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Optimized Stokes polarimeters based on a singletwisted nematic liquid-crystal device for the
minimization of noise propagation
Alba Peinado,1,* Angel Lizana,1 Josep Vidal,1,2 Claudio Iemmi,3 and Juan Campos1
1Departamento de Física, Universitat Autònoma de Barcelona, Bellatera 08193, Spain2ALBA Synchrotron Light Source Facility, Cerdanyola del Vallès 08290, Spain
3Departamento de Física, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, Buenos Aires 1428, Argentina
*Corresponding author: [email protected]
Received 29 April 2011; revised 26 July 2011; accepted 29 July 2011;posted 1 August 2011 (Doc. ID 146799); published 26 September 2011
This work evidences the suitability of applying a single twisted nematic liquid-crystal (TN-LC) device toobtain dynamic polarimeters with high accuracy and repeatability. Different Stokes polarimeter setupsbased on a TN-LC device are optimized, leading to the minimization of the noise propagated fromintensity measurements to the Stokes vector calculations. To this aim, we revise the influence of workingout of normal incidence and of performing a double pass of the light beam through the LC device.In addition, because transmissive TN-LC devices act as elliptical retarders, an extra study is performed.It analyzes the influence of projecting the light exiting from the TN-LC device over elliptical states ofpolarization. Finally, diverse optimized polarimeters are experimentally implemented and validatedby measuring different states of partially and fully polarized light. The analysis is conducted both formonochromatic (He–Ne laser) and LED light sources, proving the potential of polarimeters based on asingle TN-LC device. © 2011 Optical Society of AmericaOCIS codes: 120.5410, 120.2130, 230.3720.
1. Introduction
Polarimetric information is useful in a large numberof applications [1–4]. Recently, many works have pro-posed polarimeters based on liquid-crystal displays(LCDs) [5–12], which present some operational ben-efits with respect to mechanical polarimeters. Somestudies have designed complete polarimeters basedon parallel aligned liquid-crystal (LC) devices [5,7],showing that for the experimental implementa-tion of such devices, the use of two parallel alignedLCD elements is required as a minimum [5]. Inaddition, polarimeters based on the ferroelectricliquid crystal (FLC) are present in the literature.For example, in Refs. [8,9], complete Stokes and
Mueller polarimeters based on two FLCs are anal-yzed, respectively, whereas in Ref. [10], an imaginglinear polarimeter based on a single FLC is studied.Other authors have proposed polarimeters basedon twisted nematic liquid crystals (TN-LCs), as forinstance in Ref. [11] where a polarimeter based ontwo TN-LC devices, in which only the purely un-switched and fully switched states are applied, isprovided.
Taking advantage that TN-LC devices enable bothintroducing a retardance and rotating the polariza-tion ellipse orientation, a complete Stokes polari-meter based on a single LC device is achieved inRef. [12]. Nevertheless, such a polarimeter setup isstrongly affected by experimental noise, becominga significant drawback in the application of polari-meters based on a single TN-LC device. These polari-meters are restricted to a certain set of projection
0003-6935/11/285437-09$15.00/0© 2011 Optical Society of America
1 October 2011 / Vol. 50, No. 28 / APPLIED OPTICS 5437
states of polarization (SOPs), i.e., the availableprojection SOPs describe a specific curve into thePoincaré sphere [13]. Because projection SOPs con-figurations enclosing large volumes into the Poincarésphere are related to lower error propagation values[5], the restriction of a specific projection SOPs curvelimits the efficiency of the polarimeter.
In this paper, we propose to vary different physicalparameters and to apply an optimization process inorder to achieve diverse polarimeter designs basedon a single TN-LC device, leading to projection SOPcurves enclosing higher volumes into the Poincarésphere (i.e., lower noise propagation). First, we anal-yze the influence of working out of normal incidenceon the TN-LC polarimeter performance. Besides, theeffect of performing a double pass of the light beamthrough the LC device is revised as well. Finally, bytaking into account the fact that transmissive TN-LCdevices act as elliptical retarders, we analyze theeffect of including in the polarimeter design a quar-ter-wave plate (QWP). This allows us to project thelight exiting from the TN-LC device over the ellip-tical SOP.
