Electromagnetic transmittance in alternating material-metamaterial layered structures
RRevista Mexicana de F´ısica 63 (2017) 402-410
Electromagnetic transmittance in alternating material-metamaterial layered structures
V.H. Carrera-Escobedo and H.C. Rosu ∗ IPICyT, Instituto Potosino de Investigacion Cientifica y Tecnologica,Camino a la presa San Jos´e 2055, Col. Lomas 4a Secci´on, 78216 San Luis Potos´ı, S.L.P., Mexico
Using the transfer matrix method we examine the parametric behavior of the transmittance of
T E and
T M electromagnetic plane waves propagating in frequency range which are far fromthe absorption bands of a periodic multilayered system. We focus on the dependence of thetransmittance on the frequency and angle of incidence of the electromagnetic wave for the casein which the periodic structure comprises alternating material-metamaterial layers of variouspermittivities and permeabilities. A specific example of high transmittance at any angle ofincidence in the visible region of the spectrum is identified.
Keywords : Transfer matrix method; transmittance; metamaterial; multilayer; periodic.
I. INTRODUCTION
Studies of planar multilayer structures with alternat-ing material and metamaterial layers are motivated bythe known common feature of planar periodic systems togenerate transparency bands. Some time ago, Banerjee et al [1] calculated the intensity of the electromagneticfields that propagate through an array of periodically al-ternating positive index media (PIM) and negative indexmedia (NIM). However, they do not present their resultsin terms of either the angle of incidence or the frequencyof the plane wave, which are the important parameterswhen one is interested in the directional and frequencyselectivities of such structures. They display the field in-tensity along the direction of propagation and comparethe transfer matrix method to the finite element methodconcluding that the transfer matrix method provides thesame results as the latter. Their results motivated usto use the transfer matrix method with the main goalof studying the effect of both the angle of incidence andfrequency of the propagating plane wave on the trans-mittance spectrum. We compute the values of the trans-mittance for different values of (cid:15) and µ of the alternatingmaterial-metamaterial layers and search for those valuesthat provide wide windows of transmittance for some re-gions of frequency as well as the complementary trans-mittance gaps that may be useful for making frequencyfilters [2]. II. THE TRANSFER MATRIX METHODA. Waves at an interface
To develop the TMM we must first know how the elec-tromagnetic waves behave at the interface between two ∗ Electronic address: [email protected] dielectrics. For plane waves at an interface, the electricand magnetic fields are given by (cid:126)E = (cid:126)E e i [ (cid:126)k · (cid:126)r − ωt ] (1) (cid:126)H = (cid:126)H e i [ (cid:126)k · (cid:126)r − ωt ] , (2)respectively. The two media separated by the interfacewill be characterized by permittivities (cid:15) , (cid:15) and perme-abilities µ , µ , and the geometry of the wave vectors atthe interface is illustrated in Fig. (1). In this figure, weassume that the wave is propagating along the z -axis andthe wavevector has two components (cid:126)k = ( k x , , k z ) . (3) Figure 1: Electromagnetic wave vectors at the interface be-tween two different media.
