Abstract

A Au nanofin array embedded in SiO2 was designed and fabricated to achieve an achromatic half waveplate with high transmittance at visible wavelengths. On the basis of the waveguide theory of nanogaps and the Fresnel reflection theory, nanofin array is calculated to have ideal properties for an achromatic half-waveplate in the visible band from 560 to 660 nm with the transmittance of around 50%. A Au nanofin array with a height of 830 nm and a period of 400 nm was fabricated through a sidewall-deposition process and overcoating with spin on glass. The polarization microscopy results showed that both transmittance greater than 50% and retardation of 165° at broadband wavelengths ranging from 600 to 800 nm were simultaneously achieved. It was also demonstrated that retardation had little dependence on the incident angle.

© 2016 Optical Society of America

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References

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    [Crossref]

2015 (4)

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

M. Ishii, K. Iwami, and N. Umeda, “An Au nanofin array for high efficiency plasmonic optical retarders at visible wavelengths,” Appl. Phys. Lett. 106, 021115 (2015).
[Crossref]

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

A. Motogaito, Y. Morishita, H. Miyake, and K. Hiramatsu, “Extraordinary optical transmission exhibited by surface plasmon polaritons in a double-layer wire grid polarizer,” Plasmonics 10(6) 1657–1662 (2015).
[Crossref]

2014 (3)

Y. Dai, W. Ren, H. Cai, H. Ding, N. Pan, and X. Wang, “Realizing full visible spectrum metamaterial half-wave plates with patterned metal nanoarray/insulator/metal film structure,” Opt. Express 22(7), 7465–7472 (2014).
[Crossref] [PubMed]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

2013 (4)

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

Y. Zhao and A. Alù, “Tailoring the dispersion of plasmonic nanorods to realize broadband optical meta-waveplates,” Nano Lett. 13(3), 1086–1091 (2013).
[Crossref] [PubMed]

A. Pors and S. I. Bozhevolnyi, “Plasmonic metasurfaces for efficient phase control in reflection,” Opt. Express 21(22), 27438–27451 (2013).
[Crossref] [PubMed]

2012 (5)

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

2011 (4)

Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B 84, 205428 (2011).
[Crossref]

W. Kubo and S. Fujikawa, “Au double nanopillars with nanogap for plasmonic sensor,” Nano Lett. 11(1), 8–15 (2011).
[Crossref]

L. Gao, F. Lemarchand, and M. Lequime, “Comparison of different dispersion models for single layer optical thin film index determination,” Thin Solid Films 520(1), 501–509 (2011).
[Crossref]

A. Pors, M. G. Nielsen, G. Della Valle, M. Willatzen, O. Albrektsen, and S. I. Bozhevolnyi, “Plasmonic metamaterial wave retarders in reflection by orthogonally oriented detuned electrical dipoles,” Opt. Lett. 36(9), 1626–1628 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (2)

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

2007 (1)

2006 (2)

R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006).
[Crossref]

S. Fujikawa, R. Takaki, and T. Kunitake, “Fabrication of arrays of sub-20-nm silica walls via photolithography and solution-based molecular coating,” Langmuir 22(21), 9057–9061 (2006).
[Crossref] [PubMed]

2005 (1)

1965 (1)

Aieta, F.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Albrektsen, O.

Alù, A.

Y. Zhao and A. Alù, “Tailoring the dispersion of plasmonic nanorods to realize broadband optical meta-waveplates,” Nano Lett. 13(3), 1086–1091 (2013).
[Crossref] [PubMed]

Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B 84, 205428 (2011).
[Crossref]

Aoust, G.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

Awatsuji, Y.

Bai, B.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Blanchard, R.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Boyd, R. W.

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Bozhevolnyi, S. I.

Briggs, D. P.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Brolo, A. G.

Cai, H.

Capasso, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

Cheah, K.-W.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Chen, S.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Chen, W. T.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Chen, X.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Dai, Y.

De Leon, I.

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Delaunay, J.-J.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Della Valle, G.

Ding, F.

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

Ding, H.

Dong, J.

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Fujikawa, S.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

W. Kubo and S. Fujikawa, “Au double nanopillars with nanogap for plasmonic sensor,” Nano Lett. 11(1), 8–15 (2011).
[Crossref]

S. Fujikawa, R. Takaki, and T. Kunitake, “Fabrication of arrays of sub-20-nm silica walls via photolithography and solution-based molecular coating,” Langmuir 22(21), 9057–9061 (2006).
[Crossref] [PubMed]

Gaburro, Z.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

Gao, L.

