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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 18 — Sep. 9, 2013
  • pp: 21273–21284
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Reverse design of a bull’s eye structure based on the plasmonic far-field pattern

Akira Yamada and Mitsuhiro Terakawa  »View Author Affiliations


Optics Express, Vol. 21, Issue 18, pp. 21273-21284 (2013)
http://dx.doi.org/10.1364/OE.21.021273


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Abstract

We present a novel concept on designing a bull’s eye structure for a single-wavelength optical source. The plasmonic far-field around a subwavelength aperture on a thin gold film is calculated by finite-difference time-domain method. Based on the annular field intensity distribution on the film surface, we present a method for determining a fairly optimal first groove radius and a periodicity of the grooves that show enhanced transmission. By additionally fine-tuning groove width and groove depth, we have achieved a transmission factor of 9.74. Our novel method has high potential in applications such as silicon infrared sensors.

© 2013 OSA

1. Introduction

The focusing of optical energy within the subwavelength scale through the excitation of surface plasmons has been receiving growing interest. The focusing of energy is mediated through the combination of scattering far field and enhanced near field light on nanostructures. Energy focusing metallic nanostructures such as dimers [1

1. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004). [CrossRef] [PubMed]

], which are pairs of nanospheres each with a nanogap in between, and subwavelength apertures with various shapes [2

2. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

] have been reported. These metallic nanostructures have the potential of focusing even more energy than it can alone by periodically arranging itself, or by creating nanostructures in its vicinity. Concentric Necklace Nanolenses (CNNL) [3

3. A. J. Pasquale, B. M. Reinhard, and L. Dal Negro, “Concentric Necklace Nanolenses for Optical Near-Field Focusing and Enhancement,” ACS Nano 6(5), 4341–4348 (2012). [CrossRef] [PubMed]

] is an example of a structure composed of arrangements of nanospheres for far-field scattering and near field enhancement, which funnels energy to the dimer at the center. Self-similar nanolens [4

4. Z. Li, Z. Yang, and H. Xu, “Comment on “self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett. 97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]

] is a chain of nanospheres with different radii that induces giant optical near field in its smallest gap. Subwavelength apertures surrounded with periodic groove structures on metal films are also known as nanostructures that enhance field intensity at the central aperture. These structures are mainly used to demonstrate surface-enhanced Raman scattering for applications in the area of sensing and spectroscopy.

As of periodic groove structures surrounding a subwavelength aperture on a metal film, the structures can be categorized in to two groups by its dimensions. A slit surrounded by parallel periodic grooves [5

5. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of Highly Directional Emission from a Single Subwavelength Aperture Surrounded by Surface Corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003). [CrossRef] [PubMed]

7

7. C. K. Chang, D. Z. Lin, C. S. Yeh, C. K. Lee, Y. C. Chang, M. W. Lin, J. T. Yeh, and J. M. Liu, “Similarities and differences for light-induced surface plasmons in one- and two-dimensional symmetrical metallic nanostructures,” Opt. Lett. 31(15), 2341–2343 (2006). [CrossRef] [PubMed]

] is classified as one dimensional. A bull’s eye structure [7

7. C. K. Chang, D. Z. Lin, C. S. Yeh, C. K. Lee, Y. C. Chang, M. W. Lin, J. T. Yeh, and J. M. Liu, “Similarities and differences for light-induced surface plasmons in one- and two-dimensional symmetrical metallic nanostructures,” Opt. Lett. 31(15), 2341–2343 (2006). [CrossRef] [PubMed]

19

19. A. Agrawal, H. Cao, and A. Nahata, “Time-domain analysis of enhanced transmission through a single subwavelength aperture,” Opt. Express 13(9), 3535–3542 (2005). [CrossRef] [PubMed]

] which consists of a subwavelength central aperture with concentric annular grooves is classified as two dimensional. When light is incident on a bull’s eye structure, the focusing of energy takes place at the central aperture due to the surface plasmon polaritons propagating from the surrounding annular groove structure. As a result, the transmission of light through the central aperture increases, which is a phenomenon known as the extraordinary optical transmission (EOT). The bull’s eye structure not only focuses but transmits the focused energy through the aperture, hence possessing the possibility of various promising and potential applications. Much higher performance can be withdrawn by implementing this structure on nanodevices such as mediums of data storage [8

8. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

] and nano-photodiodes [9

9. T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si Nano-Photodiode with a Surface Plasmon Antenna,” Jpn. J. Appl. Phys. 44(12), L364–L366 (2005). [CrossRef]

].