Because the condition of continuity of the tangentialcomponents of both the electric and magnetic fields (cid:126)E t = (cid:126)E t , (cid:126)H t = (cid:126)H t (4)must be satisfied for any point of the interface, the tan-gential components of (cid:126)k have to be equal k x = k x = k x , (5) a r X i v : . [ phy s i c s . op ti c s ] A ug while the dispersion relation n ω /c = k as written forthe two media leads to n ω c = k = k x + k z , n = √ µ (cid:15) , (6) n ω c = k = k x + k z , n = √ µ (cid:15) . (7)If k x and k z are real, we can define the angles of incidenceand refraction as tan θ = k x k z (8)and tan θ = k x k z , (9)respectively. From Eq. (5), we can see that k x = k sin θ = k sin θ (10)and if we apply the dispersion relationship, we obtain k x = n ωc sin θ = n ωc sin θ . (11)If we now take the ratio of the last two equations weobtain Snell’s law sin θ sin θ = n n . (12)Considering the case of the TE polarization (s-typepolarization), then (cid:126)E is parallel to the interface, i.e. , wehave (cid:126)E = (0 , E,
0) (13)and (cid:126)H = ( H x , , H z ) . (14)We also have the boundary condition that says that thecomponent of the electric field which is parallel to theinterface is the same on both sides of the interfaces E +1 + E − = E +2 + E − , (15)which holds for the magnetic field H +1 x + H − x = H +2 x + H − x (16)as well as.Using Maxwell’s equation (cid:126)k × (cid:126)E = + µωc (cid:126)H , (17)in Eq. (16), we obtain µ ωc (cid:126)H +1 x = − k z E +1 , (18) µ ωc (cid:126)H − x = k z E − . (19) Inserting (18) in (16) leads to − k z cµ ω E +1 + k z cµ ω E − = − k z cµ ω E +2 + k z cµ ω E − . (20)Now we can write Equations (20) and (15) in matrixform (cid:18) − k z µ k z µ (cid:19) (cid:18) E +1 E − (cid:19) = (cid:18) − k z µ k z µ (cid:19) (cid:18) E +2 E − (cid:19) (21)or (cid:18) E +2 E − (cid:19) = M ( s ) (cid:18) E +1 E − (cid:19) (22)where the transfer matrix of the interface M ( s ) = 12 (cid:32) µ k z µ k z − µ k z µ k z − µ k z µ k z µ k z µ k z (cid:33) (23)is introduced.By a similar procedure, we can obtain the transfer ma-trix for the case of the TM polarization (p-type polariza-tion) (cid:18) E +2 E − (cid:19) = M ( p ) (cid:18) E +1 E − (cid:19) (24) M ( p ) = 12 (cid:32) (cid:15) k z (cid:15) k z − (cid:15) k z (cid:15) k z − (cid:15) k z (cid:15) k z (cid:15) k z (cid:15) k z (cid:33) . (25)For materials with only real dielectric constants we canexpress the wave vectors as k z = k cos θ and k z cos θ , (26)so the transfer matrices for the TE and TM cases arewritten as M ( s or p ) = 12 (cid:32) z
21 cos θ cos θ − z
21 cos θ cos θ − z
21 cos θ cos θ z
21 cos θ cos θ (cid:33) , (27)where z = µ k /µ k for the s-polarization and z = (cid:15) k /(cid:15) k for the p-polarization. B. The transfer matrix for a slab
For a set of interfaces, the systems behaves as a slab(sandwich of media). A simple example is given inFig. (2). For such systems, it is enough to apply thecomposition law for transfer matrices [3]. In this way,the relationship of the coefficients of entry and exit isgiven by (cid:18) E +3 E − (cid:19) = M (cid:18) e ik z (cid:96) e − ik z (cid:96) (cid:19) M (cid:18) E +1 E − (cid:19) , (28) Figure 2: Illustration of wave propagation through a slab com-posed of three different media. where (cid:96) is the length of the slab in the direction of prop-agation.From the last equation, one can obtain the transfermatrix for the slab as M sslab = M (cid:18) e ik z (cid:96) e − ik z (cid:96) (cid:19) M , (29)which can be used for the generalization to the multilayercase. C. Wave propagation through a multilayeredsystem
The next step is to consider a multilayered system as il-lustrated in Fig. (3). For a system of this class, the trans-
Figure 3: A system formed by two types of slabs. The firstone is a PIM (blue) labeled as A in the text, and the secondis a NIM (red) labeled as B in the text. fer matrix is obtained by applying the composition lawagain. In this way, we have an interface matrix M i , i + for every interface and a propagation matrix of the form P = (cid:18) e ik z (cid:96) e − ik z (cid:96) (cid:19) (30)for every slab in which the electromagnetic wave propa-gates. In this manner, for a ten multilayer slab system(5 A and 5 B layers) the transfer matrix is written as M = M A [ P A M AB P B M BA ] P A M AB P B M B (31)where M X is the interface matrix between air andmedium X , P X is the propagation matrix of medium X , and M XY is the interface matrix between medium X and medium Y . The M and M XY matrices are trans-fer matrices for the s-polarization or the p-polarization,depending on the nature of the incident wave.Once the transfer matrix for the multilayered systemhas been defined, one can proceed with the calculationof the transmission amplitude based on its definition forthe case of electromagnetic waves given by [3] t = det MM , (32)where det M stands for the determinant of M , and thetransmittance as its square modulus T = | t | . (33) III. NUMERICAL SIMULATIONS
The computation of the transmittance is performedby using a simple python code [4]. The system whichwe model has up to ten slabs, alternating a PIM witha NIM. The PIM is characterized by µ + and (cid:15) + , whilethe NIM by µ − and (cid:15) − . The frequency of the incidentwave is in c/ ( (cid:96) √ (cid:15) ) units, where the chosen numericalvalue for (cid:96) is 1.0 µ m, and the angle of incidence θ goesfrom − π/ < θ < π/ θ = π/ π/ A. The effect of the number of layers
TE Mode n b =5n b =3n b =2 T r a n s m i tt a n c e ε ω ℓ )/c0 1 2 3 4 TM Mode n b =5n b =3n b =2 T r a n s m i tt a n c e ε ω ℓ )/c0 1 2 3 4 Figure 4: Plots corresponding to the case of variable numberof blocks. Left panel: the TE mode. Right panel: the TMmode. In these graphs we can see that the valleys of thetransmittance get deeper as we increase the number of blocks.This is congruent with the formation of transmission bandsin the case of superlattices.
