Download Light Absorption and Light Amplification

Light Absorption and Light
Amplification
To answer the questions in the previous
section, we consider a collimated beam of unit
cross-sectional area passing through a medium
∆x → 0
dI
∆I = −αI ( x ) ∆x ⇒
= −αI
dx
Absorption coefficient
• Absorption coefficient is a measurable
quantity
• It is a macroscopic parameter
• How does it relate the microscopic
parameters such as the Einstein
coefficients?
Irradiance
• In experiment, the irradiance is observable
• It is defined as the light energy crossing unit
area in unit time
• It is related the light energy density, ρ
by multiplying the light speed
→
I = ρ c/n
Light energy density
• Light energy density, ρ may be expressed by
ρ = N hv g(v)
• N is the total photon number density
• g(v) is the probability of finding photons in the
frequency range from v to v+dv
• For blackbody radiation, g(v) →Planck’s formula
In an absorptive medium, the irradiance
dI
= − αI
dx
1 dI
1 dI 1
1 dI 1
⇒α=−
=−
=−
I dx
I dt dx
I dt c / n
dt
c
d (ρ )
1 dI
1
n dρ
n
=−
=−
=−
c dt
c c dt
cρ dt
I
(ρ )
n
n n
n d [ Nhvg ( v )]
n
dN
=−
= − hvg ( v )
cρ
dt
cρ
dt
Change in the photon number
•
The rate of change in the photon number is
caused by
1. Absorption
2. Spontaneous emission
3. Simulated emission
•
•
Both absorption and stimulated emission occur in
the light traveling direction
Spontaneous emission occurs in all the directions
Neglecting spontaneous emission
• It spreads over the whole solid angle, 4π
• In the traveling direction, the fraction
~ 1/4π ~ 0.03
• And spontaneous emission < absorption
• Hence it is negligible in comparison
dN
= − N1B12ρ + N1 A12 + N 2 B21ρ
dt
= −( N1B12ρ − N1 A12 ) + N 2 B21ρ
≈ − N1B12ρ + N 2 B21ρ
B12
= −( N1
− N 2 ) B21ρ
B21
g2
= −( N1 − N 2 ) B21ρ
g1
g2
1 dN
⇒
= −( N1 − N 2 ) B21
ρ dt
g1
Relation between the measurable absorption
coefficient and the Einstein coefficients
1 dN
g2
= −( N1 − N 2 ) B21
ρ dt
g1
⇒
n
g2
α = −( N1 − N 2 ) B21hvg ( v )
c
g1
Distribution in thermal equilibrium
Non-thermal Equilibrium
• If we want α to be positive, then N2 must be
greater than g2N1/g1
• And it must be in a non-thermal equilibrium state
• We define the small-signal gain coefficient, k
g2
n
k = ( N 2 − N1 ) B21hvg ( v )
g1
c
Reason why called “small-signal”
• Small-signal gain coefficient: k = -α
• α is derived under the condition of thermal
equilibrium
• When there is a gain, it must be non-equilibrium,
so the above expression is invalid
• But, if the non-equilibrium state is not far away
from the equilibrium state, it is valid
approximately
• Hence, the term “small-signal” is applied to k
Pumping and Population Inversion
The First Laser
• The first laser is made of ruby crystal
• Ruby is an Al2O3 crystal in which there is
about 0.05% Cr2O3
• Cr3+ ions offer energy levels for lasing
• It may be approximated to a three-level
system
• Electrons are pumped from the ground state
into its blue or green absorption bands
• Without pumping, electron population
obeys the Boltzmann distribution
• External energy from flashlight excites
electrons from E0 to E2
• E2 is a short-lived state
• Rapid decays from E2 to E1
• E1 is a matestable (long-lived) state
• Population builds up at E1
• There is a population inversion between E1
and E0
Example: ruby crystal
Ruby: energy levels
Laser invention
Three-level system
Ni
Another example: He-Ne gas
Four-level system
Generation of population
inversion by pumping
• By applying pumping, a population
inversion is created
• The exact mathematical relation between
(N2 - g2/g1N1) and the pumping power
depends on the details of pumping method
• The degree of population inversion must be
greater than losses
Optical Resonators and Lasing
Threshold
Multiple passes
• Two ways to generate intense light in one
direction
• Single pass → amplification is little
• Many passes → amplification is substantial
Amplification
n
20
2 = 2 ~ 10
6
Oscillation
• Many passes is achieved by oscillation →
photons bounces back and forth between
two mirrors
• The two mirrors form an optical cavity or
Fabry-Perot resonator
• The two mirrors have high reflectance →
high finesse → low loss → photon number
can grow
Light Losses
•
•
The gain is usually very small
Therefore, it is essential to minimize all losses
1.
2.
3.
4.
Diffraction loss
Transmission loss
Absorption and scattering losses at mirrors
Absorption and scattering loss in the laser medium
Diffraction Losses
• Diffraction effect make some of light spread
out of the cavity
Minimizing diffraction losses
• Using non-plane mirrors may minimize
diffraction losses but not eliminate them
Transmission losses
• Un-intentional loss → no one can make
mirrors with 100% reflectance
• Intentional loss → laser light must get out
Losses due to absorption and
scattering at mirrors
• Mirror is usually made of
glasses coated with silver
• Light may be absorbed by
glass and silver
• Light may be scattered by
glass and silver
• To minimize, other
dielectrics may be used
instead of silver
Losses due to laser medium’s
scattering
• Light may be scattered by laser medium
itself
Losses due to laser medium’s
absorption
• Light may be absorbed by transitions other
than the desirable transition
Single effective loss coefficient
• In experiment, it is hard to distinguish these losses
except the transmission loss
• Therefore, we define an effective loss coefficient
which includes all the losses except the
transmission losses
• Let R1 = irradiance reflectance of the 1st mirror
• Let R2 = irradiance reflectance of the 2nd mirror
Calculate the change in the
irradiance after a round trip
I=I
e
l
a
i
t
n
i
From M1 to M2
( k −γ ) L
( k −γ ) L
I = R2 I
e
From M2 to M1
I = R2 I
e2( k −γ ) L
l
a
i
t
n
i
I = R1R2 I
e
l
a
i
t
n
i
After the reflection by M1
l
a
i
t
n
i
After the reflection by M2
2( k −γ ) L
Threshold
• Round-trip gain, G
Final irradiance
G=
Initial irradiance
= R1R2e 2 ( k − γ ) L
• If G > 1, photons at the laser frequency will
undergo a net amplification → the oscillation
will grow
• If G < 1, the oscillation will die out
• Therefore, G = 1 is the threshold condition
• kth= threshold small-signal coefficient
Final irradiance
G=
Initial irradiance
= R1R2e
2( k −γ ) L
=1
1
1
ln
kth = γ +
2 L R1R2
g2
c
( N 2 − N1 )th =
kth
g1
B21hvng (v )
Pumping
→ Population inversion
→ Spontaneous emission
→ Stimulated emission in a particular direction
Summary
• According to the Boltzmann statistics, the
population at the lower energy level is
higher at thermal equilibrium
– Absorption transition rate ~ N1
– Stimulated transition rate ~ N2
– Therefore, absorption > stimulated emission
Non-thermal equilibrium
• To make stimulated emission > absorption,
non-thermal equilibrium should be
considered
• To do so, pumping is needed
– pumping is a technique to provide external
energy to a system
Selecting suitable materials
• To make simulated emission > absorption,
materials have been selected according to
the following criteria
– efficiency to make N2 – g2 N1 / g1
– minimisation of undesirable losses
• How to achieve it depends on knowledge of
material science