1. Standing Waves, Beats, and Group Velocity
Superposition again
Standing waves: the
sum of two oppositely
traveling waves
Beats: the sum of two different frequencies
Thanks to Craig Siders, UCF, for the cool movies!
Group velocity: the speed of information
Going faster than light...
Group-velocity dispersion
Prof. Rick Trebino
Georgia Tech

2. Superposition allows waves to pass through each other.
Otherwise they'd get screwed up while overlapping

3. Now we’ll add waves with different complex exponentials.
It's easy to add waves with the same complex exponentials:
where all initial phases are lumped into E1, E2, and E3.
But sometimes the complex exponentials will be different! ~ ~ ~
Note the plus sign!

4. Adding waves of the same frequency, but opposite direction, yields a standing wave.
Waves propagating in opposite directions:
Since we must take the real part of the field, this becomes:
(taking E0 to be real) ~
Standing waves are important inside lasers, where beams are
constantly bouncing back and forth.

5. A standing wave
Nodes
The points where the amplitude is always zero are called nodes.
The points where the amplitude oscillates maximally are called anti-nodes.
Anti-nodes

6. A standing wave: experiment
3.9 GHz microwaves
Mirror
Input beam
The same effect occurs in lasers.
Note the node at the reflector at left (there’s a phase shift on reflection).

7. Beams crossing at an anglex q k2 z k1
Fringe spacing, L:
L = 2p/(2ksinq)
L = l/(2sinq)

8. Laser beams crossing at an angle
Finite-size (laser) beams yield fringes only where the beams overlap.
Fringe spacing L = l/(2sinq)
Interference fringes x

9. Two point sources emitting spherical waves
Different separations. Note the different node patterns.

10. When two waves of different frequency interfere, they produce beats.
Take E0 to be real.

11. When two waves of different frequency interfere, they produce "beats."
Indiv-
idual
waves
Sum
Envel-
ope
Irrad-
iance:

12. When two light waves of different frequency interfere, they also produce beats.
Take E0 to be real.
For a nice demo of beats, check out:

13. Group velocity vg º dw /dk Light-wave beats (continued):
Etot(x,t) = 2E0 cos(kavex–wavet) cos(Dkx–Dwt)
This is a rapidly oscillating wave: [cos(kavex–wavet)]
with a slowly varying amplitude: [2E0 cos(Dkx–Dwt)]
The phase velocity comes from the rapidly varying part: v = wave / kave
What about the other velocity—the velocity of the amplitude?
Define the group velocity: vg º Dw /Dk
In general, we define the group velocity as:
carrier wave
amplitude
vg º dw /dk

14. Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies).

15. The group velocity also applies to a single pulse.
Our derivation assumed beats and hence an infinite train of pulses. But if we extend the idea from just two discrete frequencies to a finite and continuous range of them, we have a single pulse. w S
Spectrum E t
And the group velocity will be the velocity of the pulse amplitude or irradiance.

16. When the group and phase velocities are different…
More generally, vg ≠ vf, and the carrier wave (phase fronts) propagates at the phase velocity, and the pulse (irradiance) propagates at the group velocity (usually slower).
The carrier wave:
The envelope (irradiance):
Now we must multiply together these two quantities.

17. Group velocity (vg) vs. phase velocity (vf)

18. The group velocity is the velocity of the envelope or irradiance: the math.
The carrier wave propagates at the phase velocity.
And the envelope propagates at the group velocity:
Or, equivalently, the irradiance propagates at the group velocity:

19. Calculating the group velocity
vg º dw /dk
Now, w is the same in or out of the medium, but k = k0 n, where k0 is
the k-vector in vacuum, and n is what depends on the medium.
So it's easier to think of w as the independent variable:
Using k = w n(w) / c0, calculate: dk /dw = ( n + w dn/dw ) / c0
vg = c0 / ( n + w dn/dw) = (c0 /n) / (1 + w /n dn/dw )
Finally:
So the group velocity equals the phase velocity when dn/dw = 0,
such as in vacuum. Otherwise, since n increases with w, dn/dw > 0,
and:
vg < vf

20. Calculating group velocity vs. wavelength
We more often think of the refractive index in terms of wavelength, so let's write the group velocity in terms of the vacuum wavelength l0.

21. The group velocity is less than the phase velocity in non-absorbing regions.
vg = c0 / (n + w dn/dw)
In regions of normal dispersion, dn/dw is positive. So vg < c for these frequencies. w

22. The group velocity can exceed c0 when dispersion is anomalous.
vg = c0 / (n + w dn/dw)
dn/dw is negative where dispersion is anomalous dispersion (near a resonance). So vg can exceed c0 for these frequencies!
Unfortunately, absorption is strong in these regions. Also, dn/dw is only steep when the resonance is narrow, so only a narrow range of
frequencies has vg > c0. Frequencies outside this range have vg < c0.
Pulses of light (which are broadband) therefore break up into a mess.

23. Beating the speed of light
To exceed c, we need a region of negative dn/dw over a fairly large
range of frequencies. And the slope should not vary much—to avoid
pulse break-up. And absorption should be minimal.
One trick is to excite the medium in advance with a laser pulse, which
creates gain (instead of absorption), which inverts the curve.
Then two nearby resonances have a region in between with minimal
absorption and near-linear negative slope: 2

24. Prisms, phase velocity, and group velocity
Phase-fronts are always perpendicular to the propagation direction.
Because the group velocity is usually less than phase velocity, the pulse front will lag behind the phase-fronts in the thicker part of the prism, causing the pulse to tilt when light traverses a prism.
Phase-fronts
Pulse-front tilt
Angular dispersion causes pulse-front tilt.
Prism
Pulse
Pulse-front

25. Group velocity dispersion is the variation of group velocity with wavelength
GVD means that the group velocity will be different for different wavelengths in the pulse. So GVD will lengthen a pulse in time.
vg(blue) < vg(red)
Because short pulses have such large ranges of wavelengths, GVD is a bigger issue than for nearly monochromatic light.

26. Group-velocity dispersion is undesirable in telecommunications systems.
Train of input telecom pulses
Dispersion causes short pulses to spread in time and to become long pulses.
Many km of fiber
Dispersion dictates the wavelengths at which telecom systems must operate and requires fiber to be very carefully designed to compensate for dispersion.
Train of output telecom pulses

27. Pulse Compressor
This device has negative group-velocity dispersion and hence can compensate for propagation through materials.
It’s routine to stretch and then compress short light pulses by factors of >1000.

28. Adjusting the prism maintains alignment.
Any prism in the compressor can be translated perpendicular to the
beam path to add glass and reduce the magnitude of negative GVD.
Remarkably, this does
not misalign the beam.
The output path is independent of prism position.
Input beam
Output beam