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Abstract
Interferometry is one of the central organizing principles of optics. Key to interferometry is the concept of optical delay, which facilitates spectral analysis in terms of time-harmonics. In contrast, when analyzing a beam in a Hilbert space spanned by spatial modes – a critical task for spatial-mode multiplexing and quantum communication – basis-specific principles are invoked that are altogether distinct from that of ‘delay’. Here, we extend the traditional concept of temporal delay to the spatial domain, thereby enabling the analysis of a beam in an arbitrary spatial-mode basis – exemplified using Hermite-Gaussian and radial Laguerre-Gaussian modes. Such generalized delays correspond to optical implementations of fractional transforms; for example, the fractional Hankel transform is the generalized delay associated with the space of Laguerre-Gaussian modes, and an interferometer incorporating such a ‘delay’ obtains modal weights in the associated Hilbert space. By implementing an inherently stable, reconfigurable spatial-light-modulator-based polarization-interferometer, we have constructed a ‘Hilbert-space analyzer’ capable of projecting optical beams onto any modal basis.
Highlights
Interferometry is the cornerstone of fundamental investigations and precise measurements in optics[1]
The input beams are prepared by a single several large components (SLMs) (SLM0) that imprints a phase-only pattern on a Gaussian-mode laser beam, which is imaged to SLM1 that constitutes the input plane to the generalized interferometer
We have demonstrated that optical interferometry can be generalized to apply for any modal basis by replacing the traditional temporal delay with a generalized delay (GD): an optical transformation that ‘delays’ the beam in a Hilbert space spanned by the modal basis of interest
Summary
Interferometry is the cornerstone of fundamental investigations and precise measurements in optics[1]. The nature of light – both classical[2,3] and quantum4–6 – was unraveled largely through interferometric experiments, and the exquisite precision inherent in optical interferometry has been instrumental in metrology[7], bio-imaging[8], devising ultra-sensitive systems for the detection of gravitational waves[9], and enabling novel lithographic schemes[10] These examples share a common feature: interference results from combining beams with relative phases engendered by optical delays. We present a unifying principle for modal analysis by addressing the following question: can the traditional optical delay – one of the most fundamental concepts in optics – be extended beyond its implementation in the time domain to apply to Hilbert spaces associated with discrete spatial-mode bases? Sweeping the order of a fractional transform corresponds to varying a temporal delay in traditional interferometry – each in its own Hilbert space
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