International Journal of the Physical Sciences Vol. 6(35), pp. 7857 - 7864, 23 December, 2011 Available online at http://www.academicjournals.org/IJPS DOI: 10.5897/IJPS11.743 ISSN 1992 - 1950 © 2011 Academic Journals

Full Length Research Paper

A data driven model of laser light scattering in metallic rough surfaces

Arturo Moreno-Báez1, Gerardo Miramontes-de León1*, Claudia Sifuentes-Gallardo1, Ernesto García-Domínguez1 and Jorge Adalberto Huerta-Ruelas2

1Universidad Autónoma de Zacatecas, López Velarde 801, C.P. 98000, Zacatecas, México.

2Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Instituto Politécnico Nacional, Cerro Blanco 141,

Colinas del Cimatario C.P. 76090, Querétaro, México.

Accepted 05 December, 2011

A new model of laser light scattering was obtained from experimental profilometer data and ray casting technique. Modelling was based on variables common to an angle resolved scatterometer and geometric optics. With this model, it was possible to obtain the scattering patterns, masking and shadowing effects, multiple lobes and backscattering. As advantage of the model, it was unnecessary to make statistical assumptions about the slope distributions. Satisfactory results were obtained from

very small Ra values (0 m) to relatively large values (7.53 m). With no modifications or additions to the model, for different surface profiles, off-specular and multiple lobes were obtained. A complete set of equations is given, detailing the construction of the model. In addition to reproducing the scattering patterns reported in the literature, it was found that, if measured profilometer data are used, for large roughness values and large incidence angles, the scattering pattern does not hold the Helmholtz principle. Key words: Modelling, geometric optics, roughness, reciprocity principle.

INTRODUCTION The study of scattering of light incident on a surface is of great interest and is useful in the characterization of surfaces. For well polished surfaces, light scattering not only contains roughness information but also can provide a non-contact method for the demanding quality control in modern industry (Zhenrong et al., 2010).

The intensity profile of the scattered light from a single coated sphere can be calculated using the Mie-Aden-Kerker theory (Suzuki and Lee, 2008). However, the modelling and measurement of the scattering that produces a laser light source on a surface usually is based on assumptions about the surface properties. One assumption made is about the distribution of the surface defects. Some studies assume a Gaussian distribution *Corresponding author. E-mail: gmiram@ieee.org. Tel: +52 (492) 9239407 ext. 1502.

while Le Bosse et al. (1999) have found such distribution invalid. Most of the theoretical models are based on statistical parameters, since the use of detailed information about the surface had been computationally expensive.

In a recent work, Schröder et al. (2011) divide the approaches for modelling the light scattering from rough surfaces in analytical scattering models and rigorous treatments. At the same time, they recognize that analytic scattering theories and approaches make use of the statistical properties of stochastic surfaces instead of relying on the knowledge of the exact topography. Schröder et al. (2011) also states that rigorous theories are often performed in one dimensional periodic surfaces, otherwise, for additional degrees of complexity, it required a tremendous computational power.

In this work, we report the construction of a light scattering model which uses profilometer data and geometrical optics. Profilometer data were experimentally

7858 Int. J. Phys. Sci. measured on various surfaces with a commercial profilo-meter (Mitutoyo Model SJ400).

The model was primarily based on Snell's law. The results are comparable with those scattering patterns obtained experimentally and reported in the literature (Le Bosse et al., 1999; Lu and Tian, 2006). It is also shown that, for large incidence angles and big values of roughness, the scattering pattern does not satisfy the Helmholtz reciprocity principle. The Helmholtz reciprocity principle is defined as the capability of the spatial distributions of incident and reflected flux to interchange completely without alteration of the measured reflectance (Tan et al., 2007).

It is known that the distribution and characteristics of the defects determine the angular scattering and therefore, it is related to the characteristics of the surface (Rao and Lakshmi, 1998). The technique for measuring laser light scattering is to project a laser beam on the sample to analyze and measure the light reflected at different angles. As the scattering is the change in spatial distribution of the reflected beam, it is important to detect these distribution changes through geometric arrange-ments, either by placing an array of photo-detectors or moving a single detector to along an arc that covers all the possible scattering space.

