Polarization properties of polymer-dispersed liquid-crystal film with small nematic droplets

Polarization properties of polymer-dispersedliquid-crystal film with small

nematic droplets

Polina G. Lisinetskaya, Alexander A. Konkolovich, and Valery A. Loiko*B. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus,

68 Nezalezhnasti Avenue, Minsk 220072, Belarus

*Corresponding author: [email protected]‑net.by

Received 12 January 2009; revised 10 April 2009; accepted 2 May 2009;posted 7 May 2009 (Doc. ID 106256); published 1 June 2009

The polarization state of light transmitted through a polymer-dispersed liquid-crystal film with small,spherical, nonabsorbing, partially oriented nematic droplets is theoretically investigated. The modelused is based on the effective medium approach. Scattering properties of a single droplet are describedby the Rayleigh–Gans approximation. Propagation of coherent light is described within the framework ofthe Twersky theory. To describe the orientation of liquid-crystal molecules inside droplets and liquid-crystal droplets in a sample, the concept of multilevel order parameters is employed. Conditions for cir-cular and linear polarization of the transmitted light are determined and investigated. © 2009 OpticalSociety of America

OCIS codes: 160.2100, 160.3710, 290.5855, 310.6860.

1. Introduction

Nowadays a great deal of attention is given to theinvestigation of electro-optic materials based onpolymer-dispersed liquid-crystal (PDLC) materials,which consist of liquid-crystal (LC) droplets em-bedded in a solid polymeric binder [1]. The materialis placed between two transparent plates with trans-parent electrodes deposited upon them. When an ex-ternal electric field is applied to the layer, themolecules of the LC change their orientation insidethe droplets. As a result, optical properties of the LCcell are changed. Thus characteristics of light trans-mitted through a PDLC sample vary with the appliedfield. By selecting proper materials and methods ofPDLC cell preparation, different electro-optic devicesbased on this effect can be designed. Using PDLClayers with large (compared with the wavelength)LC droplets, many structures to modulate the inten-sity of light in the visible [2,3] and infrared [4] rangeshave been demonstrated. Of particular interest are

PDLC films with LC droplets that are small com-pared with the wavelength. These films possess hightransmittance [5] and enable one to modulate thephase [6,7] and polarization [8] of transmitted light.PDLC cells thus possess a wide range of applicationsfrom lenses [9,10], lens arrays [11] with a tunable fo-cal length, and other optical elements to diffusiverandom lasers [12] and vertical cavity surface-emitting lasers [13].

Although the principles of PDLC operation arerather well known, a rigorous theory that establishesconnection between the morphological properties of aPDLC sample and the characteristics of transmittedlight has not yet been developed. Different approxi-mate approaches have been developed for thispurpose [1,5,14–17]. Almost all the approaches areeither sufficiently complicated and allow obtaininganalytical results only in a limited number of casesor take into consideration only certain aspects of theproblem. As a rule, the scattering cross section,transmission, and phase retardation are investi-gated. There are fewer studies on the polarizationof transmitted light, although there are several

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3144 APPLIED OPTICS / Vol. 48, No. 17 / 10 June 2009

reports on the development of PDLC-based polari-zers [18,19]. Based on our model for light transmis-sion through a PDLC layer with fine nonabsorbingLC droplets [8,20,21], we theoretically investigatethe polarization state of transmitted light in detail.The model is based on the Rayleigh–Gans approx-

imation [22] to describe the scattering properties of asingle droplet and the Twersky theory [23,24] to de-scribe coherent field propagation in the layer. To de-scribe numerically the degree of orientational orderof LC molecules inside droplets and LC droplets in asample, the concept of multilevel order parameters isemployed. To take into consideration the size and or-ientation distribution of LC droplets the effectiveamplitude scattering matrix approach is used. Themain advantage of the model is that it is sufficientto use the sample order parameter to determine ex-tinction indices of a sample or characteristics oftransmitted light based on the strength of an appliedelectric field. We pay particular attention to the con-ditions wherein circular or linear polarization oftransmitted light is implemented. We analyze the ef-fects of the size and orientation of LC droplets.

