Abstract

We show how the phase modulation depth in twisted nematic liquid crystal displays (TNLCDs) can be increased dramatically by selecting a polarization configuration with a reduced mean intensity transmission. This phenomenon, which we have validated with various devices, is shown here for a device that presents a phase-only modulation only slightly over π radians in our classical rotated eigenvector configuration, but it is capable of producing close to a 2π phase depth for a configuration with 5% mean intensity transmission. A quantitative explanation is presented by means of a phasor analysis of the TNLCD eigenvector projections over input and output polarization states. The proposed technique can be a very useful solution in modern TNLCDs that have a very thin liquid crystal layer and a reduced maximum achievable phase modulation.

© 2010 Optical Society of America

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References

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  1. H. K. Liu, J. A. Davis, and R. A. Lilly, “Optical-data-processing properties of a liquid-crystal television spatial light modulator,” Opt. Lett. 10, 635–637 (1985).
    [CrossRef] [PubMed]
  2. J. L. Bougrenet de la Tocnaye and L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
    [CrossRef] [PubMed]
  3. T. H. Barnes, T. Eiju, K. Matsuda, and N. Ooyama, “Phase-only modulation using a twisted nematic liquid crystal television,” Appl. Opt. 28, 4845–4852 (1989).
    [CrossRef] [PubMed]
  4. K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
    [CrossRef]
  5. J. L. Pezzaniti and R. A. Chipman, “Phase-only modulation of a twisted nematic liquid crystal TV by use of eigenpolarization states,” Opt. Lett. 18, 1567–1569 (1993).
    [CrossRef] [PubMed]
  6. J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
    [CrossRef]
  7. I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
    [CrossRef]
  8. J. Nicolás, J. Campos, and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
    [CrossRef]
  9. V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
    [CrossRef]
  10. V. Durán, J. Lancis, E. Tajahuerce, and M. Fernández-Alonso, “Phase-only modulation with a twisted nematic liquid crystal display by means of equi-azimuth polarization states,” Opt. Express 14, 5607–5616 (2006).
    [CrossRef] [PubMed]
  11. A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
    [CrossRef]
  12. J. A. Davis, J. Nicolas, and A. Marquez, “Phasor analysis of eigenvectors generated in liquid crystal displays,” Appl. Opt. 41, 4579–4584 (2002).
    [CrossRef] [PubMed]
  13. I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed onto a restricted phase-mostly modulation display,” Appl. Opt. 43, 6278–6284 (2004).
    [CrossRef] [PubMed]
  14. A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
    [CrossRef]
  15. A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
    [CrossRef]
  16. I. Moreno, J. L. Martinez, and J. A. Davis, “Two-dimensional polarization rotator using a twisted nematic liquid crystal display,” Appl. Opt. 46, 881–887 (2007).
    [CrossRef] [PubMed]
  17. Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
    [CrossRef]
  18. C. R. Fernández-Pousa, I. Moreno, N. Bennis, and C. Gómez-Reino, “Generalized formulation and symmetry properties of anisotropic devices: Application to liquid crystal displays,” J. Opt. Soc. Am. A 17, 2074–2080 (2000).
    [CrossRef]

2007 (2)

V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
[CrossRef]

I. Moreno, J. L. Martinez, and J. A. Davis, “Two-dimensional polarization rotator using a twisted nematic liquid crystal display,” Appl. Opt. 46, 881–887 (2007).
[CrossRef] [PubMed]

2006 (1)

2004 (1)

2002 (2)

2001 (1)

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

2000 (2)

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

C. R. Fernández-Pousa, I. Moreno, N. Bennis, and C. Gómez-Reino, “Generalized formulation and symmetry properties of anisotropic devices: Application to liquid crystal displays,” J. Opt. Soc. Am. A 17, 2074–2080 (2000).
[CrossRef]

1998 (2)

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

1997 (1)

1994 (1)

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

1993 (2)

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

J. L. Pezzaniti and R. A. Chipman, “Phase-only modulation of a twisted nematic liquid crystal TV by use of eigenpolarization states,” Opt. Lett. 18, 1567–1569 (1993).
[CrossRef] [PubMed]

1990 (1)

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

1989 (1)

1985 (1)

Allison, D. B.

