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NBS TECHNICAL NOTE 902

•'ew of *

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

valuation, Revision an

Application of the NBS

tylus/Computer System for the

easurement of Surface Roughness

10

5753

b.902

376

.2

-132
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

and commerce. The Institute consists of the Office of Measurement Services, the Office of Radiation Measurement and the following Center and divisions:

Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center

for Radiation Research: Nuclear Sciences; Applied Radiation — Laboratory Astrophysics 2 — Cryogenics 2 — Electromagnetics 2 — Time and Frequency 2

. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized materials needed by industry, commerce, educational institutions, and Government; provides

advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard

Reference Materials, the .Office of Air and Water Measurement, and the following divisions:

Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor

Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote

the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations leading to the development of

technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Insti- tute consists of the following divisions and Centers: Standards Application and Analysis — Electronic Technology — Center for Consumer

Product Technology: Product Systems Analysis; Product Engineering — Center for Building

Technology: Structures, Materials, and Life Safety; Building Environment; Technical Evaluation and Application — Center for Fire Research: Fire Science; Fire Safety Engineering.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing

equipment, techniques, and computer languages. The Institute consists of the following

divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units: Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations — Office of International Standards.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

3 Located at Boulder, Colorado 80302.
BNAL BU& u: igJARQg

APR •; 4 1976 Evaluation, Revision and Application

of the MBS Stylus/Computer System

for the Measurement of Surface Roughness

7

Zj>M _0 foU V^ SO •

E. Clayton Teague

Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

J*? °V*

o-s

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued April 1976
National Bureau of Standards Technical Note 902

Nat. Bur. Stand. (U.S.), Tech. Note 902, 151 pages (Apr. 1976) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 03.46:902). Price $2.70 (Add 25 percent additional for other than U.S. mailing).
TABLE OF CONTENTS

INTRODUCTION

Page

1.0 System Operation 3

1.1 Data Acquisition 5

1.2 Data Analysis of Step Heights 7

1.3 Data Analysis of Surface Roughness 7

1.4 Calibration of Step Height and AA values 7

1.5 Example of System Operation 9

2.0 Analytical and Empirical Studies of the New

Calibration Procedure's Accuracy 12

2.1 The Effect of Geometry on Step Height Measurement 17

2.2 The Effect of Surface Texture on Step Height

Measurement 22

2.3 Effects of Stylus-Transducer and Interface

Hardware Non-Linearities on Arithmetic Average

and Step-Height Measurements 25

2.4 Uncertainties in Arithmetic Average Measurements

Due to Data Acquisition Processes 27

2.5 System Short Term Stability 28

2.6 Summary of the System Uncertainty Budget for

Roughness Calibrations 28

2.7 Comparison of the AA Values Measured Relative to

the NBS-1 Master Artifact with Those Measured

with the Step-Calibrated Minicomputer System 29

3.0 Further Utilization and Application of the

Computerized Roughness Measurement System 35

3.1 Mean Slope and Average Wavelength 36

3.2 Amplitude Density Function, Skewness and Kurtosis 38

111
3.3 Autocorrelation Function, Mean Square Value and

Power Spectral Density Function

3.4 Applications

REFERENCES

Appendix A System Electronics and Interfacing

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

Appendix G

Computer Software

Time Study of the Four Most Common Measurement Procedures

The Linear Least Squares Curve Fit

Step Location

Step Height Calculation

The AA Calibration and American National

Standard B46.1

Appendix H: Statistical Evaluation of the Long Term

Stability of Step Calibrated Roughness

Measurements

Appendix I: Stylus Tip Radius Measurements

Appendix J: Noise Effect in AA Measurement

Appendix K: Effects of Surface Finish on Step Height

Measurements from Stylus Profiles

Appendix L: Measurements of the Stylus-Transducer and

Interface Hardware Non-linearities

39

45

51

53

62

63

66

67

72

80

82

99

101

104

110

Appendix M: The Effect of Finite Record Length on Measured

Root-Mean-Square and Arithmetic Average Values 115

Appendix N: Some Considerations of the System's Transient

Response Characteristics 120

Appendix 0: Measurements of the Record Length and

Traversing Speed 124

Appendix P: Sample Test Report Containing Calculations

of Statistical Parameters and Functions 126

Appendix Q: Operator's Instructions 135

Appendix R: Checklist for Proper Electrical Operation 142

iv
LIST OF FIGURES

PAGE

1. Flowchart of Overall Operation of the Computerized

System for Surface Roughness Measurement 4

2. Function Diagram of Stylus Instrument/Minicomputer

System 6

3. Step Height Calculation 8

4. Teletype Output of Step/Surface Roughness Calibration 10

5. Plot of Roughness Calibration Accuracy as a Function

of Specimen Roughness 16

6. Profile of a 0.501-von Step 17

7. The Effect of Geometry Errors on the Measurement

of Step Heights 18

8. Representative Interferogram of a Step Specimen 19

9. Schematic of Interferometric Step Height Measurement 21

10 and 11. Profiles of Specimens Used for Comparing Old

and New Calibration Procedures 33-3^

12. Definition of the Amplitude Density Function 40

13. Definition of Profile Skewness 4l

14. Definition of Profile Kurtosis 42

15. Evaluating the 20-yin Area of Precision Roughness

Specimens Conforming to American National Standard

B46.1 47

16. Evaluating the 125-yin Area of Precision Roughness

Specimens Conforming the American National Standard

B46.1 48

17. Eight Characterizations of a Polished Steel Substrate,

a Bright Nickel Electroplated Surface and a Cloudy

Nickel Electroplated Surface ^9

18. Block Diagram of Waveform Receiver 57

19. Schematic of Interface Electronics 58

20. Schematic of Interrupt Circuitry 59
21. Filter-Amplifier Gain-Bandpass Characteristics 60

22. Observed Amplifier-Filter Phase Delay 6l

23. Step Locator Chart 68

24. Proper Location of Step Relative to Event Marker 69

25. Selection of Al and A2 Locations 70

26. Proper and Improper Selection of Al and A2 71

27. Schematic Strip Chart Records of Highly Sloped

Profiles 73

28. Co-Ordinate Systems for Step Height Calculation 75

29. Profiles of 12.8-ym Step Used to Obtain Statistical

Data on Possible Variation of Calculated Step

Height with Changes in Slope 76

30. Profile of Stylus Tip 100

31. Noise Effect Measurement 102

32. Results of Noise Measurement 103

33. Profiles of Steps Used for Taking Statistics on Step

Height Measurements 105

34. Step Response of System 121-122

35. Internal Lead Connections 136-138

vx
LIST OP TABLES

TABLE PAGE

1. Uncertainty Budget for Measurements of 0.5-ym (20-yin)

Precision Roughness Specimens 30

2. Uncertainty Budget for Measurements of the 3.17-ym

(125- yin) Precision Roughness Specimens 31

3. Comparison of AA Values Obtained from Roughness Meter

Corrected to NBS-1 with Values Obtained with the

Step-Calibrated Stylus/Computer System 32

4. Components List for Filter-Amplifier 55

5. Characteristic Parameters of the Talysurf 4/Interdata

3 System 56

6. Statistical Data from Step Height versus Slope

Measurement 77-79

7. Measurements on Height of 12.8-ym Step with Surface

Finish of 13-nm AA 106-107

8. Measurements on Height of 503-nm Step with Surface

Finish of 8-nm AA 108-109

9. Voltage Output versus Displacement for Stylus

Instrument Using the X1000 Magnification Scale 113

10. Voltage Output versus Displacement for Stylus

Instrument Using the X50,000 Magnification Scale 114

VI

1
EVALUATION, REVISION AND APPLICATION OF THE NBS STYLUS/COMPUTER

SYSTEM FOR THE MEASUREMENT OF SURFACE ROUGHNESS

E. Clayton Teague

This report describes in detail the hardware and software used at

NBS to implement on a stylus instrument/minicomputer system the process

of calibrating the system with an interferometrically measured step and

the calculation of important characterizations of surface profiles. The

characterizations of a profile which may be calculated include the

arithmetic average value, the mean square value, the amplitude density

function, the autocorrelation function and the average wavelength. The

report also includes a statistical evaluation, using empirical and

analytical techniques, of the calibration procedure's long term stability.

Key words: Amplitude density function; arithmetic average; autocorrelation function; average wavelength; kurtosis; minicomputer software;

random error; skewness; surface microtopography; surface roughness;

surface texture; systematic error.

VI 11
INTRODUCTION

In February 197^, NBS announced a new procedure for calibrating

precision roughness specimens which utilized a computerized system for

measuring surface roughness . The purpose of developing a computerized

system was to increase the accuracy and reliability of the arithmetic

average, AA, measurements made by NBS for industrial consumers and to

expand NBS's capability for characterizing surface roughness with

parameters and functions other than AA. The computerized system has

improved the accuracy of AA measurement, relative to American National

Standard B46.1-1962 and the International Length Standard, from 7% to

5$ in the 0.25 ym AA range and from 5% to 2% in the 2.5 ym AA range.

Reliability of the AA value assigned to a particular specimen has been

enhanced because the system may be conveniently calibrated at each use

with an interferometrically measured step and because the computer's

operational speed enables one to perform roughness measurements at a

large number of positions on the specimen's surface. With the

computerized system, NBS now has the capability to calculate the

amplitude density function, the autocorrelation function, the average

wavelength, the average slope and the RMS value for the profile of any

arbitrary roughness specimen.

Precision roughness measurements at NBS and throughout the world

in their earliest form followed the same conceptual procedure used in

NBS's computerized system. An interferometrically measured step was used to calibrate the stylus instrument by relating the height

produced in the graphical output to the measured step height. The

resulting calibration constant for the graphical output was then

coupled with manual planimetry of surface profiles to calculate AA

values of reasonably high accuracy. The technique, before the

availat ' lity of computers , was however very time consuming and limited

in accuracy by visual estimations of step heights, by the precision of

the planimeter and by operator skill.

Due to these difficulties, the procedure evolved to one in which a roughness specimen under test was compared to a master artifact with

the use of AA values produced by an analog integrating meter. The

master artifact employed by NBS for calibrations was measured at NBS

using the step-calibrated planimetry and corrobrated in a blind

round robin with other laboratories where similar measurements were

made. Even though this one step transfer decreased the time

required for calibrating a given specimen, its overall systematic

error was greater than the earlier procedure. Increased

systematic error arises from a 3% of full scale error for the

integrating meter and from the error produced by subjective

interpolation between scale divisions. In addition, the method is

limited because individual calibrations are referred to the

historical values of roughness assigned to the master artifact which

is subject to loss, damage and deterioration with ordinary use. With

the availability of dedicated minicomputers, the earlier and more

basic calibration procedure may be implemented to bypass the
comparison method's errors and to reduce the "planimetry" calculations

to fractions of a second.

This report describes in detail the hardware and software used at

MBS to implement on a minicomputer the calculation of step-calibrated

AA values and the other parameters mentioned earlier. The report also

describes an evaluation of the new procedure and its correlation with

the one previously used. Initial development of the stylus/computer

system was performed by Dennis A. Swyt and is described in his report

on the system, NBSIR 73-106. To make the present report selfcontained, relevant parts of his report, hereafter referred to as

DASR, have been included, primarily as appendices.

Following development, the stylus/computer system was subjected to a thorough evaluation for iH months. The satisfactory results which

were obtained after completing the evaluation and its indicated

revisions of the system served as a basis for announcing the new

procedure. The evaluation consisted of:

1) checking the dynamic and static operation of the system

software

,

2) statistically evaluating the calibration procedure's long

(months) and short (hours) term stability,

3) studying both empirically and analytically various

components of the procedure's systematic error,

4) measuring the system hardware's operating characteristics,

and

5) examining the correlation between roughness values obtained

from the master-artifact-comparison procedure and the

values obtained with the step-calibrated stylus/computer

system.

The final system described in this report is the product of many

iterations of an evaluate-revise process. As an explanation to those

readers who received the announcement of an availability date for

this report of April 197^, two delays in the writing of the summary

report were due to major revisions of the system software and to

subsequent evaluations which were then necessary. Other than

the addition of software for the new parameters, the

principal change from the system described in DASR is that made in the

step height calculations. Here it was found that the metric (in a geometrical context) transformation from computer coordinates to

spatial co-ordinates was incomplete. When this conversion is

carried out completely, one also finds that the slope factors given

in DASR are not needed at the accuracies of the present system. The

other revisions of system hardware and software will not be discussed

any further since they are dominantly software details. Appropriate

changes to the extracted parts of DASR have been made.
1.0 SYSTEM OPERATION

The NBS computerized system for measuring' surface roughness

consists of a Talysurf 4 stylus instrument* used to generate a profile

of the test specimen's surface, an Interdata 3 minicomputer* with data

acquisition circuitry for processing the profile in digital form, a

digital to analog converter with the associated electronics needed to

drive a strip chart recorder used for displaying the results of

computer analysis and a Teletype model 33* for system control and

output. Appendix A gives a complete description of the electronics

necessary for interfacing the system components.

Operations which may be performed with the system are represented

by the flowchart in figure 1. To bring the system hardware (appendix

A) to this operational state the set of instructions described in

appendix B must be loaded into the minicomputer. Convenient use of

the minicomputer is then made possible by the properties of the

instructional subset named, Monitor. The system is put in a "ready"

state by addressing the minicomputer to Monitor's command interpreter

located at the hexadecimal address, 3C80. In this state the system

accepts commands from the teletype (TTY) and may be directed to perform any of the operations indicated in the flowchart by entering the

appropriate letter at the teletype. If additional inputs, from either

the stylus instrument or the operator, are required the minicomputer

is programmed to type a command or question at the teletype.

To avoid unnecessary usage of minicomputer memory, only the paths

through the calibration, roughness and step height measurement

procedures are at a near conversational level (appendix Q) . Correct

execution of the other operations is more dependent on the operator's

understanding of the system. Two of the most important examples of

this operator dependence are in measuring average wavelength and in

calculating the amplitude density function (ADP) and the

autocorrelation function (ACF) . In both of these cases, the sequence

of operations executed, with respect to figure 1, must follow a general top to bottom and left to right pattern. The top to bottom

sequence is required since the five operations in the lower half of

figure 1 assume that a calibration constant has been measured and that

at least one roughness profile has been obtained. A left to right

order is necessary since the average wavelength and slope calculation

destroys the profile data required for both the amplitude density and

*Certain commerical equipment, instruments, or materials are idendified

in this paper in order to adequately specify the experimental

procedure. In no case does such identification imply recommendation

or endorsement by the National Bureau of Standards, nor does it imply

that the material or equipment identified is necessarily the best

available for the purpose.
SYSTEM AT MONITOR

3C80

ENTER "H"

AT TTY

ENTER "0"

AT TTY

INPUT

CALIBRATING

STEPS

MEASURE

DESIRED

NUMBER OF

STEP HEIGHTS

(RETURN ~^\

TO MONITOR^/

MEASURE ROUGH4

NESS AT DESIRED NUMBER

OF POSITIONS

S RETURN—

\

V^TO MON I TOR J

(RETURN "N

TO MONITOR J

CALCULATE

ADF OF

LAST PROFILE

STORED

CALCULATE

ACF OF

LAST PROF I LE STORED

PLOT

ADF ON

STR I PCHART

RECORDER

PLOT

LAST

PROFILE

STORED

CALCULATE

AVERAGE

WAVELENGTH

AND SLOPE

CALCULATE

MEAN AND STD.

DEV. OF

STORED AA'S

C

RETURN "N

TO MON I TOR J

PRINT

MEAN SQ.

VALUE

AND PLOT

ACF

Figure I. Flowchart of Overall operation of the Computerized

System for Surface Roughness Measurement.
autocorrelation function calculations. The operator dependence of

this procedure is balanced by its flexibility. Within the sequence

just outlined the operator has, following the input of a calibrating

step, as many as five correct operations which may be executed. The

most common operational sequences are:

I *+S+*+H->*,

II **S*-*+Q**+M»-*

,

III *+S-**-»<^*+J-**+V+*->Y-**

,

and

IV k+s+*->J+*->Q-v*+J+*,

where a letter symbolizes the keyboard entry and its associated

operations and, *, symbolizes a return to Moitor's command

interpreter. In the first three sequences, the paths from S+H and S->Q

may utilize the conversational routes as an alternative to bypass

returns to monitor. The fourth sequence is used to obtain a stepcalibrated display of the amplitude density function. A rough

timetable for these four operational sequences is given in appendix C.

1.1 DATA ACQUISITION*

The means by which topographical profiles are converted to surface

roughness data is represented by the functional schematic in figure 2.

When the stylus is traversed across a surface, the profile is

indicated on strip chart recorder A; the signal driving the recorder,

amplified and filtered, appears at the analog-to-digital converter

(A/D)

.

At a particular point in the stroke, the stylus arm activates a relay, which in turn generates two pulses. The first pulse, one second long, appears at the common input of the recorder and interface

amplifier. (The source of the event pulse is in the information

channel only for the duration of the pulse). Coincident with the

leading edge of the event pulse is a fifty-microsecond interrupt pulse

to the A/D converter. On receipt of the interrupt at the A/D, a programmed one-second delay is executed, and the analog signal at the

filter output is converted to binary data points and stored

sequentially in computer memory. (Thus the event mark, recorded on

the strip chart, does not appear in the digital data.)

^Sections 1.1, 1.2, 1.3, and 1.4, are slightly revised parts of DASR.
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1.2 DATA ANALYSIS OF STEP HEIGHTS

In order to efficiently handle the discontinuity in the trace

which comprises the step height and to eliminate restrictions on the

alignment of the step artifact, a mathematical model of the step data

in the computer memory has been devised. Figure 3a represents a general step profile in which the traces on opposite sides of the

discontinuity are wavy and parallel neither to each other nor to the

sides of the strip chart.

Linear least-sequares curves are fitted to the segments designated

by LI and L2 in figure 3b (appendix D); the segments are located at

the operator's discretion relative to the event mark (appendix E), are

of equal lengths and constitute one-half of the total trace. From the

slopes and intercepts of the two computed lines, SI and S2 in figure

3C, are calculated the distances Hi, i = 1, 2...., 8 (appendix F);

these heights are simply taken as the ordinate differences at equally

spaced intervals along the horizontal distance between the ends of the

line segments selected by the operator. HI, H3, H5, and H8 are

printed at the teletype to inform the operator of the two line

segment's relative slopes. The distance HM in figure 3d is the

arithmetic mean (or H4) of the distances between the two calculated

lines over the region separating the line segments.

1.3 DATA ANALYSIS OF SURFACE ROUGHNESSES

The AA roughness for a surface profile is computed from the data

in memory in a manner analogous to the operation of an integrating

meter. Since the profile signal has wavelength cut-off restrictions

imposed by analog filtering (appendix A) , no digital filtering or signal conditioning is involved. The value corresponding to the

center line of the profile is the arithmetic average of all points in

the record. The AA value is the arithmetic average of the magnitudes

of the deviations of each from the mean.

1.4 CALIBPATION OF STEP HEIGHTS AND AA VALUES

when the binary value, HM, of the calibration step profile has

been computed and the decimal value, HO, of the interferometrically

measured calibration step entered at the Teletype, a conversion constant, KCAL, is computed:

KCAL = HO (decimal)/HM (binary)

.
a)

L2

b)

LI

c) H4 H8

d) HM

Figure 3: STEP HEIGHT CALCULATION

a) Step Prof i le I nput

b) Least Squares Line Segments

c) Calculated Distances at Xm Between the two Lines

d) Mean Distance of Step Height

8
The calibrated values of unknown steps and surface roughnesses are then of the form:

and

HM (decimal) = HM (binary) x KCAL

AA (decimal) = AA (binary) x KCAL

1.5 EXAMPLE OF SYSTEM OPERATION

An example of the procedure for step-calibrating the system

followed by the measurement of an unknown step height and a surface

roughness may be illustrative. A sample teletype output is given in

figure 4. Further details appear in appendix Q. This example

assumes the operator follows a conversational route through the

procedure

After an appropriate magnification has been selected, the

corresponding calibration step is aligned on the Talysurf and program

execution begun by entering an S at the teletype. At the proper point

in the program the operator is informed at the teletype to begin the

stylus stroke in order to enter the calibration step profile. The

signal from the stylus instrument is amplified by a factor of 10,

filtered (appendix A) , converted to 512 twelve-bit data points over a

25 second record length and stored in the computer memory. The

operator is then instructed to enter HO, the interfere-metrically

measured value of the calibration step height, at the teletype;

this four-digit decimal number is converted to binary and stored.

