A Error signal when locking a cavity

[Twigg]

Sorry I am a bit confused. What I meant was, don't I need the linewidth of the laser to be smaller than the cavity linewidth in order to get a lock? Are you saying that there is a chance that at 964 nm the linewidth of the laser would be smaller than 30 kHz?

Sorry for the confusion. I'm saying at 964nm, the linewidth of the cavity will be bigger than 30kHz. Since you have finesse of 10,000 at 1064nm, that means your reflectivity must be at least 99.99%, and probably much better. It depends what kind of mirror coating you use, but the reflectivity will probably decrease significantly as you go farther from the design wavelength. Lower reflectivity means broader linewidth.

I recommended changing wavelengths not as a workaround, but just as a diagnostic tool to see if you could lock the cavity with lower finesse and a different (presumably well-behaved) laser. After all, at the non-ideal wavelength your finesse will be lower so the built-up optical power will be less.

To be honest, in hindsight my suggestion sounds like a lot of work just to do a diagnostic test.

I also tried to measure in my cavity, when applying the servo on the mirror piezo, the transmission (pink line below) and reflection signal (green light below). Each horizontal line in the grid on the oscilloscope corresponds to about 300 kHz (so the linewidth of the peak would be ~150 kHz, which is not too far from what I would expect theoretically?) and the actual time scale is 20 microsecond per grid.

You're scanning the laser much too quickly. Remember that the cavity has a finite bandwidth and acts like a lowpass filter. You're scanning 300kHz in 20us, so your (ideally) 30kHz cavity linewidth will be fully scanned out in $30\mathrm{kHz} \times \frac{20\mathrm{\mu s}}{300\mathrm{kHz}} = 2\mathrm{\mu s}$. Now ask, will a $2\mathrm{\mu s}$ pulse be distorted by passing through the cavity (or equivalently, a 30kHz lowpass filter)? Well, to not distort the pulse you need a Fourier-limited bandwidth of at least $\frac{1}{2\mathrm{\mu s}} = 500\mathrm{kHz}$. What you're seeing in your oscilloscope trace is actually a really crude cavity ringdown. You can tell because your transmission line has an exponential tail (totally not a Lorentzian or Gaussian shape, not even symmetric). Slow your scanning rate or your scanning amplitude by an order of magnitude or two (keep turning it down until you get that symmetric Lorentzian shape).

As far as I understand, these beats happen when the 2 lasers have different frequencies, so that plot shows that the 2 lasers do indeed have different frequencies, but I am not sure how does that reflect the temporal stability of each one of them individually (which is what I care about).

Essentially, the company tuned the two lasers at some frequency difference and left them there. If the lasers were perfectly stable, you'd get a pure sinusoid beat signal. On a spectrum analyzer, like the image they sent you, a pure sinusoid would look like a single, very narrow peak. The width of this peak would be limited only by the resolution of the spectrum analyzer, which is 10kHz (that's what's meant by "RBW = VBW = 10kHz" at the top of that image). However, you can see that their beat spectrum is a few hundred kHz broad (remember FWHM means the width after a 3dB drop, not halfway down the curve since it's log scale). That means that the relative frequency difference between the lasers is drifting back and forth by a few hundred kHz. Assuming the two lasers drift by a comparable amount (since they're the same make and model of laser), you can conclude that each lasers drifts by an amount that is roughly the same few hundred kHz divided by $\sqrt{2}$.

I would argue that their measurement isn't definitive here because they have the wrong timescale. According to the image they sent, they did a 50ms acquisition time to get this few hundred kHz broadened beatnote. That doesn't really matter because your servo can handle variations on a 50ms timescale. What matters is the frequency variations at timescales beyond your servo's bandwidth. If they did this same measurement at $33\mathrm{\mu s} = \frac{1}{30\mathrm{kHz}}$ acquisition time (note: this acquisition time is not possible on a spectrum analyzer), I bet the result would be less. (Note: I chose $33\mathrm{\mu s}$ because that's the fastest time that the transmission photodiode signal could change, and that's what you've been locking to in the past.)

The reason why the laser spectrum depends on the acquisition time is because laser frequency noise isn't white noise, it's usually some kind of diffusion process and is pink (1/f). This is why in frequency metrology, we don't just quote linewidths for atomic clocks or ultrastable reference cavities, we give the whole Allan variation curve versus acquisition time (or equivalently, a power spectral density of phase noise).

