Temporal coherence in Computational ghost imaging

[nandan]

In pseudo-thermal ghost imaging using a laser beam passing through a rotating ground glass plate, the temporal coherence is important since the intensities in both scanning and bucket detectors have to be measured within the coherence time of the beam. However, in computational ghost imaging this requirement is obviously not enforced (since the scanning detector intensity is calculated using the phase added by the spatial light modulator). I wanted to know what is the basic difference between these two approaches? Why is it that the phase fluctuations of laser (giving rise to finite coherence time) are important in conventional pseudo-thermal ghost imaging but is not important in computational ghost imaging.
Any help would be wonderful.
Thanks.

 

[Cthugha]

In computational ghost imaging you can change the phase deterministically (you use random images on the SLM, but you can calculate what the field looks like). In conventional pseudo-thermal ghost imaging you do not have that information directly. You can measure the pseudothermal field in one arm and as the fields in the two arms are classically correlated that gives you the necessary information about about the field in the other arm. Now the time scale on which the field and its phase change is given by the coherence time. So to resolve the changes you need to be faster.

In computational ghost imaging, this timescale is simply set by the frame rate of your SLM. You can determine when and how the phase changes. So you can always choose this time scale long enough.

Please note that these coherence times are introduced by the modulation either at the SLM or the rotating ground glass disk. They are not related to the coherence time of the bare laser

 

[nandan]

But since the input laser beam also has some phase fluctuations, the total phase after passing through the SLM should also have similar phase fluctuations over and above the phase added by SLM. Will this phase fluctuation not affect the coherence time of resulting beam? Also, is there any difference in coherence times of the spatially incoherent beams produced using ground glass and SLM?

 

[Cthugha]

But since the input laser beam also has some phase fluctuations, the total phase after passing through the SLM should also have similar phase fluctuations over and above the phase added by SLM.

You add an overall offset. But as you are interested in the changing pattern in the intensity, this is not really of interest.

Will this phase fluctuation not affect the coherence time of resulting beam?

No, what you do here is rather related to the stuff people do in intensity correlation experiments in order to find the size of beads or something. You introduce some scatterer and how quickly the scattering pattern changes determines your coherence time.

Also, is there any difference in coherence times of the spatially incoherent beams produced using ground glass and SLM?

Well, for a ground glass disk the coherence time of course depends on the roughness of the ground glass disk, the spot size on the disk and on how fast you rotate it. Also, the change in the pattern is continuous as the scattering surface just moves around. For the SLM, the changes are discrete as any new image can introduce a completely different scattering pattern completely unrelated to the last one. But besides that it really just depends on how fast the ground glass disk rotates or how quickly you change frames. If I remember correctly, I got coherence times in the low microsecond range for ground glass disks. I tried a SLM once, but the high resolution ones operate at about 60 Hz only. For high speed operation DMD devices should work, too.

 

[nandan]

Thanks a lot @Cthugha for your replies. If I understand correctly, what you are saying is that the intensity correlations between bucket and scanning detectors are calculated between average intensities at these two detectors for each phase realization of SLM. If that is the case and the laser's inherent temporal coherence is not taken into consideration, then I don't see how computational ghost imaging is ghost imaging at all. I mean instead of calculating intensity correlations between scanning and bucket detectors, we could very well apply a suitable phase profile using SLM such that the beam is focused to one point on object plane (the required phase profile can be calculated using the input field profile). Now, the object profile can be deduced by scanning the point at which the light is focused on the object plane and detecting the corresponding bucket detector signal. When the object mask will be transparent at a point, then all the light will be collected by bucket detector and vice versa.

I don't see how this kind of arrangement can be called ghost imaging which relies on the detection of two correlated photons-one in object arm and the other in reference arm. The laser's temporal coherence guarantees that within the coherence time of laser any two photons at the same spatial point in the two arms will be correlated with each other. Whereas in computational ghost imaging it's just a known intensity profile that is used to deduce the image from bucket detector signal.

 

[Cthugha]

If I understand correctly, what you are saying is that the intensity correlations between bucket and scanning detectors are calculated between average intensities at these two detectors for each phase realization of SLM.

