Pulse calculator: time-bandwidth product¶

A stand-alone calculator for the time-bandwidth product (TBP) of a pulse, its transform limit, how far it is from it (the stretch factor) and the group-delay dispersion (GDD) that would compress it. It is the same code behind the CLI (ultrafast pulse), Python (ultrafast.calc) and the C API (uf_pulse_calc), like the phase-matching calculator. Every simulation also reports timeBandwidthProduct in summary.json; this calculator is for numbers you have at hand.

Durations and bandwidths are intensity FWHM. The bandwidth is in Hz, or in metres together with the centre wavelength (Δν = c Δλ / λ², first order in Δλ / λ).

Shapes (TBP = Δt · Δν, intensity FWHM)¶

shape	TBP	exact form
`gaussian`	0.4413	2 ln 2 / π
`sech2`	0.3148	(2 ln(1+√2))² / π²
`rectangular`	0.8859	2 x / π with sin x / x = 1/√2

With the root-mean-square widths of the intensities, a Gaussian has σ_t σ_ν = 1/(4π) = 0.0796 (timeBandwidthProductRms).

Command line¶

ultrafast pulse shapes
ultrafast pulse limit --shape sech2 --duration 100e-15                 # transform-limited bandwidth
ultrafast pulse limit --shape gaussian --bandwidth-m 40e-9 --center 1030e-9
ultrafast pulse analyze --duration 300e-15 --bandwidth-m 10e-9 --center 1030e-9
ultrafast pulse measure --request measured.json --json                 # arrays: timeS, intensity, wavelengthM | frequencyHz, spectrum

analyze prints (300 fs, 10 nm at 1030 nm): TBP 0.848, 1.92 × the Gaussian limit, transform-limited duration 156 fs and a GDD magnitude of 14 400 fs².

Python¶

from ultrafast import calc
calc.tbp("sech2")                                                        # 0.3148
calc.limit("gaussian", duration=100e-15, center=800e-9)                  # bandwidth in Hz and m
r = calc.analyze(duration=300e-15, bandwidth_m=10e-9, center=1030e-9)
r["timeBandwidthProduct"], r["stretchFactor"], r["gddMagnitudeFs2"]
calc.measure(time_s, intensity, spectrum, wavelength_m=wl)               # from measured arrays (FWHM and rms products)

Results¶

key	meaning
`timeBandwidthProduct`	`durationFwhmS × bandwidthFwhmHz`
`transformLimitProduct`, `ratioToLimit`	the product of the shape, and the measured product divided by it
`transformLimitedDurationS`, `stretchFactor`	the shortest duration for this bandwidth, and `durationFwhmS` divided by it
`gddMagnitudeFs2`	Gaussian only: `\\|GDD\\| = τ₀² / (4 ln 2) · √(stretch² − 1)` for a linearly chirped Gaussian; the sign needs a phase measurement
`timeBandwidthProductRms`	`measure` only: σ_t σ_ν of the intensity curves

A stretch factor below one means the shape, the widths or the units are wrong; the result says so. A product above the limit does not distinguish a linear chirp from higher-order phase or a structured spectrum: use the instruments or a retrieval for that.

Tests¶

PulseCalculatorTests (closed-form products, the unit conversion, the chirped-Gaussian GDD, a sampled Gaussian with the rms limit 1/(4π)) and python/tests/test_setup_and_calc.py.