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N NAVAL POSTGRADUATE SCHOOL Monterey, California DTIC ElE'ECTEN THESIS Em A COMPUTER AIDED METHOD FOR THE MEASUREMENT OF FIBER DIAMETERS BY LASER DIFFRACTION by Mark Gerald Storch September 1986 uAJ Thesis Advisor: Professor Edward M. 'u Approved for public release; distribution is unlimited. 81 - 1 ,
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Page 1: NAVAL POSTGRADUATE SCHOOL · 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

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

Approved for public release; distribution is unlimited.

81

- 1 ,

Page 2: NAVAL POSTGRADUATE SCHOOL · 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

sECuAIrY CLASSIFICATION OF THIS PAGE

REPORT DOCUMENTATION PAGEla REPORT SECURITY CLASSIFICATION lb RESTRICTIVE MARKINGS

UNCLASSFIED__2a SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILABILITY OF REPORT

,IDOWNGRADING WHEDULE Approved for public release;2b ECLASSIFICATION N N Udistribution is unlimited.

4 PERFORMING ORGANIZATION REPORT NUMBER(S) S MONITORING ORGANIZATION REPORT NUMBER(S)

6a NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a NAME OF MONITORING ORGANIZATIONOf a .pabie)

Naval Postgraduate School Code 67 Naval Postgraduate School

6C ADDRESS (Oty. State. and ZIPCode) 7b ADDRESS City. State, and ZIP Code)

Monterey, California 93943 - 5000 Monterey, California 93943 - 5000

8a NAME OF FUNDINGISPONSORING 6ab OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT ;OiDTIFICATION NUMBERORGANIZATION Of $oale)

8c ADORESS (City, Sate. and ZIP Codo) 10 SOURCE OF FUNDING NUMBERSPROGRAM PROJECT TASK WORK UNIT

ELEMENT NO INO NO ACCESSION NO

TITLE (Include Security ClIsficatIon)

A COMPUTER AIDED METHOD FOR THE MEASUREMENT OF FIBER DIAMETERS BY LASER DIFFRACTION

• PERSONAL AUTHOR(S)STORCH, Mark G.

l3a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year. Month. Oay) IS PAGE COuN.T

Master's Thesis FROM TO 1986 September 13a

*6 SUPPLEMENTARY NOTATION

COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

; 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.

20 DS*n3UTION,AVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION

E'NCLASSIFIEDAJNLIMITED 0 SAME AS RPT 0 oTIC USERS UNCLASSIFIED

22a %AME OF RESPONSIBLE INDIVIDUAL 22b TELEi-HONE (Imiude Area Code) 22c CFFi-ICE SYMBOL

Professor Edward M. Wu (408) 646 3459 67Wt

DO FORM 1473,84 MAR 83 APR edition may be used until exhausted SECURITY CLASSIFICATION OF ,41S PACE

All Other editons are obiolete

Page 3: NAVAL POSTGRADUATE SCHOOL · 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

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

Page 4: NAVAL POSTGRADUATE SCHOOL · 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

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

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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

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L IST OF TABLES

1. DIGITIZATION ERROR------------------------------ 53

11I. ACCURACY VERSUS RESOLUTION--------------------- 70

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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

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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

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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

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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

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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

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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

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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

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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

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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

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-~-.,.cc-e:< ! - -

3t . . °.o. -o.4° °°.... ° ° • ° . . ° • ° .. . ° .$- °*t .

• : .°...: °. •.-° ° -

Figure 3. Tgpical Micron Ege Photograph of Interference Nodes

15

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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

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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

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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

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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

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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

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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

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Figure 7. Effect of Diameter on Intensity Profiles (3-D)

22

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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

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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

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LEGEND7.254 IMICRONS

~LEGEND

.5.803 MICRONS

~LEGEND

/8.70- MICRONS

Figures 9, 10, 11. Three Points to Define a Curve

25

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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 - -- - -- - - - - - - - -

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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

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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

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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

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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

- - .% ... . "

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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

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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

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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

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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

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ThresholdIntensity

Ratio

1.

ITIR

Ii+

Oe eO4 ei+ e I _

2Figure 30. Averaging of Theta

511f

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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

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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

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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

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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

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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

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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

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bteStij Promi

E e2 e3 e r4)

Figure 33. Finding the Theta Locations

58

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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

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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

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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)

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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

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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-- - -

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

............-

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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

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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

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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

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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

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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

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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

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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

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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

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INTENSITY PROFILES

Ell I

Figure 51. Effect of Different Diameters

85

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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

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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

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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

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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

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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

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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

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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_....-. -

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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

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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

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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

" . ... . . . . . .

