Lecture 5
1
3-Level Laser
ˆ E1 is the ground state, and E3 represents all
upper states.
ˆ Atoms are pumped from E1 to E3 , but τ3 is
very short so the atoms quickly decay to E2 ,
so we can consider that we pump from E1 to
E2 .
1.1
No Input Signal or Small Signal
dN2
N21
=R−
dt
τ2
2
At steady state, dN
dt = 0, so
N2 = Rτ21
1
If we use dN
dt , we will get the same result, so we use Na = N1 + N2 . To get population inversion:
N2 − N1 = N2 − (Na − N2 ) = 2N2 − Na
= 2Rτ21 − Na
For population inversion, N2 > N1 , so
R>
1.2
Na
2τ21
Large Input Signal
Assume steady state, so:
dN2
N2
=R−
− σϕν N2 + σϕν N1 = 0
dt
τ21
N2
dN1
= −R +
+ σϕν N2 − σϕν N1 = 0
dt
τ21
Using N1 = Na − N2 :
N2
+ 2σϕν N2
τ21
R + σϕν Na
⇒ N2 = 1
τ21 + 2σϕν
R + σϕν Na =
population inversion (N = N2 − N1 = 2N2 − Na ):
N=
2R + 2σϕν Na
− Na
1
τ21 + 2σϕν
2R − τN21a
2Rτ21 − Na
= 1
=
1
+ 2σϕν τ21
+
2σϕ
ν
τ2
No
=
1 + ϕϕs
1
where No = 2Rτ21 − Na and ϕs = 2στ1 21 .
Since E1 is the ground level, it has a large population, so to achieve population inversion, we use
very large power which is inefficient. However, in 4-level lasers, both E1 and E2 have small number of
atoms, so pumping is more efficient.
2
2-Level Laser
ˆ Note that in 3-level & 4-level lasers, we pump
from ground level to upper level (E3 ), so
we only consider spontaneous emission during
pumping. While the input signal has energy
∆E = E2 − E1 so we consider all light matter
interaction.
ˆ In 2-level lasers, we only have 2 levels, so during pumping, we have to consider all light
matter interaction (spontaneous emission +
stimulated emission + absorption).
2.1
No Input Signal or Small Signal
We can replace the pumping rate R by the absorption rate. We also have stimulated emission resulted
from the light matter interaction:
dN2
N21
=−
− σϕν N2 + σϕν (Na − N2 ) = 0
dt
τ2
σϕν Na
⇒ N2 = 1
τ2 + 2σϕν
Population inversion:
2σϕν Na
− Na
1
τ21 + 2σϕν
− Na
= 1 τ21
<0
τ2 + 2σϕν
N=
We can never make a laser system with 2 levels when using optical pumping.
3
Introduction to Semiconductor Lasers
3.1
Travelling Wave Amplifier (TWA)
ˆ We pump the semiconductor with current till we reach transparency current Itr .
ˆ Trancparency current is the current at which the input power is equal to the output power.
ˆ We put anti-reflection coating on the front and back of the semiconductor to prevent reflection
which prevents oscillation.
ˆ We can consider it as an open-loop amplifier.
ˆ Small gain but high BW
3.2
Fabry-Perot Laser (FP)
ˆ We put a mirror on the front and back of the semiconductor.
ˆ We can consider it as a positive feedback amplifier.
2
ˆ Adjusting the mirrors (reflections) will determine wether the laser will oscillate or not (amplifier
or LASER).
ˆ High gain but low BW
3.3
Erbium Doped Fiber Amplifier (EDFA)
ˆ Regular fiber doped by Erbium, so it is a 3-level laser.
ˆ The input signal is at 1550 nm, and the pump is at 980 nm.
ˆ We use a directional coupler to combine the input signal and the pump.
4
Amplifier Noise
ˆ Spontaneous emission is the main source of noise in amplifiers.
