In depth
Concept 01
Dispersion & GDD
Why a pulse stretches in time, and how a compressor puts it back.
In a dispersive medium the refractive index — and therefore the phase accumulated by light —
depends on frequency. Expanding the propagation constant around the carrier frequency gives a series
whose first meaningful term for pulse shape is the group-velocity dispersion (GVD),
written β₂, followed by third-order dispersion β₃.
The flat carrier phase and the constant group delay do not change the envelope; the curvature does.
A positive β₂ makes the redder (lower) frequencies travel ahead of the bluer ones,
spreading the pulse in time and imposing a frequency sweep, or chirp, across it. The total accumulated
quadratic phase is the group-delay dispersion (GDD), the quantity a compressor targets.
Because a chirp is just spectral phase, it can in principle be undone. A pure-GDD compressor applies a
quadratic spectral phase of the opposite sign; if the pulse’s phase is dominated by that quadratic
term, the pulse recompresses toward its shortest possible duration. Any leftover cubic or higher-order
phase is what a simple GDD compressor cannot remove.
In FibDisp: both β₂ and β₃ are propagated for the
selected gas and capillary, the Phase Analysis tab fits the output spectral phase, and the GDD Compressor
Design tab both applies a manual GDD and numerically searches for the value that minimizes the compressed
duration. A predominantly positive output phase is compensated by a negative applied GDD, and vice versa.
Concept 02
Kerr nonlinearity & self-phase modulation
How an intense pulse rewrites its own spectrum.
At high intensity the refractive index acquires an intensity-dependent part,
n = n₀ + n₂I, known as the optical Kerr effect. Because the intensity of a pulse
varies in time, the extra index — and the phase the pulse imprints on itself — also varies in time.
This is self-phase modulation (SPM).
A time-varying phase is, by definition, an instantaneous frequency shift. The rising edge of the pulse
is pushed toward lower frequencies and the falling edge toward higher ones, so the pulse generates
new frequencies that were not present at the input. The spectrum broadens roughly in
proportion to the peak power and the interaction length. A convenient single-number measure of the
accumulated nonlinear phase is the B-integral, the peak Kerr phase integrated along
the fiber, expressed in radians.
This is the key to hollow-fiber compression: SPM manufactures the extra bandwidth, and a compressor then
converts that bandwidth into a shorter pulse. The nonlinear coefficient γ that sets the
strength of the effect scales with n₂ and the carrier frequency and inversely with the
effective mode area.
In FibDisp: SPM is the iγP₀|U|²U term of the propagation
equation and can be toggled independently. The Results tab reports a spectral-broadening factor comparing
output and input spectral widths, so you can see directly how much new bandwidth a given pressure, energy,
and length produce.
Concept 03
Gas-filled hollow-core fibers
The workhorse geometry for post-compression of energetic pulses.
A hollow capillary guides an intense pulse through a controlled length of gas while keeping it confined
to a well-defined mode. The gas supplies the Kerr nonlinearity for spectral broadening, and its pressure
is a convenient knob: raising the pressure increases both the nonlinear coupling and the gas contribution
to dispersion, while the fundamental mode stays intact. This is the technique behind the generation of
few-cycle pulses from longer input pulses.
In the large-core limit the guided mode has a slightly reduced effective index relative to the bulk gas,
with a small wavelength-dependent waveguide correction set by the capillary radius. Differentiating that
effective index around the carrier gives the β₂ and β₃ the
model propagates, so the displayed dispersion already contains both the gas and the waveguide. The same
geometry also imposes a frequency-dependent propagation loss that grows rapidly as the radius shrinks.
- Higher pressure → stronger broadening, but more dispersion to compensate afterward.
- Larger radius → lower loss, but weaker intensity and less broadening for the same energy.
- Longer fiber → more accumulated nonlinear phase, up to the point where loss and over-broadening dominate.