Page 97: NAVAL POSTGRADUATE SCHOOL · 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

Io

U- U) .z

00 cr0

0 W E-4 00 z.' 0C4

N' r)

0 w-i1z

:.) U )n H I'-4g.- E- ~

g~'4W H-- E-4-

am ZOOO W4 W C0t 09 00r4CN "-4HOZ rz4 o 0HA 0 Ur4..-, % 0 0g

0 -4 4Y00 WOO0 0 0 0

ZOZ: 0 4z)... Pw~ H- E-1 H -0010 2zIu HE4 Z E-4

U)n1I0> 0 , ".Z"- -0 ce14 0 W-4- , a) W-O a4 E1N1OZ04H 0-~- WAU)

OQ04Z >4000Z)CiWz 0 %D 0 A -U)-4 W H400H>XW H- - Z H- 0 1

E -4 Z 4 -NN 4 -4 E- U) 1-f :: Q >

4"(4 -F P I - N - 0't

::W %Z-O> F-W" U- AC " oo.00-4E-4" H.J~- H-Z *-n *l' U) * 00

QXoc)U) Z'-ti.-- o U) --- X uc0 UQ00n

gC CZ)W % -~0 '-HC-4 E *-- tri 0-i* 4 H 0 Cw H 400-0 U HE- -'- H-N, 11 rIOC)

to~ l-I4. .. 4 . . D 'D0cr~0 0 1-4 11

H'' ui woz)-) 5 0. 4 En %-t4-&i U...4 11 c'r)0EH0 00 N 11'4=>4U)Z 0 Z>40:: MwwZ -44 '-WCz Q* -i .r-I H

WO cN:) zU)CI O--iE- E-1 11CM U) E-U2

44H2I-M4 i- O4z0"C Ms- 11-1 1111 OZ

* 0 0'i00 000000 00 ri 00U 0 U 0 U 0

96

Page 98: NAVAL POSTGRADUATE SCHOOL · 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

E-

4'4'4'44' 0'4'4

4'~ D"4'4(n E-4'4'4

4' Z 4' ' 4

4' 4 0 '4 z'440 - E-44I..i4

E-1 * E-' U 4

Cl) 4' C 4 4' 0 4'

U ' A ) 4' )

EA - 4' Cl) 4*~~~> w m'- 4 i -4

0 *l 4' ) ZWE K

0 'Ci- Z' 0 4' E4 0W"- " -4~ m WW-4 0'nii

w' 1.1- 4' 1%

tn~nZ 4- x E-1

E- Ho E4 -0 W-4Q ~ 4

oc 4'l acrul 4' l)IC) 4 w' 0-11 4 i C'- 4

4'E-4-4 O K- 014> E- "CEC4 4

*~o A - * 0 K>4 CiZ 4

4' C.) 0'C ~ Ci4 Z - + 4* 14 ZJ ' Y~ 4 ' Ci'4Ci H

Cii)UO L)R *-'C)LLC C- Ci1 UO* )O C L)

97 ' i-~i-4 L~ 4

Page 99: NAVAL POSTGRADUATE SCHOOL · 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

z z

"Z * WK

% z * En 04z '4= 0U) 0 -

+K'0K 'K Z

P- * * 0K' z

W U 4 'K4 Hz z

'K E4 "K "o4 . KE- * w EKC)l 4 2 'K M

$- * 1 00% Z Z 0 0"

o-z0 "K * -L) '1K m 'K PH 0 M w u

4--4"Z *K U) 00 00 0 '>4ri 'K,-. 0 L) 'K 'K 0K

'K oar) u- K~a'

U)0 'Ko O- ' 0.' 'c

eIN-n~ 'Kft *- U 'OL)U u U)L))L)C L)U UU000 CjL) 0000

'K~ K 1- 0 'E-4 K98

Page 100: NAVAL POSTGRADUATE SCHOOL · 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

K 'K'K'K ''K U 'K0

'K ~ KK '''K (. K'K 'K''K 'K K'K e 0

'K 'K 'K'K E-4-1c U~'' 84'

'K 'K'K *K''K 'KK 'K0

'K Z KI' s'KKEA

'CA w.'' *''K 'KK '-4'K' E

04 KK' -

XK 0~ 0' 0

'K ~ ~ 0 W- KKrIKC s

UK U)) 'K'K 0z KK(44U 0KK'KK

0K + 0 . wK'

E-K w% '

'Kr -z 0 K' w. H 11' Cza'K 'KK R W Z 'K',K 0 :

z'KA Z E- 'K 0 E -4 EA' E- (4

mm 00 4 'K' w 0 ' 0 Pgmr 'K z 0 >4 1-4 0f-I 01%00 on

8-0-4~~~P 'KUfZ' 4 Ci-

'K 0K *K' H 0 Z 'KK

~U)% 0 * 'Ki 000'

'K~ * 'KK ) K''K00c 00 uK' 00 UU* 'K'K

ZZ 'K'K'Kr~ - 'K' 099

Page 101: NAVAL POSTGRADUATE SCHOOL · 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

0H 000

Z W-Z IM1

-E- HZt ZH~0

0 0 '-0 0 ~00U2"E-1o E-4 0 U 0U~U

UZJ0 OH Oc) E-0k~0-wo W0 E4 4

Hl U)-4 m Ot m 41004

> ~z 0 00 Z l Q0 0 .

