N
NAVAL POSTGRADUATE SCHOOLMonterey, California
DTICElE'ECTEN
THESIS Em
A COMPUTER AIDED METHOD FOR THE MEASUREMENT OF
FIBER DIAMETERS BY LASER DIFFRACTION
by
Mark Gerald Storch
September 1986
uAJThesis Advisor: Professor Edward M. 'u
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A COMPUTER AIDED METHOD FOR THE MEASUREMENT OF FIBER DIAMETERS BY LASER DIFFRACTION
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; IED GROUP SUB-GROUP Fiber Diameter Measurement by Laser Diffraction
9 ABSTRACT (Continue on reverie if neceoary and ilentify by block number)
This thesis investigates the computer aided measurement of fiber diameters by laser
diffraction. The proposed system consists of a light sensitive Random Access Memory (RAM)
chip which collects light intensity data from the laser diffraction pattern. MeasuremLnts
of the spatial location of the nodes of the diffraction pattern enables the calculation of
the fiber diameter. These measurements may be performed manually which is tedious and
requires subjective judgement of the nodes. The alternative method of direct processing
of the intensity pattern was investigated. Simulation is conducted to examine the
feasibility of this method. Results show such a system to be capable of providing one
order of magnitude greater accuracy than optical microscopy measurements (with a shearing
eyepiece) and double the accuracy of manual laser diffractio iethods with the added
advantage of permitting the option of total computer automation in data interpretation.
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Professor Edward M. Wu (408) 646 3459 67Wt
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Approved for public release; distribution is unlimited.
A Computer Aided Method for the Measurement ofFiber Diameters by Laser Diffraction
by
Mark Gerald StorchLieutenant, United States Navy
B.S., Miami University, 1979
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING
from theNAVAL POSTGRADUATE SCHOOL
September 1986
Au'Zhor _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Mark Gerald Storch
Approved by: _
M. Wu, Thesis Advisor
M. F. Platzer, Chairman, Department of Aeronautics
John N. Dyer, Dean of Science and Engineering
2
ABSTRACT
This thesis investigates the computer aided measurement of fiber
diameters by laser diffraction. The proposed system consists of a light
sensitive Random Access flemory (RAM) chip which collects light intensity
data from the laser diffraction pattern. Measurements of the spatial
location of the nodes of the diffraction pattern enables the calculation of
the fiber diameter. These measurements may be performed manually which
is tedious and requires subjective judgement of the nodes. The alternative
method of direct processing of the intensity pattern was investigated.
Simulation is conducted to examine the feasibility of this method. Results
show such a system to be capable of providing one order of magnitude
greater accuracy than optical microscopy measurements (with a shearing
eyepiece) and double the accuracy of manual laser diffraction methods
with the added advantage of permitting the option of total computer
automation In data interpretation.Accession For
NTIS GRA&IDTIC TAB
Unannowiced EJusti±'icatio
1 ByDistribution/
Availability Codes
IAvai and/ar
Dist Sp'eca
Vol3
TABLE OF CONTENTS
I. INTRODUCTION - 11
It. BACKGROUND -------------------------------- 13
III. DIFFRACTION ANALYSIS ------------------------- 16
IV. MICRON EYE THEORY AND OPERATION ---------------- 28
V. SIMULATION OF THE MICRON EYE ------------------- 34
VI. SIMULATION: THE PERFECT AND IMPERFECT DATA SETS ---- 49
VII. DISCUSSION OF RESULTS ------------------------ 59
VIII. CONCLUSIONS -------------------------------- 72
IX. RECOMMENDATIONS ---------------------------- 73
APPENDIX A. FRAUNHOFER DIFFRACTION THEORY ------------ 77
APPENDIX B. COMPUTER PROGRAMS --------------------- 86
LIST OF REFERENCES ------------------------------ 131
INITIAL DISTRIBUTION LIST ------------------------- 132
4
L IST OF TABLES
1. DIGITIZATION ERROR------------------------------ 53
11I. ACCURACY VERSUS RESOLUTION--------------------- 70
LIST OF FIGURES
1. Fiber Diameter Measurement by Laser Diffraction ------ 13
2. Micron Eye Array with Two Interference Nodes -------- 14
3. Typical Micron Eye Photograph of Interference Nodes ---- 15
4. Three Dimensional Diffraction Pattern ------------- 17
5. Typical Intensity Profile --------------------- 18
6. Features of the Intensity Profile Cure ------------ 20
7. Effect of Diameter on Intensity Profiles (3-D) -------- 22
8. Effect of Diameter on Intensity Profiles (2-D) -------- 24
9. Three Points to Define a Curve ----------------- 25
10. Three Points to Define a Curve ----------------- 25
11. Three Points. to Define e Curve ----------------- 25
12. Intensity Profile with Superimposed Derivative
Curve -------------------------------- 27
13. Micron Eye Array -------------------------- 28
14. Micron Eye Physical Organization ---------------- 30
15. Macintosh and the Micron Eye --------------- 31
16. Exposure and Its Relation to Intensity Profile -------- 32
17. Calibrating the Micron Eye -------------------- 33
18. Simulation Intensity Profiles (5.803Mm) ----------- 36
19. Simulation Intensity Profiles (7.254Mim) ----------- 36
20. Simulation Intensity Profiles (8.705Mm) ----------- 37
6
21. Exposure for Three Data Points -------------------- 38
22. Micron Eye Placement in Diffraction Pattern ----------- 39
23. Micron Eye Position ----------------------------- 41
24. Short and Long Exposures------------------------- 42
25. Endpoints of Major Axis-------------------------- 43
26. Three Points on Major Axis ----------------------- 43
27. Rel ativYe I ntensi ty Prof ilIes ----------------------- 45
28. Region of Similar Maximum Intensities-------------- 46
29. Region of Similar Intensity Profile Derivatives--------- 46
30. Averaging of Theta------------------------------ 51
31. Intensity Variation at 5%Error-------------------- 54
32. Three Point Spacing Ratio----:--------------------- 57
33. Finding the Theta Locations----------------------- 58
34. Si r.ul ati on I ntensi ty Prof ilIes --------------------- 59
35. 5.803 gim f iber (I% error) ------------------------ 61
36. 7.254Lgm f iber ( I error)------------------------ 62
37. 6.705 jim fiber (1% error)------------------------ 63
38. 5.803 jim f iber (2% error)------------------------ 64
39. 7.254 jim f iber (2% error)------------------------ 65
40. 8.705 jim f iber (2% error)------------------------ 66
41. 5.803 jim fiber (5% error)------------------------ 67
42. 7.254 jim f iber (5% error)------------------------ 66
43. 8.705 jim fiber (5% error)------------------------ 69
44. Typical Single Slit Diffraction Pattern -------------- 74
7
45. Single Slit Path Difference Relation to the Interference
Nodes -------------------------------- 76
46. Locations of the Diffraction Maxima -------------- 78
47. Fourier Transf orm for the Single Slit ------------- 79
48. Complement of the Fourier Slit Transform ---------- 79
49. Fourier Transform for Two Parallel Slits ----------- 80
50. Effect of Number of Bessel Terms --------------- 4
51. Effect of Different Diameters ------------------ 85
52. DATAMAKR Matrix Algebra -------------------- 89
53. Residual Versus Diameter for 7jim Fiber ----------- 92
54. Residual Versus Diameter for 5jim Fiber ----------- 93
8
LIST OF SYMBOLS
Argument of the Bessel Functions
a Width of a slit in Fraunhofer diffraction theory
bn Bessel coefficient (Jn/Hn (2))
d Diameter of a fiber
e Angular position in diffraction pattern (radians)
"Jn Bessel function of the first kind
H (2) Hankel function of the second kind
ITR Threshold Intensity Ratio
Ko Constant associated with the real fiber intensity equation
7 Wavelengh of laser (632.8 nm for He-Ne)
L Longitudinal position defining distance from fiber to
diffraction pattern
m Integer indicating interference node number
P Geometric center of diffraction pattern
Ax Width of Micron Eye array (approximately 4.4 mm)
x Lateral position in diffraction pattern (perpendicular to laser
beam)
Yn Bessel function of the second kind
9
ACKNOWLEDGEMENTS
I am grateful to the following individuals for their assistance during
this rpqiarch.
