Abstract

Subwavelength gratings are gratings with periods smaller than the incident wavelength. They possess form birefringence, which depends on the grating parameters. This paper presents the results of an experimental method designed to measure the birefringent properties of diamond subwavelength gratings in the mid-infrared. The method consists of monitoring the intensity transmitted through one polarizer, a subwavelength grating, and a second polarizer for various orientations of the first polarizer. By fitting the intensity variation as a function of the orientation of the first polarizer, one can compute the phase shift induced by the grating, its local fast axis orientation, and the ratio of the transverse electric and transverse magnetic transmission efficiencies. The paper describes the method principle and its mathematical model. Then, several numerical simulations of different subwavelength gratings are presented and their results are discussed. Finally, the optical setup is described and the measurements of one subwavelength grating are displayed and compared with the values expected from the manufacturing process.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (2)

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

2013 (2)

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

2006 (1)

C. Oh and M. J. Escutti, “Time-domain analysis of periodic anisotropic media at oblique incidence an efficient FDTD implementation,” Opt. Exp. 14, 11870–11884 (2006).
[Crossref]

2005 (3)

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[Crossref]

D. Mawet, P. Riaud, J. Surdej, and P. Beaudrand, “Subwavelength surface-relief gratings for stellar coronagraphy,” Appl. Opt. 44, 7313–7321 (2005).
[Crossref]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astron. Astrophys. 633, 1191–1200 (2005).

2003 (2)

2002 (2)

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

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, 45–54 (2002).
[Crossref]

2001 (1)

1997 (1)

C. Yang and P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[Crossref]

1995 (1)

Absil, O.

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astron. Astrophys. 633, 1191–1200 (2005).

Baudoz, P.

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Beaudrand, P.

Biener, G.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[Crossref]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of space-variant subwavelength grating,” J. Opt. Soc. Am. A 20, 1940–1948 (2003).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

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, 45–54 (2002).
[Crossref]

Boccaletti, A.

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Bokor, N.

Bomzon, Z.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

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, 45–54 (2002).
[Crossref]

Carlomagno, B.

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Chauvin, G.

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

Christiaens, V.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Davidson, N.

Defrère, D.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Delacroix, C.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Escutti, M. J.

C. Oh and M. J. Escutti, “Time-domain analysis of periodic anisotropic media at oblique incidence an efficient FDTD implementation,” Opt. Exp. 14, 11870–11884 (2006).
[Crossref]

Femenía Castellá, B.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Forsberg, P.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Frisem, A. A.

Gaylord, T. K.

Girard, J.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

Gómez González, C. A.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Grann, E. B.

Habraken, S.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Hasman, E.

Hecht, E.

E. Hecht, Optique, 4th ed. (Pearson, 2005).

Hinz, P. M.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Huby, E.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

Jolivet, A.

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Karlsson, M.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

M. Karlsson and F. Nikolajeff, “Diamond micro-optics: microlenses and antireflection structured surfaces for the infrared spectral region,” Opt. Express 11, 502–507 (2003).
[Crossref]

Kleiner, V.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[Crossref]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of space-variant subwavelength grating,” J. Opt. Soc. Am. A 20, 1940–1948 (2003).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

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, 45–54 (2002).
[Crossref]

Kuittinen, M.

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Lagrange, A.

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

Matthews, K.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Mawet, D.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astron. Astrophys. 633, 1191–1200 (2005).

D. Mawet, P. Riaud, J. Surdej, and P. Beaudrand, “Subwavelength surface-relief gratings for stellar coronagraphy,” Appl. Opt. 44, 7313–7321 (2005).
[Crossref]

Milli, J.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

Moharam, M. G.

Nikolajeff, F.

Niv, A.

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[Crossref]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Near-field Fourier transform polarimetry by use of space-variant subwavelength grating,” J. Opt. Soc. Am. A 20, 1940–1948 (2003).
[Crossref]

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, 45–54 (2002).
[Crossref]

Oh, C.

C. Oh and M. J. Escutti, “Time-domain analysis of periodic anisotropic media at oblique incidence an efficient FDTD implementation,” Opt. Exp. 14, 11870–11884 (2006).
[Crossref]

Orban de Xivry, G.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Pantin, E.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Piron, P.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Pommet, D. A.

Reggiani, M.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Riaud, P.

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astron. Astrophys. 633, 1191–1200 (2005).

D. Mawet, P. Riaud, J. Surdej, and P. Beaudrand, “Subwavelength surface-relief gratings for stellar coronagraphy,” Appl. Opt. 44, 7313–7321 (2005).
[Crossref]

Ruane, G. J.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Schechter, R.

Serabyn, E.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Surdej, J.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astron. Astrophys. 633, 1191–1200 (2005).

D. Mawet, P. Riaud, J. Surdej, and P. Beaudrand, “Subwavelength surface-relief gratings for stellar coronagraphy,” Appl. Opt. 44, 7313–7321 (2005).
[Crossref]

Tristram, K. R. W.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Vargas Catalán, E.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

Vartiainen, I.

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

Wertz, O.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Wizinowich, P.

