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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 5 — Feb. 28, 2011
  • pp: 4583–4594
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Jones matrix treatment for optical Fourier processors with structured polarization

Ignacio Moreno, Claudio Iemmi, Juan Campos, and Maria J. Yzuel  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 4583-4594 (2011)
http://dx.doi.org/10.1364/OE.19.004583


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Abstract

We present a Jones matrix method useful to analyze coherent optical Fourier processors employing structured polarization. The proposed method is a generalization of the standard classical optical Fourier transform processor, but considering vectorial spatial functions with two complex components corresponding to two orthogonal linear polarizations. As a result we derive a Jones matrix that describes the polarization output in terms of two vectorial functions defining respectively the structured polarization input and the generalized polarization impulse response. We apply the method to show and analyze an experiment in which a regular scalar diffraction grating is converted into equivalent polarization diffraction gratings by means of an appropriate polarization filtering. The technique is further demonstrated to generate arbitrary structured polarizations. Excellent experimental results are presented.

© 2011 OSA

1. Introduction

The generation of two dimensional polarization distributions has received great attention in the last years, and different methods to generate light beams with structured polarization have been theoretically proposed and experimentally probed, including the use of liquid crystal spatial light modulators [1

1. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39(10), 1549–1554 (2000). [CrossRef]

], sub-wavelength gratings [2

2. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). [CrossRef]

], polarization intereferometric setups [3

3. V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).

], or linear polarizers with spatial variation [4

4. A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).

]. Such polarization encodings have been successfully exploited in a variety of applications ranging from the formation of radially or azimuthally polarized beams [5

5. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996). [CrossRef] [PubMed]

], for producing sharper focusing [6

6. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

,7

7. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

], the formation of polarization sensitive computer generated holograms [8

8. J. A. Davis, G. H. Evans, and I. Moreno, “Polarization-multiplexed diffractive optical elements with liquid-crystal displays,” Appl. Opt. 44(19), 4049–4052 (2005). [CrossRef] [PubMed]

,9

9. M. Fratz, D. M. Giel, and P. Fischer, “Digital polarization holograms with defined magnitude and orientation of each pixel’s birefringence,” Opt. Lett. 34(8), 1270–1272 (2009). [CrossRef] [PubMed]

], or the realization of the so called polarization diffraction gratings [10

10. G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003). [CrossRef]

], gratings with a periodic variation of the polarization component, with applications in polarimetry [11

11. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). [CrossRef]

,12

12. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26(9), 587–589 (2001). [CrossRef]

] or in the design of high efficient gratings [13

13. C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008). [CrossRef] [PubMed]

]. Polarization encoding has also been used for security systems either to hide [14

14. J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008). [CrossRef]

] or to encrypt information [15

15. B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000). [CrossRef]

,16

16. H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004). [CrossRef]

].

Recently, some other approaches to produce polarization maps have been proposed for different applications [17

17. M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002). [CrossRef]

20

20. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007). [CrossRef]

], which employ optical setups that can be considered as different variations of the classical optical Fourier processor. For instance, in Ref [17

17. M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002). [CrossRef]

]. a Wollaston prism was employed to create linear phases with opposite sign for two orthogonal polarizations, in order to illuminate a ferroelectric liquid crystal modulator that displays a binary phase pattern. The proper adjustment of the phase displayed by the modulator permits to create a proper polarization structure around a diffraction order in Fourier plane, which can be filtered to produce at the final output plane the desired polarization map. In Ref [18

18. K. C. Toussaint Jr, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 (2005). [CrossRef] [PubMed]

]. radially and azimuthally polarized vector beams were generated by means of a diffractive optical element (DOE) interferometer, where the two first diffracted orders are directed to two half-wave plates with different orientation. A second identical DOE recombines the two transmitted beams to create the desired polarization distribution. Similar techniques were employed to create more complicated polarized beams in Refs [19

19. X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef] [PubMed]

]. and [20

20. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007). [CrossRef]

], where different types of polarization filtering were applied to two modulated beams that were then recombined. All these works treat the polarization transformations step by step, usually employing Jones vectors to DESCRIBE the states in each plane (input mask, diffraction Fourier plane, recombination output plane).

The paper is organized as follows: in section 2 we introduce the Jones matrix – Fourier transform formalism, and extend it to the optical processor. In section 3 we present how it can be applied to generate polarization diffraction gratings from a regular scalar diffraction grating filtered in the Fourier plane by means of polarization elements. Finally in section 4 we extend this processor to create arbitrary maps of linear polarizations with variable orientation, controlled from the phase introduced by a phase-only spatial light modulator located at the input plane. In all cases, the theoretical analysis based on the combined Jones matrix – Fourier transform formalism is accompanied with experimental demonstration of the generated polarization patterns.