F . , I want to thank Mr. Jim Nageotte for his technical advice in the
laboratory. Also, the outstanding machine shop work of Mr. Glen Middleton
Is greatly appreciated.
For their advice and direction concerning mathematics, I am grateful
to Professor G. Morris and Dr. Shi Hau Own.
Finally, and most sincerely, I am indebted to my thesis advisor,
Professor Edward M. Wu. His warm encouragement, patience, and
creativity have made this a memorable learning experience. It has been
both a privilege and a pleasure to have completed this work under his
direction.
10
I. INTRODUCTION
Fiber reinforced composites are replacing structural metal
components in today's aircraft. The high strength and reduced weight of
composites results in less drag, increased payload, and longer fatigue life.
Additionally, the directional properties of composite materials provide
unique design advantages over conventional materials.
As with many developing technologies, the reasons for the success of
fiber composites was not initially appreciated. Tsai [Ref. 1:p. 21 states
that fortunately the modern composite was so strong it was reliable and
competitive in spite of less than optimum design practice. Over the last
twenty years, much progress has been made in understanding the micro and
macro mechanics of composite materials. As this knowledge matures and
is incorporated into the design process, the full potential of these
materials can be realized.
An important contribution to the design process is modeling
structural reliability. Development of probabilistic models for a
composite must take into account the complex relationships that exist
between fiber and matrix.
11
One important parameter in the probabilistic model is fiber diameter,
since variations in fiber diameter affect fiber failure density. Therefore,
more accurate fiber diameter measurement results in an enhanced
reliability model and a more quantitative prediction of structural
reliability.
The purpose of this research is to investigate a computer aided
method of fiber diameter measurement. The existing procedure, of
diameter measurement by laser diffraction is accurate to within 0.5%
[Ref. 2:p. 2101. It is desired to improve this accuracy by interpreting the
diffraction pattern with a light sensitive RAM chip.
The presentation begins with a discussion of diffraction pattern
analysis. This is followed by an introduction to the Micron Eye's theory
and operation. Next, the simulation is discussed in depth followed by the
results and conclusions.
12
It.8BCKGROUND
Fiber diameter measurement by laser diffraction is conducted by
obstructing a collimated laser beam with a fiber sample. A diffraction
pattern results and is characterized by alternating maxima and minima
symmetric about a central maximum. (See Figure 1)
..mbal
Figure 1. Fiber Diameter Measurement by Laser Diffraction
In previous work, Perry, Ineichen, and Elias.on, [Ref. 31 and Bennett,
[Ref. 41, interpretation of the diffraction pattern consisted of finding the
distance between interference nodes (minima). This distance is related to
the fiber diameter.
13
Bennett used the slit approximation (Appendix A) to relate the
distance between nodes to the fiber diameter. Perry, et al., introduced a
more exact solution by Kerker, [Ref 5:p. 2601 and compared it to the slit
approximation. Perry, et al., concluded that the slit approximation should
be treated with caution. Therefore, Kerker's solution has been adopted for
this study.
In his peoer, Bennett successfully demonstrated the feasibility of
using the light sensitive RAM chip for diameter measurements [Ref 41. An
IS32 OPTICRAM MICRON EYE, manufactured by Micron Technology, Inc.,
was connected as a peripheral device to an Apple I1+ computer. By
positioning the Micron Eye so that two interference nodes fit on its
surface (see Figure 2), an "exposure" of the diffraction pattern could be
printed by the computer (see Figure 3). Analysis of the printed diffraction
pattern gave the diameter of the fiber using the slit approximation of
Fraunhofer diffraction theory (see Appendix A).
Figure 2. Micron Eje Array with Two Interference Nodes
14
-~-.,.cc-e:< ! - -
3t . . °.o. -o.4° °°.... ° ° • ° . . ° • ° .. . ° .$- °*t .
• : .°...: °. •.-° ° -
Figure 3. Tgpical Micron Ege Photograph of Interference Nodes
15
1II. DIFFRACTION PATTERN ANALYSIS
Measuring fiber diameters by laser diffraction, requires an
understanding of the diffraction pattern. This chapter introduces some
features of diffraction patterns which will be useful in later analysis.
A. FUNCTIONAL DESCRIPTION OF THE DIFFRACTION PATTERN
The diffraction pattern of a fiber can be described as follows:
1/10 = (2/KeLiv) I b. + 2 1 b. cs(ne) F (I)
where E = the scattering angle
Ke = 21f 1 ( 7% = laser wavelength)
b,= J*(a /
and c= jdf/7 (da = fiber diameter)
J.(&) are Bessel functions of the first kind,
HP() are Hankel functions of the second kind.
16
A formal introduction of equation (1) and the related Fraunhofer
diffraction theory is presented in Appendix A.
Figure 4 depicts the three dimensional nature of the diffraction
pattern. Equation (1) describes the intensities along one linear position,
or slice, of the three dimension pattern.
Contra] Maximum
Figure 4. Three Dimensional Diffraction Pattern
A plot of equation (1) is the Intensity Profile which shows the
Intensity Ratio (I/1) versus the angle theta, for a given fiber diameter
and a given fiber to screen distance L. Figure 5 shows a typical Intensity
Profile for a fiber diameter of 6.5 microns.
17
INTENSITY PROFILEo .TYPICAL
0
C/
ZoE--z
0
0.00 0.05 0.10 0.15 0.20 0.25
THETA (RADIANS)
Figure 5. Typical Intensity Profile
18
B. FEATURES OF THE INTENSITY PROFILE CURVES
The intensity profile has several features worth noting. The first is
the central (or zeroth order) maximum. The central maximum is the
largest peak in Figure 5 and it is many times more intense than the
subsequent maxima. It should also be obvious that the profile is
symmetric bout the central maximum.
Other features of the curves are shown in Figure 6. Figure 6 is an
enlarged view of one of the higher order maxima and associated minima.
As discussed in Appendix A, the higher order maxima are not centrally
located between the minima, rather they are displaced slightly towards
the central maximum. Therefore, the higher order peaks are asymmetric
about their maxima. This asymmetry means the maximum derivative on
the upslope side of the curve will be greater than the maximum derivative
on the downslope side of the curve. This difference is dealt with later
when tuning for the optimum exposure is discussed.
The minima in Figure 6 are also distinctive features of the curves. In
Appendix A, it is shown that the location of the minima are explicitly
related to the diameter of the fiber using the slit approximation:
m = of interference node
Ob sin [ mXid N laser wavelength (2)
d =diameter of fiber
19
INTENSITY PROFILE (IPERFECT DATA)
080
'Io.
0 U0 0a o
0 aa
0 0
0 oo
o
0
0.150 0.175 0200 0225 0 250 0275THi rETA (RADIANS)
Figurf, 6. Features of the Intensity Profile Curve
20
Equation (2) is only an approximation for a fiber, but it is useful for
preliminary analysis. Such analysis includes:
(1) determining where to place the Micron Eye for data collection and,
(2) providing and initial guess of fiber diameter from diffractionpattern data.
C. EFFECT OF DIAMETER ON THE INTENSITY PROFILE CURVES
Figure 7 shows the effect of diameter on the Intensity Profiles. Note
that successive maxima and minima are further from the central maximum
for fibers of smaller diameter. This behavior will be important to later
analysis.
D. THE MAXIMUM DERIVATIVE AND THE MINIMUM NUMBER OF POINTS
The goal of automated fiber diameter measurement is to find the
diameter rapidly and accurately. Speed and accuracy can be exclusive. One
can imagine that collecting all the points on the Intensity Profile curve
will result in an almost exact determination of fiber diameter, at the
expense of time. Two questions must be answered. First, what is the
minimum number of points to uniquely describe an intensity profile?
Second, where on the curve should these points be obtained?