2. Basic Relations

Let a PDLC sample with small nonabsorbing ne-matic LC droplets be illuminated by a plane mono-chromatic linearly polarized wave. The samplecontains spherical nonabsorbing droplets with bipo-lar configuration of LC molecules with positive bire-fringence. Let LC droplets in a polymer binder bepartially aligned along some direction described byan eigenvector hedi. Choose the laboratory frame sothat the z axis is perpendicular to the sample planeand the vector hedi lies in the x–z plane. Define thescattering plane as the plane in which the wave vec-

tors of incident and scattered waves lie (see Fig. 1). Itis reasonable to divide the incident and scatteredwaves into two components, perpendicular and par-allel to the scattering plane. Directions of polariza-tion of these components are denoted in Fig. 1 byeigenvectors e⊥ ¼ ðsin−1 δÞei × es for both incident andscattered waves, ei

‖¼ e⊥ × ei for the incident wave,

and es‖¼ e⊥ × es for the scattered wave. Here δ is the

angle between vectors ei and es (see Fig. 1). Thetwo components of the wave scattered by a single dro-plet can be determined in terms of an amplitude scat-tering matrix [22]:

�Es · es‖Es · e⊥

�¼ expðikrÞ

−ikr

�S2 S3

S4 S1

��Ei · ei‖Ei · e⊥

�: ð1Þ

Here Es andEi are the electric vectors of the scatteredand incident waves, respectively; k is the module ofthe wave vector of the incident light in the polymer; ris the distance to the point of observation; Sj (j ¼1:::4) are the elements of the amplitude scatteringmatrix. Under the Rayleigh–Gans approximation,the elements of the amplitude scattering matrix canbe written in the form [25]

S1 ¼ −ik3

4π f

�εdoεp

− 1þΔεdεp

ðe⊥ · edÞ2�; ð2Þ

S2 ¼ −ik3

4π f

��εdoεp

− 1

�ei · es þ

Δεdεp

ðes‖· edÞðei‖ · edÞ

�;

ð3Þ

S3 ¼ −ik3

4π fΔεdεp

ðes‖· edÞðe⊥ · edÞ; ð4Þ

S4 ¼ −ik3

4π fΔεdεp

ðei‖· edÞðe⊥ · edÞ; ð5Þ

εdo ¼13ð2εo þ εe −ΔεdÞ; ð6Þ

Δεd ¼ ðεe − εoÞSSd; ð7Þwhere εe and εo are the permittivities of the LC forextraordinary and ordinary waves, respectively; εpis the permittivity of the polymer; ed is a unit vectorthat specifies the optical axis of a LC droplet (oftenreferred to as a droplet director); S is the molecularorder parameter of the LC [14]; Sd is the orderparameter of a LC droplet [14,26]; and f is a functionof the LC droplet size and shape and the direction oflight scattering [21]. For spherical droplets this func-tion is [22]

Fig. 1. Schematic representation of the laboratory frame, thescattering plane, and the eigenvectors relevant to the problem un-der consideration.

10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS 3145

f ¼ 4πa3 ðsinaR − aR cosaRÞ: ð8Þ

Here a ¼ k½2ð1 − ei · esÞ�1=2 and R is the radius of a LCdroplet. It should be noted that, for forward scatter-ing, the function f is equal to droplet volume V.The PDLC film under consideration consists of a

large number of LC droplets embedded in a polymermatrix, which makes it possible to use statisticalmethods to describe characteristics of transmittedlight. The transmitted radiation can be representedas the sum of coherent (average) and incoherent(fluctuating) components. The coherent componentis determined by means of the Foldy–Twerskyequation [23]. We deal with polarized light and there-fore use the Foldy–Twersky integral equation in vec-tor form:

� hEs · es‖ihEs · e⊥i

�¼

�Ψ2 Ψ3

Ψ4 Ψ1

��Ei · ei‖Ei · e⊥

�; ð9Þ

where functions Ψj (j ¼ 1::4) are [8]

