Statistical characteristics of the orbital angular momentum (OAM) of a Laguerre-Gaussian laser beam propagating through a turbulent atmosphere have been studied in the Monte Carlo numerical simulation. We have analyzed variations in the probability density of OAM versus its initial value, turbulent conditions along the propagation path, the diffraction parameter of the beam, and the receiving aperture size. This distribution is shown to be symmetrical near-Gaussian for an infinite receiving aperture and to be well approximated by an Edgeworth series. The probability density function significantly changes for finite-size apertures. The range of sizes of the receiving aperture has been found where fluctuations of the moment of energy flux density and fluctuations of light flux recorded can be considered uncorrelated.
We study how to control the orbital angular momentum, the intensity distribution, and the coherence length of a laser beam based on fiber array coherent combining. The features of the formation of nonzero orbital angular momentum of the beams when the controls the piston phase shift of the fiber array were theoretically and experimentally studied. A new method for controlling the distribution of the intensity of a light field synthesized on the basis of fiber array coherent combining were proposed. The possibilities of controlling the degree of spatial coherence of the synthesized laser beam were studied theoretically and experimentally.
The intermediate results of the experiment on vortex laser beam synthesis based on coherent combining laser subbeams formed by a fiber collimators array are presented. The wave field results from interference of individual subbeams in this case and is determined by the phase relationships between them. A functional diagram of the experimental setup is given. The experimental results of synthesis of a laser beam with an orbital angular momentum are described. It is suggested to use a spiral phase plate to determine the sign of the topological charge of an optical vortex; the efficiency of this approach is estimated.
The distribution laws of fluctuations of the field amplitude and intensity of laser beams propagating through the atmosphere are studied based on numerical simulation. The calculations are performed for the beams having the orbital angular momentum (vortex beams) at different points of their cross section. The transformation of probability densities of fluctuations of the field amplitude and intensity of the beams is studied for the relative variance of intensity fluctuations (scintillation index) varying from unity to three. It is shown that if the scintillation index has the same values at different points of the beam, the probability density of intensity has the same form, whereas the probability density of the real and imaginary parts of the field amplitude can differ.
Statistical parameters of laser beams counter propagating in the atmosphere are analyzed on the basis of numerical simulation, as well as a possibility of implementing the reciprocity principle of the counter propagation of optical waves from point sources with the aim of cryptographic keying with the help of finite-size receiving-transmitting apertures. The dependences of the degree of correlation of signals from coupled wireless optical communication laser systems on the distance and aperture sizes are studied. The quantitative dependence of the aperture size on the space between the laser systems is derived with the aim of maintaining the correlation degree at a level above 0.9999.
The efficiency of using an Airy axisymmetric beam for energy transfer and concentration to a given point in a randomly inhomogeneous medium is studied. Capabilities of this beam are assessed in comparison with beams of other types. The variance of beam wandering is calculated. For the given parameters of the transmitting system, path length, and degree of turbulent distortions, the intensity at the axis of the considered Airy beam is shown to be much lower than that of other beams, the Airy beam is compared with. In addition, the Airy beam is characterized by smaller wandering.
The generation of optical beams with a possibility of quick variations in the orbital angular momentum (OAM) and the degree of spatial coherence is shown in laboratory experiments. The methods for OAM and coherence control are based on the phase control in the fiber array optical channels. The approach suggested allows one to change the OAM (the topological charge of a vortex beam) with a high speed determined by the phase shifter operation speed. The generation of a vortex beam is shown for six coherent Gauss-like beams arranged in a circle and having a constant phase shift between neighboring beams, providing the total phase shift equal to 2π around the circle. It is shown that the far field is characterized by an annular intensity distribution and a spiral-like distribution of the Poynting vector. In addition, the features of the OAM and the topological charge of a fiber-array-based vortex beam in a homogeneous medium are investigated in numerical experiments. The method for controlling the length of spatial coherence of the beam synthesized is based on introducing pseudo-random phase fluctuations in a fiber array. The value of the coherence length which exceeds the subbeam size is set by the correlation function of pseudo-random phase fluctuations in neighboring subbeams. The value of the coherence length smaller than beamlet size is set by the divergence of the subbeams with delta correlated phase fluctuations. The effect of the number of pseudo-random realizations of the fiber array phase on the average intensity distribution is studied in the laboratory experiments. The influence of the spatial coherence of the laser beam on the bit error rate of FSO communication systems in a turbulent atmosphere is studied theoretically.
