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Fourier transform (FT) plays an important role in many fields of optics and signal processing. As its fundamental topics, various sampling theories have been established, of course, which include multidimensional (MD) sampling in the FT domain. However, given the deficiency of the existing sampling methods associated with arbitrary lattices sampling, we introduce a new simple method to get alias-free sampling matrix. Then under the premise of alias-free sampling, we drive an easy-to-compute reconstruction formula for MD bandlimited signal, which provides us with a simple way to reconstruct original signal from its sampled vision in terms of an arbitrary frequency support and has never been presented before. Consequently, the MD uniform sampling theorem about arbitrary lattices is proposed. Furthermore, in order to get optimal sampling matrix, we improve this sampling method by applying the idea of segmentation to the bandlimited area. Finally, simulations are presented to prove the rationality of the new sampling method. [ABSTRACT FROM AUTHOR]

High-frequency noise in an output optical pulse severely affects amplitude stability and pulse within the high-power acousto-optic Q-switched diode-pumped solid-state laser. In this study, we assessed the output optical pulse via fast Fourier transform (FFT) method under high pump power conditions to find that high-frequency noise in the output optical pulse is chiefly concentrated in the 0.25–1.8 GHz frequency band. High-frequency noise within 0.25–1 GHz and 1–1.8 GHz is mainly caused by high transverse and multi-longitudinal modes in the laser resonator. Our experimental and theoretical analysis results are in close accordance; both indicate that mode competition is the main factor affecting the output optical pulse high-frequency noise. [ABSTRACT FROM AUTHOR]

Shape-From-Focus method (SFF method) is a method for recovering depth from an image series which are taken with different focus settings. For method presented in this paper, the series must be taken with an optical device (optical microscope, CCD or classic camera) with very small depth of focus, different images must be focussed to different planes and may be taken with inconsiderable angle of view. Proposed method is capable to registered images with different scaling, it makes possible to construct full sharpened 2D image and also the 3D model of scanned object. Accuracy of the method is tested by comparing with 3D models obtained by confocal microscope in hardware supported confocal mode. [ABSTRACT FROM AUTHOR]

An optimized method is employed for measuring the phase retardation of wave plate based on normalized polarization modulation technique. Using the zero-order quartz quarter wave plate as the testing sample, the main factors that influence the measuring accuracy were proposed and analyzed. Optimized procedure for compensating the errors was also developed which showed excellent accordance to the theoretical calculation result. We believe that the method reported in this paper may shed light on the fast and accurate measurement of the wave plate. [ABSTRACT FROM AUTHOR]

The spectral crosstalk from neighboring peaks introduces phase ripple errors in Depth-Resolved Wavenumber-Scanning Interferometry (DRWSI), due to the sidelope of window functions of Fourier Transform (FT). To address such a problem, an algorithm, Scale-Frequency Transform (SFT), is proposed for data evaluation of DRWSI. The simulations show that the sidelope decreases from −12.2 dB to −25.5 dB after using SFT. The experimental results in four-surface interferometry as well as six-surface interferometry demonstrate that SFT can reduce the phase ripple errors effectively without spoiling the depth resolution of DRWSI. Therefore, SFT minimizes the effect of spectral crosstalk. It is practical to replace FT with SFT in DRWSI. [ABSTRACT FROM AUTHOR]

The effect of spectrum distribution from CCD nonlinearity in Fourier transform profilometry is analyzed. Firstly, the spectrum distribution expression from deformed fringe acquired by actual CCD after Fourier transform is given. Secondly, the nonlinearity condition of output deformed fringe from CCD under the actual conditions is discussed, the relationship of spectrum overlapping between first-order spectra which contains object height information and higher-order spectra is analyzed, and its separation condition is discussed. Finally, the discrete deformed fringe expression after sampling aim at pixels signal intensity from CCD is deduced, the Fourier spectrum expression of deformed fringe is obtained, spectrum overlapping and separation caused by sampling are analyzed, and the sampling condition must be satisfied with sampling frequency is given. The results of computer simulation and experiment verify the correctness of the theory analysis. [ABSTRACT FROM AUTHOR]

The optical path difference is produced by means of swinging mirrors in the oscillating scanning Fourier transform spectrometer, which decreases the influence of curve error of the moving mirror. Based on the principle of swinging interference spectroscopy, this paper presents a theoretical analysis on the optical path difference of the swinging scanning Fourier transform spectrometer. Furthermore, expressions of the optical path difference are provided, and factors affecting the optical path difference are also pointed out. This study can provide theoretical bases for the optimization design of the Fourier transform spectrometer. [ABSTRACT FROM AUTHOR]

