Partial Spatial Coherence Research Articles

Making the link between ADF and 4D STEM: Resolution, transfer and coherence

Abstract Steve Pennycook is a pioneer in the application of high-resolution scanning transmission electron microscopy (STEM) and in particular the use of annular dark-field (ADF) imaging. Here we show how a general framework for 4D STEM allows clear links to be made between ADF imaging and the emerging methods for reconstructing images from 4D STEM data sets. We show that both ADF imaging and ptychographical reconstruction can be thought of in terms of integrating over the overlap regions of diffracted discs in the detector plane. This approach allows the similarities in parts of their transfer functions to be understood, though we note that the transfer functions for ptychographic imaging cannot be used as a measure of information transfer. We also show that conditions of partial spatial and temporal coherence affect ADF imaging and ptychography similarly, showing that achromatic interference can always contribute to the image in both cases, leading to a robustness to partial temporal coherence that has enabled high-resolution imaging.

Scattering of partially coherent electromagnetic beams by a sphere.

We examine the scattering of a partially coherent, partially polarized electromagnetic beam by a homogeneous sphere (Mie scattering). The degree of polarization and the Stokes parameters in the far zone are found to be strongly dependent on the state of coherence and polarization of the incident beam. In particular, we demonstrate the emergence of polarization singularities and show that partial spatial coherence gives rise to significant depolarization effects. In addition we explore the symmetry properties of the scattered field.

Recovery of polarization entanglement in partially coherent photonic qubits.

Partially coherent photonic qubits, owing to their robustness in propagation through random media compared to fully coherent qubits, find applications in free-space communication, quantum imaging, and quantum sensing. However, the reduction of spatial coherence degrades entanglement in qubits, adversely affecting entanglement-based applications. We report the recovery of entanglement in the partially coherent photonic qubits generated using a spontaneous parametric downconversion process despite retaining their multimode nature. This study utilizes an electron multiplying charge-coupled device (EMCCD) to perform coincidence measurements, eliminating the need for raster scanning of single-pixel detectors, which simplifies optical alignment, enhances precision, and reduces time consumption. We demonstrate that the size of apertures used to select biphotons substantially impacts the visibility and S-parameter of polarization-entangled partially coherent qubits. The entanglement is recovered with partial spatial coherence properties by choosing small sizes of the apertures in the captured image plane. This study could help in the advancement of free-space quantum communication, quantum imaging, and quantum metrology.

Temporal coherence envelope function of field emission in electron microscopy

Imaging in electron microscopy is adversely affected by partial electron spatial and temporal coherence. Temporal coherence has been treated theoretically in the past using the method pioneered fifty years ago by Hanßen and Trepte, who assumed a Gaussian energy distribution. However, state-of-the-art instruments employ field emission (FE) sources that emit electrons with a non-Gaussian energy distribution. We have updated the treatment of temporal coherence to describe the effects of an arbitrary energy distribution on image formation. The updated approach is implemented in Fourier optics simulations to explore the effect of FE on image formation in conventional, non-aberration-corrected (NAC) and aberration-corrected (AC) low energy electron microscopy. It is found that the resolution that can be achieved for the FE distribution is only slightly degraded compared to a Gaussian distribution with the same energy spread. FE also produces a focus offset. These two effects are weaker for AC than for NAC microscopy. These and other insights may be relevant to the selection of the aperture size that optimizes resolution and to analyses that make use of focal image series. The approach developed here is also applicable to transmission electron microscopy.

Investigating the robustness of a deep learning-based method for quantitative phase retrieval from propagation-based x-ray phase contrast measurements under laboratory conditions