2. Jones matrix analysis of an optical Fourier polarization processor

Let us consider a generalized optical Fourier processor as sketched in Fig. 1
Fig. 1 Scheme of the optical Fourier transform polarization processor.
, where both the input plane and the Fourier filter plane are spatially dependant polarization masks. For simplicity we consider the classical 4f configuration with two convergent lenses with equal focal length. The input polarization mask can be described by a Jones matrix which depends on the spatial coordinates (x,y) as

f(x,y)=(fxx(x,y)fxy(x,y)fyx(x,y)fyy(x,y)).
(1)

The polarization mask introduced in the Fourier plane (the generalized polarization transfer function) is also characterized by a polarization transmission described by a spatially variant Jones matrix defined as

H(u,v)=(Hxx(u,v)Hxy(u,v)Hyx(u,v)Hyy(u,v)).
(4)

The Jones matrix h(x,y) defined as
h(x,y)=(hxx(x,y)hxy(x,y)hyx(x,y)hyy(x,y)),
(5)
where
hij(x,y)=FT-1{Hij(u,v)},
(6)
is obtained by inverse Fourier transforming each element of the Jones matrix in Eq. (4), and can be interpreted as the vectorial (polarization) generalization of the scalar impulse response of the classical scalar optical processor.

The passage of light through the input mask, propagation to the Fourier plane, and passage through the polarization filter, is given by the usual Jones matrix product, i.e.:
H(u,v)F(u,v)=(Hxx(u,v)Hxy(u,v)Hyx(u,v)Hyy(u,v))(Fxx(u,v)Fxy(u,v)Fyx(u,v)Fyy(u,v))==(HxxFxx+HxyFyxHxxFxy+HxyFyyHyxFxx+HyyFyxHyxFxy+HyyFyy),
(7)
where we omitted (u,v) dependence for clarity. The propagation to the final output plane implies another Fourier transformation and, therefore, the whole transformation from the input to the output plane can described by a Jones matrix m(x,y) obtained by Fourier transforming each element of the matrix in Eq. (7), i.e.:
m(x,y)=(hxxfxx+hxyfyxhxxfxy+hxyfyyhyxfxx+hyyfyxhyxfxy+hyyfyy),
(8)
where the symbol ⊗ denotes the convolution operation, and where, for clarity, we omitted the (x,y) dependence of the hij and fij functions. This equation can be written in a compact way as
m(x,y)=f(x,y)¯h(x,y),
(9)
where the symbol ¯ denotes a generalized convolution between the two Jones matrices as defined by Eq. (8).

The interest of the proposed formalism comes from the direct application of the Jones matrix calculus to obtain the output polarization maps. If the generalized polarization processor is illuminated with a polarized plane wave with a polarization state described by the Jones vector V 0, the polarization maps V 1(u,v) just before the polarization filter at the Fourier plane, and V 2(x,y) at the final output plane, are given respectively by

V1(u,v)=F(u,v)V0,V2(x,y)=m(x,y)V0.
(10)

3. Polarization gratings generated from scalar diffraction gratings

As a first example we present a very simple method to generate polarization diffraction gratings from a regular scalar diffraction grating through Fourier polarization filtering. For simplicity, we consider a diffraction grating that generates only the ±1 diffraction orders. The Jones matrix in Eq. (1) describing the input polarization mask can be written in this case as f(x,y)=cos(πax)⋅1 where 1 denotes the 2×2 identity matrix, being p=2/a the period of the grating. The complex amplitude generated at the Fourier plane is given by

F(u,v)=12(δ(ua,v)+δ(u+a,v))1.
(11)

The state of polarization in the two generated diffraction orders is the same as that of the incoming beam. The realization of the spatially periodic polarization structure that characterizes polarization diffraction gratings is obtained by modifying the polarization of each of the two diffraction orders in a different manner. For instance, let us consider two quarter-wave plates (QWP) located on each diffraction order, but oriented at 0 and 90° respectively (i.e., one QWP has the fast axis horizontal, while the second one has the fast axis vertical). The Jones matrix H(u,v) in Eq. (4) describing this simple Fourier filter is given by
H(u,v)=P(ua,v)QWPθ=90º+P(u+a,v)QWPθ=0º==P(u,v){δ(ua,v)(+i001)+δ(u+a,v)   (100+i)}==P(u,v)(iδ(ua,v)+δ(u+a,v)00δ(ua,v)+iδ(u+a,v)),
(12)
where QWP θ denotes the Jones matrix for a the quarter wave plate oriented at angle θ, i.e.:
QWPθ=R(θ)(100+i)R(+θ),
(13)
being R(θ) the 2x2 rotation matrix. P(u,v) in Eq. (12) denotes a binary amplitude function describing the physical aperture of each QWP, which we assume identical and having a size smaller than aλf. The generalized Jones matrix impulse response of this simple polarization Fourier filter is given by inverse Fourier transforming the elements in Eq. (12), leading to
h(x,y)=p(x,y)(ie+iπax+eiπax00e+iπax+ieiπax)==p(x,y)2eiπ4(cos(πax+π4)00sin(πax+π4)),
(14)
where p(x,y) = FT −1{P(u,v)}, and where we employed the trigonometrical relations cos(x)sin(x)=2cos(x+π4) and cos(x)+sin(x)=2sin(x+π4). Note that this polarization impulse response is similar to the generalized two aperture polarization interference that was analyzed in Ref [24

