We study a tunable-focus lens in which the key element is a gradientpolymer-stabilized liquid crystal (G-PSLC) structure. In this paper, we further develop the theoretical model [1, 2] that describes the dependence of the G-PSLC lens’ focal length on the applied voltage and presents a theoretical study of lens aberrations. According to Fermat’s principle, we minimize the optical path of a test light beam and calculate the angles of a ray exiting from the cell. Using these results, the lateral and longitudinal aberrations are estimated. The obtained results can be used to optimize the G-PSLC lenses.У данiй роботi вдосконалено теоретичну модель лiнзи [1, 2], утвореної в нематичному рiдкокристалiчному кристалi в процесi фотополiмеризацiї в неоднорiдному свiтловому полi гаусового пучка. Знайдено чисельно кут переорiєнтацiї директора нематичного рiдкого кристала та фокусну вiдстань лiнзи в залежностi вiд величини напруги прикладеної до нематичної комiрки. Використовуючи принцип Ферма, мiнiмiзовано оптичний шлях свiтлового пучка, що проходить крiзь утворену лiнзу. Отримано напрямок поширення свiтла на виходi з комiрки, що дозволило оцiнити поздовжню та поперечну аберацiї рiдкокристалiчної нематичної лiнзи. Зi збiльшенням прикладеної напруги, величина аберацiй зменшується. Отриманi в статтi результати дозволяють оптимiзувати якiсть зображення, що утворює лiнза

Bielykh, S.P.

Galstian, T.

Reshetnyak, V.Y.

Subota, S.L.

Ukrainian Journal of Physics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

