Gaussian Schell-model Array Beams Research Articles

Propagation of Gaussian Schell-Model Array Beams through a Jet Engine Exhaust

Here a Gaussian Shell Model Array (GSMA) beam is used to investigate the propagation characteristics in the jet engine exhaust region. It has great significance to improve various optical systems for wide application in trapping cold atoms, creating gratings, and atmospheric optical communication. We calculate analytical formulas for the spectral density (SD) and the propagation factors Mx2 and My2 of a GSMA beam. The influence of inner scale of turbulence in the jet engine exhaust region on its power spectrum has been also analyzed. According to these results, the influence of turbulence in a jet engine exhaust on a GSMA beam has been reduced by changing the parameters of light source and turbulence. For example, it is an excellent tool for mitigation of the jet engine exhaust-induced anisotropy of turbulence to increase the source coherence length, the root-mean-squared (rms) beam width, the wavelength or reduce the outer scale of turbulence.

The Combined Effect of Jet and Atmospheric Turbulence on the Propagation of Airborne Partially Coherent Array Beams

AbstractThe exhaust from engines close to the fuselage can cause extreme turbulence and have a significant impact on the performance of airborne laser systems. In this paper, the combined effect of the jet engine and atmospheric turbulence on propagation of a Gaussian Schell‐model (GSM) array beam over long distances is studied in detail. Based on the two‐step propagation method, the analytical expressions of the cross‐spectral density (CSD) function, mean‐squared beam width and spectral degree of spatial coherence (DOC) of GSM array beams are derived. It is shown that the jet engine turbulence has a significant effect on the beam broadening even at a very short distance. As the propagation distance in the jet engine turbulence increases, the beam width is less effected by the spatial coherence. Meanwhile, the resistance of lower spatially coherent beam to the turbulence increases as the distance of the jet engine turbulence or the strength of the atmospheric turbulence increases. It is demonstrated that the effect of jet engine turbulence on the beam spreading can be equivalent to moderate or even severe atmospheric turbulence. Moreover, the DOC is a function with oscillatory phenomenon when the propagation distance is short or the strength of turbulence is low.

Properties of the Rotation and Mergence of Twisted Gaussian Schell Model Array Beams Propagating in Turbulent Biological Tissues

In this paper, an analytical expression for describing propagation properties of twisted Gaussian Schell model array (TGSMA) beams through turbulent biological tissues is derived based on the extended Huygens Fresnel integral. With the help of the formulae, properties of the rotation and mergence for the TGSMA beams in turbulent biological tissues are researched in detail. It is found that the TGSMA beams go through the distinct mergence period in the far field besides phenomena of abruption and rotation in the near field, and turbulent biological tissues play a dominated role in mergence of the TGSMA beams. These novel results may be helpful in optical trapping.

Propagation of twisted EM Gaussian Schell-model array beams in anisotropic turbulence.

The behavior of the twisted electomagnetic (EM) Gaussian Schell-model array beams in anisotropic random turbulence is investigated. An example illustrates that a twisted EM source can produce lattice-like patterns in degree of polarization with rotation or not, which depends on the setting of the initial twist phase. One also finds that the anisotropy of the medium leads to an anisotropic beam spreading, and we can effectively limit such turbulence-induced effects by optimizing the initial twist and source correlation widths. Moreover, after transmitting through the turbulence for sufficiently long distances, the intensity and coherence are mainly affected by turbulence statistics; however, for the case of polarization, the initial twist plays a dominant role in determining its distribution profile.

