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The effect of the width of the angular spectrum (numerical aperture) of a broadband-frequency wave-field probing a layered object on the signal of an autocorrelation low-coherence interferometer (ALCI) under spatially coherent and incoherent illumination of the entrance pupil is considered. It is found that under incoherent illumination an increase in the width of the angular spectrum of the field leads to a decrease in the amplitude, a change in the shape and position of the measuring signals of the interferometer due to decorrelation of the object field partial components which have reflected from various interlayer boundaries inside the object. In the case of coherent illumination, the ALCI signal is formed in a confocal mode, which leads to the amplitude extraction of the measurement signals are determined by the mutual correlations between a partial component reflected from the boundary on which the probing field was focused, and partial components of this field which have reflected from other boundaries within the object. This effect makes it possible to determine parameters of the internal layered structure of an object doing without apriori structure-related information. In this case, an increase in the numerical aperture of the probing light beam leads to an increase in the systematic error in determining the optical thicknesses of the layers, which can be estimated on the basis of the obtained expressions.

Increasing the number of laser beams that can be coherently combined requires accurate and fast algorithms for compensating phase and alignment errors. The paper proposes to use a Fully Connected Artificial Neural Network (FCANN) to correct the beam positioning perturbations by evaluating the beam shifts and tilts from two images taken at slightly different locations. Then, since it is practically impossible to have a large enough experimental dataset to train the neural network, this approach required developing an accurate and fast simulation method to evaluate the beam propagation in arbitrary directions, overcoming the limitations occurring when the computation must be repeated a large number of times. The numerical approach is a variant of the Angular Spectrum (AS) method, called Non Uniform ADaptive Angular Spectrum (NUADAS) method, which relies on the combination of non-uniform and adaptive Fourier transform algorithms to allow the computation of an arbitrary field distribution in a plane that is shifted and tilted with respect to the source. The parallel implementation of the NUADAS method is discussed and the numerical and experimental validations are presented. Then, an FCANN is trained using the synthetic dataset generated with the NUADAS method and the results are discussed, demonstrating the viability of the proposed approach not only for coherent beam combing, but also in other beam alignment applications.

We present a computationally highly efficient full-wave spectral model of OCT-scan formation with the following features: allowance of arbitrary phase-amplitude profile of illuminating beams; absence of paraxial approximation; utilization of broadly used approximation of ballistic scattering by discrete scatterers without limitations on their density/location and scattering strength. The model can easily incorporate the wave decay, dispersion, measurement noises with given signal-to-noise ratios and arbitrary inter-scan displacements of scatterers. We illustrate several of such abilities, including comparative simulations of OCT-scans for Bessel versus Gaussian beams, presence of arbitrary aberrations at the tissue boundary and various scatterer motions. The model flexibility and computational efficiency allow one to accurately study various properties of OCT-scans for developing new methods of their processing in various biomedical applications.

Optical interference is not only a fundamental phenomenon that has enabled new theories of light to be derived but it has also been used in interferometry for the measurement of small displacements, refractive index changes and surface irregularities. In a two-beam interferometer, variations in the interference fringes are used as a diagnostic for anything that causes the optical path difference (OPD) to change; therefore, for a specified OPD, greater variation in the fringes indicates better measurement sensitivity. Here, we introduce and experimentally validate an interesting optical interference phenomenon that uses photons with a structured frequency-angular spectrum, which are generated from a spontaneous parametric down-conversion process in a nonlinear crystal. This interference phenomenon is manifested as interference fringes that vary much more rapidly with increasing OPD than the corresponding fringes for equal-inclination interference; the phenomenon is parameterised using an equivalent wavelength, which under our experimental conditions is 29.38 nm or about 1/27 of the real wavelength. This phenomenon not only enriches the knowledge with regard to optical interference but also offers promise for applications in interferometry., Comment: 16 pages, 6 figures

Laser beams are described through the diffraction eigenfunctions or the so-called modes. The principal fundamental Gaussian mode TEM00 is well described analytically in the paraxial approximation of the diffraction integral, and is applied here as the reference for checking the accuracy of the numerical angular spectrum technique. It has been shown that the beam angle divergence larger or smaller influences the Gaussian beam diffraction properties and affects the beam near-field and the far-field. The angular spectrum technique, has been used, with two fast Fourier transformations applied, which turns out to be accurate and fast. The results were simulated with the help of a dedicated computational program written in Borland C++Builder and Matlab software. Some numerical simulation examples of diffraction patterns are also given to illustrate the propagation characteristics of Gaussian beams, where the shape and the intensity distributions of Gaussian beams in the near-field and in the far-field can change.

