Rayleigh Resolution Criterion Research Articles

Lateral resolution enhanced interference microscopy using virtual annular apertures

The lateral resolution in microscopic imaging generally depends on both, the wavelength of light and the numerical aperture of the microscope objective lens. To quantify the lateral resolution Ernst Abbe considered an optical grating illuminated by plane waves. In contrast, the Rayleigh criterion holds for two point sources or point scatterers separated by a lateral distance, which are supposed to emit spherical waves. A portion of each spherical wave is collected by the objective lens and results in an Airy disc corresponding to a diffraction limited intensity point spread function (PSF). If incoherent illumination is employed the intensity PSFs related to different scatterers on an object are added resulting in the well-known Rayleigh resolution criterion. In interference microscopy instead of the intensity the electric field scattered or diffracted by an object will be affected by the transfer function of the optical imaging system. For a reflective object the lateral resolution of an interference microscope can be again characterized by the Abbe limit if the object under investigation is a grating. However, if two irregularities on a flat surface are being imaged the resolution no longer obeys the Rayleigh criterion. Instead, it corresponds to an optical system with an annular aperture and thus surpasses the prediction given by the Rayleigh criterion. This holds true for both, amplitude as well as phase objects, as it will be elucidated in this study by theoretical considerations, simulation results and an experimental proof of principle.

Orientation-selective sub-Rayleigh imaging with spatial coherence lattices.

The Rayleigh resolution criterion sets the minimum separation for two-point objects to be distinguishable in a classical optical imaging system. We demonstrate that the sub-Rayleigh resolution can be achieved in a telecentric imaging system with the help of a partially coherent illumination whose spatial coherence has lattice-like distribution. We show that the orientation-selective sub-Rayleigh imaging can be realized by controlling the spatial distribution of the coherence lattice into different symmetries. We carry out a proof-of-principle experiment to demonstrate the orientation-selective sub-Rayleigh imaging for a 1951 USAF resolution target. Our results indicate a flexible orientation-selective high-resolution imaging with spatial coherence engineering of the partially coherent light.

Basic Spatial Resolution Metrics for Satellite Imagers

Six basic criteria used to estimate the Ground Resolution Distance of satellite imagers are considered: Ground Sample Distance (GSD), Rayleigh diffraction limit, ground spot size, generalized Rayleigh resolution criterion, Sparrow limit, and the full-width at half-maximum (FWHM) of the Point/Line Spread Function (PSF/LSF). Theoretical arguments and experimental results that support the use of the LSF FWHM as a spatial resolution metric are presented. The procedure to compute the LSF FWHM from the system modulation transfer function is reviewed and is used to compute the FWHM of the three ideal sensors as a function of their optical factor, contrasting these theoretical results with 22 in orbit measurements. Within this context, the limited range of validity of the GSD and the Rayleigh diffraction limit as spatial resolution estimators is assessed. The limitations of the LSF FWHM as a resolution estimator are also noted.

Lateral and axial resolution criteria in incoherent and coherent optics and holography, near- and far-field regimes.

This work presents an overview of spatial resolution criteria in classical optics, digital optics, and holography. Although the classical Abbe and Rayleigh resolution criteria have been thoroughly discussed in the literature, there are a few issues that still need to be addressed, e.g.,the axial resolution criteria for coherent and incoherent radiation (which is a crucial parameter in 3D imaging), the resolution criteria in the Fresnel regime, and the lateral and the axial resolution criteria in digital optics and holography. This work discusses these issues and provides a simple guide on which resolution criteria should be applied for a particular imaging scheme: coherent/incoherent, far- and near-field, lateral and axial resolution. Different resolution criteria such as two-points resolution and the resolution obtained from the image spectrum (diffraction pattern) are compared and demonstrated with simulated examples. It is shown that, for coherent light, the classical Abbe and Rayleigh resolution criteria do not provide accurate estimation of the lateral and axial resolution. The lateral and axial resolution criteria based on the evaluation of the spectrum of the diffracted wave provide more precise estimation of the resolution for coherent and incoherent light. It is also shown that the resolution criteria derived in approximation of the far-field can be applied for the near-field (Fresnel) imaging regime.

Influence of the wind turbine blade model on echo's Doppler characteristics

Wind turbines have serious electromagnetic interference to nearby radar stations. Accurately obtaining the Doppler characteristics of wind turbine's radar echo is the key technology and precondition to solve this problem. Based on the point scattering model of the wind turbines, the expression of the echo of the wind turbine blade scattering point model which is located at any point in the space is given, and the Doppler characteristic of the wind turbine blade echo is analysed by combining the existing integral model, which proves that the existing integral model is a special case of the scattering point model. Based on the radar resolution and Rayleigh resolution criteria, the scattering point spacing of the wind turbine blade is given. The simulation model is established according to the actual size of the wind turbines. After the analysis and comparison of the scattering point model and the integral wind turbine blade model, the simulation results of the wind turbine radar Doppler results are consistent with the theoretical analysis, which is of great significance to obtain the Doppler characteristics of the wind turbine blades.

Resolution and super-resolution.

Many papers have claimed the attainment of super-resolution, i.e. resolution beyond that achieved classically, by measurement of the profile of a feature in the image. We argue that measurement of the contrast of the image of a dark bar on a bright background does not give a measure of resolution, but of detection sensitivity. The width of a bar that gives an intensity at the center of the bar of 0.735 that in the bright region (the same ratio as in the Rayleigh resolution criterion) is λ/(13.9×numericalaperture) for the coherent case with central illumination. This figure, which compares with λ/(numericalaperture) for the Abbe resolution limit with central illumination, holds for the classical case, and so is no indication of super-resolution. Theoretical images for two points, two lines, arrays of lines, arrays of bars, and grating objects are compared. These results can be used a reference for experimental results, to determine if super-resolution has indeed been attained. The history of the development of the theory of microscope resolution is outlined.

High resolution systems: The resolution argument

Resolution comes from the Latin word resolvo, which literally means “untie a knot.” The meaning evolved, in optics, toward the capability to distinguish two scattering points. For the MCM (Mine Counter Measure) problem, the introduction of synthetic aperture systems, and the resulting drastic increase in resolution, has been a game changer, as the useful information shifts from shadows to highlights. If SAS resolution has been well studied, new SAS reacquisition patterns have emerged, in particular circular acquisitions known as CSAS. In this talk, we will derive the theoretical resolution of such systems based on a PSF (point spread function) energy leakage interpretation rather than the traditional Rayleigh resolution criteria. We will compare the CSAS resolution capability with the newly developed MIMO paradigm. The statistical framework of MIMO systems greatly differs from the deterministic approach of synthetic aperture systems, and offers certain advantages in term of resolution. All these coherent systems approach the imaging problem without any priors, other than the presence of scatterers. Going back to the original meaning of resolvo however, we will show that, by introducing priors such as target shape, wideband sonars shift the analysis into the phase domain and are able to outperform imagery.

Multi-image acquisition-based distance sensor using agile laser spot beam.

We present a novel laser-based distance measurement technique that uses multiple-image-based spatial processing to enable distance measurements. Compared with the first-generation distance sensor using spatial processing, the modified sensor is no longer hindered by the classic Rayleigh axial resolution limit for the propagating laser beam at its minimum beam waist location. The proposed high-resolution distance sensor design uses an electronically controlled variable focus lens (ECVFL) in combination with an optical imaging device, such as a charged-coupled device (CCD), to produce and capture different laser spot size images on a target with these beam spot sizes different from the minimal spot size possible at this target distance. By exploiting the unique relationship of the target located spot sizes with the varying ECVFL focal length for each target distance, the proposed distance sensor can compute the target distance with a distance measurement resolution better than the axial resolution via the Rayleigh resolution criterion. Using a 30mW 633nm He-Ne laser coupled with an electromagnetically actuated liquid ECVFL, along with a 20cm focal length bias lens, and using five spot images captured per target position by a CCD-based Nikon camera, a proof-of-concept proposed distance sensor is successfully implemented in the laboratory over target ranges from 10 to 100cm with a demonstrated sub-cm axial resolution, which is better than the axial Rayleigh resolution limit at these target distances. Applications for the proposed potentially cost-effective distance sensor are diverse and include industrial inspection and measurement and 3D object shape mapping and imaging.

A Resource for Using Real-World Examples in the Physics Classroom

Physics Teaching for the 21st Century (://c21.phas.ubc.ca) is a free online resource for teachers who are interested in teaching physics concepts in real-world contexts. The materials on this site were developed by a team of physics faculty and graduate and undergraduate students at the Department of Physics & Astronomy, University of British Columbia, based on issues of great current concern—reusable energy, climate change, and medical advancement. Topics on the website also focus on applications of physics in the natural world around us. There are currently about 70 different topics on the website and it is not possible to justly give a sense of the website in total here. Instead we will present one complete example of the resources available on our website and show how it can be used in the classroom or in lecture. The example discussed here1 is suitable for a first-year university course and focuses on diffraction through a circular aperture and Rayleigh's resolution criterion by looking at the effect of pupil size on the minimum angle of resolution. The original idea came from reading a book on zoological physics,2 and a short example was later found in a first-year physics textbook.3

