Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere

The propagation properties of flattened Gaussian beam with aperture in turbulent atmosphere have been studied by using the extended Huygens-Fresnel principle. From the study and numerical calculation, the effects of aperture on the propagation of flattened Gaussian beams in turbulent atmosphere have been illuminated. It shows that when the value of the truncation parameter δ is bigger, for example δ?2, the effects of aperture on the propagation properties are too small to be neglected. But when the truncation parameter δ is smaller, for example δ<2, the effects of aperture are complex. The peak value of the average intensity descends more rapidly and the beam spot spreads quicker with aperture than that without aperture when the propagation distance increases. Meanwhile, with the propagation distance increasing, the average intensity profiles of flattened Gaussian beams gradually convert into Gaussian average intensity profiles. In addition, some limiting cases are also discussed. It agrees with the existing results.     

Propagation analysis of flattened circular Gaussian beams with a misaligned circular aperture in turbulent atmosphere

The propagation properties of flattened Gaussian beam with a misaligned circular aperture in turbulent atmosphere have been studied by using the extended Huygens-Fresnel formula. From the study and numerical calculation, the effects of aperture parameters on the propagation properties of flattened Gaussian beams in turbulent atmosphere have been illuminated. The results show that angle misalignments and lateral displacements of aperture create unsymmetrical average intensity distribution at any cross section. The intensity distributions are much more sensitive to the lateral displacement than to the angle misalignments. And the propagation properties of different flattened Gaussian beams in turbulent atmosphere are also compared.     

Long- and short-term average intensity for multi-Gaussian beam with a common axis in turbulence

With the help of the extended Huygens--Fresnel principle and the short-term mutual coherence function, the analytical formula of short-term average intensity for multi-Gaussian beam (MGB) in the turbulent atmosphere has been derived. The intensity in the absence of turbulence and the long-term average intensity in turbulence can both also be expressed in this formula. As special cases, comparisons among short-term average intensity, long-term average intensity, and the intensity in the absence of turbulence for flat topped beam and annular beam are carried out. The effects of the order of MGB, propagation distance and aperture radius on beam spreading are analysed and discussed in detail.     

Comparison between relay propagation and direct propagation of Gaussian-Schell-model beam in turbulent atmosphere along a slant path

The relay propagation of Gaussian-Schell-model in turbulent atmosphere along a slant path is studied in this paper. Based on the extended Huygens-Fresnel principle and a quadratic approximation, an analytical formula of average intensity for Gaussian-Schell-model beams in turbulent atmosphere along a slant path is derived, and some special cases are discussed. From the study and the comparison with the direct propagation, we can see that the relay propagation has an advantage over the direct propagation. When the altitude of the target is low, the peak intensity of relay propagation is much larger than that of direct propagation. However, because of the limitation of the relay system aperture for relay propagation and the variation of coherence length for direct propagation, the difference in peak intensity between the two propagations decreases with the increase of the target altitude.     

Double-distance propagation of Gaussian beams passing through a tilted cat-eye optical lens in a turbulent atmosphere

By using the extended Huygens-Fresnel diffraction integral and the method of expanding the aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula of the double-distance propagation for Gaussian beam passing through a tilted cat-eye optical lens and going back along the entrance way in a turbulent atmosphere has been derived. Through numerical calculation, the effects of incidence angle, propagation distance, and structure constant on the propagation properties of a Gaussian beam in a turbulent atmosphere are studied. It is found that the incidence angle creates an unsymmetrical average intensity distribution pattern, while the propagation distance and the structure constant can each create a smooth and symmetrical average intensity distribution pattern. The average intensity peak gradually deviates from the centre, and the central average intensity value decreases quickly with the increase in incidence angle, while a larger structure constant can bring the average intensity peak back to the centre.     

Axial intensity distribution of truncated Bessel-Gauss beams in a turbulent atmosphere

By means of the extended Huygens-Fresnel integral formula in the paraxial approximation and based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical expression for Bessel-Gauss beams propagating in a turbulent atmosphere truncated by an aperture has been derived. The result is more convenient for studying the axial intensity distribution properties of Bessel-Gauss beams with different topological charges in a turbulent atmosphere than the usual way by using diffraction integral directly. Utilizing the analytical expression, the axial intensity distribution of such kind of beam in a turbulent atmosphere is investigated numerically in detail. Results show that the intensity distribution on the axis changes with the variation of the beam topological charge m, the wavelength and beam waist width of the initial beam and the turbulent strength.     

Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere

Cylindrical vector (CV) partially coherent beam was introduced (Dong et al. Opt Express 19:5979–5992, 2011) and generated (Wang et al. Appl Phys Lett 100:051108, 2012) recently. In this paper, we derive the realizability conditions for a CV partially coherent beam, and we derive the analytical formula for the cross-spectral density matrix of a CV partially coherent beam propagating in turbulent atmosphere based on the extended Huygens-Fresnel formula. The statistical properties, such as the average intensity, the complex degree of coherence and the degree of polarization of a CV partially coherent beam in turbulent atmosphere are studied in detail and compared with those in free space. It is found that the statistical properties of a CV partially coherent beam in turbulent atmosphere are much different from those in free space, and are affected by the initial beam parameters.     

