## Use of polar decomposition of Mueller matrices for optimizing the phase response of a liquid-crystal-on-silicon display

Optics Express, Vol. 16, Issue 3, pp. 1965-1974 (2008)

http://dx.doi.org/10.1364/OE.16.001965

Acrobat PDF (2579 KB)

### Abstract

We provide experimental measurement of the Mueller matrices corresponding to an on-state liquid-crystal-on-silicon display as a function of the addressed voltage. The polar decomposition of the Mueller matrices determines the polarization properties of the device in terms of a diattenuation, a retardance and a depolarization effect. Although the diattenuation effect is shown to be negligible for the display, the behavior of the degree of polarization as a function of the input polarization state shows a maximum coupling of linearly polarized light into unpolarized light of about 10%. Concerning the retardation effect, we find that the display behaves as a retarder with a fast-axis orientation and a retardance angle that are voltage-dependent. The above decomposition provides a convenient framework to optimize the optical response of the display for achieving a phase-mostly modulation regime. To this end, the display is sandwiched between a polarization state generator and a polarization state analyzer. Laboratory results for a commercial panel show a phase modulation depth of 360° at 633 nm with a residual intensity variation lower than 6 %.

© 2008 Optical Society of America

## 1. Introduction

1. D. J. McKnight, K. M. Johnson, and R. A. Serati, “256×256 Liquid-Crystal-on-Silicon Spatial Light-Modulator,” Appl. Opt. **33**, 2775–2784 (1994). [CrossRef] [PubMed]

2. H. T. Dai, K. X. Y. Liu, X. Wang, and J. H. Liu, “Characteristics of LCoS phase-only spatial light modulator and its applications,” Opt. Commun. **238**, 269–276 (2005). [CrossRef]

2. H. T. Dai, K. X. Y. Liu, X. Wang, and J. H. Liu, “Characteristics of LCoS phase-only spatial light modulator and its applications,” Opt. Commun. **238**, 269–276 (2005). [CrossRef]

5. S. T. Tang and H. S. Kwok, “3×3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A **18**, 2138–2145 (2001). [CrossRef]

7. S. Stallinga, “Equivalent retarder approach to reflective liquid crystal displays,” J. Appl. Phys. **86**, 4756–4766 (1999). [CrossRef]

7. S. Stallinga, “Equivalent retarder approach to reflective liquid crystal displays,” J. Appl. Phys. **86**, 4756–4766 (1999). [CrossRef]

8. S. T. Tang and H. S. Kwok, “Measurement of reflective liquid crystal displays,” J. Appl. Phys. **91**, 8950–8954 (2002). [CrossRef]

*et al.*have reported a complete polarimetric characterization of an LCoS panel showing a non-negligible depolarization effect [9

9. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. **45**, 1688–1703 (2006). [CrossRef] [PubMed]

10. J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimeter”, Opt. Eng. **34**, 1558–1568 (1995). [CrossRef]

11. A. De Martino, Y. Kim, E. Garcia-Caurel, and B. Laude, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. **28**, 616–618 (2003). [CrossRef] [PubMed]

11. A. De Martino, Y. Kim, E. Garcia-Caurel, and B. Laude, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. **28**, 616–618 (2003). [CrossRef] [PubMed]

12. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am A **13**, 1106–1113 (1996). [CrossRef]

13. J. M. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vis. Res. **40**, 3791–3799 (2000). [CrossRef] [PubMed]

14. J. Chung, W. Jung, M. J. Hammer-Wilson, P. Wilder-Smith, and Z. Chen, “Use of polar decomposition for the diagnosis of oral precancer,” Appl. Opt. **46**, 3038–3044 (2007). [CrossRef] [PubMed]

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

2. H. T. Dai, K. X. Y. Liu, X. Wang, and J. H. Liu, “Characteristics of LCoS phase-only spatial light modulator and its applications,” Opt. Commun. **238**, 269–276 (2005). [CrossRef]

