The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.

On the diffraction theory of optical images

The feasibility of transcranial passive acoustic mapping with hemispherical sparse arrays (30 cm diameter, 16 to 1372 elements, 2.48 mm receiver diameter) using CT-based aberration corrections was investigated via numerical simulations. A multi-layered ray acoustic transcranial ultrasound propagation model based on CT-derived skull morphology was developed. By incorporating skull-specific aberration corrections into a conventional passive beamforming algorithm (Norton and Won 2000 IEEE Trans. Geosci. Remote Sens. 38 1337–43), simulated acoustic source fields representing the emissions from acoustically-stimulated microbubbles were spatially mapped through three digitized human skulls, with the transskull reconstructions closely matching the water-path control images. Image quality was quantified based on main lobe beamwidths, peak sidelobe ratio, and image signal-to-noise ratio. The effects on the resulting image quality of the source’s emission frequency and location within the skull cavity, the array sparsity and element configuration, the receiver element sensitivity, and the specific skull morphology were all investigated. The system’s resolution capabilities were also estimated for various degrees of array sparsity. Passive imaging of acoustic sources through an intact skull was shown possible with sparse hemispherical imaging arrays. This technique may be useful for the monitoring and control of transcranial focused ultrasound (FUS) treatments, particularly non-thermal, cavitation-mediated applications such as FUS-induced blood–brain barrier disruption or sonothrombolysis, for which no real-time monitoring techniques currently exist.

/pdf/transcranial-passive-acoustic-mapping-with-hemispherical-1psg9vfw8u.pdf

Transcranial passive acoustic mapping with hemispherical sparse arrays using CT-based skull-specific aberration corrections: a simulation study

A new class of optimum amplitude filters

We show that, at any Fresnel number, a suitable one-dimensional Fourier transform relates the complex-amplitude distribution along the optical axis with the zero-order circular harmonic of the amplitude transmittance of a two-dimensional diffracting screen. First, our general result is applied to recognize that any rationally nonsymmetric screen generates an axial-irradiance distribution that exhibits focal shift. In this way we identify a wide set of two-dimensional screens that produce the same focal shift as that produced by the clear circular aperture. Second, we identify several apodizers for shaping the axial-amplitude distribution. We discuss some examples for achieving high-precision focusing, axial hyperresolution, or high focal depth.

/pdf/on-axis-diffractional-behavior-of-two-dimensional-pupils-35q8zn5aa7.pdf

On-axis diffractional behavior of two-dimensional pupils

The resolution of two points having unequal intensity has been investigated under the partially coherent illumination by using the newly modified Sparrow criterion. The resolution is discussed as functions of the degree of spatial coherence of the illumination and the intensity difference of two object points. The critical resolution increases as the degree of coherence decreases, irrespective of the intensity difference of two points. As the difference between the two point intensities increases, the critical resolution decreases. The two ratios of the measurable to the real point separation and of the measurable maximum intensity to the real object intensity are studied in some detail as functions of the degree of coherence of the illumination and the difference of two point intensities.

http://iopscience.iop.org/article/10.1088/0335-7368/5/3/304/pdf

Resolution of two unequally bright points with partially coherent light

Walsh functions provide a viable and interesting tool in the treatment of problems of optical imagery. A reformulation of Walsh functions in polar coordinates facilitates the analysis which is presented, along with a restatement of the basic properties of these functions pertinent to the purpose. Some numerical results of aberrational diffraction images are cited to illustrate the compatibility of the proposed approach.

Walsh Functions in Problems of Optical Imagery

This paper presents a complete analytical solution of the problem of far-field diffraction by apertures of a partially space-coherent wave-field. The effect of atmospheric turbulence on the far-field diffraction pattern has been studied as regards the intensity distribution, the factor of encircled energy distribution and the effective central illumination on the same. Numerical results are presented for a few representative aperatures. Indications are given regarding the convenient amenability of the approach to the problem of optimization.

Apodization of Telescopes Working in a Turbulent Medium

The classical approach to the study of two-point resolution has considered the case of two infinitesimal pinholes in an opaque screen. Since partially coherent imagery is non-linear with respect to either intensity or amplitude, the question of resolution needs to be re-examined for a pair or point-like particles embedded in a medium of finite but different transmission. This paper deals with this situation in ordinary microscopy. It has been found that the limit of resolution may be expressed as  0 + j , where  0 is the limit obtainable on the basis of the Sparrow criterion, and is seen to be independent of the original object contrast. The second term j  is an incremental term that depends on the object contrast and the final required image contrast. Both terms, however, depend on the degree of coherence between the object particles. A simple semi-empirical expression K s =  0 sin f / u = 0·475 + 0·26 o gives the Sparrow limit of resolution, where o is the coherence between the pair of points. Curv...

Two-point Resolution in Partially Coherent Light

A general solution of the problem of imagery of phase discs in partially coherent phase contrast microscopy is given. The solution has been obtained in a form very much amenable to analysis of optimum phase plate characteristics.

Phase Contrast Imagery of Circular Phase Objects in Partially Coherent Light

This paper extends the analysis of optical imagery on the basis of Walsh functions to the determination of the diffraction-based optical transfer function (OTF) for on-axis systems. A semi-analytic and simplified expression for OTF is obtained which is directly amenable to subsequent operations which are often called for in optimization and tolerancing analyses. The computational speed for OTF evaluation is greatly increased where repetitive calculation of the OTF for different pupil plane parameters is required. Some computational results are presented and compared with the results obtained by other methods.

M. De

Papers

Walsh Functions in Problems of Optical Imagery

Apodization of Telescopes Working in a Turbulent Medium

Two-point Resolution in Partially Coherent Light

Phase Contrast Imagery of Circular Phase Objects in Partially Coherent Light

Walsh functions in lens optimization: I. FEE-based criterion