Refraction of space-time wave packets: I. theoretical principles.

Space-time (ST) wave packets are pulsed optical beams endowed with precise spatio-temporal structure by virtue of which they exhibit unique and useful characteristics such as propagation invariance and tunable group velocity. We study in detail here, and in two accompanying papers, the refraction of ST wave packets at planar interfaces between non-dispersive, homogeneous, and isotropic dielectrics. We formulate a law of refraction that determines the change in the ST wave-packet group velocity across such an interface as a consequence of a newly identified optical refractive invariant that we call the "spectral curvature". Because the spectral curvature vanishes in conventional optical fields where the spatial and temporal degrees of freedom are separable, these phenomena have not been observed to date. We derive the laws of refraction for baseband, X wave, and sideband ST wave packets that reveal fascinating refractive phenomena, especially for the former class of wave packets. We predict theoretically, and confirm experimentally in the accompanying papers, refractive phenomena such as group-velocity invariance (ST wave packets whose group velocity does not change across the interface), anomalous refraction (group-velocity increase in higher-index media), group-velocity inversion (change in the sign of the group velocity upon refraction but not its magnitude), and the dependence of the group velocity of the refracted ST wave packet on the angle of incidence.

Refraction of space-time wave packets: II. experiments at normal incidence.

The refraction of space-time (ST) wave packets offers many fascinating surprises with respect to conventional pulsed beams. In the first paper in this sequence [J. Opt. Soc. Am. A38, 1409 (2021)10.1364/JOSAA.430105], we theoretically described the refraction of all families of ST wave packets at normal and oblique incidence at a planar interface between two nondispersive, homogeneous, isotropic dielectrics. Here, in this second paper in the sequence, we present experimental verification of the refractive phenomena predicted for baseband ST wave packets upon normal incidence on a planar interface. Specifically, we observe group velocity invariance, normal and anomalous refraction, and group velocity inversion leading to group delay cancellation. These phenomena are verified in a set of optical materials with refractive indices ranging from 1.38 to 1.76, including MgF2, fused silica, BK7 glass, and sapphire. We also provide a geometrical representation of the physics associated with anomalous refraction in terms of the dynamics of the spectral support domain for ST wave packets on the surface of the light cone.

Refraction of space-time wave packets: III. experiments at oblique incidence.

The refraction of space-time (ST) wave packets at planar interfaces between non-dispersive, homogeneous, isotropic dielectrics exhibits fascinating phenomena, even at normal incidence. Examples of such refractive phenomena include group-velocity invariance across the interface, anomalous refraction, and group-velocity inversion. Crucial differences emerge at oblique incidence with respect to the results established at normal incidence. For example, the group velocity of the refracted ST wave packet can be tuned simply by changing the angle of incidence. In the third paper, we present experimental verification of the refractive phenomena exhibited by ST wave packets at oblique incidence that were in the first paper of this sequence [J. Opt. Soc. Am. A38, 1409 (2021)10.1364/JOSAA.430105]. We also examine a proposal for "blind synchronization," whereby identical ST wave packets arrive simultaneously at different receivers without a priori knowledge of their locations except that they are all located at the same depth beyond an interface between two media. A first proof-of-principle experimental demonstration of this effect is provided.

Geometric localization of thermal fluctuations in red blood cells.

The thermal fluctuations of membranes and nanoscale shells affect their mechanical characteristics. Whereas these fluctuations are well understood for flat membranes, curved shells show anomalous behavior due to the geometric coupling between in-plane elasticity and out-of-plane bending. Using conventional shallow shell theory in combination with equilibrium statistical physics we theoretically demonstrate that thermalized shells containing regions of negative Gaussian curvature naturally develop anomalously large fluctuations. Moreover, the existence of special curves, "singular lines," leads to a breakdown of linear membrane theory. As a result, these geometric curves effectively partition the cell into regions whose fluctuations are only weakly coupled. We validate these predictions using high-resolution microscopy of human red blood cells (RBCs) as a case study. Our observations show geometry-dependent localization of thermal fluctuations consistent with our theoretical modeling, demonstrating the efficacy in combining shell theory with equilibrium statistical physics for describing the thermalized morphology of cellular membranes.

