Since the discovery of the influence of the electricfield on liquid-crystal (LC) molecular orientation[1], LC phase retarders have been widely used aselectrically tunable wave plates. Some of these mod-ulators were based on the classical bipolar configura-tion using a large range of LC structures, e.g., planarnematic (PN) [2], twisted nematic (TN) [3,4], electri-cally controlled birefringence (ECB) [5], or smecticelectroclinic (SmA�) [6]. However, these technologieshave limitations, such as slow dynamic behavior(< 0:3°=μs) or limited total phase variation due tothe small LC layer thickness (eLC < 10 μm).Multipolar structures with planar alignment [7],
vertical alignment [8], homeotropic SmA� [9], orpolymer stabilized [10] liquid crystal also have beeninvestigated to fabricate rotatable wave plates.Contrary to the initial electrically tunable waveplates, the electric field is applied in the azimuthalplane and the birefringence can be oriented in an ar-
bitrary azimuthal direction with endless rotation.The drawback of these devices is that the effectivearea, where the electric field is uniform, is verysmall (≃10 μm× 10 μm) and is, therefore, difficult toprocess and couple, thus mostly limited to fiberapplications.
A quadripole architecture with a transverse-to-longitudinal continuous electric field transitionwas already studied [11,12]. Unfortunately, at thetime, no material both transparent and conductiveenough was readily available. Hence, the authorsdid not manage to make a quadripole electrodedesign with a large transverse aperture becausethe electric field was limited to the vicinity of the con-ductive electrodes.
The recent progress in conducting and trans-parent polymer overcomes these difficulties [13].By using a poly(3,4-ethylenedioxythiophene) poly(styrenesulfonate) (PEDOT-PSS) thin film, a hightransparency is obtained in the vis-IR spectrum(>90%), with an adequate sheet resistance range(100KΩ=□−1MΩ=□) and good mechanical proper-ties. This material is low cost and can be depositedby a spin-coating process.
We demonstrate that a LC quadripolar retarderimproves the maximum optical path variation(eLCΔn > 4 μm) by using a thick LC layer (eLC ¼20 μm) with high birefringence (Δn ¼ 0:224), and in-creases the LC rotation speed (0:4°=μs) by driving theLC director with a control of the electric field orien-tation in order to optimize the electric torque.
2. Principle and Theory
A. Electric Field Orientation Control
The principle of the quadripolar liquid crystal (QLC)device consists of driving the electric field orientationby modifying the voltages applied to four electrodes,as detailed in Fig. 1. Electrodes A and C are kept atconstant potentials of þV0 and −V0, respectively,while control voltages −=þ V are applied on B andC. In this configuration, the angle Ψ of the electricfield only depends on the V=V0 ratio and the celldimensions [11]:
Ψ�VV0
�¼ tan−1
�l
eLC
ð1 − V=V0Þð1þ V=V0Þ
�; ð1Þ
where l is the interelectrode distance and eLC is theLC layer thickness.
B. Electric Field Uniformity
To avoid the confinement of the electric field near theelectrodes when a longitudinal electric field (parallelto the light beam) is desired, the area between twoelectrodes of each substrate needs to be conductive[11]. To do that, a transparent conductive polymerlayer is coated between the couples of electrodes(A,B) and (C,D) as illustrated in Fig. 1. The cellcan be compared to a resistance–capacitance (RC)circuit. To allow a uniform repartition of the electricfield in the whole cell volume, the time constantτ ¼ 2πRC, where R is the conductive polymer resis-tance layers and C is the LC layer capacitance, mustbe smaller than the period of the signal. τ can then bewritten as
τ ¼ 2π lρpLep
εLCLleLC
¼ 2πR□εLCl2
eLC; ð2Þ
where εLC is the LCmaximum dielectric permittivity,ρp is the PEDOT-PSS resistivity, ep is the PEDOT-PSS layer thickness, L is the electrode length alongthe z direction, and R□ is the sheet resistance of thePEDOT-PSS layer (in Ω=□) corresponding to theρp=ep ratio. With a given cell dimension, all the para-meters but R□ are fixed.
If τ−1 > f , where f is the signal frequency, we canconsider the electric potential repartition similar tothe static case and simulate it by solving the Laplaceequation in the cell domain. Figure 2 shows the com-putation of this equation by using GetDP, a free soft-ware for the treatment of discrete problems by thefinite element method [14].
