The document discusses the concepts of phase velocity and group velocity of light, highlighting that the phase velocity can vary while the front of a pulse propagates at the speed of light, c. It also explores the implications of group velocity, including the possibility of negative group velocities and their relationship to causality. Ultimately, it concludes that information cannot travel faster than the speed of light in a vacuum.

1. FASTER THAN C
? INFORMATION
He is
gaining on
me

3. ATOMS
LIGHT
LORENTZ MODEL

4. ATOM
LORENTZ MODEL
ELECTRON
NUCLEUS
small
Damped Harmonic Oscillator

5. LORENTZ MODEL
where
Solution :

6. LORENTZ MODEL
Using Maxwell’s Equations :
Refractive index is a function of frequency!!

7. • PHASE VELOCITY :
Rate at which the phase of the wave
propagates in space.
VELOCITY OF LIGHT
PHASE VELOCITY

8. Any waveform can be decomposed as a
weighted sum of different monochromatic
components
VELOCITY OF LIGHT

9. • GROUP VELOCITY :
Different monochromatic components travel
at different phase velocities and the velocity
of the maximum of the pulse is called group
velocity.
VELOCITY OF LIGHT
GROUP VELOCITY

10. VELOCITY OF LIGHT
Group velocity is given by :

11. SLOW AND FAST LIGHT
By changing group velocity
can be made anything from
negative infinity to infinity

12. SLOW AND FAST LIGHT
NEGATIVE GROUP VELOCITY?
WHAT? THE PULSE LEAVES
BEFORE ENTERING??

13. CAUSALITY ???
THE BOSS IS NOT HAPPY !

14. CAUSALITY ???
FABIENNEGUESTS

15. CAUSALITY ???
FABIENNE
FABIEN
V= 0.75 c
x
t
t’
X’
(0,0)
(0,0)
Boundary of light cone
v1=c
v2=2cNegative time

16. CAUSALITY ???
BUT IS GROUP VELOCITY WELL
DEFINED IN SPACE AND TIME?

17. PULSE DISTORTION
Normalized field amplitude in a dispersive
medium is governed by :
Where U(z,T)is the normalized amplitude
of the electric field in the dispersive medium.
1

18. PULSE DISTORTION
To solve the PDE Fourier Transform method
can be used :
Substituting in 1 :
2

19. PULSE DISTORTION
Solving 2 :
Frequency dependence in phase
Pulse gets distorted! Group velocity
does not make sense

20. INFORMATION IN A PULSE
(1868-1951)
Arnold Sommerfeld
In 1907, predicted that the front of a step
modulated pulse always propagates with
a velocity of c irrespective of the dispersion
of the medium.
Velocity of front = c
front

21. INFORMATION IN A PULSE
Later investigations in this field report that information is
stored only in points of a pulse that are non-analytic.
(Parker et al, Optics Communications , 229(2004), 23-27)
I is the total amount of differential information. p(x) is any
arbitrary continuously varying function of space. R
denotes the residues of the function.

22. INFORMATION IN A PULSE
c
Non-analytic points propagate with velocity=c.

23. EXPERIMENTAL SETUP
K-vapour was used to create anomalous dispersion
And achieve superluminal group velocity.

24. RESULTS
Information is detected later for the advanced pulse.

25. MODEL
Comparison of delay with a theoretical model based
On Maxwell’s equations.

26. CALCULATION
where
L is the length of the cell
vi is information velocity
Delta t is detection latency
small

27. CALCULATION
Using vi vac = c

28. SUMMARY
Information cannot travel faster than the vacuum speed
of light.
So forget about changing the past…
Be in the present…
And decorate the future…