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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.[

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

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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

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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

12

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.

18

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 allfor 1

c2c22

221

πωωω

πωωω

εμ

c

ck

a

c

ck

cc

ac

ac

rr

=−=

=−=

==

2

22

221

1

III and Ifor 1 ;1

,1

III and Ifor 1 ;1

−=

≠=

=−=

==

r

ac

rr

ac

ac

rr

vk

a

c

ck

εωω

εμ

πωωω

εμ

zikDe 1

zCe 2κ−

zBe 2κzike 1

zikAe 1− acω

ccω

ω

IRegion IIRegion IIIRegion IRegion IIRegion IIIRegion

zikDe 1

zikCe 2−

zikBe 2zike 1

zikAe 1−

acω

ccω

ω

For TE10 mode

(A) (B)

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

Reduce to 1-D casePotential-like diagram

20

Transmission amplitude for two systems

)sin()()cos(2

2

22

22

1221

211

LkkkiLkkk

ekkD

Lik

+−=

−*DDT ×≡

(B)(A)

Disagree!Why?

Transmission amplitude

< 1rε

> 1rε

21

Group delay for two systems

(A) (B)

+= −

21

22

22

11

2

)tan()(tan

kk

Lkkkδφωδφτ

d

d

v

L

gg ==

Disagree!Why?

< 1rε

> 1rε

22

Modal Effect

L(e)

(d)

(c)

(b)

(a)

L

L

Region I Region II Region III

Region I Region II Region III

EV0

V

ω ω c

a

eik1z

Ae-ik1z

Beik2z

Ce-ik2z

Deik1z

eik1z

ΣAne-iknz

ΣBneiknz

ΣCne-iknzΣDneiknz

ω cb

It is a 3-D problem.Modal effect should be considered.

23

Complete wave functions and boundary conditions

−=

=

∞

=

−

∞

=

−

1

) (III

1

) (III

sin

sin

IIIRegion

n

tzkin

anx

n

tzkiny

an

an

ea

xnDkB

ea

xnDE

ω

ω

π

π

+

−=

+

=

∞

=

+−−

∞

=

+−−

1

) () (1I

1

) () (I

sin

sin

sin

sin

IRegion 1

1

n

tzkiann

tzkiax

n

tzkin

tzkiy

an

a

an

ea

xnkAe

a

xkB

ea

xnAe

a

xE

ωω

ωω

ππ

ππ

′

+

′

−=

′

+

′

=

∞

=

+−∞

=

−

∞

=

+−∞

=

−

1

) (

1

) (II

1

) (

1

) (II

c

sin

c

sin

c

sin

c

sin

IIRegion

n

tzkin

cn

n

tzkin

cnx

n

tzkin

n

tzkiny

cn

cn

cn

cn

exn

Ckexn

BkB

exn

Cexn

BE

ωω

ωω

ππ

ππ

tizkn

n

an

tzkia an

a

ea

xDie

a

xDk

2

) (11

sin

sin 1 ωω πκπ +−

∞

=

−

−

−

LzyLzy

LzyLzy

zxzx

zyzy

BB

EE

BB

EE

==

==

==

==

=

=

=

=

IIIII

IIIII

0II0I

0II0I

.4

.3

.2

.1

0 .4

0 .3

0 .2

0 .1

III

III

0I

0I

=

=

=

=

=

=

=

=

Lzy

Lzy

zx

zy

B

E

B

E

221 acn

an c

k ωω −=

2c21cn

cn c

k ωω −=

a

cnacn

πω =

cc cncn

πω =ax ≤≤00

2<≤−

xca

2

caxa

+≤<

(a)

- b2 x

y

- a2 b2 a2

h

(b)

EyI, HxI

EyII, HxII

EyIII, HxIII

24

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

Tra

nsm

issi

on,

T

N=3N=1

HFSSN=21N=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

Gro

up

dela

y (n

s)

N=3N=1

HFSSN=21N=11

(b)

Potential well

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

Tra

nsm

issi

on, T

HFSSN=9N=3N=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

Gro

up D

elay

(ns)

