Abstract

Three types of polarization gratings have been recorded in azopolymer films by the symmetrical superposition of different orthogonal pairs of polarized beams. The inscribed holographic elements have been analyzed microscopically in a Mueller polarimeter in order to image the optical anisotropies photoinduced in the film. In the most of cases, the spatial modulation of diattenuation, birefringence, and optical rotation reproduced quite well previous results reported in the literature. Nevertheless, in the particular case of coherent superposition of p- and s-polarized beams, the spatial frequency for optical rotation (related to the Stokes parameter V) was different from the one observed in linear anisotropy (related to the Stokes parameter U). It is shown by theory and experiment that, in the polarized field used to record this polarization grating, the fourth-Stokes parameter changes sign, which implies a change in circular polarization handedness, practically once between two adjacent maxima.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]

2014 (3)

H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Polarization-sensitive diffraction in vector gratings combined with form birefringence in subwavelength-periodic structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 31, 11–19 (2014).
[Crossref]

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

2013 (3)

U. Ruiz, P. Pagliusi, C. Provenzano, K. Volke-Sepúlveda, and G. Cipparrone, “Polarization holograms allow highly efficient generation of complex light beams,” Opt. Express 21, 7505–7510 (2013).
[Crossref] [PubMed]

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

L. Nedelchev, D. Nazarova, and V. Dragostinova, “Photosensitive organic/inorganic azopolymer based nanocomposite materials with enhanced photoinduced birefringence,” J. Photochem. Photobiol. A 261, 26–30 (2013).
[Crossref]

2012 (5)

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

A. Emoto, E. Uchida, and T. Fukuda, “Optical and physical applications of photocontrollable materials: azobenzene-containing and liquid crystalline polymers,” Polymers 4, 150–186 (2012).
[Crossref]

M. W. Kudenov, M. J. Escuti, N. Hagen, E. L. Dereniak, and K. Oka, “Snapshot imaging Mueller matrix polarimeter using polarization gratings,” Opt. Lett. 37, 1367–1369 (2012).
[Crossref] [PubMed]

Y. Li, J. Kim, and M. J. Escuti, “Orbital angular momentum generation and mode transformation with high efficiency using forked polarization gratings,” Appl. Opt. 51, 8236–8245 (2012).
[Crossref] [PubMed]

2011 (1)

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

R. M. Tejedor, J. L. Serrano, and L. Oriol, “Photocontrol of supramolecular architecture in azopolymers: Achiral and chiral aggregation,” Europ. Polym. J. 45, 2564–2571 (2009).
[Crossref]

2008 (2)

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

A. Sobolewska and A. Miniewicz, “On the inscription of period and half-period surface relief gratings in azobenzene-functionalized polymers,” J. Phys Chem B 112, 4526–4535 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

S. G. Cloutier, “Polarization holography: orthogonal plane-polarized beam configuration with circular vectorial photoinduced anisotropy,” J. Phys. D: Appl. Phys. 38, 3371–3375 (2005).
[Crossref]

2004 (1)

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

2002 (1)

M. J. Kim, B. G. Shin, J. J. Kim, and D. Y. Kim, “Photoinduced supramolecular chirality in amorphous azobenzene polymer films,” J. Am. Chem. Soc. 124, 3504–3505 (2002).
[Crossref] [PubMed]

2000 (1)

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

1998 (1)

1996 (1)

1995 (3)

A. Natansohn, P. Rochon, M-S. Ho, and C. Barrett, “Azo polymers for reversible optical storage. 6. Poly[4-[2-(methacryloyloxy)ethyl]azobenzene,” Macromolecules 28, 4179–4183 (1995).
[Crossref]

P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66, 136–138 (1995).
[Crossref]

D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995).
[Crossref]

1974 (1)

S. D. Kakichashvili, “Method for phase polarization recording of holograms”, Sov. J. Quant. Electron. 4, 795–798 (1974).
[Crossref]

Arenas, G.

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

Barrett, C.

A. Natansohn, P. Rochon, M-S. Ho, and C. Barrett, “Azo polymers for reversible optical storage. 6. Poly[4-[2-(methacryloyloxy)ethyl]azobenzene,” Macromolecules 28, 4179–4183 (1995).
[Crossref]

Batalla, E.

P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66, 136–138 (1995).
[Crossref]

Bliznyuk, V. N.

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

Boilot, J. P.

