Faraday and Kerr Effects in Right and Left-Handed Films and Layered

Rev. Adv. Mater. Sci. 2020; 59:243–251

Research Article

Josh Lofy, Vladimir Gasparian, Zhyrair Gevorkian, and Esther Jódar1* Faraday and Kerr Effects in Right and Left-Handed Films and Layered Materials https://doi.org/10.1515/rams-2020-0032 tive nature of the parameters ϵ and µ at first sight should √ Received Sep 22, 2019; accepted May 13, 2020 not affect the refractive index n = ϵµ, as it appears that n should remain positive. However, if complex parameters ϵ Abstract: In the present work, we study the rotations of the and µ have negative non-zero real parts and positive non- polarization of light propagating in right and left-handed zero imaginary parts (where the positive imaginary por- films and layered structures. Through the use of complex tion is required for passive materials) the real portion of values representing the rotations we analyze the transmis- n will also be negative. Indeed, the permittivity ϵ and the sion (Faraday effect) and reflections (Kerr effect) of light. permeability µ, as complex quantities, can be written in It is shown that the real and imaginary parts of the com- iθ iϕ the form ϵ = rϵ e and µ = rµ e . Accordingly, the re- plex angle of Faraday and Kerr rotations are odd and even √ i(θ+ϕ)/2 fractive index n has the form n = rϵ rµ e . The re- functions for the refractive index n, respectively. In the quirement that the imaginary part of the refractive index thin film case with left-handed materials there are large n must be positive for a non-absorbing medium leads to resonant enhancements of the reflected Kerr angle that double inequalities: 0 ≤ (θ + ϕ)/2 < π. Taking into ac- could be obtained experimentally. In the magnetic clock count the mentioned double inequalities and using the approach, used in the tunneling time problem, two charac- fact that the real parts of ϵ and µ must be both negative teristic time components are related to the real and imag- (i.e. cos θ < 0 and cos ϕ < 0) we get another inequalities: inary portions of the complex Faraday rotation angle θ. π/2 ≤ (θ + ϕ)/2 < π. The latter indicates, that the double The complex angle at the different propagation regimes negative nature of the parameters ϵ and µ affects the refrac- through a finite stack of alternating right and left-handed tive index n in the way that the real part of n of a metama- materials is analyzed in detail. We found that, in spite of √ terial, that is Ren ≡ ϵµ cos(θ + ϕ)/2, is negative (see, e.g. the fact that Re(θ) in the forbidden gap is almost zero, the Ref. [4] for more details). LHM have multiple uses that in- Im(θ) changes drastically in both value and sign. clude: the ability to resolve images beyond the diffraction Keywords: left-handed materials, metamaterials, Faraday limit [5, 6], act as electromagnetic cloaks for particular fre- rotation angle, Kerr effect, polarization of light quencies of light [7–9], enhance quantum interference [10] or yield to slow light propagation [11]. LHM used also for digital applications possess various exotic functionalities, 1 Introduction such as anomalous reflections, broadband diffusion, po- larisation conversion [12] and encoding information [13]. The presence of negative indices of refraction in one- Negative refractive index magneto-optical metamaterials, dimensional (1D) disordered metamaterials strongly sup- also called left-handed materials (LHM), are a new type of presses Anderson localization due to the lack of phase ac- artificial material characterized by having a permittivity ϵ cumulation during wave propagation, which thus weak- and permeability µ both negative [1–3]. The double nega- ens interference effects necessary for localization [14]. As a consequence, an unusual behaviour of the localization length ξ at long-wavelengths λ has been observed [14– *Corresponding Author: Esther Jódar1: Dpto. Física Aplicada, An- 16]. This is unlike the well-known quadratic asymptotic tiguo Hospital de Marina, Campus Muralla del Mar. UPCT, Cartagena behaviour ξ ∼ λ2 for standard isotropic layers (see, e.g. 30202 Murcia, Spain; Email: [email protected]; [17]). It can be seen that the metamaterial configurations Tel.: +34 968338925; Fax +34968325337 have an effect on the magneto-optical transport properties Josh Lofy, Vladimir Gasparian: Department of Physics, California of the electromagnetic waves. Particularly, the sign of the State University, Bakersfield, CA 93311, United States of America Zhyrair Gevorkian: Yerevan Physics Institute, Yerevan, Armenia plane polarization rotation angle in a left-handed medium and Institute of Radiophysics and Electronics, Ashtarak-2, 0203, (LHM) is opposite to the sign of the rotation angle in a Armenia

