# Transform-limited pulse calculator

## About these calculators

The first calculator computes the transform-limited, i.e. minimum possible, pulse duration of a Gaussian or sech² pulse with a given spectral width either in wavelength or frequency domain. The second calculator computes the inverse of that, in other words, the minimum spectral width required to obtain a given pulse duration. In both cases, the calculation is based on the time-bandwidth product, which is a constant of the order of unity for transform-limited pulses and depends slightly on the pulse shape.

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

$TBP_{Gaussian} = \dfrac{2 \log2}{\pi} \approx 0.441$

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

Since time-bandwidth product is 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), the transform-limited spectral width can be computed from pulse duration as$\Delta \nu = \dfrac{TBP}{\Delta \tau}$

and, conversely, pulse duration would be$\Delta \tau = \dfrac{TBP}{\Delta \nu}$

If required, the spectral width can be converted from wavelength to frequency before the operation or from frequency to wavelength after the operation.