Abstract

Optical components manipulating both polarization and phase of wave fields find many applications in today’s optical systems. With modern lithography methods it is possible to fabricate optical elements with nanostructured surfaces from different materials capable of generating spatially varying, locally linearly polarized-light distributions, tailored to the application in question. Since such elements in general also affect the phase of the light field, the characterization of the function of such elements consists in measuring the phase and the polarization of the generated light, preferably at the same time. Here, we will present first results of an interferometric approach for a simultaneous and spatially resolved measurement of both phase and polarization, as long as the local polarization at any point is linear (e.g., for radially or azimuthally polarized light).

© 2014 Optical Society of America

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References

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  1. S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt 32, 5222–5229 (1993).
    [CrossRef]
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    [CrossRef]
  3. Z. Ghadyani, I. Vartiainen, I. Harder, W. Iff, A. Berger, N. Lindlein, and M. Kuittinen, “Concentric ring metal grating for generating radially polarized light,” Appl. Opt. 50, 2451–2457 (2011).
    [CrossRef]
  4. Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
    [CrossRef]
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    [CrossRef]
  7. D. Goldstein and D. H. Goldstein, Polarized Light (Dekker, 2011).
  8. H. Schreiber and J. H. Bruning, Phase Shifting Interferometry (Wiley, 2006).
  9. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
    [CrossRef]
  10. C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
    [CrossRef]
  11. A. Berger, V. Nercissian, K. Mantel, and I. Harder, “Evaluation algorithms for multistep measurement of spatially varying linear polarization and phase,” Opt. Lett. 37, 4140–4142 (2012).
    [CrossRef]
  12. S. Dmitriev, I. Harder, and N. Lindlein, “Artificial wave plates made from sub-wavelength structures,” in DGaO Proceedings (2012), http://www.dgao-proceedings.de/download/113/113_p16.pd .

2013 (1)

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

2012 (1)

2011 (2)

Z. Ghadyani, I. Vartiainen, I. Harder, W. Iff, A. Berger, N. Lindlein, and M. Kuittinen, “Concentric ring metal grating for generating radially polarized light,” Appl. Opt. 50, 2451–2457 (2011).
[CrossRef]

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

2010 (1)

2002 (2)

2000 (1)

1993 (1)

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt 32, 5222–5229 (1993).
[CrossRef]

Berger, A.

Biener, G.

Bomzon, Z.

Borchardt, J.

Bruning, J. H.

H. Schreiber and J. H. Bruning, Phase Shifting Interferometry (Wiley, 2006).

Cottrell, D. M.

Davis, J. A.

Dmitriev, S.

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

Dudley, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

Duparré, M.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[CrossRef]

Flamm, D.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[CrossRef]

Forbes, A.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

Ghadyani, Z.

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

Z. Ghadyani, I. Vartiainen, I. Harder, W. Iff, A. Berger, N. Lindlein, and M. Kuittinen, “Concentric ring metal grating for generating radially polarized light,” Appl. Opt. 50, 2451–2457 (2011).
[CrossRef]

Goldstein, D.

D. Goldstein and D. H. Goldstein, Polarized Light (Dekker, 2011).

Goldstein, D. H.

D. Goldstein and D. H. Goldstein, Polarized Light (Dekker, 2011).

Harder, I.

Hasman, E.

Iff, W.

Kaiser, T.

Kim, G. H.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt 32, 5222–5229 (1993).
[CrossRef]

Kimura, W. D.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt 32, 5222–5229 (1993).
[CrossRef]

Kleiner, V.

Kuittinen, M.

Leuchs, G.

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

Lindlein, N.

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

Z. Ghadyani, I. Vartiainen, I. Harder, W. Iff, A. Berger, N. Lindlein, and M. Kuittinen, “Concentric ring metal grating for generating radially polarized light,” Appl. Opt. 50, 2451–2457 (2011).
[CrossRef]

Mantel, K.

McNamara, D. E.

Nercissian, V.