24. I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004). [CrossRef]

].

Since the diffraction orders are located on the center of the corresponding QWP, the product H(u,v)F(u,v) in Eq. (7) is simplified to:

H(u,v)F(u,v)=12(iδ(ua,v)+δ(u+a,v)00δ(ua,v)+iδ(u+a,v)).
(15)

Either inverse Fourier transforming this Jones matrix or equivalently calculating the result in Eq. (8), the whole polarization Fourier processor is described by the Jones matrix m(x,y):

m(x,y)=eiπ4(cos(πax+π4)00sin(πax+π4)).
(16)

Let us now assume that the polarization Fourier processor is illuminated with linearly polarized light oriented at 45°, so the input Jones vector is given by

V0=12(11).
(17)

The vectorial (polarization) distribution at the output plane of the optical processor is given by

V2(x,y)=m(x,y)V0=eiπ42(cos(πax+π4)sin(πax+π4)).
(18)

The result is a one dimensional periodic distribution of linear polarization states, where the orientation θ of the linear polarization varies as θ(x) = πax+π/4, i.e., with the same period 2/a as the grating in the input plane. This polarization distribution is basically equivalent to the one generated with the polarization diffraction grating proposed by Gori in Ref [11

11. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). [CrossRef]

]. Figure 2(a)
Fig. 2 Scheme of the optical Fourier transform polarization processor to generate polarization diffraction gratings. (a) Illumination with linearly polarized light oriented at 45° produces an output with a periodic map of linear polarizations with variable orientation. (b) Illumination with circularly polarized light produces an output with a periodic map of elliptical polarizations with fixed azimuth and variable ellipticity.
shows a scheme of this proposed polarization optical Fourier processor and the output when it is illuminated with linear polarization at 45°.

Another interesting case is obtained when the same polarization Fourier processor is illuminated with right handed circularly polarized light (Fig. 2(b)). In this case V 0 is
V0=12(1+i),
(19)
and the vectorial (polarization) distribution at the output plane of the optical processor is now given by

V2(x,y)=m(x,y)V0=eiπ42(cos(πax+π4)+isin(πax+π4)).
(20)

Now the result is again a one dimensional periodic distribution of polarization states along the x-direction, with the same period 2/a, but now the polarization states are always elliptical, with ellipses centered on the x-y axes (i.e., the azimuth is fixed at 0-90°), being now the ellipticity the parameter that is periodically changed.

We have experimentally built such a polarization Fourier processor. We employed regular phase grating at the input plane, and two QWPs with small aperture at the Fourier plane. We blocked all but the ±1 generated diffraction orders, and placed two QWPs on their location with the adequate orientation. We used a CCD camera with a microscope objective to visualize the final output plane. In order to visualize the polarization nature of the periodic distribution in the output plane, we placed a linear polarizer analyzer just before the objective.

4. Generation of arbitrary structured linear polarization maps

f(x,y)=exp{i[α(x,y)+πax]}.
(22)

This phase-only function generates, in the Fourier plane, the function A(u,v)=FT{exp[iα(x,y)]}, centered at the spatial frequency u=a. If the phase-only function in Eq. (22) is binarized to phase values 0 and π, the resulting function, f BIN(x,y)=BIN{f(x,y)}, can be directly implemented in a binary phase modulator. Since this is a scalar phase function, the input mask does not affect the polarization and it can be written as

f(x,y)=fBIN(x,y)1 .
(23)

Thus, it generates a diffraction pattern whose main contributions are the direct expected Fourier transform function A(u,v) centered at the expected location u=a, and its complex conjugated version located at the location u=−a (each one carrying (2/π)2=40.3% of the total energy in the Fourier plane) [28

28. I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995). [CrossRef]

]. Ignoring this amplitude factor, it can be expressed as:

F(u,v){A(ua,v)+A*(ua,v)}1.
(24)

We assume that the spatial bandwidth of the function A(u,v) fits within the aperture P(u,v) of the QWP’s. Therefore, the product H(u,v)F(u,v) in Eq. (7) can be now written as:
H(u,v)F(u,v)=A(ua,v)QPW45P0+A*(ua,v)QPW45P90.
(25)
where P 0 and P 90 denote the Jones matrices for linear polarizers oriented at 0° and 90°, i.e.:
P0=(1000),P90=(0001),
(26)
and where QWP 45 adopts the form:

QWP45=R(45º)(100+i)R(+45º)=eiπ42(1ii1).
(27)

The explicit calculation of Eq. (25) leads to

H(u,v)F(u,v)=eiπ42(A(ua,v)iA*(ua,v)iA(ua,v)A*(ua,v)).
(28)

Therefore, the Jones matrix m(x,y) describing the complete polarization processor is obtained by inverse Fourier transforming each component in the above matrix, i.e.,

m(x,y)=eiπ42(ei[α(x,y)+πax]iei[α(x,y)+πax]iei[α(x,y)+πax]ei[α(x,y)+πax]).
(29)

Let us assume that the input polarization to the optical processor is linearly polarized oriented at 45°. Then the final polarization distribution can be obtained in a direct way from the product V 2(x,y) = m(x,y)⋅V 0, where m(x,y) and V 0 are given by Eqs, (29) and (17) respectively. The result is
V2(x,y)=(cos{α(x,y)+πax+π4}sin{α(x,y)+πax+π4}),
(30)
where the same trigonometric relations used to derive Eq. (14) are employed here. Note that this result in Eq. (30) basically provides the desired result in Eq. (21), except for the linear term πax and the π/4 phase factor within the sine and cosine functions. While the π/4 phase factor implies this additional rotation, which can be simply compensated in the design of the function α, the linear phase terms in Eq. (30) contribute to modify the orientation of the polarization states in the final plane by adding a periodic variation. Therefore, it is necessary to eliminate these linear phases. The system in Ref [19

19. X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef] [PubMed]

]. is a diffraction interferometer, where a diffraction grating was introduced on the final plane in order to eliminate these linear phase terms. Here we use an alternative setup as shown in Fig. 4, where the two beams coming from the two diffraction orders at the Fourier plane are made collinear with the help of a mirror (M) and the polarization beam splitter (PBS). Once the two beam are collinear, they are Fourier transformed by the second converging lens and the SLM is imaged onto the CCD camera (note that the final Fourier transformation in the experiment is a direct transform, as opposite to the inverse transform considered to derive Eq. (29), but the result is equivalent except for an inversion).

As input binary phase SLM we employ a liquid crystal display sandwiched between two linear polarizers. Thus, this is a polarization device which must be described with a corresponding Jones matrix. However, if the polarization emerging from the SLM is split with equal power by the PBS, the simpler above results are reproduced. Figure 5
Fig. 5 (a) Desired polarization map. Experimental results obtained at the output plane for (b) absence of the analyzer and analyzer oriented at (c) 0°, (d) 45°, (e) 90° and (f) 135°.
shows the experimental results obtained with this setup. Figure 5(a) shows a simple distribution of linear polarizations that we want to generate. This is encoded by properly selecting the angle α in the input image. Figure 5(b) shows the CCD capture of the output plane of the optical processor, when no final analyzer is placed. In this situation the different orientations of the linear polarization in each region is not visible. In order to visualize them we placed the analyzer just before the output plane, and captured images when rotating it. Then, the area of the final image with a linear polarization crossed to the transmission axis of the polarizer appears as dark. When the analyzer is oriented horizontal (Fig. 5(c)), the circle appears bright and the background appears dark. The situation is reversed when the analyzer is oriented vertical (Fig. 5(e)). Also note that the circle is confused with the background when the analyzer is oriented at 45° or 135° (Figs. 5(d) and 5(f)), since the horizontal and vertical linear polarizations are transmitted with equal energy through the analyzer.

5. Conclusions

In summary, we presented a Jones matrix based formalism to analyze an optical Fourier processor. This formalism extends our previously reported method to analyze a polarization Fourier transform difractometer [24

24. I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004). [CrossRef]

], and it is revealed here as a useful tool to analyze optical Fourier processors were local changes in the polarization can occur either in the input plane, in the Fourier plane, or in both planes. We extend classical Fourier optics concepts to vectorial polarization functions, like for instance the Jones matrix h(x,y) which generalizes the impulse response of frequency Fourier filter. As a result of this whole Fourier analysis, we derive a spatial Jones matrix m(x,y) which describes the actuation of the complete polarization processor on the Jones vector V 0 describing the state of polarization launched to the processor in its input plane. The final polarization pattern in the output plane is described as the spatially dependant Jones vector V 2(x,y) directly calculated applying the usual Jones calculus as V 2(x,y) = m(x,y)⋅V 0.

