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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13263–13268
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Radiation pattern of plasmonic nano-antennas in a homogeneous medium

Takafumi Sugita, Kaori Yanazawa, Satoshi Maeda, Holger F. Hofmann, and Yutaka Kadoya  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13263-13268 (2014)
http://dx.doi.org/10.1364/OE.22.013263


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Abstract

Radiation patterns from plasmonic nano-antennas formed on a glass substrate were investigated using index-matching oils. It was confirmed that the pattern from single nano-antennas for various cases of index-mismatching between the substrate and the oil is explained well by the patterns of infinitesimal electric dipoles. We found that for an angular resolution of 2°, the index mismatch must be smaller than 0.001 to realize isotropic radiation. By using the appropriate condition, the radiation patterns of nano Yagi-Uda antennas in a quasi-homogeneous medium were obtained experimentally.

© 2014 Optical Society of America

1. Introduction

Controlling the efficiency and directivity of fluorescence from nano-emitters is a key step towards nano-photonic integrated circuits, quantum devices, as well as the sensitive detection and assay of chemical compounds. As a promising approach, plasmonic nano-particles serving as antenna structures have recently been intensively explored, either as single elements [1

1. J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95(1), 017402 (2005). [CrossRef] [PubMed]

6

6. I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014). [CrossRef] [PubMed]

], or in arrays such as Yagi-Uda antennas [7

7. H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9(7), 217 (2007). [CrossRef]

19

19. L. Tong, T. Pakizeh, L. Feuz, and A. Dmitriev, “Highly directional bottom-up 3D nanoantenna for visible light,” Sci. Rep. 3, 2311 (2013). [CrossRef] [PubMed]

].

In this work, the effect of index mismatch on the radiation pattern of metal nano-antenna was investigated experimentally using a liquid immersion method. We confirmed that the observed radiation pattern of a single element nano-antenna is described quite well by that of an infinitesimal dipole and found that the index mismatch must be smaller than 0.001 to realize a nearly isotropic radiation. Using the quasi-homogeneous medium, the radiation patterns of Yagi-Uda antennas free from the index mismatch were successfully demonstrated. With the present method, the mismatch can be controlled after device fabrication, significantly shortening the time needed for the experiment.

2. Devices and measurement setup

The antennas consisting of evaporated 50 nm-thick Au with adhesive 2 nm-thick Ti were fabricated on a glass substrate (Matsunami Glass, S1111) by electron-beam lithography and lift-off. As an example of the antenna used in this work, a scanning electron microscope (SEM) image of a 5-element (feed + reflector + 3 directors) Yagi-Uda antenna is shown in Fig. 1(a)
Fig. 1 (a) The SEM image of the 5-element Yagi-Uda antenna. (b) The layout of the antenna array for the measurement of the radiation patterns. (c) The extinction spectra of the elements. The shaded area indicates the range out of the CCD detector. (d) The scattering spectra obtained by the FDTD simulation.
. The length of the reflector, feed, and director elements are about 200, 140, and 100 nm, respectively and the widths of all the elements are about 60 nm. They are arranged with a 150 nm distance between them. The feed element is tilted by 45° for selective excitation [10

10. T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4(5), 312–315 (2010). [CrossRef]

]. For the measurement of the radiation patterns, the antennas were one-dimensionally arrayed in the z-direction perpendicular to the antenna axis with a pitch of 1150 nm, as shown Fig. 1(b).

The resonant wavelength of each element was evaluated from the transmission spectra of the elements arrayed in a square with a pitch of 800 nm. In the measurement, the samples were immersed in an oil with a refractive index (noil = 1.52) close to that of the substrate (nsub = 1.523). Figure 1(c) shows the measured extinction spectra. Unfortunately, the resonance peak of the reflector (green symbols) could not be observed because it would appear in the spectral range out of our CCD detector. Depicted in Fig. 1(d) are the corresponding spectra obtained by a finite difference time domain (FDTD) simulation, where the single element is placed in a homogeneous medium of refractive index n = 1.52. For the permittivity of Au, the values tabulated in [23

23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] were used. The resonance of the feed (blue symbols) and director (red symbols) elements are consistent with the FDTD simulation.