The operator now enters the information at the teletype which locates

the step discontinuity in the stored data; the information is

derived from- the chart trace and the step locator chart (appendix E)

The step heights are then computed, converted to decimal values

and printed. The calibration constant is also calculated with the

hexadecimal value being printed. To obtain a more complete sampling

of the data output from the step measurement process, the procedure

described in these paragraphs is repeated five times to produce an

average calibration constant AVG. KCAL for subsequent use.

The operator now aligns the specimen to be measured on the stylus

instrument. On being queried at the teletype, the operator indicates

the type of artifact by entering command characters at the teletype

keyboard. If it is a step, an H is entered and the procedures for

entering a step profile and its related information repeated. If

a roughness is to be measured, the proper filter cutoff and stoke

speeds are selected, an R entered at the teletype and the stroke

begun. In roughness measurement, the signal from the stylus

instrument is amplified by a factor of 10, filtered, converted to 4096

twelve-bit data points over a 2.5 second record length and stored in
>s STEP/SURFACE ROUGHNESS CALIBRATION

ENTER DATA/ IN ENTER HO 5105 ENTER UNITS -7 E

HI H3 H5 H7 HM

00005123 00005113 00005102 00005091 0Q0Q5104

ENTER DATA/ IN ENTER HO 5105 ENTER UNITS -7 E

HI H3 H5 H7 HM

00005102 00005103 00005105 00005107 00005105

ENTER DATA/ IN ENTER HO 5105 ENTER UNITS -7 E

HI H3 H5 H7 HI4! 00005096 00005101 00005106 00005110 00005104

ENTER DATA/ IN ENTER HO 5105 ENTER UNITS -7 E

HI H3 H5 H7 HM

00005121 00005111 00005102 00005093 00005105

ENTER DATA/ IN ENTER HO 5105 ENTER UNITS -7 E

HI H3 H5 H7 HM

00005100 00005103 00005106 00005108 00005105

AVG KCAL =00003170

MORE? YH OR R? H

ENTER DATA/ IN ENTER HO ENTER UNITS -7 ENTER

HI H3 H5 H7 HM

00005091 00005096 00005101 00005105 00005099

MORE? YH OR R? H

ENTER DATA/ IN ENTER HO ENTER UNITS -7 ENTER

HI H3 H5 H7 HM

00000000 00000000 00000000 00000000 00000000

MORE? YH OR R? H

ENTER DATA/ IN ENTER HO ENTER UNITS -7 ENTER

HI H3 H5 H7 HM

00005077 00005076 00005075 00005074 00005075

R

00000920 -8 MM 00000914 -8

00001009 -8 MM 00000854 -8

00000878 -8 MM 00000953 -8

00000846 -8 MM 00000835 -8

00000827 -8 MM 00000817 -8

00000887 -8 MM 00000806 -8

NTER Al

UNITS

-7 MM

NTER Al

UNITS

-7 MM

NTER Al

UNITS

-7 MM

NTER Al

UNITS

-7 MM

NTER Al

UNITS

-7 MM

11C2 ENTER

KCAL =

11C2 ENTER

KCAL =

11C2 ENTER

KCAL = UC2 ENTER

KCAL=

11C2 ENTER

KCAL =

A2 124A

00003 1B2

A2 124A

00003 18A

A2 124A

00003 IOC

A2 124A

00003158

A2 124A

00003190

MORE? YH OR R?

AA UNITS

00000935 -8 MM

00000873 -8 MM

00000856 -8 MM

00000864 -8 MM

00000885 -8 MM

00000912 -8 MM

00006408 -8 MM

00006391 -8 MM

Al 11C2

UNITS

-7 MM

Al 11C2

UNITS

-7 MM

Al 11C2

UNITS

-7 MM

MM

MM

MM

MM

MM

MM

ENTER A2 124A

ENTER A2 124A

ENTER A2 124A

MORE?

MORE?

MORE?

MORE?

MORE?

MORE?

00006279 -8 MM

00006491 -8 MM

00006294 -8 MM

00006495 -8 MM

MORE? Y

MORE? N

FINI

Figure 4: SAMPLE TELETYPE OUTPUT

10
the computer memory. The computer calculates either a step height or roughness AA, prints the result, and asks if more measurements are

to be made from the same calibration. Roughness values are obtained

in groups of three and may be stored for future statistical

analysis by entering a space at the teletype; the roughness value is

not stored if a K is entered. If more measurements are to

be made from the same calibration, a Y is entered at the teletype

and the measurement of the unknown as described in this

section repeated. If no more measurements or those at a different

magnification are to be made, an N is entered to return the system to

Monitor

.

11
2.0 ANALYTICAL AND EMPIRICAL STUDIES OP THE NEW

CALIBRATION PROCEDURE'S ACCURACY

Following the achievement of a reliably operating software system

for measuring step heights and roughness (appendix B), the measurement

system's roughness data output was subjected to a preliminary

evaluation. Data for the statistical evaluation was extracted from

the results of four calibration studies which NBS had performed for

industry. The calibration studies had employed the stylus/computer

system simply as a more accurate readout than the Talysurf 4's AA

meter, with the reported AA values being referenced to the NBS master.

The pattern of observations made on a precision roughness specimen at

this time was that of making 6 measurements at each of 5 positions on

a roughness area. Results of this preliminary evaluation along with

the increased accuracy of the computer calculation showed that

position-to-position variations in specimen roughness and variations

in day-to-day transfers from the NBS master were the dominant sources

of random error.

Based upon the preliminary evaluation, the complete

interfere-metrically calibrated stylus/minicomputer system was used to

obtain roughness measurements over a three month period. In these

measurements the pattern of observations was changed at first to that

of 3 traverses at each of 6 positions, then later to 3 traverses at

each of 10 positions over each roughness area (appendix H) . Only a small area of each roughness patch was used for the study in order to

minimize the effect of positional variations in the surface roughness

on the resultant data. The data was analyzed by Joseph Cameron of

the Office of Measurement Services at NBS. Results of this analysis

(appendix H) show that:*

1) the computed standard deviation, aw , of the average of three

measurements at one surface location on a particular day

was 0.028 yin for the 20-yin AA patch and 0.31 yin for the

125-yin AA patch,

2) the computed standard deviation, ag, of the day-to-day com- ponent of the average of measurements at 10 (or 6) surface

positions was 0.132 yin for the 20-yin AA patch and 0.337 yin

for the 125-yin AA patch,

3) assuming a uniform roughness specimen i.e., one with no position-to-position variations in roughness, the estimated 3

standard deviations limit for assessing the uncertainty of

roughness measurements employing an average from measurements

at 10 positions was 0.40 yin for the 20-yin AA patch and 1.1

yin for the 125-yin AA patch, or 2.0% and 0.9% of the respective mean roughness values.

^English units are used in referring to the precision roughness specimens in deference to their conventional use.

12
4) no significant correlation between the magnitude of KCAL, the

calibration constant, and the daily averages was apparent.

This last conclusion is important since correlation between KCAL and

roughness values would imply that the step calibration procedure was

not properly correcting for gain drifts in the system electronics.

Overall uncertainty in a measurement process may be described in

terms of two categories; systematic uncertainty and random

uncertainty. Systematic uncertainties describe those properties of

the measurement process which are fixed prior to and during the

procedure of obtaining data. Random uncertainties describe the

variations in a measurement process's results during repetitions of

the procedure to obtain data.

The aspects of the measurement process which are classed as being

parts of a repetition is always arbitrary to some extent. For

instance, should one repeat a roughness measurement with the use of

two or more stylus tips of nominally the same radius to determine the

uncertainty in the roughness value produced by this aspect of the

process? Should one repeat a computation of KCAL with the use of two

or more interferometrically measured steps approximately equal in

height? A consistent definition of a repetition for the roughness

measurement process results when one fixes all artifacts of the

measurement apparatus i.e. one roughness specimen, one calibrating

step, one stylus tip, one roughness filter and one environment (noise

level) . A "repetition" is then defined as a completely new and

independent measurement with this fixed set of apparatus and artifacts

including a recomputation of KCAL, repositioning the test specimen,

etc . , such as would be involved if the values were obtained one or more days apart.

Based on this definition, all the uncertainty components revealed

by the three month evaluation would be random uncertainties. Possible

systematic uncertainties describing the difference between an AA value

obtained with the use of the NBS stylus/computer system and that

obtained with a similar system are as follows:

1) Uncertainty in the interferometrically measured height of the

calibrating step. An estimated 3 standard deviation of these

measurements is 25 nm (section 2.1).

2) Uncertainty in stylus tip radius. The resultant error in an

AA measurement is dependent on the profile's waveform but an

estimate based on the highly sloped 20-yin (0.508-ym)

precision roughness specimen's waveform is + 5-nm AA with the

stylus tip in current use at NBS, (appendix I). An estimate

of the relationship between tip radius uncertainty and AA

uncertainty was made through the use of a correction chart

given in appendix C of American National Standard B46. 1-1962.

13
3) Lack of conformity of the NBS system's roughness filter

characteristics to those specified in American National

Standard B46.1. The filter-amplifier bandpass

characteristics given in appendix A and the uniformity and

magnitude of the stylus traversal speed given in appendix

demonstrate that the NBS system performance conforms to the

American National Standard B46.1 specifications. Since

tolerances on the filter characteristics of the standard

allow greater variations in the cutoff points etc . , than the

electrical component specifications of the NBS system no uncertainty estimates on this effect will be made.

h) A bias error produced by the system's mechanical and

electrical noise. The AA value of the combined system noise

is 5 to 6 ran (appendix J). The positive bias of the noise on

AA measurements would be additive according to the expression

y (roughness AA) 2 + (noise AA) 2 since no correlation between

the noise and the signal from the specimen roughness should

be present . For roughness values above 100 nm the effect of

noise on the calculated AA value is small e.g.

V (100) 2 + (5) 2 - 100 = 100

0.125*

5) A component to account for the deviation of the profile

obtained with the specified tip radius from the true profile,

i.e., the profile which would be obtained with a tip radius

approaching zero. Whitehouse [1]* has explored this problem

and concludes that if the tip radius is small compared to the

correlation length (section 3.3) of the profile, AA values

will be reduced only one to two percent from the true profile

AA. This particular value is based on the use of a tip

radius of 2-3 ym and correlation lengths of 25 ym or greater.

The systematic uncertainty from this effect is therefore not

given further consideration. All calibration and test

reports issued by NBS do however report measured AA values

relative to a specified tip radius.

Analytical and further empirical studies of the components of the

measurement process random uncertainty are presented in the following

sections. These studies show that the net random uncertainty may be

attributed to three sources: (1) the variations, due to the surface

finish of the calibrating step, in KCAL, (2) the variations, due to

sampling and digitizing processes, software computations and

^Figures in brackets indicate the literature reference.

14
nonlinearities in the stylus instrument transducer and in the

interface hardware, in KCAL and (3) in measured AA values, for a fixed

KCAL.

Section 2.2 shows that the 3 standard deviations of step heights

calculated from profiles of a specimen with an RMS surface finish of q

is 1.08 q. An estimate of a 3 standard deviations values for the

second source of uncertainty based on the results of section 2.2 and

2.3 is 1.0$ of the measured AA value. A 3 standard deviations value

of lA% of the measured AA value for the third component is obtained

from the three month evaluation (see appendix H) . Since profiles of

typical roughness specimens have peak-to-valley heights of three to

eight times the profile AA value, step heights approximately six times

the expected AA value are used for calibrating the system. Thus the

net random uncertainty in the measurement process is given by the

relation:

, ,, „ .1/2

Net Random

Uncertainty

/2Xjm\ 2 + (0>Q1 M m)2 + ( 0aQ1n M nm)2

where an RMS surface finish of 25 nm has been assumed. This relation

holds since the components are uncorrelated

.

The systematic uncertainties that are included in an NBS statement

of calibration accuracy are the components 1, 2 and 4 as described

earlier. The net systematic uncertainty is therefore given by the

relation:

Net Systematic Uncertainty =1 [ -^—j + (5 nm) 2

|

Effects of the fourth component of systematic uncertainty are

corrected for in stated AA values.

The calibration uncertainty of the NBS system for measuring

surface roughness is taken as the sum of the system's random

uncertainty and its systematic uncertainty. The calibration

uncertainty (CU) as a function of the specimen AA roughness is

therefore

:

CU(AA nm) = 6.5 nm + [20.25 nm2 + 2.96 x 10"4 (AA nm) 2

] 1/2

.

A plot of this equation is given in figure 5.

Calibration and test reports issued by NBS briefly describe these

systematic and random uncertainties and give a net value based on the

calibrating step height used for the measurements; the usual height is

approximately six times the AA value. Finally, an overall measurement

uncertainty is stated which is the sum of the calibration uncertainty

and a three-standard-deviation limit of the data obtained in 3 stylus

traverses at each of 10 positions of the test specimen's surface.

15
1

o

A1NIV1U33NH N0I1VU8I1V3

o

g

i

3

CO

1

PL| m

fe

16
The results of the study of the correlation between roughness

values obtained from the master-artifact comparison procedure and the

values obtained with the step calibrated stylus/computer system are

also presented in the following sections. The study confirms that a

smooth transition between the two measurement processes was made.

2.1 THE EFFECT OF GEOMETRY ON STEP

HEIGHT MEASUREMENT

Step artifacts used to calibrate the stylus/minicomputer system

typically have many unavoidable geometrical errors; surfaces on both

sides of the step have a microscopic texture, the surfaces are not

flat and the two sides of the step are not parallel. One crosssection of such a step obtained from a stylus trace at a 5000 to 1

vertical to horizontal distortion is shown in figure 6. These

geometrical errors affect both the precision of measuring step heights

interferometrically and the precision of transferring the measured

value to the stylus instrument.

Figure 6: Profile of a 0.501 ym step. The vertical

to horizontal magnification ratio is 5000

to 1.

17
INTERFERENCE FRINGES

l&Jtittomiilrt&MBttmritii&ailM-na***** \

| vu^^fl^^^^^^|^^^^^^^^^^l<^|ujAu^^^^l^^^^iu^H ^L A \

V i\

\

y

LINE OF STYLUS TRAVERSE

Figure 7- The Effect of Geometry Errors on the Measurement of Step

Heights

.

As an introduction to some effects of geometrical errors on the

interferometric measurement of step heights, consider the illustration

in figure 7. In the usual two beam or multiple beam interferometry

the measured height will be some function of the fringe spacing, t;

the lateral displacement of the fringe, A; and properties of the

illuminating radiation. Assuming one has well collimated light of

wavelength A and an ideal step artifact (flat and parallel sides) the

height, h, is given by the well known equation:

h = (" + !)A/2, Equation 1,

18
The order of the displaced fringe, n, is determined by "following the

fringe" or by deduction from measurements with several different

wavelengths .of illumination.

Figure 7 was composed to show that the heights measured on a step

artifact with interferometry and with a stylus instrument may differ

significantly. This is particularly true if interferometry at

magnifications of about 10X is employed. Here the lateral displacement

of the two fringes used in measuring A may be as much as 5 mm;

assuming no magnification and 5 fringes in a 2.5 cm field of view.

Thus, with reference to figure 7, some mean of z-j_ and Z2 would be

measured with interferometry, while the value measured with a stylus

instrument with the indicated stylus traverse would be less than this

mean.

Figure 8 is an interferogram obtained from a specimen with

some typical geometry errors; the planes on the two sides of the step

are not parallel in either the direction perpendicular or parallel to

the step's edge. From the bottom of the interferogram, fringe displacements at the step edge are respectively, 1.7*1, 1.77 and 1.8l of

the mean fringe spacing. Magnification of this interferogram (^ 20X)

is such that the field of view contains approximately the distance

traversed, in the fringe direction, by the stylus during step height

measurement, (3.8 mm). Notice that the lateral displacement of the

fringes at the step is ^ 2 mm.

Figure 8: Representative Interferogram of a Step

Specimen. Wavelength of the Light Used

was 589 nm.

19
For an ideal artifact the fractional error in a step height

measurement using equation 1 is given by:

^<i (i + |

+ t)(-^),

where e is the error in measuring either A or t. This expression

includes the second order error e

2 /At and assumes that the error

produced by uncertainty in X is negligible. For real step artifacts

additional error terms must be added to account for the artifact'

s

imperfect geometry and for the lateral averaging of interferometry

.

One possible way of incorporating these errors into equation 1 is to

let h be a function of the dispersion, s = A/t, such that:

h = n X/2 + g(s) X/2,

where for small geometrical errors, g(s) = s + 6(s). With these

assumptions the errors may be expressed by the equation:

dh = ds X/2 + ~ ds X/2.

ds

The first term in this equation is the error for an ideal artifact

discussed before. The second term is the error arising from geometry

and lateral averaging.

An illustration of this mixing of geometrical errors and averaging

by lateral fringe displacement such that the height measured is a

function of dispersion is given in figure 9.. For the particular

geometry shown the height measured with interferometry could therefore

change by as much as 0.12 X/2. The approximate mean deviation from

planarity for the example is 0.15 X/2. If the amplitude of this

deviation is halved to approximately 0.08 X/2 then the difference

measured for the dispersions shown is also ^0.08 X/2. The close

correspondence between the difference measured and the mean deviation

from planarity is a coincidence generated by the two dispersions used

for the example. Other dispersions do not show this correspondence.

Experimentation with other geometrical errors shows that the variation

of height measured is strongly dependent on the particular geometry

and its general slope relative to that of the reference surface.

Geometrical errors in the lower part of the step and in the reference

surface would also contribute to the total uncertainty of a height

measurement in a manner similar to the one surface considered in the

example.

Carrying the discussion further to obtain an explicit relationship

between 6(s) and a particular geometry is not fruitful. The principal

point of the discussion was to demonstrate that to properly measure a

step height with interferometry and to characterize the measurement's

uncertainty the geometry of the surfaces on both sides of the step in

20
Re

Fr

iference plane and") Position

inge locations J Position

Top surface

of step

Figure 9a. Side View of Step and Reference Plane with Fringe Locations Shown for

Two Reference Plane Positions Assuming X = 3-5 cm. Position I. Step height measured = A| /t| = 0.836 X/2

Position2. Step height measured s A2 / t 2

= 0.720 X/2

t 2

Figure 9b. Top View of Fringe Pattern Across the Step Shown in Figure 9a.

21
the neighborhood of the measured region must be determined. Step

artifacts used to calibrate the NBS stylus/computer system are therefore routinely checked for flatness and parallelism. The 3

standard deviation uncertainty of 25 nm assigned to this aspect of the

calibration procedure reflects in part the errors produced by

imperfect geometry.

2.2 THE EFFECT OF SURFACE TEXTURE ON

STEP HEIGHT MEASUREMENT

The microscopic surface texture on both sides of the step also

affects the measurement and transfer of a step height. Surface

texture affects the measurement of a step height by increasing the

fringe width and therefore increasing e, the error in measuring A

and/or t. The microscopic texture is usually unresolved at the

customary magnifications used for step height measurement in the 10-um

range. The primary effect of the surface texture is to limit the

overall accuracy of transferring a step height into the

stylus/computer system.

As described in section 1.2, the step height is transferred to the

stylus/computer system by performing a linear least-squares fit to the

data taken from each side of the step, then calculating the step

height from the position of the two resulting lines. Least-squares

methods for estimating the coefficients of a relation such as, y = ax

+ b, are strictly valid only when[2]:

a) the random variations in the y's, for many traverses across a

step, have a zero mean and a common variance;

and

b) the random variations in the y's are mutually independent in

the statistical sense.

In addition for strict validity of the confidence interval estimations

given in the following discussion, an additional assumption must be

satisfied:

c) the random variations affecting the y's have a normal

distribution.

Studies of the amplitude density function and the autocorrelation

function for profiles of highly polished surfaces similar to the ones on the calibrating step specimens have been performed[3]. The results

of these studies show that assumptions a and c are realized and that

assumption b is approximated. Thus, while these studies do not

completely justify the application of least-squares methods for

fitting the step profiles and estimating confidence intervals,

they do provide a basis for using a model of the surfaces which has

the necessary properties required by a, b, and c.

22
Therefore, assume a model of the calibrating step which has no

geometrical errors other than surface texture whose profile meets

requirements a, b, and c. If the texture has. an RMS value (-AA

value) of q, what is the 3 standard deviation of the step heights

calculated when the height routines are applied to profiles obtained

from such a step? Let the line y = ax + b be least square fit to ' N

data pairs (Xj_,Yj_). Then the estimated variance, a

2

, of a point (x',

y'), on the predicted line is given by the equation [4, 5]:

where

X =

a

2 = S2y

1 . (x' - X) 2 N N

£ (X

± - X) 2 i=l

i

N

± Z X.

