1. Quantum Super-resolution Imaging in Fluorescence Microscopy
Osip Schwartz, Dan Oron, Jonathan M. Levitt, Ron Tenne, Stella Itzhakov and Dan Oron
Dept. of Physics of Complex Systems
Weizmann Institute of Science, Israel
FRISNO 12, Ein Gedi (February 2013)

2. Microscopy and resolution
Resolution of far-field optical microscopes is limited by about half wavelength.
(Ernst Abbe, 1873)
Workarounds:
Nonlinear optical methods: use nonlinear optical response to produce narrower point spread function
Stochastic methods: use fluorophores turning on and off randomly
Quantum optics?
It has been 150 years since Abbe discovered that the resolution of optical microscopes is limited by diffraction to approximatrely half wavelength of light. This barrier held for a very long time, but about twenty years ago several approaches were developed that enable superresolution imaging by use of either nonlinear optical methods, or by imaging fluorophores that fluctuate with time. There is another
Multi-photon interference Afek et al., Science 328 (2010)
Walther et al., Nature 429 (2004)
Entangled images Boyer et al., Science 321 (2008)
Sub shot noise imaging Brida et al., Nat. Photonics 4 (2010)
Resolution enhancement? 2

3. Quantum super-resolution
Quantum Limits on Optical Resolution
Wolf equations for two-photon light
Quantum Imaging beyond the Diffraction Limit by Optical Centroid Measurements
Quantum spatial superresolution by optical centroid measurements
Quantum imaging with incoherent photons,
Sub-Rayleigh quantum imaging using single-photon sources
Sub-Rayleigh-diffraction-bound quantum imaging,
Sub-Rayleigh Imaging via N-Photon Detection,
Kolobov, Fabre, PRL2000
Saleh et al., PRL 2005
M.Tsang PRL 2009
Shin et al., PRL 2011
Thiel et al., PRL 2007
Thiel et al., PRA 2009
Giovannetti, PRA 2009
Guerrieri et al., PRL 2010
Object
Light detector
Imaging system
Quantum light
The idea of achieving superresolution with quantum optics is very attractive, so people have been working on it. Here there is a sample of recent works on the subject. The general direction of thought seems to be, to take an object, an absorptive or a phase mask, illuminate it with some non-classical state of light, then detect the photons that pass through using some sort of coincidence detection. However, this does not seem to work out, I mean there has been no experimental demonstration so far. I think the very concept here is problematic, and however quantum your light source is, as long as the interaction of light with the object is linear, like absorption or phase shift, there can be no resolution improvement. 3

4. Quantum emitters Classical light Quantum light Quantum emitters
Light detector
Imaging system
Quantum light
Classical light
An alternative approach could be, let us image quantum objects. Let us have some kind of emitters that naturally produce quantum light, and analyze the light that comes out. This may sound a little esoteric, because everybody wants to image real things, such as biological cells, and these do not produce much quantum light. But bear with me, I will show that this is not esoteric at all.
Such a scheme was first considered by Stephen Hell and colleagues almost 20 years ago. The idea, which they called “Multi-photon detection microscopy”, was to have a fluorophore that would always emit photons in pairs, instantaneously or at least with a very short interval between them.
What if we had an emitter that would always emit photon pairs?
S.W. Hell et al., Bioimaging (1995) 4

5. Multi-photon detection microscopy
cascaded emitters
Photon pair detector
Imaging system
Photon pair τ1 τ2
τ1>>τ2
Point spread function: h2phot(x) = h2(x)
They were thinking of something like a cascaded light emitter shown here. After you excite it to the highest level, it has a long decay time to the intermediate level, and then it quickly relaxes to the ground state. The photons do not have to be exactly the same frequency, do not have to be coherent in any sense, just time-entangled. Then they would use a detector sensitive only to pairs, like a pair of single photon detectors counting coincidences.
In this situation, these two photons are created as a pair and detected as a pair, so there is a two-photon object, a bi-photon, propagating. It can carry up to twice the momentum of a single photon, so clearly resolution can be increased.
So Hell and colleagues found that the effective point spread function is a square of the regular point spread function, which is narrower, of course. The argument is readily generalized to groups of N photons, so you can get your point spread function to the power of N.
To the best of my knowledge, this scheme has never been realized experimentally, I suppose because it requires that very special fluorophore that no one has.
Spatial distribution of photon pairs carries high spatial frequency information (up to double resolution)
Similarly, in N-photon detection microscopy hNphot(x) = hN(x) 5

