6 Progress In Electromagnetics Research Symposium 6, Cambrige, US, March 6-9 Wave Propagation in Groune Dielectric Slabs with Double Negative Metamaterials W. Shu an J. M. Song Iowa State University, US bstract In this paper, the wave propagation in a groune ielectric slab with ouble negative (DNG) metamaterials is stuie. Dramatically ifferent evanescent surface moes (electromagnetic fiels exponentially ecay both in air an insie the slab) are observe. They are highly epenent on meium parameters. n infinite number of complex surface moes are foun to be existing which have proper fiel istribution in the air region. The investigations on the Poynting vectors show that they o not carry away energy in both transverse an longituinal irections.. Introuction The guie ielectric slab with a DNG meium has been stuie by several groups. Various novel properties are observe: [] an [] foun that there are special regions for TM (transverse magnetic) moes where two ifferent propagation constants exist. [] theoretically consiere the properties of a planar two-layere waveguie, whose one layer is a ouble positive (DPS) meium an the other is a DNG meium. Super slowwaves with extremely short wavelengthes were foun whose fiels exponentially ecay from the interface of the two slabs insie both layers. These guie moes, terme as evanescent surface moes, were also foun by [] an [], respectively. P. Baccarelli an his colleague suggeste the concept of surface wave suppression that ensures the absence of both orinary an evanescent surface moes. This is very attractive in view of taking DNG meium as a potential substrate caniate to reuce ege iffraction effects an enhance raiation efficiency for microstrip antennas [6]. However, so far as the authors are aware no stuy on the complex moes an Poynting vectors has been reporte. This makes the moe spectra of DNG meia unpleasantly incomplete. In this paper, the authors focus on the properties of the evanescent surface moes an the complex moes, both of which belong to the proper moe spectra of the groune ielectric slab with a DNG meium. It is foun that the evanescent surface moes are highly epenent on the meium parameters an an infinite number of complex moes exists which have exponentially ecaying fiels in the air region. They are terme complex surface moes. The stuy on the Poynting vectors shows that they have zero power flows in both transverse an longituinal irections.. Eigen Equations an Graphical Solutions The structural setup of interest here is a groune ielectric slab of thickness (see Figure ). Region one is a DNG meium an region two is air. It is well known that to ensure a positive store energy in the ielectric layer, passive DNG meia must be ispersive [7]. However, for simplicity we assume that they are isotropic, losseless, an non-ispersive. This assumption is foun to be acceptable since a small ispersion of ɛ an µ can satisfy the constraints. y ε r µ r z x y = ε r µ r PEC y = Figure : Geometry structure of a groune ielectric slab with DNG meium (ɛ r <,µ r < ).
Progress In Electromagnetics Research Symposium 6, Cambrige, US, March 6-9 7 α y α y α y k y α y k β y y - - - (a) TE TE TE TE TE 7 - (b) TM TM TM TM TM 6 Figure : Graphical solutions for TE an TM moes. Soli lines in the first an fourth quarants represent () or (); soli lines in the secon quarant represent () or (); ashe line in the first an fourth quarants represents (); ashe line in the secon an thir quarants represents (6). The meium parameters are: ɛ r =.,µ r =.,ɛ r =,µ r =. Using the well-known transverse resonance metho [8], the eigen equations for orinary (γ y = jk y ) real moes are: µ r (k y )cot(k y ) = α y µ r for TE () ɛ r (k y )tan(k y ) = α y ɛ r for TM () The eigen equations for evanescent (γ y = α y ) real moes are: (k y ) + (α y ) = (k ) (ɛ r µ r ɛ r µ r ) () µ r (α y )coth(α y ) = α y µ r for TE () ɛ r (α y )tanh(α y ) = α y ɛ r for TM () (α y ) (α y ) = (k ) (ɛ r µ r ɛ r µ r ) (6) where k = ω µ ɛ. γ y, γ y are the y-irection wave constants of the two layers. Their relationship to the longituinal wave constant (z-irection) γ is written as: γ yi = k ɛ ri µ ri γ (i =,) (7) Graphical representations of the above equations are shown in Figure. The moe inex notation here follows [9]. Notice that in the first an secon quarants, α y is positive an the fiels exponentially ecay in the air region (proper); in the thir an fourth quarants, α y is negative an the fiels exponentially increase in the air region (improper). The x-axis is ivie into two segments. The right half is for k y an the fiels in the ielectric layer are sine/cosine staning waves (orinary), while the left half is for α y an the fiels in the ielectric layer are exponentially istribute (evanescent). Therefore, the intersection in the secon quarant represents the proper evanescent surface moe, which oes not exist for a DPS meium. nother important ifference for a DNG meium that can be rea from Figure is that the orinary surface moe solutions are no longer monotonic. It is clear from the subfigure in the left corner of Figure (a) that there are two intersections as the raius of the ashe circle ecreases, which correspons to a ecrease of frequency. Once the circle has only one tangential point with the soli line, further ecreasing frequency will cause this moe to be cutoff. The same thing happens to TM moes in Figure (b) in a more obvious way. These two possible moes have two ifferent power flow istributions. One has more power flowing in the air region than in the ielectric region, making the total power flow in the same irection as the phase velocity. The other is in the opposite way an isplays a backwar property. More etails on the Poynting vectors are aresse in Section.
