Explicit Fresnel formulae for the absorbing double-negative metamaterials
EExplicit Fresnel formulae for the absorbing double-negative metamaterials
Igor Tralle
College of Natural Sciences, Institute of Physics, University of Rzeszów, Pigonia 1, 35-310 Rzeszów, Poland
Levan Chotorlishvili
Institut für Physik, Martin-Luther Universität Halle-Wittenberg, D-06120 Halle/Saale, Germany
Paweł Zi¸eba
Energy Business Intelligence Systems (energyBIS), Piłsudskiego 32, 35-001 Rzeszów, Poland
Abstract
We inspect the optical properties of dissipative double-negative metamaterials (DNMM) and find explicit expression forthe total reflection angle and the correct Fresnel formulae describing the reflection and refraction for the DNMM at theoblique incident of the electromagnetic wave on the interface for TE as well as TM electromagnetic wave polarization.The reflectivity and transmissivity of DNMM film embedded in a positive refraction index (PIM) surrounding arepresented and discussed. Keywords: absorptive metamaterial, ingomogeneous EM-wave, Fresnel formulae
1. Introduction
In recent years, we have been witnessed of the explosionof interest in a field of research, which is termed metama-terials. This area of research is characterized by an expo-nential growth of a number of publications, to mention justa few, there are two monographs [1, 2] and the referencestherein. According to [1], the term “metamaterials” can beused in a more general, as well as in a more specific sense.In the more general sense, these are materials possessing“properties unlike any naturally occurring substance” orsimply “not observed in nature.” More specifically, theseare the materials with a negative refractive index, whoseexistence and properties were discussed for the first timeby Veselago [3].It is worth mentioning that most of the proposed eversince designs of metamaterials were characterized by everincreasing sophistication of fabrication methods. Contraryto these, in our previous publications [4, 5], we proposeda relatively simple way to fabricate a three-component ar-tificial composite metamaterial and demonstrated by nu-merical simulations, what are the domains of its existence.It means, that we set seven independent parameters suchas temperature, external magnetic field, relative concen-tration of ingredients and some others to be controlled atthe numerical simulations. In other words, we have seven-dimensional parameter space to search through, in order to
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Corresponding author: I. Tralle,email:[email protected] (Paweł Zi¸eba) establish the frequency domain where this composite mate-rial becomes metamaterial with negative refractive index.For the readers’ convenience, here we outline brieflythe main ideas. Let us assume we have a mixture of threematerials, and each has granular or powder form, suchthat the grain sizes are much smaller than the electromag-netic wavelength propagating in the medium. We wish tomatch the properties of ingredients in such a way that theeffective dielectric permittivity of the composite would bedetermined by the three components, while its magneticpermeability only by one of them, responsible for the mag-netic properties of the mixture. This third ingredient bythe assumption should determine the effective permeabil-ity of the hypothetical material. Suppose it to be metalmagnetic nano-particles (or grains; we shall use these twowords interchangeably). We treat these metallic grains asimmersed or dispersed in a weakly conducting matrix. Ifthe metallic particles are supposed to be single-domain,then we can take into account only the orientation align-ment of their intrinsic magnetic moments and do not needto take into account their induced magnetic moments, asit can be proved (see [6], Chap. 82). The sizes of thesingle-domain particles depend on the material and con-tributions from different anisotropy energy terms. If weassume nano-particle shape to be spherical, then typicalvalues for the critical radius a are about 15 nm for Fe and35 nm for Co, for g – Fe O it is about 30 nm, while forSmCo , it is as large as 750 nm [7]. Now we can treat thesuspension of metallic grains as a kind of "frozen paramag-netic macromolecules," where the metallic nano-particles Preprint submitted to Elsevier August 21, 2020 a r X i v : . [ phy s i c s . op ti c s ] A ug lay the role of "macromolecules."The magnetic moments of these single-domain nano-particles at room temperature are randomly distributedand we can describe their behavior in the framework ofLangevin theory of paramagnetism. Note that the ’swarm’of magnetic nanoparticles immersed into another mediumwas already considered in scientific literature and eventhe term for describing this situation was already coined,namely, superparamagnetism. The point is that such sys-tem behaves like a paramagnet, with one notable excep-tion that the independent moments are not that of a singleatom, but rather of a single-domain ferromagnetic particle,which may contain more than 10 atoms. In the absenceof an external magnetic field their magnetic moments aredistributed at random, but being placed in magnetic field,magnetic moments of individual grains treated in termsof classical physics start to precess, that is why the fre-quency range in which Re [ µ eff( ω )] is negative, appears inthe vicinity of resonance ω ≈ ω , where ω is the frequencyof the electromagnetic wave incident of the medium and ω = γH . Here γ - is the gyromagnetic ratio and H -external magnetic field.It turns out, for the composite to become a metamate-rial it is important that the sizes of ferromagnetic nanopar-ticles, their magnetic moments, the relative concentrationof the ingredients, Cd- or Sn-content in the semiconduc-tor compounds, the temperature, and the external mag-netic field have to have certain definite values. In ourprevious works it was shown that the mixture composedof a ’swarm’ of single-domain ferromagnetic nanoparticles,small metallic particles (Ag, or Al or Cu) and the smallsemiconductor particles of Hg − x Cd x Te, or Pb − x Sn x Te)attains double-negative metamaterial properties in the fre-quency range 10 −