2. Polarimeter Design
Different complete polarimeter designs are pre-sented in this work, all of them based on a singleoff-the-shelf monopixel TN-LC device and a linearpolarizer (LP). In the first proposed design (nowdenoted as A), a polarizer and a TN-LC device areset perpendicular to the incident beam [it is sketchedin Fig. 1(a)]. In the second design (B), the TN-LC istilted at an angle of α2 [see Fig. 1(b)] with the purposeof working out of normal incidence. Next, a thirdpolarimeter setup (C) is conducted by doubling theoptical path into the LC device. It is achieved bymeans of a reflection on a mirror [see Fig. 1(c)].
The Mueller matrices of the TN-LC device forconfigurations A, B, and C have been calibrated.For the particular case of setup C, we have calibratedthe Mueller matrix that takes into account the
double pass of the light through the TN-LC deviceand the reflection on the mirror. These matricesare calibrated as a function of the addressed voltageto the TN-LC device by applying the method devel-oped in Ref. [14]. Afterward, we have simulatedthe set of possible projection SOPs that can be imple-mented in the three different setups (A, B, and C) bysending a sequence of voltages from 1 to 5:5V (i.e.,the LC operative range). The projection of the lightexiting from the TN-LC device over a linear SOP(i.e., over the polarizer), defines a specific projectionSOP curve that is strongly dependent on the polari-zer orientation θ. For this reason, we have optimizedthe angle θ by minimizing the figure-of-merit so-called equally weighted variance (EWV) [5,15]. Thisquality indicator (QI) expresses the variance propa-gation from intensity measurements, I, through thecharacteristic polarimeter matrix, A, to the Stokesvector calculation, S, according to Eq. (1) describingthe measurement principle:
S ¼ A−1I: ð1ÞThe definition of the EWV is given by Eq. (2),
where all the singular values different from zero,σj, of the characteristic polarimeter matrix A contri-bute in the summation:
EWVðAÞ ¼Xj
1
σ2j: ð2Þ
Once the projection SOP curve has been optimized(i.e., the best LP orientation is found), we havesearched for the optimum set of four projection SOPsby carrying out a second minimization of the EWV.Note that this is exactly the minimum number ofintensity measurements required to implement acomplete polarimeter. The results achieved in theoptimization process are shown in Table 1 (columnsA, B, and C correspond to the three analyzed
Fig. 1. (Color online) Different Stokes polarimeter setups containing a single TN-LC, an LP, and a radiometer: (a) normal incidence,(b) oblique incidence at αi on the TN-LC, (c) double pass through the TN-LC device bymeans of a reflection on amirror, (d) normal incidenceand inserting a QWP [WPðλ=4Þ] between the TN-LC and the polarizer, (e) TN-LC oblique incidence including a QWP, and (f) TN-LC doublepass including a QWP.
5438 APPLIED OPTICS / Vol. 50, No. 28 / 1 October 2011
polarimeters up to now). Although the EWV is the QIemployed in the optimization process, we have alsocalculated another widespread used QI, the condi-tion number [5,7] (CN). The definition of the CN isnot unique: in this work it is calculated as the ratioof the maximum over the minimum singular value ofthe characteristic polarimeter matrix A:
CNðAÞ ¼ σmax
σmin: ð3Þ
Regarding polarimeters A and B, it is noticeablethat lower CN and EWV values are obtained for ob-lique incidence. Thus, an enlargement of the opticalpath causes a modification of the LC birefringence,which leads to alternative projection SOPs curves.The same idea is present in polarimeter C. In fact,the double pass performed by the light beam intothe TN-LC device leads to a significant enlargementof the optical path, achieving lower QI values than inpolarimeters A and B.