The thing is, we kind of already have reason to suspect that your laser noise exceeds your locking range above the transmission error signal bandwidth. That's because we see your noise (as in post #11) shows up on timescales comparable to $33\mathrm{\mu s}$.

I'm optimistic about your prospects for locking to the reflection signal. I found a theory paper that discusses the difference between the transient response of transmission locking and reflection locking. It turns out (different than I remembered) that the reflection signal still behaves a first-order lowpass filter with cutoff frequency at the linewidth, but the transmission signal behaves as a second-order lowpass filter.

 

[kelly0303]

Sorry for the confusion. I'm saying at 964nm, the linewidth of the cavity will be bigger than 30kHz. Since you have finesse of 10,000 at 1064nm, that means your reflectivity must be at least 99.99%, and probably much better. It depends what kind of mirror coating you use, but the reflectivity will probably decrease significantly as you go farther from the design wavelength. Lower reflectivity means broader linewidth.

I recommended changing wavelengths not as a workaround, but just as a diagnostic tool to see if you could lock the cavity with lower finesse and a different (presumably well-behaved) laser. After all, at the non-ideal wavelength your finesse will be lower so the built-up optical power will be less.

To be honest, in hindsight my suggestion sounds like a lot of work just to do a diagnostic test.


You're scanning the laser much too quickly. Remember that the cavity has a finite bandwidth and acts like a lowpass filter. You're scanning 300kHz in 20us, so your (ideally) 30kHz cavity linewidth will be fully scanned out in $30\mathrm{kHz} \times \frac{20\mathrm{\mu s}}{300\mathrm{kHz}} = 2\mathrm{\mu s}$. Now ask, will a $2\mathrm{\mu s}$ pulse be distorted by passing through the cavity (or equivalently, a 30kHz lowpass filter)? Well, to not distort the pulse you need a Fourier-limited bandwidth of at least $\frac{1}{2\mathrm{\mu s}} = 500\mathrm{kHz}$. What you're seeing in your oscilloscope trace is actually a really crude cavity ringdown. You can tell because your transmission line has an exponential tail (totally not a Lorentzian or Gaussian shape, not even symmetric). Slow your scanning rate or your scanning amplitude by an order of magnitude or two (keep turning it down until you get that symmetric Lorentzian shape).Essentially, the company tuned the two lasers at some frequency difference and left them there. If the lasers were perfectly stable, you'd get a pure sinusoid beat signal. On a spectrum analyzer, like the image they sent you, a pure sinusoid would look like a single, very narrow peak. The width of this peak would be limited only by the resolution of the spectrum analyzer, which is 10kHz (that's what's meant by "RBW = VBW = 10kHz" at the top of that image). However, you can see that their beat spectrum is a few hundred kHz broad (remember FWHM means the width after a 3dB drop, not halfway down the curve since it's log scale). That means that the relative frequency difference between the lasers is drifting back and forth by a few hundred kHz. Assuming the two lasers drift by a comparable amount (since they're the same make and model of laser), you can conclude that each lasers drifts by an amount that is roughly the same few hundred kHz divided by $\sqrt{2}$.

I would argue that their measurement isn't definitive here because they have the wrong timescale. According to the image they sent, they did a 50ms acquisition time to get this few hundred kHz broadened beatnote. That doesn't really matter because your servo can handle variations on a 50ms timescale. What matters is the frequency variations at timescales beyond your servo's bandwidth. If they did this same measurement at $33\mathrm{\mu s} = \frac{1}{30\mathrm{kHz}}$ acquisition time (note: this acquisition time is not possible on a spectrum analyzer), I bet the result would be less. (Note: I chose $33\mathrm{\mu s}$ because that's the fastest time that the transmission photodiode signal could change, and that's what you've been locking to in the past.)

The reason why the laser spectrum depends on the acquisition time is because laser frequency noise isn't white noise, it's usually some kind of diffusion process and is pink (1/f). This is why in frequency metrology, we don't just quote linewidths for atomic clocks or ultrastable reference cavities, we give the whole Allan variation curve versus acquisition time (or equivalently, a power spectral density of phase noise).

The thing is, we kind of already have reason to suspect that your laser noise exceeds your locking range above the transmission error signal bandwidth. That's because we see your noise (as in post #11) shows up on timescales comparable to $33\mathrm{\mu s}$.

I'm optimistic about your prospects for locking to the reflection signal. I found a theory paper that discusses the difference between the transient response of transmission locking and reflection locking. It turns out (different than I remembered) that the reflection signal still behaves a first-order lowpass filter with cutoff frequency at the linewidth, but the transmission signal behaves as a second-order lowpass filter.

