Optical Phase Function Research Articles

Single-shot digital holography of multiple-beam Fizeau holograms for optical phase reconstruction through the focal depth of optical fibers

Multiple-beam Fizeau interference (MBFI) is characterized by its ability to represent the fine features of phase objects. Single-shot digital holographic interferometry (SSDHI) is used for numerical focusing of the recorded holograms. In this work, the two methods are gathered and employed for solving the problem of studying the thick phase objects. The problem of thick phase objects, such as optical fibers, in MBFI is the full focusing of the fiber's optical phase in one-time single shot phase map. The proposed method is designed to collect the optical phase map due to the optical fiber through the focal depth. The main idea is based on the analysis of the MBFI to several two-beam interference sets. Each two-beam interference is treated by SSDHI method to get the optical phase at different planes of focus inside the object. Then, the resultant optical phase due to the optical fiber is collected from these planes of focus. Retrieving the optical phase maps of the multiple-beam holograms through the focal depth, for the first time, is possible using the angular spectrum method. In this case, a laser source with quite long coherence length is used as a light source.The proposed method is applied on two thick samples of, single mode and graded index, optical fibers to reconstruct the resultant optical phase maps through the focal depth. The reconstructed optical phase maps are determined for the harmonics of MBFI holograms’ angular spectra. The method proves its ability to provide more accurate, numerical focusing and phase maps reconstruction. Where, MBFI analyses indicate the magnification of optical phase differences.

Optical phase retrieving of a projected object by employing a differentiation of a single pattern of two-beam interference

In this work, we present a new approach to retrieve the optical phase map of an object which is projected by a single differentiated two-beam interference pattern. This approach is based on the differentiation of the intensity equation of the two-beam interference with respect to the carrier’s phase angle. Therefore, two interference patterns which are shifted by a very small phase angle can be obtained. Then, these two patterns are projected on the object. By exploiting the definition of the mathematical differentiation, the optical phase object’s variations are retrieved from the recorded intensity distributions of both projected patterns. According to this method, the extracted optical phase angles are raised as an inverse “sin” function. This means that the unwrapping process of this function limits the recovered phase angles between − π/2 and π/2. So, the unwrapping process of these unusual wrapped phase angles is explained. The proposed method is applied on (a) two objects which are simulated by combinations of multiple Gaussian functions and (b) a 3D real object. It is found that the inclination of the projected interference pattern on the object redistributes the intensity distribution due to the Lamber’s “cos” aw of illumination. This effect is considered in the retrieving process of the object’s phase map. The limitations of the presented method are discussed and the obtained results are found promising.

Monitoring Human Neutrophil Differentiation by Digital Holographic Microscopy

We report on the usefulness of digital holographic microscopy (DHM) for the assessment of human neutrophil differentiation from myeloid cells. The cell and nuclear regions have been designated by image segmentation of the optical phase function, and the changes of the cell nucleus morphology in relation to the whole cell morphology have been examined during the process of granulocytic differentiation into mature neutrophils in PLB-985 cell line. Nucleus phase volume and circularity and the ratios between the nucleus and the cell projected area and volume provide a reliable set of parameters to characterize the maturation process. As control, cell differentiation has been monitored in parallel using standard nucleus staining and fluorescence imaging. From this research, it emerged that DHM can be used as a valid label-free solution, alternatively to the standard staining technology.

Efficient vortex beam generation using gradient refractive-index microphase plates.

Vortex beams were theoretically demonstrated by patterning a fiber facet with $N$-segment microphase plates. By changing the aluminum oxynitride material composition of each segment, gradient refractive-index phase plates (GRPs) were designed and introduced a ${{2}}\pi l$ azimuthal optical phase difference. The gradient index profile was able to convert a fiber Gaussian mode to a Laguerre-Gaussian mode with varieties of topological charge $l$. A three-dimensional finite-difference time-domain method was applied to calculate the near-field optical phase maps and the far-field beam profiles projected from the micro-GRPs. A uniform vortex beam with a symmetrical doughnut shape was obtained by optimizing the GRPs' radii and the number of segments. The micro-GRPs enabled flat optical components for efficient vortex beam generation.