100 GHz being placed in an exter-nal magnetic field. The reasoning behind this choice ofHg − x Cd x Te, or Pb − x Sn x Te is the following. The elec-trical properties of these materials crucially depend oncadmium (the same is about Sn, but for definiteness letspeak of Hg − x Cd x Te and concentration of cadmium x . If x = 0, that is in case of HgTe, the material is semimetalwith energy gap E g <
0, while in case of x = 1 (CdTe)material becomes semiconductor with wide energy gap ofabout 1.5 eV at 300 K. Thus, changing the concentrationof cadmium, one can change the energy gap, and hence theconcentration of free electrons. In terms of our model, itmeans that one can pass smoothly and continuously fromLorentz model for dielectric permittivity, where the elec-trons are almost tightly bounded to Drude model, wherethey are almost free to move. As a result, cadmium con-centration becomes an important parameter of the model;by means of it— among others— one can control the fre-quency range where the real part of dielectric permittivitycan be made negative and force it to overlap with the fre-quency domain, where magnetic permeability is negative.Here we use the term ’double-negative’ to emphasizethe negativity of the real parts of permittivity and perme-ability,that implies the composite refractive index Re[˜ n ] is negative [3]. The negativity of the real part of refractionindex entails that the permittivity and the permeabilityare complex-valued functions [8, 9], meaning the refrac-tion index is also complex-valued.Metamaterials may be engineered to exhibit a negativerefraction [10, 11, 12, 2, 1] but they tend to be absorptiveand narrow-band for the fundamental reasons [14, 15], al-beit the imaginary part can be relatively small as long asit is allowed by the causality condition.Despite of this, many authors who explore metamate-rials often treat them as non-absorbing (see, for instance,[3, 17, 18]). To have non-absorbing metamaterials is verydesirable since they promise numerous possibly very inter-esting and exciting applications [16]. It is however logicallyinconsistent to treat them as non-absorbing for the verysimple reason: as it was mentioned above, one can assignminus sign to the refraction index only if permittivity andpermeability are complex-valued functions. In fact, theterm negative refraction index is used for short and oneshould keep in mind that it in fact, negative real part ofrefraction index is talked about, and that its imaginarypart is positive and metamaterial is always absorbing.In the classical optics of non-absorbing media one of-ten deals with plane periodic electromagnetic waves whoseplanes of constant phase and amplitude are normal to thewave vector. Such waves were designated by Voight ashomogeneous waves. In absorbing media another type ofwaves appears for which the planes of constant phase andamplitude are no longer parallel; Voight designated themas inhomogeneous waves. It is interesting to note thata relatively small number of papers were devoted to thetreatment the reflection and refraction of electromagneticwave at oblique incidence on the interface between non-absorbing and absorbing media, among them the papersby [19, 20]. It is also worth mentioning the paper devotedto oblique surface waves at an interface [21].There are many papers and textbooks devoted to thepropagation of electromagnetic waves in metamaterails (seefor example, [8], and the books mentioned above[12, 2]),but there is very little information if ever, concerning thereflection and refraction of electromagnetic wave at obliqueincidence on the interface between positive refraction in-dex material (PIM) and metamaterial. So, the main goalof present work is to derive the explicit Fresnel formulaefor the gyrotropic, magnetic, birefringent and absorbingmetamaterials and the study of optical properties of suchmaterial at the oblique incidence of electromagnetic waveon the interface between them and the positive refractionindex material.At the end of Introduction we would like to add somecomments concerning terminology we use throughout thepaper. Till now there is no unanimity as for this subjectis concerned: some authors use the term ’negative groupvelocity materials’, some others prefer the term ’negativephase velocity materials’. This is because the phase veloc-ity and group velocity are directed against each other incase of such materials, and which direction is positive and2hich is negative is a matter of convention. For that reasonwe use the term double-negative metamaterial (DNMM)throughout, in order to emphasize that in this case the realparts of dielectric permittivity as well as magnetic perme-ability are simultaneously negative in some frequency do-main. This paper is organized as follows. In section II wediscuss the magnetic birefringence and Faraday effect inDNMM, in section III we treat the wave propagation inDNMM and in section IV we derive Fresnel formulae for itwhile in V we consider the reflectivity and transmittanceof the DNMM-films.
2. Magnetic birefringence and Faraday effect inDNMM