Parameter Value (cid:15) (cid:15) + (cid:15) − -1.0 µ µ + µ − -1.2 n b We begin by analyzing the effect of the number of lay-ers upon the transmission spectrum. With this task inmind, we vary the number of periods “AB” crossed bythe propagating wave using the values of the parame-ters given in Table I, where the parameters (cid:15) and µ correspond to the respective values for the relative per-mittivity and permittivity of the medium (usually air) infront and at the end of the “AB” multilayer structure.We call the “AB” period a block. In Fig. (7), the num-ber of blocks is increased from two to five, (in terms ofinterfaces, from five to eleven), and each panel is labeledby the corresponding number of blocks. The regions ofhigh transmittance show up as three symmetrical bub-bles. Upon increasing the number of blocks in the sys-tem, one can see that the number of peaks inside eachof the bubbles increases. The number of peaks, n peaks ,is equal to twice the number of blocks, n b , minus one, n peaks = 2 n b −
1. This might be seen alike to the caseof quantum systems if the number of quantum wells isput in correspondence with the number of peaks in thetransmittance. One can also see that when the number ofblocks increases the transmittance between the bubblesof high transmittance becomes smaller. This is a con-sequence of the change in the transmittance values thatone can see in Fig. (4), where we can also notice that thetransmittance bands get wider as the number of blocksincreases. Moreover, in Fig. (7), we can see that we havea bubble of high transmittance inside the limits of thevisible frequencies (black horizontal lines), which couldbe useful for camouflage purposes. In addition, as weincrease the number of blocks, this high transmittanceregion entails a wider range of angular values. On theother hand, if we increase the number of blocks enoughthe simulation ends up in a completely opaque materialwith zero transmittance, as expected.
B. The effect of (cid:15) + and (cid:15) − Knowing that the number of blocks generates a seriesof transmittance bands, one naturally may ask if there isa way to control the width of these bands by means ofthe electric permittivities of the slabs. The values of theparameters that we use for this computation are given in
TE Mode ε + =3.0 ε + =2.0 ε + =1.5 T r a n s m i tt a n c e ε ω ℓ )/c0 1 2 3 4 TE Mode ε - =-3.0 ε - =-2.0 ε - =-1.5 T r a n s m i tt a n c e ε ω ℓ )/c0 1 2 3 4 Figure 5: Plots corresponding to the case of fixed angle ofincidence θ = π/
3. The left panel corresponds to the caseof variable (cid:15) + and the right panel corresponds to the case ofvariable (cid:15) − . In these graphs, we can see the shifting of thepeaks in transmittance only for the case with increasing (cid:15) + .Parameter Value (cid:15) (cid:15) + (cid:15) − -1.5, -2.0, -3.0 µ µ + µ − -1.2 n b Table II.The results are presented in Fig. (8). We can see thatthe effect of increasing (cid:15) + while keeping (cid:15) − fixed at − . (cid:15) + . When (cid:15) + is high enough, the bubbles lose theconnection between them and become independent ovals.For this case, we see that the effect is similar for both theTE and TM modes, but for the TM mode the bubblesare more separated and there appears a series of smallerbubbles of high transmittance. Moreover, for both typesof modes there is a shift to lower frequencies of thesebubbles of high transmittance. This can be seen in theleft panel of Fig. (5).On the other hand, if we increase the value of | (cid:15) − | while keeping (cid:15) + fixed at 2.0 we can see from Fig. (9)that the bubbles of high transmittance are absent andin their place a wide band with horizontal spikes