Open Access. © 2020 J. Lofy et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 244 Ë J. Lofy et al. right-handed medium (RHM). The Faraday and Kerr rota- The work is organized as follows. In section 2 we for- tions (FR and KR) are non-reciprocal polarization rotation mulate the problem with appropriate analytical expres- effects in which the sign of the rotation is always related sions for the complex Faraday angle of transmitted light. to the direction of the magnetic field. This is different in In section 3 we analyze the Kerr effect and calculate the optically active media where the rotation of the polariza- real and imaginary angles of reflection. In section 4we tion is related to the direction of the wave vector. Thus, the study the real and imaginary portions of the complex non-reciprocity of the Faraday and Kerr effects allow light Faraday rotation angle at different propagation regimes to accumulate rotations of the same sign and magnitude through a finite stack of alternating right and left-handed for both forward and backward propagation and can be materials which is analyzed in detail. enhanced even further by additional round-trip reflections through the medium. If we assume no absorption and neglect the influence 2 Right-handed and left-handed of the boundaries of the system, then in bulk materials for a given ϵ, µ, Verdet constant V and length of medium L, for dielectric slab a constant linear magnetic field B, the Faraday rotation an- √︁ µ gle is given by θ = LVBω . When the reflections within Let us consider a slab confined to the segment 0 ≤ x ≤ L, 2c ϵ √︀ the boundaries are relevant the outgoing reflected wave with a positive impedance Z = µ/ϵ for either RHM or is generally elliptically polarized, with or without absorp- LHM, and characterized by a permittivity ϵ = n/Z and a tion. The major axis of the ellipse is rotated with respect permeability µ = nZ. Both n and Z, and therefore ϵ and to the original direction of polarization and the maximum µ, are frequency dependent complex functions that satisfy FR (KR) angle does not necessarily coincide with angular certain requirements based on causality. For passive mate- frequencies ω of light at which zero ellipticity can be mea- rials, Re(Z) and Im(n) must be greater than zero. sured (we will come back to this question in section 2). The two semi-infinite media outside the slab are equal We represent the linear and elliptical polarizations as and are characterized by the dielectric constant ϵ1 or by √︀ the real and imaginary portions of a complex quantity. The the impedance Z1 = µ/ϵ1. A linearly polarized electro- real value of the rotation angle describes the change of po- magnetic plane wave with angular frequency ω enters the larization in linearly polarized light. The imaginary value slab from the left with normal incidence. We take the di- describes the ellipticity of transmitted or reflected light. rection of propagation as the x axis, and the direction of ⃗ Once we know the scattering matrix elements r and t on a the electric field E0 of the incident wave as the z axis. A ⃗ one-dimensional light propagation problem, then the two weak magnetic field B, that does not violate the linearity characteristic parameters of Faraday/Kerr rotation (Real) of Maxwell’s equations, is applied in the positive x direc- and Faraday/Kerr ellipticity (Imaginary) of the magneto- tion and is confined to the slab. This magnetic field causes optical transmission/reflection measurements can be writ- the direction of linear polarization to rotate while light ten in complex form as the real and imaginary parts of a propagates through the medium. As a consequence, the well-defined complex angle of θT and θR (see Eqs.(6) and dielectric tensor develops non-zero off-diagonal elements. (19)). Magneto-optic effects are related to the off-diagonal com- In the present paper, we theoretically consider the ponents ϵij (i, j ∈ {1, 2}) , whereas optical properties are Faraday rotation of light passing through a RHM/LHM film related to the diagonal components ϵii . The magnitude of of thickness L taking into account multiple reflections in the off-diagonal components ϵij is two orders of magnitude the boundaries without absorption. This exactly solvable smaller than that of the diagonal components ϵii. The gen- simple model is chosen to present different aspects of RHM eralized principle of symmetry of kinetic coefficients im- ⃗ ⃗ and LHM. It will be shown that the real part of the com- plies that ϵij(B) = ϵij(−B). The condition that absorption is plex angle of Faraday rotation is an odd function with re- absent requires that the tensor should be Hermitian ϵij = * spect to the refractive index n, while the imaginary part of ϵji. The latter implies that the real and imaginary parts of the angle is an even function of n. We have obtained the ϵij must be symmetrical and antisymmetrical, respectively. rotation angle of backscattered light (Kerr effect) from the That is: Re(ϵij)=Re(ϵji) and Im(ϵij)=-Im(ϵji). By combining ⃗ ⃗ RHM/LHM film as well. In the limit of ultra thin LHMfilm these conditions with the relation ϵij(B) = ϵij(−B), one can under specific circumstances we will see a large resonant show that the diagonal components of the dielectric tensor enhancement of the reflected KR angle. are even functions of an applied magnetic field, and the off-diagonal components are odd functions and have first- order magnetic field dependence. The dielectric tensor of Faraday and Kerr Effects in Right and Left-Handed Films and Layered Materials Ë 245 the slab is given by [18]: As it has been proven, there is a linear relation be- (︃ )︃ tween the real and imaginary parts of t± and between ln t± ϵ +ig ϵij = , (1) and ψ±. These are the known linear Kramers-Kronig rela- −ig ϵ tions, that can be rewritten in terms of localization length where ⃗g is the gyration vector in the magnetic-field direc- and density of states [19]. The complex FR angle with the e.g. tion. We include the external magnetic field ⃗B into the gy- imaginary and real parts is introduced as (see, , [20]): ⃗g ϵ 1/2 rotropic vector for our ij. This way our calculations are T i t ψ − ψ i T T T θ = − ln + = + − − ln + ≡ θ + iθ . (6) valid for the cases of a dielectric in an external magnetic t 1/2 1 2 2 − 2 2 T− field and for magneto-optical materials. As we can see from Eq. (6), if T = T , then θT ≡ θT The components Ez and Ey, Hz and Hy in the film are + − 1 not constant and depend only on the coordinate x. As well, and would only have real component; this signifies that ⃗E when a magnetic field is applied in the x direction, the off- the wave remains linearly polarized with vector rotated an angle θT to the initial direction. In the Faraday geom- diagonal elements ϵij cause coupling between the electric etry (a magnetic field is applied parallel to the direction field (Ez and Ey) and the magnetic field (Hz and Hy) components. The linearly polarized incident electromagnetic of light propagation) and in the absence of material losses R T R wave can be presented now as the sum of circularly po- within a thin film, ( + = 1, where is the reflection T T larized waves with opposite directions of rotation, which coefficient), we have that + = − if: (i) the sample is infinite (no boundaries), (ii) for certain thicknesses where to- propagate through the slab with a different wave vector 1/2 1/2 tal transmission occurs (T = 1) or (iii) n ∂T = Z ∂T . k± = ωn±/c. ∂n ∂Z E Ey iEz This third condition implies that at certain thicknesses the For circularly polarized waves ± = ± , the T imaginary portion θ2 becomes zero (the solutions follow- Maxwell equations have the form (time variation of optical Z2 iωt x +1 x field is in the form e− ) [18]: ing from the transcendental equation, 0 = Z2−1 tan 0). At these points the transmission coefficient T is not one, and 2 2 ∂ E± ω ϵ± x + E± = 0, (2) its value decreases with increasing 0, having a saturated ∂x2 c2 2 2 2 value of 4Z /(Z + 1) when x0 tends ∞, in contrast to the where ϵ± = ϵ ± g. two previous cases. This saturated value corresponds ex- The reflectance and transmittance amplitudes can be actly to one-quarter wavelength. obtained using the continuity of the tangential compo- If T+ ≠ T−, the light is not simply linearly polarized. It nents of the electric (magnetic) fields at the two interfaces, has an elliptical polarization, being the ratio of the ellipse x = 0 and x = L. Solving the equations with the appropri- semi-axis determined by the relation (b < a): ate boundary conditions at x = 0 and x = L we obtain for ⃒ ⃒ ⃒ ⃒ E′ E′ ⃒T1/2 − T1/2⃒ the transmitted waves + and − b ⃒ + − ⃒ | θT| ′ = tan 2 = ⃒ ⃒ , (7) E E t a ⃒ 1/2 1/2⃒ ± = 0 ±, ⃒T + T− ⃒ ⃒ + ⃒ where t± is the transmission amplitude for right and left where the angle between the large axis of the ellipse and circularly polarized light and can be presented in the form the y axis is: [18]: T ψ+ − ψ− 1/2 iψ± θ = . (8) t± = T± e . (3) 1 2

