Let us consider the scattering of a plane wave by a finite periodic surface,^[4,6] as shown in Fig. 1. The surface corrugation is written as 1 where is the periodic function with a period P, and W is the total length of the periodic surface. Here, 2 and is a gate function with the width W 3 For convenience, k_P and k_W are set as 4

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Scattering of a plane wave by a periodic dielectric surface.

We denote the y component of the electric field by , which satisfies the Helmholtz equation in the region and the Dirichlet condition on the surface z = f(x): 5 6

Here, is the wavenumber, and λ is the wavelength. The incident plane wave is written as 7 8 where is the incident angle, Im stands for the imaginary part, and the electric field amplitude of the incident wave is E = 1. Since the surface is flat for , the y component of the electric field is shown as 9 which is the sum of the incident plane wave, specular reflection, and scattered wave due to surface deformation. By use of the periodic Fourier transform,^[26] the scattering field has an extended Floquet form 10 11 where equations (10) and (11) hold for . A(s) is the spectral amplitude. Equation (11) is the extended Floquet form, where is the m-th order diffraction beam given by^[26] 12 Here, it is worthwhile to note that is related to the spectral amplitude A(s) as follows: 13 By substituting Eq. (13) into Eq. (11), we obtain 14 It is worth noting that equation (14) satisfies Eq. (5) and the radiation condition at term by term. is a complex amplitude of a plane wave propagating in the direction : 15 In other words, is the amplitude of a plane wave scattered into the direction, where is given by 16 If s = 0, equation (16) degenerates the famous grating formula 17 where m stands for the diffraction order. Since m is an integer, and because the incident angle, wavelength, and surface period length are constant, it can be deduced that the scattering intensity will take an extreme value at . Equation (14) is a product of the exponential phase factor exp( ) and a harmonic series that is a “Fourier series” given by band-limited Fourier integrals. When W goes to infinity and equation (1) becomes infinitely periodic, the amplitude function turns into 18 where is a diffraction amplitude, and is Dirac’s delta function. Hence, equation (14) degenerates into the famous Floquet form for the periodic grating. Therefore, equation (14) is regarded as an extension of the Floquet form and is called the extended Floquet form^[6,9] 19 When W is much larger than the wavelength, however, is expected to be a localized function taking a large value only at . This implies that is a beam whose main lobe scatters along the direction determined by Eq. (17). Thus, is called the m-th order diffraction beam.

Physically, the diffracted waves are radiated from the corrugated part of the surface, and hence they exist only at limited regions in space. Considering the far field case, the scattering cross section can be obtained by solving the integrals in Eq. (14) with the saddle point method.

We denote a scattering angle by (see Fig. 1) in the polar coordinate: 20 Then, using the saddle point method, the approximated form of Eq. (14) is 21 From this relation, the scattering cross section is obtained and expressed as 22 which is a non-dimensional quantity divided by the corrugation width W. Let , and n_W is a positive integer. When , W tends to infinity. From Eqs. (18) and (22), the bistatic scattering coefficient of the periodic surface in a large area can be obtained: 23 For the convenience of calculation, the Dirac function is approximated as follows:^[6] 24 As we can see in Eq. (23), the scattering field amplitude is an important factor in the formula of the scattering cross section of the infinite periodic surface.

2.3. Calculation of scattering field amplitude coefficient

When , by substituting Eq. (18) in Eq. (12) and utilizing the selectivity of the Dirac function, we can obtain 25 Using Floquet’s theorem and Huygens’s principle, the scattering propagation models are derived. The scattering amplitude , or diffraction amplitude, is acquired by the EBCM and the T-matrix method^[9,10] 26 where matrix is upward-going field amplitudes and is downward-going field amplitudes. and can be obtained by the following matrix equations: 27 28 where and are Dirac matrices and Neumann matrices, respectively, under different boundary conditions: 29 30 When the periodic surface is a sine function, 31 where the amplitude is h, and the period is P. Then the Q-matrix can be written as 32 According to the above method, we can determine and obtain the bistatic scattering coefficient of the periodic surface ∣ from Eq. (23).

The following work is a validation of the method described in Section 2.

