Deriving Irradiance Expression

[blorpinbloo]

Homework Statement

Electric field for a single pulse of a laser (operating at central wavelength of 800nm):

E = E0e^(-(t^2/2a^2))e^(iwt)

(the a is sigma and w is omega)

Derive an expression for the spectral intensity (irradiance)

The Attempt at a Solution

One way I know to calculate irradiance is the time average of the poynting vector: (1/u0)*ExB
But that would involve taking the time average of e^(-(t^2)) which is non-integrable.

I've also seen the function E integrated over an area, then squared to give an irradiance formula, but then again I run into problems with that function that's difficult to integrate. Any suggestions?

 

[mark726]

Dear fellow scientist,

Thank you for your post. To derive an expression for the spectral intensity (irradiance) of a single pulse of a laser, we can use the relation between electric field and irradiance, given by:

I = (1/2*u0)*|E|^2

where I is the irradiance, u0 is the permeability of free space, and E is the electric field. In your case, the electric field is given by:

E = E0e^(-(t^2/2a^2))e^(iwt)

To calculate the spectral intensity, we need to take the time average of the squared magnitude of the electric field. This can be done using the following steps:

1. Take the squared magnitude of the electric field:

|E|^2 = E0^2e^(-t^2/a^2)e^(-iwt) * E0^2e^(-t^2/a^2)e^(iwt)

2. Use the properties of exponents to simplify the equation:

|E|^2 = E0^4e^(-2t^2/a^2)

3. Take the time average by integrating over time from -∞ to ∞ and dividing by the total time:

<I> = (1/T)*∫-∞∞ E0^4e^(-2t^2/a^2) dt

4. Use the substitution u = t/a to simplify the integral:

<I> = (1/T)*∫-∞∞ E0^4e^(-2u^2) du

5. Use the Gaussian integral identity to solve the integral:

<I> = (1/T)*(E0^4*√(π/2))

6. Substitute back for u and T to get the final expression for the spectral intensity:

I = (E0^4*√(π/2))/(T*a)

I hope this helps you in your calculations. Please let me know if you have any further questions. Best of luck with your research!