ELECTRO-OPTICAL CHARACTERISTICSELECTRO-OPTICAL CHARACTERISTICS OF A LIQUIDCRYSTAL LENS WITH POLYMER NETWORKS.P. BIELYKH,1 S.L. SUBOTA,1 V.Y. RESHETNYAK,1 T. GALSTIAN21Taras Shevchenko National University of Kyiv, Faculty of Physics(2, Academician Glushkov Prosp., Kyiv 03680, Ukraine; e-mail: sveta_ pavl@ ukr. net )2Center for Optics, Photonics and Laser, Physics department, Laval University(Pavillon d Optique-Photonique, 2375 Rue de la Terrasse, Quebec, Canada G1V 0A6)PACS 42.70.Df, 61.30.Gd,42.15.Dp, 42.15.Frc©2010We study a tunable-focus lens in which the key element is a gradient-polymer-stabilized liquid crystal (G-PSLC) structure. In this pa-per, we further develop the theoretical model [1, 2] that describesthe dependence of the G-PSLC lens’ focal length on the appliedvoltage and presents a theoretical study of lens aberrations. Ac-cording to Fermat’s principle, we minimize the optical path of atest light beam and calculate the angles of a ray exiting from thecell. Using these results, the lateral and longitudinal aberrationsare estimated. The obtained results can be used to optimize theG-PSLC lenses.1. IntroductionAdaptive imaging elements are more and more requiredfor a broad range of applications, from astrophysics tosecurity. Variable optical power is one of the key ele-ments in such applications. In recent years, there hasbeen much interest in lenses with variable focal length,based on liquid crystals [1–8]. These lenses are of varioustypes and are made using manifold methods. For exam-ple, among them, there are the lenses that are createdby using electrodes with holes [3] or by non-planar elec-trodes [4]; lenses employing Fresnel zones [5]; gradient-polymer-stabilized liquid crystal (G-PSLC) lenses [6–8],and others. But the operation of all these lenses is basedon the electrical control over the refractive index distri-bution in a thin layer of Nematic Liquid Crystal (NLC).In this paper, we present a theoretical model of theG-PSLC lens [1] and describe its spherical aberrations.The essence of the idea to create such a lens is the fol-lowing [6,7]. The illumination of the mixture of a planarpre-oriented NLC containing few percent up to 3% wt)of a photopolymerizable monomer by a laser beam withthe Gaussian spatial intensity distribution may inducean inhomogeneous polymer network. The electro-opticalresponse of this system to a uniform electric field is in-homogeneous, but centrally symmetric. The refractiveindex lateral profile is defined by the spatial distributionof the director (the average orientation of long molecularaxes of the NLC). Such a cell represents a liquid crystallens. A change of the applied voltage varies the profile ofthe refractive index and, hence, controls the focal lengthof the lens.The paper is organized as follows. In the first section,we describe the cell geometry of the problem, find thedirector reorientation under the action of an externallyapplied voltage in the cell with a non-uniform polymernetwork and the focal length of the corresponding lens.In the second section, we study the spherical longitudinaland lateral aberrations of the lens. Finally, we drawsome conclusions.2. Director Reorientation in the Cell with aPolymer Network Under the Action of anExternally Applied VoltageLet us consider a NLC cell of the thickness L with pla-nar and uniform boundary conditions at each wall. Thegeometry of the problem is shown in Fig. 1. We as-sume that there is a strong director anchoring at the cellwalls. The cell is illuminated by a photopolymerizinglaser beam, the intensity distribution of which isI(ρ) = I0 exp(−αρ2), (1)where we denote α = 12$2 , and $ is a half-width of theGaussian beam.We can analyze this system when the polymerizationprocess is over in the cell. We assume there is no signifi-cant light attenuation (due to the absorption or scatter-ing) through the cell during the polymerization process,and the light profile is not changed noticeably. This as-sumption is supported by the experimental observationof a rather good fidelity of the phase profile due to theobtained gradient of the polymer network and of the spa-tial distribution of the polymerizing light intensity [7].DC electric field may be applied across the cell (alongISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 