N

i=1 i

and Sy is the sample standard deviation of the set of Yj_ about the predicted line y = ax + b. It is calculated from the equation:

N

* (Y,-y,) 2 2 _ .1=1 X 1 s

z =

y N-2

Thus with the assumed profile properties, S^ = q

z and

a

2 = q

2 1 (x' - X) 2 N N

E (X± - X) 2 1=1

The data format for the step height routines uses N=128 and X. = (constant) i, where i is an integer, so that for a particular abcissa

specified by i '

,

r 2 _ 1 (i' - 64) 2

N N

z (i - 64) 2 1=1

The position of the point used for step height calculations may

vary over a range such that; 64<|i'-64| < 192, depending on where the

operator chooses Al and A2 . A typical value for |

i

T -64 1 is 100

.

23
Substituting this value and evaluating the summation gives the result:

a

2 = q

2 (0.0650)

or:

a = 0.255 q.

If one assumes that the fits to the two sides of the steps are uncorrelated, a 3 standard deviation uncertainty, 3cr, for the

calculated step height is therefore:

3a = 3^* + a* = 1.08q.

Two other random error components are produced by the system

electrical and mechanical noise and by the quantization error

resulting from digitizing the step profile. Measurements of the

first of these gave an estimated value of q^ = 6 nm (see appendix J).

The quantization error gives rise to an equivalent RMS error of

[6] 5

q _ 29 Ful1 Scale Range 50mm^[agnification

^Q ' Number of quantized intervals "" y

4096"

For a magnification of 1000, the quantization thus produces an RMS

error of 3.2 nm. These three random error components are uncorrelated

so that a net RMS error q is:

% - (Q

2 + Q2

, + q

2

,)*

5

•

Assuming q = 12 nm and a magnification of 1000, q

= 14 nm. For

higher magnifications the quantization error becomes insignificant and

q drops to 13 nm, with q = 12 nm. An experimental study of the random

error of calculated step heights for a 12 . 7— \M step with a surface finish of 12-nm AA and a 0.51 ym step with a surface finish of

6-nm AA is described in appendix K. The results are in good

agreement with the analytical calculations given here.

Surface finish of calibrating step artifacts is typically 6-12-nm

AA. Profiles of the calibrating step often contain the cross section

of discrete scratches or dust particles which greatly increases the

uncertainty of transferring the step height. Step profiles are carefully examined before being used for calibration, but small

discrete imperfections probably escape detection. A conservative

value of the calibration step's surface finish of 25 —nm AA is

therefore used for estimating the system's systematic error.

24
2.3 EFFECTS OF STYLUS-TRANSDUCER AND INTERFACE HARDWARE NON-LINEARITIES

ON ARITHMETIC AVERAGE AND STEP HEIGHT MEASUREMENTS

To consider the effects on roughness, let the overall nonlinearity

of the signal path from the transducer to the computer memory be

described by the equation;

V(y) = by + cy 2

, (3)

where V is the number stored in memory for a tip displacement y. Then

the AA calculated from V, exclusive of digital data acquisition

effects, may be related to the AA of the profile y(x) by means of the

amplitude density function, ADF (section 3.2). Given the ADF of a profile y(x), the profile's AA may be calculated from the equation:

+00 /

AAy(x)

= / |y| ADF(y) dy. (4)

Similarly the AA of V(x) is given by:

+0O MV(x)= / |V| ADF(V) dv. (5)

Expanding equation 5 in terms of y requires a relation between ADF(V)

and ADF(y). With the assumed quadratic relationship between V and y one may show that [ 7 ]

Substituting equations (6) and (3) into equation (5) then yields:

+0O AAvr^ =

I l fey + cy

2 |ADF(y) dy. Mx) /

Since

|b| |y|-|c|y 2 <|by + cy 2

| |b||y| + |c|y 2

,

the fractional error, eLQ, between the AA value measured with a linear

system, AAl, and that measured with a quadratic system, AAq, is:

+

°° AA. - AAn bT - bn I, f

eLQ " ZIIA^ " ^T^ t bflr J

y

2 ADF(y)dy.

GO

25
The integral in the last term of this equation is the mean square

value, MS, of the profile. For most profiles the MS/AA ratio is less

than 4AA; for a profile with a Gaussian ADP the ratio is 1.57 AA.

Thus a maximum expected normalized error is:

LQ b

L

b

L

y

Using the measured values for bL, bQ

, and c for the transducer which

are given in appendix L, the maximum value for each term for AA

values less than 10 urn is:

e = 0.0027 + 0.0018.

For the interface hardware both terms are less than 0.001.

In relating these nonlinear!ties to the increase in random error

resulting from sample repositioning in the long term stability study,

the best association of the error is with a + 3 standard deviation

limit equal to the maximum value of eLQ. This association, in-stead

of considering the component as a known bias error , follows since the

coefficient's uncertainties are comparable with the difference, bL~bQ,

and with the coefficient c. If the nonlinearity uncertainties from

the transducer and interface hardware are simply added, the expected

error in AA measurements due to the system nonlinearities analyzed

thus far is about 0.6$ while the difference in random error due to

repositioning was 0.7 to 1.0$. The remainder of the random error may

be attributed to nonlinearities in the analog-to-digital conversion

process . These nonlinearities probably were not characterized by the

simple quadratic relationship assumed in the analysis given here

because of their random nature.

The effects of system nonlinearities on step height measurements

may be analyzed by again assuming the relationship given in equation

3. Then, if the profile mean values for each side of the step are respectively y^ and y2, the fractional error, SLq, between the value

calculated with a linear system and with a quadratic system is:

Using the measured values for b^, bn and c given in appendix L:

S

LQ

= 0.0027 - 4.6 x 10-5 ym-l

(y2 + Yi ).

Since y2 and y1 may be of either sign, Slq may range from 0.38$ to

0.15% of the linear values when a typical step height of one half

full scale is used on the Talysurf »s 1000 magnification. For

the same reason given in the roughness considerations, the maximum

26
value of Slq should again be taken to estimate a + 3 standard

deviation limit for the uncertainty in step height generated by

transducer nonlinear!ty . Adding the nonlinearity of the interface

hardware then gives a + 0.5% uncertainty in the step height

measurements due to these nonlinearities

.

The arguments then confirm that for these two measurements the

system nonlinearity as defined in appendix L and the uncertainties

generated from the nonlinearity are approximately equal.

2.4 UNCERTAINTIES IN ARITHMETIC AVERAGE MEASUREMENT

DUE TO DATA ACQUISITION PROCESSES

Three aspects of the data acquisition process which produce

uncertainties in the roughness measurements are:

1. record lingth is finite,

2. data are sampled,

and

3. data are digitized to a finite precision.

The arithmetic average value will change in magnitude from record

to record when it is calculated from a finite record length of

profiles with a random shape or when it is calculated from a periodic

profile which is randomly phased with respect to the finite record.

Arguments are given in appendix M to show that the 3 standard

deviation limit expected from AA measurements of precision roughness

specimens with the standard 3.8 mm record length will be:

0.05 to 0.3$ of the mean AA value for the 0.5-ym (20-yin) patch,

and

0,3$ of the mean AA value for the 3.17-ym (125-yin) patch.

How accurately does the sampled sequence | y-j_| represent the

continuous profile y(t)? The sampling theorem [8] states that for

physically realizable profiles with a power spectral density equal to zero for frequencies greater than B, the use of sampling intervals

At<l/2B enables one to reconstruct the continuous function y(x)

uniquely. The bandpass of the NBS system (appendix A) extends to 300

Hz as the -3db point for a 12dB/octave attenuation filter for all

higher frequencies. The sampling interval for roughness measurements

is 610 ys (length equivalent is 0.93 ym) or 1640 Hz. At B = l/2At = 820 Hz the system electronics has a transmission factor of

approximately 0.09. Thus, the possibility exists that up to 9% of the

power of frequency components > 820 Hz will be aliased into the AA

value measured. The only frequency components > 820 Hz which will

likely be present are electrical noise frequencies since the sampling

rate already gives 3 to 4 points across the stylus tip radius of 3.4

ym. Any effect from the power folded back into the measured frequency

range would be accounted for in the usual system noise check. As

discussed in section 2.0 to total, true plus any possible aliasing, is

negligible for AA values greater than 100 ran. 27
The accuracy with which the sequence |y-j_} represents y(t) is also

dependent on how closely the sampling process approaches a Dirac-delta

function sampling process. The error produced by obtaining a discrete

value for the signal by averaging for a finite time is known as aperture error. It is minimized by making the aperture or averaging

time small with respect to the sampling interval. In the present NBS

system aperture time is 50 ys while the sampling interval is 610 us.

The digitization of each sampled datum point to a finite precision

produces an equivalent RMS noise of 0.29 scale unit. (Section 2.2)

With the 12 bit analog to digital conversion used in the NBS system

this aspect of the data acquisition process only produces 3.2 nm RMS or AA on the Talysurf's 1000 magnification and decreases

proportionally with increasing magnification.

2.5 SYSTEM SHORT TERM STABILITY

The time between step-calibration of the NBS system and the

completion of 3 traverses at each of 10 positions is less than 30

minutes. Thus, short term relative to the NBS system implys times of

this magnitude.

During early checks of the system, repetitive step height

measurements showed that irregular changes of 5 to 10$ in the overall

gain of the system took place during the first one to two hours after

turning the system on at the beginning of a workday. After this

period of "warmup" the system remained stable. Confirmation of this

stability is given in the step height measurements reported in

appendix K. Each step height measurement requires

approximately 1 minute, so the measurements given in the appendix

cover the 30 minutes required to investigate the system's short term

stability. No overall or random change in system gain is observable

in the data. However, any effect from the short term variations in

system gain would be included in the experimental confirmations of the

uncertainty produced by the calibrating step's surface finish. Since

reasonable agreement was obtained between the analytical and

experimental effects of surface finish, the effect must have been

negligible.

All calibrations are now performed only after a two hour system

warmup or after the system has remained "on" overnight.

2.6 SUMMARY OF THE SYSTEM UNCERTAINTY BUDGET FOR

ROUGHNESS CALIBRATIONS

The motivation for the error analysis given in sections 2.0

through 2.5 was threefold:

1. to firmly establish experimentally the overall system

uncertainty for calibrations at the most commonly used

roughness values;

28
2. to extend the range of known uncertainties so that it covers

in a continuous form the operating bounds of the system (the

extension was based on the empirical and ' analytical studies

described in these sections )

.

3. to analyze the system's uncertainty so that major components

could be reduced in future system improvements.

A comparison of the analytical and experimental values for the

various uncertainty components of the calibration procedure for 0.5 ym

and 3.17 yni precision roughness specimens is given in tables 1 and 2.

The computational accuracies were computed as + one bit out of 12 bits

or 1/4096 for step height measurements and + the quantization error

for roughness measurements. The differences between the RMS-sum of

the components of 3ag and the value measured in the three-month study

is 0.005 and 0.002 for the 0.5 ym and the 3.17 ym AA measurements,

respectively. Due to the existence of this difference the uncertainty

assigned to nonlinearities , computations and other parts not taken

into account is + 1%. As stated in section 2.0 the net random

uncertainty is assigned a value based on this 1% and the other

experimental values from the three-month study. The sources of the

remaining 0.7% random uncertainty are unknown. A study of the

system's transient response was made in search of an explanation for

this remainder, (see appendix H) . No significant source of

uncertainty was revealed by the investigation.

2.7 COMPARISON OP THE AA VALUES MEASURED RELATIVE TO THE NBS-1 MASTER

ARTIFACT WITH THOSE MEASURED WITH THE STEP-CALIBRATED MINICOMPUTER

SYSTEM.

To insure that a smooth transition between these two measurement

processes was made, five different type surfaces with roughness values

covering the instrument's range and the two roughness areas of the

NBS-1 master artifact were measured with both processes. Figures 10

and 11 give profiles of the specimens used for the study. Table

3 summarizes the results of the measurements.

The procedure for the study was to:

1. measure each of the specimens and record the values read

from the roughness meter,

2. correct each of these roughness readings by the same fraction

needed to bring the NBS-1 values to those established by

planimetry and round-robin measurements,

3. calibrate the computerized system with an interferometrically

measured step,

4. remeasure each specimen and record roughness values

printed by the teletype.

29
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Figure |oc. Specimen A; Vertical Magnification = 10,000

Figure 10. Profiles of Specimens Used for Comparing Old and New

Calibration Procedures. All Profiles are at a Horizontal Magnification = 100.

33
Figure fla. Specimen B; Vertical Magnif icanon = 20/000

Figure Mb. Ground Glass; Vertical Magnification = 20,000

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Calibration Procedures. All Profiles are at a Horizontal Magnification = 100.

34
Only the two magnifications of the stylus instrument on which the two areas of NBS-1 could be conveniently measured were used for the

measurements . The mean and 3s values were calculated according to

the definitions given on page 32 from the data obtained in 3 traverses

at each of 5 or more positions on each specimen.

Let Rj_ and 6^ represent the mean roughness value and the 3s limit

respectively for the measurements on the 1TM specimen. Then a statistically significant difference between the two measurement

procedures is present only when:

R

A

- R

C

> /(6A

)

2 + (6?)2 i l i l

where the superscripts A and C represent the artifact and computerized

procedures respectively. This relation is not true for any of the

data given in table 3. The relatively uniform specimens A and B

impose the tightest constraints on the comparison between the two

procedures

.

Therefore, this study confirms that there is no significant

difference between the roughness values obtained by the two

measurement processes.

3.0 FURTHER UTILIZATION AND APPLICATION OF THE COMPUTERIZED

ROUGHNESS MEASUREMENT SYSTEM

The capabilities of the minicomputer have facilitated the on-line

implementation of many new statistical methods of characterizing

surface topography. The following sections give a brief summary of

the definitions and properties of the statistical parameters and

functions which may be calculated en the NBS system or those for which

off-line calculational capabilities have been developed. The reader

should consult recent textbooks on statistical functions [5,9] or conference proceedings [10,11] for a more complete understanding of the new characterizations

.

In addition to the AA value, the NBS system has the capability to

calculate on line the average slope and average wavelength, the

amplitude density function (ADF) and the autocorrelation function

(ACF). Numerical results may be displayed at the teletype or in

graphical form on a strip-chart recorder (for the functions). A

by-product of the ACF calculation is the calculation of the profile's

mean square value. A punched paper tape containing the ACF and ADF

values may be punched from data stored in the minicomputer during

their calculation. The ADF and ACF data may then be transfered to a larger computer to calculate the skewness and kurtosis of the profile

from the ADF and the power spectral density from the ACF.

Because all these characterizations are readily calculated after a specimen profile has been stored in the minicomputer memory, they are

35
available upon request as a standard test service.

3.1 MEAN SLOPE AND AVERAGE WAVELENGTH

Calculation of the mean slope and average wavelength of a profile

is readily performed with routines from the AA calculations once the

following observations have been made. Consider the profile as being

represented in the computer memory by the equation:

yn

= f(n) n = 1,2, ,N.

Then as demonstrated in appendix B, the AA value of the profile is

calculated by using the equations: - 1

N

y = £

z f (n)

and

1 N

m = n

i

lyn -yi

If now one calculates the differences,

Ayn

= f(n) - f(n-l),

stores the differences as a new "profile," and then performs the same AA calculation on the new profile, the result:

1

N

AA of profile differences = rpy z | Ay - Ay |

,

is obtained. Thus, if Ay~ = it follows that:

AA of profile differences = mean absolute differences,

1

N

, , since mean absolute differences = ?r-y £ | Ay I .

This condition is very closely satisfied. Consider the yn now as the original profile function, y(n), where n is a multiple of the

sample spacing along the profile. Then the following relations hold:

7— = i_ /

dy(n')

Ay * N-l J dn'

2

dn'

36
The latter equation results from the fundamental theorem of integral

calculus. Thus a worst case value for Ay would be:

n

3^ (ax^onj) =0.0267,

which would occur using the lowest magnification of the stylus

instrument. More commonly used magnifications are 2000 or greater

such that worst case errors would be 0.0066 or less, which correspond

to angular slopes of 0.38 degrees or less.

The mean slope program given in the listings therefore utilizes

the AA routines as explained. The problem of having one less

difference point than profile data points is solved by taking N+l

profile data points into the computer memory. Thus, profile data is

stored from memory addresses 1000 through 3000, while the AA

calculation uses only addresses 1002 through 3000.

A conservative range for the differences is taken as twice the

profile data range, i.e., the profile data points; -X, +X, -X; give

differences; 2X, -2X. The difference range of the profile data

however is ultimately determined by the stylus tip shape and the

sample spacing. The NBS stylus/computer system employs a 90°

pyramidal tip with a radius of ^3. 4 ym and a sample spacing of ^1 ym.

Thus, profiles with peak to valley heights >4 ym would produce maximum

data differences of approximately 1/3 the profile range. Profiles

with peak to valley heights <4 ym produce data differences up to 1.25

of the profile range at a gain of 100,000. Both of these estimates

may be obtained by considering the profiles produced as the stylus is

traversed across a sharp edge with a height chosen to give an on scale

trace

.

The AA of the profile slope or the mean profile slope is then

calculated by using the relation:

nn „ ~. n

., mean absolute difference x KCAL

AA of profile slope = =: : ^ * mean sample spacing

The mean absolute difference is scaled to the same physical dimensions

as the mean sample spacing by multiplying by KCAL and the necessary

power of 10. The power of 10 is obtained from the computer memory by

recalling the value entered at the teletype after the units query

during step-calibration. Determination of the mean sample spacing is

discussed in appendix 0. Substituting the mean sample spacing,

therefore, gives the final equation for the average slope:

mean Ay x KCAL

n i

average slope = x 10 i~P

9300

37
where p is power of ten entered for the calibrating step and the mean

sample spacing is 9300 x 10-7 mm.

Spragg and Whitehouse [12] have shown that, for most profiles

encountered in practice, the average wavelength of a profile may be

calculated from the equation:

.. , 0n AA Value

average wavelength = AM = 2n M of profiie slope.

The mean slope program also calculates and prints out at the teletype

the value of this parameter. It is important to notice that the

average wavelength is independent of KCAL and that it is a direct

function of the sample spacing used for digitizing the profile. This

point is demonstrated by the following equations:

AA x KCAL x 10-P

x M

= 2H x

(

AA slope x KCAL x 1Q-P \

9300 x 10-7 mn /

2n x mi^t x 93 -° x 10"5 -

The listing and flowchart of the mean slope program is given in

appendix B.

3.2 AMPLITUDE DENSITY FUNCTION, SKEWNESS AND KURTOSIS

A functional characterization of the amplitude properties of a

waveform or profile in a detailed statistical manner is given by the

probability or amplitude density function, ADF, Figure 12.

Numerically the ADF is defined as the probability that at a given

ordinate value, the profile amplitude is within an interval Ay in a record length L. The probability is calculated using elementary

geometrical ideas and is effectively a histogram of the profile as a function of the chosen interval's mean ordinate. The ADF is usually

specified as a decimal fraction per unit interval length as a function

of the interval's mean ordinate. Before computers, its calculation

was very tedious, but the necessary bookkeeping procedures are very

fast and efficient for a computer to perform.

Calculation of the ADF, as one may imply from the definition and

the flowchart in appendix B, primarily involves sorting the profile

data into the respective ordinate intervals . The present calculation

employs 512 ordinate intervals each with a "width" of 8 units. The

calculations performed by a computer to obtain the ADF are simpler and

38
fewer in number than those in an M computation. For a 512 interval

ADF and 4000 profile data points only about 60,000 operations are

needed. The AA calculation involves approximately twice this number

of operations and double precision arithmetic to avoid overflow during

summations. Computation time with the present minicomputer is 6

seconds but with recent generation minicomputers this time could be re- duced to 0.1 second.

A slight alteration of the software for the ADF calculation

enables the user to perform signal averaging on the ADFs obtained from

one or a group of positions on a specimen surface.

Many of the traditionally used parameters are measures of the size

of the ADF and may be easily obtained from it as various averages of

the ordinate weighted by the ADF. Some examples are:

AA = / y ADF(y) dy, /I

(RMS) 2 / y

2 ADF(y) dy.

In addition to these size dependent parameters several new

parameters have been introduced which are measures of the ADF's shape.

The skewness of the ADF is a measure of the symmetry of the profile

about its mean, (figure 13). Skewness offers a convenient way to

differentiate between the load carrying capacities of various

surfaces. As one may conclude from the examples in figure 13,

positively skewed surfaces are more suitable to carry loads than

negatively skewed surfaces. A second shape dependent parameter the

kurtosis, figure 14, is a measure of the amplitude density function's

sharpness or the profiles fourth moment. In addition to

characterizing this aspect of a profile, it has utility in

quantitatively describing the randomness of a profile's shape relative

to that of a perfectly random surface which has a kurtosis of 3.