21
Figure 7. Effect of Diameter on Intensity Profiles (3-D)
22
1. Minimum Number of Points
Equation (I) consists of an infinite series. Determining the
minimum number of points to uniquely define an infinite series is not a
straightforward procedure. Cursory inspection of Figure 8 leads to the
conclusion that one point will not uniquely define the curve, since
intensity curves for many diameters pass through the same point (e.g., at e
= .04 radians). If the absolute intensity is not known, 2 points will not
provide a unique solution either. For the case of non-absolute intensity
measurements (as is the case in this experimental set-up) 3 points
appears to uniquely define a curve.
The curve in Figure 9 illustrates this observation. An imaginary
line is fixed through three points (i.e., the experimental measurements).
The absolute intensities of these three points are not known, and neither
are the absolute spatial locations with .respect to the central maximum
(i.e., e). Only the relative spatial locations among these points are
known. Iteration of the diameter is equivalent to moving this line (with
the x marks fixed relative to each other) along different locations on the
curve, trying to match all three points. This iteration can be repeated on
curves of different diameters (Figures 10 and 1I) and no other match can
be found. Thus, this argument defines three points as the minimum
required to uniquely determine the intensity profile curve.
23
INTENSITY PROFILE
00
0.
z
LEGENDz o 6.5 MICRONS
675 MICRONS+ qWMizoff8,x 8.5 MICRONfS
0.00 0.05 0.10 0.15 0 20 0.25
THETA (RADIANS)
Figure 8. Effect of Diameter on Intensitg Profiles (2-D)
24
LEGEND7.254 IMICRONS
~LEGEND
.5.803 MICRONS
~LEGEND
/8.70- MICRONS
Figures 9, 10, 11. Three Points to Define a Curve
25
2. Maximum dervative
Having heuristically established that three points are required, the
next step is to determine the optimum location from which to select the
points. For example, if points are selected at the mimima (or maxima) a
wide variation in theta results from a small change in Intensity Ratio.
This variation is defined by the derivative of the Intensity Ratio with
respect to Theta, and at the extrema this derivative is very small. It can
be shown that the points on the curve where the derivative is a maximum
will have the least variation error. Therefore, it is desirable to use the
Intensity Ratios corresponding to the maximum derivatives as the
optimum intensity ratios for the three points. These Intensity Ratios will
be referred to as the Threshold Intensity Ratios.
Figure 12 shows a portion of an Intensity Profile with a derivative
curve (absolute value) superimposed. Vertical lines drawn through the
derivative curve maxima intersect the Intensity Profile showing the
location of the optimum (threshold) intensity points.
26
INTENSITY PROFILE (PERFECT I)ATA)
2: 2- /o
0.150 0.175 0 000.22 200 7
TIHETA (RAIDIANS)
Figure 12. Intensity Profile with Superimposed Derivative Curve
27
IV. MICRON EYE
The MICRON EYE image sensor is an optically sensitive Random Access
Memory (RAM) chip capable of sensing an image and translating it to
digital computer compatible signals. The Micron Eye was selected to
introduce automation to the existing techniques of fiber diameter
measurement by laser diffraction. Automation is expected to increase the
speed and accuracy of diameter measurements. This chapter introduces
the theory and operation of the Micron Eye.
A. PHYSICAL LAYOUT AND DIMENSIONS
The Micron Ege (IS32 OpticRAr, 1") has two arrays each containing 128
rows x 256 columns of sensors. This application will use only one of the
arrays. The size of an array is 4420 gm by 877gm. (See Figure 13)
v 4420pm
129 rows
Figure 13. flicrlm Ege Arrai
28
Each sensor is a light sensitive element is called a pixel. The
physical organization of the pixels is shown in Figure 14. The 128 x 256
elements actually map into a 129 x 514 "cell placement grid". This
arrangement leaves "space plxels' in between each pixel in the row
direction. The space pixels can be set high, low, or to the level which
agrees with the maiority of its nearest neighbors.
0. THEORY OF OPERATION
The pixels are capacitors which discharge a preapplied voltage at a
rate proportional to both the intensity and duration of the impinging light.
The voltage in an exposed capacitor is read and digitally compared to the
fixed threshold voltage. If the voltage is below threshold the pixel is read
as WHITE. If the voltage is above threshold the pixel is read as BLACK.
The digital comparison concept will be an important part of the simulation
in the next chapter.
After a pixel is read, the row containing that pixel is refreshed.
Refreshing sets all pixels which are below threshold to 0 volts, and all
pixels which are above threshold to +5 volts. [Ref. 61
29
IS32 OpticRAMTM TOPOLOGICAL INFORMATION
Um" "AiY L I-- T
C. cm ce *a . .. -" -
.. C lC C" C C C, C1 C,
c' "I - mC c. a 'F C. C, l . l,
Al~~~ -T Al , II.-I~ 3 3
.1 .=a a a a a
0-1 aS a a " a au - u fill A1I ll A I 1 a
AIIA~ .I 313 11 *3 3I A
m,, m . , Ca 0A? ° *_ a
,d a am C• CA= C" " j'
C , 1 -n Iic- c IS- -- - -, - I
112 I'M 1I'll I' m1 "mm ,
r 4 i E a Orana
..... A A A * *3
* A A * * A * A A ~ .
Figure 1. Micron2 Ey hscal Oranizat ioni
Ca. Ca. C3 Ca Ci. m C.O
1. Ooeration of the Micron Eye
Operation of the Micron Eye is simple. The current research
configuration uses an Apple Macintosh (512K) computer. Control software
is provided by Micron Technology, Inc. The Micron Eye is connected
through either the modem or printer port. (See Figure 15) The computer
also acts as the power source for the Micron Eye.
Figure 15. Macintosh and Micron Ege
2. Exoosure
An exposure of a portion of the diffraction pattern can be made by
varying exposure times. A sample exposure is shown in Figure 16. It is
important to visualize that this exposure represents a cross section of the
intensity profile curve.
31
- 1 t 4 *..36
lte=*l Prefib
Figure 16. Exposure and Its Relation to Intensitg Profile
Longer and shorter exposure times will vary the size of the cross
section. Varying exposure time is equivalent to moving up or down the
intensity ratio axis of the intensity profile curve.
C. INTENSITY MEASUREMENTS WITH THE MICRON EYE
Bennett [Ref. 41 used the Micron Eye to measure the distance between
Interference nodes. This approach was compatible with the slit
approximation requiring only the theta locations of the Interference nodes
to give the diameter. In Kerkers equation [Ref. 5:p. 2601, one cannot solve
explicitly for d. The Solution requires knowledge of the Intensity Ratios
32
end their respective theta locations. For the Micron Eye, Intensity Ratios
correspond to the user controlled exposure time.
I. Two Approaches to Intensity Calibration
The Micron Eye must be calibrated to a reference intensity level.
One method to accomplish this would require two exposures. One at 11
A and another at 12 + A. Delta represents an unknown level above the
zero (or absolute) intensity. (See Figure 17) The difference between
these two values eliminates delta.
A second method requires a fiber of known diameter. One exposure of
this "calibration" fiber will allow determination of the absolute intensity.
INTENSITY PROFILE
Z
Figure 17. Calibrating the Micron Eye
33
V. SIMULATION
This chapter introduces the simulation of the Micron Eye / Laser
Diffraction diameter measurement system. The simulation answers many
questions relating to the real system:
(1) where should the Micron Eye be placed?(2) what exposure is best?(3) what accuracy can be expected?(4) is the computer code valid?
The simulation combines diffraction pattern analysis with Micron Eye
operation. The worth of the results will depend on the accuracy of the
simulation.
A. COMPUTER PROGRAMS
Several computer programs have been written to conduct this
simulation. These are:
(1) DATAMAKR(2) EXPOSURE(3) DIAFIND
Appendix B discusses these programs in detail.
34
Briefly:(1) DATAMAKR is used to produce Intensity Ratio versus Theta data
for a given fiber diameter d, and screen to fiber distance 1. Theuser controls the number of points produce and their spacing.
(2) EXPOSJRE reads the Intensity Profile data generated byDATAM1AKR and recommends the optimum exposure based on theaverage of the intensity ratios corresponding to the maximumderivatives. Exposure will also introduce random error into thedata, as desired.
