Features of Light Scattering by Surface Fractal Structures

The average coefficient of light scattering by surface fractal structures is calculated within the scope of the Kirchhoff’s method. Two-dimensional bandlimited Weierstrass function is used to simulate a scattering surface. On the basis of numerical calculations of average scattering coefficient,...

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Дата:	2009
Автори:	Semchuk, O.Yu., Vodopianov, D.L., Kunitska, L.Yu.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут металофізики ім. Г.В. Курдюмова НАН України 2009
Назва видання:	Наносистеми, наноматеріали, нанотехнології
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/76535
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Цитувати:	Features of Light Scattering by Surface Fractal Structures / O.Yu. Semchuk, D.L. Vodopianov, L.Yu. Kunitska // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2009. — Т. 7, № 3. — С. 793-801 . — Бібліогр.: 16 назв. — англ.

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spelling	irk-123456789-765352015-02-11T03:03:25Z Features of Light Scattering by Surface Fractal Structures Semchuk, O.Yu. Vodopianov, D.L. Kunitska, L.Yu. The average coefficient of light scattering by surface fractal structures is calculated within the scope of the Kirchhoff’s method. Two-dimensional bandlimited Weierstrass function is used to simulate a scattering surface. On the basis of numerical calculations of average scattering coefficient, the scattering indicatrixes for various surfaces and incidence angles are calculated. The analysis of the indicatrixes leads to the following conclusions: the scattering is symmetric about the incidence plane; with the increase of surface calibration degree, the scattering pattern becomes more complicated; the greatest intensity of the scattered wave is observed along the mirror direction; there are other directions, in which the intensity bursts are observed. В рамках Кирхгоффової методи розраховано середній коефіцієнт розсіяння світла поверхневими фрактальними структурами. Для моделювання розсіювальної поверхні використовувалася двовимірна, обмежена смугою Вейєрштрассова функція. Виконано чисельні розрахунки середнього коефіцієнта розсіяння та побудовано індикатриси розсіяння для різних типів поверхонь та кутів падіння. Аналіза індикатрис розсіяння призводить до наступних висновків: розсіяння є симетричним відносно площини падіння; зі збільшенням ступеня калібрування поверхні картина розсіяння ускладнюється; найбільша інтенсивність розсіяної хвилі спостерігається в дзеркальному напрямку і, крім того, існують напрямки, в яких спостерігаються сплески інтенсивности. В рамках метода Кирхгофа рассчитан средний коэффициент рассеяния света поверхностными фрактальными структурами. Для моделирования рассеивающей поверхности использовалась двумерная, ограниченная полосой функция Вейерштрасса. Произведены численные расчеты среднего коэффициента рассеяния и построены индикатрисы рассеяния для различных поверхностей и углов падения. Анализ индикатрис рассеяния приводит к следующим заключениям: рассеяние является симметричным относительно плоскости падения; с увеличением степени калибровки поверхности картина рассеяния усложняется; наибольшая интенсивностьрассеянной волны наблюдается в зеркальном направлении и, кроме того, существуют другие направления, в которых наблюдаются всплески интенсивности. 2009 Article Features of Light Scattering by Surface Fractal Structures / O.Yu. Semchuk, D.L. Vodopianov, L.Yu. Kunitska // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2009. — Т. 7, № 3. — С. 793-801 . — Бібліогр.: 16 назв. — англ. 1816-5230 PACS numbers: 05.45.Df,41.20.Jb,42.25.-p,61.43.Hv,68.35.Ct,78.68.+m,81.70.Fy http://dspace.nbuv.gov.ua/handle/123456789/76535 en Наносистеми, наноматеріали, нанотехнології Інститут металофізики ім. Г.В. Курдюмова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The average coefficient of light scattering by surface fractal structures is calculated within the scope of the Kirchhoff’s method. Two-dimensional bandlimited Weierstrass function is used to simulate a scattering surface. On the basis of numerical calculations of average scattering coefficient, the scattering indicatrixes for various surfaces and incidence angles are calculated. The analysis of the indicatrixes leads to the following conclusions: the scattering is symmetric about the incidence plane; with the increase of surface calibration degree, the scattering pattern becomes more complicated; the greatest intensity of the scattered wave is observed along the mirror direction; there are other directions, in which the intensity bursts are observed.
