Fractional Fourier Transform Plane Research Articles

Fractional Fourier transforms of vortex Hermite-cosh-Gaussian beams

In this paper, we investigate the propagation properties of a vortex Hermite-cosh-Gaussian beam (vHChGB) passing through a fractional Fourier transform (FRFT) optical system. The analytical expression of a vHChGB in FRFT plane is derived based on the Huygens–Fresnel diffraction integral. From the obtained expression, the evolution of the intensity and phase distributions of output beam are illustrated numerically as a function of the order of the FRFT and under different beam parameters conditions. The results show that a vHChGB in FRFT plane evolves gradually and periodically versus the order of the FRFT optical system. The beam shape depends strongly on the fractional order, and it is affected by the beam order, the decentered parameter and the topological vortex charge. The results obtained may be beneficial to applications in optical trapping, micro-particle manipulation, and beam shaping.

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.

PROPERTIES OF AIRY-GAUSS BEAMS IN THE FRACTIONAL FOURIER TRANSFORM PLANE

An analytical expression of an Airy-Gauss beam passing through a fractional Fourier transform (FRFT) system is derived. The normalized intensity distribution, phase distribution, centre of gravity, effective beam size, linear momentum, and kurtosis parameter of the Airy-Gauss beam are demonstrated in FRFT plane, respectively. The influence of the fractional order p on the normalized intensity distribution, phase distribution, centre of gravity, effective beam size, linear momentum, and kurtosis parameter of the Airy-Gauss beam are examined in FRFT plane. The fractional order p controls the normalized intensity distribution, phase distribution, centre of gravity, effective beam size, the linear momentum, and kurtosis parameter. The period of the normalized intensity, phase, and centre of gravity versus the fractional order p is 4. The period of effective beam size, linear momentum, and kurtosis parameter versus the fractional order p is 2. The periodic behaviors of the normalized intensity distribution, phase distribution, centre of gravity, effective beam size, linear momentum, and kurtosis parameter can bring novel applications such as optical switch, optical micromanipulation, and optical image processing.

Numerical Simulations of Fractional Fourier Transform of Hypergeometric-Gaussian Beam

The fractional Fourier transform (FRFT) of a new type of laser beams called the hypergeometric-Gaussian beam (HyGGB) is investigated in detail. The analytical expression for the FRFT of a HyGGB is derived. The properties of a HyGGB in the FRFT plane with different parameters are illustrated. The results show that the intensity distribution of a HyGGB in the FRFT plane strongly depends on the fractional order, the lens focal length and the initial beam width.

Experimental study of the fractional Fourier transform for a hollow Gaussian beam

Abstract Hollow Gaussian beam (HGB) was introduced in [Cai et al., Optics Letters 2003;28:1084–1086 [12] ] and the fractional Fourier transform (FRT) for a HGB was studied theoretically in [Zheng, Physics Letters A 2006;355:156–161 [53] ] . In this paper, we derive the analytical formula for the truncated FRT for a HGB, and we report experimental observation of the FRT and the truncated FRT for a HGB. The influences of the fractional order and the truncation parameter on the intensity distribution of the HGB in the FRT plane have been studied in detail both theoretically and experimentally. It is found that the FRT optical system provides an efficient way for modulating the beam profile of a HGB. Our experimental results agree well with the theoretical predictions.

Fractional Fourier transform of Lorentz-Gauss vortex beams

An analytical expression for a Lorentz-Gauss vortex beam passing through a fractional Fourier transform (FRFT) system is derived. The influences of the order of the FRFT and the topological charge on the normalized intensity distribution, the phase distribution, and the orbital angular momentum density of a Lorentz-Gauss vortex beam in the FRFT plane are examined. The order of the FRFT controls the beam spot size, the orientation of the beam spot, the spiral direction of the phase distribution, the spatial orientation of the two peaks in the orbital angular momentum density distribution, and the magnitude of the orbital angular momentum density. The increase of the topological charge not only results in the dark-hollow region becoming large, but also brings about detail changes in the beam profile. The spatial orientation of the two peaks in the orbital angular momentum density distribution and the phase distribution also depend on the topological charge.

Fractional Fourier transform of Airy beams

An analytical expression of an Airy beam passing through a fractional Fourier transform (FRFT) system is presented. The effective beam size of the Airy beam in the FRFT plane is also derived. The influences of the order of FRFT, the modulation parameter, and the transverse scale on the normalized intensity distribution and the effective beam size of an Airy beam in the FRFT plane are examined. The order of FRFT controls the effective beam size and the orientation of the beam spot. The modulation parameter affects the effective beam size and the number of lateral side lobes. The normalized intensity distribution and the effective beam size are both proportional to the transverse scale.

