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Indian Journal of Science and Technology, Vol 8(16), DOI: 10.17485/ijst/2015/v8i16/64930, July 2015 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 Length Scales Analysis of Wave Scattering from Rough Surfaces M. Salami∗ Department of Physics, Shahrood Branch, Islamic Azad University, Shahrood, Iran; [email protected] Abstract The aim is to study surface characteristics through scattered wave intensity in frame work of the Kirchhoff wave theory. There are three length scales which scaling behavior between them play role in rough surface scattering, wave-length (λ) of the incident wave, the roughness (σ) and the correlation length (ξ) of the surface. In this work we show the effective role of the correlation length in surface scattering. Up to now, some of the reports for wave scattering from rough surfaces are based on the product of the wave number and surface roughness kσ without consideration to correlation length. For λ x, correlation length has no effects on the properties of the scattered wave and kσ is the appropriate parameter to study the rough surfaces scattering problem. But, if ξ and λ are in comparable range, scattered wave depends on how kσ is chosen by k or σ. In this case, kσ is not suitable to present wave scattering from rough surface. In other word, for a constant ka, scattered wave could depend on any of the parameters k or σ. To justify our statements we compare our theoretical results with experimental data for the intensity profile obtained from a self-affine rough surface. We show that by changing the parameters ξ, λ, σ, and the Hurst exponent, the intensity profiles obtained by theory and numerical estimation overlap when λ is much longer than ξ. But interestingly, the profiles start to diverge when ξ tends to λ. This provides a good understanding of the role of the characteristics of the surface on the profile of scattered intensity. Keywords: Correlation Length, Kirchhoff Theory, Roughness, Wave Scattering 1. Introduction Roughness effects on scattering intensity have been investigated by many researchers for the past few decades. Kirchhoff theory is one of the most used methods to study wave scattering problem1-5. It is suitable for rough surface with roughness is same order and smaller than wavelength, which is based on electromagnetic principles. Also, the surfaces with slight roughness are approximated by Rayleigh-Rice theory, which is a perturbed boundary condition method6. The scattered intensity depends on the properties of the rough surface and the incident field. Evaluating the intensity from a rough surface with known properties is called the direct problem7-9. The opposite question of the direct problem is called the inverse problem which is determining the properties of a surface by using the information embedded in measured scattered intensity10-15. *Author for correspondence Some of the scattering studies are based on variations in dimensionless kσ parameter; where k and σ are two length scale in the system3,16-18. These studies they do not consider the effects of the third length scale in the system which is correlation length ξ. Although, some reports showed how correlation length plays effective role in wave scattering in Rayleigh-Rice framework6,19-22, it could be useful to consider this effect in Kirchhoff theory, too. Schiffer indicated when ξ comparable with λ, the effect are losing in reflection, shifting in Brewster angle to smaller ones and slight reddening of scattering light6 in rayleighRice theory. Also, Yanguas-Gil et al.22 showed when the correlation length is smaller than the wavelength (ξ < λ), 2 the single parameter wa contains the information of wave x scattering phenomena in Rayleigh-Rice framework. These three length scales (λ, σ and ξ) depend on dimensions will affect the scattered wave. When λ x, the correlation length has no effect on the scattered wave d a monochromatic wave illuminates the rough surface with finite dimensions and R 0 ity. According to Dirichlet boundary Length Scales Analysis of Wave Scattering from Rough Surfaces size of the runner shoes can be interpreted as observation scale or wavelength of incident beam. If the runner wants to run fast, the size of the rocks and the height mean squared of the rocks are important. Really small rocks with small height root mean squared have no effect in the speed of the runner. This situation in a scattering system is equivalent to the wavelength larger than the surface roughness (σ) which means the diffuse scattering intensity is small. On the other hand if the size of his shoes is comparable to the size of the rocks, he has to slow down to be able to go through the troughs made by rocks. If the height mean squared are big, the runner shoes will be trapped between the rocks that makes him to slow down. Figure 1. The geometry usedused for for wave scattering from a Figure 1. The geometry wave scattering from This means that by decreasing wavelength, the fluctuation rough surface. HereHere 0\ isθ1the angle between a rough surface. is the angle betweenthe theincident incident of the surface is more intuitive and the diffuse scatterbeam andand thethe normal the scattered beam normalofofthe thesurface. surface. θ02 2isisthe scattered