Fractional Fourier Transform System Research Articles

Fractional Fourier transform of Airyprime beams

Analytical expressions of the electric field, the centriod, the beam half width, and the linear momentum of the fractional Fourier transform (FrFT) of an Airyprime beam are derived, respectively. The relation between the centriods or the beam half widths of an Airyprime beam in the exit and the input planes of a FrFT system is also presented. The effect of the fractional order p on the FrFT of an Airyprime beam is investigated in detail and graphically presented. The cycle of the light intensity, the centriod, and the linear momentum of an Airyprime beam in the exit plane of a FrFT system versus p is 4, while the cycle of the beam half width versus p is 2. This research in the FrFT of an Airyprime beam has potential applications. First, special laser beams with different shapes can be generated by controlling p. Then, the FrFT of an Airyprime beam can be used as an optical switch due to the reversal of the positive and the negative signs of the linear momentum in the first and the last half cycle. Finally, the mirror transformation of the spatial orientation of the beam spot for the FrFT of an Airyprime can be used in optical micro-manipulation.

Fractional Fourier transforms of circular cosine-hyperbolic-Gaussian beams

The propagation properties of a circular cosh-Gaussian beam (CiChGB) in a fractional Fourier transform (FRFT) optical system are investigated theoretically. The analytical expressions for a CiChGB propagating through apertured and unapertured FRFT systems are derived based on the Collins formula and the expansion of the hard-aperture function into a finite sum of Gaussian functions. From the obtained expressions, the evolution of the intensity distribution at the output plane is analysed with illustrative numerical examples. It is shown that the intensity distribution of the CiChGB propagating in FRFT is closely dependent on the decentred beam parameter and the aperture size besides the fractional order of the FRFT system. The results obtained may be beneficial to applications in laser beam shaping, optical trapping, and micro-particle manipulation.

Opto-Video Encryption Based on Logistic Adjusted Sine map in FrFT

In the last few years, videos became the most common form of information transmitted over the internet, and a lot of the traffic is confidential and must be protected and delivered safely to its intended users. This introduces the challenges of presenting encryption systems that can encode videos securely and efficiently at the same time. This paper presents an efficient opto-video encryption system using Logistic Adjusted Sine map (LASM) in the Fractional Fourier Transform (FrFT). In the presented opto-video LASM-based FrFT scheme, the encoded video is split into distinct frames and transformed into optical signals utilizing an optical supply. Each of the developed optical video frames is ciphered by utilizing the LASM in optical FrFT system using two-phase modulation forms on the video frame, the first in the time-domain and the second in the FrFT domain. In the end, the ciphervideo frame is spotted utilizing a CCD digital camera and transformed into a digital structure that can be managed using a computer. We test the proposed opto-video LASM-based FrFT scheme using various security tools. The outcomes demonstrate that the presented scheme can effectively encrypt and decrypt video signals. In addition, it encrypts videos with a high level of encryption quality without sacrificing its resistance to noise immunity. Finally, the test outcomes demonstrate that the presented scheme is immune to known attacks.

Propagation properties of Hollow higher-order cosh-Gaussian beams in quadratic index medium and Fractional Fourier transform

Hollow higher-order cosh-Gaussian (HhCG) beam is a new mathematical model to describe the dark-hollow beam. An analytical expression for HhCG beams propagating through a paraxail ABCD optical system is derived and used to investigate its propagation properties in a quadratic index medium (QIM) and a Fractional Fourier transform (FRFT) optical system. By using the derived expressions, the properties of HhCG beams of both systems are graphically illustrated with numerical example. Several influence parameters on the evolution of HhCG in the both systems QIM and FRFT, respectively, are discussed in detail. Results show that the intensity distribution of HhCG evolves periodically during propagation in the QIM. In FRFT system, it is found that the normalized intensity distribution of HhCG beams plane depends not only on the fractional order but also on the beam order and the truncation parameter. The results obtained in this work are valuable to provide a convenient way for HhCG beams shaping, optical fibers, lenses, and other fields.