L. Gao, F. Lemarchand, and M. Lequime, “Comparison of different dispersion models for single layer optical thin film index determination,” Thin Solid Films 520(1), 501–509 (2011).
[Crossref]

Genevet, P.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Giessen, H.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Gordon, R.

R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006).
[Crossref]

R. Gordon and A. G. Brolo, “Increased cut-off wavelength for a subwavelength hole in a real metal,” Opt. Express 13(6), 1933–1938 (2005).
[Crossref] [PubMed]

He, S.

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

Hiramatsu, K.

A. Motogaito, Y. Morishita, H. Miyake, and K. Hiramatsu, “Extraordinary optical transmission exhibited by surface plasmon polaritons in a double-layer wire grid polarizer,” Plasmonics 10(6) 1657–1662 (2015).
[Crossref]

Honkanen, S.

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

Hsu, S.-Y.

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

Huang, L.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Huang, Y.-W.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Hunter, W. R.

D. W. Lynch and W. R. Hunter, “Gold (Au),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).

Ida, K.

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

Ishii, M.

M. Ishii, K. Iwami, and N. Umeda, “An Au nanofin array for high efficiency plasmonic optical retarders at visible wavelengths,” Appl. Phys. Lett. 106, 021115 (2015).
[Crossref]

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

Ito, K.

Iwami, K.

M. Ishii, K. Iwami, and N. Umeda, “An Au nanofin array for high efficiency plasmonic optical retarders at visible wavelengths,” Appl. Phys. Lett. 106, 021115 (2015).
[Crossref]

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

Jin, G.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Kakue, T.

Kang, G.

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

Karimi, E.

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Kats, M. A.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

Kildishev, A. V.

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

Kobayashi, Y.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Koizumi, M.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Kravchenko, I. I.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Kubo, W.

W. Kubo and S. Fujikawa, “Au double nanopillars with nanogap for plasmonic sensor,” Nano Lett. 11(1), 8–15 (2011).
[Crossref]

Kubota, T.

Kunitake, T.

S. Fujikawa, R. Takaki, and T. Kunitake, “Fabrication of arrays of sub-20-nm silica walls via photolithography and solution-based molecular coating,” Langmuir 22(21), 9057–9061 (2006).
[Crossref] [PubMed]

Kuramochi, Y.

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Lee, K.-L.

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

Lee, M.-C.

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

Lee, Y.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Lemarchand, F.

L. Gao, F. Lemarchand, and M. Lequime, “Comparison of different dispersion models for single layer optical thin film index determination,” Thin Solid Films 520(1), 501–509 (2011).
[Crossref]

Lequime, M.

L. Gao, F. Lemarchand, and M. Lequime, “Comparison of different dispersion models for single layer optical thin film index determination,” Thin Solid Films 520(1), 501–509 (2011).
[Crossref]

Li, J.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Lin, E.-H.

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

Lin, J.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Lynch, D. W.

D. W. Lynch and W. R. Hunter, “Gold (Au),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).

Maeda, E.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Malitson, I. H.

Matoba, O.

Miyake, H.

A. Motogaito, Y. Morishita, H. Miyake, and K. Hiramatsu, “Extraordinary optical transmission exhibited by surface plasmon polaritons in a double-layer wire grid polarizer,” Plasmonics 10(6) 1657–1662 (2015).
[Crossref]

Moitra, P.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Morishita, Y.

A. Motogaito, Y. Morishita, H. Miyake, and K. Hiramatsu, “Extraordinary optical transmission exhibited by surface plasmon polaritons in a double-layer wire grid polarizer,” Plasmonics 10(6) 1657–1662 (2015).
[Crossref]

Moritani, Y.

Motogaito, A.

A. Motogaito, Y. Morishita, H. Miyake, and K. Hiramatsu, “Extraordinary optical transmission exhibited by surface plasmon polaritons in a double-layer wire grid polarizer,” Plasmonics 10(6) 1657–1662 (2015).
[Crossref]

Mühlenbernd, H.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Nielsen, M. G.

Nishio, K.

Pan, N.

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Pors, A.

Qassim, H.

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Qiu, C.-W.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Rahomäki, J.