The coefficient of transmission T± and the phase ψ± For bulk (isotropic) samples or optical devices, where one- are given by the following expressions, respectively: way light propagation is important, ψ± = ωLn±/c (see Eq. 5 when Zrel → 1). Then, for Zrel ≈ 1 the angle θT reads: [︂ (︂ )︂2 ]︂−1 ± ± 1 1 rel 1 2 √ (︂ )︂ T± = 1 + Z± − sin (ωLn±/c) , (4) T Lω µ √ √ 4 Zrel θ ≈ ϵ + g − ϵ − g . (9) ± 1 2c (︂ )︂ 1 rel 1 tan ψ± = Z + tan(ωLn±/c). (5) It can be increased the rotation angle of the linear polar- 2 ± Zrel T ± ization of θ1 on a small length scale in some ways: (i) by √︁ Zrel Z± ≡ ϵ1 taking into account the multiple reflections in a finite layer Here ± = Z ϵ±g is the "relative" impedance of 1 or resonant structures that can lead to an enhancement of a planar dielectric slab of thickness L, characterized by √︁ µ the FR angle (for example Fabry-Perot cavities filled with Z± = , surrounded by two semi-infinite media with ϵ±g a magneto-optic material [21]), (ii) tuning the optical prop- √︁ µ Z ϵ µ g positive 1 = ϵ1 . erties of , and by the modification of the structure, 246 Ë J. Lofy et al.

