Abstract

We analyze the performance of a ferroelectric liquid crystal on silicon display (FLCoS) as a binary polarization diffraction grating. We analyze the correspondence between the two polarization states emerging from the displayed grating and the polarization and intensity of the diffracted orders generated at the Fourier diffraction plane. This polarization-diffraction analysis leads, in a simple manner, to configurations yielding binary amplitude or binary phase modulation by incorporating an analyzer on the reflected beam. Based on this analysis, we present two useful variations of the polarization configuration. The first is a simplification using a single polarizer, which provides equivalent results for amplitude or phase modulation as the more general operational mode involving two polarizers. The second variation is proposed to compensate the reduction of the diffraction efficiency when the operating wavelength differs from the design one (for which the FLCoS liquid-crystal layer acts as a half-wave plate). In this situation we show how the ideal grating performance can be recovered in spite of the phase-shift mismatch originated by chromatic dispersion. In all cases, we provide experimental results that verify the theoretical analyses.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
    [CrossRef]
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    [CrossRef]
  21. I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43, 6278-6284 (2004).
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    [CrossRef] [PubMed]
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2009 (2)

M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
[CrossRef]

J. L. Martinez, A. Martinez-Garcia, and I. Moreno, “Wavelength-compensated color Fourier diffractive optical elements using a ferroelectric liquid crystal on silicon display and a color-filter wheel,” Appl. Opt. 48, 911-918(2009).
[CrossRef] [PubMed]

2008 (1)

A. Martínez-García, I. Moreno, and M. M. Sánchez-López, “Comparative analysis of time and spatial multiplexed diffractive optical elements in a ferroelectric liquid crystal display,” Jap. J. Appl. Phys. 47, 1589-1594 (2008).
[CrossRef]

2007 (1)

A. Martínez, M. M. Sánchez-López, and I. Moreno, “Phasor analysis of binary amplitude gratings with different fill factor,” Eur. J. Phys. 28, 805-816 (2007).
[CrossRef]

2006 (1)

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

2005 (1)

W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical metrology,” Opt. Pura Apl. 38, 71-81 (2005).

2004 (4)

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

J. A. Davis and G. H. Evans, “Polarizing binary diffraction grating beam splitter,” Opt. Lett. 29, 1443-1445 (2004).
[CrossRef] [PubMed]

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

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43, 6278-6284 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (2)

I. G. Manolis, T. D. Wilkinson, M. M. Redmond, and W. A. Crossland, “Reconfigurable multilevel phase holograms for optical switches,” IEEE Photon. Technol. Lett. 14, 801-803 (2002).
[CrossRef]

M. Johansson, S. Hard, B. Robertson, L. Manolis, T. Wilkinson, and W. Crossland, “Adaptive beam steering implemented in a ferroelectric liquid-crystal spatial-light-modulator free-space, fiber-optic switch,” Appl. Opt. 41, 4904-4911 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

1998 (1)

M. Le Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321-330 (1998).
[CrossRef]

1997 (1)

1995 (2)

P. Pellat-Finet, and M. Le Doucen, “Polarization properties of birefringence gratings,” Optik (Jena) 100, 159-166 (1995).

S. T. Warr and R. J. Mears, “Polarisation insensitive operation of ferroelectric liquid crystal devices,” Electron. Lett. 31, 714-716 (1995).
[CrossRef]

1994 (2)

1992 (1)

Beaudoin, N.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

Birch, M.

Bougrenet de la Tocnaye, J. L.

Brich, M.

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

Campos, J.

Cincotti, G.

G. Cincotti, “Polarization gratings: design and applications,” IEEE J. Quant. Electron. 39, 1645-1652 (2003).
[CrossRef]

Crain, J.

Crossland, W.

Crossland, W. A.

I. G. Manolis, T. D. Wilkinson, M. M. Redmond, and W. A. Crossland, “Reconfigurable multilevel phase holograms for optical switches,” IEEE Photon. Technol. Lett. 14, 801-803 (2002).
[CrossRef]

Davis, J. A.

de Bougrenet de la Tocnaye, J. L.

