22. Lasers

1
22. Lasers

• Stimulated Emission and Gain
• The Laser
• Threshhold
• Four-level System
• Laser Frequencies
• "Longitudical Modes"
• Gausssian Beams
• Laser Resonators
• Spatial or "Transverse Modes"

2
The Laser
A laser is a medium that stores energy,
surrounded by two mirrors. Photons entering the
medium undergo stimulated emission. As a result,
the irradiance exiting from the medium exceeds
that entering it. A partially reflecting output
mirror lets some light out.
A laser will lase if the beam increases in
irradiance during a round trip that is, if
The laser achieves "Threshold" if
3
Laser Gain

• The Gain, G, is the amount by which a beam is
  amplified when it traverses a medium. Neglecting
  absorption
• The solution is
• There can be exponential gain or loss in
  irradiance. Normally, there is loss. But if
  there is gain, we define the gain, G

Stimulated emission minus absorption
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How to achieve laser threshold

• In order to achieve threshold, G gt 1, and
  stimulated emission
• must exceed absorption
• B N1 I gt B N0 I
• Or, equivalently,
• This condition is called "Inversion."
• It does not occur naturally.
• In order to achieve inversion, we must hit the
  laser medium hard in some way.

N1 gt N0
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The Ruby Laser
Invented in 1960 by Ted Maiman at Hughes Research
Labs, it was the first laser.
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Four-levelSystem
fast
Molecules accumulate in this level, leading to an
inversion with respect to this level.
The four-level system is the ideal laser system.
slow
Laser transition
fast
7
The Helium-Neon Laser
Energetic electrons in a glow discharge collide
with and excite He atoms, which then collide with
and transfer the excitation to Ne atoms, an ideal
4-level system.
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Diode Lasers
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Laser frequencies modes

• An infinite train of identical pulses can be
  written
• E(t) III(t/T) f(t)
• where f(t) represents a single pulse and T is the
  time between pulses. The Convolution Theorem
  states that the Fourier Transform of a
  convolution is the product of the Fourier
  Transforms. So

If this train of pulses results from a single
pulse bouncing back and forth inside a laser
cavity of round-trip time T. The spacing between
frequencies is then dw 2p/T or dn 1/T.
10
Laser modes in a real laser

• A finite train of identical pulses can be
  written
• where g(t) is a Gaussian envelope over the pulse
  train.

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Laser Modes
A lasers frequencies are often called
longitudinal modes.
12
Gaussian Beams

• Real laser beams are localized in space at the
  laser and hence
• must diffract as they propagate away from the
  laser.

The beam has a waist at z 0, where the spot
size is w0. It then expands to w w(z) with
distance z away from the laser. The beam radius
of curvature, R(z), also increases with distance
far away.
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Gaussian Beam Math

• The expression for a real laser
• beam's electric field is given by
• w(z) is the spot size vs. distance from the
  waist,
• R(z) is the beam radius of curvature, and
• y(z) is a phase shift.
• This equation is the solution to the wave
  equation when we require that the beam be well
  localized at some point (i.e., its waist).

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Gaussian Beam Spot, Radius, and Phase

• The expressions for the spot size,
• radius of curvature, and phase shift
• where zR is the Rayleigh Range (the distance over
  which the beam remains about the same diameter),
  and it's given by

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Gaussian Beam Collimation

• Twice the Rayleigh range is the
• distance over which the beam
• remains about the same size,
• that is, remains "collimated.
• _____________________________________________
• .225 cm 0.003 km 0.045 km
• 2.25 cm 0.3 km 5 km
• 22.5 cm 30 km 500 km
• _____________________________________________
• Tightly focused laser beams expand quickly.
• Weakly focused beams expand less quickly, but
  still expand.
• As a result, it's very difficult to shoot down a
  missile with a laser.

Collimation
Collimation Waist spot Distance
Distance size w0 l 10.6 µm
l 0.633 µm
Longer wavelengths expand faster than shorter
ones.
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Gaussian Beam Divergence

• Far away from the waist, the
• spot size of a Gaussian beam will be
• The beam 1/e divergence half angle is then w(z) /
  z as z
• The smaller the waist and the larger the
  wavelength, the larger the divergence angle.

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Focusing a Gaussian Beam

• A lens will focus a collimated Gaussian beam to a
  new spot size
• d0 2 f l / D
• So the smaller the desired focus, the BIGGER the
  input beam should be!

18
The Guoy Phase Shift

• The phase factor yields a phase shift relative
  to the phase of a
• plane wave when a Gaussian beam goes through a
  focus.

Phase relative to a plane wave Irradiance
(for reference)
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Laser Spatial Modes

• Also referred to as Transverse Electro-magnetic
  (TEM) modes

Electric field
The 00 mode is the Gaussian beam. Higher-order
modes involve multiplication of a Gaussian by a
Hermite polynomial.
Irradiance
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Laser Resonators

• Mirror curvatures play a big role in lasers.

21
Q-switching

• Q-switching involves
• Preventing the laser from lasing until the flash
  lamp is finished flashing, and
• Abruptly allowing the laser to lase.
• The pulse length is limited by the round-trip
  time of the laser and yields pulses 10 - 100 ns
  long.

Several kV of applied voltage makes the Pockels
cell a quarter-wave plate. Abruptly switching it
to zero turns off the effect.
22
Stable vs. Unstable Laser Resonators

• Stable
• resonator

Unstable resonator
Unstable resonators have much bigger beam sizes
(although they have a hole), and so are better
for high-power lasers.