Olga V. Tikhomirova

Theory of singular-phase reconstruction for an optical speckle field in the turbulent atmosphere

Analytical expressions are derived and computational algorithms are constructed for retrieving optical-field phase distribution under strong scintillation. The input data for the phase reconstruction are the wave-front slopes registered by a Hartmann sensor or shearing interferometer. The theory is based on representing the slope-vector field as the sum of its potential and solenoid components; it introduces the concept of phase-source and phase-vortex density and uses strict integral expressions relating these quantities to the wave-front slopes. To overcome the difficulties arising from the singular character of phase distribution, use is made of regularization of the wave-front slopes. The slopes can be measured with an ideal point wave-front sensor. It is shown that the slopes measured at the output of a nonideal sensor can be treated as regularized values of these slopes. Numerical simulation of phase unwrapping from the reference values of the wave-front slopes has shown that the algorithm designed for visualization of local phase singularities and those for phase reconstruction are very helpful in eliminating the measurement noise.

Potential and vortex features of optical speckle fields and visualization of wave-front singularities

The feasibility of noninterferometric methods to measure phase distribution in a laser beam cross section for visualization of the vortex dislocations of an optical speckle-field wave front is analyzed. Peculiarities of the phase retrieved from the measured intensity distribution (the phase problem in optics) and from the wave-front slopes measured by a Hartmann sensor are discussed. A concept of the vortex and the potential parts of the phase is introduced. An analytic formula to retrieve the potential phase from the measured intensity has been obtained. We show that the considered means of measurements allow the positions of the dislocation centers to be sensed and the spatial configuration of the intensity zero lines to be reconstructed.

Wave and Ray Spatial Dynamics of the Light Field in the Generation, Evolution, and Annihilation of Phase Dislocations

The spatial dynamics of the light field during the generation, evolution, and annihilation of screw phase dislocations in Gaussian optical beams is analyzed. The regularities of transformation of an aberration wave front of a beam into a singular front are exposed. The structure of the field of directions of the energy-flux density vector is analyzed on the basis of a solution of a parabolic wave equation for the complex amplitude of a monochromatic light field with the construction of streamlines and investigation of the full set of critical points in the corresponding dynamic system. It is established that the energy redistribution in the beam and appearance of dislocations occur in accordance with the transformation of singularities: local focusing of a streamline in the vicinity of a node is a precursor of a generated pair of screw dislocations, and the streamline takes on a helix shape near an unstable focus.

Reconstruction of the singular phase of optical speckle field from the measurements of wave-front slopes

The algorithms of reconstruction of singular phase from the measurements of wave front slopes by Hartmann sensor, shear interferometer, and from the interference measurements of Knox-Thompson are proposed. Based on the hydrodynamic approach to description of the wave front spatial dynamics we have developed the algorithms which make it possible the singular phase of an optical speckle field to be reconstructed with the use of analytic representation of the phase in terms of its partial derivatives with the allowance for the vortex nature of the phase gradient vector field.

Singular phase dynamics in vortical optical beam

The spatial dynamics of vortex dipole nested in a Gaussian optical beam is analyzed with scalar diffraction theory. It is shown that the generation and annihilation of vortices are accompanied by extreme wave front distortions. Process of nucleation and annihilation of vortex dipole is considered to confirm the vortex aftereffect in the process of transformation of an aberration wave front into a singular one. After the vortex dipole annihilation the absolute values of the average and Gaussian curvatures increase greatly in the local areas of wave front. The phase reconstruction error increases in these areas in the process of wave front sensing. To investigate the features of the vortex aftereffect for simple field the light rays which take part in the combined translational and rotary motion of energy around the field zero-lines are constructed. The phase is computed as a potential of the optical field using the light rays. Integration of the phase gradient along the ray or energy-stream line allows a unique value of the phase to be connected with a point of the ray. Wave front of the beam is constructed from the computed phase. As in the vicinity of the vortex core the energy-stream line takes a spiral shape the phase incursion along the line increases. After the vortex annihilation and vanishing of the spiral shape the incursion remains and creates the vortex aftereffect in the form of the extreme wave front distortions. The revealed effect should be taken into account when constructing the systems of adaptive optics aimed at functioning in strong turbulence conditions.

Theory of singular phase reconstruction in speckle-field

Phase reconstruction under the conditions of strong scintillation is limited by the presence of phase discontinuities (singularities) that accompany the intensity nulls in speckle field. The singularities are hidden for the wave front sensors of conventional adaptive optics systems. Rigorous theory has been developed in the paper to reconstruct the singular phase with allowance for the vortex nature of the phase gradient vector field. Singular functions of the wavefront slope (phase gradient) are exposed to an appropriate regularization; the divergence and rotor of the regularized phase gradient are determined. It has been established that the divergence of the phase gradients contains singularities of the same type as gradient and, therefore, proper phase reconstruction can be executed using the regularized slopes only. Topological transformation of the wave front in the process of appearance and annihilation of the discontinuities is manifested as changing in relief of the regularized rotor, the relief characteristics correspond to the discontinuity parameters. With the use of Pompeiu integral the phase gradient is presented as a sum of divergent and rotor parts added by the term depending on the receiving aperture boundary conditions. The obtained wavefront slopes are unwrapped into the singular phase ignoring the noise and singularities of the phase gradient divergence and rotor. Derived analytical expression connecting the phase reconstruction from the measurements of wavefront slopes. Examples of such a reconstruction are given. The developed approach can be base for creating the discrete phase reconstruction algorithms.

Visualization of the phase singularities in wavefront sensors

Feasibility of noninterferometric methods to measure the phase distribution in laser beam cross-section for visualization of the vortex dislocations of optical speckle-field wavefront is analyzed. Peculiarities of the phase retrieval from the measured intensity distribution (the phase problem in optics) and from the wave front slopes measured by the Hartman sensor are discussed. A concept of the vortex and potential parts of the phase is introduced. An analytic formula to retrieve the potential phase from the measured intensity distribution of optical speckle-field has been obtained. We show that the considered ways of measurements allow the positions of the dislocations centers to be sensed and spatial configuration of the intensity zero-lines to be reconstructed.

Reconstruction of optical fields with wavefront singularities from intensity distribution

Wave front dislocations generated by the vortex flow of the light energy in vicinity of the points where the field intensity equals zero are studied on the basis of obtained relation between the phase and intensity in optical beam (the phase problem solution). Reasons of an ambiguity of the phase problem solution are discussed. An analytic formula for retrieving the potential phase from the known intensity distribution has been obtained. A possibility to use an information on the potential phase for correction of singular phase distortions by the optical adaptive systems is discussed.

Reconstruction of 2D fields of atmospheric parameters from lidar signals reflected by the earth's surface

Use of airborne DIAL return signals from the Earths surface in combination with a tomographic data processing technique enables one to essentially lower the sounding radiation power sufficient for effective sensing of the atmosphere. An optical arrangement of sounding and an algorithm of lidar data inversion aimed at reconstruction of two-dimensional fields of atmospheric parameters are proposed. Some results of numerical simulations of the technique are presented.