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
The Fastest Person
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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
.[
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.
Highest Particle Speed: LEP Collider
The 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
0 2 2
(1 1 ) 0.999999999988
just millimeters per second slower than .
1 v c
c
E m c
β γ
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 c
v k n k
d c
v dk
d d kL L
d d
n dn d
v τ φ
ω ω
ω
ω ω ω
ω
≡ =
≡
≡ =
= +
≈
Superluminal Mechanism: Anomalous dispersion
( ) ( )
n k ck
ω k
=
Anomalous Dispersion: Waves in a dielectric medium
Properties of ε
:When
ω
is near eachω
j (binding frequency of the jth group of electrons),ε
exhibits resonant behavior in the form of anomalousdispersion
and resonant absorption.2 0
0 (bound) 2 2 0
2
( )
ε ε
= + ω
−ω
−ωγ
+ω γ
−ω
j
j j j
f Ne f
i m i
i Ne
m
(7.51)
negligible (
f0 =0 or very small)
ω Re ε
Im ε 0
9
PA: Polyamides are semi-crystalline polymers.
The data was measured with a THz-TDS system.
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 V ?
v 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
Part II. Analogies Between Schrödinger’s Equation and
Maxwell’s Equation
13
Analogies Between Schrodinger and Maxwell Equations
Maxwell’s wave equation for a TE waveguide mode Time-independent
Schrodinger’s equation
0 ) ( 2 ]
) 2 (
[
2 2 22
= +
∂ −
∂ m E z
z m V
z ϕ
( ( )
2) 0
2 2
2 2
2
= +
∂ −
∂
z
c
B
z c c z
με ω με ω
2 2
2
k
zc = με ω
2 2
2
k
zm E
=
) 2 (
2
V z
m
2( )
2
c z ω
cμε
Anything else? Transmission and reflection coefficients
Probability and energy velocities
Group and phase velocities
Transmission for a Rectangular Potential Barrier
2 2 2 2 2
2 2
0
2 2 2 2 2
0
( ) ( )
1 1
: 1 sinh (2 ), where
4 ( )( )
c c c
c
c c
EM a
T c
ω ω ω ω
ω ω κ κ
ω ω ω ω
− −
< = + − − =
By analogy, the transmission parameter of an electromagnetic wave can be expressed as
2 2 2
0 0 2
( )
1 1 2 ( )
: 1 sinh (2 ), where
4 ( )( )
V V m V E
E V QM a
T V E E V−
κ 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(
1 2
2 2 2
* 2
2
Γ +
Γ +
−
z − zΓ
c
e e
c
κ
ω
κω με )]
Re(
2 [
) Im(
2 )
( 2
2 2
*
Γ +
Γ +
Γ
−
− x
x
e
e m
E V
κ κ
ψ
2 x probv = J
U v
E= P
V E <
(ˆz ) P=
A e ⋅S da1 ( )
16 A
U E D B H da
= π
⋅ + ⋅ ω ω <
cQM: Tunneling Time Calculation Δ =
av
probt dx
2
0
V E <
Γ +
− Γ
− Γ −
= −
Γ +
Γ Γ +
= − Δ
−
−) Re(
4 )) 1 (
) 1 2 ((
1 ) Im(
2 1 )
( 2
)]
Re(
2 ) ) [(
Im(
2 1 )
( 2
2 4 4
* 2
0
2 2 2
*
a e
E e V
m
dz e
E e V t m
a a
a
z z
κ κ
κ κ
κ
EM: Tunneling Time Calculation Δ =
av
Et dx
2
0 2 2
2 2 2
2 2 *
0
2 4 2 4
2 2 *
1 [( ) 2 Re( )]
2 Im( )
1 1
(( 1) ( 1)) 4 Re( )
2 Im( ) 2
a
z z
c
a a
c
t e e dz
c
e e a
c
κ κ
κ κ
μεω
ω ω
μεω ω ω κ
−
−
Δ = + Γ + Γ
− Γ
= − Γ − − Γ − + Γ
ω ω <
c17
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.
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.
