Section C01: Lasers
Some basics
Speed of light in a medium is:
where c is the speed of light in vacuum, and
is the index of refraction
|
Medium
|
|
|
Air
|
approx. 1
|
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Water
|
1.33
|
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Glass
|
1.5 to 1.8
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Oil
|
1.4 to 1.6
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| |
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Wavelength 
and frequency 
:
.
Example: For the strongest green light from an argon
ion laser,
= 514.5 nanometers, speed of light in vacuum is 0.3 billion meters/second
(3E+08 m/s), and refractive index of air is approximately 1. So, the frequency
of this light is: 5.83E+14 cycles per second.
Energy of a Photon
The energy of a photon depends on the frequency of the
light.
where h is Planck's constant, 6.6262E-34 Joule-seconds. For the above example,
the photon energy of the green laser light is:
38.6E-20 Joules.
If the green beam has a power of 1 watt (1Joule per second),
then in 1 second, there must be about 2.5886E+18 photons flying merrily
out of the laser aperture every second.
In the folllowing, we will assume that the refractive index
is 1 for simplicity, unless we need to worry about refraction. So we will
use c for speed of light in calculating wavelength and frequency.
Overview of Lasers
Features of lasers: why people pay $$ and spend years aligning
them.
1. Coherence
2. Spectral Purity
3. Power
4. Short pulse duration
1. Coherence:
Laser beams come close to the idealized "ray optics"
which we studied in high school. They appear to originate from "point sources",
without destroying most of their power by passing them through pinholes.
The wavefronts maintain uniform phase and shape. So the beams can be collimated
to have a very low divergence angle over long distances. Lasers are used,
for example, to reflect off seismographs placed on the moon (read the book
which became the movie "Independence Day"). Actually the seismographs were
placed there by the Apollo astronauts.
2. Spectral Purity:
It is easy to obtain laser radiation which has only a
very narrow range of linewidths (range of colors or wavelengths or frequencies
present). Laser linewidths are much narrower than the linewidth of radiation
from a typical molecular energy transition. Thus if a laser beam is sent
through a prism, it will end up in a spot much narrower than a given color
in the spectrum formed by sending a beam of sunlight, of the same beam-width
as the laser beam. In fact, the laser linewidth will be much narrower than
any of the "absorption" or emission bands inside the spectrum of sunlight.
Because of this spectral purity, lasers can be "tuned" to deliver energy
to molecules at precisely-specified wavelengths.
Lines From an Argon Ion Laser
Source: SpectraPhysics, Inc., Model 2030 High Power
Ion Laser Brochure, p. 12
| Line Wavelength, nanometers |
Rank in Power (1: Highest) |
| 275.4 |
15 |
| 305.5 |
14 |
| 334.0 |
13 |
| 351.1 |
8 |
| 363.8 |
12 |
| roughly 450 |
10 |
| 457.9 |
6 |
| roughly 460 |
11 |
| roughly 465 |
7 |
| 476.5 |
3 |
| 488.0 |
2 |
| 496.5 |
4 |
| 501.7 |
5 |
| 514.5 |
1 |
| 528.7 |
9 |
Going back to the above example, if you send white light
through a prism, you get a continuous spectrum (forget about the absorption
lines). If you send the full "all lines" output from an argon ion laser
through a prism, you get a few narrow, discrete beams of slightly different
colors (ranging from deep blue through bright green), with darkness in
between. So it is easy to obtain truly "monochromatic" light from lasers.
3. Power
Because of the coherence, spectral purity and the short
pulse duraction discussed below, it is possible to concentrate very high
intensity of laser radiation, of very specific wavelength, within extremely
small volumes. This enables us to induce several "nonlinear" molecular
phenomena, which are otherwise found only in nuclear explosions or lightning
bolts.
