Rayleigh-Sommerfeld Formulation Research Articles

Optical reconstruction of transparent objects with phase-only SLMs

Three approaches for visualization of transparent micro-objects from holographic data using phase-only SLMs are described. The objects are silicon micro-lenses captured in the near infrared by means of digital holographic microscopy and a simulated weakly refracting 3D object with size in the micrometer range. In the first method, profilometric/tomographic data are retrieved from captured holograms and converted into a 3D point cloud which allows for computer generation of multi-view phase holograms using Rayleigh-Sommerfeld formulation. In the second method, the microlens is computationally placed in front of a textured object to simulate the image of the textured data as seen through the lens. In the third method, direct optical reconstruction of the micrometer object through a digital lens by modifying the phase with the Gerchberg-Saxton algorithm is achieved.

Light trapping in solar cells: When does a Lambertian scatterer scatter Lambertianly?

We derive scaling laws for the Rayleigh-Sommerfeld formulation we recently developed to describe light scattering from nanotextured interfaces. These scaling laws provide precious intuition on how to link scattering from different interfaces. In particular, we answer the question how to obtain a Lambertian scatterer into silicon, starting from a Lambertian scatterer into air relevant to the development of light trapping schemes in thin-film silicon solar cells. We also define a Lambertionality factor which measures how close an arbitrary scatterer approaches Lambertian scattering and extend the fundamental 4n2 light trapping limit to arbitrary scattering distributions.

Spatio-temporal characterization of ultrashort pulses diffracted by circularly symmetric hard-edge apertures: theory and experiment

We carry out a complete spatio-temporal characterization of the electric field of an ultrashort laser pulse after passing through a diffractive optical element composed of several binary amplitude concentric rings. Analytical expressions for the total diffraction field in the time and spectral domain are provided, using the Rayleigh-Sommerfeld formulation of the diffraction. These expressions are experimentally validated. The spatio-temporal amplitude and phase structure of the pulse are measured at different planes beyond the diffractive optical element using spatially-resolved spectral interferometry assisted by an optical fiber coupler (STARFISH). Our results allow corroborating theoretical predictions on the presence of multiple pulses or complex spectral distributions due to the diffraction-induced effects by the hard-edge ring apertures.

Near-field diffraction of gratings with surface defects

Diffraction gratings produce self-images in the near field. Defects on the surface of the grating may occur due to the manufacturing process. These devices are often placed in dirty industrial environments. Dust particles or drops of liquid can be deposited over their surface. In this work, we analyze the effect of surface defects placed over the grating on the self-imaging process. We analytically show how the self-images gradually recover as we separate from the grating when one defect is present. Also a random distribution of surface defects over the grating is analyzed. In particular, we focus on how the contrast of the self-images decreases in terms of the density of the defects. Analytical expressions for the near field are derived, considering a stochastic description of the spatial distribution of defects. In addition, numerical simulations based on the Rayleigh-Sommerfeld formulation are performed to validate the analytical results.

Spectral analysis of femtosecond pulse diffraction through binary diffractive optical elements: theory and experiment

We report on the changes in the spectrum of a femtosecond pulse originated by diffraction of the ultrashort waveform through a circularly symmetric binary diffractive optical element. The analysis is performed in the framework of the Rayleigh-Sommerfeld formulation of the diffraction, where an analytical expression for the monochromatic amplitude distribution close to the optical axis is obtained. To corroborate our results, we experimentally measure the variations of the pulse spectrum within the collecting area of a spectrometer located at the output plane. Multiple splitting of the pulse spectrum in the vicinity of a focal position and a phase singularity are shown.

Focusing and spectral characteristics of periodic diffractive optical elements with circular symmetry under femtosecond pulsed illumination

The analytical solution is derived, within the Rayleigh-Sommerfeld formulation of diffraction, for the on-axis spectral irradiance of a broadband source after diffracting through a circular symmetric hard aperture. By using this solution, and within the paraxial approximation, we investigate several diffraction-induced effects originated by binary diffractive optical elements made up of a set of annular apertures with equal areas and periodic in the squared radial coordinate. In particular, the ability to focus femtosecond pulses is investigated. In addition, the analysis of the spectral modifier function associated with these elements allows us to simulate spectral shifts at focus positions. Finally, we introduce a relatively simple and low-cost technique to slice the spectrum of a broadband source in order to generate narrow bands or wavelength channels.