Statistical properties of twisted Gaussian Schell-model array beams in anisotropic ocean

The effect of anisotropic oceanic turbulence on propagation characteristics of twist Gaussian Schell-model array beams is investigated. The analytical expression for the cross-spectral density function of such beam propagating through turbulent ocean is derived and used to explore the evolution behavior of the spectral intensity and the degree of coherence. An example illustrates the fact that, the most important oceanic effect is destroying the far zone lattice-like beam profile, and the beam shape eventually converges into Gaussian profile. Besides, after passing through the turbulence at sufficiently large distances from the source, the twist phase modulation of the second-order statistical properties is shown to be suppressed by the turbulence fluctuation, the spectral intensity evolves in the same way with different twist strength, and so is the case for the degree of coherence. The impacts, which arise from the source coherence and the turbulence parameters on beam characteristics are analyzed in detail.

Statistical properties of electromagnetic twisted Gaussian Schell-model array beams during propagation.

An analytical model for electromagnetic twisted Gaussian Schell-model array (EM TGSMA) beams is introduced. We derive the analytical expression for the cross-spectral density matrix (CSDM) of such beam propagating in free space and investigate the spectral density, degree of polarization (DOP) and degree of coherence (DOC) in detail. Twist effects of the evolutions of DOC and DOP during propagation are also demonstrated. It is found that the twist effects of DOC and DOP can be influenced by the beam width and coherence length. In particular, senses of rotation can also be controlled by adjusting the twist strength. Furthermore, we find that twist phase always causes DOC to rotate in an opposite direction compared with that of DOP. Our results might be beneficial for free-space communications of the partially coherent beams endowed with twist.

Evolution properties of a Gaussian Schell-model Array beam in a uniaxial crystal orthogonal to the optical axis

The analytical formulas of the Gaussian Schell-model Array (GSMA) beams propagating in a uniaxial crystal orthogonal to the optical axis are derived. The evolution properties of the spectral density, the spectral degree of coherence (SDOC), and the effective beam width of the GSMA beams in a uniaxial crystal are investigated in detail. It is shown that a GSMA beam exhibits splitting property inside a uniaxial crystal, and the propagation properties are closely related with parameters of the uniaxial crystal and beam parameters. The results provide a way for the modulation of the GSMA beam.

Propagation properties of Gaussian Schell-model array beams in non-Kolmogorov turbulence

A stochastic beam generated by a recently introduced Gaussian Schell-model array (GSMA) source (2015 Opt. Lett. 40 5662) is investigated. We derive the analytical propagation formulae for the spectral density and the propagation factor in non-Kolmogorov turbulence by utilizing the extended Huygens–Fresnel principle and second-order moments of the Wigner distribution function. Numerical results show that the lattice patterns of GSMA beams, which keep propagation-invariant in free space, are destroyed by the turbulence at sufficiently large distances. The GSMA beams have significant advantage over the Gaussian Schell-model beam in the robustness of the destructive effect of non-Kolmogorov turbulence, especially for the GSMA beam with more lattice elements and bigger relative separation distance of each lattice element. The effects of beam parameters and non-Kolmogorov turbulence on the propagation factor are analyzed in detail.

Propagation of Gaussian Schell-model Array beams in free space and atmospheric turbulence

Based on the extended Huygens-Fresnel principle, the evolution behavior of the spectral density and the spectral degree of coherence of the beam produced by a recently introduced novel class of Gaussian Schell-model Arrays (GSMA) source in free space and turbulence atmospheric are explored and comparatively analyzed. And the influence of the fractal constant of the atmospheric power spectrum and refractive-index structure constant on the spectral density and the spectral degree of coherence of beams are analyzed. It is shown that the optical lattice profile is stable when beams propagate in free space, but the spectral density eventually is suppressed and transformed into a Gaussian profiles when it passes at sufficiently large distances through the turbulent atmosphere. The distributions of the spectral degree of coherence in far field eventually transformed into a shrink Gaussian profile relative to free space which means that the degree of spatial coherence turns worse.

Beam wander of partially coherent array beams through non-Kolmogorov turbulence.

