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 Interfaces Waves through DNG Slabs Slabs with 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 mediumDispersive 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)

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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.

12 Fermat Principle in DNG Media For n = -1, optical length ( source to F 1,F 2 ) = 0  All rays are recovered at the focus.  Focus points Phase: the same Intensity: weak (due to reflection) 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 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 Z 1 +Z 2 =0 !

17 Waves through DNG Slabs Transmission and reflection coefficients  Transmission matrix for a left-handed slab with width d Z 1 =Z 3  For a small value of d, phase advance is positive!

18 Waves through DNG Slabs Guided waves  Consider the imaginary values of k x,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

20 Slabs with and  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 I ncidence of evanescent waves  Evanescent plane waves are amplified inside the DNG slab  But evanescent modes do not carry energy. and

21 Slabs with and 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 Perfect tunneling  Waveguide 1,5 : above cutoff  Waveguide 2,3,4 : below cutoff  Fundamental mode : TE 10 mode  Incidence by an angle higher than the critical angle Excitation of evanescent modes in waveguide 2-4.

23 Slabs with and Perfect tunneling  TE 10 mode is incident from waveguide 1  Evanescent TE 10 modes are generated in waveguide 2-4  Some power may tunnel to waveguide 5

24 Slabs with and Perfect tunneling  In the limit Total transmission is obtained for the appropriate waveguide length

25 Slabs with and If  The amount of power tunneled through the devices decreases.  The sensitivity is higher for larger slabs.

26 Slabs with and  Perfect tunneling when  Maximum of power transmission  Field amplitude when  Dash line : the amplitude when waveguide 3 is empty

27 Slabs with and Perfect lens  The fields are exactly reproduced at x=2d  Amplitude pattern

28 Slabs with and 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.