Skewness and kurtosis are calculated directly from the definitions

given in the figures. The RMS is calculated from the ADF according to

the relation given above.

3.3 AUTOCORRELATION FUNCTION, MEAN SQUARE VALUE AND

POWER SPECTRAL DENSITY FUNCTION

The fundamental statistical function which characterizes the

wavelength properties of a profile is the autocorrelation function,

ACF. The ACF is a quantitative measure of the similarity between a laterally shifted and an unshifted version of the profile. Assuming

the profile is specified by N ordinates Yj_, the ACF is defined by the

39
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LUQ

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42
equation:

N-S

ACF(S ) =^ z Y.Y

i=l

Thus for a particular shift its value is obtained by multiplying the

shifted and unshifted waveforms, over the overlapping length, ordinate

by ordinate, then calculating the average product.

The ACP contains information about the characteristic lengths,

short and long range, which describe a profile. The characteristic

long range parameter is known as the correlation length; it is usually

defined as the distance a profile must be shifted for the ACF or its

envelope to drop to 10$ of the zero shift value. Two points on the

profile which are separated by more than a correlation length may be

considered as uncorrelated or independent, i.e., portions of the

surface represented by these points were produced by separate surface

forming processes. Correlation lengths possible with a variety of

waveforms may range from the infinite correlation length of a perfectly periodic waveform to zero for a completely random waveform.

If the profile has any periodic behavior the ACP will exhibit a periodicity with a wavelength equal to the mean wavelength present in

the profile. The wavelength of the short range periodicity of the ACF

is known as the correlation period.

The computations involved in the ACP calculation, are only

slightly more complex than those for the AA calculation. However,

since about 4000 multiplications are required for each shift, a total

of 512 shifts results in at least 2 x 10° operations . The actual

number is approximately 10? since summations, register initializations

and data shifting are also involved in the computation. The number of

programming instructions is smaller than the program required for the

AA calculations. With the present NBS minicomputer approximately 13.5

minutes is required for an ACP calculation. The use of more recent

minicomputers would reduce the time, with the same algorithm, to

approximately 20 seconds. Readers who may feel that even this time is

excessive will find that fast-Fourier-transform hardwired processors

are available which can cut the time for an ACF calculation to less

than 1 sec.

The ACP for a zero shift is:

1

N

ACF(o) = ± I Y. Y.

,

N . , l i+o i=l

43
Thus, the mean square value of the profile is easily obtained by

simply taking ACF(o), converting the binary number to a decimal value

and printing it at the teletype.

The power spectral density (PSD) function of a profile describes

the general frequency composition of the waveform in terms of the mean

square value of each component. For a narrow frequency band, Af,

about some frequency, f , the PSD(f) is defined by the equation:

L

PSD(f) = ~ I. i

J

y

2 (x,f ,Af) dx

Where L is the profile record length and y

2 (x,f ,Af) is the mean square

value of the portion of y(x) in the frequency range from f to f + Af

As a result of this definition, the PSD(f ) is always a real valued,

non-negative function and

,2 = /

(RMS)^ / PSD(f) df

o

The PSD is closely related to the autocorrelation function as expressed by the equation:

/

PSD(f) = 4 / ACP(s) cos 2nfs ds. (7)

Put in mathematical terminology the PSD and the ACF are Fourier pairs,

namely, one is the Fourier transform of the other. The ACF expresses

the length properties of a profile while the PSD expresses the

frequency properties of a profile. The PSD representation of a profile is useful when one wishes to put a machine into a process loop

where a knowledge of the transfer function of the machining process is

required. A normalized form of the PSD is usually given:

PSD (f) = £SD(fi

, n

(RMS) 2

such that one obtains the PSD(f ) as a fractional value plotted versus

frequency in cycles/mm or cycles/ym. Equation 7 is presently employed

for all PSD calculations since it only involves the transfer of 512

data points . Calculations directly from the profile would require the

transfer on paper tape of 4096 points. At standard teletype punching

speeds the transfer of 4096 points requires approximately 70 minutes.

44
3 . 4 APPLICATIONS

Since the development of the software for the calculation of

statistical parameters and functions in January 1974, NBS has

performed 10 studies involving the use of this capability. A- test

report which contain calculations of the ADF, average wavelength and

average slope is given in appendix P.

Two other examples will be given here to demonstrate the results

of employing the stylus instrument/minicomputer system's capabilities

and the improved measurements provided by the statistical

characterizations. The first study is an example of how more

controlled surface generation may be possible by comparing the

statistical characterizations of a fabricated specimen's roughness

profile with those of an ideal roughness profile. This approach was used in work to evaluate how closely typical precision roughness

specimens (PRS) conformed to the specifications in American National

Standard B46.1. The procedure for test was to obtain a specimen

profile and then to calculate its AA, mean slope, ACF, and ADF. This was followed by adjusting the simulation waveform such that its AA and

frequency were the theoretical values, then to calculate the

simulation waveform's ADF and ACF. Results for a typical specimen are

summarized in figures 15 and 16.

Some conclusions are:

1. the dominant peak to valley heights of the PRS are 13% and

29% lower than the theoretical peak-to-valley heights of the

125 yin and 20 yin areas, respectively;

2. both areas have significant amplitude and frequency

modulation of their waveforms so that the correlation lengths are less than the record length used for AA calculations, 3-8

mm (0.15 in); and

3. the mean spacing of the rulings as observed in the

correlation periods is equal to the theoretical period to

within the system's uncertainty.

The second study is an example of how one may gain knowledge about

the relationship between part function and surface texture by

experimentally taking bad, medium and good parts and then searching

for correlation between one of the parameters and part quality. A

summary of the measurements for three surfaces (a polished steel

substrate, a cloudy nickel electroplated surface, and a bright nickel

electroplated surface) is presented in figure 17. Some qualitative

conclusions which may be drawn from this limited data are;

1. brighter surfaces produce longer correlation lengths and

Gaussian ADFs (skewness approaches ) and kurtosis approaches

3),

45
and

2. increasingly bright surfaces produce ADFs which, for a fixed

ordinate range, approach a Dirac-delta function.

46
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49
ACKNOWLEDGEMENTS

Many people have contributed significantly to the development of

this system and to the preparation of this report. Those I would

especially like to thank are Dennis A. Swyt for helping me to

understand his programs and the minicomputer's operation and for

working co-operatively with me during the early stages of system

evaluation, Margie Johnson for carefully and patiently typing the

report, Fredric Scire for taking much of the data required for

statistical evaluation, and Joseph M. Cameron for thoroughly analyzing

the calibration data. Other contributors to the software are acknowledged in appendix B.

50
REFERENCE

[I] D. J. Whitehouse, Theoretical Analysis of Stylus Integration , Annals of the International Institution for Production Engineering

Research (CIRP) Volume 23, pp. 181-2 (1974).

[2] Mary Gibbons Natrella, Experimental Statistics , National Bureau

of Standards Handbook 91, 1963, pp. 4-6, U.S. Government Printing

Office, Superintendent of Documents.

[3] D. J. Whitehouse and J. F. Archard, The Properties of Random

Surface of Significance in Their Contact, Proceeding of the

Royal Society of London A316, 97-121 (1970).

[4] Natrella, op cit p. 5-15.

[5] Julian S. Bendat and Allan G. Piersol, Random Data: Analysis and

Measurement Procedures , Wiley Interscience , 1971, pages 129f.

[6] Ibid, p. 232.

[71 Ibid, p. 60.

[8] R. K. Otnes and L. Enochsen, Digital Time Series Analysis , John

Wiley and Sons, 1972.

[9] D. C. Champeney, Fourier Transforms and Their Physical

Applications . Academic Press, 1973.

[10] Proceedings of the International Conference on Surface

Technology , 1973 Society of Manufacturing Engineers Conference in

Pittsburgh, Pa.

[II] Properties and Metrology of Surfaces , Proceedings of the International Conference organized by The Institution of Mechanical

Engineers, Volume 182, part 3K, 1968.

[12] R. C. Spragg and D. J. Whitehouse, A New Unified Approach to Surface Metrology , Proceedings of the Institution of Mechanical

Engineers 1§5, 47-71 (1970).

[13] R. D. Deslattes, Optical and X-ray Interferometry of a Silicon

Lattice Spacing. Applied Physics Letters 15 386 (1969).

[14] Bendat and Piersol, op cit Chapters 3 and 6.

[15] D. K. Cheng, Analysis of Linear Systems , Addison Wesley

Publishing Co., 1959, p. 244.

[16] R. C. Spragg, The Interdisciplinary Nature of Surface

Measurement . Joint Measurement Conference, Boulder, Colorado, June

1972.

51
[17] Precision Measurement and Calibration . Edited by H. H. Ku; pp.

331-3^2, NBS Special Publication 300, Volume 1, U.S. Government

Printing Office, Superintendent of Documents, 1969,

[18] Teletype Waveform Receiving System, NBS Report 9887 by Louis J.

Palombo, National Bureau of Standards, Washington, D.C. 20234,

1967.

[19] Private Communication; Mr. Prank Flaherty, Director of Advanced

Process and Control; The Gillette Company; Boston Research and

Development Laboratory; Gillette Park; S. Boston, Massachusetts

02106.

52
APPENDIX A*

SYSTEM ELECTRONICS AND INTERFACING

The system components as shown schematically In figure 2 are:

1. An Interdata Model 3 Minicomputer

2. A teletype Model 33

3. A Talysurf Model 4 stylus instrument

4. A 12 bit analog to digital converter, A/D

5. An 8 bit digital to analog converter, D/A

6. Two strip chart recorders,

and

7. An electronic filter.

The minicomputer, teletype and A/D were interfaced by the minicomputer

manufacturer; one of the chart recorders is also a standard accessory

for the Talysurf 4. Thus, the only interfacing needed was between the

stylus instrument and the minicomputer-A/D subsystem, and between the

minicomputer and an output strip chart recorder.

The latter was accomplished with a system designed by Louis J.

Palombo [18] of NBS to transform the digital teletype output to an

analog signal. With this system two modes of output from the

minicomputer's teletype interface are available. In a normal mode,

the usual keyboard-teleprinter operations are provided. In a plot

mode, the binary value of each 8-bit character is converted to an

analog voltage with the D/A and input to the second strip chart

recorder. Several commercial systems are available which will perform

the same operations; the Palombo system was employed because of

its availability. A block diagram of the waveform receiver is shown

in figure 18. In addition to an analog signal output

proportional to the 8-bit character input of the teletype data

line, the receiver controls a set of relays which, when energized by

the plot character, lower the pen (to the paper) and start the

chart motor of the recorder. They also disable the keyboard and

paper tape punch on the teletype. In the block diagram SI is the

teletype enabling line.

The electronics for the analog filter and interfacing are shown schematically in figures 19 and 20. The gain-bandpass

characteristics of the filter circuits, shown in figure 21, have

been designed to conform with ANSI Standards when the device is used

in conjunction with a Talysurf 4. The filter 3 db down points are related to the conventional wavelength cut-off through the stylus

speed:

f(Hz) = .06 in/sec * A (in).

53
The filters are active 2-pole Butterworths with selectable high-pass

cut-off frequencies as indicated by figure 19- Roughness measurements

are made with one of three high-pass filters in series with the single

low-pass filter. Step height measurements are made with only the X10

amplifier in the circuit.

The conformity of the filters to specified operation is indicated

by the bandpass curves in figure 21 and the observed phase delay of

the low-pass filter in figure 22. Components of the filter-amplifier

circuits are listed in Table 4.

An additional part of the interfacing, interrupt circuitry, is

shown in figure 20. This circuitry generates at a signal from the

Talysurf 4 an appropriate pulse to the A/D converter and a pulse to serve as an event marker on step profile data. Both pulses have the

standard TTL voltage levels; the event maker pulse is one second long;

the A/D pulse is fifty microseconds long. As shown in appendix Q

the signal from the Talysurf 4 is produced from a relay actuated by

motion of the stylus driving mechanism.

Block diagrams of the system component interconnections are given

in appendix Q.

A compilation of characteristic parameters of the entire system is

given in Table 5.

54
Table 4

Components List for Filter-Amplifier

F.6 Frequency Devices, Inc. (FDI) 0.6 Hz

high-pass filter Model Number 708H2B

F2 FDI 2.0 Hz high-pass filter Model Number

710H2B

F6 FDI 6.0 Hz high-pass filter Model Number

710H2B

F300 FDI 300 Hz low-pass filter Model Number

706L2B

ASIG and AREC Analog Devices Model 40J Operational

Amplifiers

Table 5

Characteristic Parameters

Talysurf 4

Profile Output (nominal) + 1 V

Relay Output (nominal) - 4V

10 5 Magnification (vertical, Max. ) Stylus Speed (4X) 3.6 in

(20X) 0.72! in

(100X) 0.14i in

Scan Distance (.030 in cutoff 4X) .150 in

Chart Speed L4.4 in

Filter-Ampin'.f ier

Input (limited) + 1 V

Input Impedance 100 kti Gain (nominal) 10X

High Frequency Cut-off 300 Hz

Low Frequency Cut-off 6

2.6

Hz

Hz

Hz

Filter Roll-off -12 db/octave

Flatness + .02 db

Overall Linearity .02%

Output (nominal) + 10V

Noise Output < 5 mV.

55
Table 5 (continued)

Interface Circuitry

Pulse Output + 4V

Event Marker Output +1.0 V (Is)

Interrupt Signal Output + 3V (50 us)

Data Input; Step Mode

Record Length (time) 25 s (chart) 5.9 in

(surface, 100X) .060 in

(surface, 20X) .300 in

Number of data points 512

Time per data point 50 ms Horizontal Resolution (100X) 120 yin

(20X) 600 yin

Vertical Resolution (A/D resolution

at highst Talysurf gain). .005 yin

Data Input: AA Mode

Record Length (time) 2.5 s (chart) .6 in

(surface, 4X) .150 in

Number of data points 4096

Time per data point 600 ys

Horizontal resolution (4X) 37 yin

Vertical resolution (at 10^ X) .005 yin

Short Wavelength cut-off 200 yin

Long Wavelength cut-off .010 in, .030 in,

.100 in

and

A/D Converter

Input Voltage + 10V

Input Impedance 100 Mfi Output (including sign) 12 bit

Resolution (voltage) 5 mv

(relative) .025% FS

Linearity .010%

Waveform Receiving System

Input (including sign) 8 bit (Teletype logic

levels)

Output 0-10 volt

56
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61
APPENDIX B

COMPUTER SOFTWARE

This appendix contains the following information about the

computer software used with the minicomputer/stylus Instrument system:

A. Allocation of the Interdata Model 3 memory during use by the

step/roughness measurement software,

B. Flow diagrams of major programs or routines,

C. A memorandum by Philip G. Stein which describes the

properties and the use of an operating system he developed

for the Interdata Model 3,

D. An annotated listing of all the step/roughness measurement

software and of the monitor system designed by Philip Stein,

and

E. An instruction set listed by Op-code, for the interdata

model 3 with an excerpt on instruction word formats taken

from the Interdata Model 3 manual.

The detailed information about the computer software which is

given herein is presented so that the reader will have available a

concrete implementation of a total software system. Such

documentation of computer programs is impossible without making

reference to specific equipment and instruction sets by brand name. However, no judgement as to the quality or suitability of the

equipment discussed here has been made by the National Bureau of

Standards, and no recommendation, favorable or other wise, should be

implied by this report. Listings of the software in machine language

should be of use to the reader whose minicomputer memory size is

limited and thus, is unable to support a compiler and to the reader

considering the implementation of parts of the software system on a

recent generation microcomputer. For those with either assemblers

or compilers, the flow diagrams and listings should enable the

efficient writing of necessary programs. The incorporation of this

appendix was motivated by the desire that one or all three of

these routes be encouraged in every possible way.

Due to the length of this appendix (approximately 70 pages) it was issued as a separate volume. Interested readers may obtain a copy

of the appendix by sending a request for NBSIR 75-924 to:

E. Clayton Teague

National Bureau of Standards

Met. Bldg., Room A-123

Washington, D. C. 20234

62
APPENDIX C

TIME STUDY OF THE FOUR MOST COMMON- MEASUREMENT PROCEDURES

The times given in the following charts assume that only one step

or roughness value is measured and calculated. Statistics on the

input-calibrating step or roughness measurements would add to these

times multipes of only the times for data acquisition and computation

since realignment at different positions is not required for a good

quality artifact. Times for the autocorrelation and amplitude density

function calculations (and all other calculations), are determined by

the Interdata 3 computation times; the use of recent generation

minicomputers would reduce the calculation times to between l/10th to

l/100th the values given. The symbols are those described in section

1.0.

Time Study of Measurement Procedure Step

Operator Function : Machine Function

Align Calibration Artifact; Begin

Program ( 5 min .

)

Select Al and A2 from Chart & Scale, enter (1 min.)

Align Unknown Step

( 5 min .

)

Select Al and A2 from Chart & Scale , enter ( 1 min .

)

Input Step Data (0.5 min.)

Computer Step Height (0.25 min.)

Input Step Data (0.5 min.)

Computer Unknown: Exit (0.5 min.)

Operator Function Time Machine Function Time

(12 min.) (1.5 min.)

Total Time

13.5 min.

63
Time Study of Measurement Procedure: Roughness

Operator Function

Align Calibration Artifact;

Begin Program ( 5 min .

)

Select Al and A2 from Chart

Scale ; Enter ( 1 min .

)

Allign Unknown Roughness

(5 min.

)

Operator Function Time

(11 min.

)

Machine Function

Input Step Data (0.5 min .

)

Compute Step Height (0.25 min.)

Input Roughness Data (0.05 min.)

Compute AA; Exit (0.25 min.)

Machine Function Time

(1 min.

)

Total Time

12 min.

Time Study of Measurement Procedure: Statistical Parameters

(continuing from Roughness)

Operation Function

Enter "J" at teletype

(0.1 min.)

Enter "V" at teletype

(0.1 min.)

Enter "Y" at teletype

(0.1 min.)

Operator Function Time

(11.3 min.)

Machine Function

Calculate and Plot Amplitude

Density Function (0.5 min.)

Calculate and Plot Autocorrelation

Function (13.5 min.)

Calculate and Print Average

Wavelength and Slope (0.25 min.)

Machine Function Time

(14.8 min.)

Total Time

26.1 min.

64
Time Study of Measurement Procedure: Step

Calibrated Amplitude Density Function

Operator Function

Align Calibration Artifact;

Begin Program ( 5 min .

)

Select Al and A2 from Chart

and Scale ; Enter ( 1 min .

)

Enter "J" at teletype

(0.1 min.)

Align Roughness Unknown

( 5 min .

)

Enter "J" at teletype

(0.1 min.)

Operator Function Time

(11.2 min.)

Machine Function

Input Step Data (0.5 min .

)

Compute Step Height (0.25 min.)

Compute and Plot Amplitude Density

Function

Input Roughness Data (0.05 min.)

Compute and Plot Amplitude Density

Function (0.5 min.)

Machine Function Time

(2.0 min.)

Total Time

13-2 min.

65
Appendix D*

THE LINEAR LEAST SQUARES CURVE FIT

For the general case of a linear least squares (LSQ) curve fit of

the form:

y = a x + b,

where y is the dependent variable, x the independent variable, a the

slope, and b the intercept, the following formulae apply.

(1) b = - E y. - - I x.

N E x. y. - (S x.) (E y.)

(2) a = 3l_J: i i_

N E x? - (Z x.) 2 l l

where E indicates the summation over the n index from 1 to N.

For the special case of equally spaced x increments, x. may be

replaced by an integer n corresponding to the value of i and the

summations over x. and x? evaluated as standard series.

i i

(3) E x

±

= 2 n = | (N + 1)

(4) E x

2 = E n2 = |

(N + 1) (2 N + 1)

Substitution of equations (3) and (4) into equations (1) and (2),

with some simplification, leads to:

(5) b = l^y. - |

(N + 1)

12 E i y. - 6 (N + 1) I y.

(6) a = N(N + 1)(N - 1)

A flowchart and listing of the program for the least squares calculation is given in appendix B.

*From DASR.

66
Appendix E*

STEP LOCATION

At the interrupt signal generated near the beginning of the

stylus stroke, a one second delay is executed by the computer and

data is read into memory for twenty-five seconds at the rate of

approximately twenty points per second. Since the event marker is in

the information channel, the delay is necessary to avoid reading the

event maker as data. The signal which is recorded in those twentyfive seconds is thus entered as 512 data points in memory locations

1000]_6 to l400i6. The memory locations of selected points and the

corresponding positions on the strip chart are indicated in figure 23.