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.

Broadband space-time wave packets propagating 70 m.

The propagation distance of a pulsed beam in free space is ultimately limited by diffraction and space-time coupling. "Space-time" (ST) wave packets are pulsed beams endowed with tight spatio-temporal spectral correlations that render them propagation-invariant. Here we explore the limits of the propagation distance for ST wave packets. Making use of a specially designed phase plate inscribed by gray-scale lithography and having a laser-damage threshold of ∼0.5 J/cm2, we synthesize a ST light sheet of width ≈700 µm and bandwidth ∼20 nm, and confirm a propagation distance of ≈70 m.

Meters-long propagation of diffraction-free space-time light-sheets.

Space-time (ST) wave packets are pulsed beams in which the spatial frequencies and wavelengths are tightly correlated. Proper design of the functional form of these correlations results in diffraction-free and dispersion-free axial propagation; that is, propagation invariance in free space. To date, observed propagation distances of such ST wave packets has been on the order of a few centimeters. Here we synthesize an ST wave packet in the form of a pulsed optical sheet of transverse spatial width ∼200 µm and spectral bandwidth of ∼2 nm, and observe its diffraction-free propagation for approximately 6 meters. For such ST wave packets, we identify the spectral uncertainty - the precision in associating the spatial and temporal frequencies - as a critical parameter in determining the propagation-invariant distance. We present a design strategy and an experimental methodology that enables further increase in the diffraction-free length.

Measuring the Nonuniform Evaporation Dynamics of Sprayed Sessile Microdroplets with Quantitative Phase Imaging.

We demonstrate real-time quantitative phase imaging as a new optical approach for measuring the evaporation dynamics of sessile microdroplets. Quantitative phase images of various droplets were captured during evaporation. The images enabled us to generate time-resolved three-dimensional topographic profiles of droplet shape with nanometer accuracy and, without any assumptions about droplet geometry, to directly measure important physical parameters that characterize surface wetting processes. Specifically, the time-dependent variation of the droplet height, volume, contact radius, contact angle distribution along the droplet's perimeter, and mass flux density for two different surface preparations are reported. The studies clearly demonstrate three phases of evaporation reported previously: pinned, depinned, and drying modes; the studies also reveal instances of partial pinning. Finally, the apparatus is employed to investigate the cooperative evaporation of the sprayed droplets. We observe and explain the neighbor-induced reduction in evaporation rate, that is, as compared to predictions for isolated droplets. In the future, the new experimental methods should stimulate the exploration of colloidal particle dynamics on the gas-liquid-solid interface.

Nanoscale topography and spatial light modulator characterization using wide-field quantitative phase imaging.

We demonstrate an optical technique for large field of view quantitative phase imaging of reflective samples. It relies on a common-path interferometric design, which ensures high stability without the need for active stabilization. The technique provides single-shot, full-field and robust measurement of nanoscale topography of large samples. Further, the inherent stability allows reliable measurement of the temporally varying phase retardation of the liquid crystal cells, and thus enables real-time characterization of spatial light modulators. The technique's application potential is validated through experimental results.

Effects of spatial coherence in diffraction phase microscopy.

Quantitative phase imaging systems using white light illumination can exhibit lower noise figures than laser-based systems. However, they can also suffer from object-dependent artifacts, such as halos, which prevent accurate reconstruction of the surface topography. In this work, we show that white light diffraction phase microscopy using a standard halogen lamp can produce accurate height maps of even the most challenging structures provided that there is proper spatial filtering at: 1) the condenser to ensure adequate spatial coherence and 2) the output Fourier plane to produce a uniform reference beam. We explain that these object-dependent artifacts are a high-pass filtering phenomenon, establish design guidelines to reduce the artifacts, and then apply these guidelines to eliminate the halo effect. Since a spatially incoherent source requires significant spatial filtering, the irradiance is lower and proportionally longer exposure times are needed. To circumvent this tradeoff, we demonstrate that a supercontinuum laser, due to its high radiance, can provide accurate measurements with reduced exposure times, allowing for fast dynamic measurements.