The setup for the radiation pattern measurement is illustrated in Fig. 2(a)
Fig. 2 (a) The schematic diagram of the measurement setup. (b) The structure of the detection unit. (c) The CCD images taken at various angle of detection during the alignment. The scattering from the longer (120 μm) edges of the rectangular test pattern is seen as the pair of bright rods.
. The antennas formed on the glass substrate were immersed in an index-matching oil (Shimadzu device corp.) filled in a hyper-hemicylindrical glass vessel and illuminated by a Ti:Sapphire laser beam at 850 nm through an objective lens of NA = 0.26. The incidence angle of the excitation beam was adjusted for the wavelength and the period of the one-dimensional antenna array toresult in an interference maximum in the direction normal to the substrate. The polarization of the excitation beam was set to be parallel to the x-axis (perpendicular to the major axis of the passive elements) to excite only the tilted feed element. The detection unit was made of a pair of lenses and a pinhole-mounted Si photodiode (PD) in a confocal geometry as shown in Fig. 2(b). The full angle of the detection set by the iris is 2°. In the unit, the polarizer selects the polarization along the z-axis (parallel to the major axis of the passive elements) emitted by the Yagi-Uda antenna. The center of the detector rotation was carefully aligned to the position of the antenna array, by observing the image of a test pattern (120 μm x 50 μm rectangle) on the pinhole plane using a CCD camera as shown in Fig. 2(c). The circles in the images (pinhole of 150 μm in diameter) correspond to a circular region of 750 μm in diameter on the sample. As seen in the images, the position of the test pattern moves slightly to right or left when the detector is rotated, indicating a presence of a small but finite misalignment between the centers of the rotation and the test pattern. The maximum movement of ~50 μm corresponds to an error of 0.1° in the angle which is sufficiently smaller than the angular resolution of 2° achieved in the measurement.

The refractive index noil of the oil is specified at the D-line (587.6 nm) while the index nsub = 1.524 of the glass substrate is given for the E-line (546.1 nm) by the suppliers. Since the index at the working wavelength (850 nm) could be different depending on the temperature, we did not find the index-matching condition from these values but judged it by the radiation pattern from the single dipoles as described below. In the following, we designate noil by the nominal values.

3. Results and discussion

3.1 Radiation from single dipoles

First, we measured the radiation patterns from the single 140 nm-long elements (feed-only antenna) for various noil. The results are shown by the red symbols in Figs. 3(a)
Fig. 3 Radiation patterns from single dipoles for various value of noil; (a) noil = 1.510, (b) noil = 1.530, (c) noil = 1.523, (d) noil = 1.522, (e) noil = 1.524, and (f) theoretical prediction of radiation pattern for small index mismatch. The intensity was normalized by the value at 90° in each plot. The angle range 0°<θ <180° (θ <0° and θ >180°) corresponds to the oil (substrate) side. In (a)-(c), the red symbols and the blue lines are the experimental results and the theoretical prediction, respectively. The shaded area indicates the range where the detection is not possible in our setup. In the simulation, the angle resolution of 2° was taken into account.
3(e). The detected intensity was normalized by the value at 90° for each case. The range 0°<θ <180° (θ <0° and θ >180°) corresponds to the oil (substrate) side and the shaded area indicates the range where the detection is not possible in our setup. In the case where the index of the oil is noil = 1.510 [Fig. 3(a)], one can see sharp peaks in the substrate side, while they appear in the oil side for noil = 1.530 [Fig. 3(b)], indicating that the index of the substrate is between the two values. The blue lines in the figures are the theoretical predictions calculated using a reciprocity theorem [24