1.- E-z 0 4Z 0 .H .W NP

a- N~j z IZwri- .U)i Q E-4000z 04 H 4 or-

& -- 0 P E-IZX U~ HU)H ~ 91

Eno w~0 0 04 H 4 "H .

z ~ e~ j ) J~ U) 1% 9ZOW3 I'c:z

X>4W 140U)z4Z X 0 V O ) .XEAU$4 x P "W -E-Z4

MU) w E 0 9 D-41(

HHH4EI U)") 0 >o0 WOZ< 9wi WE =Hu~gI "-40-- U) r)CW4 HZ~ HZEAW -z W ~ 0 0 c

00-4 w ~ w E Co 1 000 WCZ~ * -EAO~~ 0 0Z~QC4 E~Z- ~zc4z) .H 0 -e

cn HU go ww C4%% r- H -w I) rzl c-cl~z w q q w& 1-m m U)OW - "94 .ZlI0)XIIDCM2N

%~~~~~ ~ ~ ~ sC -4Wnx)o 4 w O431

z~~~ 4404 Pr"C= w 4- H4Ww~~ ~ ~ U) wom~zC C .W 0=4WOA Wr4 P4 %-EA 1-) 409-0 Z* cq o::cMz I-) -

000000000000000000000000000000000 0nC1-

W W OWW OW 1 1 1 1111 -4 w w U W 0100l

Page 102: NAVAL POSTGRADUATE SCHOOL · 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

00H1

0

0 0C0e4

Cf N.l 00 H- U) U

cc H z>j H- w

0 1 0co 0 0 Nmc 0 ~- c0

Z0. +- H z w

eq4 HO E-N1 ~ I -c q. 0NltCY D 1:4I- E Zq E-1 gL +X +P.Nxo-, 4 % X.- M.-c OD .4c H- < c E

+ E-1 E- 0 'WD . E 44 Nl '0 HOH M.U) U)LA +X UCflI1 -B 1 q C14 0 H16, r4 0E %0

owz 0 -. 0x" w~z)0 . ' - w 1 1 614

lIZ Z H-4 11 E ~ -4 ll'- t-1 P- ZO II e4 .r-1 iE-E-4II .-H411II1nUE-4 I y0+ > 3 1+0 .+0 : 1101 Wa H'I HO I w IC '- 4 Z ')

(flz11 3IIzwi A4i XXHHZP -4 11 W -rfrZ14 1 C 1ZQPOWn -404D It IZ E 11 1 11 Wni EH r H- HI-l 04 1 11 11i II4It~. 1r II HP4A4- 1~I14IV11

14DOD0 0 00 0 0 00 00 0 00M r f 4 )% ,0 r-1cqm qjn %D c-cc

H- H -r- Hl HH H HHl

000 000000

101

Page 103: NAVAL POSTGRADUATE SCHOOL · 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

* >4

0 0oon0

P -4E-

-4 ~ E- 04-40 -

0 Ccn)rz 0 w -4< j9 r49 - 0

Z 00

0W on Zl 0 ZODO> '-4 0'- 0ooM

*-I w- H4UZ

*t 00 H0 OD

WO E- 0 0N4 l40*Z - M 4- Zop4e

w 04 9z N 0Oo >0r-4uz OW4w &- ~i0 zmE- H013 U2 4 >4Q P4~ 0(r-iU . >4ww on Mc NO

m*r W rrzO N0pW ow V) p:>

MMi2>4wzo >>Wl xxz 4WO

w HHO Eu MEHQ 0 .ioW Wi HW4>0 04XOHHE Z',)U)U-H w~

* U) WO0H 0~00 H - 0. z. -W* U) *OOW U) C4134zCz U 0 I-~~ 0* M:

W >-4 PHru9ZZZU) Wz0zZ HE-1 "H-EIMm m ~z0' WWHH E-P- >4HHE-lU) M Z

W* >4 >4 >04C P4 -"

000 044*z w E- M - )4 4> 4 '400-0 14L: WC -4ow"2-1H *>4 0

*Uz ::0HC-cc) OrZm ( ZIZP-4 cUo0* WWWMIzI 11 11 11 0o wi WO0U)0 ' * %

*Z H m H m wP C 444M-4HH Z M IX4~~IE -4 D4 4 H HCII ... CnHJU)U) PW :DQUEH-z W

*: H) V. 63 p~ain-- xpnzzz =r Q~u z"0O00 W< >4wu Z= 00 rE4U)0:DZ " 0

m 9 No) 00 0xzm"- >x 0Z =4H4E-lf- *D H- W

> ::) *cni) U) ww 0 *r 4

0 0m 0

102

Page 104: NAVAL POSTGRADUATE SCHOOL · 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

E- (Nq C* -x E- i (

H 0 MA U

kD W U)

o No rO-O

HO'O(UI 0 HmoKUH M .140O'clirilACV cq (N4o

PN 0'10 0U)OE-,'40% 4 -z E-

LO (Y) -<14 O -V - H

on +AOU+ Uri.- C1 l 1 -0 " (NIICl N 0U 0-0 E-N0

r-EHH041O(NOkO 04 n -* 4

r *qq 10 U) -4(-4''-- "D 0 OoIHH(NLAH~l:

H H 0C'040000 ~ '40* ~ ~ OOLOO % NU)(.0 -HO

O 0 '-4 0 H HH.0 .0+0H ulA-iC Howl+~zxI 1

-0-- 0 0 Nr 1HO...0..-0 .4 01 NNO1 110 0CN 0

0 0 0 0 00qNM 0 -

H (N4 C9) LAI 0*N~-qqo -fr l

,000 0 oeo-oo- -0 00 2 EiW4

0 m x >- 0140MOO'44. ZO >4 HD"' 103--

Page 105: NAVAL POSTGRADUATE SCHOOL · 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

