Abstract

An idealized polarizer model that works without the structural and material information is derived in the spatial frequency domain. The non-paraxial property is fully included and the result takes a simple analytical form, which provides a straight-forward explanation for the crosstalk between field components in non-paraxial cases. The polarizer model, in a 2 × 2-matrix form, can be conveniently used in cooperation with other computational optics methods. Two examples in correspondence with related works are presented to verify our polarizer model.

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

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References

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    [Crossref] [PubMed]
  5. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
    [Crossref]
  6. P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE 0307, 13–21 (1982).
    [Crossref]
  7. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982).
    [Crossref]
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    [Crossref]
  10. R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
    [Crossref] [PubMed]
  11. Z. Wang, S. Zhang, and F. Wyrowski, “The semi-analytical fast Fourier transform,” Proc. DGaO, P2 (2017).
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    [Crossref]
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    [Crossref] [PubMed]
  19. Fast physical optics software, “Wyrowski VirtualLab Fusion,” LightTrans GmbH, Jena, Germany.
  20. The simulation example on polarizer in focal region, containing the source codes and supplementary materials, can be downloaded using the link below [retrieved 19 March 2018]. https://www.lighttrans.com/index.php?id=439
  21. F. Wyrowski, “Unification of the geometric and diffractive theories of electromagnetic fields,” Proc. DGaO, A36 (2017).
  22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc.A253 (1959).
  23. The simulation example on Stokes parameters measurement behind a tilted polarizer, containing the source codes and supplementary materials, can be downloaded using the link below [retrieved 19 March 2018]. https://www.lighttrans.com/index.php?id=441
  24. S. Zhang, D. Asoubar, C. Hellmann, and F. Wyrowski, “Propagation of electromagnetic fields between non-parallel planes: a fully vectorial formulation and an efficient implementation,” Appl. Opt. 55(3), 529–538 (2016).
    [Crossref] [PubMed]

2017 (3)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
[Crossref] [PubMed]

S. Zhang, C. Hellmann, and F. Wyrowski, “Algorithm for the propagation of electromagnetic fields through etalons and crystals,” Appl. Opt. 56(15), 4566–4576 (2017).
[Crossref] [PubMed]

2016 (1)

2013 (1)

2011 (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

2009 (1)

2003 (1)

1996 (2)

1993 (1)

1984 (1)

1982 (2)

P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE 0307, 13–21 (1982).
[Crossref]

P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72(4), 507–513 (1982).
[Crossref]

1972 (1)

1969 (1)

L. Rabiner, R. Schafer, and C. Rader, ”The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[Crossref]

1941 (1)

Aiello, A.

Asoubar, D.

Banzer, P.

Berreman, D. W.

Carnicer, A.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
[Crossref] [PubMed]

Fainman, Y.

Gu, C.

Hellmann, C.

Jones, R. C.

Juvells, I.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
[Crossref] [PubMed]

Kolb, T.

Korger, J.

Kuhn, M.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Landry, G. D.

Leuchs, G.

Li, L.

Maldonado, T. A.

Maluenda, D.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
[Crossref] [PubMed]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017).
[Crossref]

Marquardt, C.

Martínez-Herrero, R.

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017).
[Crossref]

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
[Crossref] [PubMed]

Rabiner, L.

L. Rabiner, R. Schafer, and C. Rader, ”The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[Crossref]

Rader, C.

L. Rabiner, R. Schafer, and C. Rader, ”The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc.A253 (1959).

Schafer, R.

L. Rabiner, R. Schafer, and C. Rader, ”The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[Crossref]

Shamir, J.

Wang, Z.

Z. Wang, S. Zhang, and F. Wyrowski, “The semi-analytical fast Fourier transform,” Proc. DGaO, P2 (2017).

Wittmann, C.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc.A253 (1959).

Wyrowski, F.

S. Zhang, C. Hellmann, and F. Wyrowski, “Algorithm for the propagation of electromagnetic fields through etalons and crystals,” Appl. Opt. 56(15), 4566–4576 (2017).
[Crossref] [PubMed]

S. Zhang, D. Asoubar, C. Hellmann, and F. Wyrowski, “Propagation of electromagnetic fields between non-parallel planes: a fully vectorial formulation and an efficient implementation,” Appl. Opt. 55(3), 529–538 (2016).
[Crossref] [PubMed]

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Z. Wang, S. Zhang, and F. Wyrowski, “The semi-analytical fast Fourier transform,” Proc. DGaO, P2 (2017).

F. Wyrowski and C. Hellmann, “The geometric Fourier transform,” Proc. DGaO, A37 (2017).

F. Wyrowski, “Unification of the geometric and diffractive theories of electromagnetic fields,” Proc. DGaO, A36 (2017).

Yeh, P.

Zhang, S.

