Phase Space Tomography Research Articles

Characterization of the coherence properties of different optical sources

Non-interferometric phase-space tomography is used to characterize the spatial coherence properties of scalar quasi-monochromatic partially coherent optical fields. Three sources were investigated and characteristic projections through the correlation functions determined. It was seen that a single-mode fibre-coupled source exhibited a much longer coherence length than a multi-mode fibre-coupled source. Both sources exhibited a high degree of symmetry. A collimated light emitted diode, on the other hand, was shown to have a short spatial coherence length and clear asymmetries were evident.

Transverse beam matching under the influence of space charge

The matching of a particle beam to specific transverse parameters, imposed by various operational, functional and diagnostic reasons, is an essential procedure in most types of accelerators. While well-established methods can easily match beams of negligible self fields, their accuracy is significantly reduced in the presence of strong space-charge forces. In a pursuit of time-efficient matching solutions for space-charge influenced beams, two approaches are demonstrated in this paper: a faster and more approximative one for symmetric beam transport along periodic and dense lattices, and a more analytic one for broader beam and lattice conditions. Beam dynamics simulations and experimental measurements are used to validate the performance of the suggested methods for the matching requirements of the transverse phase–space tomography at the Photo Injector Test Facility at DESY, Zeuthen site (PITZ).

Experimental Phase-Space Tomography of Semiconductor Laser Dynamics.

We perform phase-space tomography of semiconductor laser dynamics by simultaneous experimental determination of optical intensity, frequency, and population inversion with high temporal resolution. We apply this technique to a laser with delayed feedback, serving as prominent example for high-dimensional chaotic dynamics and as model system for fundamental investigations of complex systems. Our approach allows us to explore so far unidentified trajectories in phase space and identify the underlying physical mechanism.

Exact and Stable Covariance Estimation From Quadratic Sampling via Convex Programming

Statistical inference and information processing of high-dimensional data often require an efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the acquisition devices, it is desirable to extract the covariance structure from a single pass over the data and a small number of stored measurements. In this paper, we explore a quadratic (or rank-one) measurement model which imposes minimal memory requirements and low computational complexity during the sampling process, and is shown to be optimal in preserving various low-dimensional covariance structures. Specifically, four popular structural assumptions of covariance matrices, namely, low rank, Toeplitz low rank, sparsity, jointly rank-one and sparse structure, are investigated, while recovery is achieved via convex relaxation paradigms for the respective structure. The proposed quadratic sampling framework has a variety of potential applications, including streaming data processing, high-frequency wireless communication, phase space tomography and phase retrieval in optics, and noncoherent subspace detection. Our method admits universally accurate covariance estimation in the absence of noise, as soon as the number of measurements exceeds the information theoretic limits. We also demonstrate the robustness of this approach against noise and imperfect structural assumptions. Our analysis is established upon a novel notion called the mixed-norm restricted isometry property (RIP-ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), as well as the conventional RIP-ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> for near-isotropic and bounded measurements. In addition, our results improve upon the best-known phase retrieval (including both dense and sparse signals) guarantees using PhaseLift with a significantly simpler approach.

Time-resolved phase-space tomography of an optomechanical cavity

We experimentally study the phase space distribution (PSD) of a mechanical resonator that is simultaneously coupled to two electromagnetic cavities. The first one, operating in the microwave band, is employed for inducing either cooling or self-excited oscillation, whereas the second one, operating in the optical band, is used for displacement detection. A tomography technique is employed for extracting the PSD from the signal reflected by the optical cavity. Measurements of PSD are performed in steady state near the threshold of self-excited oscillation while sweeping the microwave cavity detuning. In addition, we monitor the time evolution of the transitions from an optomechanically cooled state to a state of self excited oscillation. This transition is induced by abruptly switching the microwave driving frequency from the red-detuned region to the blue-detuned one. The experimental results are compared with theoretical predictions that are obtained by solving the Fokker-Planck equation. The feasibility of generating quantum superposition states in the system under study is briefly discussed.

Publisher's Note: Phase space tomography of cold-atom dynamics in a weakly corrugated potential [Phys. Rev. A90, 033620 (2014)

Publisher's Note: Phase space tomography of cold-atom dynamics in a weakly corrugated potential [Phys. Rev. A<b>90</b>, 033620 (2014)

Phase space tomography of cold-atom dynamics in a weakly corrugated potential

We demonstrate tomographic reconstruction of the phase space distribution of atoms oscillating in a harmonic trap with weak potential corrugation caused by nanoscale imperfections in an atom chip. We find that deformations in these distributions are highly sensitive to anharmonic components of the potential. They are explained in terms of angular velocity dispersion of isoenergetic phase space trajectories. We show that the method is applicable for probing classical and quantum dynamics of cold atoms, and we note its importance for future technological applications.