Metamaterial proposed in [4, 5] is interesting from sev-eral points of view. First, it is not a complicated engi-neering construction, but the mixture of three ingredi-ents. Second, despite the fact that it is not a crystal, it isanisotropic, optically bi-axial medium displaying Faradayeffect. Third, its effective dielectric permittivity (we callit effective, because it is the permittivity of mixture) canbe considered as a complex scalar, while effective magneticpermeability relates to the permeability of the third ingre-dient of the mixture, that is the ’swarm’ of ferromagneticnano-particles via the tensor represented by 3-by-3 non-Hermitian matrix. As it was shown in [4], this matrix isof the form:˜ µ = πχ π ( iG ) 0 − π ( iG ) 1 + 4 πχ
00 0 1 , where χ = χ ω i Γ (cid:18) ω − ω − ω + ω (cid:19) ,G = χ γω i Γ (cid:18) ω − ω − ω + ω (cid:19) H . Here Γ = τ − , ˜ ω = − i Γ + p ω − , ˜ ω = − i Γ − p ω − and ω = γH , where τ is the magnetic mo-ment relaxation time (see [4] for details) and H is externalmagnetic field.If the wave vector of incident electromagnetic wavealigned arbitrary with respect to external magnetic field,effective permeability is a tensor and the medium is ani-sotropic. However, if we consider the simplest case when k k H , then two circularly polarized waves can propa-gate in such medium for which the magnetic permeabil-ity and hence, the refraction indices are different (one forleft-polarized wave, while the other for the right-polarizedone). If one introduces the following auxiliary quantities χ ± = χ ± G , then the refraction indices for these twowaves are ˜ n ± = p ˜ (cid:15) eff ˜ µ eff , ± , where ˜ µ eff , ± and ˜ (cid:15) eff standfor the effective magnetic permeability and effective di-electric permittivity of metamaterial, respectively. The effective dielectric permittivity and magnetic permeabilitywere calculated in the framework of Bruggeman approxi-mation often called the effective medium theory [22]. Itsmain asset is that all ingredients of a mixture by assump-tion are treated on the same footing in a symmetric wayand none of them plays a privileged role. For example,the effective dielectric permittivity ˜ (cid:15) eff is calculated as theroot of the following third-order algebraic equation: f (cid:15) ( ω ) − ˜ (cid:15) eff ( ω ) (cid:15) ( ω ) + 2˜ (cid:15) eff ( ω ) + (1)+ f (cid:15) ( ω ) − ˜ (cid:15) eff ( ω ) (cid:15) ( ω ) + 2˜ (cid:15) eff ( ω ) + f (cid:15) ( ω ) − ˜ (cid:15) eff ( ω ) (cid:15) ( ω ) + 2˜ (cid:15) eff ( ω ) = 0 , where (cid:15) i , i = 1 , , f i is the volume filling fractionof the i − th material in the mixture. Obviously, thesethree quantities in a natural way obey the following addi-tional condition: f + f + f = 1. We calculated the rootsof the equation numerically, because they depend on theconcentrations of the constituent components of mixture.They were not known beforehand and it was more conve-nient from computational point of view to solve this equa-tion numerically. Since we consider absorbing medium,we always chosen the root that had positive imaginarypart. Since the magnetic permeabilities of two compo-nents other than ’swarm’ of ferromagnetic nanoparticles,are equal to 1 in wide frequency range, the expression foreffective magnetic permeability of the mixture takes moresimple form, namely µ eff , ± = f + (1 − f (1 + 4 πχ ± ( ω )),where f = f + f .It is interesting and worth noting that in this case realpart of refraction index is negative only for one of twowaves propagating in a medium. The case of arbitrary k -vector alignment with respect to H , is more compli-cated and will be considered elsewhere. here we simplystate that propagation of the electromagnetic wave pro-ceeds with two different phase velocities v + = c/ | Re[˜ n + ] | and v − = c/ | Re[˜ n − ] | , where c is light velocity in vacuum.Assuming Re[˜ n + ] > Re[˜ n − ] we conclude that it takes moretime for “slower” wave to traverse the plate made of ourmaterial. The time delay between the two waves whiletraversing the plate or slab made of such material, is equalto ( d is the slab thickness)∆ t = d (cid:18) v + − v − (cid:19) = dc ( | Re[˜ n + ] | − | Re[˜ n − ] | ) . (2)For both waves the total revolution of E or H vectorlasts for the wave period T = 2 π/ω , meaning that the re-tarded wave (assuming | Re[˜ n + ] | > | Re[˜ n − ] | ) arrives at theopposite surface of the slab end with the E − vector re-volved at a larger angle than the other E + . The differencein rotation angle is∆ α = 2 π ∆ tT = 2 πdλ ( | Re[˜ n + ] | − | Re[˜ n − ] | ) , (3)3here λ = cT . Note that the rotation of the polariza-tion plane in this case is linearly proportional to the mag-netic field parallel to the direction of wave propagationand hence, it is nothing else but the Faraday effect. Hav-ing in mind that ˜ n ± = p ˜ (cid:15) eff ˜ µ eff , ± and ˜ (cid:15) eff = (cid:15) eff + i(cid:15) eff ,˜ µ eff , ± = µ eff , ± + iµ eff , ± , where ˜ (cid:15) eff , ˜ µ eff , ± stand for the ef-fective permittivity and permeability of the material weare talking about, we drop henceforth the subscript eff andfind the expressions˜ n ± = q | ˜ n ± | exp( iφ ˜ n ) , (4) q | ˜ n ± | = (cid:20)(cid:16) (cid:15) µ ± (cid:17) + (cid:16) (cid:15) µ ± (cid:17) (cid:21) / , (5)and φ ˜ n = 12 arctan µ ± (cid:15) + (cid:15) µ ± µ ± (cid:15) − (cid:15) µ ± ! ∈ ( π/ , π ) , (6)from which the explicit expressions for Re[˜ n + ] and Re[˜ n − ]follow. Since the functions χ ( ω ) , G ( ω ) are complex-valued,the absorption of left-polarized and right-polarized wavesare a bit different; this different absorption of the right andleft circularly polarized light is known as magnetic-ciculardichroism. As a result, the initially linearly polarizedwave which is the superposition of left- and right-polarizedwaves acquires during its propagation within such mediumsome ellipticity that is, becomes elliptically polarized.