HFSSN=9N=3N=1

(b)

Potential barrier

26

Model effect plays an important role 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

Mag

nitu

de √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

Pha

se (

π)

0

1

2

3

4

5

Rou

nd-t

rip

phas

e (π

)

φ r

φ t = φ 't

φ 'r

(b)

(c) (d)

B1

eikz

√Reikz+φ r

√Teikz+φ t

II III

eikz

√R'eikz+φ 'r

√T'eikz+φ 't

b

a

ha

bh

B2

28

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

Group Delay Measurement

Pulsegenerator

Signal generator

Scope

PIN switch

Divider

Reference

DUT

Adaptorsequallength

τg

L

(b)

(a)

h

a

b

I

II

III

B1

B2

FT

29

30

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

Gro

up d

elay

(ns

) TD simulation

Experiment

fast

0 4 8 12 16Time (ns)

0

1

Nor

mal

ized

am

plit

ude

0

11.684ω c (slow)

0 4 8 12 16Time (ns)

0

1

0

0.16

7 8 9

63 ps

ref.

trans.

1.467ω 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 Normalized Height (h/b)

0

2

4

6

8

10

12

App

aren

t Gro

up

Vel

ocit

y ( c

)

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

issi

on

f = 1.467ω c

47 %

32

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

3.0G

roup

del

ay (

ns)

1.44 1.46 1.48 1.50f

0.0

0.1T

L=10 cm

30

50

70

90

9050

103.2%

10 %

0 5(ωc)

(a)

33

L

h

a

b

I

II

III

B1

B2

FT

34

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

36

Manipulate the Group Delay

37

Group Delay Analysis

38

( ) ( )0 02 2 2Tg d d MR t r MR Rf fφ φτ τ τ τ τ τ= + + + +

( )( )

2

2

cos 2

1 2 cos 2

eff

MR

eff

R k L Rf

R k L R

′ ′−=

′ ′− +

Multiple-reflection factor:

( )( )

0II

2

,

sin 2

1 2 cos 2

dg

t rt r

effR

eff

L

v

d d

d d

k L dR

R k L R d

φ φ

τ

φ φτ τω ω

τω

=

′= =

′ = ′ ′− +

Group Delay Analysis II

39

( ) ( )0 02 2 2Tg d d MR t r MR Rf fφ φτ 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 (B1 and 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

( )0 02d d MRfτ τ+

2 tφτ

2 r MRfφτ

Rτ

Slow Wave and Fast Wave Criteria

40

Is it possible that the group delay becomes negative?

On-resonance constructive interference: Slow wave

( )

II

12

1 1T on t t rg

g

d dR L d R

R v d d d R

φ φ φτω ω ω

′′ ′ ′+ = + + + ′ ′− −

Off-resonance destructive interference: Fast wave

( )

II

12

1 1T off t t rg

g

d dR L d R

R v d d d R

φ φ φτω 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 33 340

0.2

0.4

0.6

0.8

Nor

mal

ized

mag

nitu

de |T

p|

1

1.5

2

2.5

3

3.5

Pha

se φ

T p (

radi

us)

Expt.BS theoryHFSS

30 31 32 33 34Frequency (GHz)

-0.8

-0.6

-0.4

-0.2

0

0.2

Gro

up d

elay

τgTp

(ns

)

0

0.25

0.5

0.75

1

Ass

igne

d sp

ectr

um

S(ω

) (a

rb. u

nits

)

NGDregion

(a)

(b)

6 8 10 12 14Time (ns)

0

0.04

0.08

0.12

0.16

Out

put p

ulse

pro

file

|Eou

t p

(t)|

(arb

. uni

ts)

0

0.2

0.4

0.6

0.8

1

Inpu

t pul

se p

rofi

le |E

inp (t)

| (ar

b. u

nits

)(c)

The black dots are the measured data,while the blue squares represent thetheoretical results. The red curves arethe 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 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

xprob

E

vkd

vdk

Jd

v

Pv

U

dφω

ω

ω

ψ

τ

≡

≡

≡

≡

≡

44

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