17. V. Duran, J. Lancis, E. Tajahuerce, and M. Fernandez-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]

9. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. **45**, 1688–1703 (2006). [CrossRef] [PubMed]

## 2. Theory of polar decomposition

**M**, firstly proposed by Lu and Chipman, allows one to decompose the action of any polarizing device as the product of three elementary devices; a pure diattenuator,

**M**

_{D}, a pure retarder,

**M**

_{R}, and a pure depolarizer,

**M**

_{Δ}[12

12. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am A **13**, 1106–1113 (1996). [CrossRef]

9. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. **45**, 1688–1703 (2006). [CrossRef] [PubMed]

**M**

_{LCoS}

**M**

_{R}results

**m**

_{R}is the three-dimensional rotation matrix,

*δ*is the Kronecker delta,

_{ij}*ε*is the Levi-Cevità tensor,

_{ijk}*R*is the retardance angle, and (1,

*a*

_{1},

*a*

_{2},

*a*

_{3})

^{T}is the normalized Stokes vector corresponding to the eigenstate along the fast axis of the retarder. Inversely,

*R*and the components

*a*,

_{i}*i*=1,2,3, are obtained from

**M**

_{R}as

*tr*(

**M**

_{R}) the trace of the retarder matrix. Finally we account for the coupling of polarized light into unpolarized light through the depolarization matrix

**M**

_{Δ}

**m**

_{Δ}is a 3×3 symmetric matrix,

**m**

^{T}

_{Δ}=

**m**

_{Δ}, and

*P*

_{Δ}is the so-called polarizance vector, which, in the absence of diattenuattion, is given by the first column of

**M**

_{LCoS}. On the other hand, the averaged depolarization capability of the equivalent depolarizer, the so-called depolarization power Δ, is then given by

**m**

_{Δ}and

**m**

_{R}matrices from the experimental Mueller matrix of an LCoS display

**M**

_{LCoS}. To this end, we begin by recognizing that, by taking into account Eqs. (3) and (7), Eq. (2) can be rewritten as

**m**′≡

**m**

_{Δ}

**m**

_{R}. Taking into account that

**m**

_{Δ}is a symmetric matrix, it is straightforward to show that

**m**

_{Δ}

^{2}=

**m**′(

**m**′)

^{T}. To extract the depolarization matrix we follow the Cayley-Hamilton theorem. In this way,

## 3. Mueller matrix measurement

### 3.1. Experimental set-up

*M*,

_{αβ}*α*,

*β*=0,1,2,3, of the LCoS display is shown in Fig. 1. The radiation coming from a He-Ne laser (LS) emitting at 633 nm is spatially filtered and collimated by the lens L

_{1}to provide a homogeneous beam with a diameter of 15 mm. The LCoS display is an Aurora panel, with XGA resolution (1024×768 pixels) and a size of 19.6×14.6 mm, commercialized by Holoeye. The TNLC cells have a twist angle of 45° and the pixel array has a period of 19 µm with an inactive gap of 1µm. The voltage is applied to the pixels by displaying an image codified into a 2

^{8}gray-level scale. For our display, the voltage increases monotonically with the gray level. The radiation impinges onto the LCoS cells at a quasi-normal incidence (

*α*=4°) so that the input and the reflected beams are spatially separated and the Stokes formalism can be applied. Note that as a result of the pixelated structure of the display, the outgoing energy splits into several diffraction orders. The zero order was isolated for all intensity measurements. We used a PSG to generate the different SoPs required to measure the Mueller matrix of the display. The PSG comprises a linear polarizer P

_{1}followed by a zero-order quarter-wave plate QWP at 633 nm. On the other hand, the PSA is a home-built Stokesmeter constituted by two nematic liquid crystal variable retarders (LCVR

_{1}and LCVR

_{2}), a linear polarizer P

_{2}. A focusing lens L

_{2}is used to focus the light into the photometer PM. The LCRV retardance is controlled by the application of a voltage. The Stokes vector is extracted from the least-square fitting of the intensity data collected at the photometer as the retardance value of each LCRV is sequentially changed by application of a voltage sweep [20]. A set of twenty-five intensity measurements allows us to achieve a maximum uncertainty for the Stokes parameters of 0.03. Provided that the PSG is composed of non-ideal optical elements, we used our Stokesmeter to measure the actual polarization states of the light impinging onto the liquid crystal cells to increase the precision of the experiment.

