Abstract

We show that a liquid crystal spatial light modulator (LCOS-SLM) can be used to display amplitude images, or phase holograms, which change in a pre-determined way when the display is tilted, i.e. observed under different angles. This is similar to the tilt-effect (also called ”latent image effect”) known from various security elements (”kinegrams”) on credit cards or bank notes. The effect is achieved without any specialized optical components, simply by using the large phase shifting capability of a ”thick” SLM, which extends over several multiples of 2π, in combination with the angular dependence of the phase shift. For hologram projection one can use the fact that the phase of a monochromatic wave is only defined modulo 2π. Thus one can design a phase pattern extending over several multiples of 2π, which transforms at different readout angles into different 2π-wrapped phase structures, due to the angular dependence of the modulo 2π operation. These different beams then project different holograms at the respective readout angles. In amplitude modulation mode (with inserted polarizer) the intensity of each SLM pixel oscillates over several periods when tuning its control voltage. Since the oscillation period depends on the readout angle, it is possible to find a certain control voltage which produces two (or more) selectable gray levels at a corresponding number of pre-determined readout angles. This is done with all SLM pixels individually, thus constructing different images for the selected angles. We experimentally demonstrate the reconstruction of multiple (Fourier- and Fresnel-) holograms, and of different amplitude images, by readout of static diffractive patterns in a variable angular range between 0° and 60°.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  6. W. Harm, A. Jesacher, G. Thalhammer, S. Bernet, and M. Ritsch-Marte, “How to use a phase-only spatial light modulator as a color display,” Opt. Lett. 40, 581–584 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  11. J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
    [Crossref]
  12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  13. B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
    [Crossref]
  14. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).
  15. C. Lingel, T. Haist, and W. Osten, “Optimizing the diffraction efficiency of SLM-based holography with respect to the fringing field effect,” Appl. Opt. 52, 68776883 (2013).
    [Crossref] [PubMed]

2014 (5)

2013 (3)

2011 (1)

2009 (1)

2006 (1)

2002 (1)

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[Crossref]

1976 (1)

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Albero, J.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Bernet, S.

Bowman, R.W.

Campos, J.

del Mar Sánchez-López, María

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

Estapé, M.

Fernández, E.

Floyd, R. W.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

García-Martínez, P.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Haist, T.

Harm, W.

Iemmi, C.

Jesacher, A.

Kohler, C.

Kozacki, Tomasz

Kress, B. C.

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
[Crossref]

Lingel, C.

Lizana, A.

Love, G.D.

Márquez, A.

Martín, N.

Martínez, J. L.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Meyrueis, P.

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
[Crossref]

Moreno, I.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17, 8491–8505 (2009).
[Crossref] [PubMed]

Osten, W.

Padgett, M.J.

Ritsch-Marte, M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schwab, X.

Sinzinger, S.

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[Crossref]

Steinberg, L.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

Thalhammer, G.

Vargas, A.

J. L. Martínez, I. Moreno, María del Mar Sánchez-López, A. Vargas, and P. García-Martínez, “Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM,” Opt. Express 21, 25866–25879 (2014).
[Crossref]

Yzuel, M. J.

Appl. Opt. (3)

Opt. Commun. (1)

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (1)

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[Crossref]

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SID (1)

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc. SID 17, 75–77 (1976).

Other (1)

B. C. Kress and P. Meyrueis, Applied Digital Optics: From Micro-Optics to Nanophotonics (Wiley, 2009).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Illustration of the orientation of liquid crystal molecules in the active layer of an LCOS-SLM at applied voltages (a) U = 0 and (b) U > 0. The polarization direction of the electric field vector E⃗ is in the same plane as the long axis of the LC molecules. The total phase shift results from the passage of the incoming, and the reflected beam, whose wavevectors k⃗ (indicated in (c) and (d)) have an angle of ±α with respect to the SLM surface. For the case of an applied voltage (b) the director of the LC molecules changes by an angle of β, which results in a total angle between LC director and light polarization of α + β for the incoming beam, and βα for the reflected beam (see (d)).

