Single-shot Interferogram Research Articles

A High Precison Phase Extraction Algorithm from a Single Shot Interferogram

Abstract. Phase shifting interferometry is an optical metrology with a high precision, but it usually requires the expensive high precision phase shifter. Therefore, low cost methods of extracting phase from a single shot interferogram were very valuable, but available algorithms are usually effective only in a limited conditions. To solve these problems, a novel method is presented in this work. Based on this method, the interference fringe is retrieved to a wrapped phase and divided into different regions and the index of the pixel is calculated. The pixels in the same region have the same parity and the PSF’s gravity center of part wavefront so as to solve the sign ambiguity. The theoretical simulation results indicate that the PV of wavefront error is 0.00054λ and the rms is 0.000125λ,which is much better than the results from the Fast Fourier Transformation method. The experimentally measured interferogram is also used to validate the method. It has the advantages of simplicity, high precision and effective for both open and closed interferometer fringes.

Single shot polarimetric characterization of the polymer based flexible polyethylene terephthalate substrate

AbstractConducting polyethylene terephthalate (PET) substrates used for the flexible electronics and photonics are birefringent in nature. Birefringence of the PET substrate increases with the external applied stress. In this paper, we have studied the state of the polarization of the light transmitted through the flexible PET substrate. Polarimetric characterization of flexible substrate was done for red and yellow wavelengths of light. Single shot three beams interference was recorded by the polarizing Mach–Zehnder interferometer. Interferograms were recorded without bending and different external bending applied to the substrate which increases the stress. The recorded interferograms was analyzed by the Fourier transform fringe (FTF) analysis method. The single shot interferograms have the information of the amplitude and phase of orthogonally polarized field components. Stokes parameters were calculated from orthogonal polarization component extracted from the single shot interferograms analyzed by FTF. Calculated Stokes parameters were plotted on the Poincare sphere to see the change in the state of polarization (SOP) of the light transmitted through these substrates. The change in the SOP of the transmitted light was observed without and with the external stress applied to the PET substrate due to its birefringence characteristic.

Radius of curvature measurement based on iterative correlation coefficient matching strategy with a single-shot interferogram

We proposed an iterative correlation coefficient matching strategy for the high-precision measurement of the radius of curvature (ROC), which only needs a single-frame interferogram. This strategy includes two parts: one is to obtain an initial ROC by using the relationship between the ROC and the diameter of the dark ring; other is to further solve a high-precision value by an iterative method. In the iterative part, we simulated a series of interference patterns with each ROC in the setting range, and calculated the correlation coefficient between the each simulated interferogram and the measured interferogram, and searched the maximum of the correlation coefficient that would be used to confirm a new ROC in the next iteration. When the calculated ROCs in the adjacent iterations meet the termination condition, the high-accuracy ROC can be solved. The feasibility of our proposed method has been verified by simulations and experiments. The errors of the test mirrors with the ROC of 101.6 mm and 1000 mm are 0.013% and 0.024% in experiments, respectively. This research provides a simple and feasible method for realizing the high-precision ROC measurement.

Dynamic Speckle Illumination Digital Holographic Microscopy by Doubly Scattered System

The coherent noise always exists in digital holographic microscopy due to the laser source, degrading the image quality. A method of speckle suppression using the dynamic speckle illumination, produced by double-moving diffusers, is presented in digital holographic microscopy. The space–time correlation functions are theoretically analyzed from the statistics distribution in the doubly and singly scattered system, respectively. The configuration of double-moving diffusers is demonstrated to have better performance in speckle suppression compared with the single diffuser and moving-static double diffusers cases. The experiment results verify the feasibility of the approach. The presented approach only requires a single shot interferogram to realize the speckle reduction, accordingly it has the potential application in real-time measurement.

Fast demodulation of single-shot interferogram via convolutional neural network

In this report, a method is proposed to demodulate a single-shot interferogram based on deep learning using a convolutional neural network (CNN). The time required to demodulate a single interferogram is on the order of milliseconds. The root mean square (RMS) of the residual wavefront between the original wavefront and the estimated wavefront reached the order of 10−2λ. Moreover, the cost was lower than using commercial dynamic and phase-shifting interferometers. This method yielded superior real-time performance and high precision. The results indicated that this approach based on deep learning is robust and feasible for demodulating single-shot interferograms.

