Elsevier

Optik

Volume 124, Issue 24, December 2013, Pages 6760-6764
Optik

Speckle statistical properties of Gaussian beam from a semi-rough target in the atmospheric turbulence

https://doi.org/10.1016/j.ijleo.2013.05.081Get rights and content

Abstract

Based on extended Huygens–Fresnel principle and Goodman model for target surface, analytic expression is developed for the mutual coherent function (MCF) of a reflected Gaussian-beam from a semi-rough target in single-pass atmospheric turbulence. Then according to the MCF we derive expression about the mean intensity and average speckle size at the receiver. The analysis indicates that the mean intensity is closely related to the ratio of root mean square (rms) height to the lateral correlation length; in addition, the speckle size is associated with turbulence strength and roughness of target. Our results agree with well-known the limiting cases of perfectly smooth and Lambertian target.

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 writtenU0(r)=u0expr2w02ikr22Fwhere w0 and F are the characteristic beam radius and focal length, respectively. r = (x2 + y2)1/2, r=(x,y) 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 effectsu(p)=keikL2πiLdρur(ρ)expik|pρ|22L+ψ(p,ρ)where ψ(p,ρ) means the random part of the complex phase of a spherical wave propagating in the turbulent medium from (0,ρ) to (L,p).

The mutual coherent function of the speckle field at the receiver plane can be writtenΓ(p1,p2)=u(p1)u(p2)=k2πL2dp1dρ2ur(p1)ur(ρ2)×expik2L(|p1ρ

Variance of intensity and speckle size

For the saturated turbulence case, the correlation function of intensity is defined asΓl(p1,p2)=I(p1)I(p2)=I(p1)I(p2)+|Γ(p1,p2)|2

It follows that the covariance of intensity is given byCl(p1,p2)=I(p1)I(p2)I(p1)I(p2)=|Γ(p1,p2)|2Utilizing the mutual coherence formulations, the variance of intensity is easily derived from Eq. (23)σI2=Cl(p,p)=k2R2w02u0232L2a2exp4p2(1+(2σφ2/β2))w02+(16L2/k2ρ02)=I(p)2

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)

References (26)

  • S. Sahin et al.

    Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams

    Opt. Commun.

    (2010)
  • M. Zribi et al.

    Validation of a rough surface model based on fractional Brownian geometry with SIRC and ERASME radar data over orgeval

    Remote Sens. Environ.

    (2000)
  • D.P. Greenwood

    The scattering from curved rough surfaces of an EM wave transmitted through a turbulent medium

    IEEE Trans. Antennas Prop.

    (2004)
  • C.N. Kurtz

    Transmitted characteristics of surface and the design of nearly band-limited binary diffusers

    J. Opt. Soc. Am.

    (1972)
  • C.S. Gardner

    Target signature for laser altimeters: an analysis

    Appl. Opt.

    (1982)
  • K. Nagata et al.

    The determination of rms roughness and correlation length of rough surface by measuring spatial coherence function

    Appl. Phys.

    (1973)
  • R.A. Sprague

    Surface roughness measurement using white light speckle

    Appl. Opt.

    (1972)
  • H.L. Myung et al.

    Statistics of speckle propagation through the turbulent atmosphere

    J. Opt. Soc. Am.

    (1976)
  • J.F. Holmes et al.

    Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere

    J. Opt. Soc. Am.

    (1980)
  • C.M. McIntyre et al.

    Enhanced variance of irradiance from target glint

    Appl. Opt.

    (1979)
  • L.X. Guo et al.

    Light scattering from a diffuse target in the turbulent atmosphere

    J. Electron. Magn. Inf. Technol.

    (2001)
  • H.Y. Wei

    Scattering from a diffuse target in the slant atmospheric turbulence

    Chin. Phys.

    (2008)
  • Z.S. Wu et al.

    Q, Scattering of a partially coherent Gaussian-Schell beam from a diffuse target in slant atmospheric turbulence

    J. Opt. Soc. Am. A

    (2011)
  • Cited by (3)

    • Dependence of speckle contrast on the light spectral broadening and the roughness root mean square

      2017, Optik
      Citation Excerpt :

      Since the statistical properties of speckle patterns carry information about a rough surface, considerable attention has been paid to the statistical properties of speckle patterns [1–10].

    View full text