Guiding of optical fields in a smectic B liquid crystal cylindrical fiber

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1 REVISTA MEXICANA DE FÍSICA S NOVIEMBRE 006 Guiding of optial fields in a smeti B liquid ystal ylinial fibe L.O. Palomaes and J.A. Reyes Instituto de Físia Univesidad Naional Autónoma de Méxio Apatado postal 0364, Méxio D.F. 0000, Méxio Reibido el 5 de febeo de 006; aeptado el 7 de mazo de 006 The popagation of optial fields in a ylinial fibe of smeti B liquid ystal with banana-shaped moleules is studied. We onside a ylinde of banana-smeti liquid ystal whose tilt angle is assumed to be unifom, suounded by an infinite homogeneous isotopi ladding. We pefom an analyti desiption of the optial field popagation within the guide fo Tansvese Magneti modes TM fo a low intensity optial beam. We take into aount the oupling between the optial field and the oientational onfiguation of the smeti. Eikonal equations ae deived and ay tajetoies ae analyzed in thei oesponding Hamiltonian epesentation, in the optial limit. Finally, by solving Maxwell s equations analytially, we obtain the eletomagneti modes and the spatial eletomagneti enegy distibution within the guide. Keywods: Liquid ystals; smeti B; banana-shaped moleules. Se estudia la popagaión de ampos óptios en una fiba óptia de istal líquido esmétio on moléulas en foma de banana. Consideamos un ilino de esmétio B uyo ángulo de inlinaión es unifome, el ual está odeado po una ubieta unifome isotópia e infinita. Nuesta desipión analítia la ealizamos paa haes de baja intensidad y úniamente paa los modos Tansvesos Magnétios. En el límite óptio, deduimos las euaiones de la Eikonal y las tayetoias de ayo en su epesentaión Hamiltoniana oespondiente. Finalmente esolvemos las euaiones de Maxwell analítiamente paa obtene los modos nomales y la distibuión espaial de la densidad de enegía eletomagnétia dento de la guía. Desiptoes: Cistales líquidos; esmétios B; moléulas en foma de banana. PACS: 4.65.jx; 6.30.Gd; 78.0.Jq. Intodution Liquid ystals, due to thei ih phase behavio, play an impotant ole in biologial systems and tehnologial appliations, in patiula in eleto-optial displays. Liquid ystals ae omplex fluids with a wide ange of popeties: elasti, visous, eletial, et. Liquid-ystalline polymes ae visoelasti and an easily fom fibes, just like onventional polymes. Calamiti ba-like liquid ystals usually fom thin films like membanes. In this sense, it was an outstanding fat to find eently that bend-shape moleules, speifially the phases B 7 and B, may fom stable fibes instead of films []. This featue of self-fomation of fibes may be useful in designing and onstuting optial fibes and atifiial musles, taking into aount that smetis, like othe themotopi liquid ystals, ae suseptible to extenal fields, and themal and mehanial petubations. Smeti liquid ystals have peulia popeties suh as spontaneous eletial polaization, whih is shown in thei anti and feoeletiity and spontaneous beaking of hial symmety in one-dimensional phases smeti phases; nevetheless, they ae fomed by moleules whih ae not intinsially hial [3]. We stated a pogam to study the eletomagneti esponse of these fibes. We begin by onsideing low intensity beams, so that they do not distot the smeti onfiguation when they popagate though the fibe. The pupose of this pape is to investigate the optial field within a ylinial fibe of smeti B liquid ystal oe suounded by an infinite, homogeneous, isotopi ladding. The outline of this pape is as follows. In Ses. and 3, we set up the govening equations of ou physial system. In Se. 4, we use these equations to find the eikonal equations and to alulate the ay tajetoies in the Hamiltonian epesentation fo the tansvese magneti waves. In Se. 5, we solve Maxwell s equations, find the TM mode distibution and thei oesponding eigenvalues fo a fibe woking in the infaed ange of wavelengths. We also find the distibution of the enegy density fo these modes. Finally, in Se. 6, we disuss and summaize ou esults.. Model and govening equations We onside a nonmagneti optial fibe of smeti liquid ystal onsisting of banana-shaped moleule oe onfined within a vey long ylinial egion of adius R, suounded by an infinite homogeneous isotopi ladding with dieleti onstant ɛ 0. The fibe is fomed by a seies of onenti ylinial smeti shells whee the dieto, n, foms an angle φ with the nomal to the shells and is given by n = os φê + senφê θ, whee ê and ê θ ae unit vetos in the ylinial oodinates, θ, z. We an define the dieto s lines in analogy with the eleti field lines as the lines paallel to the field at evey point, that is dy dx = n y n x.

