- #1

CDL

- 20

- 1

## Homework Statement

Laser probes are being used to examine the states of atoms and molecules at high temporal resolution. A laser operating at a wavelength of 400 nm produces a 1 femtosecond pulse. Compute the mean frequency and frequency spread, ∆ν, of this laser pulse.

## Homework Equations

[tex]c = f \lambda, \ \Delta \nu \sim \frac{1}{\tau_c}, \ \text{where} \ \tau_c \ \text{is the coherence time.}[/tex]

3. The Attempt at a Solution [/B]

At first glance, I decided to find the mean frequency using [tex]\bar{\nu} = \frac{c}{\bar{\lambda}}.[/tex] After further consideration, I don't think this necessarily correct since if we think of the wavelengths present in the pulse as following a statistical distribution with ##\mathbb{E}(\lambda) = 400## nm, then by Jensen's inequality, we would have [tex]\mathbb{E}\left(\frac{c}{\lambda}\right) = \mathbb{E}(\nu) \geq \frac{c}{\mathbb{E}(\lambda)}.[/tex] I.e I expect the mean frequency ##\bar{\nu}## to be greater than ##\frac{c}{\bar{\lambda}}.##

I am not sure how to proceed, as I can't compute the exact mean frequency without knowing the distribution of the wavelengths, and I have only found a lower bound for the mean frequency. I could assume that the wavelength is gaussian distributed, but I'd also need to specify the variance of the wavelength in that case.

I took the coherence time ##\tau_c## to be ##10^{-15}## s, so I expect the frequency spread to approximately be ##\Delta \nu \sim 10^{15} ## Hz. Using ##\bar{\nu} = \frac{c}{\bar{\lambda}}##, I get ##\bar{\nu} = 7.5 \times 10^{14}##, which is less than the spread, meaning that according to this calculation, the minimum frequency is ##0##.