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

Cipparrone, G.

Cloutier, S. G.

S. G. Cloutier, “Polarization holography: orthogonal plane-polarized beam configuration with circular vectorial photoinduced anisotropy,” J. Phys. D: Appl. Phys. 38, 3371–3375 (2005).
[Crossref]

Dereniak, E. L.

Dragostinova, V.

L. Nedelchev, D. Nazarova, and V. Dragostinova, “Photosensitive organic/inorganic azopolymer based nanocomposite materials with enhanced photoinduced birefringence,” J. Photochem. Photobiol. A 261, 26–30 (2013).
[Crossref]

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

Emoto, A.

A. Emoto, E. Uchida, and T. Fukuda, “Optical and physical applications of photocontrollable materials: azobenzene-containing and liquid crystalline polymers,” Polymers 4, 150–186 (2012).
[Crossref]

Escuti, M. J.

Fabbri, F.

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

Fukuda, T.

A. Emoto, E. Uchida, and T. Fukuda, “Optical and physical applications of photocontrollable materials: azobenzene-containing and liquid crystalline polymers,” Polymers 4, 150–186 (2012).
[Crossref]

Galante, M. J.

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

Garrot, D.

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

Gharagozloo-Hubmann, K.

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

Goldstein, D. H.

D. H. Goldstein, Polarized light (CRC Press, 2011).

Hagen, N.

Ho, M-S.

A. Natansohn, P. Rochon, M-S. Ho, and C. Barrett, “Azo polymers for reversible optical storage. 6. Poly[4-[2-(methacryloyloxy)ethyl]azobenzene,” Macromolecules 28, 4179–4183 (1995).
[Crossref]

Holme, N. C. R.

Hristov, B.

Huey, B. D.

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

Hvilsted, S.

Izawa, M.

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

Kakichashvili, S. D.

S. D. Kakichashvili, “Method for phase polarization recording of holograms”, Sov. J. Quant. Electron. 4, 795–798 (1974).
[Crossref]

Kawatsuki, N.

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Polarization-sensitive diffraction in vector gratings combined with form birefringence in subwavelength-periodic structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 31, 11–19 (2014).
[Crossref]

Kim, D. Y.

M. J. Kim, B. G. Shin, J. J. Kim, and D. Y. Kim, “Photoinduced supramolecular chirality in amorphous azobenzene polymer films,” J. Am. Chem. Soc. 124, 3504–3505 (2002).
[Crossref] [PubMed]

D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995).
[Crossref]

Kim, J.

Kim, J. J.

M. J. Kim, B. G. Shin, J. J. Kim, and D. Y. Kim, “Photoinduced supramolecular chirality in amorphous azobenzene polymer films,” J. Am. Chem. Soc. 124, 3504–3505 (2002).
[Crossref] [PubMed]

Kim, M. J.

M. J. Kim, B. G. Shin, J. J. Kim, and D. Y. Kim, “Photoinduced supramolecular chirality in amorphous azobenzene polymer films,” J. Am. Chem. Soc. 124, 3504–3505 (2002).
[Crossref] [PubMed]

Kim, S.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Kudenov, M. W.

Kulikovska, O.

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

Kumar, J.

D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995).
[Crossref]

Kurihara, S.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Kuwahara, Y.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Lahlil, K.

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

Lai, N. D.

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

Larios-Lopez, L.

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

Lassailly, Y.

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

Ledoux-Rak, I.

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

Li, L.

D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995).
[Crossref]

Li, Y.

Martinez-Ponce, G.

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

Matsui, S.

Mazzulla, A.

Miniewicz, A.

A. Sobolewska and A. Miniewicz, “On the inscription of period and half-period surface relief gratings in azobenzene-functionalized polymers,” J. Phys Chem B 112, 4526–4535 (2008).
[Crossref] [PubMed]

Nakamura, T.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Natansohn, A.

P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66, 136–138 (1995).
[Crossref]

A. Natansohn, P. Rochon, M-S. Ho, and C. Barrett, “Azo polymers for reversible optical storage. 6. Poly[4-[2-(methacryloyloxy)ethyl]azobenzene,” Macromolecules 28, 4179–4183 (1995).
[Crossref]

Navarro-Rodriguez, D.

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

Naydenova, I.

Nazarova, D.

