Wave PhysicsPHYS 2023

Tim Freegarde

2

Wave Physics

WAVE EQUATIONS & SINUSOIDAL SOLUTIONS

wave equations, derivations and solution

sinusoidal wave motions

complex wave functions

WAVE PROPAGATION

Huygens’ model of wave propagation

interference

general wave phenomena

Fraunhofer diffraction

longitudinal waves

BEHAVIOUR AT INTERFACES

continuity conditions

boundary conditions

SUPERPOSITIONSlinearity and superpositions

Fourier series and transforms

FURTHER TOPICSwaves from moving sources

operators for waves and oscillations

waves in three dimensions

http://www.avcanada.ca/albums/displayimage.php?album=topn&cat=3&pos=7

further phenomena and implications

3

Doppler effect

source

• wave speed• source

speed• observer stationary

• frequency

observer

source

observer

• wave speed

• observer• source

speedstationary

• frequency

•

4

Doppler effect

•

momentum

• energy, momentum conserved• Doppler shift appears automatically:

THE DOPPLER SHIFT REVISITED

• superposition phase slips

• transition involves photon absorption/emission

• photons have slope

• include kinetic energy

DIPOLE-ALLOWED TRANSITION

internal electronic states linked to momentum states

55

Beating

TWO DIFFERENT FREQUENCIES

tttt2

cos2

coscoscos 212121

•

6

Group velocity

• 2 sinusoidal components:

• 10 sinusoidal components:• spreading of

wavepacket

• this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation

• generally: the group velocity

= speed of energy propagation= speed of information propagation

•

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

• if not, wavepacket changes shape as it propagates

3 assumes nearly monochromatic – i.e. that etc. can be neglected

• 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

2211 sinsin • Snell’s law:

10

Total internal reflection

2211 sinsin • Snell’s law:

1

90

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:

http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20061130.shtml

Wave PhysicsPHYS 2023

Tim Freegarde

• carries momentum

• ‘comes in lumps’ - PHOTONS

• LIGHT…

• ‘scattering force’

• imparts impulse upon absorption/emission

17

Radiation pressure

•

• carries momentum

• ‘comes in lumps’ - PHOTONS

• LIGHT…

• ~½mg – a few grains of salt

• imparts impulse upon absorption/emission

18

Radiation pressure

•

19

Radiation pressure

•

absorption

emission

2

1

• carries momentum

• ‘comes in lumps’ - PHOTONS

• LIGHT…

• ~½mg – a few grains of salt

• imparts impulse upon absorption/emission

www.ifa.hawaii.edu/faculty/jewitt/tail-HB.html

Hale-Bopp (1997) – Malcolm Ellis

• atoms see only particular wavelengths• Doppler effect changes wavelength seen

20

Doppler cooling

•

Hänsch & Schawlow (1975)

ω0ω0 – Δω

v = c Δω/ω0

• VELOCITY SELECTION

• Doppler cooling (Rb) to ~1mK• (in our lab) sub-Doppler cooling to

~10μK• (evaporative cooling) ~few pK• Bose-Einstein condensation

21

Doppler cooling

•

10 million atoms20 μK<1 mm

• atoms see only particular wavelengths• Doppler effect changes wavelength seen

• VELOCITY SELECTION

• Doppler cooling (Rb) to ~1mK• (in our lab) sub-Doppler cooling to

~10μK

22

Acousto-optic modulation

dia sinsin

• Doppler shift

• Fraunhofer diffraction condition

• Bragg diffraction condition di

a

id

transducer

crystal

aid

aid kkk

dk

ik

ik

dk

ak

• energy and momentum are conserved

k

phonon

23

Diffracting atoms

E M Rasel et al, Phys Rev Lett 75 2633 (1995)nm811

-1m.s850v

Ar40

nm012.0Ar

rad32

m25.1

• stimulated Raman transitions equivalent to Bragg scattering from moving standing wave

2424

Michelson interferometer

• interference by division of amplitude

beamsplitter detector

source

δx

25

Inertial sensing using light

•

• Mach-Zehnder interferometer• quantum wavefunction split and recombined

• phase depends upon rotation

• laser-cooled atoms sense inertial Coriolis acceleration

2626

Wave Physics

• for handouts, links and other material, see http://phyweb.phys.soton.ac.uk/quantum/phys2023.htm

Wave PhysicsPHYS 2023

Tim Freegarde