Thin θ-films optics Luis Huerta1,2,∗ 1Facultad de Ingenier´ıa, Universidad de Talca, 3340000 Curic´o, Chile. 2P4-Center for Research and Applications in Plasma Physics and Pulsed Power Technology, 7600713 Santiago, Chile. Abstract A Chern-Simons theory in 3D is accomplished by the so-called θ-term in the action, 7 (cid:82) (θ/2) F ∧F, which contributes only to observable effects on the boundaries of such a system. 1 0 When electromagnetic radiation interacts with the system, the wave is reflected and its polar- 2 ization is rotated at the interface, even when both the θ-system and the environment are pure n a vacuum. These topics have been studied extensively. Here, we investigate the optical properties J of a thin θ-film, where multiple internal reflections could interfere coherently. The cases of 7 1 pure vacuum and a material with magneto-electric properties are analyzed. It is found that ] the film reflectance is enhanced compared to ordinary non-θ systems and the interplay between h t magneto-electric properties and θ parameter yield film opacity and polarization properties - p whichcouldbeinterestinginthecaseoftopologicalinsulators, amongothertopologicalsystems. e h [ Keywords: Chern-Simons theories, θ-vacuum, topological field theories, topological insulators. 1 v 0 7 7 4 0 . 1 0 7 1 : v i X r a ∗ [email protected]; http://www.pppp.cl 1 I. INTRODUCTION Chern-Simons (CS) theories have found applications in various areas, from gauge field theories and three-dimensional gravity [1, 2] to condensed matter physics. In the latter case, CS formalism has been applied to phenomena that exhibit topological properties, a remarkable example of that being the quantum Hall effect, occuring in two spatial dimensions(2D)[3](seealso[4]). But, alsoin3Dspace, CStheorieshavebecomerelevant. Theso-calledθ-termintroducedinQCDbyPecceiandQuinn,andgeneralizedbyWilczek, represent the 3+1 spacetime dimension version of CS forms.[5, 6]. In recent years, the novel 3D topological insulators have become a new example of the above, and intensive research has been developed in the field [7, 8]. (cid:82) A 3D CS theory is characterized by the term (θ/2) F ∧ F in the action, which is a border term and it is relevant only on the system boundaries. As a consequence, electromagnetic radiation inciding on the interface between an ordinary medium and a θ-system, matter or even pure vacuum, exhibits a polarization rotation both for reflected and refracted waves. The effect is modulated by the interplay between magnetic and dielectric properties of the system and the value of the θ parameter. This phenomenon has been reported as the Kerr-Faraday rotation present in topological insulators [9] (see Ref. [7] for a review). Apart from the effect of polarization, the interface reflectance increases compared to ordinary systems. In a previous paper we reported in detail the optical properties of a θ-material interface [10]. The above phenomena are even stronger in the case of pure vacuum, where the only difference between the θ-vacuum and the surrounding medium is a nonzero value for θ inside the system [11]. At both sides of the interface, however, electromagnetic propagation is the same. Although the θ-vacuum case is the clearest manifestation of the peculiarity involved in these systems, applications to material systems are interesting and useful, since nontrivial topology is an extended issue in physics. In particular, the study of thin θ-films properties could be relevant for applications, since they are often seen in such a geometry [12]. We study here the optical properties of a generic thin θ-film, surrounded by ordinary