These data are measured in mm relative to the trailing edge of the

event marker as illustrated in figure 24.

Figure 24a illustrates a properly positioned stepr-one which is

approximately centered within the 40 mm - 110 mm region. A

representation of the data recorded in memory is given in figure

24b.

The length of the segment on the chart, corresponding to the 128

points over which a least-squares straight line is computed, is 37.5

mm. The address A]_ (input during program execution) must correspond

to a location in the region between the 40 mm position and the step

edge; similarly, A2 must correspond to a location in the region

between the step edge and the 110 mm position. These positions are illustrated in figure 25. The line segments corresponding to the A]_

and A2 selected are indicated in figure 25b. Choices for A]_ and Ap are limited to pairs of points situated equally distant to the left and

right of the step location, S, such that:

S(mm) - A]_(mm) = Ap(mm) - S(mm)

.

A}_ and Ap determine individually over which segments the lines are

to be computed by least squares fitting; the mean of A]_ and Ap

determines the location at which a step height is to be computed

from the two lines. The effects on the calculations of the step

height of proper and improper selections of A]_ and Ap may be

significant; these effects are illustrated in figures 26a and 26b,

respectively

.

*From DASR.

67
Distance on Strip Distance on Strip

Chart From Event Chart From Event

Marker to Step in Hexadecimal Marker to Step in Hexadecimal

MM Address

1116

MM Address

0040 0075 1206

0041 HIE 0076 120E

0042 1124 0077 1214

0043 112A 0078 121A

0044 1132 0079 1222

• 0045 1138 0080 1228

0046 1140 0081 1230

0047 1146 0082 1236

0048 114E 0083 123E

0049 1154 0084 1244

0050 115A 0085 124A

0051 1162 0086 1252

0052 1168 0087 1258

0053 1170 0088 1260

0054 1176 0089 1266

0055 117E 0090 126E

0056 1184 0091 1274

0057 118A 0092 127A

0058 1192 0093 1282

0059 1198 0094 ' 1288

0060 11A0 0095 1290

0061 11A6 0096 1296

0062 11AE 0097 129E

0063 11B4 0098 12A4

0064 11BA 0099 12AA

0065 11C2 0100 12B2

0066 11C8 0101 12B8

0067 11D0 0102 12C0

0068 1ID6 0103 12C6

0069 11DE 0104 12CE

0070 11E4 0105 12D4

0071 11EA 0106 12DA

0072 11P2 0107 12E2

0073 11F8 0108 12E8

0074 1200 0109 12F0

Figure 23: Step Locator 'Chart

68
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69
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FIGURE 25. SELECTION OF Al AND A2 LOCATIONS

70
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.

71
APPENDIX F

STEP HEIGHT CALCULATION

From the two lines determined by a least squares fit to the

profile data from the two opposite sides of the step profile, the

step height is calculated as the mean ordinate difference along the

horizontal distance between the line segment ends, selected by the

operator as described in appendix E. Thus, calculations as presently

implemented in the system software assume that the errors introduced by non-parallel sides of a step and non-parallelism of the

step relative to the reference datum (the reference surface plane) are insignificant with respect to the overall system accuracy . This

assumption is valid because the large vertical to horizontal

magnification ratios used with the stylus instrument allow only very

small angles to be on scale. Two extreme examples for a step height

approximately equal to one-fourth of full scale are presented in

figure 27, as they would appear on the stylus instrument's

chart recorder; figure 27a also shows, by a dashed line the output

for a step height of approximately one-tenth full scale.

Vertical to horizontal magnification ratios employed with the

stylus instrument are 50 or greater for inputs to the step height

calculations. Thus, even though the angles on the strip chart

recorder, in the extreme cases illustrated, are 15° and 30°, the true

angles are 0.31° and 0.66°, respectively, for a magnification ratio

of 50.

Errors introduced by this approach are most obvious from the

examples shown in figure 27a. For these examples the natural

definition of the specimen's step height is the perpendicular distance

between the two lines defining the step's opposite sides.

Measurements performed in this manner directly from the strip chart

recorder output would be in error unless corrected for the vertical to

horizontal magnification ratio. The true error between calculating

the step height as the ordinate difference and calculating it as the

perpendicular distance is:

Ah = y - / y

2 + x2

,

where y and x are as shown in figure 27a. For a magnification ratio

of 50 the fractional error is:

Ah _ I 1 r , tan 15°"h

y

I [_ 2 5°°

J

= 1.4 x 10 5

,

The definition of the step height whose profile is as shown in

figure 27b is not obvious . The approach taken in DASR was to define

the step height as the "mean perpendicular distance" between the two

least-squares fitted lines. This mean perpendicular distance,

hjyrpD, was calculated as the arithmetic mean of the two distances

between the point A located on the bottom of the step and the points of

72
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Figure 27. Schematic Strip Chart Records of Highly Sloped Profiles.

73
intersection of the perpendicular lines with the top of step as shown

in figure 27b. The line perpendicular to the right hand side of the

step is coincident with the step profile and is defined as the

ordinate difference, hop. The difference between the "mean

perpendicular distance", hjypp, and the ordinate difference, hQD> for

this example is:

^d - hoD

= h Cy + /y2 + x2 ] - y

or |h^

D

- h

QD | = h |Ah|.

For an angle of 30° on the strip chart recording and a magnification

ratio of 50, the fractional difference between the two definitions is

3.4 x 1(T5.

Implementing the ordinate difference definition of step height

requires only that the two lines fitted by the least squares procedure

be transformed to a common co-ordinate system. In figure 28, the line

fitted to the step's low side, S^, is referenced to the primed system

and the line fitted to the high side Sfj, is in the unprimed system; k-\_ and A2 are the step locator points entered by the operator. Prom the

least squares fitting operation one has for the representations of

lines Sl and S^:

K

= aL

x' + b

L

and

yH

= aH

X + V

The CO--ordinate transformations are:

y' = y

X' = X + c + z, where c = 128 = 80 (hexadecimal) and Z = A2 - A-j_. By transforming

both lines to the unprimed system, the result:

and

yL

= a~ (x + c + z) + b

T

yH

= a^ x + bR

,

is obtained. The step height, H, at a particular abscissa, x, is

therefore

n
x' Al A2 x

Figure 28: Co-ordinate System for Step Height Calculation.

H = yH

- yL

= (^ - aL

) x + (bR

- b ) - (Z + c) a^.

The step height routine given in appendix B calculates H for values of

X from -Z to in increments of Z/8.

To confirm that the step height calculation was insensitive to

changes in the slope of the step profile with parallel horizontal

surfaces, a sequence of measurements at the slopes shown in figure 29 was made. The slopes of the profiles in figures 29b and 29c are the

maximum possible positive and negative slopes for which on scale profile data may be obtained. Results of 10 measurements at each

of these cases is given in table 6. The format of the data is

explained in appendix K. These results show that the maximum

difference between the mean step height calculated from a

level profile and one from a highly sloped profile is 0.2$. The 3s

value (see page 32) of this calculated step height is 0.2$ and is

primarily due to surface finish (section 2.2). Thus the uncertainty,

3s, in the difference between step height calculated for the two

cases would be 0.3$. The difference between uns loped and sloped

profile calculations being less than its 3s uncertainty therefore

verifies that to within the limits imposed by surface finish effects

no statistically significant relationship between calculated step

height and input profile slope is present. The uncertainties of these

measurements are larger than the expected 0.2$ found experimentally in

appendix K. This increased uncertainty is produced by the scratches

or pits which were unfortunately present in the right side of the

step.

75
Figure 29a. Slope ^ £)•

Figure 29b. Slope = Maximum Positive Value

Figure 29c. Slope = Maximum Negative Value.

Figure 29. Profiles of 12.8 y m Step Used to Obtain Statistical

Data on Possible Variation of Calculated Step Height

with Changes in Slope. Horizontal Magnification X20;

Vertical Magnification % 1000.

76
2500G

IN00002920 FB1F1800 FFFFECAO 00905800 00001756 055B5830

HI H3 H5 H7 HM

30001004 00001002 00001000 00000997 00001000

IN

00002820 FB1E9800 FFFFE920 008E8800 00001756 05532F00

HI H3 H5 H7 HM

00001004 00001002 00000999 00000997 00001000

IN

000028E0 F31C0800 FFFFEF60 003B4300 00001756 05593680

HI H3 H5 H7 HM

00001002 00001000 00000993 00000996 00000998

IN

00002F60 FB1BA800 FFFFE7A0 C08B1800 00001756 05566F40

HI H3 H5 H7 HM

00001001 00000998 0000099 6 00000993 0000099 6

IN

000035A0 FB135300 FFFFF620 00833800 00001756 0551B480

HI H3 H5 H7 HM

00000997 00000995 00000992 00000990 00000993

IN

00003020 FB13F800 FFFFE320 00383800 00001756 055A7 100

HI H3 H5 H7 HM

00001004 00001001 00000999 00000996 00000999

IN

00003060 FB030800 FFFFEE60 007B0800 0000 1756 05556400

HI H3 H5 H7 HM

00001000 00000998 0000099 5 0000099 3 OC 00099

6

IN

00002C20 FB0ED800 FFFFEEEO 007F6800 00001756 0558AFC0

HI H3 H5 H7 HM

00001002 00001000 00000998 0000099 5 00000998

IN

000039A0 FB0EB800 FFFFF600 003380CO 00001756 05594SE0

HI H3 H5 H7 HM

00001 003 00001000 00000998 0000099 6 00000999

IN

000C31A0 FB103S00 FFFFECEO 00876800 00001756 055C5240

HI H3 H5 H7 HM

00001005 0000 1003 00001000 00000998 0000 1001

Mean = 998 3 Standard Deviation =7

Table 6a. Statistical Data for Height Measurement of

12.8 ym Step with Surface Finish of 13 nm

AA. Slope Relative to Datum % 0.

77
IN

00052F60 F6576300 0004FE80 043E0000 00001756 055S2DA0

HI H3 H5 H7 HM

00001003 00001001 00001000 00000993 00001000

IN

000540A0 FB53D800 0004FD60 043E2800 00001756 0554F9C0

HI H3 H5 H7 HM

00000999 0000099 7 0000099 5 0000099 3 0000099 5

IN

000529E0 FB0A8800 0004F3C0 03EF5000 00001756 055DB360

HI H3 H5 H7 HM

00001005 00001003 00001001 00000999 00001002

IN

00052F80 FB0A4000 0004EEC0 03F17000 00001756 055D5C40

HI H3 H5 H7 HM

00001005 00001003 00001001 00000999 00001002

IN

00052840 FAFFB000 0004F8C0 03E31000 00001756 055C36S0

HI H3 H5 H7 HM

00001004 00001002 00001001 00000999 00001001

IN

000532C0 FAFA1000 0004FC60 03DFA800 0000 1756 0553702C

HI H3 H5 H7 HM

00001001 00000999 00000997 0000099 6 0C00099S

IN

00052BA0 FAFED300 0004F240 03E59000 00001756 055EC720

HI H3 H5 H7 HM

00001006 00001004 00001002 00001000 00001003

IN

00052E20 FAF8D800 0004EE40 03E0D000 00001756 055EFEA0

HI H3 H5 H7 HM

00001007 00001004 00001002 00001000 00001003

IN

00053220 FAFA3800 0004F500 03DF3000 00001756 05594460

HI H3 H5 H7 HM

00001002 00001000 00000998 0000099 6 00000999

IN

00052CAO FAFAF800 0004F540 03DED000 00001756 05530 120

HI H3 H5 H7 HM

00001003 00001001 00000999 00000997 00001000

Mean = 1000 3 Standard Deviations = 7

Table 6b. Statistical Data for Height Measurement of

12.8 pm Step with Surface Finish of 13 nm

AA. Slope Relative to Datum t= Maximum Positive

Value
IN

00000000 07FF0000 00000000 07FF0000 00001756 00000000

HI H3 H5 H7 HM

00000000 00000000 00000000 00000000 00000000

IN

FFF6F8E0 01F20800 FFF6C600 01130000 00001756 055333A0

HI H3 H5 H7 HM

00001003 00001001 00001000 00000998 00001000

IN

FFF70180 01F3E000 FFF6C380 01174000 00001756 0558E200

HI H3 H5 H7 HM

00001002 00001000 00000998 00000996 00000998

IN

FFF707A0 01F01800 FFF6CDC0 01117000 00001756 055234A0

HI H3 H5 H7 HM

00000997 00000995 00000993 0000099 1 00000993

IN

FFF70400 01F48000 FFF6DC20 01133800 00001756 05502CA0

HI H3 H5 H7 HM

00000994 00000993 00000992 00000990 00000992

IN

FFF70360 01DEC800 FFF6D760 00FEE800 00001756 05527200

HI H3 H5 H7 HM

00000996 00000995 00000993 00000992 00000994

IN

FFF6FCC0 008F7000 FFF6C1C0 FFB21000 00001756 055S1500

HI H3 H5 H7 HM

00001003 00001001 00000999 00000997 00001000

IN

FFF6F430 009 1A000 FFF6C700 FF324000 00001756 055D54S0

HI H3 H5 H7 HM

00001004 00001003 00001001 00001000 00001001

IN

FFF6FFA0 0090D800 FFF6C640 FFB29000 00001756 05580720

HI H3 H5 H7 HM

00001001 00000999 00000997 00000995 00000998

IN

FFF70920 008AF800 FFF6C830 FF30A000 00001756 055632E0

HI H3 H5 H7 HM

00001000 00000998 00000996 00000994 0000099 6

IN

FFF6FC60 00920800 FFF6CF40 FF313000 00001756 05566860

Hi H3 H5 H7 HM

00000999 00000998 00000996 00000995 00000996

> Mean = 997 3 Standard Deviations = 9

Table 6c. Statistical Data for Height Measurement of 12.8 urn Step with Surface Finish of 13 nm AA. Slope

Relative to Datum = Maximum Negative Value

79
APPENDIX G*

'

THE AA CALCULATION AND AMERICAN NATIONAL

STANDARD B46.1

The AA roughness value of a surface profile is defined as the

arithmetic average deviation of the surface profile from its center

line; this center line in turn is defined by American National

Standard B46.1 as "the line parallel to the general direction of the

profile within the limits of the roughness width cut-off, such that

the sums of the areas contained between it and those parts which lie on either side of it are equal"

.

The specific details involved in the implementation of these

definitions in the operation of analog or digital devices are not

delineated in the standard. While band-pass characteristics in terms

of half-power points and roll-off rates are specified, the exact

nature of center-line, to which a profile is instantaneously

referenced, is not clearly specified (again see appendix A for filter

characteristics)

.

In the operation of an integrating-meter stylus instrument, the

filtered signal represents the surface profile with wavelength cut-off

restriction imposed. The continuous analog computations of the center

line and average deviation from the center line are made by

integrating circuitry. Specifically, the stylus instrument stroke is

begun at time T]_ and the resulting signal averaged up to time t to

establish a center-line; beginning at a particular point, T2, in the

stroke, the instantaneous signal is compared to the established center

line and the magnitude of the difference is averaged over a further

length of the stroke to time T->. Mathematically, the operation

corresponds to the integral:

AA = Ti - T

3

± 2

T

3

/

T

Abs <f(t) -

" T

l

/ f(t')dt'

I

T

l

dt,

This equation reflects the necessity in an analog device of comparing

the instantaneous signal to a center line which does not bracket the

signal, but rather represents a segment of the profile of which the

signal is the trailing edge.

In the operation of the present digital system, the filtered

signal is digitized and stored in the computer memory; the conversion

rate (1.64 kHZ) is sufficiently high to reproduce the analog

information in the filter bandpass (-3dB frequency is 300 HZ), with no

loss of fidelity. The record length of the data corresponds to five

wavelength cut-off widths . Since wavelength cut-off restrictions have

*From DASR. (Revised)

.

80
been imposed on the signal by filtering, the center line is computed

over the record length. Mathematically , the operation corresponds to

the integral:

T T

| J

Abs (f(t) -| Jf(t')dt')

«-

Although the mathematical descriptions of the analog and digital

computations differ, it is believed that the digital computations are in accord with the present American National Standard B46.1 on surface

texture, in terms of the fundamental definitions of AA and the center

line. Since the effective center line is computed as the mean over

the record length and the record width is five times the cut-off

width, the effect of a fractional roughness wavelength in the record

length is greatly diminished.

81
APPENDIX H

STATISTICAL EVALUATION OP THE LONG TERM STABILITY OF STEP CALIBRATED

ROUGHNESS MEASUREMENTS

In cooperation with Joseph M. Cameron of the Office of Measurement

Services at NBS, a statistical evaluation of the procedure for

measuring surface roughness was performed. The measurements for this

study were made over the period extending from November 9, 1973 to

January 30, 1974. The specific days of measurement were: operator 1;

November 9 and 12, 1973; operator 2; November 12, 13 and 14, 1973 and

January 4, 21, 25, 28 and 30, 1974. Each day the procedure consisted

of calibrating the system at two magnifications with an

interferometrically measured step then measuring the specimen

roughness for three traverses at the number of positions indicated in

the enclosed memorandum. The specimen used for the evaluation was a precision roughness specimen conforming to American National Standard

B46.1-1962. Only a small area of each patch, approximately 3 mm by

25 mm, was used for the study to minimize the effect of positional

variations on the resultant data. Random positions within this area

were used for the indiated measurements.

The following memorandum presents Joseph M. Cameron's analysis of

the data.

82
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington. D.C. 20234

Date: July 19, 1974

Clayton Teague

To: Dimensional Technology Section

Optical Physics Division

From. J. M. Cameron, Chief

Office of Measurement Services

subject: Analysis of Roughness Data

A. Structure of the Data

Two patterns of observations were made.

1. Triplicate measurements were made on each of 6 positions on

5 different days.

2. Triplicate measurements were made on each of 10 positions

on 5 different days subsequent to those in (1).

Both patterns were carried out on specimens of nominal value 20(AA)

and 120(AA). For each day a new correction factor was used to

normalize the results.

Table I and Table II give the triplicate values and their average

and standard deviation based on the differences among the triplicate (d.f. = 2) for 20(AA) and 120(AA) respectively.

B. Variability Within Triplicate

Figure 1 and figure 2 plot the deviations of the triplicate measurements from their averages and show quite clearly that triplicate

numbers 4, 56, and 68 on the 20(AA) data and number 61 on the 120(AA)

data are not consistent with the remainder of the data. If these

triplicates are omitted, then the "within triplicate" standard

deviations are:

Nominal d.f. s.d. s.d. of Average of Three

20(AA) 151 .0486 .0280

120(AA) 157 .5320 .3070

83
These values of standard deviation would serve as "accepted" values

for control of the measurement process. Because the other components

of variation are much larger (as will be seen later), the addition of more runs with only duplicates instead of triplicates would be the

better partition of a fixed number of observations.

C. Day-to-Day Component of Variation

Let y denote the average of the triplicate measurements, then its value

on the i tn day on the j

tn position can be written as

y = u . + 6 . + c.

where \ii is the value for the j

tn position; 6i is the random error

due to day-to-day variations--Var(6-j) = o£; £-jj is the random error

of the average about the value for the position and day V(e-jj) = a

2

.

The variance of y can be represented as

Var(y) = o^ + o

z

The daily average of n positions will have the structure

,th

average for the i VC^) = a

2 + a2 /n

day = y i = u + 5

i + Se^/n

Table III gives the daily averages for the two nominal roughness

specimens and table IV shows the estimates of the variance component.

TABLE III: Deviation of Daily Average From Overall Average

20 AA 120 A

6 positions

Overall

Average 1888.64

Day 12

3

4

5

.1674

-.1526

-.0209

-.1064

.1124

6 positions

Overall

Average 1196.35

-.047

.009

-.297

.081

.253

10 positions

Overall

Average 1887.78

Day 12

3

4

5

.1036

-.2297

.0383

.0066

.0813

10 positions

Overall

Average 1194.94

.349

.003

.629

-.474

-.507

84
TABLE IV: Estimation of Variance Components

20 AA

s.d. of Daily

Averages

(d.f. = 4)

Between Day

Component

a

B

~2 a

6 positions .1376 .017956 .0058 .1340 .0764

10 positions .1338 .016824 .0108 .1297 .1040

Combined .1350 (10 days) .017390 .0083 .1319 .0912

120 AA

6 positions .237 * .3367 * .5800

10 positions .500 .2238 .2624 .4730 .5120

Combined .368 (10 days) .1138 .2930 .3370 .5470

*Between component not present.

