Accounting for Astigmatism in Bow Tie Cavity: Finding the Optimal Cavity Length

[kelly0303]

Hello! I am a bit confused about how to account for the astigmatism in a bow tie cavity. I did the math for the resonant frequency of the tangential and sagittal direction of a Gaussian beam, and I got in my case (the angle of the mirrors is 3 degrees) a difference of about $862.6$kHz. However the cavity I am trying to build has a linewidth of about $25$kHz. Initially I had the image in my head that in order to align the cavity I just need to make the resonant frequency of the cavity match the laser frequency. But now I have 2 cavity frequencies. Does this mean that I need to find a length of a cavity such that both the sagittal and the tangential frequency match a (different) resonant frequency of the cavity? But in my case the free spectral range is about 200 MHz, so it seems like I won't be able to match the 2 at the same time. What am I missing. How do people get around this issue? Thank you!

 

[Twigg]

Can you explain what you mean by calculating tangential and sagittal frequencies? Or better yet, share whatever reference you read about it in?

If it's anything like the polarization modes in Fabry-Perot cavities with anisotropic mirrors, then you would expect to see both peaks in the spectrum, each 25kHz broad and space 862.6kHz apart.

 

[kelly0303]

Can you explain what you mean by calculating tangential and sagittal frequencies? Or better yet, share whatever reference you read about it in?

If it's anything like the polarization modes in Fabry-Perot cavities with anisotropic mirrors, then you would expect to see both peaks in the spectrum, each 25kHz broad and space 862.6kHz apart.

Hello! Sorry, I should have given more details. I attached below the document with the relevant equations (not sure from what book it is). Table 2.1 on page 25, gives the formula for the resonant frequency of a bow tie cavity, which depends on the length of a cavity $L$, a (very large) integer, q, and the product $g_1g_2$ (I assume m=n=0). Now $g_1$ and $g_2$ depend on the cavity dimensions and mirrors radius. However below, in section 2.8, they say that due to astigmatism, the effective radii for the tangential and sagittal directions are given by: $Rcos{\theta}$ and $R/cos{\theta}$. In my case $R=0.25$m and $\theta = 3$deg so the new radii are very slightly different. But that difference leads to the tangential and sagittal frequencies, using the formula mentioned above to be different. As far as I understand, this is not related to the polarization, but with the shape of the beam itself and based on my calculations the 2 frequencies are apart by ~800kHz. I am not sure how I can amplify both at the same time. If my laser is on resonance with only one, it seems like an initially circular spot size will become almost like a narrow slit, if the perpendicular direction will be off resonance. Am I missunderstanding this?

EDIT: The file was too large, I am attaching screenshots of the relevant pages.

 

[Twigg]

Thanks for the clarification.

You are right, it's very different from a FP cavity with asymmetric mirrors. I understand better what you're asking about now. Sorry for the misdirect.

Based on a quick literature search, it seems that the TEM00 mode is unaffected by the 862.6kHz splitting. The astigmatism shows up by lifting the degeneracy of the TEM01 and TEM10 modes. I get this idea from this paper. Specifically this part:

The cavity is slightly astigmatic, with the horizontal transverse mode spacing (440 MHz) slightly less than the vertical mode spacing (490 MHz).

This is much larger than their cavity linewidth of <50kHz, so you will see distinct spectra for TEM01 and TEM10.

 

[kelly0303]

Thanks for the clarification.

You are right, it's very different from a FP cavity with asymmetric mirrors. I understand better what you're asking about now. Sorry for the misdirect.

Based on a quick literature search, it seems that the TEM00 mode is unaffected by the 862.6kHz splitting. The astigmatism shows up by lifting the degeneracy of the TEM01 and TEM10 modes. I get this idea from this paper. Specifically this part:

This is much larger than their cavity linewidth of <50kHz, so you will see distinct spectra for TEM01 and TEM10.

Thank you! I will read that paper. But why is the math not right for the TEM00 mode, too? Based on the equation in that table, I should still get that splitting. Am I understanding that equation the wrong way?

 

[Twigg]

Thank you! I will read that paper. But why is the math not right for the TEM00 mode, too? Based on the equation in that table, I should still get that splitting. Am I understanding that equation the wrong way?

I'm afraid I don't have an answer off-hand.

How did you calculate the resonant frequencies on each axis? I would've thought that the optical path length should still be the same for tangential and sagittal, just their beam waists would differ (and all would be well as long as they both meet the stability criteria).

 

[kelly0303]

I'm afraid I don't have an answer off-hand.

How did you calculate the resonant frequencies on each axis? I would've thought that the optical path length should still be the same for tangential and sagittal, just their beam waists would differ (and all would be well as long as they both meet the stability criteria).

For the formula for $\nu_{nmq}$ from the table above, we have a dependence on $g_1$ and $g_2$, which in turn depend on $R$. So given that the tangential and sagittal components have different effective $R$'s, I assumed they will have different $\nu_{nmq}$, too ($\nu_{00q}$ in this case). Am I using that wrong?

 

[Twigg]

I see what you mean. Can you provide the parameters for your cavity? 3 degrees angle, how about g1 and g2 in each direction, and the path length L?

 

[kelly0303]

I see what you mean. Can you provide the parameters for your cavity? 3 degrees angle, how about g1 and g2 in each direction, and the path length L?

In my case I have $d_1 = 0.40$, $d_2 = 0.337$, $d_3 = 0.27$, $R=0.25$ and $L = 1.344$

 

[Twigg]

Sorry, I get the same result and confusion that you do. My last thought is that maybe the expression in the book isn't fully general for astigmatic beams, and you need to take a step back to the paraxial wave equation to get the full picture. I haven't had time to do that analysis myself though. If you do find the answer, let me know! I'm curious as well.