Terahertz Harmonic Operation of Microwave Fresnel Zone Plate Lens and Antenna: Frequency Filtering and Space Resolution Properties

This paper examines the binary Fresnel zone plate (FZP) lens frequency-harmonic and space-resolution focusing, and its application as a FZP lens antenna. A microwave FZP lens antenna (FZPA) radiates both at design (90 GHz) and terahertz (THz) odd harmonic frequencies. Frequency and space domain antenna operation are studied analytically by use of the vector diffraction integral applied to a realistic printed FZPA. It is found that all harmonic gain peaks are roughly identical in form, bandwidth, and top values. At each harmonic frequency, the FZPA has a beamwidth that closely follows the Rayleigh resolution criterion. If the lens/antenna resolution is of prime importance and the small aperture efficiency is a secondary problem the microwave-design FZP lens antenna can be of great use at much higher terahertz frequencies. Important feature of the microwave FZP lens is its broader-zone construction compared to the equal in resolution terahertz-design FZP lens. Thus, unique and expensive microtechnology for the microwave FZP lens fabrication is not required. High-order harmonic operation of the FZP lens or lens antenna could find space resolution and frequency filtering applications in the terahertz and optical metrology, imaging tomography, short-range communications, spectral analysis, synchrotron facilities, and so on.

Two-dimensional optical micro-scanner on silicon technology

A two-dimensional optical micro-scanner, which main components are two mobile flat and a concave micro-mirrors, is designed such that, all optical components can be fabricated on the same substratum. The optical parameters, which physical dimensions are between 50 and 500 μm, are obtained within the geometrical optics. The optical performance is evaluated by means of the MTF and Rayleigh resolution criteria, given 80% of modulation for a frequency of 8 cycles/mm with a Gaussian source, the resolution limit is 30 μm.

Scatterfield microscopy for extending the limits of image-based optical metrology

We have developed a set of techniques, referred to as scatterfield microscopy, in which the illumination is engineered in combination with appropriately designed metrology targets to extend the limits of image-based optical metrology. Previously we reported results from samples with sub-50-nm-sized features having pitches larger than the conventional Rayleigh resolution criterion, which resulted in images having edge contrast and elements of conventional imaging. In this paper we extend these methods to targets composed of features much denser than the conventional Rayleigh resolution criterion. For these applications, a new approach is presented that uses a combination of zero-order optical response and edge-based imaging. The approach is, however, more general and a more comprehensive set of analyses using theoretical methods is presented. This analysis gives a direct measure of the ultimate size and density of features that can be measured with these optical techniques. We present both experimental results and optical simulations using different electromagnetic scattering packages to evaluate the ultimate sensitivity and extensibility of these techniques.

Resolution Quality and Atom Positions in Sub-Ångström Electron Microscopy

Ability to determine whether an image peak represents one single atom or several depends on resolution of the HR-(S)TEM. Rayleigh's resolution criterion, an accepted standard in optics, was derived as a means for judging when two image intensity peaks from two sources of light (stars) are distinguishable from a single source. Atom spacings closer than the Rayleigh limit have been resolved in HR-TEM, suggesting that it may be useful to consider other limits, such as the Sparrow resolution criterion. From the viewpoint of the materials scientist, it is important to be able to use the image to deter-mine whether an image feature represents one or more atoms (resolution), and where the atoms (or atom columns) are positioned relative to one another (resolution quality). When atoms and the corresponding image peaks are separated by more than the Rayleigh limit of the HR-(S)TEM, it is possible to adjust imaging parameters so that relative peak positions in the image correspond to relative atom positions in the specimen. When atoms are closer than the Rayleigh limit, we must find the relationship of the peak position to the atom position by peak fitting or, if we have a suitable model, by image simulation. Our Rayleigh-Sparrow parameter QRS reveals the resolution quality of a microscope image. QRS values greater than 1 indicate a clearly resolved twin peak, while values between 1 and 0 mean a lower-quality resolution and an image with peaks displaced from the relative atom positions. The depth of the twin-peak minimum can be used to determine the value of QRS and the true separation of the atom peaks that sum to produce the twin peak in the image. The Rayleigh-Sparrow parameter can be used to refine relative atom positions in defect images where atoms are closer than the Rayleigh limit of the HR-(S)TEM, reducing the necessity for full image simulations from large defect models.

Computation of a spectrum from a single-beam fourier-transform infrared interferogram.

A new high-accuracy method has been developed to transform asymmetric single-sided interferograms into spectra. We used a fraction (short, double-sided) of the recorded interferogram and applied an iterative correction to the complete recorded interferogram for the linear part of the phase induced by the various optical elements. Iterative phase correction enhanced the symmetry in the recorded interferogram. We constructed a symmetric double-sided interferogram and followed the Mertz procedure [Infrared Phys. 7,17 (1967)] but with symmetric apodization windows and with a nonlinear phase correction deduced from this double-sided interferogram. In comparing the solution spectrum with the source spectrum we applied the Rayleigh resolution criterion with a Gaussian instrument line shape. The accuracy of the solution is excellent, ranging from better than 0.1% for a blackbody spectrum to a few percent for a complicated atmospheric radiance spectrum.

Rayleigh Resolution Criterion for Light Sources of Different Spectral Composition

The results of very simple experiments to evaluate Lord Rayleigh Resolution Criterion validity are discussed in cases of quasimonochromatic sources of small angular dimensions (LEDs) and monochromatic sources (Lasers), the emissions of which have different or equal spectral compositions. Visual observations as well as color photographs and color video recording were utilized in the experiments. When LEDs and lasers of different color were used, better resolutions than those of Rayleigh Criterion were obtained owing to the non-spectral yellow false color resulting from the overlapping of the red and green spectral colors. Therefore, the observation of the non-spectral false color implies the super-resolution process. The consideration of the non-spectral false color is a new approach in super-resolution studies. Finally, an illumination and reading system of high density CD-ROM-s (9 GB) based on the obtained results is suggested.

Correlated Ricean noise and resolution in coherent imaging

This paper presents a probabilistic resolution criterion for quantifying the performance of a coherent imaging system incorporating within its formulation the effects of noise. A deterministic field is corrupted by noise whose statistics are assumed to have a correlated Gaussian distribution, giving intensities at the detection plane described by a joint-correlated Ricean random process. The joint-correlated Ricean distribution is obtained and used to formulate a resolution criterion in terms of the joint probability of making two intensity measurements at different locations at the detector plane. Results are presented for two special cases that are equivalent to probabilistic analogues of the familiar Sparrow and Rayleigh resolution criteria used in incoherent imaging. Qualitative comparisons between these criteria are made in terms of the signal-to-noise ratio.

Interference and Fresnel diffraction

It is shown that a surprising number of traditional topics, such as the Rayleigh resolution criterion, spatial filtering, bandlimited signals, the sampling theorem, phase contrast microscopy, and white light holography, can be explained using simple interference theory. These basic results are then easily extended to the N-source case to introduce diffraction theory, and facilitate the teaching of the elements of modern optics to junior and senior-level students. >

Accuracy of period determination

Periods of oscillation are frequently found using one of two methods: least-squares (LSQ) fit or power spectrum. Their errors are estimated using the LSQ correlation matrix or the Rayleigh resolution criterion δv R = 1/ΔT, respectively. It is demonstrated that both estimates are statistically incorrect. On the one hand the LSQ covariance matrix does not account for correlation of residuals from the fit. Neglect of the correlations may cause large underestimation of the variance. On the other hand the Rayleigh resolution criterion is insensitive to signal-to-noise ratio and thus does not reflect quality of observations. The correct variance estimates are derived for the two methods.

“Resolution” in atomic-resolution Electron microscopy

The Rayleigh resolution criterion was developed for incoherent imaging conditions and cannot, in general, be applied to coherent high-resolution transmission electron microscopy (HRTEM). In fact, the Rayleigh criterion may lead to paradoxical results since it considers only two scatterers and does not account for the signal-to-noise ratio. It may, however, be used for dark-field HAAD STEM imaging. For the case of strong multiple scattering in HRTEM lattice imaging, the only general relations that can be assumed between the image of a specimen and its projected crystal potential are those imposed by symmetry, and the “local” HRTEM column approximation. The effects of limited resolution may produce an image of lower symmetry than that of the object. Also, misalignment along symmetry axes may preserve some symmetry elements. Under coherent conditions the ability to distinguish atoms in an image depends on their scattering phases, which are a property of the sample.

Formulation for quantitative performance evaluation of holographic imaging

In this article, an alternative approach is proposed to quantitatively evaluate image quality of holographic reconstructions. Instead of the conventional approach outlined by the Rayleigh resolution criterion, this approach describes the image quality in terms of the certainty of the estimation of the source distribution. Given the classical imaging parameters, this formulation gives results in exact agreement with the Rayleigh criterion. In addition, this formulation can be extended to quantitatively evaluate the degradation factors such as aperture offset, wave field sampling, quantization error, noise level, and phase-only data acquisition that are not considered in the formulation of Rayleigh resolution criterion. This approach can be also used for quantitative performance evaluation of enhancement algorithms for comparison purposes.