Alexei V. Popov

Application of the parabolic wave equation to X-ray diffraction optics

A new computational approach is suggested to the analysis of imaging properties of realistic X-ray Fresnel zone plates with high aspect ratio. Its mathematical basis is the parabolic wave equation (PWE) describing diffraction inside the zone plate body as well as wave propagation and focusing throughout the optical system. A finite-difference method is used for solving PWE in a truncated free-space domain with a perfectly transparent artificial boundary. The results of global wave field calculation are visualized by means of color graphics. Numerical estimates of resolution, diffraction efficiency and field of view enables one to compare imaging performance of realistic X-ray zone plates with an idealized plane Fresnel diffraction lens.

Accurate modeling of transparent boundaries in quasi‐optics

A general approach to reduction of the infinite computational domain to a strip or a cylinder is developed for two- and three-dimensional narrow-angle diffraction problems. Our method is based on constructing exact solutions of the parabolic wave equation in open subdomains free of nonuniform diffractive objects. As a result, an exact nonlocal boundary condition is derived providing full transparency of the artificially introduced computational border. This technique leads to an essential reduction of the computing time without any loss of accuracy, which is illustrated by practical examples from radio propagation and X ray diffractive optics. Extension of the method onto quasi-stratified environments and modified parabolic equations is discussed.

Diffraction phenomena inside thick Fresnel zone plates

The parabolic wave equation method is used to describe the complex wave field inside the body of a thick zone plate used to focus X ray radiation. Two analytical approaches are applied: (1) Diffraction of a plane wave incident onto a separate interface between opaque and open zones is considered. We construct an approximate analytical solution to the classical problem of diffraction by a dielectric wedge in terms of the Fresnel integral and a new special function. (2) Coupled wave theory is used to describe collective effects of diffraction by many interfaces. The zone plate is considered as thick diffraction grating with slowly varying period. An analytical solution is found for three interacting modes. A possibility to optimize the zone plate performance is shown. Taken together, these results describe all the main features observed in the output field of realistic X ray zone plates.

Model problem of Bragg fiber design

A new analytical approach to direct and inverse problems of wave propagation in periodic media is developed. By regarding the advancing wave phase as a new independent variable we construct a continuum of exact solutions of the 1D wave equation for arbitrary index profiles. This parametric solution is applied to the study of evanescent Bloch waves in forbidden zones. An explicit formula for the decay decrement is derived, giving an exact quantitative description of the parametric resonance in periodic media. The results are used for constructing optimal smooth models of multilayer dielectric mirrors and Bragg waveguide periodic claddings.

Parametric resonance and waves in periodic media

The method of phase parameter is used for analytical description of parametric resonance in linear and nonlinear systems. By analogy, the results are applied to wave propagation in dielectric media with periodically varying parameters, such as multilayer mirror or Bragg waveguide. The problem of analytical treatment of chromatic dispersion is discussed.

Beam propagation through straight and bent Bragg waveguides: Numerical simulation

Computational aspects of the parabolic equation method are discussed: transparent boundary conditions for straight and bent Bragg waveguides, 3D finite-difference scheme in polar coordinates, estimation of radiation losses and fundamental mode stabilization.

Wave field transformation at coherent imaging of a tilted reflection mask

The coherent beam wave field transformation is considered in the case of slanting illumination of a reflection mask. The parabolic wave equation is used to find the resulting field evolution in free space and after transmission by an ideal lens. Possible applications are lensless imaging, coherent X-ray diffraction imaging (CXDI), X-ray optics and X-ray laser based lithography and reflection microscopy, and high resolution X-ray topography.

On the explicit parametric description of waves in periodic media

A method for the parameterization of the one-dimensional wave equation is proposed that makes it possible to find its solution by quadratures under an arbitrary dependence of the refraction index on the current wave phase. The form of the solution found is used to investigate the structure of the wave function for a periodic refraction index. Explicit expressions for the fundamental system of solutions and for the Floquet index are obtained. Examples of applying the proposed method to the optimal synthesis of multilayer interference mirrors and Bragg waveguides are discussed.

Computer simulation of diffraction and focusing processes in quasioptics

Here, we describe numerical modeling of the wave fields produced by complex quasioptic elements. The parabolic equation method and exact transparency boundary conditions are used in the computational algorithm. The computer code provides global field visualization using color graphics.

Parametric representation of traveling waves in periodic media

As shown in our previous works, the linear oscillator equation with varying eigenfrequency can be integrated explicitly by introducing a new independent variable defined as the phase of non-harmonic oscillations. Such a parametrization considerably simplifies the theory of parametric resonance. By analogy, this analytical result describes exponential wave field damping in 1D photonic crystal band-gaps and can be used for optimization purposes. A similar approach can be applied to the transmission bands of a periodical medium. However, the problem proves to be more difficult compared with the stop-bands. To overcome the difficulty, one has to correctly define the phase of a complex wave function. We discuss the analytical relations following from the integrability conditions and particular solutions of the corresponding differential equations.