matter or vacuum. We also include the possibility of pure vacuum in both the film and the medium outside. In particular, we investigate theoretically the system behavior under electromagnetic radiation. Since in (3+1)D, CS term becomes a boundary term, it will not contribute to field equations in the bulk of a system, but will only affect the system at its boundaries. For thin films, interference effects are interesting, those resulting from the multiple coherent superposition of waves emerging on both sides of the film, and that is a relevant 2 topic in the present paper. This report is organized as follows. First, we present the basics of the theory and recall some results for a single interface, from Ref. [10]. In the following sections we develop our approach to the film problem and present our findings. II. THE θ-SYSTEM INTERFACE For a θ-system interacting with electromagnetic fields, as in the case of radiation, the (cid:82) action is written as S +θ F ∧F (S is the Maxwell action), and field equations are M M modified only at the boundaries. We denote the boundary surface by Σ, with a locally defined normal unit vector nˆ pointing inside the system. In terms of noncovariant electric and magnetic fields, the field equations are [13] ε∇·E = θδ(Σ)B·nˆ (2.1) 1 ∇×B−∂ E = θδ(Σ)E×nˆ, (2.2) t µ where ε and µ represent the electric permittivity and magnetic permeability of the ma- terial, respectively.1 The delta function in the RHS of the above equations stands for restricting the effects to the surface, with the corresponding terms representing the sur- face charge and current densities built from the electromagnetic field itself. The set of equations is completed with the homogeneous Maxwell equations ∇ · B = 0 and ∇ × E + ∂ B = 0. By the standard procedure, we obtain a pair of discontinuity con- t ditions at the interface Σ: [εE ] = θB (2.3) n n (cid:20) (cid:21) 1 B = −θE . (2.4) τ τ µ Subindices n and τ stand for normal and tangent components of the fields, respectively, with respect to the interface Σ. The symbol [] must be interpreted as the difference between the fields evaluated immediately inside and immediately outside the θ-material. Additionally, we have, from the homogeneous equations, the standard continuity of E τ and B components. n We consider a linearly polarized planar electromagnetic wave inciding on the boundary surface of a θ-system, with an angle ϕ with respect to the normal. The standard laws of reflectionandrefractionhold, independentlyofthepolarization, andthetransmittedwave into the θ-medium is refracted with an angle ψ. By applying the boundary conditions 1Wechoosec=1. Consequently,thevacuummagnetoelectricpropertiesareε =µ =1,andthesystem’s 0 0 ε, µ are dimensionless. 3 derivedabove, wefindtheelectromagneticfieldamplitudesforthetransmitted(refracted) and reflected waves. We adopt the usual decomposition of the fields in parallel ((cid:107), p) and perpendicular (⊥, s) components with respect to the incidence plane, and introduce the adimensional field amplitudes in terms of the incident wave field E : e ≡ E /E , i i(cid:107) i(cid:107) i e ≡ E /E and e ≡ E /E , with E = (E 2 +E 2)1/2, and similar definitions for the r(cid:107) r(cid:107) i t(cid:107) t(cid:107) i i i(cid:107) i⊥ corresponding s components.2 Then, for the transmitted wave, we find [10] (cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33) e 2 ηs+1 −θ e t(cid:107) = i(cid:107) (2.5) e D θs η +s e t⊥ i⊥ and, for the reflected wave, (cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33) e 1 (ηs+1)(η −s)+θ2s 2θs e r(cid:107) = i(cid:107) , (2.6) e D 2θs −(ηs−1)(η +s)−θ2s e r⊥ i⊥ where