Near-field ptychography provides advantages over far-field ptychography that large field-of-view (FOV) can be imaged with fewer diffraction images, and with weaker requirements on the detector dynamic range and beam coherence, which has drawn attention recently. However, the propagation distance of traditional angular spectrum (AS) method is limited and the reconstruction of the smallest resolvable object detail is restricted by the sensor’s pixel size. To this end, we propose an adaptive-window angular spectrum (AWAS) algorithm to solve both problems by adding the window adaptively associated with the propagation distance and avoid the extra computations via extra scaling factors. Meanwhile, it features validity for the independent sample size and the sample number on the observation plane. This algorithm is strictly deduced from the Rayleigh–Sommerfeld formula and based on the linear convolution, which can be evaluated by fast Fourier transform effectively. The burden of calculations is comparable to traditional AS method. The performance has been achieved both in two-dimensional and three-dimensional near-field ptychography with simulations and experiments. This method will make near-field ptychography more practical and can be used in X-ray or electron-microscopy and other computational imaging techniques.

The shifted band-extended angular spectrum method (Shift-BEASM) is proposed to calculate free-space diffraction between two parallel planes with an off-axis offset. Off-axis numerical propagation is useful for simulating non-paraxial and large-scale fields. The proposed Shift-BEASM allow us to calculate the off-axis diffraction in a wide propagation range by extending the effective bandwidth using the nonuniform fast Fourier transform. The calculation accuracy is higher than that of existing techniques, such as the shifted-Fresnel method and shifted band-limited angular spectrum method, not only in the near field but also in the far field. Numerical examples and accuracy as well as theoretical formulation are presented to confirm validity of the proposed method.

The influence of the numerical aperture of a light beam that probes a layered object on the magnitude of the quantity measured with a confocal interference microscope upon determining the object thickness has been studied theoretically and experimentally for the case in which the signal is significantly affected by the spherical aberration arising in the probing beam as it passes through the object. The dependence of the quantity to be measured on the numerical aperture is obtained taking into account the influence of the spherical aberration on the signal. The criterion of the “nonaberrational” propagation of the beam probing the object is determined and the condition under which the influence of the spherical aberration on the signal of the confocal microscope can be neglected is found. An approximate analytical expression is obtained, which makes it possible to determine the “nonaberrational” width of the angular spectrum of the light beam probing the object, taking into account an empirical correction factor. The applicability limits of the analytical dependence of the quantity to be measured on the numerical aperture, which has been previously obtained for the case of a negligibly weak influence of the spherical aberration on the signal from the confocal microscope, have been determined.

To enhance the light out coupling and stabilize the angular spectrum of white organic light emitting diodes (WOLEDs), we formed a randomly distributed corrugation directly on the external glass surface. Using a metallic master, which was prepared using laser ablation, we have imprinted the corrugation pattern directly on the glass surface. Our method yields 32% increase in the external quantum efficiency, widened luminance distribution and low angular spectra variation. Considering WOLED as a lighting source, this feature is particularly appealing. Besides the implications of the spectrally stable efficient WOLEDs through random corrugation, our work suggests a new structural approach for various light applications in which efficiency and spectral stability matter.

Lens-free microscopes can utilize holographic reconstruction techniques to recover the image of an object from the digitally recorded superposition of an unperturbed plane wave and a wave scattered by the object. Image reconstruction most commonly relies on the scalar angular spectrum method (ASM). While fast, the scalar ASM can be inaccurate for nanoscale objects, either because of the scalar approximation, or more generally, because it only models field propagation and not light-matter interaction, including inter-particle coupling. Here we evaluate the accuracy of the scalar ASM when combined with three different light-matter interaction models for computing the far-field light scattered by random arrays of gold and polystyrene nanoparticles. Among the three models—a dipole-matched transmission model, an optical path length model, and a binary amplitude model—we find that which model is most accurate depends on the nanoparticle material and packing density. For polystyrene particles at any packing density, there is always at least one model with error below 20%, while for gold nanoparticles with 40% or 50% surface coverage, there are no models that can provide errors better than 30%. The ASM error is determined in comparison to a discrete dipole approximation model, which is more computationally efficient than other full-wave modeling techniques. The knowledge of when and how the ASM fails can serve as a first step toward improved resolution in lens-free reconstruction and can also be applied to other random nanoparticle array applications such as lens-based super-resolution imaging, sub-diffraction beam focusing, and biomolecular sensing.