Based on the theory of second moments and non-Kolmogorov spectrum, the beam wander theory is extend to non-Kolmogorov turbulence, the general analytical expression of beam wander in non-Kolmogorov turbulence is derived. Beam wander depends on the non-Kolmogorov turbulence parameters and the initial second moments of the laser beam at the input plane. Taking the Gaussian Schell model array beams as an example, the effects of the generalized exponent parameter, inner scale, and outer scale of non-Kolmogorov turbulence and the beam separation distance, beam number, and coherence degree on the beam wander are studied in detail. It has been shown that the beam wander varies non-monotonically with increasing generalized exponent parameter of the turbulence. Furthermore, it increases as the inner scale decreases or outer scale increases, and decreases as the beam separation distance and beam number increase and the coherence of the beam becomes weaker. Our results also indicate that the beam wander could be reduced by adjusting the beam parameters appropriately.

Intensity distribution and coherence property for the broadband Gaussian Schell-model array beams in free space

The intensity distribution and the equal-time complex degree of coherence for the broadband Gaussian Schell-model array beams with different bandwidth propagating in free space are analyzed by the numerical calculation. Results show that the intensity distribution for the broadband Gaussian Schell-model array beams by uncorrelated superposition is rarely affected by the bandwidth; the intensity distribution in the far field for the broadband Gaussian Schell-model array beams by correlated superposition is affected by the bandwidth more obviously than that in the near field. With the increasing of the propagation distance, differences in the equal-time complex degree of coherence for the broadband Gaussian Schell-model array beams with different bandwidth become prominent.

Range of turbulence-negligible propagation of Gaussian Schell-model array beams

For linear Gaussian Schell-model (GSM) array beams, the range of turbulence-negligible propagation, in which all of the spatial and angular spreading and the beam propagation factor increasing due to turbulence can be neglected, has been investigated in detail. It is shown that this range of GSM array beams increases with decreasing turbulent parameter and coherent parameter, and depends on the beam number, the waist width, and the relative beam separation distance. This range of a GSM array beam is larger than that of a coherent Gaussian array beam, and this range of a GSM array beam with a large relative beam separation distance is larger than that of a single GSM beam, implying that a GSM array beam may be more appropriate to be used in atmospheric optical communication links than a coherent Gaussian array beam or a GSM beam.

Theoretical study on the spectral shifts of partially coherent array beams passing through ABCD optical systems

According to the extended Huygens–Fresnel integral, the expressions for the on-axis spectrum of partially coherent Gaussian Schell-model (GSM) rectangular array beams passing through ABCD optical systems have been derived. The generalized Fresnel number of the system, the spatial coherent parameter of array beamlets and the array beam parameters including the number of beamlets and the separation distance between beamlets have been taken as the characteristic parameters to compare the spectral shifts of GSM array beams for the two types of the superposition, i.e. the correlated superposition and the uncorrelated superposition. In particular, the effect of characteristic parameters on the on-axis relative spectral shifts has been discussed in detail. The results show that the spectral intensity of GSM array beams for the two types of the superposition passing through ABCD systems depends on the source spectral density S 0(ω), the spatial coherent parameter of array beamlets β, the generalized Fresnel number of the system F and the array beam parameters. Furthermore, for the uncorrelated superposition, the spectrum of GSM array beams only exhibit the blue-shift, whereas for the correlated superposition, GSM array beams exhibit the spectral switch and the number of spectral switches increases with the increase of array beam parameters in the near field due to the interference between beamlets. In particular, the effect of the array beam parameters on the on-axis relative spectral shift is more obvious.

Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams

The analytical expression for the beam propagation factor ( M 2-factor) of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the M 2-factor of GSM array beams depends on the beam number, the relative beam separation distance, the beam coherence parameter, the type of beam superposition, and the strength of turbulence. The turbulence results in an increase of the M 2-factor. However, for the superposition of the intensity the M 2-factor is less sensitive to turbulence than that for the superposition of the cross-spectral density function. The M 2-factor of GSM array beams is larger than that of the corresponding Gaussian array beams. However, the M 2-factor of GSM array beams is less affected by turbulence than that of the corresponding Gaussian array beams. For the superposition of the cross-spectral density function a minimum of the M 2-factor of GSM array beams may appear in turbulence, which is even smaller than that of the corresponding single GSM beams.