As stated before, the projection SOPs curvestrongly depends on the linear SOP on which thelight exiting from the TN-LC device is projected(i.e., the orientation of the LP). As transmissive TN-LC devices act as elliptical retarders [16] (i.e., theireigenvectors are elliptical [17]), we have also con-ducted a study of the influence on the QIs obtainedwhen light exiting from TN-LC device is projectedover elliptical SOPs. To this aim, we have modifiedthe three polarimeter setups A, B, and C by introdu-cing a QWP between the LP and the TN-LC. Thisleads to the setups sketched in Figs. 1(d)–1(f),henceforth denoted as polarimeters D, E, and F, re-spectively. Afterward, we have carried out a first op-timization process equivalent to that previouslydescribed, but in this case, the QWP angle is opti-mized as well.
Subsequently, a second EWV minimization is con-ducted with the aim of selecting the four optimal pro-jection SOPs. In the three last columns of Table 1, theQIs corresponding to polarimeters D, E, and F, re-spectively, are shown.Whereas a significant decreaseof the QIs is observed at normal incidence (compar-ing polarimeters A and D), the improvement atoblique incidence is less relevant (from B to E). Thisfact can be understood if it is kept in mind that anelliptical retarder can be described as the productof a linear retarder and a rotator of a given angle
[16]. In this situation, when the light impinges out ofnormal incidence, the effective rotator angle may bedecreased. In such a case, projecting the SOP exitingfrom the TN-LC over an elliptical polarizationstate has less influence in the optimization process.Finally, concerning polarimeter F, a slight improve-ment of QIs is obtained when comparing with polari-meter C. In this case, the double pass of the light intothe TN-LC device may drastically decrease or evencancel the effective rotator [18]. Therefore, the mostbeneficial contribution of introducing a QWP in thesetup is achieved for normal incidence. We want toemphasize that the diverse improvements appliedto the polarimeter setups have led to a progressivereduction of the EWV indicator, reaching a minimumvalue of 2.78 for polarimeter F (and 1.98 of CN). Thisis an excellent result if one takes into account thefact that the minimum theoretical values that canbe achieved for the EWV and CN are 2.5 and
ffiffiffi3
p,
respectively [5,15]. In this ideal case, the optimizedprojection SOPs are plotted on the vertices of a reg-ular tetrahedron inscribed into the Poincaré sphere.
In Fig. 2, we represent the optimized projectionSOP curve upon the Poincaré sphere correspondingto the different polarimeters (from A to F). In eachgraph, the 4 optimal projection SOPs (solution ofthe complete optimization process) are placed at thevertices of an irregular tetrahedron. As said before,a TN-LC-based polarimeter restricts the availableprojection SOPs, leading to a specific curve upon thePoincaré sphere (see Fig. 2). In general, it is not cer-tain that the TN-LC specific curve includes four pro-jection SOPs forming a regular tetrahedron inscribedinto the Poincaré sphere, and so, the polarimetricsetup based on a single TN-LC device most probablyis not able to implement the best optimized polari-meter [5,15]. However, the applied optimizationshave maximized the projection SOPs curves, leadingto irregular tetrahedrons whose volumes are similarto that given by the regular tetrahedron [15].
3. Implementation
In this section, we have experimentally implementedthe optimized polarimeters. Two different illumina-tion sources have been used. First, the polarimeterwas illuminated by using a monochromatic lightsource. Second, a red LED was used. In both cases,we have calibrated the implemented polarimeters byfollowing the method given in Ref. [5], to reduce theinfluence of possible small deviations with respect tothe theoretical configurations. Then, the implemen-ted polarimeters were tested by measuring threedifferent incident states fully polarized: a linearSOP at 30°, an elliptical SOP, and a right-handedcircular SOP.
The measurement process of the six implementedpolarimeters is as following: the voltage signal is sentto the TN-LC device, we wait 500ms to achieve astationary position of the LC molecules, and then100 measurements of the intensity are acquired ina rate of 100Hz. Because each SOP measurement
Table 1. Optimization Results by Minimizing the EWV IndicatorCorresponding to Polarimeters A–F (Using Four Projection SOPs)a
A B C D E F
LP θ 15° 136° 94° 31° 120° 109°WP θ — — — 49° 43° 0°EWV 6.82 4.88 3.70 3.50 3.88 2.78CN 4.66 3.61 2.69 2.69 3.06 1.98aIn the first two rows are the optimum LP orientation and QWP
angle. In the last two rows are QI (CN and EWV) values corre-sponding to the optimized polarimeters.
1 October 2011 / Vol. 50, No. 28 / APPLIED OPTICS 5439
performed by the polarimeter needs four intensitymeasurements, the process is repeated for the fourvoltages (in other words, using the four projectionSOPs). Afterward, the Stokes vector is calculatedby using the mean value of the 100 intensity mea-surements in Eq. (1). That SOP measurement takesaround 6 s. Finally, the SOP measurement process isrepeated 100 times (it takes approximately 10 min).The experimental data provided in this work cor-respond to the mean and standard deviation ofthese 100 measurements of the same SOP. The ob-tained results are compared with the measurementsprovided by a commercial polarimeter (Thorlabs,Polarization Analyzer System PAN 5710VIS, S/N:M60217605). Also, 100 measurements for each SOPare performed with the commercial polarimeter, andso themean and the standard deviation are provided.In addition, we indicate the expected values of theStokes coefficients to have a reference, although ineach measurement the polarization is checked withthe commercial polarimeter.
Finally, because the Mueller–Stokes formalism isused, the polarimeter is able to measure partiallypolarized light. For this reason, we have measuredtwo additional SOPs with a certain degree of polar-ization (DOP) [13], defined as
DOP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS21 þ S2
2 þ S23
q
S0; 0 ≤ DOP ≤ 1: ð4Þ
A. Monochromatic Illumination
The six optimized polarimeters previously describedwere implemented by using as a light source an He–Ne laser (633nm). The EWVs of the matrices experi-mentally calibrated are very close to the theoreticalEWV values shown in Table 1. Therefore, in spite ofthe presence of experimental inaccuracies, the imple-mented polarimeters were still well optimized whenthey were illuminated with monochromatic light.
The accuracy of the implemented polarimeters isanalyzed in Table 2, where the measures obtainedwith the implemented polarimeters can be comparedwith the information provided by the commercialpolarimeter. The accuracy is calculated by regard-ing the difference between the measurements pro-vided by the implemented polarimeter and thecommercial polarimeter (reference value). Almost allthe measurement differences are in the second dec-imal, except for a specific case (when polarimeter Ameasured the S3 parameter in the elliptical SOP, theaccuracy got 1 order of magnitude less).
The repeatability of the implemented polarim-eters was studied as well. In Fig. 3, the average ofthe Stokes parameters variances for the differentpolarimeters is represented. The rhombi in Fig. 3shows the variance average for polarimeters A, B,and C. When the exiting SOP from TN-LC is pro-jected over a linear SOP, the best result is achievedfor the double-pass setup (polarimeter C). Thesquares in Fig. 3 show the variance averages forpolarimeters E, F, and G. When the exiting SOP fromthe TN-LC is projected over an elliptical SOP, the
Fig. 2. (Color online) Optimized projection SOPs curve by addressing a sequence of voltages (1–5:5V) for polarimeters (a) A, (b) B, (c) C,(d) D, (e) E, and (f) F. The vertices of each inscribed irregular tetrahedron are the four optimal projection polarization states achieved as asolution of the EWV minimization process.
5440 APPLIED OPTICS / Vol. 50, No. 28 / 1 October 2011
Tab
le2.
