Error signal when locking a cavity

[Tom.G]

Looking back at post #11 again, I noticed the timescale of your noise events is really fast (roughly 10 microseconds), so I think you're right. This is way too fast to be environmental fluctuations.

Post #29 indicates around 20kHz, well within the range of electronic ballasts in common LED and Fluorescent lamps, switching power supplies, etc.

these peaks seems to come and go every 50 $\mu s$ or so.

That reminds me, some switching power supplies are terrible for leaking switching noise, both conductive to the output and radiative.

 

[kelly0303]

Another wild guess here, just from the troubleshooting aspect.

What do the error signal and the drive signal look like open-loop, that is with either
1) the Laser control signal set with a fixed external voltage

or​

2) the Laser turned off/blocked and a DC, battery, powered incandescent lamp illiminating the detector? A cheap LED flashlight may be adequate if it doesn't have any electronics in it; a lense may be needed to get enough power density on the sensor.

That should help determine if it is a servo loop tuning problem, faulty electronics, or perhaps outside interference (either mechanical, electrical, or possibly acoustic [air conditioning with an ultrasonic whistle ]).

Thank you for your reply. I am not sure I understand what I should do. Do you mean to keep the laser frequency constant instead of ramping it? I would have no drive signal in that case, no? And the error signal would just be random peaks when the cavity happens to be on resonance with the laser. I think I am missunderstanding what you mean.

 

[kelly0303]

You're right about the reflection being at an angle. I forgot about that quirk of bowtie cavities.Looking back at post #11 again, I noticed the timescale of your noise events is really fast (roughly 10 microseconds), so I think you're right. This is way too fast to be environmental fluctuations. Sorry for not noticing that sooner.

The problem could very well be electronic noise. After all, with the humongous 100MHz/V scaling factor on the laser modulation port, it only takes 10mV to push you off a 1MHz line (if you line is actually 30kHz, then it only takes a mere 300 microvolts). Because electronic noise can be very fast, it's possible that your servo can't keep up. I'm curious what will happen if you engage the differentiator on your D2-125 and set the corner frequency to about 10kHz or 20kHz or somewhere around there. Alternatively, you could try adding a 10 or 20kHz lowpass filter between your photodiode and your servo. (Using the battery-powered photodiode was a good choice!). Another possibility is that the origin of your noise could be from ground loops (remember, your oscilloscope is a connection to ground).

If it's not electronic, then it could be vibrations in the upper audio band. Does your lock get messed up if you start talking or clapping your hands near the cavity?

The differentiator should help regardless if the noise is electronic or vibrational. But the lowpass filter should only help if the noise is electronic and from your photodiode.

I tried the clapping, and the lock is stable still. Thank you for the suggestion with the electronic noise. However I just placed a piezo mirror, such that I am now forcing the cavity to follow the frequency of the stable laser and I am getting exactly the same result (regardless of the differential value, or the servo used). I really don't know what to do and also I am confused by the fact that no one (and I talked to quite a few people with experience in cavities) seems to have seen this before, given that I am using mostly commercial components that most people have also used. It's so unlikely that no one has ever seen this behavior before.

 

[Twigg]

I am getting exactly the same result (regardless of the differential value, or the servo used)

Just to clarify, you engaged the differentiator, and the result was that the error still remained just as noisy and the transmitted power did not increase? That's really bizarre. Did you try the lowpass filter on the photodiode?

I really don't know what to do and also I am confused by the fact that no one (and I talked to quite a few people with experience in cavities) seems to have seen this before, given that I am using mostly commercial components that most people have also used.

I know the feeling. I'm sorry it's been such a slog. You've really put in a solid effort. This is some cursed stuff.

If you haven't already, I would take some time and try perturbing the cavity/laser/servo in every (harmless) way you can think of. For some examples, you might try pulling on/shaking coax cables, tapping on photodiodes, tapping on the cavity baseplate, tapping on the table near the laser head, adding ferrite cores to various coax cables, turning off the lights, and opening/closing the room door a few times. See if anything you do will make your error signal noise get worse. What you're looking for is anything that exacerbates the existing noise (as opposed to creating new noise).

You've tried two different servos and you've tried two different actuators (laser PZT and cavity mirror PZT), with the same result. This is a bit drastic, but maybe you can try the same thing with your laser (hear me out).

I know you don't have another 1064 laser, but you can try locking another near-infrared laser to your cavity. The finesse won't be as high (because the mirror coatings' reflectance will fall off away from 1064), but I routinely use a laser 100nm away from the design wavelength to align my cavity and I still see resolved TEM modes (not yet broadened into LG modes). Do you have access to any other NIR lasers? Ideally, you'd want something close to 1064nm.

If you try a different laser and you don't get the same noise problems, then you know that the issue was your 1064 laser jumping around.