(3) DIAFIND uses the data generated by EXPOSURE to find the-diameterof the fiber. The programs begins with a user input guess of thediameter and conducts an iterative search / comparison routine tofind the actual diameter of the data.
B. OVERVIEW
The general approach in examining the proposed system will be to consider
a typical carbon fiber. This fiber has been assigned the arbitrary diameter
of 7.254im. Two additional fibers t20% of the 7.254xm fiber are also
considered. These fibers define the range of diameters from 5.803im to
8.705gm. See Figures 18, 19, and 20 for Intensity Profiles curves for
these diameters.
35
SIMULATION INTENSITY PROFILE0O .
LEGEND5.803 MICRONS
&-4
ZM-
S I I I
0.075 0.125 0.175 0.225 0.275 0.325 0.375THETA (RADIANS)
Figure 18. Simulation Intensitg Profile (5.803jum)
SIMULATION INTENSITY PROFILECO-
0.-
LEGEND>-4 7.254 MICRONS
-ow0~
I I i II
0.075 0.125 0.175 0.225 0.275 0.325 0.37
THETA (RADIANS)
Figure 19. Simulation IntensitU Profile (7.254pm)
36
SIMULATION INTENSITY PROFILE-,t,
o
LEGEND8.705 MICRONS
CO0
Ii I II0.075 0.125 0.175 0.225 0.275 0.325 0.376
THETA (RADIANS)
Figure 20. Simulation Intensity Profile (8.705jim)
C. POSITIONING OF THE MICRON EYE
In order to measure the diameter of the typical fiber, a method of
postitioning the Micron Eye must be defined. Further, from this same
position, it is desired to also measure the ±20% fibers. This requires that
one position of the Eye permit fiber diameter measurements over the
specified range of fiber diameters.
37
Such a position is defined such that three data points will be obtained
for any fiber in the diameter range, as in Figure 21. First, a simpler case
will be discussed to introduce positioning for a given diameter fiber.
btesity Profile
IP11
1 2 3 Teta (radials)
fItr.. Ege
Figure 21. Exposure for Three Data Points
30
1. Positioning for a Single Fiber
Suppose An exposure of the first through the second interference
nodes (2nd maximum) was desired for the 7.2541im fiber. Figure 22 shows
the relationship between the fiber and the Micron Eye:
L
ei~
~X
Xi
Figure 22. Micron Eye Placement in Diffraction Pattern
ei =tant- I [x i / L ]1 (3)
e,+, tant- I [xj+! / L (4)
'59
=. T w . -ir wnM
or tan +1-tan ei = IlL (xi+i - xi) Ax/L (5)
take Ax = 4.4 mm (width of Micron Eye array)
With Ax fixed, one can adjust L so that any desired e,+, - ei will
fit on the Micron Eye's array. Recall that e.h = sin [ mA/ d ]
equation (2), which will give the theta locations of the first and second
interference nodes.
emin = sin (m,/d) where m = node*
= laser wavelength (632.8nm)
d = fiber diameter
for 7.254j1m: e = .00712 radians
e,+ 'I7459 radians (for i = 1)
with e. and ei+1 fixed, L is defined from equation (5):
L = Ax / Itane,, - tane i ] 49mm
40
L is the distance of the M*cron Eye from the fiber in the
longitudinal direction. Referring back to equation (3), the lateral
placement of the Eye is given (see Figure 23):
x,. = L tan(e) = 4.27 mm
FIEUI
L
A : ' -" ' eXte
x-tw
Figure 23. Micron Ego Position
Thus, equations (3) through (5) give the Micron Eye position for any
desired d, e+1,, Li, I and m.
41
This position is not yet optimum for single diameter case. This
analysis has captured only two interference nodes and the intervening
maximum. Any exposure will yield information somewhere between the
two extremes shown in Figure 24. The Micron Eye images correspond to
the shaded areas of the intensity profile curve on the right side of Figure
24.
.... ... .. ,..., • / Exlpes e
hLe"Expse °*
IIl
Figure 24. Short and Long Exposures
One method of finding the points would be to search the Micron Eye
array data for a major axis. The endpoints of this major axis are the
points required for the analysis. (See Figure 25)
42
4 :4 - -- - -- - - - - - - - -
Major Axis
Figure 25. Endpoints of Major Axis
Recall from diffraction analysis (Chapter 3) that three points are
required to define the intensity profile curve. Therefore, more of the
diffraction pattern mL'tt be seen by the Micron Eye. A reduction in L will
yield the third point. (See Figure 26) L should not be reduced any more
than is necessary, since fewer pixels are being used to describe the date,
degrading the resolution of the Micron Eye.
Major Wxs
Figure 26. Three Points on Major Axis
43
2. Positioning the Micron Eye for a Range of Diameters
Consider the positioning of the Micron Eye so that three data points
can be collected for a range of diameters. A "window' for the array must
be defined which will ensure three points of data for any diameter in the
specified range.
Examine the Relative Intensity Profile plot in Figure 27. e begins
at .075 radians which excludes the the central maximum. The central
maximum is so intense, the Micron Eye's fastest exposure cannnot prevent
overexposure. Therefore, the central maximum is not considered as a
location for the window.
The window should also be located in a region where the maximum
intensity corresponding the the largest diameter fiber is close to the
maximum intensity of the smallest diameter fiber. This ensures the
Threshold Intensity Ratio will be more representative for all diameter in
the range. This condition occurs at the higher order nodes (2 or greater),
as illustrated in Figure 28. The plot of intensity ratio derivatives also
shows this condition in Figure 29.
44
RELATIVE INTENSITY PROFILES
LEGEND7.254 MICRONS5.803 MICRONS
z ~
/ . . . . . . . .. . . .. . . .
. --o" o
0.075 0.125 0.175 0.225 0.275 0.325 0.375
THETA (RADIANS)
Figure 27. Relative Intensity Profiles
45
SIMULATION INTENSITY PROFILES
lei LEGEND
•Q- , ,7.254 MICRONS........................... .........58 MICRONS, , .....5803 MICRONS
0-
P- .. . ....... %. ..
I I I I I
0.075 0.125 0.175 0.225 0.275 0.325 0.375THETA (RADIANS)
Figure 20. Region of Similar Maximum Intensities
DI/D(THETA) VS THETA
o ,LEGEND8.705 DERIVATIVES
E-<, 5. 8-63DERIVATIVES-S.......... ................ .... . .... .... ........
0.075 0.125 0.175 0.225 0.275 0.325 0.375THETA (RADIANS)
Figure 29. Region of Similar Intensity Ratio Derivatives
46
• :I i
- - .% ... . "
To obtain three points, at about 1.5 maxima are required. Since
the distance between nodes is the greatest for the smallest diameter, find
the 0 locations for the smallest diameter which gives 1.5 maxima.
for 5.8034jm:
0i = sin (21/5.803p1) = .216 rid
Q. = sIn (31/5.O03m) = .321 rais
Half of this range is about .050 radians and 1.5 maxima can be
approximated by .165 ----- > .321 radians. Admittedly, this is not a
sophisticated method. The method is somewhat liberal and it was later
discovered that a range of .165 to .300 radians was better than the larger
range. This is because the minima are not of interest so the right hand
minimum at .321 radians was discarded, and the range reduced to .300
radians. Only the part of the curve where the slope is a maximum needs to
be "seen" by the array.
A quick check can be made to insure that the curve representing
the maximum diameter in the range fits inside the window.
for 8.705jim:
emin2 = sin (2 1 / 8.705) = .150 raf
em =sin (3 1 / 8.705) = .216 r&
Emi = sn (4 1 / 8.705) = .286 r&
47
Thu next step is to determine the distance L at which the Micron
Eye will cover the 0 range of .165 --- > .300 radians. Equation (5)
defines the longitudinal position and equation (3) defines the
corresponding lateral position:
L = AX/ Iten(O,) - tan( e i ) ] = 30 mm
xI L tan(O1) z 5 mm (inboard position of array)
48
VI. SIMULATION: THE PERF-ECT AND IMPERFECT DATA SETS
Now that a window has been defined (0 range) for thc diameter range,
it is necessary to construct perfect data for the simulation diameters: d,
d2, and d3. DATAMAKR generates this data for a given d and L, over the
specified range of ). Additionally, the number of points over the specified
range must be chosen. The number of points defines the 0 interval and
corresponds to the physical spacing of the pixels in the Micron Eye array
which is on the order of 1 Ojim. To match this spacing, 400 points were
generated over the interval .165 to .300 radians.
