Metamaterials with Negative Parameters

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

2. Introduction Wave Propagation Energy Density and Group Velocity Negative Refraction Other Effects Waves at Interfaces Waves through DNG Slabs Slabs with Outline 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 Time-averaged density of energy Nondispersive medium Dispersive medium • nonphysical result • physical requirement : • physical media ：dispersive

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)

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 F1,F2) = 0 • All rays are recovered at the focus. • Focus points • Phase: the same • Intensity: weak (due to reflection) if Wave impedances Match! The source is exactly reproduced at the focus.

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

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

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 • For a small value of d, phase advance is positive! Z1=Z3

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 (inside the slab) Surface waves

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

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 : TE10 mode • Incidence by an angle higher than the critical angle • Excitation of evanescent modes in waveguide 2-4.

23. Slabs with and • 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 • 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.