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Laser amplifier
A review of quantum mechanical result of emission and absorption
Symmetric relation between absorption and spontaneous emission.
Spontaneous
emission
Absorption
Stimulated
emission
V
c pab =
V
c pst = n
V
c psp =
V
c pab = n
Interaction of single mode light with a single atom
Spontaneous emission: stimulated emission with zero incident photon
In a cavity of volume V , the probability density or rate ( per second) of a ( single atom ) spontaneous emission into a single mode is
σ( ν ) V
c psp = (1)
where σ ( ν ) is the transition cross section, which is a narrow line function centered at ν 0. The probability of decay between t and t+ ∆ t is psp ∆ t. The inverse of psp is the lifetime of the excited state emitting in a particular mode only. The actual lifetime of atomic system is much shorter because there are many modes to emit into.
Absorption
The probability density for the absorption of a photon from a given mode of frequency ν in a cavity of volume V is governed by the same rule
σ( ν ) V
c pab = (2)
If there are n photons present in one mode, the probability density that the atom absorbs one photon is n times larger or
σ( ν ) V
c pab = n (3)
Stimulated emission The probability of creation of photons when only one photon present is govern by the same cross section as (1)
σ( ν ) V
c pst = (4)
If there are already n photons in a mode, the probability of emitting one more photon is
σ( ν ) V
c pab = n (5)
Thus the pst is the same as pab , the probability for spontaneous and stimulated emission of one photon is the same. Since the spontaneous emission occurs in addition to the stimulated emission, the total probability of an atom emitting into one mode in the presence of n photons
is ( 1 ) σ( ν) V
c n + , while the total probabilyt of absorbing one photon in the presence of n
photons σ( ν) V
c n
Oscillator strength, S, (cm^2 s -^1 ) is the area of the probability density as a function of frequency
∞
0
S σ (ν ) d ν (6)
Line shape function g ( ν)is a normalized function of σ ( ν )
σ ( ν)= Sg ( ν ) (7)
Spontaneous emission probability density (into all modes)
σ ν VM ν dv c σν M ν d ν
V
c
Pab = ( ) ( ) = ( ) ( ) (8)
sp sp
i (^) t
n ht
cS h
d V
c h
V
W ≈ = =
∞ ( ) 8
0
3
0
0 0
ρ ν π
λ ν
ρν ν
σ ν
ρ ν (13)
where ρ(ν) is the spectral energy density per unit bandwidth per unit volume for the incident photons and n is the average photons per mode. The probability of stimulated emission is enhanced by n times in the presence of photon flux.
Note that Wi is the same for stimulated emission and absorption.
Of historical interest: Einstein’s A and B coefficients
Einstein obtained the relation using a different approach, by considering the interaction of photons and atoms in thermal equilibrium.
Consider the interaction between thermal radiation with atoms with energy levels at E 1 , E2, where E 2 >E 1. The ratio of the numbers of atoms in each state is
e E E^ kT N
N ( )/
2
The number of spontaneous emission from atoms at E 2 is N (^) 2 A 21 where A 21 is the transition
probability per unit time for transition from 2 to 1 and is know as the A coefficient.
Let B 12 and B 21 denote the proportionality constants for stimulated emission. Then the numbers of stimulated downward transition (emission) and upward (absorption) per second in the presence of photon density u are N 2 B 21 u and N 1 B 12 u. The B’s are the B coefficients.
Under thermal equilibrium, the following relation should hold: N (^) 2 A 21 + N 2 B 21 u = N 1 B 12 u (15)
By solving for u:
1 12 2 21
2 21 NB N B
N A
u −
Using (14), we can write
/ 21 12 21
21 −
= (^) h kT B B B e
A
u ν (17)
In order to agree with the plank radiation formula, the following equations must hold:
3
3
21
21
12 21 8 c
h B
A
B B
π ν
Eq(18) governs the relation between the A and B coefficients.
The Einstein’s coefficients can be related to the results based on the probability arguments by
t sp
A
ht sp
B
π
λ 8
3 = (20)
Thus the probability in Eq (15) becomes Wi = B ρ( ν 0 ) (21)
Line Broadening
Discussions on homogenous broadenings vs inhomogeneous broadenings
Homogenous broadening— the broadenings shared by all atoms of a medium
(1) Lifetime broadening --caused by the spread of the energies of the terminal states due to their finite lifetime
)
h h E (22)
where τ 1 and τ 2 are the lifetime of the upper and lower states and τ is the effective lifetime. The line shapes of these lifetime broadened transitions are Lorentzian given by
2 2 ( 0 ) ( / 2 )
g = (23)
A simple way of explaining the line shape based on single electron damped oscillator driven by an sinusoidal electric field.
