PUBLICATIONS. Radio Science. Roughness effects on absorption and scattering of left-handed materials RESEARCH ARTICLE 10.

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1 PUBLICATIONS RESEARCH ARTICLE 1.12/213RS5253 Key Points: Exponentially correlated roughness has negative effects on absorption of LHM Exponentially correlated roughness reduces LHM absorption dramatically SPM2 for TE case of exponential correlated LHMs surfaces is invalid at any time Correspondence to: P. Xu, Citation: Xu, P., and J.-C. Shi (214), Roughness effects on absorption and scattering of left-handed materials, Radio Sci., 49, 4 414, doi:1.12/213rs5253. Received 27 JUN 213 Accepted 22 MAY 214 Accepted article online 28 MAY 214 Published online 17 JUN 214 Roughness effects on absorption and scattering of left-handed materials Peng Xu 1 and Jian-Cheng Shi 2 1 School of Electronic Information, Wuhan University, Wuhan, China, 2 Institute of Remote Sensing Applications, Chinese Academy of Sciences, Beijing, China Abstract Bistatic scattering and absorptivities from both Gaussian and exponentially correlated rough surfaces of the left-handed materials (LHMs) are studied numerically for 2-D geometries in a numerical Maxwell model. The multilevel UV method is employed to accelerate the matrix equation solver. Accuracy is ensured by energy conservation check. Using a set of physical surface parameters of root-mean-square heights and correlation lengths, numerical results are illustrated for absorptivities, bistatic scattering, and bistatic transmission of the LHMs. The numerical results show that the presence of the second opposite transmission, from lower medium to upper one for exponentially correlated LHM surfaces, dramatically reduces the absorptivities and enhances the bistatic scattering remarkably. In particular, the exponentially correlated roughness has negative effect on absorption for transverse electric (TE) case. Then all absorptivities for TE case are below that of the planar interface and decrease continuously with the increasing roughness. Due to both the Brewster angle effect and the opposite transmission across exponentially correlated interface, the absorptivities for the transverse magnetic (TM) case of the LHMs increase at first and decrease subsequently down below that of the planar interface. Both polarimetric absorptive characteristics are in completely contrary to those of the right-handed materials (RHMs), where the absorptivities for both the TE and TM cases are monotonic increasing with the increase of roughness because of its fine-scale surface features and the increasing effective surface areas. For Gaussian correlated LHM surfaces, however, the second opposite transmission does not exist, thus does not show any evident differences from RHMs, except for the maximum transmission in different directions and beam broadening. In addition, comparisons are also made with the second-order small perturbation method and Kirchhoff s approximation. In contrary to RHM, the second-order small perturbation method is no longer valid for TE case for both absorptivity and scattering from exponentially correlated LHM surfaces even for very small perturbation cases. Similar to RHM surfaces, the Kirchhoff s approximation can be only valid to the LHM surfaces with Gaussian correlation function. 1. Introduction Veselago [1968] predicted that electromagnetic plane waves in a left-handed material (LHM) having simultaneously negative permittivity and permeability should propagate in a reversed direction with respect to that of energy flow. Such LHM is used to obtain super resolution well below the diffraction limit as superlens proposed by Pendry [2], and the later works verified this claim [Shelby et al., 21; Ziolkowski and Heyman, 21; Zhang et al., 22; Cui et al., 24; Chew, 25; Lei and Cui, 26]. Later, it has become an attractive research topic of negative refractive index materials [Hu and Chui, 22; Smith and Kroll, 2] due to their dramatically different electrodynamic properties, including anomalous refraction, reversed Doppler shift, negative Cerenkov radiation, and the exciting perfect lens. The smooth LHM scattering was also discussed by Houck et al. [23], Wu et al. [25], and Yao et al. [27]. Recently, Narimanov et al. [213] found that roughness reduced reflection from metamaterial with negative permittivity and positive permeability. However, to the best of our knowledge, past literatures did not take the surface roughness into account for LHMs. How would their properties be if imposing roughness upon LHMs? In this paper our main goal is to build behaviors of absorption, scattering, and transmission in rough LHM surfaces. Previously, it was analyzed for the scattering and microwave emission from the right-handed materials (RHMs) surfaces with exponential correlation function by characterizing its dependences on polarization, incident angle, and frequency [Xu and Tsang, 27; Xu et al., 21]. It is well known that exponential correlation functions do not have root-mean-square (RMS) slopes when the spectrum is extended to infinity. XU AND SHI 214. American Geophysical Union. 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2 1.12/213RS5253 So it cut off the high-frequency components of the exponential spectrum by cubic spline interpolation technique, for such truncation has less effect on scattering and emission. The obtained results demonstrate that the emissivities from exponentially correlated RHM surfaces increase with the increasing roughness and have larger backscattering, because the effective surface areas seen by the microwaves also increase due to its fine-scale features with large slopes of such surfaces. This is different from Gaussian correlated roughness which has almost no contribution to emissivities and has stronger forward scattering. In this paper, the surface roughness effects are investigated on the absorptivity, which is equal in value to emissivity, bistatic scattering, and bistatic transmission of the LHMs. It conducted the numerical Maxwell model with 2-D simulations (NMM2D) for absorptivities, scattering, and transmission of LHM surfaces with both Gaussian and exponential correlation functions in a manner similar to RHMs [Xu and Tsang, 27]. The