We analyze the Crown of Light (COL) cut using an angular spectrum evaluation tool. Several light performing features of the COL are discussed. In particular, it is found that the angular spectrum of the COL tends to be concentrated and that this maximizes the brilliance and sparkle probability when the COL is aimed at localized light sources. It is contended that the COL represents a novel paradigm in diamond cuts. A distinctive feature of the COL cut is its dome shaped crown.

Fast Fourier transform (FFT) is the most commonly used mathematical method in numerical calculation, and the FFT-based angular spectrum method (ASM) is also used widely in diffraction calculation. However, the frequency and spatial sampling rules in FFT limit the effective propagation distance and the observation window range of ASM. A novel method for calculating the angular spectrum based on the matrix product is proposed in this Letter. This method realizes the fast calculation of discrete Fourier transform (DFT) based on the matrix product, in which the sampling matrix is orthogonally decomposed into two vectors. Instead of FFT, angular spectrum diffraction calculation is carried out based on the matrix product, which is named the matrix product ASM. The method in this Letter uses a simple mathematical transformation to achieve maximum compression of the sampling interval in the frequency domain, which significantly increases the effective propagation distance of the angular spectrum. Additionally, the size of the observation window can be enlarged to obtain a wider calculation range by changing the spatial sampling of the output plane.

With this study, we are introducing a new parameter, ultimate spectral peculiarity Lu, to be used in solution of the inverse Mie problem. This parameter is evaluated from a power spectrum of an angle-resolved light-scattering profile (LSP). Direct correlation of single sphere size and Lu allows sizing a homogeneous sphere from 0.6 µm. In order to determine the refractive index (RI) of the sphere we utilized the integral of the LSP within the particular angular range providing the largest uniqueness domain where both size and RI can be unambiguously determined. The integral and Lu were used in the spectral solution of the inverse Mie problem providing the real-time analysis of milk fat globules, spherized red blood cells, polymer and silica beads with the scanning flow cytometer.

An effect of the angular spectrum width of a polychromatic illuminating field on the interference pattern color in optical microscopy of stratified objects is considered. Equations describing the intensity of interference patterns of thin stratified objects are obtained with due account for the influence of both frequency and angular spectra of the illuminating field. The angular spectrum width of the illuminating field is shown to affect the mutual coherence and phase difference of the interfering fields. Computer-simulated interference patterns for a thin layer under monochromatic and polychromatic illumination with different widths of the angular spectrum are presented. Quantitative analysis of the effect of angular aperture of white light on the colors of the resulting interference pattern is carried out. Changes in the colors of the layer interference patterns are experimentally studied when changing the width of the angular spectrum of the illuminating field. Color variations of the interference patterns obtained experimentally and theoretically are compared.

Entangled quantum states play an important role in quantum information science and also in quantum mechanics fundamental investigations. Implementation and characterization of techniques allowing for easy preparation of entangled states are important steps in such fields. Here we generated entangled quantum states encoded in photons transversal paths, obtained by pumping a non-linear crystal with multiple transversal Gaussian beams. Such approach allows us to generate entangled states of two qubits and two qutrits encoded in Gaussian transversal path of twin photons. We make a theoretical analyses of this source, considering the influence of the pump angular spectrum on the generated states, further characterizing those by their purity and entanglement degree. Our experimental results reveals that the generated states presents both high purity and entanglement, and the theoretical analysis elucidates how the pump beams profile can be used to manipulate such photonic states., 6.5 pages +1 page refs. Comments are welcome!

To improve the detection sensitivity of porous silicon microcavity biosensors, CdSe/ZnS quantum dots are used to label complementary DNA molecules for the refractive index amplification and angular spectrum method for detection. In this method, the TE mode laser is used as the detection light and the light source is changed into a parallel beam by collimating and expanding the beam, which illuminates the PSM surface and receives the reflected light from the PSM surface through the detector. The angle corresponding to the weakest reflected light intensity before and after the biological reaction between probe DNA and complementary DNA of different concentrations labeled by quantum dots was measured by the detector, and the relationship between the angle change before and after the biological reaction and the complementary DNA concentration labeled by quantum dots was obtained. The experimental results show that the angle change increases linearly with increasing complementary DNA concentration. The detection limit of the experiment, as determined by fitting, is approximately 36 pM. The detection limit of this method is approximately 1/300 of that without quantum dot labeling. Our method has a low cost because it does not require the use of a reflectance spectrometer, and it also demonstrates high sensitivity.