*M*

_{00}=1 for any addressed gray level. Concerning the matrix elements in the first row,

*M*

_{01}is obtained as the difference between the normalized reflectance values for linearly polarized input light in the horizontal and the vertical direction. Analogously,

*M*

_{02}and

*M*

_{03}result, respectively, from the differences for linearly polarized input light at 45° and 135°, and for right and left circularly polarized light. Note that, as

*M*

_{00}=1, the matrix elements

*M*

_{0i}(

*i*=1, 2, 3) coincide with the diattenuation coefficients

*D*

_{H},

*D*

_{45}and

*D*

_{C}, respectively [19]. In our measurements, we found that all of these coefficients are always less than 0.016 in absolute value, with a mean value along the entire gray level range below 0.008. In addition to the diattenuation coefficients, a generally accepted indicator for characterizing the degree of diattenuation of an optical component is the polarization dependent loss (PDL). This parameter is defined as PDL=10 log (

*R*/

_{max}*R*), where

_{min}*R*and

_{max}*R*correspond, respectively, to the maximum and minimum reflectance values, which can be calculated for any gray-level from the first row of the Mueller matrix [12

_{min}12. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am A **13**, 1106–1113 (1996). [CrossRef]

**45**, 1688–1703 (2006). [CrossRef] [PubMed]

### 3.2. Depolarization effect

**M**

_{LCoS}(

*g*) matrices show the structure in Eq. (8) with a nule polarizance vector,

*P*

_{Δ}(g). From the 3×3 submatrix

**m'**, the depolarization Mueller matrices

**M**

_{Δ}(

*g*) were obtained through Eqs. (7) and (10). Figure 2(a) shows the experimental values for the depolarization power, Δ(

*g*), calculated in accordance with Eq. (8). This parameter shows a maximum value of about 5% along the entire pixel dynamic range. Thus, the depolarization effect due to the LCoS can not be neglected. From a practical point of view, the coupling of polarized light into unpolarized light depends on the input SoP so that the depolarization power only provides a rough estimate. This dependence is observed in the degree of polarization (

*DoP*) which is defined from the Stokes parameters

*S*(

_{i}*i*=0,…,3) as [19]

*DoP*is given by the distance between the origin and the point in the Stokes-parameter space associated to the SoPs. Points over the surface of the Poincaré sphere represent polarized light whereas interior points correspond to partially polarized light. It is well-known that a pure depolarizer transforms the set of SoPs over the Poincaré sphere into an ellipsoid [21

21. S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. **146**, 11–14 (1998). [CrossRef]

*g*=200. Results are shown in Fig. 2(b). Here, the shrinkage of the equatorial circle of the Poincaré sphere is clearly noticeable. The

*DoP*reaches a maximum value of approximately 10% for input azimuths around 22.5° and 112.5° (recall that azimuths are doubled in the Poincaré sphere representation). Note that the azimuth of the input linearly polarized light corresponding to the maximum depolarization effect depends on the orientation of the depolarization ellipsoid, which, in general, changes with the gray-level

*g*.