19
Geometric and material discontinuities
, c 1
1 ,
regions all
for 1
2 c c 2 2
2 2 1
ω π ω ω
ω π ω ω
ε μ
c k c
a c k c
c c
a c a c r r
=
−
=
=
−
=
=
=
2 2
2
2 2 1
1
III and I for 1 ; 1 1 ,
III and I for 1 ; 1
−
=
≠
=
=
−
=
=
=
r a c r
r
a c a c r r
k v
a c k c
ε ω ω
ε μ
ω π ω ω ε μ
z
Deik1
Ce−κ2z
Beκ2z z
eik1 z
Ae−ik1 a
ωc c
ωc
ω
I
Region Region IIIII Region I
Region IIRegion IIIRegion
z
Deik1 z
Ce−ik2 z
Beik2 z
eik1 z
Ae−ik1
a
ωc
c
ωc
ω For TE10mode
(A) (B)
What is the difference between (A) and (B)?
Reduce to 1-D case Potential-like diagram
Transmission amplitude for two systems
) sin(
) (
) cos(
2
2
2 2
2 2 1 2
2 1
2
1 1
L k k
k i L k k
k
e k D k
L ik
+
= −
− T≡
D×
D*(B) (A)
Disagree!
Why?
Transmission amplitude
< 1 εr
> 1 εr
21
Group delay for two systems
(A) (B)
+
=
−2 1
2 2
2 2 1 1
2
) tan(
) tan (
k k
L k k
δφ k ω
τ δφ
d d v
L
g
g
= =
Disagree!
Why?
< 1 εr
> 1 εr
Modal Effect
(e) L
(d) (c)
(b) (a)
L L
Region I Region II Region III
Region I Region II Region III
E V0
V
ω ωca
e
ik1zA e
-ik1zB e
ik2zC e
-ik2zD e
ik1ze
ik1z ΣA
ne
-iknzΣ
B
ne
iknzΣ
C
ne
-iknz ΣD
ne
iknz ωcbIt is a 3-D problem.
Modal effect should be considered.
23
Complete wave functions and boundary conditions
−
=
=
∞
=
−
∞
=
−
1
) ( III
1
) ( III
sin sin III
Region
n
t z k i n
a n x
n
t z k i n
y
a n a n
a e x D n
k B
a e x D n
E
ω ω
π π
+
−
=
+
=
∞
=
+
−
−
∞
=
+
−
−
1
) ( )
( 1
I
1
) ( )
( I
sin sin
sin sin
I Region
1 1
n
t z k a i
n n t
z k i a
x
n
t z k i n
t z k i y
a n a
a n
a e x k n
A a e
k x B
a e x A n
a e E x
ω ω
ω ω
π π
π π
′
+
′
−
=
′
+
′
=
∞
=
+
∞ −
=
−
∞
=
+
∞ −
=
−
1
) ( 1
) ( II
1
) ( 1
) ( II
sin c sin c
sin c sin c
II Region
n
t z k i n
c n n
t z k i n
c n x
n
t z k i n
n
t z k i n
y
c n c
n
c n c
n
x e C n
k x e
B n k B
x e C n
x e B n
E
ω ω
ω ω
π π
π π
t i z k n
n a n t
z k i
a na
a
a e D x
i a e
D x k
2 ) ( 1
1 sin π 1 ω ∞ κ sin π − +ω
=
−
−
−
L y z L y z
L y z L y z
x z x z
y z y z
B B
E E
B B
E E
=
=
=
=
=
=
=
=
=
=
=
=
III II
III II
II 0 I 0
II 0 I 0
. 4
. 3
. 2
. 1
0 .
4
0 .
3
0 .
2
0 .