 

[Tom.G]

Thank you for your reply. I am not sure I understand what I should do. Do you mean to keep the laser frequency constant instead of ramping it? I would have no drive signal in that case, no? And the error signal would just be random peaks when the cavity happens to be on resonance with the laser. I think I am missunderstanding what you mean.

Sorry for my lack of clarity, I have not actually worked with a setup like yours so I fell back on troubleshooting approaches that I have found effective.

My overall intent was to open the closed control loop to get clues on possible problem areas, internal or external, thus narrowing the search.

For instance if the error signal shows the same chaotic pattern with the loop open, external interference, extraneous feedback, or defective equipment would be prime areas to investigate.

As @Twigg suggested, changing the laser would be one approach since that, the cavity, the detector, and power supplies (if any) are the only obvious candidates remaining.

 

[kelly0303]

You're right about the reflection being at an angle. I forgot about that quirk of bowtie cavities.Looking back at post #11 again, I noticed the timescale of your noise events is really fast (roughly 10 microseconds), so I think you're right. This is way too fast to be environmental fluctuations. Sorry for not noticing that sooner.

The problem could very well be electronic noise. After all, with the humongous 100MHz/V scaling factor on the laser modulation port, it only takes 10mV to push you off a 1MHz line (if you line is actually 30kHz, then it only takes a mere 300 microvolts). Because electronic noise can be very fast, it's possible that your servo can't keep up. I'm curious what will happen if you engage the differentiator on your D2-125 and set the corner frequency to about 10kHz or 20kHz or somewhere around there. Alternatively, you could try adding a 10 or 20kHz lowpass filter between your photodiode and your servo. (Using the battery-powered photodiode was a good choice!). Another possibility is that the origin of your noise could be from ground loops (remember, your oscilloscope is a connection to ground).

If it's not electronic, then it could be vibrations in the upper audio band. Does your lock get messed up if you start talking or clapping your hands near the cavity?

The differentiator should help regardless if the noise is electronic or vibrational. But the lowpass filter should only help if the noise is electronic and from your photodiode.

I just got a reply from the laser company:

For side of fringe locking wo/modulation you assumed that the AM error signal (between transmitted and fixed reference ) is linear (at 50 % transmission point) and proportional to a frequency fluctuations ~dn. Such assumption is prone to errors caused by the intensity fluctuations which are interpreted as frequency fluctuations. Usually to correct such crosstalk with DFB or DBR lasers it is necessary to use LF dither frequency with small amplitude Dn << fringe LW and use a lock-in detection. Your fringe has a very narrow LW which makes this approach (wo/intensity monitor) even more sensitive to the intensity noise

And then basically they mention that changing the laser frequency induces a change in power which would affect a side of fringe lock. I don't really understand everything they suggested above (I will need to do some google about the technical terms), but if anyone here can give me a dumbed down version I would really appreciate.

However, while I agree that while scanning the frequency over 300 MHz (the FSR of the cavity), the power might change, when I am locked, the frequency shouldn't need to change by more than a few kHz. Can it really be that such a small freq change lead to a power change big enough to mess up my lock?

 

[Twigg]

What they're suggesting is that your servo is trying to control the laser frequency but ends up getting confused by fluctuations in the laser output power.

For example, if your servo locks when the transmission photodiode reads 1V, but then your laser power drops to 80%, then your transmission photodiode will read 0.8V (so your error signal will be -0.2V), and the servo will compensate (even though the laser frequency had not changed).

What they suggest is to stop side-locking your laser and switch to a lock-in scheme. I drew a quick sketch below.

The key here is that your error signal lockpoint will become independent of laser power. If you want a mathematical argument for why this is, I would recommend reading about the math behind PDH locking. The main difference between the above scheme and PDH locking is that this scheme uses the transmission photodiode (PDH typically uses reflection photodiode, for higher bandwidth) and the strength of the modulating signal.

As far as jargon, "dither" usually means a small amount of frequency modulation used to generate the error signal, and "lock-in detection" likely means using the demodulated photodiode signal instead of the DC photodiode signal as your error signal. Were there other terms that were confusing?

P.S. It only just occurred to me that because you've been locking with the transmission photodiode, your servo never had a chance of dealing with the high frequency noise on your error signal. The transmitted power out of the cavity is band-limited by the cavity linewidth, so if your linewidth is 30kHz, you will never be able to stabilize noise faster than 30kHz while locking to the transmission photodiode. If you want to go faster than that, you need to use the reflection signal (which is not limited by the cavity linewidth). Sorry it took me so long to put that together!

Edit: If you're looking for a good place to learn about PDH locking, I suggest Holger Mueller's "Electronics for Pros" lecture #18, see here.

