We consider the inverse problem of calculating a refracting surface that generates a prescribed irradiance distribution in the far field for a collimated incident beam. This problem can be formulated as a mass transportation problem (MTP) with a quadratic cost function. To solve this problem, we propose a version of the supporting quadric method (SQM), in which the calculation of the quadric parameters is reduced to the problem of minimizing a convex function. We obtain simple analytical expressions for the second derivatives of this function, making it possible to calculate the quadric parameters using second-order optimization methods. This allows us to refer to the proposed method as the second-order SQM. We demonstrate the high efficiency of this approach by designing several optical surfaces that generate complex irradiance distributions. We also consider the application of the second-order SQM to nonimaging optics problems described by MTPs with a non-quadratic cost function.

We consider a spatiotemporal optical differentiator consisting of a single-layer antireflection coating that performs the optical computation of a differential operator corresponding to the sum of the partial derivatives of the incident pulse envelope with respect to time and a spatial coordinate. We demonstrate theoretically and numerically that such a differentiator makes it possible to generate an optical pulse possessing a spatiotemporal optical vortex, as well as to implement the edge detection operation in the spatiotemporal domain with high quality. The obtained results may find application in the development of analog optical computing and optical information processing systems.

Fizeau fringes are pronounced asymmetric field distributions formed by wedged Fabry–Pérot interferometers also known as the Fizeau interferometers. In this paper, we show that similar effects arise in guided-mode resonant gratings with a wedged waveguide layer. For describing these effects, we develop a formulation of the coupled-mode theory (CMT) with varying parameters, which is shown to be in very good agreement with the results of rigorous electromagnetic simulations of wedged gratings. At the same time, the calculations using the proposed CMT turn out to be by several orders of magnitude faster than solving the Maxwell’s equations numerically. We demonstrate that in addition to the Fizeau fringes, the developed CMT captures intricate optical effects such as the appearance of quasi-bound states in the continuum (quasi-BIC) originating from the symmetry-protected BIC supported by the non-wedged grating. We also extend the proposed CMT formulation to the case of stacked wedged gratings exhibiting flat-top reflection peaks. We believe that the obtained results may find application in the design of variable optical filters and resonators based on gratings with spatially varying parameters.

Bimodal Fabry-P & eacute;rot interferometer is a model generalizing the conventional Fabry-P & eacute;rot interferometer, in which not one but two kinds of waves propagate between the interfaces. Here, we study coherent perfect absorption (CPA) and lasing at threshold in bimodal Fabry-P & eacute;rot interferometers. We show that CPA and lasing appear only in certain "allowed" regions in the parameter space, which, as we demonstrate, are described by closed-form inequalities imposed on the elements of the scattering matrix of the interferometer interfaces. We demonstrate topologically governed annihilation of CPA points as they approach the boundary of a CPA-allowed region. In the particular case when the absorption losses tend to zero, the presented model describes the formation of bound states in the continuum exactly at the CPA annihilation points. The presented analytical model is in perfect agreement with the rigorous numerical simulation results of high-contrast gratings and ridge resonators implementing the bimodal Fabry-P & eacute;rot interferometer model.

We investigate the possibility of the optical computation of the Laplace operator in the oblique incidence geometry using a layered structure consisting of a set of homogeneous thin films. For this, we develop a general description of the diffraction of a three-dimensional linearly polarized optical beam by a layered structure at oblique incidence. Using this description, we derive the transfer function of a multilayer structure consisting of two three-layer metal-dielectric-metal structures and possessing a second-order reflection zero with respect to the tangential component of the wave vector of the incident wave. We show that under a certain condition, this transfer function can coincide up to a constant multiplier with the transfer function of a linear system performing the computation of the Laplace operator. Using rigorous numerical simulations based on the enhanced transmittance matrix approach, we demonstrate that the considered metal-dielectric structure can optically compute the Laplacian of the incident Gaussian beam with the normalized root-mean-square error of the order of 1%. We also show that this structure can be effectively utilized for optical edge detection of the incident signal.

One of the most important properties of diffraction gratings is their ability to direct the incident radiation to a desired diffraction order. Here, we investigate the optical properties of dielectric diffraction gratings separated by a homogeneous layer from a perfect mirror and operating in the Littrow mounting. We obtain closed-form conditions in the form of inequalities imposed on the elements of the scattering matrix of the grating, which are necessary and sufficient for the structure to possess zeros of the 0th or-1st reflected diffraction orders, i.e., to exhibit perfect retroreflection or perfect specular reflection. We also derive simple sufficient conditions for perfect retroreflection and specular reflection. We show that if both of these conditions are satisfied, the reflector-backed grating also supports bound states in the continuum. The obtained theoretical results are fully confirmed by the results of rigorous electromagnetic simulations.

