Abstract

Optical properties of two-dimensional (2D) high-contrast gratings are investigated. We analyze the mechanisms for high-contrast gratings to function as various high-performance optical components. Our top-down design procedure allows us to efficiently obtain initial structural parameters and engineer them for a wide range of applications, such as reflectors, filters, resonators, waveplates, and even 2D phase plates. Simulation results of our designed structures show ultra-high power efficiency, and excellent agreement with our predicted functionalities.

© 2015 Optical Society of America

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    [Crossref]
  5. B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
    [Crossref]
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    [Crossref]
  7. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
    [Crossref]
  8. Y. Zhou, M. Moewe, J. Kern, M. C. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2015 (1)

2014 (4)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

A. Maurel, S. Felix, J.-F. Mercier, A. Ourir, and Z. E. Djeffal, “Wood’s anomalies for arrays of dielectric scatterers,” J. Europ. Opt. Soc. Rap. Public. 9, 14001 (2014).
[Crossref]

2012 (2)

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

2010 (2)

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18, 16973–16988 (2010).
[Crossref] [PubMed]

2008 (1)

2007 (1)

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

2005 (1)

2004 (1)

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

1999 (1)

1989 (1)

1981 (1)

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

1965 (1)

1954 (1)

1952 (1)

1941 (1)

1907 (1)

L. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60 (1907)
[Crossref]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. 4, 396–402 (1902).
[Crossref]

Beausoleil, R. G.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Berggren, J.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge University Press, 1999).
[Crossref]

Brodbeck, S.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

Chang-Hasnain, C. J.

J. Ferrara, W. Yang, L. Zhu, P. Qiao, and C. J. Chang-Hasnain, “Heterogeneously integrated long-wavelength VCSEL using silicon high contrast grating on an SOI substrate,” Opt. Express 23, 2512–2523 (2015).
[Crossref] [PubMed]

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18, 16973–16988 (2010).
[Crossref] [PubMed]

Y. Zhou, M. Moewe, J. Kern, M. C. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008).
[Crossref] [PubMed]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

Chuwongin, S.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Deng, H.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Deng, Y.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

Djeffal, Z. E.

A. Maurel, S. Felix, J.-F. Mercier, A. Ourir, and Z. E. Djeffal, “Wood’s anomalies for arrays of dielectric scatterers,” J. Europ. Opt. Soc. Rap. Public. 9, 14001 (2014).
[Crossref]

Fano, U.

Fattal, D.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Felix, S.

A. Maurel, S. Felix, J.-F. Mercier, A. Ourir, and Z. E. Djeffal, “Wood’s anomalies for arrays of dielectric scatterers,” J. Europ. Opt. Soc. Rap. Public. 9, 14001 (2014).
[Crossref]

Ferrara, J.

Fiorentino, M.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Friesem, A. A.

Gaylord, T. K.

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Golub, M. A.

Hammar, M.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Hessel, A.

Höfling, S.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Huang, M. C.

Huang, M. C. Y.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

Indebetouw, G.

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Kamp, M.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Karagodsky, V.

Kern, J.

Li, J.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Ma, Z.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Mait, J. N.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Mateus, C. F. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

Maurel, A.

A. Maurel, S. Felix, J.-F. Mercier, A. Ourir, and Z. E. Djeffal, “Wood’s anomalies for arrays of dielectric scatterers,” J. Europ. Opt. Soc. Rap. Public. 9, 14001 (2014).
[Crossref]

Mcleod, J. H.

Mercier, J.-F.

A. Maurel, S. Felix, J.-F. Mercier, A. Ourir, and Z. E. Djeffal, “Wood’s anomalies for arrays of dielectric scatterers,” J. Europ. Opt. Soc. Rap. Public. 9, 14001 (2014).
[Crossref]

Mirotznik, M. S.

Moewe, M.

Moharam, M. G.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Neureuther, A. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

Oliner, A. A.

Ourir, A.

