Computer simulation of laser backscattering from fiber surfaces

Computer Simulation of Laser Backscattering from Fiber Surfaces

CHENG LUO and RANDALL R. BRESEE, Center for Materials Processing, 230 Jessie Harris Building, University of Tennessee,

Knoxville, Tennessee 37996-1900

Synopsis

The application of laser backscattering to fiber surface characterization was studied theoretically. Computer simulations were employed to investigate the theoretical influence of root-mean-square fiber surface roughness, fiber cross-sectional shape, fiber diameter, and surface asperity shape on scattering patterns. Results from this work indicate that laser backscattering may be useful for obtaining fiber structural information in a variety of applications.

INTRODUCTION

Typical textile fibers have large surface-to-volume ratios, so the surface structure of fibers greatly influences many properties of materials made from them. Additionally, the surface structure of individual fibers often significantly affects their processing behavior during manufacture into other materials.

Since the advent of lasers, scattering phenomena have received considerable attention both theoretically and experimentally. In recent years greater interest is being shown with regard to metallic surface roughness research using optical Fourier transform methods.',' Recently, we attempted to extend the Fourier transform method to fiber surface chara~terization.~ A computer simulation study is reported to investigate relationships between backscattering patterns and fiber surface structure. Multireflections were assumed to be negligible for these computations. The effects of fiber color were neglected because it would affect scattering intensity, but not the scattering pattern as long as color distribution was uniform.

Analytical methods used to measure surface roughness can be classified into contact and noncontact methods. A common example of a contact method which has been used for many years is the stylus-profilometer. The optical Fourier transform method, on the other hand, is a relatively recent noncontact method, which is based on diffraction theory.

The surface profile of a rough surface along one dimension is represented in Figure 1 by its coordinates with respect to an ideal smooth reference plane. Surface profiles are expressed in a statistical manner by the root-mean-square roughness, defined as

Journal of Polymer Science: Part B: Polymer Physics, Vol. 28, 1755-1770 (1990) 0 1990 John Wiley & Sons, Inc. CCC 0887-6266/90/01001755-016$04.00

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hi \

Fig. 1. One-dimensional surface profile.

where crrm is the RMS roughness, N is the number of height data points, hi is the height of point i with respect to the reference plane denoted by the horizontal straight line in Figure 1, and 6 is the mean of all height data points4

THEORY

Consider the geometrical situation in Figure 2. If the distance between the aperture and observation plane, 2, is much greater than the aperture itself,

where k = 27r/X is the wave number, X is the wave length, and x and y are the aperture width and height, respectively, then the electromagnetic field at the

Observation Plane

Fig. 2. Diffraction geometry.

COMPUTER SIMULATION OF BACKSCATTERING 1757

observation plane, E ( xo, yo), can be described by the well-known Fraunhofer diffraction expression

where E (x, y ) denotes electromagnetic field a t the aperture plane. This expression is exactly the Fourier integral except for some extra multiplicative phase and amplitude factors outside the integral sign, if spatial frequencies are expressed as

u = x o / x Z and u = y 0 / U (4)

The Fraunhofer expression can be applied to ba~kscattering~ if the electromagnetic field at the aperture plane, E (x, y ) , is replaced with a reflectance function, R ( x , y ) . The power of the illumination is not dependent on phase, and therefore, the intensity of a scattering pattern is a scaled version of the squared magnitude of the Fourier transform.

A computer program was written to compute the theoretical scattering pattern from a variety of single fiber models. Computations were based on the Fraunhofer diffraction expression provided in eq. (10). A flowchart of this program is provided in Figure 3. The general flow of the program is as follows: First, one of three fiber cross-sectional shapes and one of four surface asperity shapes were selected. A RMS surface roughness was determined for the model and then the reflectance function was computed. A two-dimensional Fourier transform of the reflectance function was performed using a FFT algorithm6 modified to operate in two dimensions as explained below. Finally, the theoretical backscattering pattern from each single-fiber model was plotted as intensity versus spatial frequency and a roughness coefficient was computed from the scattering pattern.

