What is the maximum differential group delay achievable by a space-time wave packet in free space?

The group velocity of 'space-time' wave packets - propagation-invariant pulsed beams endowed with tight spatio-temporal spectral correlations - can take on arbitrary values in free space. Here we investigate theoretically and experimentally the maximum achievable group delay that realistic finite-energy space-time wave packets can achieve with respect to a reference pulse traveling at the speed of light. We find that this delay is determined solely by the spectral uncertainty in the association between the spatial frequencies and wavelengths underlying the wave packet spatio-temporal spectrum - and not by the beam size, bandwidth, or pulse width. We show experimentally that the propagation of space-time wave packets is delimited by a spectral-uncertainty-induced 'pilot envelope' that travels at a group velocity equal to the speed of light in vacuum. Temporal walk-off between the space-time wave packet and the pilot envelope limits the maximum achievable differential group delay to the width of the pilot envelope. Within this pilot envelope the space-time wave packet can locally travel at an arbitrary group velocity and yet not violate relativistic causality because the leading or trailing edge of superluminal and subluminal space-time wave packets, respectively, are suppressed once they reach the envelope edge. Using pulses of width ∼ 4 ps and a spectral uncertainty of ∼ 20 pm, we measure maximum differential group delays of approximately ±150 ps, which exceed previously reported measurements by at least three orders of magnitude.

Interferometry-based modal analysis with finite aperture effects.

We analyze the effects of aperture finiteness on interferograms recorded to unveil the modal content of optical beams in arbitrary bases using generalized interferometry. We develop a scheme for modal reconstruction from interferometric measurements that accounts for the ensuing clipping effects. Clipping-cognizant reconstruction is shown to yield significant performance gains over traditional schemes that overlook such effects that do arise in practice.

Compressive optical interferometry under structural constraints.

Compressive sensing (CS) combines data acquisition with compression coding to reduce the number of measurements required to reconstruct a sparse signal. In optics, this usually takes the form of projecting the field onto sequences of random spatial patterns that are selected from an appropriate random ensemble. We show here that CS can be exploited in 'native' optics hardware without introducing added components. Specifically, we show that random sub-Nyquist sampling of an interferogram suffices to reconstruct the field modal structure despite the structural constraints of the measurement system set by its limited degrees of freedom. The distribution of the reduced (and structurally constrained) sensing matrices corresponding to random measurements is provably incoherent and isotropic, which helps us carry out CS successfully. We implement compressive interferometry using a generalized Mach-Zehnder interferometer in which the traditional temporal delay is replaced with a linear transformation corresponding to a fractional transform. By randomly sampling the order of the fractional transform, we efficiently reconstruct the modal content of the input beam in the Hermite-Gaussian and Laguerre-Gaussian bases.

Basis-neutral Hilbert-space analyzers.

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.

Efficient modal analysis using compressive optical interferometry.

Interferometry is routinely used for spectral or modal analysis of optical signals. By posing interferometric modal analysis as a sparse recovery problem, we show that compressive sampling helps exploit the sparsity of typical optical signals in modal space and reduces the number of required measurements. Instead of collecting evenly spaced interferometric samples at the Nyquist rate followed by a Fourier transform as is common practice, we show that random sampling at sub-Nyquist rates followed by a sparse reconstruction algorithm suffices. We demonstrate our approach, which we call compressive interferometry, numerically in the context of modal analysis of spatial beams using a generalized interferometric configuration. Compressive interferometry applies to widely used optical modal sets and is robust with respect to noise, thus holding promise to enhance real-time processing in optical imaging and communications.