The bull’s eye structure has various structural parameters that control the transmission of the central aperture, giving rise to many researches in this field [8

8. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

,10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

20

20. F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500 (2003). [CrossRef]

]. When light is incident on a bull’s eye structure, the incident light couples to surface plasmons on the groove structure. The coupling efficiency of the surface plasmons is greatly affected by the geometrical parameters. Since the period of the grooves determines the resonant wavelength of the exciting surface plasmons, the wavelength of maximum transmission through the aperture also changes [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

14

14. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

]. The groove depth [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

12

12. O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. 99(4), 043902 (2007). [CrossRef] [PubMed]

,14

14. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

,15

15. N. Bonod, E. Popov, D. Gérard, J. Wenger, and H. Rigneault, “Field enhancement in a circular aperture surrounded by a single channel groove,” Opt. Express 16(3), 2276–2287 (2008). [CrossRef] [PubMed]

] and groove width [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

,12

12. O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. 99(4), 043902 (2007). [CrossRef] [PubMed]

,15

15. N. Bonod, E. Popov, D. Gérard, J. Wenger, and H. Rigneault, “Field enhancement in a circular aperture surrounded by a single channel groove,” Opt. Express 16(3), 2276–2287 (2008). [CrossRef] [PubMed]

] affects the propagating plasmonic modes within the grooves, altering the focusing efficiency of the energy. The first groove radius [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

,12

12. O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. 99(4), 043902 (2007). [CrossRef] [PubMed]

,13

13. S. Carretero-Palacios, O. Mahboub, F. J. Garcia-Vidal, L. Martin-Moreno, S. G. Rodrigo, C. Genet, and T. W. Ebbesen, “Mechanisms for extraordinary optical transmission through bull’s eye structures,” Opt. Express 19(11), 10429–10442 (2011). [CrossRef] [PubMed]

,15

15. N. Bonod, E. Popov, D. Gérard, J. Wenger, and H. Rigneault, “Field enhancement in a circular aperture surrounded by a single channel groove,” Opt. Express 16(3), 2276–2287 (2008). [CrossRef] [PubMed]

17

17. H. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004). [CrossRef] [PubMed]

] holds the key to enhanced transmission, since the radius determines the phase of the surface plasmon polaritons at the central aperture, determining the state of interference of the waves. With all of these effects known to date, however, optimal parameters for greater transmission is mainly obtained through empirical methods by experiments and calculations.

In studies that utilized the plasmonic far field around subwavelength sized gold nanospheres on a silicon substrate [21

21. G. Obara, N. Maeda, T. Miyanishi, M. Terakawa, N. N. Nedyalkov, and M. Obara, “Plasmonic and Mie scattering control of far-field interference for regular ripple formation on various material substrates,” Opt. Express 19(20), 19093–19103 (2011). [CrossRef] [PubMed]

,22

22. G. Obara, Y. Tanaka, N. N. Nedyalkov, M. Terakawa, and M. Obara, “Direct observation of surface plasmon far field for regular surface ripple formation by femtosecond laser pulse irradiation of gold nanostructures on silicon substrates,” Appl. Phys. Lett. 99(6), 061106 (2011). [CrossRef]

], finite-difference time-domain (FDTD) method has been carried out to calculate the plasmonic far-field around the subwavelength structure when 800 nm light was incident. The plasmonic far field stemming from the nanosphere interfered with the incident light generating concentric periodic electric field intensity distributions on the surface of the silicon substrate.

In this paper, we describe a novel method for designing the bull’s eye structure based on the plasmonic far field around the central aperture. We predict that an isolated subwavelength aperture on a plane substrate at the center of the bull’s eye structure should generate similar plasmonic far field properties, as shown in [23

23. L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85(3), 467 (2004). [CrossRef]

]. We have considered designing a groove structure accordingly to the annularly distributed high electric field intensities, in order to design an efficient energy focusing bull’s eye structure. That is to say, we propose a novel concept of designing an efficient bull’s eye structure by starting off from observing the energy flowing out from the central aperture, and ending up with creating groove structures to propagate energy in the reversing direction. The designed periodic groove structure having multiple concentric grooves propagates the surface plasmon polaritons and they would constructively interfere at the central aperture, leading to much higher transmission rates. This novel designing method is one that is not dependent on trials and errors, but on the phenomenon arising from the central aperture.