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

Ren, W.

Schulz, S. a.

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Shalaev, V. M.

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

Shimozato, Y.

Søndergaard, T.

Sun, G.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Taino, A.

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Takaki, R.

S. Fujikawa, R. Takaki, and T. Kunitake, “Fabrication of arrays of sub-20-nm silica walls via photolithography and solution-based molecular coating,” Langmuir 22(21), 9057–9061 (2006).
[Crossref] [PubMed]

Tan, Q.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Tsai, D. P.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Tsai, W.-Y.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Turunen, J.

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

Umeda, N.

M. Ishii, K. Iwami, and N. Umeda, “An Au nanofin array for high efficiency plasmonic optical retarders at visible wavelengths,” Appl. Phys. Lett. 106, 021115 (2015).
[Crossref]

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

Upham, J.

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Ura, S.

Valentine, J.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Wang, C.-M.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Wang, W.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Wang, X.

Wang, Z.

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

Wei, P.-K.

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

Weiss, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Willatzen, M.

Wu, P. C.

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Yang, Y.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Yu, N.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Zentgraf, T.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Zhang, H.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Zhang, S.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Zhao, Y.

Y. Zhao and A. Alù, “Tailoring the dispersion of plasmonic nanorods to realize broadband optical meta-waveplates,” Nano Lett. 13(3), 1086–1091 (2013).
[Crossref] [PubMed]

Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B 84, 205428 (2011).
[Crossref]

ACS Nano (1)

F. Ding, Z. Wang, S. He, V. M. Shalaev, and A. V. Kildishev, “Broadband high-efficiency half-wave plate: A supercell-based plasmonic metasurface approach,” ACS Nano 9(4), 4111–4119 (2015).
[Crossref] [PubMed]

Appl. Phys. Lett. (5)

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

K. Iwami, M. Ishii, Y. Kuramochi, K. Ida, and N. Umeda, “Ultrasmall radial polarizer array based on patterned plasmonic nanoslits,” Appl. Phys. Lett. 101, 161119 (2012).
[Crossref]

S.-Y. Hsu, K.-L. Lee, E.-H. Lin, M.-C. Lee, and P.-K. Wei, “Giant birefringence induced by plasmonic nanoslit arrays,” Appl. Phys. Lett. 95, 013105 (2009).
[Crossref]

M. Ishii, K. Iwami, and N. Umeda, “An Au nanofin array for high efficiency plasmonic optical retarders at visible wavelengths,” Appl. Phys. Lett. 106, 021115 (2015).
[Crossref]

G. Kang, J. Rahomäki, J. Dong, S. Honkanen, and J. Turunen, “Enhanced deep ultraviolet inverse polarization transmission through hybrid Al-SiO2 gratings,” Appl. Phys. Lett. 103, 131110 (2013).
[Crossref]

J. Opt. Soc. Am. (1)

Langmuir (1)

S. Fujikawa, R. Takaki, and T. Kunitake, “Fabrication of arrays of sub-20-nm silica walls via photolithography and solution-based molecular coating,” Langmuir 22(21), 9057–9061 (2006).
[Crossref] [PubMed]

Light: Science & Applications (1)

E. Karimi, S. a. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Science & Applications 3, e167 (2014).
[Crossref]

Nano Lett. (5)

Y. Zhao and A. Alù, “Tailoring the dispersion of plasmonic nanorods to realize broadband optical meta-waveplates,” Nano Lett. 13(3), 1086–1091 (2013).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. a. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

W. Kubo and S. Fujikawa, “Au double nanopillars with nanogap for plasmonic sensor,” Nano Lett. 11(1), 8–15 (2011).
[Crossref]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

Y.-W. Huang, W. T. Chen, W.-Y. Tsai, P. C. Wu, C.-M. Wang, G. Sun, and D. P. Tsai, “Aluminum Plasmonic Multicolor Meta-Hologram,” Nano Lett. 15(5), 3122–3127 (2015).
[Crossref] [PubMed]

Nanotechnology (1)

E. Maeda, Y. Lee, Y. Kobayashi, A. Taino, M. Koizumi, S. Fujikawa, and J.-J. Delaunay, “Sensitivity to refractive index of high-aspect-ratio nanofins with optical vortex,” Nanotechnology 23, 505502 (2012).
[Crossref] [PubMed]

Nature (London) Commun. (1)

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nature (London) Commun. 4, 2808 (2013).