T θ1 size and shape of the material; varying the composition of 12 alloyed and intermetallic nanostructures, (iii) using meta- 10 Bulk Mat. Z=1 materials to tailor the optical properties of the host system Z=0.8 [22] (iv) and changing the dielectric permittivity tensor of 8 ϵ g 4 a medium with time, etc. The time dependence case may 6 concern both the diagonal and non-diagonal permittivity Z=0.3 in units of 4 terms of ϵij (see Ref. [23]). In Ref. [22] the permittivity tensor of a magneto-optical 2 metamaterial is tailored by embedded wire meshes. These 0 wires can only tune the diagonal element of the permit- 0 1 2 3 4 x tivity tensor in terms of topological parameters and mate- Figure 1: θT x ωnL c rial properties. This reduces ϵ to the value of g, creating Faraday Rotation angle 1 as a function of = / for a near zero epsilon (NZE) metamaterial [24]. For such fre- Z = 0.3 and Z = 0.8 (colour online). T quencies the second term of Eq. (9) becomes zero and θ1 in the magneto-optical metamaterial can be enhanced by al- Hence, most an order of magnitude [22]. As well, when Faraday ro- ψ ψ gn ∂ψ gZ ∂ψ tation has a time dependent dielectric permittivity tensor, θT + − − 1 1 1 = = |ϵ| ∂n − |ϵ| ∂Z (12) g g Ωt Ω 2 2 2 where = o cos( ), (being the angular frequency of (︂ )︂ 1 g ∂ψ ∂ψ the gyrotropic vector), it can be shown that the time depen- = n − Z . 2 |ϵ| ∂n ∂Z dent Faraday rotational angle contains an extra term (pro- ω portional to time t and to the frequency ratio Ω ) which in- ∂ψ ∂ψ ⃗ Evaluating the derivatives ∂n and ∂Z at B = 0 from Eq. 5, creases faster than the stationary term and becomes dom- substituting these expressions into Eq. 12 and introducing inant in short time spans, provided that ωt > 1 [23]). the new parameter x = ωnL/c, we get

2 2 T g x(Z + 1) + (1 − Z ) sin x cos x θ = . (13) 2.1 Real part of FR in RHM/LHM: 1 4|ϵ|Z (︂ )︂2 1 Z 1 2 x Transmission 1 + 4 − Z sin

