3.1.

Scattering Intensity Distribution

Figure 1 shows the incoherent scattered intensity distribution predicted by the SPM (blue-dashed line), KA (green-dotted line), GHS (red solid line), and MoM (black asterisk) for two different sets of surface parameters. In Fig. 1(a), the case of ${\widehat{\sigma }}_{\mathrm{tot}}=0.01$, ${\widehat{l}}_c=0.2$, and ${\theta }_i=30\mathrm{deg}$, which is a smooth surface with a short-correlation length, is considered. The predictions of both the SPM and GHS agree well with the MoM prediction, and the KA result agrees near the specular direction but fails in the wide-angle regime. In particular, it does not converge to zero at $\pm 90\mathrm{deg}$ scattering angles, which is a nonphysical situation. The case of ${\widehat{\sigma }}_{\mathrm{tot}}=0.2$, ${\widehat{l}}_c=2$, and ${\theta }_i=0\mathrm{deg}$, which is moderately rough with a long-correlation length, is illustrated in Fig. 1(b), and both the intensity profiles from KA and GHS methods show a good agreement to the MoM prediction; however, as expected, the SPM fails for this moderately rough surface.

Fig. 1

Scattered intensity of the SPM (dashed line), KA (dotted line), GHS (solid line), and MoM (asterisk) for (a) ${\hat{\sigma }}_{\mathrm{tot}}=0.001$, ${\hat{l}}_c=0.2$, and ${\theta }_i=30\mathrm{deg}$, (b) ${\hat{\sigma }}_{\mathrm{tot}}=0.2$, ${\hat{l}}_c=2$, and ${\theta }_i=0\mathrm{deg}$.

Figures 2(a) and 2(b) illustrate two different situations where the GHS model has reasonable accuracy for moderately rough surfaces, but the other two fail to predict an accurate intensity distribution. However, they fail for different reasons; the SPM fails because the smooth-surface approximation is violated and the KA fails because the small correlation lengths produce large angle scattering that violates the paraxial limitation. Figures 2(c) and 2(d) show for even rougher surfaces and thus, the SPM predictions have been omitted. Figure 2(c) shows the predicted intensity distribution by the KA and GHS for the case of ${\widehat{\sigma }}_{\mathrm{tot}}=0.4$, ${\widehat{l}}_c=2.5$, and ${\theta }_i=50\mathrm{deg}$. The GHS predictions agree well with the rigorous MoM predictions over the entire range of scattered angles; however, the KA prediction exhibits a significant error near the peak and in the forwarding scattering direction due to the large (nonparaxial) incident angle. In Fig. 2(d), a very rough surface (${\widehat{\sigma }}_{\mathrm{tot}}=1$) is considered, and the overall angular behavior predicted by the GHS theory is quite close to that predicted by the rigorous MoM except near the specular direction where the MoM predictions exhibit irregularities due to the considerable multiple scattering.²⁶ There is no specular beam for such a rough surface, i.e., the total integrated scatter is approximately unity. The KA method fails badly due to the large surface roughness. Extensive empirical scatter predictions with the KA method have indicated that it remains quite accurate, at this correlation width, for ${\widehat{\sigma }}_{\mathrm{tot}}\le 0.5$. This statement will be validated later in Fig. 3(b).

Fig. 2

Scattered intensity for (a) ${\hat{\sigma }}_{\mathrm{tot}}=0.1$, ${\hat{l}}_c=0.25$, and ${\theta }_i=0\mathrm{deg}$, (b) ${\hat{\sigma }}_{\mathrm{tot}}=0.3$, ${\hat{l}}_c=0.8$, and ${\theta }_i=0\mathrm{deg}$, (c) ${\hat{\sigma }}_{\mathrm{tot}}=0.4$, ${\hat{l}}_c=2.5$, and ${\theta }_i=50\mathrm{deg}$ (d) ${\hat{\sigma }}_{\mathrm{tot}}=1$, ${\hat{l}}_c=2$, and ${\theta }_i=0\mathrm{deg}$. The SPM prediction is not plotted in (c) and (d).

Fig. 3

Error contour maps at normal incidence: (a) the SPM, (b) KA, and (c) GHS.

3.2.

