# Time-bandwidth product calculator

Input
Output

✕0.441

✕0.315

fs²

This calculator computes mainly the time-bandwidth product of a laser pulse and how far the value is from the transform limit. Additionally, this calculator computes the expected autocorrelation widths given the pulse duration as well as the Gaussian chirp parameter $C$ and the accumulated GDD. The time-bandwidth product is unitless parameter defined as

$TBP = \Delta \nu \Delta \tau$

where $\Delta \nu$ is the spectral width (in Hz) and $\Delta \tau$ is the pulse duration (in s). If the spectral width is not given in Hz, the calculator makes the conversion before calculating the time-bandwidth product.

The time-bandwidth products of transform-limited Gaussian and sech² pulses are:

$TBP_{Gaussian} = \dfrac{2 \log2}{\pi} \approx 0.441$
$TBP_{sech^2} = \left(\dfrac{2 \log(1+\sqrt{2})}{\pi}\right)^2 \approx 0.315$

The calculator compares the computed time-bandwidth product to these values to give an estimate of how far the pulse is from transform limit. Next, the expected autocorrelation widths are calculated by dividing the supplied pulse duration by the deconvolution factors for Gaussian and sech² pulses. The deconvolution factors are $0.707$ for Gaussian and $0.647$ for sech².

Finally, the calculator computes the chirp parameter $C$ and the accumulated group delay dispersion (assuming a Gaussian shape). The chirp parameter is

$C = \sqrt{\dfrac{T}{T_{min}} - 1}$

where $T$ is the $1/e$ pulse duration:

$T = \dfrac{\Delta\tau}{2\sqrt{\log2}}$

and $T_{min}$ is the transform-limited $1/e$ spectral width:

$T_{min} = \dfrac{TBP_{Gaussian}}{2\sqrt{\log2}\Delta\nu}$

The accumulated GDD is then:

$GDD = CT_{min}^2$

The sign of the chirp parameter and accumulated dispersion remains ambiguous since it cannot be deduced from spectral width and pulse duration only.