D ≡ (ηs+1)(η +s) + θ2s. We have introduced the convenient definitions s ≡ (cid:112) cosψ/cosϕ, and η ≡ (n /n )(µ /µ ) = (ε µ )/(ε µ ), which describes the dielectric 2 1 1 2 2 1 1 2 and magnetic properties of the system. Also, to simplify notation we have redefined θ/η → θ.3 1 Unlike the normal systems, for θ-systems the p and s components of the reflected and refracted waves mix each other when crossing the system interface. Thus, both waves experience changes in their polarization. We find, for the transmitted wave, a polarization angle, measured with respect to the incidence plane, given by4 (cid:20) (cid:21) θs+(η +s)tanα α = tan−1 I (2.7) T ηs+1−θtanα I and, for the reflected wave, (cid:20)2θs−[(ηs−1)(η +s)+θ2s]tanα (cid:21) α = tan−1 I , (2.8) R (ηs+1)(η −s)+θ2s+2θstanα I with α representing the incident wave polarization angle. I Figure1showsthepolarizationangleversusθ fordifferentvaluesofthemagnetoelectric parameter η (for p-polarized incident wave). Included also is the θ-vacuum (η = 1) case 2As is known, at each medium magnetic and electric fields are related one to each other by B=nkˆ×E, where n=(µε)1/2 is the respective refraction index of the medium and kˆ is the corresponding unit wave vector. 3If the surrounding medium is vacuum space, the redefined θ coincides with the parameter in the action. 4The right-hand rule is followed in the definition of the polarization angle, with the corresponding wave vector kˆ as the reference axis. 4 90 90 ϕ= 30o ϕ = 30o 60 60 OLARIZATION ANGLE -33000 p-incidence 2100θ....0051 OLARIZATION ANGLE -33000 p-incidence 2100θ....0051 P s-incidence P s-incidence -60 -60 -90 -90 1.0 2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0 η η (a) (b) FIG. 1: Polarization of transmitted (left) and reflected (right) waves by a single θ-interface. Although for ordinary materials (θ = 0) there is no polarization rotation, for rather small nonvanishing values of θ, a significant change in polarization is found which is larger for smaller magnetoelectric properties. Curves for p and s incidence look similar, but there are slight differences in the actual values of the respective polarization angles. for comparison. Because of p-polarization incidence (α = 0), the curves represent the I effective polarization rotation experienced by the outgoing waves. The interplay between magnetoelectric properties, η, and the θ parameter unveils different qualitative situations. Forthetransmittedwavetothematerial, η diminishestheeffect(exceptforvaluescloseto 1) with respect to the θ-vacuum. For reflected radiation the situation is more interesting, but, similar to transmission, only for values of η close to 1, there is an enhancement in polarization rotation For large η (not shown in Fig. 1), only for very large values of θ, for the transmitted wave, or θ large enough, for the reflected wave, there will be a significant polarization rotation. An example of this is the existence of a maximum for reflected wave polarization for η > s. A more detailed analysis is found in Ref. [10]. III. θ-FILM OPTICS Letusnowconsideraninfiniterectangularthinfilmofaθ-systemimmersedinanormal non-θ medium. Both the film and the surrounding medium have different dielectric and magnetic properties, so that η (cid:54)= 1. When the electromagnetic wave reaches the film, it 5 experiencesaseriesofinternalreflectionsbetweenthetwointerfacesthatseparatethefilm from the medium outside. After each of those internal reflections, a wave emerges outside the film, from each side alternatively. Fig. 2 illustrates the geometry and definitions we use here. ei! er! et(!2) ei⊥ ⊙ ⊙e ⊙e(2) ϕ ϕ r⊥ ϕ t⊥ e e(2) ψ t! ψ r! θ"film& et ⊙⊥ er(1!) er(2⊥ )⊙ er(!3) ψψ ⊙e(1) ⊙e(3) r⊥ ψ r⊥ ϕ ϕ e(1) e(3) t! t! e(1) ⊙ ⊙ t⊥ e(3) t⊥ FIG. 2: Film geometrical optics. Internally reflected waves yield refracted and reflected waves with field amplitudes expressedintermsofthearrivingwavefield. Thediscontinuityattheinterfaceforinternal waves is given by (2.3) and ((2.4), but provided the substitution θ → −θ. We define (cid:32) (cid:33) 2ηs ηs+1 θs T ≡ (3.1) D −θ η +s as the matrix which represents the transformation of field amplitudes when the wave goes out from the film (in any boundary). For internal reflections, we have the transformation matrix (cid:32) (cid:33) 1 θ2s−(ηs+1)(η −s) −2θηs R ≡ . (3.2) D −2θηs −θ2s+(ηs−1)(η +s) It is readily seen that the transmitted and reflected wave amplitudes are given by e(l) = T Rl−1e , (3.3) t t e(l) = Rle , (3.4) r t for l = 1, 2, ... (e is the field amplitude of the wave refracted at the incident interface). t The number l represent the lth time that a film interface is reached by the internal 6 traveling wave, with field amplitudes e(l) and e(l) for the transmitted and reflected waves, t r respectively. We note that e(l), for l even, represent the field amplitudes of the waves t reflected back by the film. Of course, for l odd, we have the radiation waves passing through the film. IV. COHERENT SUPERPOSITION OF WAVES If the film is thin enough, compared to the radiation wavelength, coherent waves will interfere with each other, for both the transmitted and reflected beams going out from the film. Each wave carries a phase shift ei2nk¯d¯, for the optical path back and forth inside the film. There, n = n /n is the relative refraction index of the film with respect to 2 1 ¯ ¯ the surrounding medium, 2nkd thus representing the round optical path across the film. ¯ ¯ The corresponding wave number is n k = nk, with k ≡ n k, n being the refraction 2 1 1 index for the medium outside and k, the wave number in empty space. The distance ¯ d ≡ d/cosψ = d/scosϕ is the effective path length inside the film, which depends on the angle of incidence ϕ (d is the film thickness). No additional phase shift is introduced in the internal reflections, since we assume that n > n . 2 1 For the radiation passing through the film, the resulting field amplitude E becomes t (cid:16) (cid:17) E = T 1+R2ei2nk¯d¯+R4ei4nk¯d¯+... e , (4.1) t t and, for the radiation reflected back, the field amplitude is (cid:16) (cid:17) E = e +T Rei2nk¯d¯ 1+R2ei2nk¯d¯+... e . (4.2) r r t The geometric series, represented by the matrix M ≡ 1+R2ei2nk¯d¯+R4ei4nk¯d¯+..., (4.3) can be summed up by diagonalizing the reflection matrix R, to obtain  1  0 1−λ 2ei2nk¯d¯ M = U  + U−1. (4.4)  1  0 1−λ 2ei2nk¯d¯ − λ are the eigenvalues of R, given by ± η 1 (cid:0) (cid:1) λ = s2 −1 ± Q, (4.5) ± D D 7 and U is the corresponding unitary transformation matrix in the diagonalization process: (cid:32) √ √ (cid:33) 1 δ +Q −δ +Q U = √ √ √ , (4.6) 2Q − −δ +Q δ +Q (cid:113) where Q ≡ (D−η(s−1)2)(D−η(s+1)2) and δ ≡ θ2s−s(η2 −1). V. REFLECTANCE FOR COHERENT RADIATION ¯ ¯ For constructive interference, 2nkd = 2π. Therefore, the field amplitude of radiation transmitted through the film is given by E(C) = T M(C)e = e . (5.1) t t i Consequently, E(C) = 0, (5.2) r and the film becomes completely transparent, without any effect in the radiation polar- ization. ¯ ¯ For destructive interference, 2nkd = π, and we obtain for the transmitted radiation through the film (cid:32) (cid:33) 2η s2(D−2η)−s(s2 −1)(ηs+1) θs(s2 −1) E (D) = (e ). t Λ θs(s2 −1) (D−2ηs2)+(s2 −1)(η +s) i (5.3) For the radiation reflected by the film, the field amplitude is given by (cid:18) (cid:19) E (D) = 1 D2−2(s(ηs+1)+η)D+2η(η+s)(s2+1) 2θs[D−η(s2+1)] (e ), (5.4) r Λ 2θs[D−η(s2+1)] −D2+2[s(ηs+1)+η]D−2ηs(ηs+1)(s2+1) i with Λ ≡ [D−η(s2 +1)]2 +η2(s2 −1)2. (cid:12) (cid:12)2 (cid:12) (cid:12)2 Therefore, the film transmittance T ≡ (cid:12)E (D)(cid:12) and reflectance R ≡ (cid:12)E (D)(cid:12) , computed (cid:12) t (cid:12) (cid:12) r (cid:12) for p incidence, are, respectively, 4η2s2 (cid:104) (cid:105) T(D) = (cid:0)θ2s2 +η2s2 +1(cid:1)2 +θ2(cid:0)s2 −1(cid:1)2 , (5.5) Λ2 1 (cid:110) (cid:111) R(D) = (cid:2)θ2s2(cid:0)θ2 +2η2(cid:1)+(cid:0)η2s2 +1(cid:1)(η +s)(η −s)(cid:3)2 +4θ2s4(cid:2)θ2 +η2 +1(cid:3)2 . Λ2 (5.6) Figure 3 shows the θ-film reflectance for two different angles of incidence on the film. Thesolidcurverepresents,inbothgraphs,thenon-θ,ordinarymaterialresults. Compared 8 to ordinary materials, θ-films are more opaque to radiation, particularly for a small η where normal materials become more transparent. The larger η is, the more reflective the film. But what is interesting here is that for topological materials the effect is enhanced, with an interplay that leads to a local minimum value in terms of η. 1.0 1.0 p-incidence ϕ= 60o 0.8 0.8 θ θ E E C 0.6 C 0.6 N 2.0 N 2.0 A 1.0 A 1.0 ECT 00..50 ECT 00..50 EFL 0.4 p-incidence EFL 0.4 R R ϕ= 30o 0.2 0.2 0.0 0.0 1.0 2.0 3.0 4.0 5.0 1.0 2.0 3.0 4.0 5.0 η η (a) (b) FIG. 3: θ-film reflectance, in the case of destructive interference and for two angles of incidence. The solid curve is the results for ordinary non-θ systems. In both cases, the θ-film reflectance is much higher than for ordinary materials. In Fig. 4a below, the curves for film reflectance versus θ, and for different values of η parameter, show the strong influence of the parameter θ in energy distribution (in the case of destructive interference). It is illustrative to note the curve for η = 1 (solid curve), which represents the case where the medium inside the film and in the surroundings is the same, except for a nonvanishing value of θ in the film. This case can be also considered as representative of the pure vacuum case and it demonstrates the significatively different behavior of θ-systems. VI. POLARIZATION Only for destructive interference, is there a change in the radiation polarization. For a linear polarization of the incident wave, a polarization rotation appears for the reflected back and transmitted radiation by the film. For transmitted radiation beyond the film, (cid:0) (cid:1) we obtain, for the polarization angle α ≡ tan−1 E /E (α being the incident wave T T⊥ T(cid:107) i 9 1.0 90 p-incidence ϕ= 30o 0.8 E L G TANCE 0.6 p-incϕid=e n3c0eo ON AN 60 3η.0 REFLEC 0.4 η ARIZATI 211...050 21..05 OL 30 0.2 1.0 P 0.0 0.0 1.0 2.0 3.0 4.0 0 θ 0.0 1.0 2.0 3.0 4.0 5.0 θ (a) (b) FIG. 4: θ-film (a) reflectance and (b) polarization of reflected radiation versus θ, for destructive interference. The solid curve (η = 1) represents the case where the magnetoelectric properties are the same for the θ-system and its surroundings (the full vacuum case is also included in such a situation). polarization angle), (cid:20) θs(s2 −1)+[(D−2ηs2)+(η +s)(s2 −1)]tanα (cid:21) α = tan−1 i . (6.1) T s2(D−2η)−s(ηs+1)(s2 −1)+θs(s2 −1)tanα i For the reflected radiation, we find for the polarization angle α , R (cid:20)2θs[D−η(s2 +1)]+[−D2 +2[s(ηs+1)+η]D−2ηs(ηs+1)(s2 +1)]tanα (cid:21) α = tan−1 i . R [D−(s(ηs+1)+η)]2 −η2(s4 −1)−s2 +2θs[D−η(s2 +1)]tanα i (6.2) For p incidence, (6.2) and (6.1) become, respectively, (cid:20) θ(s2 −1) (cid:21) α(p) = tan−1 (6.3) T s(D−2η)−(ηs+1)(s2 −1) and (cid:20) 2θs[θ2s+s(η2 +1)] (cid:21) α(p) = tan−1 . (6.4) R s2(θ2 +η2)2 −η2(s4 −1)−s2 10