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Yang, C.

C. Yang and P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[Crossref]

Yeh, P.

C. Yang and P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[Crossref]

Appl. Opt. (2)

Astron. Astrophys. (4)

O. Absil, J. Milli, D. Mawet, A. Lagrange, J. Girard, G. Chauvin, A. Boccaletti, and J. Surdej, “Searching for companions down to 2 AU from β Pictoris using the L′-band AGPM coronagraph on VLT/NACO,” Astron. Astrophys. 559, L12 (2013).
[Crossref]

D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astron. Astrophys. 633, 1191–1200 (2005).

E. Vargas Catalán, E. Huby, P. Forsberg, A. Jolivet, P. Baudoz, B. Carlomagno, C. Delacroix, S. Habraken, D. Mawet, J. Surdej, O. Absil, and M. Karlsson, “Optimizing the subwavelength grating of l-band annular groove phase masks for high coronagraphic performance,” Astron. Astrophys. 595, A127 (2016).
[Crossref]

C. Delacroix, O. Absil, P. Forsberg, D. Mawet, V. Christiaens, M. Karlsson, A. Boccaletti, P. Baudoz, M. Kuittinen, I. Vartiainen, J. Surdej, and S. Habraken, “Laboratory demonstration of a mid-infrared AGPM vector vortex coronagraph,” Astron. Astrophys. 553, A98 (2013).
[Crossref]

J. Appl. Phys. (1)

C. Yang and P. Yeh, “Artificial uniaxial and biaxial dielectrics with the use of photoinduced gratings,” J. Appl. Phys. 81, 23–29 (1997).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. 251, 306–314 (2005).
[Crossref]

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, 45–54 (2002).
[Crossref]

Opt. Exp. (1)

C. Oh and M. J. Escutti, “Time-domain analysis of periodic anisotropic media at oblique incidence an efficient FDTD implementation,” Opt. Exp. 14, 11870–11884 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

O. Absil, D. Mawet, M. Karlsson, B. Carlomagno, V. Christiaens, D. Defrère, C. Delacroix, B. Femenía Castellá, P. Forsberg, J. Girard, C. A. Gómez González, S. Habraken, P. M. Hinz, E. Huby, A. Jolivet, K. Matthews, J. Milli, G. Orban de Xivry, E. Pantin, P. Piron, M. Reggiani, G. J. Ruane, E. Serabyn, J. Surdej, K. R. W. Tristram, E. Vargas Catalán, O. Wertz, and P. Wizinowich, “Three years of harvest with the vector vortex coronagraph in the thermal infrared,” Proc. SPIE 9908, 99080Q (2016).
[Crossref]

Other (1)

E. Hecht, Optique, 4th ed. (Pearson, 2005).

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Figures (18)

Fig. 1.
Fig. 1. Representation of the grating parameters for a beam with an incident angle of ω i .
Fig. 2.
Fig. 2. Representation of the optical setup: a fixed linear polarizer P 0 and a quarter-wave plate λ / 4 aim to create a circular polarization to avoid intensity modulation on the sample for different orientations of P 1 . P 1 and P 2 are computer-controlled linear polarizers, S represents the component to be measured, and D stands for the detector. The polarization states at different places of the optical setup are represented by blue arrows and the beam is shown in red.
Fig. 3.
Fig. 3. Representation of Γ as a function of θ with a sampling step of 10°, for a half-wave plate component with perfect transmission of TE and TM components, and a local orientation of 49° of its fast axis.
Fig. 4.
Fig. 4. Representation of the simulated intensities for the ideal case, for three values of θ , namely, 0°, 30°, and 60°. Panels (a–c) present I and panels (d–f) stand for I .
Fig. 5.
Fig. 5. Representation of the fast axis orientation for the ideal case. The fast axis rotates around the center of the component from 180 ° to 180°. Only the introduced fast axis is shown since there is no observable difference between the introduced one and the computed one.
Fig. 6.
Fig. 6. Representation of the simulated intensities normalized by the maximum values of the six pictures, for the chromatic phase shift case at 5.7 μm ( ϕ in = 139.7 ° ) and perfect transmission of both TE and TM components, for three values of θ , namely, 0°, 30°, and 60°. Panels (a–c) present I and panels (d–f) stand for I .
Fig. 7.
Fig. 7. Representation of ϕ obtained after the fitting routine in the chromatic ϕ case. Error bars are not visible on the graph due to their small values. The blue dots represent the averaged fitted phase shift ϕ and the black line represents the phase shift obtained due to the RCWA computations ϕ RCWA and used to compute the Γ curves for each pixel.
Fig. 8.
Fig. 8. Representation of the simulated intensities normalized by the maximum values of the six pictures, for the chromatic phase shift and efficiency ratio case at 5.7 μm ( ϕ in = 139.7 ° ) and q in = 0.9348 for three values of θ , namely, 0°, 30°, and 60°. Panels (a–c) present I and panels (d–f) stand for I .
Fig. 9.
Fig. 9. Representation of q obtained after the fitting routine in the chromatic ϕ and q case. Error bars are not visible on the graph due to their small values. The blue dots represent the averaged fitted efficiency ratio q and the black line represents the efficiency ratio obtained due to the RCWA computations q RCWA and used to compute the Γ curves for each pixel.
Fig. 10.
Fig. 10. Picture of the experimental setup (QCL not present).
Fig. 11.
Fig. 11. Representation of the measured intensities normalized by the maximum values of the six pictures, for the chromatic case at 5.7 μm ( ϕ RCWA = 139.7 ° ) and q RCWA = 0.935 for three values of θ , namely, 0°, 30°, and 60°. Panels (a–c) present I and panels (d–f) stand for I . Contrast and luminosity were modified compared with the previous pictures to enhance visibility.
Fig. 12.
Fig. 12. Absolute value of the difference between the binned phase retard ϕ and the introduced one ϕ in as a function of the radius of the component for the binned case.
Fig. 13.
Fig. 13. Absolute value of the difference between the binned phase retard Q and the introduced one q in as a function of the radius of the component for the binned case.
Fig. 14.
Fig. 14. Measured values of ϕ . The error bars represent σ ϕ . The green values correspond to the maximum and minimum ϕ obtained using RCWA with a maximum error of 2.5% on the grating parameters.
Fig. 15.
Fig. 15. Measured values of q . The error bars represent σ q . The green values correspond to the maximum and minimum q obtained using RCWA with a maximum error of 2.5% on the grating parameters.
Fig. 16.
Fig. 16. Representation of α . Panel (a) shows the expected values resulting from the intensity visual interpretation. Panel (b) shows the values obtained by the measurement process. The ring form originates from our intensity and binning criteria, where only the pixels present in the ring are taken into account in the fitting computation.
Fig. 17.
Fig. 17. Scatter plot of ϕ as a function of distance to the component center at 5.7 μm.
Fig. 18.
Fig. 18. Scatter plot of q as function of distance to the component center at 5.7 μm.