U) 1-4

U) 04HP z-

-% 0U) U) I-l U

H >4 >1 U)0 H4I~

0 400 000 0 0

X0 H >0 0

0 01 t00 H~ NN 4-> H 0

IIHXHIp+ 0 0 E- r-4>4 0 0

,-IU)WHi-OH': rT -Ij W1 0 r-4> r II I0zi I 0>4 144 ~ II H>4P.I>4>4 0' - - rC)-I H z >4>4E >4'-p4>4 -'P

S) % l 0 000 0 0- U 000

104

Page 106: NAVAL POSTGRADUATE SCHOOL · 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

C4 4 w E-4

E- H-XZ

014304Z u

E-=0 0 H

""H C4 H~14U ~ ~ ~ U) C414E4 w0(

CzI U p4p4: 0

"W(OP zH ) H '4 -

0 E, I

OzjHH- 0Q 0 H 0- 0 4

ow wZQi -i- W '4 04

ww0H0ZI40 W-' E- Zz W 0

tEV~U 0 H Q UZ P -4 H" U: &A F-

" -Cr 1 0I~4 N40 0 w~H 00r - H-

P4 MHW4 m~ 0 0,... -1I 0

H00HHH0IIH0 -- E-iU) (nEi

4 ~ ~ ~ ~ 00 OW 04W 0Q 4 P 40 H0 4-

M IE 0 OZ (1 g L4 E-4' - ~-4 0

uO0 w04400 0 0 44 r00 0 0f 0w 0 00

IEI~r:E~L)40" 74E-4E-1-0 0 105

Page 107: NAVAL POSTGRADUATE SCHOOL · 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

U) H

* z r4 V)3-P r4

U) E- -izo U o

r4'

E-i P44 t- E-4r44

rzI U)i P

H -H4 0~

UU)U) U)

H- 0z 00-

0 ) 04 z-4E-4 E- EI

0 - :0) 0 H4O - - 0P

z- - r404 .0 HI

0 - -- U --&4"34 H 00 4

U)-' - I 0 EU.-4 0 11 H CiWiG

W 4 - W4 1 H W< OQ- 00I W I ~ c

opC'PEJ4

P234 OH P2cI- li o)zl OWW OC'0 1-4 -- I "1%04 O ~- H-: .1 0- -:) ~ F-

H H H H H0 UC) 00 00 000 000

1 C6

Page 108: NAVAL POSTGRADUATE SCHOOL · 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

H) H- H

U)~U W)H

H- Z Hz

E- -4 E- r-

z z

H- H

+ +~E4 Z )i Q

W r-I 4 P4 2-4 H- + -

W + '.-[ W I I fx-r-

o ~ H-..-P4 "a) ++ 4

U+ H 1-1 0 0 W ~ E- Ei "II C*

w i<-,P4 w ~ 41 -1

m 0 0 0 P HH4 * L)c'q- H- f

4Z 0

0- 0. 0- -4,4 E- 0 Iz zZ Z z 1340 w 0

w 1-4 w- w- 4wI -

CV V) CV) ('*)4

107

Page 109: NAVAL POSTGRADUATE SCHOOL · 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

z

z

0 0

0 0. 1 1

'.)0 H 1 01 0 40 l w 0 I I

m~ W4 lI 0 0-4Z- E- 0 0 .0

rxY H rI .r-0 00 z

0 .0f .1 11 .1 +

6t oI o <-C~ w Q <<W~ 4W EC-) Hw

w 4 4 4 PHHQO HHQO HHQOH'-q W(

0 0 z0 900 0 w- W- F-4 W z

U) @Ei . U) (n U) U) C0 1II D

4 ~z0 w- L0 z -0~ w z

00(Y 0000 0 00 0 0 00

108

Page 110: NAVAL POSTGRADUATE SCHOOL · 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

N E-4

-C " -- E-4

- E-4

*n E-4 -HE-

H H 9 04

E-4 H6t II I xliw

W D o0 -U) H

0 -n U) '-i Cl- U, H H 0x Z'--

4c0 UE4 4Z 0 0 :C

E--4 wil E- E- %

.gEg - - - U) 1% Q 01 OHE- 1

'-4 gC4C~ ) E.I 4I Wx E4 H-0cK U) E-4 uIl C

K4W- W- - E-4C-4 E) 0 -4

N% E-4 'CI 0 w U H4

Z E10 2+ - Pe-~ -. - - Z-

-n 0n LoH, U)) C 0) =oH

z w I - E 1E4 -4 H -0) M OH Z

0 E- HC)P H -*4 H" .

Q E-4'4H I "~ "C H

-~~~ 0 - u-H H)0~ - .1 CV) qo O

U )UC)C)0Z) L L n L) UL iU U0 UL

H ~- a'c ~o-'-----H'~--O' 010HC9

Page 111: NAVAL POSTGRADUATE SCHOOL · 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