24. D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30(4), 535–540 (1982). [CrossRef]

]. In the prediction, since the element is assumed to be an infinitesimal electric dipole, the particle size and the tilt of the element were not taken into account. One can see a good correspondence between the experimental and theoretical radiation patterns. In particular, the sharp peak in the case of noil = 1.530 is almost perfectly reproduced. In addition, in both cases of noil = 1.510 and noil = 1.530, the measured radiation intensity is nearly zero at 0° and 180° (along the interface) as predicted, indicating that the element can be modeled well by an infinitesimal dipole. The results also confirm that the antenna is adequately set at the centers of the vessel and the detector rotation. The disagreement between the experiment and the theory in the substrate side comes from the scattering by the roughness at the substrate edge, which results in an unpredictable effect when the index is not matched. To find an oil for which noil is matched to nsub, we mixed the oils of different indices. The result for noil = 1.523 is shown in Fig. 3(c), where the radiation is nearly isotropic with neither an apparent peak nor the suppression of emission at 0° and 180°, indicating that the indices of the oil and the substrate are matched very well.

In order to know how small the index mismatch should be for the realization of a quasi-homogeneous environment in our setup, we measured the radiation pattern with small mismatches between the substrate and the oil. The results are shown in Figs. 3(d) and 3(e). From the result, a difference in the index of 0.001 can affect the radiation pattern visibly, which is consistent with the theoretical prediction shown in Fig. 3(f).

3.2 Radiation from Yagi-Uda antennas

Shown in Fig. 4(a)
Fig. 4 Radiation patterns of the Yagi-Uda antennas; (a) experimental result with noil = 1.523, (b) FDTD simulation for homogeneous medium, (c) experimental result with noil = 1.521, (d) magnified plot of (c) only in the rage 0°<θ <180° for clarity. The green, blue and red symbols or lines represent the results in the feed-only, 2-element (feed + reflector) and 5-element (feed + reflector + 3 directors) antennas, respectively.
are the radiation patterns of the feed-only, 2-element (feed + reflector), and 5-element (feed + reflector + 3 directors) Yagi-Uda antennas depicted by green, blue, and red symbols, respectively. The measurement was carried out using the mixed oil of noil = 1.523 which represents a quasi-homogeneous medium as shown in Fig. 3(c). Here the excitation power measured at the input of the objective lens was 8 mW and the detected intensity was not normalized. Except for the disturbance around 0°, which could stem from a contamination on the substrate or scratches on the vessel, one can find that the radiation is well directed along the antenna axis (0°) in the 5-element antenna. Also, the suppression of the backward radiation in both the 2- and 5-element antennas is clearly observed. The radiation patterns predicted by the corresponding FDTD simulation are shown in Fig. 4(b). One can notice that the experimental results are reproduced well by the simulation. Therefore, we can conclude that the radiation patterns of the Yagi-Uda antenna in a quasi-homogeneous medium were successfully measured.

By comparison with the feed-only antenna, the directional gain is evaluated experimentally to be about 2.2 and 4.7 in the 2- and 5-element antennas, respectively. This is qualitatively consistent but somewhat larger than the gain values obtained in the simulation, which are 1.5and 3.4 in the 2- and 5-element antennas, respectively. The difference could be due to weaker radiation from the feed-only antennas caused by a lower yield of the fabrication. On the other hand, the half angle at half the maximum of the radiation in the experiment (simulation) was 104° (125°) and 38° (35°) in the 2- and 5-element antennas, respectively, showing a better consistency. This evaluation of the antenna properties would not be possible without the realization of the homogeneous environment.

Finally, we measured the radiation pattern of the 5-element antenna with an index-mismatched oil (noil = 1.521 < nsub = 1.523) for comparison. The result is presented in Figs. 4(c) and 4(d). Note that even such a small mismatch (0.002) results in a visible disturbance of the radiation pattern, similarly to the result of the feed-only antenna [Fig. 4(e)]. Nevertheless, in the oil (low-index) side, one can see a radiation pattern similar to that observed in the index-matched case except in the region close 0° and 180°, as previously reported in [10

10. T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4(5), 312–315 (2010). [CrossRef]

].