293S.P. BIELYKH, S.L. SUBOTA, V.Y. RESHETNYAK et al.Fig. 1. LC cell geometrythe OZ-axis) by means of transparent electrodes, e.g.,Indium Tin Oxide (ITO). That field would reorient thedirector only in the xz-plane. The obtained refractiveindex profile varies, as the field is applied and inducesa lens-like behavior (thanks to the corresponding gra-dient of the polymer network), so the focal length andaberrations are also field-dependent.From the symmetry reason, the director field is givenbyn = (cos θ(ρ, z), 0, sin θ(ρ, z)). (2)In order to investigate the director reorientation in theelectrical field, we minimize the free energyFel =12∫K1(5n)2 dV +12∫K2(5× n)2 dV++12∫K3(n×5× n)2 dV−−W2∫Np(ρ)(ne)2 dS −12∫DE dV. (3)The first three terms of this equation represent the usualelastic deformation contribution to the LC total free en-ergy; in the fourth term, Np is the density of the polymernetwork, W is the anchoring energy between the polymerand a liquid crystal, and e is the direction of the easy axisat the bottom substrate. In the last term, D is the elec-tric displacement vector, K1, K2, and K3 are the elasticconstants of the pure LC, and Np is a term whose mi-croscopic investigation is a rather complicated task andrequires a separate study. To obtain polymer network’sprofile, we should solve the rate equations of chemicalreactions. This task is quite complicated. To simplifythis problem, we introduce a new parameter w = WNp,representing the local effective bulk anchoring energy perunit volume. The magnitude of the parameter w can bedirectly determined from an experiment.We the assume that NLC is the ideal dielectric. Sincethe characteristic length of the director inhomogeneity inthe z-direction is determined by the cell thickness, andit is much less than the characteristic size of the directorinhomogeneity in the x-direction, we neglect the deriva-tive ∂∂ρin comparison with the derivative ∂∂z. Using thesolution of the equation ∂Dz∂z= 0 and combining it withthe equation 5 × E = 0 with the boundary conditionsEx = Ey = 0 at the cell walls, we obtain the voltage Uacross the nematic cell asU =L∫0E dz = DzL∫0(ε⊥ + εα sin θ(z)2)−1 dz. (4)The thermodynamic functional then takes the formF =K2∫[(θz)2 + (θρ)2]dV−−W2∫Np(ρ)(cos θ(ρ, z))2dV−−12∫D2z(ε⊥ + εα sin θ(z)2)−1dV. (5)Here, we used the one elastic constant approximation:K1 = K2 = K3 = K.By minimizing functional (5), we get the Euler–Lagrange equation with boundary conditions [1]θ′′uu −w(ν)L2Kb2sin θ cos θ++ D2zL2εaK(ε⊥)2b2sin θ cos θ(1 +εaε⊥sin2 θ)2 = 0,θ(u = 0, ν) = 0,θ(u = b, ν) = 0.(6)Here, we introduced the dimensionless parameters ν =ρ/L and u = zLb, b = 200 is a scale coefficient. System(6) was solved numerically.294 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3ELECTRO-OPTICAL CHARACTERISTICSabFig. 2. Inhomogeneous director reorientation angle at dimension-less voltages V = 1.2 (a) and V = 2.5 (b)As mentioned already, while considering this model,we don’t consider the polymer diffusion during the poly-merization process. So, we suggest that the UV lightintensity profile and the polymer network’s profile are ofthe same form. Thus, we use a Gaussian as the initialtrial polymer concentration profile:w̃(ν) = w0 exp(−βL2ν2). (7)2,5 3,0 3,5 4,0 4,51,001,051,101,151,201,25V1/f,m-1Fig. 3. Lens power dependence upon the applied voltageIn our previous theoretical work [1], we estimated a valueof w0 v 11 using experimental data and ρ0 = 90, whereρ0 is determined from an alternative parametrization ofthe polymer concentration profilew(ν) = w0(1− ν2ρ20), ν ≤ ρ0,0, ν > ρ0,(8)where ν = ρL .In this paper, we will study how the lens character-istics depend upon the dimensionless NLC-polymer an-choring interaction parameters at w0 = 33 and ρ0 = 100.We have solved system (6) numerically using the fol-lowing parameters for the NLC mixture E7 [9] (fromMerck): the principal components of the low frequencydielectric tensor ε‖ = 19 and ε⊥ = 5.2, the principal op-tical refractive indices ne = 1.738 and n0 = 1.518, elasticconstant K u 10−11 N, and a cell thickness L = 10 µm.In Fig. 2,a and Fig. 2,b, one can see the dependences ofthe director reorientation angle on the coordinates of thenematic cell at some normalized voltages V = UU0, whereU0 is the Frederic’s threshold voltage for the pure NLC,and U is the voltage which is applied to the nematic cell.Using the director reorientation angles and the Fresnelapproximation, we obtain the lens power dependence onthe applied voltage [10] (Fig. 3). Note that the obtainedfocal lengths have been based on the paraxial approxi-mation. We have assumed that all rays are paraxial, thatis, that they are very close to the optic axis and makevery small angles with it. But this condition is neverobeyed exactly. So, in the next section, we consider theaberrations of this lens.ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 295S.P. BIELYKH, S.L. SUBOTA, V.Y. RESHETNYAK et al.öparaxial focal planeF”TSALSAlongitudinal spherical aberrationtransverse spherical aberrationFig. 4. Illustration of the spherical aberration3. Aberration of G-PSLC LensIn an ideal optical system, all rays of light from a pointin the object plane should converge to the same pointin the image plane, forming a clear image. But real op-tical systems, such as lenses, don’t form perfect images,and there is always some degree of aberration introducedby the lens, which causes the image to be an imperfectreplica of the object. The influences which cause differ-ent rays to converge to different points are called aber-ration. There are different types of aberration that canaffect the image quality. In this paper, for the G-PSLClens, we consider spherical ones. In Fig. 4, we showthose spherical aberration definitions.It appears from Fig. 4 that a spherically aberratedlens has no well-defined focus. Further, from the opticalaxis, the ray enters the lens and focuses nearer to thelens (crosses the optical axis). The distance along theoptical axis between the intercept of the rays that arenearly on the optical axis (paraxial rays) and the raysthat go through the edge of the lens (marginal rays) iscalled longitudinal spherical aberration (LSA) [11]. Theheight at which these rays intercept the paraxial focalplane is called transverse spherical aberration (TSA).These quantities are related by TSA=LSAtanϕ [12].Theoretically, the simplest way to obtain LSA is tofind the beam trajectory in an inhomogeneous liquidcrystal cell. So, first we calculate the optical path lengthof a ray inside the layer from its entrance point A to itsexit point B.Using the eikonal equation [13](gradS)2 = n2eff ,OsrzñõtõtFig. 5. Paths of two rays and the angles formed by them with theOZ axiswhere neff is the effective refractive index given byneff =none√n2e cos2 ψ + n2o sin2 ψ, (9)ψ is the angle between the director and the wave vectorinside the medium, we can writegradS = neffs,where s is the unit vector directed along the propagationof the beam. Then, integrating along the beam, one canobtain the optical path length of a ray inside the layerfrom its entrance point A to its exit point B given bythe integralS =B∫Aneff s dr,where dr is an elementary path section along the ray.Using sdr = cos(φ − ψ)dz, where φ = θ + ϑ, ϑ is theangle between OZ and s (Fig. 5), and relation (9), wepresent the last integral asS = neL∫0√1− α sin2 φcosϑdz, (10)where L is the cell thickness, and α = 1−(none )2. Accord-ing to the principle of Fermat, the actual optical pathof the ray must be minimal. This means that integral(10) should be minimized. Using the Euler–LagrangeequationS′ρ −ddzS′ρ′ = 0,296 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3ELECTRO-OPTICAL CHARACTERISTICSFig. 6. Longitudinal aberration versus the applied voltageFig. 7. Lateral aberration versus the applied voltagewhere S′ρ = −α sin 2φ2g cos 2ϑ∂θ∂ρ , S′ρ′ =1√1+(ρ′)2[ρ′g − α sin 2φ2g ],ddzS′ρ′ = −θ′′ρ tan θ cos3 θ, g =√1− α sin2 φ, tan θ = dρdz ,we can find the equation of the actual ray [14]ρ′′=1cos2 ϑg31−α{[(g− 1−αg3)tanϑ−α sin 2φ2g]∂θ∂ρ++[g − 1− αg3+α sin 2φ2gtanϑ]∂θ∂z}. (11)After these lengthy calculations, we find the angle ϑt ofthe beam propagation after the liquid crystal cell, usingthe Snell law for the wave vector k, and then obtain lensaberrations.Fig. 8. Relative longitudinal aberration versus the applied voltageFig. 