# Mean Frequency and Frequency Spread of a Laser Pulse

• CDL
In summary, the mean frequency and frequency spread of a 400 nm laser pulse are 7.5 × 1014 Hz and 10-15 Hz, respectively.
CDL

## 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

$$c = f \lambda, \ \Delta \nu \sim \frac{1}{\tau_c}, \ \text{where} \ \tau_c \ \text{is the coherence time.}$$
3. The Attempt at a Solution [/B]
At first glance, I decided to find the mean frequency using $$\bar{\nu} = \frac{c}{\bar{\lambda}}.$$ 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 $$\mathbb{E}\left(\frac{c}{\lambda}\right) = \mathbb{E}(\nu) \geq \frac{c}{\mathbb{E}(\lambda)}.$$ 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##.

I think you are overthinking the problem. From the lack of details given in the problem statement, the appears to be a simple introductory exercise, so you can take the pulse to be symmetric around its peak frequency and to be transform-limited and thus use the time-energy uncertainty relation to get a lower bound on the frequency spread.

While you are correct that E[1/X] is not 1/E[X], it's an OK approximation to use when the associated probability distribution is narrow, as you're supposed to assume here.

RPinPA said:
While you are correct that E[1/X] is not 1/E[X], it's an OK approximation to use when the associated probability distribution is narrow, as you're supposed to assume here.

It's a good thing to know when you're using an approximation and how good the approximation is. So I explored that a little. Let's assume for simplicity in the calculation that ##\lambda \sim U[a, b]##, that ##\lambda## is uniformly distributed between some values ##a## and ##b##, and we'll define a fractional width ##\epsilon## by ##b = a(1 + \epsilon)##.

So ##b - a = a\epsilon## and ##b + a = a(2 + \epsilon)##.

Clearly the mean value of ##\lambda## is halfway between a and b, ##E[\lambda] = \frac{b + a}{2} = a(1 + \frac{\epsilon}{2})##.

Now what is ##E[1/\lambda]##?

$$E \left [\frac{1}{\lambda} \right ] = \frac{1}{b - a} \int_a^b \frac{d\lambda}{\lambda} = \frac{\ln(b) - \ln(a)}{b-a} = \frac{\ln(b/a)}{b - a} \\ = \frac{\ln(1 + \epsilon)}{a\epsilon} = \frac{1}{a \epsilon} \left (\epsilon - \frac{\epsilon^2}{2} + \frac{\epsilon^3}{3} - \frac{\epsilon^4}{4} + ...\right )\\ = \frac{1}{a} \left (1 - \frac{\epsilon}{2} + \frac{\epsilon^2}{3} - \frac{\epsilon^3}{4} + ... \right )$$
For comparison,
$$\frac {1}{E[\lambda]} = \frac{1}{a} \frac{1}{1 + \frac{\epsilon}{2}} \\ = \frac{1}{a} \left (1 - \frac{\epsilon}{2} + \frac{\epsilon^2}{4} - \frac{\epsilon^3}{8} + ... \right )$$
So as you can see, they are the same to first order in ##\epsilon##. If ##\epsilon^2## is a lot smaller than ##\epsilon## (as it would be for, say, ##\epsilon < 0.01##) then these expressions are approximately the same.

## 1. What is the mean frequency of a laser pulse?

The mean frequency of a laser pulse is the average frequency of all the individual frequencies present in the pulse. It is calculated by adding up all the frequencies and dividing by the total number of frequencies.

## 2. Why is the mean frequency of a laser pulse important?

The mean frequency of a laser pulse is important because it determines the color or wavelength of the laser. It also affects the properties of the laser, such as its coherence and coherence length.

## 3. How is the frequency spread of a laser pulse measured?

The frequency spread of a laser pulse can be measured using a spectrometer or an interferometer. These instruments can analyze the different frequencies present in the pulse and determine their spread or distribution.

## 4. What factors can affect the mean frequency and frequency spread of a laser pulse?

The mean frequency and frequency spread of a laser pulse can be affected by the characteristics of the laser itself, such as its gain medium and cavity design. It can also be influenced by external factors such as temperature, pressure, and external optical sources.

## 5. How can the mean frequency and frequency spread of a laser pulse be controlled?

The mean frequency and frequency spread of a laser pulse can be controlled by using different types of laser gain media, adjusting the laser cavity design, and using external optical elements such as filters or modulators. Temperature and pressure can also be controlled to some extent to influence the laser's frequency properties.

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