How a wave packet propagates at a speed faster than the speed of - - PDF document

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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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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 .[

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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 .[

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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

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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.

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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.

1 2

2 2 2

1 (1 ) 0.999999999988 just millimeters per second slower than . 1 v c c m E 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 φ τ ω ω ω ω ω ω ω ≡ = ≡ ≡ = = + ≈

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Superluminal Mechanism: Anomalous dispersion

( ) ( ) ck n k k ω = See waves in a dielectric medium [Jackson Chap. 7]

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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.

2 2 2 (bound)

2

( ) ε ε ω γ ω ω ω ωγ



= + + − − −

j j j j

f Ne f i m i i

Ne m

(7.51)    

negligible ( 0 or very small) f

=



ω Reε Imε

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PA: Polyamides are semi-crystalline polymers. The data was measured with a THz-TDS system.

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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.

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 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

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Part II. Analogies Between Schrödinger’s Equation and Maxwell’s Equation

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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

= + − ∂ ∂ z E m z V m z ϕ   ) ) ( (

2 2 2 2 2 2

= + − ∂ ∂

z c

B c z c z ω µε ω µε

2 2 2 z

k c = ω µε

2 2

2

z

k E m =  ) ( 2

2

z V m 

) (

2 2

z c

c

ω µε

Anything else? Transmission and reflection coefficients Probability and energy velocities Group and phase velocities

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Transmission for a Rectangular Potential Barrier

2 2 2 2 2 2 2 2 2 2 2 2

( ) ( ) 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 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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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( 2 1

2 2 2 * 2 2

Γ + Γ + Γ −

− z z c

e e c

κ κ

ω ω µε )] Re( 2 [ ) Im( 2 ) ( 2

2 2 *

Γ + Γ + Γ −

− x x

e e m E V

κ κ

2

ψ

x prob

J v = U P vE =

V E <

ˆ ( )

z A

P S da = ⋅



e 

1 ( ) 16

A

U E D B H da π = ⋅ + ⋅



   

c

ω ω <

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QM: Tunneling Time Calculation



= Δ

a prob

v dx t

2

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

4 2 4 * 2 2 2 2 *

a e e E V m dz e e E V m t

a a a z z κ κ κ κ

κ

EM: Tunneling Time Calculation



= Δ

a E

v dx t

2

2 2 2 2 2 2 2 * 2 2 4 4 2 2 *

1 [( ) 2Re( )] 2 Im( ) 1 1 (( 1) ( 1)) 4 Re( ) 2 2 Im( )

a z z c a a c

t e e dz c e e a c

κ κ κ κ

µεω ω ω µεω κ ω ω

− −

Δ = + Γ + Γ − Γ   = − − Γ − + Γ     − Γ



c

ω ω <

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 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.

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Part III. Modal Analysis:

Effect of high-order modes

• n 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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Geometric and material discontinuities

c , 1 , 1 regions all for 1

c 2 c 2 2 2 2 1

π ω ω ω π ω ω ω ε µ c c k a c c k

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

v k a c c k ε ω ω ε µ π ω ω ω ε µ

z ik

De

1

z

Ce

2

κ − z

Be

2

κ z ik

e

1

z ik

Ae

1

− a c

ω

c c

ω ω

I Region II Region III Region I Region II Region III Region

z ik

De

1

z ik

Ce

2

− z ik

Be

2

z ik

e

1

z ik

Ae

1

− a c

ω

c c

ω ω

For TE10 mode

(A) (B)

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

Reduce to 1-D case Potential-like diagram

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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 k D

L ik

+ − =

− *

D D T × ≡

(B) (A)

Disagree! Why?

Transmission amplitude < 1

r

ε > 1

r

ε

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Group delay for two systems

(A) (B)

      + =

− 2 1 2 2 2 2 1 1

2 ) tan( ) ( tan k k L k k k δφ

ω δφ τ d d v L

g g

= =

Disagree! Why?