Ψ1;2 ¼ exp�ikl

�1þ i

2πNk3

hS1;2ð0Þi��

; ð10Þ

Ψ3;4 ¼ 0: ð11Þ

Here l is the path length of the incident light insidethe film; N is the concentration of the LC droplets inthe film; hS1;2ð0Þi are the elements of the amplitudescattering matrix [Eqs. (2) and (3)] at forward scat-tering, averaged over the size and orientation ofthe LC droplets in the PDLC film. Hereafter angularbrackets mean averaging over LC droplet size and/ororientation. Since the scattering plane is undefinedat forward scattering, we can assign directions to vec-tors e⊥, ei‖, and es

‖arbitrarily. Let vector e⊥ be perpen-

dicular to the principal plane (ei, hedi) and vector ei‖

(ei‖¼ es

‖in the case of forward scattering) belongs to

this plane. Under this geometry, ei‖¼ ee and e⊥ ¼ eo.

Vectors ee and eo are unit vectors that denote the di-rection of the extraordinary and ordinary waves po-larization, respectively. Using Eqs. (9)–(11) we findthe extraordinary and ordinary components of trans-mitted light:

Ees ¼ Re

� hEs · eeiEi expðiklÞ

�

¼ 1Ei

exp�−12γel

�cosΦeðEi · eeÞ; ð12Þ

Eos ¼ Re

� hEs · eoiEi expðiklÞ

�

¼ 1Ei

exp�−12γol

�cosΦoðEi · eoÞ: ð13Þ

Here γe;o are extinction indices andΦe;o are phases ofthe extraordinary and ordinary waves, respectively,

γe;o ¼4πNk2

RehS2;1ð0Þi; ð14Þ

Φe;o ¼2πNl

k2ImhS2;1ð0Þi: ð15Þ

By means of Eqs. (12)–(15) characteristics oflight transmitted through a PDLC film can beinvestigated.

3. Phase Shift

The phase shift between extraordinary and ordinarycomponents of transmitted light can be determinedon the basis of Eq. (15) to be

ΔΦ ¼ Φo −Φe ¼Lkcv2 cos θi

Δεdεp

½hðee · edÞ2i − hðeo · edÞ2i�:

ð16Þ

Here θi is an incident angle of light; L is the filmthickness; cv ¼ NhVi is a volume concentration ofdroplets. Generally the volume fraction of LC dro-plets is less than the volume fraction of a LC in thesample, because some portion of the LC is dissolvedin the polymer matrix. In the case of normal inci-dence, Eq. (16) is reduced to

ΔΦn ¼ Lkcv2

Δεdεp

½hsin2 θd cos2 φdi − hsin2 θd sin2 φdi�;

ð17Þ

where θd and φd are the tilt and azimuth angles ofdroplet director ed. Generally the droplet director dis-tribution in a sample is unknown and is somewhatdifficult to establish. To describe the degree of orien-tation we employ the multilevel order parameterconcept. The molecular order parameter S and thedroplet order parameter Sd were introduced in Sec-tion 2. Both parameters describe the orientation ofLC molecules inside a droplet. We consider the para-meters to be identical for each droplet in a film and tochange negligibly with an applied electric field. Todescribe the orientation distribution of LC droplet di-rectors, we use a tensor order parameter of a PDLCfilm [8,14] with components

Sij ¼12h3ðedÞiðedÞj − δiji; ð18Þ

where ðedÞi, i ¼ 1:::3, are the components of thedroplet director. Within the chosen laboratory framethis tensor is diagonal:


S ¼ 12

24 3hsin2 θd cos2 φdi − 1 0 0

0 3hsin2 θd sin2 φdi − 1 00 0 3hcos2 θdi − 1

#: ð19Þ

The components of the order parameter tensor inEq. (19) numerically characterize the degree of order-ing of LC droplet directors in the sample according tothe axes of the laboratory frame. Therefore we denotethem as S11 ¼ Sx, S22 ¼ Sy, and S33 ¼ Sz. Theseparameters are determined by the orientation ofLC droplet directors at zero applied field and dependon an applied external field. Here we consider aPDLC sample with LC droplets aligned in the follow-ing way: at zero applied field the droplet directors areoriented mainly in the sample plane. The angle oftheir maximal deviation from the plane is denotedby θdmax. Within the sample plane, the droplet direc-tors are aligned along the x axis of the laboratoryframe (we chose such a frame; see Section 2). The an-gle of droplet director maximum deviation from the xaxis in the sample plane is denoted by φd

max. As longas an external electric field is applied normally to thesample plane, we assume that angle φd

max does notdepend on its strength. Also, we assume that directorazimuth φd is uniformly distributed over the interval½−φd

max;φdmax�∪½π − φd

max; π þ φdmax�. We can then

rewrite components Sx and Sy in terms of Sz:

Sx ¼12

�ð1 − SzÞ

sin 2φmaxd

2φmaxd

− Sz

�; ð20Þ

Sy ¼12

�ðSz − 1Þ sin 2φmax

d

2φmaxd

− Sz

�: ð21Þ

Under an applied field, the LC droplets in the sam-ple tend to align along the direction of the field (weconsider LC with a positive birefringence). So long asthe field is directed normal to the sample, the changein degree of orientation of LC droplets can be de-scribed by means of the order parameter tensor com-ponent Sz. On the basis of the results published in[15] we derive

Sz ¼14

þ 34

�u

4E2 þ3E4 − 2E2 − 1

16E3 sin θmaxd

ln��uþ 2E sin θmax

d

u − 2E sin θmaxd

��;

ð22Þ

u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE2 − 1Þ2 þ 4E2 sin2 θmax

d

q: ð23Þ

Here E is the reduced strength of an external electricfield. Using the order parameter tensor component

Sz, the phase shift between extraordinary and ordin-ary components of the transmitted light at normalincidence of light [see Eq. (17)] can be rewritten asfollows:

ΔΦn ¼ Lkcv3

Δεdεp

sin 2φmaxd

2φmaxd

½1 − Sz�: ð24Þ

The dependence of the phase shift on the strength ofthe applied electric field is displayed in Fig. 2. Thecalculations were carried out for the sample withthe following parameters, which are typical for aPDLC cell [1,14]: extraordinary and ordinary refrac-tive indices of the LC (at the wavelength of incidentlight λ ¼ 0:6328 μm) ne ¼ 1:74, no ¼ 1:511 (εe ¼ ne

2,εo ¼ no

2), respectively; refractive index of the poly-mer np ¼ 1:524 (εp ¼ np

2); the order parameters S ¼0:6 and Sd ¼ 0:7; volume concentration of the LC indroplets cv ¼ 0:075; the film thickness L ¼ 50 μm; thethreshold value of the strength of the electric fieldEth ¼ 2V=μm; the initial ordering of LC droplets inthe sample is described by means of angles θdmax andφd

max specified in Fig. 2. Analysis shows that themaximal value of the phase shift is determined byboth angles θdmax and φd

max. Meanwhile, the slopeof the curve depends only on angle θdmax. This im-plies that a PDLC sample with small partially or-iented droplets can be utilized as a phase plateworking either in the sharp regime (the phase shift

0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

Strength of external field (V/ m)µ

∆Φn

(deg

rees

)

1

2

3

4

Fig. 2. Phase shift ΔΦn between extraordinary and ordinarywaves versus the applied electric field strength at 1, θdmax ¼φd

max ¼ 5°; 2, θdmax ¼ 45° and φdmax ¼ 5°; 3, θdmax ¼ 5° and

φdmax ¼ 45°; 4, θdmax ¼ φd

max ¼ 45°.


decreases abruptly frommaximum value to zero withapplied electric field) or in the smooth regime (thephase shift is changed gradually with applied electricfield).Equation (16) can be reduced to oblique light inci-

dence on a PDLC sample with randomly oriented LCdroplets. The expression for the phase shift coincideswith that which we previously obtained [8] and is ingood agreement with known experimental data [26](see Ref. [8], Figs. 3 and 4).