L. Nedelchev, D. Nazarova, and V. Dragostinova, “Photosensitive organic/inorganic azopolymer based nanocomposite materials with enhanced photoinduced birefringence,” J. Photochem. Photobiol. A 261, 26–30 (2013).
[Crossref]

Nedelchev, L.

L. Nedelchev, D. Nazarova, and V. Dragostinova, “Photosensitive organic/inorganic azopolymer based nanocomposite materials with enhanced photoinduced birefringence,” J. Photochem. Photobiol. A 261, 26–30 (2013).
[Crossref]

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

Nguyen, C. T.

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

Nguyen, T. T. N.

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

Nikolova, L.

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

T. Todorov, L. Nikolova, G. Stoilov, and B. Hristov, “Spectral Stokesmeter. 1. Implementation of the device,” Appl. Opt. 46, 6662–6668 (2007).
[Crossref] [PubMed]

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

I. Naydenova, L. Nikolova, T. Todorov, N. C. R. Holme, P. S. Ramanujam, and S. Hvilsted, “Diffraction from polarization holographic gratings with surface relief in side-chain azobenzene polyesters,” J. Opt. Soc. Am. B 15, 1257–1265 (1998).
[Crossref]

L. Nikolova and P.S. Ramanujam, Polarization holography (Cambridge, 2009).
[Crossref]

Nishi, M.

Nishioka, E.

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

Noda, K.

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Polarization-sensitive diffraction in vector gratings combined with form birefringence in subwavelength-periodic structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 31, 11–19 (2014).
[Crossref]

Ogata, T.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Oka, K.

Okada, M.

Ono, H.

H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Polarization-sensitive diffraction in vector gratings combined with form birefringence in subwavelength-periodic structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 31, 11–19 (2014).
[Crossref]

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

Oriol, L.

R. M. Tejedor, J. L. Serrano, and L. Oriol, “Photocontrol of supramolecular architecture in azopolymers: Achiral and chiral aggregation,” Europ. Polym. J. 45, 2564–2571 (2009).
[Crossref]

Orofino, A. B.

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

Oyanguren, P. A.

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

Pagliusi, P.

Peretti, J.

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

Petrova, T.

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

Provenzano, C.

Ramanujam, P. S.

Ramanujam, P.S.

L. Nikolova and P.S. Ramanujam, Polarization holography (Cambridge, 2009).
[Crossref]

Rochon, P.

P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66, 136–138 (1995).
[Crossref]

A. Natansohn, P. Rochon, M-S. Ho, and C. Barrett, “Azo polymers for reversible optical storage. 6. Poly[4-[2-(methacryloyloxy)ethyl]azobenzene,” Macromolecules 28, 4179–4183 (1995).
[Crossref]

Rodriguez-Gonzalez, R. J.

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

Ruiz, U.

Sasaki, T.

H. Ono, M. Nishi, T. Sasaki, K. Noda, M. Okada, S. Matsui, and N. Kawatsuki, “Polarization-sensitive diffraction in vector gratings combined with form birefringence in subwavelength-periodic structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers,” J. Opt. Soc. Am. B 31, 11–19 (2014).
[Crossref]

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

Serrano, J. L.

R. M. Tejedor, J. L. Serrano, and L. Oriol, “Photocontrol of supramolecular architecture in azopolymers: Achiral and chiral aggregation,” Europ. Polym. J. 45, 2564–2571 (2009).
[Crossref]

Shin, B. G.

M. J. Kim, B. G. Shin, J. J. Kim, and D. Y. Kim, “Photoinduced supramolecular chirality in amorphous azobenzene polymer films,” J. Am. Chem. Soc. 124, 3504–3505 (2002).
[Crossref] [PubMed]

Sobolewska, A.

A. Sobolewska and A. Miniewicz, “On the inscription of period and half-period surface relief gratings in azobenzene-functionalized polymers,” J. Phys Chem B 112, 4526–4535 (2008).
[Crossref] [PubMed]

Solano, C.

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

Stoilov, G.

Stumpe, J.

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

Tejedor, R. M.

R. M. Tejedor, J. L. Serrano, and L. Oriol, “Photocontrol of supramolecular architecture in azopolymers: Achiral and chiral aggregation,” Europ. Polym. J. 45, 2564–2571 (2009).
[Crossref]

Todorov, T.

