\(

\def\ket#1{{\left|{#1}\right\rangle}}

\def\bra#1{{\left\langle{#1}\right|}}

\def\braket#1#2{{\left\langle{#1}|{#2}\right\rangle}}

\)

## [tutorial] 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 a 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, that 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 constraint 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 [tutorial]

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.

Most exciting phenomenons that occurs in complex media arises from interference effects. Controlling the phase of an incident field with a spatial light modulator is what made the field of wavefront shaping possible. Nevertheless, the measurement of the phase is a crucial step for many applications. In particular, recording both the amplitude and the phase for a set of input wavefront is necessary to record the transmission matrix of a linear medium. The knowledge of the transmission matrix of a scattering medium allows for example to use it as a lens [1], a controllable phase plate [2,3] or polarizer [4,5].

In such experiments, the phase of the output optical field for different input illuminations has to be recorded with the same phase reference. For this reason, one uses interferometric methods to measure the complex field; Phase Shifting Digital Holography (tutorial to come) or Off-Axis Holography (tutorial to come). In both cases, the unknown optical field interfere with a reference wavefront. The intensity of the intereference is measured using a CCD to reconstruct the phase image. Phase Shifting Digital Holography requires 4 different measurements to obtain one phase image, leading to longer acquisition times and making the method more sensitive to interferometric instabilities. Off-Axis Holography allows to measure the complex field in one shot but at the cost of a loss of resolution.

[1] S.M. Popoff et al., *Phys. Rev. Lett.*, 104, 100601, (2010)

[2] Y. Guan et al., *Opt. Lett.*, 37, (2012)

[3] J.H. Park et al., Opt. Express, 20, (2012)

[4] J.H. Park et al., Opt. Lett., 37, (2012)

[5] E. Small et al., Opt. Lett., 37, (2012)