multilevel UV method is employed to accelerate the matrix equation solver [Tsang et al., 24a, 24b]. Accuracy is ensured by energy conservation check. Using a set of physical surface parameters of RMS heights and correlation lengths, numerical results are illustrated for absorptivities, bistatic scattering, and bistatic transmission of the LHMs and compared with those of RHMs. For exponentially correlated LHM surfaces, in completely contrary to those of RHMs, they have abnormal characteristics of absorptivities and scattering, not only the bistatic scattering coefficients for both polarizations become larger remarkably, the bistatic transmission coefficients beam is broadened, but the absorptivities for transverse electric (TE) case reduce dramatically below that of the planar interface. Namely, exponential roughness for LHMs has negative effects on absorptivities for TE case, and they decrease continuously with the increase of roughness. In addition, their absorptivities for transverse magnetic (TM) case also show distinct behavior, which go up first and then down below that of the planar interface with the increasing roughness. The explanation for these fascinating characteristic in exponentially correlated LHM surfaces is also given. The negative index of refraction in LHMs can make wave transmit across such exponentially correlated rough interfaces with large slopes once again from lower LHM to upper air, then its absorptions for TE case decrease remarkably. Combined with the Brewster angle effect, this opposite second transmission results in the reduction of absorptivities for TM case below that of the planar interface for only large roughness with large slopes. Such new physical phenomenon in rough LHM surfaces may have important implications on the usage of LHMs as special absorbing materials. For Gaussian correlated LHM surfaces, however, there are no obvious differences from those of RHMs except for the transmission beam broadening and the maximum transmission in different directions. In section 2, the formulations of scattering and transmission of LHMs are given with discretization of method of moment (MOM) equation. In section 3, numerical results including absorptivity, bistatic scattering, and bistatic transmission are illustrated for both Gaussian and exponentially correlated LHM surfaces at different roughness. Results are compared with the corresponding results of RHMs, and the physical explanations are given for completely different absorption properties between LHMs and RHMs with exponential correlation function, between Gaussian and exponential correlation functions of LHMs, and between TE and TM cases from exponentially correlated LHM surfaces. Simulation results are also compared with the second-order small perturbation method (SPM2) and Kirchhoff s approximation (KA). It is found that the SPM2 scattering results are no longer valid for TE case from exponentially correlated LHM surfaces even for very small perturbation cases, although it can be valid for the corresponding RHM cases. The SPM2 absorptivity results for both polarizations actually do not exist for exponentially correlated LHM surfaces due to divergent spectral integral, while they still exist for TE case from exponentially correlated RHM surfaces. The KA can be only valid to the LHMs with Gaussian correlation function. Finally, conclusion is drawn. 2. Wave Scattering and Transmission Formulations for LHM Similar to Xu and Tsang [27], consider a tapered plane wave, ψ inc (x, z), from upper air impinging upon a rough LHM surface with a random height profile z = f(x). Let ψ and ψ 1 denote the wave function for upper air medium and lower dielectric medium, respectively. Let E ¼ ^yψ for the TE case and H ¼ ^yψ for the TM case. These are the following dual surface integral equations [Tsang et al., 1985]: s dxgðr; rþuðr Þ- s ds^n ψðr Þ gðr; rþ þ 1 2 ψðþ¼ψ r incðþ r ρ s dxg 1 ðr; rþuðr Þ- s ds^n ψðr Þ g 1 ðr; rþ 1 2 ψðþ¼ r (1a) (1b) XU AND SHI 214. American Geophysical Union. All Rights Reserved. 41

3 1.12/213RS5253 where denotes a principle value of integral and g and g 1 are the 2-D Green s functions of the upper and lower medium, respectively. In equation (1b), ρ = μ 1 /μ = 1 and ε 1 /ε for the TE and TM cases of LHMs, respectively. And, ψðþ¼ψ r ðx; fþ ¼ ψ 1 ðx; fþ (2a) UðÞ¼ r qffiffiffiffiffiffiffiffiffiffiffiffi 1 þ f 2 xð^n ψðx; fþþ (2b) The MOM with rooftop basis function and Galerkin s method are used to discretize the integral equations. The multilevel UV method [Tsang et al., 24a, 24b] with a preconditioner of the diagonal blocks inverses was employed to accelerate solution. Accuracy is also improved by near-field integration, fine discretization, and cubic spline interpolation of surfaces [Xu and Tsang, 27]. Absorptivity and reflectivity are calculated in a manner similar to Xu and Tsang [27]; the simulation accuracy is ensured by energy conservation check of a(θ i )+r(θ i ) = 1. All the energy tests in this paper satisfy energy conservation to less than 1%. The bistatic scattering coefficient and the bistatic transmission coefficient are given, respectively: σðθ s Þ ¼ ψ ð s NÞ ðθ s Þ 2 16πηkP inc ψ ðnþ t ðθ t Þ 2 k 1 tðθ t Þ ¼ Re expð2imðk 1 ÞrÞ 16πηkP inc ρjk 1 j ¼ ψ ðnþ t ðθ t Þ 2 if Imðk 1 Þ ¼ 16πηkP inc jρj (3a) (3b) where ψ ð s NÞ ðθ s Þ ¼ s dx½u x ðþψðþik x ðf x sin θ s cos θ s ÞŠexpðikðsin θ s x þ cos θ s fðþ x ÞÞ (4a) ψ ðnþ t ðθ t Þ ¼ s dx ρu x ½ ðþψðþik x 1 ðf x sin θ t þ cos θ t ÞŠexpðik 1 ðsin θ t x cos θ t fðþ x ÞÞ (4b) and p k 1 ¼ signðμ 1 Þk ffiffiffiffiffiffiffiffiffi μ 1 ε 1 (5) 3. Numerical Results and Discussion In this section, the numerical simulation results are illustrated for the absorptivities, bistatic scattering coefficients, and bistatic transmission coefficients from LHMs rough surfaces with both exponential and Gaussian correlation functions. In the past, the RHMs rough surfaces have been studied extensively. However, LHMs negative index of refraction would bring distinct electromagnetic characteristics of absorption and scattering from rough surfaces. In the simulations, the tapering parameter of the incident wave is selected to be one fourth. The frequency is set as 1.5 GHz, and the permitivities are selected based on the experimental verification by Shelby et al. [21]. The stabilized biconjugate gradient iterations are stopped when the error (L2 norm) is less than.1%. The bistatic scattering coefficients γ(θ s ) = 2πσ(θ s ) are plotted in the plane of incidence with specular scattering at θ s = θ i and backscattering at θ s = θ i. The bistatic transmission coefficients Γ(θ t )=2πt(θ t ) are plotted in the lower half space with specular transmission at negative refractive angle for LHMs and at positive refractive angle for RHMs. XU AND SHI 214. American Geophysical Union. All Rights Reserved. 42