4.1.1. Verification of the method by the energy conservation law

Figure 2 shows the variation of the scattering power and transmission power with the incident angle under different conditions, and the detailed parameters are shown in Table 1. From the general trend of change, the scattering efficiency decreases with the increase in incident angle. Conversely, transmittance increases with the increase in incident angle in the range of . However, the scattering efficiency varies locally at partial incident angles. For example, in Fig. 2(d), the efficiencies of scattering and transmission reaches their extremes when , 37°, and 45°. As the “total” curve shows, the sum of scattering and transmission is 1, which satisfies the energy conservation law.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Variation of scattering power and transmission power with incident angle.

Table 1.

Table 1.

Table 1. The parameter list. . Fig. 2(a) Fig. 2(b) Fig. 2(c) Fig. 2(d) E 1 1 1 1 h/λ 0.1 0.4 0.4 0.4 P/λ 2.5 2.5 2.5 10 1 1 1 1 4 4 35 35

Table 1. The parameter list. .

It is worth noting that the periodic surface will degenerate to a plane when h = 0. The simulation data can be compared with the consequence of the Fresnel reflection formula to further verify the correctness of the algorithm.

The coefficients of Fresnel reflection and transmission are expressed as 38 where , k_z and are the components of the incident and transmitted wave numbers along the normal direction of the plane, respectively (see Ref. [27] for details). The corresponding power reflectivity and power transmissivity are given as 39 Equation (39) can be used to compare with the above method. As shown in Fig. 3, where , the results of the two methods are in good agreement, and the variation of scattering (reflection) and transmission power still conform to the law of energy conservation. Thus, the above algorithm is effective.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. Comparison of the method of this paper with the Fresnel method.

4.1.2. Verification of the method by comparing with experimental data

The effect of the row structure on the microwave emission from a bare agricultural field has been reported together with the soil moisture contents for the measured data.^[7,28] The theoretical results are illustrated for a sinusoidal surface in Fig. 4 at a frequency of 1.4 GHz. The periodic surface has a height h = 0.1 m and a period P = 0.95 m, and can be approximated by a sinusoidal function. For the upper medium, let and . T_B is the brightness temperature, and by definition. P_r denotes the reflection power or scattering power.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Brightness temperature as a function of incident angle .

In Fig. 4, the radiometer observation angles are perpendicular to the row direction. The soil moisture content is 29% by dry weight at the top 0 to 5 cm and becomes drier with depth. The complex permittivity is approximately . We see that the simulation data of brightness temperatures T_B for the horizontal polarization are similar to the experimental data from 0° to 60°.

Besides, we observe that the T_B curves for the periodic surface are not smoothly varying. There are kinks appearing at observation angles θ near 6°, 19°, 34°, and 51°. The corresponding change in T_B may be as high as 10 K. Such a phenomenon is caused by the redistribution of the scattered power during the course of the disappearance and appearance of the propagating Floquet modes.

Theoretically, when using Floquet’s theory to calculate the scattering of an infinite periodic surface, the scattering wave propagates in discrete Floquet modes. This means that the distribution of the scattering intensity has values just at the angles corresponding to the Floquet mode (as shown in Table 2). However, the distribution of scattering intensity of the finite periodic surface is continuous.

Table 2.

Table 2.

Table 2. The m-th modes and the corresponding angles . . m −3 45.57° −2 72.54° −1 95.74° 0 120° 1 154.16°

Table 2. The m-th modes and the corresponding angles . .

As shown in Fig. 5, under the same incident condition, as n_W increases, the periodic surface length W increases, and the distribution of the scattering intensity is only concentrated at the angle corresponding to the Floquet modes. For example, when , although the scattering cross section is continuous with the change in the scattering angle, the scattering intensity is obviously concentrated at the corresponding Floquet mode angles, and the scattering intensity values at other scattering angles are so small that they can be negligible.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Scattering cross section under different lengths of periodic surface .

Therefore, by using the above method in practical calculation, regardless of a finite or an infinite periodic surface, reasonable results can be obtained through taking the appropriate value of W. This is also one of the points of this work.

Figure 6 presents the comparison of the relative scattering power of the periodic surfaces under different conditions, with incident angle.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. The variation of scattering power with incident angle under different conditions. (a) Scattering power under different amplitudes h. (b) Scattering power under different periods P. (c) Scattering power under different permittivity .

Figure 6(a) compares the relative scattering power of different periodic surface amplitudes with the incident angle. It can be seen that the relative scattering power at different amplitudes h has little difference when and . However, the difference is obvious at the region of , that is, the larger the amplitude h, the smaller the scattering power at the same incident angle .