Diffraction phase microscopy: monitoring nanoscale dynamics in materials science [invited].

Quantitative phase imaging (QPI) utilizes the fact that the phase of an imaging field is much more sensitive than its amplitude. As fields from the source interact with the specimen, local variations in the phase front are produced, which provide structural information about the sample and can be used to reconstruct its topography with nanometer accuracy. QPI techniques do not require staining or coating of the specimen and are therefore nondestructive. Diffraction phase microscopy (DPM) combines many of the best attributes of current QPI methods; its compact configuration uses a common-path off-axis geometry which realizes the benefits of both low noise and single-shot imaging. This unique collection of features enables the DPM system to monitor, at the nanoscale, a wide variety of phenomena in their natural environments. Over the past decade, QPI techniques have become ubiquitous in biological studies and a recent effort has been made to extend QPI to materials science applications. We briefly review several recent studies which include real-time monitoring of wet etching, photochemical etching, surface wetting and evaporation, dissolution of biodegradable electronic materials, and the expansion and deformation of thin-films. We also discuss recent advances in semiconductor wafer defect detection using QPI.

Digital projection photochemical etching defines gray-scale features.

We demonstrate a maskless photochemical etching method that is capable of performing one-step etching of multi-level structures. This method uses a digital projector to focus an image onto the sample and define the etching pattern. By combining digital projection photochemical etching with diffraction phase microscopy, etch heights can be measured in situ in a non-destructive manner. This method is single shot, eliminating the need for expensive gray-scale masks or laser scanning methods. The etch rate is studied as a function of the wavelength and irradiance of the projected light. A lateral etch resolution of 2 µm is demonstrated by etching selected portions of the USAF-1951 target. Micropillars, multi-level plateaus, and an Archimedean spiral are etched, each in a single processing step, to illustrate the unique capabilities.

Derivative method for phase retrieval in off-axis quantitative phase imaging.

We present a method for phase retrieval in off-axis interferometric systems. By numerically calculating the transverse 1st and 2nd order derivatives of the interferogram, we show that one can directly retrieve the quantitative phase image, without the need for Fourier or Hilbert transformations. Because of this, the method is significantly faster than the current approaches. We illustrate our method using biological specimen data from three different off-axis quantitative phase imaging techniques.

Spectroscopic diffraction phase microscopy.

We present spectroscopic diffraction phase microscopy (sDPM) as a method capable of measuring quantitative phase images at multiple wavelengths. sDPM uses a spatial light modulator at the Fourier plane of a lens to select desired wavelengths from the white light illumination of a grating. The quantitative phase information at different wavelengths allows us to decouple the refractive index and the thickness from the phase shift induced by biological cells. We demonstrate the capability of the setup by dispersion measurements of microsphere beads and RBCs.

Diffraction phase microscopy with white light.

We present white light diffraction phase microscopy (wDPM) as a quantitative phase imaging method that combines the single shot measurement benefit associated with off-axis methods, high temporal phase stability associated with common path geometries, and high spatial phase sensitivity due to the white light illumination. We propose a spatiotemporal filtering method that pushes the limit of the pathlength sensitivity to the subangstrom level at practical spatial and temporal bandwidths. We illustrate the utility of wDPM with measurements on red blood cell morphology and HeLa cell growth over 18 hours.

Motion detection using extended fractional Fourier transform and digital speckle photography.

Digital speckle photography is a useful tool for measuring the motion of optically rough surfaces from the speckle shift that takes place at the recording plane. A simple correlation based digital speckle photographic system has been proposed that implements two simultaneous optical extended fractional Fourier transforms (EFRTs) of different orders using only a single lens and detector to simultaneously detect both the magnitude and direction of translation and tilt by capturing only two frames: one before and another after the object motion. The dynamic range and sensitivity of the measurement can be varied readily by altering the position of the mirror/s used in the optical setup. Theoretical analysis and experiment results are presented.