Slant path average intensity of finite optical beam propagating in turbulent atmosphere

The average intensity of finite laser beam propagating through turbulent atmosphere is calculated from the extended Huygens Fresnel principle. Formulas are presented for the slant path average intensity from an arbitrarily truncated Gaussian beam. The new expressions are derived from the modified von Karman spectrum for refractive-index fluctuations, quadratic approximation of the structure function,and Gaussian approximation for the product of Gaussian function and Bessel function. It is shown that the form of average intensity is not a Gaussian function but a polynomial of the power of the binomial function, Gaussian function, and the incomplete gamma function. The results also show that the mean irradiance of a finite optical beam propagating in slant path turbulent atmosphere not only depends on the effective beam radius at the transmitting aperture plane, propagation distance, and long-term lateral coherence length of spherical wave, but also on the radius of emit aperture.     

Propagation properties of cosh-squared-Gaussian beam through fractional Fourier transform systems

Based on Collins integral formula and the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the propagation properties of cosh-squared-Gaussian beam passing through ideal and apertured FRFT systems have been studied, and some comparisons between using the methods of analytical formula and diffraction integral formula have been done. Further, the studies indicate that the normalized intensity distributions on FRFT plane depend on the fractional order, truncation parameter and initial beam parameter Ω. Variations of normalized intensity distributions with FRFT order are periodic: when the impact of aperture cannot be ignored, the variation period is 4; and when the impact of aperture can be ignored, the variation period is 2.     

Average intensity and spreading of an elliptical gaussian beam propagating in a turbulent atmosphere

An analytical formula for the average intensity of an elliptical Gaussian beam (EGB) propagating in a turbulent atmosphere is derived. The spreading properties of an EGB in a turbulent atmosphere are studied. It is found that an EGB will eventually become a circular Gaussian beam in a turbulent atmosphere. This interesting phenomenon is quite different from the propagation of an EGB in free space. The evolution properties are closely related to the parameters of the beam and the turbulent atmosphere.     

Average intensity and spreading of super Lorentz–Gauss modes in turbulent atmosphere

The super Lorentz–Gaussian (SLG) modes has been introduced to describe the radiation emitted by the multi-mode diode lasers. Here the propagation properties of SLG modes in turbulent atmosphere are investigated. Based on the extended Huygens–Fresnel integral and the Hermite–Gaussian expansion of a Lorentzian function, analytical formulae for the average intensities and the effective beam sizes of SLG₀₁ and SLG₁₁ modes are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of SLG₀₁ and SLG₁₁ modes in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of SLG₀₁ and SLG₁₁ modes in turbulent atmosphere are also discussed in detail.     

The on-axis average intensity for Gaussian beam excitation limited by an annular aperture in turbulent atmosphere

Starting with a Gaussian beam excitation limited by an annular aperture, the on-axis average intensity is presented in turbulent atmosphere. The on-axis average intensity profile is evaluated by altering the outer and inner radius of an annular aperture, the propagation distance and wavelength. The results show that the outer radius of an annular aperture within a certain size impacts the on-axis intensity distributions. When the propagation distance or the outer radius of an annular aperture is large enough, the on-axis average intensity distributions is not affected by the size of the annular aperture. By calculation and analyses, the variation of the inner radius of the annular aperture and the wavelength of the Gaussian beam that impact on the axial average intensity distributions is also discussed.     

The relay propagation of partially coherent cosh–Gaussian–Schell beams in turbulent atmosphere

We have studied the relay propagation of a partially coherent cosh–Gaussian–Schell beam in turbulent atmosphere. Analytical expressions for both the cross-spectral density at the relay system and average intensity at the target are derived. By using the analytical expressions some special cases are studied and some numerical simulation comparisons are made, especially the effects of the coherence of the beam, turbulence strength, aperture and its size on the relay propagation. Our study shows that the effects of diffraction and coherence of initial beam on the intensity profiles at the relay system are so small that they can be neglected when the effects due to turbulence are large enough. Even though the correction to the receiving beam at relay system is important, it is not necessary to improve the receiving beam when the effect of turbulence over the travel path is strong. A high peak intensity at target can be obtained by optimizing these factors, such as propagation distance, aperture and spatial correlation length.     

Spectral properties of stochastic electromagnetic twist anisotropic Gaussian-Schell model beam truncated by a slit aperture propagating in turbulent atmosphere

With the help of the tensor method, the cross-spectral density matrix for the stochastic electromagnetic twist anisotropic Gaussian-Schell model (ETAGSM) beam truncated by a slit aperture propagating in turbulent atmosphere are derived. The spectral properties of this kind of beam are investigated in detail. It is shown by numerical results and analysis that the affection of the slit aperture on the spectral properties of the stochastic ETAGSM beam is obvious in the near field; while in the far field, the atmospheric turbulence plays an important role; the source beam's coherence can weaken the affection of the slit aperture and the atmospheric turbulence on the spectral properties of the stochastic ETAGSM beam truncated by a slit aperture propagating in turbulent atmosphere, while the twist properties of the source beam can strong the affection of the slit aperture on the spectral properties in the near field. Also, the spectral degree of polarization and normalized spectral density distributions and corresponding contour graphs of the stochastic ETAGSM beam truncated by a slit aperture propagating in turbulent atmosphere and free space at different propagation distances are investigated in detail.     