Beam quality of radial Gaussian Schell-model array beams

Taking the Rayleigh range z R and the M 2-factor as the characteristic parameters of beam quality, the beam quality of radial Gaussian Schell-model (GSM) array beams is studied. The analytical expressions for the z R and the M 2-factor of radial GSM array beams are derived. It is shown that for the superposition of the cross-spectral density function z R is longer and the M 2-factor is lower than that for the superposition of the intensity. For the two types of superposition, z R increases and the M 2-factor decreases with increase in beam coherence parameter, and both z R and the M 2-factor increase with increase in inverse radial fill-factor. For the superposition of the cross-spectral density function, z R increases and the M 2-factor decreases with increase in beam number, while for the superposition of the intensity both the z R and M 2-factor are independent of the beam number.

Turbulence distance of radial Gaussian Schell-model array beams

The effect of turbulence on the spreading of radial Gaussian Schell-model (GSM) array beams is studied quantitatively by examining the mean-squared beam width. The analytical expression for the turbulence distance z T of radial GSM array beams is derived by using the integral transform technique, which indicates within what ranges radial GSM array beams will be less affected by turbulence. It is shown that the effect of turbulence on the spreading of radial GSM array beams can be reduced by choosing the suitable array beam parameters and the type of the beam superposition. In addition, a comparison with the previous work is also made.

Spatial correlation properties and the spectral intensity distributions of focused Gaussian Schell-model array beams

The spatial correlation properties and the spectral intensity distributions of focused Gaussian Schell-model (GSM) array beams are studied in detail. The closed-form expressions for the spectral degree of coherence and the spectral intensity of focused GSM array beams are derived. It is shown that the spectral degree of coherence of focused GSM array beams is the same as that of focused GSM beams in the focal plane. On the other hand, it is found that, in the focal plane the spectral intensity distribution of focused GSM array beams is the fringe pattern when the value of the coherence length is small. However, it becomes one peak located at the center as the value of the coherence length is large enough. In the focal plane, the spectral intensity maximum increases and the width of the normalized spectral intensity distribution decreases as the beam number increases. In general, for GSM array beams, the width of the modulus of the spectral degree of coherence in the focal plane always exceeds that of the normalized spectral intensity distribution, which is different from the behavior of focused GSM beams. In addition, the power in the bucket (PIB) and the beam propagation factor ( M 2 factor) are also discussed. The main results are explained physically.

The far-field divergence angle and the far-field radiant intensity distribution of Gaussian Schell-model array beams

The analytical expressions for the mean-squared beam width, the far-field divergence angle and the far-field radiant intensity of Gaussian Schell-model （GSM） array beams are derived, and the condition under which GSM array beams have the same far-field divergence angle as a single Gaussian beam. It is shown that GSM array beams and a Gaussian beam which have the same far-field divergence angle may have different far-field radiant intensity distribution, which is quite different from the behavior of GSM beams.

Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence

The closed-form expression for the angular spread of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the angular spread θ sp of GSM array beams for the superposition of the cross-spectral density function is smaller than of those for the superposition of the intensity. However, the θ sp of GSM array beams for the superposition of the intensity is less sensitive to turbulence than that for the superposition of the cross-spectral density function. For the superposition of the cross-spectral density function, θ sp of GSM array beams with smaller coherence length σ 0, smaller waist width w 0, smaller beam number N, and larger separation distance x d are less affected by turbulence than of those with larger σ 0,w 0,N, and smaller x d ; while, for the superposition of the intensity, the effect of turbulence on θ sp is independent of N and x d . In addition, the angular spread is nearly the same for the two types of superposition when σ 0 or w 0 is small enough, or x d is large enough. On the other hand, it is found that there exist equivalent GSM array beams for the two types of superposition which may have the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence.