Left-handed materials• ordinary right-handed (RH) materials:

E

H, B

k, S

• left-handed (LH) materials:

E

H

S

k

• LH materials first theoretically considered by V. Veselago in 1968 (Sov. Phys. Usp., v.10 p.509 (1968)), first experimentally demonstrated by D. Smith in 2000 (Phys. Rev. Lett., v.84 p.4184 (2000))

Split ring resonator. c = 0.8 mm, d = 0.2 mm, and r = 1.5 mm. Resonance at about 4.845 GHz, Q = f / f = 600

*D. Smith, et al. (op. cit) (left), Shelby, et al. (above), D. Smith, et al. (right)

B

LH = “anti-space”

Canonical MIM structure and method of analysis

• Fields assumed harmonic time dependent, analytic solution by matching boundary conditions at each interface.

• Modes can be classified as TE (Hx, Ey, Hz) or TM (Ex, Hy, Ez).

• Modes further classified:– Surface modes: evanescent fields in core

– Oscillating guided modes: oscillatory fields in core

• Dispersion relation relates = k0neff and guidewidth 2a for a given set of core and cladding materials:

– Geometric dispersion: variation of with guidewidth at fixed frequency

– Frequency dispersion: variation of with frequency at fixed guidewidth

• Global power flux:

0 2a

z

x

clad core

region 2 region 1 region 3

clad

dxHEdxSP zz Re2

1 *

Note: Systematic study of modes supported by structure first done by Prade, et al., with mention of left-handed behavior (PRB 44 13556 (1991)).

Geometric vs. frequency dispersion

Geometric: Frequency:

• Region of negative group velocity indicates left-handed behavior of surface mode.• Parity of TM modes determined by H, parity of TE modes determined by E.

• Ag/Si/Ag at 580nm, core = 16, clad = -14.6 for geometric dispersion.

• core = 16, 2a = 30nm, lossless Drude parameters for Ag for frequency dispersion.

Geometric dispersion and global power flux

Above are typical geometric dispersion and global power flux plots

(Ag/Si/Ag at 580nm, core = 16, clad = -14.6)

The origin of left-handed behavior

• Power flux for TE modes:

cladding)or (core region each in 2

1Re

2

1 2

,eff* iEnHES iyziiiz

• Power flux for TM modes:

iHn

HES iyi

ziiiz region each in 2

1Re

2

1 2

,eff*

The TM odd surface mode can be left-handed; however, the TM even surface mode can be shown to be never left-handed.

The higher order TM oscillating guided modes can be left-handed for some material parameters and some guide widths near their cutoffs.

Conclusion: TE modes always right-handed.

Conclusion: power flux in metal regions is negative and TM modes can be left-handed.

Determination of regimes for left-handed behavior

• Structure is first assumed to be lossless.• LH modes are those for which the global power flux is in the opposite direction to the

propagation constant. Below, k0a is the normalized core half-width, and

Mode type Regime 1 Regime 2 Regime 3 Regime 4

odd surface

(always TM)always left-handed

left-handed for some k0a for

always right-handed

always right-handed

even surface

(always TM)doesn’t exist always right-handed

TMO0 always left-handedalways left-handed left-handed for

some k0a foralways right-handed

TME1

left-handed for some k0a for always right-handed

TMO1

left-handed for some k0a for always right-handed

TME2

left-handed for some k0a for always right-handed

TMO2

left-handed for some k0a for always right-handed

TE modes always right-handed

coreclad

10

35.1

245.0

085.0

039.0

025.0

22.1

22.11 35.122.1 35.1

A closer look at left-handed behavior

X

TM even mode of order 1 (TME1)

dxSP z

EvWS z

Addition of loss to the cladding regions

• Addition of loss significantly changes power flux in metal regions• In particular, if the loss is high enough, the power flux in the metal can even become

positive.• The power flux in the metal regions is now:

Mode typeMinimum loss (104 cm-1)

TMO0/surface 0.792

TME1 2.99

TMO1 2.92

TME2 2.90

2

2

clad

effcladeffclad* ImImReRe

2

1Re

2

1H

nnHES zz

• Parameters shown are for semiconductor core (n = 3.4) and plasma doped cladding at mid-IR frequencies (visible and THz frequencies were also investigated).

• Oscillating guided modes have high loss near cutoff guidewidths, and in some cases even become right-handed.

• High gain for mid-IR QC laser would be g ~102 cm-1, corresponding to a Q=200; thus, a low-loss left-handed mode would be difficult, but not impossible to realize.

