Optical phase spaces represent fields of any spatial coherence and are typically measured through phase-retrieval methods involving a computational inversion, optical interference, or a resolution-limiting lenslet array. Recently, a weak-values technique demonstrated that a beam's Dirac phase space is proportional to the measurable complex weak value, regardless of coherence. These direct measurements require raster scanning through all position-polarization couplings, limiting their dimensionality to less than 100 000 [C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.070405]. We circumvent these limitations using compressive sensing, a numerical protocol that allows us to undersample, yet efficiently measure, high-dimensional phase spaces. We also propose an improved technique that allows us to directly measure phase spaces with high spatial resolution with scalable frequency resolution. With this method, we are able to easily and rapidly measure a 1.07-billion-dimensional phase space. The distributions are numerically propagated to an object in the beam path, with excellent agreement for coherent and partially coherent sources. This protocol has broad implications in quantum research, signal processing, and imaging, including the recovery of Fourier amplitudes in any dimension with linear algorithmic solutions and ultra-high-dimensional phase-space imaging.
Bibliographical noteFunding Information:
S.H.K. thanks G. A. Howland for insightful discussions. S.H.K, D.J.L, and J.C.H acknowledge support from the Air Force Office of Scientific Research, Grant No. FA9550-16-1-0359, and from Northrop Grumman, Grant No. 058264-002. J.S. acknowledges support from the National Research Council Research Associate Programs and funding from the OSD ARAP QSEP program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of AFRL.
© 2018 American Physical Society.