6. Antibunching microscopy
Observations of antibunching:
Number of photons emitted
after excitation:
Organic dyes: W. Ambrose et al. (1997)
Quantum dots: B. Lounis et al. (2000).
NV centers: R. Brouri et al. (2000).
Fluorescence intensity autocorrelation g(2)
10 μs interval between pulses
On the other hand, nearly all fluorophores that are used in fluorescence microscopy have a property which is somewhat related: antibunching. It is observed at room temperature, under ambient conditions, in organic dyes, quantum dots, nitrogen vacancy color centers in diamond, you name it. So the situation I was describing before is not esoteric at all - in fact, the light detected in fluorescence microscopy is always non-classical.
But can we use it for super-resolution, like Stephen Hell’s pair-emitting fluorophores? Yes, we can. Let us look at this histogram here, that shows the probability distribution for emitting no photons, one photon and so on for a classical source, which has Poissonian statistics. Compare it to Hell’s two-photon emitter – there is a probability to emit zero, two, but never one photon. In Hell’s scheme, we would be looking at the excess of pairs over classical light in every point of the image plane, and that would be our signal. What we have for antibunched emitter is, we have no pairs. So instead of the excess of pairs we can quantify the shortage of pairs in every point, and this signal is equivalent to Hell’s. The good thing is, there are no three-photon events either, so we can also quantify missing three-photon events and so on.
This is just an illustration of how antibunching is usually observed. The graph here shows intensity autocorrelation of fluorescence signal from a quantum dot, under pulsed excitation. This peaked structure is due to pulsed illumination, and the zero-delay peak is much smaller than the rest, which is the signature of antibunching.
Instead of actual photon pairs, consider ‘missing’ pairs. 6

7. Antibunching-induced correlations
Two adjacent detectors in the image plane: x0
For multiple fluorophores:
For individual fluorophore:
It is not very convenient to work with photon numbers when you have more than one fluorophore, so instead we look at correlation functions. If we collect fluorescence from a single emitter, and it can either hit one detector or the other, we never get a coincidence event, just as we saw. So if we look at cross-correlation between these detectors, which would be zero classically, it turns out to be this. So antibunching induces negative correlation between detectors. Now, if we have many fluorophores forming our object, the correlation is additive in the sense that correlation signal you get in the image plane is just a sum over fluorophores, like the regular fluorescence intensity is sum of fluorescence intensities from all fluorophores.
Sum over fluorophores 7

8. Emitters Fluorescence saturation CdSe / ZnSe / ZnS quantum dots
We work with colloidal quantum dot fluorophores, here is an electron micrograph of a sample. This is their spectrum.
One problem with detecting antibunching is that we need a lot of signal, because the difference with classical statistics are very small unless we have a significant probability of detecting a photon after every photoexcitation. So we need to work near saturation. But usually fluorophores do not react very well to saturation, they start flickering rapidly, which is known as blinking, and then undergo irreversible photodamage. This is a general problem with fluorophores, that they bleach when you try to saturate them. So we had to address this before doing actual imaging. Here, we were looking at a single quantum dot under pulsed excitation, and the pulse energy was going up and down as shown here. Here, the repetition rate of the excitation pulses was 100kHz, 200, 500kHz, and 1 MHz. The shape of these is not triangular but flattened due to saturation. We found that while at high repetition rate the fluorophores were blinking frantically and turning off, at low repetition rate they were much more photostable.
Schwartz et al.,ACS Nano 6 (2012) 8

9. At 1 kHz: Schwartz et al.,ACS Nano 6 (2012)
This is more data on the same – these scatter plots are saturation curves from individual dots, and you can see how blinking decreases at low repetition rate. At the lowest repetition rate of 1kHz we reached a saturation factor of 50 without inflicting noticeable photobleaching for over 1 hour. This is quite surprising because fluorescence decay time in these QDs is about 30 ns, and it is not very clear how a quantum dot remembers its history for many many microseconds. We know it is not thermal.
But anyway, we can saturate quantum dots when excitation is pulsed at low repetition rate.
At 1 kHz:
Schwartz et al.,ACS Nano 6 (2012) 9

