View of A Mathematical Approach for Evaluation of Surface Topography Parameters

1 Introduction

Nominally flat surfaces are widely used in practice. These can be mathematically described by the composition of the deterministic and random components of irregularities in the given Cartesian coordinate system:

( ) ( ) ( )

h x y, =S x y, +z x y, (1) whereS(x,y) is the deterministic function of the surface (x,y) coordinates andz(x,y) is the homogeneous random normal field. The parameters of surface irregularities measured over the whole surface (the topographic parameters) characterize more positively the functional properties of the surface than thehprofile parameters. Surface deviations are functions of two coordinates (x, y) and therefore the profile evaluation gives incomplete information about the surface.

2 Surface topography parameter measurement

For surface topography parameter measurement it is nec- essary to determine the actual value of the parameter and to know the accuracy of the measurements. In this case the analogue mean value is taken for the actual value of the parameter, and[s] the measurement error is determined by the systematic and random components. The series of the surface topographic parameters can be represented as an averaging operator of the generalized transformation G{h(x,y)} of the surface coordinates on the given rectangular area of the surfaceL₁×L₂with sidesL₁andL₂[1]:

{ ( ) }

P_s L L G h x y x y

L L

= ^{1 21}

ò

₀¹

ò

₀^{2 , d d (2)}

Since there is a random component on the measured sur- face, the topographic parameter measured is a random value, which is characterized by the mathematical expectationE(P_s) and the variance D(P_s). Therefore, one of the problems in measuring the topographic parameter is the determination of its probability characteristics, i.e., the mathematical expec- tation E(P_s) and the variance D(P_s). It is known that the mathematical expectation of the parameters given by equa- tion (2) can be derived by integration of the mathematical expectationE(G) of the transformationG{h(x,y)} in equation (2) by thexandyvariables. As an example of the application E(G) of the transformation G{h(x, y)} in equation (2) by

thexandyvariables. As an example of the application of mea- suring methods of the topographic parameters theP_s=R_as parameter is used. This is the arithmetic mean deviation of the surface coordinates of the mean plane.

( )

{ ( ) }

E R L L E h x y x y

L L

as = ^{1 21}

ò

₀¹

ò

₀^{2 ,} d d ⁽³⁾ where^{h x y}

( )

^, is the absolute value of theh(x,y) surface coor- dinate. This expression can also be extended for theh(x,y) surface:

{ ( ) } { ^{( )} } _{( )} ( )

E h x y S x y

S x y S x y

, exp ,

, ,

= æèç ö

ø÷ é- ë êê ê

ù û úú ú 2 +

2

1 2 2

P s 2 F

s s2^{1 2}

ìí ï îï

üý ï þï(4)

where ^f( )^{z =P21 2}

ò

₀^zexp

( )

^{-t2 dt} is the Laplace function.

The generalized transformation G{h(x, y)} is the random field which has the correlation function [3] defined as

( )

( ) ( )

( )

{

( )

}

K x x y y G h G h f h h h h

E G h G h

G 1 2 1 2 1 2 1 2 1 2

1

, , , = , +

- ^-¥

+¥

-¥

+¥

ò ò

^{d d}

{

( )²

}

^,

(5)

where^{h1=h x y}

(

^{1 1,}

)

^{,h2 =h x y}

(

^{2 2,}

)

are the coordinates of the surface at the

(

^{x y1 1,}

)

^and

(

^{x y2 2,}

)

^{points,f h h}

(

^{1 2,}

)

distribution density expands into a series in terms of Hermite polynomi- als. The correlation function (5) can be represented as [3]:

( ) ( ) ( ) (

{

)

}

K x x y y C x y C x y

G n n n

n

1 2 1 2 1 1 1 1 1 2 1 2

1

, , , , , ,

= + + !