A Toolbox for Isophase-Curvature Guided Computation of Metrology Hologram.

We describe the algorithmic foundations of an open-source numerical toolbox, written in the Octave language, for the creation of computer-generated binary and multi-level holograms used in interferometric form error measurements of complex aspheric and free-form precision surfaces and wavefronts. In a typical measurement setup for this type of surface, a hologram is used to generate a test wavefront that has the design shape of the surface, which is then compared to a fabricated part using an imaging laser interferometer. The optical function of the hologram in the measurement is generally modeled with optical ray-tracing software and it can be encapsulated by a scalar optical phase function φ : R2 →R. The toolbox converts phase functions into equivalent binary holograms that generate the desired test wavefronts for an interferometric form error measurement. The algorithms in this toolbox take advantage of the relationship between the local properties of phase functions and the local geometry (curvature) of isophase lines. It forms the core of an effcient algorithm for the computation of optical holograms. Holograms are created in a format that can be processed by most laser-or e-beam lithography systems. While the toolbox is chiefy aimed at the creation of hologram layouts needed for measurements of precision surfaces and wavefronts, we show that the isophase-following algorithm is easily extended to phase functions with singularities and discontinuities. Such phase functions result in holograms with zone bifurcation and they can be used to generate helical wavefronts. Light beams with helical wavefronts have applications beyond surface and wavefront metrology. The toolbox also includes a family of functions for the effcient estimation and evaluation of Zernike polynomials, which are widely used in optical applications.

Employment of Fizeau fringe disintegration to magnify small phase variations

When an object produces a small phase variation not less than the minimum phase variation detectable by the used system and more than the system’s noise, the object is considered as a tiny (or small) object. The measurement of interferometric phase differences evolved from a tiny optical phase object has gained attention in recent years due to its usefulness in accurate detection of nanometric dimensions. In the current work, a Fizeau wedge interferometer is used as an aberrationless method to amplify phase variations of a tiny object. The phase object is inserted between the two semi-reflecting/partially transmitting plates making a small wedge angle in between. The fast Fourier transformation (FFT) spectrum of the obtained interferograms is characterized by some frequency equidistant peaks. The number of these peaks is related to the number of the interfered rays and each peak represents a set of two-beam interferences. We applied a mask to select one of these peaks and reproduced its corresponding optical phase map via employing inverse fast Fourier transformation. The produced map contains the magnified phase variation with a magnification factor equal to 2 times the selected peak’s order. Experimental verification of two different phase object samples to amplify their phase variations with different amplification factors is performed. An integrated magnified phase by the object is calculated to test the validity of the proposed method. It is found that the magnification according to the selection of different FFT peaks is a little bit deviated from the expected magnification factors which is referred to ignoring refraction of the optical rays by the object. In spite of these deviations, good results are obtained.

Radial polarizing phase-shifting interferometry with applications to single-shot n interferogram measurements and potential usage for white light interferogram analysis.

In this research, we present an interferometric system to analyze transparent samples using interferograms generated by a phase-shifting radial shear grating interferometer for two cases: the first obtaining n simultaneous phase-shifting interferograms using a coherent light source and the second one using sequential phase steps with a white light source. For the first case, the simultaneous interferograms are generated using two optical systems: the first one generates the polarized pattern while the second one consists of a 4f system creating replicas of the output interferograms. By using a 2D sinusoidal phase grating, we have the advantage of obtaining up to nine replicated interferograms, all of them with comparable intensities and having amplitudes modulated by the 2D sinusoidal phase grating diffraction orders as zero-order Bessel's functions. To obtain the optical phase map, several phase shifts are generated by placing a polarizing filter covering each replicated interferogram. We highlight the advantage of using n simultaneous interferograms by comparing resulting optical phases processed by a conventional four-step algorithm against those obtained by an implemented n=N+1 method, reducing errors with noisy interferograms. Results for n=7 and n=9 cases are presented. In addition, we have tested the setup with white light interference techniques by employing the polarizer radial shearing interferometer; for this case, the optical phase is calculated with the four-step and the three-step algorithms. Results of testing the developed system to examine static and dynamic phase objects are also included.