Appl. Opt. (4)

IEEE Trans. Audio Electroacoust. (1)

L. Rabiner, R. Schafer, and C. Rader, ”The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[Crossref]

J. Mod. Opt. (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Opt. Lett. (1)

Optics and Lasers in Engineering (1)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizers on highly focused spirally polarized fields,” Optics and Lasers in Engineering 98, 176–180 (2017).
[Crossref]

Proc. SPIE (1)

P. Yeh, “Generalized model for wire grid polarizers,” Proc. SPIE 0307, 13–21 (1982).
[Crossref]

Sci. Rep. (1)

R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Polarisers in the focal domain: theoretical model and experimental validation,” Sci. Rep. 742122 (2017).
[Crossref] [PubMed]

Other (7)

Z. Wang, S. Zhang, and F. Wyrowski, “The semi-analytical fast Fourier transform,” Proc. DGaO, P2 (2017).

F. Wyrowski and C. Hellmann, “The geometric Fourier transform,” Proc. DGaO, A37 (2017).

Fast physical optics software, “Wyrowski VirtualLab Fusion,” LightTrans GmbH, Jena, Germany.

The simulation example on polarizer in focal region, containing the source codes and supplementary materials, can be downloaded using the link below [retrieved 19 March 2018]. https://www.lighttrans.com/index.php?id=439

F. Wyrowski, “Unification of the geometric and diffractive theories of electromagnetic fields,” Proc. DGaO, A36 (2017).

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc.A253 (1959).

The simulation example on Stokes parameters measurement behind a tilted polarizer, containing the source codes and supplementary materials, can be downloaded using the link below [retrieved 19 March 2018]. https://www.lighttrans.com/index.php?id=441

Supplementary Material (1)

NameDescription
» Visualization 1       A polarizer is placed in the focal plane behind a high-NA aspheric lens. The polarizer is rotated for 180 degrees, from parallel orientation (with respect to the linearly polarization direction of the input field) to orthogonal, and again to parallel

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Figures (6)

Fig. 1
Fig. 1 Uniaxial crystal model for a polarizer. A Cartesian coordinate system x-y-z is set up as shown in (a), with the x axis along the optic axis (o.a). The polarizer plate has a permittivity tensor ∊̱ and a thickness of d. The embedding medium has a permittivity . A rotated polarizer can be treated with the help of coordinate transformations, as shown in (c).
Fig. 2
Fig. 2 Simulation of diverging Gaussian field propagating through crossed polarizers: input Gaussian field in (a) the k domain and (b) the spatial domain; output field behind the polarizer in (c) the k domain and (d) the spatial domain. The simulation of this example took 2 s.
Fig. 3
Fig. 3 A linearly polarized (along the x axis) plane wave is focused by an aspheric lens, and a linear polarizer is placed at the focal plane, making an angle of α with respect to the x axis.
Fig. 4
Fig. 4 Electric field components and intensity distribution: (a) in front of the polarizer, (b) behind the polarizer parallel to the x axis, and (c) behind the polarizer orthogonal to the x axis. The amplitudes of the field components are displayed with respect to the individual maximum, labeled with m in each sub-figure. All sub-figures share the same scale of the x and y axes. The change in the intensity when α changes from 0° to 90°, and to 180° can be visualized in the Visualization 1). The simulation from input field to the results in (a) took 2.5 s; from (a) to the results in (b) or (c) it took 3 s; and the animation generation over 180 different angles took 185 s with multi-core processing enabled.
Fig. 5
Fig. 5 A linearly polarized (along the y axis) plane wave propagated through a titled polarizer. The polarizer is tilted around the y axis by a variable angle of θ as sketched in (a), and the absorbing axis of the polarizer makes a fixed angle of ϕ = 94.5° with respect to the y axis within the polarizer plane as in (b).
Fig. 6
Fig. 6 Normalized Stokes parameters in the detector plane behind the tilted polarizer, with S0 = (|Ex|2 + |Ey|2). The whole simulation over 89 different tilt angles took 46 s with multi-core processing enabled.

Equations (23)