In the angular-spectrum method-based computer-generated hologram, the zero-padding method is used to convert circular convolution into linear convolution. However, it will increase the calculation time and memory usage significantly. Therefore, a fast and simple method is proposed to solve the issue of the numerical convolution in the process of hologram generation by using the intermediate angular-spectrum method in this paper. Through replacing numerical Fourier transform by optical Fourier transform in the hologram generation, the calculation speed is approximately 6 times faster than that of the zero-padding method. And due to the scaling factors introduced by the Fourier lens and without the cropping operation, the reconstruction quality of the proposed method is improved significantly compared with the zero-padding method. Moreover, the optical reconstruction system is more compact than the 4-f filter system in the on-axis holographic reconstruction. Both numerical simulations and optical experiments have validated the effectiveness of the proposed method.

Diffractive axicon is lightweight and easy to be manufactured, it is also efficiently used to produce diffraction-free Bessel beam. But, deducing its expression of the field distribution of light is difficult in theory and the theoretical equation expression has approximation by using the Huygen-Fresnel diffraction integral. Here, the rigorous optical angular spectrum theory that follows the Helmholtz equation is used. The propagation theory of angular-spectrum with fewer approximation, will result in more accurate results. A mathematical model to analyze the field distribution of light of the diffractive axicon is set up. The diffraction intensity distribution of the diffractive axicon with the monochromatic Gaussian beam illumination and multicolor illumination is numerically calculated and discussed. The results of numerical calculation with a monochromatic laser source show that not all the intensity distribution in the focal depth of field has the characteristics of the diffraction-free Bessel beam, and the on-axis diffraction intensity shows direct correlation of the lateral intensity distribution of the incident waves and the period of the diffractive axicon. The extended depth of field of the diffractive axicon with multicolor illumination is increasing as the wavelength reducing. It is expected that our study can provide very useful information for analyzing the diffractive axicon which exists in the optical trapping systems and laser-processing applications and imaging systems.

We presented modified forms of full-vector angular spectrum formula without singular points for longitudinal spectral components. In paraxial limitation, the modified angular spectrum formula has the same form as usual angular spectrum formula. Since there are not any singular points at the angular spectrum formula, we can apply it to analyze the propagation of non-paraxial beams with subwavelength-size waist and give results that strictly satisfy Maxwell’s equations because the calculating process includes contributions from all field components and angular spectral components.

In this paper we will analyze the propagation of the angular spectrum of the electromagnetic field through a finite length uniaxial crystal. We solve the boundary conditions of the fields at an interface of an isotropic medium and an uniaxial anisotropic medium with the optical axis in an arbitrary direction and soon after we choose the optical axis in a plane formed by the direction of propagation (z-axis) and the x-axis . We show how the couplings occur between the components of the field and in what situations we can give a vector or scalar treatment for the propagation, Comment: 8 pages, 9 figures

In the recent works on Mathieu beams, the angular spectrums commonly used are even and odd angular Mathieu functions with an angular variation vary between 0 to 2 π . In this work, we propose to introduce a new index m characterizing the rotation repetitions (m-times) of the structure of the original angular spectrum of Mathieu beams on the total circle in the frequency space. With this index, the angular spectrum of Mathieu beams rotation can be controlled to change the beam symmetry. We study the effect of these repetitions of the angular spectrum on the even and odd Mathieu modes features when they propagate through a paraxial optical system. We show that rotated repetitions of the angular spectrum does not affect the nondiffracting property of the beams. To illustrate the rotation repetitions of the angular spectrum, the Fourier transform of the beam is calculated in the focal plane of a thin lens. From the obtained results, we can conclude that the rotation repetitions of the angular spectrum can be used to produce others Mathieu beams shapes and can enrich the applications of these kinds of beams.

Accurate reconstruction of digital holograms that are large in the x direction and small in the y direction, known as horizontal parallax only digital hologram (HPO-DH), must be carried out by non-paraxial propagation approaches such as the classical angular spectrum (AS) method. However, the required space–bandwidth product (SBP) for reconstruction of HPO-DHs requires billions of pixels, which is computationally intensive. Moreover, application of zero-padding for removing aliasing components would generate an unbearable computational burden. In this work, a novel AS technique that reconstructs non-paraxial HPO-DHs with low SBP is proposed. The proposed technique first employs the multi-Fourier transform plane propagation method, which avoids the increase of size in the vertical direction of the HPO-DH to be processed. The second ingredient for field calculation is coherent superposition of vertical tiles formed from the multi-Fourier transform calculations. The described methodology enables reconstruction of HPO-DHs with the AS method and reduced SBP. Efficient managing of the SBP allows implementing zero-padding strategies in the x direction. It is shown that the padding strategies can be implemented in the frequency, space, and space–frequency domains. Hence, suppression of aliased components and increase of the spatial resolution is possible at the same time. Finally, the accuracy and utility of the developed technique is proved by both numerical simulations and experiments.