4. Extinction Indices

To calculate the extraordinary and ordinary compo-nents of transmitted light by use of Eqs. (12) and (13),one needs to know the extinction indices of a PDLCsample. Since the elements of the amplitude scatter-ing matrix (2)–(5) are pure imaginary values (in themodel we consider a nonabsorbing LC), one cannotuse Eq. (14) to calculate the extinction indices. Thuswe face the problem of determining the extinctionindices.In the independent scattering regime implementa-

tion, the extinction indices are [27]

γe;o ¼ NZ4π

�dσdΩ ðe ¼ ee;oÞ

�sin θsdθsdφs: ð25Þ

Here θs and φs are the tilt and azimuth angles of vec-tor es; e is a unit vector in the direction of incidentlight polarization; dσ=dΩ is the differential scatter-ing cross section for a single LC droplet [22,28]:

dσdΩ ¼ 1

k2ðjS2e · ei‖ þ S3e · e⊥j2 þ jS4e · ei‖ þ S1e · e⊥j2Þ:

ð26ÞBy means of Eqs. (25) and (26) the extinction indicescan be calculated only numerically. To obtain simplerand more convenient formulas, we carried out sev-

eral simplifications. Instead of averaging differentialscattering cross sections over LC droplet size and or-ientation in Eq. (26), we substituted the elements ofthe amplitude scattering matrix (2)–(5) averagedover these values [25]. Therefore, we replaced thePDLC cell with different LC droplets by an effectivecell with monodisperse and ordered LC droplets.

Consider two limiting cases of relative position ofvectors ei, es, and hedi. In the first case scatteringplane ðei; esÞ coincides with the principal plane(ei, hedi). Then the differential scattering cross sec-tions are

�dσdΩ

�ð1Þ

e¼

�dσdΩ ðe ¼ eeÞ

�ð1Þ¼ 1

k2ðjhSð1Þ

2 ij2 þ jhSð1Þ4 ij2Þ;

ð27Þ

�dσdΩ

�ð1Þ

o¼

�dσdΩ ðe ¼ eoÞ

�ð1Þ¼ 1

k2ðjhSð1Þ

3 ij2 þ jhSð1Þ1 ij2Þ:

ð28Þ

In the second limiting case the scattering plane isperpendicular to the principal plane. The differentialscattering cross sections are

�dσdΩ

�ð2Þ

e¼

�dσdΩ ðe ¼ eeÞ

�ð2Þ¼ 1

k2ðjhSð2Þ

3 ij2 þ jhSð2Þ1 ij2Þ;

ð29Þ

�dσdΩ

�ð2Þ

o¼

�dσdΩ ðe ¼ eoÞ

�ð2Þ¼ 1

k2ðjhSð2Þ

2 ij2 þ jhSð2Þ4 ij2Þ:

ð30ÞFig. 3. Schematic representation of the experimental setup in [7].

Fig. 4. Output power versus applied electric field strength. The-oretical results obtained by our model (curves) in comparison withthe experimental data of [7] (symbols).


We determine effective extinction indices of the cellas follows:

γeffe;o ¼ πNZπ0

��dσdΩ

�ð1Þ

e;oþ�dσdΩ

�ð2Þ

e;o

�sin θsdθs: ð31Þ

At normal incident light, Eq. (31) transforms to [25]

γeffe;o ¼ Nk4g16π

�εdoεp

− 1þΔεd3εp

ð1þ 2Sx;yÞ�

2; ð32Þ

g ¼Zπ0

hf i2ðcos2 θs þ 1Þ sin θsdθs: ð33Þ

Here hf i is the function given by Eq. (8), averagedover the LC droplet size. Assume that the LC dropletradii are distributed according to a gamma distribu-tion with a probability density of

ρμðRÞ ¼μμRμ−1

hRiμΓðμÞ exp�−μ R

hRi�; ð34Þ

where hRi is the mean radius of LC droplets and μ isthe distribution parameter. Then for hf i we obtain

hf i ¼ 4πa3 ðt2 þ 1Þ−μþ1

2

�sinððμþ 1Þarctan tÞ

−tðμþ 1Þffiffiffiffiffiffiffiffiffiffiffiffiffit2 þ 1

p cosððμþ 2Þarctan tÞ�; ð35Þ

t ¼ ahRiμþ 1

: ð36Þ

We used Eq. (32) to calculate the ellipticity and azi-muth of the polarization state of transmitted light.Numerical analysis shows that the calculation erroris approximately 1%. Equation (32) allows one to de-termine the effective extinction indices when onlythe order parameter tensor components are known.Note that function g depends on the average diffrac-tion parameter khRi and the degree of LC dropletpolydispersity. These quantities do not change withexternal field. The order parameter tensor compo-nents Sx;y depend on the external field and enter intoEq. (32) in a square-degree form, which enabled us toperform analytic transformations and obtain the re-sults presented in Section 6.

5. Experimental Verification

For experimental verification of the obtained theore-tical results we use the experimental data of [7], inwhich the following experiment was performed. APDLC film was prepared using LC BL24 from Merck

Industrial Chemicals, Darmstadt, Germany (ne ¼1:7174, no ¼ 1:5132 at 589nm) and polymer NOA81from Norland Products, Cranbury, New Jersey (np ¼1:5662 at 589nm).The volume concentration of LCdroplets cv was approximately 1% and the mean ra-dius of droplets was measured to be 50nm. Geometryof the experiment is displayed in Fig. 3. The samplewas illuminated with infrared radiation (λ ¼ 1:3 μm)along its plane (along the y axis in Fig. 3). Normal tothe sample plane (along the z axis in Fig. 3) an elec-tric field was applied. The incident light was polar-ized at angle α ¼ 45° to the z axis by a polarizer.An analyzer was placed behind the sample andwas oriented in two ways: parallel and perpendicularto the polarizer. The power of the transmitted lightwas measured for both orientations of the analyzer.

We consider this experiment in terms of our model.Since the LC possesses positive birefringence, underthe applied field LC droplets tend to align along itsdirection. So the extraordinary wave is polarizedalong the z axis and the ordinary wave is polarizedalong the x axis. Using Eq. (16), we find the phaseshift between extraordinary and ordinary waves un-der this geometry:

ΔΦ ¼ lkcv2

Δεdεp

½hcos2 θdi − hsin2 θd cos2 φdi�: ð37Þ

LC droplets in the sample were oriented in such away that their φd angular distribution was random(in our model it corresponds to φd

max¼π=2). But Mat-sumoto et al. [7] observed a phase shift between ex-traordinary and ordinary waves in the absence ofapplied voltage. Thus we conclude that angle θdmax

was less than π=2.Using the order parameter concept for the phase

shift we finally obtain

ΔΦ ¼ lkcv2

Δεdεp

Sz: ð38Þ

Under the approach described in Section 4 we findextinction indices for extraordinary and ordinarywaves:

γeffe ¼ Nk4

32π2�εdoεp

− 1þΔεd3εp

ð1þ 2SzÞ�

2Z4π

ðhf 1i2 sin2 θs

þ hf 2i2ÞdΩ; ð39Þ

γeffo ¼ Nk4

32π2�εdoεp

− 1þΔεd3εp

ð1þ 2SxÞ�

2Z4π

ðhf 1i2

þ hf 2i2 sin2 φsÞdΩ: ð40Þ

Here f 1;2 are the functions given by Eq. (8) with pa-rameters a:


a1 ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1 − sin θsÞ

p; ð41Þ

a2 ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1 − sinφsÞ

p: ð42Þ

Finally, for transmittance coefficients at parallel andcrossed polarizer and analyzer we obtain

T‖ ¼ expð−γeffe lÞ cos4 αþ expð−γeffo lÞ sin4 α

þ 12exp

�−12ðγeffe þ γeffo Þl

�sin2 2α cosΔΦ; ð43Þ

T⊥ ¼ sin2 α cos2 αðexpð−γeffe lÞ þ expð−γeffo lÞÞ

−12exp

�−12ðγeffe þ γeffo Þl

�sin2 2α cosΔΦ: ð44Þ

To calculate transmittance by means of Eqs. (43) and(44) we need more parameters in addition to thosestated in [7]. To calculate Sz based on the strengthof the applied field through Eq. (22) we need to de-termine angle θdmax and threshold strength Eth.We identify these parameters through the initialphase shift and the position of maximum T‖: θdmax ¼40° and Eth ¼ 7:4V=μm. The only parameter wemust assign is the product SSd, which we chose as0.79 to obtain good agreement with the experimentaldata. Matsumoto et al. [7] pointed out that part of thetransmitted light was depolarized because of imper-fection in the input and output faces, lack of unifor-mity of initial birefringence, and diffuse scattering.From a maximum value of the output power at theparallel polarizer and analyzer we estimated thispart to be approximately 20%.The results of our calculations are displayed in

