We present an iterative design method for the coupling and the mode conversion of arbitrary modes to focused surface plasmons using a large array of aperiodically randomly located slits in a thin metal film. As the distance between the slits is small and the number of slits is large, significant mutual coupling occurs between the slits which makes an accurate computation of the field scattered by the slits difficult. We use an accurate modal source radiator model to efficiently compute the fields in a significantly shorter time compared with three-dimensional (3D) full-field rigorous simulations, so that iterative optimization is efficiently achieved. Since our model accounts for mutual coupling between the slits, the scattering by the slits of both the source wave and the focused surface plasmon can be incorporated in the optimization scheme. We apply this method to the design of various types of couplers for arbitrary fiber modes and a mode demultiplexer that focuses three orthogonal fiber modes to three different foci. Finally, we validate our design results using fully vectorial 3D finite-difference time-domain (FDTD) simulations.
© 2014 Optical Society of America
Over the last decade, surface plasmon polaritons have generated a substantial amount of research interest because of their unique property of being confined to a metallic surface and the associated ability of metallic structures to confine light to sub-wavelength volumes. Surface plasmons have been used in a broad area of applications such as subwavelength focusing [1, 2], light-harvesting , and photodetection . Among those applications, mode coupling and demultiplexing are very interesting as any linear optical component can be described as a converter between two sets of orthogonal optical modes . This problem of mode splitting or conversion has recently become very topical because of the possibility of using multiple modes in optical fibers. Various techniques have been explored in the literature to separate modes optically, including fiber or waveguide coupler-based devices [6–11], phase plates or spatial light modulators with free-space optics [12–18], spatial sampling into planar light-wave circuits or silicon photonics with subsequent waveguide interferometers [19, 20], and holographic approaches . Previous work with dielectric structures has shown the possibility of arbitrary mode coupling in photonic-crystal-like structures with custom regions [22, 23]. With the additional incorporation of detectors and local feedback loops together with phase shifters and interferometers, schemes have been proposed that can automatically convert between arbitrary modes [24–27].
Structures that couple and demultiplex surface plasmons have mostly been based on metallic films perforated with subwavelength apertures or grooves [see Fig. 1]. Those have been used to show a variety of physical phenomena such as extraordinary optical transmission (EOT) , beam steering , beam steering with polarization control [30, 31] and negative refraction . While most work on these arrays has made use of periodic arrays, aperiodic arrays of metallic slits have also been used to achieve binary surface plasmon polariton focusing  and spectrally discrete wavelength demultiplexing [34, 35], which is not readily possible with periodic approaches.
In an array of slits, mutual coupling between the slits occurs as surface plasmons generated by one slit can be scattered by another slit and couple back into the first one. This phenomenon is numerically tractable for periodic arrays through the use of periodic boundary conditions  and for 1-dimsional (1D) and quasi-1D slit arrays [37–39]. However, an accurate modeling of the mutual coupling in aperiodic arrays of slits has often required rigorous exact 3-D fully vectorial methods such as Finite Difference Time Domain (FDTD) , Finite Elements Method (FEM) , rigorous coupled wave analysis  or Green’s Tensor approaches [42, 43]. Consequently, previous work on aperiodic arrays was operated in a regime where the slits are located relatively sparsely, so that the mutual coupling between slits could be safely neglected .
Recently, we have developed a novel semi-analytic model, namely the modal source radiator model, which allows us to compute the mutual coupling among arbitrarily located apertures with a significantly reduced computational cost . We present a design method based on the modal source radiator model that, in contrast to the schemes used in , can account for the scattering by the slits of both the direct excitation and the surface plasmons all the slits generate. This knowledge allows us to calculate the optimal position of slits that scatter the surface plasmon waves more than they scatter the excitation beam. In addition, we are able to detect when a slit is “blocking” a focused surface plasmon and is basically scattering the field one is trying to focus; a better device is achieved by removing this particular slit.
