Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems

Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems is investigated based on the generalized Collins formula for treating the propagation of a partially coherent beam through such optical systems. Analytical formulas for the cross-spectral density of a partially coherent Hermite-Gaussian beam propagating through such optical systems are derived. As an application example, we derive the propagation formulas for a partially coherent flattened Gaussian beam by expressing it as a superposition of a series of partially coherent Hermite-Gaussian beams by using polynomial expansion. The focusing properties of a partially coherent Hermite-Gaussian beam focused by a thin lens are studied as a numerical example.     

Evolution properties of a partially coherent Lorentz-Gauss vortex beam in a uniaxial crystal

The propagation properties of a partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal orthogonal to the optical axis were studied. The analytical expressions of the partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal were derived. Based on the derived formulae, the average intensity properties and evolution properties of the coherent vortex of the partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal were illustrated using numerical examples. The influences of the uniaxial crystal and the parameters of the beam at the source plane of z?=?0 on the average intensity and evolution properties of the coherent vortex for the partially coherent Lorentz-Gauss vortex beam are analyzed in detail.     

Reliability analysis of the flat-topped array beam FSO communication link

Free-space optical communication systems are affected by turbulent atmosphere. The atmospheric transmission is affected by absorption, scattering and turbulence. In this paper, the effects of absorption and scattering are taken into account using Beer’s law and the effects of turbulence are considered in calculating the average intensity distribution. An analytical expression for the average intensity distribution of a partially coherent flat-topped array (PCFTA) beam in turbulent atmosphere is derived based on the extended Huygens–Fresnel principle. The average intensity, power in bucket, signal to noise ratio, and bit error rate of this kind of beam are investigated in details. It is shown by numerical results and analytical methods that the average intensity and link parameters of PCFTA beams change during propagation and these changes are dependent upon both source parameters and weather conditions.     

Evolution properties of the complex degree of coherence of a partially coherent Laguerre–Gaussian beam in turbulent atmosphere

Evolution properties of the complex degree of coherence of a partially coherent Laguerre–Gaussian beam (LGB) on propagation in free space and turbulent atmosphere are studied comparatively with the help of the general propagation formula for such beam. It is found that the behavior of the complex degree of coherence of a partially coherent LGB on propagation in turbulent atmosphere is much different from that in free space and is closely related to the initial beam parameters and the structure constant of the turbulent atmosphere. The distribution of the modulus of the complex degree of coherence of the partially coherent LGB finally becomes of Gaussian distribution at long propagation distance in turbulent atmosphere, and it becomes of Gaussian distribution more slowly with the increase of the mode orders, beam width and wavelength. Our results will be useful in long-distance free-space optical communications.     

Two-wavelength ghost imaging with two-lens system in atmosphere

Based on the extended Huygens–Fresnel principle and the propagation law of a real ABCD optical system in turbulent atmosphere, an expression for two-wavelength ghost imaging with two-lens system is derived and the numerical simulations are made. The results show that the two-lens system can remarkably improve the image for a long propagation distance. The influence of the turbulence strength, the parameters of the radiation source and the focal length of the lenses on the image quality are discussed in detail.     

Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere was investigated. Analytical propagation formulae were derived for the cross-spectral densities of partially coherent Lorentz and Lorentz–Gauss beams. As an application example, the focusing properties of partially coherent Gaussian, Lorentz and Lorentz–Gauss beams in a turbulent atmosphere and in free space were studied numerically and comparatively. It is found that the focusing properties of such beams are closely related to the initial coherence length and the structure constant of turbulence. By choosing a suitable initial coherence length, a partially coherent Lorentz beam can be focused more tightly than a Gaussian or Lorentz–Gauss beam in free space or in a turbulent atmosphere with small structure constant at the geometrical focal plane.     

Analysing the behaviour of partially coherent divergent Gaussian beams propagating through oceanic turbulence

Theoretical study of propagation behaviour of partially coherent divergent Gaussian beams through oceanic turbulence has been performed. Based on the previously developed knowledge of propagation of a partially coherent beam in atmosphere, the spatial power spectrum of the refractive index of ocean water, extended Huygens–Fresnel principle and the unified theory of coherence and polarization, analytical formulas for cross-spectral density matrix elements are derived. The analytical formulas for intensity distribution, beam width and spectral degree of coherence are determined by using cross-spectral density matrix elements. Then, the effects of some source factors and turbulent ocean parameters on statistical properties of divergent Gaussian beam propagating through turbulent water are analysed. It is found that beam’s statistical propagation behaviour is affected by both environmental and source parameters variations.     

Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems

A new mathematical model called hollow elliptical Gaussian beam (HEGB) is proposed to describe a dark-hollow laser beam with noncircular symmetry in terms of a tensor method. The HEGB can be expressed as a superposition of a series of elliptical Hermite-Gaussian modes. By using the generalized diffraction integral formulas for light passing through paraxial optical systems, analytical propagation formulas for HEGBs passing through paraxial aligned and misaligned optical systems are obtained through vector integration. As examples of applications, evolution properties of the intensity distribution of HEGBs in free-space propagation were studied. Propagation properties of HEGBs through a misaligned thin lens were also studied. The HEGB provides a convenient way to describe elliptical dark-hollow laser beams and can be used conveniently to study the motion of atoms in a dark-hollow laser beam.     

Scintillation properties of a rectangular dark hollow beam

The scintillation properties of a rectangular dark hollow beam (DHB) in a weak turbulent atmosphere are investigated. Explicit expression for the on-axis scintillation index of a rectangular DHB is derived. It is found that the scintillation index value of a rectangular DHB can be smaller than that of Gaussian, elliptical Gaussian and rectangular flat-topped beams in a weak turbulent atmosphere under certain conditions. Our results will be useful in long-distance free-space optical communications. The scintillation properties of a rectangular DHB are closely controlled by its initial beam parameters.     

Decentred elliptical flat-topped light beams and their coherent and incoherent combinations

A new kind of laser beam called the decentred elliptical flat-topped light beam (DEFTB) is introduced in this paper by using a tensor method. Based on the generalized Collins integral, the propagation formula for a DEFTB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formulae for the generalized decentred elliptical Gaussian beam and elliptical flat-topped light beam under certain conditions. As a typical application example of the derived formula, the propagation characteristics of a DEFTB in a thin lens system are calculated and discussed. Furthermore, the generalized laser beam arrays with rectangular symmetry and radial symmetry by using DEFTBs as fundamental modes are constructed and the propagation properties of both a coherent and an incoherent combination of DEFTBs in the thin lens system with numerical examples are investigated.     

Partially coherent flat-topped beam and its propagation

A partially coherent beam with flat-topped profile is proposed. The cross-spectral density of this beam can be expressed as a finite series of the cross-spectral density of partially coherent Gaussian-Schell-model beams with different parameters. Analytical propagation formulas for partially coherent flat-topped beams are derived through aligned and misaligned optical systems. The propagation property of partially coherent flat-topped beams in free space is illustrated numerically. The fractional Fourier transform of partially coherent fiat-topped beams is also studied. Our method provides a convenient way to describe partially coherent flat-topped beams and treat their propagation and transformation.     

Evolution properties of a twisted Gaussian Schell-model beam in a uniaxial crystal

Paraxial propagation of a linearly polarised twisted Gaussian Schell-model (TGSM) beam in a uniaxial crystal is investigated. With the help of a tensor method, an analytical formula for the cross-spectral density (CSD) of a TGSM beam propagating in a uniaxial crystal orthogonal to the optical axis is derived. The evolution properties of the intensity distribution, the coherence widths and the twist factor of a TGSM beam in a uniaxial crystal are illustrated numerically. It is shown that the TGSM beam becomes an anisotropic TGSM beam in a uniaxial crystal and its evolution properties are closely related with its initial coherence, twist factor and the parameters of the crystal. The uniaxial crystal provides an effective way for modulating the properties of a TGSM beam, which will be useful in some applications, such as free-space optical communications and nonlinear optics, where a partially coherent beam with controlled beam profile and twist factor is required.     

Beam shaping of focused partially coherent beams by use of the spatial coherence effect

We demonstrate that when a partially coherent beam with a Gaussian intensity distribution is focused by a lens, the desired partially coherent flat-topped intensity distribution or doughnut-shaped intensity distribution at the geometrical focus can be generated by choice of appropriate form of spectral degree of coherence. We provide a novel approach to beam shaping of a partially coherent beam and offer new schemes for their potential applications such as material processing, optical therapy, and optical tweezers.     