Figure 6(b) compares the relative scattering power with the incident angle under different periods P. It can be seen that the scattering power decreases at and with an increase in period P. However, when , the trend is the opposite.

According to Figs. 6(a) and 6(b), the larger the amplitude h and the smaller the period P, which means that the periodic surface is rougher, the effect of scattering is more obvious, and the oscillation of the relative scattering power curve is more evident.

Figure 6(c) compares the relative scattering power of different relative permittivity with the incident angle. is the relative permittivity of region 1. It can be seen that has a positive correlation with the scattering power. When other conditions remain unchanged, the larger the permittivity, the greater the scattering power.

Using the formulae in Section 3, the scattering field of the dielectric periodic surface is calculated by Eq. (23). As shown in Fig. 7, the normalized bistatic scattering cross section (σ under different surface amplitudes is calculated, where , and P=2.5λ, and h takes 0, 0.05λ, 0.1λ, 0.2λ, 0.3λ, and 0.4λ, respectively.

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. Scattering cross section under different amplitudes h.

When h = 0, that is, when the interface is a plane, as shown in Fig. 1(a), the scattering intensity is concentrated only in the direction of , which means that there is no scattering, only specular reflection. This is in line with the actual situation.

Figure 7(b), 7(c), and 7(d) correspond to the cases of h=0.05λ, 0.1λ, and 0.2λ, respectively. It can be seen that the maximum peak value of the scattering cross section is still at , which is the specular direction. However, the number of peaks increases sequentially: 3, 4, and 5, respectively. This shows that as the amplitude h of the periodic surface increases, the number of peaks of the scattering cross section increases, and the scattering is more obvious.

Besides, by comparing Figs. 7(d), 7(e), and 7(f), we can see that the maximum scattering cross section is no longer located in the specular direction ( ) when h continues to increase (h=0.2λ, 0.3λ, and 0.4λ), even though the numbers of peaks are the same.

Generally, a larger amplitude h causes the surface to be rougher, thus enhancing the scattering effect. Furthermore, the corresponding scattering peaks and their positions (angles) are shown in Table 2, which agree with diffraction angles calculated by the grating formula (Eq. (17)).

Figure 8 compares the normalized bistatic scattering cross section (σ ( ∣ ) of periodic dielectric surfaces at different incident angles, where P=5λ, h=0.4λ, , and .

Fig. 8.

	Figure Option ViewDownloadNew Window
	Fig. 8. Scattering cross section under different incident angles .

It is evident from Figs. 8(a) and 8(b) that the scattering intensity is mainly concentrated on the specular direction when the incident angle is small (or in the case of small grazing angle). For example, when , the maximum value of the peak value of the scattering cross section is at the angle of .

With the increase in incident angle , the distribution of scattering intensity changes significantly. As shown in Figs. 8(c)–8(e), the number of peaks of the scattering cross section increases obviously, and the position of the maximum peak is no longer the specular position of the incident angle. For example, when , the position of the maximum peak is at , not 135^◦, which is in the specular direction of the incident angle. When , the position of the maximum peak is at , not 120^◦, which is in the specular direction of the incident angle.

When continues to increase until it is close to the vertical incidence, the intensity distribution of the scattering cross section is stable, which means that the scattering effect on the periodic dielectric surface is significant.

Table 3 records the scattering wave modes m and positions of peaks at different incident angles shown in Fig. 5.

Table 3.

Table 3.

Table 3. Angular locations of scattering cross section peaks. . 5° 15° 45° 60° 75° 90° −9 36.5° 33.5° −8 52.9° 50.6° 26.8° −7 66.2° 64.3° 46.1° 25.8° −6 78.2° 76.5° 60.5° 45.6° 19.7° −5 89.9° 88° 73° 60° 42.2° 0.01° −4 101.3° 99.6° 84.7° 72.5° 57.2° 36.9° −3 113.3° 111.5° 96.1° 84.3° 70.1° 53.1° −2 126.6° 124.5° 107.9° 95.7° 81.9° 66.4 −1 142.8° 134° 120.5° 107.5° 93.4° 78.5° 0 175° 165° 135° 120° 105° 90° 1 155.1° 134.4° 117.3° 101.5° 2 154.2° 131.2° 113.6° 3 149.2° 126.9° 4 143.1°

Table 3. Angular locations of scattering cross section peaks. .