Speckle statistical properties of Gaussian beam from a semi-rough target in the atmospheric turbulence
Introduction
Radar serves for the extraction of target information such as the refractive index, size, and surface roughness. There are numerous applications, e.g., remote sensing using the interaction of laser with turbulent atmosphere, laser radar system, coherent imaging through the atmosphere, and COAT systems. A statistical characterization of the scattered field in the receiver, when the transmitted beam illuminates the target and in its turn the return wave passing through the atmosphere, is important. We should study the first- and second-order moments of receiver intensity so as to asses speckle effect in the receiver.
Since 1970s, working on theoretical studies of speckle was carried out concerned with the case of fully developed speckle (i.e. a fully diffuse target). Many researchers had been concentrated on the nature and statistics of the target surface without considering turbulence [1], [2], [3], [4]. For example, Sprague utilized white light speckle measure surface rough but neglected atmospheric effects [5]. Then Myung et al. [6] studied the statistics of speckle of Gaussian beam from a fully diffuse target through turbulence. Holmes [7] discussed the effect of the log-amplitude covariance function about the effect of speckle propagation through turbulence. McIntyre [8] analyzed statistics of irradiance scattered from a diffuse target containing multiple glints with experimental and theoretical method. In former analysis, laser beam propagated in horizontal path. Then followed by Guo [9] discussed amply the effect of log-amplitude on speckle propagation. Wei [10] studied scattering of a Gaussian beam from a diffuse target in slant optical turbulence. Wu [11] discussed the partially coherent Gaussian Schell-model beam from a diffuse target in slant path. As mentioned above, most researches were focused on the fully diffuse target. It is well known that radar is concerned with the extraction of target information contained in the echo signal; therefore, the target surface plays an important role in determining the nature of the return beam. In reality, most targets have surfaces that are considered rough on the scale of an optical wavelength; it is modeled as partially diffuse, somewhere between the limiting cases of a smooth reflector and a fully diffuse target. The classic summary of theoretical model for the scattering from rough surface was made in [12], [13]. Experimental results are still appearing in the literature for wave in free space [14]. Then Korotkova and Andrews developed a new model for the speckle size and the scintillation index of the lowest order Gaussian beam from a finite target, based on ABCD ray matrix theory and a random phase screen model for the target surface [15], [16], [17]. Murphy, Gatt proposed a model combined effect of rough surface and turbulence double passage reciprocal path scattering (RPS), provided an excellent approximation to the effect of RPS on the average total observed intensity and found the coherent RPS enhancement factor [18], [19]. Sahin utilized coherence and polarization properties of illumination stochastic electrometric beams and return beam to predict the typical of the target surface [20].
In this paper we make an attempt to develop the model for the speckle to the situation of the arbitrary roughness surface. Following Goodman model [21], based on the extended Huygens–Fresnel principle, we derive the speckle effect of reflecting beam through atmospheric turbulence. In Section 3, the analytical expressions are derived for the mean intensity and correlation function of the speckle field at the receiver, and then numerical results are presented here. In Section 4, we have obtained explicated formulas for the intensity correlation function and the covariance of the intensity. In Section 5, average speckle size is deduced on basis of the complex degree of coherence. In order to show our model is proper in this paper, we compare our result with other researchers adopting different methods.
Section snippets
Statistical model for target
The source amplitude distribution for a single-mode Gaussian beam can be writtenwhere w0 and F are the characteristic beam radius and focal length, respectively. r = (x2 + y2)1/2, denotes a two dimensional transverse vector perpendicular to the direction of the beam propagation in the transmitter plane. k = 2π/λ is the optical wave number in free space. A schematic diagram for the laser radar system under is shown in Fig. 1. Our present analysis is confined to the
Mutual coherence function
The field at the receiver is written by reapplying the extended Huygens–Fresnel principle to the scattered field considering the turbulent atmosphere effectswhere means the random part of the complex phase of a spherical wave propagating in the turbulent medium from to .
The mutual coherent function of the speckle field at the receiver plane can be written
Variance of intensity and speckle size
For the saturated turbulence case, the correlation function of intensity is defined as
It follows that the covariance of intensity is given byUtilizing the mutual coherence formulations, the variance of intensity is easily derived from Eq. (23)
The coherence properties of the beam can be inferred from the
Discussion
In this paper we have proposed combing the extended Huygens–Fresnel principle with Goodman theory for modeling speckle propagation through atmospheric turbulence. For convenience, we have considered the case of single-passage and an infinite target. This model leads to expressions for the MCF associated with a typical lidar system from which we deduce the average speckle size in receiver plane. Where possible, we have shown that some limiting cases of our expressions reduce to those previously
Acknowledgments
This works is partly supported by the National Natural Science Foundation of China (Grant No. 61271110), New Scientific and Technological Star of Shaanxi Province funded project (Grant No. 2011KJXX39)
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