Beam tomography research at Daresbury Laboratory

Beam tomography research at Daresbury Laboratory has focussed on the development of normalised phase space techniques—starting with the idea of sampling tomographic projections at equal phase advances. This idea has influenced the design and operation of the tomography sections at the Photo Injector Test Facility at Zeuthen (PITZ) and at the Accelerator and Lasers in Combined Experiments (ALICE) at Daresbury. We have studied the feasibility of using normalised phase space to measure the effect of space charge. Quadrupole scan measurements are carried out at two different parts of a beamline. Reconstructions at the same location give results that are clearly rotated with respect to each other in normalised phase space. We are able to show that a significant part of this rotation can be attributed to the effect of space charge. We show how the normalised phase space technique can be used to increase the reliability of the Maximum Entropy Technique (MENT). While MENT is known for its ability to work with just a few projections, the accuracy of its reconstructions has seldom been questioned. We show that for typical phase space distributions, MENT could produce results that look quite different from the original. We demonstrate that a normalised phase space technique could give results that are closer to the actual distribution. We also present simpler ways of deriving the phase space tomography formalism and the Maximum Entropy Technique.

Phase-space tomography for characterization of rotationally symmetric beams

The experimental measurement of light field correlations is a difficult problem because, even in the monochromatic scalar case, the spatial coherence state of light is described by four-dimensional functions. Additional information about the field symmetry or coherence state allows reduction of the complexity of the problem. Here, we present a simplified coherence-agnostic phase-space tomography method for the experimental characterization of the widely used class of rotationally symmetric beams, which includes as a particular case partially coherent vortices. It is based on the reconstruction of the beam ambiguity function from the intensity distributions measured in the antisymmetric fractional Fourier transform domains. The experimental data can be acquired using an optical setup consisting of four cylindrical lenses and a digital camera located in fixed positions. The feasibility of the proposed method is experimentally demonstrated.

Optical coherenscopy based on phase-space tomography

Partially coherent light provides attractive benefits in imaging, beam shaping, free-space communications, random medium monitoring, among other applications. However, the experimental characterization of the spatial coherence is a difficult problem involving second-order statistics represented by four-dimensional functions that cannot be directly measured and analyzed. In addition, real-world applications usually require quantitative characterization of the local spatial coherence of a beam in the absence of a priori information, together with fast acquisition and processing of the experimental data. Here we propose and experimentally demonstrate a technique that solves this problem. It comprises an optical setup developed for automatized video-rate measurement and a method -phase-space tomographic coherenscopy- allowing parallel data acquisition, processing, and analysis. This technique significantly simplifies the spatial coherence analysis and opens up new perspectives for the development of tools exploiting the degrees of freedom hidden into light coherence.

Temporal fluctuations in the bosonic Josephson junction as a probe for phase space tomography

We study the long-time dynamics of the reduced one-particle Bloch vector S of a two-mode Bose–Hubbard model in the Josephson interaction regime, as a function of the relative phase and occupation imbalance of an arbitrary coherent preparation. We find that the variance of the long-time fluctuations of S can be factorized as a product of the inverse participation number 1/M that depends only on the preparation, and a semiclassical function C(E) that reflects the phase space characteristics of the pertinent observable. Temporal fluctuations can thus be used as a sensitive probe for phase space tomography of quantum many-body states.

A study of the maximum entropy technique for phase space tomography

We study a problem with the Maximum Entropy Technique (MENT) when applied totomographic measurements of the transverse phase space of electron beams, and suggestsome ways to improve its reliability. We show that the outcome of a phase spacereconstruction can be highly sensitive to the choice of projection angles. It isquite likely to obtain reconstructed distributions of the phase space that areobviously different from the actual distributions. We propose a methodto obtain a ``good'' choice of projections angles using a normalised phase space.We demonstrate that the resulting reconstructions of the phase space canbe significantly improved.

