Analysis of electromagnetic propagation in birefringent thin film

Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No.5 513- -520513Analysis of electromagnetic propagationin birefringent thin filmQI Hongji, WANG Jianguo, SHAO Jianda & FAN ZhengxiuShanghai Institute of Optics and Fine Mechanics, Shanghai 201800, ChinaCorrespondence should be addressed to Qi Hongji (email: qhj@mail.siom.ac.cn)Received July 22, 2005Abstract In general, the propagating behavior of extraordinary wave in anisotropicmaterials is different from that in isotropic materials. With the tangential continuity ofMaxwell s equations, the electromagnetic propagating behaviors have been investigatedat the incident and exit interfaces of the uniaxial anisotropic thin film. The emphasis wasplaced on two interesting optical phenomena such as homolateral refraction behavior andwide-angle Brewster s phenomenon, which occurred at the interfaces of uniaxialanisotropic thin film.Keywords: birefringent thin film, interface, homolateral refraction behavior, wide-angle Brewster' s phe-nomenon.DOI: 10.1360/ 142004-53Recently, a series of thin films with particular microstructure have been depositedsuccessfully' ! 2. The thin films with regular microstructure give optical anisotropy inmacroscopic view, i.e. birefringent feature. The theoretical analysis and deposition ofbirefringent thin film inject activity into traditional optical thin film5-0'. Basing on thebirefringent thin film, the researchers have successfully prepared electro-optic tunablefilters', antireflection coatings', phase compensator!', and polarizer'0. Besides, thetheoretical study of electromagnetic propagation behaviors across birefringent thin filmhad gone further. With the tangential continuity of Maxwell' s equations, Holmes andFeucht n considered multilayer anisotropic media with an obliquely propagating inci-dent wave, where two of the principal axes are arbitrarily aligned with respect to the in-terface axes. Schesser and Eichmann' " 2 ! discussed a general theory of wave propagationin three-layer anisotropic media in which the waves travel obliquely to the interfaces andthe principal axes of the three media are oriented arbitrarily. The most generalapproaches to describe the propagating behaviors in birefringent thin film are 4x4 matrixtechniquesl13- -18]. Besides, using the 2x2 extended Jones matrix method, Cojocarul19,20]discussed electromagnetic traveling characteristics in dilectricallv anisntrnpic, homo-geneous layered media at oblique incidence. Addit中国煤化工: a generalexact 2x2 matrix method for computing reflectionMHCNMHGntsofelec-Copyright by Science in China Press 2005514Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No.5 513- -520tromagnetic waves from the interface between arbitrarily oriented biaxial medialn, andconsidered multiple reflection and transmission in a biaxial slab between two anisotropicmedial2 24. The complication of mathematical treatment for anisotropic thin film with theprincipal axes oriented arbitrarily covers some interesting physical phenomena occurringat the particular conditions. By now, a lttle work dealt with such interesting physicalphenomena at the interface of birefringent thin film. Here, we limit our discussion touniaxial layered transparent thin film with the principal refractive indices no and ne andthe optic axis in the incidence plane.As is well known, there are two orthogonal polarizations to be treated, namely, p- ands-polarized wave. When the optic axis lies in the incidence plane, the incident light po-larized in the p sense will not produce any s component of reflected and refracted lightand vice versa, so that we can treat each of the polarizations separately. For s-polarizedwave, the results of transmission through and the reflection from the anisotropic mediumare identical to those derived from isotropic medium. Hence, we do not consider thepropagation of s-polarized wave in this work. In this paper, basing on Maxwell' s equa-tions, we investigated the electromagnetic propagation behavior at oblique incidence,and gave critical conditions of total reflection at the interface of thin film. The emphasiswas placed on some interesting optical phenomena occurring at the interfaces, such