Introduction of a new vortex cosine-hyperbolic-Gaussian beam and the study of its propagation properties in fractional Fourier transform optical system

Based on the framework of the Huygens–Fresnel diffraction, we investigate theoretically the propagation properties of a new vortex beam, which is referred to as vortex-cosh-Gaussian beam (vChGB), through a paraxial ABCD optical system. A closed-form formula of vChGB passing through the above system is derived. We show by analytical and numerical calculations that the decentered parameter and the topological vortex charge affect strongly the characteristics of the considered beam upon propagating in free space. In a fractional Fourier transform system (FrFT), it is found that the intensity and the phase distributions of the propagating vChGB evolves gradually and periodically versus the order of the FrFT. The shape of the vChGB depends on the parameters of the beam and the fractional order of the FrFT system. The results obtained may be beneficial to applications in optical trapping, optical micromanipulation and beam shaping.

Finite-energy Airy-Lorentz-Gaussian beam and its paraxial propagation.

We theoretically propose a new form of laser beam, named the finite-energy Airy-Lorentz-Gaussian (ALG) beam, which can be decomposed into an exponentially modified Airy function in the x direction and a Lorentz-Gaussian function in the y direction. Based on the Huygens diffraction integral formula, an analytical propagation expression for the finite-energy ALG beam passing through a paraxial ABCD optical system has been derived. For special cases, the propagation properties of the finite-energy ALG beam through the free space, a thin lens, and a fractional Fourier transformation (FrFT) system are studied numerically and graphically. Our results provide an effective and alternative way to describe the finite-energy non-diffracting light beams.

Polarization characteristics of partially coherent beam passing through astigmatic fractional Fourier system

The fractional Fourier transform is a powerful tool for analyzing optical resonators. The selection of fractional number and astigmatism coefficient determines the size of the resonator, and influences the polarization, far-field emission angle and focusing power of the laser beam. Researchers often pay more attention to the change of light intensity after the fractional Fourier transform system. However, the effect of polarization changes on the system is also critical. In this article, we took partially coherent electromagnetic Gaussian-Schell model beam as a research object. Based on Collins formula and the fractional Fourier transform system ABCD matrix, we deduced the polarization degree of electromagnetic Gauss Schell model beam passing through the astigmatic lens fractional Fourier transform, elliptical azimuth and ellipticity; and analyzed the distribution of polarization degree, elliptic azimuth and ellipticity output plane of the astigmatism fraction Fourier transform system by numerical simulation. The results show that the distribution of polarization degree, ellipse azimuth and ellipticity on the output plane are no longer uniform after the beam passing through the fractional Fourier transform system with an astigmatic lens. In the non-focal plane, affected by astigmatism, beam waist position in x and y directions do not coincide, and the bigger astigmatism is, the farther separation is. Therefore, the polarization properties are elliptically symmetrical, and the greater the astigmatism is, the greater the eccentricity of the ellipse is. In the focal plane, the beam coincides with the waist position in the x and the y direction, and the polarization characteristics are circularly symmetric. The larger the astigmatism coefficient is, the larger the radius is. The greater the astigmatism coefficient is, the faster the polarization changes on the axis. The astigmatism of the lens leads to the difference of the beam width between the x direction and the y direction. Therefore, the point polarization of the shaft will appear in doublet or double valleys, and the greater the astigmatism coefficient is, the greater the difference between the value with and without astigmatism is. The results of this study have some reference value for the selection of fractional Fourier transform order and astigmatism coefficient under different polarization characteristics, and have important guiding significance for the design of optical resonator.

Fractional Fourier transform of double-half inverse Gaussian hollow beams

In this paper, we study an appropriate mathematical model of dark-hollow beam by concerning double-half inverse Gaussian hollow (DHIGH) beams passing through fractional Fourier transform (FRFT) optical systems. Based on Collins integral formula and using the expression of the hard-edged aperture function into a finite sum of complex Gaussian functions, analytical expressions for the propagation properties of DHIGH beams through the FRFT optical systems without and with apertures have been derived in detail. Some numerical simulations are given to illustrate the normalized intensity distribution of DHIGH beams in a FRFT system by studying the important factors of both incident beams order and the FRFT order. Our results obtained by using an approximate analytical formula, are in a good agreement with those obtained by using numerical integral method. The results show that the FRFT optical system is a convenient way for DHIGH beams shaping.