Fig. 2
Fig. 2

Topview illustration of the experimental setup used to measure the phase response of an LCOS-SLM at multiple angles of illumination α, and for multidirectional Fresnel hologram projection. For the projection of Fourier holograms a lens (not shown) with focal length of f = 40 cm is placed between beam splitter cube (BS) and polarization filter.

Fig. 3
Fig. 3

Interferometric measurements (solid lines) and simulations (dashed lines) of the phase responses for (a) λ = 445 nm and (b) λ = 638 nm as a function of the a applied voltage level U for three different angles of illumination α. For an overview, in (c) and (d) the simulated phase responses at the same set of incidence angles is shown for the blue (c) and the red (d) wavelength, respectively.

Fig. 4
Fig. 4

Scale factors of the phase response bi and ri as a function of the tilt angle for (a) λ = 445 nm and (b) λ = 638 nm. The solid lines correspond to the cosine of the tilt angle.

Fig. 5
Fig. 5

(a) Phase response in radians versus the applied voltage level at different angles of incidence. (b) Phase response taken modulo 2π. With each voltage level a triplet of phase shifts can be addressed simultaneously for different angles of incidence.

Fig. 6
Fig. 6

Algorithm for the optimization of a single pixel in the multiplexed hologram H*. For each voltage level k the deviations Δαi,Uk between the respective SLM induced phase-shift, and the ideal phase values in the individual computer-generated holograms Hi (which have been previously calculated with a GS-algorithm), are calculated according to Eq. (7). In the multiplexed hologram H* the pixels are assigned with the voltage levels Uk for which the variance Vk (calculated according to Eq. (8)) is minimal. The procedure can be also performed for all pixels of H*(nx, ny) in parallel by assigning two dimensional arrays to the respective variables Δi, Vk, Vmin, and H*.

Fig. 7
Fig. 7

Optical reconstructions from a multiplexed Fourier hologram at different angles of illumination. The colored rectangles indicate the areas that have been integrated to estimate relative diffraction efficiencies.

Fig. 8
Fig. 8

Projection of Fresnel holograms from a multiplexed hologram at read-out angles (a) 0°, (b) 30°, and (c) 60°.

Fig. 9
Fig. 9

(a): Amplitude modulation as a function of the applied voltage level U with a diagonally oriented polarization filter attached to the LCOS-SLM for two angles of incidence α. Lower part: Experimental performance of an LCOS-SLM as an angle sensitive display. Upper row: Original images. Lower row: Experimental reproductions. (f) and (h) are captured at zero angle of incidence and (g) and (i) are captured at α = 30°.

Equations (9)

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ϕ ( U , λ ) = 2 π d λ n ( U , λ ) ,
ϕ 0 ( β ¯ ) = 2 2 π d n λ 2 2 π d n λ cos β ¯ = 4 π d n λ ( 1 cos β ¯ ) ,
ϕ ( α , β ¯ ) = 2 2 π d n cos α λ 2 π d n λ [ cos ( β ¯ + α ) + cos ( β ¯ α ) ] = 4 π d λ ( n cos α n cos α cos β ¯ ) = 4 π d n λ cos α ( 1 cos β ¯ ) .
I ( α , λ 1 ) = 1 2 ( cos [ ϕ 0 ( U , λ 1 ) b ( α ) ] + 1 )
I ( α , λ 2 ) = 1 2 ( cos [ ϕ 0 ( U , λ 2 ) r ( α ) ] + 1 ) ,
ϕ ( U , λ , α ) = ϕ 0 ( U , λ ) cos α .
Δ α i , U k = H i ϕ 0 ( U k ) cos α i 2 π round [ H i ϕ 0 ( U k ) cos α i 2 π ] .
V U k = Δ α 1 , U k 2 + Δ α 2 , U k 2 + Δ α 3 , U k 2 .
H i , Fresnel ( n x , n y ) = { H i ( n x , n y ) + 2 π λ f [ ( x cos α ) 2 + y 2 ] } , mod 2 π ,

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