2. Designing concept and transmission calculations

The conceptual scheme of our study is shown in Fig. 1
Fig. 1 The conceptual scheme of reverse designing of a bull’s eye structure based on the plasmonic far field around from the central aperture. (a) Light is irradiated on a system only with a central aperture on a gold film. (b) The electric field intensity distribution generated from the interference of the incident light and the scattered far-field from the central aperture is calculated. (c) A bull’s eye structure is designed so that grooves are structured with the peak intensities positioned at the center of the grooves, and its transmission is also calculated. (d) By additionally adjusting the parameters based on the structure designed in (c), a bull’s eye structure with higher transmission has been designed.
. First, we have calculated the interference of scattered far field from an isolated subwavelength aperture and uniformly incident 800 nm light on a gold film surface. According to the constructive interference of the scattered far field and incident light generating in a concentric plasmonic pattern around the central aperture, we have designed grooves to form a bull’s eye structure. We have assumed that the periodicity of the plasmonic far-field generating outwards from the central aperture is equivalent to the wavelength of surface plasmon polaritons propagating inwards from the grooves to the central aperture of the bull’s eye structure. The transmission of the bull’s eye structure with the groove structure designed from this idea was calculated. In addition, we have tuned other structural parameters in detail based on the structure designed from the idea mentioned above to achieve higher transmission.

3D finite-difference time-domain (FDTD) method has been carried out to calculate the electric field intensity distributions generated when light is irradiated over a subwavelength aperture on a gold film. Mur’s first order absorbing boundary condition [24

24. G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” IEEE Trans. Electromagn. Compat. EMC-23(4), 377–382 (1981). [CrossRef]

] was applied to the boundary perpendicular to the z axis, and periodic boundary conditions were applied to the boundaries perpendicular to the x and y axis. Continuous plane waves with linear polarization parallel to the y axis was set as the light source, propagating in –z direction. Incident wavelength is set to 800 nm, with electric field amplitude of 1 V/m.

We have used 280 nm thick gold (λ = 800 nm, n = 0.1808, κ = 5.117, εm = −26.15 + i1.85, ε = 9.0685 [25

25. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

]) film for the calculations. Thickness was chosen so as to avoid being smaller than the penetration depth of gold, but thin enough to exhibit extraordinary transmission using [13

13. S. Carretero-Palacios, O. Mahboub, F. J. Garcia-Vidal, L. Martin-Moreno, S. G. Rodrigo, C. Genet, and T. W. Ebbesen, “Mechanisms for extraordinary optical transmission through bull’s eye structures,” Opt. Express 19(11), 10429–10442 (2011). [CrossRef] [PubMed]

] as a reference. Transmission is known to decrease exponentially as the film thickness increases because no propagative modes exist in the aperture [18

18. A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12(16), 3694–3700 (2004). [CrossRef] [PubMed]

]. Calculation size was 5 μm × 5 μm × 1.7 μm, with minimum cell size of 10 nm × 10 nm × 5 nm discretized over the bull’s eye structure. We have calculated through smaller cell sizes of 5 nm × 5 nm × 5 nm, ending up with calculation results within difference of a few % of that discretized to 10 nm × 10 nm × 5 nm. This confirms that the discretization size is small enough for valid calculations.

The transmission T of the bull’s eye structure was calculated by the power component of the propagating direction, normalized to the incident power over the hole area of the aperture, expressed by the following equation.
T=ASzdAI0πd2
(1)
Here, S-z is the time-averaged Poynting vector perpendicular to the bottom surface A of the central aperture at z = 0. I0 is the power of incident light, and d is the diameter of the central aperture. Poynting vector was averaged over one period from 26.8 fs, where the transmission was in steady state.

3. Results and discussion

3.1 Designing a bull’s eye structure based on the interference of incident light and scattered far-field from the central aperture

The diameter of the subwavelength aperture was varied from 100 nm to 600 nm to find the optimal diameter for 800 nm incident light. The electric field intensity over the film surface (x = 0 nm, z = 275 nm) is shown in Fig. 2(a)
Fig. 2 (a) The electric field intensity distribution generated by the interference of incident light and scattered far-field from an aperture d in diameter. The positions of the peak intensities are the same when d is 400 nm or more in diameter. When d is 300 nm or smaller, the peak of the electric field intensity shifts away from the central aperture. (b) The transmission spectrum for d = 400 nm.
. The interference of incident light and scattered far-field generated electric field intensities on the film surface being both periodic and concentric. Maximum intensity amplitude was observed when d = 400 nm. As the diameter gets smaller for d = 300 nm, 200 nm, 100 nm, the intensity amplitude decreases, with the peaks shifting away from the central aperture. The intensity was maximum at d = 400 nm because the surface plasmons were in resonance at 400 nm. The period of the peak intensities was independent of the aperture diameter, being 800 nm. For apertures smaller than d = 400 nm, the field intensity decreases due to the cut off function of the hole. Subwavelength apertures have a cut off function, which has great effect on the transmission when incident wavelength is greater than approximately twice the diameter [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

,18

18. A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12(16), 3694–3700 (2004). [CrossRef] [PubMed]

]. In Fig. 2(b) we show the change in transmission when d = 400 nm according to the incident wavelength. The transmission exponentially decreases once the wavelength is greater than 800 nm. At d = 400 nm, the resonant wavelength of 800 nm shows the highest transmission. We have set the diameter of the central aperture to d = 400 nm for the following simulations.