0 U

'sq 1-4>4f~4~E--4 Mul

Icz: E-4

Z4 0

0 0z N

H. WCO-4 944 04Or- - opz > ~ Z-iU94: o:

"%IIII E-0'W E-4 W oc4 wHWH U)z A EzKWOE-4 oczo0E-4E-

cj Z -4..E- 4'- ZEz44 -4ZDE-" 4

ri)~~~:)z: E-1 0E4r~ rI-~111u E-4 E-4 * '-'' -U W& (El-

r4' rU)4E-M " W)~ 0U w wq-4

14 ~ mU-)cr C40 m1w 0 :)MZ

OH I4 .% Z E- . H Et U 4:r:'-4'-5-. +g4 0Z W W4 E-4DU)Z

0U 0'C 40 w'n U) E 0 * zP4HE-4

1-W c-q-L~4II-4 .4.0~~-I4J E-4 MOM 4.3

0- Qrir 14 W0C:)4 E-L '- .g

0 rl4 + .K -41 . go-o.,' p U) r~Z J4'0~C) r >4'4c p E-40E---I:P-"-><<

C4 E-OC0 - E--- * cC4 :C) )

z- %-) W4 .P:)C4~r .4'., EIZ): -4 ME-4 - EI4

t- C ssJ} 4z)' .9c.K riri C)''' U)irI

E-4 E4N E-4 Z* 11. .,J E-4 1-1 E+-0. z 9" 1W-41-4 -41-1-

M', 4-44: ZE4Q0.~Cw0 '%-E-

r4 U)W Q)-*~?)n 0 04~a -- U0 .'

2: +f 63E4Q 94 .w'.<

U) 0.30'.,'U CE

00 ~ ~ ~ W~-Y 00 00000000000C)) 0000

"=w110n t E-

Page 112: NAVAL POSTGRADUATE SCHOOL · 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

N 4H -49

E-:: .E-4 0 V

h 0l~ 4 E-4

0 ~ -WHH4 -

OHW O

E-4 WW X OH-Ifl 0 C

HZ4zrz4 --OH EnU) >ZO -4 0H-HZ Z zW ) rZ4 . - Jfl 9 4 W24

mm 0 N~ .Q0 - W-E-V0q (A o 0E

6 2 X: -OH - - 1.- 4C 1%x CQHc-00Z W U

H WI H HO 0 E-4 E-4 W

W~I04 - -O0 W~ P

=2>1 +LAOO 0 W Q

(-.EZ-00 , '.H e4c w 0

00:314 = '-i 0: -0* H 4rl):Zz1- t-40 , 0

rzwCJ (7ul 0141 0IHCC-W 0 >40 OE- L) P-

EZ 00 )-4I-4U 1-A -4 P4-O:

=H< -C) 4 440- ) 61 <ZW

0000 0 000 U000 00000000 0

tnul Mu , 111

Page 113: NAVAL POSTGRADUATE SCHOOL · 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

0'

(4'K'

E-4 E-4 E-

4 11 O' * U

M *K K

wK N* %0 * E-4

K E-1> E-4 U '

K 'WC C) w 'K

tK M3~~ E-4 * '

'K '-H U K K *'

4I P)~*- +KK *'

' 4~i) -4 *K' ~''K O 0zE-H E-'K4~'

WK~IM 'K4 'O C4 0 K U

O E-4 ZK' - 'K K C

C4 WW WK ZK-4 'K K

'K PzC'4 'K W~' z''K " ~ ~ ' *K W'-40 'K E-4 E-4K

*K IzH0U) 'K1 . w4L1 111 0 I

Z K ~E-4 ' +. 'K :D *'K Z

'Kq >czz 'K 0 0 'K " >lqr EI

'KP z ~ zv ZK Z" "Z 0 " Z*4 00 )04,-T4 0K- 0zz4 00 4 O, 0 U O

'- ' -'~~I00 0~''K 0~) 0 ONO * -(I4 K "~'K C~4' Oo *.~K' I'

'K~O)L(U 0il~ + K u11 iK' ' '

'K Q~C~ 'K ~'K112

Page 114: NAVAL POSTGRADUATE SCHOOL · 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

4. 4. 4. -4 .

*~~ -~* E-4

z >

4. * . (.44. " -4"

W. E-4 W UE-. 4 U) 4.

-4 4. ~r.

4. - H 4q En 0

zz o U) E E-40. 4 E- E-z4

4. "I 4 U) W. '4 1-4

II -4 4. 4. UU * + 4

4.00 4.P * 4. , QH:4.C1 * -.I1 Cl E-

4.) +11 11 111 * :11:) E

4. U) rr4. L . r 4 -nz1

FIZ X Z a zz ZZ 4. 4.- U2H

0. -" 4.0q H) 0 U) - W F

* ~ 4.0 4) uU) O=D.U)U)zz0 z0HZ %DUU. UU) 00 4 H E-. 00 *

0. 4. 04) -0)u * z Q z )

Q 0~4 . .~ rt .~ H-

00 4 DJU0(LU) 00 * .i 0 H 00 4

0 04 ~UQ~U004.Z t0044. 0 E. U U 4.c~oL

4. Cl 4 Lz113

Page 115: NAVAL POSTGRADUATE SCHOOL · 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

00000000000000000000000000000

.~ . ....

* 0

** .~ to U)H:: ri ,-4

.0 .. U)til w. 0

W. E- -

* Zw 0 n

> '-4 W 0 1* I-' Z ZE- W4 OZ

* 0 UO'"0 14

gm IUiC0 MZx 0 m* ZE-4 E-HE W4

* 0 E-4 0w.. r4Uai OE*-4 .. UH

E-4 * 0 WW~rI W N N l 0

U * ',zil

E- M4 g >4111 "Wl O ~'AK wC E-Du-W CC -

.O~zjt~En 04 WOE- *00L': C400

0Zn P. 4c *U)P..WWW 9 N

0 4Wc~ H4*~c0

0 U) K gc~w Z~ a'C4 E-4* -"02 1

M E-4

Page 116: NAVAL POSTGRADUATE SCHOOL · 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

En co E c )ncntl)))UUU))) m)I toU U) roc nm( htAu) u) u) U) u) ui r u) uU)UU) U) U) U) ui)U U)U U) EEnc qc u) U) nu) U)n

E-4034

00)H.O

z I

W-4 E- HI.1.