Comparative Visualization of the Domain of Validity

To quantitatively illustrate the relative accuracy of the three approximate surface scatter models, we calculate the linear-error values and illustrate the value as a corresponding gray level in a 2-D surface parameter space. Figure 3 illustrates contour maps of this linear error over the surface parameter space for each of the approximate theories with the normal incidence. Note that, for our simulation of scattered intensity with the MoM, one thousand realizations were used and the averaged intensity of the MoM predictions still exhibits minor fluctuations.

The SPM has a high accuracy over the entire range of correlation widths; however, the domain is severely restricted to very smooth surfaces as shown in Fig. 3(a). The KA can be considered to have a high degree of accuracy over a much larger fraction of the 2-D surface parameter space, increasing almost linearly to moderately rough surfaces for an increasing correlation length as shown in Fig. 3(b). However, the KA prediction is quite inaccurate ($\epsilon >0.2$) over a significant fraction of the domain represented by the upper left corner of Fig. 3(b). Meanwhile, the GHS theory exhibits slightly less accuracy than the SPM and KA in some regions of their domain of validity, but it has a much broader valid domain in general. For example, the entire illustrated domain is valid if $\epsilon <0.2$ is considered as the criterion as illustrated in Fig. 3(c).

It is widely believed that the validity of the KA method depends on the curvature of the surface irregularities making up the surface profile.^6,10,26 Due to the tangential plane approximation, the KA is considered to be accurate when the curvature is less than the wavelength of interest. In Fig. 4(a), the numerically calculated contour lines of the rms slope of the surface are plotted on the top of the error map previously shown in Fig. 3(b). In Fig. 4(b), contour lines of the rms curvature are plotted on the top of the same error map. From the two figures, it is clear that the validity of the KA method is more strongly correlated to the rms slope than to the rms curvature of the surface.

Fig. 4

Contour lines of (a) the rms slope and (b) rms curvature of random surfaces superposed on the top of the error map for the KA method with the normally incident light.

Likewise, in Fig. 5(a), the numerically calculated contour lines of the rms slope of the random surface are plotted on the top of the error profile map shown in Fig. 3(c). For normally incident light, the GHS method is relatively accurate when the rms slope of the randomly rough surfaces is less than unity. Figure 5(b) indicates that for a 40-deg incident angle, the value of the error increases for a given value of the rms slope. This degradation in accuracy with an increasing incident angle may be caused by the fact that the GHS theory does not consider shadowing effects and multiple scattering.

Fig. 5

Contour lines of the rms slope superposed on the top of the error map of the GHS theory for (a) ${\theta }_i=0\mathrm{deg}$ and (b) ${\theta }_i=40\mathrm{deg}$.

In order to more clearly show the incident angle dependency of the region of validity for the three approximate theories, error maps similar to those shown in Fig. 3 at normal incidence are provided in Fig. 6 for the incident angles of 20, 40, and 60 deg. The shifting of the error contour lines with an increasing incident angle can be studied; however, if an error value of $\epsilon <0.2$ is maintained as a criterion for validity, the entire domain of 2-D surface parameter space illustrated remains valid for the GHS theory at ${\theta }_i=20\mathrm{deg}$ and at ${\theta }_i=40\mathrm{deg}$. Not until ${\theta }_i=60\mathrm{deg}$ does a portion of this domain (upper central) exhibit an error value greater than $\epsilon =0.2$. However, within the small slope regime (lower right), it still shows a small error value, $\epsilon <0.05$, for very large incident angles. It can be seen in Fig. 6 that, if a different criterion for validity is chosen, or if a particular region in the domain is of interest, the KA theory may actually be more accurate than the GHS theory.

Fig. 6

Error contour maps for the SPM with (a) ${\theta }_i=20\mathrm{deg}$, (d) ${\theta }_i=40\mathrm{deg}$, and (g) ${\theta }_i=60\mathrm{deg}$, for the KA with (b) ${\theta }_i=20\mathrm{deg}$, (e) ${\theta }_i=40\mathrm{deg}$, and (h) ${\theta }_i=60\mathrm{deg}$, and for the GHS with (c) ${\theta }_i=20\mathrm{deg}$, (f) ${\theta }_i=40\mathrm{deg}$, and (i) ${\theta }_i=60\mathrm{deg}$.