Well, what you actually do is getting rid of the scanning detector. You do not have it there, but you calculate what it would detect as you know the light field exactly. But basically you are right. In thermal ghost imaging you do not know the field a priori and you need the scanning detector to get information about it.

If that is the case and the laser's inherent temporal coherence is not taken into consideration, then I don't see how computational ghost imaging is ghost imaging at all. I mean instead of calculating intensity correlations between scanning and bucket detectors, we could very well apply a suitable phase profile using SLM such that the beam is focused to one point on object plane (the required phase profile can be calculated using the input field profile). Now, the object profile can be deduced by scanning the point at which the light is focused on the object plane and detecting the corresponding bucket detector signal. When the object mask will be transparent at a point, then all the light will be collected by bucket detector and vice versa.

Yes, you can do that. I actually did a slighltly changed version of your experiment. The reason why you still use random matrices instead lies in a method called compressive sensing. If you use random matrices, you can reconstruct the image in question using far fewer measurements than there are pixels in the image. You can save about one order of magnitude. The idea is that you get an underdetermined system of equations and find the solution (the images) that minimizes the l1-norm. This is typically the least noisy solution and the image you want.

I don't see how this kind of arrangement can be called ghost imaging which relies on the detection of two correlated photons-one in object arm and the other in reference arm. The laser's temporal coherence guarantees that within the coherence time of laser any two photons at the same spatial point in the two arms will be correlated with each other. Whereas in computational ghost imaging it's just a known intensity profile that is used to deduce the image from bucket detector signal.

Well, these are just two different ways of getting some knowledge about the light field at the bucket detector. The terminology is historically grown and as such sometimes a bit questionable, yes.

 

[nandan]

If in computational ghost imaging, the correlation is calculated between average intensities (for each spatial phase realization of SLM) of bucket and scanning detectors (since the intensity data is acquired over the time for which each SLM phase profile is maintained to be constant), then the intensity correlation function between bucket and detector plane should not be broken down to field correlation functions (using gaussian state moment factoring). The reason I say this is following: The intensities recorded at the two detectors can be given by $I_s=\int_{-T_0/2}^{T_0/2}dt\, \left|E_s(x,y)\right|^2$. In conventional ghost imaging, the intensity is acquired in a time much shorter than the coherence time i.e. $T_0 \ll T_{coherence}$ and therefore, $I_s(x,y,t)=E_s^\dagger(x,y,t) E_s(x,y,t)$. In case of computational ghost imaging such an expression of intensity in terms of field cannot be written. A classic example to highlight this point and differentiate between these two ghost imaging methods would be to use a superposition of two independent sources of same wavelength as the source for ghost imaging experiment. Therefore, $E_s(x,y,t) = E_1(x,y,t) + E_2(x,y,t)$. The intensity recorded in conventional ghost imaging will be $I_s(x,y,t)=\left|E_1(x,y,t)\right|^2+\left|E_2(x,y,t)\right|^2 + 2Re(E_1^\dagger(x,y,t)E_2(x,y,t))$. In case of computational ghost imaging, the third term would be missing because the intensity is recorded over a longer period of time and the interference term will be averaged out to zero. So, while correlating the intensities of bucket and scanning detectors, the conventional ghost imaging will have one additional nonzero term (can be calculated by moment factoring theorem) in the intensity correlation expression which will be missing in computational ghost imaging case and now in fact the correlation terms should be something like $\left<\overline{E_{mo}^\dagger (x,y,t)}\overline{E_{nr}(x,y,t)}\right>$. In the literature on this subject, this point is not dealt with and the two schemes are pretty much always shown to be analogous in every way except the fact that the speckle pattern in computational ghost imaging can be calculated beforehand.

 

[Cthugha]

In case of computational ghost imaging such an expression of intensity in terms of field cannot be written.

Oh, but of course. In fact, it is only the phase of the incoming light beam which gets modulated in order to get the pattern you want. "Good" SLMs are phase-only.

The intensity recorded in conventional ghost imaging will be $I_s(x,y,t)=\left|E_1(x,y,t)\right|^2+\left|E_2(x,y,t)\right|^2 + 2Re(E_1^\dagger(x,y,t)E_2(x,y,t))$. In case of computational ghost imaging, the third term would be missing because the intensity is recorded over a longer period of time and the interference term will be averaged out to zero.

