Transcript Some quantum properties of light

Some quantum properties of light
Blackbody radiation to lasers
Density of Field Modes in a Cavity
z
Perfectly
conducting
walls.
Tangential
component of
E must
vanish at
bondary
y
L
x
kx 
nx
ky 
L
nx , n y , nz  0,1,2,3
n y
L
kx 
nz
L
kz
Lattice points
in the positive
octant of
k-space
kx , k y , kz  0
ky
kx
Density of States for Radiation
 k dk 
k2

2
dk or
2
  2 3 d
 c
The number of field modes per unit volume having their
wavenumber between k and k + dk
or angular frequency between  and  + d
Each mode has total energy of:
1
Ek  (nk + ) k
2
nk is the number of photons in a specific mode
Quantization of electromagnetic modes produces
energy levels just like that of the harmonic oscillator.
1

Note that if nk = 0 the energy is:
2
This is called the “energy of the vacuum.
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
Pn 
exp  En k BT 

 exp  E
n 0
n
k BT 
where k B  Bolzmann' s constant  1.38 10
Pn  e
 n k BT
1  e
  k BT

 23
J /K
The mean number,
of photons excited
in a particular
mode at
temperature T is:
n


k BT
n

k BT


e
n   nPn   n1  e


n 0 
n 0


 Planck thermal excitation function
 kT
 1  e B  1






What is the mean energy density of the radiation in these modes at
temperature T?
W  d  n   d
2

1
 d 


   k B T
  2 3 
1    c 
e
3

d
 2 3  k B T
 c e
1
Planck’s Law for radiative energy density
Energy Density of Radiation



d
0 W  d  0  2c3 e kBT  1

 k
2
4
B
3 3
15c 
3
T  urad
4
Units of J/m3
Stefan-Bolzmann Law
q
What is intensity of
radiated blackbody light.
Maximum of distribution found by taking derivative with
respect to , and equate to zero.
max T  2.898 10 m  K
3
Wien’s Displacement Law
Cosmic Microwave Background
Einstein A & B Coefficients
E2, N2

A21
spontaneous
emission
B12W  
absorption
B21W  
stimulated
emission
Consider N atoms in a cavity where the energy density
of the radiation is: W  
E1, N1
From Planck:
 3
1
W    2 3  k BT
 c e
1
Thermal Equilibrium implies:
dN1
dN 2

 N 2 A21 + N 2 B21W    N1 B12W    0
dt
dt
A21
A21
 W   
  k B T
N1
B12  B21
B12  B21 e
N2
B12  B21
and
 3
B21  A21
2 3
 c
Note that:
  3 
1
1
B21 2 3   k BT
 A21  k BT
1
e
1
 c  e
 B21W    A21n
Thermal-stimulated emission rate is equal to the spontaneous
emission rate multiplied by mean number of photons of
frequency .
Total Emission Rate  B21W   + A21
 A21n + A
 A21 n + 1
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
dN1
dN
  2  N 2 A +  N 2  N1 BW
dt
dt
Say at t=0 N=N1 (all atoms in ground state)
N2

A + BW 
N t   N
 A + 2 BW 
1
Lasers
Pump
R~.95
R~.999
Gain Medium
L
  2L nc
n  some integer
Standing light wave between mirrors.
If 1W exits laser, 20W inside cavity
2
E2
1
E1
For laser to work we need pump to
provide “Population Inversion”
Normally relative populations are small:
N2
 e  E2  E1  k BT
N1
Say DE~2eV, T=300K => N2/N1=3x10-34
Way small!
Real laser: pumping process typically something like
E3 Pump State
fast decay
E2 Metastable state with lifetime ts
hf0
pump
input
hf laser output
E1 Ground State
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 Medium
Periodic
Switch
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 shaft
2) Voltage signal to an electro-optic crystal. Optical
properties change with applied voltage.
Mode Locking
Periodic
loss
Periodic loss modulated at frequency equal to time for
pulse to travel distance of 2L.
Pulse propagates when losses are minimal.