The following back-of-the-envelope calculations do not lead to any useful result, but you might be interested in reading through them if you want to get a better understanding of what happens during two-photon excitation microscopy.
The basic idea of two-photon microscopy is to direct so many photons onto a single confined location in the sample that two photons interact with a fluorophore roughly at the same time, leading to fluorescence. The confinement in time seems to be given by the duration of the laser pulse (ca. 50-500 fs). The confinement in space is in the best case given by the resolution limit (let’s say ca. 0.3 μm in xy and 1 μm in z).
However, since the laser beam is moving around, I wondered whether this may influence the excitation efficiency (spoiler: not really). I thought that his would be the case if the scanning speed in the sample is so high that the fs-pulse is stretched out so much that it spreads over a distance that is greater than the lateral beam size (0.3 μm FWHM).
For normal 8 kHz resonant scanning, the maximum speed (at the center of the FOV) times the temporal pulse width is, assuming a large FOV (1 mm) and a laser pulse that is strongly dispersed through optics and tissue (FWHM = 500 fs):
Δx1 = vmax × Δt = 1 mm × π × 8 kHz × 500 fs = 0.01 nm
This is clearly below the critical limits. Is there anything faster? AOD scanning can run at 100 kHz (reference), although it can not really scan a 1 mm FOV. TAG lenses are used as scanning devices for two-photon point scanning (reference) and for two-photon light sheet microscopes (reference). They run at up to 1000 kHz sinusoidal. This is performed in the low-resolution direction (z) and usually covers only few hundred microns, but even if it were to cover 1 mm, the spatial spread of the laser pulse would be
Δx1 = 1 mm × π × 1000 kHz × 500 fs = 1 nm
This is already in the range of the size of a typical genetically expressed fluorophor (ca. 2 nm or a bit more for GFP), but clearly less than the resolution limit.
However, even if the infrared pulse was smeared over a couple of micrometers, excitation efficiency would still not be decreased in reality. Why is this so? It can be explained by the requirement that the two photons arriving at the fluorophor have to be absorbed almost ‘simultaneously’. I was unable to find a lot of data on ‘how simultaneous’ this must be, but this interaction window in time seems to be something like Δt < 1 fs (reference). What does this mean? It reduces the true Δx to a fraction of the above results:
Δx2 = 1 mm × π × 1000 kHz × 1 fs = 0.003 nm
Therefore, smearing the physical laser pulses (Δx1) does not really matter. What matters, is the smearing of the temporal interaction window Δt over a spatial distance larger than the resolution limit (Δx2). This, however, would require a line scanning frequency in the GHz range – which will never, ever happen. The scan rate must always be significantly higher than the repetition rate of pulsed excitation. The repetition rate, however, is limited to <500 MHz due to fluorescence lifetimes of >1-3 ns. Case closed.
You’re right, the “interaction window in time” is about a fs. This is the virtual state lifetime. The 2p cross-section = the 1p cross-section squared times the virtual state lifetime (i.e., CS_2p = CS_1p * CS_1p * tau). Since 1p cross-sections are about 10^-17 (cm^2), 2p cross-sections end up being about (10^-17) * (10^-17) * (10^-15) = 10^-49 cm^4-s, or 10 GM (Göppert-Meyer units).
Another consideration when it comes to scan speed is the number of pulses per pixel and the number of photons (detected fluorescence photons) per pulse.
Thanks for the confirmation about the interaction window in time (aka virtual state lifetime)!
Since we’re talking about numbers: I wonder what is a typical number of detected photons per pulse for the GCaMP family and a typical microscope? I guess you might know that (or at least the order of magnitude), since you are using -if I’m not mistaken- photo-counting for detection.
Usually < 10, and it's often < 1 photon per pulse. Brighter samples and higher laser power will increase the photons per pulse, of course.
Driscoll et al. 2011 has a nice analysis of photon counting and analog integration (this is an excellent paper, btw). https://neurophysics.ucsd.edu/publications/J_Neurophysiol_2011_Driscoll_3106_13.pdf
In that paper, they were getting about 10 photons per pixel, with 383 pulses per pixel = 0.026 photons per pulse. That was a dim sample, but not entirely out-of-bounds.