Objective. Quantitative phase retrieval (QPR) in propagation-based x-ray phase contrast imaging of heterogeneous and structurally complicated objects is challenging under laboratory conditions due to partial spatial coherence and polychromaticity. A deep learning-based method (DLBM) provides a nonlinear approach to this problem while not being constrained by restrictive assumptions about object properties and beam coherence. The objective of this work is to assess a DLBM for its applicability under practical scenarios by evaluating its robustness and generalizability under typical experimental variations. Approach. Towards this end, an end-to-end DLBM was employed for QPR under laboratory conditions and its robustness was investigated across various system and object conditions. The robustness of the method was tested via varying propagation distances and its generalizability with respect to object structure and experimental data was also tested. Main results. Although the end-to-end DLBM was stable under the studied variations, its successful deployment was found to be affected by choices pertaining to data pre-processing, network training considerations and system modeling. Significance. To our knowledge, we demonstrated for the first time, the potential applicability of an end-to-end learning-based QPR method, trained on simulated data, to experimental propagation-based x-ray phase contrast measurements acquired under laboratory conditions with a commercial x-ray source and a conventional detector. We considered conditions of polychromaticity, partial spatial coherence, and high noise levels, typical to laboratory conditions. This work further explored the robustness of this method to practical variations in propagation distances and object structure with the goal of assessing its potential for experimental use. Such an exploration of any DLBM (irrespective of its network architecture) before practical deployment provides an understanding of its potential behavior under experimental settings.

Experimental optimization of lensless digital holographic microscopy with rotating diffuser-based coherent noise reduction.

Laser-based lensless digital holographic microscopy (LDHM) is often spoiled by considerable coherent noise factor. We propose a novel LDHM method with significantly limited coherent artifacts, e.g., speckle noise and parasitic interference fringes. It is achieved by incorporating a rotating diffuser, which introduces partial spatial coherence and preserves high temporal coherence of laser light, crucial for credible in-line hologram reconstruction. We present the first implementation of the classical rotating diffuser concept in LDHM, significantly increasing the signal-to-noise ratio while preserving the straightforwardness and compactness of the LDHM imaging device. Prior to the introduction of the rotating diffusor, we performed LDHM experimental hardware optimization employing 4 light sources, 4 cameras, and 3 different optical magnifications (camera-sample distances). It was guided by the quantitative assessment of numerical amplitude/phase reconstruction of test targets, conducted upon standard deviation calculation (noise factor quantification), and resolution evaluation (information throughput quantification). Optimized rotating diffuser LDHM (RD-LDHM) method was successfully corroborated in technical test target imaging and examination of challenging biomedical sample (60 µm thick mouse brain tissue slice). Physical minimization of coherent noise (up to 50%) was positively verified, while preserving optimal spatial resolution of phase and amplitude imaging. Coherent noise removal, ensured by proposed RD-LDHM method, is especially important in biomedical inference, as speckles can falsely imitate valid biological features. Combining this favorable outcome with large field-of-view imaging can promote the use of reported RD-LDHM technique in high-throughput stain-free biomedical screening.

Aberration-corrected transmission electron microscopy with Zernike phase plates

We explore the possibility of applying physical phase plates (PPs) in combination with aberration-corrected transmission electron microscopy. Phase-contrast transfer characteristics are calculated and compared for a thin-film based Zernike PP, a hole-free (HF) or Volta PP and an electrostatic Zach PP, considering their phase-shifting properties in combination with partial spatial coherence. The effect of slightly converging illumination conditions, often used in high-resolution applications, on imaging with PPs is discussed. Experiments with an unheated Zernike PP applied to various nanomaterial specimens and a qualitative analysis clearly demonstrates the general compatibility of PPs and aberration-corrected transmission electron microscopy. Calculations and experiments show the benefits of the approach, among which is a strong phase-contrast enhancement of a large range of spatial frequencies. This allows the simultaneous imaging of atomic-resolution structures and morphological features at the nanometer scale, with maximum phase contrast. The calculations can explain why the HFPP damps contrast transfer at higher spatial frequencies.

Spatial coherence of electrically pumped random terahertz lasers

Light sources with high radiance and tailored coherence properties are highly desirable for imaging applications in the mid-infrared and terahertz (THz) spectral regions, which host a large variety of molecular absorptions and distinctive fingerprints to be exploited for sensing and tomography. Here, we characterize the spatial coherence of random multimode THz quantum cascade lasers (QCLs) emitting > mW optical power per mode and showing low divergence (10°–30°), performing a modified Young’s double-slit experiment. Partial spatial coherence values ranging between 0.16 and 0.34 are retrieved, depending on the specific degree of disorder. These values are significantly lower than those (0.82) of conventional Fabry–Perot THz QCLs exploiting an identical active region quantum design. We then incorporate the devised low spatial coherence random lasers into a confocal imaging system with micrometer spatial resolution and demonstrate notable imaging performances, at THz frequencies, against spatial cross talk and speckles.