I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Barnes, T. H.

Bennis, N.

Bougrenet de la Tocnaye, J. L.

Campos, J.

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

J. Nicolás, J. Campos, and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Chipman, R. A.

Climent, V.

V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
[CrossRef]

D’Nelly, K. G.

I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Davis, J. A.

I. Moreno, J. L. Martinez, and J. A. Davis, “Two-dimensional polarization rotator using a twisted nematic liquid crystal display,” Appl. Opt. 46, 881–887 (2007).
[CrossRef] [PubMed]

J. A. Davis, J. Nicolas, and A. Marquez, “Phasor analysis of eigenvectors generated in liquid crystal displays,” Appl. Opt. 41, 4579–4584 (2002).
[CrossRef] [PubMed]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

H. K. Liu, J. A. Davis, and R. A. Lilly, “Optical-data-processing properties of a liquid-crystal television spatial light modulator,” Opt. Lett. 10, 635–637 (1985).
[CrossRef] [PubMed]

Dupont, L.

Durán, V.

V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and M. Fernández-Alonso, “Phase-only modulation with a twisted nematic liquid crystal display by means of equi-azimuth polarization states,” Opt. Express 14, 5607–5616 (2006).
[CrossRef] [PubMed]

Eiju, T.

Fernández-Alonso, M.

Fernández-Pousa, C. R.

Gómez-Reino, C.

Gregory, D. A.

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

Iemmi, C.

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

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Lancis, J.

V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and M. Fernández-Alonso, “Phase-only modulation with a twisted nematic liquid crystal display by means of equi-azimuth polarization states,” Opt. Express 14, 5607–5616 (2006).
[CrossRef] [PubMed]

Lilly, R. A.

Liu, H. K.

Lu, G.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

Lu, K.

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Marquez, A.

Márquez, A.

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

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Martinez, J. L.

Matsuda, K.

Moreno, A.

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Moreno, I.

I. Moreno, J. L. Martinez, and J. A. Davis, “Two-dimensional polarization rotator using a twisted nematic liquid crystal display,” Appl. Opt. 46, 881–887 (2007).
[CrossRef] [PubMed]

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

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

C. R. Fernández-Pousa, I. Moreno, N. Bennis, and C. Gómez-Reino, “Generalized formulation and symmetry properties of anisotropic devices: Application to liquid crystal displays,” J. Opt. Soc. Am. A 17, 2074–2080 (2000).
[CrossRef]

J. A. Davis, I. Moreno, and P. Tsai, “Polarization eigenstates for twisted-nematic liquid crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

Nicolas, J.

Nicolás, J.

Ooyama, N.

Pezzaniti, J. L.

Robert, A.

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Saleh, B. E. A.

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Tajahuerce, E.

V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and M. Fernández-Alonso, “Phase-only modulation with a twisted nematic liquid crystal display by means of equi-azimuth polarization states,” Opt. Express 14, 5607–5616 (2006).
[CrossRef] [PubMed]

Tanone, A.

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

Tsai, P.

Uang, C.-M.

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

Yu, F. T. S.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

Yzuel, M. J.

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

J. Nicolás, J. Campos, and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Zhang, Z.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

Appl. Opt. (6)

J. Disp. Technol. (1)

V. Durán, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré sphere method for optimizing the phase modulation response of a twisted nematic liquid crystal display,” J. Disp. Technol. 3, 9–14 (2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (6)

A. Tanone, Z. Zhang, C.-M. Uang, F. T. S. Yu, and D. A. Gregory, “Phase-modulation depth for a real-time kinoform using a liquid crystal television,” Opt. Eng. 32, 517–521 (1993).
[CrossRef]

I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted nematic liquid crystal displays,” Opt. Eng. 37, 3048–3052 (1998).
[CrossRef]

K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40, 2558–2564 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

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Figures (7)

Fig. 1
Fig. 1

Polarization ellipse generated by a linear polarizer followed by a QWP. The azimuth is fixed by the orientation (θ) of the QWP; the ellipticity is fixed by the angle of the polarizer (ε) relative to the QWP.

Fig. 2
Fig. 2

General polarization configuration to operate the TNLCD with phase-only modulation.