Ouzounov and Chris Xu's lab's recent 3p paper gives some numbers that we can use to look at photons per pulse:
They say there are about 100 pixels per cell body, and the photons/s counts are ~100 (per cell). So converting that to photons/pulse: 100 photons / (8.49 frames/s*100 pixels per cell*3 pulses per pixel) = 0.039 photons/pulse. So for 2-4 pulses per pixel, there’s about a 8-16% chance of getting a single photon. That seems low, but since there are ~100 pixels on a cell body, the overall signal-to-noise is good enough for many measurements.
In both of these cases, the photons per pulse are on the low side, but still within the normal range.
So when using a 12 kHz resonant scanner, each line is scanned in 42 us. In that amount of time, the laser pulses 3360 times. If the line is split into 512 pixels, then there are < 10 pulses per pixel. Given the numbers above, there could be < 1 photon per pixel even in bright regions. This is why when scanning very fast, temporal averaging is often necessary to clearly make out the image. There are a lot of pixel samples that are zero (or just noise).
It's very easy to calculate the number of pulses per pixel as I did above. Granted, for resonant scanning the pulses per pixel vary over the line, because the scan speed is not constant, but it can still be easily estimated (and precise measurements are not difficult either).
It's also easy to measure the number of photons per pixel on a system, even if you're not using photon counting:
(assuming you have low-level control of your instrumentation– see the comments on that blog post)
Thus, even when not explicitly using photon counting, it is possible to estimate the photons per pulse of a 2p imaging system.
That’s very interesting – thank you for all the references!
Based on the method on your blog that you mention, I had done some estimates which gave something clearly below 1 photon/pulse for (silent) GCaMP6f, and I was a bit hesitant to believe the low numbers. But according to your references, this seems to be realistic values.
A very interesting discussion! At Pablo Blinder’s Lab we’ve lately been working on a photon counting add-on for two-photon microscopes, and we’ve just released the preprint (https://www.biorxiv.org/content/early/2018/05/09/316125). This discussion actually inspired us to add a supplementary figure (S2) with more detailed calculations using real data acquired during calcium imaging (GCaMP6f) and fast volumetric imaging.
We found that on average, a dF/F of 100% corresponds to 5.28 photons per second per um^2. We also see a variability in the ratio of dF/F to photons between different cells, probably due to different GCaMP concentrations in each individual cell and collection efficiency. We also directly quantified the different “photon budget” regimes, since we showcase work with the ultrafast TAG lens, which scans 330 um in the Z-axis approximately every 2.6 us (line rate of about 384 kHz) – leading to a displacement of each laser pulse by over 1 um.
I’d be happy to hear your thoughts on the manuscript, and\or suggestions for other interesting calculations we can make user our photon-counted data.
Thanks for sharing your manuscript, it’s very interesting, and I’m sure that the PySight software could be used in many different settings!
I’m surprised by the huge performance difference between the classical analog processing and your version using the MCS6A. For the Femto preamplifier, which settings did you use (bandwidth, gain, etc.), and which sampling rate did you use with the FlexRio system? The comparison between the two systems is very impressive, so it would be good to have all the technical details that make it transparent where the difference comes from.
I’m also wondering what the price tag for this system would be. I remember that a similar device (the TimeHarp) is quite expensive… can you give an order of magnitude for MCS6A and TA1000B, is it more around 1k, 10k or 50k $?
For Figure S2, it would be helpful if you could provide a bit more technical detail which would allow to connect the numbers to the sentence in the main text (” dF/F of 100% corresponds to 5.28 photons per second per um^2″). (Pixel size in um, integration time for Fig. S2.)
Thanks for your helpful comments, we’ll update our manuscript with the missing technical details.
The Femto preamps are DC-coupled, with a 80 MHz bandwidth and high gain settings. We use the output signal’s full bandwidth.
The FlexRIO’s sampling frequency is 120 MHz, the highest available. As for the price – the total should be about 10k USD. There are more expensive mutliscaler variants that can double this price, but they’re not necessary to achieve the same level of performance as shown in the article, at least for calcium, GEVI and volumetric imaging modalities.
For the planar calcium imaging, to which figure S2 refers, the pixel size was about 300 nm with a dwell time of 44 ns. This means an average of about 3.5 laser pulses per pixel.