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( E x out ( ρ , z out ) E y out ( ρ , z out ) ) = ( 𝒞 x x 𝒞 x y 𝒞 y x 𝒞 y y ) ( E x in ( ρ , z in ) E y in ( ρ , z in ) ) ,
_ = ( e 0 0 0 o 0 0 0 o ) ,
d d z ( E ˜ x ( κ , z ) E ˜ y ( κ , z ) η 0 H ˜ x ( κ , z ) η 0 H ˜ y ( κ , z ) ) = i k 0 ( 0 0 n x n y / o 1 n x 2 / o 0 0 1 + n y 2 / o n x n y / o n x n y o + n x 2 0 0 e n y 2 n x n y 0 0 ) ( E ˜ x ( κ , z ) E ˜ y ( κ , z ) η 0 H ˜ x ( κ , z ) η 0 H ˜ y ( κ , z ) ) ,
( E ˜ x ( κ , z ) E ˜ y ( κ , z ) η 0 H ˜ x ( κ , z ) η 0 H ˜ y ( κ , z ) ) = ( 1 0 1 0 W B W D W B W D 0 1 0 1 W C W B W C W B ) ( C + TM exp ( γ TM z ) C + TE exp ( γ TE z ) C TM exp ( γ TM z ) C TE exp ( γ TE z ) ) ,
W B ( κ ) = n x n y 0 + n x 2 , W C ( κ ) = 1 1 n x 2 / o 1 i k 0 γ TM , W D ( κ ) = 1 o + n x 2 1 i k 0 γ TE ;
γ TM ( κ ) = i k 0 e n y 2 n x 2 e / o , γ TE ( κ ) = i k 0 e n x 2 n y 2 .
V ˜ ( κ , z ) = ( E ˜ x , E ˜ y , η 0 H ˜ x , η 0 H ˜ y ) T ,
W + TM ( κ ) = ( 1 , W B , 0 , W C ) T , W + TE ( κ ) = ( 0 , W D , 1 , W B ) T , W TE ( κ ) = ( 1 , W B , 0 , W C ) T , W TE ( κ ) = ( 0 , W D , 1 , W B ) T ,
V ˜ ( κ , z ) = W + TM C + TM exp ( γ TM z ) + W + TE C + TE exp ( γ TE z ) + W TM C TM exp ( γ TM z ) + W TE C TE exp ( γ TE z ) .
| exp ( γ TM d ) | 0 , | exp ( γ TE d ) | 1
| exp ( γ TM d ) | 1 , | exp ( γ TE d ) | 0
V ˜ in ( κ , z ) = W ¯ + TM C + in , TM exp ( γ ¯ z ) + W ¯ + TE C + in , TE exp ( γ ¯ z ) , V ˜ out ( κ , z ) = W ¯ + TM C + out , TM exp ( γ ¯ z ) + W ¯ + TE C + out , TE exp ( γ ¯ z ) ,
W ¯ + TM ( κ ) = ( 1 , W ¯ B , 0 , W ¯ C ) T , W ¯ + TE ( κ ) = ( 0 , W ¯ D , 1 , W ¯ B ) T ,
W ¯ B ( κ ) = n x n y + n x 2 , W ¯ C ( κ ) = 1 1 n x 2 / 1 i k 0 γ ¯ , W ¯ D ( κ ) = 1 + n x 2 1 i k 0 γ ¯ ,
γ ¯ ( κ ) = i k 0 n x 2 n y 2 .
( C + out , TM ( κ , z out ) C + out , TE ( κ , z out ) ) = ( s 11 + + ( κ ) s 12 + + ( κ ) s 21 + + ( κ ) s 22 + + ( κ ) ) ( C + in , TM ( κ , z in ) C + in , TE ( κ , z in ) ) .
( C + out , TM ( κ , z out ) C + out , TE ( κ , z out ) ) = ( 0 0 0 1 ) ( C + in , TM ( κ , z in ) C + in , TE ( κ , z in ) )
( C + out , TM ( κ , z out ) C + out , TE ( κ , z out ) ) = ( 1 0 0 0 ) ( C + in , TM ( κ , z in ) C + in , TE ( κ , z in ) )
( C + in / out , TM ( κ , z in / out ) C + in / out , TE ( κ , z in / out ) ) = ( 1 0 W ¯ B ( κ ) W ¯ D ( κ ) ) 1 ( E ˜ x in / out ( κ , z in / out ) E ˜ y in / out ( κ , z in / out ) ) .
( E ˜ x out ( κ , z out ) E ˜ y out ( κ , z out ) ) = ( 1 0 W ¯ B ( κ ) W ¯ D ( κ ) ) ( 0 0 0 1 ) ( 1 0 W ¯ B ( κ ) W ¯ D ( κ ) ) 1 ( E ˜ x in ( κ , z in ) E ˜ y in ( κ , z in ) ) = ( 0 0 W ¯ B ( κ ) 1 ) ( E ˜ x in ( κ , z in ) E ˜ y in ( κ , z in ) ) .
( E x out ( ρ , z out ) E y out ( ρ , z out ) ) = ( 1 0 0 1 ) ( 0 0 W ¯ B ( κ ) 1 ) ( 0 0 ) ( E x in ( ρ , z in ) E y in ( ρ , z in ) ) = ( 0 0 1 W ¯ B ( κ ) 1 ) ( E x in ( ρ , z in ) E y in ( ρ , z in ) ) ,
( E ˜ x out ( κ , z out ) E ˜ y out ( κ , z out ) ) = ( 0 0 0 1 ) ( E ˜ x in ( κ , z in ) E ˜ y in ( κ , z in ) )
( E ˜ x out ( ρ , z out ) E ˜ y out ( ρ , z out ) ) = ( 0 0 0 1 ) ( E ˜ x in ( ρ , z in ) E ˜ y in ( ρ , z in ) )

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