Diffraction Integral Research Articles

Acquiring the Lefschetz thimbles: efficient evaluation of the diffraction integral for lensing in wave optics

ABSTRACT Evaluating the Kirchhoff–Fresnel diffraction integral is essential in studying wave effects in astrophysical lensing, but is often intractable because of the highly oscillatory integrand. A recent breakthrough was made by exploiting the Picard–Lefschetz theory: the integral can be performed along the ‘Lefschetz thimbles’ in the complex domain where the integrand is not oscillatory but rapidly converging. The application of this method, however, has been limited by both the unfamiliar concepts involved and the low numerical efficiency of the method used to find the Lefschetz thimbles. In this paper, we give simple examples of the Lefschetz thimbles and define the ‘flow lines’ that facilitate the understanding of the concepts. Based on this, we propose new ways to obtain the Lefschetz thimbles with high numerical efficiency, which provide an effective tool for studying wave effects in astrophysical lensing.

Synthesis and study of nonlinear optical properties of an enaminone derived from dibenzoylmethane and N,N-diethylaminoaniline

The compound, (Z)-3-((4-(diethylamino)phenyl)amino)-1,3-diphenylprop-2-en-1-one, is synthesized by the reaction of dibenzoylmethane and 4-N,N-diethylaniline. The relative stabilities of the possible tautomers of the molecule are studied via the DFT B3LYP-D3BJ, CAM-B3LYP, M062X, and ωB97XD functionals in conjunction with the 6-311++G(d,p) basis set. The results showed that the enaminone tautomer with the intramolecularly hydrogen bonded chelated ring is the most stable. This is further confirmed by the Car–Parrinello MD calculations in the gas phase as well as the NCI analysis. The electronic spectrum is calculated by the TD DFT B3LYP/c-pVDZ level in ethanol, and the hole–electron analysis is carried out for the interpretation of the bands, which revealed that the longest one at 430 nm is of charge transfer origin while the others are of local transition origin. Atoms-in-molecules calculations in several media and levels of theory predicted that the ρBCP at the hydrogen bond in the gas phase to be 0.03791–0.04255 e/a0 3 which is a characteristic of a medium strong hydrogen bond. Researchers investigated the enaminone’s nonlinear optical (NLO) characteristics when it was exposed to a low power (<1 Watt), single fundamental transverse mode laser beam at 473 nm. By using diffraction patterns (DPs) and Z-scan methods, we calculated the nonlinear refractive index (NLRI) of the enaminone up to 4.597 × 10−11 m2 W−1 using DPs. The resulting DPs are numerically investigated using the Fraunhofer (F.) approximation and the Fresnel-Kirchhoff (F.K.) diffraction integral, showing excellent agreement with experimental findings. We successfully explored all-optical switching (AOS) in enaminone using two laser beams.

An Efficient Hybrid Method for Calculating the Focal Field of a Cassegrain Antenna

The evaluation of the focal field of Cassegrain antennas is crucial for the design and optimization of the complex feeds in quasi-optical systems. However, employing traditional physical optics methods generates high computational complexity and is inefficient. An accurate and efficient calculation method of Cassegrain antennas’ focal fields that involves a range of incidence angles is proposed, which integrates ray tracing and vector diffraction integration (RT-VDI) theories. It can calculate the focal field in any given or predefined incident direction, not limited to the case of axial incidence. In addition, the use of the equivalent parabolic theory greatly simplifies the process of integral calculation. Moreover, ray tracing and integration operations are executed upon the calculation of the reflector to further improve efficiency. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Resampling the optical transmission matrix to accelerate the training of the diffractive deep neural network

The diffractive deep neural network (D2NN) enables all-optical implementation of machine learning tasks. During the training, the Rayleigh–Sommerfeld (RS) diffraction integral is employed for connecting neurons between neighboring hidden layers. The RS formula can be rewritten as a transmission matrix (TM), which allows for the parallel computation of multiple vectorized light fields. However, the TM has a large size, demanding substantial computational resources, and resulting in long training time. In this paper, we propose to resample the TM in free space based on the propagation invariant modes (PIMs), thereby reducing the size of the matrix, and accelerating the propagation simulations. This method enables the training of the large-scale D2NN with reduced memory requirements and fast speed.

