Rayleigh-Sommerfeld Diffraction Formula Research Articles

Fast scaled cylindrical holography based on scaled convolution

Recently, the 360°display of cylindrical holography has garnered significant attention. However, existing studies have been limited to concentric cylindrical surfaces with equal heights, which restricts the height of the objects. In this paper, a fast scaled cylindrical holography is proposed based on scaled convolution to break this constraint. Firstly, the scale cylindrical diffraction is derived using the Rayleigh–Sommerfeld diffraction formula for scaled cylindrical holography, in which the object’s height can be adjusted. Then, a method of fast scaled cylindrical diffraction is presented by scaled convolution based on scaled Fourier Transform, and it significantly reduces computational time, enhancing its practical feasibility. Furthermore, the proposed method is extended to 3D objects using a layer-oriented approach. Moreover, the effectiveness and feasibility of the proposed method are verified through numerical simulations and optical experiments. Therefore, the issue of the constraint on the object’s height in cylindrical holography is thoroughly discussed and effectively addressed for the first time. This breakthrough paves the way for the development of a zoom-able cylindrical holographic display with immense potential for applications in mixed reality and beyond.

Conical holographic display to expand the vertical field of view.

Recently, cylindrical holographic display technology as a 360-degree display technology has attracted much attention. However, all the studies are based on the field of view (FOV) in the azimuth direction, and the issue of FOV in the vertical direction has never been discussed. In this paper, a new holographic display is proposed to expand the vertical FOV by a conical holographic diffraction model, in which the object plane is the outer cylinder and the observation plane is a part of the cone. In this proposed method, the proposed diffraction model is firstly established by the Rayleigh-Sommerfeld diffraction formula, and then the convolution and FFT are used for a fast diffraction calculation. The correctness and effectiveness of our proposed method are verified by the simulation of Young's interference and the numerical reconstructions from the complex amplitude and encoded holograms, respectively. In addition, an accurate relationship between the conical inclination angle and the vertical FOV expansion is analyzed, and the simulation results show that our proposed method can significantly expand the vertical FOV by 0.4 of the original object. Therefore, the issue of the vertical FOV in cylindrical holography is deeply discussed and successfully addressed for the first time.

Nonparaxial propagation and the radiation forces of the chirped annular Bessel Gaussian beams

We propose a new class of abruptly autofocusing beams, the chirped annular Bessel Gaussian beams (CABGBs), which are based on the radial Bessel profile. By using the Rayleigh–Sommerfeld diffraction formula, the nonparaxial propagation of the CABGBs is investigated. We find that the initial chirp with negative value tends to intensify the abruptly autofocusing property by shortening the focal length, shrinking the focal spot size, and increasing the peak intensity contrast. The optical trapping effect of the CABGBs acting on a Rayleigh particle is analyzed, and we find that the particle can be stably trapped in three dimensions if the parameters of the CABGBs are appropriate. If the absolute value of the negative chirp increases, the trapping position is shifted toward the input plane, and both the longitudinal and transverse trapping forces rise. A higher magnitude of the radiation forces can be reached if the incident power increases.

Fourier propagation tool for aberration analysis and a point spread function calculation of systems with curved focal planes

This paper describes a new Fourier propagator for computing the impulse response of an optical system with a curved focal plane array, while including terms ignored in Fresnel and Fraunhofer calculations. The propagator includes a Rayleigh-Sommerfeld diffraction formula calculation from a distant point through the optical system to its image point predicted by geometric optics on a spherical surface. The propagator then approximates the neighboring field points via the traditional binomial approximation of the Taylor series expansion around that field point. This technique results in a propagator that combines the speed of a Fourier transform operation with the accuracy of the Rayleigh-Sommerfeld diffraction formula calculation and extends Fourier optics to cases where the receiver plane is a curved surface. Bounds on the phase error introduced by the approximations are derived, which show it should be more widely applicable than traditional Fresnel propagators. Guidance on how to sample the pupil and detector planes of a simulated imaging system is provided. This report concludes by showing examples of the diffraction patterns computed by the new technique compared to those computed using the Rayleigh-Sommerfeld technique in order to demonstrate the utility of the propagator.

Huygens-Fresnel principle: Analyzing consistency at the photon level

Typically the use of the Rayleigh-Sommerfeld diffraction formula as a photon propagator is widely accepted due to the abundant experimental evidence that suggests that it works. However, a direct link between the propagation of the electromagnetic field in classical optics and the propagation of photons where the square of the probability amplitude describes the transverse probability of the photon detection is still an issue to be clarified. We develop a mathematical formulation for the photon propagation using the formalism of electromagnetic field quantization and the path-integral method, whose main feature is its similarity with a fractional Fourier transform (FrFT). Here we show that, because of the close relation existing between the FrFT and the Fresnel diffraction integral, this propagator can be written as a Fresnel diffraction, which brings forward a discussion of the fundamental character of it at the photon level compared to the Huygens-Fresnel principle. Finally, we carry out an experiment of photon counting by a rectangular slit supporting the result that the diffraction phenomenon in the Fresnel approximation behaves as the actual classical limit.

Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation

This paper describes a Fourier propagator for computing the impulse response of an optical system, while including terms ignored in Fresnel and Fraunhofer calculations. The propagator includes a Rayleigh–Sommerfeld diffraction formula calculation from a distant point through the optical system to its image point predicted by geometric optics. The propagator then approximates the neighboring field points via the traditional binomial approximation of the Taylor series expansion around that field point. This technique results in a propagator that combines the speed of a Fourier transform operation with the accuracy of the Rayleigh–Sommerfeld diffraction formula calculation and extends Fourier optics to cases that are nonparaxial. The proposed propagator facilitates direct calculation of aberration coefficients, making it more versatile than the angular spectrum propagator. Bounds on the phase error introduced by the approximations are derived, which show that it should be more widely applicable than the Fresnel propagator. Guidance on how to sample the pupil and detector planes of a simulated imaging system is provided. This report concludes by showing examples of diffraction calculations for a laboratory setup and comparing them to measured diffraction patterns to demonstrate the utility of the propagator.

Diffraction by random Ronchi gratings

In this work, we obtain analytical expressions for the near-and far-field diffraction of random Ronchi diffraction gratings where the slits of the grating are randomly displaced around their periodical positions. We theoretically show that the effect of randomness in the position of the slits of the grating produces a decrease of the contrast and even disappearance of the self-images for high randomness level at the near field. On the other hand, it cancels high-order harmonics in far field, resulting in only a few central diffraction orders. Numerical simulations by means of the Rayleigh-Sommerfeld diffraction formula are performed in order to corroborate the analytical results. These results are of interest for industrial and technological applications where manufacture errors need to be considered.

A computer algebra derivation of the Rayleigh–Sommerfeld diffraction formula by a plane screen

The Rayleigh–Sommerfeld diffraction formula is derived without resorting to generalized or Green functions starting from the plane wave or angular spectrum representation of the scalar field after a plane screen. All the steps are clarified by means of computer algebra software.

Calculation of the volumetric diffracted field with a three-dimensional convolution: the three-dimensional angular spectrum method

The first Rayleigh-Sommerfeld diffraction formula is treated in an exact form as a three-dimensional (3D) convolution in the spatial domain. Therefore, a 3D Fourier transform can be employed to convert the 3D diffracted electromagnetic field to the reciprocal space without approximations, which we call the 3D angular spectrum (3D-AS) method. It is also demonstrated that if evanescent waves are neglected, the 3D-AS method can be readily implemented numerically, with the results in good agreement with theoretical predictions.

Rayleigh–Sommerfeld diffraction formula in k space

An angular spectrum representation in three dimensions is used to develop three-dimensional Fourier forms of the first and second Rayleigh-Sommerfeld diffraction formulae and the Kirchhoff diffraction formula. For forward-propagating waves, these reduce to three-dimensional Fourier representations for diffraction in the forward half-space.

Fractional second-harmonic Talbot effect

We demonstrate the fractional second-harmonic (SH) Talbot effect in a hexagonally poled LiTaO3 crystal. We carefully record the SH Talbot images at 1/2, 1/3, and 1/4 Talbot lengths, which are well matched with the simulated images by using the modified Rayleigh-Sommerfeld diffraction formula. A simplified model with a hexagonal array is adopted in the simulations. Also, we use a modified reciprocal vector theory to analytically explain the evolution of the SH array at fractional Talbot lengths. Our results show that the images are sensitive to the duty circle and the background of the array.

Optimized 3-D Simulation Method for Modeling Out-of-Plane Radiation in Silicon Photonic Integrated Circuits

We present an accurate and fast 3D simulation scheme for out-of-plane grating couplers, based on two dimensional rigorous (finite difference time domain) grating simulations, the effective index method (EIM), and the Rayleigh-Sommerfeld diffraction formula. In comparison with full 3D FDTD simulations, the rms difference in electric field is below 5% and the difference in power flux is below 3%. A grating coupler for coupling from a silicon-on-insulator photonic integrated circuit to an optical fiber positioned 0.1 mm above the circuit is designed as example.