angle ing intensity increases. By decreasing the wavelength, the angle between the incident and planes. angle 3 isangle andand θ3 is0the between the incident and scattered the diffuse scattering intensity has some reduction. This andnumbers k sc are the numbers for scattered k incwave kinc andplanes. ksc are the for wave incident and scattered resembles the situation where the sizes of the rocks or the incident scattered beam respectively. beam and respectively. height mean squared are larger than the size of the runλ=1000ξ λ=ξ ner’s shoes. In this situation the parameters of the rocks σ σ λ λ has less effect in the speed of the runner. This means if the ns, R 0 = -1 for such surfaces3. wavelength is smaller than the fluctuations of the rough surface, the scattering intensity decrease. In addition to h these assumptions we can obtain that the coherent part of scattered light wave is: these parameters, the complexity of the surface roughness σ σ is another key determining point in the diffuse scatter(a) (b) Figure 2: Dependence of diffuse scattered intensity per k ξ on kσ for angles θ = θ = 0 , θ = 5 and ing intensity. This means a surface that is covered by the H = 1, (a) λ = 1000ξ, (b) λ = ξ. Figure 2. Dependence of diffuse scattered intensity per periodical peaks and troughs will have different diffuse 2 2 on kσ for angles θ1 = θ33 = 0°, θ2 = 5° and H = 1, (a) λ = 4.2k ξExperimental approach scattering intensity from a surface with height fluctua1000ξ, (b) λ = ξ. In this section we study the diffuse scattering intensity with the presence of three length scale for rough tions. This parameter of a rough surface is described as andWe kσsuppose is a suitable present wave scattering surfaces. that ξ and λ parameter are in comparableto range. As mentioned above to change kσ we can Hurst exponent, H. from the rough surface (see Figure 2(a)). But when ξ and either change k or σ. We discuss about the effects of changing each of these parameters individually. λ are inwe the same range dimensionless Furthermore, will show our results withanother different values of correlation length,parameter ξ, and Hurst exponent, 2. Kirchhoff Theory kξ will appear as well. We need to know how kσ is changed H, on self-affine surfaces. Any incident field can be written as yinc (r ) = exp ( −ik inc .r ) , (see Figure 2(b)). To make any changes in kσ, we can 4.2.1 Effect of variation in kσ where kinc is the wave number of the incident beam and r either change k or σ. Different values of k or σ may result Fig. 3 and Fig. 4 show the diffuse scattering intensity as a function of scattering angle for different kσ is the position vector. The incident beam is scattered from in the same value for kσ. In this paper, we focus on scatat two different values of Hurst exponent. In the former one the wavelength is constant, λ = 633nm a rectangular rough surface area with following conditering problem in framework of Kirchhoff approximation andwhen the σ is variable and in the latter one the height root mean squared is constant, σ = 50nm, there are three length scales in the system, while ξ when tions: –X < x0 ≤ X and –Y < y0 ≤ Y. The scattered field sc the and wavelength canin modify a variable. In both figures we found the diffuse scattering intensity at from the mentioned rough surface is shown by y . By λ are theassame range. different values kσ betweenthe 0.2 andconcept 2.0 with the intervals of 0.2.relation These values are in agreement with using Kirchhoff theory one can determine the distribuTo ofclarify of the between tion of coherent and diffuse parts of scattered wave from roughness, and wavelength in a scatexperimental values incorrelation previous works [13,length 14]. 3 tering system we can represent the problem with a runner In both figures the diffuse scattering intensity is calculated for two different values of Hurst exponent. a rough surfaces . Kirchhoff theory is calculated based on three a surface up the ofones rocks with (Hdifferent By running comparing the on two graphs on the topmade (H = 1) with on the bottom = 0.5), we can see 3 h ypotheses . (1) The roughness of each point of the sursizes. The longitudinal size of a rock and the height of that that the diffuses scattering intensity has smaller standard deviation at H = 0.5. Also, the peak of the face is assumed to have the same optical behavior as its indicate correlation length, ξ, and the height root mean diffuse scattering intensity is higher for smaller value of the Hurst exponent. tangent plane. Fresnel laws can thus be locally applied. (2) squared, σ, respectively. The speed of the runner depends 8 height root mean squared The surface reflectivity R0 is independent of the position on the longitudinal size and the on the rough surface and the local angle of incidence. (3) of the rocks in addition to the size of his shoes. Here the (b) 0.25 Diffuse scattering intensity per k 2ξ 2 Diffuse scattering intensity per k 2ξ 2 (a) 0.3 0.25 =constant =constant 0.2 0.15 0.05 0.1 0.05 0.5 1 k 1.5 2 0 2 2 2 =constant =constant 0.2 0.15 0.1 0 0.3 Vol 8 (16) | July 2015 | www.indjst.org 0.5 1 1.5 k 1 3 2 ◦ 2 ◦ Indian Journal of Science and Technology M. Salami Calculations are performed in the far-field, which works when the surface area S0 is small. For all the calculations in this study, we assumed a monochromatic wave illuminates the rough surface with finite dimensions and R0 reflectivity. According to Dirichlet boundary conditions, R 0 = –1 for such surfaces3. Through these assumptions we can obtain that the coherent part of scattered light wave is: where AM = 4XY is the area of the mean scattering surface and Cor(R) is the height-height correlation function and it is defined as Cor ( R ) = h (x + R) h ( x ) s2 . 