2' The Generation of Ultrashort Laser Pulses

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2. The Generation of Ultrashort Laser Pulses
The importance of bandwidth More than just a
light bulb Laser modes and mode-locking Making
shorter and shorter pulses Pulse-pumping Q-swit
ching and distributed-feedback lasers Passive
mode-locking and the saturable absorber Kerr-lens
ing and TiSapphire Active mode-locking Other
mode-locking techniques Limiting
factors Commercial lasers
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Continuous beams vs. ultrashort pulses

• A constant and a delta-function are a
  Fourier-Transform pair.

Continuous beam Ultrashort pulse
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Long vs. short pulses of light

• The uncertainty principle says that the product
  of the temporal
• and spectral pulse widths is greater than 1.

Long pulse
Short pulse
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For many years, dyes have been the broadband
media that have generated ultrashort laser pulses.
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Ultrafast solid-state laser media have recently
replaced dyes in most labs.

• Solid-state laser media have broad bandwidths and
  are convenient.

Laser power
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But a light bulb is also broadband. What else
is required to make an ultrashort
pulse?Answer Mode-lockingOkay, what are
modes and what does it mean to lock them?
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The Shah Function

• The Shah function, III(x), is an infinitely long
  train of equally
• spaced delta-functions

The symbol III is pronounced shah after the
Cyrillic character III, which is said to have
been modeled on the Hebrew letter (shin)
which, in turn, may derive from the Egyptian
a hieroglyph depicting papyrus plants along
the Nile.
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The Fourier Transform of the Shah Function

• If w 2np, where n is an integer, the sum
  diverges
• otherwise, cancellation occurs.
• So

The Shah function is its own Fourier transform!
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The Shah Function and a train of pulses

• An infinite train of identical pulses (for
  example, from a laser) can
• be written
• where f(t) is the shape of a single pulse and T
  is the time between
• pulses.

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An Infinite Train of Pulses and its Fourier
Transform

• 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

F E(t) µ III(wT/2p) F(w)
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 (the modes) is then dw 2p/T or dn
1/T.
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Generating short pulses mode-locking

• Locking the phases of the laser modes yields an
  ultrashort pulse.

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Mode-locked vs. non-mode-locked light
Mode-locked pulse train
A train of short pulses
Non-mode-locked pulse train
Random phase for each mode
A mess
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Locked modes
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Numerical simulation of mode-locking
Ultrafast lasers often have thousands of modes.
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A generic ultrashort-pulse laser

• A generic ultrafast laser has a broadband gain
  medium, a pulse-shortening device, and two or
  more mirrors

Many pulse-shortening devices have been proposed
and used.
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Pulsed Pumping

• Pumping a laser medium with a short-pulse flash
  lamp yields a short pulse. Flash lamp pulses as
  short as 1 µs exist.
• Unfortunately, this yields a pulse as long as the
  excited-state lifetime of the laser medium, which
  can be considerably longer than the pump pulse.
• Since solid-state laser media have lifetimes in
  the microsecond range, it yields pulses
  microseconds to milliseconds long.

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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.
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Distributed-feedback lasers
When two beams cross at an angle, their intensity
is sinusoidal.
Intensity fringes
When energy is deposited sinusoidally in space,
the actual gain goes quadratically with the
energy deposited, yielding a type of very fast
Q-switching. Using several stages, fs pulses have
been created this way.
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Passive mode-locking the saturable absorber

• Like a sponge, an absorbing medium can only
  absorb so much. High-intensity spikes burn
  through low-intensity light is absorbed.

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Passive mode-locking the saturable absorber

• High-intensity spikes (i.e., short pulses) see
  less loss and hence can lase while low-intensity
  backgrounds (i.e., long pulses) wont.

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Passive Mode-locking with a Slow Saturable
Absorber

• What if the absorber responds slowly (more slowly
  than the pulse)?
• Then only the leading edge will experience pulse
  shortening.

This is the most common situation, unless the
pulse is many ps long.
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Gain Saturation shortens the pulse trailing edge.

• The intense spike uses up the laser gain-medium
  energy, reducing the gain available for the
  trailing edge of the pulse (and for later pulses).