The standard deviation of the daily averages* would have to be taken

as the appropriate variability for assessing the uncertainty of the

roughness measurements. This leads to a 3 s.d, limit of .40 AA on a mean of 18.88 (or a percentage uncertainty of 2.1%) for the 20 AA

specimen and to a 3 s.d. limit of 1.1 AA on a mean of 119.5 for a percentage uncertainty of 0.9% for the 120 AA specimen.

Figures 3 and 4 plot the deviations of the daily values from the

averages for the positions for the 20 AA specimen and 120 AA specimen

respectively. Figures 5 and 6 show the values for the specimens on successive days.

For the 20 AA specimen, statistically significant position-to-position

variation exists whereas it does not for the 120 AA specimen. These

are shown in Table V as deviations from the grand average.

*s.d. of average of ten values = /a§ + a

2 /10 give combined value of

.135 for 20 AA and .368 for 120 AA.

85
TABLE V: Deviations of Position Averages From Overall Average*

20 AA 120 AA

Position 1 .1469 .411

2 -.0798 -.069

3 -.0244 -.002

4 -.0364 -.249

5 .0176 -.169

6 -.0238 .078

Position 1 .2209 .213

2 -.0664 .153

3 -.0984 -.107

4 -.0677 .139

5 .0363 -.094

6 .0443 -.001

7 -.0391 -.001

8 .0703 -.121

9 -.0437 -.081

10 -.0564

1

-.101

*s.d. of position average =

/(

- and .210 for 120 AA.

v *

86
D. Correction Factor

Each day a correction factor was determined and used with the

results of that day. Table 6 shows these correction factors along

with the daily averages and their corresponding rankings. Figure 6

is a plot of the rankings of the daily averages as a function of

the rankings of the correction factor. No significant dependence

of the results on the magnitude of the correction factors is apparent,

1

2

3

4

5

6

78

9

10

TABLE VI: Correction Factors

20 AA

Correction

Factors

45778

46070

45680

45394

45834

45148

45032

45164

45016

45732

R

A

NK

Daily

Average

RA

N

K

120 AA

Correction

Factors

6 Positions

.167 5

-.153 1 -.021 3

-.106 2

.112 4

10 Positions

.104 5

-.230 1 .038 3

.007 2

.081 4

44184

44230

44142

44052

44290

43902

43856

43630

43316

43664

R

A

NK

Daily

Average

R

A

NK

3 -.047 2

4 .009 3

2 -.297 1

1 .081 4

5 .253 5

5 .349 4

4 .003 3

2 .629 5

1 -.474 2

3 -.507 1

Figure 5: Ranking of Daily Average as a Function of the Ranking

of Correction Factors

5 * * * *

4 * * * *

3 * * * *

2 * * * *

1 * * * * 12345 12345 12345 12345

6 pos. 10 pos.

20 AA

6 pos. 10 pos.

120 AA

87
Conclusions: The following conclusions emerge from the analysis.

(1) The standard deviation of the average of k values is

0.0486/ZF-yin for the 20 yin AA and 0.5320/A-yin for the

120-yin AA specimens which turns out to be 0.028 yin and

0.31 yin respectively for k=3. One would use these values to

establish process control on the range of duplicate or triplicate

measurements.

(2) If an average of values from 10 positions is used to establish a value for a test specimen, then the standard deviation associated

with such averages is .135 for 20 AA and .378 for 120 AA ignoring

position-to-position (i.e., if one could return to the same 10

positions) .

(3) If there are significant position-to-position variations, then

the question as to how well the average of 10 positions represents

the specimen is brought into focus. Th e value for a single

position will have standard deviation /ag + a* which turns out

to be .427 for 20 AA and .6378 for 120 AA. If one regards the

position variation as having a variance component ap, then the

s tandard devia tion based on ten positions has the long-run value

/an + aS + a

2"

is if the 10 positions are selected at random, the average, y, _

II have a standard deviation that can be estimated by s = K {y-yf/9,

Thus

wil

i (It would not be proper to use the triplicate values and compute

s using 30 values and with divisor 29 because of the correlation

of the triplicate values with each other.)

88
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APPENDIX I

STYLUS TIP RADIUS MEASUREMENTS

Periodically the stylus tip radius used in the NBS roughness

measurement system is measured by obtaining a profile of the tip as it

is traversed across an edge whose radius of curvature is very small

compared to that of the stylus tip. Most razor-blades manufactured

after about i960 have edges whose radii of curvature are less than 0.1

ym [19]. The radius of curvature observed in the profile obtained

from the motion of a stylus tip across a razor-blade edge is equal to

the sum of the tip radius and any curvature present in the razor-blade

edge. Thus, the observed radius of curvature is an upper bound on the

tip radius i.e., the tip radius is less than or equal to that observed

in the profile.

For graphical determinations of the tip radius large, equal

vertical and horizontal magnifications should be used. The profile in

figure 30 was obtained in the most recent measurements of the stylus's

tip radius. A convenient method for estimating the radii of inscribed

and circumscribed circles for tips with a 90° included angle is to

measure the distance from the point of intersection of lines drawn

along the tip sides to the estimated points of tangency with the

measured profile. The results of this procedure are illustrated in

figure 30.

Based on the two radii values shown in figure 30, an estimated tip

radius would be (3.^+0.6 ym) . This estimate does not include any

uncertainty due to elastic or plastic motion of the razor-blade edge

during the motion. However, profiles of the tip were obtained at

several different positions along the razor-blade edge and by

traversing the tip across the same position several times. All the

profiles measured were very similar in shape and had approximately the same radii of curvature as the one shown in figure 30.

A further confirmation of this radius value and of the measurement

technique was obtained by making a photomicrograph of one face of the

pyramidal tip. The photomicrograph is shown as an insert in figure

30; the included angle is approximately 70° which is the angle expected

when looking in a direction perpendicular to one face of a 90°

pyramid

.

99
cd

100
APPENDIX J*

NOISE EFFECT IN AA IVEASUREMENT

It is known that measurements of ultrafine finishes are limited in

accuracy by the inherent noise level of the stylus instrument itself.

The specific effect of this noise has been established by a study of

the computer calculated AA values as a function of the signal-tonoise ratio. The experimental set-up is shown in figure 31. An

operational amplifier was employed as an inverting adder without

weighting. The apparatus allowed the stylus instrument output to be

measured separately and combined with a known signal. Measurements

were made of the inherent electronic noise in the system (with the

stylus stationary), the mechanical and electronic noise (stylus

resting on a plane which traversed with the stylus), and the stylus

output while traversing an ultrafine surface (a float-formed glass

substrate). The results for each case, normalized to the zero signal

input level, are indicated in figure 32. The smooth curve represents

the equation:

Signal Output - [(Signal Input) 2 + (Noise) 2

] 1/2

.

The actual "noise" levels associated with the three cases above

indicated: (1) an electronic noise equivalent to .13 yin. AA, (2) a

mechanical and electronic noise equivalent to .16 yin. AA, and (3) a

probable minimum AA reading of .25 yin. It has not been determined

what portion of the minimum AA reading is due to the highly

polished surface. Therefore, if is assumed that the full minumum

reading is due to the instrument; thus; the total inherent noise is

taken to be .25 yin.

*From DASR

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Results of Noise Measurement

Figure 32

103
APPENDIX K

EFFECTS OF SUFFACE FINISH ON STEP HEIGHT

1VEASUREMENTS FROM STYLUS PROFILES

To check the analytical predictions of section 2.2, measurements

on two step-specimens were performed. Profiles of the two specimens

used in the measurements are shown in figure 33. Twenty-eight (28)

measurements of each specimen in approximately the same location were

made to obtain a sufficient amount of data to statistically

characterize the relationship between surface finish and calculated

step height. The procedure for the measurements was to employ each of

the steps as a calibrating step, labelling its height as 10,000, then

to repetitively remeasure its height using the calibration constant,

KCAL, from the first measurement.

Analysis of the profile data followed the procedure outlined in

section 1.2. Computer printouts of the analysis are given in tables 7

and 8. In each of the 56 printouts the following results are given

after the word IN:

line 1; left slope - left intercept - right slope - right

intercept - KCAL - step height (all in hexadecimal)

,

line 2; step height headings,

line 3; calculated step heights.

For the 12.8 ym step with a surface finish of 13 nm AA, the

3s value* of the data was + . 16% of the mean step height . For the

503 ran step with a surface finish of 8 nm AA, the 3s value* of

the data was +2.0$ of the mean step height. Expected values, based

on the predictions of section 2.2, for the two specimens were 0.11$

and 1.7%, respectively. The predicted 3s values* would be slightly

larger if system noise and quantization were included, but not by

the amount needed to give exact agreement with the measured values.

Agreement between the analytical and measured values is however

sufficient to confirm that the major source of statistical

variation in calculated step heights is the step specimen's surface

finish.

*See page 32 for the definition of s.

104
~ -

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. ^xzn20

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Vertical Magnification - 2000; Horizontal Magnification

- 20.

i '20

'- h

'

i- n

a ^*- KJtaj, »«,«« V 1

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Figure 33b: Profile of 503 nm Step with Surface Finish of 8 nm AA.

Vertical Magnification - 100,000; Horizontal Magnification - 20.

Figure 33: Profiles of Steps Used for Taking Statistics on Step

Height Measurements.

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INFFFFE760 FD148800 FFFFD320 04A0B800 0000A488 079E4D80

HI H3 H5 H7 HM . 00010035 00010031 00010027 00010023 00010028

Table 7a: Measurements on Height of 12.8 ym

Step with Surface Finish of 13 nm

AA.

106
Mean of all 28 = 10032 ± (3a = 16 or 0.16% )

INFFFFE320 FD14B800 FFFFD200 04A04000 0000A488 07A02A50

HI H3 H5 H7 HM

00010043 00010040 00010036 00010033 00010037

INFFFFE680 FD13A000 FFFFD400 049F2000 0000A488 079E0840

HI H3 H5 H7 HM

00010033 00010029 00010025 00010022 00010026

INFFFFE620 FD125800 FFFFD2E0 049D6800 0000A488 079DE940

HI H3 H5 H7 HM

00010032 00010029 00010025 00010021 00010026

INFFFFE300 FD1 1A000 FFFFD180 049CC000 000GA488 079FE040

HI H3 H5 H7 HM

00010042 00010038 00010035 00010031 00010036

INFFFFE180 FD1 12000 FFFFD160 049B4800 0000A488 079FC570

HI H3 H5 H7 HM

00010041 00010038 00010034 00010031 00010035

INFFFFE3A0 FD0FF8 00 FFFFD0 60 049BA8 00 0000A488 07A02FC0

HI H3 H5 H7 HM

00010044 00010040 00010036 00010033 00010037

INFFFFE2C0 FD10B000 FFFFD480 049B8000 0000A488 079F6E20

HI H3 H5 H7 HM

00010038 00010036 00010033 00010030 00010033

INFFFFE320 FD125800 FFFFD560 049E6800 0000A488 07A06300

HI H3 H5 . H7 HM

00010043 00010040 00010038 00010035 00010038

INFFFFE540 FD1 17000 FFFFD3E0 049E8800 0000A488 07A059 10 HI H3 H5 H7 HM

00010044 00010041 00010037 00010034 00010038

INFFFFE400 FD11C000 FFFFD420 049DF800 0000A488 07A029 10 HI H3 H5 H7 HM

00010043 00010040 00010036 00010033 00010037

INFFFFE2E0 FD128800 FFFFD180 049E8000 0000A488 07A0CA70

HI H3 H5 H7 HM

00010047 00010043 00010040 00010036 00010C40

INFFFFE480 FD 1 16000 FFFFD3C0 049D9000 0000A488 079FE120

HI H3 H5 H7 HM

00010042 00010038 00010035 00010032 00C10036

INFFFFE4E0 FD104800 FFFFD440 049CF000 0000A488 07A0 1 6D0

HI H3 H5 H7 HM

00010043 00010039 00010036 0C010C33 00010037

INFFFFE3E0 FD10A800 FFFFD360 049CC800 0000A488 07A034E0

HI H3 H5 H7 HM

00010043 00010040 00010037 00010033 00010037

Table 7b: Measurements on Height of 12.8 ym Step

with Surface Finish of 13 nm M.

107
2534G

INFFFF5C40 F8E7F000 FFFEC3A0 057F7800 00005FS4 0D0B1400

HI H3 H5 H7 HM

OOOlOOll 00009997 00009983 00009968 0000998 6 INFFFFB060 F92E4800 FFFFABCO 06577000 00005FB4 0D5BE7C0

HI H3 H5 H7 HM

00010229 00010228 0001*0228 00010227 00010228

IN00003920 F9361800 FFFF67C0 06741000 00005FB4 0D2A2D40

HI H3 H5 H7 HM

00010114 00010094 00010074 00010055 00010079

INFFFFF120 F94C9800 FFFF8640 0685F000 00005FB4 0C4B1840

HI H3 H5 H7 HM

00010195 00010185 00010175 00010165 00010178

INFFFFF260 F96E4800 00000700 06738000 00C05FB4 0D143AC0

HI H3 H5 H7 HM

00010010 00010012 00010014 00010016 00010014

IN000047C0 F9431000 FFFF6080 068C4000 00005FB4 0D2DD980

HI H3 H5 H7 HM

00010128 00010106 00010085 00010063 00010090

IN00000220 F9515800 FFFF8880 066D4000 00005FB4 0D241040

HI H3 H5 H7 HM

00010081 00010070 00010058 00010047 00010061

INFFFFFAOO F945E000 FFFFACEO 0663C800 00005FB4 0D27BA80

HI H3 H5 H7 HM

00010085 00010077 00010070 00010063 00010072

INFFFFB980 F9440000 FFFF82E0 065CC800 C0005FB4 0D49A980

HI H3 H5 H7 HM

00010182 00010177 00010172 00010167 00010173

IN00004000 F8FB8000 FFFF60E0 0659C800 00005FB4 0D473680

HI H3 H5 H7 HM

00010203 00010182 00010161 00010140 00010166

INFFFFFF20 F909 1800 FFFF6A60 0636E800 00005FB4 0D39FCC0

HI H3 H5 H7 HM

00010151 00010137 00010123 00010109 00010127

INFFFFD680 F9378000 FFFF6C20 066FD800 00005FB4 0D5AEA80

HI H3 H5 H7 HM

00010242 00010232 00010223 00010213 00010225

IN000006C0 F9053000 FFFF6FE0 063AE800 00005FB4 0D3D3C00

HI H3 H5 H7 HM

00010161 00010147 00010133 00010119 00010136

IN00000880 F8F4E000 FFFF3760 0637A800 00005FB4 0D4DBD80

HI H3 H5 H7 HM

00010220 00010200 00010181 00010161 00010186

Table 8a. Measurement on Height of 503nm Step

with Surface Finish of 8nm AA.

108
M = 10143 + (3a = 207 or 2.0%)

IN00002460 F8E8 6800 FFFF5AA0 06397800 00005FB4 0D49CE40

HI H3 H5 H7 HM

00010207 00010188 00010169 00010150 00010174

INFFFFEF20 F8EFD800 FFFF4D40 06417000 00005FB4 0D68E840

HI H3 H5 H7 HM

00010293 00010278 00010263 00010248 00010267

INFFFFB480 F8FD4000 FFFF9 6E0 061BC800 00005FB4 0D509F80

HI H3 H5 H7 HM

00010199 00010196 00010194 00010191 00010194

IN00000500 F9 1CA000 FFFFAFEO 0647C800 00005FB4 0D2EA480

HI H3 H5 H7 HM

00010107 00010099 00010091 00010083 00010093

IN00001340 F8FEF000 FFFF4680 063BA000 00005FB4 0D408080

HI H3 H5 H7 HM

00010180 00010160 00010141 00010122 00010146

INFFFFCDOO F90C0000 FFFF4E40 063AS000 00005FB4 0D58DD00

HI H3 H5 H7 HM

00010240 00010228 00010216 00010204 00010219

IN00001CC0 F8FED000 FFFFBECO 062C1000 00005FB4 0D226680

HI H3 H5 H7 HM

00010071 00010063 00010054 00010045 00010056

INFFFFD3C0 F903D000 FFFF9220 0638 1800 00005FB4 0D556900

HI H3 H5 H7 HM

00010219 00010213 00010207 00010201 00010209

INFFFFF420 F8FAD800 FFFF7600 062A8000 00005FB4 0D410640

HI H3 H5 H7 HM

00010168 00010156 00010145 00010133 00010143

INFFFFECCO F8D97000 FFFF87C0 06039000 00005FB4 0D3E3280

HI H3 H5 H7 HM

00010156 00010146 00010137 00010127 00010139

INFFFFB620 F8E1B800 FFFF7FA0 0604B300 00005FB4 0D5601C0

HI H3 H5 H7 HM

00010219 00010214 00010209 00010204 C0010210

INFFFFEC80 F8E02000 FFFF5740 061F5000 00005FB4 0D573000

HI H3 H5 H7 HM

00010238 00010224 00010210 00010196 00010214

IN00001680 F8F02000 FFFF9520 061BB800 00005FB4 0D277680

HI H3 H5 H7 HM

00010092 00010080 00010068 00010056 00010071

IN000039C0 F9 1F5000 FFFF3D60 0672A800 00005FB4 0D428400

HI H3 H5 H7 HM

00010193 00010170 00010146 00010123 00010152

Table 8b. Measurements on 503nm ste P w f +h

Surface Finish of 8 nm AA.

109
APPENDIX L

MEASUREMENTS OP THE STYLUS-TRANSDUCER AND

INTERFACE HARDWARE NON-LINEARITIES

Stylus-transducer non-linearity was measured by recording the

transducer's voltage output as a function of an interferometrically

measured displacement input. Displacement inputs were generated

either by driving a linear-translation-stage with a differential screw

or directly from the differential screw. The differential screw was constructed so that a 360° rotation of the instrument's barrel

produced a 6.4 ym displacement of the output screw. The lineartranslation-stage [13] employed parallel-arm flexure pivots to obtain

a reduction-ratio between input and output displacements of 69.

Displacements input to the linear-translation-stage or directly to the

stylus were measured with a Hewlett-Packard 5526 laser measurement

system including its associated remote interferometer and

retroflector . Direct displacements were therefore measured with a resolution of 25 nm; displacements from the linear-translation-stage

were measured with a resolution of 0.4 nm.

All components used in generating and measuring the displacements

were clamped together and to the Talysurf 's base to form a stiff

structure. With the stylus tip resting on the displacement-stage

mechanical vibrations had approximately twice the arithmetic average

value as found with the tip resting on a solid specimen i.e., 12 nm.

Output voltages were measured with a digital voltmeter having a

non-linearity of + one digit on the readout over the range from

+ 0.100 to + 1.000 volts. Thus, this roundoff non-linearity ranged in

the measurements from +0.1% to .+ 1% of the readout values, typically

0.2% or less . Analytic non-linearities of this magnitude and less

may however be extracted from the measurements by statistical analysis

since the data subjected to roundoff were random i.e., the next

digit was representative of the system noise level.

Non-linearity for purposes of these measurements is defined as the

maximum deviation of a quadratic least-squares fit from a linear

least-squares fit to the experimental data. (This definition was adopted after an unsuccessful search for a commonly accepted

definition of non-linearity.) Computer printouts of the data on

stylus displacement versus transducer output voltage and the results

of analyses are given in tables 9 and 10. Table 9, gives the results

obtained while using the Talysurf 's sensitivity of "1000"; table 10

gives the results obtained while using the Talysurf 's sensitivity of

"50,000". In both tables: column 4 is the output voltage in millivolts;

column 42 is the deviation of the data from a linear-least squares fit;

column 52 is the deviation of the data from a quadratic-least-squares

fit; column 53 is the difference between a linear and a quadratic least

squares fit; and column 54 is this difference expressed as a

percentage of its respective datum value. In table 10 column 2 is

110
interferometrically measured displacements in microinches input to the

linear-translation-stage; stylus displacements are therefore, the

numbers given in column 2 divided by 69. In table column 2 gives

the direct stylus displacements in microinches.

The measurements given in table 9 cover the entire range of the

Talysurf on the "1000" sensitivity so that the maximum deviation

between the linear and quadratic least-squares fit is the nonlinearity of the instrument's transducer. The "500" sensitivity is

rarely employed for measurements at NBS and thus was not used for the

non-linearity tests. Measurements of non-linearity, on the "50,000

magnification" were performed to verify that the non-linearity of the

transducer would be less over the limited range and that smaller scale

variations not resolved at the "1000 magnification" were not present.