Eq. (13) is a general expression and is valid for any con- Let us consider the FR for transmission from a slab. Since tinuous material with arbitrary parameters L, n and Z. As the Faraday effect is typically very small the effective in- expected, θT is odd in n and there is a change of the sign dices of refraction and impedance for the two circular po- 1 of n in LHM. Below we analyze a few of the limits for these larizations of the first order in g can be presented inthe parameters. form √ 1 gn When L tends to zero (kL ≪ 1) , the above equation n± = ϵ±µ ≈ n ± , 2 |ϵ| reduces to g gωL ϵ θT x ≡ √︀ 1 gZ 1 ≈ ϵZ c |ϵ| , (14) Z± = µ/ϵ± ≈ Z ∓ , 2 2 2 |ϵ| which coincides with the thin-film result of Ref. [25] for n where (refractive index of a homogeneous material) and RHM (ϵ > 0). Z (impedance of a homogeneous material) are calculated If Z = 1, i.e. when light propagates in an homogenous ⃗g n → when gyration vector is zero. Note that by replacing medium, we get −n we can use the above expressions for LHM. θT θT T g gωL √ One can simplify the analysis of 1 and 2 by expand- θ = x ≡ µϵ, (15) 1 2|ϵ| 2c|ϵ| ing ψ± around n and Z of the slab in the absence of the ⃗B T1/2 ψ magnetic field . Then the Taylor series of ± and ± in which coincides with the result of Refs. [18, 25] in the thick n Z the neighborhood of and becomes: film limit where kL ≫ 1, for µ = 1 in RHM and for the range Z2+1 of all optical frequencies. At the points x = 2 tan x , 1 gn ∂T1/2 1 gZ ∂T1/2 0 Z −1 0 T1/2 = T1/2(n, Z) ± ∓ , (10) θT ± 2 ϵ ∂n 2 ϵ ∂Z where the ellipticity is zero ( 2 = 0), as was mentioned previously, we get for the real part of FR: 1 gn ∂ψ 1 gZ ∂ψ ψ± = ψ(n, Z) ± ∓ . (11) T gZ 2 |ϵ| ∂n 2 |ϵ| ∂Z θ1 = x0 (16) |ϵ|(Z2 + 1) Faraday and Kerr Effects in Right and Left-Handed Films and Layered Materials Ë 247

In Figure 1 we show the FR angle of transmission vs x = is very different: The transmission coefficient is not 1and 2 2 2 ωnL/c for RHM, using Eq. (13) for three different values of instead T approaches 4Z /(Z + 1) as x0 tends ∞. This surface impedance: Z = 0.3, Z = 0.8, and bulk material saturated value corresponds exactly to one-quarter wave- with no reflections where Z = 1 (dashed line). The angle length. T ⃗ steadily increases and oscillates around the line θ1 = 2x Note that in the limit of a small magnetic field B the g π b θT (in units 4|ϵ| with certain periodicity of or on the scale expression for a , Eq. (7), coincides with 2 defiend by −1 T L ∼ k ). The oscillations in θ1 are due to interference ef- Eq. (17). fects in the plane-parallel slab and the amplitude of the os- Figure 2 shows the imaginary angle of the FR in Eq. (17) cillating part depends on x. At xl = π(l + 1/2) (l = 1, 2 ... ) for a RHM (n > 0) versus x, for materials with Z = 0.3 T θT gZ xl x πl (solid) and Z = 0.8 (dashed). θ in the interval [0, π] in- we have for FR angle 1 = |ϵ| Z2+1 , and for l = we 2 T g Z2+1 creases with x, reaches a peak value and then drops to have θ1 = |ϵ| Z xl. We were not able to find an analyt- 4 a minimum. This pattern repeats as x increases. Let us fi- ically closed-form solution for the maximum of θT, and 1 nally remark that from Figure 1 and Figure 2 it follows that Eq. (13) could not calculate the maximum increase of FR the maximums of the real portions of the Faraday effect do angle. However, for the estimated increase we used points T not coincide with the simultaneously zero imaginary por- xl = πl, because the maximum value of θ for each period 1 tions. of oscillation is located very close to that points (see Fig. 1 T θ T where the vertical grid line appears). Then the ratio of 1 θ2 T at xl = πl to the θ1 in homogeneous media, Eq. (15), reads (Z2 + 1)/2Z ≥ 1. For materials with relative impedance 10 ∼ 0.3, such as semiconductors with zero extinction coef- T for Z=.8 5 ϵ

ficients in the near or mid infrared range (like tellurium g T 8 θ for Z=0.3 or aluminum gallium arsenide), the ratio can be almost 2 T for Z=.3

