Theory of Ultrashort Laser Pulse Generation

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Theory of Ultrashort Laser Pulse Generation
Active mode-locking Passive mode-locking Build-u
p of mode-locking The Landau-Ginzberg
Equation The Nonlinear Schrodinger
Equation Solitons
gain
gain gt loss
time
Reference Hermann Haus, Short pulse
generation, in Compact Sources of Ultrashort
Pulses, Irl N. Duling, ed. (Cambridge University
Press, 1995). (Some slides thanks to Dan
MIttleman)
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Mode-locking yields ultrashort pulses.
Recall that many frequencies (modes) oscillate
simultaneously in a laser, and when their phases
are locked, an ultrashort pulse results.
Sum of ten modes with the same relative phase
Sum of ten modes w/ random phase
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Mode Locking
Active Mode Locking
Insert something into the laser cavity that
sinusoidally modulates the amplitude of the
pulse. ? mode competition couples each mode to
modulation sidebands ? eventually, all the modes
are coupled and phase-locked
Passive Mode Locking
Saturable absorber
Insert something into the laser cavity that
favors high intensities. ? strong maxima will
grow stronger at the expense of weaker ones ?
eventually, all of the energy is concentrated in
one packet
loss
gain
gain gt loss
time
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Active mode-locking the electro-optic modulator
Applying a voltage to a crystal changes its
refractive indices and introduces birefringence.
A few kV can turn a crystal into a half- or
quarter-wave plate.
Polarizer
If V 0, the pulse polarization doesnt change.
Pockels cell (voltage may be transverse or
longitudinal)
V
If V Vp, the pulse polarization switches to its
orthogonal state.
Applying a sinusoidal voltage yields sinusoidal
modulation to the beam. An electro-optic
modulator can also be used without a polarizer to
simply introduce a phase modulation, which works
by sinusoidally shifting the modes into and out
of the actual cavity modes.
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Active mode-locking the acousto-optic modulator
An acoustic wave induces sinusoidal density, and
hence sinusoidal refractive-index, variations in
a medium. This will diffract away some of a light
waves energy.
Acoustic transducer
Pressure, density, and refractive-index
variations due to acoustic wave
w
Output beam
Input beam
w
Quartz
Diffracted Beam (Loss)
Such diffraction can be quite strong 70.
Sinusoidally modulating the acoustic wave
amplitude yields sinusoidal modulation of the
transmitted beam.
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The Modulation Theorem The Fourier Transform of
E(t)cos(wMt)
F E(t)cos(wM t)
F E(t)cos(wMt)
Multiplication by cos(wMt) introduces side-bands.
F E(t)cos(wMt)
F E(t))
If E(t) sinc2(t)exp(iw0t)
w
w0
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Active mode-locking
In the frequency domain, a modulator introduces
side-bands of every mode.
For mode-locking, make sure that wM mode
spacing. This means that wM 2p/cavity
round-trip time 2p/(2L/c) pc/L
Each mode competes for gain with adjacent
modes. Most efficient operation is for phases to
lock. Result is global phase locking. n coupled
equations En ? En1, En-1
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Modeling laser modes and gain
Gain profile and resulting laser modes
Lasers have a mode spacing
w0
Let the zeroth mode be at the center of the gain,
w0. The nth mode frequency is then
where n , -1, 0, 1,
Let an be the amplitude of the nth mode and
assume a Lorentzian gain profile, G(n)
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Modeling an amplitude modulator
An amplitude modulator uses the electro-optic or
acousto-optic effect to deliberately cause losses
at the laser round-trip frequency, wM.
A modulator multiplies the laser light (i.e.,
each mode) by M1-cos(wMt)
where the superscript indicates the kth round
trip.
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Solve for the steady-state solution
In steady state,
Also, the finite difference becomes a second
derivative when the modes are many and closely
spaced
This differential equation has the solution
with the constraints
In practice, the lowest-order mode occurs
A Gaussian!
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Fourier-transforming to the time domain
Recalling that multiplication by -w2 in the
frequency domain is just a second derivative in
the time domain (and vice versa)
becomes
which has the solution
This makes sense because Hermite-Gaussians are
their own Fourier transforms. The time-domain
will prove to be a better domain for modeling
passive mode-locking.
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Passive mode-locking
Saturable gain
gain saturates during the passage of the
pulse leading edge is selectively amplified
Saturable absorption
absorption saturates during the passage of the
pulse leading edge is selectively eroded
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Kerr-lensing is a type of saturable absorber.
If a pulse experiences additional focusing due to
high intensity and the nonlinear refractive
index, and we align the laser for this extra
focusing, then a high-intensity beam will have
better overlap with the gain medium.
High-intensity pulse
TiSapph
Low-intensity pulse
This is a type of saturable absorption.
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Saturable-absorber mode-locking
Intensity
Neglect gain saturation, and model a fast
saturable absorber
Saturation intensity
The transmission through a fast saturable
absorber
where
Including saturable absorption in the
mode-amplitude equation
Lumping the constant loss into l
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The sech pulse shape
In steady state, this equation has the solution
where the conditions on t and A0 are
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The Master Equation including GVD
Expand k to second order in w
After propagating a distance Ld, the amplitude
becomes
Ignore the constant phase and vg, and expand the
2nd-order phase
Inverse-Fourier-transforming
where
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The Master Equation (continued) including the
Kerr effect
The Kerr Effect
so
The master equation (assuming small effects)
becomes
(were converting a(k) to a(t))
In steady state
This important equation is called the
Landau-Ginzberg Equation.
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Solution to the Master Equation
It is
where
The complex exponent yields chirp.
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The pulse length and chirp parameter
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The spectral width
The spectral width vs. dispersion for various SPM
values.
A broader spectrum is possible if some positive
chirp is acceptable.
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Stability of solutions
The laser will be stable when the gain is less
than the loss just before and just after the
pulse.
Instability for high SPM
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The Nonlinear Schrodinger Equation
First, imagine raster-scanning the pulse vs. time
like this
Intensity
Long time (round trips) (ns)
Notice that the weak pulses are suppressed, and
the strong pulse shortens and is amplified.
Well call the long-time round-trip-number axis
z.
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The Nonlinear Schrodinger Equation
Recall the master equation
which becomes (when we assume many round trips
and let k be z)
Neglecting every effect in this equation except
for the Kerr effect and GVD, we have the
Nonlinear Schrodinger Equation
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The Nonlinear Schrodinger Equation
The solution to the nonlinear Shrodinger equation
is
where
and Dw is the detuning from w0.
Note that d /D lt 0, or no solution exists. But
note that, despite dispersion, the pulse length
and shape do not vary with distance.
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Solitons and solitary waves
A solitary wave is a wave that retains its
shape, despite dispersion and nonlinearities. A
soliton is a pulse that can collide with
another similar pulse and still retain its shape
after the collision, again in the presence of
both dispersion and nonlinearities.
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Collision of two solitons
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Evolution of a soliton from a square wave
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Higher-order soliton
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Other mode-locking techniques
FM mode-locking
produce a phase shift per round
trip implementation electro-optic
modulator similar results in terms of
steady-state pulse duration
Synchronous pumping
gain medium is pumped with a pulsed laser, at a
rate of 1 pulse per round trip requires an
actively mode-locked laser to pump your laser
() requires the two cavity lengths to be
accurately matched useful for converting long AM
pulses into short AM pulses (e.g., 150 psec
argon-ion pulses ? sub-psec dye laser pulses)
Additive-pulse or coupled-cavity mode-locking
external cavity that feeds pulses back into main
cavity synchronously requires two cavity
lengths to be matched can be used to form
sub-100-fsec pulses