Fiber Model Generation

In order to investigate the influence of different fiber cross-sectional and asperity shapes on scattering, three fiber cross-sectional shapes (rectangular, circular, and triangular) and four asperity shapes (square, triangular, sine', and random) were defined. Each fiber surface model was generated by adding points from an asperity shape function to points from a fiber cross-sectional shape function so that surface height points Hi, of a fiber model were given by

where Si represents points from a fiber cross-sectional shape function and S: represents points from an asperity shape function. C is a variable used to control RMS surface roughness. Figure 4 illustrates four fiber models where each model

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- Per form two-dimensional

Fourier t ransform

Scale the scat ter ing pat te rn

I

LUO AND BRESEE

I

Calculate RC f r o m scat ter ing pat tern

I

Sqlect f i be r cross- sectlonal shape model

Rectanqul ar Clrcu lar Trlangular cross section cross oectlon cross section

Select asperity shape model

I I

I Sqlect f i be r cross- I sectlonal shape model

Rectanqul ar cross section cross oectlon cross section

Select asperity shape model

I I

Calculate RMS surface roughness I o f f l be r model 1

Calculate ref lectance function

Fig. 3. Flowchart of the program.

is composed of a circular fiber cross-sectional shape with the same RMS surface roughness but each model is composed of a different asperity shape function.

The Reflectance Function

If optical absorption is neglected, the amplitude of incident illumination of light at a fiber surface is not changed by reflection. On the other hand, the phase of incident light is changed when it encounters a rough surface. Referring to Figure 5, the reflectance function can be derived as follows: The optical axis lies along the fiber diameter and an expression for the phase perturbation of an incident wave, W, caused by a fiber surface which is represented by a height function, h ( x , y ) , is obtained.


Fig. 4. Fiber models with circular cross-sectional shapes and (a ) random asperity shapes, ( b ) square asperity shapes, ( c ) triangular asperity shapes, ( d ) sine' wave asperity shapes. RMS surface roughness is the same for all models.

At an arbitrary position in the (x, y ) plane such as ( 0 , O ) , the zero-roughness phase retardation is determined by the distance L1 + L2, this sum being in- dependent of the position (x, y ) in the plane. The surface roughness point, h ( x , y ) , changes the optical path length to L; + LL. To determine the reflectance function, R (x, y ) , the differences in path length, AL1 + AL2, are needed. We write the actual path L; + L', as

L; + L', = ( L , + L,) - (AL1 + AL,) (6)

where

Now, for the phase term exp(jwt) using Eqs. (6) and ( 7 ) and dropping unessential constant phase delays, the reflectance function R (x, y ) can be defined by

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Fig. 5. Rough surface representation of fiber cross section and surface asperities ( n axis is perpendicular to the plane of the paper and coincides with the fiber axis).

R ( x , y ) = exp[j2kh(x,y)cos 191 ( 8 )

in which k = 2r/X is the wave number, and X is the wavelength of the illumination.

Equation (8) gives us an expression for the perturbed wave front W' in the space domain. We can write the scalar component of the field E ( xo, yo) at the observation plane, W", in the frequency domain, if it is located at a distance great enough to meet the requirement of Fraunhofer diffraction as discussed in eq. (2) . The paraxial form is given by2

in which A is a constant, R(x, y ) is given by eq. (a), and Z is the distance between the fiber surface and the observation plane.

Substituting u = x o / ( x Z ) and u = yo/(xZ) into eq. (9), considering the case of normal illumination, where 0 = 0, and substituting eq. (8) for R ( x , y ) , the result becomes

r r + m

E ( u , u ) = A J J exp[j2kh(x,y)]exp[-j27r(ux + uy)] dxdy (10) -m

This expression shows that the scattering pattern, E (u, u ) can be calculated by Fourier transformation of the exponential height function.


Two-Dimensional Discrete Fourier Transform

The two-dimensional discrete Fourier transform pair is defined as

for u = 0, 1 , 2 , . . . , M - 1, and u = 0,1 ,2 , . . . , N - 1.

forx = 0, 1 , 2 ,..., M - 1 ,andy = 0, 1, 2, . . . , N - 1. In the computation, the principle of separability and an FFT algorithm6 was

used. The principal advantage of the separability property is that F ( u, u ) can be obtained in two steps by successive applications of the one-dimensional Fourier transform as follows:

, N-l

F ( u , u ) = F ( x , u)exp[- j2~ux/N] N x=o

where

For each value of x , the expression inside the brackets of eq. (14) is a one- dimensional transform with frequency values u = 0, 1,2, . . . , N - 1. Therefore, the function F ( x , u ) is obtained by taking a transform along each row of f( x , y ) and multiplying the result by N. The desired result, F ( u, u ) , is then obtained from eq. ( 13) by taking a transform along each column of F ( x, u ) .