Variational Hilbert Quantitative Phase Imaging

Utilizing the refractive index as the endogenous contrast agent to noninvasively study transparent cells is a working principle of emerging quantitative phase imaging (QPI). In this contribution, we propose the Variational Hilbert Quantitative Phase Imaging (VHQPI)—end-to-end purely computational add-on module able to improve performance of a QPI-unit without hardware modifications. The VHQPI, deploying unique merger of tailored variational image decomposition and enhanced Hilbert spiral transform, adaptively provides high quality map of sample-induced phase delay, accepting particularly wide range of input single-shot interferograms (from off-axis to quasi on-axis configurations). It especially promotes high space-bandwidth-product QPI configurations alleviating the spectral overlapping problem. The VHQPI is tailored to deal with cumbersome interference patterns related to detailed locally varying biological objects with possibly high dynamic range of phase and relatively low carrier. In post-processing, the slowly varying phase-term associated with the instrumental optical aberrations is eliminated upon variational analysis to further boost the phase-imaging capabilities. The VHQPI is thoroughly studied employing numerical simulations and successfully validated using static and dynamic cells phase-analysis. It compares favorably with other single-shot phase reconstruction techniques based on the Fourier and Hilbert–Huang transforms, both in terms of visual inspection and quantitative evaluation, potentially opening up new possibilities in QPI.

Rapid quantitative interferometric microscopy using fast Fourier transform and differential–integral based phase retrieval algorithm (FFT-DI-PRA)

Phase retrieval in quantitative interferometric microscopy is a time-consuming and computationally expensive process, thus limiting the applications on cell dynamic detections. Here, we propose Fast Fourier Transform and Differential–Integral based Phase Retrieval Algorithm (FFT-DI-PRA) focusing on quantitative cell phase imaging to significantly accelerate the phase retrieval process. The proposed method first obtains the wrapped phase from a single-shot interferogram using the FFT based phase extraction method; then, its phase derivatives are computed along two orthogonal directions; finally, the unwrapped specimen phase is reconstructed by phase integration from corrected phase derivatives. Since the FFT based phase extraction method can effectively suppress the noise, the phase wraps can be easily recognized in phase derivatives avoiding the noise influences, thus supporting the rapid and accurate phase wrapping. As a perfect combination of the FFT based phase extraction and the differential–integral based phase wrapping, FFT-DI-PRA only takes ∼0.03 second for phase imaging from one single-shot interferogram of 512×512 pixels using a typical desktop; moreover, it provides high-accurate specimen phase distributions as shown here by both simulations and experiments, suggesting that FFT-DI-PRA is an effective tool used in rapid quantitative live cell phase imaging and display.

Mean gradient descent: an optimization approach for single-shot interferogram analysis.

Complex object wave recovery from a single-shot interference pattern is an important practical problem in interferometry and digital holography. The most popular single-shot interferogram analysis method involves Fourier filtering of the cross term, but this method suffers from poor resolution. To obtain full pixel resolution, it is necessary to model the object wave recovery as an optimization problem. The optimization approach typically involves minimizing a cost function consisting of a data consistency term and one or more constraint terms. Despite its potential performance advantages, this method is not used widely due to several tedious and difficult tasks such as empirical tuning of free parameters. We introduce a new optimization approach, mean gradient descent (MGD), for single-shot interferogram analysis that is simple to implement. MGD does not have any free parameters whose empirical tuning is critical to the object wave recovery. The MGD iteration does not try to achieve minimization of a cost function but instead aims to reach a solution point where the data consistency and the constraint terms balance each other. This is achieved by iteratively progressing the solution in the direction that bisects the descent directions associated with the error and constraint terms. Numerical illustrations are shown for recovery of a step phase object from its corresponding off-axis as well as on-axis interferograms simulated with multiple noise levels. Our results show full pixel resolution as evident from the recovery of the phase step and excellent rms phase accuracy relative to the ground truth phase map. The concept of MGD as presented here can potentially find applications to a wider class of optimization problems.

Angle-resolved spectral Fabry–Pérot interferometer for single-shot measurement of refractive index dispersion over a broadband spectrum

An angle-resolved spectral Fabry–Pérot interferometer is reported for fast and accurate measurement of the refractive index dispersion of optical materials with parallel plate shape. The light sheet from the wavelength tunable laser is incident on the parallel plate with converging angles. The transmitted interference light for each angle is dispersed and captured by a 2D sensor, in which the rows and the columns are used to simultaneously record the intensities as a function of wavelength and incident angle, respectively. The interferogram, named angle-resolved spectral intensity distribution, is analyzed by fitting the phase information instead of finding the fringe peak locations that present periodic ambiguity. The refractive index dispersion and the physical thickness can be then retrieved from a single-shot interferogram within 18 s. Experimental results of an optical substrate standard indicate that the accuracy of the refractive index dispersion is less than 2.5 × 10−5 and the relative uncertainty of the thickness is 6 × 10−5 mm (3σ) due to the high stability and the single-shot measurement of the proposed system.

Characterization of temporal coherence of hard X-ray free-electron laser pulses with single-shot interferograms.