Simultaneous measurement of translation and tilt using digital speckle photography.

A Michelson-type digital speckle photographic system has been proposed in which one light beam produces a Fourier transform and another beam produces an image at a recording plane, without interfering between themselves. Because the optical Fourier transform is insensitive to translation and the imaging technique is insensitive to tilt, the proposed system is able to simultaneously and independently determine both surface tilt and translation by two separate recordings, one before and another after the surface motion, without the need to obtain solutions for simultaneous equations. Experimental results are presented to verify the theoretical analysis.

Free-space optical delay line using space-time wave packets.

An optical buffer featuring a large delay-bandwidth-product-a critical component for future all-optical communications networks-remains elusive. Central to its realization is a controllable inline optical delay line, previously accomplished via engineered dispersion in optical materials or photonic structures constrained by a low delay-bandwidth product. Here we show that space-time wave packets whose group velocity is continuously tunable in free space provide a versatile platform for constructing inline optical delay lines. By spatio-temporal spectral-phase-modulation, wave packets in the same or in different spectral windows that initially overlap in space and time subsequently separate by multiple pulse widths upon free propagation by virtue of their different group velocities. Delay-bandwidth products of  ~100 for pulses of width  ~1 ps are observed, with no fundamental limit on the system bandwidth.

Simultaneous cell traction and growth measurements using light.

Characterizing the effects of force fields generated by cells on proliferation, migration and differentiation processes is challenging due to limited availability of nondestructive imaging modalities. Here, we integrate a new real-time traction stress imaging modality, Hilbert phase dynamometry (HPD), with spatial light interference microscopy (SLIM) for simultaneous monitoring of cell growth during differentiation processes. HPD uses holographic principles to extract displacement fields from chemically patterned fluorescent grid on deformable substrates. This is converted into forces by solving an elasticity inverse problem. Since HPD uses the epi-fluorescence channel of an inverted microscope, cellular behavior can be concurrently studied in transmission with SLIM. We studied the differentiation of mesenchymal stem cells (MSCs) and found that cells undergoing osteogenesis and adipogenesis exerted larger and more dynamic stresses than their precursors, with MSCs developing the smallest forces and growth rates. Thus, we develop a powerful means to study mechanotransduction during dynamic processes where the matrix provides context to guide cells toward a physiological or pathological outcome.

Ratiometric analysis of optical coherence tomography-measured in†vivo retinal layer thicknesses for the detection of early diabetic retinopathy.

Influence of diabetes mellitus (DM) and diabetic retinopathy (DR) on parafoveal retinal thicknesses and their ratios was evaluated. Six retinal layer boundaries were segmented from spectral-domain optical coherence tomography images using open-source software. Five study groups: (1) healthy control (HC) subjects, and subjects with (2) controlled DM, (3) uncontrolled DM, (4) controlled DR and (5) uncontrolled DR, were identified. The one-way analyses of variance (ANOVA) between adjacent study groups (i. e. 1 with 2, 2 with 3, etc) indicated differences in retinal thicknesses and ratios. Overall retinal thickness, ganglion cell layer (GCL) thickness, inner plexiform layer (IPL) thickness, and their combination (GCL+ IPL), appeared to be significantly less in the uncontrolled DM group when compared to controlled DM and controlled DR groups. Although the combination of nerve fiber layer (NFL) and GCL, and IPL thicknesses were not different, their ratio, (NFL+GCL)/IPL, was found to be significantly higher in the controlled DM group compared to the HC group. Comparisons of the controlled DR group with the controlled DM group, and with the uncontrolled DR group, do not show any differences in the layer thicknesses, though several significant ratios were obtained. Ratiometric analysis may provide more sensitive parameters for detecting changes in DR. Picture: A representative segmented OCT image of the human retina is shown.