Niv, A.

Rusina, O.

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

Schmidt, O. A.

Schreiber, H.

H. Schreiber and J. H. Bruning, Phase Shifting Interferometry (Wiley, 2006).

Schröter, S.

Schulze, C.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[CrossRef]

Sonehara, T.

Tidwell, S. C.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt 32, 5222–5229 (1993).
[CrossRef]

Vartiainen, I.

Appl. Opt (1)

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt 32, 5222–5229 (1993).
[CrossRef]

Appl. Opt. (2)

J. Eur. Opt. Soc. (1)

Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011).
[CrossRef]

New J. Phys. (1)

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[CrossRef]

Opt. Lett. (4)

Other (3)

D. Goldstein and D. H. Goldstein, Polarized Light (Dekker, 2011).

H. Schreiber and J. H. Bruning, Phase Shifting Interferometry (Wiley, 2006).

S. Dmitriev, I. Harder, and N. Lindlein, “Artificial wave plates made from sub-wavelength structures,” in DGaO Proceedings (2012), http://www.dgao-proceedings.de/download/113/113_p16.pd .

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Figures (12)

Fig. 1.
Fig. 1.

Interferogram of an object beam J⃗O with radial polarization and spiral phase and a linearly polarized reference beam J⃗R with a wedge-shaped phase. In the regions, where the polarizations of J⃗O and J⃗R are orthogonal, the visibility of the fringe pattern vanishes.

Fig. 2.
Fig. 2.

Sketch of a Mach–Zehnder interferometer adapted to the measurement task. Wollaston prism as a beam splitter, reference arm with an additional HWP, moveable mirror M1 in object arm, phase grating as a beam combiner.

Fig. 3.
Fig. 3.

Fresnel coefficients as a function of the angle of incidence.

Fig. 4.
Fig. 4.

Schematic representation of the effect of a radPOL. RadPOL with an additional phase vortex is generated from right and left circular input polarizations, respectively.

Fig. 5.
Fig. 5.

Functional sketch of the generation of radial (top), azimuthal (bottom) polarization and an “intermediate state” (middle) from linearly polarized light.

Fig. 6.
Fig. 6.

Orientation of polarization (left) and relative phase (right) measured with N5b algorithm for the element radPOL.

Fig. 7.
Fig. 7.

Orientation of polarization (left) and relative phase (right) measured with N5b algorithm for the element radHWP.

Fig. 8.
Fig. 8.

Orientation of polarization after addition of reverse spiral term (left) and phase distribution after addition of a measured phase with an illumination of reverse circular polarization (right) measured with N5*,4 algorithm of radPOL.

Fig. 9.
Fig. 9.

Orientation of polarization after addition of reverse spiral term (left) and phase (right) measured with N5*,4 algorithm of radHWP.

Fig. 10.
Fig. 10.

Orientation of polarization (left) and relative phase (right) measured with N3,3 algorithm of radPOL after addition of reverse spiral term.

Fig. 11.
Fig. 11.

Orientation of polarization (left) without the spiral term and relative phase (right) measured with N3,3 algorithm.

Fig. 12.
Fig. 12.

Difference of results measured with N3,3 and N6,6 algorithms showing polarization (left) and phase (right), rms values are 0.005 and 0.01, respectively.

Equations (12)

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J⃗O=uO(x,y)eiΦ(x,y)(cos(Ω(x,y))sin(Ω(x,y))),
J⃗R=uR(x,y)eiφ(cos(ω)sin(ω)).
I=I0[1+Vcos(Φφ)cos(Ωω)],
I0=uO2+uR2,V=2uOuRuO2+uR2.
φi=i·δφ;i{0;;Nφ1},ωi=i·δω;i{0;;Nω1}.
δφ=2πN,δω=4πN.
δφ,ω=2πNϕ,ωorδφ,ω*=2πNϕ,ω*1.
δφ=2π5
δω=4π5.
δφ*=π2
δω=π2.
δφ,ω=2π3

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