Roughness Coefficient Calculation

A perfectly smooth surface will show scattering intensity located only at zero spatial frequency (for Oo incidence of light source). On the other hand, a fiber surface which is not smooth will exhibit reduced zero frequency intensity and the intensity at other frequencies will be increased according to the nature of the surface roughness. Therefore, the integral of all frequency components with respect to zero frequency will give a measure of fiber surface roughness without the influence of diameter if frequency components paralled to the fiber axis are analyzed. A quantity called roughness coefficient, RC , is thus defined as

where I. is the intensity at zero frequency and It represents the total intensities scattered at other frequencies. The It term will be zero for a perfectly smooth

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surface and its RC value is equal to zero. The rougher a fiber surface becomes, the greater It becomes and RC will increase.

RESULTS AND DISCUSSION

The fiber models and their transforms were plotted three dimensionally. Figure 6 shows typical plots of one fiber model and its Fourier transform represented as scattering intensity versus spatial frequency.

Spatial frequencies computed with the FFT are defined by

u = n / ( N A x ) n = - N / 2 , . . . , 0 , . . . + N / 2 ( 1 6 )

u = n / ( N A y ) n = - N / 2 , . . . , O , . . . + N / 2

where N = 256 was the total number of sample points, A x was the sample spacing in the x direction equal to the total interval length divided by the number of sample points N , and Ay was defined similarly but in they direction. For sampling intervals A x and Ay equal to 1 pm (total sampled length along fiber axis was 256 pm) , Au and Au were 1/256 pm-' so the maximum frequencies calculated by the FFT were equal to 0.5 pm-'. The maximum frequencies at- tainable decrease as the sampling intervals increase if the number of sample points remains constant. For example, if the sampling intervals are 4 pm (total sampled length along fiber axis is 1024 pm) , the maximum frequency will be 0.125 pm-'so Au and Au are equal to 1/1024 pm-'.

The scattered intensity at the origin of the u axis, I ( 0 , u ) , is the specular reflection term. I ( 0, u ) data primarily contain information about the fiber diameter and cross-sectional shape although surface roughness also contributes to this data. Data points above or below this line are due primarily to fiber surface roughness. That is, intensities along u directions in the spatial frequency

Fig. 6. Fiber model and its theoretical scattering pattern: ( a ) fiber model; circular fiber cross- sectional shape, fiber diameter = 16 pm, random asperity shape, RMS asperity height = 0.09 pm, ( b ) scattering pattern computed from the model; RC = 14.87.


domain [ e.g., I ( 0.3, v)] are mainly from surface waviness along y directions in the space domain, whereas intensities along u directions in the spatial frequency domain [ e.g., I ( u , O . l ) ] are mainly from surface waviness along the x directions in the space domain. The relationship between surface roughness and scattered intensity can be explained better by comparing one-dimensional profiles in space and spatial frequency domains. For example, a one-dimensional profile in the x direction (fiber axis) of a nearly smooth fiber would exhibit intense scatter a t u = 0 in the spatial frequency domain. In the perfectly smooth case, the scattered intensity would consist of an impulse located at u = 0 and no scatter could appear at other u coordinates. On the other hand, if a fiber’s surface were relatively rough, the zero frequency intensity would decrease and spread to more u values.

For the purpose of investigating the theoretical influence of RMS asperity height on scattering patterns, fibers were modeled to have the same fiber diameter, cross-sectional shape, and asperity shape, but different RMS asperity height. RMS height ranged from quite smooth (RMS = 0.02) to rough (RMS

Fig. 7. Fiber models having the same fiber diameters, circular cross-sectional shapes, and random asperity shape, but different RMS asperity height: ( a ) RMS = 0.02 pm, (b) RMS = 0.05 pm, ( c ) RMS = 0.08 pm, (d) RMS = 0.16 pm.

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= 0.16). Figure 7 shows four fiber models and Figure 8 shows their corresponding Fourier transforms.

Figure 7 ( a ) is a relatively smooth fiber (RMS = 0.02) and its Fourier transform pattern is quite simple as shown in Figure 8 ( a ) and is characterized by RC = 0.10. There is little theoretical scatter above or below the u = 0 spatial frequency and scatter a t u = 0 is very intense. A perfectly smooth fiber surface would exhibit no scatter above or below the u = 0 frequency. The theoretical scattering pattern of Figure 8 ( a ) indicates that the surface along the fiber axis is quite smooth.

