# Explaining linear spectroscopy from first principles

1. Jul 24, 2012

### photon stew

Hi! I'm a little confused about simple spectroscopy. In typical absorption experiments, we scan the laser frequency w and get a anti-peak/dip in the transmitted intensity I(w) around a resonance frequency. The size of the dip, i.e. the population of the excited level, usually depends linearly on the laser intensity. How can this be understood from first principles?

I know about Rabi oscillations in optically driven two level systems, but these are of course quite different. I have also studied more advanced descriptions -density matrices, master equations like the optical Bloch equations, etc.- but I'm still kind of stuck with this simple question.

photon stew

Last edited: Jul 24, 2012
2. Jul 24, 2012

### Cthugha

What is a "typical" absorption experiment from your point of view?

Are you just interested in the absorption of a single light beam or is it rather something like saturated absorption spectroscopy using pump-probe geometry?

3. Jul 24, 2012

### photon stew

At first, I'm interested in the former.

4. Jul 24, 2012

### Cthugha

In this case I am a bit puzzled. In all simple absorption experiments I ever did, the relative absorption (transmitted intensity divided by incoming intensity) is constant with laser power unless one exits the regime where saturation or carrier-carrier interaction needs to be taken into account.

Or are you interested in the absolute absorbed intensity (which is indeed proportional to the excited state population) varying linearly with the incoming intensity?

5. Jul 24, 2012

### photon stew

Yes. This is the reason why this kind of spectroscopy is called linear, isn't it? Ultimately, I'm interested in nonlinear stuff but I realized that I don't understand linear spectroscopy very well, so I wanted to learn more about it first.

6. Jul 24, 2012

### Cthugha

I was rather asking whether you are interested in relative or absolute absorption.

Never mind. The intensity is proportional to the photon number per time interval present in the beam. Each photon has some chance of being absorbed on or near resonance. In the linear regime you can consider each absorption process as statistically independent of each other and the amount of absolute absorbed photons is just the absorption probability times the number of photons present which is necessarily linear in intensity.

Is this the kind of explanation you seek or is it something deeper?

7. Jul 24, 2012

### mikeph

I believe it is linear under some basic assumptions, one of them being that the absorption process is random, so that the amount of absorption is proportional to the amount of incident radiation. The constant of proportionality should be the linear absorption coefficient, and you get some differential equation like dI = -k*I*dL, for intensity I, coefficient k and path length L. Integrating this gives you Beer's law which is valid for linear absorption, I/I0 = exp(-kL) for some uniform absorber of length L.

To get to this stage you need to assume the absorber is in thermal equilibrium, which means it has a Boltzmann state population and is surrounded by blackbody radiation of the same temperature. Once you start using very high radiation intensities, you alter the state population of the material, so the above assumption fails and Beer's law becomes inaccurate.

At least that's my understanding!

8. Mar 4, 2014

### rrbbss

Pump probe spectroscopy in two level system

Hi,
Can someone tell me how to calculate the probe absorption when both pump and probe beams are addressing to same levels.