scholarly journals Multi-Incidence Holographic Profilometry for Large Gradient Surfaces with Sub-Micron Focusing Accuracy

Sensors ◽  
2021 ◽  
Vol 22 (1) ◽  
pp. 214
Author(s):  
Moncy Sajeev Idicula ◽  
Tomasz Kozacki ◽  
Michal Józwik ◽  
Patryk Mitura ◽  
Juan Martinez-Carranza ◽  
...  
Keyword(s):  
Digital Holography ◽  
Fourier Transforms ◽  
Element Approximation ◽  
Fourier Space ◽  
Manual Adjustment ◽  
Large Gradient ◽  
Gradient Surfaces ◽  
Digital Holograms ◽  
Focus Plane ◽  
Numerical Generation

Surface reconstruction for micro-samples with large discontinuities using digital holography is a challenge. To overcome this problem, multi-incidence digital holographic profilometry (MIDHP) has been proposed. MIDHP relies on the numerical generation of the longitudinal scanning function (LSF) for reconstructing the topography of the sample with large depth and high axial resolution. Nevertheless, the method is unable to reconstruct surfaces with large gradients due to the need of: (i) high precision focusing that manual adjustment cannot fulfill and (ii) preserving the functionality of the LSF that requires capturing and processing many digital holograms. In this work, we propose a novel MIDHP method to solve these limitations. First, an autofocusing algorithm based on the comparison of shapes obtained by the LSF and the thin tilted element approximation is proposed. It is proven that this autofocusing algorithm is capable to deliver in-focus plane localization with submicron resolution. Second, we propose that wavefield summation for the generation of the LSF is carried out in Fourier space. It is shown that this scheme enables a significant reduction of arithmetic operations and can minimize the number of Fourier transforms needed. Hence, a fast generation of the LSF is possible without compromising its accuracy. The functionality of MIDHP for measuring surfaces with large gradients is supported by numerical and experimental results.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

Phase-shifting digital holography with an unknown reference beam

2021 ◽  
Author(s):  
Roghayeh Yazdani ◽  
Hamidreza Fallah
Keyword(s):  
Digital Holography ◽  
Phase Retrieval ◽  
Reference Beam ◽  
Phase Shifting ◽  
Retrieval Algorithm ◽  
Reference Field ◽  
Digital Holograms ◽  
Noisy Conditions ◽  
Phase Retrieval Algorithm

In digital holography, errors of the reference field degrade the quality of the reconstructed object field. In this paper, we propose an effective method in phase-shifting digital holography in which the reference field does not need to be known and perfect. The unknown complex amplitudes of both reference and object fields are derived simultaneously. The method employs only five digital holograms and a single execution of a phase retrieval algorithm. So, the required measurements and algorithm executions in this method are fewer than those in other methods; it suggests a simpler and faster method. The effectiveness of the suggested method is indicated by simulation, under noise-free and noisy conditions. Moreover, the capability of the method to extract full information about the phase singularities in both fields is demonstrated.

scholarly journals Fast and spectrally accurate evaluation of gyroaverages in non-periodic gyrokinetic-Poisson simulations

2017 ◽  
Vol 83 (4) ◽  
Author(s):  
J. Guadagni ◽  
A. J. Cerfon
Keyword(s):  
Time Complexity ◽  
Numerical Scheme ◽  
Fourier Transforms ◽  
Hankel Transform ◽  
Log P ◽  
Fourier Space ◽  
Compactly Supported ◽  
Geometric Convergence ◽  
Spatial Grid ◽  
Grid Points

We present a fast and spectrally accurate numerical scheme for the evaluation of the gyroaveraged electrostatic potential in non-periodic gyrokinetic-Poisson simulations. Our method relies on a reformulation of the gyrokinetic-Poisson system in which the gyroaverage in Poisson’s equation is computed for the compactly supported charge density instead of the non-periodic, non-compactly supported potential itself. We calculate this gyroaverage with a combination of two Fourier transforms and a Hankel transform, which has the near optimal run-time complexity$O(N_{\unicode[STIX]{x1D70C}}(P+\hat{P})\log (P+\hat{P}))$, where$P$is the number of spatial grid points,$\hat{P}$the number of grid points in Fourier space and$N_{\unicode[STIX]{x1D70C}}$the number of grid points in velocity space. We present numerical examples illustrating the performance of our code and demonstrating geometric convergence of the error.