8 Progress In Electromagnetics Research Symposium 6, Cambrige, US, March 6-9. Evanescent Moe Evanescent Moe β/k. Moe Improper Leaky Moe 6 k (a) ɛ r =., µ r =., ɛ r =, µ r =. β/k Moe Improper Leaky Moe.. k (b) ɛ r =., µ r =., ɛ r =, µ r =. Figure : Two possible ispersion curves for TE proper surface moes (soli lines) an TE improper leaky moes (otte lines). The ashe line, representing ɛ r µ r, is the watershe for evanescent surface moe an orinary surface moes.. Evanescent Moe s state in Section, the proper evanescent surface moe oes exist with a DNG meium. It is the intersection in the secon quarant. The normalize effective ielectric constant ɛ eff = (β/k ) for evanescent surface moe is larger than both ɛ r µ r an ɛ r µ r. Therefore the transverse propagation constant in the ielectric layer γ y = k ɛ rµ r γ = k ɛeff ɛ r µ r is a pure real number. The electromagnetic fiels are no longer sine/cosine staning waves, but have the form of e αyy + Be αyy. It is foun, however, that the ispersion curves for evanescent surface moes are very complicate, an they are highly epenent on the meium parameters. Figure shows two ispersion iagrams for TE moe with ifferent meium parameters. The ispersion curves represent the intersection points of the ashe line an the first soli branch in Figure (a), incluing the part in the secon quarant. The soli line in Figure is for proper moes, while the otte line is for improper moe, which is the set of intersections in the fourth quarant in Figure (a). The ashe lines in both figures epict the value of ɛ r µ r. They are the watershes by which one can tell the evanescent surface moe from orinary ones. In Figure (a), the evanescent surface moe has low cutoff frequency. s the frequency increases, the orinary surface moe becomes an evanescent surface moe an its effective ielectric constant, ɛ eff, keeps increasing. In Figure (b), however, the situation is reverse. The evanescent surface moe has a high cutoff frequency above which it becomes the orinary surface moe. t the low frequency range, the evanescent surface moe has an extremely large ɛ eff, which ecreases rapily as the frequency increases. One can refer to the subfigures of Figure to check the valiations. The reason for such ramatically ifferent ispersion curves is that with DNG metamaterials, one can not only make ɛ an µ simultaneously negative but also let their absolute values be less than one []. From () an Figure (a), it is easy to see that the crossing point of the first soli branch TE with the x-axis is fixe at (π/, ), while the crossing point with the y-axis note as in Figure (a) is (, µ r /µ r ). With a conventional DPS meium, µ r is always equal to unity, or slightly greater or smaller than unity as in the case of paramagnetic or iamagnetic materials. With metamaterials, however, µ r is not confine near unity any more an the intercept with the y-axis may change a lot. This change affects the possible intersections of the first soli line an the ashe line in Figure (a) an finally results in ramatically ifferent ispersion curves.. Complex Moes an Poynting Vectors It is well known that the complete proper moe spectra for a DPS ielectric slab inclue iscrete surface moes an continuous raiation moes, both of which are real moes [8]. With a DNG meium, however, it is prove by the authors that the complex roots of the eigen equations are exclusively on the top Riemann sheet []. These solutions, terme complex surface waves, form another set of proper moes since they have exponentially ecaying fiels in the air region an satisfy the bounary conitions at infinity. Unlike real surface moes, complex surface moes have high cutoff frequencies below which they exist.
