steps (Figure 5c). Nevertheless, the image pattern starts to appear, still asymmetrically, at 150,000 time steps. It is straightforward that an electrically larger device needs more time to converge, which is the case for Figures 5c and 5d. Note that the same effect in the FDTD code's convergence has been observed when more nanorods are used to construct the nanolens (not shown here).

Figure 5.

Amplitude distribution of the electric field component (Ex). (a and b) Image formation of an asymmetric nanolens with nanorod length L = 50 nm after 60,000 and 150,000 time steps, respectively. (c and d) Image formation of an asymmetric nanolens with nanorod length L = 80 nm after 60,000 and 150,000 time steps, respectively. Nonsymmetrical image results are obtained due to the slow converge time of the structure.

4. Conclusion

[17] To conclude, a parallel dispersive FDTD method was applied to simulate a silver nanolens operating at optical frequencies. The full wave simulation of the nanostructure was used to thoroughly explore the physics of the device, leading to a better understanding of its performance. It was found that this nanodevice can achieve the desirable subwavelength resolution (λ/12) only for a particular length of the silver nanorods (L = 80 nm). Moreover, the dynamical behavior of the image formation was studied numerically. It was found to vary, depending on whether the device has a symmetric/asymmetric or larger cross section. As a result, more robust and faster performance can be achieved if a symmetric design is chosen for future practical implementations of the nanolens.