Elegant Laguerre-Gaussian beam in a turbulent atmosphere

Paraxial propagation of an elegant Laguerre-Gaussian beam (ELGB) in a turbulent atmosphere is investigated in detail. Analytical formulae for the average intensity and effective beam size of an ELGB in a turbulent atmosphere are derived based on the extended Huygens-Fresnel integral. The average intensity and spreading properties of an ELGB in a turbulent atmosphere are studied numerically and compared with those of a standard Laguerre-Gaussian beam (SLGB). Our results indicate that the propagation properties of an ELGB in a turbulent atmosphere are much different from its properties in free space, and are closely related to its beam parameters and the structure constant of the atmospheric turbulence. The ELGB with higher mode orders is less affected by the turbulence. The SLGB spreads more rapidly than the ELGB in a turbulent atmosphere under the same conditions. Our results will be useful in long-distance free-space optical communications. ©2010 Elsevier Science B.V. All rights reserved.     

Generalized tensor ABCD law for an elliptical Gaussian beam passing through an astigmatic optical system in turbulent atmosphere

The propagation of an elliptical Gaussian beam (EGB) through an astigmatic ABCD optical system in a turbulent atmosphere is investigated. An analytical formula for the average intensity of an EGB and a generalized tensor ABCD law for the generalized complex curvature tensor are derived. As an application example, we derived an analytical formula for the average intensity of an elliptical flat-topped beam propagating through an astigmatic ABCD optical system in a turbulent atmosphere. As a numerical example, the focusing properties of an EGB focused by a thin lens in a turbulent atmosphere are studied. It is found that the focused beam at the focal plane becomes a circular Gaussian beam when the atmospheric turbulence is strong enough, and the beam width of the circular Gaussian beam is determined by atmospheric turbulence strength, focal length of the thin lens, and wavelength of the initial beam but is independent of the initial beam widths (i.e., initial intensity distribution).     

Spectral shifts and spectral switches of elliptical Gaussian beam propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of elliptical Gaussian beam (EGB) propagating through turbulent atmosphere can be derived analytically by tensor method. It can avoid time-consuming numerical integral that was commonly used in the previous study of spectral changes. Analytical results show that the on-axis normalized spectrum S(ω) of EGB propagating through turbulent atmosphere is different from the original spectrum S₀(ω) and there exist spectral shifts and spectral switches for EGB propagating through turbulent atmosphere. Besides, spectral shifts and spectral switches of EGB are closely related with the structure constant of the turbulent atmosphere, the beam parameters and the coordinate of observation point. Compared with the Gaussian beam, there are two spectral switches for EGB propagating through turbulent atmosphere.     

Propagation properties of a non-circular partially coherent flat-topped beam in a turbulent atmosphere

The propagation of partially coherent flat-topped (PCFT) beams with non-circular symmetries in a turbulent atmosphere is investigated and an analytical formula for the average intensity is derived. Detailed analyses of PCFT beams in a turbulent media led to the derivation of analytical formulae. These formulae demonstrated that higher-order PCFT beams are less broadened in comparison with those with smaller orders and also demonstrated that PCFT beams with smaller correlation lengths are more broadened than the others. The Strehl ratio for PCFT beams is also calculated and the results show that PCFT beams with a higher order of flatness and smaller correlation length are less affected in turbulent media.     

Spatial correlation properties of apertured partially coherent radially polarized beams in turbulent atmosphere

Based on the extended Huygens–Fresnel integral, the analytical formulae for the cross-spectral density matrix of the partially coherent radially polarized beams diffracted at a circular aperture in turbulent atmosphere are derived. The unapertured and free-space cases can be viewed as the special cases of our general result. By using the degree of coherence formula, the spatial correlation properties of the apertured partially coherent radially polarized beams in turbulent atmosphere are studied. The analyses indicate that the spatial correlation of the apertured partially coherent radially polarized beams are more affected by the atmospheric turbulence with the larger structure constant, the smaller truncation parameter, the larger coherence length, and the farther propagation distance.     

Beam spreading of truncated Gaussian beams in Kolmogorov turbulence

Based on the extended Huygens-Fresnel principle and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian function, analytical expression for average intensity of truncated Gaussian beam in Kolmogorov turbulence is derived, and some limiting cases are discussed. The influences of many factors, such as Fresnel number, wavelength, truncation parameter and structure constant on beam spreading are studied with the help of the average intensity formula. We found that the peak value of average intensity decreases and the beam spot spreads with the decrease of Fresnel number. The change of peak intensity against Fresnel number is slower with large aperture than that with small aperture. When Fresnel number is not small enough the influence of turbulence on intensity profiles is so small that can be neglected if structure constant is small.