Modal Interactions in Lossy Dielectric Metamaterial Slabs

Transcription

1 Modal Interactions in Lossy Dielectric Metamaterial Slabs A. B. Yakovlev (), G. Lovat (), P. Burghignoli (), and G. W. Hanson () () University of Mississippi () La Sapienza University of Rome () University of Wisconsin

2 Outline Motivation and background Metamaterial slab model Dispersion equation Lossless case: Characterization of solutions Magnetic resonance and losses Perturbation of TE solutions Critical losses: Modal interaction Magnetic and electric resonances for TM modes Conclusions

3 Motivation and Background

4 Dielectric Metamaterial Slab Model () N r µ r p ε r p ε rh µ rh p /

5 Dielectric Metamaterial Slab Model () r p µ r ε r Holloway et al., IEEE Trans. Antennas Propag., 5, Oct. Jylhä et al., J. Appl. Phys., 99, Feb. p ε rh µ rh Clausius-Mossotti mixing rules εreff εrh αe( f ) = n ε + ε ε reff rh rh µ reff µ rh αm( f ) = n µ + µ µ reff rh rh Cubic lattice n = p Mie theory α e m F() ξε r rh f = πεrhr F () ξεr + εrh ( ) F() ξ µ µ r rh f = rhr F () ξ µ r + µ rh α ( ) πµ ε sin ( t t cost) Ft () = ( t )sint + t cost πf ξ = r µ rεr c

6 Dispersion Equations z Z h ε reff, µ reff ρ k zs Z S PEC Characteristic impedances Z Z ωµ ωµ µ TE TE reff =, ZS = kz kzs k TM z TM zs =, ZS = ωε ωεεreff k Transverse wavenumbers z = k k k ρ zs = µ reff εreff k k k ρ TE/TM Dispersion equation TE/TM S TE/TM zs Z kρ f = jz k h + Z = (, ) tan( )

7 Real Modes kρ = βρ > k k =± k β =± jα substrate air z ρ z growing (improper) attenuated (proper) k ± β =± = ± jα zs k µ reff εreff βρ zs zs ordinary evanescent absent in SNG slabs, but present in double positive (DPS) and DNG slabs absent in DPS slabs, but present in both SNG and DNG slabs ordinary proper ordinary improper evanescent proper evanescent improper

8 Lossless Case: Complex Modes jµ k k h + k = tan( ) jk tan k h + ε k = TE: reff z zs zs TM: zs ( zs ) rff e z µ ε reff reff DPS > TE Improper > TM Improper ENG MNG µ ε reff > reff < TE Improper TM Proper TE Proper TM Improper µ ε reff < reff > DNG TE Proper µ reff < TM Proper ε reff < Evolution and different modal behavior with frequency

9 Effects of Losses

10 Lossless Metamaterial Slab ε r = ( j tan δe), µ r = r = mm ε rh =, µ rh = p = mm N = 5 Yuan et al., J. Appl. Phys., 87, 5 5 ε reff 5 µ reff 5 tan δ e =

11 TE and TM Dispersion Curves ε reff =, µ reff = e tan δ = β/k.5 TM.5 TE TM TE.5.5 TM TE 5 5

12 TE Modes: Magnetic Resonance Lossless case: tan δ e = kρ = β jα β/k 5 α/k 5 TE TE TE TE 5 TE 5 TE 5 5

13 TE Modes () Lossless case: tan δ e = kρ = β jα 9 β/k TE with µ reff = TE 9 α/k

14 TE Modes () Pole dynamics Lossless case: tan δ e = α/k TE.8.98 TE..95 β/k 5 TE TE TE 5 TE β/k

15 TE Modes () Lossy cases Pole dynamics α/k.77.5 α/k tan δ e =.5 tan δ e =. 8 8 β/k 8 5 β/k

16 TE Modes (5) Modal interaction (modal exchange) occurs with varying dielectric loss α/k α/k tan δ e =.5 β/k.59 tan δ e =.55 β/k

17 TE Modes () Lossy cases Phase constant tan δ e =.5 tan δ e =.55 β/k β/k

18 TE Modes (7) Lossy cases Attenuation constant tan δ e =.5 tan δ e =.55 α/k α/k

19 Complex Frequency-Plane Branch Points Type branch points () ω n - positive and negative solutions of dispersion equation meet forming a second-order zero Type branch points () ω n - separate branches of complex-conjugate solutions of dispersion equation (leaky-wave cutoff) Type branch points () ω n/ n + Complex frequency-plane branch points: - connect n and n+ different modes within a given class (TE or TM) ( ρ ω) k ( ρ ω) Z k, = Z k, = ( ω) kk ( ω) Z k, Z k, ω ρ ρ ρ ρ ρ Hanson and Yakovlev, Radio Sci.,, Nov.-Dec. 999 Hanson et al., IEEE Trans. Antennas Propag., 5, Feb. Yakovlev and Hanson, IEEE Trans. Antennas Propag., 5, Apr. () ω + A complete rotation about n/ n in the complex frequency plane results in the smooth interchange of the n and n+ TE or TM modes of the metamaterial slab

20 Complex Frequency-Plane Branch Points () ω /5 Evolution of branch point of TE modes and 5 parameterized by loss factor.5 Im[f ] (GHz) tan δ e = Re[f ] (GHz)

21 Interaction of TE modes: Lossy Cases The branch point is below, but close to, the real frequency axis The branch point is slightly above the real frequency axis α/k α/k tan δ e =.5 β/k.59 tan δ e =.55 β/k

22 TE Modes: Electric Resonance () Lossless case: tan δ e = kρ = β jα 5 β/k TE 5 α/k TE TE 5 TE TE TE

23 TE Modes: Electric Resonance () Lossless and lossy cases: TE mode α/k tan δ e = tan δ e = tan δ e =. tan δ e =. 5 7 β/k

24 TM Modes () Lossless case: tan δ e = kρ = β jα β/k 5 β/k 5 TM TM TM TM TM TM Qualitatively similar modal interactions occur for TM modes close to magnetic and electric resonances WORK IN PROGRESS

25 Conclusions

26 Conclusions Modal interactions of TE (TM) modes in the presence of loss are studied in the homogenized metamaterial slab composed of spherical dielectric inclusions by investigating a pole dynamics in the complex wavenumber plane It is observed that in the vicinity of electric and magnetic resonances of metamaterial slab the modal behavior is significantly perturbed and for some loss modal interactions occur resulting in the modal exchange Evolution of branch-point singularities in the complex frequency plane across the real frequency axis results in modal interactions (modal exchange) of TE (TM) modes, leading to the non-uniqueness of modal spectrum classification in lossy metamaterials