2 7 L.O. PALOMARES AND J.A. REYES Then, witing the omponents of Eq. in Catesian oodinates and inseting them into the last equation, we get the following expession afte integation: = Ae θ/ tan[φ], 3 whee A is an integation onstant. Notie that this expession defines a plane spial depating fom the oigin. We shall assume that the optial field has low eletomagneti intensity; thus the dieto field is not distoted by the optial field. Unde these iumstanes, the dieleti tenso of the smeti an be ompletely desibed by a uniaxial tenso ɛ = ɛ I + ɛ a n n, 4 whee ɛ is the dieleti pemittivity pependiula to the long axis of the moleule and ɛ a is its dieleti anisotopy. Thus ɛ an be expessed in tems of the tilt angle as: ɛ = ɛ ɛ 0 ɛ ɛ ɛ = ɛ + ɛ a os φ ɛ a os φ sin φ 0 ɛ a os φ sin φ ɛ + ɛ a sin φ ɛ. 5 Hee we have assumed that the tilt angle is unifom fo the entie fibe. Notie that the dieleti tenso exhibits expliitly the anisotopy whih is ontolled by ɛ a. Sine the medium is nonmagneti, its magneti suseptibility tenso is µ = δ ij. Following the usual poedue, fom Maxwell s equations in the fequeny spae we an obtain, the wave equation fo both E and H, whih eads [ µ E ] ω + ɛ E = 0 6 and [ ɛ H ] ω + µ H = 0. 7 Hee ω = k 0, ω is the fequeny, k 0 the wave numbe and is the speed of light in vauum. 3. Optial Limit If the wavelength of the light is smalle than the typial length of the system l, we an use the optial limit k 0 l to simplify the equation govening the eletomagneti field. Fo isotopi media, this poedue leads to a sala eikonal equation [4]. In ontast, fo anisotopi media whose tansvese eleti modes ae deoupled fom the eleti field, the optial field an be desibed with only two eikonal equations [6], one fo the tansvese eleti modes and the othe fo the tansvese magneti ones. Nevetheless, fo the geneal anisotopi ase, Bakovskii [5] has poposed a genealization of the well-known tial funtion used to find the eikonal equation: E j = B i e ik 0lψ ij, 8 whee ψ ij ae the omponents of the eikonal tenso and B i ae omplex amplitudes whih may be funtions of the oodinates. Hee, we have used the onvention of the sum ove epeated indexes. Afte inseting this expession into Eq.7 and assuming the optial limit, we obtained the following genealized eikonal equation: e af ɛ b e ψ dh ψ hl bgd + µ al = 0, 9 g f whee e af is the Levi-Civita pseudotenso. Fom Eq. 5 we an see that the dieleti tenso ouples the and θ omponents of the involved field. Thus, we assume an eikonal tenso of the same stutue, that is, Ψ = ψ ψ 0 ψ ψ ψ. 0 Then, expanding Eq.8 fo the E x, E y and E z omponents, we have expliitly E x = B e iψ + B e iψ, E y = B 3 e iψ + B 4 e iψ, E z = B 5 e iψ. 3 Fo a positively anisotopi smeti, the eleti field tends to be aligned to the smeti dieto n o dieto lines, beause the pojetion of ɛ is maximum in that dietion. Consequently, a oupling between the optial field and the oientational onfiguation of the smeti ous in suh way that the polaization of the optial field otates following n. It is woth noting that the fat that n is onstained to the x y plane means that only E x and E y ae oupled, wheeas E z emains always pependiula to n and theefoe deouples fom it. If we wite expliitly a otation of the tansvesal omponents of the eletial field E x and E y, we aive at E = A os αe iw A sin αe iw A sin αe iw + A os αe iw A 3 e iw, 4 whee α and w denote the position dependent polaization and phase of the eleti field. Howeve, this expession fo E should be onsistent with Eqs. -3. Then, expessing the tigonometi funtions in tems of omplex exponentials, equating amplitudes and phases of both expessions, we get the following elations: w t + α = ψ = ψ, 5 w t α = ψ = ψ, 6 w l = ψ, 7 Rev. Mex. Fís. S