4 1.12/213RS5253 absorptivity absorptivity TE, LHM.2 TE, RHM TM, LHM.1 TM, RHM rms height (cm) Figure 1. Comparison of the average absorptivities and absorptivity differences versus RMS heights from Gaussian correlated surfaces at 1.5 GHz between LHMs and RHMs. Correlation length: l =6.cm, incident angle: θ i =4, andε 1 = ± 7.29, μ 1 =±1forRHMand LHM, respectively. Absorptivity. Absorptivity difference from planar interfaces Roughness Effects on Absorptions Gaussian correlated surface is much smoother than other types of surfaces, because it contains much less highfrequency components of spectral density. Xu et al. [21] showed that Gaussian roughness has small contributions to surface absorptivity for RHMs. In Figure 1a, it is shown for the absorptivities of LHMs as a function of RMS heights with an identical correlation length of 6 cm. They are almost consistent with RHMs for both polarizations as expected. To better observe the roughness effect, their differences from those of planar interfaces are also plotted as shown in Figure 1b. It clearly reveals that relatively small contributions are generated for TM case from Gaussian correlated surfaces; even at moderate RMS heights, the absorptivity differences are almost negative. And, there are contributions for TE case only at large roughness. absorptivity absorptivity (rough planar) TE, LHM TE, RHM.4 TM, LHM TM, RHM TE, LHM TE, RHM TM, LHM TM, RHM rms height (cm) Figure 2. Comparison of average absorptivities and absorptivity differences versus RMS heights from exponentially correlated surfaces at 1.5 GHz between LHMs and RHMs. Correlation length: l =6.cm, incident angle: θ i =4, andε 1 = ± 7.29, μ 1 =±1forRHMand LHM, respectively. Absorptivity. Absorptivity difference from planar interfaces. Different from Gaussian spectra, exponential spectral density has much stronger high-frequency components corresponding to many fine structures of the surface profile, resulting fast fluctuations of the induced surface current when excited by the incident wave. Xu and Tsang [27] and Xu et al. [21] reveal that exponentially correlated RHM surfaces have large backscattering due to their fine-scale structures, and their absorptivities increase constantly with increasing roughness for both the TE and TM cases. To find how the LHMs exponential roughness affecting on absorptivities, in Figure 2, it is shown for the absorptivities and their differences from those of planar interfaces as a function of RMS heights with an identical correlation length of 6 cm by comparison with those of RHMs. For RHMs, the absorptivities and absorptivities differences for both polarizations constantly increase with the increase of RMS heights as stated by Xu et al. [21]; the exponential roughness always has positive effect on RHM absorption. With the increase of RMS height, the absorption shows saturation with the increase of XU AND SHI 214. American Geophysical Union. All Rights Reserved. 43

5 1.12/213RS5253 Table 1. Comparison of the Average Absorptivities at 1.5 GHz Between Lossless and Lossy Dielectrics for Three Kinds of Surfaces: Exponentially Correlated RHM Surfaces, Exponentially Correlated LHM Surfaces, and Gaussian Correlated LHM Surfaces, All With h = 1. cm, l = 6. cm, and θ i =4 Absorptivity Material Correlation Function k 1 k TE TM RHM exponential i LHM exponential i LHM Gaussian i roughness. However, for LHMs as shown in Figure 2, the trend for TE case is reversed dramatically. The absorptivities and absorptivities differences constantly decrease with increase of RMS heights, and all the absorptivities are less than the planar interface, which means that the exponential roughness always has negative effect on absorption for TE case. On the other hand, for TM case, they increase at first and decrease subsequently with increase of RMS heights; in particular, after moderate roughness, the absorptivities for TM case also become less than that of the planar interface. In other words, not only absorptivities behaviors of LHMs are different from those of RHMs, but LHM absorptivity behaviors for TM case are also different from those for TE case. Now we introduce relatively small permittivity loss in lower media from ε 1 = ± 7.29 to ε 1 = ± i, and then the wave numbers change from k 1 = ± 2.7k to k 1 = (± i)k. In Table 1, it is shown for the comparisons of absorptivities between lossless and lossy dielectrics for three cases: exponentially correlated RHM surface, exponentially correlated LHM surface, and Gaussian correlated LHM surface. Because the given imaginary part of the wave number is very small, the differences for both polarizations between lossless and lossy ones are almost negligible for the first and third cases. However, there are huge differences for the second case, exponentially correlated LHM surface as shown in Table 1, which indicates that its absorptivities are very sensitive to the permittivity, and there must be a distinct physical behavior of absorption and scattering for this case. Next the physical explanations will be given. Figure 3. Comparison of the second opposite transmission from the lower medium to the upper one among exponentially correlated RHM surface, exponentially correlated LHM surface, and Gaussian correlated LHM surface. There is the second transmission as like E t2 y only for the case of exponentially correlated LHM surface. Exponentially correlated RHM surface. Exponentially correlated LHM surface. (c) Gaussian correlated LHM surface Physical Insights on Fascinating Properties of Absorptivity From LHMs Rough Surfaces With Exponential Correlation Function The large increases in absorptivity for exponentially correlated RHM surfaces are due to the increase in surface area. The effective surface area seen by the wave also increases with the increase of XU AND SHI 214. American Geophysical Union. All Rights Reserved. 44

