1

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

**How a wave packet propagates at a speed**

**faster than the speed of light**

**A novel superluminal mechanism **

**A novel superluminal mechanism**

**with high transmission and broad bandwidth**

Tsun-Hsu Chang (張存續)
**with high transmission and broad bandwidth**

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

**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

.^{[}

**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.

**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*

=

**Anomalous Dispersion: ** Waves in a dielectric medium

*Properties of* ε

^{: }

When

### ω

is near each### ω

*j*

*(binding frequency of the j*

^{th}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 (

*f*0 =

### 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*

*V*_{0}

**I** **II** **III**

### 2( ) *E V* ?

*v* *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 *

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

### Maxwell’s Equation

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

^{−}

< = + − − =

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* <

(ˆ* _{z}* )

*P*=

###

*A*

**⋅**

^{e}*S 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*

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.**

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*

*De**ik*^{1}

*Ce*^{−}^{κ}^{2}*z*

*Be*^{κ}^{2}*z*
*z*

*e**ik*^{1}
*z*

*Ae*^{−}*ik*^{1} ^{a}

ω*c*
c

ω*c*

ω

I

Region Region IIIII Region I

Region IIRegion IIIRegion

*z*

*De**ik*^{1}
*z*

*Ce*^{−}*ik*^{2}
*z*

*Be**ik*^{2}
*z*

*e**ik*^{1}
*z*

*Ae*^{−}*ik*^{1}

*a*

ω*c*

c

ω*c*

ω
For TE_{10}mode

### (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
ε*r*

> 1
ε*r*

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
ε*r*

> 1
ε*r*

*Modal Effect*

### (e) *L*

### (d) (c)

### (b) (a)

*L* *L*

Region I Region II Region III

Region I Region II Region III

*E*
*V*^{0}

*V*

ω ω^{c}^{a}

*e*

^{ik}^{1}

^{z}*A* *e*

^{-ik}^{1}

^{z}*B* *e*

^{ik}^{2}

^{z}*C* *e*

^{-ik}^{2}

^{z}*D* *e*

^{ik}^{1}

^{z}*e*

^{ik}^{1}

*Σ*

^{z}^{A}

^{A}

^{n}*e*

^{-ik}

^{n}

^{z}Σ

^{B}

^{B}

^{n}*e*

^{ik}

^{n}

^{z}Σ

^{C}

^{C}

^{n}*e*

^{-ik}

^{n}*Σ*

^{z}^{D}

^{D}

^{n}*e*

^{ik}

^{n}*ω*

^{z}^{c}

^{b}### It is a 3-D problem.

### Modal effect should be considered.

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* *n*^{a}

*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)

### -

^{b}_{2}

*x* *y*

### -

^{a}_{2}

^{a}_{2}

^{b}_{2}

*h*

### (b)

*E*_{yI}*, H*_{xI}

*E*_{yII}*, H*_{xII}

*E*_{yIII}*, H*_{xIII}

**Modal Effect Corrects the Problems (I)**

**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 well25

**Modal Effect Corrects the Problems (II)**

**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 *

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)**

**B**_{1}

*e*^{ikz}

### √

^{Re}

^{ikz+}^{φ}

^{r}### √

^{Te}

^{ikz+}^{φ}

^{t}### II III

*e*^{ikz}

### √

^{R'e}

^{ikz+}

^{φ '}

^{r}### √

^{T'e}

^{ikz+}

^{φ '}

^{t}*b*

*a*
*h*

*a*
*b*
*h*

**B**_{2}

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

### 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**

**B**

_{1}**B**

**B**

_{2}*F*

_{T}29

### 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

### Gr ou p del ay (n s)

TD simulationExperiment

### 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

**B**_{1}

**B**_{2}*F*_{T}

### 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

**B**_{1}

**B**_{2}*F*_{T}

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

### Manipulate the Group Delay

37

### Group Delay Analysis

### (

^{0}

^{2}

^{0}

### ) ( ^{2} ^{2} )

*T*

*g* *d* *d*

*f*

*MR*

_{φ}

*t*

_{φ}

*r*

*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

=

= = ′

′

= − ′ + ′

### Group Delay Analysis II

39

### (

^{0}

^{2}

^{0}

### ) ( ^{2} ^{2} )

*T*

*g* *d* *d*

*f*

*MR*

_{φ}

*t*

_{φ}

*r*

*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 (B*1*and B*2) 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}

^{d}^{0}

^{+} ^{2} ^{τ}

^{d}^{0}

^{f}

^{f}

^{MR}### )

### 2 τ

_{φ}

_{t}### 2 τ

_{φ}

_{r}*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.

### 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 *|T**p*|

1 1.5 2 2.5 3 3.5

Phase φ*T**p* (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 τ*g**T**p* (ns)

0 0.25 0.5 0.75 1

Assigned spectrum *S*(ω) (arb. units)
**NGD**

**region**
(a)

(b)

6 8 10 12 14

0 0.04 0.08 0.12 0.16

Output pulse profile *|**E*out*p**(t*)| (arb. units)

0 0.2 0.4 0.6 0.8 1

Input pulse profile *|**E*in*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= 45^{o}

(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 thepolarization 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.

45

### Acknowledgement

Hsin-Yu Yao (姚欣佑) Herbert Winful,

Univ. of Michigan