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.     

Scaled Fractional Fourier Transform for Partially Coherent Beams

The definition of scaled fractinal Fourier transform(SFRT) is extended to partially coherent beams based directly on the cross-spectral density.Based on this formula,an equivalent tensor ABCD law for the SFRT of partially coherent twisted anisotropic Gaussian-Schell model beams is derived..The derived formulae provide a powerful tool for analysing and calculating the SFRT of partially coherent beams.An application example is provided.     

Can two partially coherent cosh-Gaussian beams generate far fields with the same spectral degree of coherence？

Taking partially coherent cosh--Gaussian (ChG) beams as an example of more general partially coherent beams, we have studied the spectral degree of coherence of partially coherent ChG beams in the far field. It is shown that, unlike Gaussian Schell-model (GSM) beams, in the strict sense there do not exist two partially coherent ChG beams which can generate far fields with the same spectral degree of coherence. However, under certain conditions it is possible to find two partially coherent ChG beams with the same spectral degree of coherence in the far field.     

Scintillation of partially coherent Gaussian–Schell model beam propagation in slant atmospheric turbulence considering inner- and outer-scale effects

Based on the modified Rytov theory and the international telecommunication union-radio （ITU-R） slant atmospheric structure constant model, the uniform scintillation index of partially coherent Gaussian-Schell model （GSM） beam propa- gation in the slant path is derived from weak- to strong-turbulence regions considering inner- and outer-scale effects. The effects of wavelength of beams and inner- and outer-scale of turbulence on scintillation are analyzed numerically. Compar- ison between the scintillation of GSM beams under the von Karman spectrum and that of beams under the modified Hill spectrum is made. The results obtained show that the scintillation index obtained under the von Karman spectrum is smaller than that under the modified Hill spectrum. This study can find theory bases for the experiments of the partially coherent GSM beam propagation through atmospheric turbulence.     

Turbulence-induced changes in degree of polarization, degree of coherence and spectrum of partially coherent electromagnetic beams

Based on a recently formulated unified theory of coherence and polarization, a method is described to study turbulence-induced changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic beams on propagation. The electromagnetic Gaussian Schell-model beam is taken as a typical example of partially coherent electromagnetic beams, and the closed-form expressions for the degree of polarization, the degree of coherence and the spectrum of electromagnetic Gaussian Schell-model beams propagating through atmospheric turbulence are derived in the quadratic approximation of Rytov's phase structure function. Some interesting results are obtained, which are illustrated by numerical examples and are explained in physics.     

Study correlation vortices in the near-field of partially coherent vortex beams diffracted by an aperture

We have derived the analytical expression of the electric cross-spectral density in the near-field of partially coherent vortex beams diffracted by an aperture. Taking the Gaussian Schell-model vortex beam as a typical example of partially coherent vortex beams, the spatial correlation properties and correlation vortices in the near-field of partially coherent vortex beams diffracted by a rectangle aperture are studied. It is shown that the off-axis displacement, spatial degree of coherence parameter, propagation distance, and the opening factor of the aperture affect the spectral degree of coherence and positions of correlation vortices. With the optimization algorithm, we obtain the symmetric distributing coherent vortex.     

Design of phase plates for shaping partially coherent beams by simulated annealing

Taking the Gaussian Schell-model beam as a typical example of partially coherent beams, this paper applies the simulated annealing （SA） algorithm to the design of phase plates for shaping partially coherent beams. A flow diagram is presented to illustrate the procedure of phase optimization by the SA algorithm. Numerical examples demonstrate the advantages of the SA algorithm in shaping partially coherent beams. An uniform flat-topped beam profile with maximum reconstruction error RE 〈 1.74% is achieved. A further extension of the approach is discussed.     

Consistency of the directionality of partially coherent beams inturbulence expressed in terms of the angular spread and the far-field average intensity

Under the quadratic approximation of the Rytov's phase structure function, this paper derives the general closed-form expressions for the mean-squared width and the angular spread of partially coherent beams in turbulence. It finds that under a certain condition different types of partially coherent beams may have the same directionality as a fully coherent Gaussian beam in free space and also in atmospheric turbulence if the angular spread is chosen as the characteristic parameter of beam directionality. On the other hand, it shows that generally, the directionality of partially coherent beams expressed in terms of the angular spread is not consistent with that in terms of the normalized far-field average intensity distribution in free space, but the consistency can be achieved due to turbulence.     

Characteristics of a partially coherent Gaussian Schell-model beam propagating in slanted atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle and the model of the refractive-index structure constant in the atmospheric turbulence proposed by the International Telecommunication Union-Radio Communication Sector,the characteristics of the partially coherent Gaussian Schell-model(GSM) beams propagating in slanted atmospheric turbulence are studied.Using the cross-spectral density function(CSDF),we derive the expressions for the effective beam radius,the spreading angle,and the average intensity.The variance of the angle-of-arrival fluctuation and the wander effect of the GSM beam in the turbulence are calculated numerically.The influences of the coherence degree,the propagation distance,the propagation height,and the waist radius on the propagation characteristics of the partially coherent beams are discussed and compared with those of the fully coherent Gaussian beams.     

Numerical Simulation for Coherent and Partially Coherent Beam Propagation through Atmospheric Turbulence

Propagation properties of coherent and partially coherent beams in atmospheric turbulence are investigated respectively by using numerical simulation It is found that a partially coherent beam has a spreading larger than a coherent beam. However, from the point view of relative beam spreading and intensity scintillation, a partially coherent beam is less affected than the corresponding coherent beam, which may be the most important virtue of partially coherent beams that could be utilized to improve the performance of laser engineering. The beam wandering is almost independent of the degree of the source coherence. More aperture averaging occurs when beam becomes more coherent.     