T. Todorov, L. Nikolova, G. Stoilov, and B. Hristov, “Spectral Stokesmeter. 1. Implementation of the device,” Appl. Opt. 46, 6662–6668 (2007).
[Crossref] [PubMed]

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

I. Naydenova, L. Nikolova, T. Todorov, N. C. R. Holme, P. S. Ramanujam, and S. Hvilsted, “Diffraction from polarization holographic gratings with surface relief in side-chain azobenzene polyesters,” J. Opt. Soc. Am. B 15, 1257–1265 (1998).
[Crossref]

Tomova, N.

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

Tripathy, S. K.

D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995).
[Crossref]

Uchida, E.

A. Emoto, E. Uchida, and T. Fukuda, “Optical and physical applications of photocontrollable materials: azobenzene-containing and liquid crystalline polymers,” Polymers 4, 150–186 (2012).
[Crossref]

Ujiie, S.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Volke-Sepúlveda, K.

Wolf, E.

E. Wolf, Theory of coherence and polarization of light (Cambridge, 2007).

Wu, X.

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

Yagi, R.

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Zucchi, I.

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

T. Sasaki, M. Izawa, K. Noda, E. Nishioka, N. Kawatsuki, and H. Ono, “Temporal formation of optical anisotropy and surface relief during polarization holographic recording in polymethylmethacrylate with azobenzene side groups,” Appl. Phys. B 114, 373–380 (2014).
[Crossref]

Appl. Phys. B. (1)

X. Wu, T. T. N. Nguyen, I. Ledoux-Rak, C. T. Nguyen, and N. D. Lai, “UV beam-assisted efficient formation of surface relief grating on azobenzene polymers,” Appl. Phys. B. 107, 819–822 (2012).
[Crossref]

Appl. Phys. Lett. (3)

L. Nikolova, L. Nedelchev, T. Todorov, T. Petrova, N. Tomova, V. Dragostinova, P. S. Ramanujam, and S. Hvilsted, “Self-induced light polarization rotation in azobenzene-containing polymers,” Appl. Phys. Lett. 77, 657–659 (2000).
[Crossref]

P. Rochon, E. Batalla, and A. Natansohn, “Optically induced surface gratings on azoaromatic polymer films,” Appl. Phys. Lett. 66, 136–138 (1995).
[Crossref]

D. Y. Kim, S. K. Tripathy, L. Li, and J. Kumar, “Laser-induced holographic surface relief gratings on nonlinear optical polymer films,” Appl. Phys. Lett. 66, 1166–1168 (1995).
[Crossref]

Europ. Polym. J. (1)

R. M. Tejedor, J. L. Serrano, and L. Oriol, “Photocontrol of supramolecular architecture in azopolymers: Achiral and chiral aggregation,” Europ. Polym. J. 45, 2564–2571 (2009).
[Crossref]

J. Am. Chem. Soc. (1)

M. J. Kim, B. G. Shin, J. J. Kim, and D. Y. Kim, “Photoinduced supramolecular chirality in amorphous azobenzene polymer films,” J. Am. Chem. Soc. 124, 3504–3505 (2002).
[Crossref] [PubMed]

J. Opt. A: Pure Appl. Opt. (2)

G. Martinez-Ponce, C. Solano, R. J. Rodriguez-Gonzalez, L. Larios-Lopez, D. Navarro-Rodriguez, and L. Nikolova, “All-optical switching using supramolecular chiral structures in azopolymers,” J. Opt. A: Pure Appl. Opt. 10, 115006 (2008).
[Crossref]

G. Martinez-Ponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. Nikolova, “Investigations on photoinduced processes in a series of azobenzene-containing side-chain polymers,” J. Opt. A: Pure Appl. Opt. 6, 324–329 (2004).
[Crossref]

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

J. Photochem. Photobiol. A (1)

L. Nedelchev, D. Nazarova, and V. Dragostinova, “Photosensitive organic/inorganic azopolymer based nanocomposite materials with enhanced photoinduced birefringence,” J. Photochem. Photobiol. A 261, 26–30 (2013).
[Crossref]

J. Phys Chem B (1)

A. Sobolewska and A. Miniewicz, “On the inscription of period and half-period surface relief gratings in azobenzene-functionalized polymers,” J. Phys Chem B 112, 4526–4535 (2008).
[Crossref] [PubMed]

J. Phys. Chem. B (1)

F. Fabbri, D. Garrot, K. Lahlil, J. P. Boilot, Y. Lassailly, and J. Peretti, “Evidence of two distinct mechanisms driving photoinduced matter motion in thin films containing azobenzene derivatives,” J. Phys. Chem. B 115, 1363–1367 (2011).
[Crossref] [PubMed]

