cos-Gaussian Beams Research Articles

Evolution of Cos–Gaussian Beams in the Periodic Potential Optical Lattice

The evolution of Cos−Gaussian beams in periodic potential optical lattices is theoretically and numerically investigated. By theoretical analysis, a breathing soliton solution of the Gross–Pitaevskii equation with periodic potential is obtained, and the period of the breathing soliton is solved. In addition, the evolution of Cos−Gaussian beams in periodic potential optical lattices is numerically simulated. It is found that breathing solitons generate by appropriately choosing initial medium and beam parameters. Firstly, the effects of the initial parameters of Cos−Gaussian beams (initial phase and width) on its initial waveform and the propagation characteristics of breathing soliton are discussed in detail. Then, the influence of the initial parameters (modulation intensity and modulation frequency) of a photonic lattice on the propagation characteristics of breathing solitons is investigated. Finally, the effects of modulation intensity and modulation frequency on the width and period of the breathing soliton are analyzed. The results show that the number of breathing solitons is manipulated by controlling the initial parameters of Cos−Gaussian beams. The period and width of a breathing soliton are controlled by manipulating the initial parameters of a periodic photonic lattice. The results provide some theoretical basis for the generation and manipulation of breathing solitons.

Propagation Properties of an Astigmatic Cos-Gaussian Beam through Turbulent Atmosphere

We use the extended Huygens-Fresnel integral to investigate the propagation properties of a cos-Gaussian beam (cosGB) with astigmatism in atmospheric turbulence. The intensity distribution behaviour along the propagation distance for an astigmatic cosGB in atmospheric turbulence are analytically and numerically demonstrated. Some novel phenomena are presented graphically, indicating that the intensity distribution and the on-axial intensity closely depend on the astigmatic parameter and the turbulent structure constant of the cosGBs in the atmospheric turbulence.

Effects of underwater turbulence on average transmittance of cos-Gaussian and cosh-Gaussian optical beams

The impacts of underwater turbulence on the average optical transmittance are investigated when optical cosine-Gaussian and optical cosine-hyperbolic-Gaussian (cosh-Gaussian) beam propagate in weakly turbulent underwater medium. Findings obtained by using MATLAB reveal that when the displacement parameter is smaller (larger) for the cos-Gaussian (cosh-Gaussian) beam, the average transmittance increases. Also, an increase in the source size results in elevated average transmittance values. Rate of dissipation of kinetic energy per unit mass of fluid is found to be directly proportional to the average transmittance. However, the rate of dissipation of the mean-squared temperature is inversely proportional to the average transmittance. Average transmittance is found to be essentially the same when the temperature-induced optical turbulence is dominant. An increase in the salinity-induced optical turbulence suddenly reduces the average transmittance of the cos-Gaussian and cosh-Gaussian beam in underwater turbulence. Finally, it is observed that for the same sized beams in the same underwaterturbulent medium, cosh-Gaussian beam will have larger average transmittance than the corresponding cos-Gaussian beam. As the influence and value of our research conclusion on the development of our research, we comment that cosh-Gaussian beams can be preferred over the cos-Gaussian and Gaussian beams in obtaining larger average transmittance in underwater turbulence.

Propagation of annular cos-Gaussian beams through turbulence.

The propagation characteristics of annular cos-Gaussian beams and Bessel-Gaussian beams in turbulence have been widely investigated. However, the expression of the average intensity of Bessel-Gaussian beams involves a complex double integral, which requires large computational resources and does not easily ensure accuracy. Under this background, through the general cos-Gaussian beam rotation, a new annular cos-Gaussian beam is proposed which not only has some properties of the general cos-Gaussian beam, but also properties similar to those of the Bessel-Gaussian beam. Moreover, the annular cos-Gaussian beam is simpler in form than the general one and much more time-saving in calculation than the Bessel-Gaussian beams. Based on the extended Huygens-Fresnel principle, the average intensity on a receiver plane of the annular cos-Gaussian beams can be simplified to a summation of several polynomials and it is in good agreement with that of low-order Bessel Gaussian beams. The average radius and the power in the bucket are also derived. In summary, the annular cos-Gaussian beam has promising applications for its quick calculation and simple form among annular beams and hollow beams.

