some quantum properties of light blackbody radiation to lasers
Post on 19-Dec-2015
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Some quantum properties of light
Blackbody radiation to lasers
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Density of Field Modes in a Cavity
x
y
zPerfectly conducting walls. Tangential component of E must vanish at bondary
L
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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
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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
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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
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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
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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
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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
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Energy Density of Radiation
radB
Tk
uTc
k
e
d
cdW
B
433
42
032
3
0
15
1
Units of J/m3
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Stefan-Bolzmann Law
What is intensity of radiated blackbody light.
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Maximum of distribution found by taking derivative with respect to , and equate to zero.
KmT 3max 10898.2
Wien’s Displacement Law
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Cosmic Microwave Background
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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
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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
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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 .
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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?!?
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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
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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
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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!
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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.
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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.
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Mode Locking
Periodicloss
Periodic loss modulated at frequency equal to time for pulse to travel distance of 2L.
Pulse propagates when losses are minimal.