Figure 7 ( b ) represents a slightly rougher fiber surface with a RMS asperity height of 0.05. The theoretical scattering pattern from the fiber model in Figure 8 ( b ) shows that the scattered intensity at u = 0 is reduced compared with Figure 8 ( a ) , and scattering humps have emerged in the uu plane, although their intensity is small. The RC value associated with this scatter was computed to be 1.78, which indicates that the fiber surface is slightly more rough than that of Figure 7 ( a ) .

Fig. 8. Fourier transforms (theoretical scattering patterns) and computed roughness coefficient values corresponding to the fiber models with different RMS asperity height in Fig. 7. (a ) RC = 0.10, ( b ) RC = 1.78, ( c ) RC = 8.68, (d) RC = 55.93.


Figure 7(c, d ) represents further increases in RMS asperity height to 0.08 and 0.16, respectively. The Fourier transforms illustrated in Figure 8(c, d ) show further reductions in scattered intensity a t u = 0 and further increases in the number and intensity of scattered humps throughout the scattering region. The RC values for these two fibers were computed to be 8.68 and 55.93. These results show that theoretical backscattering patterns from fibers are greatly influenced by the height of asperities on the fiber surfaces.

In order to investigate the influence of asperity shape on theoretical scattering patterns, fiber models were used which had the same fiber diameter, fiber cross- sectional shape, and RMS asperity heights, but had various asperity shapes. Asperity shapes included random, square, triangular, and sine2 wave shapes and are illustrated in Figure 9 in one dimension. Figure 10 shows theoretical scattering patterns computed for fibers having diameters of 16 pm, circular cross-sectional shapes and RMS asperity shapes = 0.15 pm. Despite the same RMS asperity height for each of the four fiber surfaces, the theoretical scattering patterns differed substantially. These differences can be seen easily in a qual- itative spectrum, and are expressed quantitatively in the RC values computed from each of the scattering patterns, ranging from 3.89 to 57.93.

The square asperity shape exhibited the least high frequency scatter and the greatest amount of zero frequency scatter. On the other hand, the random

n . 1 21 41 81 81 (01 Q1 I41 181 181

X

1 21 41 61 81 (01 121 I61 161 m1 X

( c )

Fig. 9. One-dimensional surface profiles along the fiber axis for (a ) square, (b ) triangular, ( c ) sine', (d ) random wave asperity shape models.

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Fig. 10. Fourier transforms (theoretical scattering patterns) and computed RC values for fiber models with fiber diameters = 16 fim, circular fiber cross-sectional shapes, RMS asperity height = 0.15 pm, but different asperity shapes: ( a ) square, RC = 3.89, ( b ) triangular, RC = 11.36, ( c ) sine', RC = 12.93, ( d ) random, RC = 57.93.

asperity shape exhibited the greatest high frequency scatter and least zero frequency scatter. These differences can be explained qualitatively by examining the one-dimensional asperity shapes in Figure 9. The square-shaped asperities are composed mostly of flat tops, which would produce scatter corresponding to low frequency components, whereas only a small fraction of the asperity area is composed of steeply sloped sides that would produce scatter corresponding to high-frequency components. In contrast, the randomly shaped asperities have little flat top surface but have a large amount of steeply sloped side surface which would result in little low frequency but much high frequency scatter. The triangular and sine' wave asperities can be seen to be intermediate between these two extremes. In conclusion, these results show that theoretical backscattering patterns from fibers are sensitive to the shape of asperities on fiber surfaces.

We pointed out earlier that scattering at the u = 0 frequency provided information about fiber diameter. Modeling studies were used again to investigate


how scattering patterns and RC values change with fiber diameter. Figure 11 shows fiber models with the same circular fiber cross-sectional shape, random asperity shape, and RMS asperity height = 0.08, but with different fiber diameters. The diameter of the fiber in Figure ll ( a ) is 12 pm, whereas that of Figure I l ( b ) is 32 pm. The corresponding Fourier transforms of these fibers are shown in Figure l l ( c ) and (d) . Because the space and spatial frequency domains are inversely related, the peaks seen at the spatial frequency u = 0 move closer to the spatial frequency origin, ( 0 , O ) , as the fiber diameter increases. A similar effect was noted with each of the two fiber cross-sectional shapes. One may conclude that it should be possible to determine the diameter of fibers from backscattering measurements a t the u = 0 spatial frequency.