2. That is, an impedance z < 1 causes multiple reflec- in units of 0 T tions that increase the overall time the light spends within θ2 for Z=0.8 the system, showing an increase in Faraday rotation [20]. A similar increase of Faraday rotation was also found in -5 0 1 2 3 4 [22, 26]. However, for a composite system (dielectric with x metamaterials or super lattice systems) the effective ϵ can Figure 2: θT x ωnL c be reduced up to 10−2 and the ratio can thus be increased Faraday Rotation angle 2 as a function of = / for Z=0.3 and Z=0.8. Transmission is multiplied by 5 to highlight the re- by an order of magnitude or greater. lationship between the Faraday ellipticity angle and corresponding transmission value, where T = 1 is the norm (colour online). 2.2 Imaginary part of FR in RHM/LHM: Transmission

T1/2 n Z Expanding ± around and of the slab in the absence 3 Real and Imaginary parts of KR in of the magnetic field ⃗B (see Eq. (10) by using the Taylor series for ln(1 + x) centered at 0, we can similarly derive RHM/LHM: Reflection 1/2 T 1 T+ the expression for θ2 = ln 1/2 for the imaginary portion 2 T− of Faraday rotation as: When linearly polarized light is reflected from the surface of a magnetized material, the direction of polarization T g (1−Z2) sin x (Z2+1) sin x+x(1−Z2) cos x changes and the light can be elliptically polarized. This is θ [ ] (17) 2 = 8|ϵ|Z2 1 Z 1 2 2 x . 1+ 4 ( − Z ) sin the Kerr effect, very similar to the Faraday effect with the exception that the Kerr effect refers to the reflection of light This is again a general expression valid for the arbi- T and the Faraday effect refers to its transmission. trary parameters L, n and Z. As expected, θ2 is even in n, T Before analizing the complex Kerr effect in more detail, and θ2 → 0 when L tends to zero. As it was previously men- T let us note that if we ignore the losses there are some use- tioned, θ2 becomes 0 at Z = 1 (no boundaries), at x = πl T R Z2+1 ful results which relate the θ and θ which follow from (complete transmission) and at x0 = 2 tan x0. In the two Z −1 the general expressions of the scattering matrix elements former cases the coefficient of transmission T becomes 1 in terms of the transmission and reflection probabilities when an external magnetic field ⃗B is zero. The third case 248 Ë J. Lofy et al.

R θ2 and the scattering phases ψ and ψ ± ψa. Here ψ is the to- 80 tal phase accumulated in a transmission event and ψ ± ψa 60 are the phases accumulated by a particle which is incident from either face of the material when it is reflected. The 40 ϵ g scattering-matrix elements can be written in the form: 8 20 Z=0.8

(︃ )︃ (︃ √ √ )︃ Z=0.3 i(ψ+ψa) i(ψ) r t −i Re Te in units of 0 S = ′ = √ √ t r Tei(ψ) −i Rei(ψ−ψa) -20 . For a spatially symmetric barrier the phase asymmetry -40 ′ 0 1 2 3 4 ψa vanishes and r = r . It is clear that for any symmetric x structure with no material losses, including the slab we are Figure 3: θR x ωnL c discussing, Kerr Rotation angle 2 as a function of = / for R T Z = 0.3 and Z = 0.8 (colour online). θ1 = θ1 .

Whereas t± describes the transmission amplitude of the wave, we now describe r± as the reflection amplitude. 4 Alternating right and left-handed It can be shown for the slab that the reflection amplitude materials: layered structures is given by [18]:

t± 1 r± = −i (Z± − ) sin(n±ωL/c), (18) In this section we shall analyze in detail the real and imag- 2 Z ± inary portions of the complex Faraday rotation angle at where t± was defined in Eq. (3). Using an expression similar the different propagation regimes through a finite stack of to that of FR to describe the Kerr effect complex rotation alternating right and left-handed materials (n1 > 0 and angle we find that, n2 < 0, respectively). In such systems, the total Faraday R r+ R R rotation is defined as the rotation of the resultant super- θ = −i ln = θ1 + iθ2 , (19) r− position of the transmitted electric fields E± that experi- and, ence multiple reflections, as discussed previously for a sin- (︂ 2 )︂ gle film. The mentioned multiple reflections along differ- R T g (nkL) cos(nkL) Z + 1 θ = θ − + , (20) ent trajectories, corresponding to different traversal times, 2 2 2ϵ sin(nkL) 1 − Z2 can lead to evanescent modes of so-called "superluminal T where θ2 is defined by Eq. (17). velocities" on microwave transmission through an under- When L tends to zero (the thin film approximation), sized waveguide [27, 28] or periodic dielectric heterostru- the above expression reduces to cures [29, 30]. It is widely believed that the evanescent R ϵ g modes take almost zero time to cross the forbidden region θ ≈ , 2 |ϵ| ϵ − µ and for an opaque barrier, where there is a strong expo- where epsilon has a sign change from RHM to LHM. nential decay of the wave function, the tunneling time be- T comes independent of the barrier’s length (known as the As it can be seen in the above expression, θ2 is proportional to the extremely small parameter g and in RHM, Hartman effect [31]). However, a very fast tunneling, ora where µ ≪ ϵ, it is too difficult to measure θR. However, the zero tunneling time holds a serious consequence: the tun- situation is very different for LHM, where µ and ϵ can be of neling velocity or the average velocity may become higher c the same order of magnitude for some frequency ranges (as than the velocity of light . Indeed, following the standard t dφ for NZE metamaterials). For these frequencies it can be ver- definition of the phase time, = ~ dE (where time elapses ified experimentally that a narrow resonantly enhanced re- between the peak of the wavepacket entering the barrier flection angle can be found for the Kerr effect. and leaving it), it is easy to be convinced that for evanes- Figure 3 shows the imaginary angle of the KR, Eq. (20), cent modes one will get a very fast tunneling time. for two different RHM of different surface impedance Z ver- Without pretending to give an exhaustive review on R the theory of the traversal time problem of electromagnetic sus x . θ2 at xl = πl shows a discontinuity. We also note T R waves, we just mention that two characteristic times have that zeroes for both θ2 and θ2 coincide and are the solu- x Z2+1 x arisen in many approaches (see for example, Refs. [32, 33] tions to the transcendental equation 0 = Z2−1 tan 0. At these points there is linearly polarized light for both the and references therein). These times are related as a conse- reflected and transmitted light. quence of the analytical properties of the complex quantity Faraday and Kerr Effects in Right and Left-Handed Films and Layered Materials Ë 249

whose real and imaginary components are the two charac- n2 = −2.12 and length L2 = 39.38 nm) materials. The teristic times. However, as we know, any experiment must wave-numbers in the layers of both types are ki = ωni /c, measure a real quantity, and so the outcome of any mea- where ω is the frequency and c the vacuum speed of light. surement of the interaction time must be a real quantity, Due to the high contrast between the two dielectrics n1 possibly involving the two characteristic times. It depends and |n2|, 10 primitive cells are already enough to formu- on the experimente which of the two components of this late allowed and forbidden band structures. In Figure 4a interaction time is the most relevant. we present the transmission coefficient T as a function of We can see in the experiment performed in Ref. [34] the frequency ω for single, five, ten and fifty primitive cells. an example of a two components interaction time. There it was experimentally investigated the tunneling times as- sociated with frustrated total internal reflection of light. It was shown that the two characteristic times correspond, respectively, to the spatial and angular shifts of the beam. As a second example, we can mention the magnetic clock approach [20] (Faraday rotation), where the time that characterizes the interaction of a classical electromagnetic wave with a barrier must always be described by two- components of a complex time: τ = τ1 − iτ2. The real portion of complex Faraday angle θ is proportional to the traversal time τ1 of light through the magnetic materials or dielectric structures in an external weak magnetic field [20]. At the same time the imaginary component of complex θ is proportional to the degree of ellipticity, τ2. Remarkably, the two times τ1 and τ2 are related via Figure 4: (a)The transmission coeflcient and (b) real and imaginary Kramers-Kronig integral dispersion relation [35] that pre- parts of the Faraday Rotation angle θ as a function of ω for an alter- vents a violation of the principle of causality. Indeed, the nating structure described in the text. The parameters are: number validity of the Kramers-Kronig relations for the complex of primitive cells M = 50, n1 = 1.58, n2 = −2.12, L1 = 52.92 nm, L = 39.38 nm and n L = |n |L (colour online). interaction time has a rather deep significance because it 2 1 1 2 2 may be demonstrated that these conditions are a direct result of the causal nature of physical systems by which the The transmission coefficient was calculated by using response to a stimulus never precedes the stimulus [35]. the characteristic determinant method [36, 37], that allows Based on this remark, note that the experiments with, e.g., one to express the transmission coefficient of a wave prop- undersized waveguides [27, 28] or periodic dielectric het- agating in a one-dimensional structure through the deter- erostrucures [29, 30], where the so called "superluminal" minant T = |D|−2. The latter depends on the amplitudes of have been observed for the barrier tunneling time need to reflection of a single scatterer only. This characteristic de- be interpreted very carefully. terminant approach is compatible with the transfer matrix To shed some light on the obstacles which have per- method and has been widely used to calculate the average sisted in the tunneling time problem, we analyze the real density of states over a sample, the energy spectrum of el- and imaginary portions of the complex Faraday rotation ementary excitations, or the characteristic barrier tunnel- angle in the forbidden bands of a finite stack of alternating ing time, among others (Ref. [38]). right and left-handed materials. This type of alternating Having a close look into forbidden and allowed band structure, assuming that the optical paths of two slabs are regions in Figure 4a, one observes that practically the equal to each other (n1L1 = |n2|L2), exhibits a broad for- entire transmission spectrum, starting from M = 10, is bidden bands spectrum (see Figure 4a below). This serves formed by forbidden gaps. Further increase of cells will be as a good candidate and a basis for a qualitative under- narrowing the allowed bands, and in the limit of an ideal standing of the peculiarities of complex Faraday rotation, infinite crystal one gets a set of periodically distributed as well as the two-components of the interaction time in a Lorentzian resonances. The centers of the m − th allowed barrier. band, ωc can be easily determined via the dispersion rela- For our numerical simulations we have chosen a finite tion obtained from the Bloch-Floquet theorem for an ideal stack of alternating right (refractive index n1 = 1.58 and length L1 = 52.92 nm) and left-handed (refractive index 250 Ë J. Lofy et al. infinite crystal (see, for example, Ref. [15]) We found that the rotation and ellipticity of the transmit-

