Microwave Electronics Lab

Transmission Line (TL) Approach ofLeft-Handed (LH) Materials

Christophe Caloz, Hiroshi Okabe, Taisuke Iwai and Tatsuo Itoh

Electrical Engineering DepartmentUniversity of California, Los Angeles

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LH-TL as the Dual of the Conventional TL (1)Conventional RH-TL (lossless) LH-TL (lossless)

LCjZYjβγ ω===

linear→= LCωβ

>==

>==

01

01

LCddv

LCv

g

p

βω

βω

distortion no→== cstevv gp

( )LCjZYjβγ ω−===

( ) nonlinear1 →−= LCωβ

>+=

<−=

0

02

2

LCv

LCv

g

p

ω

ω

distortion)( →=−= ωfctvv gp

dzLjZ )( ω=

dzCjY )( ω=pass-low

( )dzCjZ ω1=

( )dzLjY ω1=pass-high

( ) ( ) CLCjLjYZ === ωωη ( ) ( ) CLLjCj == −− 11 ωωη

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LH-TL as the Dual of the Conventional TL (2)Conventional (RH) TL LH-TL

( )LCωβ 1−=LCωβ =

β

0slope >= gv 0slope <= pv

prop. z+(energy)

prop.z−(energy)

ωprop.z+prop. z−

β

pv=slope0>= gv

ω

[ ]matrix thefrom S21 ABCDϕ

(MHz) f

p-βS ⋅=21ϕ [ ]matrix thefrom S21 ABCDϕ

(MHz) f

0→ϕ∞→ωas

0 as →ω∞→ϕ

= m 10:length

p

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Determination of LH Material Parameters ε & µ

)1(, ωεωωµω jCjYjLjZ ====• Relations TL / Field for a plane wave in an (RH) medium:∞

( ) ( ) )2(1,1 LjYCjZ ωω ==• Impedance / admittance in terms of LH-TL parameters:

( )

( )C

C

21!0

1

ωµµ

µωωµω

−=<

⇒

−==−

• Equating (1) and (2) yields the dispersive ε & µ:

( )

( )L

L

21!0

1

ωεε

εωωεω

−=<

⇒

−==−

→ Demonstration of negative ε and µ in the TL-LH material→ Explicit expressions for ε and µ in this material

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Demonstrations for ε, µ and n

• ε & µ satisfy entropy conditions for dispersive materials: (L. D. Landau, E. M. Lifshitz and L.P. Pitaevskii, “Electrodynamics of Continuous Media”, Pergamon, 1984)

( )

( )

>=∂

∂

>=∂

∂

01

01

C

L

ωωµωωε

√

√

( ) ( )

( ) ( )

∀>∂

∂>

∂∂

⇒

>∂

∂+

∂∂

=

ωωωµ

ωωε

ωωµ

ωωε

,0and0

0 :energy Total 22 HEW→

• Demonstration of negative index of refraction& determination of its explicit expression:

( )( ) LCcn

n

cjjjZY

LCjZY

rr

rr2

0

0

!0

:Material

1j :TL-LH

ω

εµ

εµωωεωµ

ωβ

−=

<=⇒

+==

−==

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Lumped-Element Approx. of the Physical (RH) LinePhysical Line Lumped-el. Approx.Infinitesimal model

( )( )( ) ( ) ωγωβ

ωωγ

βω

Im 2)

1)

:diagram

=

++=

−•

CjGLjR

pGGpRRpCCpLL

p

S

FH

⋅=⋅=⋅=⋅=

•

Ω ,,,

:length of line

line coaxial :e.g. •

pδRR roi ,,tan,,, :parameters physical

σε•

iRoRrε

δtan

σ

p

,ln2

:parameters TL

i

o

RRL

πµ

=

•

( ) etc. ,ln

'2

io RRC πε=

[ ]( ) ( ) p

Sωϕωβ

βω

21S )2matrix scattering 1)

:diagram

−=

−•

:circuitladder •…1 2 N

:cellunit •uR uL

uG

Nppu =

≡

( ) etc. ,NpLpLL uuH ⋅=⋅=

uC

0→dz

:model TL •( )mHL

( )mSG

( )mR Ω

( )mFC

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Lumped-Element Approx. of the LH LinePhysical Line Lumped-el. Approx.Infinitesimal model

( )( )( ) ( ) ωγωβ

ωωγ

βω

Im 2)

1)

:diagram

=

++=

−•

CjGLjR

!!!naturally existingNot •

C,L,R,G

:yarbitraril prescribed becan

length)unit (per parameters TL •

[ ]( ) ( ) p

Sωϕωβ

βω

21S )2matrix scattering 1)

:diagram

−=

−•

:circuitladder •…1 2 N

:cellunit •uR uC

uG

Nppu =

≡

( ) etc. ,NpCpCC uuF ==

uL

0→dz

:model TL •( )mFC ⋅

( )mSG

( )mR Ω

( )mHL

⋅

:length of line p•

pGGpRR S ⋅=⋅=Ω ,,, pCCpLL FH ==

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Results for the Physical Model of a RH-TL:line coaxial Physical •

(MHz)frequency

parameters-S of Phase Cumulative

β

pv=slope0>= pv

parameters-S of Magnitude

(MHz)frequency

(some mismatch was intentionallyintroduced to create S11 peaks:

Zin = 71 Ω with ports Zin = 50 Ω ).