Stokes parameters of optical singularities of random electromagnetic beams

In this paper,the relation between the spectral degree of coherence and degree of polarization of random electromagnetic beams is derived by the Stokes parameters.And the concept of polarization singularity is extended from spatially fully coherent beams to partially coherent electromagnetic beams.Theoretical analysis shows that correlation vortices are linearly polarized singularities.The results are illustrated by numerical examples.     

Generation of partially coherent vortex bottle beams

Intensity distribution of the partially coherent Bessel vortex beams focused by an aperture lens is investigated. It is found that the intensity distribution in the neighborhood of the geometrical focus is not only dependent on the topological charge and the radial frequency of the incident partially coherent Bessel vortex beam, but also on its coherence length. Based on this, the desired partially coherent vortex bottle beams can be obtained by choosing appropriate values of parameters. Because such bottle beams possess characteristics of low coherence and vortex, it may be used in microscopic particles guiding, trapping, and inducing rotation.     

Generation of partially coherent vortex bottle beams

Intensity distribution of the partially coherent Bessel vortex beams focused by an aperture lens is inves- tigated.It is found that the intensity distribution in the neighborhood of the geometrical focus is not only dependent on the topological charge and the radial frequency of the incident partially coherent Bessel vortex beam,but also on its coherence length.Based on this,the desired partially coherent vortex bottle beams can be obtained by choosing appropriate values of parameters.Because such bottle beams possess characteristics of low coherence and vortex,it may be used in microscopic particles guiding,trapping,and inducing rotation.     

Coherence vortices of partially coherent beams in the far field

Based on the propagation law of cross-spectral density function, studied in this paper are the coherence vortices of partially coherent, quasi-monochromatic singular beams with Gaussian envelope and Schell-model correlator in the far field, where our main attention is paid to the evolution of far-field coherence vortices into intensity vortices of fully coherent beams. The results show that, although there are usually no zeros of intensity in partially coherent beams with Gaussian envelope and Schell-model correlator, zeros of spectral degree of coherence exist. The coherence vortices of spectral degree of coherence depend on the relative coherence length, mode index and positions of pairs of points. If a point and mode index are kept fixed, the position of coherence vortices changes with the increase of the relative coherence length. For the low coherent case there is a circular phase dislocation. In the coherent limit coherence vortices become intensity vortices of fully coherent Laguerre--Gaussian beams.     

Characterization of tightly focused partially coherent radially polarized vortex beams

<正>Tight focusing properties of partially coherent radially polarized vortex beams are studied based on vectorial Debye theory.We focus on the focal properties including the intensity and the partially coherent and polarized properties of such partially coherent vortex beams through a high numerical aperture objective. It is found that the source coherence length and the maximal numerical aperture angle have direct influence on the focal intensity,as well as coherence and polarization properties.This research is important in optical micromanipulation and beam shaping.     

A resistance formula for coherent multi-barrier structures

Using the Landauer formula and the quantum S-matrix scattering theory, we derive a resistance formula for multi-barrier structure under phase coherent transmission condition. This formula shows that when the transport is coherent, the potential wells of the structure are just like conductors contributing to the overall resistance. And because the resistance formula is derived based on the scattering theory, the barrier resistance will change with the number of scattering centres (i.e. the number of barriers) in the structure.     

Elastic properties of spherically anisotropic piezoelectric composites

Effective elastic properties of spherically anisotropic piezoelectric composites, whose spherically anisotropic piezo-electric inclusions are embedded in an infinite non-piezoelectric matrix, are theoretically investigated. Analytical solutions for the elastic displacements and the electric potentials under a uniform external strain are derived exactly. Taking into account of the coupling effects of elasticity, permittivity and piezoelectricity, the formula is derived for estimating the effective elastic properties based on the average field theory in the dilute limit. An elastic response mechanism is revealed, in which the effective elastic properties increase as inclusion piezoelectric properties increase and inclusion dielectric properties decrease. Moreover, a piezoelectric response mechanism, of which the effective piezoelectric response vanishes due to the symmetry of spherically anisotropic composite, is also disclosed.     

Effect of astigmatism on spectral switches of partially coherent beams

A detailed study of the spectrum of partially coherent beams diffracted at an astigmatic aperture lens is performed. Considerable attention is paid to the effect of astigmatism on spectral switches of polychromatic Gaussian Schell-model beams. It is shown that the spectral switch can also take place in the vicinity of intensity minimum in a geometrical focal plane for the astigmatic case, but the astigmatism of the lens and the spatial correlation of the beam affect the critical position u_c, spectral minimum S_{min}, and transition height Δ of spectral switches.     

High-Order Bessel-Gaussian Beam and its Propagation Properties

A hlgh-order Bessel-Gaussian mode is introduced to describe hollow beams. The results for high-order Bessel-Gaussian beams propagating through lens focusing system and free space are derived in terms of Collins integral formula. The diffraction patterns and profile for high-order Bessel-Gaussian beams propagating through the above-mentioned optical systems are illustrated.     

A Transverse-Longitudinal Cross-Spectral Density Matrix of Partially Coherent Electromagnetic Beams

A transverse-longitudinal cross-spectral density matrix （TLCSDM） of partially coherent electromagnetic beams is proposed. It can extend the traditional Stokes parameters and polarization singularities from the paraxial field to the more general situation. The impact produced by the atmospheric turbulence on polarization singularities of the partially coherent electromagnetic vortex beams is analyzed with the TLCSDM.