1
III III I 0 I 0
=
=
=
=
=
=
=
=
L y z
L y z x z y z
B E B E
2 2
1 a
cn a
n c
k = ω −ω
c2
1 2
cn c
n c
k = ω −ω
a c
a n
cn
ω = π c
c n c
cn
ω = π a
x≤
≤
0 0
2 ≤ <
−c x a
2 c x a a< ≤ +
(a)
-
b2x y
-
a2 a2 b2h
(b)
EyI, HxI
EyII, HxII
EyIII, HxIII
Modal Effect Corrects the Problems (I)
2.0 2.4 2.8 3.2 3.6
Frequency (GHz)
0.7 0.8 0.9 1.0 1.1
Tran sm ission , T
N=3 N=1 HFSS
N=21 N=11
(a)
2.0 2.4 2.8 3.2 3.6
Frequency (GHz)
0.4 0.6 0.8 1.0 1.2 1.4
Gr ou p de lay (n s)
N=3 N=1 HFSS
N=21 N=11
(b)
Potential well25
Modal Effect Corrects the Problems (II)
2.0 2.4 2.8 3.2 3.6 4.0
Frequency (GHz)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Transmission, T
HFSS N=9 N=3 N=1
(a)
2.0 2.4 2.8 3.2 3.6 4.0
Frequency (GHz)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Group Delay (ns)
HFSSN=9 N=3 N=1
(b)
Potential barrier Model effect
plays an importantrole for a 3D discontinuity. To achieve a better agreement between the theory and
experiment in a quantum tunneling system, the model effect
should be considered.
Summary #3
27
Part IV. Superluminal Effect:
Theoretical and Experimental Studies a new mechanism
H. Y. Yao and T. H. Chang, Progress In Electromagnetics Research, PIER 122, 1-13 (2012).
Transmitted/Reflected Properties due to Modal Effect
I II
(a)
1 1.2 1.4 1.6 1.8 2
Frequency ( ω /ω
c)
0.52 0.54 0.56 0.82 0.84 0.86
Magnitude
√
R= √
R'√
T= √
T'1 1.2 1.4 1.6 1.8 2
Frequency ( ω /ω
c)
0 0.04 0.08 0.96 1.02 1.08
Phase(π)
0 1 2 3 4 5
Round-trip phase (π)
φ
rφ
t= φ
'tφ
'r(b)
(c) (d)
B1
eikz
√
Reikz+φr√
Teikz+φtII III
eikz
√
R'eikz+φ 'r√
T'eikz+φ 't ba h
a b h
B2
The existence of the higher order modes (evanescent waves) will modify the amplitude and phase of the dominant mode.
Group Delay Measurement
Pulse generator
Signal generator
Scope PIN switch
Divider
Reference
DUT
Adaptors equal length
τ
gL
(b) (a)
h
a b
I
II
III
B
1B
2F
T29
Experiment data and analysis
We can get the information from oscilloscope!
Experimental Result
1 1.2 1.4 1.6 1.8
Frequency ( ω /ω
c)
-1 0 1 2 3
Gr ou p del ay (n s)
TD simulationExperiment
fast
0 4 8 12 16
Time (ns)
0 1
No rma lize d am pl it ude
0
1.684 ω
c(slow)
10 4 8 12 16
Time (ns)
0 1
0 0.16
7 8 9
63 ps
ref.
trans.
1.467 ω
c(fast)
L/c = 0.33 nsslow (a)
(b) (c)
(c)
(b)
+30 ps
31
Effect of Waveguide Height
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8 10 12
Appar ent Group Velocity ( c)
0.0 0.2 0.4 0.6 0.8 1.0
Transmi ssion
f = 1.467 ω
c47 %
L h
a b
I
II
III
B1
B2 FT
Effect of Waveguide Length
1.3 1.4 1.5 1.6
Frequency ( ω /ω
c)
0.0 1.0 2.0 3.0
Gro up de la y (n s)
1.44 1.46 1.48 1.50
f
0.0 0.1
T
L=10 cm 30
50 70 90
50 90 3.2% 10 10 %
(ω )
c0 5
(a)
33
L h
a b
I
II
III
B1
B2 FT
A new mechanism of the superluminal effect has been theoretically analyzed and experimentally demonstrated.
In contrast to the two traditional mechanisms which all involve evanescent waves, this mechanism employs
propagating waves.
This mechanism features high transmission and broad
bandwidth.
