Figure from [1]

I really like this paper: http://www-ee.stanford.edu/~dabm/430.pdf

Imagine you have an incoming laser beam. It will have some kind of spatial profile, for example it might be a gaussian beam that's brighter in a hump in the middle and falls off towards the edges. Or it might have two "humps". Now imagine you want to perform an arbitrary linear transformation, M, on this profile. Like converting a beam with one hump to one with two, one with two humps to one with one, and leaving anything orthogonal to those unchanged. Can you build an optical component to do this?

Any complex matrix M has a singular value decomposition as U*DV where U and V are unitary and D is diagonal. This elegant paper shows how to physically realise this decomposition as a pair of "matrices" of mirrors and phase shifters with a spatial light modulator acting as the diagonal matrix. (Mirrors and phase shifters are unitary.) Even better, it shows how if you have dynamically adjustable mirrors and phase shifters you can use a method akin to a matrix factorization algorithm to configure everything by using a "training set" of beams without having to calculate a thing yourself.

It hasn't been built yet, but everything is buildable in principle. There's lots of related work here: http://www-ee.stanford.edu/~dabm/Selfalign.html

*. Thank you Dan for the pointer !*

__.It's a whole new world__
Figure from [2]

Figure from [3]

References:

We show how to design an optical device that can perform any linear function or coupling between inputs and outputs. This design method is progressive, requiring no global optimization. We also show how the device can configure itself progressively, avoiding design calculations and allowing the device to stabilize itself against drifts in component properties and to continually adjust itself to changing conditions. This self-configuration operates by training with the desired pairs of orthogonal input and output functions, using sets of detectors and local feedback loops to set individual optical elements within the device, with no global feedback or multiparameter optimization required. Simple mappings, such as spatial mode conversions and polarization control, can be implemented using standard planar integrated optics. In the spirit of a universal machine, we show that other linear operations, including frequency and time mappings, as well as nonreciprocal operation, are possible in principle, even if very challenging in practice, thus proving there is at least one constructive design for any conceivable linear optical component; such a universal device can also be self-configuring. This approach is general for linear waves, and could be applied to microwaves, acoustics, and quantum mechanical superpositions.

We show how multiple optimal orthogonal channels for communicating or interconnecting with waves between two objects can be aligned and optimized automatically using controllable beamsplitters, detectors and simple local feedback loops, without moving parts, without device calibration, without fundamental beam splitting loss, and without calculations. Optical applications include multiple simultaneous orthogonal spatial communication channels in free space or multimode optical fibers, automatically focused power delivery with waves, multiple channel communication through scattering or lossy media, and real-time-optimized focused channels to and from multiple moving objects. The approach physically implements automatic singular value decomposition of the wave coupling between the objects, and is equivalent in its effect to the beam forming in a laser resonator with phase-conjugate mirrors with the additional benefits of allowing multiple orthogonal channels to be formed simultaneously and avoiding the need for any nonlinear optical materials.

[3] Self-configuring universal linear optical component, David Miller

We show how to design an optical device that can perform any linear function or coupling between inputs and outputs. This design method is progressive, requiring no global optimization. We also show how the device can configure itself progressively, avoiding design calculations and allowing the device to stabilize itself against drifts in component properties and to continually adjust itself to changing conditions. This self-configuration operates by training with the desired pairs of orthogonal input and output functions, using sets of detectors and local feedback loops to set individual optical elements within the device, with no global feedback or multiparameter optimization required. Simple mappings, such as spatial mode conversions and polarization control, can be implemented using standard planar integrated optics. In the spirit of a universal machine, we show that other linear operations, including frequency and time mappings, as well as non-reciprocal operation, are possible in principle, even if very challenging in practice, thus proving there is at least one constructive design for any conceivable linear optical component; such a universal device can also be self-configuring. This approach is general for linear waves, and could be applied to microwaves, acoustics and quantum mechanical superpositions.

but also:

- Optics in 2013 Special Issue of Optics and Photonics News
- "All linear optical devices are mode converters," Opt. Express 20, 23985-23993 (2012)
- "How complicated must an optical component be?" J. Opt. Soc. Am. A 30, 238-251 (2013)
- "Self-aligning universal beam coupler," Opt. Express 21, 6360-6370 (2013)
- "Self-configuring universal linear optical component," Photon. Res. 1, 1-15 (2013)
- "Establishing optimal wave communication channels automatically," J. Lightwave Technol. 31, 3987“ 3994 (2013)
- "Reconfigurable add-drop multiplexer for spatial modes," Opt. Express 21, 20220-20229 (2013)

Here is a video of David Miller explaining the concepts:

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