In this section, scattering intensity predictions by the SPM, KA, and GHS are compared to the rigorous MoM predictions for surfaces having Gaussian surface PSD for the 1-D TE polarization case. Using the linear-error value, the region of validity of those three approximate theories is obtained in surface parameter space. The domain of validity of the SPM is largely restricted to the smooth surface criterion, but the domain is broadened as the angle of incidence is increased. The domain of the KA depends on the rms slope of the random surfaces and it shrinks as the angle of incidence is increased. Regarding the GHS method, its domain of validity is broader than the other two approximate methods and it becomes smaller when the incident angle is larger.

4.1.

Scattering Intensity Distribution

Figure 7 shows the incoherent scattered intensity distributions predicted by the SPM (blue dashed), KA (green dotted), GHS (red solid), and MoM (black asterisk) for the case of (a) ${\widehat{\sigma }}_{\mathrm{tot}}=0.025$, $c=1.4$, and ${\theta }_i=40\mathrm{deg}$ and (b) ${\widehat{\sigma }}_{\mathrm{tot}}=0.6$, $c=2.8$, and ${\theta }_i=0\mathrm{deg}$ in the logarithmic scale. In Fig. 7(a), since the rms roughness is small, both the SPM and MoM results agree with each other as expected. The log-error value of the SPM is $\epsilon *=0.03$ which is quite a smaller value than those of the KA ($\epsilon *=0.47$) and GHS ($\epsilon *=0.07$). In Fig. 7(b), the surface is quite rough and the SPM is out of its valid domain, but the KA shows a good agreement near the specular direction. However, the KA prediction fails at the large scattering angles leading to a log-error value $\epsilon *=0.20$. In both cases, the GHS prediction shows a quite good agreement not only near the specular direction and but also over the whole range of scattering angles.

Fig. 7

Scattering intensity distributions by the SPM, KA, GHS, and MoM for the case of (a) ${\hat{\sigma }}_{\mathrm{tot}}=0.025$, $c=1.4$, and ${\theta }_i=40\mathrm{deg}$ and (b) ${\hat{\sigma }}_{\mathrm{tot}}=0.6$, $c=2.8$, and ${\theta }_i=0\mathrm{deg}$ in the logarithmic scale.

Figure 8 shows the predicted scattered intensity for three different cases of surface parameter sets. In Fig. 8(a), the scattering intensity distributions for the case of ${\widehat{\sigma }}_{\mathrm{tot}}=0.1$, $c=1.4$, and ${\theta }_i=40\mathrm{deg}$ are plotted in the logarithmic scale and the SPM shows a good agreement to the MoM except near specular direction leading to $\epsilon *=0.07$. It is quite interesting that the SPM prediction agrees well at the large scattering angles even when the rms roughness is not quite small. Regarding the GHS, its prediction is superior near the specular direction than the prediction by the SPM, but it predicts little stronger scattering at large scattering angles leading to $\epsilon *=0.09$. If the distribution is plotted in linear scale as in Fig. 8(b), it is obvious that the SPM overestimates the scattering near the specular direction, but the GHS and KA predict quite accurate values at that direction. This results in a linear-error value of the SPM, KA, and GHS of ${\epsilon }_{\mathrm{SPM}}=0.12$, ${\epsilon }_{\mathrm{KA}}=0.09$, and ${\epsilon }_{\mathrm{GHS}}=0.02$, respectively. Figure 8(c) shows the scattering predictions for the case of ${\widehat{\sigma }}_{\mathrm{tot}}=0.3$, $c=2$, and ${\theta }_i=0\mathrm{deg}$, whose rms roughness is moderately large. The SPM fails to predict the scattered intensity properly near specular direction and the KA fails to accurately predict the scattered intensity both near specular direction and at large scattering angles. In Fig. 8(d), the rms roughness is the same as Fig. 8(c) but the slope is larger ($c=2.8$). Again, the SPM overestimates scattering near the specular direction, and the KA prediction deviates at the large scattering angles. However, in both the cases of Figs. 8(c) and 8(d), the GHS has good accuracy over all scattering angles for both small and relatively large incident angle cases.