Pulsed TV holography measurement and digital reconstruction of compression acoustic wave fields: application to nondestructive testing of thick metallic samples

This paper describes a technique that numerically reconstructs the complex acoustic amplitude (i.e. the acoustic amplitude and phase) of a compression acoustic wave in the interior volume of a specimen from a set of full-field optical measurements of the instantaneous displacement of the surface. The volume of a thick specimen is probed in transmission mode by short bursts of narrowband compression acoustic waves generated at one of its faces. The temporal evolution of the displacement field induced by the bursts emerging at the opposite surface is measured by pulsed digital holographic interferometry (pulsed TV holography). A spatio-temporal 3D Fourier transform processing of the measured data yields the complex acoustic amplitude at the plane of the surface as a sequence of 2D complex-valued maps. Finally, a numerical implementation of the Rayleigh–Sommerfeld diffraction formula is employed to reconstruct the complex acoustic amplitude at other planes in the interior volume of the specimen. The whole procedure can be regarded as a combination of optical digital holography and acoustical holography methods. The technique was successfully tested on aluminium specimens with and without an internal artificial defect and sample results are presented. In particular, information about the shape and position of the defect was retrieved in the experiment performed on the flawed specimen, which indicates the potential applicability of the technique for the nondestructive testing of materials.

Packed domain Rayleigh-Sommerfeld wavefield propagation for large targets

For applications in the domain of digital holographic microscopy, we present a fast algorithm to propagate scalar wave fields from a small source area to an extended, parallel target area of coarser sampling pitch, using the first Rayleigh-Sommerfeld diffraction formula. Our algorithm can take full advantage of the fast Fourier transform by decomposing the convolution kernel of the propagation into several convolution kernel patches. Using partial overlapping of the patches together with a soft blending function, the Fourier spectrum of these patches can be reduced to a low number of significant components, which can be stored in a compact sparse array structure. This allows for rapid evaluation of the partial convolution results by skipping over negligible components through the Fourier domain pointwise multiplication and direct mapping of the remaining multiplication results into a Fourier domain representation of the coarsly sampled target patch. The algorithm has been verified experimentally at a numerical aperture of 0.62, not showing any significant resolution limitations.

Propagation properties of Gaussian beams through the anamorphic fractional Fourier transform system with an eccentric circular aperture

As known, it is important for the propagation of Gaussian beams in optics. In this paper, based on the expanding a hard-edge circular aperture function as a finite sum of complex Gaussian functions and the scalar Rayleigh–Sommerfeld diffraction formula, an approximation analytical solution for Gaussian beams propagating through the anamorphic fractional Fourier transform system with an eccentric circular aperture is performed. Then, the detailed numerical calculation for the two-dimensional Gaussian beams in the above-mentioned optical system is presented. The simulation also shows that different location and size of aperture result in the change of diffracted field, including location, intensity and width. All these characteristics help us understand Gaussian beams propagation better.

Fast computation algorithm for the Rayleigh-Sommerfeld diffraction formula using a type of scaled convolution

We describe a fast computational algorithm able to evaluate the Rayleigh-Sommerfeld diffraction formula, based on a special formulation of the convolution theorem and the fast Fourier transform. What is new in our approach compared to other algorithms is the use of a more general type of convolution with a scale parameter, which allows for independent sampling intervals in the input and output computation windows. Comparison between the calculations made using our algorithm and direct numeric integration show a very good agreement, while the computation speed is increased by orders of magnitude.

Shaping tight-focusing patterns of linearly polarized beams through elliptic apertures

We propose elliptic deformation of apertures for shaping focusing patterns of tightly focused linearly polarized beams. Numerical integration of the vectorial Rayleigh-Sommerfeld diffraction formula predicts the formation of symmetric focusing patterns by shrinking the aperture shape in the direction perpendicular to the polarization of the beams. The elliptic deformation is also applied to a focused linearly polarized beam through an annular aperture to demonstrate the formation of smaller symmetric focal spot than that through a simple oval aperture.

A circular aperture-array structure optical system for digital sun sensor

Based on the first type of Rayleigh Sommerfeld diffraction formula, an imaging model of circular aperture-array structure digital sun sensor optical system is developed. Then a 6×6 circular aperture-array structure optical system is designed. The results of numerical simulation show that the optical system is designed well and is conformed to the requirements of miniaturization and high accuracy of sun sensor.

Class of basis functions for use in optical systems analysis

A method is described for modeling the effects of spatial apertures on optical sensor systems. The method consists of defining a set of basis functions that is obtained by partitioning the aperture image plane into a series of rectangular regions and replacing the field in each rectangular subregion with an orthogonal function series approximation. Each orthogonal function has a finite extent that is matched to the aperture image. The individual functions are propagated by application of the Fresnel approximation of the Rayleigh-Sommerfeld diffraction formula to other ranges, and the resultant functions are shown to be valid basis functions for defining a field at any other range. The technique is applied to a scattering problem using complex Fourier series.

Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula

The numerical calculation of the Rayleigh-Sommerfeld diffraction integral is investigated. The implementation of a fast-Fourier-transform (FFT) based direct integration (FFT-DI) method is presented, and Simpson's rule is used to improve the calculation accuracy. The sampling interval, the size of the computation window, and their influence on numerical accuracy and on computational complexity are discussed for the FFT-DI and the FFT-based angular spectrum (FFT-AS) methods. The performance of the FFT-DI method is verified by numerical simulation and compared with that of the FFT-AS method.