Nature Mater. (1)

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater. 8(9), 758–762 (2009).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. B (2)

R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006).
[Crossref]

Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B 84, 205428 (2011).
[Crossref]

Plasmonics (1)

A. Motogaito, Y. Morishita, H. Miyake, and K. Hiramatsu, “Extraordinary optical transmission exhibited by surface plasmon polaritons in a double-layer wire grid polarizer,” Plasmonics 10(6) 1657–1662 (2015).
[Crossref]

Proc. Nat. Acad. Sci. (1)

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Nat. Acad. Sci. 109(31), 12364–12368 (2012).
[Crossref]

Thin Solid Films (1)

L. Gao, F. Lemarchand, and M. Lequime, “Comparison of different dispersion models for single layer optical thin film index determination,” Thin Solid Films 520(1), 501–509 (2011).
[Crossref]

Other (1)

D. W. Lynch and W. R. Hunter, “Gold (Au),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985).

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Figures (5)

Fig. 1
Fig. 1

(a) Schemtaic of the conventional free-standing Au nanofin array. Tall, thin gold walls (fins) are arrayed along the x-axis. TM and TE polarizations are defined perpendicular and parallel to the nanofins, respectively. (b) Schemtaic of the Au nanofin array embedded in a dielectric medium. Slits between the nanofins and the top are filled with a dielectric medium with permittivity identical to that of the substrate.

Fig. 2
Fig. 2

Optical characteristics of the Au nanofin array. In the calculation, slit width (w), height (h), and period (p) are set to 310, 800, and 400 nm, respectively. (a) Real part of the calculated refractive index. (b) Imaginary part of the calculated refractive index. (c) Calculated retardation induced by the Au nanofin array. Solid lines show free-standing nanofin array and dashed lines show the results of SiO2-embedded one. (d) Calculated transmittances through Eq. (4). Solid lines show the resultant free-standing nanofin array and dashed lines show the results of the glass-embedded one. Blue and red lines indicate the TE and TM polarizations, respectively.

Fig. 3
Fig. 3

Schematic drawing of the fabrication process. (a) Cr patterning with EB lithography and a lift-off process. (b) Silica glass RIE. (c) Cr mask removing. (d) Au sputtering. (e) Milling of the top and bottom of a Au film by Ar RIE. (f) Spin-on-glass coating and annealing. (g) Cross-sectional SEM image of the fabricated Au nanofin array. (h) Surface topography of the sample measured using AFM.

Fig. 4
Fig. 4

(a) Schematic of the measurement setup based on polarization microscopy. (b) Theoretical and measured transmittance spectra. Red and blue lines show the transmittances under TM and TE polarization, respectively. Solid and dashed lines correspond to the measured and theoretical values derived from Eq. (4). Thick and thin lines show the averages and standard deviations of measurement(n = 5). (c) Transmittance as the function of the analyzer angle θ with the constant polarizer angle ψ = 90° at a wavelength of 633 nm. Black and gray plots correspond to transmittances measured with and without the sample and red line shows the direction of the sample. (d) Retardation spectra. The thick solid line shows the measured retardation, whereas the thin lines show the standard deviations(n = 5). The dashed line shows the theoretically calculated retardation.

Fig. 5
Fig. 5

Measured spectra of transmittance and retardation under various incident angles. Transmittance spectra when the substrate was tilted around the x- (a) and y-axes (b). Retardation spectra when the substrate was tilted around the x- (c) and y-axes (d).

Equations (6)

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tan ( k 0 2 ε d k TE 2 w / 2 ) = k TE 2 k 0 2 ε m k 0 2 ε d k TE 2 ,
tanh ( k TM 2 k 0 2 w / 2 ) = k TM 2 k 0 2 ε m ε m k TM 2 k 0 2 .
t 321 = A TE , TM t 32 t 21 exp ( 2 i π N 2 h 2 / λ ) 1 r 23 r 21 exp ( 4 i π N 2 h 2 / λ )
t 3210 = t 321 t 10 1 r 10 r 12 exp ( 4 i π n 1 h 1 / λ ) ,
S = LP θ X 45 ° LP 90 ° S unp
S 0 = p 1 2 + p 2 2 2 p 1 p 2 cos Δ cos 2 θ + ( p 1 2 p 2 2 ) sin 2 θ 4 ,

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