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

Doroski, D.

Dupont, L.

Evans, G. H.

Fracasso, B.

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

García-Martínez, P.

M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gori, F.

Gourlay, J.

Hard, S.

Heggarty, K.

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

W. J. Hossack, E. Theofanidou, J. Crain, K. Heggarty, and M. Birch, “High-speed holographic optical tweezers using a ferroelectric liquid crystal microdisplay,” Opt. Express 11, 2053-2059 (2003).
[CrossRef] [PubMed]

Hossack, W. J.

Huard, S.

S. Huard, Polarization of Light (Wiley, 1997).

Iemmi, C.

Johansson, M.

Johnson, K. M.

Kohler, C.

W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical metrology,” Opt. Pura Apl. 38, 71-81 (2005).

Krüerke, D.

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

Le Doucen, M.

M. Le Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321-330 (1998).
[CrossRef]

P. Pellat-Finet, and M. Le Doucen, “Polarization properties of birefringence gratings,” Optik (Jena) 100, 159-166 (1995).

Letort, C.

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

Liesener, J.

W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical metrology,” Opt. Pura Apl. 38, 71-81 (2005).

Manolis, I. G.

I. G. Manolis, T. D. Wilkinson, M. M. Redmond, and W. A. Crossland, “Reconfigurable multilevel phase holograms for optical switches,” IEEE Photon. Technol. Lett. 14, 801-803 (2002).
[CrossRef]

Manolis, L.

Márquez, A.

Martinez, J. L.

Martínez, A.

A. Martínez, M. M. Sánchez-López, and I. Moreno, “Phasor analysis of binary amplitude gratings with different fill factor,” Eur. J. Phys. 28, 805-816 (2007).
[CrossRef]

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

Martinez-Garcia, A.

Martínez-García, A.

M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
[CrossRef]

A. Martínez-García, I. Moreno, and M. M. Sánchez-López, “Comparative analysis of time and spatial multiplexed diffractive optical elements in a ferroelectric liquid crystal display,” Jap. J. Appl. Phys. 47, 1589-1594 (2008).
[CrossRef]

McOwan, P.

Mears, R. J.

S. T. Warr and R. J. Mears, “Polarisation insensitive operation of ferroelectric liquid crystal devices,” Electron. Lett. 31, 714-716 (1995).
[CrossRef]

Moreno, I.

M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
[CrossRef]

J. L. Martinez, A. Martinez-Garcia, and I. Moreno, “Wavelength-compensated color Fourier diffractive optical elements using a ferroelectric liquid crystal on silicon display and a color-filter wheel,” Appl. Opt. 48, 911-918(2009).
[CrossRef] [PubMed]

A. Martínez-García, I. Moreno, and M. M. Sánchez-López, “Comparative analysis of time and spatial multiplexed diffractive optical elements in a ferroelectric liquid crystal display,” Jap. J. Appl. Phys. 47, 1589-1594 (2008).
[CrossRef]

A. Martínez, M. M. Sánchez-López, and I. Moreno, “Phasor analysis of binary amplitude gratings with different fill factor,” Eur. J. Phys. 28, 805-816 (2007).
[CrossRef]

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

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

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43, 6278-6284 (2004).
[CrossRef] [PubMed]

Osten, W.

W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical metrology,” Opt. Pura Apl. 38, 71-81 (2005).

Pellat-Finet, P.

M. Le Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321-330 (1998).
[CrossRef]

P. Pellat-Finet, and M. Le Doucen, “Polarization properties of birefringence gratings,” Optik (Jena) 100, 159-166 (1995).

Redmond, M. M.

I. G. Manolis, T. D. Wilkinson, M. M. Redmond, and W. A. Crossland, “Reconfigurable multilevel phase holograms for optical switches,” IEEE Photon. Technol. Lett. 14, 801-803 (2002).
[CrossRef]

Robertson, B.

Samus, S.

Sánchez-López, M. M.