6 1.12/213RS5253 roughness. And, the absorptivity shows saturation with the increase of roughness as shown in Figure 2a. However, in contrary to RHMs, the large decreases in absorptivity instead for exponentially correlated LHM surfaces are because of the opposite second transmission from lower LHM space to upper air space. It is clearly seen by comparison between Figures 3a and 3b that the refraction wave E t 1y can cross back to the upper half space only for case of exponentially correlated LHM surface. Just because it can cut across the rough interface again due to its refractions in opposite direction, the second opposite transmission wave E t2 y results in more bistatic scattering coefficients. That is also why the absorptivities are larger than the planar one for exponentially correlated RHM surfaces, while they can be much less than the planar one for exponentially correlated LHM surfaces as shown in Figure 2a. Due to large surface slopes, Figure 4. Comparison of the reflection at Brewster angle between at grazing incidence, it seems that there may the TE and TM cases from exponentially correlated LHM surfaces. be also the second opposite transmission on At incident Brewster angle, there is still reflection E r y for TE case, the RHM surface such as the place of P 1 as while H r y vanishes for TM case. For TE case. For TM case. shown in Figure 3a, where the wave E i y is the incident on the left-hand side of the local normal, but these possible opposite transmission can be negligible because it is nonline of sight at P 1.In addition, on the surface points with negative slopes such as P 2, it is impossible for RHMs to have opposite transmission even at grazing incidence, while it is still possible for LHMs at grazing incidence as shown in Figure 3b. On the other hand, not all the types of LHM surfaces can make absorptivities less. The Gaussian correlated surface is much smoother than the exponentially correlated one. As shown in Figure 3c, there is also no second transmission back to upper medium for Gaussian correlated LHMs surfaces, although its refraction wave is in the opposite direction. This is why there are not obvious differences in the absorptivities between RHMs and LHMs for Gaussian correlated surfaces as shown in Figure 1a. Corresponding to differences shown in Table 1, that is also why only absorptivities from exponentially correlated LHM surfaces are sensitive to the imaginary part of permittivity. Hence, there are two opposite factors affecting the absorptivity for the case of exponentially correlated LHM surfaces. One is the large increases of effective area seen by the wave with the increasing roughness, which results in the increase of absorptivity and the decrease of scattering; the other is the large increase of opposite re-refraction from lower medium to upper one with the increasing roughness too, which results in the decrease of absorptivity and the increase of scattering. Just such opposite second transmission E t2 y makes absorptivities for TE case of exponentially correlated LHM surfaces, less than the planar interface and decrease continuously with the increase of roughness. For the TM case of exponentially correlated LHM surfaces, the two factors still exist. However, comparing TM case with TE case as shown in Figure 4, there is less reflection H r y for TM case than the reflection E r y for TE case because of the Brewster angle effect, which brings more absorption and less scattering. Thus, the absorptivities for TM case do not show continuous decrease, but increase at first, and decrease subsequently with the increasing roughness. Only after moderate roughness, does the factor of the opposite second transmission become predominated, and then it has negative absorptivity differences too as shown in Figure 2b. So the exponential roughness of LHMs usually shows negative effect on absorption for TE case, while it shows negative effect for TM case only after moderate roughness. With the increase of incident angle θ i, both the local incident angle θ li and the local refraction angle at the place of P 1 by Snell s law in Figure 3b get smaller, then it also become less for the refracted waves cutting XU AND SHI 214. American Geophysical Union. All Rights Reserved. 45

7 1.12/213RS5253 absorptivity absorptivity h=.6 cm, TE, RHM.2 h= cm, TE, planar h=.6 cm, TE, LHM h=1.5 cm, TM, RHM.6 h= cm, TM, planar h=1.5 cm, TM, LHM incident angle (deg.) Figure 5. Comparison of the average absorptivities between RHMs and LHMs exponentially correlated surfaces relative to planar interfaces. RMS heights: h =.6 cm for TE case and h = 1.5 cm for TM case, both l = 6. cm at 1.5 GHz, and ε 1 = ± 7.29, μ 1 = ± 1 for RHMs and LHMs, respectively. Absorptivities are plotted as a function of incident angles. For TE case. For TM case. across the rough interface. On the contrary, they become larger and more at the place of P 2 unless it is no longer line of sight. In Figure 5, we compare absorptivities as a function of incident angles, between LHM and RHM surfaces with exponential correlation function. As shown in Figure 5a with a fixed RMS height of.6 cm and a correlation length of 6 cm, it clearly reveals that it is positive effects on RHM absorptivities (hollow circles) for TE case in the range of all incident angles, which denotes that the opposite transmission can be negligible even at grazing incidence. However, it has negative effects on LHMs (solid circles) in a wide range of incident angles up to 7. Although the differences from planar interfaces (solid line) decrease with the increase of incident angles, the second opposite transmission still plays an important role in reducing absorptivities even at grazing incidence, where the LHM absorptivities are still much less than the RHMs. In Figure 5b, we fix a moderate RMS height of 1.5 cm and a correlation length of 6 cm, where the opposite transmission is predominated on LHMs exponentially correlated surfaces, the absorptivities from LHMs for TM case are all much less than RHMs in all range of incident angles