A. Maurel, S. Felix, J.-F. Mercier, A. Ourir, and Z. E. Djeffal, “Wood’s anomalies for arrays of dielectric scatterers,” J. Europ. Opt. Soc. Rap. Public. 9, 14001 (2014).
[Crossref]

Palmer, C. H.

Peng, Z.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466–470 (2010).
[Crossref]

Prather, D. W.

Qiao, P.

Rayleigh, L.

L. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60 (1907)
[Crossref]

Schneider, C.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Sedgwick, F. G.

Seo, J.-H.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Shuai, Y.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Sorin, W. V.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Tran, T.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

Vo, S.

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

Wang, Z.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge University Press, 1999).
[Crossref]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylos. Mag. 4, 396–402 (1902).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yang, H.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Yang, W.

J. Ferrara, W. Yang, L. Zhu, P. Qiao, and C. J. Chang-Hasnain, “Heterogeneously integrated long-wavelength VCSEL using silicon high contrast grating on an SOI substrate,” Opt. Express 23, 2512–2523 (2015).
[Crossref] [PubMed]

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

Zhang, B.

B. Zhang, Z. Wang, S. Brodbeck, C. Schneider, M. Kamp, S. Höfling, and H. Deng, “Zero-dimensional polariton laser in a subwavelength grating-based vertical microcavity,” Light Sci. Appl. 3, e135 (2014).
[Crossref]

Zhao, D.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Zhou, W.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, and W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6, 615–620 (2012).
[Crossref]

Zhou, Y.

Y. Zhou, M. Moewe, J. Kern, M. C. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16, 17282–17287 (2008).
[Crossref] [PubMed]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007).
[Crossref]

Zhu, L.

Adv. Opt. Photonics (1)

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

IEEE Photonics Technol. Lett. (2)

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004).
[Crossref]

S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26, 1375–1378 (2014).
[Crossref]

J. Europ. Opt. Soc. Rap. Public. (1)

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Figures (15)

Fig. 1
Fig. 1

(a) Schematic of a 2D high-contrast grating on a rectangular lattice with a substrate. The plane-wave incident polar and azimuthal angles are θ and ϕ, respectively. Parameters: Λ x and Λ y (periods in x ^ and ŷ), Dx and Dy (grating widths in x ^ and ŷ), ng and ns (grating and substrate indices), and tg and ts (grating and substrate thicknesses). (b) Schematic of a 2D high-contrast grating on a hexagonal lattice with a substrate. Parameters: period Λ, rod diameter d, grating and substrate indices ng and ns, grating and substrate thicknesses tg and ts.

Fig. 2
Fig. 2

(a) Comparison among the (0,0)-th order reflectivity spectra of a 2D high-contrast grating (HCG) under normal incidence calculated using finite-element method (blue star), finite-difference time-domain (red dots), and rigorous coupled-wave analysis (RCWA) using N = 289 (magenta), N = 441 (green), and N = 625 (blue). HCG parameters are Λ x = 1μm, Λ y = 0.5μm, Dx = 0.6μm, Dy = 0.3μm, and tg = 0.5μm. (b) Spectra of normalized scattered power in the z ^ -direction for (0,0), (+1,0), and (1,0) spectral orders (blue, red, and black, respectively), and the sum of the three spectra (green).

Fig. 3
Fig. 3

(a) Incident polar angle dependent reflectivity (black), transmissivity (magenta), and their sum (green) for the (0,0) fundamental order under p-polarized incidence with λ = 2μm calculated using rigorous coupled-wave analysis, and comparison with finite-element method (red and blue crosses). The same grating structure is solved under (b) s-polarized and (c) p-polarized incidence with wavelength being λ = 0.6μm. Normalized scattered power fluxes in the z ^ -direction for the (0,0)-th, (+1,0)-th, (1,0)-th, (2,0)-th, and (3,0)-th spectral orders are shown as the blue, red, black, green, and cyan lines, respectively. The magenta line indicates the total normalized scattered power flux in the z ^ -direction. Arrows indicate the cutoff angles for spectral orders to appear or disappear.