When a longitudinal field is applied [Fig. 2(a)],VA ¼ VB ¼ V0 and VC ¼ VD ¼ −V0, the voltage is
Fig. 1. (Color online) (a) Quadripolar cell cross section that illus-trates the control of the electric field by tuning the electrode vol-tages (black thick lines). The polarized light propagation directionis along y. (b) Top view of the cell.
Fig. 2. (Color online) Simulation of the equipotential lines in thecell for (a) longitudinal, (b) transient, and (c) transverse electricfields. We used the following parameters for the simulation:V0 ¼ 10V, eLC ¼ 20 μm, l ¼ 100 μm, d ¼ 5 μm, εLC ¼ 20 ε0.
constant on the top surface (þV0) and on the bottomsurface (−V0), and the surface current density ~Js ¼ 0on both PEDOT-PSS layers.In the opposite, for the transverse field case
[Fig. 2(c)], VA ¼ VD ¼ V0 and VB ¼ VC ¼ −V0, thereis a linear variation of the voltage from V0 to −V0 onthe two PEDOT-PSS layers. The surface currentdensity in the PEDOT-PSS layers is
~Js ¼1R□
VA − VB
L~ex ¼
2V0
R□L~ex: ð3Þ
In Fig. 2(b), the electric field is oriented betweenthe longitudinal and transverse positions, VA ¼ V0,VB ¼ −V0=2, VC ¼ −V0, and VD ¼ V0=2. It is ob-served that potential lines are always nearly paralleland, hence, produce a uniform LC director orienta-tion in the cell.
C. Electric Torque Optimization
In our device structure, no alignment layers areused. The surface forces are negligible compared tothe viscosity η and the electric torque [15]. For a fieldapplied with angle Ψ, the dynamic response of theLC is described by the Erickson–Leslie equation:
dθdt
¼ ΔεE2
2η sin 2ðΨ − θÞ; ð4Þ
where θ is the LC director orientation (see Fig. 1) andΔε is the anisotropic dielectric permittivity.The electric torque ~ΓE induced by the electric field
is oriented in the ~z direction normal to the ð~x;→ yÞplane and is described by
~ΓE ¼ ~P∧~E ¼ ΔεE2
2sin 2ðΨ − θÞ~z; ð5Þ
where ~P is the induced polarization:
~P ¼ �ε~E; ð6Þand �ε is the dielectric permittivity tensor:
�ε ¼�εo þΔεcos2θ Δε
2 sin 2θΔε2 sin 2θ εo þΔεsin2θ
�: ð7Þ
Thus, the director rotation speed depends only onviscosity and electric torque. Electric torque is max-imum if the electric field is at 45° of the director andnull if both vectors are parallel or perpendicular(Ψ − θ ¼ 0 or π=2). This property is now used in thecoming experiments in order to decrease the re-sponse time of the modulator.
3. Experiment and Results
A. Design and Realization
QLC cells are achieved as follows. First, chromiumelectrodes are etched on glass substrates by the
photolithographic process. In the active area, theelectrode pattern consists of two parallel chromiumstrips spaced with a l ¼ 100 μmgap. Each electrode is0:5 μm thick, d ¼ 200 μm wide, and L ¼ 500 μm long.The 100 μm × 500 μm area between electrodes consti-tutes the optical aperture. Substrates are thencoated with 30nm of an appropriate layer of PED-OT-PSS by spin coating, with a 30 min post curingat 180 °C. Finally, the two substrates are assembledand pressed using a modified MJB3 mask aligner toprecisely align the electrodes and then glue themwith a UVadhesive at a thickness controlled by eLC ¼20 μm plastic ball spacers. The cell is then filled withnematic LC BL004 [Δn ¼ 0:224 at 589nm, no ¼1:5180, Δε ¼ 15:6, ε∥ ¼ 20:9, and viscosity (at 23 °C)η ¼ 250mPa:s].
B. PEDOT-PSS Characterization
The PEDOT-PSS aqueous dispersion used here is asolution composed with 15 wt. % of Baytron P fromHCStarck, 25% of ethylene glycol, 47% of isopropa-nol, and 2% of tetraethoxysilane. In this solution,ethylene glycol is used for conductivity enhancement[16] and tetraethoxysilane for increasing film tough-ness and adhesion. For achieving optimum hardness,a curing temperature of about 180 °C is required. Thefilm resistance is constant until 200V and does notshow any diode effect or electrical leakage. The opti-cal transmission spectrum ismeasured with aUV/visspectrophotometer (Lambda 900 from Perkin Elmer)and the optical losses are less than 5% over a 400 to1700nm spectrum. The sheet resistance is chosenclose to 1MΩ=□ to limit the electric power dissi-pated in the conducting polymer layer, but this pa-rameter is a compromise with the time constant asseen above. With this value and the cell dimensionsdescribed above, the electric power dissipated in thePEDOT-PSS layer correspond to 1W:mm−2 at 100Vand the cell impedance is similar to a RC circuit witha cutoff frequency superior to 100KHz (τ ≤ 10 μs).