### 3.3. Retardance effect

**M**

_{R}(

*g*) were obtained through Eq. (11). From the above set of data, the retardance angle

*R*(

*g*) and the azimuth and the ellipticity angles corresponding to the fast axis orientation,

*θ*(

*g*) and

*ε*(

*g*), were calculated in accordance with Eqs. (5) and (6), respectively. Note that for each gray level

*θ*=(1/2) arctan (

*a*

_{2}/

*a*

_{1}) and

*ε*=(1/2) arcsin (

*a*

_{3}) [19]. Results are shown in Fig. 3. For

*g*=0 the fast axis is oriented along the vertical direction while for g=255 the azimuth

*θ*is close to −45° and the equivalent retarder behaves approximately as a half-wave plate. These facts suggest the underlying mechanisms of the behavior of the LCoS display as an intensity modulator. It also should be noted that the magnitude of the ellipticity angle

*ε*(g) raises along the gray level range, which points out that the fast axis becomes slightly located outside the equator of the Poincaré sphere. In other words, the retardance properties of the LCoS display for high values of the gray level are best fitted by an elliptic retarder [21

21. S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. **146**, 11–14 (1998). [CrossRef]

7. S. Stallinga, “Equivalent retarder approach to reflective liquid crystal displays,” J. Appl. Phys. **86**, 4756–4766 (1999). [CrossRef]

## 4. Optimization of the LCoS phase modulation response

*ζ*

_{1}and

*ξ*

_{1}from the horizontal direction in the laboratory framework, respectively. Analogous angles are denoted by

*ζ*

_{2}and

*ξ*

_{2}for the PSA. The Stokes vector

**S**’ corresponding to the light emerging from the last polarizer is given by

**S**=(1, cos 2

*ζ*

_{1}, sin 2

*ζ*

_{1}, 0)

^{T}and

**M**

_{P}and

**M**

_{QWP}are, respectively, the conventional Mueller matrices for a linear polarizer and a quarter wave plate [19],[22].

*S*’(

_{0}*g*). It is worth mentioning that no information about the phase of the outgoing light is obtained using the Mueller matrix formalism. However, for some applications, it is also important to evaluate the phase modulation of the totally polarized light component emerging from the LCoS display. This task can be accomplished by use of the Jones calculus as we will show next.

*χ*as

*θ*and

*ε*are, respectively, the azimuth and the ellipticity angle of the polarization ellipse, and

*j*is the imaginary unit. This characterization was first introduced by Poincaré and later developed by Azzam and Bashara [22]. Within this framework, the Jones matrix for an elliptic retarder with a retardance angle

*R*and a fast eigenstate described by

*χ*is [22]

_{ef}**J**is a unitary matrix. By using the values for

*θ*(

*g*),

*ε*(

*g*), and R(

*g*) in Fig. 3, we obtain the Jones matrices

**J**

_{LCoS}(

*g*) as a function of the gray level. The real and imaginary parts of

*J*

_{11}and

*J*

_{12}are represented in Fig. 4. Next, the components of the electric field at the output of the PSG-LCoS-PSA system are

**J**

_{P}and

**J**

_{QWP}correspond to the conventional Jones matrices of a polarizer and a quarter wave plate, respectively (see Ref. [19] and [22]). The angle

*β*is the birefringence parameter defined as

*β*=

*π d*Δ

*n*/

*λ*, where

*d*is the cell thickness, Δ

*n*is the difference between the extraordinary and the ordinary refraction index, and

*λ*is the wavelength. Four our display,

*d*=5.5

*µm*and Δ

*n*~0.1 in the off-state [3

3. X. Wang, H. T. Dai, and K. Xu, “Tunable reflective lens array based on liquid cristal on silicon,” Opt. Express **13**, 352–357 (2004). [CrossRef]

23. K. H. Lu and B. E. A. Saleh, “Complex amplitude reflectance of the liquid-crystal light valve,” Appl. Opt. **30**, 2354–2362 (1991). [CrossRef] [PubMed]

*a*and

*b*are quantities that depend on the gray level g and on the angular configurations of the PSG and PSA. From the above equation the total phase shift