J. Phys. D: Appl. Phys. (1)

S. G. Cloutier, “Polarization holography: orthogonal plane-polarized beam configuration with circular vectorial photoinduced anisotropy,” J. Phys. D: Appl. Phys. 38, 3371–3375 (2005).
[Crossref]

Macromolecules (1)

A. Natansohn, P. Rochon, M-S. Ho, and C. Barrett, “Azo polymers for reversible optical storage. 6. Poly[4-[2-(methacryloyloxy)ethyl]azobenzene,” Macromolecules 28, 4179–4183 (1995).
[Crossref]

Nanotechnology (1)

O. Kulikovska, K. Gharagozloo-Hubmann, J. Stumpe, B. D. Huey, and V. N. Bliznyuk, “Formation of surface relief grating in polymers with pendant azobenzene chromophores as studied by AFM/UFM,” Nanotechnology 23, 485309 (2012).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Polym. Int. (1)

S. Kim, T. Nakamura, R. Yagi, Y. Kuwahara, T. Ogata, S. Ujiie, and S. Kurihara, “Photo-response orientation behaviors of polyethylene imine backbone structures with azobenzene side chains,” Polym. Int. 63, 733–740 (2014).
[Crossref]

Polymer (1)

A. B. Orofino, G. Arenas, I. Zucchi, M. J. Galante, and P. A. Oyanguren, “A simple strategy to generate light-responsive azobenzene-containing epoxy networks,” Polymer 54, 6184–6190 (2013).
[Crossref]

Polymers (1)

A. Emoto, E. Uchida, and T. Fukuda, “Optical and physical applications of photocontrollable materials: azobenzene-containing and liquid crystalline polymers,” Polymers 4, 150–186 (2012).
[Crossref]

Sov. J. Quant. Electron. (1)

S. D. Kakichashvili, “Method for phase polarization recording of holograms”, Sov. J. Quant. Electron. 4, 795–798 (1974).
[Crossref]

Other (3)

E. Wolf, Theory of coherence and polarization of light (Cambridge, 2007).

D. H. Goldstein, Polarized light (CRC Press, 2011).

L. Nikolova and P.S. Ramanujam, Polarization holography (Cambridge, 2009).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

Result of coherent superposition in typical configurations for the recording of polarization gratings in photoanisotropic materials.

Fig. 2
Fig. 2

a) Numerical evaluation and b) experimental measurement of Mueller matrix images given by the photoanisotropic modulation in the film induced by the superposition of left and right circularly polarized beams. Simulation has been performed using Δnl = 0.0075, Δαl = 0.0025, Δd = 100 nm, and d0 = 4 μm.

Fig. 3
Fig. 3

Mueller imaging polarimeter based on rotating compensators.

Fig. 4
Fig. 4

Polarization pattern formed when recording and diffracted beams are superimposed coherently.

Fig. 5
Fig. 5

Images obtained by polar decomposition of the images obtained by Mueller imaging. a) Diattenuation, b) Depolarization, c) Retardance, and d) Optical rotation.

Fig. 6
Fig. 6

Spatial modulation of Stokes parameters U and V when two light beams, s- and p-polarized, are superimposed.

Fig. 7
Fig. 7

Same as in Fig. 5 but using the second setup.

Fig. 8
Fig. 8

Same as in Fig. 5 but using the third setup.

Equations (53)