format	Article
author	Semchuk, O.Yu. Vodopianov, D.L. Kunitska, L.Yu.
spellingShingle	Semchuk, O.Yu. Vodopianov, D.L. Kunitska, L.Yu. Features of Light Scattering by Surface Fractal Structures Наносистеми, наноматеріали, нанотехнології
author_facet	Semchuk, O.Yu. Vodopianov, D.L. Kunitska, L.Yu.
author_sort	Semchuk, O.Yu.
title	Features of Light Scattering by Surface Fractal Structures
title_short	Features of Light Scattering by Surface Fractal Structures
title_full	Features of Light Scattering by Surface Fractal Structures
title_fullStr	Features of Light Scattering by Surface Fractal Structures
title_full_unstemmed	Features of Light Scattering by Surface Fractal Structures
title_sort	features of light scattering by surface fractal structures
publisher	Інститут металофізики ім. Г.В. Курдюмова НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/76535
citation_txt	Features of Light Scattering by Surface Fractal Structures / O.Yu. Semchuk, D.L. Vodopianov, L.Yu. Kunitska // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2009. — Т. 7, № 3. — С. 793-801 . — Бібліогр.: 16 назв. — англ.
series	Наносистеми, наноматеріали, нанотехнології
work_keys_str_mv	AT semchukoyu featuresoflightscatteringbysurfacefractalstructures AT vodopianovdl featuresoflightscatteringbysurfacefractalstructures AT kunitskalyu featuresoflightscatteringbysurfacefractalstructures
first_indexed	2025-07-06T00:55:40Z
last_indexed	2025-07-06T00:55:40Z
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fulltext	793 PACS numbers: 05.45.Df, 41.20.Jb, 42.25.-p, 61.43.Hv, 68.35.Ct, 78.68.+m, 81.70.Fy Features of Light Scattering by Surface Fractal Structures O. Yu. Semchuk, D. L. Vodopianov, and L. Yu. Kunitska O. O. Chuiko Institute of Surface Chemistry, N.A.S. of Ukraine, 17 General Naumov, 03164 Kyyiv, Ukraine The average coefficient of light scattering by surface fractal structures is cal- culated within the scope of the Kirchhoff’s method. Two-dimensional band- limited Weierstrass function is used to simulate a scattering surface. On the basis of numerical calculations of average scattering coefficient, the scattering indicatrixes for various surfaces and incidence angles are calculated. The analysis of the indicatrixes leads to the following conclusions: the scattering is symmetric about the incidence plane; with the increase of surface calibration degree, the scattering pattern becomes more complicated; the greatest inten- sity of the scattered wave is observed along the mirror direction; there are other directions, in which the intensity bursts are observed. В рамках Кирхгоффової методи розраховано середній коефіцієнт розсіян- ня світла поверхневими фрактальними структурами. Для моделювання розсіювальної поверхні використовувалася двовимірна, обмежена смугою Вейєрштрассова функція. Виконано чисельні розрахунки середнього ко- ефіцієнта розсіяння та побудовано індикатриси розсіяння для різних ти- пів поверхонь та кутів падіння. Аналіза індикатрис розсіяння призводить до наступних висновків: розсіяння є симетричним відносно площини па- діння; зі збільшенням ступеня калібрування поверхні картина розсіяння ускладнюється; найбільша інтенсивність розсіяної хвилі спостерігається в дзеркальному напрямку і, крім того, існують напрямки, в яких спосте- рігаються сплески інтенсивности. В рамках метода Кирхгофа рассчитан средний коэффициент рассеяния света поверхностными фрактальными структурами. Для моделирования рассеивающей поверхности использовалась двумерная, ограниченная по- лосой функция Вейерштрасса. Произведены численные расчеты среднего коэффициента рассеяния и построены индикатрисы рассеяния для раз- личных поверхностей и углов падения. Анализ индикатрис рассеяния