Lasers and Laser Light — Question 7

Question 7

A Gaussian pulse cannot have arbitrarily small duration and spectral width

Use the transform-limited Gaussian intensity relation $\Delta\nu\,\Delta t=2\ln2/\pi$, where both widths are full widths at half maximum. A pulse has $\Delta t=100\,\mathrm{fs}$ and central wavelength $\lambda_0=800\,\mathrm{nm}$. For narrow relative bandwidth use $\Delta\lambda\simeq\lambda_0^2\Delta\nu/c$ with $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the minimum spectral frequency width.

2. Estimate the wavelength width and test the small-relative-bandwidth condition.

Question 7 - Solution

A Gaussian pulse cannot have arbitrarily small duration and spectral width

(a) Method and calculation

$$\Delta\nu=\frac{2\ln2}{\pi\Delta t}.$$$$\boxed{\Delta\nu=4.413\times10^{12}\,\mathrm{Hz}=4.413\,\mathrm{THz}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta\lambda\simeq\frac{(800\times10^{-9})^2(4.413\times10^{12})}{3.00\times10^8}=9.414\,\mathrm{nm}}.$$$$\boxed{\Delta\lambda/\lambda_0\simeq.01177\ll1}.$$

Physical check

The equality is the minimum for an unchirped Gaussian pulse. A chirped pulse may have a larger time-bandwidth product; short laser pulses are not strictly monochromatic.