Optical cross-correlation analog computing based on dielectric metasurface at mid-infrared wavelengths

Compared to inefficient, energy-intensive digital computing, optical analog computing has significant advantages in terms of computational speed and power consumption. Traditional optical analog computing is typically based on Fourier lenses and spatial frequency filters, which are bulky and difficult to integrate. These disadvantages can be overcome by replacing Fourier lenses and spatial frequency filters in traditional optical analog computing via a metasurface. We showcase a real-time compact optical cross-correlation analog computing based on a dielectric metasurface. An extra phase profile is introduced in the metasurface to approximate the Fresnel diffraction integral as the Fourier transform to realize a more compact scheme. The optical cross-correlation analog computing is derived by directly modulating the impinging optical field. A reference circular pattern of a circular target is highlighted recognized and identified by the operation. The recognition accuracy was analyzed by comparing the recognition peaks with the shifted center of the circle, showing that the operation is affected by the side flap of the other circle pattern in the background. Optical analog computing with a compact and powerful kernel may find potential solutions for bio-detection, pattern recognition, information retrieval, and electron tomography.

Improvement of Fresnel Diffraction Convolution Algorithm

With the development of digital holography, the accuracy requirements for the reconstruction phase are becoming increasingly high. The transfer function of the double fast transform (D-FFT) algorithm is distorted when the diffraction distance is larger than the criterion distance dt, which reduces the accuracy of solving the phase. In this paper, the Fresnel diffraction integration algorithm is improved by using the low-pass Tukey window to obtain more accurate reconstructed phases. The improved algorithm is called the D-FFT (Tukey) algorithm. The D-FFT (Tukey) algorithm adjusts the degree of edge smoothing of the Tukey window, using the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) to remove the ringing effect and obtain a more accurate reconstructed phase. In a simulation of USAF1951, the longitudinal resolution of the reconstructed phase obtained by D-FFT (Tukey) reached 1.5 μm, which was lower than the 3 μm obtained by the T-FFT algorithm. The results of Fresnel holography experiments on lung cancer cell slices also demonstrated that the phase quality obtained by the D-FFT (Tukey) algorithm was superior to that of the T-FFT algorithm. D-FFT (Tukey) algorithm has potential applications in phase correction, structured illumination digital holographic microscopy, and microscopic digital holography.

Propagation characteristics of the Bessel-Gaussian beams in turbulent plasma sheath surrounding the hypersonic aerocraft

To investigate the transmission characteristics of the Bessel-Gaussian beams in a turbulent plasma sheath, the flow field around a hypersonic vehicle is numerically simulated and analyzed according to the Navier-Stokes (N-S) equation and turbulence model. Then, utilizing the characteristics of the thickness of the plasma flow field around the supersonic vehicle at the centimeters level, the double fast Fourier transform (D-FFT) algorithm based on the Fresnel diffraction integral is used for the first time to predict the propagation characteristics of the Bessel-Gaussian beams in the turbulent plasma sheath. The results show that the lower the height and the higher the velocity, the smaller the thickness of the plasma sheath shock layer. The refractive index variance, the wavelength and the waist-radius are essential factors affecting the transmission quality of the Bessel-Gaussian beams. When compared to the Gaussian beams, the Bessel-Gaussian beams have smaller drift and scintillation index, which have the advantage in suppressing turbulence.

Optical Force Effects of Rayleigh Particles by Cylindrical Vector Beams.

High-order cylindrical vector beams possess flexible spatial polarization and exhibit new effects and phenomena that can expand the functionality and enhance the capability of optical systems. However, building a general analytical model for highly focused beams with different polarization orders remains a challenge. Here, we elaborately develop the vector theory of high-order cylindrical vector beams in a high numerical aperture focusing system and achieve the vectorial diffraction integrals for describing the tight focusing field with the space-variant distribution of polarization orders within the framework of Richards-Wolf diffraction theory. The analytical formulae include the exact three Cartesian components of electric and magnetic distributions in the tightly focused region. Additionally, utilizing the analytical formulae, we can achieve the gradient force, scattering force, and curl-spin force exerted on Rayleigh particles trapped by high-order cylindrical vector beams. These results are crucial for improving the design and engineering of the tightly focused field by modulating the polarization orders of high-order cylindrical vector beams, particularly for applications such as optical tweezers and optical manipulation. This theoretical analysis also extends to the calculation of complicated optical vortex vector fields and the design of diffractive optical elements with high diffraction efficiency and resolution.

