Abstract

In this article, we demonstrate that a subwavelength metal slit surrounded by dielectric surface gratings possesses a directional beaming effect. We propose a surface plasmon diffraction scheme to explain the three kinds of beaming conditions. The numerical simulations of the illustrative structures undertaken used a Finite Difference Time Domain (FDTD) Method and a Rigorous Coupled Wave Analysis (RCWA) Method. Our simulations were found to be consistent and in agreement with the experimental results. In comparison with other metal structures, we find that dielectric metal structures offer better performance as well as the advantage of being able to be efficiently mass produced for large volume industrial applications.

© 2006 Optical Society of America

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References

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  1. H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
    [CrossRef]
  2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
    [CrossRef] [PubMed]
  3. F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
    [CrossRef]
  4. L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
    [CrossRef] [PubMed]
  5. L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
    [CrossRef]
  6. J. Dintinger, A. Degiron, T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bulletin 30, 381-384 (2005).
    [CrossRef]
  7. Z. B. Li, J. G. Tian, Z. B. Liu, W. Y. Zhou, C. P. Zhang, "Enhanced light transmission through a single subwavelength aperture in layered films consisting of metal and dielectric," Opt. Express 13, 9071-9077 (2005). http://www.opticsexpress.org/abstract.cfm?id=86076
    [CrossRef] [PubMed]
  8. D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
    [PubMed]
  9. W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
    [CrossRef]
  10. S. Shinada, J. Hashizume, and F. Koyama, "Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating," Appl. Phys. Lett. 83, 836-838 (2003).
    [CrossRef]
  11. D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
    [CrossRef]

2005

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

J. Dintinger, A. Degiron, T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bulletin 30, 381-384 (2005).
[CrossRef]

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

Z. B. Li, J. G. Tian, Z. B. Liu, W. Y. Zhou, C. P. Zhang, "Enhanced light transmission through a single subwavelength aperture in layered films consisting of metal and dielectric," Opt. Express 13, 9071-9077 (2005). http://www.opticsexpress.org/abstract.cfm?id=86076
[CrossRef] [PubMed]

2004

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

2003

S. Shinada, J. Hashizume, and F. Koyama, "Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating," Appl. Phys. Lett. 83, 836-838 (2003).
[CrossRef]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

2002

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

1944

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Bethe, H. A.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Chang, Y. C.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Chen, Y. C.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Degiron, A.

J. Dintinger, A. Degiron, T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bulletin 30, 381-384 (2005).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Dintinger, J.

J. Dintinger, A. Degiron, T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bulletin 30, 381-384 (2005).
[CrossRef]

Ebbesen, T. W.

J. Dintinger, A. Degiron, T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bulletin 30, 381-384 (2005).
[CrossRef]

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Fang, N.

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

Garcia-Vidal, F. J.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Haes, A. J.

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

Hashizume, J.

S. Shinada, J. Hashizume, and F. Koyama, "Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating," Appl. Phys. Lett. 83, 836-838 (2003).
[CrossRef]

Hicks, E. M.

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

Huang, K. T.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Koyama, F.

S. Shinada, J. Hashizume, and F. Koyama, "Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating," Appl. Phys. Lett. 83, 836-838 (2003).
[CrossRef]

Lee, C. K.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

Lezec, H. J.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Li, Z. B.

Liaw, J. W.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Lin, C. L.

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

Lin, D. Z.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Liu, J. M.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Liu, Z. B.

Luo, Q.

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Martín-Moreno, L.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

Shinada, S.

S. Shinada, J. Hashizume, and F. Koyama, "Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating," Appl. Phys. Lett. 83, 836-838 (2003).
[CrossRef]

Srituravanich, W.

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

Stuart, D. A.

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

Sun, C.

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

Tian, J. G.

Van Duyne, R. P.

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

Yeh, C. S.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

Yeh, J. T.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Yonzon, C. R.

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

Yu, L. B.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

Zhang, C. P.

Zhang, X.

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

Zhou, W. Y.

Appl. Phys. Lett.

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, ‘Focusing light with subwavelength aperture flanked by surface corrugations,’ Appl. Phys. Lett.,  83, 4500-4502 (2003).
[CrossRef]

S. Shinada, J. Hashizume, and F. Koyama, "Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating," Appl. Phys. Lett. 83, 836-838 (2003).
[CrossRef]

IEE Proc. Nanobiotechnol.

D. A. Stuart, A. J. Haes, C. R. Yonzon, E. M. Hicks, and R. P. Van Duyne, "Biological applications of localised surface plasmonic phenomenae," IEE Proc. Nanobiotechnol. 152, 13-32 (2005).
[CrossRef]

MRS Bulletin

J. Dintinger, A. Degiron, T. W. Ebbesen, "Enhanced light transmission through subwavelength holes," MRS Bulletin 30, 381-384 (2005).
[CrossRef]

Nano Lett.

W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic Nanolithography," Nano Lett. 4, 1085-1088 (2004).
[CrossRef]

Opt. Express

Phys. Rev.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[CrossRef]

Phys. Rev. B

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang, J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005).
[CrossRef]

Phys. Rev. Lett.

L. Martín-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

Scanning

D. Z. Lin, L. B. Yu, C. K. Lee, C. S. Yeh, and C. L. Lin, "Simulation and fabrication of subwavelength structures for a nanometer feature enabled lens-less laser writers," Scanning 26, I73-I77 (2004).
[PubMed]

Science

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Supplementary Material (6)

» Media 1: MPG (1704 KB)     
» Media 2: MPG (1704 KB)     
» Media 3: MPG (1704 KB)     
» Media 4: MPG (1704 KB)     
» Media 5: MPG (1704 KB)     
» Media 6: MPG (1704 KB)     

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Figures (9)

Fig.1.
Fig.1.

Atomic Force Microscope (AFM) image of a metal surface structure milled by FIB.

Fig. 2.
Fig. 2.

FDTD simulation conditions for three DM structures. (period of Device 1 = 465m, Device 2 = 625nm and Device 3 = 530nm)

Fig. 3.
Fig. 3.

FDTD simulations (Sz) of (a) Device 1 (b) Device 2 (c) Device 3 and (d) close-up view of the dielectric surface structures.

Fig. 4.
Fig. 4.

(a) Picture of a prepared sample (three color band represent the three different periods of the DM structure) and AFM topography data of (b) Device 1 (P=465nm), (c) Device 2 (P=625nm), and (d) Device 3 (P=530nm).

Fig. 5.
Fig. 5.

Movies (low resolution~200KB & high resolution~1.7MB) of beaming phenomenon: (a) Device 1 [Media 1], (b) Device 2 [Media 2], (c) Device 3 [Media 3], and (d) Device 1 with filter [Media 4] (e) Device 2 with filter [Media 5], and (f) Device 3 with filter [Media 6].

Fig. 6.
Fig. 6.

Calculations of the beaming angles from experimental data of Device 2 (Fig. 5(e)). (left figure indicates the definition of the coordinates)

Fig. 7.
Fig. 7.

RCWA simulations of resonance conditions for the three devices. Using Helmholtz reciprocal theorem, the minimum reflection (i.e. maximum absorption) indicates the beaming angle.

Fig. 8.
Fig. 8.

FDTD simulations of surface plasmon propagation path for a MM structure (left) and a DM structure (right).

Fig. 9.
Fig. 9.

RCWA simulation of DM and MM structures. (Simulation parameters are detailed)

Tables (1)

Tables Icon

Table 1. Design and actual geometric parameters for the three devices.

Equations (1)

Equations on this page are rendered with MathJax. Learn more.

k sp ± m 2 π Λ = k 0 sin θ .

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