3. Scattering from Self-affine Surface One of the main groups of the rough surface is represented by self-affine fractal scaling, defined by Mandelbrot in SM terms of fractional Brownian motion23. Lets consider a 1 Aa surface with+ aBb single-valued height function, z(r) of + c Where the phase function is f ( x0 + y0 ) = Ax0 + By0 + Ch ( xrough 0 , y0 ) , F = r = (x, y). All rough surfaces C C vector, 2 positional the in-plane 1 Aa Bb are characterized by two parameters, the mean-square Ax0 + By0 + Ch ( x0 , y0 ) , F = + + c and C 2 C 2 1 roughness s =< z (r ) > 2 and z (r ) = h (r ) − < h (r ) > . As y sc (r ) = −ikeikr 2F (q1 , q2 , q3 ) 4pr ∫ exp ikf (x , y ) dx dy , (1) 0 0 A = sin q1 − sin q2 cos q3 B = − sin q2 sin q3 C = − ( cos q1 + cos q2 ) a = sin q1 (1 − R0 ) + sin q2 cos q3 (1 + R0 ) b = sin q2 sin q3 (1 + R0 ) c = cos q2 (1 + R0 ) − cos q1 (1 − R0 ) 0 0 mentioned earlier, h(r) is the height function and < … > is the spacial average over a planar reference surface. We assume that z(r) – z(r’) is a stochastic Gaussian variable whose distribution depends on the difference of the two position vectors r and r’, (x’ – x, y’ – y). Using the height function we can define the structure function as S(R) =<[z(r’) – z(r)]2 >, where R =| r’ – r |. The average is taken over all pairs of points on the surface that are separated horizontally by the distance R. S(R) could be defined in terms of the height-height correlation function that was < z ( R ) z (0 ) > defined earlier, Cor ( R ) = , as, s2 The coherent field could be derived from the average amplitude of the scattered field and the intensity of the coherent field for a surface with a Gaussian height distribution, I coh = y sc y sc∗ = e − g I 0 , (2) where g = k 2C 2a 2 and Io is the coherent scattered intensity from a smooth surface with the same size as the rough surface. The average diffuse field intensity can also be calculated using ysc, I d = y sc y sc∗ − y sc = k2 F 2 ∞ ( ) y sc∗ gCor R AM e − g J 0 kR A2 + B 2 e ( ) − 1 RdR, (3) 2pr 2 0 ∫ Vol 8 (16) | July 2015 | www.indjst.org g ( R ) = 2 < z (r ) > −2 < z (r ) z (r ′ ) >= 2s 2 − 2 2 < z ( R ) z (0) >= 2s 2 − 2s 2Cor ( R ) . (4) If the surface exhibits self-affine roughness, S(R) will scale as S(R) ∝α R2H, 24, where 0 < H < 1 is referred to as Hurst exponent 25. The Hurst exponent represents the degree of the surface irregularity. Large values of H correspond to smooth height-height fluctuations, while the small values shows the jagged and irregular surfaces at the short length scale. The meanSquare Roughness, S(R), of any physical self-affine surface will saturate at sufficiently large horizontal lengths. Thus S(R) is characterized by a correlation length, S ( R ) aR2 H , For R x, S ( R ) = 2s 2 , For R x (5) Indian Journal of Science and Technology 3 Length Scales Analysis of Wave Scattering from Rough Surfaces By using the diffuse reflectivity data26,27 the correlation function for a self-affine surfaces can be written as28, − Rx Cor ( R ) = e ( ) 2H (6) When R x, there will be no correlation, Cor(R) = 0 and for R x correlation function can be written as 2H R Cor ( R ) 1 − This indicates for short length scale x r x, the correlation function shows power-law behavior29. At the largest possible value for Hurst exponent, smooth height-hight fluctuations, the correlation function is Gaussian and Eq. 3 for the diffuse scattered intensity can be written as3: < I d >= k 2 F 2 x2e − g 4pr 2 ( ) k 2 A2 + B 2 x 2 gn (7) exp AM ∑ 4n n =1 n ! n ∞ For a Gaussian noise surface with H = 0.5, we can obtain the following expression for the diffuse scattered intensity from Eq. 318: < I d >= k2 F 2 2pr 2 gn n =1 n ! x2 ∞ AM e − g ∑ ( ) 3 k 2 A2 + B 2 x 2 2 n2 1 + n2 . (8) 4. Result and Discussion It was stated earlier, when the system is characterized by the three length scales k, σ, and ξ, the dimensionless parameter kσ is unable to give a complete description of the scattering problem from the rough surface. Hence, a new dimensionless parameter kξ is introduced which enables a comprehensive study on the scattering phenomena. Hence, the combined effects of the wavelength, roughness and correlation length on the scattered waves need to be studied. The formalism presented here would provide basis for both theoretical studies and experimental observations, in a sense that two approaches are proposed based on the physical applications. To comply with the theoretical modeling needs, the chosen variable would be kξ which is more convenient due to its dimensionless nature, and to comply with the observers needs in order to calculate the scattered intensity, values of the wave-length and roughness have 4 Vol 8 (16) | July 2015 | www.indjst.org been chosen close to expected measurements. These two approaches are p resented in the following sections. 