Fig. 8

Scattering intensity distributions by the SPM, KA, GHS, and MoM for the case of (a) ${\hat{\sigma }}_{\mathrm{tot}}=0.1$, $c=1.4$, and ${\theta }_i=40\mathrm{deg}$ in the logarithmic scale, (b) the surface parameters are the same to those of (a) but scattering intensity is plotted in linear scale, (c) ${\hat{\sigma }}_{\mathrm{tot}}=0.3$, $c=2$, and ${\theta }_i=0\mathrm{deg}$ in the logarithmic scale (d) ${\hat{\sigma }}_{\mathrm{tot}}=0.3$, $c=2.8$, and ${\theta }_i=60\mathrm{deg}$ in the logarithmic scale.

Figure 9 shows the dependency of the validity of the SPM, KA, and GHS with the variations in incident angle for the two surface parameter sets. In Fig. 9(a), the scattered intensity distributions for the 0, 30, and 60 deg of incident angle for the case of ${\widehat{\sigma }}_{\mathrm{tot}}=0.15$ and $c=2$ are provided. The SPM predicts overly stronger scattering at the specular direction and the amount of incorrectness is reduced when the angle of incidence becomes larger. The KA prediction shows a good agreement to the rigorous MoM result near the specular direction but its large scattering angle behavior is very inaccurate regardless of incident angle. The GHS predictions agree quite well with the rigorous MoM result over all scattering angles and for all the three incident angles; however, the predictions are slightly high at large scattering angles. Figure 9(b) illustrates the scattering predictions for the surface whose roughness and the slope parameter $c$ values are larger than the previous one (${\widehat{\sigma }}_{\mathrm{tot}}=0.3$ and $c=2.4$) and the overall shape of the scattering distribution is less sharply peaked in the specular direction than the case of Fig. 9(a). Similarly to Fig. 9(a), the KA prediction is valid only near the scattering angle, but the GHS predicts quite accurately not only at the specular ray direction but also at large scattering angles. Note that in Fig. 9(b), the SPM results are not plotted.

Fig. 9

Scattering intensity distributions by the SPM, KA, GHS, and MoM for the case of (a) ${\hat{\sigma }}_{\mathrm{tot}}=0.15$, $c=2$, and ${\theta }_i=0\mathrm{deg}$ and (b) ${\hat{\sigma }}_{\mathrm{tot}}=0.3$, $c=2.4$, and ${\theta }_i=0\mathrm{deg}$.

4.2.

Comparative Visualization of the Domain of Validity

To reveal the dependency of the validity of the SPM according to the surface parameters, both the log- and linear-error values for the SPM with the normal incidence are calculated and shown in Figs. 10(a) and 10(b), respectively. Figure 10(b) shows that the validity of the SPM is largely restricted to smooth surface criteria regardless of the slope parameter. However, if the log-error values are calculated as shown in Fig. 10(a), the valid region of the SPM is dramatically broadened. This can be explained by the fact that, as shown in the previous section, the scattering intensity distribution at large angles predicted by the SPM agrees very well to the rigorous MoM prediction not only for the surface whose rms roughness is very small but also moderately large.

Fig. 10

(a) SPM log-error map and (b) SPM linear-error map for the normal incidence.

Figure 11 shows error maps for the KA using (a) the log-error values and (b) the linear-error values at normal incidence. In Fig. 11(b), the KA prediction can be considered to be valid where the ratio of the rms roughness to the slope parameter is small. However, when the log-error value is calculated, Fig. 11(a) indicates that the KA fails to make accurate predictions over almost the entire surface parameter domain. This characteristic results from the fact that the KA prediction shows relatively good agreement to the prediction of the MoM near the specular direction when the ratio of the rms roughness to the slope parameter is small, but it overestimates the large angle scattering over almost the entire surface parameter domain.

Fig. 11

(a) KA log-error map and (b) KA linear-error map for the normal incidence.

The log- and linear-error values for the GHS theory are calculated and illustrated in Fig. 12. In Fig. 12(b), similar to Fig. 11(b), the calculated error values are smaller when the ratio of the rms roughness to the slope parameter is smaller. However, the valid domain of the GHS is much broader than that of the KA. In Fig. 12(a), comparing to Fig. 10(a), the log-error values for the GHS are larger than those for the SPM in some regions and it comes from the fact that the GHS overestimates the large angle scattering. However, the error map in Fig. 12(a) shows that the prediction of the GHS show a good agreement to the rigorous MoM prediction in linear scale over the larger surface parameter domain than the other two approximate methods with reasonable accuracy.