4) Continuous-wattage (cw) and Pulsed lasers
The Helium-Neon (HeNe) laser, which radiates at 632 nm,
and the argon ion laser (476 - 514.5 nm) are generally used as "cw" lasers,
with electrical energy being continuously converted into light (with very
poor efficiency! Well over 99% of the electrical energy is removed as waste
heat using air or water coolant). In some kinds of lasers, devices such
as thyratrons or Q-switches are used to discharge stored energy in a short
pulse of light. Thus as much as 1 Joule of light may be obtained as a pulse
lasting a few nanoseconds. Such pulses may be repeated a few times a second.
BASIC CONCEPTS OF LASER OPERATION
Active Medium
At equilibrium, most substances have their internal energy
distributed over a large number of discrete ("quantized") levels of energy.
In general, the vast majority of molecules occupy the "average" energy
levels. This is a "Boltzmann" type of distribution. Look at the high-energy
portion: the levels with higher energy have lower "populations".
In the active medium of the laser, by putting in energy
of the right amount, we can cause a "population inversion", where many
more molecules go into a higher level. This is an unstable situation. If
we now "hit" this medium with photons of the right frequency (energy),
we can knock these molecules back down to the lower energy levels. In the
process, many more photons are emitted. This is called "Stimulated Emission"
of radiation.
Spontaneous Emission
This process occurs all the time: molecules collide, with some absorbing
energy, and others releasing energy. Sometimes the energy release occurs
through emission of a photon. The emission occurs in random directions,
and the energy release occurs with random phase relationships. As gas temperature
increases, the emission occurs with photons distributed over a wider and
wider range of energies.
Stimulated Emission
The nice thing is that in Stimulated Emission, the emitted
photons have the same phase and direction of propagation as the incident
photon. In spontaneous emission, which occurs all the time, the emission
may occur in any direction. Thus, in Stimulated Emission, you get Light
Amplification, which you can use if you keep down all the losses due to
absorption and scattering. The "gain" per pass of the light through the
medium is generally very small, so you have to align the mirrors nearly
perfectly , and keep things very clean, to get the laser to work.
Note: As seen above, "LASER" stands for Light Amplification
by Stimulated Emission of Radiation".
So we see that there are 2 essential steps to "lasing":
a) Pump molecules or ions into an excited state, producing
a Population Inversion
b) Stimulate emission of photons, forcing relaxation
of the molecules or ions towards equilibrium.
4-Level Laser
In practice, the simple 2-step process described above
does not work: the gain is simply not high enough. Some other tricks are
used to:
c) Keep pumping up the population of the excited state
d) Keep draining the population of the lower state.
in the figure, the ground state of molecular energy (corresponding
to the equilibrium distribution of energy states) is Level 1. The
Excited State is Level 4. The actual lasing occurs in the transition between
Levels 3 and Level 2. To keep the gain high, the population inversion between
Levels 2 and 3 must be maintained. As molecules fall down from Level 3,
more molecules must fall into Level 3. This occurs by sponstaneous emission
from Level 4. As molecules fall into Level 2, the population of Level 2
must be depleted, so that the population inversion is maintained. This
is achieved by spontaneous emission, down to the ground level, or by collisions
with something else.
An added feature shown in the figure is a "reservoir" of molecules with
energy very close to that of Level 4. For example, in a CO2 Gas Dynamic
laser, the excited state of CO2 has energy very close to that of the nitrogen
molecules, which are present in the gas in very large numbers. This
allows a "resonant exchange" between Level 4 and this "reservoir": there
are so many reservoir molecules that there are numerous collisions, bumping
molecules from other levels back into Level 4.
The Quantum Efficiency of the process in the 4-level laser shown in
the figure is given by: 
= laser energy emitted per unit energy added to the molecular states.
Optical Cavity
If the stimulated emission is greater than absorption,
then the light amplifies every time it passes through the active medium.
The optical cavity containing the active medium is designed to enable amplification
of the modes which we want. Mirrors coated with susbtances that filter
out other wavelengths, prisms for total internal reflection, or other means
("aerodynamic windows" to produce refractive index gradients with minimal
losses) are used to achieve the required reflection. There are "stable
resonator" cavities and "unstable resonators".
Stable Cavity Resonators
The light can bounce back and forth inside the laser
cavity indefinitely, with a small amount allowed to leak out as the laser
beam.