Summary #4
35
Part V. Manipulate the Group Delay
H. Y. Yao, N. C. Chen, T. H. Chang, and H. G. Winful, Phys. Rev. A 86, 053832 (2012).
Superluminality
in a Fabry-Pérot Interferometer
Manipulate the Group Delay
37
Group Delay Analysis
(
02
0) ( 2 2 )
T
g d d
f
MR φt φrf
MR Rτ = τ + τ + τ + τ + τ
( )
( )
2 2
cos 2 1 2 cos 2
eff MR
eff
R k L R
f R k L R
′ − ′
= − ′ + ′
Multiple-reflection factor:
( )
( )
0
II
2
,
sin 2 1 2 cos 2
d
g
t r
t r
eff R
eff
L v
d d
d d
k L dR
R k L R d
φ φ
τ
φ φ
τ τ
ω ω
τ ω
=
= = ′
′
= − ′ + ′
Group Delay Analysis II
39
(
02
0) ( 2 2 )
T
g d d
f
MR φt φrf
MR Rτ = τ + τ + τ + τ + τ
Dwell time:
Effective time for the signal staying within the system excluding boundary dispersion effect.
Lifetime of stored field energy escaping through the both ends (B1and B2) of FP cavity excluding boundary dispersion effect.
Boundary transmission times:
Effective transmission time for the signal passing through the both boundaries of FP cavity.
Boundary reflection time:
Effective reflection time accumulated from signal reflecting on the both boundaries of FP cavity (modified by multiple-reflection factor).
Dispersive time:
due to frequency-dependent reflectivity
( τ
d0+ 2 τ
d0f
MR)
2 τ
φt2 τ
φrf
MRτ
RSlow Wave and Fast Wave Criteria
Is it possible that the group delay becomes negative?
On-resonance constructive interference: Slow wave
( )
II
1 2
1 1
T on t t r
g
g
d d
R L d R
R v d d d R
φ φ φ
τ ω ω ω
′
′ ′ ′
+
= − ′ + + + − ′
Off-resonance destructive interference: Fast wave
( )
II
1 2
1 1
T off t t r
g
g
d d
R L d R
R v d d d R
φ φ φ
τ ω ω ω
′
′ ′ ′
−
= + ′ + + − + ′
Yes, it is possible in a birefringent waveguide system.
Negative Group Delays in a Birefringent Waveguide
41
Negative Group Delays
42
30 31 32 33 34
0 0.2 0.4 0.6 0.8
Normalized magnitude |Tp|
1 1.5 2 2.5 3 3.5
Phase φTp (radius) Expt.
BS theory HFSS
30 31 32 33 34
Frequency (GHz) -0.8
-0.6 -0.4 -0.2 0 0.2
Group delay τgTp (ns)
0 0.25 0.5 0.75 1
Assigned spectrum S(ω) (arb. units) NGD
region (a)
(b)
6 8 10 12 14
0 0.04 0.08 0.12 0.16
Output pulse profile |Eoutp(t)| (arb. units)
0 0.2 0.4 0.6 0.8 1
Input pulse profile |Einp (t)| (arb. units) (c)
The black dots are the measured data, while the blue squares represent the theoretical results. The red curves are the simulation results.
(a) Transmission and phase (b) Group delay when Φ= 45o
(c) The time-domain profiles of the incident and transmitted pulses.
Adjustable Group Delays & Summary #5
43
We have demonstrated a
negative group delay
in an anisotropic waveguide system. This study provides a means
to control the group delay by
simply changing thepolarization azimuth of the incident wave.
g
2
Group delay: apparent group velocity or phase tim
Phase velocity:Group velocity:
Probability velocity:
Energy veloci
e
ty:
p
g
prob x
E
v k
v d
dk
J d
v v P
U d φ
ω ω
ω
ψ τ
≡
≡
≡
≡
≡
Conclusions
Information velocity: The speed at which information is
transmitted through a particular medium.
Signal velocity: The speed at which a wave carries information.
45
Acknowledgement
Hsin-Yu Yao (姚欣佑) Herbert Winful,
Univ. of Michigan