The electric field intensity distribution when light is incident on a gold film surface with an aperture 400 nm in diameter is shown in Fig. 3
Fig. 3 The electric field intensity distribution generated when light was incident uniformly over the gold film surface with an aperture 400 nm in diameter in the center. The peak intensity is generated by a period of 800 nm, with the closest peak to the central aperture being 540 nm from the center.
. The peak intensity closest to the central aperture had a radius of 540 nm. The following peak intensities were generated every 800 nm from the first peak. The surface plasmon polariton wavelength on a gold film surface is determined as λspp = 784 nm from the following equation [23

23. L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85(3), 467 (2004). [CrossRef]

].
λspp=λincεm+1εm
(2)
Here, λinc is the wavelength of incident light, εm is the real part of the dielectric constant of gold. The term 1 within the square root is the dielectric constant of air. The surface plasmon polariton wavelength slightly changes from the free-space wavelength (800 nm), which is theoretically calculated to be 784 nm by Eq. (2). The period of the peak intensities shown in Fig. 3 shows little difference to the 784 nm, probably due to the minimum cell size at the discretization in the FDTD calculation.

Here, we have designed a bull’s eye structure with the center of the grooves overlapping the peaks aiming to increase transmission by efficiently propagating surface plasmon polaritons to the central aperture. The radius of the central aperture d = 400 nm, first groove radius a1 = 540 nm, and groove period p = 800 nm was set. Groove depth was set to h = 90 nm by reference to [13

13. S. Carretero-Palacios, O. Mahboub, F. J. Garcia-Vidal, L. Martin-Moreno, S. G. Rodrigo, C. Genet, and T. W. Ebbesen, “Mechanisms for extraordinary optical transmission through bull’s eye structures,” Opt. Express 19(11), 10429–10442 (2011). [CrossRef] [PubMed]

]. Later in our study, we will tune the positions of the grooves to investigate the change in transmission. We have set the groove width to g = 50 nm which is quite narrow in order to make it easier to analyze the effects of groove position on the phenomenon arising on the film surface. For the number of grooves N, a small number should be chosen to make the analysis of the structure simple, however not too small that the properties do not allow valid evaluation. N = 3 was chosen for the number of grooves applied to the following calculations, since it possesses periodicity, and the contributions of each groove should be clearly analyzed.

Figure 4(a)
Fig. 4 (a) The electric field intensity distribution of the bull’s eye structure designed according to the interfered far-field when light is incident on the surface of the structure. (a1 = 540 nm, d = 400 nm, g = 50 nm, h = 90 nm, t = 280 nm, N = 3). (b) The electric field intensity distribution of the same structure with additional grooves in between.
shows the electric field intensity of the cross section at x = 0 of the designed bull’s eye structure with the parameters we have defined. The transmission of the bull’s eye structure was 6.01 calculated using the Eq. (1). High electric field intensities were observed at the grooves, which can be attributed to the surface plasmon polaritons [13

13. S. Carretero-Palacios, O. Mahboub, F. J. Garcia-Vidal, L. Martin-Moreno, S. G. Rodrigo, C. Genet, and T. W. Ebbesen, “Mechanisms for extraordinary optical transmission through bull’s eye structures,” Opt. Express 19(11), 10429–10442 (2011). [CrossRef] [PubMed]

]. There is a slight enhancement of intensity between each of the grooves, which is due to the far-field scattering from the grooves. Considering the slightly generated enhancements, we have compared the structure designed here with a bull’s eye structure with additional grooves exactly in the middle of the two grooves. The transmission dropped to 1.66 for this structure. The electric field intensity of the cross section is shown in Fig. 4(b). The structure with additional grooves shows little charge propagation between the grooves, decreasing the transmission. The enhanced fields within the grooves are generated from the surface plasmons strongly localized within the grooves, showing greater intensities within the grooves compared to the original structure. On the other hand, we have compared the original structure to a structure with the second groove removed from the original, which allowed a transmission of 4.75. In this case, the period of the grooves is approximately twice the surface plasmon polarition wavelength, which makes it a resonant period for coupling. Transmission is fairly high, but lower than the original structure because the attenuation of the polaritions increases as the groove period is increased.

Therefore, by considering the periodically interfering electric field intensities of single-wavelength incident light and scattered far-field from the central aperture, we have succeeded in designing a bull’s eye structure showing high transmission. This designing procedure was carried out on the idea of effectively using the plasmonic far-field pattern to propagate energy in the reversing direction, which eliminated the step of trial and error for choosing the first groove radius a1 and periodicity p.

