Hi forum members,

Could you please shed some light for me on the measurement of TM, for the case of a multimode fibre?

1. Is it absolutely critical to use a orthorgonal basis in the input? As for the case of a MMF, can I use random patterns as per the paper below. My thinking is along this
2. Use some set of random patterns and measure a TM.
3. Apply SVD on the TM, find SVs (say 98% of the total power)
4. Use a bigger set of random patterns, measure again the TM
5. Apply SVD again, check the SVs... when the number of SVs is stable it indicates that all of the modes has been activated (assuming large NA input)

Can it be done that way? Here the paper they used the random pattern but they just say that 900 patterns is enough.
https://arxiv.org/ftp/arxiv/papers/1806/1806.09315.pdf

1. The TM for a multimode fibre should ideally has the rank of N (number of macro pixel on the SLM)? It is because I think the basic vectors of both left and right singular matrices are the same fibre modes, so that the TM would be an square vector of rank N.

My question is: After obtaining the TM, should I apply SVD on such TM and try to reduce it to a leaner version (less noisy) of the TM before testing the stuffs such as focusing or imaging?

1. I was told that if I do the SVD directly on the intensities obtained from different input (basically not applying the 4-phase or off-axis technique to extract the complex field pattern but just use the intensity vectors as column of the "intensity TM"), the number of SVs will be roughly twice the number of modes. I could not find any paper which actually did this but just wonder if it is something obvious that someone here can give me some light on? For the case of TM matrix measured on the complex field pattern then the number of SVs is the number of modes as shown in this paper
https://www.osapublishing.org/ol/fulltext.cfm?uri=ol-37-21-4558&id=244834

Thank you very much for your help.

Hi,

Let's take one question at a time.

First:

``Is it absolutely critical to use a orthogonal basis in the input?``

No, it's not!

If you do it, the advantage is that you directly measure the TM. Meaning that if X is the matrix representing the stack of input vectors, and X is unitary (i.e. the input vectors form an orthogonal basis of same norm), then the stack Y of output directly gives an estimation of the TM in this basis.

If your inputs are random, then X is a random matrix. The first consequence is that because they are not orthogonal, you need more inputs. So if you have N input pixels, you need enough input vectors so that X in of rank N, the more the better as it mitigates the effect of noise (typically, 5 x N is a good choice).
More importantly, the stack of Y does not directly gives you an estimation of the TM! You need to convert the data into an orthogonal basis, typically the basis of the pixels. To do so, you need to use X⁺ the pseudo-inverse of X:

TM = Y x X⁺

We have detailed this step in our last paper:

Learning and avoiding disorder in multimode fibers

It is in the Supplementary section S1.4, we will update the version soon and it will be in Appendix B of the final paper.

The TM for a multimode fibre should ideally has the rank of N (number of macro pixel on the SLM)? It is because I think the basic vectors of both left and right singular matrices are the same fibre modes, so that the TM would be an square vector of rank N.

The rank of the matrix cannot be higher than the number of propagating modes. Physically, light can only propagate through those modes. If the rank of the TM is higher, that means that there is more orthogonal channels to transmit information than the number of modes, which is not possible.

However, for the rank of the matrix to be equal to the number of modes, which is the case if you do things correctly, the number of input pixels should be high enough, much higher than the number of modes.

This is because if you do not have a good spatial resolution, some modes, typically the highest order modes, will not be sampled correctly and their contributions would be lowered or killed.

My question is: After obtaining the TM, should I apply SVD on such TM and try to reduce it to a leaner version

It is not a bad idea, and something that I sometimes do.
You can do the SVD, keep only the singular values/vectors corresponding to the modes, and come back to the pixel basis. To know where to cut, you look at the singular values and you should see a drop.

However, in our experiments, I noticed that if we use enough random input vectors, this step was not necessary as we already efficiently mitigated the effect of noise. Of course, that may not be true for another experiment so it is worth trying anyway.

quI was told that if I do the SVD directly on the intensities obtained from different input (basically not applying the 4-phase or off-axis technique to extract the complex field pattern but just use the intensity vectors as column of the "intensity TM"), the number of SVs will be roughly twice the number of modes.

Never heard of that. It may be true though, but I do not know any paper discussing that point.

Hi Sebastien,

About the SVD on intensity, it is also my problem that I cannot find any formal discussion of that but I was told by a professor from Yale that should be the case. My (guessed) understanding of its physical meaning is that the left SV vectors mapping to the intensity can be considered as a summation in the Fourier domain with each pair of modes interfering is represented by a FFT frequency. Even though in that case the total number of FFT components would be larger than 2N (with N the number of modes), for large enough N many of the constituent interferences will have overlap FFT components and average to 2N.

But I cannot prove it theoretically yet.

Hi,

I was told by a professor from Yale

Is it Hui Cao? Doug Stone?

Not sure I understand your point about interferences of pairs of FFT components (and by the way modes do not corresponds to FFT components).
But you can quite easily try numerically to see if it works.
We have a simple simulation code in Python in which you can simulate mode matrices and transmission matrices:
github.com/wavefrontshaping/pyMMF

sebastien.popoff wrote:

Hi,

I was told by a professor from Yale

Is it Hui Cao? Doug Stone?

Not sure I understand your point about interferences of pairs of FFT components (and by the way modes do not corresponds to FFT components).
But you can quite easily try numerically to see if it works.
We have a simple simulation code in Python in which you can simulate mode matrices and transmission matrices:
github.com/wavefrontshaping/pyMMF

Yes Hui Cao.

Thanks for the Python code I will check it out. We have a Matlab code that we simulate the effective indices of the fiber, and assuming random coupling to each mode we could also estimate the SVD but we haven't run it for the case of many modes because it is too slow. Our measurement kind of showed the SVs in the vicinity of 2N (with N is the mode number) as well. So I think it is likely the case.

Yeah sorry I wasn't clear. I meant the FFT component corresponds to a pair of modes that interferes with each other. The intensity is the square of summation of all the modes and therefore it could be decomposed into FFT components with each FFT peak corresponding to the interference of 2 modes. I think that when the number of modes become large (~100), there will be many overlapping FFT frequencies because it is basically the difference between the effective indices of 2 modes and the ultimate SVs will average to 2N.

For the case of optical field I can understand that a TM linearly maps the input fields to output fields. Doing an SVD on the TM of a MMF can reveal the steps in which the input field (SLM-pixel vector) is projected onto the fiber mode space, scaling along each mode axis by its SV, and sum up the fiber mode vectors back to the CCD-pixel vector. But intensity is nonlinear which depends on the input fields in a quadratic way so I am struggling with the meaning of an "Intensity TM" and the SVD on such matrix.