M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
[CrossRef]

A. Martínez-García, I. Moreno, and M. M. Sánchez-López, “Comparative analysis of time and spatial multiplexed diffractive optical elements in a ferroelectric liquid crystal display,” Jap. J. Appl. Phys. 47, 1589-1594 (2008).
[CrossRef]

A. Martínez, M. M. Sánchez-López, and I. Moreno, “Phasor analysis of binary amplitude gratings with different fill factor,” Eur. J. Phys. 28, 805-816 (2007).
[CrossRef]

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

Sharp, G. D.

Stockley, J. E.

Tervo, J.

Theofanidou, E.

Turunen, J.

Underwood, I.

Vargas, A.

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

Vass, D. G.

Velásquez, P.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

Waring, M. A.

Warr, S. T.

S. T. Warr and R. J. Mears, “Polarisation insensitive operation of ferroelectric liquid crystal devices,” Electron. Lett. 31, 714-716 (1995).
[CrossRef]

Wilkinson, T.

Wilkinson, T. D.

I. G. Manolis, T. D. Wilkinson, M. M. Redmond, and W. A. Crossland, “Reconfigurable multilevel phase holograms for optical switches,” IEEE Photon. Technol. Lett. 14, 801-803 (2002).
[CrossRef]

Worboys, M.

Yzuel, M. J.

Appl. Opt. (6)

Electron. Lett. (1)

S. T. Warr and R. J. Mears, “Polarisation insensitive operation of ferroelectric liquid crystal devices,” Electron. Lett. 31, 714-716 (1995).
[CrossRef]

Eur. J. Phys. (1)

A. Martínez, M. M. Sánchez-López, and I. Moreno, “Phasor analysis of binary amplitude gratings with different fill factor,” Eur. J. Phys. 28, 805-816 (2007).
[CrossRef]

Ferroelectrics (1)

K. Heggarty, B. Fracasso, C. Letort, J. L. de Bougrenet de la Tocnaye, M. Brich, and D. Krüerke, “Silicon blackplane ferroelectric liquid crystal spatial light modulator for uses within an optical telecommunications environment,” Ferroelectrics 312, 39-55 (2004).
[CrossRef]

IEEE J. Quant. Electron. (1)

G. Cincotti, “Polarization gratings: design and applications,” IEEE J. Quant. Electron. 39, 1645-1652 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

I. G. Manolis, T. D. Wilkinson, M. M. Redmond, and W. A. Crossland, “Reconfigurable multilevel phase holograms for optical switches,” IEEE Photon. Technol. Lett. 14, 801-803 (2002).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. A: Pure Appl. Opt. (2)

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Characterization and optimization of the constrast-ratio of a ferrroelectric liquid crystal optical modulator,” J. Opt. A: Pure Appl. Opt. 8, 1013-1018 (2006).
[CrossRef]

M. M. Sánchez-López, P. García-Martínez, A. Martínez-García, and I. Moreno, “Poincaré sphere analysis of a ferroelectric liquid crystal optical modulator: application to optimize the contrast ratio,” J. Opt. A: Pure Appl. Opt. 11, 015507 (2009).
[CrossRef]

Jap. J. Appl. Phys. (1)

A. Martínez-García, I. Moreno, and M. M. Sánchez-López, “Comparative analysis of time and spatial multiplexed diffractive optical elements in a ferroelectric liquid crystal display,” Jap. J. Appl. Phys. 47, 1589-1594 (2008).
[CrossRef]

Opt. Commun. (1)

M. Le Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321-330 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Opt. Pura Apl. (1)

W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical metrology,” Opt. Pura Apl. 38, 71-81 (2005).

Optik (Jena) (1)

P. Pellat-Finet, and M. Le Doucen, “Polarization properties of birefringence gratings,” Optik (Jena) 100, 159-166 (1995).

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

S. Huard, Polarization of Light (Wiley, 1997).

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Figures (8)

Fig. 1
Fig. 1

(a) Scheme of the operating system. (b), (c) Oscilloscope traces corresponding to the detected optical intensity for λ = 568 nm with the two polarizers oriented at 15 ° . (b) FLCoS displays a uniform white screen. (c) FLCoS displays a uniform black screen.