A. DIGITIZATION OF DATA
The EXPOSURE program is used to digitize the perfect data. Digitizing
the data is the simulative analog of the Micron Eye threshold voltage
comparison. In the simulation, any intensity ratios above a certain ratio
(call it the threshold ratio) are assigned e value of 1 and any intensity
ratio below the threshold ratio are assigned a value of 0.
49
1. Determining the Threshold Intensity Ratio
Before EXPOSURE can digitize the perfect data, the threshold ratio
must be determined. EXPOSURE calls the subroutine DERIV which
calculates the derivatives of the intensity profile data at each theta
location.
DERIV proceeds to search for all the local maximum derivatives.
The average of the maximum derivatives is returned to the EXPOSURE
program and is used as the threshold intensity.
After the data is digitized with respect to the threshold intensity,
the digital data is searched to find the theta locations where the intensity
changes from 0 -- > I or I -- > 0.. The values for theta at these locations
are averaged and this average is taken to be the theta location where the
threshold intensity is located. The averaging of theta is based on the fact
the Intensity Profile curve is approximately linear in the region of the
Threshold Intensity Ratio. Figure 30 shows the general location of the
Threshold Intensity Ratio.
In the simulation, EXPOSURE finds three average theta locations
corresponding to the threshold intensity ratio. These data are stcred in a
data file which is subsequently input to the program DIAFIND. DIAFIND
prompts the user to input an initial guess of the diameter and proceeds to
find the diameter associated with the EXPOSURE data.
50
ThresholdIntensity
Ratio
1.
ITIR
Ii+
Oe eO4 ei+ e I _
2Figure 30. Averaging of Theta
511f
!
3. INTRODUCTION OF ERROR INTO THE PERFECT DATA
EXPOSURE prompts the user for an error input. If error is desired,
EXPOSURE calls the subroutine RANDOM. RANDOM introduces error based on
the maximum intensity ratio in the perfect data:
let = le + [ II X *ERROR*RND j
where ERROR is the error to be introduced and RND is a random number
between ±I generated by the NONIMSL subroutine RANDU. RANDOM returns
the now "imperfect" intensity ratios to the EXPOSURE program.
C. DIGITIZATION PROBLEMS WITH THE IMPERFECT DATA
Digitizing the imperfect datb -rsults in local regions where the
intensity oscillates between 0 and 1, as shown in Table 1. The
introduction of random error changes the smooth curve to erratic points.
Loccily, variations above and below the threshold intensity ratio cause a
series of 0-->I and I-->0 oscillations. (See Figure 31) Ideally, only one
local theta location is to be associated with the threshold intensity ratio.
There are two approaches to this problem. First, the average of the
local theta locations can be calculated. This method results in three
theta locations for input into the DIAFIND program (the same as perfect
data).
52
TABLE 1. DIGITIZATION ERROR
THETA LOCATION INTENSITY RATIO
0.2321624000E 00 O.35443080O0E-04 10.2324997000E 00 O.3196243000E-04 10. 2328373000E 00 0. 3160309000E-04 10.2331748000E 00 O.3246925000E-04 10.2335124000E 00 0.3372986000E-04 10.2338498000E 00 O.3785736000E-04 00.2341873000E 00 0.3090432000E-04 10.2345249000E 00 0.33422190O0E-04 10.2348623000E 00 O.3387519000E-04 10.2351998000E 00 0.3205373000E-04 10.2355374000E 00 O.3516222000E-04 10. 2358747000E 00 0. 3860490000E-04 0O.2362123000E 00 0.3421140000E-04 10. 2365499000E 00 0. 3468163000E-04 10. 2368872000E 00 0. 3580747000E-04 10.2372248000E 00 O.3904779000E-04 00.2375622000E 00 0.3578593000E-04 10.2378997000E 00 0.3912353000E-04 00. 2382373000E 00 0. 3599435000E-04 10.2385748000E 00 O.3474336000E-04 10.2389124000E 00 O.3538676000E-04 10. 2392498000E 00 0. 3470230000E-04 10.2395873000E 00 O.3952176000E-04 00.2399249000E 00 0.3923$-57000E-04 00. 2402623000E 00 0. 3944175000E-04 00.2405998000E 00 O.3966210000E-04 00.2409374000E 00 0.4409726000E-04 00.2412747000E 00 0.4269459000E-04 00. 2416123000E"00 0.437 1968000E-04 00. 2419497000E 00 0. 3904802000E-04 0
53
ERROR SIMULATION
00
C]
C00
0
z 0:0:
o o
LEGEND0] 5% ERROR
PERFECT DATA
I I I
0.224 0.225 0.226 0.227 0.228 0.229
THETA (RADIANS)
Figure 31. Intensitg Variation at 5X Error
54
The second method is to accept the theta locations as they are and
enter them into the DIAFIND program. The latter method will pass a
variable number of data points to DIAFIND.
Both approaches were tested (at 5X error for 7.254 jim) and the some
diameter was recovered by DIAFIND in each case. The only difference
between the two methods is that the averaging method runs one second
faster (out of 22 seconds on the IBM 360) than the other method.
The averaging method was adopted for this simulation for two
reasons:(1) The data input to DIAFIND will always be a constant number of
points so that DIAFIND can be easily adapted to run actual MicronEye data without simulation related logic buried in the code.
(2) The averaging method returns the same diameter as the othermethod.
D. CHOOSING THE CORRECT EXPOSURE (TUNING)
The optimum exposure for this simulation was calculated by the
program EXPOSURE. It was an average value of the intensity ratios
associated with the maximum derivatives locations for the 7.2541m fiber
data. This same exposure was used throughout the simulation for all the
fibers.
Using the same exposure introduces realism into the simulation since
one cannot tune the exposure for every fiber that is to be measured.
Because the Micron Eye window is located at least two nodes out from the
55
central maximum the exposure is closer to optimum for all diameters over
the range. (In contrast to a window location closer to the central
maximum.)
The process EXPOSURE uses to determine the threshold intensity ratio
can be called "tuning'. The Micron Eye analog of tuning would consist of
three steps:
(1) Determine the approximate diameter for the fiber to be measuredusing an optical shearing eyepiece.
(2) Determine the Micron Eye location parameters L and x by themethods in Chapter 5.
(3) Vary exposure to obtain a defined relationship between the threepoints on the Micron Eye photograph.
As an example, return to the case where L = 30 mm and x. = 5mm.
(x,. is the inboard x location of the Micron Eye array.) Figure 32 shows
the "tuned" relationship between three points on the Micron Eye array.
This relationship can be expressed as a ratio of a : b : c. The theta
locations at the inboard and outboard edges of the Micron Eye are knewn.
e.g., e,.=.165 radians and 0 .300 radians.
56
*1
oI
b C
X humq" Xl X2 X3 X aster
01 E) 62 60 3 )
Figure 32. Three Point Spacing Ratio
This means that '9inm 4 09 C 02 < 093 < '9te r -
To find E6 , 62, and e3 , run EXPOSURE for the diameter being tuned.
EXPOSURE outputs the theta values "or the optimum exposure. (See Figure
33)
The theta values are related to x locations by:
x = L tan(e0)X2 = L tan(e2)x3 = L tan(e3 )
where a = xi -xiMW
b = x2 -xI
c=x5-x 2
57
bteStij Promi
E e2 e3 e r4)
Figure 33. Finding the Theta Locations
58
VII. DISCUSSION OF RESULTS
The simulation was conducted for three diameters:
di = 5.803 jm
d2 = 7.254 im
d3 = 8.705 jim (where dI and d3 are ±20% d2)
For each d,, three levels of error were introduced: 1%, 2% and 5%.
Figure 34 shows the three intensity profile curves for the di.