Fig. 4
Fig. 4

(a) Reflectivity (blue), transmissivity (red), and the normalized scattered power (green) of a normal incident plane wave with λ = 1.55μm as functions of the azimuthal polarization angle. The arrows indicate the components parallel and perpendicular to the incident wave. (b) Phase differences between the parallel and perpendicular components for reflection (red) and transmission (blue) as functions of the azimuthal polarization angle. The 90° phase differences correspond to polarization angles of 27.7° and 52.7° for reflection and transmission types, respectively.

Fig. 5
Fig. 5

Design procedures of 2D high-contrast gratings.

Fig. 6
Fig. 6

(a) HE-like even eigenmodes in a 2D rectangular grating on a rectangular lattice. The two dashed lines indicate the kz = ω/c and kz = ngω/c light lines. (b) EH-like even eigenmodes in a 2D rectangular grating on a rectangular lattice. (c) ℜe[Hy] for the EH00-like eigenmode and EH20-like eigenmode at ω = 0.8 2 π c Λ x , as indicated by the green dots in (b).

Fig. 7
Fig. 7

(a) Circular high-contrast grating on hexagonal lattice. The yellow rectangle indicates a choice of the unit cell. Parameters: Λ = 1μm, η = 0.6. Dispersion curves of (b) Hx-dominant and (c) Hy-dominant eigenmodes possessing symmetry (red) and anti-symmetry (blue). ℜe[Hy] for (d) the first symmetric, (e) the first anti-symmetric, and (f) the second symmetric modes at frequency ω = 0.5 2 π c Λ , as indicated by the green dots in (c).

Fig. 8
Fig. 8

(a) Reflectivity (red) and transmissivity (blue) of a 2D circular high-contrast grating on a hexagonal lattice with Λ = 750nm, λ = 1.55μm, and tg = 713nm. Blue dashed lines indicate the wavelengths for perfect transmission. (b) Back-coupling magnitude from the first (red), second (black) symmetric eigenmodes and all other eigenmodes (green) in the grating to the (0,0) mode in the incident region. The dashed lines in (b) are at the same wavelengths as in (a). Blue solid line is the phase difference between the reflected waves back-coupled from the two eigenmodes.

Fig. 9
Fig. 9

(a) Half-trip phases of the supermodes 1 (blue) and 2 (red) in the same grating as in Fig. 2 under λ = 2.1μm normal incidence. Empty boxes and solid circles indicate half-trip phases being odd and even multiples of π, respectively. (b) Resonance conditions for the grating thicknesses at given wavelengths. Red and blue lines indicate the half-trip phase of supermode 1 being odd and even multiples of π, respectively. Black and green lines indicate the half-trip phase of supermode 1 being odd and even multiples of π, respectively.

Fig. 10
Fig. 10

(a) Reflectivity contour plot of a rectangular high-contrast grating (same as in Fig. 2) as a function of the thickness tg and wavelength λ. The solid lines indicate λ = 2.55Λ x and λ = 1.21Λ x , corresponding to the cutoff frequencies ω c EH 20 = 0.392 2 π c Λ x and ω c EH 02 = 0.827 2 π c Λ x for the EH20- and EH02-like modes, as shown in Fig. 6(b). (b) Overlap between the resonance contour plot (white) and the reflectivity contour plot.

Fig. 11
Fig. 11

(a) Dual-mode dispersion curves in a hexagonal-lattice grating and the dualmode window (same as in Fig. 7) is indicated by the blue dashed lines at ω = 0.44 2 π c Λ and ω = 0.6 2 π c Λ . (b) Transmission contour plot as a function of the grating thickness tg and wavelength λ. The dual-mode window indicated by the two black lines. (c) Overlap between the resonance lines (white) and the transmission contour plot.

Fig. 12
Fig. 12

Contour plots as functions of the grating period Λ and duty cycle η for the (a) magnitude and (b) phase of the transmission through a hexagonal-lattice grating under λ = 1.55μm normal incidence. The black lines indicates a 2π phase range at Λ = 750nm. The white lines indicate the contour for 90% transmission.

Fig. 13
Fig. 13

(a) Tuning of the transmission magnitude (solid) and phase (dots) by grating duty cycle with thickness being 0 (blue), 258nm (black), and 517nm (red). Same structure as in Fig. 7 and λ = 1.55μm. (b) Transmission magnitude (blue) and phase (green) as functions of the substrate thickness with 48% duty cycle.

Fig. 14
Fig. 14

e[Ex] in the xz-plane for an x-polarized normal-incident wave toward + z ^ at λ = 1.55μm transmitting through HCG phase plates designed for Gaussian beams to: (a) be deflected by 15°; (b) focus at a 15-μm distance; (c) be converted to Bessel beams. Black solid lines indicate the grating layer. Blue dashed lines indicate the source locations. Other dashed lines are for visual aids. All field intensities are normalized by the incident field intensity.

Fig. 15
Fig. 15

(a) Structure of the 2D high-contrast grating phase plate for focusing at f = 20μm and generating + 1 h ¯ orbital angular momentum. Field intensities at z = f − 4λ, z = f, and z = f +4λ are shown in (b), (c), and (d), respectively. (e) Gaussian source intensity with a beam waist of 12μm. Phase distributions (in π) of the transmitted wave at z = f − λ/3, z = f, and z = f +λ/3. are shown in (f), (g), and (h), respectively.

Equations (12)

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k ( m , n ) = x ^ k x + y ^ k y + z ^ k z , k x = k i x + G x , m , k y = k i y + G y , n , G x , m = m π Λ x , m = 0 , ± 1 , ± 2 , , ± N x , G y , n = n π Λ y , n = 0 , ± 1 , ± 2 , , ± N y , k z = ± 4 π 2 n r 2 λ 2 k x 2 k y 2
E t ( i ) = e i K z ( i ) z ( m , n ( x ^ E ˜ x , ( m , n ) ( i ) + y ^ E ˜ y , ( m , n ) ( i ) ) e i G x , m x + i G y , n y ) e i k i x x + i k i y y
M ¯ E ˜ t ( i ) = ( K z ( i ) ) 2 N ¯ E ˜ t ( i )
E ˜ t ( i ) = ( E ˜ x , ( N x , N y ) ( i ) , , E ˜ x , ( m , n ) ( i ) , , E ˜ x , ( N x , N y ) ( i ) , E ˜ y , ( N x , N y ) ( i ) , , E ˜ y , ( m , n ) ( i ) , , E ˜ y , ( N x , N y ) ( i ) ) T
E ˜ t = i A i E ˜ t ( i ) = ¯ t A
[ E ˜ t ( z ) H ˜ t ( z ) ] = [ ¯ t ¯ t ¯ t ¯ t ] [ A + ( z ) A ( z ) ] = [ ¯ t ¯ t ¯ t ¯ t ] [ e i K ¯ z z 0 0 e i K ¯ z z ] [ A + A ]
[ A II + A II ] = [ ¯ t II ¯ t II ¯ t II ¯ t II ] 1 [ ¯ t I ¯ t I ¯ t I ¯ t I ] [ A I + A I ] = [ T ¯ I II ] 4 N × 4 N [ A I + A I ]
( k 0 sin θ cos ϕ + m π Λ x ) 2 + ( k 0 sin θ sin ϕ + n π Λ y ) 2 = k 0 2
θ c = sin 1 ( 1 | m | λ Λ x ) for ( + | m | , 0 ) to disappear θ c = sin 1 ( | m | λ Λ x 1 ) for ( | m | , 0 ) to appear
E ˜ t ¯ t DM A DM
| Ω | e i φ A II + ( 0 ) = R ¯ II I A II ( 0 ) = R ¯ II I e i K ¯ z . ( t g ) A II ( t g ) = R ¯ II I e i K ¯ z t g R ¯ II III A II + ( t g ) = R ¯ II I e i K ¯ z t g R ¯ II III e i K ¯ z t g A II + ( 0 ) = M ¯ ( λ , t g ) A II + ( 0 )
[ M ¯ DM ] 2 × 2 ( λ , t g ) [ A p A q ] = | Ω DM | e i φ DM [ A p A q ]

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