Fig. 2
Fig. 2

Normalized transmittances i a and i b of the system polarizer–FLCoS display–analyzer, as a function of the input polarizer angle. The analyzer is always crossed to the input polarizer. Two Ar Kr laser wavelengths are used: λ = 568 nm and λ = 647 nm , respectively.

Fig. 3
Fig. 3

Oscilloscope traces corresponding to the detected optical intensity for λ = 568 nm , when the FLCoS displays a white uniform screen and with the input polarizer oriented at 0 ° . The output polarizer is oriented at (a)  0 ° , (b)  45 ° , (c)  90 ° , and (d)  135 ° .

Fig. 4
Fig. 4

Polarization states involved in the ideal half-wave plate PDG. V in is the incident polarization. V a and V b are the two states emerging from the PDG. V ˜ 0 and V ˜ n are the states at the DC and at the odd diffracted orders. (a)  α = 0 ° . (b) α is arbitrary. (c) Analyzer orientation (discontinuous blue arrow), relative to the two states emerging from the PDG, to obtain amplitude modulation, phase modulation, or absence of modulation.

Fig. 5
Fig. 5

Experimental diffraction pattern of the binary HWP–PDG obtained for λ = 568 nm . Input polarizer is oriented at θ P 1 = 0 ° . (a) Case without analyzer. (b)–(e) Cases with the analyzer oriented at (b)  θ P 2 = 0 ° , (c)  θ P 2 = 45 ° , (d)  θ P 2 = 90 ° , and (e)  θ P 2 = 135 ° . PDG, AmDG, and PhDG indicate the configurations for polarization, amplitude, and phase diffraction grating. No DG indicates the configuration for the absence of grating.

Fig. 6
Fig. 6

Experimental diffraction pattern of the binary grating obtained for the wavelength λ = 568 nm . Configuration with a single polarizer oriented at (a)  θ P = 0 ° , (b)  θ P = 22.5 ° , (c)  θ P = 45 ° , and (d)  θ P = 67.5 ° .

Fig. 7
Fig. 7

(a) Polarization states involved in a nonideal PDG ( ϕ 180 ° , Δ θ = 45 ° ). The input polarizer is oriented at θ P 1 = 0 ° . (b)–(e) Experimental diffraction patterns obtained for λ = 647 nm with different orientations of the analyzer ( θ P 2 = 0 ° , 45 ° , 90 ° , and 135 ° ), which is indicated on the right.

Fig. 8
Fig. 8

(a) Polarization states involved in a compensated nonideal PDG ( ϕ 180 ° , Δ θ = 45 ° ). (b)–(e) Experimental diffraction patterns obtained for λ = 647 nm with different orientations of the analyzer ( θ P 2 = 12 ° , 33 ° , 78 ° , and 123 ° ), which is indicated on the right.

Tables (4)

Tables Icon

Table 1 Angular Orientation of the Linear Polarization States Involved in the HWP PDG

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Table 2 Elliptical States Involved in the Standard Configuration Corresponding to Fig. 7 ( ϕ = 105 ° )

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Table 3 Elliptical States Involved in the Optimized Configuration Corresponding to Fig. 8

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Table 4 Absolute (η) and Relative ( η rel ) Diffraction Efficiencies for the Considered Binary PDG Configurations a

Equations (29)