We achieved a coating with better optical trans-mission than ITO in the whole vis-IR spectrum (upto 99%), an adequate resistivity (≃1MΩ=□), a suffi-cient bandpass for control voltages, good mechanicalproperties, and chemical compatibility with LC.
C. Electro-Optical Curve
The QLC cell is then mounted between crossed polar-izers, with polarizer and analyzer oriented at 45°from the x axis. Two spherical lenses are used to cou-ple a 632nm He–Ne laser beam into the slit formedby the electrodes. The transmitted intensity modula-tion is measured by a Si photodiode connected to anoscilloscope. To drive the electric field, we apply fourindependent 2kHz square signals on each electrodewith a high slew rate (1 μs) and 0 to 90V amplitudeas explained above.
Then, the electro-optic response of the cell is mea-sured for a complete transverse-to-longitudinal elec-tric field transition (Fig. 3). To do this, voltages on Aand C are kept constant (VA ¼ V0, VC ¼ −V0) while
voltages on D and B are varied with a constant rampduring 1 s (VD ¼ V, VB ¼ −V, V ¼ þV0…−V0). With-in this period, it is assumed that θ ¼ Ψ because theresponse of the LC is far quicker than the rotation ofthe electric field.The optical phase variation ϕ between the x polar-
ization and the z polarization is [17]
ϕðθÞ ¼ 2πeLCðnðθÞ − noÞλ ; ð8Þ
where no and ne are the LC ordinary and extraordin-ary refraction indices and nðθÞ is the relative index infunction of θ, such as
The relation between the outgoing light intensity Iafter the analyzer and the optical phase variationϕ is given by
I ¼ I0sin2
�ϕðθÞ − ϕ0
2
�; ð10Þ
where I0 is the input light intensity and ϕ0 is theoptical phase for θ ¼ 0.Now, the light intensity depending on V=V0 can be
calculated using the expression of θ given in Eq. (1),and it can be compared to the measured optical out-put as described in Fig. 3. The experimental andtheoretical curves present the same number of peaksand valleys, corresponding to the same optical pathvariation. However, the first minimum and maxi-mum do not have a good contrast and there is adifference in the peak position. This may be due toa too-high sheet resistance of the polymer layer,which may prevent a fully uniform longitudinal elec-tric field. Moreover, as explained above, we did notuse any alignment layers because we completely con-trol the LC orientation with the electric field. How-ever, the lack of alignment layers could result inscattering due to small defects on the PEDOT-PSSsurfaces or to the transient states in the first turn-on and final field-off processes. Thus, in further workwe shall reduce the sheet resistance value to obtain amore uniform longitudinal electric field and use somealignment layers to avoid scattering and improve the
QLC cell accuracy. The innovation of this drivingmethod is that the total phase variation is controlledwith a voltage range of −V0 to V0 (if l=eLC ≤ 5), incomparison with classical bipolar cells, which dependon the threshold voltage and where this range isclose, from 0 to 2V.
D. Dynamic Studies
To study the dynamic behavior of the QLC phasemodulator, we measure the transition time betweenthe first maximum intensity I1max and the secondminimum intensity I2min of the curves in Fig. 3, cor-responding to, respectively, θ1 ¼ 77° and θ2 ¼ 68:5°and driving voltages V1 ¼ 1V and V2 ¼ 6V withV0 ¼ 20V. Figure 4 shows that the switching timesfrom θ1 to θ2 and from θ2 to θ1 are not equal. Thisdifference is due to the geometry of the cell. Indeed,for the same voltage amplitude, the electric field isstronger in the longitudinal direction than in thetransverse direction because the transverse dimen-sion is bigger than the longitudinal one.
In Eq. (4), the response time depends on E2. How-ever, in our case, the dimension ratio between thelongitudinal and the transverse dimensions is 1=5so that the switching time should be 25 times shorterin the transverse case than in the longitudinal one.Here it does not exactly correspond because the elec-tric field switches between two alternative positions.Moreover, the total amplitude voltage is bigger in theθ2 orientation. This is why τ2−1 (3ms), the transitiontime from θ2 to θ1 (solid curve in Fig. 4), is 6 timeslarger than the switching time τ1−2 (500 μs) from θ1to θ2 (dashed curve in Fig. 4).