The angular spectrum decomposition (ASD) is a powerful tool for analyzing the interaction between spherical particles and shaped beams, in which homogeneous plane waves and/or evanescent waves are expanded into series of vector spherical harmonics (VSH). The paper presents a numerical study on the VSH expansion of the evanescent wave and its behavior in the field reconstruction of the shaped beam. It is revealed that the beam shape coefficients (BSCs) of the evanescent waves increase exponentially along the increasing partial wave number, consequently extending the convergence of the series and restricting the field reconstruction into a limited space.

The band-extended angular spectrum method (ASM) is proposed in this Letter for both near- and far-field diffraction calculation with high accuracy. Due to the aliasing problem of the transfer function (TF), the ASM is not suitable for far-field diffraction calculation. For band-limited ASM, the non-aliased bandwidth of the TF would shrink rapidly with the increase of the propagation distance, which would reduce the calculation accuracy in the far field. For the proposed band-extended ASM, the non-aliased bandwidth is significantly extended by rearranging the sampling points in the spatial frequency domain. Therefore, more frequency components of the TF contribute to the wave-field calculation, leading to a much wider propagation range and a higher computational accuracy.

Holographic near-eye displays present true three-dimensional images with full monocular depth cues. In this paper, we propose a technique to expand the eyebox of the holographic near-eye displays. The base eyebox of the holographic near-eye displays is determined by the space bandwidth product of a spatial light modulator. The proposed technique replicates and stitches the base eyebox by the combined use of a holographic optical element and high order diffractions of the spatial light modulator, achieving horizontally and vertically expanded eyebox. An angular spectrum wrapping technique is also applied to alleviate image distortions observed at the boundaries between the replicated base eyeboxes.

The present paper may be viewed as built from comments concerning a recently published paper in Journal of Quantitative Spectroscopy and Radiative Transfer, and actually constitutes as well a rebuttal of some of the claims it contains. The issue considered concerns the computation of scattering properties of the interaction between an arbitrary electromagnetic shaped beam and a homogeneous sphere. Our discussion of the issue may be easily extended as well to more general cases, beyond the simplest case of a homogeneous spherical scatterer, up to the case of arbitrary shaped particles in the framework of T-matrix formulation. Finally, we take the opportunity of the present paper to review the use of the angular spectrum decomposition in light scattering.

In the lithography process, overlay describes the position errors of placing one mask layer pattern over an existing pattern on the wafer, and overlay metrology is one of the key challenges in semiconductor manufacturing. With the increase of process complexity especially for multiple patterning technologies in advanced nodes, the demand for accuracy, precision and process robustness in overlay metrology continues to tighten. In this paper, we have proposed and demonstrated a novel overlay metrology method with high process robustness. A straightforward theory called interference of diffraction light is developed for explanations why this method works for single layer overlay metrology. The overlay signals with continuous wavelengths and angles are simultaneously acquired in the pupil plane. Therefore, compared with the current monochrome measurement technology, this work can gather and utilize as much overlay information as possible for measurements. Experiments illustrate that this technique can achieve better measurement repeatability, faster measurement speed and better process robustness than current monochrome method, and thus this method has great potential to meet the higher node requirements for measurement accuracy. What’ more, our work provides a novel idea for scatterometry technology in the field of micro-nano measurement.

Two-index Bessel beams (TIBBs) was introduced by Ornigotti and Aiello (2014) theoretically. In this paper, we propose a simple experimental scheme for generation of two-index Bessel–Gauss beams (TIBGBs), as an extension of the TIBBs. The scheme is based on manipulating the amplitude and phase in the Fourier plane with the use of a spatial light modulator and a spiral phase plate. Furthermore, we experimentally report the generation of the several examples of the TIBGBs based on the proposed optical system. The focusing properties of the TIBGB with indices p = 1 and l = 2 passing through a single lens are investigated both theoretically and experimentally. The experimental results agree well with theoretical predictions.

A modified angular spectrum algorithm is presented for the diffraction calculation of partially coherent beams propagating in optical systems. The proposed algorithm can directly calculate the cross-spectral density of partially coherent beams at each surface of the optical system and possesses much higher computational efficiency for low coherent beams compared with that of the common modal expansion methods. Then, a Gaussian–Schell model beam propagating in a double-lens array homogenizer system is introduced to carry out a numerical simulation. Results show that the proposed algorithm can obtain an identical intensity distribution as the selected modal expansion method but with a much higher speed, thus verifying its accuracy and high efficiency. However, it’s worth noting that the proposed algorithm is only suitable to the optical systems in which the partially coherent beams and optical components have no coupling effects in the x and y directions and can be dealt with individually.

International audience; The Maggi-Rubinowicz method (MRM) is a useful tool to compute diffraction patterns from opaque planar objects. We adapted the MRM to planar rectangles. In the first part of this study, differences between diffraction patterns, both the intensity and the phase distributions, from a tilted rectangle and from the square having the same orthogonal projection on the observation plane, have been analyzed. In the second part, we compared results obtained with the MRM to those obtained with angular spectrum theory (AST) coupled to fast Fourier transform (FFT). The main novelty of this work is the fact that MRM is particularly well suited for evaluating anti-aliasing procedures applied to AST-FFT calculations.

A controllable energy method, which considers the undersampling issue of the transfer function and valid spectral energy of a source signal, is proposed to implement angular spectrum diffraction calculation in near and far fields. The proposed method provides an optimized frequency boundary $f_{CE}$ within which it always keeps controllable energy to be diffracted. The controllable energy angular spectrum method significantly reduces the number of samples while having the same accuracy as previous angular spectrum methods, implying a higher calculation efficiency. The new perspective of analyzing spectral energy is shown to improve the performance of relevant diffraction calculations.

The exact modeling of light transport in scattering media is critical in biological imaging, free-space communication, and phosphor-converted lighting. Angular spectrum is proved to be a fast and effective approach to reconstructing the wavefront dynamics during the propagation in scattering media, however, finding it difficult in acquiring the wavefront and energy change simultaneously. Besides, conventional methods for energy tracing, such as the Monte Carlo method, are inefficient in speed and hard to simulate the wavefront change. Here, we propose an extended angular-spectrum modeling (EASM) approach using tenuous scattering approximate solutions to obtain a time-efficient and accurate method for reconstruction of energy and wavefront dynamics in various scattering media. The generality of our method is numerically simulated and experimentally verified with a set of scattering media with different properties. EASM has a time advantage under the guarantee of calculation accuracy, especially when calculating several thickness changes after the calculation model is established. Furthermore, multi-layered media can also be simulated by EASM with a good precision. The results suggest that EASM performs certain computations more efficiently than the conventional method and thus provides an effective and flexible calculation tool for scattering media.

Digital holographic microscopy (DHM) is a future three-dimensional (3D) microscopy due to its high-resolution and high-precision 3D images. Thus, it is getting attention in bioinformatics, semiconductor defect detection, etc. However, some limitations still exist. Especially, high-speed holographic imaging requires high-power lasers, which are difficult to image on highly absorbent or light-sensitive samples. To overcome these issues, we propose a new, to the best of our knowledge, digital hologram recovery algorithm called angular spectrum matching (ASM), which achieves hologram imitation to recover holograms in digital holography at low light intensities. The hologram used for the background phase comparison is recorded without objects; thus, no power limitation is required. The ASM utilizes this background hologram to recover dark holograms. We present experimental results showing improved DHM numerical reconstructions and recovered holograms under extremely low light conditions.

A stepwise angular spectrum method (SASM) for curved interfaces is presented to calculate the wave propagation in planar lens-like integrated optical structures based on photonic slab waveguides. The method is derived and illustrated for an effective 2D setup first and then for 3D slab waveguide lenses. We employ slab waveguides of different thicknesses connected by curved surfaces to realize a lens-like structure. To simulate the wave propagation in 3D including reflection and scattering losses, the stepwise angular spectrum method is combined with full vectorial finite element computations for subproblems with lower complexity. Our SASM results show excellent agreement with rigorous numerical simulations of the full structures with a substantially lower computational effort and can be utilized for the simulation-based design and optimization of complex and large scale setups.