cπm 16 ted or reflected light of the Faraday and Kerr effects areodd ωc = ≡ m1.13 × 10 rad/s n n1L1 and even functions with respect to the refractive index . These odd and even functions are not just the properties In Figure 4b we present the real and imaginary por- of a thin film, but apply just as well to the case of any sys- tions of Faraday complex angle θ for M = 50 primitive tem of arbitrary length. cells. The maximums of Re(θ) are centered in the allowed 16 For a spatially symmetric film with no material losses bands and coincide with ωc ≡ m1.13×10 rad/s. At these the real portion of Faraday and Kerr rotations are equal for resonance peaks T− = T+ and we deal with the pure Fara- RHM and LHM. However, the imaginary portion of Kerr ro- day rotation of linearly polarized wave, i.e. with no wave tation for LHM has a peculiar behaviour when the symmet- ellipticity. Within any forbidden bands, Re(θ) is an almost ric film’s length tends to zero. In the limit of an ultrathin flat function with a very small value. This is what weex- LHM film, where ϵ and µ can be of the same order of mag- pect according to the above discussion on "superluminal nitude for some frequency ranges, it was experimentally velocities" and taking into account the existing deep con- detected a large resonant enhancement of the reflected KR nection between the Re(θ) and the tunneling time τ . How- 1 angle. It has been shown that the maximums of the real ever, the situation is completely different for Im(θ) (propor- portions of the Faraday effect do not coincide with the si- tional to the degree of ellipticity) or for τ . First, the lat- 2 multaneously zero imaginary portions, in contrast to the ter is zero at any allowed band at ωc where the condition case of bulk materials. T− = T+ is satisfied, and jumps from zero to some positive To shed some light on the obstacles which have per- value at the end of the allowed band. Second, in the for- sisted in the tunneling time problem, we have analyzed the bidden gap Im(θ) starts to decrease monotonically with in- real and imaginary portions of the complex Faraday rota- creasing frequency ω changing the sign from positive to tion angle in the forbidden bands of a finite stack of alter- negative, and sharply becomes zero in the next allowed nating right and left-handed materials. It has been shown band. In forbidden gaps, at the frequencies where Im(θ) that in spite of the fact that Re(θ) in the forbidden gap is al- becomes zero, T− is exactly equal to T+. At these frequen- most zero, the Im(θ) changes drastically in both value and cies the electromagnetic waves remain linearly polarized. sign. At the rest of frequencies in the forbidden gap the wave is elliptically polarized and its axis is rotated either in the right or left direction. The dependence of Im(θ) on ω is similar to a sawtooth function with increasing amplitude. 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