too. On the other hand, although the RHM absorptivities are also less than those of the planar interfaces at incident angles greater than 55, that is not because of the existence of the opposite transmission on RHM surface but because the planar interfaces have very large absorptivities in the neighborhood of Brewster angle of θ B = Roughness Effects on Scattering bistatic scattering coefficient (db) h=1 cm, TE, LHM h=1 cm, TE, RHM h=3 cm, TE, LHM -6 h=3 cm, TE, RHM bistatic scattering coefficient (db) h=1 cm, TM, LHM h=1 cm, TM, RHM h=3 cm, TM, LHM -6 h=3 cm, TM, RHM Figure 6. Comparison of the average bistatic scattering coefficients γ(θ s ) between the LHMs and RHMs Gaussian correlated surfaces at 1.5 GHz. Correlation length: l = 6. cm, incident angle: θ i =4, and ε 1 = ± 7.29, μ 1 = ± 1 for RHM and LHM, respectively. For TE case. For TM case. As mentioned before, the absorptions for Gaussian correlated LHM surfaces are consistent with those for RHMs; accordingly, their bistatic scattering coefficients are also consistent with those for RHMs as shown in Figure 6. At a moderate RMS height of 1 cm, both polarizations show strong forward scattering. With the RMS heights up to 3 cm, the forward scattering coefficients become weaker, while their backscatterings get stronger. However, exponential spectral density has much stronger high-frequency components corresponding to many fine structures of the surface profile. Figure 7 illustrates the comparisons of bistatic scattering coefficients between XU AND SHI 214. American Geophysical Union. All Rights Reserved. 46

8 1.12/213RS5253 bistatic scattering coeff. (db) h=.2 cm, TE, LHM h=.2 cm, TE, RHM h=.6 cm, TE, LHM h=.6 cm, TE, RHM h=1. cm, TE, LHM h=1. cm, TE, RHM (c) (e) (g) -2 h=1.5 cm, TE, LHM h=1.5 cm, TE, RHM bistatic scattering coeff. (db) h=.2 cm, TM, LHM h=.2 cm, TM, RHM h=.6 cm, TM, LHM h=.6 cm, TM, RHM (d) (f) -2 h=1. cm, TM, LHM h=1. cm, TM, RHM (h) -2 h=1.5 cm, TM, LHM h=1.5 cm, TM, RHM Figure 7. Comparison of the average bistatic scattering coefficients γ(θ s ) between LHMs and RHMs exponentially correlated surfaces for both the TE and TM cases at 1.5 GHz. Correlation length: l = 6. cm, incident angle: θ i =4, and ε 1 = ± 7.29, μ 1 = ± 1 for RHM and LHM, respectively. RMS heights: (a and b) h =.2 cm, (c and d) h =.6 cm, (e and f) h = 1 cm, and (g and h) h = 1.5 cm. RHMs and LHMs, for both polarizations from exponentially correlated surfaces at different RMS heights with a fixed correlation length of 6 cm. For TE case, the backscattering at θ s = 4 and the forward scattering at specular angle of 4 for LHMs are larger than those for RHMs as shown in Figures 7a, 7c, 7e, and 7g; in particular, the backscattering are much larger even at a very small roughness case as shown in Figure 7a, which is just because of the second opposite refraction in LHMs. On the other hand, the comparisons for TM case between LHMs and RHMs become complicated, only after moderate roughness are both the backscattering and the forward scattering of LHMs much larger than those of RHMs as shown in Figure 7h, where the second opposite refraction effect predominates over the Brewster angle effect. Seen from Figure 7, it is also found that their backscattering coefficients for both polarizations of LHMs become stronger with the increasing roughness, which is just the characteristic of the scattering from exponentially correlated surfaces. However, their specular scattering do not always become weaker with the increasing roughness because of the second opposite refraction to make scattering larger, although the specular scattering of RHMs decrease with the increasing roughness. When a relatively small permittivity loss in lower media was introduced from ε 1 =±7.29toε 1 =± i, where thewavenumberschangefrom k 1 =±2.7k to k 1 =(±2.7+.1i)k, itis found from Figure 8 that the difference of the bistatic scattering coefficients for RHMs is almost negligible between lossless and lossy surfaces with exponential correlation function, while the scattering for lossy LHMs become smaller for the TE case and larger for the TM case than those for the lossless LHMs. Such phenomena indicate that the second opposite refraction from LHM interface with exponential correlation function enlarges the difference of the bistatic scattering even given a relatively small permittivity loss. Similar to Table 1, it is also sensitive to permittivity for LHM surfaces with exponential correlation function Roughness Effects on Transmission Compared with the RHMs, the transmission beams broaden for the LHM Gaussian correlated surfaces as shown in Figure 9. That is because the anomalous refractions at LHM interfaces make the transmission XU AND SHI 214. American Geophysical Union. All Rights Reserved. 47

9 1.12/213RS TE, k 1 /k=-2.7 TE, k 1 /k= i -2 TE, k 1 /k= 2.7 TE, k 1 /k= i TM, k 1 /k=-2.7 TM, k 1 /k= i TM, k 1 /k= 2.7 TM, k 1 /k= i Figure 8. Comparison of the average bistatic scattering coefficients γ(θ s ) at 1.5 GHz between the lossless and lossy dielectric surfaces of both the LHM and RHM surfaces with exponential correlation function. RMS height: h = 1. cm, correlation length: l = 6. cm, and incident angle: θ i = 4. For TE case. For TM case. energy more disperse. In addition, at smaller roughness as shown in Figures 9a and 9b, their maximum transmission coefficients are at opposite directions of θ t = ± 13.77, which are, namely, their specular transmission directions for a planar interface, respectively, for RHMs and LHMs. This agrees with the Snell s law of a planar interface. sin 4 ¼ n 1 sin θ i ¼ ±n 2 sin θ t p ¼ ± ffiffiffiffiffiffiffiffiffi 7:29 sin ð±13:77 Þ (6) With the increase of RMS height, the directions of the maximum transmission coefficients for LHMs are moved to those for RHMs. In Figure 1, the bistatic transmission coefficients from exponentially correlated surfaces are also compared between LHMs and RHMs. In Figure 1a, their coherent transmission coefficients for TE case are also plotted. Similar to the Gaussian cases, it clearly reveals that the peaks for coherent part at θ t = ± are the specular transmission directions too, respectively, for RHMs and LHMs. All transmission beams for LHMs also broaden than those for RHMs. And, the directions