This research aims to study the analytical and numerical simulation of the Gaussian beams variables of the fundamental mode TEM 00 during its propagation in free space between two parallel planes, using the angular spectrum technique. A dedicated computational program in Borland C++ Builder was developed to calculate the diffraction through the angular spectrum technique with the application of two fast 2D Fourier transforms. Some 2D and 3D cross-sections of results are presented. That helped to present the complex amplitude of the Gaussian beams with its real and imaginary parts, Gouy phase shift and full phase and intensity distribution. The Gaussian beam is one of the rare cases when the diffraction problem solution, defined within the frames of scalar and paraxial approximation, is given in the exact analytical form. A comparison between that form and the results of the angular spectrum algorithm brings information about the performance quality of that algorithm.

When dealing with light scattering and propagation of an electromagnetic beam, there are essentially two kinds of expansions which have been used to describe the incident beam (i) a discrete expansion involving beam shape coefficients and (ii) a continuous expansion in terms of an angular spectrum of plane waves. In this paper, we demonstrate that the angular spectrum decomposition readily leads to two important consequences, (i) laser light beams travel in free space with an effective velocity that is smaller than the speed of light c , and (ii) the optical theorem does not hold for arbitrary shaped beams, both in the case of electromagnetic waves and scalar waves, e.g. quantum and acoustical waves.

This research aims to make a comparison between the analytical and numerical methods of simulation of the Gaussian beam propagation in free space. This comparison brings evaluation of the accuracy and applicability limitations of the angular spectrum diffraction algorithm. In this approach, planar harmonic waves are assumed as basic components of a propagating wave, so the algorithm combines the angular spectrum technique and the 2D fast Fourier transformation procedures. It means a quite different approach than the well-established Huygens–Fresnel–Rayleigh–Sommerfeld one, based on spherical waves. The Gaussian beam is a rare case of the exact analytical 3D solution of a specific wave diffraction problem, in paraxial scalar approximation. Gaussian beam form has been chosen here as an evaluation reference. It allows to avoid comparisons between two numerical procedures, both of which may suffer from imperfections. In this comparison, the respective 2D cross-sections of calculated 3D complex valued wave fields distributions are presented. Also values of specific Gaussian beam parameters like full phase and Gouy phase shift and wave-front curvature radius are presented. The comparison between that form and the results of the angular spectrum algorithm brings information about the performance quality of that algorithm.

Numerical propagation calculation is a fundamental research topic in optical engineering. The standard angular spectrum method (ASM) is accurate but time- and memory-consuming, especially for high-NA systems. In this work, we propose a fast and simple numerical propagation method, the resampling ASM (RS-ASM). Numerical propagation can be accelerated by combining a resampling technique with interpolation methods in the angular spectrum domain of a constrained object at the focal plane. RS-ASM has three main advantages: simple implementation, faster calculation than the standard ASM, and SNR enhancement. Here we validate RS-ASM using theory, simulation and experiment. Using the “bilinear” ASM with a proper resampling factor can result in a speed-up factor of up to 20x (for a transformation from the angular spectrum to the E field) and 4x (for a transformation from E field to the angular spectrum), together with a SNR improvement of approximately 2x. For an application example of Gerchberg-Saxton phase reconstruction, the “bilinear” RS-ASM can converge 2.6x faster than the standard ASM.

In this paper, we present a detailed and rigorous study of cylindrical harmonic Fresnel lenses (HFLs) using the finite difference time domain method (FDTD) and angular spectrum method (ASM). The HFL is a kind of diffractive lens that can have maximum diffraction efficiency at several discrete harmonic wavelengths, which is suitable for some broadband applications. Previous studies on HFLs were investigated mainly in the domain of paraxial approximation. By using our proposed calculation method, we have determined the efficiency, focal length, maximum focus intensity, and full width at half maximum (FWHM) of the focal spot for several harmonic numbers and for 𝐹-numbers of 0.5, 1, and 3. To compare with the paraxial approximation, we have presented the response to both 𝑠-polarized and 𝑝-polarized light with constant refractive index and real dispersive material, BK7. Moreover, we have also analyzed the cases with oblique illumination. We have shown that the harmonic wavelengths do not change with 𝐹/# and that the diffraction efficiency and FWHM of the focus increase as 𝐹/# increases. New results on harmonic wavelengths shift and oblique angle of incidence response have been detailed.

A time-division multiplexing method for computer-generated holograms (CGHs) is proposed to solve the problem of the limited space-bandwidth product. A three-dimensional (3-D) scene is divided into multiple layers at different depths. The CGH corresponding to each layer is calculated by an angular-spectrum algorithm that is effective at a wide range of propagation distances. All of the CGHs are combined into several group-CGHs. These group-CGHs are sequentially uploaded onto one spatial light modulator at a high frame rate. The space-bandwidth product can be benefited by the time-division processing of the CGHs. The proposed method provides a new approach to achieve high quality 3-D display with a fast and accurate CGH computation.

