Introduction to

Yih-Peng Chiou (邱奕鵬)

Graduate Institute of Photonics and Optoelectronics &Department of Electrical Engineering

National Taiwan University

November 7, 2011

Left-handed Materials

2

Outline

Review of Electromagnetics

Negative Refractive Index

Making Left-Handed (Meta)materials

Some Other Behaviors

Transmission Line Approach

3

Outline

Review of Electromagnetics

Negative Refractive Index

Making Left-Handed (Meta)materials

Some Other Behaviors

Transmission Line Approach

4

Review of Electromagnetics

Polarization EP eχε 0=

litysusceptibi electric :eχ

Rao, Elements of Engineering Electromagnetics

5

Some Quantities in EM Waves

Rao, Elements of Engineering Electromagnetics6

Rao, Elements of Engineering Electromagnetics

7

Review of Electromagnetics

Electric displacement

Dispersive

Anisotropic

constant) c(dielectrity permittivi relative :)1( 000

r

re EEPEDε

εεχεε =+=+=

)(ωεε rr =

8

Review of Electromagnetics

Magnetization

01 μχχ BM

m

m

+=

litysusceptibi magnetic :mχ

9Rao, Elements of Engineering Electromagnetics

10Rao, Elements of Engineering Electromagnetics

11

Review of Electromagnetics

Magnetic field intensity

Dispersive:

Anisotropic

ty permeabili relative :

)1( 000

r

rm

BBMBH

μμμχμμ

=+

=−=

)(ωμμ rr =

12

Maxwell’s Equations

22

2

d EEdt

με∇ =Wave Equation:

13

Review of Electromagnetics

Phase velocityvelocity of the propagation of an equal phase surface

Group velocityVelocity of the propagation of a wave packet

Energy velocityVelocity of the propagation of the electromagnetic energy

pv

gv

ev

14

Questions

?

?

and of the same direction (movie)

always positive

pg vv =

HESk ×= //

pv gv

n0>v pi

15

Questions

?

and of the same direction? (movie)

always positive?

pg vv =

HESk ×= //

pv gv

n

?

16

Normally not

Dispersive materials

pg vv = ?

= =p gdv vd

ω ωβ β

17

Questions

?

and of the same direction? (movie)

always positive?

pg vv =

HESk ×= //

pv gv

n

?

18

HESk ×= // ?

K. Iizuka, Elements of Photonics

19

HESk ×= // ?

Not always!

K. Iizuka, Elements of Photonics

Usually not foranisotropic materials

20

Indicatrix Methodof Anisotropy

K. Iizuka, Elements of Photonics

21

refinc θθ ≠

K. Iizuka, Elements of Photonics 22

Questions

?

?

and of the same direction (movie)

always positive

pg vv =

HESk ×= //

pv gv

n ?

?0>v pi ?

Not Always!

24

Outline

Review of Electromagnetics

Negative Refractive Index

Making Left-Handed (Meta)materials

Some Other Behaviors

Transmission Line Approach

25

Victor G. Veselago, Soviet Physics, 1968 predicted EM wave behaviors in

left-handed materials (LHM) ε <0, μ <0 => n<0

0 0r rn με μ ε

μ ε= ± = ±

26

Some Non-Intuitive Properties of Negative Refraction

A beam incident on an LHM from an RHM refracts to the same side of the normal.

A point source impinging on a flat, parallel slab of LHM would be refocused to a point on the opposite side of the material.

Of suitable index, it can produce a focus with subwavelength resolution beating the normal diffraction limit associated with positive refractive index optics.

Misc. applications: e.g. open cavity.

27

θinc θrefl

θtrans

metamaterial

Free space

Snell’s Law

28

Refocusing and Open Cavity

29

Self-imaging

30

Simulation

Finite-Difference Time-Domain MethodUsing Dispersive ModelingDispersive PML

Taflove, Computational Electrodynamics –The Finite Difference Time Domain Method, Artech House, 2000

Normal NIM NIM-slab

31

Normal incident

Division points in one wavelength = 20;n = -1;

32

Oblique incident

Division points in one wavelength = 20;n = -1;The angle of incidence = 20 degree;

Division points in one wavelength = 40;n = -2;The angle of incidence = 20 degree;

33

Negative refraction index prism

Division points in one wavelength = 20;n = 2;

Division points in one wavelength = 20;n = -1;

34

Sub-diffraction limited focusing

35

Outline

Review of Electromagnetics

Negative Refractive Index

Making Left-Handed (Meta)materials

Some Other Behaviors

Transmission Line Approach

36

Victor G. Veselago, Soviet Physics, 1968 predicted EM wave behaviors in

left-handed materials (LHM) ε <0, μ <0 => n<0

0 0r rn με μ ε

μ ε= ± = ±

37

j ze β−

2= nπβλ

38

39

Permittivity, Permeability Reflection, and Refraction

μηε

=

2 1

2 1+η ηη η

−Γ =

40

Negative Permittivity (Permeability)

Polarization

Harmonically bounded oscillationPolarization out of φ of applied field (ω<ωo)Near resonance freq.Polarization overcome applied field

