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.     

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.     

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.     

Statistical properties of a radially polarized twisted Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical formulas for the elements of the cross-spectral density matrix of a radially polarized partially coherent beam with a twist phase named radially polarized twisted Gaussian Schell-model (RPTGSM) beam propagating in a uniaxial crystal, and explore the statistical properties, such as the intensity distribution, the degree of polarization (DOP) and the state of polarization (SOP) of such beam in a uniaxial crystal with the help of the derived formulas. It is found that the statistical properties of a RPTGSM beam are closely related with the twist phase and the anisotropy of the uniaxial crystal, e.g. the twist phase leads to the rotation of the intensity, DOP and SOP distributions, and the anisotropy of the uniaxial crystal leads to the asymmetry distributions of these statistical properties. Our results will be useful for designing light field with prescribed intensity, DOP and SOP distributions, and may be useful in optical manipulations and free-space optical communications.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

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.     

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 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.     

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.     

Scattering of a partially coherent Gaussian-Schell beam from a diffuse target in slant atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle, the scattering of partially coherent Gaussian-Schell-model (GSM) beams from a diffuse target in slant double-passage atmospheric turbulence is studied and compared with that of fully coherent Gaussian beams. Using the cross-spectral density function of the GSM beams, we derive the expressions of the mutual coherence function, angle-of-arrival fluctuation, and covariance and variance of the intensity of the scattered field, taking into account the fluctuations of both the log-amplitude and phase. The numerical results are presented, and the influences of the wavelength, propagation distance, and waist radius on scattering properties are discussed. The perturbation region of the normalized intensity variance of the partially coherent GSM beam is smaller than that of the fully coherent Gaussian beam at the middle turbulence level. The normalized intensity variance of long-distance beam propagation is smaller than that of beam propagation along a short distance.     

Evolution properties of partially coherent four-petal Gaussian beams in oceanic turbulence

Based on the extended Huygens–Fresnel integral formula, the analytical expressions for partially coherent four-petal Gaussian beam propagating in oceanic turbulence are derived, and the influences of coherence length, beam order N and the parameters of oceanic turbulence (the rate of dissipation of turbulent kinetic energy per unit mass of fluid, the rate of dissipation of mean square temperature and the relative strength of temperature and salinity fluctuations) on average intensity properties are investigated using numerical examples in detail. The results show that the beam with the higher beam order N or coherence length will lose its initial four-petal profiles slower. It is also indicated that the beam will evolve into a Gauss-like beam more rapidly with increasing oceanic turbulence strength. The results have the potential application in underwater laser communication using a partially coherent four-petal Gaussian beam.     

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.     

Complex degree of coherence for partially coherent general beams in atmospheric turbulence

With the use of the general beam formulation, the modulus of the complex degree of coherence for partially coherent cosh-Gaussian, cos-Gaussian, Gaussian, annular and higher-order Gaussian optical beams is evaluated in atmospheric turbulence. For different propagation lengths in horizontal atmospheric links, the moduli of the complex degree of coherence at the source and receiver planes are examined when reference points are taken on the receiver axis and off-axis. In the on-axis case, it is observed that in propagation, the moduli of the complex degree of coherence are symmetrical and look like the intensity profile of the related coherent beam propagating in a turbulent atmosphere. For all the beams considered, the moduli of the complex degree of coherence profiles turn into Gaussian shapes beyond certain propagation lengths. In the off-axis case, the moduli of complex degree of coherence patterns become drifted at the earlier propagation lengths. Among the beams investigated, the cos-Gaussian beam is found to be almost independent of the changes in the source partial coherence parameter, and the annular beam seems to be affected the most against the variations of the source partial coherence parameter.     

Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication

A partially coherent quasi-monochromatic Gaussian laser beam propagating in atmospheric turbulence is examined by using a derived analytic expression for the cross-spectral density function. Expressions for average intensity, beam size, phase front radius of curvature, and wave-front coherence length are obtained from the cross-spectral density function. These results provide a model for a free-space laser transmitter with a phase diffuser used to reduce pointing errors.     

Effects of atmosphere on spectrum of modified partially coherent flat-topped beams

The analytical formulas for the spectrum of modified partially coherent flat-topped (MPCFT) beams propagating in a turbulent atmosphere have been derived under strong fluctuation condition of turbulence. The spectral properties of MPCFT beams propagating in a strong turbulence have been investigated, and the changes of on-axis and off-axis spectral distributions have been analyzed quantitatively. The results show that the on-axis relative spectral shift of the modified partially coherent flat-topped beam exhibits spectral switch for the beam order M > 1, and with the increasing propagation distance the blue-shift gradually decreases in atmospheric turbulence. A rapid transition for the off-axis spectrum appears at one critical position in turbulence, and the position decreases with the increasing beam order for the lower beam order and the nearer propagation distance. The spectral properties and the spectral shifts of MPCFT beams also depend on the inner scale, the spatial coherence parameter.     

Propagation of a random electromagnetic beam through a misaligned optical system in turbulent atmosphere

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system in turbulent atmosphere is derived. Some analyses are illustrated by numerical examples relating to changes in the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through such an optical system. It is shown that the misalignment has a significant influence on the intensity profile and the state of polarization of the beam, but the influence becomes smaller for the beam propagating in strong turbulent atmosphere. The method in this paper can be applied for sources that are either isotropic or anisotropic. It is shown that the isotropic sources and the anisotropic sources have different polarization properties on beam propagation.     

Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system

On the basis of the generalized diffraction integral formula for an ABCD optical system in the spatial domain, a propagation law for the generalized Stokes parameters of a stochastic electromagnetic beam passing through an ABCD optical system is obtained. We describe the Stokes parameters of the source as linear combinations of the elements of the cross-spectral density matrix, and study the changes in the spectral degree of polarization and in the state of the polarization ellipse of a stochastic electromagnetic Gaussian Schell-model beam propagating through a gradient-index fiber with the help of generalized Stokes parameters and the cross-spectral density matrix. The medium has significant effect on the change of the spectral degree of polarization. However, when the correlation coefficients of the source satisfy the relation delta(xx)=delta(yy)=delta(xy)=delta(yx), the medium does not influence the spectral degree of polarization.     

Focal shift of a focused partially coherent Laguerre-Gaussian beam of all orders

Abstract

We explore the focusing properties of a partially coherent Laguerre–Gaussian (LG) beam of all orders, particularly the focal shift (i.e. the shift of the actual focal plane away from the geometrical focal plane). We derive the analytical expressions for the average intensity and the effective beam width of a focused partially coherent LG beam, and we adopt the minimum effective beam width instead of the conventional maximum on-axis intensity to determine the actual focal plane. It is found that the focused beam shape, minimum effective beam width and the focal shift of a focused partially coherent LG beam are determined by its initial coherence width, radial mode order and azimuthal mode order (i.e. topological charge) together. Our results may be useful for optical trapping and micro-fabrication, where precise focal position and prescribed beam shape are required.     

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.