9. Relative lateral aberration versus the applied voltageFigures 6 and 7 show the expected LSA (longitudinalaberrations) and TSA (lateral aberrations) dependenceson the applied voltage for ρo = 100 and w = 33.On the next plots, we show the dependence of relativeaberrations of the G-PSLC lens on the applied voltage(Figs. 8 and 9).4. ConclusionsIn this paper, we present a theoretical model that de-scribes the electro-optical characteristic of an NLC lensbased on the G-PSLC concept. We calculated the di-rector profile of the NLC in a cell for some param-eters of the lens, particularly for the Gaussian beamprofile of a polymerizing beam. Using the geomet-ric optics approximation, we found the optical pathISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 297S.P. BIELYKH, S.L. SUBOTA, V.Y. RESHETNYAK et al.of a ray inside the NLC in the cell. According toFermat’s principle, we minimized the functional de-scribing the optical path of a ray and calculated an-gles of the ray exiting from the cell. Using these re-sults, the lateral and longitudinal aberrations were esti-mated. While the optical power of the lens first goesup, the focal length and aberrations of the G-PSLCfurther decrease with increase in the applied voltage(for the given case of a Gaussian profile of the net-work). The obtained aberration and the focal lengthdependence on the applied voltage allows optimizing thelens parameter (including the polymer network’s pro-file) to get the best image quality, for example, thehighest optical power at the lowest aberrations. Theobtained results can be applied to develop G-PSLClenses that have no moving parts and allow the electro-optical zooming of high quality. The corresponding ex-periments are under way to validate the above predic-tions.We acknowledge the NATO grantCBP.NUKR.CLG.981968. We are grateful to TimothyJ. Sluckin (Southampton, UK) for fruitful discussions.1. S.L. Subota, V.Yu. Reshetnyak, S.P. Pavliuchenko, andT. Sluckin, Mol. Cryst. Liq. Cryst. 489, 40 (2008).2. V.Yu. Reshetnyak, S.L. Subota, and T.V. Galstian, Mol.Cryst. Liq. Cryst. 454, 187/[589]-200/[602] (2006).3. F. Naumov, G.D. Love, M.Yu. Loktev, and F.L. Vla-dimirov, Optics Express 4, 344 (1999).4. B. Wang, M.Ye, M. Honma, T. Nose, and S. Sato, Jpn.J. Appl. Phys. 41, L 1232 (2002).5. Y.-H. Fan, H. Ren, and S.-T. Wu, Optics Express 11,3080 (2003).6. V. Presnyakov and T. Galstian, Polymer stabilized liq-uid crystal lens for electro-optical zoom, Centre for Op-tics, Photonics and Lasers (Universite Laval, Quebec,Canada, 1997).7. V. Presnyakov and T. Galstian, in Photonics North 2004:Optical Components and Devices, edited by J.C. Ar-mitage, S. Favard, R.A. Lessard, G.A. Lampropoulos(2004), p. 861.8. V.Y. Reshetnyak, S.M. Shelestiuk, S.L. Subota, S. Pavli-uchenko, and T.J. Sluckin, Theoretical modeling of het-erogeneous LC systems: nano-suspensions and polymerstabilized LC lens, 103101-1-103101-6 (2007).9. P.S. Drzaic and A. Muller, Liq. Cryst. 5, 1467 (1989).10. J.W. Goodman, Introduction to Fourier Optics(McGraw-Hill, New York, 2002).11. M. Born and E.W. Wolf, Principles of Optics (PergamonPress, Oxford, 1991).12. http://www.mellesgriot.com/products/optics.13. Yu.A. Kravtsov and Yu.I. Orlov, Geometrical Optics ofInhomogeneous Media (Springer, Berlin, 1990).14. J.A. Kosmopoulos and H.M. Zenginoglou, Applied Optics26, 1714 (1987). Oxford, Pergamon Press (1991).Received 26.01.10ЕЛЕКТРООПТИЧНI ХАРАКТЕРИСТИКИРIДКОКРИСТАЛIЧНОЇ ЛIНЗИЗ ПОЛIМЕРНОЮ СIТКОЮС.П. Бєлих, С.Л. Субота, В.Ю. Решетняк, Т. ГалстянР е з ю м еУ данiй роботi вдосконалено теоретичну модель лiнзи [1, 2],утвореної в нематичному рiдкокристалiчному кристалi в про-цесi фотополiмеризацiї в неоднорiдному свiтловому полi гау-сового пучка. Знайдено чисельно кут переорiєнтацiї директо-ра нематичного рiдкого кристала та фокусну вiдстань лiнзи взалежностi вiд величини напруги прикладеної до нематичноїкомiрки. Використовуючи принцип Ферма, мiнiмiзовано опти-чний шлях свiтлового пучка, що проходить крiзь утворену лiн-зу. Отримано напрямок поширення свiтла на виходi з комiрки,що дозволило оцiнити поздовжню та поперечну аберацiї рiд-кокристалiчної нематичної лiнзи. Зi збiльшенням прикладеноїнапруги, величина аберацiй зменшується. Отриманi в статтiрезультати дозволяють оптимiзувати якiсть зображення, щоутворює лiнза.298 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3

Electro-optical Characteristics of a Liquid Crystal Lens with Polymer Network

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Electro-optical Characteristics of a Liquid Crystal Lens with Polymer Network

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)