Abstract

Polarizers are key components in optical science and technology. Thus, understanding the action of a polarizer beyond oversimplifying approximations is crucial. In this work, we study the interaction of a polarizing interface with an obliquely incident wave experimentally. To this end, a set of Mueller matrices is acquired employing a novel procedure robust against experimental imperfections. We connect our observation to a geometric model, useful to predict the effect of polarizers on complex light fields.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]
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2013

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

2012

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

2011

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

2010

2009

2007

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

2006

2005

1999

1998

S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998).
[CrossRef]

1997

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

1996

1994

1992

1989

1987

1986

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta33, 185–189 (1986).
[CrossRef]

1984

1982

P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE0307, 13–21 (1982).
[CrossRef]

1980

1941

1852

A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,” Ann. Phys.162, 78–88 (1852).
[CrossRef]

Aiello, A.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Opt. Lett.34, 3160–3162 (2009).
[CrossRef] [PubMed]

A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-likelihood estimation of Mueller matrices,” Opt. Lett.31, 817–819 (2006).
[CrossRef] [PubMed]

A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett.94, 090406 (2005).
[CrossRef] [PubMed]

Anderson, D. G. M.

Armigliato, A.

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

Azzam, R. M. A.

Banzer, P.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

Barakat, R.

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

Beer, A.

A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,” Ann. Phys.162, 78–88 (1852).
[CrossRef]

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta33, 185–189 (1986).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Pr., Oxford, 1999), 7th ed.

Borrelli, N.

S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998).
[CrossRef]

Borrelli, N. F.

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

Bottiger, J. R.

Branczyk, A. M.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Chille, V.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

Chipman, R. A.

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

Compain, E.

Damask, J. N.

J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005).

Darabi, A.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Drevillon, B.

Fagherazzi, G.

S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998).
[CrossRef]

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

Fainman, Y.

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

Fry, E. S.

Gabriel, C.

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

Gil, J. J.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta33, 185–189 (1986).
[CrossRef]

Goldstein, D. H.

gyu Jang, Y.

Hong, Q.

James, D. F. V.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Jones, R. C.

Kang, W.-S.

Kim, K.

Kim, S. M.

Kolb, T.

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

Korger, J.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

Lee, C. H.

Lee, G.-D.

Lee, S. H.

Leuchs, G.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Opt. Lett.34, 3160–3162 (2009).
[CrossRef] [PubMed]

Lindlein, N.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

Lopez, A. G.

Lu, R.

Lu, S.-Y.

Mahler, D. H.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Mandel, L.

Marquardt, C.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

A. Aiello, C. Marquardt, and G. Leuchs, “Nonparaxial polarizers,” Opt. Lett.34, 3160–3162 (2009).
[CrossRef] [PubMed]

Moon, J.-W.

Poirier, S.

Polizzi, S.

S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998).
[CrossRef]

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

Puentes, G.

Riello, P.

S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998).
[CrossRef]

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

Rozema, L. A.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

Shamir, J.

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

Steinberg, A. M.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Thompson, R. C.

Voigt, D.

Wittmann, C.

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

Woerdman, J.

A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett.94, 090406 (2005).
[CrossRef] [PubMed]

Woerdman, J. P.

Wolf, E.

Wu, S.-T.

Wu, T.

Wu, T. X.

Yeh, P.

P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE0307, 13–21 (1982).
[CrossRef]

yong Han, H.

Zhu, X.

Am. J. Phys.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys.75, 163 (2007).
[CrossRef]

Ann. Phys.

A. Beer, “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten,” Ann. Phys.162, 78–88 (1852).
[CrossRef]

Appl. Opt.

Appl. Phys. B

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric Spin Hall Effect of Light at polarizing interfaces,” Appl. Phys. B102, 427–432 (2011).
[CrossRef]

J. Non-Cryst. Solids

S. Polizzi, P. Riello, G. Fagherazzi, and N. Borrelli, “The microstructure of borosilicate glasses containing elongated and oriented phase-separated crystalline particles,” J. Non-Cryst. Solids232–234, 147–154 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microsc. Microanal. M.

S. Polizzi, A. Armigliato, P. Riello, N. F. Borrelli, and G. Fagherazzi, “Redrawn phase-separated borosilicate glasses: A TEM investigation,” Microsc. Microanal. M.8, 157–165 (1997).
[CrossRef]

New J. Phys.

A. M. Brańczyk, D. H. Mahler, L. A. Rozema, A. Darabi, A. M. Steinberg, and D. F. V. James, “Self-calibrating quantum state tomography,” New J. Phys.14, 085003 (2012).
[CrossRef]

Opt. Acta

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta33, 185–189 (1986).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

A. Aiello and J. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett.94, 090406 (2005).
[CrossRef] [PubMed]

Proc. SPIE

P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE0307, 13–21 (1982).
[CrossRef]

Other

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin Hall effect of light,” arXiv:1303.6974 (2013).

J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005).

