Fractional Fourier Transform System Research Articles

A minimum distortion transmission scheme under combined effects of chromatic dispersion and SPM based on optical time-domain fractional Fourier transformation

In this paper, an optical time-domain fractional Fourier transformation (FRFT) system is proposed to achieve the minimum distortion transmission under combined effects of chromatic dispersion and self-phase modulation (SPM). In the new method, the pulses operated as FRFT will propagate in a new domain, in which the waveform in time domain will keep nearly unchanged through the transmission. The novel method achieves a 400 km optical transmission for an optical pulse with the full width at the 1/e point of peak power of 80 ps without any dispersion compensation and the pulse offers a nice performance with negligible nonlinear distortion. Compared with the soliton communication, this scheme shows more advantages on linear and nonlinear distortions without strict restriction to input pulses.

Teaching stable two-mirror resonators through the fractional Fourier transform

We analyse two-mirror resonators in terms of their fractional Fourier transform (FRFT) properties. We use the basic ABCD ray transfer matrix method to show how the resonator can be regarded as the cascade of two propagation–lens–propagation FRFT systems. Then, we present a connection between the geometric properties of the resonator (the g parameters) and those of the equivalent FRFT systems (the FRFT order and scaling parameters). Expressions connecting Gaussian beam q-transformation with FRFT parameters are derived. In particular, we show that the beam waist of the resonator's mode is located at the plane leading to two FRFT subsystems with equal scaling parameter which, moreover, coincides with the mode Rayleigh range. Finally we analyse the resonator's stability diagram in terms of the fractional orders of each FRFT subsystem, and the round trip propagation. The presented analysis represents an interesting link between two topics (optical resonators and Fourier optics) usually covered in optics and photonics courses at university level, which can be useful to teach and connect the principles of these subjects.

Fractional Fourier transform of Ince-Gaussian beams

Ince-Gaussian beams are introduced to describe the natural resonating modes produced by stable resonators, and they form the third completely orthogonal family of exact solutions of the paraxial wave equation. The fractional Fourier transform (FRFT) is applied to treat the propagation of Ince-Gaussian beams, and an analytical expression for an Ince-Gaussian beam passing through a FRFT system is derived. The normalized intensity distribution of an Ince-Gaussian beam in the FRFT plane is graphically illustrated with numerical examples, and the influences of the different parameters on the normalized intensity distribution are discussed in detail.

Propagation properties of beams generated by Gaussian mirror resonator in fractional Fourier transform plane

Based on the definition of the fractional Fourier transform (FRFT) and irradiance moments in the cylindrical coordinate system, the propagation expressions and kurtosis parameter of beams generated by Gaussian mirror resonator passing through the ideal fractional Fourier transformation systems are obtained. The propagation properties and kurtosis parametric characteristic of the beams in the FRFT plane are analyzed in detail. Some numerical examples are given to illustrate the analytical results. The influences of the fractional order on the intensity distribution and the kurtosis parameter of the beams are also investigated. The results show that the intensity distribution and the kurtosis parameter of the beams in the FRFT plane are closely related to the fractional order and beam parameters.

Fractional Fourier transform of Lorentz beams

This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent fields. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.

Fractional Fourier transform of a higher-order cosh–Gaussian beam

A higher-order cosh–Gaussian beam is an appropriate model to describe the flattened laser beam. The fractional Fourier transform (FRFT) is applied to treat the propagation of higher-order cosh–Gaussian beams. An analytical expression for a higher-order cosh–Gaussian beam passing through a FRFT system has been derived. By using the derived expression, the properties of a higher-order cosh–Gaussian beam in the FRFT plane are graphically illustrated with numerical examples.

Fractional Fourier transform of Lorentz-Gauss beams

Lorentz-Gauss beams are introduced to describe certain laser sources that produce highly divergent beams. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz-Gauss beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz-Gauss beam passing through an FRFT system has been derived. By using the derived expression, the properties of a Lorentz-Gauss beam in the FRFT plane are graphically illustrated with numerical examples.