This paper is structured as follows: first we provide a brief description of the modal source radiator model which allows us to calculate the scattered field in a quantitatively accurate way. Next we describe our design method which is built on the modal source radiator model. We conclude with some design examples: the first of these allows the focusing of a higher-order L31 fiber mode to an arbitrary location and the second is a structure that demultiplexes three orthogonal fiber modes to three different foci as depicted in Fig. 1. The latter design is confirmed with full-field FDTD simulations to prove the validity of our calculations.
2. Radiator source model
In the modal source radiator model, the field inside the slits is decomposed into a superposition of a finite number of eigenmodes, such as TE01-like and TE02-like modes [see Fig. 2], which are the guided modes that would propagate in the z-direction inside the slit if the metal was infinitely thick . Those modes can also be seen as the plasmonic equivalents of TE modes in rectangular waveguides made of a perfect conductor, where Ez = 0. In addition, in the examples we consider below, we limited this decomposition to the TE01-like and TE02-like modes shown in Fig. 2. We can therefore write the |E〉 and |H〉 fields inside the slit using the Dirac notation as
In Eq. (1), α runs over all the eigenmodes in all the slits, qα is the propagation constant of the slit mode, Aα and Bα are the complex amplitudes of the downward and the upward propagating modes respectively and represents the mode profiles of the slits. The modal source radiator model is constructed based on the fact that the transverse magnetic field at the top interface can be expressed in two different ways using the following modal basis:Eq. (2) is only valid for the area covered by the apertures of the slits. In Eqs. (2) and (3) |htα〉 is the transverse part of the magnetic field of the slit modes, is the field that would be present at the top interface in the absence of any slits and is the transverse magnetic field of the radiation pattern at the top metal interface, which is generated by the downward and upward propagating slit modes. Similar equations can be written for the bottom metal interface. Using Eqs. (2) and (3) at the top interface and analog expressions at the bottom interface, Aα and Bα can be calculated for all slits using a low-rank linear set of equations whose coefficients can be numerically calculated for arbitrary slit locations using a set of numerical simulations on a single slit . The transverse magnetic and the normal electric field can be very efficiently calculated at any point on the interface using Eq. (3) and 44].
3. Design method
Figures 3 and 4 show the optimization algorithm we employ to design the mode demultiplexer, which couples multiple orthogonal optical modes to SPPs and focuses each mode towards a different position on either the top or bottom interface of the metal layer. Our iterative design method, which is illustrated in Fig. 3, uses the modal source radiator model at every iteration step. We define ns as the total number of slits and nf as the total number of modes and aim to focus the optical mode q onto the desired location .
First, in Step 1 of Fig. 3, we compute the full field of the slit array under illumination of each optical mode q, using the modal source radiator model, which we have described in the previous section. Consequently, for a certain slit configuration, Aα and Bα are known for every mode in every slit and the electric field in the normal direction can be calculated at both interfaces in a very computationally efficient way using Eq. (4).
Then, in Step 2, the complex contribution Ψq,k of each slit to the field at each focal point is isolated by performing only a summation of all the modes α in slit k in Eq. (4) and evaluating at as shown in Eq. (5).Eqs. (5)–(7), ΔnΦq,k is calculated for all slits at step 2.
In step 3, we calculate for each slit k the position where ΔΦq,k is as small as possible, compromising between all foci q. This is done by finding the minimum of a cost function CFk(r(x, y)) that is set up to be minimal when ΔnΦq,k is small for each focus in an area close to the slit and not in the immediate vicinity of other slits. In addition, CFk needs to take the field intensity in each focus into account so as to optimize towards equal intensities in each focus. We choose CFk as :Fig. 4. Naturally, we cannot check this for all n in practice and only perform the minimization for |n|<2 throughout this work. In Eq. (8) we also introduce
To obtain a uniform intensity distribution along the nf foci we introduce a weight factor wq in Eq. (8) to give more weight to a focus with a relatively lower intensity. wq is initially defined as 1 for all q and is updated in step 4 according to the relation34]. In the examples we discuss later, we found that when ν << 0.1 we converged to a solution where one mode would be significantly favoured over the other modes, while in the case that ν >> 0.1, the algorithm would not converge at all as it would oscillate between solutions where one mode would be favoured over the others.