Parallel phase shifting radial shear interferometry with complex fringes and unknown phase shift.

We present an interferometric method to analyze transparent samples using complex fringes generated by a parallel phase shifting radial shear interferometer using two coupled interferometers. Parallel interferograms are generated using two interferometers: the first one generates the polarized base pattern, and the second system is used to generate parallel interferograms allowing the adjustment of the x-y positions of the parallel interferograms. To obtain the optical phase map, parallel phase shift is generated by collocating polarizing filters at the output of the system; the polarizers are placed at arbitrary angles since they do not require adjustment because of the phase-recovery algorithm. The optical phase was processed using a two-step algorithm based on a modified Gram-Schmidt orthogonalization method. Such an algorithm has the advantage of not being iterative and is robust to amplitude modulation. The proposed method reduces the number of captures needed in phase-shifting interferometry. We applied the developed system to examine static and dynamics phase objects.

Efficient mid-Infrared vortex beam generation using optical waveguides integrated with micro-spiral phase plates

Vortex beam generation from a mid-infrared photonic chip was numerically demonstrated by engineering the phase front of a waveguide mode. Multi-level micro-spiral phase plates (SPPs) consisting of N segments were patterned on the waveguide facet. The micro-SPP introduced an annular phase change by 2π that converted the Hermite–Gaussian (HG) waveguide mode into Laguerre–Gaussian (LG) mode at topological charge ∣l∣ = 1. Using a 3D finite difference time domain simulation, we resolved the near-field optical phase map next to the micro-SPP and its far-field beam profile projected from the waveguide surface. The vortex beam formation was spectrally dependent. A uniform and symmetric donut-shaped pattern corresponding to an ideal vortex beam was created at λ = 3.1–3.3 μm. In addition, the HG → LG mode conversion efficiency was improved by increasing the number of segments, N. Thus, our micro-SPP enabled present photonic circuits to create vortex beams, which is critical for the development of optical interconnection and broadband communication.

Dynamic Mach\u2013Zehnder interferometer based on a Michelson configuration and a cube beam splitter system

We developed simultaneous phase-shifting system based on a Mach–Zehnder interferometer and a replicating system integrated by a Michelson configuration and a cube beam splitter. The system is capable to obtain four simultaneous interferograms in a single capture, and the phase shifts are controlled by placing a linear polarizer in each replica obtained. The system retrieves four interferograms with a relative phase shift of π/2 and the optical phase map is calculated using the four-step algorithm. In addition, the configuration presents potential capabilities for generating spiral interference patterns. To show the advantage of the technique, experimental results are presented for static and dynamic samples.

Quantitative phase microscopy of red blood cells during planar trapping and propulsion.

Red blood cells (RBCs) have the ability to undergo morphological deformations during microcirculation, such as changes in surface area, volume and sphericity. Optical waveguide trapping is suitable for trapping, propelling and deforming large cell populations along the length of the waveguide. Bright field microscopy employed with waveguide trapping does not provide quantitative information about structural changes. Here, we have combined quantitative phase microscopy and waveguide trapping techniques to study changes in RBC morphology during planar trapping and transportation. By using interference microscopy, time-lapsed interferometric images of trapped RBCs were recorded in real-time and subsequently utilized to reconstruct optical phase maps. Quantification of the phase differences before and after trapping enabled study of the mechanical effects during planar trapping. During planar trapping, a decrease in the maximum phase values, an increase in the surface area and a decrease in the volume and sphericity of RBCs were observed. QPM was used to analyze the phase values for two specific regions within RBCs: the annular rim and the central donut. The phase value of the annular rim decreases whereas it increases for the central donut during planar trapping. These changes correspond to a redistribution of cytosol inside the RBC during planar trapping and transportation.