3 GUIDING OF OPTICAL FIELDS IN A SMECTIC B LIQUID CRYSTAL CYLINDRICAL FIBER 73 B = A + ia B 3 = A ia = B, 8 = B 4, 9 A 3 = B 3. 0 It is inteesting to note that Eqs. 5, 6 and 7 not only expess the elements of the eikonal tenso Eq. 0 in tems of the wave phases w t and w l of the tansvese and longitudinal omponents of the eleti fields, but also involve the polaization angle α. In this sense, the geometial optial limit is able to desibe hanges in the polaization of the eleti field suh as those involved in optial ativity phenomena. In this setion we have just followed the dynamis of the eleti field beause the magneti esponse is obtained by Maxwell s equations; hene, the phase of the eleti field is the same as the magneti one 4. Ray Tajetoies In this setion we solve the eikonal equations within the smeti B liquid ystal oe by taking into aount the oupling between the dieto and the eleti field. This allow us to find expessions fo the polaization and optial paths of the eletomagneti fields. By applying Hamilton equations to the Hamiltonian of the system fo the TM modes, we find the paameti ay tajetoies. Substitution of Eqs. 5, µ = δ ij, and 0 into Eq.9 leads us to a oupled eikonal equation system. Expanding the oesponding subindexes, we get 0 = ɛ ψ ψ ɛ 0 = ɛ 0 = ɛ ψ ψ + ɛ ψ ψ + ɛ ɛ ɛ ψ ɛ ψ ψ ψ ψ ψ ɛ + ɛ ψ ψ ɛ ψ + ɛ ψ ψ ψ + ɛ ψ + ɛ ψ ψ +, ψ, +. 3 Inseting Eqs. 5 and 6 in the fist two equations, we find two oupled equations fo the polaization and the phase of the eleti field: [ α ] [ α ] = wt ɛ + ɛ wt + ɛ [ α ] wt + ɛ [ α + + ɛ ] wt + ɛ [ [ + wt wt + α α + + α α + Then, by adding and subtating equations 4 and 5 we an deouple the system as follows: ɛ ɛ ɛ + α ɛ ɛ + ɛ ] α, 4 ] α = wt ɛ = 4, 6 α α + α ɛ = 4. 7 Rev. Mex. Fís. S

4 74 L.O. PALOMARES AND J.A. REYES It is onvenient to emak that, sine the oeffiients of these equations ae only funtions of, by applying a onvenient anonial tansfomation we an edue this system of patial diffeential equations to a simple system of odinay diffeential equations. If w t does not depend on z and θ, they ae yli oodinates and thei oesponding onjugate vaiables p z and p θ, ae onseved quantities. Consequently, it is useful to use the anonial tansfomation w t = f + p t zz + p t θθ, 8 w l = υ + p l zz + p l θθ, 9 whee f and υ ae only funtions of. Evidently, the ay omponents ae elated to w t and w l by the usual expessions p t z =, p θ =, 30 p l z = w l, p θ = w l. 3 Similaly, we an expess α in the fom α = g + P z z + P θ θ, 3 whee P z and P θ ae onstants and υ is only a funtion of. Nevetheless, P z and P θ ae no longe ay omponents, but instead the atio of hange of the polaization angle with espet to z and θ. If we inset Eqs. 8, 9 and 3 into Eqs. 3, 6 and 7, we an obtain thee odinay diffeential equations fo f, g and υ. µ 0 p l z p t z ɛ + p l z + p t z ɛ +ɛ pt θ p l θ p t θf + m g = 0, p t z ɛ ɛ + p l z ɛ + ɛ +ɛ p t θ P θ ɛ Thei solutions ae given by + p l θ + p t θf + p l θg = 0, 34 + P θɛ υ ɛ υ = f = p t θ ln + p t z ɛ ɛ 4 g = p t θ ɛ 4 ɛ + ɛ Pz ɛ ɛ P θ υ = ɛ ɛ p l θ, 36 P θ ln ɛ p l θ +, 38 Afte substituting these expessions bak into Eqs. 8, 9 and 3, we find w t, θ, z = p t zz + p t θθ + C + lnp t θ + 4p t z ɛ ɛ 8p t θ ɛ α, θ, z = P z z + P θ θ + C lnp θ + 4Pz ɛ + ɛ 8P θ ɛ, 39, 40 w