# Universal linear optics

Quite generally, any matrix can be written as a product of three matrices – a unitary one, a diagonal one, and another unitary one. This process is a standard mathematical one (singular value decomposition – SVD), but it can be used as a constructive way to make an optical system that implements an arbitrary matrix. David Miller proposed this general, universal, architecture (the “singular-value decomposition” or SVD architecture), which allows any linear optical operation at a given frequency or wavelength  . This architecture can also be self-configured and self-stabilized. The “input” unitary mesh (self-aligning input coupler) can be self-configured by shining in the vectors that are the rows of the “input” unitary matrix, and the corresponding “output” unitary mesh can be trained by shining, backwards into the output, the vectors that are the rows of the “output” unitary matrix.

A mesh of interferometers to implement any 4×4 matrix between the input and output waveguides, synthesizing the matrix using its “singular value decomposition” or “SVD”

This a practical way to implement arbitrary matrix multiplications, multiplying a vector of input input amplitudes in input waveguides on the left to give the corresponding set of output amplitudes in the waveguides on the right. This output vector of amplitudes is the result of multiplying input vector of amplitudes by the arbitrary matrix. Since any linear operation can be written as a matrix, we can implement arbitrary linear operations in optics.

This work also constitutes the first proof that any linear optical component at a given wavelength can actually be constructed in principle. Hence we can propose thought experiments that can build on this concept to construct proofs of physical laws.

This has led to new physical laws, including (i) 4 new “Kirchhoff” radiation laws  , which now include all the effects of diffraction and can cover even non-reciprocal materials and objects, and (ii) a new version of Einstein’s “A&B coefficient” argument  that links absorption, spontaneous emission and stimulated emission. This new version, which works mode by mode, only needs one coefficient for each mode.

The process of singular value decomposition also automatically leads to a description of the system in terms of a mapping from orthogonal input vectors to a corresponding set of orthogonal output vectors, which can also be described as the “mode-converter basis sets” for the system. As a result, this idea of singular value decomposition of linear optical system can be “flipped round”. We can describe the system in terms of the pairs of input and output modes. This concept is behind both of the above new physical laws, and it and its many consequences are discussed at length in . See the larger discussion of waves and modes.

 D. A. B. Miller, “Self-configuring universal linear optical component,” Photon. Res. 1, 1-15 (2013).
http://www.opticsinfobase.org/prj/abstract.cfm?URI=prj-1-1-1  http://dx.doi.org/10.1364/PRJ.1.000001

 Patent #10,534,189, “Universal Linear Components,” David A. B. Miller (Jan. 14, 2020)

 D. A. B. Miller, Linxiao Zhu, and Shanhui Fan, “Universal modal radiation laws for all thermal emitters,” PNAS 114, no. 17, 4336-4341 (2017) https://doi.org/10.1073/pnas.1701606114

 D. A. B. Miller, “Waves, modes, communications, and optics: a tutorial,” Adv. Opt. Photon. 11, 679-825 (2019) https://doi.org/10.1364/AOP.11.000679