Transmission Line (TL) Approach of Left-Handed (LH) Materials

Transcription

1 Transmission Line (TL) Aroach of Left-Handed (LH) Materials Christohe Caloz, Hiroshi Okabe, Taisuke Iwai and Tatsuo Itoh Electrical Engineering Deartment University of California, Los Angeles

2 LH-TL as the Dual of the Conventional TL () Conventional RH-TL (lossless) LH-TL (lossless) Z ( jl) dz Z ( jc)dz Y ( jc) dz low - ass Y ( jl)dz high - ass γ j ZY j γ j ZY j ( ) linear ( ) nonlinear v v v g > d d > v cste no distortion g v v v g < + > v fct( ) g distortion η Z Y ( jl) ( jc) L C η ( j C) ( jl) L C

3 LH-TL as the Dual of the Conventional TL () Conventional (RH) TL LH-TL z ro. + z ro. ( ) sloe v > sloe v < g ϕ sloe v v > g { } from the [ ABCD]matrix S ϕ { S } - + z ro. z ro. (energy) (energy) ϕ { } from the [ ABCD]matrix S length : m ϕ as ϕ as f (MHz) f (MHz)

4 Determination of LH Material Parameters ε & µ Relations TL / Field for a lane wave in an (RH) medium: Z jl jµ, Y jc jε Imedance / admittance in terms of LH-TL arameters: Z ( C) µ () ( j C), Y ( j ) () L Equating () and () yields the disersive ε & µ: µ µ µ <! ( C) ( L) ε < ε ε ε! ( L) Demonstration of negative ε and µ in the TL-LH material Exlicit exressions for ε and µ in this material

5 Demonstrations for ε, µ and n ε & µ satisfy entroy conditions for disersive materials: (L. D. Landau, E. M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon, 984) Total energy : ( ε ) ( µ ) ( ε ) ( µ ) > W and E + >, H > ( ε ) Demonstration of negative index of refraction & determination of its exlicit exression: LH - TL : j Material : ZY ZY j c ( µ ) ( jµ )( jε ) + j µ rε r L C > > n µ rε r < c n!

6 Lumed-Element Arox. of the Physical (RH) Line Physical Line Infinitesimal model Lumed-el. Arox. e.g.:coaxial line TL model: L R Ω ( m) ( H m) unit cell: R L u u σ G ( S m) C ( F m) G u C u ε r R i tanδ R o dz u N hysical arameters: Ri, Ro, ε r, tan δ, σ, TL arameters: µ Ro L ln, π Ri πε ' C ln R, o R i ( ) etc. line of L H RΩ R, G diagram : ) γ ) L length, C F S : C G ( R + jl)( G + jc) ( ) Im γ ( ) { }, L uh ladder circuit : N diagram : ) L [ S] ( ) ϕ{ ( ) } ) scattering matrix u L S ( N ), etc.

7 Lumed-Element Arox. of the LH Line Physical Line Infinitesimal model Lumed-el. Arox. Not existing naturally!!! TL model: R Ω C ( m) ( F m) unit cell: R C u u G ( S m) L ( H m) G u L u dz u N TL arameters (er unit length) can be rescribed arbitrarily : C,L,R,G line of length : L L, C C, diagram : ) γ ) H F Ω, R R GS G ( R + jl)( G + jc) ( ) Im γ ( ) { } CuF C u C ladder circuit : N diagram : ) [ S] ( ) ϕ{ ( ) } ) scattering matrix S ( N ), etc.

8 Results for the Physical Model of a RH-TL Magnitude of S- arameters Physical coaxial line : P R i.6 mm, Re 9.5 mm,., m ε r P Corresonding TL arameters : L 56 nh/m, C 47 F/m, R. 8 mω/m, G S/m L.56 µ H, C 47 F, R 8 mω, G S H F Ω S (some mismatch was intentionally introduced to create S eaks: Zin 7 Ω with orts Zin 5 Ω ). Cumulative Phase of S- arameters + z ro. hase unwraing ( ) ϕ{ S ( ) } sloe v v >

9 Lumed-Element Arox. (RH-TL) N LOW - PASS Magnitude of aroximat ion cutoff ( f ) c S- arameters f f c N arox. Cumulative Phase of S- arameters (unmatched) (linear) deendency N eaks N aroximat ion cutoff ( f ) c

10 Phys. Model vs L.E. Arox. (RH-TL) / Matched TL Magnitude of S- arameters Cumulative Phase of S- arameters Comarison Magnitude of S- arameters Cumulative Phase of S- arameters λ Matched RH - TL

11 Lumed-Element Realization of a LH-TL () N HIGH - PASS Magnitude of S- arameters f f c N arox. Cumulative Phase of S- arameters aroximat ion cutoff ( f ) c λ!!! (matched) aroximat ion cutoff ( f ) c N eaks N deendency ( )

12 Lumed-Element Realization of a LH-TL () Magnitude of S- arameters Phase of S- arameters > lossless > unlimited BW Phase and grou velocities higher frequencies behavior hase unwraing > unlimited BW > moderate disersion ϕ as - diagram ϕ as v g + ( ) ( ) lu.el.arox. cutoff ( f ) c cutoff v

13 Possible Distributed Realization of the LH Line microstri line T-junction series interdigital caacitor shunt siral inductor unit cell via to ground Features: LH bandwidth limited by Q-factor of C-L, but still broad still very low losses and moderate disersion

14 The Phase Paradox of LH Materials In general: S Paradox :Since LH <, jϕ j + ( jt e ) S e S e we might exect oosite hase variation... Conventional (RH) TL ϕ ϕ "increases negatively" ϕ LH-TL ( ) ϕ + ( ) ϕ "decreases ositively":idem! ϕ start : λ ( ) because λ π π ( ) start : λ ( ) because λ π π!!! In a LH material, hase rotates in the same sense as in a RH material.

15 General Considerations many other circuits exhibit a LH effect (e.g. eriodic) low losses, broad bandwidth, moderate disersion otential interest for microwave alications ossible transosition of the distributed LH-TL to a D/3D LH material constitutive arameters different than in other aroaches: ε F ( ), µ ( ) ε ( ), µ ( ) ossible link with the slit-rings / wires structure: E H LH - TL L C

16 Plane Wave Proagation at Interface RH-LH () Normal Incidence - diagram for the RH and LH media E field amlitude as a fct of distance LH ( ) RH ( ) RH LH LH RH C L f 47 F/m, 56 nh/m, MHz RH LH RH LH v > < RH medium LH v medium (.44) (.39) RH LH

17 Plane Wave Proagation at Interface RH-LH () Oblique Incidence (Γ) ϑ inc 45, ϑtrans 8 (Snell's law, n ± ε rµ r ) RH LH RH (.65) ε r L, µ r C LH (.34) ε /( L), µ /( C) r r ( C 47 F/m, L 56 nh/m, f 3 MHz)

18 Plane Wave Proagation at Interface RH-LH (3) Oblique Incidence ( Γ >) ϑ inc 45, ϑtrans 8 (Snell's law, n ± ε rµ r ) RH LH RH (.65) ε r L, µ r C LH (.34) ε /( L), µ /( C) r r ( C 47 F/m, L 56 nh/m, f 3 MHz)