1

Metamaterials with Negative Parameters

Advisor: Prof. Ruey-Beei Wu

Student : Hung-Yi Chien 錢鴻億 2010 / 03 / 04

2

Outline

Introduction Wave Propagation Energy Density and Group Velocity Negative Refraction Other Effects

Waves at InterfacesWaves through DNG SlabsSlabs with 0/ -1 0/ -1 and

3

What are Metamaterials?

Artificial materials that exhibit electromagnetic responses generally not found in nature.

Media with negative permittivity (-ε) or permeability (-μ)

Focus on double-negative (DNG) materials Left-handed media Backward media Negative-refractive media

4

Wave Propagation in DNG Media

Ordinary medium Left-handed medium

Energy and wavefronts travel in opposite directions.

5

Energy Density in DNG Media

Nondispersive medium Dispersive medium

nonphysical result

physical media ： dispersive

Time-averaged density of energy

physical requirement :

6

Group Velocity in DNG Media

Backward-wave propagation implies the opposite signs between phase and group velocities.

Wavepackets and wavefronts travel in opposite directions (additional proof of backward-wave propagation)

7

8

Negative Refraction in DNG Media

The angles of incidence and refraction have opposite signs.

9

Negative Refraction in DNG Media

Rays propagate along the direction of energy flow. Concave lenses -> convergent Convex lenses -> divergent

10

Negative Refraction in DNG Media

Focusing of energy

11

Fermat Principle in DNG Media

Fermat principle :

The optical length of the actual path chosen by light maybe negative or null

The path of light is not necessary the shortest in time.

/t L c

12

Fermat Principle in DNG Media

For n = -1, optical length ( source to F1,F2) = 0 All rays are recovered at the focus. Focus points

Phase: the same Intensity: weak (due to reflection)

0/ -1

0/ -1

Wave impedances Match!

The source is exactly reproduced at the focus.

if

13

Other Effects in DNG Media

Inverse Doppler effect Backward Cerenkov Radiation Negative Goos-Hänchen shift

Ordinary medium

DNG medium

14

Waves at Interfaces

For TE wave,

Wave impedance

,2Re 0xk

,1Re 0xk for ordinary media

for DNG media

15

Waves at Interfaces Transverse transmission matrix

Transmission and reflection coefficients

16

Waves at Interfaces

Surface waves Decay at both sides of the interface

General condition for TE surface waves

Surface waves correspond to solutions

of following eq.

It has nontrivial solution if Z1+Z2=0 !

17

Waves through DNG Slabs

Transmission and reflection coefficients

Transmission matrix for a left-handed slab with width d

Z1=Z3 For a small value of d,

phase advance is positive!

18

Waves through DNG Slabs

Guided waves Consider the imaginary values of kx,1

Surface waves correspond to the solution of following eq.

(the poles of the reflection coefficient)

Volume waves

Surface waves(inside the slab)

19

Waves through DNG Slabs

Backward leaky waves Power leaks at an angle θ

Power leaks backward with

regard to the guided power

inside the slab

1cos Re /zk k

20

Slabs with and 0/ -1 0/ -1

Wave impedances of left-handed medium become identical to

that of free space. The phase advance inside the slab is positive, and can be

exactly compensated by the phase advance outside the slab. Zero optical length

Incidence of evanescent waves

Evanescent plane waves are amplified inside the DNG slab But evanescent modes do not carry energy.

0/ -1 0/ -1 and

21

Slabs with and 0/ -1 0/ -1

Perfect tunneling A slab of finite thickness (not too thick) Some amount of energy can tunnel through medium 2 (slab)

Tunneling of power is due to the coupling of evanescent waves generated at both sides of the slab.

22

Slabs with and 0/ -1 0/ -1

Perfect tunneling

Waveguide 1,5 : above cutoff Waveguide 2,3,4 : below cutoff Fundamental mode : TE10 mode Incidence by an angle higher than the critical angle

Excitation of evanescent modes in waveguide 2-4.

23

Slabs with and 0/ -1 0/ -1

Perfect tunneling

TE10 mode is incident from waveguide 1

Evanescent TE10 modes are generated in waveguide 2-4 Some power may tunnel to waveguide 5

24

Slabs with and 0/ -1 0/ -1

Perfect tunneling

0/ -1

0/ -1 In the limit

Total transmission is obtained for the appropriate waveguide length

25

Slabs with and 0/ -1 0/ -1

If The amount of power tunneled through the devices decreases. The sensitivity is higher for larger slabs.

26

Slabs with and 0/ -1 0/ -1

Perfect tunneling when

Maximum of power transmission

Field amplitude when

Dash line : the amplitude when waveguide 3 is empty

27

Slabs with and 0/ -1 0/ -1

Perfect lens

The fields are exactly reproduced at x=2d

Amplitude pattern

28

Slabs with and 0/ -1 0/ -1 Comparison

Veselago lens

A point source is focused into 3-D spot.

The radius of spot is not smaller than a half wavelength.

Pendry’s perfect lens

The fields at x=0 are exactly reproduced at x=2d.

2-D spot

The size of spot can be much smaller than a square wavelength.