3. Total reflection angle in case of absorbing DNMM
Now we consider the process of wave propagation insuch material starting from Maxwell equations, in orderto study the reflection and refraction on the boundarybetween two media, the first one is of positive refractionindex (PIM) and the other one is double negative meta-material (DNMM). It should be noted that the opticalproperties of absorbing materials were considered alreadyby many authors, for example by Born and Wolf in theirclassical book [23] and using somewhat different approach,by M.A. Dupertuis, M. Proctor and B. Acklin [20], as wellas quite recently by P.C.Y. Chang, J.G. Walker and K.I.Hopcraft [24]. All these authors (Born and Wolf includ-ing) considered however absorbing, PIM and nonmagneticmaterials and hence, they assumed µ = 1. On the otherhand, the authors who considered DNMM, treated themas non-absorbing (for instance [17]).For studying the wave propagation, reflection and re-fraction at the boundaries between two media, one of pos-itive refraction index (PIM) and another one, double neg-ative metamaterial (DNMM) one has to inspect the cor-responding solutions to the the Maxwell equations. As wementioned in the previous section, in case of consideredmetamaterial when the wave vector of incident electro-magnetic wave aligned arbitrary with respect to external magnetic field, permeability is a tensor and the mediumis anisotropic. However, if we consider the simplest casewhen k k H , one can consider medium as if it wouldbe isotropic with two different values of refractive indexfor two waves. Keeping this in mind, for a homogeneous,absorbing, isotropic, linear, charge-free magnetic mediumone can write down the Maxwell equations as: ∇ × E ( r, t ) = − ˜ µ∂ t H ( r , t ) , ∇ · E ( r , t ) = 0 , (7) ∇ × H ( r, t ) = (cid:15)∂ t E ( r , t ) + σ E ( r , t ) , ∇ · H ( r , t ) = 0 . Here ˜ µ is the complex permeability, (cid:15) is the real permittiv-ity and σ stands for the conductivity of the medium. Forabsorbing media such as metals the wave vector is com-plex [23]: k = k e + ik e . Searching for a plane wavesolution E ( r , t ) = E exp( i k · r − iωt ) , H ( r , t ) = H exp( i k · r − iωt ) , (8)one gets: k × E = ˜ µω H , k · E = 0 , k × H = − ˜ (cid:15)ω E , k · H = 0 , (9)where ˜ µ = µ + iµ , ˜ (cid:15) = (cid:15) + iσ/ω = (cid:15) + i(cid:15) and k arenow complex numbers. Remembering all the time thatmaterial which we consider is birefringent, we neverthelessdropped the subscripts ± in what follows in order to makeformulae more readable. At the end of calculations onecan simply choose the corresponding subscript + or − .Complex vectors, like in Eq. (8) sometimes are called bi-vectors. Then, from Eq. (7) one can infer( k · k ) E = − ˜ µ ˜ (cid:15)ω E , (10)and thus k = k − k + 2 i k · k + 2 i cos( e , e ) . (11)The EM -waves with this property are called non-uniform(or inhomogeneous) EM -waves and of course, they werealready considered in the literature [23, 25]. The equations k · r = const and k · r = const determine the planesof equal phases and equal amplitudes, respectively. It isconvenient to introduce in what follows the relative anddimensionless complex permittivity and permeability by˜ µ ˜ (cid:15)ω = (cid:15) µ ˜ (cid:15) r ˜ µ r ω = k ˜ (cid:15) r ˜ µ r , where (cid:15) and µ are the per-mittivity and permeability of vacuum, while the subscript r is for ’relative’. If losses are negligible one can define thewave phase velocity in the medium as v p h = c/n = ω/k and the wave number as k = ( ω/c ) n . For absorbing mediait follows: ˜ k = ( ω/c )˜ n . Since ˜ n = ˜ (cid:15) r ˜ µ r , from these andEq.(11) one deduces k − k = k (cid:20)(cid:16) n (cid:17) − (cid:16) n (cid:17) (cid:21) , k · k = k k cos( e , e ) = k (cid:16) n (cid:17) (cid:16) n (cid:17) , (12)4nd (cid:16) n (cid:17) − (cid:16) n (cid:17) = (cid:16) (cid:15) r µ r − (cid:15) r µ r (cid:17) , (cid:16) n (cid:17) (cid:16) n (cid:17) = (1 / (cid:16) (cid:15) r µ r + (cid:15) r µ r (cid:17) . (13)Equations (13) imply (cid:16) n (cid:17) = (cid:16) (cid:15) r µ r − (cid:15) r µ r (cid:17) q ( (cid:15) r µ r − (cid:15) r µ r ) + ( (cid:15) r µ r + (cid:15) r µ r ) ,n = (cid:16) (cid:15) r µ r + (cid:15) r µ r (cid:17) n . (14)The next interesting issue is the reflection and refractionof EM -wave on the boundary between positive refractionindex material (PIM) and double-negative metamaterial(DNMM). In [18] the Snell’s law was assumed to be validfor the non-absorbing metamaterials. By analogy withnon-absorbing dielectric one can write the law of refractionas followssin θ t = 1˜ n sin θ i , (15)where the subscripts i and t correspond to the incident andthe refracted waves respectively. Due to the complex re-fraction index, θ t is also complex and cannot be interpretedsimply as a refraction angle. In order to use the complexrefraction index, one may resort to ansatz elaborated forthe absorbing