The dependences of the orbital angular momentum of a Laguerre–Gaussian beam propagating through a turbulent atmosphere on its azimuthal index and atmospheric turbulence strength are analyzed in the numerical simulation. The effect of errors in alignment of the beam and receiver axes on the average value and variance of orbital angular momentum fluctuations is studied. The statistics of the orbital angular momentum distribution at the end of the atmospheric propagation path is analyzed.
Statistical characteristics of vortex beams propagating in the atmosphere are analyzed on the basis of numerical simulation. The probability density function of the intensity fluctuations are compared for common vortex beams and vortex beams synthesized by combining a fiber laser array. It is shown that the intensity fluctuations are lognormally distributed on the axis of a Gaussian beam when radiation is propagating through a turbulent atmosphere under weak fluctuations (the Rytov parameter is much lower than unity), while the probability density function of the intensity fluctuations at the axes of vortex beams is well approximated to the exponential distribution typical for strong fluctuations (the Rytov parameter is much higher than unity). This function is well approximated to a gamma distribution in the cases where the scintillation index is lower than unity independently of the beam type and observation point position relative to the propagation axis.
In this study, we predict qualitatively the random wander of the Airy vortex beam propagating in a statistically homogeneous turbulent atmosphere based on the earlier revealed regularity [5] that the variances of random wander of vortex laser beams propagating in the turbulent atmosphere and having identical orbital angular momenta and identical transverse size in the source plane are different, if the effective volumes occupied by the beams in the homogeneous medium are different. The larger is the volume occupied by the beam in space, the smaller is the variance of the beam wander.
We suggest a technique for generation of optical vortex beams with a variable orbital angular momentum based on a fiber laser array. The technique uses the phase control of each single subbeam. Requirements for the number of subbeams and the spatial arrangement for the vortex beam generation are determined. The propagation dynamics of a vortex beam synthesized is compared with that of a continuous Laguerre–Gaussian beam in free space and in a turbulent atmosphere. Spectral properties of a beam synthesized, which is represented as a superposition of different azimuth modes, are determined during its free-space propagation. It is shown that energy and statistical parameters coincide for synthesized and continuous vortex beams when propagating through a turbulent medium. Probability density functions of the beam intensity fluctuations are well approximated to a gamma distribution in the cases where the scintillation index is lower than unity independently of the beam type and observation point position relative to the propagation axis.
Numerical simulation and analytical calculations of the variance of fluctuations of the total orbital angular momentum (OAM) of Laguerre—Gaussian and Gaussian laser beams propagating in the randomly inhomogeneous atmosphere have been carried out. It is shown that as such beams propagate in the weakly turbulent atmosphere, the relative variance of OAM fluctuations remains much smaller than the relative variance of intensity fluctuations.
A technique for generation of optical vortex beams with a variable orbital angular momentum on the basis of a fiber laser array is suggested. The technique is based on controlling the radiation phase of individual subapertures of the array. Requirements for parameters of a device for vortex beam generation are determined (the number and size of subapertures and their arrangement). The propagation dynamics of a vortex beam synthesized is compared with that of a continuous vortex beam in free space and in a turbulent atmosphere. Spectral properties of a beam synthesized, which is represented as a superposition of different azimuth modes, are determined during its free-space propagation. It is shown that energy and statistical parameters coincide for synthesized and continuous vortex beams when propagating through a turbulent medium.
Statistical characteristics of the vortex laser beams propagating in the turbulent atmosphere are analyzed. The vortex beams are appeared to be less affected by turbulence as compared with the conventional beams: for the vortex beam the variance of random displacements and the variance for fluctuations of the orbital angular momentum, which are induced by atmospheric turbulence less than these characteristics for the vortexless beam, beams having a larger topological charge are founded to be more stable.
