Apertured Paraxial ABCD Optical System Research Articles

Propagation of anomalous vortex beams through an annular apertured paraxial ABCD optical system

A new form of laser beam, called anomalous vortex beam (AVB), has been proposed recently. Here we study the propagation characteristics of the AVB passing through a paraxial ABCD optical system with an annular aperture. The approximate analytical expressions are derived based on the Collins integral formula. Some numerical examples are given to illustrate the intensity distribution of the AVB through a paraxial ABCD optical system with an annular aperture. It is shown that the field distribution depends on the topological charge \(m\), the waist \(w_{0}\), and the width of annular aperture.

The propagation of hypergeometric beams through an annular apertured paraxial ABCD optical system

By means of expanding the hard aperture function into a finite sum of complex Gaussian functions and based on the generalized Huygens–Fresnel diffraction integral, a novel approximate analytical expression of hypergeometric (HyG) beams passing through a paraxial ABCD optical system with an annular aperture is derived. The results could be reduced to the case of a circular aperture or a circular black screen. Some numerical simulations are also performed and illustrated for the propagation characteristics and focusing properties of a HyG beam through a paraxial ABCD optical system with an annular aperture. The results obtained from the approximate analytical formula provide more efficiency than the usual way of using diffraction integral formula directly.

The intensity distribution of Hermite–Gaussian beam through an annular aperture

Based on Collins formula and the fact that the hard-edge aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expression of Hermite–Gaussian (HG) beam passing through an annular apertured paraxial ABCD optical system is obtained. Large of numerical examples are simulated with Matlab software. The study results show that intensity distribution is closely related with HG beam mode. When the mode order of HG beam is even, the intensity on the corresponding axis for different HG beam modes has a certain proportion; when the mode order is odd, the intensity on the corresponding axis is dark fringe.The results are useful to deepen understanding the characteristics of HG bean, improve the beam quality and optimize the optical system design.

Propagation of various dark hollow beams through an apertured paraxial ABCD optical system

Propagation of a dark hollow beam (DHB) of circular, elliptical or rectangular symmetry through an apertured paraxial ABCD optical system is investigated. Approximate analytical formulas for various DHBs propagating through an apertured paraxial optical system are derived by expanding the hard-aperture function into a finite sum of complex Gaussian functions in terms of a tensor method. Some numerical results are given. Our formulas provide a convenient way for studying the propagation of various DHBs through an apertured paraxial optical system.

A detailed study of Mathieu–Gauss beams propagation through an apertured ABCD optical system

Abstract Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu–Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu–Gauss beams in (y–z) and (x–z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu–Gauss beam is similar to that of Bessel–Gauss beam. Furthermore, the focalization of the Mathieu–Gauss beams through a thin lens is illustrated and analyzed with numerical results.

Wigner distribution function of Hermite–cosine-Gaussian beams through an apertured optical system

By introducing the hard-aperture function into a finite sum of complex Gaussian functions, the approximate analytical expressions of the Wigner distribution function for Hermite-cosine-Gaussian beams passing through an apertured paraxial ABCD optical system are obtained. The analytical results are compared with the numerically integrated ones, and the absolute errors are also given. It is shown that the analytical results are proper and that the calculation speed for them is much faster than for the numerical results.

Paraxial propagation of Mathieu beams through an apertured ABCD optical system

Based on the relationship between separable solutions of the Helmholtz equation, we expanded the field’s amplitude associated with Mathieu beams in terms of Bessel beams. By using the expansion of the circ function into a finite sum of complex Gaussian functions, we derived an approximated analytical expression of the amplitude distribution of Mathieu beams at the output plane of any apertured paraxial ABCD optical system. Some numerical calculations are performed and it is shown that our approach can substitute the usual numerical integration of the Helmholtz equation or the numerical calculation of the diffraction integral.

Propagation of elegant Laguerre–Gaussian beams through an annular apertured paraxial ABCD optical system

Based on the generalized Huygens–Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of elegant Laguerre–Gaussian beams passing through a paraxial ABCD optical system with an annular aperture is derived. Meanwhile, the corresponding closed-forms for the unapertured, circular apertured or circular black screen cases are also given. The obtained results provide more convenience for studying their propagation and transformation than the usual way by using diffraction integral formula directly. Some numerical examples are given to illustrate the propagation properties of elegant Laguerre–Gaussian beams.

Approximate analytical expressions of Laguerre-Gaussian beams passing through a paraxial optical system with an annular aperture

Summary On the basis of the expansion of the hard aperture function into a finite sum of complex Gaussian functions, the approximate analytical expression of Laguerre-Gaussian beams passing through an annular apertured paraxial ABCD optical system is derived. Meanwhile, the corresponding closed-forms for the unapertured or circular apertured or circular black screen cases have also been given. Numerical examples are given to illustrate the propagation characteristics of Laguerre-Gaussian beams.

Propagation characteristics of linearly polarized Bessel-Gaussian beams through an annular apertured paraxial ABCD optical system

Summary By means of Collins diffraction integral formula in the paraxial approximation and based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical expression for linearly polarized Bessel-Gaussian beams passing through a paraxial ABCD optical system with an annular aperture has been derived. The results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly. By using the analytical expression and the diffraction integral formula some numerical simulations are done to illustrate for the propagation characteristics of a linearly polarized Bessel-Gaussian beam through an optical system with an annular aperture.

Propagation of off-axial Hermite–cosh-Gaussian laser beams

Based on the fact that a hard aperture function can be expanded by an approximate sum of complex Gaussian functions with finite numbers, the approximate analytical expressions for the off-axial Hermite–cosh-Gaussian beams and off-axial cosh-Gaussian beams passing through an apertured paraxial ABCD optical system have been derived. By doing some numerical simulations the results obtained by the approximate analytical expression are compared with those obtained by the diffraction integral formula directly. It is shown that the former has good consistency with the latter and that our method can significantly improve the numerical calculation efficiency. The approximate analytical expressions for the kurtosis parameter and beam radius of an off-axial Hermite–cosh-Gaussian beam through an apertured paraxial ABCD optical system are also derived.

Propagation of one-dimensional off-axial Hermite-cosine-Gaussian beams through an apertured paraxial ABCD optical system

Analytical expressions for the one-dimensional off-axial Hermite-cosine-Gaussian beams and one-dimensional off-axial cosine-Gaussian beams passing through an apertured paraxial ABCD optical system have been respectively derived. Our results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly.