Abstract

In this paper, the implementation of optical elements in the form of Pancharatnam-Berry optics is considered. With respect to 3D bulk and diffractive optics, acting on the dynamic phase of light, Pancharatnam-Berry optical elements transfer a phase that is geometric in nature by locally manipulating the polarization state of the incident beam. They can be realized as space-variant sub-wavelengths gratings that behave like inhomogeneous form-birefringent materials. We present a comprehensive work of simulation, realization, and optical characterization at the telecom wavelength of 1310 nm of the constitutive linear grating cell, whose fabrication has been finely tuned to get a π-phase delay and obtain a maximum in the diffraction efficiency. The optical design in the infrared region allows the use of silicon as candidate material due to its transparency. In order to demonstrate the possibility of assembling the single grating cells for generating more complex phase patterns, the implementation of two Pancharatnam-Berry optical elements is considered: a blazed grating and an optical vortices demultiplexer.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
[Crossref]

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
[Crossref] [PubMed]

2017 (5)

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017).
[Crossref] [PubMed]

I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11(5), 274–284 (2017).
[Crossref]

M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358(6367), 6367 (2017).
[Crossref] [PubMed]

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017).
[Crossref]

G. Ruffato, M. Massari, and F. Romanato, “Compact sorting of optical vortices by means of diffractive transformation optics,” Opt. Lett. 42(3), 551–554 (2017).
[Crossref] [PubMed]

2016 (3)

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016).
[Crossref] [PubMed]

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

G. Ruffato, M. Massari, and F. Romanato, “Diffractive optics for combined spatial- and mode- division demultiplexing of optical vortices: design, fabrication and optical characterization,” Sci. Rep. 6(1), 24760 (2016).
[Crossref] [PubMed]

2015 (4)

2014 (2)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

2013 (1)

2011 (3)

Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B Condens. Matter Mater. Phys. 84(20), 205428 (2011).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

A. Pozzato, G. Grenci, G. Birarda, and M. Tormen, “Evaluation of a novolak based positive tone photoresist as NanoImprint Lithography resist,” Microelectron. Eng. 88(8), 2096–2099 (2011).
[Crossref]

2010 (2)

C. Haensch, S. Hoeppener, and U. S. Schubert, “Chemical modification of self-assembled silane based monolayers by surface reactions,” Chem. Soc. Rev. 39(6), 2323–2334 (2010).
[Crossref] [PubMed]

A. Emoto, M. Nishi, M. Okada, S. Manabe, S. Matsui, N. Kawatsuki, and H. Ono, “Form birefringence in intrinsic birefringent media possessing a subwavelength structure,” Appl. Opt. 49(23), 4355–4361 (2010).
[Crossref] [PubMed]

2006 (1)

2005 (1)

2004 (1)

2002 (3)

M. Beck, M. Graczyk, I. Maximov, E. L. Sarwe, T. G. I. Ling, M. Keil, and L. Montelius, “Improving stamps for 10 nm level wafer scale nanoimprint lithography,” Microelectron. Eng. 61–62, 441–448 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27(13), 1141–1143 (2002).
[Crossref] [PubMed]

E. Hasman, Z. Bomzon, A. Niv, G. Biener, and V. Kleiner, “Polarization beam-splitters and optical switches based on space-variant computer-generated subwavelength quasi-periodic structures,” Opt. Commun. 209(1-3), 45–54 (2002).
[Crossref]

2001 (1)

Z. Bomzon, V. Kleiner, and E. Hasman, “Space-variant state manipulation with computer-generated subwavelength metal stripe gratings,” Opt. Commun. 192(3-6), 169–181 (2001).
[Crossref]

1998 (2)

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

H. Kikuta, Y. Ohira, H. Kubo, and K. Iwata, “Effective medium theory of two-dimensional subwavelength gratings in the non-quasi-static limit,” J. Opt. Soc. Am. A 15(6), 1577 (1998).
[Crossref]

1997 (1)

1996 (1)

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43(10), 2063–2085 (1996).
[Crossref]

1995 (3)

H. Kikuta, H. Yoshida, and K. Iwata, “Ability and Limitation of Effective Medium Theory for Subwavelength Gratings,” Opt. Rev. 2(2), 92–99 (1995).
[Crossref]

M. A. Green and M. J. Keevers, “Optical properties of intrinsic silicon at 300K,” Prog. Photovolt. Res. Appl. 3(3), 189–192 (1995).
[Crossref]

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995).
[Crossref]

1989 (1)

V. DePalma and N. Tillman, “Friction and Wear of Self -Assembled Trichlorosilane Monolayer Films on Silicon,” Langmuir 5(3), 868–872 (1989).
[Crossref]

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its application,” Proc. Indian Acad. Sci. Sect. A Phys. Sci. 44(5), 247–262 (1956).
[Crossref]

Agrell, E.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Aieta, F.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Alù, A.

Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B Condens. Matter Mater. Phys. 84(20), 205428 (2011).
[Crossref]

Ambrosio, A.

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017).
[Crossref] [PubMed]

Arbabi, A.

Atwater, H. A.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
[Crossref] [PubMed]

Bagheri, M.

Beck, M.

M. Beck, M. Graczyk, I. Maximov, E. L. Sarwe, T. G. I. Ling, M. Keil, and L. Montelius, “Improving stamps for 10 nm level wafer scale nanoimprint lithography,” Microelectron. Eng. 61–62, 441–448 (2002).
[Crossref]

Biener, G.

Birarda, G.

A. Pozzato, G. Grenci, G. Birarda, and M. Tormen, “Evaluation of a novolak based positive tone photoresist as NanoImprint Lithography resist,” Microelectron. Eng. 88(8), 2096–2099 (2011).
[Crossref]

Boltasseva, A.

S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
[Crossref]

Bomzon, Z.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27(13), 1141–1143 (2002).
[Crossref] [PubMed]

E. Hasman, Z. Bomzon, A. Niv, G. Biener, and V. Kleiner, “Polarization beam-splitters and optical switches based on space-variant computer-generated subwavelength quasi-periodic structures,” Opt. Commun. 209(1-3), 45–54 (2002).
[Crossref]

Z. Bomzon, V. Kleiner, and E. Hasman, “Space-variant state manipulation with computer-generated subwavelength metal stripe gratings,” Opt. Commun. 192(3-6), 169–181 (2001).
[Crossref]

Bowers, J. E.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Brandt-Pearce, M.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Briggs, R. M.

Brongersma, M.

D. Lin, M. Meli, E. Poliakov, P. ST. Hilaire, S. Dhuey, C. Peroz, S. Cabrini, M. Brongersma, and M. Klug, “Optical metasurfaces for high angle steering at visible wavelengths,” Sci. Rep.7, 2286 (2017).
[Crossref]

Brongersma, M. L.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
[Crossref] [PubMed]

Cabrini, S.

D. Lin, M. Meli, E. Poliakov, P. ST. Hilaire, S. Dhuey, C. Peroz, S. Cabrini, M. Brongersma, and M. Klug, “Optical metasurfaces for high angle steering at visible wavelengths,” Sci. Rep.7, 2286 (2017).
[Crossref]

Capasso, F.

M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358(6367), 6367 (2017).
[Crossref] [PubMed]

R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017).
[Crossref] [PubMed]

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Chaudhuri, K.

S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
[Crossref]

Cheben, P.

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
[Crossref] [PubMed]

Chen, M.L.N.

M.L.N. Chen, L.J. Jiang, and W.E.I. Sha, “Orbital angular momentum generation and detection by geometric-phase based metasurfaces,” Appl. Sci.8, 362 (2018).
[Crossref]

Choudhury, S. M.

S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
[Crossref]

Chraplyvy, A. R.

E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

DePalma, V.

V. DePalma and N. Tillman, “Friction and Wear of Self -Assembled Trichlorosilane Monolayer Films on Silicon,” Langmuir 5(3), 868–872 (1989).
[Crossref]

Desiatov, B.

DeVault, C.

S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
[Crossref]

Devlin, R.

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C. Haensch, S. Hoeppener, and U. S. Schubert, “Chemical modification of self-assembled silane based monolayers by surface reactions,” Chem. Soc. Rev. 39(6), 2323–2334 (2010).
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Keevers, M. J.