A new approach is suggested to the numerical simulation of laser beam propagation through large-scale atmospheric inhomogeneities. Within this approach, the propagation medium is represented as a continuous 3D distribution of largescale inhomogeneities and a set of common 2D phase screens. A new highly effective simulation method with accounting for the effect of large-scale inhomogeneities is suggested within the aberration-free approximation. It is shown that experimentally observed large-scale inhomogeneities can strongly affect long-distance radiation propagation.
Compensation of turbulence in atmospheric imaging is usually hampered by the lack of a reference point source. We
consider the possibility of phase compensation based on the reference wave scattered by the rough surface of an object
under review. Analysis has been made on the basis of numerical simulation. The recently developed method for
incoherent imaging through the atmosphere in anisoplanatic turbulent conditions allows simulating both long-exposure
and short-exposure images. We analyze the influence of the size of object reflecting reference wave on the efficiency of
phase compensation. Image quality enhancement is observed for the object size significantly exceeding the diffractivelimited
size with respect to the receiving aperture. Also we consider the features of phase compensation with the
reference wavelength different from visible light ones.
Modern laser tracking systems use digital video cameras for imaging the tracked objects in atmosphere. Using images
from camera for measurement lead to a measurement error of object coordinates and all other related parameters due to
atmospheric turbulence and thermal blooming of laser beam which track the object.
In our investigation the turbulent and thermal blooming features of object coordinates measurement error are considered.
The estimations in the conditions of mutual action of distorting atmospheric effects mentioned above are generated.
The calculation of error using our estimations in most cases became two orders faster. Modeling is performed for the
case of strong turbulent fluctuations as well as for the case of weak turbulence.
We present the development of a novel technique for numerical simulation and analysis of wide field-of-view (FOV)
incoherent and anisoplanatic imaging of an object through volume turbulence. This technique is based on the recently
developed brightness function method [J. Opt. Soc. Am. A, v. 22, p. 126 (2005)]. We present computer simulation
results demonstrating the anisoplanatic turbulence effects on an object image quality.
The error caused by atmospheric turbulence, in determining the orientation angle of an object (a series of reflectors) has
been studied. The orientation angle was determined by studying the image of the object. Numerical modeling was
performed involving construction of the image of a series of reflectors as if they were observed through a turbulent
medium, calculation of the coordinates of reflector mass centers, finding of the line closest to the reflector mass centers,
and determination of its slope angle. Variance of the slope angle fluctuations is calculated.
The paper considers the error caused by atmospheric turbulence, in determining the motion speed of an object by using
its successive images recorded on a matrix of a digital camera. Numerical modeling of the image of a moving object in
successive time moments is performed. Fluctuation variance of the image mass centre affecting the measurement error
is calculated. Error dependences on the distance to the object and path slope angle are obtained for different turbulence
models. Considered are the situations, when the angular displacement of the object between two immediate shots of the digital camera is greater than the isoplanatism angle as well as the situations when the angular displacement is smaller
than this angle.
The paper suggests a model for predicting the behavior of the optical characteristics of a medium that determine the character
of laser beam propagation in the atmosphere. These are first of all the refractive index and the refractive-index-structure
parameter as well as the wind velocity, kinetic turbulence energy, dissipation energy speed and other meteorological
parameters that are important in the laser propagation problem. The prediction of the refractive index and the refractive-indexstructure
parameter is based on the known equations of the theory of wave propagation in a turbulent atmosphere and a
mathematical model for short-range forecast of evolution of local meteorology and turbulent structure of the atmospheric
boundary layer (ABL).
The problem of propagation in a turbulent atmosphere of the speckle field reflected from rough surface (the scattered
field) is considered. Two methods of numerical analysis of the averaged field characteristics are compared: the Monte-
Carlo method based on solution of a parabolic equation and the method of solving a transfer equation Fourier-conjugate
to the second order field momentum equation. It is demonstrated that the use of the transfer equation allows us to
considerably (up to several orders) shorten the time taken by numerical modeling, which is important for this type of
problems. It was found that in the case of frozen (stationary) turbulence, the extent of turbulence manifestation but
insignificantly affects the mean correlation field characteristics.