4.1 Theoretical Approach The scattered intensity depends on the altitude correlation function of the rough surface, so the area of the correlated section (the part of the surface that has correlated altitudes) also plays an important role. To calculate the intensity of the scattered wave, we use r = 0 and r as the lower and upper limits of the integral respectively, where r and ξ have the same order of magnitude. For the case r x, the correlated function and the integrand will be zero. On the other hand we know that a correlated surface is proportional to ξ2. Also we should keep in mind that ξ is a characteristic length of the surface which possesses a scaling behavior. This implies consideration of the scale of observation; this indicates the importance of the incident wave length to the surface. Consequently the parameter kξ would show its importance. Since the scattered field intensity depends on ξ2, in order to gain its independence from the parameters ξ and λ, it is divided by k2ξ2. In such a condition, we obtained an expression independent of the parameters ξ, σ, λ. When the correlated regime is much smaller than the wave length, the intensity only depends on the kσ (see Figure 2(a)). However when ξ is as of the order of λ its effects on the scattered wave intensity becomes prominent and the curve indicating the variations of the diffused intensity in terms of kσ becomes obviously dependent on k or σ (see Figure 2(b)). 4.2 Experimental Approach In this section we study the diffuse scattering intensity with the presence of three length scale for rough surfaces. We suppose that ξ and λ are in comparable range. As mentioned above to change k σ we can either change k or σ. We discuss about the effects of changing each of these parameters individually. Furthermore, we will show our results with different values of correlation length, ξ, and Hurst exponent, H, on self-affine surfaces. 4.2.1 Effect of Variation in kσ Figure 3 and Figure 4 show the diffuse scattering intensity as a function of scattering angle for different kσ at two different values of Hurst exponent. In the former one the wavelength is constant, λ = 633nm and the σ is variable and in the latter one the height root mean squared is constant, Indian Journal of Science and Technology M. Salami (a) kσ <0.6, H=1, λ =633nm (a) kσ <0.6, H=1, λ =633nm (b) kσ >0.6, H=1, λ =633nm (b) kσ >0.6, H=1, λ =633nm σ = 50 nm, when the wavelength can modify as a variable. In both Figures we found the diffuse scattering intensity at different values of kσ between 0.2 and 2.0 with the intervals of 0.2. These values are in agreement with experimental values in previous works13,14. In both figures the diffuse scattering intensity is calculated for two different values of Hurst exponent. By Scattering Angel (Degree) Scattering Angel (Degree) comparing the two graphs on the top (H = 1) with the Scattering Angel (Degree) Scattering Angel (Degree) (c) kσ <0.6, (d) kσ >0.6, (a) kH=0.5, <0.6,λ =633nm H=1, λ =633nm (b) kH=0.5, >0.6,λ =633nm H=1, λ =633nm σ σ (c) kσ <0.6, H=0.5, λ =633nm (d) kσ >0.6, H=0.5, λ =633nm ones on the bottom (H = 0.5), we can see that the diffuses (a) kσ <0.6, H=1, λ =633nm (b) kσ >0.6, H=1, λ =633nm k σ =0.2 k σ =0.6 k σ =0.6 scattering intensity has smaller standard deviation at H kσ =0.2 k σ =0.4 =0.2 k σ =0.8 =0.6 k σ =0.8 kσ =0.4 k σ =0.6 =0.4 k σ =1 =0.8 kkσσ=0.6 kkσσ=0.2 =1 =0.6 k σ =0.6 k σ =1.2 =1 kkσσ=0.8 kσ =0.4 =1.2 = 0.5. Also, the peak of the diffuse scattering intensity is k σ =1.4 =1.2 kkσσ=1 kσ =0.6 =1.4 k σ =1.6 =1.4 kkσσ=1.2 =1.6 higher for smaller value of the Hurst exponent. k σ =1.8 =1.6 kkσσ=1.4 =1.8 k σ =2 =1.8 kkσσ=1.6 =2 In all different cases, we found a threshold value for k σ =2 k σ =1.8 k σ =2 kσ where the diffuse scattering intensity has its own maximum value. This threshold value is happening where the Scattering Angel (Degree) Scattering Angel (Degree)Angel (Degree) correlation length, height mean squared and the waveScattering Angel (Degree) Scattering Scattering Angel (Degree) Scattering Angel (Degree) Scattering (Degree) Scattering (Degree) (c) kσ <0.6, H=0.5, =633nm (d) kσ >0.6, H=0.5, =633nm λ Angel λ Angel length are in the same order of magnitude. In both Figure Figure 3: The scattering of scattering angleH=0.5, with constant = 633nm. (c)diffuse kσ <0.6, H=0.5, λintensity =633nm as aa function (d) kσ >0.6, =633nmλ λconstant Figure 3: The function of scattering angleangle with and λ = between 633nm. The graphs are diffuse plottedscattering for valuesintensity of kσ andasHurst exponent. The incident the angle 3 and Figure 4, the graphs on the left and right show the k =0.2 σ σ k =0.6 ◦ The graphs are for planes values are of kσ and θHurst exponent. The incident angle and the angle between the incident andplotted scattered equal, 1 = θk3σ = =0.40◦ and the correlation length is ξ = 1000nm. k σ =0.8 the Figure incident and are equal, θ1 = kθkσ3σ=0.2 = and the correlation length is ξ = 1000nm. kkσσ=0.6 3. scattered Theplanes diffuse scattering as a function of =0.60 intensity =1 diffuse scattering intensity for the kσ values smaller and k σ =0.4 kkσσ=0.8 =1.2 k σ =0.6 kkσσ=1 =1.4 In all different cases, we found a threshold value for=kσ633nm. where the diffuse scattering intensity has larger than the threshold value respectively. The rate of