приводит к следующим заключениям: рассеяние является симметричным относительно плоскости падения; с увеличением степени калибровки по- верхности картина рассеяния усложняется; наибольшая интенсивность Наносистеми, наноматеріали, нанотехнології Nanosystems, Nanomaterials, Nanotechnologies 2009, т. 7, № 3, сс. 793—801 © 2009 ІМФ (Інститут металофізики ім. Г. В. Курдюмова НАН України) Надруковано в Україні. Фотокопіювання дозволено тільки відповідно до ліцензії 794 O. Yu. SEMCHUK, D. L. VODOPIANOV, and L. Yu. KUNITSKA рассеянной волны наблюдается в зеркальном направлении и, кроме того, существуют другие направления, в которых наблюдаются всплески ин- тенсивности. Key words: fractal surface, light, wave, indicatrixe, rough surface. (Received 20 October, 2008) 1. INTRODUCTION Accurate measurement of surface roughness of machined work pieces is of fundamental importance particularly in the precision engineering and manufacturing industry. This is caused by the more stringent de- mands on material quality as well as the miniaturization of product components in these industries [1—3]. For instance, in the disk drive industry, to maintain the quality of the electrical components mounted on an optical disk, the surface roughness of the disk must be accurately measured and controlled. Hence, the surface finish, normally ex- pressed in terms of surface roughness, is a critical parameter used for the acceptance or rejection of a product. Surface roughness is usually determined by a mechanical stylus pro- filometre. However, the stylus technique has certain limitations: the mechanical contact between the stylus and the object can cause defor- mations or damage of the specimen surface and it is a point wise and time-consuming measurement method. Hence, a noncontact and faster optical method would be attractive. Different optical noncontact meth- ods for surface roughness measuring were developed. They are based on reflected light detection, focus error detection, laser scattering, speckle and interference measurements [4—10]. Some of them have a good resolution and are applied in some sectors where mechanical measurement methods previously enjoyed clear predominance. Among these methods, the light scattering method [11] is a noncontact area- averaging technique and is potentially faster for surface inspection than other profiling techniques, particularly, the traditional stylus technique. Other commercially available products such as the scanning tunnelling microscope (STM), the atomic force microscope (AFM) and subwavelength photoresist gratings [12—15], which are pointwise tech- niques, are used mainly for optically smooth surfaces with roughness in the nanometre range. In this paper, the average coefficient of light scattering by surface fractal structures was calculated in the frameworks of the Kirchhoff’s method (scalar model). A normalized band-limited Weierstrass function was used to simulate 2D fractal rough surfaces. On the basis of numeri- cal calculation of average scattering coefficient, the scattering indica- trixes for various surfaces and incidence angles were calculated. The FEATURES OF LIFHT SCATTERING BY SURFACE FRACTAL STRUCTURES 795 analysis of the indicatrixes leads us to the following conclusions: the scattering is symmetric about the incidence plane; with the increase of surface calibration degree, the scattering pattern becomes more compli- cated; the greatest intensity of the scattered wave is observed along the mirror direction; there are other directions, in which the intensity bursts are observed. 