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Metamaterials with Negative Parameters
Advisor: Prof. Ruey-Beei Wu
Student : Hung-Yi Chien 錢鴻億 2010 / 03 / 04
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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
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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
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Wave Propagation in DNG Media
Ordinary medium Left-handed medium
Energy and wavefronts travel in opposite directions.
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Energy Density in DNG Media
Nondispersive medium Dispersive medium
nonphysical result
physical media : dispersive
Time-averaged density of energy
physical requirement :
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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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Negative Refraction in DNG Media
The angles of incidence and refraction have opposite signs.
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Negative Refraction in DNG Media
Rays propagate along the direction of energy flow. Concave lenses -> convergent Convex lenses -> divergent
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Negative Refraction in DNG Media
Focusing of energy
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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
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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
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Other Effects in DNG Media
Inverse Doppler effect Backward Cerenkov Radiation Negative Goos-Hänchen shift
Ordinary medium
DNG medium
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Waves at Interfaces
For TE wave,
Wave impedance
,2Re 0xk
,1Re 0xk for ordinary media
for DNG media
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Waves at Interfaces Transverse transmission matrix
Transmission and reflection coefficients
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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 !
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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!
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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)
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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
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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
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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.
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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.
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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
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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
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Slabs with and 0/ -1 0/ -1
If The amount of power tunneled through the devices decreases. The sensitivity is higher for larger slabs.
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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
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Slabs with and 0/ -1 0/ -1
Perfect lens
The fields are exactly reproduced at x=2d
Amplitude pattern
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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.