Progress In Electromagnetics Research Symposium 6, Cambrige, US, March 6-9 9.. TM (α+jβ)/k. TE TE TE TE7 (α+jβ)/k. TM TM TM 6.. k (a) TE k (b) TM Figure : Dispersion iagrams for all moes. Soli line is for normalize β of the proper moes. Dashe line is for normalize α of the proper moes. Dotte line is for normalize β of the improper moes. The meium parameters are: ɛ r =.,µ r =.,ɛ r =,µ r =. Figure shows the ispersion iagrams for both TE an TM moes, incluing evanescent, orinary, an complex surface moes. lso inclue are improper leaky moes rawn as otte lines. When the frequency is much lower than the first cutoff frequency of the real moes, all complex moes exist with very high normalize α an β. s the frequency increases, β/k tens to ecrease rapily within a very narrow frequency range; after that it increases slowly till its cutoff frequency. Notice it is not monotonic an the value of β/k can be less than unity, which is a notable ifference compare with evanescent an orinary surface moes. The curve of α/k, however, monotonically ecreases very fast as the frequency increases. t the cutoff point, α reaches zero an β becomes the starting point of the real moe. The real surface moe bifurcates into two branches from this point. One branch has an increasing β/k as the frequency goes high, while the other has a ecreasing β/k, which will reach unity shortly. This property is expecte from Figure. Further increasing frequency makes β/k of the secon branch begin to rise. However, it is no longer a proper moe. It is foun that the complex surface moes have zero power flows []. To erive the Poynting vector for complex moes, γ y, γ y, an γ are assume to be: The Poynting vector is written as z = E xh y = γ y = a + jb γ y = u + jv γ = α + jβ (8) { S TE z, Sz TE, where is the electric fiel intensity an z an z are as follows: for < y < for y z (y,z) = β + jα ωµ r e αz [cosh(ay) cos(by)] () z (y,z) = β + jα ωµ r e u(y ) αz [cosh(a) cos(b)] () Figure shows the ispersion iagram an the integral results of Poynting vector for the TE moe. In Figure (a), only the complex moe exists (branch ) when the frequency is lower than the cutoff frequency of the real surface moe. The zero power flow in z-irection in Figure (b) shows that the complex surface moe oes not carry away any energy. s the frequency increases, the real surface moe begins. The top branch (branch B ) of the real moe carries a negative power flow an shows backwar properties. When a waveguie operates in this moe, its fiels are largely confine insie the ielectric layer. The bottom branch (branch C ) of the real moe carries a positive power flow an its fiels exten far away in the air region. Further increasing frequency causes the fiels in the air region to ecay more slowly, an eventually reach infinity. t that point, the raiation bounary conitions are violate an the moe becomes improper. (9)
Progress In Electromagnetics Research Symposium 6, Cambrige, US, March 6-9 (α+jβ)/k... B C 6 k (a) Dispersion iagram P z. -. C - 6 k (b) Power flow in z-irection Figure : Dispersion iagram an the power flow in z-irection for TE moes. is for complex surface moe; B is for top branch of the real surface moe; C is for bottom branch of the real surface moe. The meium parameters are: ɛ r =.,µ r =.,ɛ r =,µ r =. B. Conclusion In this paper, an investigation on the moe properties of a groune ielectric slab with a DNG meium has been ealt with. The graphical metho is use to fin the possible real roots. Dramatically ifferent ispersion curves of evanescent surface moes are observe, showing that they are very sensitive to the material parameters. It is foun that there is an infinite number of complex surface moes with a DNG meium an they o not carry away energy. lthough the consiere meium here is iealize an currently cannot be realize, the results of this paper still unveil some exotic properties as well as potential applications of the metamaterials. REFERENCES. Cory, H. an. Barger, -wave propagation along a metamaterial slab, Microwave Opt. Technol. Lett., Vol. 8, 9 9, Sept... Dong, H. an T. X. Wu, nalysis of iscontinuities in ouble-negative (DNG) slab waveguies, Microwave Opt. Technol. Lett., Vol. 9, 8 88, Dec... Nefeov, I. S. an S.. TretyaKov, Waveguie containing a backwar-wave slab, Raio Sci., Vol. 8, 9,.. Wu, B.-I., T. M. Grzegorczyk, Y. Zhang, an J.. Kong, Guie moes with imaginary transverse wave number in a slab waveguie with negative permittivity an permeability, J. ppl. Phys., Vol. 9, 986 988, Jun... Sharivov, I. W.,.. Sukhorukov, an Y. S. Kivshar, Guie moes in negative-refractive-inex waveguies, Phys. Rev. E, Stat. Phys. Plasmas Fluis Relat. Interiscip. Top., Vol. 67, 76 76, May. 6. Baccarelli, P., P. Burghignoli, F. Frezza,. Galli, P. Lampariello, G. Lovat, an S. Paulotto, Funamental moal properties of surface waves on metamaterial groune slabs, IEEE Trans. Microwave Theory Tech., Vol.,, pr.. 7. Smith, D. R. an N. Kroll, Negative refractive inex in left-hane materials, Phys. Rev. Lett., Vol. 8, 9 96, Oct.. 8. Collin, R. E., Fiel Theory of Guie Waves, n, E. Piscataway, IEEE Press, NJ, 99. 9. Balanis, C.., vance Engineering Electromagnetics, John Wiley & Sons, NJ, 989.. Shu, W. an J. M. Song, On the properties of a groune ielectric slab with ouble negative metamaterials, IEEE Trans. Microwave Theory Tech., to be submitte.