of the maximum transmission coefficients for RHMs are always at positive transmission angles for different RMS height cases, while the LHM ones are moved continuously from negative to positive transmission angles with the increasing roughness as shown in Figure Comparisons With SPM2 and KA bistatic transmission coefficient (db) bistatic transmission coeff. (db) h=1 cm, TE, LHM h=1 cm, TE, RHM (c) -3 h=3 cm, TE, LHM h=3 cm, TE, RHM transmission angle (deg.) bistatic transmission coefficient (db) bistatic transmission coeff. (db) h=1 cm, TM, LHM h=1 cm, TM, RHM (d) -3 h=3 cm, TM, LHM h=3 cm, TM, RHM transmission angle (deg.) Figure 9. Comparison of the average bistatic transmission coefficients Γ(θ t ) between LHMs and RHMs Gaussian correlated surfaces for both the TE and TM cases at 1.5 GHz. Correlation length: l = 6. cm, incident angle: θ i =4, and ε 1 = ± 7.29, μ 1 = ± 1 for RHMs and LHMs, respectively. RMS heights: (a and b) h = 1 cm and (c and d) h = 3 cm. Next, the results of both bistatic scattering coefficients and absorptivities for both polarizations are compared with the SPM2 and the KA. Analytic formulas of SPM2 for only TM case of RHMs were given by Gu et al. [27]. The general formulas for both the TE and TM cases of both RHMs and LHMs are given in the Appendix A, where some new terms are supplied to make it available for TE case. Analytical formulas of RHMs for the KA were given by the appendix of Xu and Tsang [27]; later, its reflection coefficient was replaced with the reflection coefficient evaluated at the stationary phase point [Johnson et al., 27], and the LHMs are similar to those of RHMs except for signs of k 1 and k 1z. For SPM2, the incoherent bistatic scattering coefficients are plotted. The coherent reflectivity exists for all the Gaussian correlated surfaces regardless XU AND SHI 214. American Geophysical Union. All Rights Reserved. 48

10 1.12/213RS5253 bistatic transmission coefficient (db) h=.2 cm, TE, LHM -3 h=.2 cm, TE, RHM coherent, TE, LHM coherent, TE, RHM (c) bistatic transmission coefficient (db) h=.2 cm, TM, LHM h=.2 cm, TM, RHM (d) bistatic trans. coeff. (db) h=.6 cm, TE, LHM h=.6 cm, TE, RHM bistatic trans. coeff. (db) h=.6 cm, TM, LHM h=.6 cm, TM, RHM bistatic trans. coeff. (db) 1-1 (e) -2 h=1. cm, TE, LHM h=1. cm, TE, RHM transmission angle (deg.) bistatic trans. coeff. (db) 1-1 (f) -2 h=1. cm, TM, LHM h=1. cm, TM, RHM transmission angle (deg.) Figure 1. Comparison of the average bistatic transmission coefficients Γ(θ t ) between LHMs and RHMs exponentially correlated surfaces for both TE and TM cases at 1.5 GHz. Correlation length: l = 6. cm, incident angle: θ i =4, andε 1 =±7.29,μ 1 = ± 1 for RHMs and LHMs, respectively. RMS heights: (a and b) h =.2cm,(candd) h =.6cm,and(eandf)h =1cm. of RHMs or LHMs. However, for exponentially correlated RHM surfaces, the coherent reflectivity exists only for TE case but not for TM case. This is because the divergent spectral integral was calculated over the roughness spectrum variable dk x. Moreover, for LHM surfaces with exponential correlation function, the coherent reflectivities for both TE and TM cases do not exist due to the divergent spectral integral. Comparisons are made for the bistatic scattering coefficients in Figure 11 and absorptivities in Table 2, between LHM and RHM for both polarizations of exponentially correlated surfaces with a small RMS height of.2 cm and a correlation length of 6 cm at 1.5 GHz. As shown in Figures 11a and 11b, small-scale features in exponentially correlated RHM surfaces still obey Bragg s scattering, making bistatic scattering by SPM2 agrees well with NMM2D for both polarizations. However, SPM2 can give reasonable absorptivity for only TE case, and its result for TM case is not convergent as shown in Table 2. As for exponentially correlated LHM surfaces, the absorptivities for two polarizations by SPM2 are both divergent as shown in Table 2, and only for TM case, do its bistatic scattering coefficients completely match NMM2D as shown in Figure 11d. The TE case by SPM2 is different from NMM2D except for forward scattering as shown in Figure 11c. This is because the exponentially correlated LHM surfaces give large second transmission from lower medium to upper one for TE case even for a very small RMS height. Namely, the SPM2 cannot reveal the second reversed transmission XU AND SHI 214. American Geophysical Union. All Rights Reserved. 49

11 1.12/213RS5253 bistatic scattering coeff. (db) bistatic scattering coeff. (db) RHM, TE, NMM2D RHM, TE, SPM2 RHM, TE, KA (c) -3 LHM, TE, NMM2D LHM, TE, SPM2 LHM, TE, KA bistatic scattering coeff. (db) bistatic scattering coeff. (db) RHM, TM, NMM2D RHM, TM, SPM2 RHM, TM, KA (d) -3 LHM, TM, NMM2D LHM, TM, SPM2 LHM, TM, KA Figure 11. Comparison of the average bistatic scattering coefficients γ(θ s ) from exponentially correlated surfaces by NMM2D with SPM2 and KA for both polarizations at 1.5 GHz. RMS height: h =.2 cm, correlation length: l = 6. cm, incident angle: θ i =4, and ε 1 = ± 7.29, μ 1 = ± 1 for RHMs and LHMs, respectively. (a and b) For TE and TM cases, respectively, of RHMs. (c and d) For TE and TM cases, respectively, of LHMs. effect. Because the exponential correlation function has fine roughness features, the scatterings for both the TE and TM cases of RHMs by the KA are erroneous and are different from NMM2D, although their absorptivities are in good agreement by accident. Both the scatterings and absorptivities for both the polarizations of LHMs by the KA are also erroneous and are different from NMM2D. Furthermore, the KA overestimates the absorptivity for TE case even above the planar interface while underestimates TM case even below the planar interface. Numerical results of the bistatic scattering and absorptivities are also compared with the SPM2 and the KA, for both the polarizations of LHM surfaces with both exponential and Gaussian correlation functions with a moderate RMS height of 1 cm and a correlation length of 6 cm. As shown in Figure 12 and Table 3, because of moderate RMS height, the scatterings for both the polarizations and the correlation Table 2. Comparison of the Average Absorptivities From Exponentially Correlated Surfaces by NMM2D With SPM2 and KA With the Parameters Same as Those in Figure 11 Material Method TE Absorptivity TM RHM NMM2D SPM NC a KA LHM NMM2D SPM2 NC NC KA Planar interface analytical a Not convergent. functions by SPM2 are all completely different from NMM2D, although the absorptivities from Gaussian correlated surfaces are in good agreement by accident. Moreover, SPM2 cannot give convergent absorptivities from exponentially correlated surface. On the other hand, the bistatic scattering and the absorptivity by the KA agree well with NMM2D only for surfaces with Gaussian correlation function. However, they disagree for exponentially correlated LHM surfaces, and the KA overestimates the absorptivities for both the polarizations even above the planar interface as shown in Table 3. XU AND SHI 214. American Geophysical Union. All Rights Reserved. 41