Fig. 4 (curves) in comparison with the experimentaldata (symbols) presented in [7]. It is obvious that ourtheoretical model is in good agreement with the ex-periment. Moreover, since the only fitting parameteris the product SSd, we expect that the proposedmethod can be used to determine the order para-meters of LCs and LC droplets by experimental mea-surements.

6. Polarization of Transmitted Light

We now use the model described in Sections 2–4 toinvestigate the polarization state of light trans-mitted through a PDLC film. Let the incident beambe parallel to the direction of the applied electric fieldand normal to the film surface, with which we candetermine the amplitudes of extraordinary and or-dinary waves. At normal illumination from Eqs. (12)and (13) it follows that

ae;o ¼ te;o

�cos αsin α

�: ð45Þ

Here α is the polarization angle of the incident waverelative to the x axis of the laboratory frame; te;o arethe transmission coefficients:

te;o ¼ exp�−12γe;oL

�: ð46Þ

Using Eqs. (12), (13), and (24)–(45) one can write theexpression for polarization ellipse:

�Ex

ae

�2þ�Ey

ao

�2− 2

ExEy

aeaocosΔΦn ¼ sin2 ΔΦn: ð47Þ

The semiaxes of the polarization ellipse are

A2 ¼ ðae cos βÞ2 þ ðao sin βÞ2 þ aeao sin 2β cosΔΦn;

ð48Þ

B2 ¼ ðae sin βÞ2 þ ðao cos βÞ2 − aeao sin 2β cosΔΦn;

ð49Þwhere

tan2β ¼ 2aeao

a2e − a2

ocosΔΦn: ð50Þ

Consider ellipticity and azimuth of the transmittedwave. Ellipticity η is a ratio of the semiminor axis ofthe ellipse of polarization to the semimajor axis. Forellipticity, from Eqs. (48) and (49) we can write

η ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðae sin ξÞ2 þ ðao cos ξÞ2 − aeao sin 2ξ cosΔΦn

ðae cos ξÞ2 þ ðao sin ξÞ2 þ aeao sin 2ξ cosΔΦn

s:

ð51Þ

Here ξ is the azimuth; an angle between the majoraxis of the polarization ellipse and the x axis of thelaboratory frame counted counterclockwise. If A > B,angle ξ ¼ β. Otherwise, if β < π=2, angle ξ ¼ β þ π=2;if β > π=2, ξ ¼ β − π=2.

7. Results and Discussion

Dependence of the ellipticity and the azimuth on thestrength of the external electric field applied to thePDLC sample is illustrated in Fig. 5. The phase shiftbetween extraordinary and ordinary waves (seedashed curve) is also displayed in Fig. 5. Calculationsare carried out for the average LC droplet radiushRi ¼ 0:05 μm; the gamma-distribution parameteris μ ¼ 15; the polarization angle of the incident lightis α ¼ 50°; angles θdmax ¼ 25° and φd

max ¼ 5°; theother parameters are the same as in Section 3.

In a general case, the transmitted light is ellipti-cally polarized but, under certain conditions, linearor circular polarization can be implemented. For thecase under consideration, when the strength of the


external field is equal to 1:44V=μm (corresponding tothe phase shift between extraordinary and ordinarywavesΔΦn ¼ 180°), ellipticity η ¼ 0 (see Fig. 5). Con-sequently, the transmitted light is linearly polarized.At this field strength, the polarization angle of thetransmitted light is characterized by the azimuthξ ¼ 128°. This means that the polarization plane ofthe transmitted light is rotated by 78° (polarizationangle of the incident light is 50°). If phase shiftΔΦn ¼ πn (n is an integer), the transmitted lightis linearly polarized at angle

