Chapter 3

Electromagnetic Theory, Photons,and Light

Lecture 6

Photons Radiation Emission of light by atoms

Example problemA laser pointer emits light at 630 nm in xy plane at =450 to axis x(counter clock-wise). The light is polarized along axis z , beam cross-section is A=1 mm2 and its power is P=1 mW.1. Write an equation of E and B components of this EM wave for the region of the beam.

x

y

z trkEE

cos0

Find : c22 Find k: sinˆcosˆ2 jik

Find E0:Irradiance: 2

00

2EcI

AP

00 2 AcPE k̂2 00 AcPE

tcrji

AcPE

2sinˆcosˆ2cosk̂2

0

Electric field:

B

E

Example problem (continued)

tcrji

AcPE

2sinˆcosˆ2cosk̂2

0

x

y

z

B trkBB

cos0Magnetic field:It is in phase with E. Need only find its amplitude and direction.

000

21/AcP

ccEB

cosˆsinˆ21

00 ji

AcP

cB

tcrjiji

AcP

cB

2sinˆcosˆ2coscosˆsinˆ21

0

Example problem (continued)

2. This laser beam is reflected backwards by a mirror. What is the average force on the mirror due to the radiation pressure?

AcP

cIt

T22 PFind average pressure

Find force: N 106.6m/s 103

W1022 128

3

cPAtF

TP

3. How much energy is contained in EM field of 1 m long beam?

Power is amount of energy per unit time. During one second, light travels c meters:

J 103.3m 1m/s 103

J/s 10)1( 128

3

LcPmEnergy

Alternatively can find u using E0 and multiply by volume

Classical EM waves versus photonsThe energy of a single light photon is E=h

The Planck’s constant h = 6.626×10-34 JsVisible light wavelength is ~ 0.5 m J 104 19

1

chhE

Example: laser pointer output power is ~ 1 mWnumber of photons emitted every second:

photons/s 105.2J/photon 104J/s10 15

19

3

1

EP

Conclusion: in many every day situations the quantum nature of light is not pronounced and light could be treated as a classical EM wave

Photons

1900: to explain black body radiation spectrum Max Planck suggested that light is emitted in small indivisible quanta of energy:

E=h (h=6.626×10-34 J.s)

1905: to explain photoelectric effect Einstein stated that EM field itself is quantized

Photons cannot be observed directly, one can only see them through interaction with matter - absorption. Photon is destroyed in the process.

Photons carry energy and momentum (atoms recoil when emit photons)

hp kp

or , where2h

propagation vector

Experimental confirmation:Compton effect

optical power

Energy of a single photon at mean frequency 0 in quasi-monochromatic beam

Energy per unit time crossing some area A

Photon

Mean photon flux:00 h

PhAI

Number of photons emitted every second from a ~1 mW laser pointer is ~1015 photons

Photons strike screen every 1 fs on average.Exact position and time of arrival for each photon cannot be predicted with absolute certainly - we can only predict the probabilities.

no. photons per unit area per unit time

Photon

At any location on a screen, the classical irradiance is proportional to the probability of detecting a photon at that location

photographic film

Light exposurelow medium high

Quantum uncertainty.Example: throw a single coin, it will fall either heads or tails up, unpredictable

but with 1015 coins - can predict result with high precision

Photon statistics

Maxwell-Boltzmann statistics: for distinguishable particles

In quantum physics for indistinguishable particles:* Bose-Einstein statistics for bosons (particles with integer spin)* Fermi-Dirac statistics for fermions (particles with integer+half spins)

Photons are bosons - many photons can simultaneously be in exactly the same state, i.e. have the same energy

When a large number of photons occupy the same state (i.e. have the same energy, polarization and direction), the inherent granularity of the light beam vanishes and the EM field appears as the continuous medium of an electromagnetic wave - monochromatic plane wave.

Different monochromatic plane waves represent different photon states

Photon counter

It is possible to detect single photons

Example: photomultiplier tube (PMT)

Photon kicks an electron out of cathodeThe electron is accelerated by an E-field toward a dynodeThe accelerated electron strikes the dynode and kicks out more electronsMany dynodes are used to get burst of ~105 electrons per single photoelectronThe burst of electron current can be detected electronically

Photon statistics

PMTlow powerlight beam

Photons arrive at random.

Poisson distribution of photons arriving at detector during time T

Radiation: accelerated charges

Electromagnetic pulse can propagate in spaceHow can we initiate such pulse?

Short pulse of transverseelectric field

Field of a moving charge

Radiation: accelerated charges

1. Transverse pulse propagates at speed of light

2. Since E(t) there must be B

3. Direction of v is given by: BE

E

Bv

Electric dipole radiationOscillating charges in dipole create sinusoidal E field and generate EM radiation

Electric dipole radiation

Dipole moment:

t

tddqd

coscos

0

0

pp

p

Electric field of oscillating dipole:

r

tkrkE

cos4

sin

0

20p

2

2

032

420 sin

32 rcI

p

Irradiance:* EM wave is polarized along dipole* I ~ 4 - higher frequency, stronger radiation* No radiation emitted in direction of dipole

Dipole antenna

Example: connect AC generator to ‘dipole’ antenna/Charges will run up and down - dipole moment will be oscillating and radiation will be emitted