12 1.12/213RS5253 bistatic scattering coefficient (db) Gau., TE, NMM2D Gau., TE, SPM2-6 Gau., TE, KA (c) bistatic scattering coefficient (db) Gau., TM, NMM2D Gau., TM, SPM2 Gau., TM, KA (d) -1-2 exp., TE, NMM2D exp., TE, SPM2 exp., TE, KA exp., TM, NMM2D exp., TM, SPM2 exp., TM, KA Figure 12. Comparison of the average bistatic scattering coefficients γ(θ s ) from LHM surfaces by NMM2D with SPM2 and KA for both polarizations at 1.5 GHz. RMS height: h = 1. cm, correlation length: l = 6. cm, incident angle: θ i =4, ε 1 = i, μ 1 = 1. (a and b) For TE and TM cases, respectively, from Gaussian correlated surfaces. (c and d) For TE and TM cases, respectively, from exponentially correlated surfaces. Nevertheless, the KA underestimates the absorptivities from exponentially correlated RHM surfaces in Xu et al. [21]. This is because only exponentially correlated LHM surfaces can make the second opposite refraction from lower medium to upper one, which results in larger decrease of absorptivity, then the KA shows overestimate for exponentially correlated LHM surfaces while underestimate for exponentially correlated RHM surfaces. For exponentially correlated LHM surfaces, it can be concluded that the KA is invalid at any roughness. The reasons for the failure of the KA are that exponentially correlated surfaces have fine-scale features and second refractions from lower LHM to upper air. It is stated by simulations in this paper that SPM2 is valid only for scattering for TM case with a very small RMS height. The fine-scale features act as strong diffuse scatterers. The large re-refraction in opposite direction makes absorption less for TE case, results in scattering and absorptivity by SPM2 not valid any longer for TE case even with a very small RMS height. But the KA can still be applied to the Gaussian correlated LHM surfaces due to its relative small slopes and no second transmission effect. Table 3. Comparison of the Average Absorptivities From LHM Surfaces by NMM2D With SPM2 and KA With the Parameters Same as Those in Figure 12 Correlation Function Method TE Absorptivity Gaussian NMM2D SPM KA Exponential NMM2D SPM2 NC NC KA Planar interface analytical TM 4. Conclusion In this paper, the effects of surface roughness are investigated on absorptivities, bistatic scattering, and bistatic transmission coefficients from LHM surfaces. Comparing with RHM surfaces, it is shown that Gaussian correlated roughness has similar effect on the cases of LHMs. However, exponentially correlated roughness has anomalous effect on absorptivities and scattering of LHMs. Because of the second opposite transmission XU AND SHI 214. American Geophysical Union. All Rights Reserved. 411

13 1.12/213RS5253 from lower medium to upper one, the scattering enhances dramatically and the absorptivity decreases remarkably for TE case with increasing roughness; in particular, they are all less than that of the planar interface. Incorporating Brewster angle effect, the absorptivity trend for TM case goes up first and then down with the increasing roughness. The SPM2 for LHMs is not valid for exponentially correlated surfaces even with a very small roughness except for scattering for TM case. The KA for LHMs is not valid for any exponentially correlated surfaces except for the Gaussian correlated surfaces with moderate slopes. Results illustrated in this paper denote that exponential roughness has a large effect on absorption from LHMs. It suggests that in application of LHMs to absorptive material, the important role played by the exponential correlation roughness should be taken into account. Appendix A: Second-Order Small Perturbation Method for Arbitrary Materials SPM2 is only conducted for the TM case of RHMs by Gu et al. [27] and invalid for TE case. In this appendix, the results are listed for the incoherent wave bistatic scattering coefficients and absorptivity for both the TE and TM cases of both LHMs and RHMs in a manner similar to Gu et al. [27]. Following Gu et al. [27], the zeroth-order solution of the surface field unknowns by SPM is where δ(k x ) is the Dirac delta function and A ðk x Þ ¼ ea δðk x k ix Þ (A1) B ðk x Þ ¼ eb δðk x k ix Þ (A2) and ea ðþ 2k iz ¼ k iz þ k 1iz =ρ eb ðþ ¼ 2ik izk 1iz k iz þ k 1iz =ρ ρ ¼ μ 1=μ ; for TE ε 1 =ε ; for TM qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1iz ¼ signðμ 1 Þ k 2 1 k2 ix (A3) (A4) (A5) (A6) Please note that ρ = 1 for the TE case of the LHM and the real parts of k 1 and k 1iz is negative for the LHM, while their imaginary parts are still positive. The first-order solution is ð Þ ðk x A ð1þ ðk x Þ ¼ ea 1 B ð1þ ðk x Þ ¼ eb 1 ð Þ ðk x where F(k x ) is the Fourier transform of f(x) and ea ð1þ ðk x Þ ¼ i ea ðþ ½ðρ 1Þk x k ix k 1iz ðk 1z þ k z Þ ρk z þ k 1z where ÞFk ð x k ix Þ (A7) ÞFk ð x k ix Þ (A8) þk 2 1 ρk2 eb ð1þ A ðk x Þ ¼ e ðþ ½ρk ix k x ðk z þ k 1z Þρ k 1z k 2 þ k z k 2 1 þ ðρ 1Þk1iz k z k 1z ρk z þ k 1z k 1z ¼ signðμ 1 q Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 1 k2 x (A9) (A1) ; Imðk 1z Þ (A11) Note that there is no underlined part, k 2 1 ρk2,ingu et al. [27]. And, k 2 1 ρk2 ¼ only for the TM case of the RHMs. XU AND SHI 214. American Geophysical Union. All Rights Reserved. 412