10. Photon counting with a CCD
threshold
Dark counts
Less noise
More signal
Read noise
Pixel signal distribution
CCD ADC units
This is a schematic of our microscope. It is a regular epi-fluorescence microscope. A laser beam excites the field of view uniformly, the signal is collected, separated using a dichroic mirror and detected with an electron-multiplying camera. The laser gives out sub-nanosecond pulses with a rate of 1KHz, and the image is read out from the ccd after every excitation pulse. So we have a cycle that lasts a millisecond, one excitation pulse – one frame. The camera was running in the photon counting mode, that is, at full gain. That means that the photoelectrons were stochastically multiplied, like in a photo-multiplying tube, before readout. This is a typical probability distribution of readings from our camera, in logarithmic scale. This parabola is due to Gaussian distribution of the read noise in all pixels that had no photoelectrons. The exponential distribution here is due to random amplification of the photoelectons or dark counts.
Every pixel in the camera can be used as a single photon detector if we put a threshold and regard every count above it as a detection event. When we move it to the right, we reduce noise, but also reduce the signal. When we move it to the left, we can keep more photoelectrons, but we also let in more noise, so there is a tradeoff between noise and effective quantum yield.
arXiv:

11. Computing correlations
2nd order:
Quantifies the missing pairs
3rd order:
Once we recorded the movies of single-exposure images, we compute correlations. For the second order, we compute cross correlations between adjacent pixels in configurations shown here. These are configurations for the three-point correlation functions. This is the expression we compute for the second order, it is the cross correlation. When counting missing three-photon events, we have to leave out the ones that are missing due to missing pairs, which we already accounted for in the second order. For this we use so-called irreducible correlation function. It is also additive.
compute correlations for all pixel configurations
Fourier-interpolate the resulting images
Sum the interpolated images
Missing 3-photon events
(except those due to missing pairs, already accounted for)
arXiv:

12. Antibunching with a CCD
Quantum dot
Classical signal
Second order autocorrelation function:
g(2)(τ)=<n(t)n(t+ τ)>
τ2, ms
τ, ms
τ, ms
This is an example of cross-correlation we are getting from a quantum dot. It has an anti-bunching peak at zero, but it is not as deep as we expected. It turns out that the noise in the camera electronics makes the signal appear bunched, even when in reality it is not. This is a correlation function measured with a classical source, and it shoes a bunching peak. In the analysis, we have to correct for camera artifacts.
This is a third order correlation function from the same quantum dot. The depressed lines at … are due to missing 2 photon coincidence events. The central pixel, is even lower due to missing 3photon events. Again, the same correlation function with a classical source mirrors this behavior, showing ridges at … and a peak at zero delay.
Third order:
g(3)(τ1, τ2)=
=<n(t)n(t+τ1)n(t+ τ2)>
τ2, ms
τ1, ms
τ1, ms
arXiv: 12

13. Fluorescence image
Images
arXiv: 13

14. Fluorescence image
2nd order antibunching
Images
arXiv: 14

15. Fluorescence image 2nd order antibunching 3rd order antibunching
Resolution:
271 nm FWHM
2nd order antibunching
216 nm FWHM
(x1.26)
Images
3rd order antibunching
181 nm FWHM
(x1.50)
arXiv: 15

16. 2nd order antibunching imaging
Optical sectioning
Defocused image of a quantum dot:
Fluorescence imaging
2nd order antibunching imaging
Defocusing, μm
Images
Optical signal integrated over the field of view: 16

17. Summary
Far-field super-resolution imaging demonstrated by using quantum properties of light naturally present in fluorescence microscopy
The experiment was performed with commercially available equipment, at room temperature, with commonly used quantum dot fluorophores
With further development of detector technology, antibunching imaging may become feasible as a practical imaging method 17

18. The team Jonathan M. Levitt Stella Itzhakov Zvicka Deutsch Dan Oron
Ron Tenne 18

19. Images 19

20. Superresolved images Reconstructed high resolution images
Regular (photon counting) image
Second order correlations
Third order correlations
Images 20

21. Superresolved images
Images
arXiv: 21

22. Superresolved images
Images 22

23. Superresolved images
Images 23

24. Images 24

25. Quantum super-resolution
Conceptual difficulty: an absorptive grating with sub-wavelength period acts as an attenuator for every photon
Done with introduction
Transmitted light contains no information on the grating phase or period
Any linear absorber mask is a superposition of gratings
High spatial frequency components of the mask are lost 25