To increase the molecular rotation speed, we needto increase the electric field amplitude and maximizethe electric torque, as explained in Subsection 2.C. To
Fig. 3. (Color online) Theoretical and experimental curves for thetransmitted intensity by the QLC cell.
Fig. 4. (Color online) Measurements of response times τ2−1 for aθ2 to θ1 switch (solid curve) and τ1−2 for a θ1 to θ2 switch (dashedcurve).
do this, the electric field must have a transient orien-tation at 45° of the LC director, hold this position dur-ing the LC rotation, and then return to the desiredorientation angle [15].To satisfy this principle for switching the LC mo-
lecules from θ1 to θ2, we have to apply an electric fieldwith a higher amplitude at a transient angle of 32°during the switching time, and then come back to thefinal orientation θ2 ¼ 68:5°. This sequence is repre-sented in Figs. 5(a) and 5(b); the curve in Fig. 5(b)represents the optical transmission through the op-tical system and the time diagram in Fig. 5(a) showsthe sequence voltage applied on the four electrodes.For τ ≤ −75 μs, VA ¼ −VC ¼ �20V, and VD ¼
−VB ¼ �1V corresponding to θ1, then for −75 ≤
t ≤ 75 μs, a higher amplitude electric field orientedat a 32° angle is applied, with voltages VA ¼ −VC ¼�85V and VD ¼ −VB ¼ �66V. Finally, voltagesVA ¼ −VC ¼ �20V and VD ¼ −VB ¼ �6V are ap-plied to maintain the LC director at θ2. With this
driving method, we obtained a switching time of120 μs, corresponding to a rotation speed of 0:07°=μs.
For the longitudinal-to-transverse switching, wecould apply the torque optimizing scheme but wedecided to apply a complete longitudinal field forthe transient state, which is not exactly at 45° ofθ2, but it is a much stronger electric field than thealternative field with a 113:5° orientation.
The time diagrams in Fig. 5(c) show the drivingvoltages applied to the four electrodes. First, theLC is oriented at 68:5° by applying the voltage de-scribed above. A short pulse of 40 μs with VA ¼−VC ¼ 85V and −VB ¼ VD ¼ −85V is then appliedand, finally, the voltages are set to obtain the 77°angle. The acquisition in Fig. 5 represents the LCresponse time, close to 20 μs, that corresponds ap-proximately to a 0:4°=μs LC director rotation speed.These performance can still be improved by usinghigher voltages and LC with lower viscosity.
Fig. 5. (Color online) LC response time measurement when driven with optimized voltage forms. (a) and (c) Time diagrams of the drivingvoltages applied on the four electrodes corresponding to, respectively, the optimized LC response time measurement in the (b) transverse-to-longitudinal case, and (d) the optimized LC response time measurement in the longitudinal-to-transverse switching.
In this paper, we have validated the use of a trans-parent conductive polymer to perform a uniform andmultidirectional electric field in a four-electrode LCdevice.Cells studied in this article used a 20 μm thick LC
layer, providing a large phase variation compared tobipolar cells. The thickness of the LC layer could beincreased up to 100 μm and, thus, create an evenlarger phase delay. This hypothesis, however, needsto be confirmed by measuring the electric and opticalparameters of these new cells.Another advantage of the QLC retarder lies in the
voltage range for a maximal phase variation. Indeed,the LC director orientation does not depend on athreshold voltage, as in the classical bipolar cells,but on the complete driving voltage range from−V0 to V0.A 0:4°=μs rotation speed of the nematic director
has been achieved with voltages of 85V and withan optimizing torque driving method. This per-formance could also be improved by increasing theapplied voltages because of the square dependencebetween the LC response time and the electric fieldamplitude, and by using LC with lower viscosity. Toimprove the size of the active window, another cellarchitecture will be studied with thin interdigitedelectrodes.The high transparency of the PEDOT-PSS layer in
the whole vis-IR spectrum allows it to be used in thewhole range of LC applications. Moreover, these thinconductive transparent electrodes are a cheaperalternative to classical ITO electrodes.Finally, this kind of device provides improvements
in total optical path variation, dynamic behavior, andpupil dimensions in comparison to other LC devices.That can be useful for applications that need high op-tical phase delay, such as polarization analysis, mi-crowave phase shifting, or optical network switching.
We would like to thank J.L. de Bougrenet de laTocnaye for making the clean room available,O. Castany for his recommendations and technicalhelp, and Association Nationale de la Recherche etde la Technologie (ANRT) for financing of this work.
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