The Fresnel Approximation and Diffraction of Focused Waves

In this paper, diffraction of scalar waves by a screen with a circular aperture is explored, considering the incidence of either a collimated beam or a focused wave, a historical review of the development of the theory is presented, and the introduction of the Fresnel approximation is described. For diffraction by a focused wave, the general case is considered for both high numerical aperture and for finite values of the Fresnel number. One aim is to develop a theory based on the use of dimensionless optical coordinates that can help to determined the general behaviour and trends of different system parameters. An important phenomenon, the focal shift effect, is discussed as well. Explicit expressions are provided for focal shift and the peak intensity for different numerical apertures and Fresnel numbers. This is one application where the Rayleigh–Sommerfeld diffraction integrals provide inaccurate results.

An advanced light scattering imaging model for total internal reflection microscopy considering a stratified medium

An advanced light scattering model for Total Internal Reflection Microscopy (TIRM) is presented. The model considers the specific TIRM geometry and deals with the scattering by an axisymmetric particle of arbitrary orientation placed in a stratified medium and the imaging of the scattered field. The scattered field is computed by truncating the scattered and internal field expansions and by using spherical and plane wave expansions for the free-space dyadic Green’s function. While the first expansion is valid outside a sphere enclosing the particle, the second one is valid outside the tangent planes bounding the particle from above and below. We demonstrate that in both cases, the results are the same, and thus, that the restrictive condition according to which the interface should not intersect the particle’s circumscribed sphere is not relevant. The image of the scattered field is computed by using the Debye diffraction integral and fast Fourier transform, while for a better reconstruction of the particle orientation, an image processing step consisting in a contour extraction and ellipse fitting is considered. The numerical simulations dealing with scattering by a prolate spheroid provide evidence of the remarkably sensitivity of the geometric parameters of the image ellipse to the particle orientation angles, as well as, of the integral response of the detector to the distance between the particle and the interface.

Gaussian to flat-top beam shaping in an off-axis reflective scenario by a millimeter-wave metasurface.

In this paper, we present a metallic reflectarray for shaping the emitted Gaussian beam of a 100GHz horn antenna in reflective and off-axis mode. The proposed reflectarray is a compact planar surface consisting of a 2D array of unit cells with a thin Rogers 5880 substrate. This reflectarray offers advantages over other beam shapers in millimeter-wave (MMW) imaging, as it generates a flat-top beam in off-axis reflective mode and is straightforward to fabricate. To conduct a feasibility study, we first introduce the 1D reflectarray. The phase profile of the 1D reflectarray is a linear combination of a beam shaping phase profile based on geometrical optics (GO) and a compensating phase profile. The design of the 1D reflectarray involves tailoring the dimensions of the unit cells in one direction. We calculate the scattered field from the 1D reflectarray using the finite element method (FEM) and then determine the generated beam using the Fresnel diffraction integral. Subsequently, we design the 2D reflectarray following the same approach by using a 2D version of the proposed phase profile. The full-wave simulation is employed to calculate the generated beam at a certain distance from the reflectarray. The results reveal the successful generation of a high-quality flat-top beam, showcasing the proposed reflectarray as an evolutionary solution among beam shapers.

Technique for enhancing the accuracy of the Rayleigh-Sommerfeld convolutional diffraction through the utilization of independent spatial sampling.

The Rayleigh-Sommerfeld diffraction integral (RSD) is a rigorous solution that precisely satisfies both Maxwell's equations and Helmholtz's equations. It seamlessly integrates Huygens' principle, providing an accurate description of the coherent light propagation within the entire diffraction field. Therefore, the rapid and precise computation of the RSD is crucial for light transport simulation and optical technology applications based on it. However, the current FFT-based Rayleigh-Sommerfeld integral convolution algorithm (CRSD) exhibits poor performance in the near field, thereby limiting its applicability and impeding further development across various fields. The present study proposes, to our knowledge, a novel approach to enhance the accuracy of the Rayleigh-Sommerfeld convolution algorithm by employing independent sampling techniques in both spatial and frequency domains. The crux of this methodology involves segregating the spatial and frequency domains, followed by autonomous sampling within each domain. The proposed method significantly enhances the accuracy of RSD during the short distance while ensuring computational efficiency.