No. Why? That would happen if you averaged over a timescale longer than several frames of the SLM. As long as you take one image per frame you are fine.

 

[nandan]

If the image is taken in time interval which is much smaller than the coherence time of the lasers (here the phase fluctuations of lasers are more important since the phase from SLM is held constant in one frame). If however, the image is acquired over a timescale longer than the coherence time of laser (even for one single frame of SLM), then the phase fluctuations of two fields $E_1$ and $E_2$ will be averaged and the term $E_{10}E_{20}exp(i(\phi_1(t)+\phi_2(t)))$ will average out to zero. Here $\phi_1$ and $\phi_2$ are the random phase noise of the two laser sources. This noise will be present over and above the deterministic spatial phase profile added by the SLM.

 

[Cthugha]

This is not the case for the same reason the interference pattern seen in a double slit stays constant for timescales longer than the coherence time of the laser light. You get a flat phase front at the double slit position. Phase fluctuations just introduce an offset. The interference pattern is determined by the phase difference created by the two different path lengths. If that phase difference is pi, you will be in a minimum, no matter whether you add zero phase pi/2 phase, pi phase or any other absolute phase offset.

A SLM does exactly the same. It introduces a phase offset to every pixel, so that you get a target pattern of constructive and destructive superpositions in the target plane. As long as the incoming beam is spatially coherent and has a flat wave front, the absolute phase offset does not matter. If the phases vary differently at different spatial positions of the beam, however, you are in trouble.

 

[nandan]

You are right when you say that an absolute additional phase will not make any difference to say a double slit interference pattern. But, if this additional phase is a time varying quantity, then it can make a difference. Since the fields from the two slits generally travel different distance to reach a point on the detection plane, the additional overall phase noise (which is time varying) will be different for the two interfering fields at that point. In such a case, if the temporal coherence is too low, the interference effect may wash out. In usual cases, the distance scales involved in double slit experiment are too small for phase fluctuations to make any difference. The same may very well be the case with SLM induced speckle pattern. But the intensity at each point in speckle should still fluctuate due to intensity/phase fluctuations of laser source (if the laser intensity fluctuates by itself, there is no reason why the speckle pattern produced by this laser will have a completely fluctuation free). This fluctuation is impossible to pre-calculate for a given SLM frame. However, the average intensity can indeed be calculated.
But when two independent fields are mixed, at each point on the detector plane, the phase between the field due to the two source components will fluctuate with time and IF THIS INTENSITY PATTERN IS AVERAGED OVER A PERIOD LONGER THAN THE COHERENCE TIME OF THE TWO LASERS (EVEN IF IT THE SLM PHASE IS CONSTANT), the averaged out pattern will not have the third term due to interference between two fields. If instead, the image is taken using a detector which has a very small response time (smaller than laser coherence time), then indeed the interference between the two fields will be recorded even with the SLM.

What I am basically saying is that since the computational ghost imaging requires the laser intensity fluctuation averaged detection in bucket and scanning detector, then the interference term between two independent sources will also be washed out. Please correct me if I am wrong in making this assumption.

 

[Cthugha]

But when two independent fields are mixed, at each point on the detector plane, the phase between the field due to the two source components will fluctuate with time and IF THIS INTENSITY PATTERN IS AVERAGED OVER A PERIOD LONGER THAN THE COHERENCE TIME OF THE TWO LASERS (EVEN IF IT THE SLM PHASE IS CONSTANT), the averaged out pattern will not have the third term due to interference between two fields. If instead, the image is taken using a detector which has a very small response time (smaller than laser coherence time), then indeed the interference between the two fields will be recorded even with the SLM.

What I am basically saying is that since the computational ghost imaging requires the laser intensity fluctuation averaged detection in bucket and scanning detector, then the interference term between two independent sources will also be washed out. Please correct me if I am wrong in making this assumption.