2. FRACTAL MODEL FOR TWO-DIMENSIONAL ROUGH SURFACES The following form of the modified two-dimensional band-limited Weierstrass function is proposed: ( ) 1 ( 3) 0 1 2 2 , sin cos sin , N M D n n w nm n m m m z x y c q Kq x y M m − − = = ⎧ ⎫π π⎡ ⎤= + + ϕ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ ∑ ∑ (1) where cw is a constant, which ensures that W(x, y) has a unit perturba- tion amplitude; q (q > 1) is the fundamental spatial frequency; D (2 < D < 3) is the fractal dimension; K is the fundamental wavenumber; N and M are number of tones, and ϕnm is a phase term that has a uni- form distribution over the interval [−π, π]. The above function is a combination of both deterministic periodic and random structures. This function is anisotropic in two directions if M and N are not too large. It has a large derivative and is self- similar. It is a multiscale surface, which has same roughness down to some fine scales. Since natural surfaces are generally neither purely random nor purely periodic and often anisotropic, the function pro- posed above is a good candidate to model natural surfaces. The phases ϕnm can be chosen determinedly or casually, obtaining ac- cordingly determine or stochastic function z(x, y). We shall further con- sider ϕnm as casual values, which are regularly distributed on an interval [−π, π]. For each particular choice of numerical value of all N×M phases, ϕnm (for example, by means of the random numbers generator), we obtain particular (with the meanings of parameters cw, q, K, D, N, M chosen in advance) realization of function z(x, y). Every possible realization of function z(x, y) forms an ensemble of surfaces. Deviation of points of a rough surface from a basic plane is propor- tional to cw; therefore, this parameter is connected with the height of surface structure irregularities. Further, a rough surface is deter- mined, specifying root-mean-square height of its structure σ, which is determined by such expression: 2 ,hσ ≡ (2) where h = z(x, y), 1 0 1 ... (...) 2 N M nm n m dπ− = = −π ϕ = π∏∏ ∫ –averaging over ensemble of 796 O. Yu. SEMCHUK, D. L. VODOPIANOV, and L. Yu. KUNITSKA surfaces. The relationship between cw and σ can be established by direct calcu- lation of integrals: ( ) ( )( ) ( )( ) 1 21 2 2 3 1 2 2 3 0 1 1 , . 2 2 1 N D N M nm w D n m M qd z x y c q −π− − = = −π ⎡ ⎤−⎡ ⎤ϕ ⎢ ⎥σ = =⎢ ⎥ ⎢ ⎥π −⎣ ⎦ ⎣ ⎦ ∏∏ ∫ (3) So, the rough surface in our model is described by function of six pa- rameters: cw (or σ), q, K, D, N, M. The influence of different parame- ters on a kind of a surface can be investigated both analytically and studying structures of surfaces constructed by results of numerical calculations of Weierstrass function. Thus, it was found out that: the wave number K determines the wavelength of the basic har- monic of the surface; the numbers, N, M, D, and q, determine a degree of surface calibra- tion at the expense of imposing of additional harmonics on the basic wave, and N and M determine the number of harmonics, which are im- posed; D determines amplitude of harmonics; q determines both amplitude and frequency of harmonics. Let us note that, with increase of N, M, D, and q, the spatial uni- formity of the surface on a large scale is also increased. 3. LIGHT SCATTERING ON SURFACE FRACTAL STRUCTURES Experiment diagram of light scattering is presented in Fig. 1. The initial light wave falls on a rough surface S under angle θ1 and is scattered in all directions. The scattering wave is registered by the de- D S � 1 � 2 � 3 Fig. 1. Experiment diagram for light scattering by fractal surface; S is a scat- tering surface; D–detector, θ1 is an incidence angle; θ2 is a polar angle; θ3 is an azimuth angle. FEATURES OF LIFHT SCATTERING BY SURFACE FRACTAL STRUCTURES 797 tector D in the direction, which is characterized by a polar angle θ2 