occurs.Therefore, we conclude that the occurrence of bubblesdepend on the chosen values of (cid:15) + . When (cid:15) − increases,we can see that the wide band is fragmented in threesections, and the separation between sections is propor-tional to the increment in | (cid:15) − | . In this case, there is noshift to lower frequencies. This fact can be observed inthe right panel of Fig. (5). The effect is similar for bothTE and TM modes, but in the TM case the decrease intransmittance is less but in a wider range of frequencies. Parameter Value (cid:15) (cid:15) + (cid:15) − -1.0 µ µ + µ − -1.5, 2.0, 3.0 n b µ values arechanged. In general, we can say that the value of (cid:15) + determinesthe form of the transmittance bands and the frequencyrange where they occur. On the other hand, the value of (cid:15) − determines the width of the horizontal transmittancespikes. C. The effect of µ + and µ − TE Mode μ + = 1.5 μ + = 2.0 μ + = 3.0 T r a n s m i tt a n c e ε ω ℓ )/c0 1 2 3 4 TE Mode μ - = -1.5 μ - = -2.0 μ - = -3.0 T r a n s m i tt a n c e ε ω ℓ )/c0 1 2 3 4 Figure 6: Plots corresponding to the case of fixed angle ofincidence θ = π/
3. Left panel: case of variable µ + . Rightpanel: case of variable µ − . One can see that the shiftingof the peaks in transmittance occurs only upon increasing µ + (left panel) and there is no shifting effect when | µ − | isincreased (right panel). We move now to the study of the effects of increasingthe values of the permeabilities of the system. For thistask, we use the values from Table III for the parametersunder control.From Fig. (10), we see that if we increase the magni-tude of µ + the bubbles of high transmittance have a shiftto lower frequencies which is proportional to the magni-tude of µ + . Unlike the case of varying (cid:15) + , we see thatthe bubbles remain connected, and as the value of µ + increases the contact area between bubbles increases aswell as. This is opposite to the case of varying (cid:15) + andoccurs for both the TE and TM modes.From the left panel of Fig. (6), we can appreciate easilya shifting effect to lower frequencies. Also, and even moreinteresting, we see that if we increase the value of µ + the valleys (regions of low transmittance) become less apparent. Thus, increasing µ + has the effect of rising thevalleys of transmittance diminishing the relative bordersof the transmittance bands.In the case of increasing the values of | µ − | , Fig. (11)shows that the main effect is to diminish the transmit-tance between the three regions of high transmittance.This effect is also visible on both the TE and TM modes.However, in this case, we notice that there is no shiftingeffect, which can be seen easily from the right panel ofFig. (6).In addition, Figs. (7) - (11) show that the regions ofhigh transmittance do not vary with the angle of inci-dence, which may be related to the independence of thepolarization angle of the incident wave as also reported in[5]. Considering that the reflectance and transmittanceare complementary properties, our findings regarding thetransmittance spectrum seem to be in agreement withthose reported in [6] for an Ag-TiO multilayer system,which is the standard multilayer candidate with hightransmittance in the visible region. IV. CONCLUSION
We have obtained the transmission properties of amultilayered structure of alternating positive index me-dia and negative index media using the TMM. We havesought for profiles of high transmittance at any angle ofincidence in the visible region of the spectrum. In thisrespect, our results suggest a system with the propertiesdisplayed in Fig. (7), i.e., with (cid:15) + = 2 . , (cid:15) − = − . , µ + =2 .
0, and µ − = − .
