Some quantum properties of light Blackbody radiation to lasers.

Some quantum properties of light

Blackbody radiation to lasers

Density of Field Modes in a Cavity

x

y

zPerfectly conducting walls. Tangential component of E must vanish at bondary

L

kx

ky

kz

3,2,1,0,,

zyx

zx

yy

xx

nnnL

nk

L

nk

L

nk

Lattice points in the positive octant of k-space

0,, zyx kkk

Density of States for Radiation

d

cdk

kdkk 32

2

2

2

or

The number of field modes per unit volume having their wavenumber between k and k + k

or angular frequency between and

Each mode has total energy of:

kkk nE )2

1(

nk is the number of photons in a specific mode

Quantization of electromagnetic modes produces energy levels just like that of the harmonic oscillator.

Note that if nk = 0 the energy is:

This is called the “energy of the vacuum.

2

1

In thermal equilibrium at temperature T the probability Pn that the mode is thermally excited to the nth state is given by the Bolzmann factor

TkTknn

B

nBn

Bnn

BB eeP

KJk

TkE

TkEP

1

/1038.1constant sBolzmann' where

exp

exp

23

0

The mean number, of photons excited in a particular mode at temperature T is:

n

11

100

Tk

Tk

n

n

Tk

nn

B

BB

e

eennPn

Planck thermal excitation function

What is the mean energy density of the radiation in these modes at temperature T?

1

1

1

32

3

32

2

Tk

Tk

B

B

e

d

c

c

d

e

dndW

Planck’s Law for radiative energy density

Energy Density of Radiation

radB

Tk

uTc

k

e

d

cdW

B

433

42

032

3

0

15

1

Units of J/m3

Stefan-Bolzmann Law

What is intensity of radiated blackbody light.

Maximum of distribution found by taking derivative with respect to , and equate to zero.

KmT 3max 10898.2

Wien’s Displacement Law

Cosmic Microwave Background

Einstein A & B Coefficients

E2, N2

E1, N1

A21

spontaneousemission

WB12 WB21

absorption stimulatedemission

Consider N atoms in a cavity where the energy density of the radiation is: W

From Planck:

1

132

3

TkBecW

Thermal Equilibrium implies:

2112

21

21122

1

21

12121221221 0

BBe

A

BBNN

AW

WBNWBNANdt

dN

dt

dN

TkB

212132

3

2112 ABc

andBB

Note that:

nAWB

eA

ecB TkTk BB

2121

2132

3

21 1

1

1

1

Thermal-stimulated emission rate is equal to the spontaneous emission rate multiplied by mean number of photons of frequency .

1

RateEmission Total

21

21

2121

nA

AnA

AWB

Spontaneous emission, or is it stimulated emission that was “stimulated” by the vacuum!??!

Atom in a cavity?!?

In general, W() tends to be small. In order to excite atoms one needs a powerful source of light, like a laser.

N atoms

WBNNANdt

dN

dt

dN122

21

Say at t=0 N=N1 (all atoms in ground state)

1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

WBA

WBANtN

21

N2

Lasers

Gain Medium

L

R~.999 R~.95

integer some 2 nncL

Standing light wave between mirrors.If 1W exits laser, 20W inside cavity

Pump

1

2

E1

E2For laser to work we need pump to provide “Population Inversion”

Normally relative populations are small:

TkEE BeN

N12

1

2

Say E~2eV, T=300K => N2/N1=3x10-34

Way small!

Real laser: pumping process typically something like

E1 Ground State

E3 Pump State

E2 Metastable state with lifetime ts

hf0

pumpinput

fast decay

hf laser output

In order for one to achieve population inversion the lifetime of the metastable state must be greater than time atom spends in ground state or pump state during the pumping process.

Pulsed Lasers

Gain MediumPeriodicSwitch

Pump gain media but do not let light flow out until you say so. All energy released as a high power pulse when “lasing” is allowed.

1) Put mirror on a rotating shaft2) Voltage signal to an electro-optic crystal. Optical

properties change with applied voltage.

Mode Locking

Periodicloss

Periodic loss modulated at frequency equal to time for pulse to travel distance of 2L.

Pulse propagates when losses are minimal.