Model-Based Optoelectronic Packaging Automation

In this paper, we present an automation technique that yields high-performance, low-cost optoelectronic alignment and packaging through the use of intelligent control theory and system-level modeling. The control loop design is based on model-based control, previously popularized in process and other control industries. The approach is to build an a priori knowledge model, specific to the assembled package's optical power propagation characteristics, and use this to set the initial conditions of the automation system. In addition to this feed-forward model, the controller is designed with feedback components, along with the inclusion of a built in optical power sensor. The optical modeling is performed with the rigorous scalar Rayleigh-Sommerfeld formulation, efficiently solved online using an angular spectrum technique. One of the benefits of using a knowledge-based control technique is that the efficiency of the automation process can be increased, as the number of alignment steps can be greatly reduced. An additional benefit of this technique is that it can reduce the possibility that attachment between optical components will occur at local power maximums, instead of the global maximum of the power distribution. Therefore, the technique improves system performance, while reducing the overall cost of the automation process.

Shape calibration of a conformal ultrasound therapy array.

A conformal ultrasound phased array prototype with 96 elements was recently calibrated for electronic steering and focusing in a water tank. The procedure for calibrating the shape of this 2D therapy array consists of two steps. First, a least squares triangulation algorithm determines the element coordinates from a 21 x 21 grid of time delays. The triangulation algorithm also requires temperature measurements to compensate for variations in the speed of sound. Second, a Rayleigh-Sommerfeld formulation of the acoustic radiation integral is aligned to a second grid of measured pressure amplitudes in a least squares sense. This shape calibration procedure, which is applicable to a wide variety of ultrasound phased arrays, was tested on a square array panel consisting of 7- x 7-mm elements operating at 617 kHz. The simulated fields generated by an array of 96 equivalent elements are consistent with the measured data, even in the fine structure away from the primary focus and sidelobes. These two calibration steps are sufficient for the simulation model to predict successfully the pressure field generated by this conformal ultrasound phased array prototype.

Electromagnetic calculation of soft x-ray diffraction from 0.1-μm scale gold structures

Because the effects of diffraction during proximity-print x-ray lithography are of critical importance, a number of previous researchers have attempted to calculate the diffraction patterns and minimum achievable feature sizes as a function of wavelength and gap. Work to date has assumed that scalar diffraction theory is applicable—as calculated, e.g., by the Rayleigh–Sommerfeld formulation—and that Kirchhoff boundary conditions (KBC) can be applied. KBC assume that the fields (amplitude and phase) are constant in the open regions between absorbers, and a different constant in regions just under the absorbers (i.e., that there are no fringing fields). An x-ray absorber is, however, best described as a lossy dielectric that is tens or hundreds of wavelengths tall, and hence KBC are unsuitable. In this report we use two numerical techniques to calculate (on a Cray 2 supercomputer) accurate diffracted fields from gold absorbers for two cases: a 30-nm-wide line at λ=4.5 nm, and a 100-nm-wide line at λ=1.3 nm. We show that the use of KBC introduces unphysically high spatial frequencies into the diffracted fields. The suppression of these frequencies—which occurs naturally without the need to introduce an extended source or broad spectrum—tremendously improves exposure latitude for mask features near 0.1 μm and below. In particular, we show that KBC should not be applied to 0.1-μm features and wavelengths near 1.3 nm for gaps below 11 μm.

Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: A comparison of exact evaluations for axial points*

The second Rayleigh–Sommerfeld (RS) diffraction integral, wherein the normal derivative is specified, is evaluated in simple closed form for all axial points when a divergent or convergent spherical wave is incident upon a circular aperture or disk. These evaluations (solutions) are compared with known corresponding solutions of the first RS diffraction integral. These sets of solutions are intercompared with their mean value, i.e., the derived solutions of the Kirchhoff diffraction integral. The three diffraction formulations are shown to be in agreement for incident divergent spherical waves when the source and observation points are equally distant from the aperture or disk. Conversely, for convergent spherical waves, the three formulations are never in exact agreement for focal and observation points located at finite distances from the aperture, though at optical frequencies the relative error at the geometric focal point is vanishingly small. The second RS formulation predicts, in the limit of plane waves incident on a disk, that the axial irradiance is everywhere equal to the incident irradiance, whereas the first RS formulation predicts that the irradiance goes to zero at the back of the disk.