Edit edit: Actually, I'm not sure if the reflection signal on a bowtie cavity is bandlimited by the linewidth or not (it definitely is not limited for Fabry-Perot cavities, but I don't know about bowtie cavities). I added strikethrough to my comments where I made this claim. Sorry!

 

[kelly0303]

What they're suggesting is that your servo is trying to control the laser frequency but ends up getting confused by fluctuations in the laser output power.

For example, if your servo locks when the transmission photodiode reads 1V, but then your laser power drops to 80%, then your transmission photodiode will read 0.8V (so your error signal will be -0.2V), and the servo will compensate (even though the laser frequency had not changed).

What they suggest is to stop side-locking your laser and switch to a lock-in scheme. I drew a quick sketch below.
View attachment 317038
The key here is that your error signal lockpoint will become independent of laser power. If you want a mathematical argument for why this is, I would recommend reading about the math behind PDH locking. The main difference between the above scheme and PDH locking is that this scheme uses the transmission photodiode (PDH typically uses reflection photodiode, for higher bandwidth) and the strength of the modulating signal.

As far as jargon, "dither" usually means a small amount of frequency modulation used to generate the error signal, and "lock-in detection" likely means using the demodulated photodiode signal instead of the DC photodiode signal as your error signal. Were there other terms that were confusing?

P.S. It only just occurred to me that because you've been locking with the transmission photodiode, your servo never had a chance of dealing with the high frequency noise on your error signal. The transmitted power out of the cavity is band-limited by the cavity linewidth, so if your linewidth is 30kHz, you will never be able to stabilize noise faster than 30kHz while locking to the transmission photodiode. If you want to go faster than that, you need to use the reflection signal (which is not limited by the cavity linewidth). Sorry it took me so long to put that together!

Edit: If you're looking for a good place to learn about PDH locking, I suggest Holger Mueller's "Electronics for Pros" lecture #18, see here.

Edit edit: Actually, I'm not sure if the reflection signal on a bowtie cavity is bandlimited by the linewidth or not (it definitely is not limited for Fabry-Perot cavities, but I don't know about bowtie cavities). I added strikethrough to my comments where I made this claim. Sorry!

Thanks a lot for the explanation and the link, that is really helpful! Trying to lock to the reflected signal is a great idea, I will try it on Monday!

About power variation, I actually sent the laser directly (after some attenuation) to the diode, and I reduced the scale on the oscilloscope from 10 V to ~10 (or even 1) mV, and the signal looked flat by eye. Doesn't this mean that the power is stable at the 0.1% levels (and this was while scanning the frequency over ~300 MHz range)? Shouldn't that be enough power stability to not affect my lock?

About frequencies higher than 30 kHz, shouldn't them average out somehow and not affect my signal? I thought that is was the low frequencies that were the main issue.

 

[Twigg]

About power variation, I actually sent the laser directly (after some attenuation) to the diode, and I reduced the scale on the oscilloscope from 10 V to ~10 (or even 1) mV, and the signal looked flat by eye. Doesn't this mean that the power is stable at the 0.1% levels (and this was while scanning the frequency over ~300 MHz range)? Shouldn't that be enough power stability to not affect my lock?

I believe the concern is about power fluctuations when the servo is closed-loop, not in the open-loop configuration (you measured it in the open-loop configuration, since you had no cavity signal). The response from the laser company suggests that your laser is a DBR or DFB. If that's true, then the signal into the modulation port will affect the diode current (unlike in an ECDL where it would affect the grating), so you can expect that any signal applied to the modulation port (like the servo control signal) will affect both power and frequency. This can be very confusing to your poor servo box, which cannot distinguish between power changes and frequency changes. It could cause some real nasty pathologies.

About frequencies higher than 30 kHz, shouldn't them average out somehow and not affect my signal? I thought that is was the low frequencies that were the main issue.

In a perfect world, they would not matter much to the optical power. However, your data suggests otherwise, as you have the transmission photodiode signal swinging from locked to 0. I'm surprised the laser doesn't stay unlocked. But hey, if your laser stays locked for a good amount of time, then maybe you shouldn't worry about it? It's not clear to me, but whatever it is it's not a normal thing.

 

[kelly0303]

I believe the concern is about power fluctuations when the servo is closed-loop, not in the open-loop configuration (you measured it in the open-loop configuration, since you had no cavity signal). The response from the laser company suggests that your laser is a DBR or DFB. If that's true, then the signal into the modulation port will affect the diode current (unlike in an ECDL where it would affect the grating), so you can expect that any signal applied to the modulation port (like the servo control signal) will affect both power and frequency. This can be very confusing to your poor servo box, which cannot distinguish between power changes and frequency changes. It could cause some real nasty pathologies.