Tables (5)

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Table 1. Table of the Results Obtained with the Fitting of Γ for the Ideal Case

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Table 2. Table of the Results Obtained with the Fitting of Γ for the Component for Several Wavelengths

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Table 3. Table of the Results Obtained with the Fitting of Γ for the Component for Several Wavelengths

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Table 4. Table of the Results Obtained with the Fitting of Γ for the Real Component

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Table 5. Comparison between the Old Results O and the Filtered Ones F

Equations (12)

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Λ λ 1 n I sin ω i + max ( n I , n II ) ,
J in = 2 2 ( 1 i ) .
P 1 = ( cos 2 ( θ ) 0.5 sin ( 2 θ ) 0.5 sin ( 2 θ ) sin 2 ( θ ) ) ,
S = ( η TM cos 2 ( α ) + η TE sin 2 ( α ) e i ϕ 1 2 sin ( 2 α ) ( η TM η TE e i ϕ ) 1 2 sin ( 2 α ) ( η TM η TE e i ϕ ) η TM sin 2 ( α ) + η TE cos 2 ( α ) e i ϕ ) ,
P 2 = ( cos 2 ( γ ) 0.5 sin ( 2 γ ) 0.5 sin ( 2 γ ) sin 2 ( γ ) ) ,
J out = P 2 × S × P 1 × J in J out = P 2 × S × P 1 × J in .
I = 1 16 ( η TM 2 + η TE 2 ) cos ( 4 α 4 θ ) + 1 4 ( η TM 2 η TE 2 ) cos ( 2 α 2 θ ) + 3 16 ( η TM 2 + η TE 2 ) 1 8 η TM η TE cos ( 4 α 4 θ ) cos ( ϕ ) + 1 8 η TM η TE cos ( ϕ ) ;
I = 1 16 ( η TM 2 + η TE 2 ) 1 16 ( η TM 2 + η TE 2 ) cos ( 4 α 4 θ ) + 1 8 η TM η TE cos ( 4 α 4 θ ) cos ( ϕ ) 1 8 η TM η TE cos ( ϕ ) .
Γ ( α , ϕ , θ , q ) = ( 1 + q ) ( 1 + q ) cos ( 4 α 4 θ ) + 2 q 0.5 cos ( 4 α 4 θ ) cos ( ϕ ) 2 q 0.5 cos ( ϕ ) ( 1 + q ) cos ( 4 α 4 θ ) + ( 1 q ) cos ( 2 α 2 θ ) + 3 ( 1 + q ) 2 q 0.5 cos ( 4 α 4 θ ) cos ( ϕ ) + 2 q 0.5 cos ( ϕ ) .
Γ ( θ i ) = I ( θ i ) 0.5 [ D a ( θ i ) + D b ( θ i ) ] I ( θ i ) 0.5 [ D a ( θ i ) + D b ( θ i 1 ) ] .
i I T ( θ i , x , y ) 0.02 × max [ x , y ] [ i I T ( θ i , x , y ) ] ,
I ( θ i , X , Y ) = k = 1 27 l = 1 27 I ( θ i , x k , y l ) , I ( θ i , X , Y ) = k = 1 27 l = 1 27 I ( θ i , x k , y l ) .

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