and an azimuthal angle θ3. The intensity of light Is scattered in (θ2, θ3) di- rection is measured. Our goal is construction of indicatrix of electro- magnetic wave scattering by a fractal surface (1). As ∗=s s sI E E (where Es is an electric field of the scattered wave in complex representation), the problem of Is finding is reduced to find- ing of the scattered field Es. We shall find the scattered field using Kirchhoff’s method [16], and considering complexity of a problem, we shall take advantage of sim- pler scalar variant of the theory, according to which the electromag- netic field is described by scalar variable. Thus, we lose an opportunity to analyze polarizing effects. The base formula of Kirchhoff’s method makes it possible to find the scattered field under such conditions: the incident wave is monochromatic and plane; a scattering surface is rough inside some rectangular (−X < x0 < X, −Y < y0 < Y) and smooth outside of its boundaries; the size of the rough site is significantly greater than length of inci- dent wave; all points of the surface have finite gradient; the reflection coefficient is identical for all points of the surface; the scattered field is observed in a wave zone, i.e. well away from the scattering surface. Under these conditions, the scattered field is given by ( )= − θ θ θ ϕ + π ∫ 0 1 2 3 0 0 0 0 exp( ) ( ) , , exp[ ( , )] ( ) 2s e S ikr E ikrF ik x y dx dy E r r r ,(4) where k is the wave number of incident wave; θ θ θ =1 2 3( , , )F ( )= − + +2 2 2 2 R A B C C is the angle factor; 1 2 3sin sin cosA = θ − θ θ ; 2 3sin sinB = − θ θ ; 1 2cos cosC = − θ − θ ; R is the scattering coefficient; 0 0 0 0 0 0( , ) ( , )x y Ax By Ch x yϕ = + + is the phase function; =0 0( , )h x y = 0 0( , )z x y ; ( )= − + π 1 2 exp( ) ( ) 4e R ikr E AI BI C r r , ( ) ( ) ( ) ( )0 0 0 0, , , , 1 0 2 0, . Y X ik X y ik X y ik x Y ik x Y Y X I e e dy I e e dxϕ ϕ − ϕ ϕ − − − ⎡ ⎤ ⎡ ⎤= − = −⎣ ⎦ ⎣ ⎦∫ ∫ (5) After calculation of integrals (4) and (5) using formula ( )sin ,iz il l l e I z e ∞ φ φ =−∞ = ∑ where I1(z) is the Bessel function of integer order, we obtain: 798 O. Yu. SEMCHUK, D. L. VODOPIANOV, and L. Yu. KUNITSKA { } ϕ∑⎧ ⎫⎡ ⎤= − ξ +⎨ ⎬⎢ ⎥π ⎣ ⎦⎩ ⎭ ∑ ∏exp( ) ( ) 2 ( ) sin ( ) sin ( ) ( ) nm nm nm uv rs i l s l u c s e l uv ikr E ikFXY I e c k X c k Y E r r r (6) where F = F(θ1, θ2, θ3), ( ){ } 0,1 0,2 1 , ... rs N Ml l l l − =−∞ ∞ ∞ ∞ =−∞ =−∞ ≡∑ ∑ ∑ ∑ , 1 1 0 N M uv u v − = = ≡∏ ∏∏ , 1 1 0 N M nm n m − = = ≡∑ ∑ ∑ , ( )3D u u wkc Cq −ξ ≡ , sin sin x cx x ≡ , π π≡ + ≡ +∑ ∑2 2 cos , sin ,n n c nm s nm nm nm m m k kA K q l k kB K q l M M ( ) ( ) ( ) ( )2 2 sin sin ikr e R e E ikXY A B c kAX c kBY C r = − + π r . Thus, expression (6) gives the solution of the problem of finding of field scattering by a fractal surface within the frameworks of Kirchhoff’s method. Now, it is possible to calculate intensity of scattered waves using formula (4) if parameters of the scattering surface cw (or σ), D, q, K, N, M, X, Y, φnm, parameter k 2 or k π⎛ ⎞λ =⎜ ⎟ ⎝ ⎠ of the incident wave, and pa- rameters θ1, θ2, θ3 of geometry of experiment are known. This intensity will characterize scattering of specific realization of the surface z(x, y) (with a specific set of casual phases φnm). For comparison of calcula- tions with experimental data, it is necessary to operate with intensity averaged over ensemble of surfaces: ∗=s s sI E E . This intensity is proportional to intensity θ⎛ ⎞= ⎜ ⎟π⎝ ⎠ 2 1 0 2 coskXY I r of the wave reflected from the corresponding smooth basic surface. Therefore, for the theo- retical analysis of results, it is more convenient to use averaged scat- tering coefficient: 0 .s s I I ρ = After calculation of sI and starting from (6), we obtain exact ex- pression: ( ) ( ) ( ) ( )⎡ ⎤θ θ θ ⎧ ⎫ρ = ξ +⎨ ⎬⎢ ⎥θ ⎩ ⎭⎣ ⎦ ∑ ∏ { } 1 2 3 2 2 2 1 , , sin sin cos uv rs s l u c s l uv F I c k X c k Y FEATURES OF LIFHT SCATTERING BY SURFACE FRACTAL STRUCTURES 799 ( ) ( ) ( ) 2 2 2 2 2 1 sin sin 2 cos R A B c kAX c kBY C ⎡ ⎤+ ⎢ ⎥+ θ⎢ ⎥⎣ ⎦ . (7) As expression (7) consists of the infinite sum, it is inconvenient to use for numerical calculations. Essential simplification is reached in the case when ξn < 1. Using series expansion of function as follows, ( ) ( ) ( ) 2 0 / 43 2 ! 