We present a new method based on Fourier transform to correct the frequency shift and compensate the phase in off-axis holography. First, we show detailed theoretical analysis about the frequency shift error caused by asynchronous sampling, and then the further errors in reconstructed phase associated with the frequency shift error are demonstrated. We use the artificial neural network algorithm to perform surface fitting with the discrete spectrum value, calculate the frequency shift error, and do the phase compensation. The necessity and validity of the phase compensation have been verified by both theoretical deduction and representative simulations. [ABSTRACT FROM AUTHOR]

In this paper, we proposed an improved method for calculating the fringe width of Fraunhofer diffraction. Because the diffraction patterns are equidistant when the diffraction angle is small, the width of fringes can be determined through finding out the spatial frequency of the fringes using Fourier transform. To make the result more precise, the experimental data are compared with the theoretical values having the same range of the image in the frequency domain to get the modified frequency. We applied this method to calculate the diameters of some filaments by measuring their Fraunhofer diffraction fringes. [ABSTRACT FROM AUTHOR]

Two-dimensional (2D) step parallel confocal image processing algorithm for synthetic aperture imaging ladar (SAIL) is proposed. The feature of the algorithm is the collected data are confocal imaged in space domain by 2D Fourier transform. The collected data are first expressed by separation of variables and time-domain data are translated into a spatial coordinate expression suitable for space optical conversion, compensated with an azimuth quadratic phase and then imaged in space domain by 2D Fourier transform. Functions of the algorithm for SAIL with rectangular apertures is described and analyzed in terms of continuous variables and functions. Then the position of the imaging point is given. The technique is a new, one-step confocal imaging method for ladar echo signal. The confocal image processing is simplified and the imaging time is shortened. The requirement of a ladar image processing system is also reduced. It has a significant advantage in the data processing of synthetic-aperture ladar target echo confocal imaging. [ABSTRACT FROM AUTHOR]

An optical spectrometer based on the Fourier transform is presented. Both the measured light and reference light are illuminated into a Michelson interferometer with angle difference between two mirrors, and the Fourier transform spectra of the interference fringes are obtained by using charge-coupled device (CCD) and the field-programmable gate array (FPGA). The peak point in the Fourier transform spectra of the reference light is used to calibrate the measured light, and the wavelength of the measured light can be detected by calculating the Fourier transform spectra between the measured light and the reference light. Due to the Michelson interferometer with angle difference between two mirrors, the Fourier transform spectrometer is realized without any mechanical movement, and the accuracy of the optical spectrometer is improved by propagating two lights into the common optical path in the same Michelson interferometer. The experiment shows that the maximum error is 1.15 nm. The proposed method possesses high performance, high reliability, and low cost. [ABSTRACT FROM AUTHOR]

A non-cube beam-splitter (NCBS) is proposed, by which an incident beam can be separated largely in a direction and then the lights from the test object and the lights from a reference surface placed adjacently to the test object can be combined to construct a simple electronic speckle pattern interferometry (ESPI) system. Two mainly useful quantitative ways, to calculate the phase change of the test object, the phase-shift and the fringe carrier method with Fourier transform, can be achieved in the ESPI system with the NCBS. Experiments with phase-shifting and fringe carrier method are completed. The experimental results show that the monolithic design of the proposed NCBS is effective in ESPI measurement and immunity to vibrations. [ABSTRACT FROM AUTHOR]

Abstract: Based on interferogram zero padding and fast Fourier transform (FFT) methods, an effective, straightforward and stable carrier-removal approach in Fourier transform (FT) based method for carrier interferogram analysis is proposed. The spatial carrier interferogram is firstly extrapolated by zero padding method, and the carrier-frequency values within a small fraction of an integral (or a pixel) are estimated from the extrapolation interferogram with FFT method. Then the carrier-phase component is removed by subtracting a pure carrier-frequency phase constructed by the estimated carrier-frequencies in the spatial domain. Numerical simulations and experiments are given to demonstrate the performance of the proposed method and the results show that the proposed method is effective and stable for suppressing the carrier-removal error in the FT method for carrier interferogram analysis. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to describe the development and application of a software tool in MATLAB based on a Graphical User Interface that allows evaluation of the frequency response of a microwave photonic filter. Microwave photonic filters have potential applications in optical communication systems. The main characteristic of this software tool is the friendly way to edit the parameters involved to determine the frequency response. In order to validate the results obtained by the use of this software tool, a series of experiments were carried out in our laboratory and the results were compared qualitatively showing a good agreement. The developed software has a potential application as a didactic tool in a course of microwave electro-optical systems. [Copyright &y& Elsevier]