Unstable Resonators
Design Aspects
1) Stable cavities are good for small volumes and low-order
modes, but not good for scale-up, due to diffraction spreading.
2) Unstable resonators provide large volumes for low-order
modes, and are thus good for scale-up. Negative Branch designs have an
intra-cavity focus, which may cause "optical breakdown" of the medium due
to extremely high intensity at the focal volume.
The ratio of the outer to inner radius of the annular output
of an unstable resonator is called "magnification". This determines the
intensity distribution of the focal point.
Cavity Modes
Only certain discrete frequencies can amplify: these
resonant frequencies belong to the permitted spatial distribution of the
electromagnetic field and energy. This these spatial distributions, called
the cavity modes, are solutions to Maxwell's equation for the electromagnetic
field, for the boundary conditions imposed by the cavity geometry and the
nature of its walls.
Longitudinal or Axial Modes are field variations along
the resonator axis. They are prescribed by their half-wavelength fitting
integrally into the cavity length L.
.
Frequency
The frequency interval between modes is:
Example:
L = 0.3m, Refractive Index = 1,
Wavelength = 500nm
Interval between modes = 0.5E+09
Hz.
Number of modes = (2)(0.3)/(500E-09)
= 1.2 million.
So, a small change in cavity length
is of no consequence: there are plenty of modes to choose from.
The linewidth is much less than the linewidth corresponding
to a molecular transition. So many modes can co-exist, and "compete" for
the available population inversion.
Linewidth of a Molecular Transition
Consider the volume of gas inside a (gas) laser. There
are zillions of molecules, which at any given instant are distributed among
many possible energy levels. Each molecule's energy state at a given instant
is a combination of its translational kinetic energy, rotational energy,
vibration energy, and electronic excitation. Lets say that the power source
has put energy into the electronic excitation, so that there are many more
molecules in an "excited" electronic state than there should be for equilibrium
with all the other kinds of energy. When collisions occur between molecules
(and a gas is bettter thought-of as a "Demolition Derby" than as a civilized
environment), the excited molecules "relax" to equilibrium by emitting
energy, and calmer molecules might get "excited" to higher energy levels
by absorbing energy. Given the above "non-equilibrium" of the electrronic
states, there is a much higher likelihood of net emission. Whenhit by photons
of the right energy (say from a laser already functioning), the excited
molecules emit photons and relax to equilibrium. Now this emission should
be all of the same wavelength, because
should correspond to the energy difference
between the excited state and the equilibrium state. So the linewidth of
the molecular emission should be very small. Unfortunately, there are many
factors which bring uncertainty into the wavelength of the emission. The
molecules are constantly in motion, so there is a small frequency change
due to the Doppler shift of the radiation relative to the observer. Also,
the molecules may be colliding with other molecules between the absorption
of the incident photon and the relase of the emitted photon, and these
interactions change the energy. The increases in line width (the uncertainty
in the frequency of the emitted photons) due to these effects are called
Doppler broadening and Collision broadening.
The laser cavity modes allow us to pick very narrow line-widths
out of this broad mess, actually forcing a lot of the energy into the narrow
line. Collision broadening does not actually reduce the laser efficiency
much, but Doppler broadening does.
Mode Selection
If the objective is to get the most power out of a laser,
one generally tries to force the laser to run in the lowest order mode
for a given range of wavelengths, where the intensity distribution comes
to a peak at the axis. The losses are expected to be lowest at the lowest-order
mode.
Higher order modes have higher diffraction spreading. They
can be suppressed (with substantial losses) using an aperture. An "etalon",
which is a narrow cavity whose small dimensions mean large separation 
is also used to suppress higher-order axial modes.
The trouble with the Gaussian distribution, above, is that
beam is not uniform in intensity. So designers sometimes seek "top-hat"
beam profiles:
Higher-order modes are possible, and indeed we often spend
a lot of energy trying to prevent them. There must be some smart applications
where these modes provide just the right intensity distribution, as well.