For decades, several models have been proposed to describe the reflection of light striking a surface, all of them with different properties and degrees of success. Among them are Phong (1975), Blinn (1977), Beckmann and Spizzchino (1963), Torrance and Sparrow (1966) and many others.

Geometrical optics makes use of the behaviour of light as rays. Torrance and Sparrow (1966) justify the use of geometrical optics and can usually explain the phenomenon of reflection of light when the wavelength is small compared with the dimensions of system, that is, it is valid if the surface roughness is large compared with

wavelength, m/>> 1, where m measures the level of

roughness and is the light wavelength. These models may be based on experimentation (empirical models) or by physical interaction of light and matter (physical models).

On the other hand, the reflectance can be described in terms of a bidirectional reflectance distribution function (BRDF), which is based on geometrical parameters. One can also have the following cases depending on the type of surface and material: a) ideal diffuse reflectance, when reflection occurs equally in all directions (called Lambert model), b) ideal specular reflectance, when reflection occurs without any scattering of the beam, and c) directional diffuse scattering. As a real surface does not behave completely as a mirror or as a Lambert surface, it disperses light from a single source in many directions. The main direction is close to the specular and presents a scattering lobe.

Lu and Forrest (2007) used geometrical optics to represent the light scattering based on the BRDF, which includes the specular reflection by a geometrical model of

Torrance-Sparrow. Several works (Torrance and Sparrow, 1996; Cook and Torrance, 1981; Oren and Nayar, 1995) used an element of surface area composed of micro-facets (or mirrors) randomly oriented. These models assume that the surface is composed of a large number of micro-facets distributed statistically with a direction given by a probability distribution function.

Unlike other models of light scattering, no assumptions are made about any slopes distribution and inter-reflections factor in this work. In either case, the distri-bution is obtained from the experimental data provided by mechanical profilometer. Additionally, it is considered that the beam is collimated, fully coherent and meets the conditions of Rayleigh.

To define the smoothness or roughness of a surface, which is commonly used by the Rayleigh criterion, that is,

define a critical height ch given by:

)cos(8 i

ch

(1)

where is the wavelength and i is the angle of

incidence. If h is defined as the height between the

maximum to minimum lump, then the surface is

considered smooth if chh and rough if chh .

Koenderink and van Doorn (1998) included Helmholtz reciprocity principle and considered at least empirically, that all materials meet this principle. However, the physical meaning of the principle of Helmholtz is given taking the symmetry of the BRDF. As shown in the results, this principle was not met when experimental surface profile was used and under conditions of large incident angles and large values of roughness.

Until now, the complexity and computational cost has prevented developing a model based entirely on geometrical optics and analyzing profile data thoroughly. So, the aim of this study is to develop a model of laser light scattering, on metallic surfaces, based on physical variables instead of statistical assumptions, taking advantage of current computational tools.

CONSTRUCTION OF THE MODEL

The reflection of light on metal surfaces can occur in two main ways, as a specular reflection, when the surface has a mirror-like surface finish, and as diffuse reflection, if the surface finish is rough.

When the roughness is low, the reflection will behave according to Snell's law. Real surfaces are not completely mirror (smooth) and always have roughness. The

roughness ( aR ) is calculated by Equation 2, where iz is

the height of the surface and the point i and N is the

total number of points taken in the surface profile.

Figure 1. Model parameters for an angle resolved

scatterometer.

N

i

ia zN

R1

||1

(2)

Model parameters The model involves, as shown in Figure 1, the following parameters: the distance from the laser to the sample

Ld , the tilt angle referred to the surface normal, the

beam diameter sd , the radius of the circular path of the

detector er defining the distance of the detector to the

sample and angle of the detector in the range

2/2/ . The length l of the sample was

divided into N points each with height iz of the

roughness profile.

