1. 1 Metamaterials with Negative Parameters Advisor: Prof. Ruey-Beei Wu Student : Hung-Yi Chien 錢鴻億 2010 / 03 / 04

2. 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. 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. 4 Wave Propagation in DNG Media Ordinary medium Left-handed medium Energy and wavefronts travel in opposite directions.

5. 5 Energy Density in DNG Media Nondispersive mediumDispersive medium  nonphysical result  physical media ： dispersive Time-averaged density of energy  physical requirement :

6. 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. 7

8. 8 Negative Refraction in DNG Media The angles of incidence and refraction have opposite signs.

9. 9 Negative Refraction in DNG Media Rays propagate along the direction of energy flow.  Concave lenses -> convergent  Convex lenses -> divergent

10. 10 Negative Refraction in DNG Media Focusing of energy

11. 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. 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. 13 Other Effects in DNG Media Inverse Doppler effect Backward Cerenkov Radiation Negative Goos-Hänchen shift Ordinary medium DNG medium

14. 14 Waves at Interfaces For TE wave,  Wave impedance for ordinary media for DNG media

15. 15 Waves at Interfaces  Transverse transmission matrix  Transmission and reflection coefficients

16. 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. 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. 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. 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. 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. 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. 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. 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. 24 Slabs with and Perfect tunneling  In the limit Total transmission is obtained for the appropriate waveguide length

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

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

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

28. 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.