2, for the case of five blocks, n b = 5,which present an almost uniform region of high transmit-tance in the visible part of the electromagnetic spectrum.In addition, we have studied the manner in whichthe permeability and permittivity parameters affect thetransmission properties in an alternation of material andmetamaterial. We have observed that positive perme-abilities may have the effect of reversing the appearanceof the transmission bands that one may see in multilay-ered structures. On the contrary, positive permittivitieshave the effect of making the transmittance bands morepronounced. Somewhat surprisingly, the negative valuesof the electromagnetic material parameters do not haverelevant effects in these settings.The results obtained in this paper are valid far fromthe absorption bands of the multilayer structure to avoidnumerical instability problems which are known to occurwhen the absorption is included [7]. However, the resultswe report for the visible region may still remain validfor some metamaterials, such as the hyperbolic ones, forwhich the absorption bands lie in the ultraviolet region[8]. Acknowledgments
The first author acknowledges the financial support ofCONACyT through a doctoral fellowship at IPICyT. [1] P.P. Banerjee, H. Li, R. Aylo, and G. Nehmetallah, Trans-fer matrix approach to propagation of angular plane wavespectra through metamaterial multilayer structures,
Proc.SPIE (2011) 3387-3389.[2] L. Solymar and E. Shamonina, “Waves in Metamaterials” (Oxford University Press, 2009).[3] P. Markos and C.M. Soukoulis, “Wave propagation fromelectrons to photonic crystals and left-handed materials”
Plasmonics (2017) 95-102.[6] P. Shekhar, J. Atkinson, and Z. Jacob, Hyperbolic meta-materials: fundamentals and applications, Nano Conver-gence (2014) 14.[7] W.-J. Hsueh and J.-C. Lin, Stable and accurate methodfor modal analysis of multilayer waveguides using a graphapproach, J. Opt. Soc. Am. (2007) 825-830.[8] T. Tumkur, Y. Barnakov, S.T. Kee, M.A. Noginov, andV. Liberman, Permittivity evaluation of multilayered hy-perbolic metamaterials. Ellipsometry vs. reflectometry, J.Appl. Phys. (2015) 103104.
TE Mode | n b = 2 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TE Mode | n b = 3 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TE Mode | n b = 5 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1TM Mode | n b = 2 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TM Mode | n b = 3 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TM Mode | n b = 5 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 Figure 7: Contour plots. The top row corresponds to the TE mode (s-polarization) while the bottom row correspond to the TMmode (p-polarization). The number of blocks takes the values n b = 2 , ,
5. The x -axis corresponds to the angle of incidence,the y -axis to the frequency of incidence in √ (cid:15) (cid:96)/c units, and the color bar corresponds to transmittance. The black horizontallines indicate the limits of the frequencies of the visible range. T r a n s m i tt a n c e TE Mode | ε + = 1.5 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TE Mode | ε + = 2.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TE Mode | ε + = 3.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TM Mode | ε + = 1.5 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TM Mode | ε + = 2.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TM Mode | ε + = 3.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 Figure 8: Contour plots. The top row corresponds to the TE mode while the bottom row corresponds to the TM mode. Thepositive permeability takes values (cid:15) + = 1 . , , (cid:15) − is fixed at -1.0. T r a n s m i tt a n c e TE Mode | ε - = -1.5 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TE Mode | ε - = -2.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TE Mode | ε - = -3.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TM Mode | ε - = -1.5 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TM Mode | ε - = -2.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 T r a n s m i tt a n c e TM Mode | ε - = -3.0 ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 Figure 9: Contour plots. The top row corresponds to the TE mode while the bottom row corresponds to the TM mode. Thepositive permeability takes values (cid:15) − = 1 . , , (cid:15) + is fixed at 2.0. TE Mode | μ + = 1.5 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TE Mode | μ + = 2.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TE Mode | μ + = 3.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1TM Mode | μ + = 1.5 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TM Mode | μ + = 2.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TM Mode | μ + = 3.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 Figure 10: Contour plots. The top row corresponds to the TE mode while the bottom row corresponds to the TM mode. µ − is fixed at -1.2. TE Mode | μ - = -1.5 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TE Mode | μ - = -2.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TE Mode | μ - = -3.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1TM Mode | μ - = -1.5 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TM Mode | μ - = -2.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 TM Mode | μ - = -3.0 T r a n s m i tt a n c e ( √ ε ω ℓ ) / c θ / θ −1 −0.5 0 0.5 1 Figure 11: Contour plots. The top row corresponds to the TE mode while the bottom row corresponds to the TM mode. µ ++