Stoke
sParam
etersofThreeDifferentSOPs(theStoke
sVec
tors
Are
Norm
alized
)Mea
suredbyEac
hIm
plemen
tedPolarimeter
(A,B,C,D,E,an
dF),when
Monoch
romatic
LightIs
Employe
da
Linea
rSOP
EllipticalSOP
Circu
larSOP
S1
S2
S3
S1
S2
S3
S1
S2
S3
Exp
ectedva
lues
0.5
0.86
60
00.86
60.5
00
1A
0:48
7�0:00
10:88
3�0:00
40:00
9�0:00
2−0:01
6�0:00
10:88
3�0:00
10:19
4�0:00
1−0:01
6�0:00
50:01
0�0:00
30:98
4�0:00
3Tho
rlab
s0:50
41�0:00
030:86
35�0:00
020:01
59�0:00
04−0:00
63�0:00
570:86
25�0:00
120:50
59�0:00
200:02
47�0:01
62−0:04
07�0:01
710:99
86�0:00
10B
0:49
2�0:00
20:87
7�0:00
20:01
9�0:00
4−0:02
7�0:00
20:87
3�0:00
10:51
0�0:00
3−0:00
9�0:00
1−0:00
5�0:00
11:00
5�0:00
2Thorlabs
0:49
72�0:00
050:86
72�0:00
030:02
65�0:00
040:00
03�0:00
070:86
67�0:00
030:49
88�0:00
050:03
67�0:00
23−0:05
15�0:00
160:99
80�0:00
01C
0:50
3�0:00
10:86
8�0:00
1−0:00
5�0:00
1−0:00
4�0:00
20:87
9�0:00
20:51
5�0:00
10:02
5�0:00
20:00
6�0:00
10:99
9�0:00
1Thorlabs
0:50
35�0:00
070:86
37�0:00
04−0:02
28�0:00
07−0:02
14�0:00
110:86
45�0:00
030:50
21�0:00
050:04
79�0:00
11−0:07
52�0:00
120:99
60�0:00
01D
0:49
3�0:00
10:87
2�0:00
1−0:00
6�0:00
1−0:02
7�0:00
10:87
4�0:00
10:51
0�0:00
1−0:00
1�0:00
10:00
0�0:00
11:00
4�0:00
2Thorlabs
0:51
95�0:00
030:85
41�0:00
02−0:02
44�0:00
04−0:03
45�0:00
110:87
11�0:00
220:48
99�0:00
620:01
66�0:00
09−0:05
74�0:00
070:99
82�0:00
01E
0:50
1�0:00
10:86
9�0:00
20:00
0�0:00
2−0:03
2�0:00
10:87
3�0:00
10:52
5�0:00
1−0:00
3�0:00
10:00
0�0:00
31:00
6�0:00
3Thorlabs
0:50
81�0:00
030:86
11�0:00
020:01
55�0:00
04−0:03
01�0:00
050:86
68�0:00
030:49
78�0:00
050:05
88�0:00
12−0:04
41�0:00
120:99
73�0:00
01F
0:48
1�0:00
10:85
0�0:00
1−0:00
4�0:00
1−0:02
7�0:00
20:86
6�0:00
10:50
8�0:00
1−0:00
4�0:00
2−0:00
5�0:00
11:00
7�0:00
1Tho
rlab
s0:49
37�0:00
110:86
93�0:00
06−0:02
35�0:00
08−0:03
09�0:00
060:86
81�0 :00
020:49
54�0:00
040:02
69�0:00
09−0:04
74�0:00
030:99
851�0:00
003
a SOPsarealso
mea
suredwiththecommercial
polarimeter
(Tho
rlab
s).
Tab
le3.