余弦高斯光束通过含球差分数傅里叶变换系统的传输

余弦高斯光束通过含球差分数傅里叶变换系统的传输

Bessel光束通过含光阑的球差分数傅里叶变换系统的传输

Bessel光束通过含光阑的球差分数傅里叶变换系统的传输

Fractional Fourier transform of astigmatic sine-Gaussian beams and generation of dark hollow light beams with elliptic geometry

In this work, we develop a novel method of creating dark hollow beam with vortex by converting a sine-Gaussian beam (SeGB) with edge-dislocation and astigmatism through using fractional Fourier transform (FrFT) optical system. On the basis of the definition of the FrFT, an analytical transformation formula is derived for an astigmatic SeGB passing through such a transform system. By use of the derived formulae, the changes of the intensity distribution and the corresponding phase properties associated with the transforming astigmatic SeGBs are analytically discussed in detail. It is found that for an input SeGB without astigmatism, there is still a dark line or an edge dislocation associated with the intensity distribution of the FrFT beam along the initial dislocation line, similar to that of the input SeGB. However, when the input SeGB astigmatically passes through an FrFT optical system, the dark line of the intensity distribution of the input SeGB can be converted into a solitary zero point, or in other words, a dark hollow beam with a single-charge vortex can be produced by SeGB with an edge dislocation. The results reveal that the astigmatism plays a critical role in transforming a SeGB into a dark hollow one through the FrFT optical system. Furthermore, some numerical calculation results based on the derived formula are presented and discussed graphically. It is shown that for appropriate beam parameters and carefully adjusting the transform angle of FrFT, dark hollow beams with single-charge vortex and elongated elliptic geometry can be realized with astigmatic SeGBs. The influences of the beam parameters and the transform angle of FrFT optical system on the generation of perfect dark hollow beams are also investigated. The results demonstrate that the linear eccentricity of the dark hollow beam, which is roughly defined as the ratio of semi-minor axis to semi-major one of the intensity pattern, mainly depends on the Fresnel number. And the optimal linear eccentricity may be relatively large under carefully selecting the beam and optical system parameters. Moreover, optimal parameter values corresponding to perfect dark hollow beam configurations which can be experimentally accessed are presented. As is well known, there are two types of pure phase defects or dislocations in the optical fields:one is screw dislocation or vortex and the other is edge-dislocation. Due to their important applications, the propagation dynamics of optical vortices or edge dislocations are extensively studied both theoretically and experimentally. The vortex-edge dislocation interaction is investigated in detail. However, there are fewer reports on the direct conversion between a single edge dislocation and a vortex. Therefore, the results obtained in this paper represent a significant step forward in understanding the transformation dynamics between beams with pure edge dislocation and vortex, and also opens possibilities for their potential applications, e.g., in generating dark hollow beams with elliptic geometry using FrFT systems.

Propagation of an Airy-Gaussian beam passing through the ABCD optical system with a rectangular aperture

Abstract By expanding the hard rectangular aperture function into a finite sum of complex Gaussian functions, we obtain the analytical formula for the propagation of an AiG beam passing through a paraxial ABCD optical system. According to the derived formulas, the propagation properties of an AiG beam through a paraxial ABCD optical system with a hard rectangular aperture are illustrated numerically. As a special case, the fractional Fourier transformation (FrFT) system is taken into consideration. The derived analytical formulas provide a convenient and effective way for studying the propagation and transformation of an AiG beam through an apertured general optical system.

Fractional Fourier transform of Airy-related beams generated from flat-topped Gaussian beams

Abstract The analytical expression for Airy-related beams generated from flat-topped Gaussian beams propagating through a paraxial ABCD optical system is derived and used to investigate its properties in the fractional Fourier transform (FrFT) optical system. The influences of the Airy-related beam order N and the fractional order p on the evolution of the beam intensity distribution in the FrFT system are examined in detail. Results show that the FrFT optical system provides a convenient way for modulating the beam profile of Airy-related beams by properly choosing optical parameters: lower order Airy-related beams have a longer non-spreading FrFT range, in which the Airy-related beams can maintain their original intensity distribution with no measurable spreading; and it is also found that the Airy-related beam intensity distribution versus the fractional order is symmetrical about p = 1 . Moreover, the Airy-related beam intensity distribution versus the fractional order is periodical, and the period is 2. The results obtained in this work are valuable for Airy-related beam shaping.