l, θ, z = p l zz + p l θθ + C 3 + ɛ ln p l θ ɛ + ɛ + p l θ ɛ p l θ ɛ + pl θ ɛ p l θ ɛ + pl θ ɛ 3/ / ɛ p l ln θ ɛ + ɛ + / ɛ + pl θ ɛ + pl θ ɛ p ] l θ ɛ + pl θ ɛ +, 4 whee C, C and C 3 ae additive integation onstants. To alulate the ay tajetoies, we an follow two poedues: one onsists in applying the Hamilton-Jaobi theoy to both w t, θ, z and α, θ, z in ode to find plane pojetions of the thee dimensional tajetoies. The othe onsists in witing and solving Hamilton s equation fo these funtions, expessed in tems of thei natual vaiables, to find paameti equations of thei ay tajetoies. We follow the seond fomalism and get the Hamiltonians H t = ɛ ɛ p t z ɛ H l = ɛ pl ɛ pl θ pt p t θ + ɛ p t θ, 4 ɛ p l θ, 43 whee p t z = df/ and p l z = dg/. Hene, using one of the Hamilton s equations fo w t, we obtain Rev. Mex. Fís. S

5 GUIDING OF OPTICAL FIELDS IN A SMECTIC B LIQUID CRYSTAL CYLINDRICAL FIBER 75 d θ dτ z = 0 ɛ ɛ 0 ɛ ɛ ɛ pt p t θ p t z. 44 We obseve that the fist and seond equations of this system an be integated staightfowadly fo τ and zτ, sine p t θ and pt z ae onseved quantities. Howeve, fo solving the seond equation, we need to inset p t = df/ fom Eq.36 and then integate the esulting expession. The above poedue allows us to wite the following paameti equations fo the thee dimensional ay tajetoy: τ = C pt θ τ ɛ, 45 { [ θτ = p t θ ɛ + C p t θ + p t ɛ a + ɛ θτ C ] ɛ ɛ ɛ a + ɛ + p t z [ɛ a + os φ + sin φ + ɛ ]C ɛ 4p t θτ p t θ ɛ ɛ a + ɛ ln C 4pt θ τ }, 46 ɛ zτ = C 3 4τp t z ɛ a ɛ a + ɛ [ + os φ + sin φ ɛ a + ɛ ], 47 whee C, C, C 3 ae onstants to be detemined by the initial position, while p t z and p t θ ae the ay omponents whih ae to be detemined by the initial ay dietion. 5. Exat solutions When we substitute 5 into 7 by assuming a nonmagneti medium, we find a system of oupled diffeential equations whose oeffiients do not depend on the vaiables z and θ. Thus it is onsistent to assume that the eletomagneti field is of the fom H = H θ, H z, H e iγθ+iβz, 48 whee β and γ ae the z and θ omponents of the waveveto. Then, inseting these expession into Eq. 7, we obtain 0 = iγɛ dh + iγɛ dh θ + i βɛ dh z H ω β + γ ɛ iγɛ H θ β + γ ɛ iγɛ, 49 0 = ɛ d H 0 = ɛ d H z + ɛ d H θ + ɛ iγɛ + iγɛ + β ɛ H + i βɛ dh dh ω i βɛ dh θ + ɛ iγɛ dh θ + iγɛ + β ɛ + ɛ dh z iβɛ + iγɛ H + H θ + βγɛ H z, 50 ω γ ɛ H z. 5 If we estit ou study to desibing the tansvese magneti modes TM, fom Eq. 50, when H θ is diffeent fom, we get 0 = ɛ d H θ + ɛ iγɛ dh θ + ω / + iγɛ + β ɛ H θ. 5 We an expess the solution of this equation in tems of the omplex ylinial Bessel funtion of omplex ode and agument as follows: Rev. Mex. Fís. S

6 76 L.O. PALOMARES AND J.A. REYES H θ x = C x iδ p J iδ ix +C x iδ J iδ ix ɛ p R ω ɛ, 53 whee δ = γɛ /ɛ, x = /R, p = βr ae the dimensionless popagation paametes, with R the adius of the fibe and J ν x the Bessel funtion [7]. We should mention that we have witten H θ x as a linea ombination of the two independent solutions with positive and negative ode, J ν and J ν, instead of using the moe ommon Neumann funtion N ν x, whih is indeed a linea ombination of the funtions mentioned. It is impotant to assue that H θ x is finite fo evey point within the guide, patiulaly at the oigin. To do this, we expand eah of the funtions involved in H θ x in a powe seies. Up to the fist ode in x, this leads to x iδ x iδ J iδ x iδ iδ! = iδ! ln x = eiδ, 54 iδ! whih, using the identity e iθ = os θ + i sin θ, takes the fom J iδ x xiδ! [ os δ ln x E z x = C ɛ {J xiδ Rωɛ 0 ɛ iδ x inside the fibe and ixj iδ x + i sin δ ln x ]. 