Optical trapping Rayleigh dielectric particles with focused partially coherent dark hollow beams

The focusing properties of coherent and partially coherent dark hollow beams (DHBs) through a paraxial ABCD optical system are theoretically investigated. It is found that the evolution behavior of the intensity distribution of focused partially coherent DHBs is closely related to their spatial coherence. The radiation forces (RFs) of focused coherent and partially coherent DHBs acting on a Rayleigh dielectric particle are also theoretically investigated. Numerical results show that the coherent and partially coherent DHBs can be focused into a tight focal spot, which can be used to stably trap a Rayleigh dielectric particle with high refractive index at the focus point. The influences of different beam parameters, including the spatial coherence, beam waist width, beam order, and hollow parameter of partially coherent DHBs, on the RFs and the trap stiffness are analyzed in detail. Finally, the stability conditions for effective trapping particles are also discussed.     

Partially coherent flattened Gaussian beam and its paraxial propagation properties

A simple model called partially coherent flattened Gaussian beam (FGB) is proposed to describe a partially coherent beam with a flat-topped spatial profile. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent FGB propagating through a paraxial ABCD optical system. The propagation factor and propagation properties of a partially coherent FGB in free space are studied in detail and found to be closely related to its coherence and beam order.     

Propagation properties of a partially coherent anomalous hollow vortex beam in uniaxial crystal orthogonal to the optical axis

The concept of a partially coherent anomalous hollow vortex beam generated by a Schell-model source was introduced in this paper. Next, the analytical expressions of a partially coherent anomalous hollow vortex beam propagating in a uniaxial crystal orthogonal to the optical axis were derived, and the evolution properties of a partially coherent anomalous hollow vortex beam propagating in a uniaxial crystal were illustrated using numerical examples. Finally, the influences of the uniaxial crystal and the beam parameters on the spreading properties of partially coherent anomalous hollow vortex beam were analysed in detail.     

Propagation of a partially coherent higher-order cosh-Gaussian beam through a paraxial ABCD optical system

Based on the generalized Huygens–Fresnel integral, analytical expressions for the mutual coherence function, the spatial complex degree of coherence, and the effective size of a partially coherent higher-order cosh-Gaussian beam through a paraxial ABCD optical system have been derived. As a numerical example, the propagation of a partially coherent higher-order cosh-Gaussian beam through an optical Fourier-transforming system with a limiting aperture is illustrated. The normalized intensity distribution, the spatial complex degree of coherence, and the effective beam size for the partially coherent higher-order cosh-Gaussian beam are numerically demonstrated in the observation plane. The influences of the spatial coherence length and the limiting aperture on the normalized intensity distribution, the spatial complex degree of coherence, and the effective beam size are also examined in detail.     

Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere

Spectral changes of a partially coherent standard or elegant Laguerre–Gaussian (LG) beam propagating in turbulent atmosphere were studied numerically. Our results show that the spectral changes of a partially coherent standard or elegant LG beam in turbulent atmosphere are determined by both the structure constant of the turbulent atmosphere and the initial beam parameters. Furthermore, it is found that a partially coherent elegant LG beam is less affected by the turbulent atmosphere than a partially coherent standard LG beam from the aspect of the on-axis spectral shift, and this advantage is enhanced for small structure constant, small beam waist size, large mode orders, and large transverse coherence length. Our results will be useful in long-distance free-space optical communications.     

Propagation properties of partially coherent anomalous hollow beams in uniaxial crystals

Analytical propagation formulae for partially coherent anomalous hollow beams in uniaxial crystals are derived. Paraxial propagation of partially coherent anomalous hollow beams in uniaxial crystals orthogonal to the optical axis is investigated based on the beam propagation equations. The propagation properties of partially coherent anomalous hollow beams in uniaxial crystals and in free space are studied numerically and comparatively. It is found that the propagation properties of partially coherent anomalous hollow beams in uniaxial crystals behave very differently from those in free space and are closely determined by the parameters of the uniaxial crystals and the initial coherence width. The uniaxial crystals provide an effective way for generating astigmatic beams.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.