SIMULATION INTENSITY PROFILES
4 ,LEGEND7.254 MICRONS
S5.803 MICRONS-,, .... 705"MICRONS.
C~r) a : ...-zc., •- ,.:\ "' ," ,,. '" ' , ... .,.. -E I I - -
0.075 0.125 0.175 0.225 0.275 0.325 0.371THETA (RADIANS)
Figure 34. Simulation Intensity Profiles
59
The s.ted Micron Eye window was determined as previously
described at L = 3 0wmm and . 5rm. The exposure was tuned with
respect to the 7.254jum perfect data. EXPOSURE recommended the
Threshold Intensity Ratio of .0000370. This Threshold Intensity Ratio
remained constant throughout the simulation, for all diameters.
Before the simulation, all diameters were tested with no error using
the 7.254jim Threshold Intensity Ratio. DIAFIND recovered d,, d2 and d3
exactly (i.e., to the three digits accuracy of the original diameters).
The simulation originally began by collecting thirty data points for
each diameter/error combination. Twenty additional points were
collected (50 total) to produce meaningful histograms.
Figures 35 through 43 are histograms depicting the results of the
simulation. In general, the expectation was to see a decrease in
resolution as more error was introduced. It was also anticipated the
resolution would be best for the 7.254gm case (for all values of error)
since the exposure was "tuned" f or this diameter. Finally, it was hoped the
method would be more accurate than existing laser diffraction
measurement methods (z 0.5%).
60
5.803 pm fiber (1Z error)20-
Is]16
14
U 12C
S 10-Cr* 0L.6_ 6
4-2
0o 0
In 10if in) tU
Diamneter (microns)interval .OO41im
Figure 35. 5.8O3pm Fiber (I1Z error)
61
7.254 Unm fiber (11 error)
16.
14.
u 12
2 10
L6 6
4-
2-
Diameter (microns)Interval =.OO42pm
Figure 36. 7.254gm Fiber (12 error)
62
8.705 pim fiber (IS error)2D0IS-
16-
14-
U 12-C
* 10-
* 8-6 6-
4-
2-0-
0 0 0 0 090 0
0 C-
Diameter (microns)interyal z.OO6im
Figure 37. 0.70O1pm Fiber (11 error)
63
0-- lo 4 :C :F tz-- - -
5.003 jim fiber (22 error)201
16j
14
( 12C
10a'* 8LI
L6 6-
4-
2
0V N - '
Diameter (microns)interval .0048pm
Figure 38. 5.803pm Fiber (21 error)
64
7.254 jim fiber (2Z error)20
18
16
14
12-
* 10
* 86L6 6-
4-
2-04
ClN 01 C
Diameter (microns)
increment =.00541im
Figure 39. 7.254pm Fiber (2Z error)
65
0.705 jim fiber (21 error)20-
16-
14-
12-
* 10-
* a-
6-
4.
2
0 i 0 6 0 0 0
Diameter (microns)increment = .OO49jim
Figure 40. 8.7O51im Fiber (2% error)
66
5.803 jim fiber (5Z error)22-
20-
16-
S 14-
12
Cr 10
6 8L6.
6
41
2
010 COC
N to#0
Diametqr (microns)interv.1 .0 1 O5jim
Figure 41. 5.803pim Fiber (5 error)
67
7.254 jim fiber (5Z error)22-
20-
18-
16-
S 14
* 12
106
L6.6-
4-
2-
0in i ) In to 0) In
0 %a o 0 rN oe V) U) WN' ' N N N N C
Diameter (microns)interval = .OO7jim
Figure 42. 7.254Lpm Fiber (51 error)
68
0.705 jim fiber (5Z error)22
2018
16
2 14UC 12S
106L 8III.
6-4-
2
0-- U) N0 N
~0 F-
Diameter (microns)
interval =.0136im
Figure 43. 8.705pm Fiber (5Z error)
69
A. ACCURACY VERSUS RESOLUTION
The histrograms show the accuracy and the resolution of the method.
The eccuracy is associated with the largest spike, and related to the size
of the interval called the resolution. The resolution is the ability of the
routine to *see" a difference between two different fibers. For example, if
the resolution is .O051im the method does not discriminate between
8.705jim and 8.700jm. Table 2 shows accuracy versus resolution for the
results.
Table 2. ACCURACY VERSUS RESOLUTION
%ERROR DIAMETERpm RESOLUTION ACCURACY (Res/dact) %
1 5.803 .004 .0697.254 .0042 .0588.705 .0060 .069
2 5.803 .0048 .0837.254 .0054 .0748.705 .0049 .056
5 5.803 .0105 .1817.254 .0070 .0968.705 .0136 .156
70
The data show a decrease in accuracy and resolution as error is
increased. It is evident in most cases that the 7.254jim results are
better because it was the "tuned" diameter. The only exception is the
accuracy for the 8.7051tm fiber (2% error) is better then the 7.254Am
fiber. Overall, the largest error is .18 percent which is less than one half
of the error associated with the manual laser diffraction method.
B. ERROR
There are two contributions to error in this simulation.
The first can result from programming/calculation errors. Many trial
runs of the software were takpn to minimize the likelihood of this kind of
error.
The next type of error is the digitizing error eni-ountered in an actual
experiment. This error is simulated based on the maximum intensity of
the profile curve, in the region of interest, to apply random error equally
to all points. Had the random error been based on each point, points with
less intensity would have less error, and points with higher intensity more
error, which is inconsistent with the physical digitizing process. Using
the maximum intensity avoided such a condition but it cannot be
ascertained that this is the optimal representation of the physical system.
71
,.1
*~> -V .*.>3
ViII. CONCLUSIO!.
The results of the simulation are encouraging. The largest error is
1 lOX or less than one half of the manual methods. Because the programs
were carefully developed and tested it is unlikely they contri' -ted to the
error. Also, much effort was directed towards accurate simulation of the
physical system so that the results would reflect what can be expected
from the actual experimental measurements.
The simulation demonstrates an increase in accuracy two to ten
times better then that currently possible by making manual measurements
with laser diffraction. The method also lends itself to automation which
makes it attractive for quality control purposes and research for
materials development.
72
IX. RECOMMENDATIONS
This study has shown the feasibilty of computer aided diameter
measurements. There are many directions future work can take.
A. SOFTWARE.
More development and testing of the software could result in
increased accuracy. It would also be valuable to implement the software
on a small computer (like the Macintosh) to allow real time processing of
actual data.
0. HARDWARE.
There remain some hardware considerations which must be resolved.
The biggest of these, perhaps, is the accurate positioning of the Micron Eye
array in the direction perpendicular to the laser beam. In an effort to
increase the resolution and accuracy of the problem, the system may
benefit from two Micron Eyes.
73
APPENDIX A. FRAUNHOFER DIFFRACTION THEORY
The following is a brief discussion of Fraunhofer diffraction theory
with emphasis on aspects of the theory which relate to this research.
A. FRAUNHOFER DIFFRACTION
Fraunhofer diffraction (Figure 44) results when light approaches and
leaves an obstacle or aperture in the form of plane wavefronts. [Ref. 4:p.
1761 The light source and the piane of observation in effect are at
infinity. A collimated laser beam is ideally capable of Fraunhofer
diffraction because its beam consists of parallel rays advancing in phase.
C, itral Mai, Iiterfereii.Nde
Figure 44. Typical Single Slit Diffraction Pattern
74
B. THE CLASSICAL SINGLE SLIT EXPERIMENT
The simplest demonstration of Fraunhofer diffraction -is the single
slit experiment. (See Figure 45) Parallel, co' imated light passes through
a slit of width a. Th diffraction pattern is visible on a screen located a
listance L from the slit. An observer at point P, moving across the
screen, sees a succession of maximum and minimum intensity points.
These extrema are the result of constructive and destructive interference
of the light. For example, a minimum occurs when the angle e produces a
phase difference of one wavelength between the rays at the upper and
lower edges of the slit. Thus, minima occur whenever
a sin(e) = mA, where m -1,2,3 ...... (A.1)
These minima are referred to as interference nodes. (For further
discussion of this subject, see Meyer-Arendt, [Ref. 41.)
