Diffraction Integral Formula Research Articles

Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems.

A new mathematical model called hollow elliptical Gaussian beam (HEGB) is proposed to describe a dark-hollow laser beam with noncircular symmetry in terms of a tensor method. The HEGB can be expressed as a superposition of a series of elliptical Hermite-Gaussian modes. By using the generalized diffraction integral formulas for light passing through paraxial optical systems, analytical propagation formulas for HEGBs passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, evolution properties of the intensity distribution of HEGBs in free-space propagation were studied. Propagation properties of HEGBs through a misaligned thin lens were also studied. The HEGB provides a convenient way to describe elliptical dark-hollow laser beams and can be used conveniently to study the motion of atoms in a dark-hollow laser beam.

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.

Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems

The apertured fractional Fourier transform (FRFT) system is introduced and applied to treat the propagation of Hermite-cosh-Gaussian beams. Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of a Hermite-cosh-Gaussian beam through apertured FRFT systems are derived and compared with those obtained from diffraction integral formulae by numerical simulations. It is shown that our method can significantly improve the numerical calculation efficiency.

Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh diffraction integral formula and without using the far-field approximation, a solution of the wave equation beyond the paraxial approximation is found, which represents vectorial non-paraxial elliptical Gaussian beams in free space. The far-field expressions for non-paraxial Gaussian beams and elliptical Gaussian beams can be regarded as special cases treated in this paper. Some basic propagation properties of vectorial non-paraxial elliptical Gaussian beams, including the irradiance distribution, phase term, beam widths and divergence angles are studied. Numerical results are given and illustrated.

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.

ANALYSIS OF EXTENDING THE FIELD-OF-VIEW OF REVERSE-MICROSCOPE IMAGING SYSTEM AT MILLIMETER-WAVELENGTHS

In this paper, focal field of reverse-microscope imaging system at millimeter wavelengths is analyzed with ray-tracing method and Stratton-Chu vector diffraction integral formula. Calculations have been compared with experiments and agreement is found. Lens profile is optimized to reduce aberrations and extend field-of-view of the imaging system. The numerical results are presented to show the field distribution of the optimized imaging system.

A comparative study of the propagation of vectorial nonparaxial beams in the use of different approaches

Starting from the vectorial Rayleigh diffraction integral formula and using a simple expansion of the nucleus in the Rayleigh formula, an analytical propagation equation of vectorial nonparaxial Gaussian beams in free space is derived, which permits us to perform numerical calculations in comparison with the expression derived by Ciattoni et al. and with the direct numerical integration of the Rayleigh formula. It is found that as usual the use of expansion of vectorial Rayleigh diffraction integral is sufficient to provide satisfactory numerical results as compared with the direct integration of the Rayleigh formula. The above two analytical expressions are valid under certain conditions, however both are applicable in the far field.

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.

Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane.

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.

Propagation of elliptical Hermite–Gaussian beam through misaligned optical system

Based on the generalized diffraction integral formula in terms of tensor form for paraxial misaligned optical systems, an analytical propagation formula is derived for the elliptical Hermite–Gaussian beam (EHGB) passing through misaligned optical systems by using vector integration. By using the derived formula, the propagation properties of EHGB through a misaligned thin lens are illustrated. As an application example, we derived the analytical propagation formula for elliptical flattened Gaussian beam through misaligned optical system by expressing it as superposition of a series of EHGB by using polynomial expansion.

Far-field properties of non-paraxial elliptical Gaussian beams

By using the Rayleigh diffraction integral formula, a far-field asymptotic solution of the wave equation is found, which represents non-paraxial elliptical Gaussian beams. The basic properties of such beams are studied, and a comparison with the previous work is also made.

Propagation of partially coherent twisted anisotropic Gaussian–Schell model beams through misaligned optical systems

The generalized diffraction integral formulae for partially coherent beams through misaligned optical systems are derived in both spatial and spatial-frequency domain. Based on these formulae, the analytical propagation formulae for partially coherent twisted anisotropic Gaussian–Schell model (GSM) beams through misaligned optical systems are derived. The derived formulae provide a powerful and convenient tool for treating the propagation and transformation of partially coherent twisted anisotropic GSM beams through misaligned optical systems in spatial domain and spatial-frequency domain. As an application example, the intensity distribution properties of partially coherent twisted GSM beams through a misaligned thin lens are studied.

Generalized diffraction integral formulae for misaligned optical systems described by fractional Fourier transforms

Summary It is shown that, under some conditions, the generalized diffraction integral formula for any misaligned optical system in spatial-domain could be described by the fractional Fourier transform. On the other hand, the generalized diffraction integral formula for angular spectra through any misaligned optical system in spatial-frequency domain could also be described by the fractional Fourier transform. A practical optical system with a misaligned lens is given to illustrate its application.

Fractional Fourier transform and the diffraction of any misaligned optical system in spatial-frequency domain

Relationships between the generalized diffraction integral formula for angular spectrum through any misaligned optical system in spatial-frequency domain and the fractional Fourier transform are bridged in this paper, which provides a new description of the diffraction of any misaligned optical system in spatial-frequency domain. Some special cases are also analyzed.

Effect of misalignment on optical fractional Fourier transforming systems

The effect of misalignment on optical fractional Fourier transforming systems is analysed in this paper. Meanwhile, a concept of perturbed fractional Fourier transform is put forward. By using the generalized misaligned diffraction integral formulae for misaligned optical systems, it is shown that the diffraction characteristics for misaligned Lohmann's fractional Fourier transforming systems and quasi-asymmetric inhomogeneous media are able to perform the perturbed fractional Fourier transform and are essentially equivalent. And also, a relative discussion is given.

Propagation of Optical Pulses and Pulsed Beams in theFrequency Domain

The diffraction integral formulae in the temporal andspatial-temporal frequency domains are derived by using a Fourier transformand tensor analysis method. Based on these formulae, the abcd law in thetemporal frequency domain and the tensor ABCD law in the four-dimensionalspatial-temporal frequency domain are derived. An application example of thederived formulae is provided.

Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domain: Short Note

Abstract Based on Fourier transforms the generalized diffraction integral formula for a beam through any misaligned optical system in spatial-frequency domain is derived in this paper. It is shown that the diffraction integral formula in spatial-frequency domain is related with the ray transfer matrix elements and the misaligned parameters. And some special cases are discussed, an example is also given to illustrate its application.

Generalized diffraction integral formula for misaligned optical systems in spatial-frequency domainShort Note

Based on Fourier transforms the generalized diffraction integral formula for a beam through any misaligned optical system in spatial-frequency domain is derived in this paper. It is shown that the diffraction integral formula in spatial-frequency domain is related with the ray transfer matrix elements and the misaligned parameters. And some special cases are discussed, an example is also given to illustrate its application.

MISALIGNED FRACTIONAL FOURIER TRANSFORMS AND THEIR OPTICAL IMPLEMENTATION

Based on the misaligned diffraction integral formula for misaligned optical systems, the diffraction for misaligned Lohmann's fractional Fourier transformation system and quasi-asymmetrically inhomogeneous medium are analysed, respectively. It is found that there exist similar expressions of misaligned diffraction integral formula and they are essentially equivalent. And also, a new concept of misaligned fractional Fourier transform is put forward.