Throughout the optimization, some slits will lie in the path of the focused surface plasmon generated by other slits and will consequently scatter the surface plasmon more than they scatter excitation source. This surface plasmon already has the correct phase and the scattering caused by this slit will not improve the merit function, as those slits are just “blocking” the propagation of the surface plasmon. Such slits can be detected as ΔnΦq,k does not tend to converge to a multiple of 2π. For this reason every nsc iterations we remove the slits for which
Finally, in step 6, we move each slit k to a new position given by
In this section, we apply our optimization method to two different cases with arrays of slits with a width ws = 60 nm and a length ls = 300 nm in a silver layer of 200 nm thick on top of a silicon oxide layer with refractive index 1.45. We use a wavelength of 800 nm. First, we illustrate that our design method can be successfully used to design slit arrays that can focus an arbitrary mode onto an arbitrary spot on bottom side of the metallic sheet at the dielectric metal interface. To this end, we design two different slit arrays that focus an L31 mode of a Corning® SMF-28e® optical fiber which has a diameter of 8.2 μm, a core index of 1.4585 and a cladding index of 1.4533 [Fig. 5(a)] onto foci at the oxide-metal interface at μm and μm respectively. In both cases we start with a 40 × 34 array of slits that is initially centered around (0, 0) and with an initial slit separation of 400 nm and 500 nm in the x and y direction respectively.
Also, in Eq. (8) we use σg = 0.5 μm, σnn = 0.05 μm ; kspp was computed using FDTD simulations and the mode profile itself was numerically calculated using COMSOL Multiphysics 3.5a. Our algorithm converged to a solution within 40 iterations and took about 50 seconds per iteration on a regular desktop computer with a Intel(R) Xeon(R) CPU X5650-2.67GHz processor and using 8 GB of RAM. In Figs. 5(b) and 6(a), the intensity of the normal electric field at the oxide metal interface is depicted for both cases. This intensity is normalized to the average intensity of the electric field inside the fiber core. The final positions of the slits are drawn as white lines in Figs. 5(b) and 6(a). The normalized field intensity at the focal point at each iteration step is depicted in Fig. 7. For both cases, it can be seen that the field intensity increases significantly throughout the optimization before saturating. The discontinuous improvement that occurs at the 30th iteration is caused by the removal of the slits for which Pk < 0, which can be thought of as “blocking” the focused surface plasmon as explained earlier. This can also be seen in Fig. 5(b), where slits have been removed around the focus and Fig. 6(a) where slits have been removed in the area where the focused surface plasmon has already reached substantial intensity as it is being focused onto the focal point.
In a second example, we apply our optimization method to design a slit array aiming at demultiplexing three orthogonal fiber modes. In addition, to validate our method, we confirm the field patterns with full-field FDTD simulations. In order to keep the FDTD simulation size reasonable, we choose a smaller fiber core and a smaller array of slits. More specifically, we choose to demultiplex the L01, L21 and L22 [see Figs. 8(a)–8(c)] modes of a multimode fiber with a diameter of 2.2 μm, a core index of 2 and a cladding index of 1.6 onto three foci located at μm, μm and μm respectively. We start with a 13 × 11 array of slits that is initially centered around (0, 0) and with a initial slit separation of 400 nm and 500 nm in the x and y direction respectively. All other parameters are equal to those used in the previous example. As can be seen from Fig. 9, our design method converged to a solution containing 48 slits after 110 iterations and three clearly distinct intensity peaks can be observed at x = 7 μm corresponding to the yellow dotted line in Figs. 8(d)–8(f). To confirm the correct implementation of the modal source radiator model, we perform a full-field 3DFDTD simulation of the area illuminated by the source. In order to limit the simulation size the focus plane was not included in the simulation area and only the area inside the yellow dotted box in Figs. 8(d)–8(f) was simulated with a cell size of 5 nm. This resulted in an FDTD grid of 1200×1200×400 cells. The FDTD simulation was performed using Belgium CAlifornia Light Machine on 16 NVIDIA Tesla M2070 GPUs simultaneously. In this configuration, a simulation of 40000 time-steps completed in less than 140 minutes. As can be seen from Fig. 10, an almost perfect match is obtained between the modal source radiator model and the full-field FDTD, which confirms the validity of our model.
5. Estimating the overall efficiency
An exact calculation of the overall efficiency is difficult, since the modes to which we couple are not well defined nor did we include a material in which the coupled light would be absorbed. Nevertheless, it is possible to estimate the efficiency from the field intensity |Ez(r)|2 as presumably most of the energy at the metal oxide interface is carried by surface plasmons. Therefore, we assume that the in-plane Poynting vectorEq. (14) P(x, y, z) is the Poynting vector and z denotes a unit vector along the z direction. We calculate the proportionality factor τ in Eq. (15) numerically using a simulated surface plasmon propagating in the x direction and then estimate the efficiency η from Figs. 6 and 8 and 4 percent for the design depicted in Fig. 5. As is shown in Figs. 5 and 6, most of the surface plasmon energy is concentrated in the focus. However, for the demultiplexing design we observe multiple “peaks” of surface plasmon energy. While the main focus still contains most surface plasmon energy, we calculated that ηp ≈ 40%, where ηp is the ratio of the surface-plasmon energy that is actually concentrated in the focus, over the integrated surface plasmon energy around the slits. For this particular integration, we use a square with side length 2xf centered at (0,0). Also, η and ηp remain stable when using a larger number of slits as the slits are already covering most of the illuminated area. In addition, η and ηp remain stable when we permute the location of the foci in our design as shown in Fig. 11, where the intensity is plotted for a design where we optimize the slit positions for demultiplexing the L01, L21 and L22 fiber modes to μm, μm, μm, respectively. The behaviour of this design is completely analogous to the one shown in Fig. 8 and we obtained efficiencies η ≈ 0.8% and ηp ≈ 40%. From Figs. 8 and 11, we observe that most of the non-focussed light is back scattered. As better unidirectional periodic grating couplers can be designed under illumination at an off-axis angle[46, 47], it may be possible to improve ηp and η by optimizing the design for illumination at an angle.
We emphasize that the goal of this paper is to explore an optimization scheme for mode couplers and demultiplexers that is based on the source radiator model for a certain slit. Consequently, the slits’ size, metal thickness and wavelength are fixed and those parameters were somewhat arbitrarily chosen to illustrate the optimization of the positions of a number of given slits. In our structures, most of the light is still coupled into the oxide or reflected back and higher efficiencies may be obtained by optimizing the slit size and metal thickness for a specific wavelength . Also we could consider using grooves instead of slits to prevent transmission when coupling at the top interface , or coating of the metal with a dielectric grating, trapping the light at the metal surface. The algorithms presented here could be used for the above-mentioned topologies, as those topologies could be modeled accurately with the source radiator model.
We can compare the complexity of our designs here with previous analysis of the complexity required to make a particular optical component. As an example, we take our second design of a mode splitter above. Ref.  gives the complexity number ND of real numbers that need to be specified for a given component to beEq. (11) of ]; in Eq. (17), MI is the number of input modes (the dimensionality of the input space), here 3, MC is the number of channels to be coupled through the device (here 3 also), and MO is the dimensionality of the output space. We can see from Fig. 9(b) that we are able to place our outputs along one line, with each output spot spaced approximately by one spot width from the neighboring spots. We might reasonably interpret this as saying we could place any one output at any of 5 distinct positions at the output along this line. In our device, with slits oriented in a “top-bottom” axis in Fig. (8), we can design to route light to the left and right faces, essentially. Light scattering into the “top-bottom” direction is forbidden by the polarization of the light and the symmetry of the slits. Since we could design this device to give its outputs along either the left or the right “faces” of the device, we could argue that our device has at least MO ∼ 10 different output modes it can choose from. Substituting these values in Eq. (17) gives a required minimum number of degrees of freedom of ND= 57. In our design, we choose the x and y positions of 48 slits, corresponding to 96 degrees of freedom, just under a factor of 2 larger than ND. We found that approximately this number of slits was the smallest that we could use and still effectively separate the modes into the desired output spots. We should expect to use more degrees of freedom than the minimum number ND in a real design because that minimum number does not account for any control over the precise form of the output beams. To control that form, we should expect to be using a larger dimensional output space in practice. Our work here is suggesting a factor of ∼ 2 increase in output dimensionality is sufficient to allow the formation of convenient forms of output beams.
In this paper we successfully used a quantitatively accurate semi-analytic modal source radiator model for arbitrary two-dimensional non-periodic arrays of slits in order to design arbitrary mode couplers and demultiplexers. The use of the modal source radiator model allows us to calculate the contribution of each slit taking both the direct scattering of the source and the scattering of the surface plasmon into account in a quantitatively accurate and computationally efficient way. This ability enables the design of large arrays of closely-spaced slits. Our design method aims at focusing an arbitrary set of orthogonal modes onto different foci, by iteratively moving slits to minimize the phase difference between the contributions of each slit in each focus. Thanks to our model, we can detect the slits that are scattering the focusing surface plasmon more than the source and remove them. We successfully applied our method to design two large aperiodic slit arrays that focus a L31 mode in the middle of the array itself or to a focal point outside they array. Also we used our design method to calculate aperiodic slit array that focuses the L01, L21 and L22 fiber modes onto 3 separated foci and confirmed the correctness of our calculations using full field FDTD simulations.
The authors acknowledge the support of the Belgian American Education Foundation, the Methusalem and Hercules Foundations, IAP, FWO-Vlaanderen. This project was supported by funds from Duke University under an award from the DARPA InPho program, and by the AFOSR Robust and Complex On-Chip Nanophotonics MURI. The authors expres their gratitude to Andrew V. Adinetz, Jiri Kraus and Dirk Pleiter for their help in running the B-CALM simulations on JUDGE, a GPU-cluster in Forschungszentrum Jülich.
References and links
1. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Sub-wavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005). [CrossRef] [PubMed]
3. Y. Akimov, W. S. Koh, and K. Ostrikov, “Enhancement of optical absorption in thin-film solar cells through the excitation of higher-order nanoparticle plasmon modes,” Opt. Express 17, 10195–10205 (2009). [CrossRef] [PubMed]
4. D. S. Ly-Gagnon, K. C. Balram, J. S. White, P. Wahl, M. L. Brongersma, and D. A. B. Miller, “Routing and photodetection in subwavelength plasmonic slot waveguides,” Nanophotonics 1, 9–16 (2012). [CrossRef]
6. N. Riesen, J. D. Love, and J. W. Arkwright, “Few-mode elliptical-core fiber data transmission,” IEEE Photon. Tech. L., IEEE 24, 344–346 (2012). [CrossRef]
8. N. Riesen and J. D. Love, “Weakly-guiding mode-selective fiber couplers,” IEEE J. Quantum Electron. 48, 941–945 (2012). [CrossRef]
9. T. Sakamoto, T. Mori, T. Yamamoto, N. Hanzawa, S. Tomita, F. Yamamoto, K. Saitoh, and M. Koshiba, “Mode-division multiplexing transmission system With DMD-Independent low complexity MIMO processing,” J. Light-wave Technol. 31, 2192–2199 (2013). [CrossRef]
10. C. P. Tsekrekos and D. Syvridis, “All-fiber broadband mode converter for future wavelength and mode division multiplexing systems,” IEEE Photon. Tech. L. 24, 1638–1641 (2012). [CrossRef]
11. A. M. Bratkovsky, J. B. Khurgin, E. Ponizovskaya, W. V. Sorin, and M. R. T. Tan, “Mode division multiplexed (MDM) waveguide link scheme with cascaded Y-junctions,” Opt. Commun. 309, 85–89 (2013). [CrossRef]
12. G. Stepniak, L. Maksymiuk, and J. Siuzdak, “Binary-phase spatial light filters for mode-selective excitation of multimode fibers,” J. Lightwave Technol. 29, 1980–1987 (2011). [CrossRef]
13. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, and P. Sillard, “Mode-division multiplexing of 2 × 100 Gb/s channels using an LCOS-based spatial modulator,” J. Lightwave Technol. 30, 618–623 (2012). [CrossRef]
14. J. Carpenter and T. D. Wilkinson, “Characterization of multimode fiber by selective mode excitation,” J. Light-wave Technol. 30, 1386–1392 (2012). [CrossRef]
15. J. A. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Optical vortex based mode division multiplexing over graded-index multimode fibre,” (Optical Society of America, 2013), OSA Technical Digest (online),OTh4G.3+.
18. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010). [CrossRef]
19. H. Bulow, “Optical-mode demultiplexing by optical MIMO filtering of spatial samples,” IEEE Photon. Tech. L. 24, 1045–1047 (2012). [CrossRef]
20. H. Bulow, H. Al-Hashimi, and B. Schmauss, “Spatial mode multiplexers and MIMO processing,” in Opto-Electronics and Communications Conference (OECC), 2012 17th, (IEEE, 2012), pp. 562–563. [CrossRef]
21. K. H. Wagner, “Mode group demultiplexing and modal dispersion compensation using spatial-spectral holography,” in IEEE Photonics Society Summer Topical Meetings, Space Division Multiplexing for Optical Communications,(2013), pp. 89–90.
22. Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. 30, 141–143 (2005). [CrossRef] [PubMed]
23. V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20, 28388–28397 (2012). [CrossRef] [PubMed]
25. D. A. B. Miller, “Self-configuring universal linear optical component,” Photonics Research 1, 1–15 (2013). [CrossRef]
26. D. Miller, “Establishing optimal wave communication channels automatically,” J. Lightwave Technol . 31, 3987–3994 (2013). [CrossRef]
28. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]
29. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002). [CrossRef] [PubMed]
30. J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013). [CrossRef] [PubMed]
31. F. Afshinmanesh, J. S. White, W. Cai, and M. L. Brongersma, “Measurement of the polarization state of light using an integrated plasmonic polarimeter,” Nanophotonics 1, 125–129 (2012). [CrossRef]
32. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]
34. T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett. 11, 2693–2698 (2011). [CrossRef] [PubMed]
36. S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13, 3150–3165 (2005). [CrossRef] [PubMed]
38. H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010). [CrossRef]
39. X. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B 84, 245120 (2011). [CrossRef]
41. E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000). [CrossRef]
42. M. Paulus, P. G. Balmaz, and O. J. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797 (2000). [CrossRef]
43. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001). [CrossRef]
44. T. Tanemura, P. Wahl, S. Fan, and D. A. B. Miller, “Modal source radiator model for arbitrary two-dimensional arrays of subwavelength apertures on metal films,” IEEE J. Sel. Top. Quant. 19, 4601110 (2013). [CrossRef]
45. P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-Source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromag. Res. 138, 467–478 (2013).
46. S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single sub-wavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007). [CrossRef]
47. B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quant. Electron. 34, 47–87 (2010). [CrossRef]
48. J. S. White, G. Veronis, Z. Yu, E. S. Barnard, A. Chandran, S. Fan, and M. L. Brongersma, “Extraordinary optical absorption through subwavelength slits,” Opt. Lett. 34, 686–688 (2009). [CrossRef] [PubMed]
49. F. López-Tejeira, S. G. Rodrigo, L. Martíin-Moreno, F. J. Garcíia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007). [CrossRef]
50. D. A. B. Miller, “How complicated must an optical component be?” J. Opt. Soc. Am. A 30, 238–251 (2013). [CrossRef]