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Kikuta, H.

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S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
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D. Lin, M. Meli, E. Poliakov, P. ST. Hilaire, S. Dhuey, C. Peroz, S. Cabrini, M. Brongersma, and M. Klug, “Optical metasurfaces for high angle steering at visible wavelengths,” Sci. Rep.7, 2286 (2017).
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Li, S.

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M. Beck, M. Graczyk, I. Maximov, E. L. Sarwe, T. G. I. Ling, M. Keil, and L. Montelius, “Improving stamps for 10 nm level wafer scale nanoimprint lithography,” Microelectron. Eng. 61–62, 441–448 (2002).
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E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
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Niv, A.

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E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
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R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017).
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Sarwe, E. L.

M. Beck, M. Graczyk, I. Maximov, E. L. Sarwe, T. G. I. Ling, M. Keil, and L. Montelius, “Improving stamps for 10 nm level wafer scale nanoimprint lithography,” Microelectron. Eng. 61–62, 441–448 (2002).
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E. Agrell, M. Karlsson, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fisher, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, I. Tomkos, J. E. Bowers, S. Srinivasan, M. Brandt-Pearce, and N. Gisin, “Roadmap of optical communications,” J. Opt. 18(6), 063002 (2016).
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P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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A. Pozzato, G. Grenci, G. Birarda, and M. Tormen, “Evaluation of a novolak based positive tone photoresist as NanoImprint Lithography resist,” Microelectron. Eng. 88(8), 2096–2099 (2011).
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Wang, D.

S. M. Choudhury, D. Wang, K. Chaudhuri, C. DeVault, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Material platforms for optical metasurfaces,” Nanophotonics 7(6), 959–987 (2018).
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Appl. Opt. (2)

Chem. Soc. Rev. (1)

C. Haensch, S. Hoeppener, and U. S. Schubert, “Chemical modification of self-assembled silane based monolayers by surface reactions,” Chem. Soc. Rev. 39(6), 2323–2334 (2010).
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Figures (13)

