Abstract
The analytical expression for the Bi-frequency correlation function of the intensity scattered from two-dimensional dielectric randomly rough surfaces obeying Gaussian distribution are presented based on the scalar Kirchhoff approximation theory with the root-mean-square (rms) slope of the surface less than 0.25 and the Gaussian moment theorem. The results show that the bi-frequency correlation properties of the scattered intensity closely depend on the incident and scattered conditions as well as on the statistical parameters and complex refractive index of the surface. Especially, the correlation function mainly comes from the specular direction, and the coherence bandwidth and the function decrease with the increase of the roughness of the rough surface. In addition, comparing with the real part, the imagery of the complex refractive index has a greater impact on the bi-frequency correlation function.
©2012 Optical Society of America
1. Introduction
The problem of the scatterings of electromagnetic waves and optics from randomly rough surfaces has always been a very important topic, and has very widely scientific and technical applications, such as in radar, SAR remote sensing, surface detection and target recognition [1–5]. In many radio and optical measurements, regardless of the radar waves or the lasers emitting pulses with some bandwidth, it is necessary to study the correlation of the scattered fields with different frequency. Ishimaru employed the two-frequency mutual coherence function to study pulse scattering from rough surfaces and discussed the pulse broadening and the enhanced backscattering effect [6]. Schertler and George derived the formulas of the two-frequency mutual correlation function of the backscattering from roughened sphere and roughened disk [7, 8]. Chen et al derived the two-frequency mutual coherent function to investigate the pulse scattering properties of the pulse plane wave and pulse beam from randomly rough surfaces [9]. The two-frequency mutual coherence function of the scattering from arbitrarily shaped rough objects were obtained and the numerical results for rough spheres and cylinders were given to analyze the dependence of the function on the shape and the size of the objects and on the roughness of the surface [10].
In practice, the intensity not the scattered field is the directly detected parameter by experiments; therefore, the correlation properties of the scattering intensity are of more considerable interest. The solution for the fourth moment equation of waves in random media was given by Xu et al [11]. The frequency spectrum of the intensity fluctuations of the scattered field was used to measure the parameters of vibrations and the surface roughness, and the experimental results were presented [12]. The fourth order moment statistical characteristics of the wave scattering from random rough surfaces were derived and the two-frequency mutual coherence function of the scattered intensity was investigated numerically [13]. The correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media was analytically derived, and its dependence on spatial Fourier transforms of both the intensity and degree of spatial correlation of scattering potentials were analyzed [14]. The expressions for the fourth-order moment of a random field incident upon deterministic and random scatterers were derived which can serve as a rigorous analytic prediction of a scattered field by scatterers and demonstrated how the intensity–intensity correlations were affected at various points in the far field [15]. Most of the discussions above simplified the correlation function of the intensity as the square of the module of the correlation function of the scattered field, since the surface considered was rougher than the incident wavelength. With Gaussian moment theorem, the correlation properties of the intensity scattered from one-dimensional and two-dimensional randomly conducting rough surfaces with RMS roughness smaller than the wavelength were discussed [16, 17] in detail. However, during the investigations the influences of the slopes and the dielectric property of the surface on the intensity correlation function were not considered for simplicity.
Based on the scalar Kirchhoff approximation theory, the bi-frequency correlation function of the scattered intensity from two-dimensional dielectric randomly rough surface with a root-mean -square slope smaller than 0.25 is investigated in this paper. Assuming the distribution of the rough surface to be Gaussian, the expression for the bi-frequency correlation function of the scattered intensity is obtained, and then the results and the influence factors are presented. It is noted that the bi-frequency correlation function of the scattered intensity closely dependent on the roughness of the surface mainly comes from the specular direction when the surface is weakly fluctuating.