Fig. 12

(a) GHS log-error map and (b) GHS linear-error map for the normal incidence.

In order to compare the incident angle dependency of the region of validity for the SPM, KA, and GHS, the error maps using log-error values for the three methods in the case of ${\theta }_i=20$, 40, and 60 deg are shown in Fig. 13. The valid domain of the SPM appears that it is not significantly affected by the change of the incident angle. Meanwhile, for both the GHS and KA, the valid area in the domain reduces with an increasing incident angle. But, in these three incident angle cases, the error maps show that the GHS theory again has a wider range of validity for large incident angles than those of the KA method.

Fig. 13

SPM log error map (a) ${\theta }_i=20\mathrm{deg}$, (d) ${\theta }_i=40\mathrm{deg}$, and (g) ${\theta }_i=60\mathrm{deg}$; KA log error map (b) ${\theta }_i=20\mathrm{deg}$, (e) ${\theta }_i=40\mathrm{deg}$, and (h) ${\theta }_i=60\mathrm{deg}$; and GHS log error map (c) ${\theta }_i=20\mathrm{deg}$, (f) ${\theta }_i=40\mathrm{deg}$, and (i) ${\theta }_i=60\mathrm{deg}$.

4.3.

Scattering Intensity Distribution for Different Shoulder Frequencies

In the previous sections, the value of the parameter $b$ is fixed to 20. In this section, the scattered intensity comparison is performed for different $b$ parameter values. Figure 14(a) shows the surface PSD function of the three surfaces. The surface A has $0.02\lambda$ of rms roughness with $b=20$, the surface B has $0.045\lambda$ of rms roughness with $b=100$, and the surface C has $0.14\lambda$ of rms roughness with $b=500$. All the three surfaces have the same slope parameter $c$ value of 2. Figure 14(b) shows the rigorous MoM predictions in log–log scale at normal incidence and it shows that different shoulder frequencies with the same slope parameter change only the small angle scattering behavior.

Fig. 14

(a) Surface PSD functions for the surface A, B, and C with the same slope parameter $c=2$ and different $b$ parameter values. (b) The scattering intensity distributions predicted by the MoM for the three surfaces in log–log scale at normal incidence.

Figure 15(a) shows the predicted scattering distribution for the surface A in the case of ${\theta }_i=0\mathrm{deg}$ and ${\theta }_i=60\mathrm{deg}$ for the four different methods. Since the rms roughness is small, the SPM predicts quite accurate results, the GHS has reasonable accuracy for entire scattering and incident angles, and the KA has a good agreement near specular direction, but fails to predict at large angle scattering behavior. This trend holds for the surface B as shown in Fig. 15(b) where the rms roughness is moderately larger and the $b$ parameter value is also moderately smaller than the case of Fig. 15(a).

Fig. 15

Predicted scattering intensity distributions by the SPM, KA, GHS, and MoM in the case of ${\theta }_i=0\mathrm{deg}$ and ${\theta }_i=60\mathrm{deg}$ for (a) the surface A and (b) the surface B.

Figure 16(a) shows the predicted scattered intensity for the surface C whose shoulder frequency is much smaller and rms roughness is much larger than surfaces A and B. The behavior of the scattered intensity is very strongly peaked in the specular direction when plotted in a linear-log scale. That the SPM fails to accurately predict the scattered intensity near the specular direction is shown in the log–log plot of Fig. 16(b). The GHS and KA theories show a good agreement with the rigorous MoM prediction as shown in Fig. 16(b). But unlike the KA, the GHS predicts reasonably accurate predictions at large scattering angles as well as near specular direction as shown in Fig. 16(a).

Fig. 16

Predicted scattering intensity distributions by the SPM, KA, GHS, and MoM for surface C in the case of (a) ${\theta }_i=0\mathrm{deg}$ and ${\theta }_i=60\mathrm{deg}$ in linear-log scale and (b) ${\theta }_i=0\mathrm{deg}$ in log–log scale.

Due to the limitation of the computer memory and computation time, the simulation for the surface with much smaller shoulder frequency is not calculated. However, as shown in this section, it is strongly possible that the trend of the region of validity obtained with $b=20$ still holds for the cases with much smaller shoulder frequency. The full analysis of the region of validity for different shoulder frequencies remains a topic for later research.