tan ξlin ¼ ð−1Þnexp

�k3gn8hVi

�εdoεp

− 1

þ 2nπφmaxd

Lkcv sinð2φmaxd Þ

��tan α

: ð52Þ

Analysis shows that for hRi tending to zero, trans-mitted light polarization angle ξlin ¼ π − α for ΔΦn ¼ð2nþ 1Þπ and ξlin ¼ α for ΔΦn ¼ 2nπ. The depen-dence of angle ξlin at ΔΦn ¼ π on average droplet ra-dius hRi at different polarization angles of e incidentlight is displayed in Fig. 6. The calculations were car-ried out using the sample parameters specifiedabove. It is evident from Fig. 6 that, at small hRi,angle ξlin ¼ π − α. Thus the polarization plane can

Fig. 6. Polarization angle ξlin of the transmitted linearly polar-ized light versus the average LC droplet radius hRi at differentpolarization angles of incident light.

Fig. 5. Phase shift ΔΦn between extraordinary and ordinarywaves (dashed curve); ellipticity η (thin solid curve); and azimuthξ (bold solid curve) of transmitted light versus applied electric fieldstrength.

Fig. 7. (a) Ellipticity η versus polarization angle α of the incidentlight and the applied electric field strength. (b) Plane ðη; αÞ projec-tion of (a).


be rotated by more than 90° by means of a PDLC cellwith fine LC droplets. The other parameters that en-ter into Eq. (52) have a rather weak influence on po-larization angle ξlin.Onemore case of linear polarization of transmitted

light is displayed in Fig. 5. For a strong applied field,ellipticity η ¼ 0 and the azimuth are equal to the po-larization angle of the incident light, which impliesthat the transmitted light maintains its initial polar-ization state. It follows from Fig. 5 that, if phase shiftΔΦn ¼ π=2 (for the sample under consideration it oc-curs at 2V=μm), the ellipticity has a peak, and theazimuth is equal to π=2. Figure 7(a) illustrates thedependence of the ellipticity on the polarization an-gle of incident light and the strength of the externalelectric field applied to the sample. The results areobtained for the same parameters of the sample andincident light as specified for Figs. 2 and 3. As is evi-dent from Fig. 7(b), there are two polarization anglesof incident light such that circular polarization oftransmitted light (the ellipticity η ¼ 1) is obtained.If incident light is polarized at angle

αcirc ¼ arctanexp

�−k3gðnþ 1=2Þ

8hVi�εdoεp

− 1

þ πðnþ 1=2Þφmaxd

lkcv sinð2φmaxd Þ

��ð53Þ

or at angle α0circ ¼ π − αcirc, and phase shift ΔΦn ¼ðnþ 1=2Þπ, circular polarization of the transmittedlight takes place.Dependence of angle αcirc on an average LC droplet

radius hRi is displayed in Fig. 8. The results are ob-tained at φd

max ¼ 5° and at the sample parametersspecified above. Angle αcirc tends to π=4 at small LCdroplets and does not exceed this magnitude for LC

materials with positive birefringence. We expectthat, by means of Eqs. (52) and (53), the size andshape of LC droplets can be estimated through opti-cal measurements.

8. Conclusions

We have considered the phase shift between extraor-dinary and ordinary components and the polariza-tion state of light transmitted through PDLC filmwith small spherical partially oriented nematic dro-plets. Analysis has shown that our approach is alsovalid for PDLC samples with randomly aligneddroplets and can be extended to films with ellipticdroplets. The model described in the work is shownto be in good agreement with an example of knownexperimental data. The dependence of the polariza-tion state of the transmitted light on the properties ofa PDLC sample and incident light is established. Ifthe special conditions described in Section 6 are ful-filled, circular or linear polarization of transmittedlight is obtained. Our results can be used to designvarious devices for light modulation based on compo-site liquid crystal materials: optical filters, polari-zers, phase plates, polarization plane rotators, andlenses with variable focal lengths. By means of theproposed model, the electro-optic properties of thesedevices can be described. Our results can also be usedto determine droplet and sample optical parametersby measurement of the characteristics of trans-mitted light.

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Polarization properties of polymer-dispersed liquid-crystal film with small nematic droplets

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