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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 3 — Feb. 4, 2008
  • pp: 1965–1974
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Use of polar decomposition of Mueller matrices for optimizing the phase response of a liquid-crystal-on-silicon display

P. Clemente, V. Durán, Ll. Martínez-León, V. Climent, E. Tajahuerce, and J. Lancis  »View Author Affiliations


Optics Express, Vol. 16, Issue 3, pp. 1965-1974 (2008)
http://dx.doi.org/10.1364/OE.16.001965


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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

Commonly designed for high-resolution projection systems, liquid-crystal-on-silicon (LCoS) displays are programmable pixelated devices that operate in reflection. An LCoS pixel consists of a liquid crystal film over an aluminium electrode mirror controlled by a silicon integrated circuit [1

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]

]. LCoS displays are characterized by a high spatial resolution, with a pixel period that can be lower than 10 microns and a fill factor that can exceed the 90% [2

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]

]. Because of the above outstanding features, the use of LCoS displays as spatial light modulators (SLMs) in non-displays applications has gained considerable attention in recent years [2-4

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]

]. We consider a commercial LCoS panel with twisted nematic liquid crystal (TNLC) cells. From an optical point of view, a TNLC film is equivalent to a system composed of a linear retarder followed by a rotator [5-6

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]

]. For a reflective display, the rotation effect is cancelled by the double pass of the light through the medium. Therefore, a reflective TNLC display behaves in essence as a linear retarder with a neutral axis orientation and a retardance angle that are voltage-dependent [7

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

]. This approach has been applied for optimizing the brightness and contrast of a reflective TNLC display [7

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

], as well as for determining the design parameters of reflective TNLC cells [8

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

].

Recently, Wolfe 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]

]. As a result, the degree of polarization of the impinging light is reduced to a certain extent, which should be considered for proper operation. A depolarizing LCoS display working as intensity SLM can suffer from a severe diminution of the contrast ratio, leading to a worsening of the display performance. The physics of light depolarization has been attributed to electric field fluctuations and to edge effects in pixels or temperature variations. To deal with this effect, it is convenient to analyze light polarization devices in terms of Mueller matrices. Then, the state of polarization (SoP) is not described by the complex amplitudes of the electric field but by the corresponding Stokes vector. Experimental Mueller matrices of a LCoS cell have been measured with several polarimetric techniques [10

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

,11

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]

]. Between them, it should be mentioned those based on nematic liquid crystal variable retarders [11

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]

]. From an optical point of view, the diattenuation, the retardance and the depolarization of the input polarized light fully characterizes the behavior of the LCoS cells. The above effects can be conveniently separated by the so-called polar decomposition of the Mueller matrix introduced by Lu and Chipmann [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]

]. Here, the action of any polarization device is decomposed as the association of a pure diattenuattor, a pure retarder, and a pure depolarizer. The polar decomposition, especially when used for polarimetric imaging, has proved to be a powerful tool for studying the polarization properties of a sample in a widespread set of applications [13

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

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]

].

In this paper, we perform, for the first time to our knowledge, the full characterization of an LCoS display, including light depolarization, for phase-mostly modulation of the wavefront. To this end, the behavior of the whole optical setup is performed by use of the Mueller-Stokes algebra. Although in principle this is a major drawback for phase-only modulation applications, as the information about the phase of the totally polarized light component emerging from the whole device is discarded, we show how the polar decomposition of the Mueller matrix provides a convenient framework. To this end, we extract the retardance parameters of the LCoS display. Therefore, combining both Jones and Stokes formalisms, the configuration of the PSG and the PSA is optimized for achieving a phase-only modulation regime. In Sec. 2, and for the sake of completeness, we review the theory of polar decomposition, defining the relevant polarization parameters. In Sec 3, we describe the laboratory set-up for measuring the Mueller matrix of the LCoS as a function of the voltage. In Sec. 4, we present the calculations, based on the polar decomposition, for optimizing the phase response of the LCoS display and show the experimental operating curve for the LCoS display. Conclusions are summarized in Sec. 5.