A new approach to formation of random turbulent screens is proposed. Simulation of temporal fluctuations of refractive index inhomogeneities helps in removing the spatial periodicity arising at the transversal displacement of the screen. This improvement allows the formation of random turbulent screens having an infinite extension. Some examples of the problems of atmospheric turbulent optics are presented along with their numerical solution obtained using the algorithm proposed. The problems under consideration involve simulating long (in time) observation series of characteristics of laser beams propagating through the atmosphere.
The energy and statistical characteristics of laser radiation reflected from an infinite surface in the form of an array of
single retroreflectors have been investigated. The study of the reflecting properties of such a surface involved the
calculation of the coherence function of the radiation in the reflection plane. Rigorous and high-accuracy approximate
expressions have been obtained for this characteristic.
The intensity in the far zone and the coherence function of the reflected radiation at an arbitrary distance from the
surface have been calculated. Approximate equations have been derived for these characteristics of the radiation. The
results of numerical simulation by the Monte Carlo technique have been compared with the rigorous and approximate
calculations. It has been shown that in the most significant cases the approximate equations proposed give a deviation
within 5% from the rigorous ones and from the results of averaging over numerical realizations.
The approximate equations obtained have been used to solve the problem of radiation propagation along sensing paths,
including the forward propagation through the turbulent atmosphere, reflection, and backward propagation.
Photon budget analysis for laser target tracking systems under atmospheric turbulence conditions is performed in the paper. This analysis includes evaluations of the effects of molecular and aerosol absorption/scattering at various propagation distances and tracking angles for the following laser tracking wavelengths: 0.53 micrometers, 1.06 micrometers and 1.55 micrometers and evaluations of tracking beam/target interaction (target light scattering, target-induced coherence degradation, influence of target shape) for targets with rough surfaces, retro-reflective tape, and a single retro-reflector.
Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related with maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing outgoing wave propagation, and the equation describing evolution of the mutual coherence function (MCF) for the backscattered (returned) wave. The resulting evolution equation for the MCF is further simplified by the use of the smooth refractive index approximation. This approximation enables derivation of the transport equation for the returned wave brightness function, analyzed here using method characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wavefront sensors that perform sensing of speckle-averaged characteristics of the wavefront phase (TIL sensors). Analysis of the wavefront phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric turbulence-related phase aberrations. We also show that wavefront sensing results depend on the extended target shape, surface roughness, and the outgoing beam intensity distribution on the target surface.
Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related with maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing outgoing wave propagation, and the equation describing evolution of the mutual intensity function (MIF) for the backscattered (returned) wave. The resulting evolution equation for the MIF is further simplified by the use of the smooth refractive index approximation. This approximation enables derivation of the transport equation for the returned wave brightness function, analyzed here using method characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wavefront sensors that perform sensing of speckle-averaged characteristics of the wavefront phase (TIL sensors). Analysis of the wavefront phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric turbulence-related phase aberrations. We also show that wavefront sensing results depend on the extended target shape, surface roughness, and the outgoing beam intensity distribution on the target surface.
The propagation of a partially coherent wave field in inhomogeneous media is investigated. The influence of linear and nonlinear refraction and refraction parameter fluctuations on target characteristics of radiation are taken into consideration. The exact solution expressed in quadrature is obtained for the partially coherent radiation propagating through a turbulent medium with the Kolmogorov spectrum of fluctuations and a parabolic distribution of the mean refractive index. On the basis of comparison with this solution the accuracy of the solution of the equation for a coherence function of the second order obtained by the effective ray-tracing technique and taking into account the influence of turbulence and a nonlinearity of a medium to the propagation of a radiation with an arbitrary initial coherence is investigated.
Investigations of the dynamics of turbulent characteristics were carried out for different tracking paths based on theoretical equations. A multi-screen model and single-screen one for turbulent atmosphere have been constructed for numerical simulation of laser beam propagation along atmospheric paths within the framework of the paraxial approximation. These models are suitable for simulation of the propagation along both homogeneous and inhomogeneous paths. Within this model, the Fried radius, the scintillation index, the effective beam radius, and the coherence length of radiation were calculated. The values obtained in the numerical experiment were compared with those calculated analytically.
