Question about bow-tie cavities

[kelly0303]

Hello! This might be a naive question and I should have probably asked this earlier, but I am a bit confused about how one can lock a laser to a bow tie cavity, by applying the locking signal to the laser (i.e. changing the laser frequency). In a 2 mirrors (Fabry-Perot) cavity, I can imagine that if one mirror moved by a given amount, the servo would change the laser frequency such that it matches the new cavity length an integer number of times. However in a bow-tie cavity, if the position of a mirror changes, the length of the cavity changes, which can be compensated by adjusting the laser frequency, but given that the mirrors are at an angle, the position on the next mirror changes, too, and that propagates quickly from one mirror to the next. This is especially a problem for curved mirrors, as they are not hit in the center anymore and for high-finesse cavities. And changing the laser frequency will never compensate for this change of actual path inside the cavity. I imagine that putting a piezo on the mirror and changing the location of the mirror instead would solve that issue but I can't see how this can be achieved at all by adjusting the laser frequency. Can someone help me? Thank you!

 

[Baluncore]

And changing the laser frequency will never compensate for this change of actual path inside the cavity.

If the light remains coherent, then there is a path length that can be compensated.

Step changes in path length may require step changes in wavelength that are limited by the bandwidth of the locking mechanism.

 

[kelly0303]

If the light remains coherent, then there is a path length that can be compensated.

Step changes in path length may require step changes in wavelength that are limited by the bandwidth of the locking mechanism.

I am not sure I understand. I attached an exaggerated diagram of what I have in mind. The blue path is when the cavity is on resonance, and the light path is self-replicating. If the mirror on the right is displaced to the right, the path of light is the red one, and the path is not self-replicating anymore (hence the enhancement is lost). I don't understand how changing the laser frequency would solve this issue (even if you'd have infinite bandwidth).

 

[Baluncore]

I don't understand how changing the laser frequency would solve this issue (even if you'd have infinite bandwidth).

Only modes that are closed can be employed. In your example, the incidence on the curved mirrors must change to again close the path with the diagonal mirrors. For each change of mirror position, there will be a wavelength change that can again close the path, although no part of that path will be identical to the previous path.

 

[kelly0303]

Only modes that are closed can be employed. In your example, the incidence on the curved mirrors must change to again close the path with the diagonal mirrors. For each change of mirror position, there will be a wavelength change that can again close the path, although no part of that path will be identical to the previous path.

I agree that a change in the incidence angle on the curved mirror would solve the issue, but how does changing the laser wavelength would change the incidence? I might be missing something, but I thought that the path is independent by the wavelength i.e. when computing the path of a laser in a cavity the wavelength doesn't appear in the equations (of course that matters for other stuff, like coating of the mirrors). Do you have any reading that you can point me towards? Thank you!

 

[Twigg]

I agree that a change in the incidence angle on the curved mirror would solve the issue, but how does changing the laser wavelength would change the incidence?

It won't.

Remember, the laser light at the input mirror is not an infinitely thin ray, rather it's a gaussian beam with some width to it. Even if the beam is slightly off-center from the cavity mode, some part of the laser gaussian beam will still be aligned to it. (Likewise, the cavity resonant mode is also an approximately gaussian mode with some girth to it.)

I think it will be easier to understand if you don't think of cavity resonance in terms of rays and closed paths. Instead, think of cavity resonance as the overlap between the gaussian mode of the incident beam and the TEM₀₀ mode of the cavity. The mode overlap doesn't have to be perfect to lock a laser, there just has to be some overlap.

When the mirror is displaced by a small amount, the TEM₀₀ mode will shift and the overlap with the incident light will become worse but not zero. Some of the incident light still matches the cavity TEM mode and is injected into the cavity. If you displace the mirror by a large amount, then of course there will no longer be mode overlap and you will lose the laser lock. ("Small" and "large" displacements here mean whether the cavity mode is displaced by less than or more than one beam waist from the incident beam, respectively.)

As a separate point, if the cavity is on the edge of stability (where the TEM₀₀ stops existing), then of course a displacement outside the stability regime will completely kill the laser lock. But you typically would avoid this situation when designing a reference cavity.

 

[kelly0303]

It won't.

Remember, the laser light at the input mirror is not an infinitely thin ray, rather it's a gaussian beam with some width to it. Even if the beam is slightly off-center from the cavity mode, some part of the laser gaussian beam will still be aligned to it. (Likewise, the cavity resonant mode is also an approximately gaussian mode with some girth to it.)

I think it will be easier to understand if you don't think of cavity resonance in terms of rays and closed paths. Instead, think of cavity resonance as the overlap between the gaussian mode of the incident beam and the TEM₀₀ mode of the cavity. The mode overlap doesn't have to be perfect to lock a laser, there just has to be some overlap.

When the mirror is displaced by a small amount, the TEM₀₀ mode will shift and the overlap with the incident light will become worse but not zero. Some of the incident light still matches the cavity TEM mode and is injected into the cavity. If you displace the mirror by a large amount, then of course there will no longer be mode overlap and you will lose the laser lock. ("Small" and "large" displacements here mean whether the cavity mode is displaced by less than or more than one beam waist from the incident beam, respectively.)

As a separate point, if the cavity is on the edge of stability (where the TEM₀₀ stops existing), then of course a displacement outside the stability regime will completely kill the laser lock. But you typically would avoid this situation when designing a reference cavity.

Thanks a lot! I agree with what you said and indeed one wouldn't lose all the power. However, if one has a high finesse cavity I assume that a slight change in the path will change to some degree the power (not sure by how much), no? And I guess my real question is, regardless of how big this effect is (i.e. by how much the power changes due to a slightly different path), the laser servo won't correct for it. So if we apply the feedback to the laser, this is an error one has to live with. However, if the correction is applied to the mirror directly (using a piezo for example), that effect can be corrected (basically by bringing the mirror back to its original position). Is this right?

 

[Twigg]

This depends what kind of signal your servo locks to.

If you're sidelocking, then yes, as you say a change in the power will cause the servo to push the laser frequency one way or the other.

If you're PDH locking, then no, a change in power will not cause the laser lock to shift. This is because the PDH error signal is two-sided, so the slope of the error signal will change but the zero-crossing won't move.