Modelling of ATR-FTIR MEMS Spectrometer Under Partially-Coherent Multimode-Fiber Illumination

Miniaturization of attenuated total reflectance (ATR) spectrometers has been an emerging field applying such powerful surface sensing method for in-situ spectroscopic analysis. In this work, we present a model for field propagation through ATR element under the illumination of multimode fiber (MMF) in a micro-electro-mechanical-systems (MEMS) based ATR Fourier transform infrared (FTIR) spectroscopic system. The core spectrometer is based on micro-fabricated Michelson interferometer using deep etching technology, in which the light propagates in-plane with respect to the silicon substrate. An ATR multiple reflection crystal is illuminated with infrared (IR) thermal blackbody radiation source through an MMF. The output light of the crystal is fiber-coupled to the MEMS interferometer then to an IR broadband photodetector. The optical system is modelled using scaler Fourier optics, where the fiber output field is represented by a group of spatially shifted elementary sources, to predict the ATR absorbance response, taking into account the partial spatial coherence nature of the MMF output. The model output leads to the requirements on the ATR measurement conditions and numerical aperture (NA) of the system. The model is compared to practical results of MEMS spectrometer which is experimentally characterized over the mid-infrared (MIR) wavenumber range from 5000 cm−1 to 2100 cm−1, lower limited by the used fiber and photodetector cut-off. Spectra of liquid samples are obtained using two different crystals and total internal reflection (TIR) angles showing good agreement with theoretical prediction.

Flexible ptychography platform to expand the potential of imaging at free electron lasers.

Ptychography, a scanning coherent diffraction imaging method, can produce a high-resolution reconstruction of a sample and, at the same time, of the illuminating beam. The emergence of vacuum ultraviolet and X-ray free electron lasers (FELs) has brought sources with unprecedented characteristics that enable X-ray ptychography with highly intense and ultra-fast short-wavelength pulses. However, the shot-to-shot pulse fluctuations typical for FEL pulses and particularly the partial spatial coherence of self-amplified spontaneous emission (SASE) FELs lead to numerical complexities in the ptychographic algorithms and ultimately restrict the application of ptychography at FELs. We present a general adaptive forward model for ptychography based on automatic differentiation, which is able to perform reconstructions even under these conditions. We applied this model to the first ptychography experiment at FLASH, the Free electron LASer in Hamburg, and obtained a high-resolution reconstruction of the sample as well as the complex wavefronts of individual FLASH pulses together with their coherence properties. This is not possible with more common ptychography algorithms.

Direct observation of diffraction-induced far-field polarization effects in electromagnetic beams

Traditional Fraunhofer diffraction theory explains the far-field diffraction of fully coherent scalar fields; however, it does not include the partial coherence and partial polarization features, which are essential characteristics of a light field. We report, to the best of our knowledge, the first experimental demonstration of effect on the degree and state of polarization of partially coherent light fields caused by Fraunhofer diffraction. We have investigated this effect by synthesizing a near-unpolarized optical field having vectorial partial spatial coherence features and diffracting it with a circular aperture. We demonstrate that the spatial coherence can cause polarization modulations at the far-field, which can be reversed/nullified by controlling the size of the diffracting element. The conceptual clarity brought out by this experimental demonstration will be useful in polarization-sensitive applications such as imaging and tomography.

Multi-frequency coherent emission from superstructure thermal emitters

Long-range spatial coherence can be induced in incoherent thermal emitters by embedding a periodic grating within a material supporting propagating polaritons or dielectric modes. However, only a single spatially coherent mode is supported by purely periodic thermal emitters. While various designs have been proposed for the purpose of allowing arbitrary emission profiles, the limitations associated with the partial spatial coherence of thermal emitters are not known. Here, we explore superstructure gratings (SSGs) to control the spatial and spectral properties of thermal emitters. SSGs have long-range periodicity but employ a unit cell that provides multiple Bragg vectors to interact with light. These Bragg vectors allow simultaneous launching of polaritons with different frequencies/wavevectors in a single grating, manifesting as additional spatial and spectral modes in the thermal emission profile. However, SSGs still have a well-defined period, which allows us to assess the role that finite spatial coherence plays in thermal emitters. We find that the spatial coherence length defines the maximum possible SSG period that can be used. This provides a fundamental limit on the degree of spatial coherence that can be induced in a thermal emitter and has broader implications for the use of techniques such as inverse design for structure optimization.