1 k k z I z k k ν ∞ ν = −⎛ ⎞= ⎜ ⎟ Γ ν + +⎝ ⎠ ∑ , and rejecting terms of orders greater than 2 nξ , we obtain the ap- proximate expression for averaged scattering coefficient ( ) ( ) ( ) ( ){⎡ ⎤θ θ θ ⎡ ⎤ρ ≈ − σ +⎢ ⎥ ⎣ ⎦θ⎣ ⎦ 2 21 2 3 2 2 1 , , 1 sin sin coss F k C c kAX c kBY ( )2 32 2 21 2 sin cos sin 2 D n n f nm m c q c kA Kq X c M − ⎡ ⎤π⎛ ⎞+ +⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ ∑ + + 2 sinn m kB Kq Y M ⎫⎡ ⎤π⎛ ⎞+ +⎬⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦⎭ + ( ) ( ) ( ) 2 2 2 2 2 1 sin sin 2 cos R A B c kAX c kBY C ⎡ ⎤ +⎢ ⎥θ⎣ ⎦ , (8) where ( ) ( ) − − ⎡ ⎤−≡ = σ ⎢ ⎥ −⎢ ⎥⎣ ⎦ 1 2 3 2 2 3 2 1 1 D f w N D q c kc C k C M q . 4. RESULTS OF NUMERICAL CALCULATIONS On the basis of numerical calculations of average factor of scattering according to formula (8), we constructed the dependence of the aver- aged scattering coefficient sρ on θ2 and θ3 (scattering indicatrixes) for different types of scattering surfaces. At these calculations, we supposed that R = 1, and, consequently, did not consider real depend- ence of reflection coefficient R on the incidence wavelength, λ, and in- cidence angle, θ1. The obtained results are presented in Fig. 2. The analysis of the indicatrixes leads us to the following conclusions: the scattering is symmetric about the incidence plane; the greatest intensity of the scattered wave is observed along the mirror direction; 800 O. Yu. SEMCHUK, D. L. VODOPIANOV, and L. Yu. KUNITSKA there are other directions, in which splashes in intensity are observed; with the increase of surface calibration degree (or with growth of its large-scale heterogeneity), the scattering pattern becomes more compli- cated. The scattering is slightly dependent on the type of scattering sur- face, and there is a dependence of the scattering coefficient on the light-wave incidence angle. With increase of the incidence angle from 30° to 60°, the number of additional peaks decreases. Their most num- ber is observed at θ1 = 30°. It is related to the influence of the height of irregularities of the surface on the scattering process. а а′ a″ b b′ b″ c c′ c″ d d′ d″ Fig. 2. Dependences of log sρ on angles θ1 and θ3 for various types of fractal surfaces: a, a′, a–the samples of rough surfaces, for which the calculation of scattering indexes is produced; from top to bottom, the change of scattering index is routine for three angles of incidence: θ1 = 30, 40, 60° (a—d, a′—d′, a″— d″) at N = 5, M = 10, D = 2,9, q = 1,1; n = 2, M = 3, D = 2,5, q = 3; N = 5, M = 10, q = 3, respectively. FEATURES OF LIFHT SCATTERING BY SURFACE FRACTAL STRUCTURES 801 The noted features of scattering are investigation of combination of chaotic state and self-similarity of the scattering-surface relief. 5. CONCLUSION In this paper, the average coefficient of light scattering by surface fractal structures was calculated within the scope of Kirchhoff’s method. A normalized band-limited Weierstrass function is presented for modelling of 2D fractal rough surfaces. 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Features of Light Scattering by Surface Fractal Structures

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