Thank you for this! So what are the small oscillations on top of the main peak? Are they expected from the fact that the frequency changes in time?

About scanning my cavity, I am currently scanning the mirror piezo at 100 Hz. I could go lower but from my past experience the peaks start behaving crazy if I go below that. I am attaching below a peak (this is actually a nice looking one compared to what I see) obtained when scanning the cavity at 10 Hz. I also suspect that at this frequency I might have some issues with mechanical vibrations? Definitely the peaks start to move left and right and change their amplitude a lot more compared to when I was scanning at 100 Hz or higher, and actually I start seeing each individual "peak" made of lots of peaks as if the peak moves left and right, while I scan over it.

 

[Twigg]

When you scan the laser with a scan rate of 10Hz, what is the range of the scan? What I really need is the frequency-per-second scan rate. For example, in post #50, it was $300\mathrm{kHz}$ in $20\mathrm{\mu s}$, or $15\mathrm{kHz / \mu s}$. What was this rate in the data in post #52?

 

[kelly0303]

When you scan the laser with a scan rate of 10Hz, what is the range of the scan? What I really need is the frequency-per-second scan rate. For example, in post #50, it was $300\mathrm{kHz}$ in $20\mathrm{\mu s}$, or $15\mathrm{kHz / \mu s}$. What was this rate in the data in post #52?

For the last figure I sent, I didn't change the voltage range, just the frequency, so the speed should be 10 times smaller than before, so that would be $1.5 kHz/\mu s$

 

[Twigg]

So what are the small oscillations on top of the main peak?

Probably fluctuations either in the laser frequency or in your cavity's optical path length. It's impossible to discriminate between the two of them with the tools at your disposal. There are tricks you could try if you had two lasers (or equivalently, a fiber EOM phase modulator for frequency-offset-locking).

Are they expected from the fact that the frequency changes in time?

I don't think so. Take a look at this paper for the full theory. It's written for Fabry-Perot cavities, but I think it's the same except the "round trip time" is $\tau = L/c$ for a bowtie cavity. I did the math for the data in post #52 and I get a $v_\omega$ of 10.6. As you can see in Figure 4d, there really shouldn't be any oscillations due to the laser scan.

However, if your cavity mirrors were contaminated with dirt causing you to have a broadened linewidth, then anything is possible. I don't really understand the transient dynamics in this regime, but they're noisy and annoying.

I also suspect that at this frequency I might have some issues with mechanical vibrations?

At 10Hz, that's very possible. Again, there isn't a good way to distinguish between laser fluctuations and cavity fluctuations when you only have the one laser and the one cavity.

For the last figure I sent, I didn't change the voltage range, just the frequency

Could you try turning the frequency back up to 100Hz and turn the voltage range down by a factor of 10x? Are the results the same? It might give some indication of the frequency-dependence of your oscillations.

 

[kelly0303]

Probably fluctuations either in the laser frequency or in your cavity's optical path length. It's impossible to discriminate between the two of them with the tools at your disposal. There are tricks you could try if you had two lasers (or equivalently, a fiber EOM phase modulator for frequency-offset-locking).I don't think so. Take a look at this paper for the full theory. It's written for Fabry-Perot cavities, but I think it's the same except the "round trip time" is $\tau = L/c$ for a bowtie cavity. I did the math for the data in post #52 and I get a $v_\omega$ of 10.6. As you can see in Figure 4d, there really shouldn't be any oscillations due to the laser scan.

However, if your cavity mirrors were contaminated with dirt causing you to have a broadened linewidth, then anything is possible. I don't really understand the transient dynamics in this regime, but they're noisy and annoying.At 10Hz, that's very possible. Again, there isn't a good way to distinguish between laser fluctuations and cavity fluctuations when you only have the one laser and the one cavity.Could you try turning the frequency back up to 100Hz and turn the voltage range down by a factor of 10x? Are the results the same? It might give some indication of the frequency-dependence of your oscillations.

I will check the effect of reducing the range tomorrow (I think I tried and didn't see a major difference).

Sorry I wasn't clear with the "So what are the small oscillations on top of the main peak?" question. This was meant for the figure the laser company sent. I understand that I get a broadened peak, because the 2 laser don't have a constant frequency difference, but what are the small oscillations on top of that peak?

 

[Twigg]

It's probably just noise. I wouldn't read too much into it.