Fractional Fourier transform for confluent hypergeometric beams

Based on the definition of the fractional Fourier transform (FRFT) in the cylindrical coordinate system, the propagation properties of a new family of paraxial laser beams named confluent hypergeometric (HyG) beams, of which intensity profile is similar to that for the Bessel modes, passing through FRFT optical systems have been studied in detail by some typical numerical examples. The results indicate that the normalized intensity distribution of a HyG beam in the FRFT plane is closely related to not only the fractional order p but also the beam parameters m,n, and focal length f.

Propagation properties of controllable dark-hollow beams through fractional Fourier transform systems

Based on the definition of fractional Fourier transform (FrFT) in the cylindrical coordinate system, the propagation properties of a controllable dark-hollow beam (CDHB) are investigated in detail. An analytical formula is derived for the FrFT of a CDHB. By using the derived formula, the properties of a CDHB in the FrFT plane are illustrated numerically. The results show that the properties of the intensity of the beam in the FrFT are closely related to not only the fractional order but also initial beam parameter, beam order and the lens focal length of the optical system for performing FrFT. The derived formula provides an effective and convenient way for analyzing and calculating the FrFT of a CDHB.

Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system

Abstract By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derive approximate analytical formulae for Lorentz and Lorentz–Gauss beams propagating through an apertured fractional Fourier transform (FRT) optical system. As an application example, properties of a Lorentz–Gauss beam in the FRT plane after propagating through a squarely or annularly apertured FRT optical system are studied numerically. The results obtained using the approximate analytical formula are in good agreement with those obtained using numerical integral calculation. The FRT optical system provides a convenient way for laser beam shaping.

Propagation of a general-type beam through a truncated fractional Fourier transform optical system

Paraxial propagation of a general-type beam through a truncated fractional Fourier transform (FRT) optical system is investigated. Analytical formulas for the electric field and effective beam width of a general-type beam in the FRT plane are derived based on the Collins formula. Our formulas can be used to study the propagation of a variety of laser beams--such as Gaussian, cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, flat-topped, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, higher-order annular Gaussian, Hermite-sinh-Gaussian and Hermite-cos-Gaussian beams--through a FRT optical system with or without truncation. The propagation properties of a Hermite-cos-Gaussian beam passing through a rectangularly truncated FRT optical system are studied as a numerical example. Our results clearly show that the truncated FRT optical system provides a convenient way for laser beam shaping.

Fractional Fourier transform for annular flat-topped beams

The fractional Fourier transform (FRFT) is applied to treat the propagation of annular flat-topped beams. Based on the definition of FRFT in the cylindrical coordinate system, analytical formulae are derived for annular flat-topped beams through the FRFT optical systems. By using the formulae, the properties of annular flat-topped beams in the FRFT plane are illustrated numerically. The results show that the intensity distribution properties in the FRFT plane are closely related to the fractional order of the FRFT optical system and initial beam parameters.

Fractional Fourier transform for an anomalous hollow beam

Based on the definition of fractional Fourier transform (FRT), the propagation properties of an anomalous hollow beam (AHB) through the FRT have been investigated in detail. An analytical formula is derived for the FRT of an AHB. By using the derived formula, the properties of an AHB in the FRT plane are illustrated numerically. The results show that the properties of an AHB in the FRT plane are closely related to the parameters of the beam and the fractional order p. The derived formula provides an effective and convenient way for analyzing and calculating the FRT of an AHB.

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.

Propagation of four-petal Gaussian beams in apertured fractional Fourier transforming systems

Based on the Collins diffraction integral formula and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation of a four-petal Gaussian beam passing through an apertured fractional Fourier transform (FRFT) optical system has been studied in detail. Some typical numerical examples are given to illustrate the properties of the four-petal Gaussian beam in the FRFT plane. The results indicate that the intensity distributions of the beam in the FRFT plane are closely related to not only the fractional order but also the beam parameters and the aperture 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 for beams generated by Gaussian mirror resonator

Based on the definition of the fractional Fourier transform (FRFT) in the cylindrical coordinate system and the fact that a hard-edged-aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of beams generated by Gaussian mirror resonator passing through FRFT optical systems have been studied in detail, and some typical numerical examples are given to compare the results obtained by the approximate analytical method with those by the numerical integral method, and it is shown that our method can significantly improve the numerical calculation efficiency. Further, the results indicate that the normalized intensity distributions in the FRFT plane are closely related to the fractional order, truncation parameter and initial beam parameters. The variation period of normalized intensity distributions with p is 2 whatever the value of the truncation parameter δ is.

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.