Proc. SPIE. 5396, Tenth Joint International Symposium on Atmospheric and Ocean Optics/Atmospheric Physics. Part I: Radiation Propagation in the Atmosphere and Ocean
The diffraction of narrow sharply focused beam propagating through the free space and nonlinear medium was studied. The validity of two approximated approaches for analysis of narrow beam diffraction was studied on the base of comparison with the solution of Kirchgof-Helmholtz diffraction. Both of these approaches is based on the particular solution of Helmholtz equation, which is corresponded with propagation of unidirectional wave. In the first of them attenuating waves are taken into account for the case where the spatial spectrum width is greater than the wave number. The second one is based on the neglect of contribution from attenuating waves. Also, validity of a parabolic wave equation for these situations was researched.
Numerical results of a propagation of the CW laser radiation for extensive slightly elevated atmospheric paths are presented in the paper. The calculations are carried out in the limits of the small-angle equation for the coherence function of the second order describing the partially coherent beam self-action. To solve the equation a new ray-tracing technique creating an effective numerical algorithm was used. The gas absorption was taken into account in frameworks of the average seasonal summer model of an atmosphere. The average cyclic and the volcanic models of an atmosphere were used for aerosol attenuation. High-altitude vertical profiles of turbulent fluctuations of a refractive index were determined for the best, average and worst turbulent conditions. Calculations were obtained for the radiation sources located at altitudes from 10 Km up to 25 Km. Paths up to 500 Km were considered. Let's note, that it is necessary to take into account the influence of the Earth surface curvature for these paths.
To studying the problem of a minimization of an atmosphere influence on an energy transfer as a numerical criterion was selected the effective power density:
Ieff = P/Seff,
where P is a power of radiation at the reception plane, Seff is an effective square of a beam at the reception plane. The initial power, the initial beam focusing and the wavelength of radiation were optimized.
The propagation problem for partially coherent wave fields in inhomogeneous media is considered in this work. The influence of refraction, inhomogeneity of gain medium properties and refraction parameter fluctuations on target characteristics of radiation are taken into consideration. Such problems arise in the study of laser propagation on atmosphere paths, under investigation of directional radiation pattern forming for lasers which gain media is characterized by strong fluctuation of dielectric constant and for lasers which resonator have an atmosphere area. The ray-tracing technique allows us to make effective algorithms for modeling of a partially coherent wave field propagation through inhomogeneous random media is presented for case when the influecne of an optical wave refraction, the influence of the inhomogeiety of radiaitn amplification or absorption, and also the influence of fluctuations of a refraction parameter on target radiation parameters are basic. Novelty of the technique consists in the account of the additional refraction caused by inhomogeneity of gain, and also in the method of an account of turbulent distortions of a beam with any initial coherence allowing to execute construction of effective numerical algorithms. The technique based on the solution of the equation for coherence function of the second order.
Proc. SPIE. 5026, Ninth Joint International Symposium on Atmospheric and Ocean Optics/Atmospheric Physics. Part I: Radiation Propagation in the Atmosphere and Ocean
Test data of a propagation of the CW laser radiation for extensive weak sloping atmospheric paths are studied in this paper. The calculations are carried out in the limits of the small-angle equation for a coherence function of the second order describing the partially coherent beam self-action. The gas absorption was taken into account in frameworks of the average seasonal summer model of an atmosphere. The average cyclic and the colvanic models of an atmosphere were used for aerosol attenuation. High-altitude vertical profiles of turbulent fluctuations of a refractive index were determined for best, average and worst turbulent conditions. Calculations were obtained for radiation sources located at altitudes from 10 Km up to 25 Km. Paths up to 500 Km were considered. Let's note, that it is necessary to take into account an influence of the Earth surface curvature for these paths. The initial power, the initial beam focusing and the wavelength of radiation were optimized.