For example, consider the "doughnut" mode below. This mode often plagued
the argon ion lasers used a few years ago in the Laser Doppler Velocimeter.
When the beam went into this mode, there was no light in the center of
the measurement volume, and the fringes were limited to such a narrow region
that it was difficult to get any particles to cross 8 successive fringes.
The solutions were (a) clean and realign all the mirrors starting with
the mirrors in the laser, or (b) turn down the output aperture, so that
the losses were increased in the outer parts of the beam, forcing what
remained into the Gaussian mode. The former option consumed days of surgery-level
anxiety and delicate operations, and the latter cut the effective power
down, usually to about 50% of the rated power of the laser.
LASER SURVEY
For flow diagnostics, the frequency range of interest
is usually in the visible (450nm - 600nm) range and into the ultraviolet
(down to about 250nm) range. The Rayleigh and Raman scattering cross-sections
of molecules are bigger in this range (i.e., molecules scatter more light
in this range). The spectral location (wavelength of photons corresponding
the energy differences between molecular states) of electonic resonances
for fluorescence are also in this range. Obviously the peak sensitivity
of optical detectors is in this range, because this is where research interest
and market interest have concentrated.
Neodymium-Yttrium Aluminum Garnet (Nd-YAG)
Lasing occurs in the Nd3+ ions housed in a YAG crystalline
host material (Y3Al5O12). Excitation is done using flashlamps or diode
lasers. YAG is better than glass, and permits cw operation. Typically,
the wavelength is 1064nm (infrared), but "frequency doubling" can convert
this to 532nm or 266nm by mixing the fundamental with harmonics.
The efficiency of flashlamp to 1064nm lasing is roughly
0.5%. The efficiency of frequency doubling is 30% to 40%.
Q-switched pulse energies of 300mJ in the 10-nanosecond
pulses can be repeated at 20 pulses per second. Nd-YAG is good for many
applications in Raman and Rayleigh scattering, for pumping dye lasers in
fluorescence / absorption work, and in combination with dye lasers for
wave mixing diagnostic techniques such as CARS (Coherent AntiStokes Raman
Scattering).
Dye Lasers
Dye lasers use molecular systems as the active medium.
A dye molecule is a complex organic molecule with 50 or even more atoms.
There are many possible energy states, consisting of combinations of rotational
energy and vibrational energy. Thus we can obtain almost any wavelength
in the range of 200nm to 1500nm by careful selection of the dye. Also,
we can tune into one of many closely-spaced frequencies. Dye lasers have
high quantum efficiency, but make a mess every time people spill the dye
in the lab.
Ruby Laser
Lasing occurs in the Cr3+ ions housed in rods of crystalline
Al2O3 host material. The wavelength is 694.3 nm.
Nitrogen Laser
Wavelength is 337.1nm. Typical output energies from a
few hundred microjoules to several millijoules, repeated at a few hundred
Hz.
Copper Vapor Laser
Wavelengths are 510.6nm (green) and 578.2nm (yellow).
Discharge is from excited copper vapor in the 2P to 2D manifold of spin-orbit
split states. Repetition rates are from 4000 to 12000 pulses per second.
Average power can be as high as 50 watts, and pulse widths 25 to 50 nanoseconds.
Excimer Lasers
Excimer = "Excited State Dimer".
Exciplex = "Excited State Complex. " Molecules of an Exciplex
are bound attractively only when one of the atoms is electonically excited.
Example: 
is stable, but 
is unstable. Examples of exciplexes are:
Q-Switching
Stimulated emission is held down (end losses are kept large),
so that the population inversion becomes large. In other words, the Q-factor
of the resonant mode is kept down. When the Q-factor is suddenly increased,
the losses are suddenly reduced and we get very high gain from the stimulated
emission. This produces an intense, short pulse.
Methods:
1) Rotating / vibrating mirror
2) polarization: A Pockels Cell is a crystal inserted
in the laser cavity, with various polarizing and polarization rotating
elements. High voltage pulse causes the polarization to rotate into a low-loss
cavity mode.