4. Conclusion

In this paper, we presented the experimental results on the radiation pattern from plasmonic nano-antennas performed using a liquid immersion method to control the refractive index of the medium covering the glass substrate. It was confirmed that the effect of the index mismatch on the radiation pattern is well represented by that of infinitesimal electric dipoles. Experimentally, the index mismatch should be less than 0.001 to realize a quasi-homogeneous medium. With this index-matched condition, the radiation patterns of Yagi-Uda antennas were obtained experimentally, including the emission along the antenna axis.

Acknowledgment

This work was supported by JSPS KAKENHI Grant Number 23310095.

References and links

1.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95(1), 017402 (2005). [CrossRef] [PubMed]

2.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule Fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef] [PubMed]

3.

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7(9), 2871–2875 (2007). [CrossRef] [PubMed]

4.

R. M. Bakker, H.-K. Yuan, Z. Liu, V. P. Drachev, A. V. Kildishev, V. M. Shalaev, R. H. Pedersen, S. Gresillon, and A. Boltasseva, “Enhanced localized fluorescence in plasmonic nanoantennae,” Appl. Phys. Lett. 92(4), 043101 (2008). [CrossRef]

5.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008). [CrossRef]

6.

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014). [CrossRef] [PubMed]

7.

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9(7), 217 (2007). [CrossRef]

8.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B 76(24), 245403 (2007). [CrossRef]

9.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Express 16(14), 10858–10866 (2008). [CrossRef] [PubMed]

10.

T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4(5), 312–315 (2010). [CrossRef]

11.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010). [CrossRef] [PubMed]

12.

A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4(6), 3390–3396 (2010). [CrossRef] [PubMed]

13.

T. Coenen, E. J. R. Vesseur, A. Polman, and A. F. Koenderink, “Directional emission from plasmonic Yagi-Uda antennas probed by angle-resolved cathodoluminescence spectroscopy,” Nano Lett. 11(9), 3779–3784 (2011). [CrossRef] [PubMed]

14.

D. Dregely, R. Taubert, J. Dorfmüller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat. Commun. 2, 267 (2011). [CrossRef] [PubMed]

15.

I. S. Maksymov, A. R. Davoyan, and Y. S. Kivshar, “Enhanced emission and light control with tapered plasmonic nanoantennas,” Appl. Phys. Lett. 99(8), 083304 (2011). [CrossRef]

16.

T. Shegai, P. Johansson, C. Langhammer, and M. Käll, “Directional scattering and hydrogen sensing by bimetallic Pd-Au nanoantennas,” Nano Lett. 12(5), 2464–2469 (2012). [CrossRef] [PubMed]

17.

W. Zhu, D. Wang, and K. B. Crozier, “Direct observation of beamed raman scattering,” Nano Lett. 12(12), 6235–6243 (2012). [CrossRef] [PubMed]

18.

F. Bernal Arango, A. Kwadrin, and A. F. Koenderink, “Plasmonic antennas hybridized with dielectric waveguides,” ACS Nano 6(11), 10156–10167 (2012). [CrossRef] [PubMed]

19.

L. Tong, T. Pakizeh, L. Feuz, and A. Dmitriev, “Highly directional bottom-up 3D nanoantenna for visible light,” Sci. Rep. 3, 2311 (2013). [CrossRef] [PubMed]

20.

W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69(11), 1495–1503 (1979). [CrossRef]

21.

B. Auguié, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010). [CrossRef]

22.

A. G. Nikitin, T. Nguyen, and H. Dallaporta, “Narrow plasmon resonances in diffractive arrays of gold nanoparticles in asymmetric environment: Experimental studies,” Appl. Phys. Lett. 102(22), 221116 (2013). [CrossRef]

23.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

24.