Well, yes, but I am a bit puzzled as there is only one light source in computational ghost imaging and you encounter the same problem in pseudothermal ghost imaging. The coherence time considered there is mostly given by the surface roughness of the ground glass disk. People do not consider the coherence of the initial laser beam there as well.

 

[nandan]

I understand your point. But what I am trying to convey is that the physics involved in conventional and computational ghost imaging is different (at least in my opinion). And that this difference can lead to different results as I have tried to explain using two component source. And if the input source field is composed of N such independent coherent laser beams, the correlation function obtained for two schemes will be vastly different (though the profile of the correlation function will look the same in both cases). And if the two schemes give different results, I feel it shows that not just the methods, but also the underlying physics has to be completely different for the two schemes.

 

[Cthugha]

I am not sure I am convinced. Sure, assuming several independent light sources, you will get different results depending on whether your time resolution is better or worse than the mutual coherence time of the light sources you use, but this applies to both schemes. It is not guaranteed that you have sufficient temporal resolution in pseudothermal imaging and it is also not necessarily true that your temporal resolution in CGI is too bad to see it. This is a question of the temporal resolution of the detector you use.

 

[nandan]

That is precisely my point. The temporal resolution of the detector is important. It makes a difference to the final result as well. In conventional ghost imaging, the temporal resolution of detector can be chosen to be faster or slower than the coherence time of source laser(s). However, in computational ghost imaging the detector has to be slow (to produce the same speckle intensity pattern at object plane as is pre-calculated for the SLM phase pattern). This restriction therefore imposes a strict condition and therefore the computational ghost image can only be thought to be equivalent to conventional ghost imaging with slow detectors.

 

[Cthugha]

Why? You can use a fast detector in CGI. You will just get the same image several times as you take several images with the SLM showing the same phase pattern (or the same basic result and some additional effects due to the coherence time of the source lasers).

That might not be the best idea in terms of signal to noise, but one could do that.

 

[nandan]

But in that case the intensity profile recorded in each image will be different due to the time dependent phase differences of the N different components of the source lasers. Such a fluctuating intensity pattern (due to phase fluctuations of lasers) cannot be calculated from just the phase pattern of SLM. Therefore, the correlation measurement will have to be performed only after averaging over all the images acquired for each SLM frame.

 

[Cthugha]

Oh, so you want to consider randomly varying phases, which are also completely unknown? Yes, that will keep you from calculating things.

However, you could still use the SLM and divide the modulated beam into two beams and send one to the bucket detector and the other to the scanning detector, just like in a normal pseudo-thermal imaging setup. That would still work.

 

[nandan]

Yes. That is exactly what I am saying. For computational part in computational ghost imaging to work, the randomly varying phases of the source lasers have to be averaged over. In this regard, it is completely different from conventional ghost imaging. And I am surprised that not enough has been said in the literature on this topic. In fact, in many papers it is not mentioned at all whether these random varying phases have been averaged or not (sometimes it is very important to make that distinction). Till today, I was so confused about the difference between conventional and computational ghost imaging systems. Everywhere I read the same things that the computational ghost imaging is same as conventional GI except ground glass is replaced by SLM to calculate the field at the object plane. I am really thankful to you @Cthugha for helping me understand the subtle but important difference between these two schemes.

 

[Cthugha]

At least in the publications I know, people use just one spatially coherent laser with a flat wave front, so the problem does not come up. I do not think these publications even considered the "bad light source" problem which is why they do not mention it. For a single spatially coherent beam, things work well.

Do you have a specific application in mind that requires several light beams? Maybe multi-color imaging or something like that?

 

[nandan]

In fact there are some papers using multi-wavelength sources e.g. PHYSICAL REVIEW A 88, 053842 talks about one such setup. I believe that in this paper authors should have carefully explained what kind of detectors are being used e.g. eqn. 2 suggests that slow detectors are used even though this not being a computational ghost imaging system (SLM is used for a different purpose), even fast detectors could have been used. In any case, the specific proposed experimental conditions should have been mentioned. It becomes very confusing for beginners like me.

And as I said in one of my earlier posts, even though the average intensity (averaged over phase fluctuations of lasers) is used for calculating correlations, eqn. 5 suggests that the correlations are calculated between fields in the two arms at the same instant of time.