Temporal coherence is one of the most fundamental characteristics of light, connecting to spectral information through the Fourier transform relationship between time and frequency. Interferometers with a variable path-length difference (PLD) between the two branches have widely been employed to characterize temporal coherence properties for broad spectral regimes. Hard X-ray interferometers reported previously, however, have strict limitations in their operational photon energies, due to the specific optical layouts utilized to satisfy the stringent requirement for extreme stability of the PLD at sub-ångström scales. The work presented here characterizes the temporal coherence of hard X-ray free-electron laser (XFEL) pulses by capturing single-shot interferograms. Since the stability requirement is drastically relieved with this approach, it was possible to build a versatile hard X-ray interferometer composed of six separate optical elements to cover a wide photon energy range from 6.5 to 11.5 keV while providing a large variable delay time of up to 47 ps at 10 keV. A high visibility of up to 0.55 was observed at a photon energy of 10 keV. The visibility measurement as a function of time delay reveals a mean coherence time of 5.9 ± 0.7 fs, which agrees with that expected from the single-shot spectral information. This is the first result of characterizing the temporal coherence of XFEL pulses in the hard X-ray regime and is an important milestone towards ultra-high energy resolutions at micro-electronvolt levels in time-domain X-ray spectroscopy, which will open up new opportunities for revealing dynamic properties in diverse systems on timescales from femto-seconds to nanoseconds, associated with fluctuations from ångström to nanometre spatial scales.

Single-shot interferogram analysis for accurate reconstruction of step phase objects.

We describe a constrained optimization approach for demodulation of single-shot interferograms corresponding to step phase objects. The demodulation problem is formulated as a minimization problem for a cost function consisting of a L2-norm squared error term and a gradient-based penalty (total variation) suitable for step objects. The optimization approach is tested with two methods that use complex (or Wirtinger) derivatives for the iterative solution update. The first simplistic method has a free parameter and thus provides a family of solutions. The second method is practically more valuable in that it is adaptive in nature and automatically leads to an appropriate solution. Both the off-axis case with straight line fringes and the on-axis case with closed fringes are treated in a unified manner. Excellent recovery of the phase of the step object suggests that the optimization approach as presented here is capable of providing full detector resolution, which is usually difficult to achieve in single-shot interferogram analysis.

Ultrafast measurements of optical spectral coherence by single-shot time-stretch interferometry.

The palette of laser technology has significantly been enriched by the innovations in ultrafast optical pulse generation. Our knowledge of the complex pulse dynamics, which is often highly nonlinear and stochastic in nature, is however limited by the scarcity of technologies that can measure fast variation/fluctuation of the spectral phase (or coherence) and amplitude in real-time, continuously. To achieve this goal, we demonstrate ultrafast interferometry enabled by optical time-stretch for real- time spectral coherence characterization with microsecond-resolution. Accessing the single-shot interferograms continuously, it further reveals the degree of second-order coherence, defined by the cross-spectral density function, at high speed-a capability absent in any existing spectroscopic measurement tools. As the technique can simultaneously measure both the high-speed variations of spectrally resolved coherence and intensity, time-stretch interferometry could create a new arena for ultrafast pulse characterization, especially favorable for probing and understanding the non-repetitive or stochastic dynamics in real-time.

Single shot interferogram analysis for optical metrology.

We report a novel constrained optimization method for single shot interferogram analysis. The unknown test wavefront is estimated as a minimum L2-norm squared solution whose phase is constrained to the space spanned by a finite number of Zernike polynomials. Using a single frame from standard phase shifting datasets, we demonstrate that our approach provides a phase map that matches with that generated using phase shifting algorithms to within λ/100 rms error. Our simulations and experimental results suggest the possibility of a simplified low-cost high quality optical metrology system for performing routine metrology tests involving smooth surface profiles.

Full-field Hilbert phase microscopy using nearly common-path low coherence off-axis interferometry for quantitative imaging of biological cells

We demonstrate single shot low coherence quantitative Hilbert phase microscopy (HPM) for the reconstruction of the two dimensional (2D) phase map of biological cells. The system is based on a compact and nearly common-path high magnification Mirau-interferometric objective lens. The spatial carrier frequency of the interference fringes was increased by means of introducing tilt in one of the arms of the interferometer, thus making the system off-axis. The system is user friendly as the interference fringes and imaging of objects with high lateral and axial resolution can be obtained quickly using a low cost commercially available microscope. Experimental results for the 2D phase map of polystyrene spheres and human red blood cells (RBCs) are presented. Hilbert transform fringe analysis was used for reconstructing the phase map and refractive index (RI) of the objects. For dynamic substances which change rapidly, single shot low coherence interferometric microscopy is an important method for obtaining the phase. Experimental results with increased field-of-view and large tilt angle are also presented. It is well known that on increasing the tilt angle for improved spatial phase sampling the object remains focused in only a small area even though the field-of-view is large. This limitation was overcome by means of vertical scanning low coherence interferometry. Due to the low coherence properties of the light source the interference occurs only at the desired location of the object, i.e., where the object is sharply focused. The object was vertically scanned and the single shot interferograms were recorded for every scan and analyzed by Hilbert transform. In this way a large area of the sample can be imaged quantitatively.