55 ɛ p ɛ ɛ p γɛ iɛ ɛ This expession eadily shows that J iδ x diveges when x 0. On the othe hand, afte expanding J iδ x nea the oigin, we get the expession ln x J iδ x e iδ iδ! = x iδ! os δ ln x i sin δ ln x, 56 whih poves that J iδ x vanishes at the oigin. Hene, to have a finite solution inside the fibe, we have to set C = 0, so the solution given by Eq.53 edues to H θ x = C x iδ J iδ x ɛ 0 µ 0 ɛ p. 57 If the wave popagating inside the fibe is to be onfined, then outside the fibe the solution must tend to zeo fa fom the guide. Thus we have H θ x = C K x p / 58 whee K is the modified of Bessel funtion of ode and seond lass. We an also alulate the eleti field fom the Faaday equation Ex = i ω ɛ Hx, 59 whose the only nonvanishing omponent is in the z dietion and is given by p + ixj iδ x ɛ p ɛ ɛ p } E z x = ic K x p p / / Rωɛ 0 6 outside the fibe. The bounday onditions to be fulfilled by the eletomagneti field ae the ontinuity of the tangential omponents of H and E aoss the intefae. This leads to the two onditions: R C J iδ ω ɛ p = C K p / 6 C ɛ {J Rωɛ iδ + ij iδ ɛ p γɛ iɛ ij iδ ɛ p ɛ ɛ ɛ p ɛ ɛ p 60 } p p ic K / / p =. 63 Rω Rev. Mex. Fís. S

7 GUIDING OF OPTICAL FIELDS IN A SMECTIC B LIQUID CRYSTAL CYLINDRICAL FIBER Notie that this is a system of linea homogenous algebai equations fo C and C. Hene, in ode to obtain a nontivial solution, we set the deteminant of this system equal to zeo. This amounts to the ondition Ez Hθ Ez Hθ = 0, 64 whih establishes a tansendental equation fo p, the popagation paamete. 6. Results In Fig. we show paameti tajetoies stating with the same initial position and value of pθ, and diffeent values of the tilt φ, and ay omponent pz. We obseve that fo lage values of pz, the tajetoies begin to get lose, but thee is not muh diffeene among the uves oesponding to diffeent values of φ. On the othe hand, fo smalle values of pz the tajetoies ae quite diffeent. Fo instane, when φ = 30, the spial tuns moe than fo φ = 90. In Fig., the paameti tajetoies ae depited fo diffeent values of the tilt φ and ay omponent pθ. We notie a simila behavio to that shown fo small values of pz. By hanging pθ, the modifiations in the ay tajetoies ae F IGURE. Thee dimensional ay tajetoies fo vaious values of φ as a funtion of the spae oodinates fo initial onditions, 0 = 0 8 m, θ0 = 0, z0 = 0, pθ = 0 4 and a pz =,, b pz =.05, pz = 5, dpz = muh smalle than those obtained when pz is vaied, as an be seen by ompaing Figs. and. The otation of all the tajetoies and a vaying polaization angle α evidently show the pesene of optial ativity in the guide even though the smeti foming the guide is not hial. In Fig. 3, we plot the amplitude magneti field Hθ thoughout the spae as a funtion of x paameteized by φ fo typial themotopi liquid ystal paametes: n =.506, nk =.735, by taking R = µm, λ = 750nm, γ =. It is impotant to point out that the efative indexes fo smeti liquid ystals in the optial spetum have not been measued yet. Beause some othe phases like holesteis and nematis pesent vey simila optial efative indexes, we have taken the values of a typial nemati. The onstants C and C ae found by imposing the nomalized ondition 8π Z Ez Dz =. 0 Notie that the allowed numbe of modes hanges fo diffeent values of the tilt φ. Fo φ = 90, thee exist thee modes, wheeas fo φ = 30 o smalle thee only exists one mode. F IGURE. The same as Fig. but fo pz = and a pθ = 0 8, b pθ = 0 6, pθ = 0 4. Rev. Mex. Fı s. S