Measurement of red blood cell characteristic using parallel phase shifting interferometry

In this research, an interferometric technique is presented for the calculation of the optical path difference (OPD) induced by a red blood cell (RBC) to measure its 3D phase profile, allowing the examination of its morphology which is very important in the biomedical field to diagnose diseases. The developed system is based on a polarized cyclic path interferometer with a nonpolarizing beam splitter in the output, which generates two interferograms with relative phase shifts of π/2. From these two interferograms, which are captured in single-shot of the camera, it is possible to measure the optical phase map using a regularized optical flow algorithm developed by Vargas–Quiroga. The presented measurement system is immune to vibrations caused by environmental disturbances, so it can be implemented outside the laboratory; additionally, the use of two interferograms simplify the number of phase steps required to find the optical phase, and it can be adapted to obtain the optical phase with four interferograms obtained in two shots. Using the proposed technique, the experimental results showing the 3D morphology of RBC are presented.

Full off-axis red-green-blue digital holographic microscope with LED illumination

We developed a new full off-axis red-green-blue (RGB) digital holographic microscope with an LED illumination. A decisive advantage of the use of LED illumination is a large image quality improvement due to its partially coherent nature. The off-axis configuration enables the fast recording of the holographic data in each spectral channel. The digital holographic refocusing and the optical phase map computation are successfully demonstrated. The multiwavelength operation provides a significant improvement of the collected information for colored samples.

Achromatic digital speckle pattern interferometer with constant radial in-plane sensitivity by using a diffractive optical element

We report on a digital speckle pattern interferometer that applies a binary diffractive optical element (DOE) to generate double illumination and radial in-plane sensitivity. The application of the DOE ensures independence on the wavelength of the laser used as an illumination source. Furthermore, in-plane sensitivity only depends on the grating period of the DOE. An experimental setup was built allowing the measurement of a set of radial in-plane displacement fields either using a red laser as a light source or a green one. When displacement fields computed from the measured optical phase maps obtained with a red or a green laser were compared, two main results were observed: (a) deviations between mean values ranged only up to 7 nm and (b) phase maps presented the same amount of fringes. In addition, phase maps measured with the red laser were processed as they were obtained with green light. For this case, deviations have ranged only up to 0.5 nm. On the other hand, a set of measurements performed changing the DOE by a conical mirror showed clearly that radial in-plane sensitivity increased when the red laser was changed by the green one.

Identification of the electroelastic coupling from full multi-physical fields measured at the micrometre scale

Metal coated microcantilevers are used as transducers of their electrochemical environment. Using the metallic layer of these cantilevers as a working electrode allows one to modify the electrochemical state of the cantilever surface. Since the mechanical behaviour of micrometre scale objects is significantly surface-driven, this environment modification induces bending of the cantilever. Using a full-field interferometric measurement set-up to monitor the objects then provides an optical phase map, which is found to originate from both electrochemical and mechanical effects. The scaling of the electrochemically-induced phase with respect to the surface charge density is modelled according to Gouy–Chapman–Stern theory, whereas the relationship between the mechanical effect and the surface charge density is analysed. An identification technique is described to determine a modelling of the electroelastic coupling and to identify the spatial charge density distribution from full-field phase measurements. Minimizing the least-squares gap between the measured phase and a statically admissible phase field, the mechanical effect is found to be charge-driven. The charge density field is also found to be singular on the cantilever edge, and the shear stress versus charge density is found to be non-linear.

Nomarski imaging interferometry to measure the displacement field of micro-electro-mechanical systems

We propose to use a Nomarski imaging interferometer to measure the out- of-plane displacement field of micro-electro-mechanical systems. It is shown that the measured optical phase arises from both height and slope gradients. By using four integrating buckets, a more efficient approach to unwrap the measured phase is presented, thus making the method well suited for highly curved objects. Slope and height effects are then decoupled by expanding the displacement field on a functions basis, and the inverse transformation is applied to get a displacement field from a measured optical phase map change with a mechanical loading. A measurement reproducibility of approximately 10 pm is achieved, and typical results are shown on a microcantilever under thermal actuation, thereby proving the ability of such a setup to provide a reliable full-field kinematic measurement without surface modification.