Next we will express the transition cross section defined earlier with the line shape function. From (6) and (7), the Oscillator Strength, S, the transition cross section, σ ( ν ), and line shape function are defined by
∞
0
S σ (ν ) d ν (6) (24)
σ ( ν)= Sg ( ν ) (7)(25)
where, from (10-1), t sp
S
2 =. Combinging (24) and (25) we have
2
σ ν g
tsp
The peak cross section at the center frequency of the line ν 0 is
2 t sp
2 0 (27)
(1) Amplifier gain From (12), the probability that an atom in the ground state absorbing a photon is
V
Wi n c (12)(29)
where, from (26), ( ) 8
2
σ ν g
tsp
=. When there are N 1 atoms in the lower level and N 2
atoms in the upper level , the average number of absorbed photons is N 1 Wi , and the number of produced photons is N 2 Wi. The net production of photons is given by ( N 2 -N 1 )Wi =NWi. , where N is the population density difference between the upper and lower states.
z z+dz
φφφφ φφφφ +d^ φφφφ
0 d
The increment of photon flux passing through a slice of gain medium is d φ = NWi dz (30)
Define the gain coefficient of a laser medium as
( ) 8
2
γ ν σν g
t
N N
sp
The photon flux along the medium follows the following equation:
( ) ( )
z dz
d z
The solution of (31) is an exponentially increasing function of z.
φ ( z )= φ( 0 )exp[γ( ν) z ] (32)
To exhibit a net gain, N must be larger than zero, the condition known as population inversion. The above treatment is also applicable to the absorption process when N<0.
The overall gain of the laser amplifer of length d is
G ( ν )= exp[γ( ν) d ] (33)
(2) Amplifier bandwidth
The amplifier bandwidth, measured in units of frequency or wavelength, is simply the width of the lineshape. For a Lorentzian shape as defined in (23), the bandwidth is ∆ν. The gain coefficient is also Lorentian
2 2 0
2 0 ( ) ( / 2 )
ν ν ν
ν γ ν γν − +∆
where γ ( ν 0 ) is the gain at the center of the line.
How does the gain bandwidth affect the amplification?
- Tunability
- Pulse duration
(3) Phase shift
∆ν ν
ν
χχχχ ’’
χχ^ χχ ’
We will use the result of Lecture 5 for the bound-electron model to show that the real and imaginary parts of the susceptibility are related.
2 2 2 2 0
2 2 0
2 0 0
' ( ) ( )
ν ν ν ν
ν ν ν χ χ − + ∆
2 2 2 2 0
2 0 0
'' (ν ν ) (ν ν )
ν ν ν χ χ − + ∆
The real part contributed to the index of refraction which in turns creates a phase shift of the wave. The imaginary part is the gain or absorption loss. The relation between the absorption and phase shift is the Kramers-Kronig relation.
Thus a laser amplifier creates a net change in the amplitude of the field and a phase change. is a complex number. The phase change is related to the gain by
Rate equations in the absence of amplifier radiation
21
2 1
1 1
1
2
2 2 2
N N
R
dt
dN
N
R
dt
dN
where R 2 is the rate of pumping into level 2 and R 1 is the rate of pumping out of level 1. The steady state (time derivative =0) population difference N 0 is
1 1 21
1 0 2 2 (^1 )^ τ τ
τ N = R τ − + R (39)
Discussions on how various factors affect the gain in two-level, three-level, four level systems.
Rate equations in the presence of amplifier radiation
i i
i
NW N W
N N
R
dt
dN
NW NW
N
R
dt
dN
2 1 21
2 1
1 1
1
2 1 2 2
2 2
2
The steady-state solution is
21
2 2 1
0
s
sWi
N
N
The population difference between the upper and lower levels decreased in the presence of stimulated emission, a phenomenon known as gain saturation. The time τ s is the saturation time constant.
Small signal regime: τ sWi <<1. –The depletion can be neglected.
Discussions: The effect of long and short lifetime of the upper and lower levels on on the saturation.
Gain coefficient
Combining (41) (31) and (29), the gain coefficient can be expressed using the unsaturated gain coefficient γ 0 (ν), the saturation flux and the saturation photon flux,
( )^0
φ φ ν
γ ν γ ν
where φ is the photon flux and
( ) 8
2
00 π^ ν
γ ν g
t
N
sp
g tsp
= s (44)
Discussion of inhomogenous broadening and hole burning
(5) Amplifier noise- (A qualitative discussion )
φ dz sp
d = + (45)
π
ξ ν 8
2
d g B t
N
sp
sp (46)
where ξ is the flux of the spontaneous emission to be amplified together with the “signal, B is the frequency bandwidth, and dΩ is the solid-angle of acceptance. The physical meaning is quite straightforward.
Exercises
- 13.3-2 (p.482) about the broadening of the saturated gain profile.
- Problem 13.2.1 (p.492)
- Problem 13.3.
- Problem 13.3.