On the "1000 magnification" with the stylus tip being displaced +

25.4 ym from an arbitrary zero the measured non-linearity, as defined

earlier, was 0.4%; on the "50,000 magnification" the non-linearity

was 0.2$. Expressed in an analytical form the results were:

aT-r-r) = (37.25 + .06) mV/ym,

y ( / linear

(IrT^r-) = [(37.15 + .10) + (3.4 + 9.3) x 10-3y] mV/ym, \ay in j quadratic quadrat

for "1000 magnification" and:

/dV out \

\dy In

J

/dV out \

\dy in

J

= (1.445 + .007) volts/ym,

linear

= [(1.443 + .035) + (0.004 + .056)y] volts/ym,

quadratic

for the "50,000 magnification." The + values after each coefficient

are the 3 standard deviations of the coefficients estimated from the

data analysis. For both the "1000 and 50,000 magnifications" the

quadratic coefficient is less than its 3 standard deviation value.

Thus, the non-linearities quoted are values based on the small

probability that a quadratic component is present in the transducer's

output voltage versus stylus displacement.

The signal path from the transducer's output to data storage in

the minicomputer was tested and analyzed in a similar manner. For

these measurements a voltage source with an accuracy of 100 yV was used as the input to the filter-amplifier (appendix A) whose

output was in turn used as the input to the analog-to-digital

111
converter , ADC . The output of the ADC was then stored in the

minicomputer memory and printed at the teletype as a function of

the voltage input. Input voltages were stepped over the range from

-1.000 to + 1.000 volts producing outputs from the filter-amplifier of

- 10.0 to + 10.0 volts. At the input to the ADC, noise was approximately 1 mv RMS, or 0.2 of the ADC's resolution. Effects

from the noise were further minimized by sampling the analog signal 16

times at 30 ms intervals to obtain the average value printed at

the teletype. An analysis of the data for memory stored values

versus filter-amplifier input voltage performed according to

the same procedure described for the transducer gave a non-linearity

of 0.06$.

112
M N I T A B

I n L U M N COLUMN COLUMN 4 2 COLU M N 52 COLUMN S3 COLUMN 54

3! 3.00000

350.00000

3ss.ooooo

489.00000

599.00000

673.00000

728.00000

962. OOOOO

1057.0000

1 17?. 0000

926.00000

890.00 n op B56.000on

757.OC00C

655.00000

585.00O00

-532.00000

3 1 2.00000

-223.00000

-1 1 2.00 n 00

I . L- 7 6 o 7 7 8 -.083549568

.80 I 76949

7. 34P r, 799

-.6744005

1

-.68935408

.27250653

1.1261183

-2.0101 7P2

. 1 8 2 8 T 5 9

1.5696773

-.5H492772

. 44549554

2. 1 707554

-.685575 19 -.60388783

. 422243 <5 -.7746^785

-1 .6098760

.6) 657279

,49355453

i421 378 16

1 35577395

, 1 7777443

,0111 24671

,08546625

1

,14 9 7 3 7 3 7

, 35 1 5 1041

,40030214

,43377 1 20

,15336 I 635

,047350305

,04 1 523509

,0234 1 1 456

,0017001 735

,014626851

,028131 71

3

,11307171

,18 11 4053

, 3866646 1

1 457.0OOQ

1554.0000

1 6 6 6.0 000

1739.0000

I 795.0000

1 58.00000

25 O.OOOOO

353.00000

422.00000

474.00000

.53°627?7

. 75427 1 4" 2.2 1 42"94

•2.283)051

3.2673974

.9265^404

1 .098007 1 •1 .9555994

2.094 1 429

3.1395838

39589677

34373562

,25870004

, 1 8896222

,12 7 8 8 5 7

,25141 191

,13 7 9 10 3 3 ,072829287

,044536824

,026779238

1 876.0000

1-938.0000

20 1 1 .0000

2074.0000

2141 .0000

553.oo n Of! 6 1 3.0Cinoo

68 1 .00000

73" .OOOQO

802.00000

-.905374 1

2

.'3 3 44738

-.6 3535460

2.2426774

2.6345918

-. 87778077

.37493643

-

.

80569O94

2.51 «601 1 3.0320 3 7"

,027593344

,0585 10948

, 1 7033634

,27592373

,39744610

,0049815989

,00955 17698

,024989364

,037224479

,0493946 7 2222.0000

2250.0000

21 68.0000

2074.0000

201 8.OC0O

883.00000

9 1 4.00000

830. OOOOO

74O.00000

687 .OOOOO

1 . 7274 1 55

6. 2352737

-.18 058938

•1 . 2476774

1 .2583937

1.17 3 4 9 2 5.61 97954

-.629697Q0

1.51 860| 1 •1 .4400465

,55706629

,6 1 547826

,44910762

,27592373

,18 1 6528 I

06321 1575

,067801 5 1 ,054097581

,037224479

,0263931 1

3

1947.0 000

1860.0000

I 79 1 .0000

1 765.0000

1 72 2.0o.on

161 3.0000

1535.0000

15 1 2.0000

114 1 .0000

1010.0000

624. OOOOO

540 . O'jOOO

473. OOO^o

449. 00° on 409.O(j0on

307 .00000

233.O00--.0

21 1 .00000

-14 0.00000

-269.00000

2.918! 1 97

1 .2329776

-.4928 0072

- . 88295 1 09

.80141323

1.931 544 !

1 . 73 I 083a

I .4924879

1 .51 3 3908

3.5412222

2.8464 3 25 1.281 4838

-. 35040360

-.72 1^5498

1 .0076747

2. 23379P2

2.0866072

1.861 2569

1 .9409997

-3.167669 1

,07 1687 192

,04849 1 1 50

, 1 32397 I 3 ,16 1 396 1 ,20621 1 48 ,302254 1

4

,35552341

, 36876905

.4275,j 888

,37855405

,01 1542309

,0090003932

,027962394

,035955 52

,0505 1 7439

,099077478

,15 3 7 2 7 2 7 ,17 6 17 10

. 30209782

, 1 4260370

9 8 3.00000

72 9. OOOOO

668.00000

520.00000

480. OOOOO

292.00000

532.00000

588.00000

728. OOOOO

-767.00 000

-.99522828

.27250653

I . 4 I 3 8 9 7 1.0712890

-. 192786785

-.6 3 128537

.42774390

1 . I 206947

.94473697

-.27 3 58097

36394291

,14973737

,079305470

, 1 270S204

, 1 90794 1 8

, 1 2506424

,0281 3 17 13

,01 3463480

,0 1 7426559

,024878 68

3 9 . CO C 280. OOOOO849.0r; n 00

958.00 n G0

3.0705335

1 .8532073

2.723" I 96

2.4 1 3^785

3466 1388

56 37620

,04067901

3

,058607757

Table 9: Voltage Output Versus Displacement for Stylus

Instrument Using the X1000 Magnification

113
M N I T A R

COLUMN C L U M N C I. U M M '4 2 COLUMN 5 2 COLUMN 53 COLUMN 51

353.C0°O">

5 11.00000

6 8 9 . D 8.25.00000

13 8 9.0000

I 59?. 00 DP

I 6 8 2 . O

t 7 8 9 . n n

1 9 5 2 . n n 2067.0000

2239. C C C

2 1 1 1 . 2575. 00 CO 2705.0000

2 839.0000

290". i r ? n

3 6 ! .OOOn

29 6 2.00

2 7 fi-7 . 2 6 1 1.0000

2 4 7 ? . 2295. 00 CO

2 1 2 ? . C r

n

I 9 3 8 . 00

1 767 . 0000

1 5 8 9 . C

r "

|, 4.3 7 . 712.000 00

503.00000

1»56.00 000

• 3 7 ? . o

r' n o •230.0""!""

2 n 8 . P •" 92.000000

1 9 9,00000

?48.ooooo

3 o ? . c o n o n

3 9 4.00000

4 60.0000

5 5 . 6 4 5.00001

7 3!.0O0Of)

7 9 R . '1 8 6 7.00000

91B.",nrr ^ 903.309 on 9?7.00

8 3 7.000^0

750.00000

67 5.00000

5 8 1 .00000

4 8 9.00000

39 0.00f! n

300.onr)'3fl

7 P 7 . :j 00

12 2.0000!}

-241.00000

-366.00000

1 ."3 1 1 200

2. 695662"

•3.8093028

•1.17 18930

•4.2638317

5

.

2756507

•4.1626614

• 7 . O 9 4 9 9 9 1 •1

.

0236995

2.9873410

1 . 4699375

3. 3563 "in

3 . 6" 1 754P

1.571 6262

- . 7 7 6 8 I 4 3 6 3.509809 3 7.898 1128

6. 222 4 n 2 n

• 3 . 1 r' 7 6 I 7

I .9407171

2 . 4957P 1 9 2.67357 4 4

3 . 7R72 1 r>9

I .625391 2 2.6107151

-.67941778

. 1 "64273

1

6.9915692

9.1571952

• 1 . 367 1 559

•2.91 45902

3.9176023

•4.2062H38

4.103 1 826

-5.08895) 5 3.97 I P853

6.9053493

1 .6402368

3. 1 43735 I

1 .583023Q

3.4097786

3.6742318"

1 .4360940

-.9<42473 1 3 3.2909721

-3.2109139

-6.47 5674

-3. 2121 178

1 ,9"3797r]

2.5253425

2. 77000 1

8

3.9 3 22937

1 .8026636

2.801 1 ?67

-. 4929334?

.365 3 4433

6.91 2350 1 8.9307333

.33603592

.21892822

. 10829955

.034310818

•. 1 6064906

•. 1 8669927

•. 1 90776 1

7

•• 1 896H982

•• 1 7546275

•• 1 5639403

-.11 388545

-.053427607

.01 7522991

.085532278

. 1 6565877

.210887 1 3 .320801 I 7 .24816513

. 1 3335660

.036920086

-.029560655

-.096427321

-. 1 4508271

-. 1 7727239

-. 19011 1 30

-. 1 8618137

-. 16891 703

.0782191 16 .22616189

.073859098

.059281216

.03921 1875

.016833212

'. 16688113

•.091395753

.075655994

••061356483

-.04432851 1 -.034220941

-.020761933

-.0083266782

.0021092935

.010738807

.019090021

.023315580

.032538977

.026592311

.015873730

.0019351190

-.0013956088

-.01667351

1

-.029900812

-.015614691

-.061027625

-.092009523

-.13867986

-.031511 165

-.060361532

Table 10: Voltage Output Versus Displacement for

Stylus Instrument Using the X50,000

Magnification

114
APPENDIX M

THE EFFECT OF FINITE RECORD LENGTH ON MEASURED

ROOT-MEAN-SQUARE AND ARITHMETIC AVERAGE VALUES

Consider first the case of estimating the uncertainty in the mean

square value, q

2

, calculated from a finite length L of the profile

y(x). By definition:

q

2 = f / y

z i ( x ) d*< J

For convenience and brevity in the following discussion, the

definitions listed below will be used [14].

Sample space = the set of points representing the possible

outcomes of a measurement.

<j>(k) E a real number, called the random variable, which

represents the outcome of a measurement indexed by

k.

,a\ = ilm Probability [>«i>(k) <.<!>+& <|>]

A<J)-K5

E[g(<£(k))] = the expected value of any real single-valued

continuous function g(<f>) of the random variable

<j>(k). It is given by:

+00

E[g(<j)(k))] = J

g((j))p(<t>) d<f>,

As an example the mean square value of y(x) is given by:

+00 E[y

2 (x)] = f y

2 (x)p(x) dx.

E[y2 (x)] will be defined asq'2

. q

2 is that value approached by q

2 as

the record length L approaches infinity or that value obtained for the

mean q

2 of a large number of finite samplings. The variance of y(x)

is defined by:

+O0 var )y[ = E[(y-y) 2

] = f (y-y) 2p(x) dx,

115
Similarly the variance in q

2 is defined by:

var |q2

} = E[(q2 -q2) 2 ].

For the special case when y(x) is a bandwidth limited Gaussian

waveform with zero mean, Bendat and Piersol [14] show that;

where L* is shift distance required for the ACF to drop by 63% of

its zero shift value. If L>>L* the relation reduces to:

var |q2

f ? L* -h L

q^

The propogration of error formulae discussed by Ku [17] may be

used to relate the variance of the root-mean-square value = RMS = ,/q2

to this variance of q

2

. Results given in table 1 of this reference

show that if:

var q

2

, , var RMS e ^— = e, then = -n- ^

q

2 4

Thus, the normalized 3 standard deviations limit, 3SD in the rootmean-square values calculated from randomly sampled length, L, of a

Gaussian profile with a correlation length L* is given by:

3SD limit of RMS = -I V^ = 2.lV~s 3SDR

Whitehouse and Archard [3] report reasonable confirmation of this

result for their test specimens which had approximately Gaussian

profiles. The importance of this result is stated very clearly by

Whitehouse and Archard [3]. "The variance of measured RMS or AA values

for the roughness of a surface may be found easily

if one knows the standard deviation of a large number of

such measurements made upon the same surface. Alternatively one may

predict the variance from a knowledge of the correlation length of a

typical profile of the test surface." The argument just given for the

3SD limit partially fulfills the "it can be shown" statement by

whitehouse and Archard.

Unfortunately these results do not apply for periodic waveforms

such as that of the precision roughness specimens. However, the prin116
ciples of calculation are still applicable. Consider the example of a sinusoidal waveform:

y(x) = A sin(o)X+e).

The waveform is sampled for a length L with the phase angle 6 considered as the random variable of the sample space. 9 will be considered as having a uniform distribution over values from to 2tt i.e.

p(6) = p- , 0<6<2 and p(e) = 0, for all other values of 6.

According to the definitions given earlier:

1 V* A \ / q

2 = jr / [A sin(cox+9) r jcos6- cos(<dL+9) > ] 2 dx.

^o

The last term under the integral is the mean value of y as a function

of e. Evaluation of the integral yields;

2 A f, sin 2(6+9) , sin 26

q

z = — 1 - -•

2 ii 2(oL 2ooL

where 6 = toL - 2nir and terms of order n~2 have been dropped. The mean

square uncertainty in q

2 is given by:

2tt «:(«wj -h f g (i - ^r^+ ^- t * «

Evaluation of the integral yields:

2 k

h (1-cos 26)

var q

z = -

16 a) 2 L

2

Thus, if the record length includes an integral number of periods the

var q

2 = 0. But, if we take the more likely case when the range of

experimentally measured wavelengths is such that 6 has a distribution like that of 9, the result averaged with respect to 6 is

/ 2^ A

4 (var q

2 )^ 1 (var q

z

) = or ^ /f> = . 6 16 W

2L2 qk 4 u) 2 L

117
Use of the same relationships between var q

2 and van RMS given earlier

yields

var RMS 1

-2 16 u) 2 L

2

or normalized 3SD limit of RMS = zJSl = 3SDP.

For the 3.17 ym (125 yin) patch of a precision roughness specimen,

wL = 80tt with the standard 3.8l ym record length; for the 0.5 ym (20

yin) patch, wL = 500tt . Thus,

3SDP

125

= °' 3% >

3SDP

2Q

= 0.05$.

These estimates neglect small effect expected from the waveform

differences between the analyzed sine wave and the triangular waveform

of a precision roughness specimen. One would also expect that the

normalized 3SD of the AA value would be approximately equal to that

just found for the RMS value since both quantities are similar

measures of the sampled profile's density function.

The major source of additional uncertainty not accounted for is

that produced by the imperfect waveforms of the precision roughness

specimens. These imperfections are illustrated in section 3.^. The

illustrations show that the distortions may be produced by either an

additive random function together with phase modulation or by the

combination of random amplitude and phase modulation. The additive

random function is the only error which can be readily analyzed. For

this case the RMS value of the sum of two uncorrelated waveforms adds

in an RMS manner. The use of the propagation of error formulae

derived by Ku [17] then yields:

3SD = 1.5 (2a L*/L + -

]

2 \ 4 W

2 L

2 /

where a Hr-H)- = II + — r , b= I 1 + -4- I and

qp" R

= mean square values of periodic and random components,

respectively. Both qp/q£ &n& L*/L are difficult to estimate from the

investigatory experiments performed so far on the precision roughness

118
specimens. However, for the 0.5 ym (20 yin) roughness waveform

preliminary estimates are L = 20L* and q^ = 0.01 q£. With these

estimates the 3SD limit, for the waveform sampled for 250 periods

would be dominated by the uncertainty from the random component;

this would increase the limit to a 3SD of 0.3$- These same values of

L* and q^ are insignificant for the 3.17 ym precision roughness

specimen since now q^ = 2.8 x 10"^ q£.

119
APPENDIX N

SOME CONSIDERATIONS OF THE SYSTEM'S TRANSIENT

RESPONSE CHARACTERISTICS

Ideally the system response, the storage of a datum point V(t)

in memory at time t, would be directly proportional to the stylus tip

displacement input y(t) at time t. The stored signal would be

unchanged from that of y(t) except for a possible delay t ; i.e.,

V(t) = Ay(t-t

Q

),

where A is a proportionality constant equal to the inverse of KCAL.

In any real system the relationship between V(t) and y(t) is typically

a complex function of the system's properties such as its mechanical

and electrical resonance frequencies, passband or cutoff frequencies

and response times . One way of characterizing systems is to

determine the step response function S(t), which is the response of an

initially relaxed system to a unit step input U(t) defined by the

relations

U(t) =ltiO,

= 0t<0.

In terms of S(t) the response to an arbitrary excitation y(t) is

given by [15]:

t

v(t) = y(o) s(t) + / y'(r) s(t-r) dr.

o Thus, the step response of an ideal system would be S(t) = A U(tt ), with A and t as defined earlier. Employing this approach to

analytically characterize the stylus instrument/minicomputer system

response is very difficult. The difficulty is in producing a realistic and known input for V(t) that has none of the undesirable

properties for which the system is being tested. However, the concept

of this technique was used to assess the magnitude of the system

transient errors and the range of response because of its simplicity

relative to other techniques such as the vibrating platform approach

discussed by Spragg [16].

The measurement procedure was as follows: the stylus was

traversed across a step formed by two closely spaced gage blocks wrung

to a platen; the generated profile was stored in the minicomputer

memory. The step input was measured and recorded by traversing

the stylus across the step at the Talysurf 's slowest speed, 12 ym/sec.

The transient response was then measured by traversing the stylus across the same position on the step at the speed used for

120
"System Input"

12.7um Step Height

Vertical to Horizontal

Mangification Ratio = 25 Traversing Speed = 12 ym/s

"System Output"

12.7 ym Step Height

Vertical to Horizontal

Magnification Ratio = 25 Traversing Speed = 1500 ym/s

Figure 3^a: Step Response of System

121
"System Input"'

12.7 vim Step Height

Vertical to Horizontal

Magnification Ratio = 25

Traversing Speed = 12 ym/s

"System Output"

12.7 vtm Step Height

Vertical to Horizontal

Magnification Ratio = 25

Traversing Speed = 1500 ym/s

Figure 34b: Step Response of System

122
roughness measurements, 1500 ym/sec. Figures 3**a and 3^b demonstrate

the results for steps near the borderline before "ringing" or

"bouncing" began to occur. Steps with a crevice between the gage

blocks were intentionally used to obtain profiles with both directions

of acceleration present.

The ringing present in the "double-crevice" profile of figure 34b

was also found in the system response to a larger step height. No

ringing or overshoot was found in the transient response to steps

smaller that the 12.7 ym step used for the measurements illustrated.

At several positions along the step edge the gradient in the step was

not as great as shown in figure 3%. Profiles at these positions also

showed no evidence of overshoot or ringing. Thus, the transients

produced by these steps are an approximate upper limit for which the

system's characteristics may be represented by the step response S(t) = A U(t-t ).

The most suspect component of the system which would cause the

ringing is the stylus instrument pickup. In use the stylus tip is

held in contact with the specimen surface by the force of a flexurepivot spring. This force, the internal damping of the spring, and the

moving mass of the pick-up determine the maximum acceleration to which

the pick-up may be subjected before the stylus tip will no longer

follow the surface profile. A quantitative estimate of the value of

acceleration at which the bouncing is initiated may be obtained from

figure 34a. Using conservative estimates for the transition time

between the two sides of the crevice and the transition time between

the drop and rise to the high side of the step, the accelerations

necessary to cause bouncing are greater than 0.25 m/s2

.