We propose a method for designing diffractive optical elements (DOEs) with a smooth phase function generalizing harmonic diffractive lenses and intended for generating a prescribed intensity distribution at several harmonic wavelengths. In this method, the phase function is represented as an expansion over a certain set of smooth and differentiable functions. The expansion coefficients are considered as optimization parameters and are calculated using a gradient method from the condition of minimizing an error function describing the deviation of the generated intensity distribution from the prescribed one. We present examples of calculating harmonic DOEs with phase functions modulo 2πM, M>1 represented as a sum of B-splines. We show that in order to obtain good performance of a harmonic DOE, the error function has to take into account not only the intensity distribution generated at the central wavelength but also the distributions formed at the other harmonic wavelengths of interest. The obtained theoretical results are confirmed by the results of an experimental investigation, including the fabrication of the designed harmonic DOE using the direct laser writing technique and measurement of the generated intensity distributions.

We theoretically describe and numerically investigate the operation of "vectorial"optical differentiation of a three-dimensional light beam, which consists in simultaneous computation of two partial derivatives of the incident beam profile with respect to two spatial coordinates in different transverse electric field components. It is implemented upon reflection of the beam from a layered structure by simultaneously utilizing the effect of optical resonance and the spin Hall effect of light. As an example of a layered structure performing this operation, we propose a three-layer metal-dielectric-metal (MDM) structure. We show that by choosing the parameters of the MDM structure, it is possible to achieve the so-called isotropic vectorial differentiation, for which the intensity of the reflected optical beam (squared electric field magnitude) is proportional to the squared absolute value of the gradient of the incident linearly polarized beam. The presented numerical simulation results demonstrate high-quality vectorial differentiation and confirm the developed theoretical description.

We theoretically describe the optical computation of the divergence of a two-dimensional vector field, which is composed of the transverse electric field components of an incident light beam. The divergence is computed in reflection at oblique incidence of light on a layered structure. We show that in the particular case of a linearly polarized incident beam, the layered structure implementing the divergence operator also allows one to compute the gradient and perform the isotropic differentiation. As an example of a layered structure computing the divergence, we propose a metal-dielectric multilayer consisting of two pairs of metal and dielectric layers on a metal substrate. The presented numerical simulation results of the designed multilayer confirm that the divergence operator is computed with high accuracy. We also demonstrate the possibility of using the designed structure for optical directional differentiation and computation of the gradient and Laplace operators.

We present a formulation of the supporting quadric method (SQM) taking into account diffraction effects and allowing one to calculate the phase function of a diffractive optical element (DOE) generating a prescribed intensity distribution. In the method, the DOE phase is represented as a piecewise smooth function constituted by the phase functions of lenses (quadrics) focusing the incident beam to the points of the required distribution. The quadric parameters are calculated using a gradient method minimizing an error function representing the difference between the generated and required intensity distributions. Importantly, we calculate the gradient of the error function with respect to the quadric parameters in the framework of the scalar diffraction theory (SDT) in an analytical form. The presented examples of DOE design demonstrate good performance of a hybrid approach based on using the geometrical-optics SQM solution as a starting point for the proposed SQM operating in the SDT framework.

We propose a method for the design of diffractive neural networks (DNNs) for image classification, which takes into account the positioning errors (transverse shifts) of phase diffractive optical elements (DOEs) constituting the DNN. In this method, the error in solving the classification problem is represented by a functional depending on the phase functions of the DOEs and on random vectors describing the transverse shifts of the DOEs. The mathematical expectation of this functional is used as an error functional in the gradient method for calculating the DNN taking into account the transverse shifts of the DOEs. Explicit expressions are obtained for the derivatives of the error functional. It is shown that the calculation of the derivatives of this functional using the Monte Carlo method corresponds to the DNN training method, in which the DOEs have random transverse shifts. By using the proposed gradient method, DNNs are designed that are robust to transverse shifts of the DOEs and enable solving the problem of classifying handwritten digits at a visible wavelength. Numerical simulations demonstrate good performance of the designed DNNs at transverse shifts of up to 17 wavelengths.

Coupled-mode theory (CMT) is a widely used approach for describing resonances and eigenmodes in various photonic structures. Here, we propose a formulation of the CMT describing resonant multilayer structures. In particular, we revisit the conventional Fabry-P & eacute;rot resonator and describe its optical properties from the point of view of the spatiotemporal formulation of the CMT. This formulation provides partial differential equations describing both temporal and spatial evolution of the field distribution, thus generalizing the conventional temporal and spatial versions of the CMT. The developed CMT takes into account the symmetry of the considered structure, energy conservation law, reciprocity, and causality. By considering the parameters of the developed CMT to be spatially dependent, we apply it to describe the optical properties of linear variable filters (LVFs) comprising two Bragg mirrors separated by a wedge-shaped (tapered) layer. In good agreement with the results of the rigorous numerical solution of Maxwell's equations, the proposed CMT accurately reproduces the broadening of the resonant peak and the appearance of Fizeau fringes when increasing the wedge angle of the LVF.