Fig. 1
Fig. 1 Comparison between a diffractive optical element (DOE) on the left and a Pancharatnam-Berry optical element (PBOE) on the right, in the case of 8 phase levels. It is possible to infer the higher flatness and ease of fabrication of 2D with respect to 3D optics.
Fig. 2
Fig. 2 TM polarization (in red) has the electric field parallel to the grating vector, whose modulus is given by K = 2π/Λ. The orthogonal TE polarization (in blue) has the electric field parallel to the grating ridges. A qualitative comparison between wavelength and period of the grating is shown, which allows treating the grating as an effective form-birefringent material.
Fig. 3
Fig. 3 (a) Comparison of effective indices for TE (blue) and TM (red) polarizations calculated with different numerical methods: zero-order effective medium theory (EMT0), second-order effective medium theory (EMT2), rigorous coupled-wave analysis (RCWA). Incident wavelength 1310 nm and different grating periods from 100 to 350 nm, step 50 nm, were considered at normal incidence. (b) Grating thickness providing π-delay between TE and TM polarizations as a function of the duty-cycle for different grating periods. (c) Optimal configurations of duty-cycle and period providing π-retardation for varying grating thickness from 510 to 560 nm, step 10 nm. (d) Optimal configurations of duty-cycle and period providing π-retardation for fixed grating thickness 540 nm, input wavelength in the telecom O-band from 1260 to 1360 nm, step 10 nm.
Fig. 4
Fig. 4 SEM images of a sample at the beginning and at the end of the fabrication process. On the left, it is possible to appreciate the homogeneity of the grating on a wide scale. On the right, detail of the final sample at the end of the pattern transfer process.
Fig. 5
Fig. 5 Experimental data of the introduced phase delay as a function of the grating depth. Inset graph: zoom around nearly-optimized Pancharatnam-Berry samples (data shown in Table 1).
Fig. 6
Fig. 6 Experimental setup employed to analyze the state of polarization exiting the sample in the transmission analysis measurements.
Fig. 7
Fig. 7 (a) Experimental transmission dependence as a function of the analyzer angle for samples fabricated with different etching times, i.e. different grating thickness, fitted by means of Eq. (8). (b) Zoom for the nearly-optimized samples.
Fig. 8
Fig. 8 SEM images of the sample labeled PBBG1 (PBBG stands for Pancharatnam-Berry blazed grating). On the left, it is possible to appreciate a wide view of pattern homogeneity and its partition in pixels of different orientation. On the right, the attention is focused on single pixels.
Fig. 9
Fig. 9 Scatterometry graphs of samples PBBG1, PBBG2 and PBBG3 illuminated with both left-handed (LH) and right-handed (RH) circular polarization, measured for co-polarized (LH-LH, RH-RH) and cross-polarized output (LH-RH, RH-LH). The three samples have been fabricated with different depths: 493 nm (PBBG1), 542 nm (PBBG2), 582 nm (PBBG3).
Fig. 10
Fig. 10 Scheme of the Pancharatnam-Berry optics working principle for OAM-beam sorting with the method of optical beam projection. When a circularly-polarized OAM beam illuminates the optical element, a bright spot appears in the far-field, at a position depending on the carried OAM and on the polarization handedness.
Fig. 11
Fig. 11 (a) Numerical phase pattern for the sorting of OAM beams in the range from −3 to + 3. 16 phase levels. Pixel size: 6.125 μm × 6.125 μm. Radius size: 256 pixels. (b) Far-field channel constellation for the given OAM set and circular polarization states.
Fig. 12
Fig. 12 SEM inspections of the fabricated PB demultiplexer on silicon substrate. Grating period Λ = 290 nm, duty-cycle 0.5, thickness 535 nm, pixel size 6.125 μm.16 rotation angles.
Fig. 13
Fig. 13 Interference patterns of the OAM beams from −3 to + 3 exploited for the sorter characterization. The number and handedness of the spirals denotes the carried OAM. Opposite-handedness far-fields for input right-handed and left-handed circular polarization states. As expected, the bright spot positions are in accordance with the theoretical channel constellation in Fig. 11(b).

Tables (1)

Tables Icon

Table 1 Phase delay of Pancharatnam-Berry cell (PBC#) samples characterized via ellipsometry (PBC1-PBC5) or transmission analysis (PBC6-PBC8). Comparison between experimental results (δexp) and numerical estimations (δth) calculated with RCWA, assuming the tabulated values of period, duty-cycle and depth.

Equations (15)

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δ= 2π λ d( n n )
T( x,y )=R[ θ( x,y ) ]τ( δ )R [ θ( x,y ) ] 1
R[ θ( x,y ) ]=| cosθ( x,y ) sinθ( x,y ) sinθ( x,y ) cosθ( x,y ) |
τ( δ )=| exp( iδ/2 ) 0 0 exp( +iδ/2 ) |
T( 1 ±i )=cos( δ 2 )( 1 ±i )isin( δ 2 ) e ±2iθ ( 1 i )
T( 1 ±i )=i e ±2iθ ( 1 i )
E out =( cos 2 θ sinθcosθ sinθcosθ sin 2 θ )( e iδ 0 0 1 )( cosα sinα sinα cosα )( i 0 0 1 )( cosα sinα sinα cosα )( 1 0 )
I=a[ 1cos( 2θ+2b ) sin 2 ( 2Δ )cos( 2Δ )sin( 2Δ )sin( 2θ+2b )cosδ+sinδ ]
ϕ( x,y )=γx
φ=arcsin( λ/L )
Ω( u,v )=arg{ j=1 n c j R j * exp[ i l j ϑ+i α j u+i β j v ] }
( x j , y j )= f k ( α j , β j )
Ω (+) ( u,v )=arg{ j=1 n c j R j * exp[ i l j ϑ+i α j u+i β j v ] } Ω () ( u,v )=arg{ j=1 n c j R j exp[ +i l j ϑi α j ui β j v ] }
β () ( l j )= β (+) ( l j ) α () ( l j )= α (+) ( l j )
β () ( l j )= β (+) ( l j ) α () ( l j )= α (+) ( l j )