Phase Analyse of Non-Vortical and Vortical Beams in Fractional Fourier Transform System

The phase of vortical beams is very different from that of non-vortical beams. The phase of non-vortical and vortical beams in fractional Fourier transform system is investigated by selecting different parameters of the anomalous vortical beam. It is found that although the intensity distribution is similar except nearby the Fourier transform plane for the non-vortical and the vortical beams, the phase distribution is very different even the beam parameters are the same except the topological charge. The different phases bring different intensity distributions especially at the Fourier transform plane, i.e the center of non-vortical beams is a very strong intensity peaks, however the center of vortical beams is a dark region.

Propagation of symmetric tunable dual airy beam through ABCD optical system

A simple mathematical model is proposed to describe symmetric tunable dual Airy beam (STDAB) [Opt. Lett. 39 (2014) 2370; Opt. Lett. 38 (2013) 1286; Opt. Lett. 37 (2012) 3402]. This beam is composed of two conventional Airy beams with a tunable parameter which have several distinguishing features. For example, the intensity distribution of the beam can be Gaussian, elliptical Gaussian, rectangular Gaussian or dumbbell shaped with difference tunable parameters. Based on the Collins integral formula, an analytical expression of the STDAB passing through a paraxial ABCD optical system is derived. For special cases, the propagation properties of the beams in free space and in chiral medium, passing a thin lens and passing a fractional Fourier transformation system are studied.

Propagation properties of hollow sinh-Gaussian beams through fractional Fourier transform optical systems

A hollow sinh-Gaussian beam (HsG) is an appropriate model to describe the dark-hollow beam. Based on Collins integral formula and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of a HsG beam passing through fractional Fourier transform (FRFT) optical systems with and without apertures have been studied in detail by some typical numerical examples. The results obtained using the approximate analytical formula are in good agreement with those obtained using numerical integral calculation. Further, the studies indicate that the normalized intensity distribution of the HsG beam in FRFT plane is closely related with not only the fractional order but also the beam order and the truncation parameter. The FRFT optical systems provide a convenient way for laser beam shaping. • We study the propagation of a HsG beam through an ideal FRFT optical system. • The properties of a HsG beam through an apertured FRFT system have also been studied. • The intensity of the HsG beam in FRFT plane is closely related some parameters. • The FRFT optical systems provide a convenient way for laser beam shaping.

Finite energy Airy-Hermite-Gaussian beam and its paraxial propagation

A new form of laser beam called finite energy Airy-Hermite-Gaussian beam (AHGB) is introduced to describe the out mode of the metal strip waveguide in this paper. In the source plane, this beam can be decomposed into a Hermite-Gaussian function in y-direction and exponentially modified Airy function in x-direction. When the exponential truncation factor is large, the intensity profile in the source plane is an approximate off-axis Hermite-Gaussian distribution. Based on the Collins integral, an analytical expression for the finite energy AHGB passing through a paraxial ABCD optical system has been derived. For special cases, the propagation properties of this kind of beam passing in free space, through a thin lens and a fractional Fourier transformation system are studied and illustrated with numerical calculation. Our results provide an effective and fast way for studying the paraxial propagation properties of the finite energy AHGB.

Propagation of Airy beams through a hard-edged aperture

Based on the fact that a hard-edged aperture function can be expanded into a finite sum of complex Gaussian functions, we obtain the analytical formula for the propagation of the Airy beams in free space and through the fractional Fourier transformation (FrFT) system. According to the derived formulas, the propagation properties of the Airy beams in free space and through the FrFT plane with a hard-edged aperture are illustrated numerically.

Analytical solution for an anomalous hollow beam in a fractional Fourier transforming optical system with a hard aperture

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 an anomalous hollow beam (AHB) passing through an apertured fractional Fourier transform (FRT) system are derived. By using the approximate analytical formulae and diffraction integral formulae, the propagation properties of an AHB in circular and rectangular apertured FRT system are studied numerically. The results show that this method provides a convenient tool for studying the propagation properties of an AHB passing through apertured FRT system, and the apertured FRT system can be applied to laser beam shaping conveniently.

Propagation of broadband Gaussian Schell-model beams in the apertured fractional Fourier transformation systems

On the basis of the fact that a hard-edged aperture function can be expressed as finite matrices with different weighting coefficients, we obtain the analytical formula for the propagation of the broadband gaussian Schell-model (BGSM) beam through the apertured fractional Fourier transformation (AFrFT) system. It is shown by numerical examples that the intensity distribution in the plane of a small fractional order is obviously influenced by the bandwidth when the BGSM beams propagate through the AFrFT system. Further extensions are also pointed out.