Quasi-non-diffracting beams are attractive for various applications, including optical manipulation, super-resolution microscopes, and materials processing. However, it is a great challenge to design and generate super-long quasi-non-diffracting beams with sub-diffraction and sub-wavelength size. In this paper, a method based on the idea of compressing a normalized angular spectrum is developed, which makes it possible and provides a practical tool for the design of a quasi-non-diffracting beam with super-oscillatory sub-wavelength transverse size. It also presents a clear physical picture of the formation of super-oscillatory quasi-non-diffracting beams. Based on concepts of a local grating and super-oscillation, a lens was designed and fabricated for a working wavelength of λ = 632.8 nm. The validity of the idea of normalized angular spectrum compression was confirmed by both numerical investigations and experimental studies. An optical hollow needle with a length of more than 100λ was experimentally demonstrated, in which an optical hollow needle was observed with a sub-diffraction and sub-wavelength transverse size within a non-diffracting propagation distance of 94λ. Longer non-diffracting propagation distance is expected for a lens with larger radius and shorter effective wavelength.

We have been investigating the angular spectrum method for analysis of single-mode and multimode optical fibers. The justification is simple because the approach requires less computational time than other methods because it is heavily dependent on the fast Fourier transform. This method should become much more important as new approaches are being studied for higher data rate transmission. We examine two basic components in the application of this technique to propagation of light in optical fibers – an absorbing layer at the outer radius and design of the interface between the cord and cladding of the optical fiber.

We examine single-mode optical fiber transmission using the angular spectrum method. We find excellent agreement with the theoretical solutions for the cylindrical single-mode optical fiber. Next, we examine fiber coupler configurations with 1×2, 1×3, and 1×5 designs. The input within the central fiber core is broadcast equally into the surrounding fiber cores and then returns with propagation distances as expected. Then we examine supermode theory for the 1×3 case. The initial energy is sent into different combinations of fiber cores where their phase relationship governs the output. Surprisingly, we find evidence of an unexpected mode in some of the multicore designs. We expect these results might be useful for exploration of more complicated fiber core arrays. However, computational times are short using this algorithm.

In many previous studies where a tightly focused beam is scattered by a spherical particle, either the experimental conditions were such that evanescent components were absent from the angular spectrum of the incident beam, or if they were present, it was assumed that their contribution to scattering was small with respect to that of the oscillatory components, and could safely be ignored. In this paper the contribution of the evanescent components is explicitly calculated and the validity of their neglect in various situations is assessed. It is demonstrated that when a particle whose size is comparable to the wavelength is located near the plane in which the strength of the evanescent components is maximized, the angular spectrum components just inside the evanescent regime, when added to the contribution of the oscillatory components, can possibly make a significant contribution to the shape coefficients of the incident beam.

We proposed a novel off-axis angular spectrum method (ASM) for simulating free space wave propagation with a large shifted destination plane. The off-axis numerical simulation took wave propagation between a parallel source and a destination plane, but a destination plane was shifted from a source plane. The shifted angular spectrum method was proposed for diffraction simulation with a shifted destination plane and satisfied the Nyquist condition for sampling by limiting a bandwidth of a propagation field to avoid an aliasing error due to under sampling. However, the effective sampling number of the shifted ASM decreased when the shifted distance of the destination plane was large which caused a numerical error in the diffraction simulation. To compensate for the decrease of an effective sampling number for the large shifted destination plane, we used a variable sampling interval in a Fourier space to maintain the same effective sampling number independent of the shifted distance of the destination plane. As a result, our proposed off-axis ASM with a variable sampling interval can produce simulation results with high accuracy for nearly every shifted distance of a destination plane when an off-axis angle is less than 75°. We compared the performances of the off-axis ASM using the Chirp Z transform and non-uniform FFT for implementing a variable spatial frequency in a Fourier space.

We propose a method that eliminates speckle artifact in the reconstruction of triangular mesh based computer generated holograms. The proposed method generates a number of holograms with different sets of interleaved plane carrier waves. The generated holograms are displayed sequentially with time-multiplexing, achieving speckle-free reconstruction without sacrificing the viewing angle or depth of focus. The proposed method is verified experimentally using viewing-window type holographic display setup. The experimental results indicate that the proposed method can achieve speckle-free reconstruction with smaller number of the time-multiplexing than conventional temporal speckle averaging techniques.