Norm
alized
Stoke
sParam
eters(Fully
Polarize
dContribution)an
dDOP
Mea
suremen
tsofTwoDifferentSOPsPartially
Polarize
dbyMea
nsofPolarimeter
Dan
dtheCommercial
Polarimeter
(Thorlab
s),when
Monoch
romatic
LightIs
Use
d
S1
S2
S3
DOP
SOP
1Polarim
eter
D0:66
9�0:00
20:08
5�0:00
40:73
9�0:00
30:74
2�0:00
1Tho
rlab
s0:65
34�0:00
080:08
12�0:00
080:75
26�0:00
060:72
4�0:04
7SOP
2Polarim
eter
D−0:24
9�0:00
2−0:03
8�0:00
20:96
8�0:00
20:63
2�0:00
1Tho
rlab
s−0:29
23�0:00
12−0:00
53�0:00
070:95
63�0:00
040:64
9�0:04
2
1 October 2011 / Vol. 50, No. 28 / APPLIED OPTICS 5441
variances are smaller than those obtained for line-arly polarized light configurations, this improvementbeing especially relevant in configuration D. Weremark that the EWV is exactly the average of theStokes parameter variances. Therefore, in Fig. 3 weare representing the experimental EWV obtainedfrom the measurements and so, evidencing experi-mentally the improvement achieved during the opti-mization process using the six setups. Note that thetendency shown in Fig. 3 is in agreement with theEWV values of Table 1.
Finally, we have measured two partially polarizedstates by using polarimeter D, being the polarimeterproviding the lowest average variance. ThemeasuredSOPs were generated by illuminating a depolarizer(Thorlabs, DPU-25). The light transmitted throughthis element has a polarization that varies spatially,and because the polarimeter performs an average of asmall area, the resulting SOP mean has a certainamount of depolarized light. TheDOPmeasurementsobtained are shown in Table 3. Notice that the DOPvalues between polarimeter D and the commercialone are very close. Moreover, we include the Stokes
parameters normalized byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS21 þ S2
2 þ S23
q, providing
the fully polarized contribution. Again, the experi-mental results are in agreement. However, small dis-crepancies between results can be attributed to thedifficulty in applying exactly the same measuringarea for the two polarimeters. Nevertheless, experi-mental results evidence the suitability of the polari-meter to measure the polarization of light partiallypolarized.
B. LED Illumination
Finally, the polarimeter was tested by illuminatingwith a nonmonochromatic light source. In particular,we have used a red LED (Thorlabs, M625L2), withnominal wavelength 625nm and a bandwidth of20nm. Because birefringence depends on the wave-length, the QWP and the TN-LC device involved insetup D [Fig. 1(d)] may introduce a different retar-dance to the different wavelengths in the LED spec-trum. For this reason, an achromatic QWP was usedin the setup. However, a study of the TN-LC achro-maticity is important to ensure the same projectionSOPs for all the LED wavelengths.
Fig. 3. (Color online) Stokes parameter variance average as afunction of different polarimeters, using a linearly polarizedSOP (rhombi) or an elliptically polarized SOP (squares) to projectover the exiting light from the TN-LC device.
Tab
le4.
Stoke
sParam
etersofT
hreeDifferentStatesofP
olariza
tion(theStoke
sVec
tors
areNorm
alized
)Mea
suredbyPolarimeter
Dan
dtheCommercial
Polarimeter
(Thorlab
s),w
hen
They
areIlluminated
withan
LED
LightSource
Linea
rSOP
EllipticalSOP
Circu
larSOP
S1
S2
S3
S1
S2
S3
S1
S2
S3
Exp
ectedva
lues
0.5
0.86
60
00.86
60.5
00
1D
0:49
4�0:00
10:87
8�0:00
1−0:00
1�0:00
10:02
5�0:00
10:83
9�0:00
1−0:41
45�0:00
040:01
7�0:00
1−0:00
9�0:00
11:00
2�0:00
1Tho
rlab
s0:49
77�0:00
030:86
73�0:00
02−0:00
73�0:00
010:04
59�0:00
030:89
404�0:00
003
−0:44
56�0:00
01−0:01
88�0:00
01−0:03
44�0:00
010:99
923�0:00
001
5442 APPLIED OPTICS / Vol. 50, No. 28 / 1 October 2011
Different optical elements were illuminated withthe LED source, and the outcoming SOP was mea-sured with the commercial polarimeter. First, a chro-matic waveplate (for 633nm) was illuminated withlinearly polarized light at 45° with respect to theQWP axis. The measured DOP was 0.44. Then, thechromatic retarder was replaced by an achromaticQWP, obtaining a DOP of 0.99. Finally, we haveanalyzed the effect of illuminating the TN-LC devicewith linearly polarized light, and the DOP of theexiting SOP was measured for different voltagesapplied to the TN-LC device. The DOP values werefrom 0.96 to 1 (i.e., practically fully polarized light).Comparing these data, we have deduced that theTN-LC device is acting as an achromatic element forthis bandwidth.