prop.z+ω

unwrapping phase

( ) ( ) pωϕωβ 21S−=

pS 10,m 128pF, 470H, 56.2 =Ω=Ω

==S

GRF

CH

L µ

:parameters TL ingCorrespondpS/m 1/m,m 812pF/m, 47nH/m, 256 =Ω=== G.RCL

P1 P2mm, 5.9 mm, 6.2 == ei RR

m 10 ,2.1 == prε

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parameters-S of Magnitude

(MHz)frequency

ionapproximat)(cutoff cf

1−= NN peaks

Lumped-Element Approx. (RH-TL)

↓⇒↑∝approx.f

Nfc

20=N

(MHz)frequency

parameters-S of Phase Cumulative

ionapproximat)(cutoff cf

dependency (linear) ωLCωβ =→

PASS-LOW

)(unmatched

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Phys. Model vs L.E. Approx. (RH-TL) / Matched TLparameters-S of Magnitude

(MHz)frequency (MHz)frequency

parameters-S of Phase Cumulative

parameters-S of Phase Cumulative

(MHz)frequency

TL-RHMatched

Comparison

parameters-S of Magnitude

(MHz)frequency

ωλ 1∝

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(MHz)frequency

parameters-S of Phase Cumulative

ionapproximat)(cutoff cf

dependency 1 ω( )LCωβ 1−=→

Lumped-Element Realization of a LH-TL (1)

20=NPASS-HIGH

parameters-S of Magnitude

(MHz)frequency

1−= NN peaks

ionapproximat)(cutoff cf

(matched)

↓⇒↓∝approx.

1f

Nfc

!!!ωλ ∝

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parameters-S of Magnitude

Lumped-Element Realization of a LH-TL (2)

behaviorsfrequencie

higher

( ) ( )LCββω 1−=

)(cutoff cf approx. el. lu.

diagram ω-β

> lossless> unlimited BW

(MHz)frequency

parameters-S of Phase

0→ϕ∞→ωas

0 as →ω∞→ϕ

ω1∝

> unlimited BW> moderate dispersion

unwrappingphase

(MHz)frequency

s velocitiegroup and Phase

(MHz)frequency

LCvg2ω+=

LCvp2ω−=

cutoff

β

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Possible Distributed Realization of the LH Linemicrostrip

lineseries

interdigitalcapacitor

shuntspiral

inductor

T-junction

via toground

unit cell

Features:• LH bandwidth limited by Q-factor of C-L, but still broad• still very low losses and moderate dispersion

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ϕ↑ω

( ) 000 :start==

=ωλω

p↑ω

ϕ↑ω

( ) 0 :start

=∞=∞=

ωλω

p

↓ω

The “Phase Paradox” of LH Materials

Conventional (RH) TL LH-TL

( )tjpjj eeSeSS ωβϕ +⋅−== 212121 :generalIn

"negatively increases" ϕωωϕωβ

↑⇒−=→= pLCLC

... variationphase oppositeexpect might we,0 Since :Paradox <LHβ

( ) ( )idem! :"positively decreases"

1ϕω

ωϕωβ↑⇒

+=→−= LCpLC

( ) ωωπβπλ 122 ∝== LCbecause

!!!22 ωπωβπλ ∝== LCbecause

In a LH material, phase rotatesin the same sense as in a RH material.

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General Considerations

• many other circuits exhibit a LH effect (e.g. periodic)• low losses, broad bandwidth, moderate dispersion

→ potential interest for microwave applications• possible transposition of the distributed LH-TL to a

2D/3D LH “material”• constitutive parameters different than in other approaches:

• possible link with the split-rings / wires structure:

( ) ( ) 20

2

2

2

2

1,1ωω

ωωµωω

ωε−

−=−=Fp ( ) ( )

CL 22

1,1ω

ωµω

ωε −=−=↔

≡ TL-LH

E

Hβ

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Plane Wave Propagation at Interface RH-LH (1)Normal Incidence

===

MHz 20 nH/m,256

pF/m,47

0fLC

β

ω

0ω

LHβ RHβ

( ) LCRH ωωβ =( ) LCLH ωωβ 1−=

media LH and RH for the diagram -ωβ

↓↑

⇒↑LH

RH

ββ

ω

distance offct a as amplitude field−E

0>RHpv 0<LH

pv

medium RH medium LH( )44.0=RHβ ( )39.2−=LHβ

RHβ LHβ

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Plane Wave Propagation at Interface RH-LH (2)

Oblique Incidence (Γ=0)) law, s(Snell'18,45 rrtransinc n µεϑϑ ±=°−=°=

( )34.20

1−=−=

LCLH ωβ( )65.00 =−= LCRH ωβ

MHz) 30 nH/m,256 pF/m,47( 0 === fLC

CL rr == 11 , µε )/(1),/(1 21

21 CL rr ωµωε −=−=

LHβRHβ

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Plane Wave Propagation at Interface RH-LH (3)

Oblique Incidence (|Γ|>0)) law, s(Snell'18,45 rrtransinc n µεϑϑ ±=°−=°=

( )34.20

1−=−=

LCLH ωβ( )65.00 =−= LCRH ωβ

MHz) 30 nH/m,256 pF/m,47( 0 === fLC

CL rr == 11 , µε )/(1),/(1 21

21 CL rr ωµωε −=−=

LHβRHβ