Fraunhofer approximation of Fresnel integrals

A method to derive the Fraunhofer diffraction integral by making use of the theory of distributions is presented. With respect to standard textbooks, the pedagogical benefit of this method is that it uniquely uses the far-field approximation of the Fresnel diffraction integral. An operator representation of Fresnel diffraction is also derived.

Exponential Fusion of Interpolated Frames Network (EFIF-Net): Advancing Multi-Frame Image Super-Resolution with Convolutional Neural Networks.

Convolutional neural networks (CNNs) have become instrumental in advancing multi-frame image super-resolution (SR), a technique that merges multiple low-resolution images of the same scene into a high-resolution image. In this paper, a novel deep learning multi-frame SR algorithm is introduced. The proposed CNN model, named Exponential Fusion of Interpolated Frames Network (EFIF-Net), seamlessly integrates fusion and restoration within an end-to-end network. Key features of the new EFIF-Net include a custom exponentially weighted fusion (EWF) layer for image fusion and a modification of the Residual Channel Attention Network for restoration to deblur the fused image. Input frames are registered with subpixel accuracy using an affine motion model to capture the camera platform motion. The frames are externally upsampled using single-image interpolation. The interpolated frames are then fused with the custom EWF layer, employing subpixel registration information to give more weight to pixels with less interpolation error. Realistic image acquisition conditions are simulated to generate training and testing datasets with corresponding ground truths. The observation model captures optical degradation from diffraction and detector integration from the sensor. The experimental results demonstrate the efficacy of EFIF-Net using both simulated and real camera data. The real camera results use authentic, unaltered camera data without artificial downsampling or degradation.

General diffraction integral for converging freeform vectorial fields

Herein it is presented a new vector diffraction integral capable to compute the diffraction patterns for aspheric and freeform wavefronts, with different pupil shapes, and different polarization states. The aforementioned vector diffraction integral is based on the energy conservation theorem, the far field approximation, and the angular spectrum representation of optical fields. The integral is validated through illustrative examples whose results are as expected and coincides with the related works.

Highly localized continuous wave optical vortex with controllable orbital angular momentum orientation and topological charge.

We report an approach to simultaneously control orbital angular momentum (OAM) orientation and topological charge in highly localized optical vortices by employing a 4π focusing system. The required continuous wave illumination field in the pupil planes is derived by superimposing the radiation pattern of only one dipole placed at the focal point of the high numerical aperture lens and the corresponding tailored spiral phase factor. The topological charge and OAM orientation of the obtained focused fields are quantitatively evaluated based on the focal field distributions calculated by the Richards-Wolf vector diffraction integration theory. Results show that the OAM of the generated optical vortices can be tailored by changing the oscillation orientation of the mimic dipole and the topological charge of the superimposed spiral phase term. The presented method may find potential applications in optical trapping, optical tweezers, light-matter interaction, etc.

Effect of primary astigmatism on the tight focusing of ellipse field singularities

The tight focusing of ellipse field singularities, namely bright and dark C-points, is studied using the Debye–Wolf diffraction integral in the presence of primary astigmatism. The effect of polarization distribution, the sign of polarization singularity index, amplitude, and handedness of the ellipse field singularities are investigated in the intensity distribution of the focal plane of a high numerical aperture lens system in the presence of primary astigmatism. Ellipse field singularities that are tightly focused can create an additional longitudinal component at the focal plane. At the focal plane, both the longitudinal and transverse components of the ellipse fields are found to be hosted with phase vortices of charge ±1. In the presence of primary astigmatism, the helicity inversion of input C-point does not rotate the focal plane intensity distributions by 180°.