scattering angle with constant λ The graphs are kkσσ=1.2 =1.6 In all different cases, we found a threshold value for kσ where the diffuse scattering intensity has kkσσ=1.4 =1.8 forvalue. values of kσ value andis Hurst The length, incident its plotted own maximum This threshold happeningexponent. where the correlation heightkkσmean changing the diffuse scattering intensity is larger for the =2 σ=1.6 its own maximum value. This threshold value is happening where the correlation length, heightk σmean =1.8 k σ =2 angle and the angle incident and3 and scattered squared and the wavelength are in thebetween same order of the magnitude. In both Fig. Fig. 4, the graphs kσ values smaller than the threshold value. squared and the wavelength are in the same order of magnitude. In both Fig. 3 and Fig. 4, the graphs planes are equal, θ1 = scattering θ3 = 0°intensity and for thethecorrelation length is In Figure 3, where the wavelength is constant, the on the left and right show the diffuse kσ values smaller and larger than the on the left and right show the diffuse scattering intensity for the kσ values smaller and larger than the ξ = 1000nm. standard deviation of diffuse scattering intensity is getting (b) kdiffuse H=1, σ =50nm Angel of (Degree) σ =50nm Scattering threshold valueH=1, respectively. The rate changing the intensity is larger for the kσ (a) kσ <1, σ >1, scattering Scattering Angel (Degree) threshold value respectively. The rate of changing the diffuse scattering intensity is larger for the kσ Scattering Angel (Degree) larger by increasing kσ. However when the height mean (b) kσ >1, H=1, σ =50nmScattering Angel (Degree) (a) smaller kσ <1, than H=1, σ =50nm values the threshold value. kσ =1 values smaller than the threshold value. kσ =0.2 Figure 3: The diffuse scattering intensity as a function of scattering angle withkσconstant λ = 633nm. kσ =0.4 =1.2 squared is constant (σ in Figure 4) the standard deviation The graphs arediffuse plottedscattering forisvalues of kσthe and Hurst exponent. The incident angle the angle between In Fig. 3, where the wavelength constant, standard deviation of diffuse scattering intensity kσ intensity =0.6 =1.4 σ Figure 3: The as a function of scattering angle withkσkand constant λ is= 633nm. kσ =0.2 =1 In Fig.the 3, where theand wavelength isplanes constant, the standard of diffuse scattering intensity is kσ =0.8 kσ =1.6 scattered arekσequal, θ1 = θexponent. = 0◦ andThe the correlation length ξ= 1000nm. 3 deviation kσ =0.4 kσ =1.2is of diffuse scattering intensity decreases as kσ increases. The incident graphs are plotted for values of and Hurst incident angle and the angle between kkσ =1 kσ=1.4 =1.8 kσ σ =0.6 the incident and scattered planes are equal, θ1 = θ3 = 0◦ and the correlation length kσ=1.6 =2is ξ = 1000nm. kσ =0.8 kσ Furthermore, by decreasing Hurst exponent due to the kσ =1 a threshold value for kσ where the diffuse kscattering σ =1.8 In all different cases, we found intensity has kσ =2 9 In all different cases, we found a threshold value for kσ where the diffuse scattering intensity has increased irregularity of the surface roughness, the diffuse 9 its own maximum value. This threshold value is happening where the correlation length, height mean increases. The curve of the diffuse intensity is getits own maximum value. This threshold value is happening where the correlation length, heightintensity mean squared and the wavelength are in the same order of magnitude. In both Fig. 3 and Fig. 4, the graphs squared and the wavelength are in the same order of magnitude. In both Fig. 3 and Fig. 4, the ting graphs sharper and the standard deviation of that decreases. on the left and right show the diffuse scattering intensity for the kσ values smaller and larger than the Scattering Angel (Degree) Scattering Angel (Degree) on the left and right show the diffuse scattering intensity for the kσ values smaller and larger than theIn some of the studies the scattering intensity has been (c) kσ <0.8, H=0.5, (d) kσ >0.8, H=0.5, =50nm σscattering threshold value respectively. the(b) diffuse intensity is larger for the kσ kScattering H=1, σ =50nm σ =50nmThe rate of changing (a) kScattering <1,=50nm H=1, σσ σ >1, Angel (Degree) Angel (Degree) studied as a function of dimensionless kσ. As discussed value respectively. The rate of changing the H=0.5, diffuseσscattering intensity is larger for the kσ (c) kσthreshold <0.8, H=0.5, =50nm (d) kσ >0.8, =50nm σ (b) kσ >1, H=1, σ =50nm kσ =0.8 σ =50nm (a)smaller kσ <1, than H=1,the values threshold value. kσ =0.2 kσ =0.2 kσ =1 before the changes in kσ could be the result of variation kσ =1 kσ =0.4 kσ =1.2 values smaller than the threshold value. kσ =0.4 kσ=0.8 =1.2 kσ kσ =0.6 =0.2 is constant, In Fig. 3, where the wavelength scattering intensity is kσ =0.6 the standard deviation of diffuse kσ =1.4 kσ =0.2 kσ =1 in the wavelength or the height mean squared paramekσ=1=1.4 kσ kσ =0.8 =0.4 kσ =0.8 kσ =1.6 kσ =0.4 the standard deviation of diffuse kσ intensity =1.2 In Fig. 3, where the wavelength is kσscattering =1.6 kσ =1.2 kσ =0.6is constant, kσ =1 kσ =1.8 kσ =0.6 k =1.4 σ kσ kσ=1.4 =1.8 kσ =0.8 kσ =2 ters. Our result show that not only the value of kσ is one kσ =0.8 kσ =1.6 kσ kσ=1.6 =2 kσ =1 kσ =1.8 kσ =1.8 k =2 of the key parameters, but also how this