I
75
• . '.L- -...,. _ .". ':..."_,, .,. ,'- ., , ,' ., ,' ,'' . ' " '".' J',,," . ,",,,,.,, P. ',,.,' ",, ' .-',,, ,,,' ',,' ' " .',' e.
m=Iinterfaerence node-
geometric center
path difference = one wavelength
L
Figure 45. Single Slit Path Difference Relationto the interference Node
1. Diffraction Minima
Exbmination of equaticn (A.) shows that as the slit becomes
narrower the angle 0 becomes larger. As this theory is extended to
approximate the diffraction phenomena of an obstruction (a fiber), one can
expect short distances between interference nodes for larger fibers and
greater distances between nodes for fibers of smaller diameters.
76
............-
Another important point concerns the distance between minima. It
appears the interference nodes are equidistant. This is true only within
the limits of the small angle approximation. The distances between
interference nodes actually increases as one moves outward from the
central maximum by the relation.
ein = sin [ m ld (A.2)
2. Diffraction Maxima
The diffraction maxima are not located midway between minima.
The locations of the maxima can be derived as by Meyer-Arendt [Ref. 7:p.
2201. These occur whenever the derivative of the intensity is equal to
zero:
dl/dB = di/dB [ i (sin B/8)2 = 0
= 21. stnB/B I -snB/B 2 +csB/B ]
or whenever: B = tanB (A.3)
77
g=tanB and g=B. (See Figure 46) The maxima are displaced slightly
from center towards the central maximum. Much further away from the
central maximum, the maxima are nearly halfway between minima.
0 r 37IF 57r
Figure 46. Locations of the Diffraction Maxima
C. FRAUNHOFER DIFFRACTION AND THE FOURIER TRANSFORM
The foregoing discussion centered on the Fraunhofer diffraction due
to a slit. The mathematics of the slit example are simple and provide
insight into diffraction physics. A more elegant derivation of Fraunhofer
diffraction shows that the diffraction pattern of an object is the Fourier
transform of that object. [Ref. 9:p. 1741
78
f(x) A 10. IxI > bi, Ixl < b
F(O)= 2sinbee
-b +b
Figure 47. Fourier Transform for the Single Si t
Thus, F(e) describes the amplitude of the diffraction pattern, and
IF(e)12 represents the intensity.
Modeling the transform for the obstruction is more complicated.
consider the complement of the slit transform: g(x) I - f(x).
g(X) = 1-f(x) g(x) = > Ixb
0, Ix
4I-b +b x
Figure 48. Complement of the Fourier Slit Transform
79
This results in the, form 1 s-2sin (bO)IO which is not
transformabl e.
An alternative Is to use a substitute function hW = g(x) - M~).
Physically, this is approximating the obstruction as two parallel slits:
h(X) = g(x)-f(x)
C -b Ib +C
igr 49 Fore TrnfrIo woPrle lt
whc ha thIouin
whhsthe soluse tion :
80
D. REJECTIO OF THE SLIT APPROXIMATION
In their paper titled "Fiber Diameter Measurement by Laser
Diffraction" [Ref. 3:p. 1378], Perry, Ineichen, and Eliasson conclude that
the diffraction pattern of a real fiber is sufficiently different from that
of a slit to warrant the slit approximation being treated with caution.
Further, they recommend a solution presented by Kerker [Ref.5:p. 260]
which has been adopted in thip study. Kerkers solution assumes the fiber
is perfectly reflecting (i.e.,the reflective index m=oo), and while this is
not completely true, Perry, et al., [Ref. 3:p. 13781 indicate some degree of
absorption is not likely to be significant.
E. INTENSITY EQUATION FOR A REAL FIBER
Kerker [Ref. 5:p. 260] gives the scattered intensity relation for a real
fiber
I1/1 - (2/K*Lii) I b0 + 2 X b. cosne) j2 (AA)
where 6 = the scattering angle
Ke = 2i /?\ (?\= laser wavelength)
b.=J,(a') / H()
81
e- - . '-le ,-P I- X
and or= iMdY.b (df = fiber diameter)
J.(&) ar, Bessel functions of the first kind,
W(a)( ) are Hankel functions of the second kind.
The real fiber equation (A.4) is somewhat obscure in its compact
form. It can be shown that:
1/11=(2/KLU) [( r.+2. rcos(ne) )2+(s.+21 s cos(n8) )2] (A.5)
where r.= j [J 2 ) + Y 2(a) (A.6)
and sV (Ja(() / [J*2 (0f) + v 2 () ] (A.7)
Now, the intensity ratios for any e location can be calculated for any
diameter riber.
1. UnjtlvityQ of the Results to the Number of Bessel Terms
The calculation of Intensity Ratios requires the computation of n
and s. Bessel terms, equations (A.6) and (A.7). Here the phrase
"Bessel terms" indicates the algebraic combinations of the J. and Y.
Bessel functions.
82S.I
a. Two competing phenomena
There exist two competing phenomena which govern the number
of Bessel terms to be used in the calculations. The first requires a
minimum number of terms for accuracy. The second, limits the number of
terms so the Y functions do not cause an underflow error during
computation.
(1) Minimum Number of Terms. As in any series, there is a
minimum number of terms required for computational accuracy. Figure 50
shows the effect of the number of Bessel terms computed for seven
curves, all with a diameter of eight microns.
Note that the 43, 50, 75, and 86 term curves are nearly identical and
the curves with fewer than 43 terms are decreasing towards a smooth
line. This research, indicates that 50 terms returns four digit accuracy
for diamqters ranging from 5 to 10 microns.
(2) Maximum Number of Terms. The maximum number of terms
depends on the value of a. As the diameter decreases, so does a. This
in turn produces very large values of y.2(o) (in the r. and s.
denon;inator) which causes an underflow error. For the diameter range
considered in this research, underflow was not a problem. In one
instance, below 5 microns, it was found ' ie number of Bessel terms had to
be reduced to 43 to prevent underflow. The effects of the minimum
number of terms was not investigated at this smaller diameter range.
83
INTENSITY PROFILE
00~0
0
o
0 1
"" I.FA;END
So .'5 TERNIS0 o 8 UIS
o - 5- 0 , • ERMS
x 12 T E.,.~-13 TERMIS
0 0-8i TERSK
0.000 0.025 0.050 0.075 0.1 00 G 0 125
Figpire 50. Effect of Number of Bessel Terms
2. Effect of Different Diameters
Recall that the diffraction pattern for a slit showed that as the
diameter decrsased, the interference nndes moved away from the center of
the pattern. It is interesting to note that the pattern ior a real fiber
behaves the same way. (See Figure 51)
84
INTENSITY PROFILES
Ell I
Figure 51. Effect of Different Diameters
85
APPENDIX B. COMPUTER PROGRAMS
The following is a list of programs discussed in this Appendix:
DATAMAKR
DIAFIND
EXPOSURE
The programs are written in Waterloo Fortran IV (WATFIV) and run on
the Naval Postgraduate School's IBM 360 computer.
A. FORMAT OF DATA
All data is formatted using exponential notation. The most
frequently manipulated files are those containing theta locations and
intensity ratios. The format for these files is: (IX, E17.10, IX, E 17. 10).
In WATFIV all data files must be of filetype "WATFIV.
To compile and run a program on the IBM 360 type " WATFIV
PROGNAME DATAFILE *(XTYPE', where PROGNAME is the filename of the
program and DATAFILE is the filename of the data file.
86
IB. COMPUTER PROGRAMS
This appendix presents the various computer programs and outlines
their logic.
1. DATAMAKR
DATAMAKR generates the data used in the simulation. The equation
for computing the Intensity Ratios of the diffraction pattern f or any value
of theta is:
1/1, (2/KOL) I b. 2 X b cos(ne) j (A.4)
where e = the scattering angle
Ke = 20l 0. (A\ laser wavelength)
be= J I() / Ha(2)(&)
and Of= Iid1/A (df = fiber diameter)
Jn(x are Bessel functions of the first kind,
H (2)(a) are Hankel functions of the second kind.lU
All computations begin bU calculating the required number of
Bessel terms. Two NONIMNSL subroutines are called: BESJ for the J,(oe) and
87
BESY for the ¥.W. Note: should double precision be desired, the NONIMSL
subroutine BINT will return double precision values for the J. and Y..
