## Jones matrix treatment for optical Fourier processors with structured polarization |

Optics Express, Vol. 19, Issue 5, pp. 4583-4594 (2011)

http://dx.doi.org/10.1364/OE.19.004583

Acrobat PDF (1192 KB)

### 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

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]

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]

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]

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]

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]

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]

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]

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]

## 2. Jones matrix analysis of an optical Fourier polarization processor

*f*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

**h**(

*x*,

*y*) defined aswhereis 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.

*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.:where the symbol ⊗ denotes the convolution operation, and where, for clarity, we omitted the (

*x*,

*y*) dependence of the

*h*and

_{ij}*f*functions. This equation can be written in a compact way aswhere the symbol

_{ij}*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

## 3. Polarization gratings generated from scalar diffraction gratings

**f**(

*x*,

*y*)=cos(π

*ax*)⋅

**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

**H**(

*u*,

*v*) in Eq. (4) describing this simple Fourier filter is given by

**QWP**

_{θ}denotes the Jones matrix for a the quarter wave plate oriented at angle θ, i.e.: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

*a*λ

*f*. The generalized Jones matrix impulse response of this simple polarization Fourier filter is given by inverse Fourier transforming the elements in Eq. (12), leading towhere

*p*(

*x*,

*y*) =

**FT**

^{−1}{

*P*(

*u*,

*v*)}, and where we employed the trigonometrical relations

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]

**H**(

*u*,

*v*)

**F**(

*u*,

*v*) in Eq. (7) is simplified to:

**m**(

*x*,

*y*):

*x*) = π

*ax*+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]

*V*

_{0}isand the vectorial (polarization) distribution at the output plane of the optical processor is now given by

*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.

## 4. Generation of arbitrary structured linear polarization maps

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]

**9**(3), 78 (2007). [CrossRef]

*x*,

*y*) is the orientation of the linear polarization at the location (

*x*,

*y*). In order to produce such distribution, we encode the angle α in the form of the phase introduced by a phase-only spatial light modulator located on the input plane as:

*A*(

*u*,

*v*)=

**FT**{exp[

*i*α(

*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

*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]

*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:where

**P**

_{0}and

**P**

_{90}denote the Jones matrices for linear polarizers oriented at 0° and 90°, i.e.:and where

**QWP**

_{45}adopts the form:

**m**(

*x*,

*y*) describing the complete polarization processor is obtained by inverse Fourier transforming each component in the above matrix, i.e.,

*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 iswhere 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 π

*ax*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]

## 5. Conclusions

**51**(14), 2031–2038 (2004). [CrossRef]

**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}.

## Acknowledgments

## 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. |

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. |

3. | V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. |

4. | A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. |

5. | M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. |

6. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

7. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

8. | J. A. Davis, G. H. Evans, and I. Moreno, “Polarization-multiplexed diffractive optical elements with liquid-crystal displays,” Appl. Opt. |

9. | M. Fratz, D. M. Giel, and P. Fischer, “Digital polarization holograms with defined magnitude and orientation of each pixel’s birefringence,” Opt. Lett. |

10. | G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. |

11. | F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. |

12. | J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. |

13. | C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. |

14. | J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. |

15. | B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. |

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. |

17. | M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. |

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. |

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. |

20. | C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. |

21. | F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. |

22. | F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. |

23. | E. Wolf, |

24. | I. Moreno, M. J. Yzuel, J. Campos, and A. Vargas, “Jones matrix treatment for polarization Fourier optics,” J. Mod. Opt. |

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. |

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. |

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. |

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. |

**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

- 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]
- 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]
- V. Ramírez-Sánchez and G. Piquero, “Global beam shaping with nonuniformly polarized beams using amplitude transmitances,” Opt.Pura Apl. 40, 87–93 (2007).
- A. Volke and G. Heine, “Bringing order into light with structured polarizers,” Photonik Int. 2, 6–9 (2008).
- 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]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]
- 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]
- 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]
- G. Cincotti, “Polarization gratings: Design and applications,” IEEE J. Quantum Electron. 39(12), 1645–1652 (2003). [CrossRef]
- F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). [CrossRef]
- 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]
- C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33(20), 2287–2289 (2008). [CrossRef] [PubMed]
- J. L. Martínez, I. Moreno, and F. Mateos, “Hiding binary optical data with orthogonal circular polarizations,” Opt. Eng. 47(3), 030504 (2008). [CrossRef]
- B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000). [CrossRef]
- 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]
- 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]
- K. C. Toussaint, 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]
- 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]
- 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]
- F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]
- 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]
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge 2007).
- 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]
- 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]
- 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]
- 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]
- 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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