Spectral Analysis and Solvability of Volterra Integro-Differential Equations

Doklady Mathematics - Integro-differential equations with unbounded operator coefficients in a Hilbert space are studied. The equations under consideration are abstract hyperbolic equations...     

Polarization properties of fluctuations of the field created by randomly located dipoles

The vector properties of the strength of a field created by an ensemble of N parallel dipoles located, on average, uniformly are considered. For N → ∞, the problem is reduced to the Poisson problem. The direction of the dipoles specifies the symmetry axis of fluctuations. The moments of probability distributions for the Cartesian components of the strength are calculated. The anisotropy of the fluctuations is approximately 15–20%. Under conditions that are of interest for spectroscopic applications, the N-particle probability distribution contains a weak broad negative background, the shape of which replicates the one-particle distribution. The analogy between the problem considered and the theory of spectral line broadening in the impact approximation, in particular, the Dicke narrowing of the Doppler profile, is analyzed. Conditions of applicability of the theory developed are discussed.     

Polarization effects in nanoaggregates of silver caused by local and nonlocal nonlinear-optical responses

A theoretical and experimental study is made into the combined manifestation of local and nonlocal optical responses in a cubic nonlinear isotropic medium such as an aggregated colloidal silver solution. The phenomenological treatment of polarization effects is performed for the general case with due regard for the frequency dispersion of both local and nonlocal nonlinearities and for the noncollinear propagation of pump and probe light waves. The inverse Faraday effect, the optical Kerr effect, and the self-rotation of the polarization ellipse in a fractal-disordered nonlinear medium are observed for the first time. The tensor components of the local and nonlocal cubic nonlinearities of colloidal silver solutions are measured for different degrees of aggregation. It is demonstrated that, as the size of silver aggregate increases, the nonlocal nonlinear response increases much more strongly than the local one. An inference is made that the mechanical motion of metal nanoparticles because of their dynamic interaction with the light wave field can contribute to the nonlinear polarization effects.     

On some properties of the Fabry-Perot etalon with negative permittivities of the medium

Specific features of the Fabry-Perot etalon with negative permittivities of the medium are analyzed. Inhomogeneous waves may exhibit, after passing through such an etalon, a giant amplitude enhancement. They remain inhomogeneous and do not transfer the field energy. Due to absorption and other losses, there may arise, in the inhomogeneous wave, running components and interference effects. The etalon cannot improve the diffraction-limited resolving power of the optical devices detecting traveling waves. The possibilities of application of the etalon in the near-field optics are noted. The giant enhancement of the inhomogeneous wave amplitude occurs due to resonance of the incident wave with eigen (surface) waves of the input and output planes of the etalon. It is shown that, with respect to inhomogeneous waves, the Fabry-Perot etalon with negative permittivities is a narrow-band filter with the peak transmission for the spatial frequency of the Fourier expansion of the boundary values on the order of the inverse wavelength. Under the resonance conditions, the running components, when passing through the etalon, experience aberrational distortions. Polarization properties of the resonance amplification are clarified.     

Saturation and self-saturation effects in the field of intense laser waves: The <Emphasis Type="Italic">J</Emphasis> = 1 − <Emphasis Type="Italic">J</Emphasis> = 1 transition

The shape of the resonance of a saturated absorption is calculated at an arbitrary intensity of a probe field. It is revealed that, in the case of orthogonal polarizations of strong and probe fields, an increase in the intensity of the probe field can invert the shape of the nonlinear resonance of clearing of the medium, changing it to the resonance of absorption. The saturation of optical transitions by intrinsic spontaneous radiation increases the contrast of the inverted resonance of saturation and narrows its width. In the case of parallel polarizations of the optical fields, an absorption peak can arise in the shape of the nonlinear resonance upon splitting of the lower state.     

Exponential Stability of Semigroups Generated by Volterra Integro-Differential Equations with Singular Kernels

Differential Equations - In a separable Hilbert space, we study abstract linear inhomogeneous second-order Volterra integro-differential equations on the positive half-line with operator...     

On the strength distribution of a field created by randomly distributed dipoles

The probability distribution of the strength of a field created by parallel dipoles arranged randomly, on average, uniformly is considered. The problem of divergence at small distances is analyzed. It is shown that, for any, including an arbitrarily small, number of dipoles ε in the cutoff sphere, the average field value is zero. In the case ε ? 1, the field distribution consists of a central part, whose half-width is of an order of the average polarization of a medium, and a very wide and structured background. An interpolation formula for the Fourier image of the field distribution applicable for arbitrary values of ε is proposed. An analogy with the theory of spectral line broadening is established. The connection with the problems of molecular optics and inhomogeneous Stark broadening of spectral lines is discussed.     

Volume Fresnel reflection of electromagnetic waves in three-dimensionally inhomogeneous media

Eikonal approximation is used to derive equations describing propagation of monochromatic electromagnetic waves in a three-dimensionally inhomogeneous medium, including volume Fresnel reflection from inhomogeneities. The analysis is based on a locality principle. Separation into reflection and transmission effects is performed. The former effects are found to be isotropic, whereas the latter are anisotropic and depend on interference phenomena. Interference effects lead to violation of the Rytov law of polarization rotation. Brewster phenomena in layered and three-dimensionally inhomogeneous media are shown to occur under different conditions.     

Representation of solutions of integro-differential equations with kernels depending on the parameter

Integro-differential equations with unbounded operator coefficients in a separable Hilbert space are studied. These equations are an abstract form of the Gurtin–Pipkin-type equation, which describes finite-speed propagation of heat in media with memory. A representation of strong solutions of these equations is derived in the form of the sums of series in exponents that correspond to the spectral points of operator-functions that are the symbols of these equations.