Abstract: An Iterative Fourier Transform Algorithm (IFTA) for digital hologram generation from a digital image using phase-only information is described in this paper. The algorithm is implemented in a Texas Instruments TMS320C64x fixed point digital signal processor (DSP). The holograms are reconstructed in real time using a liquid crystal display system and a He–Ne laser. The ideas described in this paper can be extended to digital hologram generation of three-dimensional objects and their reconstruction in real time using liquid crystal spatial light modulators, which could be a way for 3D-display systems. [Copyright &y& Elsevier]

Abstract: Phase-shifting interferometry is widely used for high-precision surface measurements, but has difficulty in dealing with parallel optical plates. In this paper, an advanced method is proposed to simultaneously measure surface distributions of parallel optical plates from multiple surface interference fringes. The basic theory behind the technique is by applying wavelength-modulated phase shifting interferometry (WMPSI) to get enough frames of multiple surface interference fringes. In the procession of wavelength-modulated phase shifting, the phase variation for one point of the surface is traced and is processed by Fourier transform, and then the frequency spectrum of every surface can be separated from each other. Therefore, it allows extraction of front surface, back surface and thickness variation from multiple surface interference fringes with high precision. [Copyright &y& Elsevier]

Abstract: In this article, two-dimensional windowed Fourier assisted Hilbert transform for fringes phase extraction has been proposed. The mathematical derivations of the filtered fringes image and its Hilbert transformations are presented using windowed Fourier transform. The proposed method has been verified by calculating the wrapped and unwrapped phase of the experimental moire fringes. In comparison to the conventional fringes phase extraction methods, the proposed method is found to be effective. [Copyright &y& Elsevier]

Abstract: Comparative studies of the different reconstruction algorithms in digital holography are presented. It shows that each method has different properties with respect to available reconstruction distances, resolution of reconstructed images and computational load. It is very important to choose an appropriate method for the reliable reconstruction in a given situation. Although the method based on Fresnel diffraction formula can still give an accurate reconstruction for smooth and slowly varying objects where the Fresnel approximation is not strictly satisfied, it cannot correctly reconstructed near wave-fields for more diffractive objects, where the higher-order terms in the expansion of the Fresnel approximation are more significant. Angular spectrum method can be a way out of those problems in Fresnel or convolution method, because of its validity for the closer region that the Fresnel region and is very useful for the measurement of microscopic samples requiring a high-numerical aperture imaging system as a necessity. We present one example of quantitative phase-contrast microscopy of small particles and its 3D prospective. [Copyright &y& Elsevier]

Abstract: A procedure for the measurement of focal length and radius of curvature of a lens using coherent gradient sensing system and Fourier transform fringe analysis technique has been demonstrated. Light from the laser illuminates the specimen and the wavefront emerging from the specimen is tested using coherent gradient sensing interferometer. The fringe pattern corresponding to the test wavefront is stored, and analyzed using Fourier fringe analysis technique. The slope of the wavefront in the direction of shear has been evaluated. It provides the information regarding exact location of points corresponding to the focus/center of curvature of the lens. High accuracy and precision has been achieved. [Copyright &y& Elsevier]

Digital holographic microscopes provide quantitative phase images of transparent objects like biological cells. These phase images can in turn provide a host of parameters based on the object morphology, which could be used for object identification. But one of the biggest challenges in digital holographic microscopy is the design of compact setups with minimal optical elements providing high quality images and nanometer level temporal stability. Self-referencing interferometers in which the reference beam is obtained from the object beam is ideal for implementing compact devices with only few optical components. But they suffer from reduced field of view and has restriction on the sparsity of object distribution. Here we describe the development of a low-cost large field of view self-referencing digital holographic microscope based on wavefront division interferometer utilizing Lloyd's mirror geometry. The device takes away the restriction on the sparsity of the sample also. The developed device does not have any translation stage and the focusing is achieved post-recording of the holograms. Off the shelf low-cost optics, a laser pointer and a webcam were used in development of the device. The developed device was tested on glass microspheres as well as was used to study human erythrocytes. To the best of our knowledge this is the first time such low-cost configuration using webcam is demonstrated for quantitative phase imaging of cells. [ABSTRACT FROM AUTHOR]