< 1

r

ε > 1

r

ε

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Modal Effect

L

(e) (d) (c) (b) (a)

L L

Region I Region II Region III Region I Region II Region III

E V0 V

ω ω c

a

eik1z

Ae-ik1z Beik2z Ce-ik2z Deik1z

eik1z ΣAne-iknz ΣBneiknz ΣCne-iknz ΣDneiknz ω c

b

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

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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

e a x n D k B e a x n D E

ω ω

π π              +       − =       +       =

 

∞ = + − − ∞ = + − − 1 ) ( ) ( 1 I 1 ) ( ) ( I

sin sin sin sin I Region

1 1

n t z k i a n n t z k i a x n t z k i n t z k i y

a n a a n

e a x n k A e a x k B e a x n A e a x E

ω ω ω ω

π π π π              ′ +       ′ − =       ′ +       ′ =

   

∞ = + − ∞ = − ∞ = + − ∞ = − 1 ) ( 1 ) ( II 1 ) ( 1 ) ( II

c sin c sin c sin c sin 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

e x n C k e x n B k B e x n C e x n B E

ω ω ω ω

π π π π

t i z k n n a n t z k i a

a n a

e a x D i e a x D k

2 ) ( 1 1

sin sin

1

ω ω

π κ π

+ − ∞ = −

      −       −



L z y L z y L z y L z y z x z x z y z y

B B E E B B E E

= = = = = = = =

= = = =

III II III II II I II I

. 4 . 3 . 2 . 1 . 4 . 3 . 2 . 1

III III I I

= = = =

= = = = L z y L z y z x z y

B E B E

2 2

1

a cn a n

c k ω ω − =

2 c 2

1

cn c n

c k ω ω − = a c n

a cn

π ω = c

c

c n

cn

π ω = a x ≤ ≤ 2 < ≤ − x c a 2 c a x a + ≤ <

(a)

• b

2

x y

• a

2 b 2 a 2

h

(b)

EyI, HxI EyII, HxII EyIII, HxIII

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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

Transmission, 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

Group delay (ns)

N=3 N=1 HFSS N=21 N=11

(b)

Potential well

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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

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 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

SLIDE 14

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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)

0.04 0.08 0.96 1.02 1.08

Phase (π )

1 2 3 4 5

Round-trip phase (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 h a b h

B2

28

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

SLIDE 15

Group Delay Measurement

Pulse generator Signal generator Scope PIN switch Divider Reference DUT Adaptors equal length

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!

SLIDE 16

Experimental Result

1 1.2 1.4 1.6 1.8

Frequency (ω /ω c)

• 1

1 2 3

Group delay (ns)

TD simulation

Experiment

fast

4 8 12 16

Time (ns)

1

Normalized amplitude

1 1.684ω c (slow) 4 8 12 16

Time (ns)

1 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

0.2 0.4 0.6 0.8 1

Normalized Height (h/b)

2 4 6 8 10 12

Apparent Group Velocity (c)

0.0 0.2 0.4 0.6 0.8 1.0

Transmission f = 1.467ω c 47 %

32

L h a b I II III B1 B2 FT

SLIDE 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

Group delay (ns)

1.44 1.46 1.48 1.50

f

0.0 0.1

T L=10 cm 30 50 70 90

90 50 10 3.2% 10 %

0 5

(ω c) (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

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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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SLIDE 19

Manipulate the Group Delay

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Group Delay Analysis

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( ) (

)

2 2 2

T g d d MR t r MR R

f f

φ φ

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:

( ) ( )

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 = ′ = =   ′   =    ′ ′ − +      

SLIDE 20

Group Delay Analysis II

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( ) (

)

2 2 2

T g d d MR t r MR R

f f

φ φ

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 (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

( )

2

d d MR

f τ τ +

2

t φ

τ

2

r MR

f

φ

τ

R

τ

Slow Wave and Fast Wave Criteria

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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.

SLIDE 21

Negative Group Delays in a Birefringent Waveguide

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Negative Group Delays

42

30 31 32 33 34 0.2 0.4 0.6 0.8 Normalized magnitude |Tp| 1 1.5 2 2.5 3 3.5 Phase φ Tp (radius) Expt. BS theory HFSS

  

30 31 32 33 34 Frequency (GHz)

• 0.8
• 0.6
• 0.4
• 0.2

0.2 Group delay

τg

Tp (ns)

0.25 0.5 0.75 1 Assigned spectrum S(ω ) (arb. units) NGD region (a) (b) 6 8 10 12 14 Time (ns) 0.04 0.08 0.12 0.16 Output pulse profile

|Eout

p (t)| (arb. units)

0.2 0.4 0.6 0.8 1 Input pulse profile

|Ein

p (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.

SLIDE 22

Adjustable Group Delays & Summary #5

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 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 x prob E

v k d v dk J d v P v U dφ ω 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.

SLIDE 23

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Acknowledgement

Hsin-Yu Yao (姚欣佑) Herbert Winful,

• Univ. of Michigan