The profile is given by a list of positions ix , and

height iz , where 10 Ni , and N is the number of

points. The positions on the x axis must be equally

spaced by a distance x , determined by the measuring

conditions of the profilometer which generates N points

on a distance l , therefore:

1

1

Nx (3)

list are the facets (Figure 2) and for every facet the slope is given by:

x

zim

. (4)

Moreno-Báez et al. 7859

Figure 2. Collision search between a ray and a facet,

),( rr zx , the laser emission point; rm , the beam

slope; i , the direction for analysis; maxz and minz ,

maximum and minimum profile values; 1n , a likely

initial collision point; 2n , the likely ending collision point.

The laser light source is modelled by a ray density rN .

The sensor is modelled as a circle of radius er ; this circle

defines the path to traverse by the sensor in an angular resolved scatterometer (Figure 1).

Algorithm

The construction of the model is based on an algorithm that traces rays and calculates the possible reflections according to surface profile. The steps of the algorithm are:

1) Maximum maxz and minimum minz heights are located

from data set iz . This will be useful to calculate the

search range and minimize the number of operations to find the collision of a ray with some of the facets. 2) To simulate the laser source and the distance

illuminated by the same, we will use a number of rays rN ,

separated by a distance r , which is given by:

1

r

sr

N

d , (5)

where rN must be at least 2. The larger the rN , the more

accurate the scattering pattern will be, but a greater number of rays represent more processing and longer calculation time. 3) Each ray has an initial position, which is given by:

2

12)cos()sin(

jNrdx rLr , (6)

7860 Int. J. Phys. Sci.

2

12)sin()cos(

jNdz r

rLr , (7)

where 10 rNj and the initial slope of each ray is

given by:

)cot(rm . (8)

4) It is necessary to calculate the range of facets where there is no possibility of a collision between the beam previously calculated and the surface, as shown in Figure 2. This will minimize the number of calculations required

to find the collision. The range 21,nn is calculated by:

2

11 max N

m

xmzzn

xr

rrr , (9)

2

12 min N

m

xmzzn

xr

rrr , (10)

Direction of travel is given by i . If 21 nn then 1i

otherwise, 1i , and if 1n and 2n is not found in the

range 0 to 2N it must conform to the nearest

boundary.

5) The scanning begins from 1n to 2n in direction i ,

calculating for each i :

ri

rrriiic

mm

zxmzxmx

, (11)

iiicic zxmxmz . (12)

If for some cx is true that 1 ici xxx , then it has found

a collision with one facet and must recalculate the source and direction of the ray, this is given by:

cr xx 1

, (13)

cr zz 1

(14)

12

22

21

rii

ririr

mmm

mmmmm , (15)

where the superscript (1) indicates next value, defining

the new range of search, where 1n is the current step,

2n is calculated using the Equation 9, when 01

immr

and also 0rm . Otherwise, Equation 10 is used and the

search directions

i as defined above, repeating the

process (step 5) until there are no collisions. 6) To calculate the angle at which the beam intercepts

the detector, it is required to calculate the point ),( ee zx

where they cross each other, so we define the

parameters 21 rma , )(2 rrrr xmzmb and

22)( errr rxmzc and:

a

acbbxe

2

42 , (16)

rrere zxxmz )( . (17)

Because Equation 16 has two solutions and the surface is being modelled as a number of reflective facets, the

solution set is that holding the condition 0ez .

The angle at which the beam will be captured by the sensor is given by:

)arctan(2 e

e

x

z

. (18)

The beam will be accumulated as an event in a vector

V .

Each element of the vector represents an angular range (the path) of the sensor. These operations will be accounted for each of the rays from step 3. 7) The final step is to normalize the vector

V by splitting

each of its elements inrN .

METHODOLOGY

A set of surface samples was prepared using different levels of roughness and different types of finishing and polishing (Figure 3). Later, the roughness profiles were obtained using a Mitutoyo profilometer, Model SJ400.