The beam dynamics for equal power ray tubes and equal coherence ray tubes is investigated for the partially coherent radiation propagating in non-linear media. The Gaussian beam propagation through media with Kerr non- linearity and thermal blooming is considered on the basis of the solution of the equation for coherence function of the second order. Calculations are adduced for the two- dimensional beam. The dimension of a coherence function decreases from five down to three for this case. Consequently, the numerical solution of this equation is possible by means of the method of a separation on physical factors, widely using for a solution of the parabolic wave equation, with using the algorithm of fast Fourier transformation. The comparable analysis of the equal power and equal coherence ray tubes behavior is carried out. The same analysis is carried out for solutions of the equation for coherence function obtaining by the ray-tracing technique allowing to create effective numerical algorithms for the three dimensional problem. The technique is asymptotically exact since it gives exact solutions at limiting cases when the coherence length tends to zero or when the distribution of complex dielectric constant of medium has the parabolic form.
The propagation problem of the partially coherent radiation in inhomogeneous media is investigated in this paper. The error of the ray-tracing technique for solving the equation for coherence function of the second order is estimated. The ray- tracing technique is asymptotically exact since it gives the exact solution in following limiting cases: within the limit of the geometrical optics ((lambda) yields 0), when the coherence length tends to zero, or when the distribution of the medium complex dielectric permeability has a parabolic form. To estimate an error of the ray-tracing technique for both coherence length and wave length not equal to zero and for an arbitrary distribution of the dielectric permeability the comparison of solutions obtained by the ray-tracing technique with exact ones is performed. The exact solutions are obtained for a model propagation problem of a two-dimensional slit beam for which the dimensionality of the equation decreases from five up to three. To solve the coherence function equation the exact splitting method is used in a combination with the fast Fourier transform. It was obtained that the relative error of the power and statistical characteristics calculating by the ray-tracing technique is no more than 15% at worst. For the majority of problems having practical meaning the relative error of the ray-tracing technique does not exceed 3 - 5%.
The propagation problem for a partially coherent wave field in inhomogeneous random media is considered in this work. The influence of the refraction, the inhomogeneity of gain (absorbing) medium properties and refraction parameter fluctuations on target radiation characteristics are taken into consideration. On the basis of the solution of the equation for the second order coherence function the ray- tracing technique makes an algorithm of the account of turbulence and medium inhomogeneity for arbitrary initial coherence.
The estimates of absorption of CO-, HF- and DF-lasers multifrequency radiation along extended atmospheric paths are presented in the paper. Peculiarities in the behavior of absorption characteristics depending on the path profile, spectrum of the laser emission, and meteorological conditions are analyzed. We also analyze the influence of thermal nonlinearity on the beam characteristics.
Radiation propagating through inhomogeneously absorbing media is investigated within of the geometric optics approach. On the basis of the equation obtained for a real ray trajectory, numerical analysis of propagation features is carried out for various symmetrical distributions of complex dielectric constant such as parabolic, Gaussian and power low distribution. On the basis of the numerical simulations the validity of the solutions constructed on real geometric rays is determined for a propagation problem of coherent and partially coherent radiation. It is shown that the solutions of the propagation problem within the geometric optics limit approach the exact solutions for partially coherent radiation faster than for the coherent one. Recognizing that the equation for the real ray trajectory is not the equation of the second order, we show that two initial conditions do not suffice the problem, namely index point and initial slope angle for the unique definition of the real ray trajectory. On the basis of the analytical solution obtained for the case of parabolic distribution of dielectric constant it was proved analytically that the ray trajectory in highly absorbing inhomogeneous media depends not only on the complex dielectric constant distribution, but also on the wave phase front distribution.
Propagation of optical radiation through media with complex dielectric constant is studied in this paper. The rigorous solution of this problems taking into account for curvature of ray trajectories caused by the inhomogeneous absorption is obtained on the basis of the ray-tracing technique. For a parabolic distribution of dielectric constant the analytical solution for a field is obtained. For this and other distributions of the complex dielectric constant such as Gaussian and power-low distribution the comparison is made for exact solutions and solutions neglecting the refraction caused by inhomogeneities of the imaginary component of the complex dielectric constant. It has been found that in the media where the imaginary component of the dielectric constant is comparable to or larger than the real component the effect of the curvature of the ray trajectories due to the inhomogeneity of the imaginary component is essential or dominate, respectively. The validity of solutions neglecting the refraction caused by inhomogeneous absorption is investigated.