For comparison, a value for this acceleration obtained from

Spragg [l6] is 5 m/s2 . This value is estimated from his curve of the

peak to peak amplitudes necessary to cause bouncing versus

frequency of vibration. The acceleration needed to cause bouncing was

found to be independent of frequency, to first order. Differences

between the two values may be attributed to the different pick-up

properties; NBS stylus force was 500 yN vs 1000 yN used by Spragg;

the NBS moving mass was larger than that of the Spragg system.

For perspective, the acceleration generated by the fundamental

sinewave component of the 0.5 ym (20 yin) AA precision roughness

specimen at a traversing speed of 1500 ym/s is only 0.01 m/s2

; that

generated by the fundamental sinewave component of the 3.17 ym (125

yin) AA precision roughness specimen is 2.4 x 10~3 m/s2. Using the

acceleration limit estimated from the step response study, the

maximum sharpness triangular waveform which the NBS system could

reliably reproduce at the 1500 ym/s traversing speed would be one with

an included angle at the peak or valley of 140° or greater. (This

does not include errors resulting from the finite tip radius.) The

included angle on a precision roughness specimen is 150°; for

typical machined surfaces it is about 160°.

123
APPENDIX

MEASUREMENTS OF THE RECORD LENGTH AND

TRAVERSING SPEED

Record Length

For meaningful average slope and wavelength measurements from the

stored digital profiles the horizontal motion of the stylus must be

calibrated to the accuracies desired for slope and wavelength. With

the computerized system the calibration may be performed easily by

adjusting the data acquisition rate so that a known length between two or more index points on a specimen occupies the correct proportion of

the record length in memory. The data acquisition rate is adjusted

by entering at the teletype the necessary count rate (appendix B)

.

The rulings of a 3.17 ym (125 yin) precision roughness specimen

proved to be very convenient for calibrating the traversing length

since they produce a clean profile in memory and their spacings may be

measured with a length measuring microscope. The particular length

measuring microscope employed was one designed so that the calibrated

scale and the cross-hair used for measuring stage, and specimen,

displacement were observed with separate microscopes. Mean spacing of

the rulings was (95+8) ym. The uncertainty stated is the sum of the

3 standard deviation limit estimated from values obtained in 20

measurements and the systematic error of the microscope estimated as 2

ym.

The traversing length required for five 762 ym (0.030 in) cutoff

lengths is therefore equivalent to 40.2 periods of the 3.17 ym

roughness waveform. Using a plot of the contents of memory the count

number was adjusted to give 40 periods in the stored profile. This

accuracy is considered sufficient for the present applications

of average wavelength and slope. With this adjustment the sampling

interval is (0.93 + 0.07) ym.

Traversing Speed

To measure accuracy and uniformity of traversing speed a retroflector was attached to the motor drive of the Talysurf by means

of modeling clay. The motion of the retroflector was then measured

with a Hewlett-Packard 5526 laser measurement system. Speed values

were recorded at rates up to 10 per sec. with a digital recorder.

Analysis of the data showed:

at nominal speed =1.52 mm/s (0.060 in/s)

measured speed = (1.52 + 0.04) mm/s,

at nominal speed = 0.305 mm/s (0.012 in/s)

measured speed = (0.305 + 0.013) mm/s.

124
A plot of the speed versus time from start of traverse exhibited some periodic behavior. The data was therefore Fourier analyzed to

determine the magnitude of any dominant components. The maximum

magnitude was only 0.7% of the mean speed; this occurred for a component with period approximately equal to 1/3 of the traverse

length

.

The primary concern in measuring the traversing speed was to

ascertain the uniformity of the samplings of the roughness profile for

use in the AA, average slope, average wavelength and the step height

LSQ fit calculations.

125
APPENDIX P

SAMPLE TEST REPORT CONTAINING CALCULATIONS

OF STATISTICAL PARAMETERS AND FUNCTIONS

126
U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

WASHINGTON, D.C. 20234

November 6, 19 7

4

REPORT OF CALIBRATION 232.08/211071

For: Two Copper Gravure Printing Plates - Segments of Cylindrical Printing Surfaces

Submitted by:

On each of the two plates roughness measurements have been made to obtain

profiles at appropriate magnifications, arithmetic average (AA) values,

amplitude density functions, average profile slopes and profile wavelengths .

Table I gives the average AA value obtained from 3 traverses in each of

10 positions on both sample A and sample D. The positions are indicated

schematically in figure 1. Average wavelength and average slope calculations for the profiles of positions 4, 8, 10 were made. The resulting

values are also shown in table I. For positions 4 and 8 profile graphs

at a horizontal magnification of 100, with vertical magnifications as indicated, and amplitude density functions were obtained. The graphs are displayed in figures 2 to 4-

.

The property of surface roughness in the 5 ym (200 microinch) AA range

and below is measured at NBS by means of a minicomputer/stylus instrument

system. Usi.ig an interfere-metrically measured step the system is calibrated on each value of magnification employed during a measurement.

Surface profiles are taken according to the American National Standard

B-46. 1-1962 using a O.76 mm (0.030 inch) cutoff length and a 3-80 ym

(150 yin) stylus tip radius. Data is stored in the minicomputer memory

employing 12 bit analog to digital conversion and a sampling rate of 1

point/ym (1 point/40 pin) over the traversing length. AA values are then calculated as described in Appendix A of the same American National

Standard. The other parameters or functions may also be calculated from

the stored profile data.

A conservative estimate* of the systematic error (3a = 3 standard deviations) resulting from stylus pickup nonlinearity , interface hardware,

software computations and analog to digital conversion is 1% for any type

of uniform roughness specimen. Other errors are due to stylus tip radius

uncertainty [3<r = 5 nm (0.2 yin)], to the surface finish of the step

specimen [3a of step transfer = 25 nm (1 yin)] and to the interferometric

*At the time this report was written the classifications of uncertainty

components was slightly different from that in section 2.6. All subsequent

reports conform to the classifications in section 2.6.

127
measurements of step heights [3a = 25 ran (1 yin)]. The use of a step

height approximately equal to the surface profile peak to valley height

then gives a net systematic error for the calibration procedure of 2.8%

(3a) for AA values in the 0.25 ym (10 yin) range and 12% (3a) for AA

values in the 0.05 ym (2 yin) range.

From the data in table I and the measured system noise level the computed

value for the average surface roughness of the specimen marked A is

0.226 ym (8.9 yin) AA with an uncertainty of 0.055 ym (2.2 yin).

Similarly, the computed value for the average surface roughness of the

specimen marked D is 0.43 ym (1.71 yin) AA with an uncertainty of

0.13 ym (0.52 yin).

The uncertainties quoted are the sum of the systematic errors just described and the 3a limit estimated from an analysis of the values from

the 10 positions on each of the specimens . Values given as the average

value have been corrected from the small effect of noise by assuming

that the noise and signal from the specimen were not correlated. With

this assumption the true value was calculated from the equation:

true value = [(measured value) 2 - (noise value) 2

]

2

.

The measured AA value of the noise during the time of roughness measure- ments was 0.0035 ym.

The amplitude density function of a surface profile or waveform describes the probability that the profile line will assume a value within

some defined range at any point along the sampling length. The probability that the profile f(x) assumes a value between the range f and

(f + Af) is obtained by taking the ratio of Lx/L, where Lx is the total

length that f(x) falls inside the range (f , f + Af) over the sampling

length L.

In equation form, for small Af, one has:

Lim Lim L„

Af-K) Prob f<f(x)<f + Af = Af+0 =± = ADF(f) df

Calculation of the amplitude density function proceeds according to the

definition given above by dividing the minicomputer data storage (12

bits) into 512 increments of 8 units width. The attached plots of the

estimated amplitude density function (ADF) obtained with these finite

increments are therefore the percentage of the profile data within the

respective intervals as a function of the average ordinate value of the

interval. Further interpretation of the ADF graphs may be obtained by

comparing the experimental forms with the "ideal" Gaussian ADF for a theoretically random waveform which has a symmetrical bell shape. In

128
addition to the information obtainable from the form of the ADF, one may

measure the maximum peak to valley height of the profile from the extension of the graph and the scales indicated in figures 2 to 4. The

scales shown in the figures are based on measurements of the ADF of a

step of known height.

In figures 3 a^d 4 the graphs of the ADF contain a flat portion for

ordinate values near the mean ordinate. These regions are due to saturating the graphing system at the magnifications chosen for the ADF

graphs. The large magnification was chosen to reveal more structure of

the ADF's trailing edge, which would have been lost at magnifications

not causing saturation.

The average profile slope is computed as the ratio of Arithmetic Average

of the ordinate difference between successively sampled points to the

mean sample spacing. Angles given in table I are the arc tangent of the

computed value. The average wavelength is calculated with the equation:

Average Wavelength = 2tt

-"-''''

----- --

' AA of profile slope

Measurements made by c , (ij/^iA^i^J ^)^# g

For the Director,

{f/sU

Russell D . Young { I Chief, Optics and Micrometrology ^Section

Optical Physics Division, IBS.

129
TABLE 1

Summary of Parameter Measurements

Sample A

Position AA(ym)

1 .1927

2 .2323

3 .233^

4 .2236

5 .2504

6 .2203

7 .2087

8 .2351

9 .2261

10 .2406

Average Slope Average Wavelength

Degrees (ym)

5.8

5.9

6.3

11.8

15.2

12.9

Sample D

Position AA(ym) Average Slope Average Wavelength

Degrees (um)

1 .0411

2 .0475

3 .0431

4 .0398 1.2 12.5

5 .0430

6 .0443

7 .0475

8 .0459 1.3 13.0

9 .0413

10 .0412 1.1 12.9

130
A

1 2 !

1

' 5 4 3

6 7 8

10 9

D

1 2

5 4 3 j 6 7 8

10 9

The points indicated on the sketch above

are not to scale but are used to represent

the approximate areas at which measurements

were taken on each specimen.

Figure 1: Schematic of Measurement Positions

131
Surface Profile

= 0.1358/nm

Profile ADF

Figure 2: Profile and ADF of Area 4 on Sample A.

132
hWnji

8 ^k*

* xlOO

^HAu^Kj^^n>

^

Surface Profile

t

'

= 0.66#/nm

1

Profile ADF

= 118 nm

1 V

Mean

Ordinate

Figure 3: Profiles and ADF of Area 4 on Sample D.

13 3
= 0.66#/nm

Profile ADF

Surface Profile

n

Mean

Ordinate

118 nm

Fi^ore 4: Profile and ADF Graphs of Area 8 on Sample D.

134
APPENDIX Q*

OPERATOR'S INSTRUCTIONS: INTERDATA 3/TALYSURF 4 SYSTEM

I. Preliminary Set-up

A. Instrumentation : the following connections are necessary:

1. those on the Talysurf for normal operation as indicated

by the Talysurf Operating Instrumentions manual;

2. those internal to the A/D converter as shown in figure 35a;

3. the power cord of the interface electronics as in figure

35b (N. B. Disconnect main power line to computer when

connecting this line).

4. The external leads between the Talysurf, interface

electronics and A/D converter as in figure 35c

B. Software : the following programs must reside in memory;

. 1. the hexadecimal monitor (3A80-3PPF);

2. the arithmetic subroutines (3010-3AOO);

3. the master program (0080-1000).

II. General System Operation

A. Calibration of Stylus Instrument

1. Magnification Selection

(a) Align unknown specimen on Talysurf;

(b) traverse stylus with chart recorder on and select

magnification on Talysurf electronic unit to obtain

approximately three-quarters of maximum on-scale

deflection of recorder pen;

(c) turn off recorder and select calibration step

artifact of appropriate size.

2

.

Input of Calibration Step : Step-up

(a) Set switches

•Revised from DASR

135
<o

PQfaOXw

p-1 MH

ED

w o

CO PQ

wQM

COo

fafao

faO

a.a i

PQ

PiwH

fa

faOo

cooHuoo9

1-3 <

IS

PC! H

in

onwoM

fa

<!

faO

fa

fa

H

J

b

<:o M H

CO

H

fa

H !=> Pi

fa 6 o

fa

c_) h-l

fa

faCO

fa

w

fa

136
REAR OF INTERFACE CHASSIS

LEFT SIDE OF INTERDATA 3 POWER SUPPLY

INTERFACE ELECTRONICS: AC POWER CONNECTION

FIGURE 35b

137
w<

Fnw

O pq

i—i VX

J t> < PM s S O M S M PL. CO H H

PL.

b

H 5 o

Piw

a

w o w CO

CJ

M Pn

u& omonw

PiOM

138
(1) Talysurf Electronic Control Unit

(1) Magnification: as determined in II.A.l

(2) Cut-off: K

(3) Operation: N

(2) Talysurf Gear Box

(1) Stroke knob: L

(2) Speed knob: X20

(3) Interface Electronics

(1) Mode Select: 10X

(2) Cut-off: Step

(b) Align artifact and traverse stylus to obtain trace

with step edge in proper position relative to the

event mark (See appendix E)

.

(c) Rotate the Talysurf Setting Lever to the left (CCW)

position (see p. 4 of Talysurf Manual and Note 1

below)

.

3. Input of Calibration Step : Program Execution

(a) Begin program executive from TTY Control (Enter an

"S")

(b) At "Enter Data": turn on the chart recorder and

gently move and Setting Lever to the right (CW);

data will be read for twenty-five seconds.

(c) At In: shut off the chart recorder and tear off

the recorded trace;

(d) Al and A2 selection:

(1) with the "Step Locator Scale," determine the

position in mm of the step edge relative to

the trailing edge of the event mark (see

appendix E)

;

(2) select Al and A2 symmetrically about the step

edge.

(3) Record on the strip chart the mm and address

locations of Al and A2;

139
(e) At "Enter HO": type in the step height value as four digits and depress the space bar on the TTY;

(f) At "Enter Units": type in a two character unit

symbol (do not press the space bar);

(g) At "Enter Al"; type in the four-digit address Al

and depress the space bar;

(h) At "Enter A2"; either A2 as above; the computer

will now print out the computed step heights HI, H2

and HM with units

.

(i) At "More?":

(1) If the computed step heights are unsatisfactory, type N and repeat the

procedure for step 3a:

(2) If the heights are satisfactory, type Y.

B. Calibration of Unknown Specimen

1. Align the specimen on the Talysurf and traverse the

stylus to obtain a properly centered trace.

2. Calibration of a Step

(a) Repeat procedure in IIA2, b and c;

(b) At "H or R?": type in H;

(c) At "Enter Data", follow procedure from IIA3, b thru

h;

3. Calibration of a Roughness

(a) Select filter cut-off on interface; select X4

stroke speed on Talysurf; type R.

(b) At "Enter Data," rotate the Setting Lever to right

position (CW); data will be read for 3.5 seconds.

(c) The computed AA and units will now be printed.

4. At "More?":

(a) If a step is to be calibrated, type Y; then at "H or R?", type H.

(b) If a roughness is calibrated, type Y, then at

"H or R?", type R.

140
(c) If no more measurements are to be made, exit from

program by typing N.

Note 1. The Event Marker appears on the chart recorder at the "Start

of cut-off average" position in the stroke. With the stroke knob on

the gear box at position L, the event mark appears when the marks on

the gear box window are aligned as indicated below. Therefore, in

returning the Setting Lever to the left (CCW) position, it is

necessary only for L mark to be to the left of the rightmost mark on

the upper scale.

141
APPENDIX R

CHECKLIST FOR PROPER ELECTRICAL OPERATION

Talysurf Outputs

(1) The profile Output should produce a voltage of nominally +1

volt when the pen is at the 2 inch mark on the strip chart

and -1 volt at the zero inch mark.

(2) The interrupt signal should be a positive - going step of

four volts near the "Start of cut-off average" position of

the stylus stroke (see Talysurf Manual).

A/D Outputs

(1) With the Talysurf s interrupt signal lead connected to the

Interrupt Input terminal on the A/D converter front panel.

the A/D output at the Pulse Output terminal should be a

zero to four volts pulse of about one second duration;

(simultaneously the computer interrupt should trigger

internally)

.

Interface Electronics Outputs

(1) With the system fully wired (figure 35) s the output at the

X10 OUT should be the amplified, filtered Profile Signal;

(2) A one-second pulse of nominally one volt should be present at

the "PROFILE IN" terminal (and a similar one recorded on the

chart recorder) at the "Start of cut-off average" stroke

position.

Internal A/D Signals (A/D Multiplex or Board) .

With the Talysurf 's interrupt signal present at the Interrupt

Input

The output of the A/D Interrupt pulse (daughter board 40)

should be a 50 microsecond positive-going pulse of about four

volts

.

142
3J BS-114A [RU.v. /-/s r

U.S. DEPT. OF COMM. 3. Recipient's Accession No. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-9022. Gov't Accession

No.

4. TITLE AND SUBTITLE 5. Publication Date Evaluation, Revision and Application of the NBS

Stylus/Computer System for the Measurement of

Surface Roughness

April 1976

6. Performing Organization Code

7. AUTHOR(S)

E. Clayton Teague

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2320184

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as No. 9

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report describes in detail the hardware and software used at NBS to

implement on a stylus instrument/minicomputer system the process of calibrating

the system with an interferometrically measured step and the calculation of

important characterizations of surface profiles. The characterizations of

a profile which may be calculated include the arithmetic average value, the mean

square value, the amplitude density function, the autocorrelative function and

the average wavelength. The report also includes a statistical evaluation, using

empirical and analytical techniques, of the calibration procedures' s long term

stability.

7. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Amplitude density function; arithmetic average; autocorrelation function; average

wavelength; kurtosis; minicomputer software; random error; skewness; surface microtopography; surface roughness; surface texture; systematic error.

i. AVAILABILITY (j£ Unlimited

I ! For Official Distribution. Do Not Release to NTIS

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NBS TECHNICAL PUBLICATIONS

PERIODICALS JOURNAL OF RESEARCH reports National Bureau

of Standards research and development in physics,

mathematics, and chemistry. It is published in two sections, available separately:

• Physics and Chemistry (Section A)

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on

standards of physical measurement, fundamental constants, and properties of matter. Issued six times a

year. Annual subscription: Domestic, $17.00; Foreign,

$21.25.

• Mathematical Sciences (Section B)

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical analysis, theoretical physics and chemistry, logical

design and programming of computers and computer

systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic, $9.00; Foreign, $11.25. DIMENSIONS/NBS (formerly Technical News Bulletin) —This monthly magazine is published to inform

scientists, engineers, businessmen, industry, teachers,

students, and consumers of the latest advances in science and technology, with primary emphasis on the work at NBS. The magazine highlights and reviews such

issues as energy research, fire protection, building tech- nology, metric conversion, pollution abatement, health and safety, and consumer product performance. In addition, it reports the results of Bureau programs in measurement standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $9.45; Foreign, $11.85.

NONPERIODICALS Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and

industrial practice (including safety codes) developed

in cooperation with interested industries, professional

organizations, and regulatory bodies.

Special Publications—Include proceedings of confer- ences sponsored by NBS, NBS annual reports, and other

special publications appropriate to this grouping such

as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists,

engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides

quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a world-wide

program coordinated by NBS. Program under authority

of National Standard Data Act (Public Law 90-396). NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical

Reference Data (JPCRD) published quarterly for NBS

by the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints,

and supplements available from ACS, 1155 Sixteenth

St. N. W., Wash. D. C. 20156.

Building Science Series—Disseminates technical infor- mation developed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and performance criteria related to the structural and environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete

in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other' government agencies.

Voluntary Product Standards—Developed under pro- cedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations.

The purpose of the standards is to establish nationally

recognized requirements for products, and to provide

all concerned interests with a basis for common understanding of the characteristics of the products. NBS

administers this program as a supplement to the activities of the private sector standardizing organizations. Federal Information Processing Standards Publications (FIPS PUBS)—Publications in this series collectively constitute the Federal Information Processing Standards Register. Register serves as the official source of information in the Federal Government regarding stand- ards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

Consumer Information Series—Practical information, based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable language and illustrations provide useful background

knowledge for shopping in today's technological

marketplace.

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS for

outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Service (Springfield, Va. 22161) in paper

copy or microfiche form.

Order NBS publications (except NBSIR's and Biblio- graphic Subscription Services) from: Superintendent of Documents, Government Printing Office, Washington,

D.C. 20402. BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service A literature survey issued biweekly. Annual subscription: Domestic, $20.00; foreign, $25.00.

Liquefied Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature

survey issued quarterly. Annual subscription : $20.00.

Send subscription orders and remittances for the preceding bibliographic services to National Bureau of Standards, Cryogenic Data Center (275.02)

Boulder, Colorado 80302. Electromagnetic Metrology Current Awareness Service Issued monthly. Annual subscription: $24.00. Send

subscription order and remittance to Electromagnetics

Division, National Bureau of Standards, Boulder,

Colo. 80302.
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