Next, polarimeter D was calibrated, achieving anexperimental EWV equal to 3.59. This result is closeto the theoretical one (3.50), and it is very similar tothat obtained by illumination with monochromaticlight (3.57). So, the current polarimeter was also wellconditioned for the LED illumination.
Then, we tested the performance when measuringfully polarized states. The SOPs were generated bymodifying the polarization of the light beam with apolarizer and an achromatic retarder, to ensure thesame incident SOP for all the wavelengths of theLED spectrum and so, generating a fully polarizedmean SOP.
In Table 4, we present the measurements of threedifferent fully polarized states. The Stokes param-eters obtained with the implemented polarimeterare in accordance to those provided by the commercialpolarimeter. The accuracy of the polarimeter, when il-luminated with the LED, is of the same order as theone obtained by applying monochromatic light.
Finally, we have completed the study bymeasuringtwo partially polarized incoming light beams. Thepartially polarized SOPs were generated by illumi-nating a polarizer with the LED light followed by achromatic QWP. Because the chromatic QWP pre-sents a retardance depending on the wavelength,each wavelength generates a different SOP. Becausethe polarimeter performs an average measurement,the detected mean SOP had a certain amount ofdepolarization (i.e., a DOP lower than 1).
In Table 5, two particular cases are provided: SOP1 and SOP 2. In the first case (SOP 1), the studiedSOP was generated by illuminating a polarizer withthe LED source followed by a chromatic QWP withits fast axis at 45° of the polarizer transmission axis.In such a case, circular light is obtained for 633nm(i.e., the QWP operational wavelength), but for theother wavelengths in the LED spectrum, differentSOPs are achieved. The measurement obtained withthe polarimeter shows a mean SOP almost circularlypolarized, accompanied with a significant unpolar-ized light value (DOP equal to 0.4). The second case(SOP 2) was analogous to the first one, but for adifferent relative angle between the polarizer trans-mission axis and the QWP fast axis (i.e., a differentmean SOP is obtained).
We want to emphasize that the results given inTables 4 and 5 prove the potential of the implemen-ted polarimeter, which is able to measure partially orfully polarized light with a nonmonochromatic lightsource with a small bandwidth (around 20nm).
4. Simulation of Polarimeters by Changing Some ofthe TN-LC Cell Features
The monopixel TN-LC cell is extracted from a solar-powered flashing LCD key ring. We emphasize thatin this work, the TN-LC characterization by means
Table 5. Normalized Stokes Parameters (Fully Polarized Contribution) and DOP Measurements of Two Different SOPs Partially Polarizedby Means of Polarimeter D and the Commercial Polarimeter (Thorlabs), when an LED Light Source Is Employed
S1 S2 S3 DOP
SOP 1 Polarimeter D −0:019� 0:002 −0:114� 0:001 0:993� 0:002 0:437� 0:001Thorlabs −0:0418� 0:0002 −0:0647� 0:0002 0:99703� 0:00001 0:410� 0:003
SOP 2 Polarimeter D −0:044� 0:001 0:686� 0:001 0:727� 0:001 0:5514� 0:0004Thorlabs −0:0500� 0:0002 0:7120� 0:0001 0:7004� 0:0001 0:546� 0:005
Fig. 4. (Color online) EWV dependence on the twist angle of theTN-LC device. (Actual twist angle value used in the implementedpolarimeters in Section 3: 93:2°.)
Fig. 5. (Color online) EWV dependence on the maximum birefrin-gence of the TN-LC device. (Actual maximum birefringence valueused in the implemented polarimeters in Section 3: 276:5°.)