Modal loss at 10 um

Design of THz QC nanoresonator structure

QC active region

Ag cladding

Ag cladding

• Operating wavelength = 100um.

• E54 = 13 meV, dipole moment, z54 = 6.1 nm

• Size of single stage of active region ~ 50 nm.

• Systematic study of cavity enhancement with scaling of device size.

• Lz = =k0neff) (fundamental mode half-wavelength) [5-10 m], while Lx and Ly varied over range spanning current laser device dimensions (~mm to 10um) to nanoresonator dimensions (1um to 0.1um).

x

y

z

Lx=2a

Lz

Ly

Veff = Leff Ly Lz = ~/2 |Ephpeak|2

B. Williams, et al.,Optics Express, v13(9), 2005.

10 m

Fabry-Perot cavity with negative index region

• Impedance matching of the two materials eliminates reflection at positive index-negative index interface:

nnn

• Cavity then behaves as an ordinary Fabry-Perot etalon, but with a potentially very large FSR given its physical size:

0

0,

0,

nn

0

0,

0,

nn

L+ L-

0

0,

0,

1

1

1

nn

0

0,

0,

1

1

1

nn

22

LLn

c

LnLn

cFSR

Frequency shift for small index/length modulation

number. mode theis where

,2

,2

2

2

m

LLnLn

mcn

nLnLn

mcL

• Frequency shift for small modulation in n+ or L+:

FSR

• For n+ = 3.4, = 1,

n- = 10.2, = 3 (impedance match),

L+ = 6m + 1um, L- = 2m,

air background: FSR = 44 THz (6.8 um in free space) Peak FWHM = FSR/F = 18 THz

• To shift one FWHM: n+ = 2.3 x 10-7, or

L+ ~ 1um

Device behaves as a large interferometer in terms of sensitivity, but has the spectrum of a wavelength-scale device

for identical electronic and magnetic Lorentizian oscillators

• In the above plots, the parameters for both electronic and magnetic resonances are taken to be identical for simplicity. All frequencies are normalized to the resonant frequency of one of the oscillators. Here both oscillators share the same resonant frequency. The parameters are:

1 ,1.0 ,1 ,1

,1 ,1.0 ,1 ,1 0

cmmrbg

ceebg

The relative permittivity and permeability response functions, dispersion relation, coefficients of refraction and extinction

iii

iii

mr

mcm

mr

rcmbg

mr

cmbg

e

ece

e

cebg

e

cebg

22222

2

22222

222

22

2

222220

2

222220

220

2

220

2

)()(

)()(

)()(

)()(

.conventionby 0 always

ocity energy vel thekeep chosen to becan root square theofsign The

)(2

1)sign(Im

,2

1Re

n

k 2 coeff. gain)(or absorption The

where,)()(

,)()(

:relation dispersion obtain the we

,equation wave theinto )](exp[ ansatz planewave thengSubstituti

2

22

2

2

22

0

kikc

ic

n

ck

ck

t

E

cEtkziEE

Dispersion diagram for identical resonances

• This dispersion diagram clearly shows the boundaries between “positive index”, “right-handed”, “negative index” and “left-handed” regimes.

• The convention is that the energy velocity (shown on the next slide) is always taken to be a positive quantity; thus the regions of negative real part of k indicate “negative index” (when group velocity is negative) or “left-handed” (when group velocity is positive) behavior.

Phase velocity, group velocity and energy velocity plots

• Note that the group velocity matches the energy velocity closely except in the immediate neighborhood of the resonance.

• Note also that all 3 velocities asymptote to the same values far away from the resonance, as would be expected (0.5c at low freq., c at high freq.).

• The locations of the singularities in the group velocity and phase velocity plots can be confirmed by looking at the dispersion relation diagram.

Both oscillators have gain

1 ,1.0 ,1 ,1

,1 ,1.0 ,1 ,1 0

cmmrbg

ceebg

• Here both the electronic and magnetic oscillators have gain. We see that there is gain for all frequencies, including the region of left-handed behavior. The discontinuities in the dispersion diagram that arose from keeping the energy velocity always positive are gone (they are only present when one material has loss and the other has gain).

Final Question…

FSR

In the balanced 8 m long half-wavelength F-P cavity, how is it that the cavity mode linewidth () is not decreasing with the physical cavity length; i.e., the photon lifetime is not getting any longer even though the end-mirrors are moving farther and farther apart? What if the linewidth did get narrower?