Broadband X-ray ptychography using multi-wavelength algorithm.

Ptychography is a rapidly developing scanning microscopy which is able to view the internal structures of samples at a high resolution beyond the illumination size. The achieved spatial resolution is theoretically dose-limited. A broadband source can provide much higher flux compared with a monochromatic source; however, it conflicts with the necessary coherence requirements of this coherent diffraction imaging technique. In this paper, a multi-wavelength reconstruction algorithm has been developed to deal with the broad bandwidth in ptychography. Compared with the latest development of mixed-state reconstruction approach, this multi-wavelength approach is more accurate in the physical model, and also considers the spot size variation as a function of energy due to the chromatic focusing optics. Therefore, this method has been proved in both simulation and experiment to significantly improve the reconstruction when the source bandwidth, illumination size and scan step size increase. It is worth mentioning that the accurate and detailed information of the energy spectrum for the incident beam is not required in advance for the proposed method. Further, we combine multi-wavelength and mixed-state approaches to jointly solve temporal and spatial partial coherence in ptychography so that itcan handle various disadvantageous experimental effects. The significant relaxation in coherence requirements by our approaches allows the use of high-flux broadband X-ray sources for high-efficient and high-resolution ptychographic imaging.

Effects of source spatial partial coherence on intensity statistics of optical beams in mono-static turbulent channels.

We investigate, via both experimental measurements and wave-optics computer simulations, the statistical characteristics of the fluctuating intensity, such as the average intensity, the beam wander and the scintillation index, of the Gaussian Schell-model (GSM) beams on passing through double-pass, monostatic turbulence channels with either a retro-reflector (RR) or a flat mirror (FM). Our experimental results reveal that the enhanced backscatter (EBS) gradually weakens as the spatial coherence of the GSM source decreases, and eventually disappears for the sufficiently low source spatial coherence states. The r.m.s beam wander remains practically invariant with the variation of the source coherence width in the range from 0.2 to 6.0 mm both in the case of the RR and the FM, which formed the RR case being much smaller. In addition, it is found that the long-term scintillation index of the untracked beam with the RR is smaller than that with the FM, while the situation is reversed for the short-term scintillation index of the tracked beam. In both cases, the scintillation index decreases as the spatial coherence of the GSM source decreases. The obtained computer simulation results agree reasonably well with the experimental results. In addition, the effects of spatial coherence on statistical characteristics of the GSM beams along a 1 km propagation distance through the double-pass monostatic turbulence are also investigated using wave-optics simulation. We also carry out evaluation and comparison of the intensity probability density functions for the RR and FM cases and for various source coherence states that are of utmost importance for free-space optical communications in retro-reflection modulation regime. In addition, our findings will be beneficial for the development of remote sensing and directed energy laser applications in the presence of air turbulence.

Polarimetric dimension and nonregularity of tightly focused light beams

Polarimetric dimension and nonregularity are newly introduced concepts that characterize three-dimensional (3D) polarization states of light. We analyze the spectral polarimetric dimension and the degree of nonregularity associated with two kinds of tightly focused beams: a radially fully polarized Gaussian Schell-model (GSM) beam and a partially polarized beam composed of an incoherent superposition of two orthogonally polarized (coherent) plane-wave modes. We show that for both beams the focal field can exhibit genuine 3D and nonregular character, with even perfect nonregularity encountered for the tightly focused two-mode beam. These features originate from the partial spatial coherence and partial polarization of the incident beams, and in the limit of full coherence and polarization the three-dimensionality and nonregularity of the focal field vanish. We also find that the GSM beam can generate a nanoscale region around the focus where the field is essentially 3D unpolarized. The results demonstrate the rich polarimetric structure of focal fields and may find uses in optical particle manipulation and sensing.