asthe homolateral refraction behavior at the incident interface and the wide -angle Brew-ster’s phenomenon at the exit interface.1 Propagation of p-polarized plane waves in anisotropic thin flmFirst of all, we discuss the propagating properties of the incident and exit interfaces ofuniaxial anisotropic thin film, labeled by the symbol a and b, as shown in Fig. 1. Thethickness of thin film is denoted by d. The incident medium and the substrate are sup-posed to be homogeneous and isotropic, with their indices of refraction no and ng, re-spectively. The xyz Cartesian coordinate system is chosen, and the xy plane is parallelto the interface of layered medium. Without loss of generality, we can let the principalsection coincide with the principal plane, i.e.the optic axis lies in the incident plane. Thenangle of the inclined optic axis with respectto y axis is denoted by φ (-90°<φ≤90°),\kwhich is also shown by the thin line in Fig. 1.CIn this case, s-polarized wave propagates in入_an ordinary fashion. However, p-polarizedwave propagates in an extraordinary fashion.↑*In Fig. 1, a monochromatic p-polarized planewave with wave vector ko+ is incident fromFig. 1. Reflection and refraction of the incidentthe ambient region at angle θn+ in the inci-p-polarized light at the iteracres of anisotropie dent pla中国煤化工b, the re-layered medium in the incident plane.flectedMYHC NMH Grs are de-Copyright by Science in China Press 2005Analysis of electromagnetic propagation in birefringent thin film515noted by kg_ ,k, and k , kg+, respectively. In the following analysis, the subscripts“+”and“”refer to the forward and backward directions of propagation, respectively.Unlike isotropic thin film, the wave vector surface in the uniaxial thin film is split intotwo shells corresponding to the ordinary and extraordinary linearly polarized lightwavesh7. The shells for ordinary and extraordinary wave are sphere with a radius ofon./c and an ellipsoid with three principal axes of on。/c, on。/c and on./c,which are tangent along the optic axis. Here c is the speed of light in free space, 0 isthe angular frequency of a monochromatic plane wave. It is useful to normalize the wavevector, and the usual way is to define a refractive index vectorn = k /0, then, the wavevector k is normalized, and the shell of wave vector is replaced with the shell of refrac-tive index vector. Therefore, the curve of intersection of the incident plane with the el-lipsoid is an inclined ellipse with two principal axes no and ne for extraordinary wave,and the angle between the principal semi-axis with length of no and axis y is φ. The re-fractive index of extraordinary wave is determined by φ, θ 0+ and no, varying between noand ne. At the inner interface a, the section of wave vector shell for extraordinary wavein the thin film is an inclined semi-ellipse, which is bounded by the interface a and tan-gent with the section of wave vector shell for ordinary wave along the optic axis, asshown in Figs. 2(a) and (b). The tangent point is labeled by E, i.e. the vector OE is par-allel to the optic axis.n8yE(a)b)Fig. 2. The relced and refracted surface of wave vector at the interface a of thin film, (a)no> ne, φ>0 ornoneφ<0° orn> ne, φ>0°.At the inner interface b, the section of wave vector shell for extraordinary wave isalso an inclined semi -llipse, which is bounded by the interface b and tangent with thesection of wave vector shell for ordinary wave along the optic axis, as shown in Figs. 3(a)and (b). The tangent point is denoted by F, i.e. O'F is parallel to the optic axis. Twosemi-ellipses of extraordinary wave vector at the interfaces a and b can construct awhole ellipse.The y components ky, of the incident, the forward and backward propagation extraor-dinary wave vectors are identical with value中国煤化工k, = no sinθo+MYHCNMHG(1)www .scichina.com516Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No.5 513- -520_CF入FAdbolea)b)Fig. 3. The reflected and refracted surface of wave vector at the interface b of thin film, (a)n> ne,0>0° or no< ne,φ<0°;(b)no>neφ<0° orn。> ne,q>0°.The z components kz of the refractive wave vector k+ and the reflected wave vector k_can be determined by the equation(-k, cosφ +k。sinq)2 1n。2 +(k, sinφ +k. cosp)21n2=1.