In FibDisp: Helium, Neon, Argon, Krypton, Xenon, and Nitrogen are built in, with
pressure, radius, length, and wavelength as inputs; the tool computes the effective mode area, the loss,
and the dispersion for that configuration, or lets you enter coefficients directly in Manual mode.
Concept 04
Self-steepening
The correction that matters once a pulse gets broadband.
Ordinary self-phase modulation treats the nonlinear response as the same at every frequency. For a very
broadband pulse that is no longer quite true: the nonlinear coupling grows slightly with frequency. The
leading correction is self-steepening, sometimes called the optical-shock term.
Its effect is that the most intense part of the pulse experiences a small intensity-dependent delay, so
the peak slips backward relative to the edges and the trailing side steepens — the temporal analogue
of a wave about to break. This asymmetry shows up as an asymmetric, blue-shifted broadening of the
spectrum, distinguishing self-steepening from pure SPM.
Numerically the steepening edge is demanding: it concentrates energy into an ever-sharper feature that
can outrun the resolution of the grid. Robust integration and adequate sampling in both time and
frequency are essential, and a sharp shock is a signal to check convergence rather than trust the first
result.
In FibDisp: self-steepening is a separate, toggleable term with two solvers — a fast
exponential scheme and an accurate adaptive Runge–Kutta scheme that subdivides the nonlinear step and
stops if an unreasonably fine grid would be required. It is the term most worth cross-checking for
convergence in strong-broadening runs.
Concept 05
High-harmonic generation
From an infrared field to attosecond bursts of XUV light.
High-harmonic generation (HHG) is the strongly nonlinear process that turns an intense infrared laser
field, focused into a gas, into coherent radiation in the extreme ultraviolet and soft-X-ray range. It is
the workhorse source of attosecond science.
Its physics is captured by the semiclassical three-step model: the strong laser field
distorts the atomic potential so that an electron tunnels into the continuum; the electron
is then accelerated by the oscillating field; and when the field reverses, it can be driven
back to recombine with its parent ion, releasing its accumulated kinetic energy plus the
ionization energy as a single high-energy photon. Because this repeats every half-cycle of the driver, the
emitted spectrum forms a comb of odd harmonics on a broad plateau that ends at a sharp cutoff, and in the
time domain the emission is a train of attosecond bursts.
The plateau ends at a cutoff photon energy of about 3.17 U₊ + Iₚ, where
Iₚ is the ionization potential and U₊ the ponderomotive energy the
electron gains in the field. Confining efficient emission to a single half-cycle — for example with a
few-cycle driver — turns the cutoff into a continuum and yields an isolated attosecond pulse rather than
a train.
Where it fits: HHG is the downstream application that few-cycle drivers enable. FibDisp
addresses the stage before it — producing and compressing the intense, broadband driving pulse —
while HHG itself is the target physics a future tool in the collection could cover.
Concept 06
The time–bandwidth limit
Why a short pulse must be a broad pulse.
A pulse’s duration and its spectral width are Fourier conjugates: they cannot both be made
arbitrarily small. The product of the temporal and spectral widths has a lower bound that depends only on
the pulse shape. A pulse that reaches that bound — one whose spectral phase is flat — is called
transform-limited and is the shortest pulse its spectrum allows.
This is why broadening the spectrum is the prerequisite for shortening a pulse: more bandwidth lowers the
transform-limited duration. But bandwidth only sets the potential. A real pulse is usually longer
than its transform limit because it carries residual spectral phase — chirp and higher-order terms.
Compression is the act of flattening that phase to approach the limit the bandwidth already permits.
Two numbers are therefore worth separating: the transform-limited duration, which asks “how short
could this spectrum ever be?”, and the actual duration, which asks “how short is it right now,
given its phase?” The gap between them is exactly what a compressor tries to close.
In FibDisp: the tool reports the transform-limited output duration alongside the actual
output, so the residual-phase gap is visible at a glance, and several duration metrics (main FWHM, RMS,
95%-energy width) characterize how clean the compressed pulse really is.