The profilometer has the following characteristics: measurement

range 800 m and a resolution of 0.000125 m (in the range of 8

m). In addition to delivering the raw data, the profilometer provides statistical processing which can be performed in multiple measurements for a roughness parameter. The length measurement selected was 2 mm; each sample was classified by

level of roughness given by aR . A total of 74 roughness profiles

were obtained from 0 to 7.53 m. The surface profile corresponding

to 0 m was generated synthetically. The model takes into account the effects of shadowing and inter-

reflections (masking) produced by the topography of the surface (Figure 4) without requiring probabilistic factors to assume the existence of them.

RESULTS AND DISCUSSION

Scattering at different incident angles

The scattering pattern was displayed in different forms,

Moreno-Báez et al. 7861

Figure 3. Set of samples with different surface finish and roughness levels.

a b

Figure 4. Zoomed images from profile data showing. a, Shadowing (left) and masking (right) effects. The reflected ray losses energy (shown in pale grey color) in each reflection.

using Cartesian coordinates with intensity in arbitrary units (a.u.) against the angle of reflection and in polar form. In addition, the filtering effect was taken into account according to the size of the sensor.

Varying the angle of incidence on a sample with a given roughness, the model showed the scattering pattern changes. Figure 5 shows the result using a profile

with a roughness level 78.3aR m. Vertical lines show

the maximum value of the pattern and the possible off-specular lobe. The angles of incidence applied were 15, 30, 45, 60 and 75°.

Higher roughness gave a greater scattering, which contains larger global effects like shadow and masking at large angles. Figures 6a and 6b show the effect of roughness at two angles of incidence; the first of which is

angled 15° with roughness of 0.26, 1.73 and 5.10 m,

while the second has 75° with the same roughness values.

The results showed in Figures 5 and 6 are comparable to those results showed by Le Bosse et al. (1999), where in their Figure 3 also show scattered intensities for different incidence angles versus the scattering angle.

Reciprocity principle

Finally in Figure 7, it is shown that exchanging the position of the source and observation point for large angles of incidence and high values of roughness, the Helmholtz reciprocity principle was not met. In theory, the scattering pattern should keep symmetry between an

angle of degrees and degrees. It is also shown

in Figure 7 (right side) a strong backscattering. By contrast,

7862 Int. J. Phys. Sci.

Figure 5. Scattering at different incident angles and aR 3.78 m.

a b

Figure 6. Incident angle fixed at 15° and roughness values of 0.26, 1.73 and 5.10 m (left), and Incident angle fixed at 75°, roughness

values of 0.26, 1.73 and 5.10 m (right).

contrast, when the roughness was small, the reciprocity principle remained valid, as shown in Figure 8.

In Koenderink and van Doorn (1998) for each reflection term, that is, Lambertian lobe, Specular lobe and back-scattered lobe, a new equation is given. In comparison, the results shown in Figures 7 and 8 appear according to the surface profile data and the incident angle without any modification of the proposed model.

Conclusion Scattering patterns were obtained from a laser light scattering model that predicted the behaviour of light on the measured profile of a reflective surface. The profile data were experimentally obtained from a commercial profilemeter. The model did not require making assume-ptions about the distribution of slopes; it used physically

Moreno-Báez et al. 7863

a b

Figure 7. Reciprocity principle fails for large incident angles and large aR . Incident angles of +60 (left) and -

60 (right)° show different BRDF. Multiple lobes and backscattering is produced with incident angle of -60°.

a b

Figure 8. Reciprocity principle remained valid (left versus right) for small roughness values at large incident angles (+60 and -60°).

permissible variables such as laser distance, spot diameter of beam, incidence angle, detector distance and wavelength of the laser beam. With the studied surfaces and parameters of the optical arrangement, geometrical optics was applied. One important result was the Helmholtz reciprocity principle which was not met for large roughness values and large incident angles. Further work is in progress to use this result in surface analysis. ACKNOWLEDGEMENTS This project was partially funded by the National Council of Science and Technology (CONACyT) under FOMIX-support agreement number Zacatecas ZAC-2007-CO1-

82136. Gerardo Miramontes received support for a leave of absence at CICATA-IPN, from CONACyT under Contract No. 290537-IPN. REFERENCES

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