1 How a wave packet propagates at a speed faster than the speed of light A novel superluminal mechanism with high transmission and broad bandwidth Tsun-Hsu Chang (張存續) Department of Physics, National Tsing Hua University Claim: The phenomena we present here do not violate the special relativity, which is a cornerstone of the modern understanding of physics for more than a century. Outline Introduction (evanescent wave) Matter wave and electromagnetic wave Modal analysis (a 3D effect) New superluminal mechanism (propagating wave) Manipulating the group delay Conclusions Acknowledgement 2
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1
How a wave packet propagates at a speed faster than the speed of light
A novel superluminal mechanism with high transmission and broad bandwidth
Tsun-Hsu Chang (張存續)Department of Physics, National Tsing Hua University
Claim: The phenomena we present here do not violate the special relativity, which is a cornerstone of the modern understanding of physics for more than a century.
Outline
Introduction (evanescent wave)
Matter wave and electromagnetic wave
Modal analysis (a 3D effect)
New superluminal mechanism (propagating wave)
Manipulating the group delay
Conclusions
Acknowledgement
2
The Fastest PersonUsain Bolt is a Jamaican sprinter widely regarded as the fastest person ever. 100 m in 9.58 s, Speed ~ 10 m/s
.[
3
Top Speed of Racing Car: Formula 1
The 2005 BAR-Honda set an unofficial speed record of 413 km/h at Bonneville Speedway. Speed ~ 115 m/s
.[
4
Flight Airspeed Record: SR-71 Blackbird
The SR-71 Blackbird is the current record-holder for a manned air breathing jet aircraft. 3530 km/h ~ 980 m/s
5
Controlled Flight Airspeed Record: Space Shuttle
Fastest manually controlled flight in atmosphere during atmospheric reentry of STS-2 mission is 28000 km/h ~ 7777 m/s.
6
Highest Particle Speed: LEP ColliderThe Large Electron–Positron Collider (LEP) is one of the largest particle accelerators ever constructed. The LEP collider energy eventually topped at 209 GeV with a Lorentz factor γ over 200,000. LEP still holds the particle accelerator speed record.
Matter cannot exceed the speed of light in vacuum.
12
2
202
1(1 ) 0.999999999988
just millimeters per second slower than .
1
v
c
c
mE c
βγ
β
= = − =
=−
How about wave?
10
7
The index of refraction n(ω) is a function of frequency.
g
( )Phase velocity: (7.88)
( )
Group velocity: (7.89
( )Grou
)( ) ( )
p delay:
p
g
g
k cv
k n kd c
vdk
d d kL Ld d
n dn d
vφτ
ω
ω
ω
ω ω ω
ω
≡ =
≡
≡ =
=+
≈
8
Superluminal Mechanism: Anomalous dispersion
( )( )ck
n kkω
=
See waves in a dielectric medium [Jackson Chap. 7]
Anomalous Dispersion: Waves in a dielectric medium
Properties of ε: When ω is near each ωj (binding frequency of the jth group of electrons), ε exhibits resonant behavior in the form of anomalous dispersion and resonant absorption.
20
0 2 2 (bound) 0
2
( )ε ε
ω γ ωω ω ωγ= + +
−− −j
j j j
f Ne fi
m iiNem
(7.51)
0negligible ( 0 or very small) f =
ω
Reε
Imε
09
PA: Polyamides are semi-crystalline polymers.The data was measured with a THz-TDS system.
10
The tunneling effect
The microwave propagating in a waveguide system seems to be analogous to the behavior of a one-dimensional matter wave.
L
E
V
V0
I II III
2( )?
E Vv
m
−= =
Comparing with the matter wave, the electromagnetic wave is much more easier to implement in experiment.
11
Anomalous dispersion and tunneling effect are the two major mechanisms for the superluminal phenomena.
Both mechanisms involve evanescent waves, which means waves cannot propagate inside the region of interest.
Summary #1
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Part II. Analogies Between Schrödinger’s Equation and
Maxwell’s Equation
13
Analogies Between Schrodinger and Maxwell Equations
Maxwell’s wave equationfor a TE waveguide mode
Time-independent Schrodinger’s equation
0)(]2
)(2
[222
2
=+−∂∂
zEm
zVm
zϕ
0))((
2
2
2
2
2
2
=+−∂∂
zc B
cz
cz
ωμεωμε
22
2
zkc
=ωμε22
2zkE
m =
)(2
2zV
m
)(
2
2
zc
cωμε
Anything else? Transmission and reflection coefficientsProbability and energy velocitiesGroup and phase velocities
14
Transmission for a Rectangular Potential Barrier
2 2 2 2 22 20
2 2 2 2 20
( ) ( )1 1 : 1 sinh (2 ), where
4 ( )( )c c c
cc c
EM aT c
ω ω ω ωω ω κ κω ω ω ω
− −< = + = − −
By analogy, the transmission parameter of an electromagnetic wavecan be expressed as
22 20
20
( )1 1 2 ( ) : 1 sinh (2 ), where
4 ( )( )
V V m V EE V QM a
T V E E Vκ κ
− −< = + = − −
15
Analogies Between Probability and Energy Velocities
Quantum Mechanics:Probability velocity
Electromagnetism:Energy Velocity
Can we use EM wave to study a long-standing debate in QM, i.e. the tunneling time?
)Re(2)(
)Im(21
222
*
2
2
Γ+Γ+Γ−
− zz
c
ee
cκκω
ωμε)]Re(2[
)Im(2)(222
*
Γ+Γ+Γ−
− xx eem
EVκκ
2ψx
prob
Jv =
U
PvE =
VE <
ˆ( )zAP S da= ⋅ e
1( )
16 AU E D B H da
π= ⋅ + ⋅
cω ω<
16
QM: Tunneling Time Calculation =Δa
probv
dxt
2
0
VE <
Γ+−Γ−−
Γ−=
Γ+Γ+Γ−
=Δ
−
−
)Re(4))1()1((2
1
)Im(2
1
)(2
)]Re(2)[()Im(2
1
)(2
424*
2
0
222*
aeeEV
m
dzeeEV
mt
aa
azz
κκ
κκ
κ
EM: Tunneling Time Calculation =Δa
Ev
dxt
2
022
22 22 2 *
0
224 4
2 2 *
1[( ) 2 Re( )]
2 Im( )
1 1(( 1) ( 1)) 4 Re( )
22 Im( )
az z
c
a a
c
t e e dzc
e e ac
κ κ
κ κ
μεωω ω
μεωκω ω
−
−
Δ = + Γ + Γ− Γ
= − − Γ − + Γ − Γ
cω ω<
17
Superluminal effect is common to many wave phenomena.
The matter wave and the electromagnetic wave share many common characteristics.
Summary #2
The moment of truth: Put the idea to the test in a 3D-EM system.
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Part III. Modal Analysis:
Effect of high-order modes on tunneling characteristics
H. Y. Yao and T. H. Chang, “Effect of high-order modes on tunneling characteristics", Progress In Electromagnetics Research, PIER, 101, 291-306, 2010.