How does filtering affect the coherence length of white light?

[tomasko789]

Hi,
I just passed a course of a quantum coherence theory (or quantum optics). However, during the course i found more questions rather than answers.
One particular:

Imagine there's a white light coming from the sun. What is the coherence length of this light, according to the quantum coherence theory predictions (we know the temperature of the source, which is the only parameter of the light coming, so there should be a particular answer - the length in micrometers).

Specifically, imagine, we take a beam of the light coming from the sun and let it pass through a diffraction grating. Then we use a slit to get the light with a particular wavelength. And then we measure the coherence length of the light that passes through the slit. Again? What is the value of it and how it depends on the wavelength.

One more: The slit has of course a finite width, so the light that passes has some finite bandwidth. Let's measure the coherence length for one particular wavelength by changing the slit width, i.e. the bandwidth of the light? What would happen? Will the coherence length of this passed beam increase up to infinity, when the slit width will approach zero??

There is maybe a problem, that when the slit is very small, almost no photon passes through and we cannot even measure its coherence length. But anyway, if somebody has any idea about how it works, i'd really appreciate if he shed some light on this. Thanks

 

[Baggio]

Yes, as you reduce the width of the slit you would increase the coherence length upto infinity as the width tends to zero. Like you mentioned your intensity would drop off substantially. By filtering the white light your essentially going from a white light to a monchromatic beam with a fixed energy phase relationship.

 

[tomasko789]

At the first, thank you very much for your reply.
Anyway, it is still a bit mysterious for me. I'll try to explain my point:

Imagine, you have a single mode oscillator, a cavity, where only one mode of light can exist. If you couple this oscillator to a termal reservoir of certain temperature, there will be an equilibrium between them and the mode inside will be in let's say thermal state(with certain photon statistics). If you'll make one side of the oscillator semitransparent, you can use this as a source of "thermal" light. If I get right, you can adjust the bandwidth of this source by adjusting the transparency of the mirror. Thus, you can have almost monochromatic light source, but still with "thermal" photon statistics.

If you'll do the same with the laser oscillator, you'll again obtain a light source, monochromaticity can be tuned by the transparency of the mirror. However, the statistics of the light will be different from the previous case.

My point is, that you cannot obtain "laser" light by simply filtering the "thermal" light. And how does fact interfere with the coherence lenght...
Maybe it's a bit chaotic, but it's hard to impose a concrete question, if I don't understand it:)

 

[Cthugha]

My point is, that you cannot obtain "laser" light by simply filtering the "thermal" light. And how does fact interfere with the coherence lenght...

The coherence length or coherence time is NOT the defining property for laser light. One needs to have a look at second order coherence properties to tell, whether a light source is thermal or lasing. The second order intensity correlation tells us - roughly speaking - the probability to detect a photon at a given time t, given the fact that another photon was already detected at time 0 divided by the same probability for a light source with the same average intensity, which emits photons in a completely independent manner.

In laser light each photon emission is statistically independent, so the second order intensity correlation is 1 for all times t. The underlying photon number distribution is Poissonian.
For thermal light bunching is observed, which means, that the photons have the tendency to arrive in pairs, which means that the photon number fluctuations around the mean value are larger. This is a trademark of spontaneous emission. In this case the underlying photon number distribution is the Bose-Einstein distribution.

However in some situations this is difficult to show experimentally. For example it is very complicated to determine the laser threshold in semiconductor lasers like quantum dot VCSELs due to the extremely short coherence time of the emission of these structures below threshold.

 

[Andy Resnick]

Hi,
I just passed a course of a quantum coherence theory (or quantum optics). However, during the course i found more questions rather than answers.
One particular:

Imagine there's a white light coming from the sun. What is the coherence length of this light, according to the quantum coherence theory predictions (we know the temperature of the source, which is the only parameter of the light coming, so there should be a particular answer - the length in micrometers).

Specifically, imagine, we take a beam of the light coming from the sun and let it pass through a diffraction grating. Then we use a slit to get the light with a particular wavelength. And then we measure the coherence length of the light that passes through the slit. Again? What is the value of it and how it depends on the wavelength.

One more: The slit has of course a finite width, so the light that passes has some finite bandwidth. Let's measure the coherence length for one particular wavelength by changing the slit width, i.e. the bandwidth of the light? What would happen? Will the coherence length of this passed beam increase up to infinity, when the slit width will approach zero??

There is maybe a problem, that when the slit is very small, almost no photon passes through and we cannot even measure its coherence length. But anyway, if somebody has any idea about how it works, i'd really appreciate if he shed some light on this. Thanks

There's two flavors of coherence: spatial and temporal. Spatial coherence is given by the size of the source (about 0.5 degree for the sun, IIRC), and the temporal coherence is given by the bandwidth of the source. Blackbody radiation, because fo the large bandwidth has a vanishingly small coherence time.

Spectrally filtering the light increases the coherence time, and spatially filtering the light increases the coherence area.