G exp N 2 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:
Stimulated laser emission.
If many molecules in a medium are excited, one photon can become many. Excited medium
Cr:forsterite 10fs w/Compression Ti:sapphire
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Continuous vs. ultrashort pulses of light
A constant and a delta-function are a Fourier-Transform pair. Irradiance vs. time Continuous beam: Spectrum
Solid-state laser media have broad bandwidths and are convenient.
Laser power
Light bulbs, lasers, and ultrashort pulses
But a light bulb is also broadband.
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
Total Gain > Total Loss This called achieving Threshold.
Calculating the Gain: Einstein A and B Coefficients
2 1
In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of irradiance, I: Absorption rate = B N1 I
This is the essence of the laser. The factor by which an input beam is amplified by a medium is called the gain and is represented by G.
The Laser
in the form of excited states
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
What exactly is required to make an ultrashort pulse?
Answer:
A Mode-locked Laser
Okay, what’s a laser, what are modes, and what does it mean to lock them?
Inversion
Inversion
Energy
4 3 2
B N2 I > B N1 I
Canceling the BI factors, N2 > N1, or:
―Negative temperature‖
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.
Rate equations for the densities of the two states:
Absorption
Stimulated emission Spontaneous emission
2
Pump
N2
Laser
1
N1
dN 2 BI p ( N1 N 2 ) AN 2 dt
Dye SHORTEST PULSE DURATION 1ps CW Dye Color Center 100fs CP M Dye S-P Dye Diode
Cr:LiS(C)AF Er:fiber
Nd:fiber Cr:YAG
The shortest pulse vs. year (for different media)
A laser is a medium that stores energy, surrounded by two mirrors. The output mirror is partially reflecting and so lets some light out.
Back mirror Output mirror
If the total number of molecules is N:
Pump intensity
dN1 BI p ( N 2 N1 ) AN 2 dt d DN 2 BI p DN 2 AN 2 dt
N N1 N2 DN N1 N2
2 N2 ( N1 N2 ) ( N1 N 2 ) N DN
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
Neglecting spontaneous emission:
Laser medium
I(0)
0 L
I ( L) z
dI dI c BN 2 I BN1I dt dz B N 2 N1 I I = I(z-ct)
The solution is:
Stimulated emission minus absorption
Proportionality constant is the absorption/gain cross-section, g and are the gain and absorption coefficients.
I ( z ) I (0) exp N 2 N1 z
d DN 2 BI p DN AN ADN dt
Why Inversion is Impossible in a Two-Level System
d DN 2 BI p DN AN ADN dt In steady-state: 0 2BI DN AN ADN p
DN is always positive, no matter how high Ip is!
It’s impossible to achieve an inversion in a two-level system!
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
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
Limiting factors Commercial lasers
Prof. Rick Trebino Georgia Tech
But first: the progress has been amazing!
10ps Nd:glass Nd:YLF
Nd:YAG
time
frequency
Ultrashort pulse:
time
frequency
Long vs. short pulses of light
The uncertainty principle says that the product of the temporal and spectral pulse widths is greater than ~1. Irradiance vs. time Long pulse Spectrum
Spontaneous emission rate = A N2