*φ*introduced by the polarization arrangement is given by

*β*(

*g*) is a decreasing function of the applied voltage [23

23. K. H. Lu and B. E. A. Saleh, “Complex amplitude reflectance of the liquid-crystal light valve,” Appl. Opt. **30**, 2354–2362 (1991). [CrossRef] [PubMed]

*S*’(

_{0}*g*); and 2) a maximum variation of the phase shift

*φ*(

*g*). Here, we also demand that an increasing behavior with the gray level for the term depending on the quantities

*a*and

*b*just to add constructively with the birefringence term -2

*β*(

*g*). Note both the Stokes and the Jones calculus are employed in our determination. Taking into account the values previously determined for

**M**

_{LCoS}and

*χ*, we perform the numerical computation of

_{ef}*S*’(

_{0}*g*,

*ζ*

_{1},

*ξ*

_{1},

*ζ*

_{2},

*ξ*

_{2}) and

*φ*(

*g*,

*ζ*

_{1},

*ξ*

_{1},

*ζ*

_{2},

*ξ*

_{2}), using Eqs. (13) and (18), respectively. The full range of values for the angular variables

*ζ*

_{1},

*ξ*

_{1},

*ζ*

_{2}, and

*ξ*

_{2}was covered in steps of 1°. In this way, we found the optimal angular configuration of our system when

*ζ*

_{1}=94°,

*ξ*

_{1}=163°,

*ξ*

_{2}=179°, and

*ζ*

_{2}=99°. With this set of angular variables the mean value for

*S*’(

_{0}*g*) is about 62% with a residual variation lower than 3%.

17. V. Duran, J. Lancis, E. Tajahuerce, and M. Fernandez-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]

24. A. Serrano-Heredia, G. W. Lu, P. Purwosumarto, and F. T. S. Yu, “Measurement of the phase modulation in liquid crystal television based on the fractional-Talbot effect,” Opt. Eng. **35**, 2680–2684 (1996). [CrossRef]

25. V. Duran, L. Martínez, Z. Jaroszewicz, and A. Kołodziejczyk: “Calibration of spatial light modulators by inspection of their Fresnel images,” European Optical Society, Topical Meetings Digest Series, L-043, (2005) (European Optical Society Topical Meeting on Diffractive Optics, 3 September - 7 September 2005, Warsaw, Poland).

*g*=0 whereas the other ranges along the entire pixel dynamic range. The relative phase shift Δ

*φ*=

*φ*(

*g*)−

*φ*(0) is determined by measuring the contrast of the Fresnel images at a quarter of the Talbot distance of the periodic pattern. Concerning the transmitted intensity

*S*’(

_{0}*g*), it was measured with a photometer by displaying onto the LCoS a uniform image. Fig. 5(a) shows a plot of the phase-shift Δ

*φ*versus the gray level

*g*in the phase-mostly configuration. We have also represented the function

*f*(

*g*)=arctan[

*b*(

*g*)/

*a*(

*g*)]-arctan[

*b*(0)/

*a*(0)] calculated from Eq. (16). The difference between Δ

*φ*(

*g*) and

*f*(

*g*) gives the variation along the gray level range of the birefringence term that appears in Eq. (18). Finally, Fig. 5(b) shows the LCoS operation curve. The radius and polar angle of each point of this curve represent, respectively, the transmitted intensity and the phase shift for a given value of the addressed gray level. The experimental results are close to a pure phase modulation regime. Further, maximum phase-modulation depth greater than 360° is achieved.