We have presented two examples of application of such a polarization optical Fourier processor. The goal of the first one is to convert a regular scalar diffraction grating at the input plane into a periodic one dimensional distribution of polarization states on the output plane (equivalent to a polarization diffraction grating). For that purpose two QWPs are introduced in the Fourier plane, with orthogonal orientations, and centered on the location on the ±1 diffraction orders. We presented how the processor is capable to generate different periodic polarization distributions depending on the input polarization. This system therefore permits to reproduce a polarization diffraction grating from a regular grating, using very simple optical components.

We have provided excellent experimental results for both presented examples, that probe the validity of the method and show its potentiality use in Fourier optics based polarization systems.

Acknowledgments

We acknowledge financial support from Ministerio de Ciencia e Innovación from Spain (ref. FIS2009-13955-C02-01 and −02). C. Iemmi gratefully acknowledges the support of the Universidad de Buenos Aires and CONICET (Argentina).

References and links

1.

J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39(10), 1549–1554 (2000). [CrossRef]

2.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). [CrossRef]

3.

V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).

4.

A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).

5.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996). [CrossRef] [PubMed]

6.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

7.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

8.

J. A. Davis, G. H. Evans, and I. Moreno, “Polarization-multiplexed diffractive optical elements with liquid-crystal displays,” Appl. Opt. 44(19), 4049–4052 (2005). [CrossRef] [PubMed]

9.

M. Fratz, D. M. Giel, and P. Fischer, “Digital polarization holograms with defined magnitude and orientation of each pixel’s birefringence,” Opt. Lett. 34(8), 1270–1272 (2009). [CrossRef] [PubMed]

10.

G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003). [CrossRef]

11.

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). [CrossRef]

12.

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26(9), 587–589 (2001). [CrossRef]

13.

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008). [CrossRef] [PubMed]

14.

J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008). [CrossRef]

15.

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000). [CrossRef]

16.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulators,” J. Opt. A, Pure Appl. Opt. 6(6), 524–528 (2004). [CrossRef]

17.

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27(21), 1929–1931 (2002). [CrossRef]

18.

K. C. Toussaint Jr, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. 30(21), 2846–2848 (2005). [CrossRef] [PubMed]

19.

X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef] [PubMed]

20.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007). [CrossRef]

21.

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

22.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000). [CrossRef]

23.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge 2007).

24.

I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004). [CrossRef]

25.

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006). [CrossRef]

26.

I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007). [CrossRef]

27.

A. Martínez-García, I. Moreno, M. M. Sánchez-López, and P. García-Martínez, “Operational modes of a ferroelectric LCoS modulator for displaying binary polarization, amplitude, and phase diffraction gratings,” Appl. Opt. 48(15), 2903–2914 (2009). [CrossRef] [PubMed]

28.

I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995). [CrossRef]

OCIS Codes
(230.6120) Optical devices : Spatial light modulators
(260.5430) Physical optics : Polarization
(070.2615) Fourier optics and signal processing : Frequency filtering

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: November 9, 2010
Revised Manuscript: January 1, 2011
Manuscript Accepted: January 25, 2011
Published: February 24, 2011

Citation
Ignacio Moreno, Claudio Iemmi, Juan Campos, and Maria J. Yzuel, "Jones matrix treatment for optical Fourier processors with structured polarization," Opt. Express 19, 4583-4594 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4583


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References

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  19. X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef] [PubMed]
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  24. I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. 51(14), 2031–2038 (2004). [CrossRef]
  25. I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Binary polarization pupil filter: Theoretical analysis and experimental realization with a liquid crystal display,” Opt. Commun. 264(1), 63–69 (2006). [CrossRef]
  26. I. Moreno, C. Iemmi, J. Campos, M. J. Yzuel, and A. Vargas, “Polarization vortices generation by diffraction from a four quadrant polarization mask,” Opt. Commun. 276(2), 222–230 (2007). [CrossRef]
  27. A. Martínez-García, I. Moreno, M. M. Sánchez-López, and P. García-Martínez, “Operational modes of a ferroelectric LCoS modulator for displaying binary polarization, amplitude, and phase diffraction gratings,” Appl. Opt. 48(15), 2903–2914 (2009). [CrossRef] [PubMed]
  28. I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423–6432 (1995). [CrossRef]

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