# How a wave packet propagates at a speed faster than the speed of light

## Full text

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1

### with high transmission and broad bandwidth

Tsun-Hsu Chang (張存續)

Department of Physics, National Tsing Hua University

### 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

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.[

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

.[

(3)

### 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

### Space Shuttle

Fastest manually controlled flight in atmosphere during

atmospheric reentry of STS-2 mission is 28000 km/h ~ 7777 m/s.

(4)

### 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.

12 2

0 2 2

### = −

10

7

The index of refraction n(

### ω

) is a function of frequency.

g

Phase velocity: ( ) (7.88) ( )

Group velocity: (7.89

( ) ( ) )

p

g g

≡ =

= +

( ) ( )

=

(5)

:

When

is near each

### ω

j (binding frequency of the jth group of electrons),

### ε

exhibits resonant behavior in the form of anomalous

### dispersion

and resonant absorption.

2 0

0 (bound) 2 2 0

( )

= +

+

j

j j j

(7.51)



f0 =

### 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.

E V

V0

I II III

### = − =

Comparing with the matter wave, the electromagnetic wave is much more easier to implement in experiment.

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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.

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13

### Analogies Between Schrodinger and Maxwell Equations

Maxwell’s wave equation for a TE waveguide mode Time-independent

Schrodinger’s equation

2 2 2

2

2

2 2

2 2

2

z

c

2 2

2

z

2 2

z

2

2

2

c

### 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

 −  −

< = +  − −  =

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

< = +  − −  = 

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15

### Analogies Between Probability and Energy Velocities

Quantum Mechanics:

Probability velocity

Electromagnetism:

Energy Velocity

2 2 2

* 2

2

z − z

c

κ

κ

2 2

*

− x

x

κ κ

2 x prob

E

z ) P=

A eS da

1 ( )

16 A

U E D B H da

= π

 ⋅ + ⋅ 

c

a

prob

2

0

2 4 4

* 2

0

2 2 2

*

a a

a

z z

κ κ

κ κ

a

E

2

0 2 2

2 2 2

2 2 *

0

2 4 2 4

2 2 *

a

z z

c

a a

c

κ κ

κ κ

### ω ω <

c

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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.

### 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.

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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

Ceκ2z

Beκ2z z

eik1 z

Aeik1 a

ωc c

ωc

ω

I

Region Region IIIII Region I

Region IIRegion IIIRegion

z

Deik1 z

Ceik2 z

Beik2 z

eik1 z

Aeik1

a

ωc

c

ωc

ω For TE10mode

### What is the difference between (A) and (B)?

Reduce to 1-D case Potential-like diagram

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*

### Why?

Transmission amplitude

< 1 εr

> 1 εr

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21

2 1

2 2

2 2 1 1

g

g

< 1 εr

> 1 εr

### LL

Region I Region II Region III

Region I Region II Region III

E V0

V

ω ωca

ik1z

-ik1z

ik2z

-ik2z

ik1z

ik1z Σ

n

-iknz

Σ

n

iknz

Σ

n

-iknz Σ

n

iknz ωcb

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23



=

=

=

=

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< +

b2

a2 a2 b2

h

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

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

N=3 N=1 HFSS

N=21 N=11

Potential well

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25

### 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

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

HFSS

N=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

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27

### Theoretical and Experimental Studies a new mechanism

H. Y. Yao and T. H. Chang, Progress In Electromagnetics Research, PIER 122, 1-13 (2012).

### I II

(a)

1 1.2 1.4 1.6 1.8 2

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

c

### )

0 0.04 0.08 0.96 1.02 1.08

Phase(π)

0 1 2 3 4 5

Round-trip phase (π)

r

t

't

'r

(b)

(c) (d)

B1

eikz

Reikz+φr

Teikz+φt

eikz

R'eikz+φ 'r

### √

T'eikz+φ 't b

a h

a b h

B2

The existence of the higher order modes (evanescent waves) will modify the amplitude and phase of the dominant mode.

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Pulse generator

Signal generator

Scope PIN switch

Divider

Reference

DUT

g

(b) (a)

1

2

T

29

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c

TD simulation

Experiment

0

c

1

0 0.16

7 8 9

63 ps

trans.

c

L/c = 0.33 ns

+30 ps

31

c

L h

a b

I

II

III

B1

B2 FT

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c

### Gro up de la y (n s)

1.44 1.46 1.48 1.50

0.0 0.1

### 50 70 90

50 90 3.2% 10 10 %

c

### (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

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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).

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37

0

0

T

g d d

MR φt φr

MR R

2 2

eff MR

eff

### = − ′ + ′

Multiple-reflection factor:

0

II

2

,

sin 2 1 2 cos 2

d

g

t r

t r

eff R

eff

φ φ

### τ ω

=

= = ′

  ′

=  − ′ + ′  

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39

0

0

T

g d d

MR φt φr

MR R

### 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

d0

d0

MR

φt

φr

MR

R

( )

II

1 2

1 1

T on t t r

g

g

### τ ω ω ω

  ′

′ ′ ′

+

    

= − ′ + + +  − ′

( )

II

T off t t r

g

g

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### 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

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) (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.

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### 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 the

polarization azimuth of the incident wave.

g

2

### Group delay: apparent group velocity or phase tim

Phase velocity:

Group velocity:

Probability velocity:

Energy veloci

ty:

p

g

prob x

E

### Conclusions

Information velocity: The speed at which information is

transmitted through a particular medium.

Signal velocity: The speed at which a wave carries information.

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45

### Acknowledgement

Hsin-Yu Yao (姚欣佑) Herbert Winful,

Univ. of Michigan

Updating...

## References

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