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

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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

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Usain 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

.[

(3)

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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.

(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.

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

=

(5)

Anomalous Dispersion: Waves in a dielectric medium

Properties of ε

:

When

ω

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

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.

(6)

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

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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

0 ) ( 2 ]

) 2 (

[

2 2 2

2

= +

∂ −

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

z

c = με ω

2 2

2

k

z

m E

 =

) 2 (

2

V z

m

2

( )

2

c z ω

c

με

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

< = +  − −  = 

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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 − z

Γ

c

e e

c

κ

ω

κ

ω με )]

Re(

2 [

) Im(

2 )

( 2

2 2

*

Γ +

Γ +

Γ

− x

x

e

e m

E V

κ κ

ψ

2 x prob

v = J

U v

E

= P

V E <

z ) P=

A eS da

1 ( )

16 A

U E D B H da

= π

 ⋅ + ⋅ 

ω ω <

c

QM: Tunneling Time Calculation Δ =

a

v

prob

t 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 =

a

v

E

t 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 <

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.

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.

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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

(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

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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

ik1z

A e

-ik1z

B e

ik2z

C e

-ik2z

D e

ik1z

e

ik1z Σ

A

n

e

-iknz

Σ

B

n

e

iknz

Σ

C

n

e

-iknz Σ

D

n

e

iknz ωcb

It is a 3-D problem.

Modal effect should be considered.

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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)

-

b2

x y

-

a2 a2 b2

h

(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 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

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)

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

should be considered.

Summary #3

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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

r

φ

t

= φ

't

φ

'r

(b)

(c) (d)

B1

eikz

Reikz+φr

Teikz+φt

II III

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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Group Delay Measurement

Pulse generator

Signal generator

Scope PIN switch

Divider

Reference

DUT

Adaptors equal length

τ

g

L

(b) (a)

h

a b

I

II

III

B

1

B

2

F

T

29

Experiment data and analysis

We can get the information from oscilloscope!

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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 simulation

Experiment

fast

0 4 8 12 16

Time (ns)

0 1

No rma lize d am pl it ude

0

1.684 ω

c

(slow)

1

0 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 ns

slow (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

c

47 %

L h

a b

I

II

III

B1

B2 FT

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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 )

c

0 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

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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).

Superluminality

in a Fabry-Pérot Interferometer

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Manipulate the Group Delay

37

Group Delay Analysis

(

0

2

0

) ( 2 2 )

T

g d d

f

MR φt φr

f

MR R

τ = 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

=

= = ′

  ′

=  − ′ + ′  

(20)

Group Delay Analysis II

39

(

0

2

0

) ( 2 2 )

T

g d d

f

MR φt φr

f

MR R

τ = 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

d0

f

MR

)

2 τ

φt

2 τ

φr

f

MR

τ

R

Slow 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.

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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) (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

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.

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45

Acknowledgement

Hsin-Yu Yao (姚欣佑) Herbert Winful,

Univ. of Michigan

Figure

Updating...

References

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