## 5. Conclusions

## Acknowledgments

## References and links

1. | D. J. McKnight, K. M. Johnson, and R. A. Serati, “256×256 Liquid-Crystal-on-Silicon Spatial Light-Modulator,” Appl. Opt. |

2. | H. T. Dai, K. X. Y. Liu, X. Wang, and J. H. Liu, “Characteristics of LCoS phase-only spatial light modulator and its applications,” Opt. Commun. |

3. | X. Wang, H. T. Dai, and K. Xu, “Tunable reflective lens array based on liquid cristal on silicon,” Opt. Express |

4. | Q. Mu, Z. Cao, L. Hu, D. Li, and L. Xuan, “Adaptive optics imaging system based on a high resolution liquid crystal on silicon device,” Opt. Express |

5. | S. T. Tang and H. S. Kwok, “3×3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A |

6. | V. Duran, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. |

7. | S. Stallinga, “Equivalent retarder approach to reflective liquid crystal displays,” J. Appl. Phys. |

8. | S. T. Tang and H. S. Kwok, “Measurement of reflective liquid crystal displays,” J. Appl. Phys. |

9. | J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. |

10. | J. L. Pezzaniti and R. A. Chipman, “Mueller matrix imaging polarimeter”, Opt. Eng. |

11. | A. De Martino, Y. Kim, E. Garcia-Caurel, and B. Laude, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. |

12. | S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am A |

13. | J. M. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vis. Res. |

14. | J. Chung, W. Jung, M. J. Hammer-Wilson, P. Wilder-Smith, and Z. Chen, “Use of polar decomposition for the diagnosis of oral precancer,” Appl. Opt. |

15. | J. L. Pezzaniti and R. A. Chipman, “Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states,” Opt. Lett. |

16. | J. A. Davis, I. Moreno, and T. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. |

17. | V. Duran, J. Lancis, E. Tajahuerce, and M. Fernandez-Alonso, “Phase-only modulation with a twisted nematic liquid crystal display by means of equi-azimuth polarization states,” Opt. Express |

18. | V. Duran, J. Lancis, E. Tajahuerce, and V. Climent, “Poincaré Sphere Method for Optimizing the Phase Modulation Response of a Twisted Nematic Liquid Crystal Display,” J. Display Technol. |

19. | D. Goldstein, |

20. | S. R. Davis, R. J. Uberna, and R. A. Herke, “Retardance Sweep Polarimeter and Method,” U. S. Patent, No. 6,744,509 (2004). |

21. | S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. |

22. | R. M. A. Azzam and N. M. Bashara, |

23. | K. H. Lu and B. E. A. Saleh, “Complex amplitude reflectance of the liquid-crystal light valve,” Appl. Opt. |

24. | A. Serrano-Heredia, G. W. Lu, P. Purwosumarto, and F. T. S. Yu, “Measurement of the phase modulation in liquid crystal television based on the fractional-Talbot effect,” Opt. Eng. |

25. | V. Duran, L. Martínez, Z. Jaroszewicz, and A. Kołodziejczyk: “Calibration of spatial light modulators by inspection of their Fresnel images,” European Optical Society, Topical Meetings Digest Series, L-043, (2005) (European Optical Society Topical Meeting on Diffractive Optics, 3 September - 7 September 2005, Warsaw, Poland). |

**OCIS Codes**

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(230.3720) Optical devices : Liquid-crystal devices

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: November 16, 2007

Revised Manuscript: January 2, 2008

Manuscript Accepted: January 2, 2008

Published: January 28, 2008

**Citation**

P. Clemente, V. Durán, Ll. Martínez-León, V. Climent, E. Tajahuerce, and J. Lancis, "Use of polar decomposition of Mueller matrices for optimizing the phase response of a liquid-crystal-on-silicon display," Opt. Express **16**, 1965-1974 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1965