Picosecond Soft-X-ray Laser Interferometer for Probing Nanometer Surface Structure

We have developed a soft-X-ray laser interferometer based on a double Lloyd's mirror and obtained a single-shot interferogram using a 7-ps-duration pulse at a wavelength of 13.9 nm. Micrometer grooves with a depth of 5 nm were successfully reconstructed from the interferogram. The lateral and depth resolutions were estimated to be 1.5 µm and better than 1 nm, respectively. This interferometer will be an attractive diagnostic device for observing transiently changing nanometer-scale deviations on solid surfaces.

3D reconstruction of the complex dielectric function of glass during femtosecond laser micro-fabrication

We report on a new technique to reconstruct the 3D dielectric function change in transparent dielectric materials and the application of the technique for on-line monitoring of refractive index modification in BK7 glass during direct femtosecond laser microfabrication. The complex optical field scattered from the modified region is measured using two-beam, single-shot interferogram and the distribution of the modified refractive index is reconstructed by numerically solving the inverse scattering problem in Born approximation. The optical configuration suggested is further development of digital holographic microscopy. It takes advantage of high spatial resolution and almost the same optical paths for both interfering beams, and allows ultrafast time resolution. © Springer Science+Business Media, LLC. 2011.

Internal frequency conversion extreme ultraviolet interferometer using mutual coherence properties of two high-order-harmonic sources

We report on an innovative two-dimensional imaging extreme ultraviolet (XUV) interferometer operating at 32 nm based on the mutual coherence of two laser high order harmonics (HOH) sources, separately generated in gas. We give the first evidence that the two mutually coherent HOH sources can be produced in two independent spatially separated gas jets, allowing for probing centimeter-sized objects. A magnification factor of 10 leads to a micron resolution associated with a subpicosecond temporal resolution. Single shot interferograms with a fringe visibility better than 30% are routinely produced. As a test of the XUV interferometer, we measure a maximum electronic density of 3x10(20) cm(-3) 1.1 ns after the creation of a plasma on aluminum target.

Ultrashort pulse characterization by spectral shearing interferometry with spatially chirped ancillae

We report a new version of spectral phase interferometry for direct electric field reconstruction (SPIDER), in which two spatially chirped ancilla fields are used to generate a spatially encoded SPIDER interferogram. We dub this new technique Spatially Encoded Arrangement for Chirped ARrangement for SPIDER (SEA-CAR-SPIDER). The single shot interferogram contains multiple shears, the spectral amplitude of the test pulse, and the reference phase, which is accurate for broadband pulses. The technique enables consistency checking through the simultaneous acquisition of multiple shears and offers a simple and precise calibration method. All calibration parameters--the shears, and the upconversionfrequency--can be accurately obtained from a single calibration trace.

Group refractive index measurement by fringes of equal chromatic order

Fringes of equal chromatic order in transmission across a thin liquid or a thin solid sample inside a wedge interferometer, followed with a grating spectrograph, are produced. A single-shot interferogram of the air and sample regions is recorded. Locations of fringes maxima in the air region are fitted in a numerical procedure based on Cauchy's dispersion function. Then it is used for measuring the interferometric gap thickness. The order of interference in the sample region is represented by a third-order polynomial in the wavenumber for deducing the sample group refractive index. An error analysis of the measured group refractive index is given. The method is applied for measuring the group refractive index of water and mica samples across the visible spectrum. The method measures both the sample thickness and its group refractive index. It is static with no moving parts and suitable for thin liquid or solid samples without immersion liquids.

Interferometric determination of refraction and dispersion of human blood-serum, saliva, sweat and urine

Multiple-beam interference fringes of equal chromatic order are produced in air and liquid sample interferometric gaps. The two gaps are of the same thickness and simultaneously enclosed in a wedge interferometer. A single shot interferogram containing fringes in the two gaps is sufficient to deduce the needed experimental data. Locations of the fringe maxima, in the two gaps, are introduced in a non-numerical procedure for determining the gap thickness and the liquid-phase refractive indices across the visible spectrum. The method has been used for measuring the phase refractive indices of human blood-serum, saliva, sweat, urine and water liquids. A third-order polynomial dispersion relation is applied for fitting the measured phase indices. Group refractive indices have been derived and fitted to the same dispersion formula.