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i = sin 2 ( 2 φ ) sin 2 ( ϕ 2 ) ,
M ( x ) = M a · ( rect ( x p / 2 ) n δ ( x n p ) ) + M b · ( rect ( x p / 2 ) n δ ( x n p p / 2 ) ) ,
rect ( x ) = { 1 if   | x | < 1 / 2 1 / 2 if   | x | = 1 / 2 0 if   | x | > 1 / 2 .
M ˜ ( u ) = 1 2 ( M a + M b exp ( i π u p ) ) sinc ( p 2 u ) · n δ ( u n p ) ,
M ˜ ( u ) = n M ˜ n · δ ( u n p ) ,
M ˜ n = 1 2 sinc ( n 2 ) ( M a + M b exp ( i π n ) ) = a n ( M a + M b exp ( i π n ) ) .
M ˜ n = 0 = M a + M b 2 .
M ˜ odd   n = ± M a M b n π .
V ˜ n = 0 = 1 2 ( cos ( 45 ° α ) sin ( 45 ° α ) ) ,
V ˜ odd   n = ± 2 n π ( cos ( 45 ° α ) sin ( 45 ° α ) ) .
i n = { 1 / 2 for     n = 0 2 ( n π ) 2 for     odd   n 0 for     even   n 0 .
r = sin 2 ( ϕ 2 ) .
M a = M W P ( ϕ , θ = 0 ° ) = ( 1 0 0 e i ϕ ) ,
M b = M W P ( ϕ , θ ) = R ( θ ) M W P ( ϕ ° , θ = 0 ° ) R ( θ ) = ( cos 2 ( θ ) + sin 2 ( θ ) e i ϕ sin ( θ ) cos ( θ ) ( 1 e i ϕ ) sin ( θ ) cos ( θ ) ( 1 e i ϕ ) sin 2 ( θ ) + cos 2 ( θ ) e i ϕ ) ,
R ( θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) .
M ˜ n = 0 = 1 2 ( 1 + cos 2 ( θ ) + sin 2 ( θ ) e i ϕ sin ( θ ) cos ( θ ) ( 1 e i ϕ ) sin ( θ ) cos ( θ ) ( 1 e i ϕ ) sin 2 ( θ ) + [ 1 + cos 2 ( θ ) ] e i ϕ ) ,
M ˜ odd   n = ± sin ( θ ) ( 1 e i ϕ ) n π ( sin ( θ ) cos θ ) cos ( θ ) sin ( θ ) ) .
M W P ( ϕ = 180 ° , θ ) = R ( θ ) M W P ( ϕ = 180 ° , θ = 0 ° ) R ( θ ) = ( cos ( 2 θ ) sin ( 2 θ ) sin ( 2 θ ) cos ( 2 θ ) ) .
M ˜ odd   n = ± sin ( θ ) ( 1 e i ϕ ) n π M W P ( ϕ = 180 ° , θ = θ 2 45 ° ) .
i odd   n = 4 n 2 π 2 sin 2 ( θ ) sin 2 ( ϕ 2 ) ,
η = i n = 1 = 4 π 2 sin 2 ( θ ) sin 2 ( ϕ 2 ) .
η D = n 0 i n = 8 π 2 sin 2 ( θ ) sin 2 ( ϕ 2 ) n = 1 , 3 , 5. .. 1 n 2 = sin 2 ( θ ) sin 2 ( ϕ 2 ) ,
i 0 = 1 η D = 1 sin 2 ( θ ) sin 2 ( ϕ 2 ) = cos 2 ( θ ) + 1 2 sin 2 ( θ ) ( 1 + cos ϕ ) .
M ˜ n = 0 = 1 2 ( + 1 + 1 + 1 1 ) = 1 2 M W P ( ϕ = 180 ° , θ = + 22.5 ° ) ,
M ˜ odd   n = ± 1 n π ( + 1 1 1 1 ) = ± 2 n π M W P ( ϕ = 180 ° , θ = 22.5 ° ) .
V a = M a · ( cos ( α ) sin ( α ) ) = ( cos ( α ) sin ( α ) ) ,
V b = M b · ( cos ( α ) sin ( α ) ) = ( cos ( 90 α ) sin ( 90 α ) ) ,
V ˜ n = 0 = M ˜ n = 0 · ( cos ( α ) sin ( α ) ) = 1 2 ( cos ( 45 ° α ) sin ( 45 ° α ) ) ,
V ˜ odd   n = M ˜ odd   n · ( cos ( α ) sin ( α ) ) = ± 2 n π ( cos ( 45 ° α ) sin ( 45 ° α ) ) .

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