Fig. 3
Fig. 3

Full complex modulation for λ = 514 nm represented as intensity and phase versus addressed gray level for (a) classical negative eigenvector configuration, (b) configuration with 50% intensity transmission, and (c) configuration 5% intensity transmission. Experimental data are represented by points and continuous lines depict theoretical predictions.

Fig. 4
Fig. 4

Simulated maximum phase modulation versus mean intensity (points) and regressions (lines).

Fig. 5
Fig. 5

Input and detected states of polarization E in and E out as a function of the selected mean intensity: (a) ellipticities and (b) azimuthal angles.

Fig. 6
Fig. 6

Positive (green, innermost line), negative [red line, middle in (a) and outermost in (b) and (c)], and global [blue line, outermost in (a) and middle in (b) and (c)] phasors for (a) averaged negative eigenvector, (b) 50% intensity transmission, and (c) 5% intensity transmission configurations.

Fig. 7
Fig. 7

Diffraction patterns obtained when addressing a blazed grating to the TNLCD for each configuration: (a) negative averaged eigenvector, (b) 50% intensity transmission, and (c) 5% intensity transmission configurations. The left column shows the results obtained with a direct gray level blaze grating addressed to the display; the right column shows the results using the LUT.

Tables (3)

Tables Icon

Table 1 Angles of the Polarizing Elements Corresponding to the Three Configurations in Fig. 3

Tables Icon

Table 2 Theoretical Relative First Order Diffraction Efficiencies η Corresponding to Fig. 7 a

Tables Icon

Table 3 Angles Leading to Equivalent Modulation E T a

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

M TNLCD = exp ( i β ) R ( α ) M .
M ( α , β ) = ( X i Y Z Z X + i Y ) ,
M · E μ ± = μ ± E μ ± = exp ( i γ ) E μ ± ,
E μ ± = 1 2 ( γ ± β ) γ ( α i ( β ± γ ) ) .
ϕ ± = β γ .
X = cos ( γ ) cos ( 2 δ ) β γ sin ( γ ) sin ( 2 δ ) ,
Y = cos ( γ ) sin ( 2 δ ) + β γ sin ( γ ) cos ( 2 δ ) ,
E T = exp ( i β ) E out · M · E in ,
E in = g + E μ + + g E μ ,
E T = t + g + exp ( i ϕ + ) + t g exp ( i ϕ ) .
Δ = k 1 σ I 2 + k 2 Δ ϕ max 2 ,
σ I 2 = 1 N g ( I ( g ) I ave ) 2 ,
η = I 1 I T ,
E ( θ , ε ) = R ( θ ) · ( cos ε i sin ε ) .
E in = QWP ( θ 1 ) · R ( ε 1 θ 1 ) · ( 1 0 ) = R ( θ 1 ) · ( cos ( ε 1 ) i sin ( ε 1 ) ) = E ( θ 1 , ε 1 ) ,
QWP ( 0 ) = ( 1 0 0 i ) .
E out = QWP 1 ( θ 2 ) · R ( ε 2 θ 2 ) · ( 1 0 ) = R ( θ 2 ) · ( cos ( ε 2 ) i sin ( ε 2 ) ) ,
E T = exp ( i β ) E ( θ 2 , ε 2 ) · M · E ( θ 1 , ε 1 ) .
E in = R ( θ 1 90 ° ) · ( cos ( ε 1 90 ° ) i sin ( ε 1 90 ° ) ) = i · R ( θ 1 ) ( cos ε 1 i sin ε 1 ) = i · E ( θ 1 , ε 1 ) .
E out = R ( θ 2 90 ° ) ( cos ( ε 2 90 ° ) i sin ( ε 2 90 ° ) ) = i · R ( θ 2 ) ( cos ε 2 i sin ε 2 ) = i · E ( θ 2 , ε 2 ) .
M t = J · M · J ,
J = ( 1 0 0 + 1 ) .
E T = exp ( i β ) { E ( θ 2 , ε 2 ) · M · E ( θ 1 , ε 1 ) } t = exp ( i β ) E t ( θ 1 , ε 1 ) · M t · E * ( θ 2 , ε 2 ) .
E T = exp ( i β ) E ( θ 1 , ε 1 ) · M · E ( θ 2 , ε 2 ) .

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