Propagation of partially coherent radiation through strongly absorbing media is studied in this paper. The equation for coherency function was used in the investigation. It was shown that the equation could be reduced to a system of ray equations, assuming a construction of an efficient numerical procedure for its solution. For parabolic distribution of the dielectric constant and an initial Gaussian distribution of the average radiation intensity the analytical solution for a field was obtained. Based on this, the influence of the refraction on inhomogeneous profiles of the refraction index, the absorptance on the statistical, and energy characteristics of radiation were investigated. The peculiarities of coherent and partially coherent radiation propagation through inhomogeneous media were found.
The paper presents the results of the investigation of space-power structure of laser radiation characterized by high amplification factors of active medium and short pulse duration. The experimental-theoretical method are proposed, which indicate that the copper vapor laser radiation is the superposition of several components, and enable one to determine the number of these components as well as the divergence and power of each component. The potentialities of the above methods are given when determining the output copper vapor laser radiation characteristics.
We analyze the effect of plasma density fluctuations on x-ray laser coherence and discuss the implications of our results for exploding-foil x-ray laser experiments. We treat propagation of coherence using a deterministic ray-tracing technique based on the radiation transfer equation, coupled with the phase approximation of Huygens-Kirchhoff method to treat the random fluctuations. First we develop the propagation technique for the zero-fluctuation case, and derive analytical solutions for the intensity distribution and coherence function in the output plane of an active medium with parabolic transverse profiles of dielectric constant and gain coefficient. We discuss under what conditions we may include only the contribution of spontaneous sources adjacent to the far face of the active medium, and show that in many practical cases it is necessary to take into account sources throughout the whole active medium. We then include density fluctuations, and obtain expressions for the coherence function for homogeneous fluctuations with both Gaussian and exponential correlation, as a function of fluctuation amplitude and correlation length. We apply our results to selenium and yttrium exploding foil x-ray lasers, where non- uniformity in the optical laser pump beams or hydrodynamic instabilities may lead to fluctuations. We show how fluctuations can cause the coherence to saturate with increasing laser length, and indicate how a moderate level of fluctuations can explain the observed coherence in experiments on selenium x-ray lasers. We also characterize the sensitivity of the coherence to fluctuations by defining critical amplitude level above which they dominate the coherence.
A ray-tracing technique based on the radiation transfer equation is used to describe the spontaneous emission gain in active media. Using this approach analytical solutions for the intensity distribution and coherence function in the output plane of an active medium with parabolic transverse profiles of dielectric constant and gain coefficient are presented. Applicability of the approximation when contribution into output emission is made by only spontaneous sources adjacent to the far face region of an active medium is analyzed. This approximation is seen not to be applicable for many real situations and it is necessary to take into account the sources in the whole active medium. The effect of dielectric constant fluctuations on output coherence is treated by using the phase approximation of Huygens-Kirchhoff method.
The propagation of partially coherent beam in refractive media (linear or nonlinear) obeys the equation for the second order coherence function. Numerical solution of this equation is encountered with difficulties, because this equation is in five independent variables. A ray method for solving this task is constructed. The advantage of this approach is that partial differential equation reduces to the set of ordinary differential equations similar to the geometric optics equations. But there are no divergences in the present method due to the presence of diffractive term in the ray equation. The accuracy of this method is discussed. It is shown that propagation of the coherent beam is exactly described by this technique. Based on the solutions obtained with this technique a comparison of self-action of a coherent and partially coherent beams with the same Fresnel number is made. Relations of this technique to the ray methods of solving small angle radiation transfer equation are discussed.
A partially coherent laser beam propagated in the far region of diffraction through a nonlinear layer adjacent to the radiation plane is considered. The propagation is described using a small-angle approximation of a transfer equation. It is demonstrated that 'slow' phase correction is effective only in the limited region of the beam energy parameters. Beyond the limits of the region even the optimum phase correction does not decrease angular expansion, and the use of nonoptimum algorithms causes deterioration of radiation characteristics. 'Fast' phase correction makes it possible to derive the angular expansion value that is less than the diffractive value, when a 'focusing lens' formed in a nonlinear layer.
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