We have obtained a1 = 540 nm, and p = 800 nm through the design method we have proposed. In order to show that the determined values are fairly optimal, we have compared the transmission of the designed structure with that of the designed structure with altered first groove radius a1 and period p.

Much have been argued considering the first groove radius, some reporting that the value should be equal to incident wavelength [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

], while some report it should be a smaller value than the incident wavelength [12

12. O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. 99(4), 043902 (2007). [CrossRef] [PubMed]

,17

17. H. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004). [CrossRef] [PubMed]

]. Our results support the latter. Firstly, we have altered the first groove radius a1 of the bull’s eye structure based on the original structure, to see how it affects the transmission. The results are shown in Fig. 5(a)
Fig. 5 Dependence of the transmission on (a) first groove radius a1 with other parameters fixed to the original structure and that on (b) groove period p with other parameters fixed to the original structure. In Fig. 5(a), transmission was maximum at a1 = 590 nm (d = 400 nm, p = 800 nm, g = 50 nm, h = 90 nm, t = 280 nm, N = 3).
. The original bull’s eye structure with a1 = 540 nm showed high transmission, and enhancement was enabled by fine-tuning to a1 = 590 nm. When a1 = 590 nm, the position of the first peak intensity generated from the interference of the incident light and the scattered far-field from the central aperture nearly corresponds to the inner radius a1in. According to Fig. 4(a), the surface plasmon polaritons constructively interfere at the edge of each groove, while intensities within the grooves stay low especially at the center of the grooves. Surface plasmon polaritons originating from the inner edge of the first groove show the least attenuation on its propagation to the central aperture, compared to ones originating from the outer grooves. Therefore, it can be hypothesized that the transmission will be maximized when the energy of the surface plasmon polaritons originating from the inner edge of the first groove is also maximized. The energy of the surface plasmon polaritons originating from the inner edge of the first groove should be maximized when the incident light and scattered far field from the central aperture constructively interfere at this inner edge, which is the case when a1 = 590 nm.

For groove period p, the wavelength that shows maximum transmission is known to be slightly larger than the period p [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

,14

14. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

]. Theoretically, maximum transmission can be obtained when the incident wavelength is equal to the resonant wavelength of the surface plasmon λres at groove period p. This is, however, when groove depth is ignored, considering only the propagation of surface plasmons over the film surface, supposing that the charge displacements do not occur in the direction parallel to the z axis. Under this condition, the resonant wavelength λres is

λres=pεmε(ω)εm+ε(ω)
(3)

Here, εm is the permittivity of the metal film, and ε(ω) is the permittivity of the surrounding environment (air, n = 1.00). We will fix the first groove radius to a1 = 540 nm and tune the period p of the original structure and compare it with theoretical calculations from Eq. (3). Results are shown in Fig. 5(b). Maximum transmission was obtained at p = 780 nm. According to Eq. (3), λres = 795 nm when p = 780 nm. This is in good agreement with studies reporting that maximum transmission is achieved when groove period is chosen so that incident wavelength is approximately the same as λres [4

4. Z. Li, Z. Yang, and H. Xu, “Comment on “self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett. 97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]

,14

14. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

]. Because the value within the square root of Eq. (3) is only slightly larger than 1 when calculated, the resonant wavelength of the surface plasmons is close to the incident wavelength. Therefore high transmission can be achieved even when the period of the grooves designed in the original structure is set to match the incident wavelength of 800 nm.

The bull’s eye structure we have first designed was p = 800 nm, which was quite optimal. By tweaking the period to match the resonant wavelength of surface plasmons, a slightly higher transmission is achieved since more energy propagates from the second and third groove as surface plasmon polaritons.

From the above discussion, a bull’s eye structure with high transmission can be designed by designing the grooves so that the incident light and the scattered far-field from the central aperture constructively interfere at the inner edges of the grooves.

3.2 Comparison of the designed bull’s eye structure and its counterparts with altered parameters

Additionally, by altering the groove width and groove depth, an even higher transmission should be achieved.

Figure 7 also shows the electric field intensity of the cross section of the grooves when h is altered at g = 50 nm and g = 300 nm. At g = 50 nm, the peak of the field intensity shifts towards the bottom of the groove with increasing groove depth h. When h becomes larger than 90 nm, the charge displacement occurs at a deeper place within the grooves making the surface plasmons more difficult to couple to incident light. As a result, the standing waves generating at the edge of the grooves weaken, leading to the loss in energy of the surface plasmon polaritons propagating towards the central aperture. On the other hand, when groove depth is small for example as in h = 50 nm, the field intensity at the edges decrease. This is because the structure gets closer to a flat plane, making it difficult for the charge displacements to occur. When g = 300 nm, the intensity at the bottom of the groove changes as the groove becomes deeper. As we have discussed when altering the groove width, the mode within the groove is more excited at g = 300 nm. Field enhancements are generated at the edges of the grooves, which is the same for g = 50 nm. However, at g = 300 nm, when the depth is larger than h = 90 nm, the standing waves shows weakening due to the long distance of the displacement.

From the above analysis, the coupling of surface plasmons to incident light is weak when the groove depth is large, resulting in low transmission as in Fig. 6(b). For deeper groove depth, the effective refractive index provided by the groove depth must be taken into consideration for optimal parameters. The design method we propose in this paper is based on the field intensity generated on a flat gold film surface. Therefore the groove depths that provide little change in the effective refractive index are suitable to be used in our design method.

Finally, we have investigated the effects of polarization (linear and circular) of incident light on the transmission. To evaluate the energy focusing efficiency of the bull’s eye structure, we have placed a silicon (n = 3.692, κ = 0.006 [25

25. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

]) substrate 500 nm in thickness under the metal film of the bull’s eye structure, assuming a silicon sensor. The field intensity distributions of the top-view (Fig. 8(a)
Fig. 8 A system of a bull’s eye structure with a Si substrate right under the structure. The electric field intensity at the bottom surface of the central aperture is shown for (a) linearly and (b) circularly polarized incident light. The electric field intensity distribution of the cross section is shown for (c) linearly and (d) circularly polarized incident light (p = 780 nm, g = 300 nm, h = 90 nm, d = 400 nm, t = 280 nm, N = 3). (e) The transmitting power through the Si substrate for linearly polarized and circularly polarized light.
, 8(b)) and the cross sectional view (Fig. 8(c), 8(d)) are shown. When calculating the energy penetrating into the silicon substrate, we have discretized the volume within the silicon substrate to 10 nm × 10 nm × 20 nm, taking into consideration only the cells whose electric field intensity was larger than 1(V/m)2. When the incident light was linearly polarized parallel to the y axis, the intensity in the direction parallel to the x axis hardly generated. This is because the electric field of the incident light continuously oscillates in the y direction, thus generating standing waves efficiently in the y direction, forcing energy to propagate along the y axis towards the central aperture. When incident light was circularly polarized, the electric field intensity distribution was point-symmetric. The electric field of circularly polarized light oscillates in a direction so that it rotates within the xy plane, generating standing waves equivalently in all directions on the bull’s eye structure. The energy focused to the central aperture is thus distributed equally in all directions. As a result, the value of integrated power of transmitted light through the silicon substrate was independent of the polarization, being 4.7 × 10−15 W in both cases (Fig. 8(e)). The introduction of the silicon substrate may change the resonant wavelength for the structure. A more precise designing can be provided based on calculations of the field scattered from the central aperture with the silicon substrate attached.

Taking all of the analyses into account, we have designed a bull’s eye structure by combining each of the parameters that showed maximum transmission. As a result, we have obtained maximum transmission when a1in = 540 nm, p = 780 nm, g = 300 nm, h = 90 nm, t = 280 nm, d = 400 nm, N = 3. The transmission reached 9.74. We compared our designing method with [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

] which optimized the transmission of a bull’s eye structure of a gold film with a central aperture 300 nm in diameter. We have performed the same calculations as section 3.1 with an aperture of 300 nm in diameter. The maximum transmitting wavelength was λ = 625 nm in the optimal structure in [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

]. We have set the incident wavelength to λ = 630 nm and calculated the interference of the incident light and the scattered far-field. The peak intensity closest to the central aperture generated 405 nm from the center of the aperture. This is in good agreement with a1in = 415 nm of the structure in [10

10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

]. This agreement supports our conclusion that a bull’s eye structure with enhanced transmission can be designed by calculating the plasmonic far-field pattern and placing the inner edge of the first groove over the peak intensity closest to the aperture.

4. Conclusion

We have presented a novel method of designing a bull’s eye structure with high transmission of a single wavelength light source based on the interference of incident light and the scattered far-field from the central aperture. By simply designing the groove structure according to the intensity peaks of the interfered far-field, a bull’s eye structure with a high transmission rate can be designed. Based on the designed structure, additional analysis of the distance between the central aperture and the first groove proved that when the incident light and the scattered far-field from the central aperture constructively interfere at the inner edge of the first groove. Moreover, by setting the period of the grooves to the resonant wavelength of the surface plasmons, a higher transmission was achieved. The inner radius of the first groove and the period can easily be determined from FDTD calculations from the idea proposed above. Based on the parameters determined from this concept, by additionally tuning the groove width and depth, an even higher transmission was achieved. Our novel method is an effective way to design a bull’s eye structure envisioned for single wavelength light sources.

Acknowledgments

The authors are grateful to Prof. Minoru Obara for fruitful comments on this paper.

References and links

1.

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004). [CrossRef] [PubMed]

2.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

3.

A. J. Pasquale, B. M. Reinhard, and L. Dal Negro, “Concentric Necklace Nanolenses for Optical Near-Field Focusing and Enhancement,” ACS Nano 6(5), 4341–4348 (2012). [CrossRef] [PubMed]

4.

Z. Li, Z. Yang, and H. Xu, “Comment on “self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett. 97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]

5.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of Highly Directional Emission from a Single Subwavelength Aperture Surrounded by Surface Corrugations,” Phys. Rev. Lett. 90(16), 167401 (2003). [CrossRef] [PubMed]

6.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple Paths to Enhance Optical Transmission through a Single Subwavelength Slit,” Phys. Rev. Lett. 90(21), 213901 (2003). [CrossRef] [PubMed]

7.

C. K. Chang, D. Z. Lin, C. S. Yeh, C. K. Lee, Y. C. Chang, M. W. Lin, J. T. Yeh, and J. M. Liu, “Similarities and differences for light-induced surface plasmons in one- and two-dimensional symmetrical metallic nanostructures,” Opt. Lett. 31(15), 2341–2343 (2006). [CrossRef] [PubMed]

8.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13(3), 429–432 (2002). [CrossRef]

9.

T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si Nano-Photodiode with a Surface Plasmon Antenna,” Jpn. J. Appl. Phys. 44(12), L364–L366 (2005). [CrossRef]

10.

O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express 18(11), 11292–11299 (2010). [CrossRef] [PubMed]

11.

K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Finite-difference time-domain studies of light transmission through nanohole structures,” Appl. Phys. B 84(1-2), 11–18 (2006). [CrossRef]

12.

O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. 99(4), 043902 (2007). [CrossRef] [PubMed]

13.

S. Carretero-Palacios, O. Mahboub, F. J. Garcia-Vidal, L. Martin-Moreno, S. G. Rodrigo, C. Genet, and T. W. Ebbesen, “Mechanisms for extraordinary optical transmission through bull’s eye structures,” Opt. Express 19(11), 10429–10442 (2011). [CrossRef] [PubMed]

14.

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26(24), 1972–1974 (2001). [CrossRef] [PubMed]

15.

N. Bonod, E. Popov, D. Gérard, J. Wenger, and H. Rigneault, “Field enhancement in a circular aperture surrounded by a single channel groove,” Opt. Express 16(3), 2276–2287 (2008). [CrossRef] [PubMed]

16.

Y. Cui and S. He, “A theoretical re-examination of giant transmission of light through a metallic nano-slit surrounded with periodic grooves,” Opt. Express 17(16), 13995–14000 (2009). [CrossRef] [PubMed]

17.

H. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004). [CrossRef] [PubMed]

18.

A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12(16), 3694–3700 (2004). [CrossRef] [PubMed]

19.

A. Agrawal, H. Cao, and A. Nahata, “Time-domain analysis of enhanced transmission through a single subwavelength aperture,” Opt. Express 13(9), 3535–3542 (2005). [CrossRef] [PubMed]

20.

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500 (2003). [CrossRef]

21.

G. Obara, N. Maeda, T. Miyanishi, M. Terakawa, N. N. Nedyalkov, and M. Obara, “Plasmonic and Mie scattering control of far-field interference for regular ripple formation on various material substrates,” Opt. Express 19(20), 19093–19103 (2011). [CrossRef] [PubMed]

22.

G. Obara, Y. Tanaka, N. N. Nedyalkov, M. Terakawa, and M. Obara, “Direct observation of surface plasmon far field for regular surface ripple formation by femtosecond laser pulse irradiation of gold nanostructures on silicon substrates,” Appl. Phys. Lett. 99(6), 061106 (2011). [CrossRef]

23.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85(3), 467 (2004). [CrossRef]

24.

G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” IEEE Trans. Electromagn. Compat. EMC-23(4), 377–382 (1981). [CrossRef]

25.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(050.2770) Diffraction and gratings : Gratings
(240.6680) Optics at surfaces : Surface plasmons
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 31, 2013
Revised Manuscript: July 23, 2013
Manuscript Accepted: August 26, 2013
Published: September 4, 2013

Citation
Akira Yamada and Mitsuhiro Terakawa, "Reverse design of a bull’s eye structure based on the plasmonic far-field pattern," Opt. Express 21, 21273-21284 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21273


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References

  1. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys.120(1), 357–366 (2004). [CrossRef] [PubMed]
  2. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys.82(1), 729–787 (2010). [CrossRef]
  3. A. J. Pasquale, B. M. Reinhard, and L. Dal Negro, “Concentric Necklace Nanolenses for Optical Near-Field Focusing and Enhancement,” ACS Nano6(5), 4341–4348 (2012). [CrossRef] [PubMed]
  4. Z. Li, Z. Yang, and H. Xu, “Comment on “self-similar chain of metal nanospheres as an efficient nanolens”,” Phys. Rev. Lett.97(7), 079701, discussion 079702 (2006). [CrossRef] [PubMed]
  5. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of Highly Directional Emission from a Single Subwavelength Aperture Surrounded by Surface Corrugations,” Phys. Rev. Lett.90(16), 167401 (2003). [CrossRef] [PubMed]
  6. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple Paths to Enhance Optical Transmission through a Single Subwavelength Slit,” Phys. Rev. Lett.90(21), 213901 (2003). [CrossRef] [PubMed]
  7. C. K. Chang, D. Z. Lin, C. S. Yeh, C. K. Lee, Y. C. Chang, M. W. Lin, J. T. Yeh, and J. M. Liu, “Similarities and differences for light-induced surface plasmons in one- and two-dimensional symmetrical metallic nanostructures,” Opt. Lett.31(15), 2341–2343 (2006). [CrossRef] [PubMed]
  8. T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, and R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology13(3), 429–432 (2002). [CrossRef]
  9. T. Ishi, J. Fujikata, K. Makita, T. Baba, and K. Ohashi, “Si Nano-Photodiode with a Surface Plasmon Antenna,” Jpn. J. Appl. Phys.44(12), L364–L366 (2005). [CrossRef]
  10. O. Mahboub, S. C. Palacios, C. Genet, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and T. W. Ebbesen, “Optimization of bull’s eye structures for transmission enhancement,” Opt. Express18(11), 11292–11299 (2010). [CrossRef] [PubMed]
  11. K. L. Shuford, M. A. Ratner, S. K. Gray, and G. C. Schatz, “Finite-difference time-domain studies of light transmission through nanohole structures,” Appl. Phys. B84(1-2), 11–18 (2006). [CrossRef]
  12. O. T. A. Janssen, H. P. Urbach, and G. W. ’t Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett.99(4), 043902 (2007). [CrossRef] [PubMed]
  13. S. Carretero-Palacios, O. Mahboub, F. J. Garcia-Vidal, L. Martin-Moreno, S. G. Rodrigo, C. Genet, and T. W. Ebbesen, “Mechanisms for extraordinary optical transmission through bull’s eye structures,” Opt. Express19(11), 10429–10442 (2011). [CrossRef] [PubMed]
  14. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett.26(24), 1972–1974 (2001). [CrossRef] [PubMed]
  15. N. Bonod, E. Popov, D. Gérard, J. Wenger, and H. Rigneault, “Field enhancement in a circular aperture surrounded by a single channel groove,” Opt. Express16(3), 2276–2287 (2008). [CrossRef] [PubMed]
  16. Y. Cui and S. He, “A theoretical re-examination of giant transmission of light through a metallic nano-slit surrounded with periodic grooves,” Opt. Express17(16), 13995–14000 (2009). [CrossRef] [PubMed]
  17. H. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express12(16), 3629–3651 (2004). [CrossRef] [PubMed]
  18. A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express12(16), 3694–3700 (2004). [CrossRef] [PubMed]
  19. A. Agrawal, H. Cao, and A. Nahata, “Time-domain analysis of enhanced transmission through a single subwavelength aperture,” Opt. Express13(9), 3535–3542 (2005). [CrossRef] [PubMed]
  20. F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett.83(22), 4500 (2003). [CrossRef]
  21. G. Obara, N. Maeda, T. Miyanishi, M. Terakawa, N. N. Nedyalkov, and M. Obara, “Plasmonic and Mie scattering control of far-field interference for regular ripple formation on various material substrates,” Opt. Express19(20), 19093–19103 (2011). [CrossRef] [PubMed]
  22. G. Obara, Y. Tanaka, N. N. Nedyalkov, M. Terakawa, and M. Obara, “Direct observation of surface plasmon far field for regular surface ripple formation by femtosecond laser pulse irradiation of gold nanostructures on silicon substrates,” Appl. Phys. Lett.99(6), 061106 (2011). [CrossRef]
  23. L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett.85(3), 467 (2004). [CrossRef]
  24. G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” IEEE Trans. Electromagn. Compat.EMC-23(4), 377–382 (1981). [CrossRef]
  25. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

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