=

å

+¥ ^{t t r t t (6)}

( )

( ) ( ) ( )

C x y G h H h S h S

n 1 1 1 2 h

1 2

2 2

, = æ - exp

èç ö

ø÷ ì- - íï îï

üý ï

P s þï

s

s

n d

-¥

+¥

ò

⁽⁷⁾

t1= x2 -x1

t2 = y2 -y1

sis the r.m.s. deviation of the random componentz(x,y) and

( )

r t t1 2, is the correlation coefficient of the random compo- nents z(x, y) , ^{h=h x y}

( )

^{, ,S =S x y}

( )

, . For the generalized transformation^{G =}

{

^{h x y}

( )

^,

}

^{= h x y}

( )

^, , which determines the R_asparameter, the coefficientsC_nin equation (7) are written as follows after the transformation:

A Mathematical Approach for Evaluation of Surface Topography Parameters

A. K. Haghi

The probability characteristics of surface topography parameters described by the composition of the deterministic component and the homogeneous random normal field were analysed. Formulae for the calculation of the mathematical expectation of the R_asparameter and the evaluation of its variance are given.

Keywords: mathematical approach, surface topography, deterministic component.

( ) ( ) { ( ) }

C x y S x x S x y S x

1 1 1

1 2

1 2 1 1

2 2

, , exp 2,

= æèç ö

ø÷ é-

ë êê

ù û úú+

P s sF

(

1 1

)

21 2 ,y s ìí î

üý þ

( ) ^{

^{( )}

^}

C x y S x y

H S x y

n 1 1

1 2 2

2

2 1

2

, exp 2,

,

= æèç ö

ø÷ é- ë êê

ù û úú´

- -

P s

s

n

(

1

) (

1 1

) (

1 1

)

s s s

ìí î

üý

þ+ ì-

íî

üý þ é

ëê ê

ù ûú - ú

S x y

H S x y

, ,

n 1

(8)

whereH_n(z) are Hermite polynomials. The variance of the parameterP_Sis determined by integration of the correlation function^KG

(

^{x x y y1 2 1 2, , ,}

)

of the generalized transformation G{h(x,y)} using variablesx₁,x₂,y₁,y₂:

( )

( )

D P L L K x x y y x x y y

L L L

S = ¹²¹²²

ò

₀¹

ò

₀¹

ò

₀² G ^{1 2 1 2, , ,} d d d d^{1 2 1 2. (9)} Calculation of the integral in equation (9) involves consid- erable difficulties; therefore, in the general case, formula (9) in not suitable for calculations. However, having^{S x y}

( )

^{, ³s}

and^{S x y}

( )

^{, £s}offers simplified evaluations of the variance.

Thus, having^{S x y}

( )

^{, ³s}from equation (8), C₁»s

C_n »0 n=2 3, ,¼ (10) The correlation function of the transformationG{h(x,y)}

from equation (6) is reported approximately as

( ) ( )

K_G t t1 2, »s r t t² 1 2, . (11) By using^{S x y}

( )

^{, £s} from equation (8) we obtain

C₁»0

( ) Cn » æ Hn

èç ö ø÷ -

2 0

1 2

P s 2 .

The correlation function of the transformationG{h(x,y)}

from equation (6) is

( ) { ( ) }

K C

G n n

n

n

t t r t t

1 2 2 1 2

2

, ,

» !

=

å

+¥ ^{. (12)}

In both cases the correlation function^KG

(

^{t t1 2,}

)

^{is a func-}

tion of only two variablest1andt2. Thus, for the above-men- tioned approximations,D(P_S) can be calculated by using the formula:

( )

( )

D P L L L L K_G

L

S = æ - d d

èç ö ø÷ -æ

èç ö ø÷

ò

4 1 1

1 2

1 1

2 2

1 2 1 2

0

2 t t t t, t t

0 L1

ò

^{. (13)}

The following notation is now used:

( ) ( )

S_KG K K_G

G

= d d

¥

¥

òò

1

0 0 ^{1 2 1 2}

0 0

, t t, t t (14)

with L₁´L₂ ³S_KGequation (13) may be written approxi- mately:

( )

( )

D PS »L L4 KG SK_G

0 0

1 2

, . (15)

( )

K_G 0 0, is the variance ofG{h(x,y)}. It is not possible to carry out analogue measurements of the topographic parameter P_S. Therefore the analogue-discrete and discrete methods are the only ones that can be used to measure the topographic

parameters. With analogue-discrete measurements the aver- aging operator is:

( )

{ }

$ ,

P N L G h i y y

L

i N

S = ^{1 12}

å

₌

ò

¹ d

1 0

1 2

D (16)

whereD1is the sampling in the sampling interval between the profiles andN₁is the number of profiles. The evaluation of the parameter in equation (2) in the general case is therefore shifted together with the task of determining the parameter probability characteristics, i.e., for its mathematical expecta- tion and variance it is necessary to determine the shift in the value of the evaluation of equation (16) with respect to that of equation (2). We can regard the probability characteristics of the evaluation (16) for that particular transformation as

( )

{ } ( )

G h iD1,y = h iD1,y . (17) R$_asrepresents the analogue-discrete evaluation of the top- ographic parameter R_as. The mathematical expectation of the transformation (17) will be determined by the expression in equation (4) with ^{S x y}

( ) (

^{, =S iD1,y}

)

. The mathematical expectation of theR$asparameter is identically:

( ) ^{{ ( ) }}

E R N L E h i y y

L

i N

$_as = ^{1 11 2}

å

₌

ò

¹, d

1 0

1 2

D . (18)

The integrals in equation (18) are not taken into account, but if the deterministic component ^{S i}

(

^D1,y

)

is linearized with a small error using theyargument on the terminal num- ber of intervals the length of which isD2then it is possible to obtain an exact expression for the evaluation^{E R}

( )

^{$as of the}

topographic parameterR_as. For this, we expand the deter- ministic component^{S i}

(

^D1,y

)

in its Taylor series by using the y argument up to the linear terms at the pointj= 1, 2, …,n:

( ) ( ) ( ) ( )

S i y S i j S i j

y y

D1 D D1 2 D D1 2 D

, , , 2

» + ¶ -

¶ . (19)

By substituting equation (19) into equation (18) and tak- ing into account equation (4), we obtain:

( )

( )

E R N L

j j

j n

i N

$

exp

as = Á

Á = æ èç ö

ø÷ -

= -

=

å ò

å

1 1

2

1 2

1 1 1 1 2

2 1 2

D D

P s

( )

( )

a b

s

a b a b

s

ij ij

ij ij

ij ij

y

y y

ì + íï îï

ü ýï þï é +

ë êê

+ + æ +

2 2

1 2

2

F 2

èç ö

ø÷ù ûúdy

(20)

where

( )

( ) ( )

a ¶

¶

b ¶

¶

ij

ij

S i j y

S i j S i j

y j

=

= -

D D

D D D D D

1 2

1 2 1 2

2

,

, ,

.

(21)

The integrals in both components of expression (20) can be tabulated [4]. Then after transformation we obtain:

( )

E R N L C

A C

A CD

j n

i N

ij

ij ij ij ij

$_as =

- +

æ è

=

=

å

å

1 1 1

2 1

4 2

1 2 1 1

1

a a

( )

çç ö

ø

÷÷ +

ìí ï îï

üý ï F A Cij Gþï

(22)

( )

G F

=æ - + P

è çç

ö ø

÷÷ +

B C

B CD B C A

ij

ij ij ij ij ij ij

a a aij

1

4 2

1 2exp

( )

( )

- +

+ -

C A

B C B

ij

ij ij

ij

2 2

1 2

2 2 a P exp where

A_ij =a_ijD1+b_ij

( )

B_ij =a_ij j-1D₁+b_ij C= 1

2s^{1 2} D_ij

ij

= 2 ²

1 2

s a P

TheF( )z and exp(-z²)and the standard calculation pro- gram for the F( )z and exp(-z²) functions can be used to calculateE R($_as)on the computer. Formula (22) shows that the mathematical expectation of the parameterR$asdepends on the characteristic of the^{¶S i}( )^D1 ^¶y, the deterministic compo- nent S i(D1, )y and the random component s². With the definite relationship between the deterministic and the ran- dom components, an approximate evaluation ofE R($_as)can be obtained.