To investigate the effect of fiber diameter on the roughness coefficient, fiber models were compared having the same cross-sectional shape, asperity shape, and RMS asperity height, but having different diameters and RC values were computed from theoretical scattering patterns. Results for fibers having circular

Fig. 11. Fiber models with circular fiber cross-sectional shape, random asperity shape and RMS asperity height = 0.08 pm, but with different fiber diameters: ( a ) d = 12 pm, (b ) d = 32 pm. Fourier transforms (theoretical scattering patterns) and computed RC values for the fiber models: ( c ) RC = 8.97, for the d = 12 pm fiber, and (d) RC = 8.48 for the d = 32 pm fiber.

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v) O D - v) a 5 -t m 3 0 b- a

CD

fiber cross-sectional shapes, random asperity shapes, and RMS asperity heights of 0.08 pm are plotted in Figure 12. The plot shows that there is little long- term dependence of the RC value on fiber diameters in the range of diameters examined.

Next, the effect of fiber cross-sectional shape on scattering patterns and RC values from fibers with the same RMS asperity height was examined. Since the spectrum in the u direction largely contains surface information along the fiber axis direction whereas u direction scatter largely contains information about fiber diameter and cross-sectional shape, we compared scattering patterns in the u direction. Figures 13-15 show three models with the same fiber diameter, asperity shape, and RMS asperity height, but with different cross-sectional shape. Theoretical scattering patterns and RC values computed from these scattering patterns also are provided in the figures. As expected, the scattering pattern changed at u = 0 with the fiber cross-sectional shape changes. These changes would be expected to affect RC values computed from the scattering patterns since RC as defined by eq. (15) depends on the intensity at zero frequency. Although RMS values of the fiber models in Figures 13 to 15 are identical (RMS = 0.08 pm) , RC values computed from the theoretical scattering patterns differed (8.68, 8.68, 5.34). These and similar results for other fiber models suggest that this effect is due partly to the rather limited scattering region computed. That is, intense scattered lobes along u = 0 may be included in the scattered region computed for one fiber diameter or cross-sectional shape but may not be included in the scattered region computed for another fiber diameter or cross-sectional shape. At any rate, this study indicates that RC values computed from scattering patterns of fibers having different cross-sectional shapes must be compared carefully.

-- I I I I I I 1 I I

v) O D - v) a 5 -t m 3 0 b- a

CD -- I I I I I I 1 I I D

Fig. 12. Roughness coefficients computed for fibers with circular fiber cross-sectional shape, random asperity shape and RMS asperity height = 0.8 pm, but with different fiber diameters.


Fig. 13. ( a ) Fiber model with rectangular fiber cross-sectional shape, random asperity shape and RMS asperity height = 0.08 pm, ( b ) Fourier transform (theoretical scattering pattern) of the fiber model and computed RC = 8.68.

SUMMARY AND CONCLUSIONS

In order to investigate the relationship between the fiber surface structure and its theoretical backscattering pattern, computer simulations of laser backscattering were performed. RMS asperity height, asperity shape, fiber cross- sectional shape, and fiber diameter were examined. The simulation studies showed that changes in fiber cross-sectional shape and fiber diameter affected the scattered intensity distribution in the direction perpendicular to the fiber axis, whereas changes in asperity shape affected the intensity distribution in the direction parallel to the fiber axis. Results also showed that theoretical

Fig. 14. ( a ) Fiber model with circular fiber cross-sectional shape, random asperity shape and RMS asperity height = 0.08 pm, (b) Fourier transform (theoretical scattering pattern) of the fiber model and computed RC = 8.68.

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Fig. 15. ( a ) Fiber model with triangular fiber cross-sectional shape, random asperity shape and RMS asperity height = 0.08 pm, ( b ) Fourier transform (theoretical scattering pattern) of the fiber model and computed RC = 5.34.

backscattering patterns from fibers are greatly influenced by the RMS height and the shape of asperities on the fiber surface. The parameter “roughness coefficient” was defined as the ratio of scattered intensities at nonzero spatial frequencies to intensities at zero spatial frequencies and provided a suitable measure of fiber surface roughness from backscattering patterns.

References

1. B. J. Pernick, Appl. Opt., 18, 796 (1979). 2. K. J. Allardyce and N. George, Appl. Opt., 26, 12 (1987). 3. C. Luo, M.S. thesis, the University of Tennessee, 1989. 4. K. H. Guenther, P. C. Wierer, and J. M. Bennett, Appl. Opt., 23,21 (1984). 5. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968. 6. R. C. Gonzalez and P. Witz, Digital Image Processing, Addison Wesley, Reading, MA, 1987.

Received February 24, 1989 Accepted December 27, 1989

Computer simulation of laser backscattering from fiber surfaces

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