2. Bi-frequency correlation function of the scattered intensity from dielectric rough surfaces
When the surface discussed is weakly rough that the roughness is smaller than the incident wavelength, the multiple scattering from different area of the rough surface and the shadowing effect can be omitted. The rms slope of the surface smaller than 0.25, according to the scalar Kirchhoff approximation theory, the scattered field from two-dimensional randomly dielectric rough surface can be expressed as [18]
where, and is the wave-number in free space,is the incident wavelength,is the incident frequency,is the light velocity. The aperture function is introduced to expand the integral domain to infinite, andis its dimension. , whereare polarization coefficients whose expressions are in reference [18], andrepresent the slope of the surface along,directions at the point , respectively.,is the fluctuating function of the surface, and , are the scattered and incident wave vectors, where andare the scattered and incident directions, respectively.The bi-frequency correlation function of the intensity scattered from randomly rough surfaces is defined as
(2)
The rms slope of the surface discussed in this paper is smaller than 0.25 and its rms roughness is less than the incident wavelength, the surface is weak rough surface, therefore, the scattering composes coherent and incoherent parts, that is
The bi-frequency correlation function of the scattered intensity can be written as follow [17]
whereand.2.1 Solution for the term
According to Eq. (1), the termis
neglecting the unnecessary factor before the integral, and. For convenience, let the vectorsandbe,. And the termcan be written asThe second number in the subscript of the polarization coefficient represents the frequency. And here the second terms of the slopes are omitted, which because when the maximum radius of curvature of the surface is much greater than the incident wavelength, the second terms of the slope will be much smaller than the first terms.
From Eq. (6), it can be seen that can be decomposed into two terms: non-slope term and slope term,
The non-slope term is
The slope term includes -slope term and -slope term
where with where.The rough surface is assumed to be Gaussian, then the joint-characteristic function has the form
where, and is the correlation coefficient of the rough surface,and are the rms roughness and the correlation length, respectively. For a Gaussian randomly rough surface, there are the relationships as follow [18] where.Making the following change of variables
Substituting Eqs. (14)and(17) into Eq. (8) gives the non-slope terms as
with.To evaluate the integral, we must expand the terminto Taylor series
Then the solution is
To solve the integrals in Eqs. (12) and (13), some new variables are introduced
The partial derivatives of the functionbecome
Inserting Eqs. (15), (17), (21) and (22) in Eq. (12) and carrying out the integration over, yields
whereand.With the complex exponential term in Eq. (23) which can be expressed as [19]
and the integral relationships [20] and, then the functionhas the solutionwhereis the nth-order Bessel function.Inserting Eq. (28) into Eq. (23) and utilizing Eq. (19), the final solution for Eq. (12) can be attained
In the same way, the solution of Eq. (13) can be obtained as
Then substituting Eq. (29) and Eq. (30) into Eq. (10) and Eq. (11), respectively, the solution for the slope termis
With Eqs. (20)and(31), the analytical expression for the termcan be achieved.2.2 Solution for the term
Similarly, from Eq. (1) the term can be written as
whereAlso can be expressed by the non-slope term and the slope term
andThe slope term is then
where andHere the joint-characteristic function is
andBy Eqs. (17)and(21), and finishing the integration over, Eqs. (35) and (39)-(40) can be transformed as follow
where with.By Eqs. (25)-(27), The functions,andhave the solutions
Noting that in Eqs. (44)-(46), because of the term, it is difficult to get the solutions analytically, a more simple approximation should be taken which we have discussed in our paper [16] before. Using the Eq. (16) in the reference [16], and inserting Eqs. (50)-(52)into Eqs. (44)-(46), respectively, we can get
whereSince the Bessel functions of integer order can be expressed explicitly as [21]
And there are the relationships [20]
The final solutions forandcan be given by
with.2.3 Mean scattered field
At last, the mean scattered fieldshould be derived. By Eq. (1), can be written as
The term in the equation above is the characteristic function of the Gaussian random variable
and by reference [18]The mean scattered field is thenThus every term needed has been derived, inserting these solutions in Eq. (4), the analytical expression for the bi-frequency correlation function of the scattered intensity from two- dimensional randomly dielectric weak rough surface can then be obtained.