materials [23]. However, one should remem-ber that the material we are dealing with is magnetic andcontrary to the case of Born and Wolf [23], permeabilityin our case ˜ µ eff , ± (cid:44) x − z plane. Then thespace-dependent part of the wave phase in our absorbingmaterial is equal to ˜ k r · e t . Here the superscript t standsfor ’transiting’, that is refracted wave, while the subscripts x and z denote the corresponding components of e t , theunit vector in the direction of the transmitted wave. Then, e tx = sin θ t = n − in ( n ) + ( n ) sin θ i ,e tz = p − sin θ t , (16)and we infer e tx = 1 − iδ Re [˜ n ] (1 + δ ) sin θ i ,e tz = s − (1 − δ ) sin θ i (Re [˜ n ]) (1 + δ ) + i n ] sin θ i (Re [˜ n ]) (1 + δ ) . (17) δ = Im [˜ n ] Re [˜ n ] is the "figure of merit". As in [23], we express e tz as e tz = cos θ t = q exp( iγ ),where q cos 2 γ = (cid:0) − δ (cid:1) sin θ i (Re [˜ n ]) (1 + δ ) , (18) q sin 2 γ = 2Im [˜ n ] sin θ i (Re [˜ n ]) (1 + δ ) . (19)From Eqs. (18-19) it follows:˜ k (cid:0) r · e ˜k (cid:1) = ωc [ x sin θ i + z Re [˜ n ] q (cos γ − δ sin γ ) (20)+ izRe [˜ n ] q ( δ cos γ + sin γ )] , (21)and for q and γ q = 2Im [˜ n ] sin θ i ∆ sin 2 γ , (22) γ = 12 arctan (cid:18) n ] sin θ i ∆ − (1 − δ ) sin θ i (cid:19) , (23)∆ = (Re [˜ n ]) (cid:0) δ (cid:1) . (24)The obtained Eqs. (18-24) generalize the classical for-mulae by Born and Wolf for metamaterials. The constantamplitude planes are defined by the relation z = const,meaning they are parallel to the boundaries between thetwo media. The planes of constant real phase are deter-mined by the equation x sin θ i + z Re [˜ n ] q (cos γ − δ sin γ ) = const . (25)These are planes with normals making an angle θ t withthe normal to the boundary planecos θ t = Re [˜ n ] q (cos γ − δ sin γ ) q sin θ i + (Re | [˜ n ]) q (cos γ − δ sin γ ) . (26)Since the amplitude plane is parallel to the boundary, theangle between the vectors e and e (cf. Eq.(11) ) is equalto θ t .As for the possible applications of metamaterials, it isworth mentioning that there are some papers publishedalready, in which the authors proposed to use these mate-rials to construct the waveguides [26, 27, 28]. The authorsof these papers considered the propagation of TE and TMmodes in the waveguide made of DNMM, but for the sakeof simplicity, they consider the MTM to be lossless, withthe permittivity and permeability tensors taking only realvalues, which is not very realistic. As it is known, theoperation principle of waveguides is the EM-wave total in-ternal reflection. That is why, it seems useful to deriveformula for the angle of total internal reflection in case ofabsorbing metamaterial. Usually, the discussion of this is-sue concerns mainly the region of the Goos-Hänchen shift5t the boundary between PIM and magnetic DNMM [29],but not the total internal reflection angle. The cause oftotal reflection in metamaterials is the same as in case ofusual dielectrics, but in case of metamaterials the formuladescribing critical angle is more complicated, as it is shownbelow.Suppose that the EM wave impinges on the bound-ary between PIM and DNMM at the side of DNMM. As-sume also that DNMM is more optically dense, that isRe [˜ n DNMM ] > n P IM . Writing the refraction law in theform n × k i = n × k t where n is the vector normal to aboundary and noting that | k i | = ˜ k DNMM , | k t | = k P IM and k ,DNMM sin α i + ik ,DNMM sin α i = k P IM sin α t , (27)we infer the expression for the critical angle α i,c (subscript c stands for ’critical’):sin α i,c = k P IM k ,DNMM . (28)With Eq.(12)-(14) and Eq.(26) one can calculate the angleof total internal reflection for our case. Using Eq.(12) wededuce k = " b + p b + 4 β r β / , k = k (cid:16) n (cid:17) (cid:16) n (cid:17) k β , (29)where β = cos ( e , e ), b = k (cid:20)(cid:16) n (cid:17) − (cid:16) n (cid:17)(cid:21) β , r = h k (cid:16) n (cid:17) (cid:16) n (cid:17)i and n , n are determined by means ofEq.(14). Keeping in mind that Eq.(27) k ,DNMM is simplyequal to k from Eq.(29) and β = cos ( e , e ) = cos θ t , onecan determine the angle of total internal reflection as: α i,c = arcsin k P IM k ,DNMM . (30)
4. Fresnel formulae in case of absorbing DNMM
