Does a Fabry Perot cavity charge in discrete steps?

[Daniel Petka]

TL;DR Summary

I'm wondering if it is possible to measure the discrete charging steps for the step response of a FP cavity with an oscilloscope.

I'm trying to understand the Fabry Perot interferometer and came across this amazing video.

Basically it all comes down to adding E-fields together with each of them being delayed by the cavity round trip time. In the extreme case, either they all interfere constructively at the second mirror and the transmission goes to 100% if the length is an even multiple of half-wavelength and close to 0% if it's an odd multiple according to the airy formula.

Now I'm wondering how the FP could react to a step response of coherent CW laser light. In my EM class we did a step response for transmission lines. The result is similar to the charging of a capacitor but it's discrete because the wave takes some time to come back and interfere (round trip time)

Would it look the same for a FP? And if yes, is this something that can be measured with an oscilloscope? For a 1m cavity, the steps should be 1/c seconds long, so approx. 3.3 ns, this shouldn't be a problem but I haven't seen anyone measure it...

I came across this paper called "Rapid-Swept CW Cavity Ring-down Laser Spectroscopy for Carbon Isotope Analysis"

Here they seem to ignore all the discrete summation and just approximate the solution as a decaying exponential. I've seen the exponential various times without addressing the individual charging steps. Is there a flaw in my model? Thanks!

 

[Charles Link]

It would seem like what you have in the optics case should be workable, if you have been able to do it with the mismatched impedances using a transmission line/coaxial cable.

One item you may find of interest is something I figured out a few years back, that has often been omitted in the textbooks on the subject, and that is to consider the case of two sinusoidal sources incident from opposite directions onto a single dielectric interface. The energy reflection coefficient $R$ is no longer a good number, but instead the energy redistribution, which in all cases has complete energy conservation, will depend on the relative phase of the two sources. The Fresnel coefficients remain good numbers. It is ok that the system sees a non-linear behavior w.r.t. the energy, because the expression for the energy is second order in the linear parameter, the electric field.

See https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/

I anticipate you might see some success with the optical case as well, but there are of course no guarantees.

 

[sophiecentaur]

TL;DR Summary: I'm wondering if it is possible to measure the discrete charging steps for the step response of a FP cavity with an oscilloscope.

Now I'm wondering how the FP could react to a step response of coherent CW laser light. In my EM class we did a step response for transmission lines. The result is similar to the charging of a capacitor but it's discrete because the wave takes some time to come back and interfere (round trip time)

The "round trip Time" determines the input input impedance of the input port. As the frequency sweeps, there will be frequencies at which there is total reflection of the incident power so the energy levels between the two ends of the line will vary between cancellation frequencies. I see no reason for the FP to behave differently from a regular RF transmission line (as long as the model is correct and the effect of geometry on spreading the path length differences can be ignored).
IMO, the 'staircase' graph is insufficient to describe what's really going to be observed because the equivalent to the drive voltage will not be constant for the FP. A better model would have to be a length of transmission line with two transitions, spaced out and where the impedance steps produce appropriate reflections along the path.

 

[DaveE]

Add my vote for optical resonant cavities behaving essentially the same as RF waveguide resonances.

 

[DaveE]

One issue with the step response measurement is that you either need a big stable cavity or a fast laser. These are absolutely doable, but not easy or cheap.
Also a fast measurement system, which isn't very hard in a good lab.

 

[DaveE]

I think there may be issues with dispersion stretching your pulse. A high finesse cavity will have lots of bounces and thus lots of smearing of your pulse.

BTW, it would be a pulse from an ultrafast laser in a normal test, not a step. Those lasers would be much harder to make.

 

[sophiecentaur]

One issue with the step response measurement is that you either need a big stable cavity or a fast laser.

One problem with this is that the two experiments involve very different fractional bandwidths and different path lengths as multiples of wavelengths. The same equations are at work but the values of the variables are so different that I think it would be very unlikely that the 'fine structure' of the responses could match.