14 1.12/213RS5253 The ensemble average of the second-order solution is where < A ðk x Þ >¼ ea δðk x k ix Þ (A12) < B ðk x Þ >¼ eb δðk x k ix Þ (A13) ea ð2þ ¼ h2 k 2 iz ea ðþ þ i 2 k 2 1 ρk2 dk e xa ð1þ k x W k x k ix ðρ 1Þk ix þ i dk e xa ð1þ k x W k x k ix k x þ k iz þ k 1iz dk e xb ð1þ k x W k x k ix (A14) eb ð2þ h 2 k 2 iz ¼ik 1iz A 2 e ðþ ρ k2 1 k iz þ k 2 k 1iz dk e xa 1 W k x k ix ð Þ k x þ ρk ixðk iz þ k 1iz Þ dk e xa ð1þ k x W k x k ix k x þ i ðρ 1Þk iz k 1iz dk e xb ð1þ k x W k x k ix (A15) k 2 1 Please note that there is also no underlined part i ρk2 ρk iz þk 1iz dk e xa ð1þ k x W k x k ix in Gu et al. [27], where W(k x ) is the spectral density, namely, the Fourier transform of the correlation function, and given by Wk ð x 8 h 2 l π 1 þ k 2 for exponential >< x Þ ¼ l2 correlation function h 2 l >: p 2 ffiffiffi exp k2 x l2 for Gaussian π 4 correlation function Now the zeroth-order and the second-order coherent scattered fields are, respectively, (A16) where ψ s Þ ðk x Þ ¼ eψ s δðk x k ix Þ (A17) < ψ ðk x Þ > ¼ eψ δðk x k ix Þ (A18) s s eψ ðþ s ¼ ρk iz k 1iz eψ ð2þ s ¼ ρ 1 h ik ix dk e xa ð1þ k x W kix k x k x k 2 1 ρk2 þ i dk e xa 1 ð Þ k x W kix k x þ k 1iz ρ dk x e B 1 ð Þ k x W kix k x (A19) (A2) Note the underlined part, which is not included in Gu et al. [27]. Accordingly, the coherent reflectivity is The first-order incoherent scattered field is r coh ¼ < S s ^z> coh < S inc ^z > ¼ eψ ðþ s 2 þ 2Re eψ ð Þ s eψ 2 s ð Þ* (A21) ð Þ ¼ eψ ð1þ ð ψ ð s 1Þ k x s k x ÞFk ð x k ix Þ (A22) XU AND SHI 214. American Geophysical Union. All Rights Reserved. 413

15 1.12/213RS5253 where eψ ð1þ ð Þ ¼ 1 A 2 e 1 s k x ð Þ k x ð Þþ i 2k z " k x k ix k 2 þ k zk 1iz ea ðþ e B ð1þ # ðk x Þ ρ ρ (A23) Accordingly, the incoherent reflectivity is π π r incoh ¼ < S s ^z> incoh < S inc ^z > ¼ 1 2 2π π dθ s γ incoh ðθ s Þ ¼ 1 2 2π π 2 2 Thus, the bistatic scattering coefficient for the incoherent wave is γ incoh ðθ s Þ ¼ 2πσ incoh ðθ s Þ ¼ 2πk cos2 θ s eψ ð1þ s ð cosθ i dθ s 2πk cos2 θ s eψ ð1þ s ð cosθ i k x k x Þ 2 Wk ð x k ix Þ (A24) Þ 2 Wk ð x k ix Þ (A25) Since SPM2 conserves energy exactly, the absorptivity is a ¼ 1 r coh r incoh (A26) SPM2 can figure out the absorptivities for Gaussian correlated surfaces. However, because the former two integrals in equation (A2) are not convergent for exponential correlation function, it cannot calculate the coherent reflection by equation (A21) for exponentially correlated surfaces except for the TE case of the RHM, where the former two integrals cancel due to ρ = 1. Thus, the absorptivities for both the polarizations of the LHMs and for the TM case of the RHMs are all divergent by SPM2. Acknowledgments The data of the results for this paper are all generated by using our new FORTRAN code to conduct the numerical Maxwell model with 2-D simulations. This work is supported in part by the National Science Foundation of China grant and in part by the Open Fund of State Key Laboratory of Remote Sensing Science (grant OFSLRSS2111). References Chew, W. C. (25), Some reflections on double negative materials, Prog. Electromagnet. Res., 51, Cui, T. J., Z. C. Hao, X. X. Yin, W. Hong, and J. A. Kong (24), Study of lossy effects on the propagation of propagating and evanescent waves in left-handed materials, Phys. Lett. A, 323, Gu, X., L. Tsang, H. Braunisch, and P. Xu (27), Modeling absorption of rough interface between dielectric and conductive medium, Microw. Opt. Tech. Lett., 49(1), Houck, A. A., J. B. Brock, and I. L. Chuang (23), Experimental Observations of a left-handed material that obeys Snell s Law, Phys. Rev. Lett., 9(13), 13741, doi:1.113/physrevlett Hu, L., and S. T. Chui (22), Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials, Phys. Rev. B, 66, Johnson, J. T., K. F. Warnick, and P. Xu (27), On the Geometrical Optics (Hagfors Law) and Physical Optics Approximations for Scattering From Exponentially Correlated Surfaces, IEEE Trans. Geosci. Remote Sens., 45(8), Lei, Z., and T. J. Cui (26), Super-resolution imaging of dielectric objects using a slab of left-handed material, Appl. Phys. Lett., 89, Narimanov, E. E., H. Li, Y. A. Barnakov, T. U. Tumkur, and M. A. Noginov (213), Reduced reflection from roughened hyperbolic metamaterial, Opt. Express, 21(12), 14,956 14,961. Pendry, J. B. (2), Negative refraction makes a perfect lens, Phys. Rev. Lett., 85(18), Shelby, R. A., D. R. Smith, and S. Schultz (21), Experimental verification of a negative index of refraction, Science, 292(5514), Smith, D. R., and N. Kroll (2), Negative refractive index in left-handed materials, Phys. Rev. Lett., 85(14), Tsang, L., J. A. Kong, and R. Shin (1985), Theory of Microwave Remote Sensing, Wiley, New York. Tsang, L., D. Chen, P. Xu, Q. Li, and V. Jandhyala (24a), Wave scattering with the UV multilevel partitioning method: 1. Two-dimensional problem of perfect electric conductor surface scattering, Radio Sci., 39, RS51, doi:1.129/23rs39. Tsang, L., Q. Li, P. Xu, D. Chen, and V. Jandhyala (24b), Wave scattering with UV multilevel partitioning method: 2. Three-dimensional problem of nonpenetrable surface scattering, Radio Sci., 39, RS511, doi:1.129/23rs31. Veselago, V. G. (1968), Electrodynamics of substances with simultaneously negative values of sigma and mu, Sov. Phys. Uspekhi-USSR, 1(4), 59. Wu, Q., F. Y. Meng, M. F. Wu, J. Wu, and L. W. Li (25), Research on the negative permittivity effect of the thin wires array in left-handed material by transmission line theory, Progress in Electromagnetics Research Symposium, Hangzhou, China. Xu, P., and L. Tsang (27), Bistatic scattering and emissivities of lossy dielectric surfaces with exponential correlation functions, IEEE Trans. Geosci. Remote Sens., 45(1), Xu, P., K. S. Chen, and L. Tsang (21), Analysis of microwave emission of exponentially correlated rough soil surfaces from 1.4 GHz to 36.5 GHz, Prog. Electromagnet. Res., 18, Yao, H. Y., L. W. Li, C. W. Qiu, Q. Wu, and Z. N. Chen (27), Scattering properties of electromagnetic waves in a multilayered cylinder filled with double negative and positive materials, Radio Sci., 42, RS26, doi:1.129/26rs359. Zhang, Y., T. M. Grzegorczyk, and J. A. Kong (22), Propagation of electromagnetic waves in a slab with negative permittivity and negative permeability, Prog. Electromagnet. Res., 35, Ziolkowski, R. W., and E. Heyman (21), Wave propagation in media having negative permittivity and permeability, Phys. Rev. E., 64(5), XU AND SHI 214. American Geophysical Union. All Rights Reserved. 414