parameter has σ 9 kσ =2 9 been changed is important too. Our results show that the scattered intensity varies by changing the wavelength and the height mean squared individually. In another word, Scattering Angel (Degree) Scattering Angel (Degree) Scattering Angel (Degree) Scattering Angel (Degree) (c) kScattering (d) kScattering >0.8,AngelH=0.5, σ <0.8,AngelH=0.5, σ (Degree)σ =50nm (Degree)σ =50nm studying the diffuse scattering intensity as a function of Scattering Angel (Degree) Scattering Angel (Degree) Figure 4: The diffuse scattering intensity as a function of scattering angle with constant σ = 50nm. (c) kσ <0.8, H=0.5, (d) kσ >0.8, H=0.5, σ =50nm σ =50nm The graphs are plotted for different values of kσ and Hurst exponent. The incident angle and the angle σ k =0.8 kσ =0.2of scattering angle with constant σ = 50nm. kσ does not depict all the features of the scattered intenFigure 4: The diffuse scattering intensity as a function kσ is =1 =0.4 θ1 = θ3 = 0◦ and the correlation length between the are incident and planesofare and The graphs plotted for scattered different values kσ equal andkkσσHurst exponent. The incident angle and the angle kσ =1.2sity and we need to consider the effects of the wavelength =0.6 kσ =0.8 ◦ as a function ξbetween = Figure 1000nm. kσ =0.2 4. and Thescattered diffuse the incident planesscattering are equal the correlation length is=1.4 kσ 1 = θ3 = 0 and kσ and =0.8 θintensity kσ =1 kσ =0.4 kσ =1.6 ξ = 1000nm. kσ =0.6 σ =1.8and the height mean squared as well. of scattering angle with constant σ = 50nm. The graphs kkσσk=1.2 =1.4 kσ =0.8 σ =2 kthe getting larger by increasing kσ. However when the height mean squared is constant (σ in Fig. 4) kσ =1.6 Furthermore, by adding more parameters to our system arelarger plotted for different of kσ and Hurst exponent. =1.8 getting by increasing kσ. However values when the height mean squared is constant (σ in Fig. 4)kσthe σ k =2 standard deviation of diffuse scattering intensity decreases as kσ increases. Furthermore, by decreasing such as correlation length, ξ, other dimensionless parameThe deviation incident angle and intensity the angle between the Furthermore, incident byand standard of diffuse scattering decreases as kσ increases. decreasing Hurst exponent due to the increased irregularity of the surface roughness, the diffuse intensity increases. ter such as kξ effects the scattering intensity. We will discuss scattered planes are equal and θ = θ = 0° and the correlation 1 surface 3 roughness, the diffuse intensity increases. Hurst exponent due to the increased irregularity of the The length curve of the intensity is getting sharper and the standard deviation of that decreases. the effect of this parameter in the next section. To increase is diffuse ξ = 1000nm Scattering Angel (Degree) Scattering Angel (Degree) 26 26 20 18 Diffuse Diffuse Scattering Scattering Intensity Intensity (ar.un.) (ar.un.) 22 20 18 16 16 14 14 12 12 10 108 68 46 24 02 0 -50 0 50 -50 0 50 k σ =0.6 k σ =0.8 =0.6 k σ =1 =0.8 k σ =1.2 =1 k σ =1.4 =1.2 k σ =1.6 =1.4 k σ =1.8 =1.6 k σ =2 =1.8 k σ =2 24 22 22 20 20 18 18 16 16 14 14 12 12 10 108 68 46 24 02 0 -50 0 50 -50 0 50 26 40 26 40 35 24 26 22 24 20 22 18 20 16 18 14 16 12 14 10 12 8 10 6 8 4 6 2 4 -50 0 2 -50 40 35 24 26 22 24 20 22 18 20 16 18 14 16 12 14 10 12 8 10 6 8 4 6 2 4 0 -50 2 -50 0 35 30 30 25 25 20 20 15 15 10 10 5 5 0 0 0 -50 0 0 50 050 -50 50 0 35 30 30 25 25 20 20 15 15 10 10 5 5 0 0 50 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 40 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Diffuse Scattering Scattering Intensity Intensity (ar.un.) (ar.un.) 26 24 kσ =0.2 kσ =0.4 =0.2 kσ =0.6 =0.4 kσ =0.6 24 22 Diffuse Diffuse Scattering Scattering Intensity Intensity (ar.un.) (ar.un.) Diffuse Diffuse Scattering Scattering Intensity Intensity (ar.un.) (ar.un.) 26 24 3025 2520 2015 1510 105 50 -50 0 -50 50 0 50 70 DiffuseDiffuse Scattering Intensity (ar.un.) (ar.un.) Scattering Intensity 5060 4050 3040 2030 1020 -20 -20 80 60 20 0 20 40 20 0 2015 1510 105 50 -50 0 -20 0 -20 0 0 -20 20 0 20 30 40 20 30 10 20 100 60 60 40 40 20 20 0 0 20 0 20 -20 0 -20 20 0 20 60 50 50 40 40 30 30 20 20 10 10 0 -20 -200 80 0 -20 0 -20 20 0 20 0 20 20 80 80 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 50 40 50 70 20 10 50 0 50 60 70 60 40 30 0 -50 70 60 0 -20 10 0 2520 80 30 20 20 3025 80 50 40 40 3530 60 70 0 60 50 60 50 50 4035 70 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 80 0 Diffuse Scattering Intensity (ar.un.)(ar.un.) Diffuse Scattering Intensity Scattering Intensity DiffuseDiffuse Scattering Intensity (ar.un.) (ar.un.) 0 6070 0 Diffuse Scattering Intensity Diffuse Scattering Intensity (ar.un.)(ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 3530 70 010 50 0 50 0 40 4035 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 40 0 -50 0 -50 80 60 60 60 60 40 40 40 40 20 20 20 20 0 -20 0 20 0 -20 0 20 The curve of the diffuse intensity is getting sharper and the standard deviation of that decreases. 