M. Born and E. Wolf, Principles of Optics (Pergamon Pr., Oxford, 1999), 7th ed.

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Figures (5)

Fig. 1
Fig. 1

Geometric interpretation of polarizer models. A plane wave with its electric field Ein in the x̂ŷ-plane interacts with a tilted polarizer not parallel to this plane. Our goal is to connect the orientation θ, ϕ of the polarizer to the direction of the transmitted field component Eout. (a) Fainman and Shamir [4] suggested to find this direction tFS by projecting a vector T interpreted as the polarizer’s transmitting axis onto the x̂ŷ-plane. (b) The polarizer in question is made of elongated particles, all with their long axes oriented in direction of A. Thus, our absorbing model makes of use of the projection a of the absorbing axis A. The field component parallel to a is scattered and eventually absorbed. Consequently, the transmitted field is polarized in direction of , orthogonal to a.

Fig. 2
Fig. 2

State of polarization transmitted across a polarizer rotated around the vertical axis ŷ by an angle θ, keeping the angle ϕ = 94.5° between the absorbing axis and ŷ constant. (a) Visualization of the FS model [4]: The projection of the polarizer’s transmitting axis T (red arrow) onto the plane of the electric field (green plane) determines the transmitted field component (green arrow). (b) Visualization of the absorbing polarizer model (6): The projection of the polarizer’s absorbing axis A (blue arrow) onto the plane of the electric field (green plane) determines the absorbed field component. (c) Experimental data points (black circles) compared to both models. The dashed red line depicts the original FS model, while the solid blue line describes the analogously constructed absorbing model. The data shows the polarizance vector Mi0 [20] acquired as a part of our Mueller matrix measurement. This is the state of polarization after transmission across the polarizer if the incident wave is unpolarized. Only the absorbing model explains the drastic change of the transmitted state of polarization observed when the polarizer is tilted.

Fig. 3
Fig. 3

(a) Scheme of the Mueller matrix measurement. Using a collimated light beam (wavelength λ = 795nm), polarizing beam splitters (PBS), quarter wave plates (QWP), and two photo detectors IH and IV, the effect of an unknown sample on the polarization can be measured. For both QWPs, we use 6 different settings αin/out of their fast axes. Our sample is a commercial glass polarizer submerged in an index-matching liquid, which can be rotated around the vertical axis such that the incident beam impinges under an angle θ. This setup allows to study the polarizing effect of the metal nano-particles, the polarizer is made of, without interference from the glass surfaces. (b) Observed depolarization index PD [21] as a function of the orientation ϕ, θ of the polarizer relative to the incident beam. PD = 1 describes a non-depolarization sample while PD = 0 indicates a total depolarizer.

Fig. 4
Fig. 4

Jones matrix representation of the operation a light beam experiences when passing across our polarizer. The polarizer’s absorbing axis A is oriented almost horizontally (ϕ = 89.2°) and rotated around the vertical axis ŷ by an angle θ. The experimental data points (black circles) are calculated from our measured Mueller matrices. Ignoring an irrelevant global phase, we set Im(J11) = 0. Our phenomenological model, described by TP, is depicted using solid green lines. Dashed blue lines show the geometric absorbing model given by TA.

Fig. 5
Fig. 5

Reduced Mueller matrices M = 1 M 00M describing the tilted polarizer for two different orientations of its absorbing axis ϕ. Our polarizer model (solid lines) agrees well with the experimental data (markers). The model, we have employed, is deterministic. The small deviation from the model occurs for large tilting angles θ, where the devices is slightly depolarizing (compare Fig. 3(b)). Depolarization effects cannot be modelled using Jones calculus as employed by our model.

Equations (14)

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( S 0 S 1 S 2 S 3 ) = ( I 0 ° + I 90 ° I 0 ° I 90 ° I + 45 ° I 45 ° I R I L ) = ( | E x | 2 + | E y | 2 | E x | 2 | E y | 2 E x E y * + E x * E y i ( E x E y * E x * E y ) ) ,
S μ = Tr [ ( J J ) σ ( μ ) ] .
σ ( 0 ) = 1 2 ( 1 0 0 1 ) , σ ( 1 ) = 1 2 ( 1 0 0 1 ) , σ ( 2 ) = 1 2 ( 0 1 1 0 ) , σ ( 3 ) = 1 2 ( 0 i i 0 ) ,
J in J out = T J in ,
S in S out = M S in ,
a ^ = P ^ A ( P ^ A z ^ ) z ^ 1 ( P ^ A z ^ ) 2 .
E in E out = E in ( E in a ^ ) a ^ .
I i j = 1 2 ( S j out ) T M S i in
ε ( M LS ) = i , j | 1 2 ( S j out ) T M LS S i in I i j E | 2
M a b = Tr [ H ( σ ( a ) σ ( b ) * ) ]
ε ( M LS ) = i , j | 1 2 ( S H , V ) T M j out M i in S H I i j cal | 2 .
E in E out = T P E in with T P = τ a a ^ a T + τ t t ^ t ^ T .
τ t ( θ ) = exp ( 0.025 / cos ( θ ) ) and
τ a ( θ ) = 0.89 exp ( 6.70 cos ( θ ) ) i 0.62 exp ( 13.6 cos ( θ ) ) .

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