 

[sophiecentaur]

it would be a pulse from an ultrafast laser

You'd have to bear in mind that a laser relies on an FP or equivalent. If it produces fast pulses, that would imply it has a wide bandwidth so I think that would blur out any measured steps in response because the bandwidth of the pulse would exceed the step width.
The only way to examine / resolve the response would need to be with a very slow sweep with an appropriately narrow frequency spike.

 

[DaveE]

You'd have to bear in mind that a laser relies on an FP or equivalent.

No, ultra-fast lasers are typically ring cavities (plus amplifiers, compressors, etc.) for exactly the reason you mentioned; they MUST be wide bandwidth.

However, I agree filters can be characterized slowly in the frequency domain. Or you can do it in the time domain with δ-like functions.

Since the question involves transient behavior, I don't think you'll see it with steady state excitation. Although you certainly could characterize the system that way. Then your physics book will tell you it must have had the step responses in the OP.

 

[Charles Link]

The OP @Daniel Petka has an interesting problem. It would be nice to get some feedback on what he thought of all of our comments. I see he has been active as recently as Friday, so he probably has read most of what we had to say.

 

[Daniel Petka]

OP here, I didn't expect to get replies after 2 weeks, had other things to do in the meantime and came back only now. Apologies for that to anyone who is still reading this.

I think there may be issues with dispersion stretching your pulse. A high finesse cavity will have lots of bounces and thus lots of smearing of your pulse.

BTW, it would be a pulse from an ultrafast laser in a normal test, not a step. Those lasers would be much harder to make.

Dispersion is only a problem if you're dealing with etalons. A free space FP shouldn't have that issue, right?
Also, I don't quite understand why it can't be a CW laser sweep. Couldn't you just place a Q-switch before the laser to toggle it? The FP cavity would have to be long enough, so that the step separation is longer than the rise time of that Q switch, but that shouldn't be an issue, as the FP cavities in LIGO are 4 km long. I can't imagine how that cavity doesn't charge in steps.

It would seem like what you have in the optics case should be workable, if you have been able to do it with the mismatched impedances using a transmission line/coaxial cable.

One item you may find of interest is something I figured out a few years back, that has often been omitted in the textbooks on the subject, and that is to consider the case of two sinusoidal sources incident from opposite directions onto a single dielectric interface. The energy reflection coefficient $R$ is no longer a good number, but instead the energy redistribution, which in all cases has complete energy conservation, will depend on the relative phase of the two sources. The Fresnel coefficients remain good numbers. It is ok that the system sees a non-linear behavior w.r.t. the energy, because the expression for the energy is second order in the linear parameter, the electric field.

See https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/

I anticipate you might see some success with the optical case as well, but there are of course no guarantees.

Yess, it took me a long time to figure out as well. That pi phase shift is actually the key to understanding all interferometers and explains exactly where the light goes if it interferes destructively (it interferes constructively somewhere else) I still don't get how this works for silvered BSs but that's another rabbit hole..

The "round trip Time" determines the input input impedance of the input port. As the frequency sweeps, there will be frequencies at which there is total reflection of the incident power so the energy levels between the two ends of the line will vary between cancellation frequencies. I see no reason for the FP to behave differently from a regular RF transmission line (as long as the model is correct and the effect of geometry on spreading the path length differences can be ignored).
IMO, the 'staircase' graph is insufficient to describe what's really going to be observed because the equivalent to the drive voltage will not be constant for the FP. A better model would have to be a length of transmission line with two transitions, spaced out and where the impedance steps produce appropriate reflections along the path.

If you work out the geometric series, you get exactly this exponential charging curve. The equivalent to voltage is the amplitude of the electric field wave.

 

[Charles Link]

@Daniel Petka See https://www.physicsforums.com/threa...where-does-the-energy-go.942715/#post-5963684 post 5 the second "link" there, for the silvered beamsplitter

 

[Daniel Petka]

@Daniel Petka See https://www.physicsforums.com/threa...where-does-the-energy-go.942715/#post-5963684 post 5 the second "link" there, for the silvered beamsplitter

Ah damn it, quantum mechanics has entered the chat. Thanks for the paper :)