Measurement of the complex amplitude of transient surface acoustic waves usingdouble-pulsed TV holography and a two-stage spatial Fourier transform method

We present a technique to measure the mechanical complex amplitude, i.e. themechanical amplitude and phase of vibration, of an ultrasonic plane wavefield ofnanometric amplitude that propagates on a surface. Our aim is to detect perturbationsof the initially smooth wavefronts that indicate the presence of flaws in thematerial. We use bursts of surface acoustic waves (SAWs) and a double-pulsed TVholography system that records two correlograms with time separations down to1.5 µs.The phases of the correlograms are calculated separately using the spatial Fouriertransform method (SFTM) and operated on to obtain the phase change between exposures.In the resultant optical phase map, the field of instantaneous displacements of the surface(that comprises several periods of the SAW) acts as a modulated spatial carrier, nowrelated to the mechanical phase and amplitude, that are extracted by applying the SFTMagain.

Measurement of the temporal evolution of periodic induced displacement derivatives using stroboscopic electronic speckle-shearing interferometry

Single pulsed subtraction electronic speckle-shearing pattern interferometry is used to measure the time evolution of deformations induced by periodic impulse loading on a metal plate. This is accomplished with a stroboscopic system based on a pulsed laser and an interline-transfer CCD camera running at 60 frames/s. In a few seconds, a sequence of hundreds of interferograms is recorded with an effective sampling time interval of some tens of microseconds. We describe the experimental setup used to generate the transient deformation on the metal plate and also the synchronization of laser pulses and image acquisition. Several correlation fringe patterns are presented, showing the time evolution of the deformation. Finally, these data are used to calculate optical phase maps and displacement derivative fields for different times after the application of the dynamic load on the object.

Topological singularities in cortical orientation maps: the sign theorem correctly predicts orientation column patterns in primate striate cortex

Optical imaging methods have revealed the spatial arrangement of orientation columns across striate cortex, usually summarized in terms of two measurements at each cortical location: (i) a ‘best’ stimulus orientation, corresponding to the stimulus orientation that elicits a maximal response, and (ii) the magnitude of the response to the best orientation. This mapping has been described as continuous except at a set of singular points (also termed ‘vortices’ or ‘pinwheels’. Although prior work has shown that vortex patterns qualitatively similar to those observed in visual area 17 of the Macaque cortex can be produced by either band-pass or low-pass filtering of random vector fields, there has been to date little further topological characterization of the structure of cortical vortex patterns. Nevertheless, much theoretical work has been done in other disciplines on mappings analogous to the cortical orientation map. In particular, a recent theorem in the optics literature, termed the sign principle, states that adjacent vortices on zero crossings of a phase (orientation) mapping must always alternate in sign. Using digitized samples of recently published optical recording data in monkey striate cortex, we show that the cortical orientation data does indeed possess 100% anti-correlation in vortex sign for nearest-neighbour vortices, as predicted by the sign theorem. This provides strong experimental support for the assumptions of continuity of cortical vortex maps which underly the sign theorem. Similar analysis predicts a lack of ‘higher-order’ vortices in the cortical orientation map, which is also found to be in agreement with optical imaging observations. It also follows from this work that cortical vortices must be created simultaneously in clockwise–anti-clockwise pairs. This suggests a possible basis for a modular (hyper-columnar) relationship among pairs of cortical vortices that originate at the same developmental time. In summary, this work indicates that primate visual cortex orientation column structure is best understood in the context of other ‘ordered continuous media’ (e.g. liquid He3, cholesteric liquid crystals, random optical phase maps, to name only a few) in which an order parameter (orientation in this case) is mapped to a physical space, and in which the topological properties of the mapping determine the observable regularities of the system. We also point out that these methods may well be applied to a variety of other cortical map systems which admit an ‘order parameter’, i.e. for which each cortical position is assigned a continuous stimulus value.

Determination of the optical phase function of Ag and Mn films

The optical phase function F = 2 γ 1 − β 1 − β 2 of silver and manganese thin films, which is used as an upper component in a multiple-beam air/wedge at reflection, and is responsible for the formation of transmission-like fringes at reflection is determined. The dependence of such a redistribution on thickness has been investigated for Ag films ranging from 80 Å to 470 Å and for Mn films from 155 Å to 685 Å at a wavelength of 5461 Å. Two methods have been performed, the first is based on computing the intensity distribution for different values of F and Δ, and comparing it with the experimental microphotometer tracing. The second method is based on computing the phase at both l max and l min.