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

{ Σ ( r , t ) } = { E 1 ( r , t ) + E 2 ( r , t ) } , = { [ E 01 e j k 1 r + E 02 e j k 2 r ] e j ω t } , = p ( x , z ) cos ω t + q ( x , z ) sin ω t ,
E 01 = E 01 ( cos ψ 2 cos θ i + sin ψ 2 e j ϕ 1 j + cos ψ 2 sin θ k ) ,
E 02 = E 02 ( cos ψ 2 cos θ i + sin ψ 2 e j ϕ 2 j cos ψ 2 sin θ k ) .
L a , b 2 = 1 2 [ p 2 + q 2 ± ( p 2 q 2 ) 2 4 ( p q ) 2 ] ,
± l i = [ 1 1 + ξ ] 1 / 2 p i L a + [ ξ 1 + ξ ] 1 / 2 q i L a , i = x , y , z ,
ξ = L a 2 q 2 L a 2 p 2 .
e = L b L a ,
α = α x .
I ( x , y ) = 2 I 0 [ 1 + cos ψ 1 cos ψ 2 cos 2 θ cos ( 2 k x sin θ ) + sin ψ 1 sin ψ 2 cos ( 2 k x sin θ 2 ϕ ) ] ,
p ( x , z = 0 ) = { E 01 cos θ i + E 02 j + E 01 sin θ k } cos [ k ( x sin θ ) ] ,
q ( x , z = 0 ) = { E 01 cos θ i + E 02 j E 01 sin θ k } sin [ k ( x sin θ ) ] .
{ Σ ( x , t ) } = ( i + j ) cos ( k x sin θ ) cos ω t ( i j ) sin ( k x sin θ ) sin ω t .
I = Σ x Σ x * + Σ y Σ y * ,
Q = Σ x Σ x * Σ y Σ y * ,
U = Σ x Σ y * + Σ y Σ x * ,
V = j ( Σ x Σ y * Σ y Σ x * ) ,
S = { 1 , 0 , cos ( 2 k x sin θ ) , sin ( 2 k x sin θ ) } T .
D U = [ 0 tanh ( U k Δ α 0 l d ) tanh ( V k Δ α 0 c d ) ] .
M D U = [ 1 D T U P m D ] ,
m D = a I 3 + b ( D U × D T U ) = [ a 0 0 0 a + b tanh 2 ( U k Δ α 0 l d ) tanh ( U k Δ α 0 l d ) tanh ( V k Δ α 0 c d ) 0 tanh ( U k Δ α 0 l d ) tanh ( V k Δ α 0 c d ) a + b tanh 2 ( V k Δ α 0 c d ) ] .
a = 1 | D U | 2 ,
b = 1 a | D U | 2 .
M R U = [ 1 0 T 0 m R ] ,
m R = [ cos ( U k Δ n 0 l d ) cos ( V k Δ n 0 c d ) cos ( U k Δ n 0 l d ) sin ( V k Δ n 0 c d ) sin ( U k Δ n 0 l d ) sin ( V k Δ n 0 c d ) cos ( V k Δ n 0 c d ) 0 sin ( U k Δ n 0 l d ) cos ( V k Δ n 0 c d ) sin ( U k Δ n 0 l d ) sin ( V k Δ n 0 c d ) cos ( U k Δ n 0 l d ) ] .
M L = M R U ( Δ n 0 c = 0 ) M D U ( Δ α 0 c = 0 ) = e k α e l d [ cosh ( k Δ α l d ) 0 sinh ( k Δ α l d ) 0 0 cos [ k Δ n l d ] 0 sin [ k Δ n l d ] sinh [ k Δ α l d ] 0 cosh [ k α l d ] 0 0 sin [ k Δ n l d ] 0 cos ( k Δ n l d ) ] ,
M c = M R U ( Δ n 0 l = 0 ) M D U ( Δ α 0 l = 0 ) = e k α e c d [ cosh ( k Δ α c d ) 0 0 sinh ( k Δ α c d ) 0 cos [ 2 k Δ n c d ] sin [ 2 k Δ n c d ] 0 0 sin [ 2 k Δ n c d ] cos [ 2 k Δ n c d ] 0 sinh ( k Δ α c d ) 0 0 cosh ( k Δ α c d ) ] ,
M I = M L M c .
p ( x , z = 0 ) = 2 2 { ( E 01 + E 02 ) cos θ i + ( E 01 E 02 ) j + ( E 01 E 02 ) sin θ k } cos [ k ( x sin θ ) ] ,
q ( x , z = 0 ) = 2 2 { ( E 02 E 01 ) cos θ i ( E 01 + E 02 ) j ( E 01 + E 02 ) sin θ k } sin [ k ( x sin θ ) ] .
{ Σ ( x , t ) } = 2 [ cos ( k x sin θ ) cos ω t i sin ( k x sin θ ) sin ω t j ] .