1 October 2011 / Vol. 50, No. 28 / APPLIED OPTICS 5443
of Mueller matrices as a function of the voltage isenough to optimize and implement the presentedpolarimeters. Nevertheless, in order to have extra in-formation about the physical properties of the LCDdevice used, we have conducted an additional experi-mental characterization by following the methodexplained in Ref. [19]. In Table 6, we show the cali-brated parameters of our particular TN-LC cell: totaltwist angle, the maximum birefringence when illu-minating with 633nm and the orientation of themolecular director at the input face.
Afterward, we simulated the simplest TN-LCpolarimeter setup, corresponding to Fig. 1(a), byapplying a mathematical model of the TN-LC celldeveloped in Ref. [20]. In particular, two studies havebeen conducted in order to analyze the influence ofthe TN-LC birefringence and the twist angle on thepolarimeter optimization. Several projection SOPcurves are simulated, each one is composed by thesame number of projection SOPs (46) and changingone of the TN-LC parameters (twist angle or maxi-mum birefringence). The parameter unchangedkeeps the calibrated value shown in Table 6.
First, the influence of the twist angle is discussed.Simulations of TN-LC device with a different twistangle, from 0° to 180°, are analyzed. The EWV of thesimulated projection SOP curves are plotted in thegraph of Fig. 4. Notice that TN-LC devices with atwist angle from 40° to 140° give low EWV values ofthe same order. In addition, the use of a supertwistednematic LC cell with a twist angle higher than 140°is not an advantage for the optimization of theinstrument.
Second, the maximum birefringence is analyzed.Note that by changing the maximum birefringencevalue, we are simulating different thickness of theTN-LC cell [20]. The EWV of the projection SOPscurve, using the specific values of TN-LC birefrin-gence, are calculated. Figure 5 shows the evolution ofthe QI when the maximum birefringence increases.Note that the EWV is stabilized after 250° of maxi-mum birefringence, approximately. Therefore, it isnot necessary to work with higher birefringences(more than 250°). Moreover, working with thickerTN-LC cell leads to a slower response time.
We remark that theEWVvalues achieved in Figs. 4and 5 are not comparable with those shown inTable 1, because the first ones correspond to charac-teristic curves (using 46 projection SOPs) and theothers provided in Table 1 are for a set of 4 projectionSOPs.
5. Conclusions
In summary, we present the optimization of sixdifferent polarimeters composed of a single TN-LCdevice. The advantage of using a single LC deviceis its simplicity and reduced cost compared with thestandard polarimeters based on two LCDs. By mod-ifying some parameters such as the optical path, theincidence angle or the projection polarization states,a significant improvement in the minimization ofnoise propagation is achieved. The optimized polari-meters were experimentally implemented, achievingexcellent results for accuracy and repeatability.Experimental tests were developed in order to provetheir correct performance for monochromatic andLED illumination. In particular, experimental mea-surements verify that the instrument is suitable tomeasure partially and fully polarized light. Finally,we have simulated polarimeters based on a TN-LCdevice with different values of maximum birefrin-gence and twist angle. We have deduced that themost suitable TN-LC cell has a twist angle from40° to 140° and a maximum value of birefringencearound 250°.
Therefore, this work proves that dynamic Stokespolarimeters with very low error propagation valuescan be performed by using a single TN-LC device.
We acknowledge financial support from theSpanish Ministerio de Ciencia e Innovación(FIS2009-13955-C02-01) and the Generalitat de Cat-alunya (2011FI_B2 00140). C. Iemmi acknowledgesthe support from Universidad de Buenos Aires andthe Consejo Nacional de Investigaciones Científicasy Técnicas (Argentina).
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Table 6. Features of the TN-LC Device Employedin the Polarimeter Setups
TwistAngle (°)
MaximumBirefringence for
633nm (°)
Orientation of the MolecularDirector at the Input
Face (°)
93.24 276.51 48.01
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