2. Theory of polar decomposition

Three-factor polar decomposition of the Mueller matrix 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]

]. In mathematical terms,

M=MΔMRMD.
(1)

Note that the order in which the matrices are multiplied is a relevant feature. The intensity reflected by an LCoS display has been shown to be independent of the input polarization state [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]

]. In other words, for an LCoS cell diattenuation is a minor effect that can be neglected to provide a simplified Mueller matrix M LCoS

MLCoS=MΔMR.
(2)

It is well-known that the action of a pure retarder on the input polarized light can be geometrically described over the surface of the Poincaré sphere by means of a rotation along the fast axis of the device [19

19. D. Goldstein, Polarized light (Marcel Dekker, 2004).

]. Thus, M R results

MR=[10T0mR],
(3)

where 0⃗ is the three-element zero vector and m R is the three-dimensional rotation matrix,

(mR)ij=δijcosR+aiaj(1cosR)+k=13εijkaksinR,i,j=1,2,3.
(4)

In Eq. (4), δij is the Kronecker delta, εijk is the Levi-Cevità tensor, 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 ai, i=1,2,3, are obtained from M R as

R={cos1[tr(MR)21]0Rπ2πcos1[tr(MR)21]πR2π,
(5)

and

ai=12sinRj,k=13εijk(mR)jk.
(6)

with 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

MΔ=[10TPΔmΔ],
(7)

where 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

Δ=1tr(mΔ)3,0Δ1.
(8)

Now we focus on the extraction of the 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

MLCoS=[10TPΔmΔmR]=[10TPΔm],
(9)

where 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,

mΔ=±[m(m)T+(λ1λ2+λ2λ3+λ3λ1)I]1
×[(λ1+λ2+λ3)m(m)T+λ1λ2λ3I],
(10)

where λ 1, λ 2, and λ 3 are the eigenvalues of m′(m′)T and I is the 3×3 identity matrix. The sign in the right-hand side of Eq. (10) follows the sign of the determinant of m′. With the above expression for m Δ, M Δ can be determined through Eq. (7) and M R can be obtained by

MR=MΔ1MLCoS.
(11)

3. Mueller matrix measurement

3.1. Experimental set-up

The laboratory set-up for the determination of the the matrix elements Mαβ, 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 L1 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 28 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 P1 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 (LCVR1 and LCVR2), a linear polarizer P2. A focusing lens L2 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

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

]. 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.

Fig. 1. Experimental set-up for measuring the LCoS Mueller matrices.

The first row of the Mueller matrix is obtained without the use of the PSA, as only reflectance measurements should be made. We normalized the measurements by using the LCoS reflectance for unpolarized input, so that 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

19. D. Goldstein, Polarized light (Marcel Dekker, 2004).

]. 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 (Rmax/Rmin), where Rmax and Rmin 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

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]

]. For our LCoS display, the mean value of PDL is 0.12 dB. As in Ref [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]

], we conclude that the diattenuation effect is negligible.

3.2. Depolarization effect

Once the LCoS Mueller matrices were determined, we applied the polar decomposition given by Eq. (2). Experimental 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 Si (i=0,…,3) as [19

19. D. Goldstein, Polarized light (Marcel Dekker, 2004).

]

DoP=S12+S22+S32S0.
(12)
Fig. 2. (a). Depolarization power provided by the LCoS display for a wavelength of 633 nm and (b) deformation of the Poincaré sphere equator due to the LCoS display for g=200.

This parameter takes values between 1 (totally polarized light) and 0 (unpolarized light). Geometrically, the 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]

]. We carried out a preliminary verification by impinging the LCoS with linearly polarized input light. We chose 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

Finally, the retardance matrices 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

19. D. Goldstein, Polarized light (Marcel Dekker, 2004).

]. 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]

]. Although, in principle, a reflective LCoS cell acts as a linear retarder by the double pass of the light through the liquid crystal [7

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

], some deviations are originated by small alterations of the SoP during the reflection inside the pixel.

4. Optimization of the LCoS phase modulation response

After the display characterization performed in Section 3, we optimize the phase response of the LCoS. To this end, the LCoS is inserted in between a PSG constituted by a linear polarizer followed by a zero-order quarter-wave plate at 633 nm and a PSA consisting of identical elements but in reversed order. Concerning the PSG, the transmission axis of the linear polarizer and the fast axis of the quarter wave plate are oriented at angles 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

Fig. 3. (a). Retardation angle and (b) orientation of the retarder fast axis for a wavelength of 633.
S(g,ζ1,ξ1,ζ2,ξ2)=MP(ζ2)MQWP(ξ2)MLCoS(g)MQWP(ξ1)S(ζ1),
(13)

where S=(1, cos 2ζ 1, sin 2ζ 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

19. D. Goldstein, Polarized light (Marcel Dekker, 2004).

],[22

22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).

].

The intensity response of the above arrangement is evaluated for any angular configuration of the PSG and the PSA by means of S0’(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.

We begin by describing the SoP of an arbitrary totally polarized light beam through the complex variable χ as

χ=tanθ+jtanε1jtanθtanε,
(14)

where 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

22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).

]. Within this framework, the Jones matrix for an elliptic retarder with a retardance angle R and a fast eigenstate described by χef is [22

22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).

