How to Correct for Bias in Off-Axis Holography for Transmission Matrix Measurements

Sébastien M Popoff¹ , Antoine Loquet¹

I presented in a previous tutorial how to reconstruct a complex field using a camera and a plane wave reference tilted with respect to the optical axis, known as off-axis holography. This works perfectly with an ideal plane wave as a reference. Of course, real life is not perfect, and the reference usually presents imperfections. While low spatial fluctuations can be compensated for afterward, high spatial frequency noise has the effect of adding a small bias to the estimation of the field. Such bias is typically small, but in transmission matrix measurements—since it is static and added to all measurements—it can affect the singular value distribution and perturb or hide transmission channels that would otherwise be visible.

ZoomFFT for speeding up off-axis computation (and more)

Sébastien M Popoff¹  and Rodrigo Gutiérrez-Cuevas¹

When performing the computation for some tasks such as off-axis holography, we often have to compute the entire FFT of a signal or an image while being only interested in a very small part of the spectrum. The rest of the information is just discarded. While the FFT algorithm and its implementation in standard computing libraries are very efficient, we can still take advantage of a slower approach which only computes what we need. The ZoomFFT algorithm does just that! And it's already available in standard packages such as SciPy for Python or in MATLAB. Using off-axis holography as an example, I will show how to save time – or not – using ZoomFFT.

Semidefinite Programming for Intensity Only Estimation of the Transmission Matrix

The possibility of measuring the transmission matrix using intensity only measurements is a much sought-after feature as it allows us not to rely on interferometry. Interferometry usually requires laboratory-grade stability difficult to obtain for real-world applications. Typically, we want to be able to retrieve the transmission matrix from a set of pairs composed of input masks and output intensity patterns. However, this problem, which corresponds to a phase-retrieval problem, is not convex, hence difficult to solve using standard techniques. The idea proposed in [I. Waldspurger et al., Math. Program (2015)] is to relax some constraints to approximate the problem to a convex one that can be solved using the semidefinite programming approach. I briefly detail the approach and provide an example of the procedure to reconstruct the transmission matrix using Python. A Jupyter notebook can be found on my Github account: semidefiniteTM_example.ipynb.

Off-axis holography

Sébastien M Popoff¹

Off-axis holography is a popular technique to reconstruct a hologram. It allows retrieving the amplitude and the phase of a field pattern by measuring only one image with a digital camera. It relies on an intereferometric setup with a non-zero angle between the reference beam and the signal beam and requires to numerically filter the spatial frequencies. I provida Matlab and Python example codes.