(2)For the forward and backward propagation waves, the roots determined by eqs. (1)and (2) must satisfy kg≤0 and k:≥0, respectively. The following analysis will give θ 1and θ2 on solution of eqs. (1) and (2). .By defining D= no sinθ2 and substituting eq. (1) into eq. (2), we have(n2 sin2φ +n。2 cos2 φ)k? + Dsin(20)(n,2 -n2)k。+D2(n2 cos2φ+n2 sin2φ)-n?°n?2=0.(3)The discriminant A of eq. (3) is given by 4n,in?(n。2 cos3φ +n。sin2φ-D). If A≥0, the z components of the wave vector k+ and k - are denoted by k . Besides, wecan give y coordinate of the intersecting point of the inclined ellipse with axis y in theincident plane, kya =(n.n2)/(n。2 sin2 φ + n2 cos2 φ)"2 . Only the positive value of kyais of interest here. For the sake of simplification, we define kvu =(n,2 cos2φ +n2 sin2 φ)/2. From the values of no, ne, φ and D, the transmission and reflection proper-ties of the electromagnetic wave at the interface a and b would be listed in Table 1.When there is not the forward wave vector propagating in thin film, the extraordinarywave is totally reflected at the interface of thin film.When D= kyb, for n。> ne,φ> 0° orn,< ne,φ < 0°, it corresponds to the critical to-tal reflection condition for p-polarized wave at the interface a, and the critical angles inthe incident medium and in the thin film are given t中国煤化(s,sk)>.When the critical total reflection occurs at the in|TYHCNMHGt thin film,Copyright by Science in China Press 2005Analysis of electromagnetic propagation in birefringent thin film517Table 1 The transmission and reflection properties of the electromagnetic wave at the interfacea and b under the different conditions of no, ne,中and DDn o,n,中k:+kFBPhenomenonD>ky .<(Any caseN N Total reflection at the interface a<0c N Critical angle less than 90° at theorn,neφ<0°>0> 0N N Total reflection at the interface aorno 0°Brewster' s phenomenon at the in-<0YNandD < kybD> kyaorn, ne,φ<0°> 0N N Total rflection at the interface aorno0°no>ne,φ>0°The rflected wave vector propa-=0YCorn。 ne,φ< 0°>0cNCritical angle equal to 90° at theorno 0, for no> ne,φ>0° orn,< ne, φ< 0, the reflected wave vec-tor does not exist at the interface b. That is to say, the reflectivity for the p-polarizedwave is zero at the interface b, as if the interface b disappeared. This case is similar tothe Brewster' s phenomenon occurring at the isotropic isotropic interface for p-polarizedwave at the oblique incidence. The curves1.68n,=1.68,n-=1.50of kya and kyb versus the inclined angle of1.64the optic axis φ are plotted in Fig. 5. When。1.601.60the tangential component D of the incident1.56wave vector lies in the leaf-like region con-1.52L.52structed by the curves of kya and kyb, theBrewster' s phenomenon will occur at the1.48 L50 80 100J1.48exit interface of the anisotropic thin film.中国煤化工phenomenonTherefore，we call this Brewster' s phe-versus.MYHCNMH Go.www .scichina.com518Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No.5 513- -520nomenon the wide-angle Brewster' s phenomenon. W ide-angle Brewster' s phenomenonin this case is not exactly the same as that at the interface between two isotropic materi-als, where the reflectivity disappears only at a special angle, i.e. the Brewster' s angle'-: ! .Besides, wide- angle Brewster' s phenomenon is also different from that occurring at theincident interface of uniaxial crystals!24- 0 , which corresponds to one or two angles withzero reflectance for p- or s-polarized wave. Here, as long as the condition of kya ne,φ>0° orno ne, φ<0° or no< ne andφ> 0°, it gives kq+ = 0, which corresponds to the critical total reflection at the interface aand gives the incident angle and the refracted angle to be arcsin(k sa /no) in the inci-dent medium and 90° in the thin film, respectively.When D0 and k+<0, it gives θ = -arctan(D1k+) and θ2 = arctan(D1k2_).There are forward and backward-traveling extraordinary waves in the uniaxial thin film.2 DiscussionFor isotropic thin film, the wave vector and the ray propagate in the same direction;but for uniaxial anisotropic thin film, their propagations for extraordinary wave are gen-erally not in the same directions. The relationship connecting them is given bytanθ =n2 tanξ /n,2 , where θ and ξ are the angle between the wave vector and the opticaxis and