scholarly journals Interactive Tools for Teaching Fourier Transforms

2020 ◽  
Vol 1 (1) ◽  
pp. 4
Author(s):  
Carlos R. Baiz
Keyword(s):  
Fourier Transforms ◽  
Lesson Plan ◽  
Real Space ◽  
Fourier Space ◽  
Core Component ◽  
Interactive Approach ◽  
Interactive Teaching ◽  
Mathematical Skills ◽  
Interactive Tools ◽  
Upper Level

Fourier transforms (FT) are universal in chemistry, physics, and biology. Despite FTs being a core component of multiple experimental techniques, undergraduate courses typically approach FTs from a mathematical perspective, leaving students with a lack of intuition on how an FT works. Here, I introduce interactive teaching tools for upper-level undergraduate courses and describe a practical lesson plan for FTs. The materials include a computer program to capture video from a webcam and display the original images side-by-side with the corresponding plot in the Fourier domain. Several patterns are included to be printed on paper and held up to the webcam as input. During the lesson, students are asked to predict the features observed in the FT and then place the patterns in front of the webcam to test their predictions. This interactive approach enables students with limited mathematical skills to achieve a certain level of intuition for how FTs translate patterns from real space into the corresponding Fourier space.

scholarly journals Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 581
Author(s):  
Seth T. Merkel ◽  
Emily J. Pritchett ◽  
Bryan H. Fong
Keyword(s):  
Fourier Analysis ◽  
Recent Work ◽  
Fourier Transforms ◽  
Decay Rates ◽  
Alternative Proof ◽  
Mathematical Framework ◽  
Fourier Space ◽  
Clifford Group ◽  
Depolarizing Channel ◽  
The Ideal

We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami \cite{GH15}, we provide an alternative proof of Wallman's \cite{Wallman2018} and Proctor's \cite{Proctor17} bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero. This framework also allows us to construct a gauge in which the average gate-set error is a depolarizing channel parameterized by the RB decay rates, as well as a gauge which maximizes the fidelity with respect to the ideal gate-set.

scholarly journals Providing a Visual Understanding of Holography Through Phase Space Representations

2020 ◽  
Vol 10 (14) ◽  
pp. 4766
Author(s):  
Tobias Birnbaum ◽  
Tomasz Kozacki ◽  
Peter Schelkens
Keyword(s):  
Phase Space ◽  
Frequency Analysis ◽  
Digital Holography ◽  
Joint Space ◽  
Visual Interpretation ◽  
Spatial Frequencies ◽  
Digital Holograms ◽  
Spatial Coordinates ◽  
Visual Understanding ◽  
Space Frequency

Digital holograms are a prime example for signals, which are best understood in phase space—the joint space of spatial coordinates and spatial frequencies. Many characteristics, as well as optical operations can be visualized therein with so called phase space representations (PSRs). However, literature relies often only on symbolic PSRs or on, in practice, visually insufficient PSRs like the Wigner–Ville representation. In this tutorial-style paper, we will showcase the S-method, which is both a PSR that can be calculated directly from any given signal, and that allows for a clear visual interpretation. We will highlight the power of space-frequency analysis in digital holography, explain why this specific PSR is recommended, discuss a broad range of basic operations, and briefly overview several interesting practical questions in digital holography.

Corrected FFT-Based Method for the Calculation of Deformations in Line Contacts

2005 ◽  
Author(s):  
Vasilios Bakolas ◽  
Wolfgang Borchers ◽  
Alexander F. Liebel
Keyword(s):  
Kernel Function ◽  
Fourier Transforms ◽  
Contact Problems ◽  
Discretization Error ◽  
New Method ◽  
Fourier Space ◽  
Numerical Examples ◽  
Line Contacts ◽  
Improved Accuracy ◽  
Calculation Grid

During the last years the Fast Fourier Transforms (FFT) technique was applied by many researchers for the calculation of the deformations in contact problems. The use of conventional convolution algorithms however led to the inclusion of a periodicity error that proved hard to correct. The correction procedures that had been suggested so far, involved the extension of the calculation grid and thus negated any time gains that the FFT would provide. In order to overcome the lack of accuracy of the current FFT methods, a new method for the calculation of deformations for line contacts was developed. In the framework of this method the so-called periodicity error has been identified as a discretization error in the Fourier space. The new method imposes a correction on the Fourier transformed kernel function in order to compensate the discretization error and achieve better accuracy. Tests have proven that the new method provides the desired accuracy while having a cost of arithmetic operations of O(N log(N)). Numerical examples are presented showing the improved accuracy of the proposed method.

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