Haidan Mao

Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system

By introducing the hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for the one-dimensional off-axial Hermite–cosine–Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system has been derived. As special cases, the corresponding closed-forms for the off-axial or non-off-axial Hermite–cosine–Gaussian beams passing through apertured or unapertured and misaligned or aligned paraxially ABCD optical systems have also been given. The results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly, which can be straightforward to the two-dimensional case. Some numerical examples are also illustrated.

Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system

Beam profiles that consist of a sum of complex-Gaussian functions, a sum of polynomial-Gaussian functions and a sum of multi-Gaussian functions offset by some fixed amount are proposed as three types of model for a hard-aperture function. By expanding an aperture function into these models, approximate analytical propagation equations for a Gaussian beam through an apertured ABCD optical system are obtained. Comparison among these models themselves and among propagation characteristics of a Gaussian beam through these models are made. It is shown that the first and third types of model for a hard-aperture function are more suitable than the second type, in terms of calculation efficiency and simulation results, for application to such diffraction problems. Moreover, there are some differences in the applicability of the first and the third models.

Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems

Based on the fact that a hard-edge aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of a flattened Gaussian beam passing through the apertured fractional Fourier transforming systems are derived. By using the approximate analytical formulae and diffraction integral formulae, some numerical simulation comparisons are done, and it is shown that our method can significantly improve the numerical calculation efficiency.

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 Hermite-Gaussian beams in apertured fractional Fourier transforming systems

Summary The apertured fractional Fourier transform (FRFT) system is introduced and applied to treat the propagation of Hermite-Gaussian beams. Based on the treatment that a rectangular function can be expanded into a approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the output field distribution of a Hermite-Gaussian beam through an apertured FRFT system are obtained and compared with those obtained from numerically integral calculation. The results show that our method can significantly improve the numerical calculation efficiency.

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 characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens

Based on the Collins diffraction integral formula and irradiance moment definition, the propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with spherically aberrated lenses are studied in detail. By introducing an efficient algorithm, numerical calculations are performed and the results show that under certain conditions the evolution characteristics of the kurtosis parameters of a flattened-Gaussian beam passing through the fractional Fourier transformation systems with spherically aberrated lenses are very similar to those of a super-Gaussian beam, but they are different from the propagation characteristics of the kurtosis parameter of a flat-topped beam defined by Yajun. It is also shown that the kurtosis parameters of a flat-topped beam passing through two optical setups are very different. It is implied that the two optical setups for implementing the fractional Fourier transformation are no longer equal in the presence of spherically aberrated lenses. We also find that the kurtosis parameters change with the fractional orders periodically and the fundamental periods for the two optical setups with spherically aberrated lenses are different.

The kurtosis parametric characterization of the passage of a standard Hermite–Gaussian beam and an elegant Hermite–Gaussian beam through a fractional Fourier transformation system with a spherically aberrated lens

The propagation characteristics of the kurtosis parameters of a standard Hermite–Gaussian (SHG) beam and of an elegant Hermite–Gaussian (EHG) beam, each passing through a fractional Fourier transformation (FRFT) system with a spherically aberrated lens, are studied in detail. Some numerical calculations are made by introducing an efficient algorithm, based on the Collins diffraction integral formula. The resulting graphs illustrate the striking difference between ideal FRFT systems and those with a spherically aberrated lens. The kurtosis parameters of both SHG and EHG beams passing through a type I Lohmann system with a spherically aberrated lens are seen to change with the fractional order periodically and the fundamental period is 4, but for type II the fundamental period is 2. Different values of spherical aberration coefficients affect the kurtosis parameters in greatly different ways. The values of the kurtosis parameters of a SHG beam passing through either type of Lohmann system with a spherically aberrated lens are no longer equal to those of an EHG beam, even when they have the same fractional orders and the same spherical aberration coefficients.

The relative phase shift of off-axial Gaussian beams through an apertured and misaligned optical system

Abstract By introducing the hard-edge aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression of the relative phase shift for an off-axial Gaussian beam passing through an apertured and misaligned paraxial ABCD optical system has been derived. It is shown that it is related with the off-axial beam parameters, the aperture size, the ABCD matrix elements and misaligned parameter. Some numerical simulations are also illustrated for its application. The obtained results could be extended to the two-dimensional case.