基于级联分数傅里叶变换系统的数字水印技术

基于级联分数傅里叶变换系统的数字水印技术

Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams

Cosh-Gaussian beams are of practical interest because they are more efficient at extracting energy from conventional laser amplifiers, and they have some important applications because their profiles can closely resemble the flat-top field distribution by choosing suitable beam parameters of cosh parts. The fractional Fourier transform (FRFT) is applied to treat the propagation of off-axial elliptical cosh-Gaussian beams (EChGBs). By the use of vector integration, the analytical expression for an off-axial EChGB passing through a FRFT system is derived in terms of tensor method. Our formula provides a convenient way for studying the FRFT of off-axial EChGBs; some numerical simulations are also given.

Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams

A new kind of laser beams named the elliptical Hermite-sinusoidal-Gaussian beams (EHSGBs) is introduced and defined by use of tensor method, and the elliptical Hermite-cosh-Gaussian beam (EHChGB) can be regarded as special case of EHSGBs. The fractional Fourier transform (FRFT) is applied to treat the propagation of EHChGBs. An analytical expression for an EHChGB passing through an FRFT system is derived by using vector integration. Some numerical simulations are illustrated for the propagation properties of EHChGBs through FRFT systems.

The propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian beams passing through fractional Fourier transformation systems

Based on the Collins diffraction integral formula and irradiance moments, the propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian (HChG) beams passing through fractional Fourier transformation systems with spherically aberrated lenses are analyzed in detail. The closed-form expressions for the propagation equation and kurtosis parameter of HChG beams passing ideal fractional Fourier transformation systems are also obtained. By using an efficient algorithm some numerical calculations are also performed in this paper. The results show that the spherically aberrated lenses greatly influence on the intensity distribution and kurtosis parameter of HChG beams passage through fractional Fourier transformation systems. The intensity distribution and kurtosis parameter of HChG beams passage through two types of Lohmann's systems with spherically aberrated lenses, respectively, are no longer equal even in the case of the same fractional order and the same spherical aberration coefficients. The fundamental periods of the two types of Lohmann's systems with spherically aberrated lenses are different, 4 for the type I and 2 for the type II. Therefore, the two optical setups for implementing the fractional Fourier transformation are no longer equivalent in the presence of spherically aberrated lenses.

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.

Fractional Fourier transform for partially coherent beam in spatial-frequency domain

By using Fourier transform and the tensor analysis method, the fractional Fourier transform (FRT) in the spatial-frequency domain for partially coherent beams is derived. Based on the FRT in the spatial-frequency domain, an analytical transform formula is derived for a partially coherent twisted anisotropic Gaussian–Schell model (GSM) beam passing through the FRT system. The connections between the FRT formula and the generalized diffraction integral formulae for partially coherent beams through an aligned optical system and a misaligned optical system in the spatial-frequency domain are discussed, separately. By using the derived formula, the intensity distribution of partially coherent twisted anisotropic GSM beams in the FRT plane are studied in detail. The formula derived provide a convenient tool for analysing and calculating the FRTs of the partially coherent beams in spatial-frequency domain.

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 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.

Moments of the Wigner distribution of rotationally symmetric partially coherent light.

The Wigner distribution of rotationally symmetric partially coherent light is considered, and the constraints for its moments are derived. Although all odd-order moments vanish, these constraints lead to a drastic reduction in the number of parameters that we need to describe all even-order moments: whereas in general we have (N + 1)(N + 2)(N + 3)/6 different moments of order N, this number reduces to (1 + N/2)2 in the case of rotational symmetry. A way to measure the moments as intensity moments in the output planes of (generally anamorphic) fractional Fourier-transform systems is presented.

Effect of spherically aberrated lens on the fractional Fourier transformation systems

The effect of lens with spherical aberration on the fractional Fourier transformation systems is introduced and analysed in this paper. Consider the fundamental mode Gaussian beam as an input function, the intensity distributions of the output field through the fractional Fourier transformation system with spherically aberrated lens are obtained by numerical calculation. The results show that different kinds or values of spherical aberration coefficient contribute to the output intensity distribution differently even for the same fractional order. It is also shown that Lohmann’s two types of optical setups with spherically aberrated lens for implementing the fractional Fourier transform are no longer equivalent.

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.

Fractional Fourier transform for partially coherent Gaussian–Schell model beams

The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.