Diffraction by a Tilted Aperture

aPIE: an angle calibration algorithm for reflection ptychography

Reflection ptychography is a lensfree microscopy technique particularly promising in regions of the electromagnetic spectrum where imaging optics are inefficient or not available. This is the case in tabletop extreme ultraviolet microscopy and grazing incidence small angle x ray scattering experiments. Combining such experimental configurations with ptychography requires accurate knowledge of the relative tilt between the sample and the detector in non-coplanar scattering geometries. Here, we describe an algorithm for tilt estimation in reflection ptychography. The method is verified experimentally, enabling sample tilt determination within a fraction of a degree. Furthermore, the angle-estimation uncertainty and reconstruction quality are studied for both smooth and highly structured beams.

Polygon-based computer-generated holography: a review of fundamentals and recent progress [Invited]

In this review paper, we first provide comprehensive tutorials on two classical methods of polygon-based computer-generated holography: the traditional method (also called the fast-Fourier-transform-based method) and the analytical method. Indeed, other modern polygon-based methods build on the idea of the two methods. We will then present some selective methods with recent developments and progress and compare their computational reconstructions in terms of calculation speed and image quality, among other things. Finally, we discuss and propose a fast analytical method called the fast 3D affine transformation method, and based on the method, we present a numerical reconstruction of a computer-generated hologram (CGH) of a 3D surface consisting of 49,272 processed polygons of the face of a real person without the use of graphic processing units; to the best of our knowledge, this represents a state-of-the-art numerical result in polygon-based computed-generated holography. Finally, we also show optical reconstructions of such a CGH and another CGH of the Stanford bunny of 59,996 polygons with 31,724 processed polygons after back-face culling. We hope that this paper will bring out some of the essence of polygon-based computer-generated holography and provide some insights for future research.

Nonuniform fast Fourier transform method for numerical diffraction simulation on tilted planes

The method, based on the rotation of the angular spectrum in the frequency domain, is generally used for the diffraction simulation between the tilted planes. Due to the rotation of the angular spectrum, the interval between the sampling points in the Fourier domain is not even. For the conventional fast Fourier transform (FFT)-based methods, a spectrum interpolation is needed to get the approximate sampling value on the equidistant sampling points. However, due to the numerical error caused by the spectrum interpolation, the calculation accuracy degrades very quickly as the rotation angle increases. Here, the diffraction propagation between the tilted planes is transformed into a problem about the discrete Fourier transform on the uneven sampling points, which can be evaluated effectively and precisely through the nonuniform fast Fourier transform method (NUFFT). The most important advantage of this method is that the conventional spectrum interpolation is avoided and the high calculation accuracy can be guaranteed for different rotation angles, even when the rotation angle is close to π/2. Also, its calculation efficiency is comparable with that of the conventional FFT-based methods. Numerical examples as well as a discussion about the calculation accuracy and the sampling method are presented.

Propagation of electromagnetic fields between non-parallel planes: a fully vectorial formulation and an efficient implementation

The propagation of electromagnetic fields between non-parallel planes based on a spectrum-of-plane-wave analysis is discussed and formulations for an efficient numerical implementation are presented in detail. It is shown that with the help of interpolation techniques, the numerical implementation can be done with the standard uniform fast Fourier transform (FFT) of easy access. Different interpolation techniques are numerically examined, and it turns out that the use of cubic interpolation, together with the uniform FFT, brings both significantly increased computational efficiency and high simulation accuracy. Apart from the aspect of computational efficiency, all formulations in this work are generalized in a fully vectorial manner in comparison to previous works.

Fast polygon-based method for calculating computer-generated holograms in three-dimensional display

In the holographic three-dimensional (3D) display, the numerical synthesis of the computer-generated holograms needs tremendous calculation. To solve the problem, a fast polygon-based method based on two-dimensional Fourier analysis of 3D affine transformation is proposed. From one primitive polygon, the proposed method calculates the diffracted optical field of each arbitrary polygon in the 3D model, where the pseudo-inverse matrix, the interpolation, and the compensation of the power spectral density are employed. The proposed method could save the computation time in the hologram synthesis since it does not need the fast Fourier transform for each polygonal surface and the additional diffusion computation. The numerical simulation and the optical experimental results are presented to demonstrate the effectiveness of the method. The results reveal the proposed method could reconstruct the 3D scene with the solid effect and without the depth limitation. The factors that influence the image quality are discussed, and the thresholds are proposed to ensure the reconstruction quality.

Noninvasive identification of subcellular organization and nuclear morphology features associated with leukemic cells using light-scattering spectroscopy

Leukemia is the most common and deadly cancer among children and one of the most prevalent cancers among adults. Improvements in its diagnosis and monitoring of leukemic patients could have a significant impact in their long-term treatment. We demonstrate that light-scattering spectroscopy (LSS)-based approaches could serve as a tool to achieve this goal. Specifically, we characterize the light scattering properties of leukemic (NALM-6) cells and compare them to those of normal lymphocytes and granulocytes in the 440-710 nm range, over ±4 deg about the exact backscattering direction. We find that the LSS spectra are well described by an inverse power-law wavelength dependence, with a power exponent insensitive to the scattering angle but significantly higher for leukemic cells than for normal leukocytes. This is consistent with differences in the subcellular morphology of these cells, detected in differential interference contrast images. Furthermore, the residual light-scattering signal, extracted after subtracting the inverse power-law fit from the data, can be analyzed assuming a Gaussian distribution of spherical scatterers using Mie theory. This analysis yields scatterer sizes that are consistent with the diameters of cell nuclei and allows the detection of the larger nuclei of NALM-6 cells compared to those of lymphocytes and granulocytes.

Method to calculate the far field of three-dimensional objects for computer-generated holography

Here, a new method for calculating the computer-generated holograms of three-dimensional (3D) objects is presented along with a review of current techniques. The method, the planar layers method (PLM), is established on the idea of representing 3D objects in discrete planar layers perpendicular to the observation plane, then calculating the total far field pattern by summing up the far field patterns of each layer. Simulation results, computational complexity, and error comparisons reveal that this new method can be used to calculate far field patterns--hence, the holograms--of computer-synthesized objects very efficiently.

Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves

A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.

Frequency analysis of light diffraction between rotated planes

The relation for the angular spectra of rotated planes is evaluated, starting from the knowledge of the mono-chromatic scalar field on a given plane. Diffracted light on a tilted plane can then be calculated in the frequency domain by fast-Fourier-transform algorithms. Unlike the Fresnel and Fraunhofer approaches, this analysis does not require approximations; as a consequence, it permits any positions in space for the planes under investigation. Digital images are generated, which show the effects of rotation.

Computer modeling wave propagation with a variation of the Helmholtz-Kirchhoff relation

We have employed a variation of the Helmholtz-Kirchhoff equation to computer model a wide range of diffracting systems. The equation leads to a straightforward, easily understood and implemented simulation. It is shown to work as well in the Fresnel as in the Fraunhofer limits and is used to model a wide variety of aperture shapes and incident wavefronts.

Computer-generated holograms of 3-D objects composed of tilted planar segments

The Fresnel diffraction of tilted planar objects is determined. It can be calculated by a Fourier transformation, a coordinate transformation, and a subsequent multiplication by a quadratic phase. Images of 3-D objects are composed by superimposing several planar segments. The resultant complex amplitude is coded and stored in a computer-generated hologram. Optical reconstructions of planar segments nearly parallel to the optical axis are demonstrated.