]

J=(1+χefχef*)1×
(exp(jR2)+χefχef*exp(jR2)2jχef*sin(R2)2jχefsin(R2)exp(jR2)+χefχef*exp(jR2)).
(15)

Here the symbol * denotes the complex conjugate. Note that 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

(ExEy)=exp(j2β)JP(ζ2)JQWP(ξ2)JLCoSJQWP(ξ1)(cosζ1sinζ1).
(16)

Here J P and J QWP correspond to the conventional Jones matrices of a polarizer and a quarter wave plate, respectively (see Ref. [19

19. D. Goldstein, Polarized light (Marcel Dekker, 2004).

] and [22

22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).

]). The angle β is the birefringence parameter defined as β=π dΔn/λ, where d is the cell thickness, Δn is the difference between the extraordinary and the ordinary refraction index, and λ is the wavelength. Four our display, d=5.5 µm and Δn~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]

]. Note the effect originated by the double pass of the light through the medium [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]

]. Of course, the Jones vector in Eq. (16) corresponds to a light beam polarized in the direction of the transmission axis of the last analyzer. More conveniently, this vector can be written in the analyzer framework as

Fig. 4. Jones matrix elements corresponding to the retardance effect of the LCoS display
(ExEy)=exp(j2β)(a+jb0),
(17)

where 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

φ(g,ζ1,ξ1,ζ2,ξ2)=2β(g)+arctan[b(g,ζ1,ξ1,ζ2,ξ2)a(g,ζ1,ξ1,ζ2,ξ2)].
(18)

It is important to note that the birefringence parameter 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]

], i.e., a decreasing function of the gray-level for our display.

Now, let us recover the phase-mostly modulation question. We look for the optimal angular configurations of the PSG and PSA satisfying: 1) a minimum variation for the transmitted intensity S0’(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 -2β(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 χef, we perform the numerical computation of S0’(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 S0’(g) is about 62% with a residual variation lower than 3%.

Finally we experimentally test the whole LCoS response. To this end, we construct the operation curve of the display, where the complex amplitude response is drawn in a polar plot as a function of the addressed gray level. The curve was constructed by experimental measurement of the transmitted intensity and the phase introduced by the LCoS. To measure the phase, we used a technique based on the fractional Talbot effect [17

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

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

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).

]. In short, we imprint onto the LCoS display a pattern consisting of a binary diffraction grating; one gray level is fixed to g=0 whereas the other ranges along the entire pixel dynamic range. The relative phase shift Δφ=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 S0’(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 Δφ 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 Δφ(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.

Fig. 5. Modulation provided by the LCoS display in the phase-mostly configuration. a) Phase shift versus gray level and b) operation curve.

5. Conclusions

We have achieved phase-mostly modulation of the wavefront by means of an LCoS display. To this end, we have inserted the display into a polarimetric arrangement consisting of a polarization state generator and a polarization state analyzer constituted by a polarizer and a zero order quarter-wave plate. Previously the LCoS was calibrated following the polar decomposition of the Mueller matrix. Although the diattenuation effect was shown to be negligible, the effect of light depolarization must be taken into account for proper operation of the cells. A depolarization effect as high as 10% was measured for certain states of polarization of the input light. Finally, the analysis of the retardance behaviour is performed within the conventional Jones formalism. Our technique allows one to optimize the angular configuration of the polarimetric arrangement to achieve a flat intensity response (a residual error lower than 6%) with a broad phase response (as high as 360°). Experimental measurements in the laboratory corroborate the optimization algorithm. This technique benefits from the high spatial resolution and excellent fill factor of commercial panels made on silicon.

Acknowledgments

This research was funded by the Conveni Fundació Caixa Castelló-Bancaixa-Universitat Jaume I, Spain, under the project P1 1B2006-29. We also acknowledge financial support from the Ministerio de Educación y Ciencia and FEDER under the project FIS2007-62217.

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. 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]

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]

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 14, 8013–8018 (2006). [CrossRef] [PubMed]

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]

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. 99, 113101–6 (2006). [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]

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]

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]

16.

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

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. 3, 9–14 (2007). [CrossRef]

19.

D. Goldstein, Polarized light (Marcel Dekker, 2004).

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. 146, 11–14 (1998). [CrossRef]

22.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).

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]

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).

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


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References

  1. 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]
  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]
  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]
  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 14, 8013-8018 (2006). [CrossRef] [PubMed]
  5. 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]
  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. 99, 113101-6 (2006). [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]
  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]
  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]
  16. J. A. Davis, I. Moreno, and T. Tsai, "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998). [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]
  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. 3, 9-14 (2007). [CrossRef]
  19. D. Goldstein, Polarized light (Marcel Dekker, 2004).
  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. 146, 11-14 (1998). [CrossRef]
  22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1st edition, (Elsevier, Amsterdam, 1987).
  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]
  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).

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