the angle between the ray and the optic axis, respectively. For the normal inci-dence (θo+ =0°), the wave vector for the extraordinary wave is normal to the interfaceof the layered medium, i.e. θ=°-φ, and then ξ = arctan(n2 /[n2 tan($)]) . Using thesimple geometric relationship, we can determine the angle between the direction of raypropagation and the normal of interface to be π 12 -φ - arctan(n2 /[n2 tan(中)]) for theextraordinary wave. When no < ne and φ>0°，it obtains π/2_φ - arctan(n2 /[n2 tan(中)]) < 0, which indicates that the ray propagates toward the lower left direction.Thus, when the incident direction slightly deviates from the normal and the ray in theincident medium propagates toward the lower right direction of the interface, the ray inthe anisotropic layered medium still propagates t中国煤化工tion of theinterface, that is, “Homolateral refraction phenome:MHCNMH Gion. WhenCopyright by Science in China Press 2005Analysis of electromagnetic propagation in birefringent thin film519n。< ne and φ< 0°, the homolateral refrac-9(tion phenomenon can also occur. This phe-80n=1.68, n-=1.5nomenon is impossible for the iso-70tropic-isotropic interface. When the homo-60; 50lateral refraction phenomenon occurs, the《40upper and lower limitations of the inclined30angle of the optic axis φ vs. the incident angle10are plotted in Fig. 6 (n。=1.68, ne=1.50). As24681012shown in Fig. 6, when the incident angle is 0',Incident angle ()he refracted extraordinary wave vectorpropagates along the normal, but the re-Fig. 6. The upper and lower limitation of the in-fracted extraordinary ray propagates towardclined angle of the optic axis φ corresponding to theoccurrence of homolateral refraction phenomenonthe lower left direction, and the homolateral versus the incident angle.refraction phenomenon occurs at the front interface for 0<φ < 90° . As the incident angleincreases, the corresponding range of the inclined angle of the optic axis φ decreasesmonotonically, when homolateral refraction phenomenon occurs. When the incident angleis more than 10.36°， the homolateral refraction phenomenon does not occur at theincident interface.Using the homolateral refraction phenomenon, we can control the propagation of rayin four quadrants constructed by the interface and normal of thin film, which providemore flexibility to control the propagation of ray. Besides birefringent thin film, thenegative refractive index material can also produce the homolateral refraction phe-nomenon. The mechanism of homolateral refraction phenomenon in the birefringentmaterial is distinct from that in the negative refractive index. The most important prop-erty of the negative refractive index material is the negative permeability and permittiv-ity, which can be archived with so called metamaterials and offers the possibility of per-fect lenses and other exotic phenomena such as a reversal of Snell's law. Here, metama-terial is constructed as arrays of current-conducting elements (such as loops of wire) thathave suitable inductive and capacitive characteristics, and gains its (electromagnetic)material properties from its structure rather than directly from the materials that it iscomposed of. The negative refractive index material exhibits a plasma-like behavior andthe equivalent frequency is near gigahertz. However, the homolateral refraction phe-nomenon in the visible region can be archived with the birefringent material. Besides, asfor the negative refractive index material, the propagation direction of wave vector isopposite to that of ray. But for the birefringent material, two propagation directions con-struct a small angle.3 ConclusionIn this paper, we give some theory analysis and numerical calculation of the electro-magnetic propagation in uniaxial thin film. The emphasis is placed on some interestingoptical phenomena. The results show that for the bime中国煤化工”φ>0°orn,< ne中< 0°, the homolateral refraction phenornt interface,when the incident angle is small. Owe to the birefriHCNMHGedangleiswww .scichina.com520Science in China Ser. G Physics, Mechanics & Astronomy 2005 Vol.48 No.5 513- -520less than 90° in the thin film, when the total reflection is obtained at the incident inter-face. Besides, as long as the condition of ksga