Thus, having^{S x y}

( )

^{, £s}and linearizing equation (4) we obtain:

( ) ^{( )}

E R N L S i y y

L

i N

$_as » ^{1 11 2}

å

₌

ò

¹, d

1 0

1 2

D . (23)

By using^{S x y}

( )

^{, £s}and linearizing equation (4):

( ( ) ) ^{

_{( )}

^{( ) }}

E h x y S x y

, ,

» æèç ö ø÷ + 2

2

1 2 2

P s P1 2

s. (24)

5,06 mm 5,06 mm

μm

20 0

0 1 2 3 4 mm

0 90 μm

Fig. 1: Representation of surface roughness measurement

2,9 mm

2,5 mm

0 0

1 2 mm

20 10 μm

Fig. 2: Representation of surface roughness measurement

Figs. 3 and 4: A 3D topographical image

Fig.5:Representationof3Droughnesstestinalargescale

Then

( )

_{( )}

^{{ ( ) }}

E R L L S x y x y

L L

$as » æ ,

èç ö

ø÷ +

ò ò

2 1

2

1 2

1 2 1 2

2 0

0

2 1

P s P

s d d. (25)

To analyse theE R($_as)dependence onS sand to deter- mine what relationship withS sis needed in order to make formulae (23) and (25) applicable to the determination of the mathematical expectation E R($_as), calculations were carried out using formulae (22), (23) and (25). The mathematical models of the composition surface and the experimental calculation of the parameter

R$ h_ij

j N

i

N _i

as =

=

=

å

å

¹_{1 1} ⁽²⁶⁾

whereh_ijis the deviation of theh(x,y)surface from the mean plane at the discrete points. The random component was

modelled with the help of the random number generator.

The deterministic component was designated by formula (19).

3 Discussion

l If the deterministic component is a piecewise linear func- tion and the random component is a homogeneous normal field, then the mathematical expectation of parameterR$_as increases with increasingS s.

l IfS sp1, whenE R($_as)is calculated the influence of the de- terministic component can be ignored and the error is less than 10 %.

l WhenS sp3, formula (25) can be used to calculateE R($_as) with an error of less than 10 %.

l IfS sf4,E R($_as)can be evaluated with an error of less than 10 % by using the deterministic component (for- mula 23).

l Formula (22) holds for the whole range ofS schanges.

l Full agreement with the theoretical expression can be demonstrated by the experimental calculations using for- mula (26).

l Some optical profilometers based on these principles are shown in figures (1–6) to measure the statistical parameters of rough surfaces. The contrast is found to be related to sur- face roughness when the length of coherence of the light [are] is comparable in magnitude.

4 Conclusion

The theoretical dependence of the mathematical expecta- tion of parameterR$_as has been determined for the case when the deterministic componentS(x,y) can be linearized on the separate sections. The dependence of the evaluation bias in parameterR$_as on the relationship between the random and deterministic components has been determined.

References

[1] Lukyan, V.:Measurement of the surface topography. ICTE, 2000, p. 103–154.

[2] Lewis, J.: Fundamentals of Microgeometry. RPI, 2001, p. 32–65.

[3] Zahedi, M.:Mesure Sans Contact de la Topographie d’une Surface. IUT de Belfort, 2000, p. 43–78.

[4] Galat, T.: Optical Surface Roughness Determination. Ap- plied Optic, 2000.

Dr. A. K. Haghi

e-mail: haghi@kadous.gu.ac.ir Guilan University

P.O. Box 3756 Rasht, IRAN Fig. 6: Microtopograph of a rough surface