Considering the Fresnel formulae for our absorbing DNMM,one may attempt to proceed as Born and Wolf [23] for themetals and simply use a complex-valued µ .Special attention should be given to the boundary con-ditions at interfaces, however [30, 31]. The point is thatin the derivation of Fresnel formulae an important role isplayed by boundary conditions, which are different for theinterface between dielectric media and for the interface be-tween dielectric and conducting (and hence, lossy) media.Namely, for an interface between dielectric media, the tan-gential components of the magnetic vector is continuous,while for dielectric-metallic (or other conducting material)interface the tangential components of the magnetic vectoris discontinuous and the discontinuity is proportional to the current surface density . It is thus important to figureout when this discontinuity can be neglected, so that theBorn and Wolf’s approach to Fresnel formulae of absorbingmaterials can be exploited. Stratton [32] pointed out thatthe discontinuity is relevant for perfect conductors only,otherwise to a good approximation the tangential compo-nents can be regarded as continuous. Hence, we need toestimate the conductivity of our composite. There are sev-eral approaches to describe the effective macroscopic char-acteristics ( conductivity, permittivity, etc.) of compos-ite media such as the Maxwell-Garnett theory (Clausius-Mosotti approximation) [33, 34, 35] and the Bruggemanapproximation (the effective medium theory) already men-tioned above[22]. We employ the last one to calculate theeffective conductivity of the composite medium, because inthis approach all components of the composite are treatedon equal footing. The effective conductivity derives as theroot of the following cubic algebraic equation (cf. [1]): f σ − σ eff σ + 2 σ eff + f σ − σ eff σ + 2 σ eff + f σ − σ eff σ + 2 σ eff = 0 , (31)where σ , σ , σ , f , f , f are the conductivities and therelative concentrations of the ingredients 1,2,3 in the com-posite, respectively, and σ eff is the effective conductivityof the composite. Generally, the equation roots depend onthe relative concentrations of ingredients. For equal con-centrations instead of Eq.(31) we find the following equa-tion:4 σ − ( σ σ + σ σ + σ σ ) σ eff − σ σ σ = 0 . (32)For an estimate, suppose that the first ingredient of ourmixture is Ag, or Cu or Al (cf. [5]). The conductivitiesof these metals depend on the EM wave frequency andtemperature; for room temperatures and our frequencyband one finds the relevant conductivities to be in thesame range, namely σ Ag = 61 . × (Ω · m) − , σ Cu =58 . × (Ω · m) − , and σ Al = 36 . × (Ω · m) − . So,assume σ = σ Cu . For iron dioxide (cf. [4, 5]) and mostof the semiconductors the conductivity is of the order of10 (Ω · m) − . The conductivity of Pb − x Sn x Te (the thirdcomponent of our mixture; see [5]) depends on the Sn-content x ; here we assume it 10 (Ω · m ) − . Then we have: p ≈ − , × , q ≈ − . × , (cid:0) p (cid:1) ≈ − . × , (cid:0) q (cid:1) ≈ . × and hence, Q <
0. As a result, threeroots of Eq.(32) are equal to: σ eff , = 2 r − p (cid:18) φ (cid:19) ,σ eff , , = − r − p (cid:18) φ ± π (cid:19) , (33)where cos φ = − q √ − ( p/ , p = − ( σ σ + σ σ + σ σ ),and q = − σ σ σ Thus, only the first root is positive,while the other two are negative and unphysical. The nu-merical value of first root is about σ ≈ . × (Ω · m ) − ,6eaning σ eff (cid:28) σ . Our composite is therefore a relativelybad conductor allowing to use the continuity of tangentialcomponents of the magnetic vector as the boundary con-ditions.For the derivation of Fresnel formulae for the absorp-tive, magnetic and gyrotropic metamaterial let us considera slab of composite metamaterial sandwiched between twolayers of dielectric PIM (see Fig.1). One can express thereflection and transmission of EM-wave in terms of param-eters called reflectivity R and transmissivity T via the co-efficients r , t and r , t associated with the reflectionand refraction at the first and second interface respectively.Considering at first TE -wave (so called s -polarization) wefind (cf. [23], §1.6, (55)-(56)) r ± = n cos θ − Z − ± ) cos θ n cos θ + Z − ± ) cos θ ,t ± = 2 n cos θ n cos θ + Z − ± ) cos θ . (34) Here n is the refraction coefficient of the first PIM and Z ± ) = q ˜ µ ± eff / ˜ (cid:15) eff is the wave impedance of the meta-material. Dropping as previously the subscript eff , we canrewrite the expressions above in an alternative form as r ± = ˜ µ ± ) n cos θ − ˜ n ± ) cos θ ˜ µ ± ) n cos θ + ˜ n ± ) cos θ ,t ± = 2˜ µ ± ) n cos θ ˜ µ ± ) n cos θ + ˜ n ± ) cos θ . (35)Here ˜ n ± ) , ˜ µ ± ) are the complex refraction index andmagnetic permeability of the metamaterial; the indices +and − refer to two values of them (remember, the ma-terial is birefringent) while the subscripts 1 and 2 refersto the order in which the media are set in this multi-layer ’sandwich’. Following [23], we introduce the notation˜ n ± ) cos θ = u ± ) + iv ± ) , where however, u ± ) and v ± ) have a different form (see below). For the reflectioncoefficient at the first interface we obtain r ( ± )12 = (cid:16) µ ± + iµ ± ) (cid:17) n cos θ − (cid:0) u ± ) + iv ± ) (cid:1)(cid:16) µ ± + iµ ± ) (cid:17) n cos θ + (cid:0) u ± ) + iv ± ) (cid:1) = ρ ( ± )12 exp (cid:16) iϕ ( ± ) (cid:17) ,ρ ( ± )12 = | r ( ± )12 | . (36)After cumbersome but straightforward calculations we arrive at (cid:16) ρ ( ± )12 (cid:17) = h ( µ ± ) ) + ( µ ± ) ) i n cos θ − (cid:16) u ± ) + v ± ) (cid:17) ( µ ± ) n cos θ + u ± ) ) + ( v ± ) + µ ± ) ) + 4 n cos θ ( µ ± ) v ± ) − µ ± ) u ± ) ) h ( µ ± ) n cos θ + u ± ) ) + ( v ± ) + µ ± ) ) i , (37)tan ϕ ( ± )12 = 2 n cos θ (cid:16) µ v ± ) − µ ± ) u ± ) (cid:17)(cid:20)(cid:16) µ ± ) (cid:17) + (cid:16) µ ± ) (cid:17) (cid:21) n cos θ − (cid:16) u ± ) + v ± ) (cid:17) , (38)2 u ± ) = (cid:0) Re[˜ n ± ) ] (cid:1) (cid:0) − δ ± (cid:1) − n sin θ + rh(cid:0) Re[˜ n ± ) ] (cid:1) (cid:0) − δ ± (cid:1) − n sin θ i + 4 (cid:0) Re[˜ n ± ) ] (cid:1) δ ± , (39)2 v ± ) = − h(cid:0) Re[˜ n ± ) ] (cid:1) (cid:0) − δ ± (cid:1) − n sin θ i + rh(cid:0) Re[˜ n ± ) ] (cid:1) (cid:0) − δ ± (cid:1) − n sin θ i + 4 (cid:0) Re[˜ n ± ) ] (cid:1) δ ± ,δ ± = n ± ) n ± ) . (40)For the transmission at the first surface we obtain: t ( ± )12 = | τ ( ± )12 | exp (cid:16) iχ ( ± )12 (cid:17) , τ ( ± )12 = | t ( ± )12 | , (41)7 t ( ± )12 | = (cid:20)(cid:16) µ ± ) (cid:17) + (cid:16) µ ± ) (cid:17) (cid:21) n cos θ + 2 n cos θ (cid:16) µ ± ) u ± ) + µ ± ) v ± ) (cid:17)(cid:16) µ ± ) n cos θ + u ± ) (cid:17) + (cid:16) v ± ) + µ ± ) n cos θ (cid:17) + 4 n cos θ (cid:16) µ ± ) u ± ) − µ ± ) v ± ) (cid:17) (cid:20)(cid:16) µ ± ) n cos θ + u ± ) (cid:17) + (cid:16) v ± ) + µ ± ) n cos θ (cid:17) (cid:21) (42)tan χ ± = 2 n cos θ (cid:16) µ ± ) u ± ) − µ ± ) v ± ) (cid:17)(cid:20)(cid:16) µ ± ) (cid:17) + (cid:16) µ ± ) (cid:17) (cid:21) n cos θ + 2 n cos θ h µ ± ) u ± ) + µ ± ) v ± ) i (43)For the reflection and refraction coefficients at the first interface of TM -wave ( p -polarization) we infer r ( ± )12 = ρ ( ± )12 exp( iϕ ( ± )12 ) = n − cos θ − Z ± ) cos θ n − cos θ = ˜ (cid:15) ± ) cos θ − n ˜ n ± ) cos θ ˜ (cid:15) ± ) cos θ + n ˜ n ± ) , (44) t ( ± )12 = τ ( ± )12 exp( iχ ± ) = 2 cos θ cos θ + n Z ± ) cos θ . (45)In a similar way we derive the explicit expressions for ρ ( ± )12 , ϕ ( ± ) , τ ( ± )12 , χ ( ± )12 in terms of (cid:15) ± ) , (cid:15) ± ) , u ± ) , v ± ) and˜ n ± for the reflection and refraction coefficients for both waves at the second interface. For example, r ( ± )23 = (cid:16) u ± ) − µ ± ) n cos θ (cid:17) + i (cid:16) v ± ) − µ ± ) n cos θ (cid:17)(cid:16) u ± ) + µ ± ) n cos θ (cid:17) + i (cid:16) v ± ) + µ ± ) n cos θ (cid:17) = ρ ( ± )23 exp (cid:0) iφ ± (cid:1) ,ρ ( ± )23 = | r ( ± )23 | . (46)By means of (46) one calculates tan φ ( ± )23 . The parameters characterizing the absorbing films (made of DNMM) whichcould be measured directly are the reflectivity R , phase shift δ r at the reflection, transmissivity T and phase shift δ t on transmission. Using the results obtained above one can derive the corresponding expression for these parameters. Itturns out that they are essentially the same as the corresponding formulae of [23] but the particular entries are differentand determined by the formulae above. For the reflectivity we have R = | r | = (cid:0) ρ ± (cid:1) e v ± ) η + (cid:0) ρ ± (cid:1) e − v ± ) η + 2 ρ ± ρ ± cos (cid:0) φ ± − φ ± + 2 u ± ) η (cid:1) e v ± ) η + (cid:0) ρ ± (cid:1) (cid:0) ρ ± (cid:1) e − v ± ) η + 2 ρ ± ρ ± cos (cid:0) φ ± + φ ± + 2 u ± ) η (cid:1) , (47)tan δ r = ρ ± (cid:16) − (cid:0) ρ ± (cid:1) (cid:17) sin (cid:0) u ± ) η + φ ± (cid:1) + ρ ± (cid:16) e v ± ) η − (cid:0) ρ ± (cid:1) e − v ( ± ) η (cid:17) sin φ ± ρ ± (cid:16) (cid:0) ρ ± (cid:1) (cid:17) cos (cid:0) u ± ) η + φ ± (cid:1) + ρ ± (cid:16) e v ± ) η + (cid:0) ρ ± (cid:1) e − v ( ± ) η (cid:17) cos φ ± . (48)Here η = πλ h where h is the thickness of the film (or slab). These formulae are valid for both waves, T E as well as
T M ;one should simply use for ρ ij , φ ij the corresponding expressions for T E and
T M waves obtained above. Similarly onecan obtain the expression for the transmissivity T and the phase shift δ t on transmission as T = n cos θ n cos θ (cid:0) τ ± (cid:1) (cid:0) τ ± (cid:1) exp (cid:0) − v ± ) η (cid:1) (cid:0) ρ ( ± ) (cid:1) (cid:0) ρ ( ± ) (cid:1) e − v ± ) η + 2 ρ ± ρ ± e − v ± ) η cos (cid:0) φ ± + φ ± + 2 u ± ) η (cid:1) , (49)tan (cid:2) δ t − χ ± − χ ± + u ± ) η (cid:3) = e v ± ) η sin (cid:0) u ± ) η (cid:1) − ρ ± ρ ± sin (cid:0) φ ± + φ ± (cid:1) e v ± ) η cos (cid:0) u ± ) η (cid:1) + ρ ± ρ ± cos (cid:0) φ ± + φ ± (cid:1) . (50). For a T M wave the factor ( n cos θ ) / ( n cos θ ) mustbe replaced by (cos θ /n ) / (cos θ /n ) and for the entries8n the last formula one should use the corresponding ex-pression for τ ± ij and χ ± ij . Note also that the informationconcerning refraction index, reflectivity and transmissivityof absorbing media can be useful for studying the multiplereflections and transmissions in a bi-axial slab sandwichedbetween two anisotropic media [36]. PIMPIMDNMM n n n = n ’ + i n ’’ q q q q Figure 1: Propagation of an electromagnetic wave through a meta-material film.
500 600 700 800 - - λ , nm R e ε ,I m ε Permittivity
500 600 700 800 900 - - - - λ , nm R e μ ,I m μ Permeability
Figure 2: Real (solid lines) and reconstructed imaginary parts(dashed lines) of the permittivity and permeability for the sampleB of [37].