8 78 L.O. PALOMARES AND J.A. REYES F IGURE 3. Amplitude of the magneti Field Hθ as a funtion of x paameteized by φ fo the a zeo ode mode b fist ode mode andq seond ode mode. Nomalized by R Hθ = Hθ /H0 / /8π 0 Bx H xxdx/µ0 H0, whee H0 is the geatest intensity of the magneti field. A fist mode B seond mode C thid mode. F IGURE 6. Eleti field phase ϕe as a funtion of x fo the same values as in Fig. 3. F IGURE 7. Nomalized eletomagneti enegy density R u u/ xudx fo the same values of Fig F IGURE 4. Amplitude of the eleti field Ez as a funtion of x fo the same values q takenr in Fig. 3. Nomalized by Ez = Ez /E0 / /8π 0 Ex D xxdx/²0 E0, whee E0 is the geatest intensity of the eleti field. Fo diffeent values of tilt. A fist mode B seond mode C thid mode. In Fig 4 we depit the amplitude of the omponent Ez fo the same paamete values. Now the uves show a maximum at the ente of the guide and one node moe, as would be expeted. In Fig. 5 and 6 we show the phase of Hθ and Ez fo the same values. Notie that all the uves ae ontinuous in Fig. 5, wheeas in Fig. 6 thee ae two disontinuity points: one due to the hange of sign of the amplitude, and the othe aused by the bounday ondition. The distibution of the oesponding eletomagneti enegy density uem x, k0 is alulated fom the equation uem x, k0 = E D + B H 8π 65 The plot of the above expession fo uem, Fig. 7, shows that uem is lage inside the fibe and is patially zeo fo lage x. 7. F IGURE 5. Magneti field phase ϕh as a funtion of x fo the same values taken in Fig. 3. Fo diffeent values of tilt. A fist mode B seond mode C thid mode. Conluding emaks We onstuted the fist model fo studying the popagation of eletomagneti waves within a smeti B ylinial guide. Unde the optial limit we found thee-dimensional ay tajetoies within the fibe of banana-shaped moleules Rev. Mex. Fı s. S

9 GUIDING OF OPTICAL FIELDS IN A SMECTIC B LIQUID CRYSTAL CYLINDRICAL FIBER 79 liquid ystal oe. We showed that the tajetoies ae spials eithe diveging o onveging fom the ente, desibed by a paaboloid in the z plane. Sine all the ays otate while they ae popagating though the fibe, and thee exists a vaying polaization angle α, we evidently exhibited the pesene of optial ativity in the guide even though the smeti foming the guide is not hial. We found analytial expessions fo desibing the exat eletomagneti equation fo the TM modes. We showed that the numbe of modes an be ontolled by the value of the smeti tilt. Speifially, we show that fo a thee modal guide in the low infaed spetum, the numbe of modes an be edued to one, if the tilt is diminished to a value equal to o smalle than 30. Finally, we eall that ou model was pefomed fo a low intensity beam suh that the smeti onfiguation is not distoted. Thus, a study of stonge intensities fo whih the optial popagation tuns out to be non linea, emains to be assessed. Also, the mehanisms unde whih the smeti tilt an be hanged by extenal agents in this onfiguation ae unde study.. A. Jakli, D. Keke, and G.N. Geetha, Phys Rev E G. Pelzl et al., Liq. Cyst G. Liao, S. Stojadinovi, G. Pelz, W. Weisflog, and S. Spunt, A. Jï li. Phys. Rev E M. Bon and E. Wolf, Piniples of Optis, Pegamon Pess, New Yok, L.M. Bakovskii and T.N. Khang Fo Opt. Spektosk J.A. Reyes, J. Phys A Math Gen Afken, Mathematial methods fo physis, Thid edition, Aademi Pess In., Oxfod, Ohio, USA., J.A. Reyes and R. F. Roíuez. Optis Comm P.G. de Gennes, The physis of Liquid Cystals Claendon Pess, Oxfod, 974. Rev. Mex. Fís. S