Lecture6 Ch3 Photons

7/27/2019 Lecture6 Ch3 Photons

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Chapter 3

Electromagnetic theory, Photons.and Light

Lecture 6

Pointing vector and Irradiance

Photons

Radiation Electromagnetic spectrum



Energy of EM wave

It was shown (in Phys 272) that field

energy densities are:

20

2 E u E

2

02

1

Bu B

Since E=cB and c=( 0 0)-1/2:

B E uu - the energy in EM wave is shared equally

between electric and magnetic fields

Total energy:2

0

2

0

1 B E uuu B E

(W/m2)



The Poynting vector

EM field contains energy that propagates

through space at speed c

Energy transported through area A in timet : uAct

EB EBcB E c E cuct A

t uAcS

0

0

00

0

2

0

11

Energy S transported by a wave through

unit area in unit time: E c2

The Poynting vector:

B E S

0

1

power flow per unit area for a

wave, direction of propagation

is direction of S .

(units: W/m2)

John Henry Poynting

(1852-1914)



The Poynting vector: polarized harmonic wave

B E S 0

1

Polarized EM wave:

t r k E E

cos0

t r k B B

cos0

Poynting vector:

t r k B E S

2

00

0cos

1

This is instantaneous value: S is oscillating

Light field oscillates at ~10 15 Hz -most detectors will see average value of S .



Irradiance

t r k B E S

2

00

0

cos1Average value for periodic function:

need to average one period only.

It can be shown that average of cos2

is: 21cos

2

T t 2

00

00

0 22

1 E

c B E S

T

And average power flow per unit time:

Irradiance:2

00

2 E

cS I

T

Alternative eq-ns:

T T B

c E c I

2

0

2

0

Usually mostly E-field component interacts with matter, and we

will refer to E as optical field and use energy eq-ns with E

Irradiance is proportional to the square of the amplitude of the E field

For linear isotropic

dielectric:

T E I 2

v

Optical power radiant flux total power falling on some area (Watts)



Spherical wave: inverse square law

Spherical waves are produced by point sources.As you move away from the source light intensity

drops

t r k r

t r vcos, A

Spherical wave eq-n:

t r k

r

E E

cos0 t r k

r

B B

cos0

t r k r

B

r

E S

200

0

cos1

2

02

0 1

2 E

r

cS I

T

Inverse square law: the irradiance from a point source drops as 1/r 2



Classical EM waves versus photons

The energy of a single light photon is E=h

The Planck’s constant h = 6.626×10-34 Js

Visible light wavelength is ~ 0.5 mJ104 19

1

c

hh E

Example: laser pointer output power is ~ 1 mW

number of photons emitted every second:

photons/s105.2J/photon104

J/s10 15

19

3

1

E

P

Conclusion: in many every day situations the quantum nature of

light is not pronounced and light could be treated as a classical

EM wave



Photon counter

It is possible to detect single photons

Example: photomultiplier tube (PMT)

Photon kicks an electron out of cathode

The electron is accelerated by an

E-field toward a dynode

The accelerated electron strikes

the dynode and kicks out more

electrons

Many dynodes are used to get

burst of ~105 electrons per single

photoelectron

The burst of electron current can

be detected electronically





Radiation pressure

Using classical EM theory Maxwell showed that radiation pressure

equals the energy density of the EM waves:

2

0

20

2

1

2 B E u

P

ucS

c

t S t P

This is the instantaneous pressure that would be exerted on a

perfectly absorbing surface by a normally incident beam

Average pressure:

c

I

c

t S t T

T P (N/m2)

* for reflecting surface pressure doubles* in quantum picture each photon has a momentum:

h p k p

or , where

2

h

propagation vector

Experimental confirmation:

Compton effect





Example problem (continued)

t

cr ji

Ac

P E

2sinˆcosˆ2

cosk ˆ2

0

x

y

z

B

t r k B B

cos0Magnetic field:It is in phase with E .

Need only find its amplitude and direction.

0

00

21/ Ac

P

cc E B

cosˆsinˆ21

0

0 ji Ac

P

c B

t

cr ji ji

Ac

P

c B

2sinˆcosˆ2

coscosˆsinˆ21

0





Radiation: accelerated charges

Electromagnetic pulse can propagate in space

How can we initiate such pulse?

Short pulse of transverse

electric field

Field of a moving charge



Radiation: accelerated charges

1. Transverse pulse

propagates at speed of

light2. Since E(t) there must

be B

3. Direction of v is given

by:

E

Bv



Electric dipole radiation

Oscillating charges in dipole create sinusoidal E

field and generate EM radiation



Electric dipole radiation

Dipole moment:

t

t d d

qd

cos

cos

0

0

p p

p

Electric field of oscillating dipole:

r

t kr k E

cos

4

sin

0

2

0 p

2

2

0

32

42

0 sin

32 r c I

p

Irradiance:

* EM wave is polarized along dipole

* I ~ 4 - higher frequency, stronger radiation

* No radiation emitted in direction of dipole

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Lecture6 Ch3 Photons