D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30(4), 535–540 (1982). [CrossRef]

OCIS Codes
(240.0240) Optics at surfaces : Optics at surfaces
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Plasmonics

History
Original Manuscript: February 12, 2014
Revised Manuscript: May 10, 2014
Manuscript Accepted: May 15, 2014
Published: May 27, 2014

Citation
Takafumi Sugita, Kaori Yanazawa, Satoshi Maeda, Holger F. Hofmann, and Yutaka Kadoya, "Radiation pattern of plasmonic nano-antennas in a homogeneous medium," Opt. Express 22, 13263-13268 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13263


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References

  1. J. N. Farahani, D. W. Pohl, H.-J. Eisler, B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95(1), 017402 (2005). [CrossRef] [PubMed]
  2. S. Kühn, U. Håkanson, L. Rogobete, V. Sandoghdar, “Enhancement of single-molecule Fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef] [PubMed]
  3. O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7(9), 2871–2875 (2007). [CrossRef] [PubMed]
  4. R. M. Bakker, H.-K. Yuan, Z. Liu, V. P. Drachev, A. V. Kildishev, V. M. Shalaev, R. H. Pedersen, S. Gresillon, A. Boltasseva, “Enhanced localized fluorescence in plasmonic nanoantennae,” Appl. Phys. Lett. 92(4), 043101 (2008). [CrossRef]
  5. T. H. Taminiau, F. D. Stefani, F. B. Segerink, N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008). [CrossRef]
  6. I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014). [CrossRef] [PubMed]
  7. H. F. Hofmann, T. Kosako, Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9(7), 217 (2007). [CrossRef]
  8. J. Li, A. Salandrino, N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B 76(24), 245403 (2007). [CrossRef]
  9. T. H. Taminiau, F. D. Stefani, N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Express 16(14), 10858–10866 (2008). [CrossRef] [PubMed]
  10. T. Kosako, Y. Kadoya, H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4(5), 312–315 (2010). [CrossRef]
  11. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010). [CrossRef] [PubMed]
  12. A. Devilez, B. Stout, N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4(6), 3390–3396 (2010). [CrossRef] [PubMed]
  13. T. Coenen, E. J. R. Vesseur, A. Polman, A. F. Koenderink, “Directional emission from plasmonic Yagi-Uda antennas probed by angle-resolved cathodoluminescence spectroscopy,” Nano Lett. 11(9), 3779–3784 (2011). [CrossRef] [PubMed]
  14. D. Dregely, R. Taubert, J. Dorfmüller, R. Vogelgesang, K. Kern, H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat. Commun. 2, 267 (2011). [CrossRef] [PubMed]
  15. I. S. Maksymov, A. R. Davoyan, Y. S. Kivshar, “Enhanced emission and light control with tapered plasmonic nanoantennas,” Appl. Phys. Lett. 99(8), 083304 (2011). [CrossRef]
  16. T. Shegai, P. Johansson, C. Langhammer, M. Käll, “Directional scattering and hydrogen sensing by bimetallic Pd-Au nanoantennas,” Nano Lett. 12(5), 2464–2469 (2012). [CrossRef] [PubMed]
  17. W. Zhu, D. Wang, K. B. Crozier, “Direct observation of beamed raman scattering,” Nano Lett. 12(12), 6235–6243 (2012). [CrossRef] [PubMed]
  18. F. Bernal Arango, A. Kwadrin, A. F. Koenderink, “Plasmonic antennas hybridized with dielectric waveguides,” ACS Nano 6(11), 10156–10167 (2012). [CrossRef] [PubMed]
  19. L. Tong, T. Pakizeh, L. Feuz, A. Dmitriev, “Highly directional bottom-up 3D nanoantenna for visible light,” Sci. Rep. 3, 2311 (2013). [CrossRef] [PubMed]
  20. W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69(11), 1495–1503 (1979). [CrossRef]
  21. B. Auguié, X. M. Bendaña, W. L. Barnes, F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010). [CrossRef]
  22. A. G. Nikitin, T. Nguyen, H. Dallaporta, “Narrow plasmon resonances in diffractive arrays of gold nanoparticles in asymmetric environment: Experimental studies,” Appl. Phys. Lett. 102(22), 221116 (2013). [CrossRef]
  23. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
  24. D. B. Rutledge, M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30(4), 535–540 (1982). [CrossRef]

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