0 0 -20 scattering 0 20 has been studied as a function In some of the studies the intensity of dimensionless -20 0 20 kσ. As Scattering Angel (Degree) Scattering Angel (Degree) In some of the studies scattering intensity has been as aoffunction of dimensionless kσ. As σ = 50nm. Figure 4: Thethediffuse scattering intensity as astudied function scattering angle with constant discussed before the changes in kσ could be the result of variation in the wavelength or the height mean The graphs are plotted for different values of kσ and Hurst exponent. The incident angle and the angle discussed before the in kσscattering could be intensity the result of are the height meanσ = length Figure 4: changes The diffuse as avariation function ofthe scattering withthe constant 50nm. is between the incident and scattered planes equal in and θ1wavelength = θ3 =angle 0◦orand correlation Vol parameters. 8 The (16) |1000nm. July 2015 | www.indjst.org squared Our result show not only theofvalue of kσ is one of the key but and alsothe angle are plotted for that different values kσ and Hurst exponent. Theparameters, incident angle ξ =graphs ◦ squared parameters. Our result show that not only the are value of kσand is one of the but also length is between the incident and scattered planes equal θ1 = θ3 =key 0 parameters, and the correlation ξ = 1000nm. getting larger by increasing kσ. However when the height mean squared is constant (σ in Fig. 4) the 10 10 Indian Journal of Science and Technology 5 Length Scales Analysis of Wave Scattering from Rough Surfaces kσ while the wavelength is constant, both the height mean squared, σ, and the correlation length, ξ, should change to keep the Hurst exponent constant. If the variation in kσ is due to the changes in the wavelength, both the height mean squared, σ, and the correlation length, ξ, remain constant for a constant value of the Hurst exponent. Even though the parameters that we used to calculate the diffuse scattering intensity is different in the top and the bottom graphs, they show the similar behavior when kσ increases. This is due to the relative magnitude of the wavelength and the roughness of the surface. In the top graphs λ is constant for the top graphs, but σ has to increase to enlarge kσ. This means the wavelength is decreasing compare to σ. On the other hand, σ is constant in the bottom graphs, so to enlarge kσ, λ has to decrease. This also results in small wavelength compare to the roughness of the surface. When the wavelength is smaller than the roughness of a surface, the diffuse scattering intensity decreases. This is similar to the runner example that we mentioned earlier. If we assume that the Hurst exponent of a surface is constant, the changes in correlation length, ξ, is coupled with that of the height mean squared, σ. Our results show that the scattering intensity for a surface with constant Hurst exponent not only depends on the wavelength of the incident beam, but also it changes by changing the σ and ξ. 4.2.2 Effect of correlation Length Figure 5 shows the diffuse scattering intensity as a function of kσ for two different values of the Hurst exponent H = 0.5, 1.0. For the top graphs λ is constant and for the ones at the bottom σ is constant. In all four graphs of Figure 5, the diffuse scattering intensity is calculated for three values of the correlation length, ξ = 500, 1000, 1500 nm. Besides, for all the graphs the angle between the incident beam and the normal of the plane and the angle between the incident and scattered planes are kept constant, θ1 = θ3 = 0°. The diffuse scattered intensity is calculated for one specific scattered angle, θ2 = 5°. In Figure 5 (a and b) the diffuse scattering intensity increases as kσ increases due to the increment of σ. Both of these graphs show a maximum value for intensity at a specific value of kσ that is the same as the threshold value mentioned in the previous section. Similar behavior can be seen in Figure. 5 (d) where σ is kept constant and kσ increases be decreasing λ. However, in Figure 5 (c), the intensity ends up to a plateau after reaching to its maximum value. This result is confirmed by Figure 4 (b), where shows that the changes of the diffuse scattering intensity is small at θ2 = 5°. 5. Conclusion The scattered wave from a rough surfaces depends on the roughness parameters such as σ, ξ and λ. In the limiting case of l x, only two characteristic lengths; λ which is the observation scale, and σ which is the scale of surface, or their ratio comes in to effect on the scattered wave intensity. This means that in this case kσ is important. This limit has been the case of consideration by other researchers. λ λ But if ξ and λ are in comparable range, the system posλ λ sesses three characteristic lengths; λ the observation scale, σ and ξ which are the scales of the rough surface. In this (b) H=0.5, λ =633nm (a) H=1, λ=633nm (b) H=0.5, λ =633nm (a) H=1, λ=633nm 30 45 30 45 case, kσ does not prove to be a suitable parameter in order ξ =500nm ξ =500nm =500nm ξ =500nm 40 40 to present wave scattering from ξξthe rough surface, unless =1000nm =1000nm ξ =1000nm ξ =1000nm ξ 25 25 =1500nm =1500nm =1500nm =1500nm ξ ξ ξ ξ 35 35 the consideration of a varying k or σ is studied for a conσ σ 20 30 λ =633nm 20 30 λ=633nm λ =633nm (b) H=0.5, (a) H=1, (b) H=0.5, λ=633nm (a) H=1, stant kσ. In other words is must be understood that which 30 45 σ σ 30 45 25 25 15 ξ =500nm ξ =500nm of the parameters and which is constant. 