5. Reflectivity and transmittance of the DNMM-films
Fig.1 represents schematically the oblique incidence of EM -wave on the surface of DNMM sandwiched betweenthe positive refraction index materials. To demonstrate how these formulae work, one should express the reflectiv-ity and R transmittance T in terms of real and imaginaryparts of (cid:15), µ and n . It should be noted, that the formu-lae derived above are valid not only for the metamaterialsproposed in [4, 5], but for every absorbtive metamaterialswhich are at the same time relatively bad conductors. Thelast condition has to be fulfilled for the boundary condi-tion used above at the derivation of Fresnel formulae tobe valid. So, in order to compare the results presentedabove with experiment, corresponding experimental dataare needed. Unfortunately, for various reasons (see [1], §4.3and the references cited therein) these data are scarce; nev-ertheless, partially some of them can be found [37]; theyare restricted however only to the real parts of (cid:15) and µ .
500 600 700 800 - - λ , nm R en ,I m n Refractive index
Figure 3: Real (solid line) and imaginary part (dashed line) of therefractive index for the sample B of [37]. n - + - + / / / / | Re ( n )| h R Reflectivity, TE polarization
Figure 4: Reflectivity of a metamaterial film as a function of itsoptical thickness for TE polarized wave.
Using Kramers-Kronig relations χ ( ω ) = 1 π VP Z ∞−∞ χ ( ω ) ω − ω d ω , (51) χ ( ω ) = 1 π VP Z ∞−∞ χ ( ω ) ω − ω d ω , (52)9 - + - + / / / / | Re ( n )| h T Transmissivity, TE polarization
Figure 5: Transmissivity of a metamaterial film as a function of itsoptical thickness for TE polarized wave. n - + - + / / / / | Re ( n )| h R Reflectivity, TM polarization
Figure 6: Reflectivity of a metamaterial film as a function of itsoptical thickness for TM polarized wave. n - + - + / / / / | Re ( n )| h T Transmissivity, TM polarization
Figure 7: Transmissivity of a metamaterial film as a function of itsoptical thickness for TM polarized wave. where χ ( ω ) = χ ( ω ) + iχ ( ω ) is complex-valued function,while χ ( ω ), χ ( ω ) both are real and VP stands for Cauchyprincipal value. Then, we were able to calculate the realand imaginary parts of the refraction index for broad fre-quency domain. The values of real parts of (cid:15) and µ were n - + - + - + - + - + - + - + / / / / / | Re ( n )| h R Reflectivity, TM polarization
Figure 8: Reflectivity of a metamaterial film as a function of itsoptical thickness for TE polarized wave. n - + - + - + - + - + - + - + / / / / / | Re ( n )| h T Transmissivity, TE polarization
Figure 9: Transmissivity of a metamaterial film as a function of itsoptical thickness for TE polarized wave. n - + - + - + - + - + - + - + / / / / / | Re ( n )| h R Reflectivity, TE polarization
Figure 10: Reflectivity of a metamaterial film as a function of itsoptical thickness for TM polarized wave. taken from the work [37]. The results of calculations arepresented in Figs. 2 and 3. To illustrate formulae (47)and (49) only two values of n = − .
064 + i × .
173 and n − .
134 + i × . - + - + - + - + - + - + - + / / / / / | Re ( n )| h T Transmissivity, TM polarization
Figure 11: Transmissivity of a metamaterial film as a function of itsoptical thickness for TM polarized wave. we would like to demonstrate how the explicit Fresnel for-mulae work in the particular case of metamaterial. As iteasily can be seen, the imaginary part of refraction indexis positive everywhere in considered frequency domain, asit of course should be in accordance with causality condi-tion. For clarity and readers convenience, we present theresults of calculations made for permeability, refraction in-dex, as well as the reflectivity and transmissivity for both,TE and TM polarizations on the separate charts. Theseresults could be checked directly by proper future experi-ments.As for the Figs. 8-11, we were interested to compare thereflectivity and transmissivity of the film made of metama-terial with these values for dielectric films considered byBorn and Wolf ([23], page 68, Fig. 1.18) in order to observewhether these parameters would periodically dependent onthe film thickness or not. To this end, we modeled somefictitious metamaterial choosing the corresponding valuesof permittivity and permeability and made the calcula-tions as it was described above. As it can be seen in theFigs. 8-11, in case of available experimental data used inour simulations, for the trasmissivity of both TE and TMwaves such quasi-periodic dependence is practically absentor at least very weak, while for the reflectivity it is clearlyseen for some values of real and imaginary parts of com-plex refraction index and as it is observed for some metalfilms. Note also, that the values in horizontal axis in Fig.4-11 are expressed as Re [ n ] × h where n and h are therefraction index and thickness of metamaterial layer sand-wiched between the positive refraction index materials asit is shown in Fig. 1. Additionally, they are presented inlog-scale in Figs. 4-7.
6. Conclusions
In this paper we studied the optical characteristics ofdouble-negative metamaterials taking into account the factthat all metamaterials are inevitably absorptive and de-rived the explicit formulae for total reflection angle as well as correct Fresnel formulae describing the reflection and re-fraction coefficients for the DNMM in case of TE as well as TM EM wave polarization. The reflectivity and transmit-tance of DNMM film embedded in the PIM-surroundingare also presented.
7. Acknowledgment
This work has been done due to the support which twoof us (IT and PZ) have got from the Centre for Innovationand Transfer of Natural Science and Engineering Knowl-edge, University of Rzeszów.
CRediT authorship contribution statementIgor Tralle : Conceptualization, Formal analysis, Writ-ing – Original draft preparation, Supervison.
Levan Cho-torlishvili : Methodolgy, Writing - Review & Editing.
Paweł Zie¸ba : Software, Visualization, Writing - Review& Editing.
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