Propagation and interaction of cos-Gaussian beams in photorefractive crystals

Investigate numerically the propagation and interaction of cos-Gaussian beams in a biased photorefractive crystal by the finite difference method. The results show that the single cos-Gaussian beam can evolve into Y-type breathing solitons when the self-focusing nonlinearity is small, and the soliton properties can be controlled by adjusting the nonlinear parameter or cos modulation parameter. The distance between two components of Y-type breathing solitons will decrease with increasing the nonlinear parameter or decreasing the cos modulation parameter. The breathing soliton with two weak sidebands can form when the self-focusing nonlinearity is big. Moreover, two internal components of two cos-Gaussian beams have obvious interaction but two outside components have tiny interaction.

Gyrator transform of generalized sine-Gaussian beams and conversion an edge-dislocation into a vortex

Abstract We proposed a scheme that converts a sine-Gaussian beam with an edge dislocation into a dark hollow beam with a vortex. Based on the gyrator transform (GT) relation, the closed-form field distribution of generalized sine-Gaussian beams passing through a GT system is derived; the intensity distribution and the corresponding phase distribution associated with the transforming generalized sine-Gaussian beams are analyzed. According to the numerical method, the distributions are graphically demonstrated and found that, for appropriate beam parameters and the GT angle, dark hollow vortex beams with topological charge 1 can be achieved using sine-Gaussian beams carrying an edge dislocation. Moreover, the orbital angular momentum content of a GT sine-Gaussian beam is analyzed. It is proved that the GT retains the odd- or even-order spiral harmonics structures of generalized sine-Gaussian beams in the transform process. In particular, it is wholly possible to convert an edge dislocation embedded in sine-Gaussian beams into a vortex with GT. The study also reveals that to obtain a dark hollow beam making use of GT of cos-Gaussian beams is impossible.

Scintillation and BER for optimum sinusoidal Gaussian beams in weak non-Kolmogorov turbulence

The scintillation index and the average bit error rate (BER) are evaluated for the optimum sinusoidal Gaussian beams in weak non-Kolmogorov turbulence. The beam parameters that minimize the scintillation index and the average BER are stated and such beams are denoted as the optimum beams. For the collimated Gaussian, cos- and cosh-Gaussian beams, the scintillations increase as the power law exponent, α increases. Cos- and cosh-Gaussian beams that have larger absolute displacement parameters are found to exhibit larger scintillations especially at small α. Larger focal length and larger source size of cos-Gaussian beams induce reduction in the scintillations. When the propagation distance is large, the power law exponent is small and the source size is large, the scintillations of the optimum beams tend to decrease. Small power law exponent and large source size reduce the average BER. The optimum beam is shown to exhibit the smallest average BER for any α.

EVOLUTION OF COS-GAUSSIAN BEAMS IN A STRONGLY NONLOCAL NONLINEAR MEDIUM

The dynamical properties of cos-Gaussian beams in strongly nonlocal nonlinear (SNN) media are theoretically investigated. Based on the moments method, the analytical expression for the root- mean-square (RMS) of the cos-Gaussian beam propagating in a SNN medium is derived. The critical powers that keep the RMS beam widths invariant during propagation in a SNN medium are discussed. The RMS beam width tends to evolve periodically when the initial power does not equal to the critical power. The analytical solution of the cos-Gaussian beams in SNN media is obtained by the technique of variable transformation. Despite the difierence in beam proflle symmetries and initial powers, a cos-Gaussian beam always transforms periodically into a cosh-Gaussian beam during propagation and the transformation between the two beams revives after a propagation distance.

Focusing properties of a general-type beam in turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, analytical propagation formulas for a general-type beam propagating through aligned or misaligned ABCD optical systems in turbulent atmosphere are derived. The derived formulas provide a convenient way for studying the focusing properties 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 in turbulent atmosphere. As an application example, the focused intensities of cos-Gaussian, Hermite-sine-Gaussian and flat-topped beams in turbulent atmosphere are studied numerically. Focal shift of a flat-topped beam in turbulent atmosphere is investigated. Effect of the misalignment of the thin lens on the focusing properties of a cos-Gaussian beam is also explored. Our results will be useful for the applications of the general-type beam in LIDAR systems and remote sensing operating in turbulent atmosphere, where optical elements such as aligned or misaligned thin lens are commonly encountered.

