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
Group-velocity dispersion

2. Leerdoelen In dit college leer je:
Dat lichtgolven kunnen interfereren (superpositie)
Hoe lichtgolven bij verschillende golflengtes samen een puls vormen
Interferentiepatronen berekenen
Wat de groepssnelheid is van een lichtpuls en hoe je die berekent
Analyseren wat de invloed van dispersie is op lichtpulsen
Een golf f = f ( x,t ). Bedenk steeds of in een plot x constant wordt gehouden (“film”) of dat t constant is (“foto”).
Hecht: 7.2

3. Sums of fields: Electromagnetism is linear, so the principle of Superposition holds.
If E1(x,t) and E2(x,t) are solutions to the wave equation,
then E1(x,t) + E2(x,t) is also a solution.
Proof: and
This means that light beams can pass through each other.
It also means that waves can constructively or destructively interfere.

4. Superposition allows waves to pass through each other.
Otherwise they'd get deformed while overlapping

5. Intensity of multiple waves
When multiple light waves are present, the intensity is calculated using the total field:
Suppose there are two light waves with fields E1 and E2: the resulting intensity becomes:
in which:
So the intensity of the total field is:
This cross-term (called the interference term) can be as large as the intensity of the individual beams!

6. Add waves with the same frequency and k but different complex amplitude.
It's easy to add waves with the same complex exponentials:
where all initial phases are lumped into E1, E2, and E3. ~
Sum the amplitude: Phasor addition. ~ ~ ~ Im ~ ~ Re

7. Now we’ll add waves with different complex exponentials.
But sometimes the complex exponentials will be different!
Now we’ll add waves with different complex exponentials.
Note the plus sign!

8. Adding waves of the same frequency and amplitude, but opposite direction, yields a standing wave.
Waves propagating in opposite directions:
And the intensity of this light wave then becomes:
(E0 is assumed to be real)

9. A Standing Wave l
Nodes
The points where the amplitude is always zero are called nodes.
The points where the amplitude oscillates maximally are called anti-nodes.
Note that the nodes and anti-nodes are each separated by l/2. E x
Anti-nodes

10. Beams Crossing at an Anglex q z
Fringe spacing, L:
L = 2p/(2ksinq)
L = l/(2sinq)

11. Big angle: small fringes. Small angle: big fringes.
The fringe spacing, L:
Large angle:
As the angle decreases to zero, the fringes become larger and larger, until finally, at q = 0, the intensity pattern becomes constant.
Small angle:
L = 0.1 mm is about the minimum fringe spacing you can see:
Fringes:

12. Laser beams crossing at an angle
Finite (laser) beams yield fringes only where the beams overlap. x

13. Two point sources
Interfering spherical waves also yield standing waves and fringes (in 2D and 3D)
Examples with different separations. Note the different node patterns.

14. Superposition of waves with different frequencies [at one point x]
Take E0 to be real.
We typically have that w1 and w2 are similar, this means wave >> Dw

15. When two cosines or sines of different frequency interfere, the result is beats.
time
2p/Dw
In phase
Out of phase
Individual
Waves
Sum
Envelope 1
Irradiance
1 beat

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

17. Traveling-Wave Beats Individual waves Sum Envelope Intensity z
In phase
Out of phase
Individual
waves
Sum
Envelope
Intensity z

18. Seeing Beats
It’s usually very difficult to see optical beats because they occur on a time scale that’s too fast to detect.
This is why we said earlier that beams of different colors don’t interfere, and we only see the average intensity.
However, a sum of many frequencies will yield a train of well-separated pulses:
In phase
Out of phase
Individual
waves
Sum
Irradiance
Pulse separation: 2p/Dwmin
2p/Dwmax
time

19. Group velocity Light-wave beats (continued):
Etot(x,t) = 2E0 cos(kave x–wave t) cos(Dkx–Dwt)
This is a rapidly oscillating wave: [cos(kave x–wave t)]
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 Dw /Dk
In general, we define the group velocity as:
carrier wave
amplitude

20. Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies).
[w = kc0]

21. 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.

22. Phase and Group Velocities
The phase velocity, vf, is that of the high-frequency oscillations. The group velocity, vg, is that of the pulse envelope.
In vacuum
Most common case

23. 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:

24. 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 [usually] increases with w, dn/dw > 0, and:
vg < vf

25. 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.

26. 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

27. 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.

28. 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.

29. 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

30. Tot slot Wat hebben we vandaag gezien:
Superpositie en interferentie van licht
Lichtpulsen zijn een superpositie van lichtgolven met verschillende golflengtes
Fase- en groepssnelheid van lichtpulsen
De invloed van dispersie op lichtpulsen

31. 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