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We demonstrate spectroscopy of incoherent light with sub-diffraction resolution. In a proof-of-principle experiment we analyze the spectrum of a pair of incoherent point-like sources whose separation is below the diffraction limit. The two…
The scope of this text is to study a process that induces another proof of the Spectral Embedding Theorem: that any densely defined symmetric operator can be extended by a multiplication operator through an embedding of the Hilbert space…
We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety…
Fluorescence microscopy is essential in biological and medical research, providing critical insights into cellular structures. However, limited by optical diffraction and background noise, a substantial amount of hidden information is still…
We propose a microscopic model to describe the scattering of light by atoms in optical lattices. The model is shown to efficiently capture Bragg scattering, spontaneous emission and photonic band gaps. A connection to the transfer matrix…
It is shown that a detailed sub microscopic consideration denies the wave-particle duality for both material particles and field particles, such as photons. In the case of particles, their \psi-wave function is interpreted as the particle's…
The life of a cell is governed by highly dynamical microscopic processes. Two notable examples are the diffusion of membrane receptors and the kinetics of transcription factors governing the rates of gene expression. Different fluorescence…
Quantum entanglement-based imaging promises significantly increased resolution by extending the spatial separation of optical collection apertures used in very-long-baseline interferometry for astronomy and geodesy. We report a table-top…
We develop a theory of light transmission through an aperture-type near-field optical probe with a dissipative matter in its semiconducting core described by a complex frequency-dependent dielectric function. We evaluate the near-field…
A transfer-matrix algorithm is presented herein as a beginning to study the transmission characteristics of coherent light through three-dimensional periodic microstructures, in which the structures are treated as two-dimensional-layer…
Point spread function (PSF) plays an essential role in image reconstruction. In the context of confocal microscopy, optical performance degrades towards the edge of the field of view as astigmatism, coma and vignetting. Thus, one should…
An integrated microscope that combines different optical techniques for simultaneous imaging is demonstrated. The microscope enables spectral-domain optical coherence microscopy based on optical backscatter, and multi-photon microscopy for…
A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $\mathcal{H}$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group $G$ on…
We propose and demonstrate a single channel interferometer that can be used to study how vortex beams propagate. The interferometer consists of a multifunctional diffractive optical element (MDOE) synthesized by the spatial random…
We consider the spectral definition of the fractional Laplace operator and study a basic linear problem involving this operator and singular forcing. In two dimensions, we introduce an appropriate weak formulation in fractional Sobolev…
A dipole-dipole scattering amplitude is calculated exactly in the first two orders of perturbation theory. This amplitude is an analytic function of the relative energy and the dipoles' sizes. The cross section of the dipole-dipole…
Spatial modulation microscopy is a technique originally developed for quantitative spectroscopy of individual nano-objects. Here, a parallel implementation of the spatial modulation microscopy technique is demonstrated based on a line…
An introductory theory of frames on finite dimensional quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart.
Spectroscopy is one of the most important tools that an astronomer has for studying the universe. This chapter begins by discussing the basics, including the different types of optical spectrographs, with extension to the ultraviolet and…
We present and analyse a model for cell signalling processes in biological tissues. The model includes diffusion and nonlinear reactions on the cell surfaces, and both inter- and intracellular signalling. Using techniques from the theory of…