S = { 1 , cos ( 2 k x sin θ ) , 0 , sin ( 2 k x sin θ ) } T .
D Q = [ tan ( Q k Δ α 0 l d ) 0 tan ( V k Δ α 0 c d ) ] .
M D Q = [ 1 D T Q D Q m D ] ,
m D = [ a + b tanh 2 ( Q k Δ α 0 l d ) 0 tanh ( Q k Δ α 0 l d ) tanh ( V k Δ α 0 c d ) 0 a 0 tanh ( Q k Δ α 0 l d ) tanh ( V k Δ α 0 c d ) 0 a + b tanh 2 ( V k Δ α 0 c d ) ] .
M R Q = [ 1 0 T 0 m R ] ,
m R = [ cos ( V k Δ n 0 c d ) sin ( V k Δ n 0 c d ) 0 cos ( U k Δ n 0 l d ) sin ( V k Δ n 0 c d ) cos ( U k Δ n 0 l d ) cos ( V k Δ n 0 c d ) sin ( U k Δ n 0 l d ) sin ( U k Δ n 0 l d ) sin ( V k Δ n 0 c d ) sin ( U k Δ n 0 l d ) cos ( V k Δ n 0 c d ) cos ( U k Δ n 0 l d ) ] .
M l = M R Q ( Δ n 0 c = 0 ) M D Q ( Δ α 0 c ) = e k α e l d [ cosh ( k Δ α l d ) sinh ( k Δ α l d ) 0 0 sinh ( k Δ α l d ) cosh ( k Δ α l d ) 0 0 0 0 cos [ k Δ n l d ] sin [ k Δ n l d ] 0 0 sin [ k Δ n l d ] cos [ k Δ n l d ] ] ,
M II = M l M c .
p ( x , z = 0 ) = 2 2 { ( E 01 + E 02 ) cos θ cos [ k ( x sin θ ) ] i ( E 01 + E 02 ) sin [ k ( x sin θ ) ] j + ( E 01 E 02 ) sin θ cos [ k ( x sin θ ) ] k } ,
q ( x , z = 0 ) = 2 2 { ( E 01 E 02 ) cos θ sin [ k ( x sin θ ) ] i ( E 01 E 02 ) cos [ k ( x sin θ ) ] j ( E 01 + E 02 ) sin θ sin [ k ( x sin θ ) ] k } .
{ Σ ( x , t ) } = 2 [ cos ( k x sin θ ) cos ω t i sin ( k x sin θ ) sin ( ω t + π 2 ) j ] .
S = { 1 , cos ( 2 k x sin θ ) , sin ( 2 k x sin θ ) , 0 } T .
M III = M L M l or M III = M l M L .
Σ ( x , t ) = ( i + j ) [ cos ( k x sin θ ) + cos ( k x sin 3 θ ) ] cos ω t ( i j ) [ sin ( k x sin θ ) + sin ( k x sin 3 θ ) ] sin ω t .
S = 1 4 [ 1 0 sin ( 2 k x sin θ ) 2 sin ( 4 k x sin θ ) sin ( 6 k x sin θ ) cos ( 2 k x sin θ ) + 2 cos ( 4 k x sin θ ) + cos ( 6 k x sin θ ) ] ,
J LD = [ e k α x d 0 0 e k α y d ] = e k α ¯ d [ e k Δ α l d / 2 0 0 e k Δ α l d / 2 ] ,
J LD = e k α ¯ d [ cos ( k Δ α l d / 2 ) j sinh ( k Δ α l d / 2 ) j sinh ( k Δ α l d / 2 ) cosh ( k Δ α l d / 2 ) ] .
J CD = e k α ¯ d [ cos ( k Δ α c d / 2 ) j sinh ( k Δ α c d / 2 ) j sinh ( k Δ α c d / 2 ) cosh ( k Δ α c d / 2 ) ] .
M = A ( J J * ) A 1 ,
A = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 j j 0 ] .
M LD = cosh ( k Δ α l d ) [ 1 tanh ( k Δ α l d ) 0 0 tanh ( k Δ α l d ) 1 0 0 0 0 sech ( k Δ α l d ) 0 0 0 0 sech ( k Δ α l d ) ] .
M LD = cosh ( k Δ α l d ) [ 1 0 tanh ( k Δ α l d ) 0 0 sech ( k Δ α l d ) 0 0 tanh ( k Δ α l d ) 0 1 0 0 0 0 sech ( k Δ α l d ) ] .
M CD = cosh ( k Δ α c d ) [ 1 0 0 tanh ( k Δ α c d ) 0 sech ( k Δ α c d ) 0 0 0 0 sech ( k Δ α c d ) 0 tanh ( k Δ α c d ) 0 0 1 ] .

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