The Bessel function values are stored in an vector J(1) and YO),
and are used to calculate values of R(I) and S(O). Next, the matrix of
cosines is contructed. The size of this matrix is M by K, where N = M- I
and N is the number of Bessel terms. K is the number of theta locations.
Values of theta at each location have previously been stored in an array
called T(K).
To visualize the program's calculations, the matrices _ymbolic of
these calculations are sketched. (See Figure 52) Note that r. and s. are
o.utside the summation term in equation (A.1) and therefore will not be
multiplied with any cosine terms. Since array subscripts must be
designated with a nonzero value, r. and s, will be represented by r(1)
and s( I).
After multiplying the two matrices, it is necessary to complete
the summation by summing the columns in the matrix. This results in an N
term value for each value of theta. A straightforward calculation then
yields the intensity ratios. The intensity ratios are stored in an array
with their corresponding theta locations.
88
Frcos e Cos e . . . . . . . cosek
z caste, cos 2e 2 Cos 2ek
r4 c0s30 1 Cs e COS 3ek
r 0 cas(m-1)e, cos(m-1)e 2. . . . .--cos(m-l)ek,
Fr7C, , .... r..CI k
r3 C21
[R) x ICI r4c.,
r.C.5.1 - - -- r ,-
r.cs4
a ek
5'.I
Fiur 5 . AAARMti Aler
5% 9
The logic of DATAMAKR is:
Input ----- > Bessel terms to be computed
ditneter of the fiber
screen to fiber distance, L
array of theta location values
Calculate: R. and S. using BESJ and BESY
Produce the Matrix of Cosines
Produce the Matrx of Products
Sum the columns of the Product Matrix
Compute and output the Intensity Ratios
2. DIAFIND
This program finds the diameter of a fiber through an iterative
process of residual comparison. The program accepts the data which has
been output by the program EXPOSURE. The user is prompted for K and df.
K is the number of theta values the program should expect and d, is the
initial guess of the fiber diameter (in microns).
90
The program first calculates intensity ratios at the diameter di,
corresponding to the input theta locations. The difference between the
input intensity ratios and those associated with di is called the residual.
The program calculates another set of intensity ratios based on a new
diameter d, + Ads, where Ad, is a small increment of diameter (usually
.05 microns to start). The intensity ratios calculated using d, + Ad are
compared with the input intensity ratios to give a second residual. The
residuals are compared and program logic determines whether or not the
Ad increment is producing convergence to the actual diameter. The
initial guess diameter is incremented and decremented as necessary until
a desired level of accuracy is achieved.A key to understanding the convergence process is the residual curve
in Figure 53. It is important that the initial guess, d,, be fairly close to
the -actual diameter.
For example, the residual curve in Figure 53 is for an actual diameter
of 7 microns. If di is greater than 9 microns, the program will not
converge to the correct diameter, but rather to a diameter just above 10
microns.
It is hard to define exactly the limits of d, for any given actual
diameter. Figure 54 is a residual curve for 5 microns and shows an upper
91
- -.4~ . . . 1
limit of 9 microns for di. A thorough study to define limits for di has
not been conducted, although convergence has always been attained by
guessing d, within I 1 micron for diameters ranging from 5 to 9 microns.
RESIDUAL VS DIAMETER'o .
0
C5,0-
o
0. 3.0 4.__ _ _ _ 0 7.
0.0 10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12 0
DIAMETER (METERS)
Figure 53. Residual Versus Diameter for 7pm Fiber
2U
U%
"1
.7_....-. -
RESIDUAL VS DIAMETER
0.
0
.6
0
0
C-
3.0 4.0 5.0 6.0 7.0 8.0 9.0 10).0 1 1.0 12.0 11.0 14.0 15.0DIAMETER (METERS) .i0"'
Figure 54. Residual Versus Diameter for 5pro FiberI9m- -
C'
l~~ l~l• • T
•% i 't% % ~lql t, • %lll ... * . . j • ,*0
3. EXPOSURE
EXPOSURE simulates the operation of the Micron Eye by taking e
photograph of intensity ratio data. EXPOSURE reads in the perfect data
generated by DATAMAKR and digitizes the data with respect to a threshold
intensity ratio.
The optimum intensity ratio for an exposure is termed the
threshold intensitg ratio. The threshold intensity ratio is located
where the absolute value of the derivative of the intensity ratio with
respect to theta is a maximum. This assures that the threshold intensity
ratio is located in a region of the curve which is closest to a straight line.
This reduces the effects of subsequent interpolation errors.
The threshold intensity r3tio is calculated by the subroutine
DERW. Since the irput intensity profile curve will have at least three
maximun derivative points, DERIV calculates the average of the three
intensity ratios. This average is then passed to the calling program,
EXPOSURE.
EXPOSURE searches the intensity ratios and compares them with
the value of the threshold intensity ratio. This process corresponds to the
Micron Eye addressing a pixel and comparing its voltage to the threshold
voltage. If an iniensity ratio is greater than the threshold value, the ratio
is assigned a digital wflue of 1. If the intensity ratio is less thatn the
threshold value, the ratio is assigned a digital value of 0.
94
The digitized data is then searched to find the theta locations at
which the digital intensities change from 0 -- > 1, or from I -- > 0. The
theta locations are averaged to produce a theta location which is very
close to the threshold intensity. Because this averaging process is
occuring in the regions of steepest slope, error is assumed to be
minimized since the curve can be approximated as a straight line (refer toFigure 30 in Chapter 6).
EXPOSURE provides the user with the option of introducing error
into the perfect data. The program prompts the user to input the desired
error and calls the subroutine RANDOM. RANDOM introduces random errorinto the intensity ratios and returns the imperfect data to EXPOSURE.
EXPOSURE digitizes the imperfect data and searches for the occurences
where 0-->l and 1-->0. Imperfect data will have many more occurences
of the digital intensities changing from I-->0 and 0-->1. EXPOSURE will
average the theta locations if they are in the same location (i.e., within
t.005 radians). This method has been tested for random error up to 5
percent.
95
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LIST OF REFERENCES
1. Tsai, Stephen W., Composites Design 1986. USAF MateralsLaboratory, Dayton, Ohio, 1986.
2. M. Koedam, Determination of Small Dimensions byDiffraction of a Laser Beam," Philips Technical Review, v. 27,p. 162, 1966.
3. Perry, AJ., Ineichen, B., and Eliasson, B., "Fiber DiameterMeasurement by Laser Diffraction," Journal of MaterialsScience v. 9, pp. 1376-1378, 1974.
4. Bennett, Thomas A., A Comparison of Two Methods for FiberDiameter Measurement and A System for the Study ofComposite Reliability, Master's Thesis, Naval PostgraduateSchool, Monterey, California, December 1965.
5. Kerker, Milton, The Scattering of Light and OtherElectromagnetic Radiation, Academic Press, New York, 1969.
6. Micron Technology, Inc., Boise, Idaho. IS32 OPTICRAMSENSOR MANUAL, (Micron Eye Operators Manual)
7. Meyer-Arendt, Jurgen R., INTRODUCTION TO CLASSICAL ANDMODERN OPTICS, 2nd ed., Prentice Hall, Inc., EnglewoodCliffs, New Jersey, 1984.
6. Houstoun, R. A., PHYSICAL OPTICS. Interscience Publishers,Inc., New York, 1957.
9. Lipson, S. G. and Lipson, H., OPTICAL PHYSICS CambridgeUniversity Press, New York, 1969.
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INITIAL DISTRIBUTION LIST
No. copies
1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22304-6145
2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, California 93942-5002
3. Dr. Edward M. Wu 20Professor of Aeronautics, Code 67WtNaval Postgraduate SchoolMonterey, California 93942-5000
4. Mark G. Starch, LT, USN 5920 Wallace AvenueMilford, Ohio 45150
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