EP eχε 0=

litysusceptibi electric :eχ

41

Extremely low frequencyplasmons in metallic mesostructure

Periodic array ofmetal thin wires (TW)tunable in the GHz range

Pendry et al, PRL 76, 4773, 1996

lossless

42

Magnetism from conductorsand enhanced nonlinear phenomena

Periodic array of metalsplit-ring resonator (SRR)tunable in the GHz range

Pendry et al, MTT 47, 2075, 1999

lossless

43

The Building Blocks of LHM

2

2( ) 1 pωε ω

ω= −

2

2 20

( ) 1 Fωμ ωω ω

= −−

Electric DipolesElectric Dipoles Magnetic DipolesMagnetic Dipoles+

44

Real partAlways small than 1

Imaginary partAlways positive

Drude modelΓ = 1e8 s-1

f0 = 30 GHzωp = 2πsqrt(2) f0

-1

f0

It will intersect at 0 whenfrequency equals to

ωp = 266 GHz

Real partAlways small than 1

Imaginary partAlways positive

Drude modelΓ = 1e8 s-1

f0 = 30 GHzωp = 2πsqrt(7) f0

-6

f0

It will intersect at 0 whenfrequency equals to

ωp = 498 GHz

))(

1()(2

0e

pe

iΓ+−=

ωωω

εωε

))(

1()(2

0m

pm

iΓ+−=

ωωω

μωμ

Drude model:

45

Drude-Lorentz Model

J. B. Pendry et al., IEEE/MTT 47, 2075(1999)

Array of wire elements

Split−ring resonator (SRR)

46

Split rings and metal strips

Rings and strips on opposite sides of a fiber glass circuit board

Cell dimension: 5mmInput: X-band (8~12 GHz)

~30 mm @ 10GHz

D. R. Smith et. al., Science, 292, 77 (2001)

47

Left-Handed Materials for Real

D. R. Smith et. al., Physics Today, 17, May (2000).

Phys. Rev. Lett. 84, 4184 (2000) ; Science, 292, 77 (2001)48

49

Metamaterial

50

LHM Measurement

51

25.6 deg n = 1.41

18.4 deg. n = -1.0

52

ω: magnetic/electronic plasma/resonance frequencies

Total internal reflection?

53OptExp200304, v11, p640, Kolinko

Refraction from a wedge

54

Stepped pattern

55 56

Outline

Review of Electromagnetics

Negative Refractive Index

Making Left-Handed (Meta)materials

Some Other Behaviors

Transmission Line Approach

57

Some Other Behaviors of LHM

Sub-diffraction limited focusingInterchange of convergence and divergence effects in convex and concave lenses, respectively, when the lens is made LHMReversal of Doppler effectReversal of Vavilov-Cerenkov radiationReversal of the boundary conditions relating the normal components of the electric and magnetic fields at the interface between a conventional/righthanded (RH) medium and a LH mediumNegative space Magic transformation => invisible cloaks

58

Sub-diffraction limited focusing

59

f = R/(n − 1)

60

Negative index focusing lens

61

Reverse Doppler Effect in LHM

In RHM, EM waves undergo blue shift when they are reflected from a approaching object, or red shift from a leaving object

In LHM, it is reversed

cnvcnvff

/1/1' 0 +

−=

(LHM)

(RHM)

(LHM)

(RHM)n < 0

cnvcnvff

/1/1' 0 −

+=

62

Reverse Cerenkov Radiation

63

Boundary Conditions

antiparallel

antiparallel

64

Localized Surface Wave

65 66

Negative Space

67 68U. Leonhardt et al., Science 323, 110 -112 (2009)

Invisible Cloak

Coordinate transformation

69

J. B. Pendry et al., Science 312, 1780 -1782 (2006)

70

Realization of Invisible Cloak

D. Schurig et al., Science 314, 977 -980 (2006)

71D. Schurig et al., Science 314, 977 -980 (2006) 72D. Schurig et al., Science 314, 977 -980 (2006)

simulation

(reduced material properties)

73U. Leonhardt et al., Science 323, 110 -112 (2009)

Wider Operating Frequency

74

U. Leonhardt et al., Science 323, 110 -112 (2009)

3D cloaking

Wider Operating Frequency

75

New invisibility cloak allows object to 'see' out through the cloak

PRL 102, 093901 (2009) 76

New invisibility cloak allows object to 'see' out through the cloak

PRL 102, 093901 (2009)

77

Outline

Review of Electromagnetics

Negative Refractive Index

Making Left-Handed (Meta)materials

Some Other Behaviors

Transmission Line Approach

78

Transmission Line (TL) Approach

Equivalent circuit model of SRR

79

Two Different Approaches

Resonant StructureResonantLossyNarrow bandEM wavesNo systematic analysis

Transmission LineNon-resonantLow lossBroad bandCircuitsSystematic analysis

Backward waves have been analyzedMost mathematics are already there

Planar config. Microwave IC

Trade-off between BW and loss

Caloz and Itoh, Electromagnetic Metamaterials80

Incremental circuit model

Caloz and Itoh, Electromagnetic Metamaterials

81

Planar TL LH Structures

Caloz and Itoh, Electromagnetic Metamaterials82

Composite Right/Left-Handed (CRLH) Metamaterials

Infinite wavelength

Caloz and Itoh, Electromagnetic Metamaterials

83

Some Remarks

Photonics crystals or photonic band gap (PBG)Size p ~ nλ/2

Effective homogeneity in LHMSize p < λ/4

Properties of photonic crystals are essentially determined by the lattice, while the (refractive) properties of MTM are determined by the nature of the unit cell

84

Terminology

Left-handed (LH)originally suggested by Veselago

Doubly-negative (DNG)Negative-refractive-index (NRI)Negative index materials (NIM)Backward-wave (BW)Veselago mediumNegative phase velocity medium (NPV)