04 M0OU)A

Z W-ZH~~ .i0 rZ)M> -

H OL) 4 *Q 0

m HE-4 1 z

Z)z1-U ,

wcz Z~4 1%

74 z ~ 0Z_ v.O O N-~~

OOOz)4Z4 U1- -n *4 to'oo)i "1 -.1--4 CiC<w~z~u) NH C ~'cVOH .z

0003:0 0

000000000 0 0 000 000

X 11t52:E-

Page 117: NAVAL POSTGRADUATE SCHOOL · 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

00000000000000000000000c0000000000000000000000000

Mu)(Ul MU) U)CO Ul) ( U) f) V)' U) U) U) U) U) U) U)U) En U) M U) Uf) U) U) M) U) Un Un En U) to U.% U) U) LO U) (n U) U) U) M)

* 0

z Q

U)w

* -4

z 00 0

1- PE-Z 44t

4 C- 0 0H

*~I * " t0 * 0 ottzjit Ff

o) *w I-Iz U):r wo

0 0 WWWW E-to m ~ U)I W It

0 * IU)WI w

0 T U) WOP Ho0ItoH w I >4 'gHIwzgzzU)

0 '-i tO U) <W E- 'CW H ItUH- X: H co >4 Z pZC-j H 0-

1 0 U) I .UWtXOOWU<ZZXM0 14 0 *It% 0WWIH

cq H 4E *4 H >4 <0

Hq Z t-4o +XO + N Z H Mr*c , < H 4N*1 Mm' : H-HHHWitWtW

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0 z~ H *c 4Z '<lc OD:.~ * 04 No 00 0>ZM-I 1- CNM40 .Hr4 HHII&411 mH4iNI < m

H041H0 OI C=z'l1)4C'2:"mm D En W)I

H0HeI0Iz-E- 1 111t

0 00 00 0 00 0 0He- cv N c)14L o rco N a 0H- H-IH H H H H-4 H- ('4

116

Page 118: NAVAL POSTGRADUATE SCHOOL · 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

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117

Page 119: NAVAL POSTGRADUATE SCHOOL · 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

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Page 120: NAVAL POSTGRADUATE SCHOOL · 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

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119

Page 121: NAVAL POSTGRADUATE SCHOOL · 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

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E130 "- 0 0 i-4H)0 E-H W0 M 0 U)EP4Z

Z 0 - 0w0W 0 W- * zrI4ZU)HO1 l H- HM)HE- Q ri E4 -0

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Wz C4 P4 U) 0 U)ZH E-4 w m wH 0 .0U)U)W W

w wwoo OPW HOH'-4 HW 0"- "Hr- H ZC40QWr C, % *4HH> z w0 -qz'-o- '-'Ej U) "W~z U) 400H

H 0::):H p r1C uw 0 w 4~ ::CzWY r-4 OZUE13 4pHw~ >4CJ0 )n- WZO) 1 >4 H Q"Ww)U) f4 z: -~ '4'~H4 HOWz >4-4U ~0>4 0P4 H 9 : D 4a

E-1 POX U)N w-'44 DH: 4H' p U OWHCC, 1

0QWM~~~z'~ ..l [z40 W -4Zf >% HZ OWz Wz H U-4'4- -Z -

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MMUHO.< UQ ' >4>44 - > < >4H Cub >40 Z t- ,

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0

120

Page 122: NAVAL POSTGRADUATE SCHOOL · 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

0

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H- ~ P En4 f4H-0 x -4-0 H-4

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0,14 Zr-ig: E-

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H- ~ ~ ~ %W H E-'4 ) 42 1 H 01x

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C12UtflU C)w E-4 4-4Cz H 0I~H H q P4 00 4H Q u

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0 zo Q 114H 0- ~- 1-4- H0i4 00

ZZZZ ~ ~ ZZ0 _-f: % Q .H -}~~ 0-V -

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zzzz~~ p E4 W410"H H 1 -HPQ

w) wC C4CCC C))C z))C C), 0- "C)-0 p W -- )

0 no H zp 0 -4(., E,4 :,121 Z-

Page 123: NAVAL POSTGRADUATE SCHOOL · 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

-4 4 'q 0-1Pz: 0 0~'

U)Ux zW 0

*zZ 0- U) 0i

P4N- -CO m) 0 11El ri-i H

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E- W'- Z '- H O

(n >4L&~ 3: H >0

Z z 0wH ZPO00 iI P

U) H Z.'- E-zr- Z1

z. 004 H0U

EH1o 01%1 Q 2- 4 W

C4 WcU)x 11 W U 0YZ i m- 04 w

0 U) Z

Z 'OO> H)-r DQWZ

x O - WH H '-4 :H

0- H(IH m a C)'ar7 U) Z ) H- 0<-O~z '- '. ZH 0 H z ~ :D4

Z0 MZZ H 0 U OH -

H U)~r~ 0= -0 0W C4 0 -C) U) qd x4 w 4 U) xW Z r+ M -1O) H > -i H- " -X ri40 HZ- f) 2

C) Z:~)<0 O %W E1 u C) 0 '4-%4 r-I L)0 -E-1 WZ4 ZC - O- W - 44 61 - E4 -g .- -4ZJ - H

SH-XX- HHZH Hs -I'sq r-I- 0 <HZQ HHM: P121 0 04

0C m -~ 0) 000 0~ -rH O

0) rrl C14 0-HZ III-U) U)i- ~X HH H r-4 H~ - <H r H~ -

~ ~ *-HQCOO~0~z ~~Oz)Z4 OC-1~Z~O122r ~

Page 124: NAVAL POSTGRADUATE SCHOOL · 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

0 0

0 0

HEE-4

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z -4

$- -4 E- *-

w~ 1-4

E-4-4 wZN-

z E- '-4

z 1- E--4

0 .+

H-

C4 . -4 o

CtI E-4 --40

~-4 ~ E-4 &-. 0- <- 4~ 00Il4H1' 1 -" 4 ZO 11 <I <~ 4 4 - OH 00-+ m'

44 o0--ZwL '-E-HH-Z oil I~uj IWWW OZr001-4 Dt-*-41 0 zZ -- '-4 4 '.., - UO OOH -

WH C4 c4 -4- W" -4 H QOO c0 1-4-I11 <

-HZM ouooQz -4u)4 fll %..- -- 0- wm o

w0z0z 0-4"-4z0 WOOZ 0 wz wz wz m HOU"z -Q-40Wi C CzOWQWU J"44CW o-)o P- - -

0%D 0

00 r-4 00 c(00 00 U0000 0

123

Page 125: NAVAL POSTGRADUATE SCHOOL · 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

* 00 E-4

H~9 H E-4 ::Z zE-4 E-4 1-4 1-4 E-0

N U) 0Z4 0

U)P H O E-41-1o N C U) 14 H -

0 '-' 0~

>4 E -4 0 m-n

ul OH- U) 'i'-I U2 Z- z Z z- X

11 0 H- 0 "-0 W0EA z- U)H- WZEnjE-

HI- 0 *i ..r 0 9nHO-

Ex2 X PWz -W " : 4Z<ZQCd 0 Z- N %-I L- E-4 0 4

zrz2 H I-x 04N 0 -4X1-4HC" - H HHA -'- OUU2Fz242

H E-4 0 CA* H EA 00 0 44.

9-In Q i E~-4 C)E4wz. 4 E4

0 H x E 3: w E- E-.u- 0- 01) H -CE40 - ZH OZ E-4

040C- - aco (: Lf-~ 04 r -- 4(:Q2)

w -.Xce' -s-HJ4x X- - - HUX 0Ci20-4Ci2t: -4 .- 144 11 --1 w H- 11 'H-i2 WCL.H:Qa HO

Z C' 3 U3. (X, 1-0- "00-:: -D O- I 1-4CO'D 4 0 H- I H-1-4 O -i ki*4CI (-I4 111-4Z '-HO4 -fH -.- Z Ci20CZ W: EllEA ZE44' 1:44" Wi HWu<1-4 CuwcH W52 .-0 HrqW~4" XQ>rntoZ 4HA 2WE-4 H; E-1 1-4H HE - nE- - -H4 -A~ Ha -iH E- 10 I~4U1 ~ I4 -4%Z "c~-, I- I "94~zO WW4O001%Oz 1%0 OC400 C40H 0 O OC00 NM

"3:ww:33WW OQwu 3CzwCz xwz z 3w

0 U) to O OH- LA r'- OHl

000 0 00- 00 00 0 000000000 0

124

Page 126: NAVAL POSTGRADUATE SCHOOL · 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

zz

z C

zl~

0 E- 44 N

W zo lo- rtr-

ZO - E4- E-

0 -X 00"~

H- 1-4v- 04 k i-4

00 Cxi >0 Cl s N- -00:i + + 004H -4 H >0

:4i N HH. E-1 E- 1 r-i -11 XHHzo- H. N :

..4.i+ w1w- (1W - E~i .-

in i ~ C 0-0~ ~~ -- '--4~--

1Nc "1 'OZ-iCJ- 'X r-f-IZ.--44 U) -4 0 W E- 0 "> >Z"M- -4 H NH -4H1 H~

rp " 11 0~ 1H1 "Z63x' r0 0k0 Z0~ H"H4<"Z- I -0OW -1 " , 1 1U)1 1t 1 1 11 O P :- ~Cxi f4< rCiz t CI 1C

W,.0E MA OH 0 - , r-i(' P N It

QQ-w~ow<2m~ z0c H 0-3-w 0 0 OI40 0r 00

125

Page 127: NAVAL POSTGRADUATE SCHOOL · 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

=4' P4' HE-' -1 0

.g'4' HZ> P4 z>4'4 stI4 0 -A

H~ 63Z Q 0

4'4' -ZU) 0

4' ~~, Z' Z"- 049 O~ z

04,- > +4' 0 OPPH MW4> -4'4 w z 044'-4) En

4'4' E-OQW"U) 0 Q

P4P 0' -4 H 04' 4'D u UC4 l 0 4 IN,

.3'~ ~ 4' .. 0~0rz

P4 2: E--4

Qrz4 Z' 4 -Tv

4'Z4' 0 0ci- um '"