Phase-space tomography of matter-wave diffraction in the Talbot regime

We report on the theoretical investigation of the Wigner distribution function (WDF) reconstruction of the motional quantum state of large molecules in de Broglie interference. de Broglie interference of fullerenes and the like already proves the wavelike behaviour of these heavy particles, while we aim to extract more quantitative information about the superposition quantum state in motion. We simulate the reconstruction of the WDF numerically based on an analytic probability distribution and investigate its properties by the variation of parameters which are relevant for the experiment. Even though the WDF described in the near-field experiment cannot be reconstructed completely, we observe negativity even in the partially reconstructed WDF. We further consider incoherent factors to simulate the experimental situation, such as a finite number of slits, collimation and particle-slit van der Waals interaction. From this we find experimental conditions to reconstruct the WDF from Talbot interference fringes in molecule Talbot–Lau interferometry.

Experimental compressive phase space tomography

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two–point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low–entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground–truth estimate from the van Cittert–Zernike theorem and verify substantial quantitative improvements in the reconstruction.

Beam tomography in transverse normalised phase space

We demonstrate that the normalised phase space can be used to improve the quality of the reconstructed image in phase space tomography. This study uses the Filtered Back Projection (FBP) for reconstruction. We simulate the tomographic projections of a Gaussian phase space distribution in real phase space and in normalised phase space. The reconstruction can be carried out in real phase space directly. It can also be done first in normalised phase space, and then the co-ordinates can be transformed to real phase space. Using FBP, we show that the latter procedure produces images that agree well with the original distribution even when there are only three angles. It also gives better resolution for a more complex distribution, which we simulate with a random collection of Gaussian spots.

Time-resolved electron beam phase space tomography at a soft x-ray free-electron laser

High-gain free-electron lasers (FELs) in the ultraviolet and x-ray regime put stringent demands on the peak current, transverse emittance, and energy spread of the driving electron beam. At the soft x-ray FEL FLASH, a transverse deflecting microwave structure (TDS) has been installed to determine these parameters for the longitudinally compressed bunches, which are characterized by a narrow leading peak of high charge density and a long tail. The rapidly varying electromagnetic field in the TDS deflects the electrons vertically and transforms the time profile into a streak on an observation screen. The bunch current profile was measured single shot with an unprecedented resolution of 27 fs under FEL operating conditions. A precise single-shot measurement of the energy distribution along a bunch was accomplished by using the TDS in combination with an energy spectrometer. Variation of quadrupole strengths allowed for a determination of the horizontal emittance as a function of the longitudinal position within a bunch, the so-called slice emittance. In the bunch tail, a normalized slice emittance of about $2\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ was found, in agreement with expectations. In the leading spike, however, surprisingly large emittance values were observed, in apparent contradiction with the low emittance deduced from the measured FEL gain. By applying three-dimensional phase space tomography, we were able to show that the bunch head contains a central core of low emittance and high local current density, which is presumably the lasing part of the bunch.

Phase space tomography reconstruction of the Wigner distribution for optical beams separable in Cartesian coordinates

We propose a simple approach for the phase space tomography reconstruction of the Wigner distribution of paraxial optical beams separable in Cartesian coordinates. It is based on the measurements of the antisymmetric fractional Fourier transform power spectra, which can be taken using a flexible optical setup consisting of four cylindrical lenses. The numerical simulations and the experimental results clearly demonstrate the feasibility of the proposed scheme.

Quantum tomography with wavelet transform in Banach space on homogeneous space

In this study the intimate connection is established between the Banach space wavelet reconstruction method on homogeneous spaces with both singular and nonsingular vacuum vectors, and some of the well known quantum tomographies, such as: Moyal-representation for a spin, discrete phase space tomography, tomography of a free particle, Homodyne tomography, phase space tomography and SU(1,1) tomography. And both the atomic decomposition and the Banach frame nature of these quantum tomographic examples are also revealed in details. Finally the connection between the wavelet formalism on Banach space and Q-function is discussed.

X-ray noninterferometric phase imaging: a unified picture

A unified theory of noninterferometric phase recovery based on the so-called ambiguity function is introduced. The theory is used to analyze previously published techniques and unify them with the methods of phase-space tomography applicable to partially coherent data. The theory is then used to propose some new approaches to noninterferometric phase recovery.

Phase-space reconstruction of focused x-ray fields

We apply the method of phase-space tomography to reconstruct x-ray beams focused using a compound refractive lens. We show that it is possible to decouple the effect of aberrations in the optical system from the field and hence measure both them and the original field. We recover the complex coherence function and find that it is consistent with expectations.