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Saturable gain and loss
Lasers lase when the gain exceeds the loss.
The combination of saturable absorption and
saturable gain yields short pulses even when the
absorber is slower than the pulse.
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The Passively Mode-locked Dye Laser
Passively mode-locked dye lasers yield pulses as
short as a few hundred fs.
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Some common dyes and their corresponding
saturable absorbers
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The Colliding-Pulse Mode-locked Laser
If one intense pulse can burn its way through a
saturable absorber and create short pulses, why
not have two pulses collide and do even better?
Colliding-pulse-mode-locked (CPM) dye lasers
produce even shorter pulses 30 fs. (Other
effects also contribute, too.)
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A lens and a lens
A lens is a lens because the phase delay seen by
a beam varies with x f(x) n k L(x)
L(x)
In both cases, a quadratic variation of the phase
with x yields a lens.
Now what if L is constant, but n varies with x
f(x) n(x) k L
n(x)
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Kerr-lens mode-locking

• A mediums refractive index depends on the
  intensity.
• n(I) n0 n2I
• If the pulse is more intense in the center, it
  induces a lens.
• Placing an aperture at the focus favors a short
  pulse.

Losses too high for a low-intensity cw mode to
lase, but not for high-intensity fs pulse.
Kerr-lensing is the basis of the TiSapphire
laser.
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Modeling Kerr-lens mode-locking
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Titanium Sapphire

• TiSapphire is currently the workhorse laser of
  the ultrafast community, emitting pulses as short
  as a few fs and average power in excess of a Watt.

It can be pumped with a (continuous) Argon laser
(450-515 nm) or a doubled-NdYAG laser (532
nm). It lases well between 700 and 1000 nm.
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Active mode-locking

• Any amplitude modulator can preferentially induce
  losses for times other than that of the intended
  pulse peak. This produces short pulses.
• It can be used to start a TiSapphire laser
  mode-locking.

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Gain switching

• Modulating the gain rapidly is essentially the
  same as active mode-locking.
• This method is a common one for mode-locking
  semiconductor lasers.

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Synchronous pumping

• Pumping the gain medium with a train of already
  short pulses yields a train of even shorter
  pulses.

The laser round-trip time must precisely match
that of the train of pump pulses!
Short pulses (ps)
Trains of 60 ps pulses from a NdYAG laser can
yield lt1 ps pulses from a sync-pumped dye laser.
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Hybrid mode-locking

• Hybrid mode-locking is any type of mode-locking
  incorporating two or more techniques
  simultaneously.
• Sync-pumping and passive mode-locking
• Active and passive mode-locking
• However, using two lousy methods together doesnt
  really work all that much better than one good
  method.

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The Soliton Laser
Nonlinear-optical effects can compensate for
dispersion, yielding a soliton, which can be
very short and remain very short, despite
dispersion and nonlinear-optical effects.
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Additive-pulse mode-locking

• Nonlinear effects in an external cavity can yield
  a phase-distorted pulse, which can be combined in
  phase with the pulse in the main cavity, yielding
  cancellation in the wings, and hence
  pulse-shortening.

Fiber lasers use this mechanism.
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Traveling-wave Excitation
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Mechanisms that limit pulse shortening
The universe conspires to lengthen pulses.

• Gain narrowing
• G(w) exp(-aw2), then after N passes, the
  spectrum will narrow by GN(w) exp(-Naw2),
  which is narrower by N1/2
• Group-velocity dispersion
• GVD spreads the pulse in time. And everything
  has GVD
• Etalon effects
• This yields multiple pulses, spreading the
  energy over time, weakening the pulses.

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Commercial fs lasers

• TiSapphire
• Coherent
• Mira (lt35 fs pulse length, 1 W ave power),
• Chamelion (Hands-free, 100 fs pulse length),
• Spectra-Physics
• Tsunami (lt35 fs pulse length, 1 W ave power)
• Mai Tai (Hands-free, 100 fs pulse length)
• Fiber
• IMRA America
• Femtolite (1560 nm up to 60 mW down to 100 fs)

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Commercial fs lasers (contd)