Sort: Year | Journal | Reset

### References

- D. J. McKnight, K. M. Johnson, and R. A. Serati, "256 x 256 Liquid-Crystal-on-Silicon Spatial Light-Modulator," Appl. Opt. 33, 2775-2784 (1994). [CrossRef] [PubMed]
- H. T. Dai, K. X. Y. Liu, X. Wang, and J. H. Liu, "Characteristics of LCoS phase-only spatial light modulator and its applications," Opt. Commun. 238, 269-276 (2005). [CrossRef]
- X. Wang, H. T. Dai, and K. Xu, "Tunable reflective lens array based on liquid cristal on silicon," Opt. Express 13, 352-357 (2004). [CrossRef]
- Q. Mu, Z. Cao, L. Hu, D. Li, and L. Xuan, "Adaptive optics imaging system based on a high resolution liquid crystal on silicon device," Opt. Express 14, 8013-8018 (2006). [CrossRef] [PubMed]
- S. T. Tang and H. S. Kwok, "3 x 3 Matrix for unitary optical systems," J. Opt. Soc. Am. A 18, 2138-2145 (2001). [CrossRef]
- V. Duran, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, "Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays," J. Appl. Phys. 99, 113101-6 (2006). [CrossRef]
- S. Stallinga, "Equivalent retarder approach to reflective liquid crystal displays," J. Appl. Phys. 86, 4756-4766 (1999). [CrossRef]
- S. T. Tang and H. S. Kwok, "Measurement of reflective liquid crystal displays," J. Appl. Phys. 91, 8950-8954 (2002). [CrossRef]
- J. E. Wolfe and R. A. Chipman, "Polarimetric characterization of liquid-crystal-on-silicon panels," Appl. Opt. 45, 1688-1703 (2006). [CrossRef] [PubMed]
- J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimeter", Opt. Eng. 34, 1558-1568 (1995). [CrossRef]
- A. De Martino, Y. Kim, E. Garcia-Caurel, and B. Laude, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003). [CrossRef] [PubMed]
- S. Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am A 13, 1106-1113 (1996). [CrossRef]
- J. M. Bueno, "Measurement of parameters of polarization in the living human eye using imaging polarimetry," Vis. Res. 40, 3791-3799 (2000). [CrossRef] [PubMed]
- J. Chung, W. Jung, M. J. Hammer-Wilson, P. Wilder-Smith, and Z. Chen, "Use of polar decomposition for the diagnosis of oral precancer," Appl. Opt. 46, 3038-3044 (2007). [CrossRef] [PubMed]
- J. L. Pezzaniti and R. A. Chipman, "Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states," Opt. Lett. 18, 1567-1569 (1993). [CrossRef] [PubMed]
- J. A. Davis, I. Moreno, and T. Tsai, "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998). [CrossRef]
- V. Duran, J. Lancis, E. Tajahuerce, and M. Fernandez-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]
- V. Duran, J. Lancis, E. Tajahuerce, and V. Climent, "Poincaré Sphere Method for Optimizing the Phase Modulation Response of a Twisted Nematic Liquid Crystal Display," J. Display Technol. 3, 9-14 (2007). [CrossRef]
- D. Goldstein, Polarized light (Marcel Dekker, 2004).
- S. R. Davis, R. J. Uberna, and R. A. Herke, "Retardance Sweep Polarimeter and Method," U. S. Patent, No. 6,744,509 (2004).
- S. Y. Lu and R. A. Chipman, "Mueller matrices and the degree of polarization," Opt. Commun. 146, 11-14 (1998). [CrossRef]
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).
- K. H. Lu and B. E. A. Saleh, "Complex amplitude reflectance of the liquid-crystal light valve," Appl. Opt. 30, 2354-2362 (1991). [CrossRef] [PubMed]
- A. Serrano-Heredia, G. W. Lu, P. Purwosumarto, and F. T. S. Yu, "Measurement of the phase modulation in liquid crystal television based on the fractional-Talbot effect," Opt. Eng. 35, 2680-2684 (1996). [CrossRef]
- V. Duran, L. Martínez, Z. Jaroszewicz and A. Kołodziejczyk: "Calibration of spatial light modulators by inspection of their Fresnel images," European Optical Society, Topical Meetings Digest Series, L-043, (2005) (European Optical Society Topical Meeting on Diffractive Optics, 3 September - 7 September 2005, Warsaw, Poland).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.