Photonic bound states in the continuum (BICs) are nonradiating eigenmodes of structures with open scattering channels. Most often, BICs are studied in highly symmetric structures with one or two open scattering channels. In this simplest case, the so-called symmetry-protected BICs can be found by tuning a single parameter, which is the light frequency. Another kind of BIC-accidental BICs-can be obtained by tuning two parameters, e.g., the frequency and a wave-vector component. For more complex structures lacking certain symmetries or having many open scattering channels, more than two parameters might be required. In the present work, we propose an algebraic approach for computing the number of parameters required to obtain a BIC by expressing it through the dimension of the solution set of certain algebraic equations. Computing this dimension allows us to relate the required number of parameters to the number of open scattering channels without solving Maxwell's equations. We show that different relations take place when the scattering matrices describing the system are symmetric or asymmetric. The obtained theoretical results are confirmed by the results of rigorous electromagnetic simulations.

Simple isotopic relations between the energy levels of 32S16O2, 33S16O2, and 34S16O2 isotopologues and of other isotopic variants are applied to calculations of vibrational-rotational energy levels. To estimate the accuracy of these isotopic relations, we calculated line centers in the microwave spectrum of 36S16O2 isotopologue and compared them with measured ones. The comparison shows their quite satisfactory agreement at a level of 10−4 cm−1. Vibrational-rotational energy levels of sulfur dioxide isotopologues XS16O2, X = 35–38, are presented for five lower vibrational states up to J = 9.

We consider the design of cascaded phase diffractive optical elements (DOEs) operating at several different wavelengths. The problem of DOE design is formulated as the problem of minimizing a certain error functional that depends on the functions of diffractive microrelief height of the cascaded DOE and evaluates its performance at different design wavelengths. Explicit expressions are obtained for the Frechet derivatives of the error functional. The presented expressions for the derivatives of the error functional constitute the basis for gradient design of cascaded multiwavelength DOEs in various problems including the beam shaping and optical classification problems. As particular example, we consider the calculation of cascaded DOEs focusing radiation of three different wavelengths into different letter-shaped areas.

We consider two analytical models describing resonant scattering of light by resonant diffraction gratings possessing a horizontal symmetry plane. For the case of oblique incidence, a multiple interference model is formulated, on the basis of which, a new approach for deriving the coupled-mode theory for the considered structures is proposed. Both considered models describe resonances in the reflectance and transmittance spectra of the studied diffraction gratings arising due to the excitation of quasiguided modes and Fabry–Pérot modes. Interaction of these modes leads to the appearance of bound states in the continuum, which are also described by the proposed models.

A wide range of practically important problems is nowadays efficiently solved using artificial neural networks. This gave momentum to intensive development of their optical implementations, among which, the so-called diffractive neural networks (DNNs) constituted by a set of phase diffractive optical elements (DOEs) attract considerable research interest. In the practical implementation of DNNs, one of the standing problems is the requirement for high positioning accuracy of the DOEs. In this work, we address this problem and propose a method for the design of DNNs for image classification, which takes into account the positioning errors (transverse shifts) of the DNN elements. In the method, the error of solving the classification problem is represented by a functional depending on the phase functions of the DOEs and on random vectors describing their transverse shifts. The mathematical expectation of this functional is used as an error functional in the gradient method for calculating the DNN taking into account the transverse shifts of the DOEs. It is shown that the calculation of the derivatives of this functional corresponds to the DNN training method, in which the DOEs have random transverse shifts. Using the proposed gradient method, DNNs are designed that are robust to transverse shifts of the DOEs and enable solving the problem of classifying handwritten digits at a visible wavelength. Numerical simulations demonstrate good performance of the designed DNNs at transverse shifts of up to 17 wavelengths.

We consider the problem of designing a diffractive neural network (DNN) consisting of a set of sequentially placed phase diffractive optical elements (DOEs) and intended for the optical solution of several given classification problems at different operating wavelengths, so that each classification problem is solved at the corresponding wavelength. The problem of calculating the DNN is formulated as the problem of minimizing a functional that depends on the functions of the diffractive microrelief height of the DOEs constituting the DNN and represents the error in solving the given classification problems at the operating wavelengths. We obtain explicit and compact expressions for the derivatives of this functional, and using them, we formulate a gradient method for the DNN calculation. Using this method, we design DNNs for solving the following three classification problems at three different wavelengths: the problem of classifying handwritten digits from the MNIST database, the problem of classifying fashion products from the Fashion MNIST database, and the problem of classifying ten handwritten letters from the EMNIST database. The presented simulation results of the designed DNNs demonstrate the high performance of the proposed method.

We establish the connection between two theoretical approaches widely used for describing resonant optical properties of guided-mode resonant gratings (GMRGs): the coupled plane waves model and the coupled-mode theory (CMT). In the coupled plane waves model (also known as the multiple interference model), the field inside and outside the structure is represented as a small number of propagating plane waves; these waves are coupled at one or several interfaces, which leads to a system of linear equations. In the CMT, the optical properties are described by several coupled differential equations, each corresponding to one of the modes supported by the structure. The approach proposed in the present work provides explicit expressions for the coupling coefficients of the CMT through “local” reflection, transmission, and diffraction coefficients of the coupled plane waves model, which is important for the design and analysis of optical resonators, filters, and sensors based on GMRGs.

An Erratum to this paper has been published: https://doi.org/10.1134/S1062873824110017