Scintillation behavior of cos, cosh and annular Gaussian beams in non-Kolmogorov turbulence

For a non-Kolmogorov spectrum, scintillation aspects of cos, cosh and annular Gaussian beams are investigated. The appropriate mathematical formulation is developed, the derived scintillation index is evaluated and its variation is plotted in graphs. We find that, when the values of the power coefficient of the spectrum are just above 3, low scintillation is encountered, then as the power coefficient is increased, rises will occur with a peak being reached around 3.21. From there onwards, scintillation will drop, as the power coefficient approaches a value of 5. For extreme off-axis positions, there will be slight increases in scintillation at high power coefficient values. At points near on-axis and when the beams have small width sizes, cosh Gaussian beam having a bigger displacement parameter will offer the lowest scintillation. At large width sizes, this advantage will switch to the side of the cos Gaussian beam. In this study, the variation of scintillation with other sources and propagation parameters is examined as well.

Minimization effects on scintillations of sinusoidal Gaussian beams in strong turbulence

Minimization effects on the on-axis scintillation index of cos Gaussian (cG) and coshGaussian (chG) beams are studied in strong turbulence. In our formulation, the unifiedsolution of the Rytov method, which imposes spatial filtering to extend the solution to thestrong turbulence regime, is applied. Our solution correctly reduces to the weak turbulencesinusoidal beam scintillations and the strong turbulence Gaussian beam scintillations. Theconditions to minimize the scintillations are found to be focused chG beams. Small scalescintillations mainly determine the overall scintillations of cG and chG beams instrong turbulence. In strong turbulence, increase in the source size decreases thescintillations of collimated cG beams but does not change the scintillations of focused cGbeams. Collimated cG beams having larger displacement parameters and large focallengths show smaller scintillations in the strong regime. Change in the displacementparameters for collimated and focused chG beams and the focal length of focusedchG beams do not considerably vary their scintillations in strong turbulence.

Scintillations of multiwavelength Gaussian, cos, cosh and annular Gaussian beams

We provide the scintillation formulation for a multiwavelength source. Within this context, the scintillation aspects of Gaussian, cos, cosh and annular Gaussian beams are investigated. For all situations examined, it is found that for a source comprising many wavelengths, there will be less scintillations as compared to a single wavelength source of the lowest wavelength and but the reverse will be true if the comparison is with respect to the single wavelength source of the highest wavelength. The same is observed at all propagation distances, source sizes, on-axis and off-axis positions considered. Additionally, it is seen that the scintillation characteristics of multiwavelength sources will follow similar trends of single wavelength sources. The analysis is based on the Rytov approximation, therefore our results are valid for conditions of weak atmospheric turbulence.

Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams inatmospheric turbulence

On-axis and off-axis scintillation indices of partially coherent collimated cos Gaussian andcosh Gaussian beams are evaluated on their propagation in a weakly turbulent atmosphere.In the coherent limits, our results are in quite good agreement with those for the existingRytov scintillation index of coherent cos Gaussian and cosh Gaussian beams,especially for very small and very large sized beams and at weaker turbulence levels.The on-axis scintillation index becomes lower as the degree of source coherencedecreases for both cos Gaussian and cosh Gaussian beams. Comparing to thepartially coherent Gaussian beam on-axis scintillations, partially coherent cosGaussian beam scintillations are generally lower and partially coherent cosh Gaussianbeam scintillations are higher. Scintillations of Gaussian beams are very closeto the scintillations of equal-power cos Gaussian and are quite similar to thescintillations of equal-power cosh Gaussian beams. Off-axis evaluations yield that thescintillations of both cos Gaussian and cosh Gaussian beams increase as the diagonaldistance from the origin increases, being valid for all degrees of partial coherence.

Intensity correlations of general type beam in weakly turbulent atmosphere

In weakly turbulent atmosphere, intensity correlations at the receiver plane are formulated for a general type optical beam. Evaluating our formula at the appropriate source parameters versus the diagonal distance, intensity correlations for cos Gaussian, cosh Gaussian, annular and flat-topped Gaussian beams are obtained. As compared to Gaussian beam, intensity correlations are found smaller for cos Gaussian and larger for cosh Gaussian beams. Intensity correlations of cos Gaussian, cosh Gaussian and flat-topped Gaussian beams become larger at large source sizes. Thicker annular beams and flatter flat-topped Gaussian beams show larger intensity correlations. Without generalizing, intensity correlations tend to increase when the link length and the structure constant decrease and the wavelength increases. Our results show that the intensity correlations are not only built up using random medium effects, but also using the diffraction pattern formed at the receiver plane for the specific incidence investigated. As a check point, for all source types and medium parameters, our evaluations indicate that intensity correlations approach zero at sufficiently large diagonal distances.

