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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 Person

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)

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.

(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

(7)

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

< = +  − −  = 

(8)

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

(9)

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.

(10)

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

(11)

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.

(12)

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

(13)

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

(14)

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.

(15)

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!

(16)

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

(17)

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

(18)

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

(19)

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.

(21)

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.

(22)

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.

(23)

45

Acknowledgement

Hsin-Yu Yao (姚欣佑) Herbert Winful,

Univ. of Michigan

參考文獻

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