Fractional Fourier Transform Optical System 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.

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.

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.

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.

THE FRACTIONAL FOURIER TRANSFORM OF HYPERGEOMETRIC-GAUSS BEAMS THROUGH THE HARD EDGE APERTURE

Based on the Collins integral formula and Lohmann optical system, we expand the hard edge aperture into complex Gauss function and derive an approximate analytic expression of intensity distribution theoretically for Hypergeometric-Gauss beams through the fractional Fourier transform (FRT) optical systems with hard edge aperture. The influences of FRT order, aperture size and other optical parameters on the light intensity distribution of Hypergeometric-Gauss beams passing through the FRT optical systems are discussed in detail. The results show that the FRT is an excellent beam- shaping method.

Fractional Fourier transform of hollow sinh-Gaussian beams

The analytical expression for hollow sinh-Gaussian (HsG) beams propagating through a paraxial ABCD optical system is derived and used to investigate its propagation properties in a fractional Fourier transform (FrFT) optical system. Several influence parameters of both the HsG beams and the FrFT optical system are discussed in detail. Results show that the FrFT optical system provides a convenient way for modulating HsG beams: HsG beams maintain their dark-centered distribution when the fractional order p is low, and low-ordered HsG beams lose their original dark-centered distribution more quickly than high-ordered ones when the value of p increases. Eventually all HsG beams’ intensities evolve into peak-centered distributions with some side lobes located sideways. Furthermore, our results also show that HsG beam intensity distribution versus the fractional order is periodical and the period is 2. The results obtained in this work are valuable for HsG 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.

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.

Propagation of Bessel–Gaussian beams through a double-apertured fractional Fourier transform optical system

Based on the scalar diffraction theory and the fact that a hard-edged aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical solution for Bessel-Gaussian (BG) beams propagating through a double-apertured fractional Fourier transform (FrFT) system is derived in the cylindrical coordinate. By using the approximate analytical formulas, the propagation properties of BG beams passing through a double-apertured FrFT optical system have been studied in detail by some typical numerical examples. The results indicate that the double-apertured FrFT optical system provides a convenient way for controlling the properties of the BG beams by properly choosing the optical parameters.

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.

The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems

Based on the generalized Huygens–Fresnel integral, the analytical expressions for Airy beams through three types of fractional Fourier transform (FRFT) optical systems (Lohmann I, Lohmann II and quadratic graded-index systems) are derived. Using the two analytical expressions, the propagation properties of Airy beams for the three types of FRFT optical systems are discussed in detail with numerical examples. Results indicate that the intensity of Airy beams periodically changes and remains form-invariant with the changing of fractional order p for the three types of FRFT optical systems.

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 of flat-topped multi-Gaussian beams

The fractional Fourier transform (FRFT) of the flat-topped multi-Gaussian beam (FMGB) is investigated based on the three kinds of FRFT optical systems: Lohmann I, Lohmann II, and quadratic graded-index systems. The analytical expressions for the FRFT of the FMGB are derived based on the propagation of the FMGB through the three systems. By introducing a hard-edge aperture function, the analytical expressions for the FRFT of the FMGB carried out by the apertured FRFT optical systems are presented. The FRFT characteristics of the FMGB for the three kinds of FRFT optical systems with and without apertures are discussed in detail. Results show that the three types of FRFT optical systems have the same function when the apertures are ignored but that significantly different characteristics are exhibited when the apertures appear.

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.

Propagation properties of Bessel and Bessel–Gaussian beams in a fractional Fourier transform optical system

The properties of Bessel–Gaussian beams (BGBs) and Bessel beams (BBs) propagating through a fractional Fourier transform (FRT) optical system have been investigated. The analytical transformation formulae for BBs and BGBs propagation through a FRT optical system are derived based on definition of the FRT in the cylindrical coordinate system. By using the derived formula, numerical examples are illustrated.

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.

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.

Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics

We report the experimental observation of the fractional Fourier transform (FRT) for a partially coherent optical beam with Gaussian statistics [i.e., partially coherent Gaussian Schell-model (GSM) beam]. The intensity distribution (or beam width) and the modulus of the square of the spectral degree of coherence (or coherence width) of a partially coherent GSM beam in the FRT plane are measured, and the experimental results are analyzed and agree well with the theoretical results. The FRT optical system provides a convenient way to control the properties, e.g., the intensity distribution, beam width, spectral degree of coherence, and coherence width, of a partially coherent beam.

The fractional Fourier transform for a partially coherent pulse

The fractional Fourier transform (FRT) is applied to a partially coherent pulse. On the basis of the definition of the FRT in the temporal domain, an analytical transformation formula is derived for the correlation function of a partially coherent Gaussian–Schell model (GSM) pulse passing through a FRT optical system by means of a tensor method. The corresponding tensor ABCD law for performing a FRT of a partially coherent GSM pulse is obtained. The properties of a partially coherent GSM pulse in the FRT plane are illustrated numerically. The derived formulae provide a powerful tool for analysing and calculating the FRT of a partially coherent GSM pulse in the temporal domain.