Investigation on the Formation of Laser Transverse Pattern Possessing Optical Lattices

Optical lattices (OLs) with diverse transverse patterns and optical vortex lattices (OVLs) with special phase singularities have played important roles in the fields of atomic cooling, particle manipulation, quantum entanglement, and optical communication. As a matter of consensus until now, the OL patterns are generated by coherently superimposing multiple transverse modes with a fixed phase difference through the transverse mode locking (TML) effect. There are phase singularities in the dark area of this kind of OL pattern, so it is also called OVL pattern. However, in our research, it is found that some high-order complex symmetric OL patterns can hardly be analyzed by TML model. Instead, the analysis method of incoherent superposition of mode intensity could be applied. The OL pattern obtained by this method can be regarded as in non-TML state. Therefore, in this article, we mainly study the distinct characteristics and properties of OL patterns in TML and non-TML states. Through intensity comparison, interferometry, and beat frequency spectrum, we can effectively distinguish OL pattern in TML and non-TML states, which is of significance to explore the formation of laser transverse pattern possessing OL.

On class (BD) Operators

In this paper, we introduce the class of (BD) operators acting on a complex Hilbert space H. An operator if T ∈ B (H) is said to belong to class (BD) if T * 2 (TD) 2 commutes with (T *TD) 2 equivalently [T * 2 (TD) 2, (T *TD) 2] = 0. We investigate the properties of this class and we also analyze the relation of this class to D-operator and then generalize it to class (nBD) and analyze its relation to the class of n-power D-operator through complex symmetric operators.

Distributions of zeros and poles of -point Padé approximants to complex-symmetric functions defined at complex points

UDC 517.5 The knowledge of the location of zeros and poles Padé and -point Padé approximations to a given function provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of complex-symmetric functions.The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of All considered cases are graphically illustrated.Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them.