I am a bit confused, isn't the signal applied during open and closed loop the same? Of course the amplitude and the pattern are different, but it is a change in the diode current in both cases. And given that the frequency change during the closed loop should be much smaller than in the open loop and given that during the open loop I see no power variation, I am not sure I understand how can I have power variations during the closed loop?

One more questions: if I am to drive the mirror piezo, instead of the laser (basically I would add a 50 Ohm resistor to the BNC cable of the laser), all these issue should be gone (up to the high frequency problem), as the power should not depend on the resonant frequency of the cavity anymore, right?

 

[Twigg]

I am a bit confused, isn't the signal applied during open and closed loop the same? Of course the amplitude and the pattern are different, but it is a change in the diode current in both cases. And given that the frequency change during the closed loop should be much smaller than in the open loop and given that during the open loop I see no power variation, I am not sure I understand how can I have power variations during the closed loop?

Well, your laser *does* have frequency (and/or power) variations when closed-loop based on your noisy error signal. If anything, your power noise might be larger in closed loop, because it could be driven by the servo compensating for frequency noise.

You said earlier that your power was stable in open loop at 0.1% levels, so if your linewidth is 30kHz then your frequency noise due to power fluctuations should be 30Hz at those timescales, which is absolutely negligible.

One more questions: if I am to drive the mirror piezo, instead of the laser (basically I would add a 50 Ohm resistor to the BNC cable of the laser), all these issue should be gone (up to the high frequency problem), as the power should not depend on the resonant frequency of the cavity anymore, right?

Good point. You are absolutely right.

In the long run, you probably want to switch to lock-in or PDH scheme anyways, because it lets you lock on resonance which will double your cavity power. However, based on your arguments I'm not sure that adding modulation will resolve the high frequency noise on its own, as the laser company claimed it would. Locking to reflection could still be worth a shot? Unclear to me.

 

[kelly0303]

Well, your laser *does* have frequency (and/or power) variations when closed-loop based on your noisy error signal. If anything, your power noise might be larger in closed loop, because it could be driven by the servo compensating for frequency noise.

You said earlier that your power was stable in open loop at 0.1% levels, so if your linewidth is 30kHz then your frequency noise due to power fluctuations should be 30Hz at those timescales, which is absolutely negligible.Good point. You are absolutely right.

In the long run, you probably want to switch to lock-in or PDH scheme anyways, because it lets you lock on resonance which will double your cavity power. However, based on your arguments I'm not sure that adding modulation will resolve the high frequency noise on its own, as the laser company claimed it would. Locking to reflection could still be worth a shot? Unclear to me.

Thank you! You mentioned earlier that I can use a different laser (even at a different wavelength) to test my setup. Regardless of the wavelength, don't I need a laser with a linewidth smaller than the cavity linewidth (~30 kHz)? Isn't this a very narrow laser linewidth i.e. should I expect to easily find one around (the reason I am using this particular laser was its 5 kHz linewidth)? We have an M Squared laser, which I think it can go up to 1000nm (so might be ok-ish in term of wavelength) and a C-WAVE OPO which can actually produce 1064 light, but their linewidths (I think) are ~100 kHz and ~1 MHz, respectively. Would they even lock to this cavity?

 

[Twigg]

Regardless of the wavelength, don't I need a laser with a linewidth smaller than the cavity linewidth (~30 kHz)?

Your linewidth won't be as narrow when you're at, say, 964nm. See if you can't get a reflectivity vs wavelength curve for your mirrors and you can estimate the new linewidth.

 

[kelly0303]

Your linewidth won't be as narrow when you're at, say, 964nm. See if you can't get a reflectivity vs wavelength curve for your mirrors and you can estimate the new linewidth.

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?

 

[kelly0303]

Your linewidth won't be as narrow when you're at, say, 964nm. See if you can't get a reflectivity vs wavelength curve for your mirrors and you can estimate the new linewidth.

@Twigg I contacted the laser company about the possibility of using my laser as the reference and applying the servo to the piezo of the mirror. They said this would not be possible given the stability of my laser and sent me the first figure below, showing a typical heterodyne beat stability of two similar lasers (same model as mine). I am a bit confused by that plot. 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). Also I am not sure why do we have this Lorentzian shape (I am not sure what the x-axis is relative, to). Could you please help me a bit with understand that, I would really appreciate it!

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. Do you have any idea why am I getting this signal? It definitely looks like beats, and I would be tempted to say that indeed the frequency of the laser changes slightly between the light reflected directly and the one spending time in the cavity then exiting (hence giving beats when overlapping). This would also be consistent with not seeing beats in the transmission signal. Does this make sense (I don't fully understand the sent heterodyne signal to check if they are indeed consistent)? Also I am not sure why in the peak of the reflection signal I see no beat-like signal.

 

[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.