Propagation of a cos-Gaussian beam in a Kerr medium

The dynamic properties of cos-Gaussian beams in the presence of Kerr nonlinearity are investigated analytically and numerically using the nonlinear Schrödinger equation (NLS). Based on the moments method, evolution of a cos-Gaussian beam width in the root-mean-square (RMS) sense is obtained analytically. The beam propagation factors and the critical powers of the cos-Gaussian beams with a uniform wavefront are calculated. Using numerical simulation, it is found that although the RMS beam width broadens, the central parts of the beam give rise to an initial radial compression and a significant redistribution during propagation. The partial collapse of central parts of the beam is observed below the threshold for a global collapse. The cos-Gaussian beams eventually convert into cosh-Gaussian type beams in Kerr media with low initial power.

Propagation of a general-type beam through apertured aligned and misaligned ABCD optical systems

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, analytical formulae for the electric field of a general-type beam propagating through apertured aligned and misaligned ABCD optical systems are derived using the generalized Collins formulae, which provide a convenient way of studying 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 such optical systems. As numerical examples, the propagation properties of a cos-Gaussian beam through an apertured aligned or misaligned thin lens are studied.

First-order approximation in studying beam spreading of cosh-Gaussian and cos-Gaussian beam in Kolmogorov turbulence

With the help of extended Huygens–Fresnel principle and the expansion of the mutual coherence function in a Taylor series, the analytical expression of the first-order approximation for the average intensity of a cosh-Gaussian and cos-Gaussian beam in Kolmogorov turbulence has been obtained. Based on the analytical expression of the first-order approximation, the variation of average intensity in Kolmogorov turbulence has been examined. Comparing with numerical integration, the relative error due to the quadratic structure function and first-order approximation of the mutual coherence function is studied and discussed. It is shown that the analytical expression of the first-order approximation provides a more precise approach to study the beam spreading of the cosh-Gaussian and cos-Gaussian beam in Kolmogorov turbulence, no matter whether the effect of turbulence is weak or strong.

Beam wander characteristics of cos and cosh-Gaussian beams

Within the context of a general beam formulation, beam wander characteristics of cos and cosh-Gaussian beams are derived and numerically evaluated. In our graphs, the fundamental Gaussian beam is used as a benchmark for comparisons. The associated plots reveal that at small source sizes, a cos-Gaussian beam has the lowest beam wander, while this property is enhanced with increasing values of the displacement parameter. At large source sizes however, this advantage is taken over by cosh-Gaussian beam. Joint examination against the changing source sizes and propagation lengths shows that the range of source sizes, where the beam wander of cos-Gaussian beam remains lower, is enlarged as we go toward higher propagation lengths. Asymmetric beams tend to exhibit higher beam wanders both at small and large source sizes, but for the intermediate source size ranges, the beam wanders of asymmetric beams will fall below those of the symmetric beams. Explanations concerning these behaviors are offered. A historical account of beam wander formulation is also included.

Bit error rates for general beams

In order to analyze the effect of beam type on free space optical communication systems, bit error rate (BER) values versus signal-to-noise ratio (SNR) are calculated for zero order and higher order general beam types, namely for Gaussian, cos-Gaussian, cosh-Gaussian, and annular beams. BER analysis is based on optical scintillation using log-normal distribution for the intensity, which is valid in weak atmospheric turbulence. BERs for these beams are plotted under variations of propagation length, source size, wavelength of operation, and order of the beam. According to our graphical outputs, at small source sizes and long propagation distances, the smallest BER value is obtained for the annular beam. On the other hand, at large source size and small propagation distance, the smallest BER value is obtained for the cos-Gaussian beam. Moreover, our study of the order of the beam shows that higher order beams have lower BER values than the zero order beams at longer propagation distances. But this drop compared with the order seems to be incremental.

Scintillation index of higher-order cos-Gaussian, cosh-Gaussian and annular beams

In this paper, via the generalized beam formulation, we evaluate the scintillation index for higher-order general beams propagating through the weak atmospheric turbulence. The investigated higher-order beam types are cos-Gaussian, cosh-Gaussian and annular beams. The scintillation indices of these beams are plotted with respect to propagation length, source size and order of the beam. According to our graphical outputs, higher-order beams have less on-axis scintillation index than zero-order beams at longer propagation distances. The on-axis scintillation exhibits a slight increase when the order of even-ordered beams is made higher. Moreover, our study on the source size variation shows that, for most of the source size range, cos-Gaussian beams have a lower scintillation index than the other beams.