Abstract: A novel Fourier transform 3D shape measurement method based on color fringe projection is proposed in order to solve the spectrum overlapping and phase unwrapping problems existed in Fourier transform profilometry (FTP). The R and G components of the color fringe are set to two sinusoidal patterns with different frequencies and the B component is set to the average value of R or G component. Then this pattern is projected to the object and the deformed fringe image is captured. Three gray patterns are separated from the color fringe, the background and high frequency noise can be eliminated using our method and the accurate unwrapped phase can be got. Only one shot color pattern is projected to get the 3D information of the object. Experiment results show that the 3D information of an object can be obtained rapidly and accurately. [Copyright &y& Elsevier]

In this paper, two-dimensional real time method of measuring the complex degree of spatial coherence of a laser light field is presented. Far-field interferogram is generated by a two-dimensional mask with non-redundant multiple apertures in the focal plane of the lens. The Fourier spectrum of the interferogram is analyzed in the term of classes of aperture pairs. Through using heights of the peaks in the magnitude spectrum, as well as intensities in the apertures, the modulus of the complex degree of spatial coherence was determined. The available experimental data of Young interferometry, confirmed the graphs obtained with the above methods. [ABSTRACT FROM AUTHOR]

Abstract: The expression of gyrator transform (GT) is rewritten by using convolution operation, from which GT can be composed of phase-only filtering, Fourier transform and inverse Fourier transform. Therefore, fast Fourier transform (FFT) algorithm can be introduced into the calculation of convolution format of GT in the discrete case. Some simulations are presented in order to demonstrate the validity of the algorithm. [Copyright &y& Elsevier]

Abstract: Causes of the bright cross appearing in the reconstructed field of lensless Fourier transform digital hologram (LFDH) are presented. Firstly, as LFDH''s reconstruction was directly performed by Fourier transform algorithm, the intensity distribution of reconstructed image plane was just LFDH''s spectrum, and three parts of the reconstructed field are imaged in the spectral plane. Meanwhile their intensities were almost the same. Hence, the ratio of signal to noise of the reconstructed image was obviously lower than that of the conventional digital hologram. Further, non-uniformity of the background intensity and random noises usually existed in a practical LFDH. Meanwhile their spectra spread through the central and the edge areas; this also led to decrease in the ratio of signal to noise of the reconstructed image. Specially, as the two-dimensional Fourier transform algorithm was performed row by row, and then column by column, the low-frequency spectrum induced by the non-uniformity of the background and random noises along the directions of rows and columns would concentrate on the central column and row, respectively, and what is gotten looked like a cross. Therefore, the cross appearing in LFDH''s reconstructed field should be attributed to the combination of the background and the Fourier transform algorithm itself. [Copyright &y& Elsevier]

Abstract: We propose an image encryption scheme based on double random amplitude coding technique by using random Hartley transform, which is defined according to the random Fourier transform. The significant feature of this algorithm is that the encrypted image is real and convenient for storage as well as transfer of the encrypted information. Moreover, the algorithm has enhanced security and the correct information of original image can be well protected under bare decryption, blind decryption and brute force attacks. Numerical simulation results are also presented in support of the proposed scheme. [Copyright &y& Elsevier]

Abstract: A method based on Fourier transform to revise the non-linear spectrum attenuation of optical fiber used as a probe in a spectrum-measuring system is proposed in this paper. Spectral curves of a normal light source are measured by a spectrum-measuring system with and without the optical fiber probe, respectively. The generalized frequency spectrum (GFS) of these spectral curves can be calculated based on Fourier transform. From them, an optical fiber spectrum correction function can be obtained and the non-linear spectrum attenuation of optical fiber can be revised with the SCF. The experiment results show that the method has the accuracy of 1% and it is more accurate than the Ratio Method (RM) and the Weighted Ratio Method (WRM). [Copyright &y& Elsevier]

Abstract: Diffraction properties of the transmission phase grating are analyzed in terms of the Fourier optics theory. The analysis results show that the diffraction intensity distribution of the transmission phase grating is closely related to the optical thickness of the grating and the wavelength of the incident light, and not only determined by the grating''s period and slit width. For monochromatic incident light, more than 80% of the diffraction energy will concentrate to the first diffraction order under certain conditions. For white incident light, the energy of the first order of diffraction spectrum may be much higher than other orders. Based on these results, some possible applications of the transmission phase grating are discussed. [Copyright &y& Elsevier]