A Systematic Summary and Comparison of Scalar Diffraction Theories for Structured Light Beams

Structured light beams have recently attracted enormous research interest for their unique properties and potential applications in optical communications, imaging, sensing, etc. Since most of these applications involve the propagation of structured light beams, which is accompanied by the phenomenon of diffraction, it is very necessary to employ diffraction theories to analyze the obstacle effects on structured light beams during propagation. The aim of this work is to provide a systematic summary and comparison of the scalar diffraction theories for structured light beams. We first present the scalar fields of typical structured light beams in the source plane, including the fundamental Gaussian beams, higher-order Hermite–Gaussian beams, Laguerre–Gaussian vortex beams, non-diffracting Bessel beams, and self-accelerating Airy beams. Then, we summarize and compare the main scalar diffraction theories of structured light beams, including the Fresnel diffraction integral, Collins formula, angular spectrum representation, and Rayleigh–Sommerfeld diffraction integral. Finally, based on these theories, we derive in detail the analytical propagation expressions of typical structured light beams under different conditions. In addition, the propagation of typical structured light beams is simulated. We hope this work can be helpful for the efficient study of the propagation of structured light beams.

Laser Beam Quality of Airy Beam in the Jet Engine Exhaust Induced Turbulence

Theoretical predictions of laser beam interactions with jet-engine-exhaust-induced turbulence are of importance for optimization of various optical systems, including LIDARs and airborne laser measurement systems. This paper is based on the extended Huygens–Fresnel diffraction integral and the Wigner distribution function, combined with the recently proposed power spectral density function of atmospheric turbulence induced by jet engine exhaust, to derive the formula for the quality factor of a partially coherent Airy beam in turbulent media induced by jet engine exhaust. Numerical calculations based on the analytical formula show that the smaller the structure constant and outer scale of the atmospheric turbulence induced by jet engine exhaust, the smaller the quality factor of a partially coherent Airy beam, indicating better laser beam quality. At the same time, the larger the generalized parameter and inner scale of the jet-engine-exhaust-induced turbulence and the longer the wavelength of the laser beam, the smaller the corresponding quality factor. The numerical results also show that reducing the coherence length of a partially coherent Airy beam and increasing its characteristic width are beneficial for reducing the quality factor in the atmospheric turbulence induced by jet engines. However, the numerical results show that increasing the coherence length of a partially coherent Airy beam and reducing its characteristic width are beneficial for reducing the quality factor in vacuum. This situation demonstrates significant differences between the interaction mechanisms of an Airy beam in vacuum and in the jet-engine-exhaust-induced turbulence. The results of our discussion contribute to clarifying the mechanism of interaction between an Airy laser beam and random media, as well as the application of a curved Airy laser beam in practical situations.

Effect of gamma ray on self-induced diffraction patterns of organic compound Poly (methyl-methacrylate films

Poly (methyl-methacrylate) (PMMA) doped organic compound (2-(2,3-dimethylphenyl-amino)-N-Phenyl benzamide) (2-NPb (films are produced on glass substrates with 5 μm thickness. The influence that gamma (γ) radiation has on the structure of the surface morphology of polymer films, the structure dependence of the optoelectronic properties and the optical properties of a prepared 2-NPb:PMMA film was studied under different doses (21, 42 and 64 KGy) of γ- rays emitted by a 137Cs source with an exposure rate of 0.56 Gy/min. The surface morphology of the 2-NPb:PMMA film for un-irradiated and irradiated sample was studied using a microscope with a digital camera along with texture, waviness and roughenss analysis of the entire surface of the films. The optical properties, together with an in-depth analysis of the dispersion energies and parameters, optoelectrical characteristics, and certain nonlinear optical parameters were comprehensively examined and addressed. In addition, the study of optical properties also included some nonlinear optical parameters. Results show that by doping organic compound (2-NPb) in PMMA, the absorption spectra of samples are decreased and superficial morphology of the film has been affected by γ-rays. As the polymer film was irradiated with a continuous-solid state laser (Solid-State-Laser −532–100 T) beam of 532 nm, self-diffraction ring patterns were seen, and it was shown that the diffraction ring patterns rely heavily on ring numbers and radiation dosage. The nonlinear refractive index,n2, for polymer film for un-irradiated and irradiated are estimated to be 0.23×10-6, 0.92×10-6, 1.64×10-6, 3.14×10-6cm2/W, respectively due to diffraction ring patterns. The diffraction ring patterns that were acquired via experimentation are then numerically estimated by making use of the Fresnel-Kirchhoff diffraction integral. The results of these calculations have satisfactory qualitative and quantitative accords.