Inversion
B N2 I > B N1 I
Canceling the BI factors, N2 > N1, or:
DN N1 N2 < 0
This condition is called inversion. It does not occur naturally (it’s forbidden by the Boltzmann distribution). It’s inherently a non-equilibrium state.
Usually, additional losses in intensity occur, such as absorption, scattering, and reflections. In general, the laser will lase if, in a round trip:
Total Gain > Total Loss
Laser medium
Output mirror
Will this intensity be sufficient to achieve inversion, N2 > N1? It’ll depend on the laser medium’s energy level system.
Rate Equations for a Two-Level System
Proportionality constant is the absorption/gain cross-section,
I(z) I(0)exp N2 N1 z
There can be exponential gain or loss in irradiance. Normally, N2 < N1, and there is loss (absorption). But if N2 > N1, there’s gain, and we define the gain, G:
Laser Gain
Laser medium
I(0)
I(L)
Neglecting spontaneous emission:
dI dt
c dI dz

BN2I BN1I
I = I(z-ct)
BN2 N1 I
The solution is:
z
0
L
Stimulated emission minus absorption
g and are the gain and absorption coefficients.
G exp N2 N1 L exp(gL)
If N2 > N1: If N2 < N1 :
g N2 N1 N1 N2
In order to achieve G > 1, stimulated emission must exceed absorption:
The Laser
in the form of excited states
A laser is a medium that stores energy, surrounded by two mirrors. The output mirror is partially reflecting and so lets some light out.
Nd:glass

Dye S-P Dye
Nd:YAG Diode
Nd:YLF
1ps 100fs
10fs
CW Dye
Color Center
CP M Dye
Cr:LiS(C)AF Er:fiber
Nd:fiber Cr:YAG
w/Compression
Cr:forsterite Ti:sapphire
The shortest pulse vs. year (for different media)
2
Pump
N2
Laser
Rate equations for the densities of the two states:
1
N1
Absorption Stimulated emission Spontaneous emission
dN2 dt

BI p (N1 N2 )
AN2
Pump intensity
This called achieving Threshold.
Calculating the Gain:
2
Einstein A and B Coefficients
1
In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of irradiance, I:

2BI p DN

AN

ADN
Why Inversion is Impossible in a Two-Level System
2
N2
Laser
d DN dt

2BI p DN

AN

ADN
1
N1
In steady-state: 0 2BI p DN AN ADN
( A 2BI p )DN AN
Limiting factors Commercial lasers
Prof. Rick Trebino Georgia Tech

But first: the progress has been amazing!
SHORTEST PULSE DURATION
10ps
Okay, what’s a laser, what are modes, and what does it mean to lock them?
Stimulated emission leads to a chain reaction and laser emission.
If many molecules in a medium are excited, one photon can become many.
Achieving Inversion: Pumping the Laser Medium
Now let Ip be the intensity of (flash lamp) light used to pump energy into the laser medium:
Back mirror
Ip
Back mirror
I0
R = 100% I3
Laser medium with gain, G
Output mirror
I1
I2 R < 100%
A laser will lase if the beam increases in intensity during a round trip:
that is, if I3 I0 .
Ultrafast solid-state laser media have recently replaced dyes in most labs.
Solid-state laser media have broad bandwidths and are convenient.
Laser power
Energy
Inversion
“Negative
4
temperature”
3
2
Molecules
1
Here, there is inversion from level 4 to levels 3 and 2.
In order to achieve inversion, we must hit the laser medium very hard in some way and choose our medium correctly.
Absorption rate = B N1 I
Spontaneous emission rate = A N2
Stimulated emission rate = B N2 I
where Ni is the number density of molecules in the ith state, and I is the irradiance.
Pulse-pumping Q-switching and distributed-feedback lasers Passive mode-locking and the saturable absorber Kerr-lensing and Ti:Sapphire Active mode-locking Other mode-locking techniques
Rate Equations for a Three-Level System
3
2
Fast decay
Assume we pump to a state 3 that rapidly decays to level 2—eliminating
The Generation of Ultrashort Laser Pulses
The importance of bandwidth
More than just a light bulb
Lasers, laser modes, and mode-locking
Making shorter and shorter pulses
Light bulbs, lasers, and ultrashort pulses