Abstract: A new kind of wavefront reconstruction algorithm is proposed in this paper, which can estimate the wavefront from its orthogonal gradients at different sample points. We model the gradients measured by wavefront sensor with the Fried geometry. Iterative discrete Fourier transforms and the correspondent Fried inverse filter in spatial frequency domain are introduced. No special boundary control is required to process the boundary condition in this algorithm. The stimulation and the experimental results show that the new algorithm has strong capability to fit the wavefront and it is more available if the sample points are few. [Copyright &y& Elsevier]

• Here, a generic method presented that redistributes optical energy among the lobes of high order Hermite –Gaussian modes, in a controlled manner. • This method modulates all the identical low intense lobes into both moderate or high intense lobes, and vice-versa. • Further, the modes which consist of only two types of intensity distribution are transformed together into all high intense lobes. • Such controlled modulation of energy may offer efficient and selective utilization of each lobe in the applications like optical Tweezers. The multiple lobes of high order Hermite-Gaussian (HG) laser modes differ in terms of shape, size, and optical energy distribution. Here, we introduce a generic numerical method that redistributes optical energy among the lobes of high order HG modes such that all the identical low intense lobes become both moderate or high intense lobes and vice-versa, in a controlled manner. Further, the modes which consist of only two types of intensity distribution among its multiple lobes are transformed together into all high intense lobes. Furthermore, in some cases, moderate intense lobes together with high intense lobes become high intense lobes, and moderate intense lobes together with low intense lobes become high intense lobes. Such controlled modulation of optical energy may offer efficient and selective utilization of each lobe of HG modes in most applications like particle manipulation, optical lithography, and the method can be used in other fields like nonlinear frequency conversion and shaping ultrafast optical pulses. [ABSTRACT FROM AUTHOR]

Time-of-flight (TOF) three-dimensional (3D) imaging is an effective means for remote sensing and target recognition. Fourier ghost imaging (GI) acquires the Fourier spectrum first, then applies an inverse Fourier transform (IFT) to retrieve the image of interest. The Fourier spectrum can be artificially processed when needed. The Fourier spectrum of some partially transmissive occluders is generally concentrated in high-frequency regions. These occluders can be removed by applying an IFT using the low-frequency spectrum coverage. This strategy is innovative in TOF 3D GI. In a TOF 3D GI system with DMDs, the high repetition rate and short-pulse laser is generally employed, which is mismatched with the lower modulation frequency of grayscale patterns. The grayscale Fourier patterns are decomposed into binary patterns. The detected intensities corresponding to the binary patterns are multiplied by the corresponding weighted coefficients, and are summed to form the Fourier spectrum slices at different distances. Then, an IFT is applied to retrieve the image cubes, furtherly to obtain the depth maps with the image cubes. Two 3D experiments are carried out to evaluate the proposed technique. The results indicate the reflectivity and depth maps can reveal the information of the 3D objects under 25 % spectrum coverage. From an imaging experiment through a partially transmissive occluder, we we find the reflectivity and the depth map obtained by the low-frequency Fourier spectral coverage are significantly better than those obtained reconstructions with complete spectral coverage. These preliminary results indicate TOF 3D Fourier GI has potential advantages in the removal of netting. [ABSTRACT FROM AUTHOR]

Abstract: Based on the transport-of-intensity equation and the Fourier transform, a rapid phase retrieval algorithm for X-ray phase contrast imaging is described in detail. This algorithm is a non-iterative phase retrieval method, and it has a potential speed for large scale X-ray phase contrast images. Then the numerical simulation is made to evaluate the rapid phase retrieval algorithm performance. Finally, a real fiber material experiment is carried out using a micro focus X-ray phase contrast imaging experiment platform, and the phase distribution image is calculated out by this algorithm. The result shows that this phase retrieval algorithm is effective and optional to practical application. [Copyright &y& Elsevier]

Spectral calibration is one of the key technologies for quantitative application of imaging spectrometer. Based on analyzing the spectral calibration principle of Fourier transform imaging spectrometer, a spectral calibration method of long-wave infrared imaging spectrometer based on material characteristic spectrum is proposed in this paper. Polypropylene film and surface blackbody are employed for spectral calibration experiment. Using surface blackbody for temperature control, the interference data of polypropylene film at different temperatures are collected and the position of absorption peaks of polypropylene film are obtained, which compared with the position of the standard absorption peaks of polypropylene film, and the result proved that the proposed method has high calibration accuracy and good stability, and has reliable reference value for spectral calibration of long-wave infrared imaging spectrometer. [ABSTRACT FROM AUTHOR]