15 =500nm ξ =500nm k or σ is varying ξ 40 40 20 ξ =1000nm ξ =1000nm 20 ξ =1000nm ξ =1000nm 25 25 =1500nm =1500nm =1500nm =1500nm ξ ξ ξ ξ This has been emphasized in this work. Besides, we study 35 35 10 15 10 15 20the scattering phenomena including all three length 30 20 30 10 10 5 5 25 25 scales, k, σ and ξ, both theoretically and e xperimentally. 5 5 15 15 20 20 0 0 To study the relation of the scattered wave intensity 0 0 0.5 1 1.50.5 2 1 1 1.50.5 2 1 1.5 2 0.5 1.5 2 kσ kσ kσ kσ 10 15 10 15 Figure 5: Dependence of diffuse scattered intensity on kσ for different correlation length (ξ = 500, 1000, and kσ when ξ and λ are comparable, we used a self-af10 θ = θ = 0 , θ = 5 . 1500nm) and for 10 (c) H=1, σ =50nm (d) H=0.5, σ =50nm (c)angles H=1, σ =50nm (d) H=0.5, σ =50nm 5 Figure 5: Dependence of diffuse scattered intensity on kσ for different correlation length (ξ = 500, 1000, 5fine rough surface. We compare two regimes for the kσ 40 100 40 100 5 5 1500nm) and for angles θ = θ = 0 , θ = 5 . wavelength is changed relative magnitude the wavelength and ξ the roughness the surface. In intensity the top graphson λ is constant =500nmis σ = cont. and Figure 5. of Dependence of=500nm diffuseof scattered kσ for 0parameter, ξone ξ =500nm ξ =500nm 0 0 0 =1000nm =1000nm 0.5 ξ 1.5 2 0.5 0.5 ξ =1000nm 2 1 1.5 ξ =1000nm 2 ξ 0.5 1 1.5 21 1 =1500nm =1500nm k= ξenlarge ξ =1500nm σ the the 1.5 other is λ k=σ const. andξ σ=1500nm is changed. The result 80 k ξ1500nm) σ wavelength σ 80 thefor top magnitude graphs, butofσthe has to increaseand to length kσ.(ξThis means the wavelength is decreasing compare relative the roughness of surface. In the top graphs λ is constant for kand different correlation 500, 1000, and 30 30 shows the diffuse scattering intensity in both of them angles θσ1hand, = σθto3σ=50nm =is 0°, θto2 enlarge =in 5°. σ =50nm to for σ.topOn the(c) other constant the kσ. bottom so H=0.5, to enlarge λ has to decrease. (d) H=0.5, the graphs, butH=1, has increase Thisgraphs, means(d) the wavelength is decreasing compare (c) H=1, σ =50nm σkσ, =50nm (b) H=0.5, =633nm (a) H=1, =633nm 30 ξ =500nm ξ =1000nm ξ =1500nm 25 15 20 10 15 5 1 kσ (c) H=1, =50nm 0 100 1.5 0.5 1 1.5 kσ 40 60 20 40 0 20 0 60 100 40 80 20 60 0 1 1.5 kσ 0.5 1 2 1.5 kσ 1 3 1 3 ◦ ◦ 2 2 2 10 15 5 10 0 5 0.5 1 1.5 (d) H=0.5, =50nm 0 0.5 1 1.5 40 kσ kσ ◦ 40 30 30 20 20 10 10 0 0 2 2 ξ =500nm ξ =1000nm ξ =1500nm (d) H=0.5, =50nm ξ =500nm ξ =1000nm ξ =1500nm 0.5 tering Intensity (ar.un.) Intensity (ar.un.) Diffuse Scattering Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 60 80 2 ξ =500nm ξ =1000nm ξ =1500nm (c) H=1, =50nm 80 100 2 15 20 ξ =500nm ξ =1000nm ξ =1500nm 0.5 1 1.5 2 0.5 1 1.5 2 kσ kσ ◦ 60 100 40 tering Intensity (ar.un.) Intensity (ar.un.) Diffuse Scattering 5 0.5 ξ =500nm ξ =1000nm ξ =1500nm 20 25 Diffuse Scattering Intensity Diffuse (ar.un.) Scattering Intensity (ar.un.) 30 20 10 0 ξ =500nm ξ =1000nm ξ =1500nm 30 ering Intensity Diffuse (ar.un.) Scattering Intensity (ar.un.) 35 25 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) ξ =500nm ξ =1000nm ξ =1500nm 40 30 Diffuse Scattering Intensity (ar.un.) Intensity (ar.un.) Diffuse Scattering Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity Diffuse (ar.un.) Scattering Intensity (ar.un.) ering Intensity Diffuse (ar.un.) Scattering Intensity (ar.un.) 45 35 (b) H=0.5, =633nm 25 Diffuse Scattering Intensity (ar.un.) Diffuse Scattering Intensity (ar.un.) 40 Diffuse Scattering Intensity (ar.un.) Intensity (ar.un.) Diffuse Scattering 45 (a) H=1, =633nm 40 This in small compare the roughness surface. kσ, When thetowavelength to σ. also On results the other hand,wavelength σ is constant in thetobottom graphs, of so the to enlarge λ has decrease. 6 20 ξ =500nm 20 ξ =500nm ξ =1000nm ξ =1000nm =1500nm is smaller than the of a surface, the diffuse scattering intensity This similar to This also results in roughness small wavelength compare to the roughness of the surface. When theiswavelength =1500nm 40 ξ decreases. 80 ξ Vol 8 (16) | July 2015 | www.indjst.org 30 30 thesmaller runnerthan example that we mentioned earlier. If we scattering assume that the10Hurst exponent of isa similar surface to is is the roughness of a surface, the diffuse intensity decreases. This 20 60 10 constant, changesthat in correlation length, ξ, is coupled with that theHurst height mean squared, σ. Our the runnerthe example we mentioned earlier. If we assume that ofthe exponent of a surface is 20 20 results show the scattering intensity forξ,aissurface with constant exponent only depends constant, thethat changes in length, coupled with that of Hurst the meannot squared, σ. Our 0 height 0 correlation 40 0 ξ =500nm ξ =1000nm ξ =1500nm ξ =500nm ξ =1000nm ξ =1500nm Indian Journal of Science and Technology M. Salami have a threshold value for kσ which happening when the correlation length, height mean squared and wave length have comparable value. It is noted that by existence of correlation length ξ, kσ is not the only dimensionless parameter. Therefore we must consider by variation of ξ which parameters (k or σ) are being changed. The role of the Hurst exponent has been investigated for self-affine rough surfaces. Such an examination was performed over a wide range of surface topographies, from logarithmic (H = 0) to a power-law self-affine rough surface, 0 < H < 1. 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