3. Numerical results and analyses
In this section, the bi-frequency correlation function of the scattered intensity will be calculated numerically to analyze its bi-frequency correlation properties and the influencing factors. In the calculation, let the center frequency beand the frequency difference be, and the medium be assumed non-dispersive or in the extent of the bandwidth the refractive index have hardly change, thus for the frequency difference, the polarization coefficients are constant. From the formulas derived above, it can be seen that the bi-frequency correlation function of the scattered intensity from two-dimensional dielectric rough surface with a root-mean-square slope smaller than 0.25 and rms roughness less than the incident wavelength is closely dependent on the incident and scattered conditions as well as on the statistical parameters and refractive index of the surface.
Taking the wavelength isand the aperture dimension is as well as the refractive index is, Figs. 1 -4 are the variation of he bi-frequency correlation functionof the scattered intensity with the scattering angleand the frequency difference. In Figs. 1-3 , ,, the rms roughnessincreases sequentially. Comparing with Fig. 3, in Fig. 4 the incident angle is, is invariable, ,the correlation length is increased to be.
It is shown that the contribution for the functionmainly comes from the specular direction while the values in other directions can be neglected. With a smaller rms slope, the decrease of the function versus the frequency differenceis much more slowly that the correlation bandwidth is bigger and the profile near the specular is sharper. While with an increase, the values of the function in specular direction decreases dramatically and others increase relatively. It is explained that when the rms slope is small, the scattering mode is similar to the plane scattering which scattered power only occurs in specular direction, while the surface becomes more roughness, the non-specular scattering enhances and the specular scattering weakens. In addition, the increase of rms height results in a narrower correlation bandwidth.
and, with different roughness, Figs. 5 -6 and Figs. 7 -8 are the variation of the bi-frequency correlation function against the scattering azimuth angle and the frequency difference under HH- and VH-polarization, respectively. Under both the polarizations, the functiondecreases monotonously with the increase of the frequency difference and the roughness. A bigger roughnessresults in a smaller correlation bandwidth. However, in HH-polarization the variable tendency of the functionwith the increase of the azimuth angle is completely different to that in VH-polarization.
To illustrate the influence of the dielectric property of the rough surface on the bi-frequency function, Fig. 9 gives the profiles of the functionversus the frequency differencewith different roughness and refractive indexes. It is can be seen easily that with a complex refractive index, the smaller the roughness, the bigger the function, while with a same roughness, the refractive index has influence on the value of the functionnot on the correlation bandwidth. Especially, the imagery of the refractive index is the main influencing factor that a smaller difference of the absolute values of the imagery of the refractive indexes results in a smaller difference of the values of the function.
4. Conclusion
Based on the scalar Kirchhoff approximation theory, the Bi-frequency correlation properties of the intensity scattered from two-dimensional dielectric randomly rough Gaussian surfaces assumed to be Gaussian with a root-mean-square (rms) slope of the surface less than 0.25 are investigated in this paper. The rms roughness of the surface is assumed to be smaller than the incident wavelength. By the Gaussian moment theorem, expanding the exponential and complex exponential function into series and with some mathematical simplification, the Bi-frequency correlation function of the scattered intensity considering the influences of the scattering from the slope of the surface has been derived in detail and calculated. The numerical results show that the incident and scattered conditions as well as on the statistical parameters and complex refractive index of the surface have great impacts on the bi-frequency correlation properties of the scattered intensity. Especially, the correlation function mainly comes from the specular direction, and the coherence bandwidth decreases with the increase of the roughness of the rough surface. In HH-polarization and VH-polarization, the functionversus with the scattering azimuth angle have different varying tendency. In addition, the refractive index can also affect the value of the function but not on the correlation bandwidth. Comparing with the real part, the imagery of the complex refractive index has a greater influence on the bi-frequency correlation function that the larger difference of the absolute values of the imagery of the refractive indexes, the larger difference of the values of the function. The work performed in this paper provide a much better knowledge about the scattering properties of dielectric rough surfaces to investigate the scattering problem and the identification of three-dimensional rough dielectric objects which we will discuss in future.
Acknowledgments
The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 61172031 and from the Fundamental Research Funds for the Central Universities under Grant No. K50510070009.
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