ˆ Note that 1 atom emits 1 photon
Rate per atom:
1
gν (ν)∆ν
τsp o
Number of atoms per second per unit volume:
1
gν (ν)∆νN2
τsp o
Energy of spontaneous emission per unit volume per unit time:
1
gν (ν)∆νN2 hν
τsp o
2
λ
Spontaneous emission is in all directions in a Gaussian shape with a solid angle of area
. However, we
are interested in the direction of the input signal, so:
2
LASER solid angle
λ
volume ∗
=
area∆z
4π
4π
area
Spontaneous power in the beam direction:
dPASE =
1
gν (ν)∆νN2 hν∆z
τsp o
λ2
4π
Ex = Ecos(θ) and Ix ∝ E 2 cos(θ)2 . For unpolarized beam, the polarization can be in any direction, so
we need to take the average:
3
Z 2π
< Ix >= Io
0
1
Io
cos2 θdθ =
2π
2
Only half the power couples the x-component, so we need to multiply by half:
2
λ
1
1
gν (ν)∆νN2 hν∆z
dPASE =
τsp o
4π 2
2
N2 −N1
λ
σ = 8πτ
gνo (ν) and N2 = N2 N
, so:
sp
2 −N1
dPASE
N2 − N1
= σN2
hν∆ν
dz
N2 − N1
γ = σ(N2 − N1 ):
N2
dPASE
=γ
hν∆ν
dz
N2 − N1
2
:
Using spontaneous emission factor nsp = N2N−N
1
dPASE
= γnsp hν∆ν
dz
The noise spectrum is very wide, so we pass it through an optical filter of bandwidth Bo so only inband
noise is passed.
dPASE
= γnsp hνBo
dz
Notes:
ˆ Net stimulated emission is proportional with N2 −N1 , but net spontaneous emission is proportional
2
is named the spontaneous emission factor.
with N2 . Thus nsp = N2N−N
1
ˆ Since N2 > N2 − N1 , spontaneous emission is dominant, but since it is random, it is called noise.
ˆ Spontaneous emission can happen between any 2 levels, unlike stimulated emission which only
happens between E2 and E1 from the input signal. Therefore, spontaneous emission has a much
larger bandwidth than stimulated emission.
dP
= γP + γnsp hνBo
dz total
Solving the differential equation assuming γ = γo (s.s gain) and neglecting the gain saturation:
Pout = Go Pin + nsp hνBo (Go − 1)
Notes:
ˆ Sources of noise:
4
– Relative Intensity Noise: Fluctuations in the LASER output from the spontaneous emission.
– Stimulated Emission Noise: Different material interactions with the same input signal.
P
Go Pin
in
ˆ Optical SNR = Psignal
, if Go >> 1, then SN R = nspPhνB
= nsp hνB
o
o (Go −1)
noise
ˆ The ν used in the equations is the center of the LASER linewidth.
5
LASER
A LASER is an oscillator, so we need to achieve the positive feedback conditions:
ˆ Starting Noise (spontaneous emission noise).
ˆ Feedback (mirrors).
ˆ Aβ > 1 which reaches Aβ = 1 due to non-linearities (A is from material amplification, and β is
from the refelectance of the mirrors).
5.1
Types of Mirrors
ˆ Simple Discontinuity:
– Γ = n−1
n+1 (real)
– The normal case in semiconductor lasers.
ˆ Dielectric Slab:
– Γ can be complex dependeing on time.
– The normal case in gas.
5
5.2
Phase Condition
Assuming constant gain (γ):
E1
γ
αs
E2 = E1 e−jβL e− 2 L e 2 L
αs
γ
E3 = r2 E1 e−jβL e− 2 L e 2 L
E4 = r2 E1 e−2jβL e−αs L eγL
E5 = r1 r2 E1 e−2jβL e−αs L eγL
For oscillation, E1 should add with E5 :
ω
2βL = 2qπ ⇒ 2 L = 2qπ
c
2πν
qc
2
L = 2qπ ⇒ ν =
c
2L
where q is an integer. Practically q is a large number ≈ 102 .
5.3
Gain Condition
r1 r2 e−αs L eγL > 1
So:
eαs L
1
⇒ γL > αs L + ln(
)
r1 r2
r1 r2
1
1
γ > αs + ln
L
r1 r2
eγL >
r1 =
√
R1 and r2 =
√
R2 , so:
1
γ > αs +
ln
2L
1
R1 R2
where:
ˆ αs is the scattering loss and other losses (apart from absorption as it is included in the gain γ).
1
ˆ 2L
ln R11R2 is the mirror loss (if R1 = R2 = 1 this term is 0).
1
ˆ αs + 2L
ln R11R2 is called the resonator loss (αr ). For amplification, γ > αr .
ˆ We assumed that both r1 and r2 are real so they only affect the gain condition, but the general
case is that they are complex, so they affect the phase condition as well.
6
6
Conclusion
For oscillation, we must satisfy both conditions:
qc
ˆ Phase condition: ν = 2L
ˆ Gain condition: γ > αr
Gain peak is lower than αr ⇒ No oscillation.
Oscillation only happens at νq , νq+1 , νq−1 :
As light intensity increases, the gain decreases, so the gain curve moves down, so some modes will stop
oscillating (γ < αr ) even if they satisfy the phase condition.
7