Effect of Partial Spatial Coherence on Image Sharpness

Effect of Partial Spatial Coherence on Image Sharpness

Development and characterization of a source having tunable partial spatial coherence and polarization features

Partially coherent fields offer more robustness to losses occurring due to atmospheric propagation, and are, therefore, considered a promising candidate for free-space optical communication (both classical and quantum). The quantification of loss immunity for such fields requires a versatile source having tunable partial coherence and polarization attributes. In this work, we report construction of an optical source having tunable polarization and spatial coherence using an interferometric arrangement along with a variable beam expender. Realization of such source first requires a thorough study of spatial coherence properties of the laser beam passing through a rotating ground glass diffuser for different sizes of the source at the diffuser plane. It is found that spatial coherence of laser beam decreases, as size of the source increases. We also investigate the effect of rotation speed of diffuser on spatial coherence of a laser beam and find that it does not change significantly with rotation speed of the diffuser. This study also validates the van Cittert-Zernike theorem for our multi-scatterer source having linear, circular and spatial polarizations confirming enhancement of coherence with free-space propagation.

Modeling of full-field optical coherence tomography in scattering media.

We develop a model of full-field optical coherence tomography (FF-OCT) that includes a description of partial temporal and spatial coherence, together with a mean-field scattering theory going beyond the Born approximation. Based on explicit expressions of the FF-OCT signal, we discuss essential features of FF-OCT imaging, such as the influence of partial coherence on the optical transfer function, and on the decay of the signal with depth. We derive the conditions under which the spatially averaged signal exhibits a pure exponential decay, providing a clear frame for the use of the Beer-Lambert law for quantitative measurements of the extinction length in scattering media.

Partially spatially coherent digital holographic microscopy and machine learning for quantitative analysis of human spermatozoa under oxidative stress condition

Semen quality assessed by sperm count and sperm cell characteristics such as morphology and motility, is considered to be the main determinant of men’s reproductive health. Therefore, sperm cell selection is vital in assisted reproductive technology (ART) used for the treatment of infertility. Conventional bright field optical microscopy is widely utilized for the imaging and selection of sperm cells based on the qualitative analysis by experienced clinicians. In this study, we report the development of a highly sensitive quantitative phase microscopy (QPM) using partially spatially coherent light source, which is a label-free, non-invasive and high-resolution technique to quantify various biophysical parameters. The partial spatial coherence nature of light source provides a significant improvement in spatial phase sensitivity and hence reconstruction of the phase of the entire sperm cell is demonstrated, which was otherwise not possible using highly spatially coherent light source. High sensitivity of the system enables quantitative phase imaging of the specimens having very low refractive index contrast with respect to the medium like tail of the sperm cells. Further, it also benefits with accurate quantification of 3D-morphological parameters of sperm cells which might be helpful in the infertility treatment. The quantitative analysis of more than 2500 sperm cells under hydrogen peroxide (H2O2) induced oxidative stress condition is demonstrated. It is further correlated with motility of sperm cell to study the effect of oxidative stress on healthy sperm cells. The results exhibit a decrease in the maximum phase values of the sperm head as well as decrease in the sperm cell’s motility with increasing oxidative stress, i.e., H2O2 concentration. Various morphological and texture parameters were extracted from the phase maps and subsequently support vector machine (SVM) based machine learning algorithm is employed for the classification of the control and the stressed sperms cells. The algorithm achieves an area under the receiver operator characteristic (ROC) curve of 89.93% based on the all morphological and texture parameters with a sensitivity of 91.18%. The proposed approach can be implemented for live sperm cells selection in ART procedure for the treatment of infertility.

Two-shot measurement of spatial coherence

We propose and demonstrate an interferometric scheme for measuring the two-dimensional two-point cross-spectral density function in a two-shot manner. Our scheme comprises a Michelson interferometer with a converging lens in one of the arms of the interferometer, and the cross-spectral density function of an input optical field gets encoded in the intensity distribution of the output interferograms. This scheme works for any cross-spectral density function that is real and that depends on the spatial coordinates only through their difference. Using this scheme, we report measurements of several lab-synthesized cross-spectral density functions with very good agreement with theory. Our measurement technique can be very important for applications that are based on utilizing the partial spatial coherence properties of optical fields.