E'-4 )-= M Q-C

U)~4 0LZ>Z E-4 WO r- E-

rA'U1)4' Z<WO 0H -4' 4' 0 U)E-iQ 0 ">M ZI-44~~i O-4~ ~4' 4'o E- x1> W< -,(-IH

4'~ 0"0 WW ZO H0H'iP - 0~~~4', :D W) O= O OZ 00 ~ (C0* % 0 Z'-l~4 zi 4

4' 044' 3z -

UUUUUUCUU UUUUUUUU 004 >0 > 0-

::)* -1 ,-0> -.26

Page 128: NAVAL POSTGRADUATE SCHOOL · 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

rz

E-4ZE-4

+~~ ~ E-4.-H> WH

I--# Ma

04W4

Z":: z >4

O E-1 I m ' -4N

'-0 z )>- DZ44

IN01-4 *-. W

EOW W

no 0 4 H--OH- > A&

'-4 zH W 14rH <

H> w w z PD'-z H- Z Z c,4 * >"- z

H4 1 -x 00C1-

E4'- :O-i .o Qs <W64 0E-10 00 O"C IY H Ho ZHCQ

+-4 Oz' N 40 mS~ m-- Z' H -'--Ct

HO" 0 W-+- N~s x- OH-4 I

E--4 + Z0 - & ..*411T ..*4- H O)-4Zrq ..qH O~w-Q..-&-. " -41 'A44-4 Hw D -1-4 I f4Cz1 n~ - Czl 0CzIt- z -<E4 H4

-1 P11 LoE- IIZ >"'-E- "W-l4iH W4"W4 Hw< WHCqFQC44 ZO OOZ En" -40 '-i~ '-0= '-'o OW400

"-)W~z eO*:40"- mzCZo wo~oozo 1:4 ozO0 Qz< ~0

0 0 aoO0 H- ON'O U) ONOUo U) co m-H HHC.

00 00000000 U N0)~ 0 00 0O

127

Page 129: NAVAL POSTGRADUATE SCHOOL · 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

E-4

** * * * 0 z

OCQ -0 OW * EE4

04oo * * * -

OCD* o4 WE .#5

04 =4"40 w* * *' E4E4flz zi 3:- 0 0

ali 0 m*n~ E- >4Z-*94 44.4 04Ea04: U)Ci)"W) 0* * * 44 Z9 4Zc C4U*W W .- E-4 -1 E4.4% m 0

44E- 44Z9t 44 E-M I 0~C): ~-C)1 i44. ~ * 4Z W.-E-4c:= ZD M Z a-

* z4 0H3 E-OU >~E4 E-4 00 -14444 4444 E-1E- >-~-:DM0 -

444 4 tE-40WC..) E) CQZH0

* 4444Cz (Cz)QE-4w4El 0 W44444 4444E-0 ruiPzm 0 %Q 0

gcoe E44W4E44 " E4Ci M ~ - Qi44(24 * *E I-4 U)W:i) 4 zW WOw C w

* Z* 00 W OW~ 00000H'-hA-4C)4 * 04.. *"*1 E4mo~ - C4~ m) PC

gEZgC*CzAPO< E2C)n W Czi>i. CiCidq40CW

Cii fxw :RC4 0-044044r)( Iwo r4>04o-It-

w44 44:.:c:i ~~- W E 6 <4<4> > P x44124 4~4Z U10~ ZCDP-4 m U

K4 41414 0:0ZP.~z 14 0- HZHX4 04

rq*i4 44 "W- "4 wi c~ zC

4444 -E-4 >'-C AZA O 4 wr44

4444* 4444E-<44

128

Page 130: NAVAL POSTGRADUATE SCHOOL · 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

OW)

+ r0

rzj -4

rzaH

ri E-4C4>0- -4 E-1

E- z:DH rzzw -

0 + ME-e -H-Z

0Z E-4HH

0 z

0-

zzrz)U2 0 0

E- w '-V U) H<-

or~o EH Z ZMEA

E--4

HOEi .H>4 Ef) 1-

>-4- *+- 0 '-4EU1H H >4-i H(UIX

"1-4 Z- -0MWH H .1 1 fli -

"Ho I HE-4 r4 ,>4 ofC -~ ilHZZ 11 1"1I M4 :z rZI- :: Z IH f

:Zcncn HrP-z Z- 11 1 w 11UH"

fIIZO Ix -.X Q Z 4 H 01-, -1 H PL4 W 04Z rirOl4H -wrzo0 0 0 z 0 0~Z0l O-Z

0

129

Page 131: NAVAL POSTGRADUATE SCHOOL · 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

-4, 4N 4141

0 E-*4 * *1 4

0 414

NO -r 1.

0 0 > 4 =1.

w E-i N- 914

O~E- 0-0 1 4rzz

E-X -4 .

X1 E-1 W1 4

NY "Z E-1O "0gr4.4 4 E -qH *44

o- ~- - r-' 11 XZ: - -- ON U24 F4 '-q W1

04 M W44- " rt H 1441414EA ..E-s *Fl 04-4 41.

0 * E- E1- H~ E-4 E-

.Z ~ I~~ Z4 ImN Z14

-- * ., w

00Z

H ~rX t- I-X--- X iiX 4130~

Page 132: NAVAL POSTGRADUATE SCHOOL · 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

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.

131

Page 133: NAVAL POSTGRADUATE SCHOOL · 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

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

132