1. Wave Physics
PHYS 2023
Tim Freegarde

2. Wave Physics WAVE EQUATIONS & SINUSOIDAL SOLUTIONS
general wave phenomena
wave equations, derivations and solution
sinusoidal wave motions
complex wave functions
WAVE PROPAGATION
Huygens’ model of wave propagation
interference
Fraunhofer diffraction
longitudinal waves
BEHAVIOUR AT INTERFACES
continuity conditions
boundary conditions
SUPERPOSITIONS
linearity and superpositions
Fourier series and transforms
FURTHER TOPICS
waves in three dimensions
waves from moving sources
operators for waves and oscillations
further phenomena and implications

3. Doppler effect frequency wave speed source speed observer stationary

4. Doppler effect transition involves photon absorption/emission
internal electronic states linked to momentum states
THE DOPPLER SHIFT REVISITED
include kinetic energy
photons have slope
energy, momentum conserved
Dipole-allowed transition
Doppler shift appears automatically:
superposition phase slips
momentum

5. Beating
TWO DIFFERENT FREQUENCIES 5 5

6. Group velocity generally: the group velocity
= speed of energy propagation
= speed of information propagation
10 sinusoidal components:
2 sinusoidal components:
spreading of wavepacket
this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation

7. Kelvin ship waves
deep-water waves:

8. Superluminal waves generally: the group velocity
= speed of energy propagation
= speed of information propagation
1 assumes energy is conserved:
not true in an absorbing medium
not true in an amplifying medium
not true in a nonlinear medium
2 assumes wave propagates:
no constraint if wave doesn’t propagate from to
3 assumes nearly monochromatic – i.e. that etc. can be neglected
if not, wavepacket changes shape as it propagates
group velocity dispersion
4 beware of resonators, e.g. atoms in a crystal
anomalous group velocities at Brillouin zone edges
single frequency  steady-state excitation
system has memory

9. Total internal reflection
Snell’s law:

10. Total internal reflection
Snell’s law:

11. Total internal reflection

12. Frustrated total internal reflection (tunnelling)

13. Superluminal waves
tunnelling & the evanescent field

14. Superluminal waves
tunnelling & the evanescent field

15. Speed of light Listen again to Melvyn Bragg’s In Our Time:

16. Wave Physics
PHYS 2023
Tim Freegarde

17. Radiation pressure LIGHT… ‘comes in lumps’ - PHOTONS carries momentum
imparts impulse upon absorption/emission
‘scattering force’

18. Radiation pressure LIGHT… ‘comes in lumps’ - PHOTONS carries momentum
imparts impulse upon absorption/emission
~½mg – a few grains of salt

19. Radiation pressure LIGHT… ‘comes in lumps’ - PHOTONS carries momentum
Hale-Bopp (1997) – Malcolm Ellis
emission
carries momentum
‘comes in lumps’ - PHOTONS
LIGHT…
~½mg – a few grains of salt
imparts impulse upon absorption/emission
absorption

20. Doppler cooling VELOCITY SELECTION
ω0 – Δω
atoms see only particular wavelengths ω0
Doppler effect changes wavelength seen
Doppler cooling (Rb) to ~1mK
v = c Δω/ω0
(in our lab) sub-Doppler cooling to ~10μK
(evaporative cooling) ~few pK
Bose-Einstein condensation
Hänsch & Schawlow (1975)

21. Doppler cooling VELOCITY SELECTION
atoms see only particular wavelengths
Doppler effect changes wavelength seen
Doppler cooling (Rb) to ~1mK
10 million atoms
(in our lab) sub-Doppler cooling to ~10μK
20 μK
<1 mm

22. Acousto-optic modulation
Fraunhofer diffraction condition
crystal
phonon
Bragg diffraction condition
Doppler shift
transducer
energy and momentum are conserved

23. Diffracting atoms
E M Rasel et al, Phys Rev Lett (1995)
stimulated Raman transitions equivalent to Bragg scattering from moving standing wave

24. Michelson interferometerδx
interference by division of amplitude
beamsplitter
detector
source

25. Inertial sensing using light
Mach-Zehnder interferometer
quantum wavefunction split and recombined
laser-cooled atoms sense inertial Coriolis acceleration
phase depends upon rotation

26. Wave Physics
for handouts, links and other material, see

27. Wave Physics
PHYS 2023
Tim Freegarde