NBSIR 75-975

'NST. OF STAND « TECH » I,c NIST

PUBLICATIONSi

The Calibration of an Optical

Flat by Interferometric

Comparison to a Master

Optical Flat

Charles P. Reeve

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

December 1975

Final

U S. DEPARTMENT OF COMMERCE

NATIONAL BUf€AU OF STANDARDS

NBSIR 75-975

THE CALIBRATION OF AN OPTICAL

FLAT BY INTERFEROMETRIC

COMPARISON TO A MASTER

OPTICAL FLAT

Charles P. Reeve

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

December 1975

Final

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, III, Under Secretary

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambier. Acting Director

Contents

Page

1. Introduction 1

2. Preliminaries 1 2.1. Physical Properties 1

2.2. "One-dimensional" Flatness 2 2.3. Choice of Profile Reference Line 2

3. Geometrical Model for 2-flat Comparison 4

3.1. Method of Support

3.2. Coordinate System 7 3.3. Gravitational Bending 7

4. Some Results of 3-flat Comparisons 7 4.1. Calibration of NBS Master Flats 7 4.2. Empirical and Theoretical Bending 9

5. Measurement Process 11

5.1. Calibration Facilities 11

5.2. Choice of Master Flat and Support Radii 11

5.3. Preparation for Measurement 13

5.4. Procedure for Taking Data 13

5.5. Mathematical Model of Fringe Profile 15

5.6. Computation of Test Flat Profile 19

5.7. Error Analysis 24

6. Graphical Display of Profiles 26

7. Example 27

8. Conclusion 31

Acknowledgement 33

Table A 34

Table B 35

References 36

ii
I
The Calibration of an Optical Flat

by Interferometric Comparison to a Master Optical Flat

by

Charles P . Reeve

1. Introduction

The Dimensional Technology Section has been involved in the calibration of optical flats for many years. Present facilities are capable of

handling any flat whose diameter does not exceed 16 inches. All test

flats are calibrated by either the "2-flat method" or the "3-flat method".

The former method is a direct comparison of the test flat to an NBS master

flat whose surface profile is well known. The latter method involves an intercomparison between three flats, none of whose profiles need be known.

Most test flats are calibrated by the 2-flat method because it is simpler

and requires fewer measurements. The 3-flat method is used only for the

calibration of NBS master flats or for special calibrations.

The purpose of this report is to present a fairly detailed description of the procedures used in calibrating an optical flat by the 2-flat

method. Emphasis is on the geometrical aspects of the setup, the method

of obtaining data, the mathematics of data reduction, and the display of

the computed results. Some useful results from previous 3-flat calibrations are also incorporated.

2 . Preliminaries

2.1. Physical Properties

Normally an optical flat is used in the laboratory as a reference

plane in connection with some measurement process. The flat is discshaped with parallel or nearly parallel surfaces, one or both of which

are of finished optical quality (within a few microinches of flatness

except for dubbing at the edges) . The most common material for an optical flat is fused quartz, but many are made of glass or one of the newer materials which has a very small coefficient of thermal expansion.

The surface of a fused quartz flat reflects only about 3% of the

incident light, so the surface is often coated to give a higher reflec1
tivity. Depending on the type and thickness of the coating, reflectivity

can be increased to nearly 100%. A flat with such a highly reflective

coating is usually referred to as a mirror. In defining the point where

a flat becomes a mirror, use is made of the fact that the closer the re- flectivity of two interfering surfaces, the more distinct the fringes.

Thus any coated flat which is too reflective to give usable fringes, in

the judgement of the observer, against an uncoated master flat is con- sidered to be a mirror and must be calibrated against a coated master

flat.

Most flats have two perpendicular diameters which are identified as A-B and C-D by markings on the side. On a one-sided flat the finished

surface is indicated and on a two-sided flat the surfaces are usually

identified as 1 and 2 or T and B.

2.2. "One-dimensional" Flatness

The objective of the 2-flat method is to measure the profile of the

test flat along the specified diameters. This is accomplished by individually comparing each test flat diameter to the known diameter of the appro

priate master flat. The measurement process yields an estimated profile f

each diameter of the test flat in the form of a discrete number of points

(x^,y^), ... '(^^'Yj^) where denotes a position along the diameter and

y. denotes the measured deviation of the surface from some reference line

1

at the position x^. A sample set of points is given in figure 2.1 where

n=7 . The set of n points corresponding to a given diameter may be plotted

and connected by a smooth curve to give a perspective of the measured profile along that diameter as shown in figure 2.2.

Although the surface of an optical flat is often used as a twodimensional reference plane, it is not calibrated in the two-dimensional

sense by the method described herein. This method is most accurately

described as a "one-dimensional" flatness measurement with the y. values

being reported as "deviations from a straight line".

2.3. Choice of Profile Reference Line

The choice of which reference line to use in reporting deviations is

arbitrary. A set of deviations relative to one straight line can easily

be converted to deviations from another. Several definitions of the ref- erence line have been used at the National Bureau of Standards in recent

years. Some of these methods of defining the reference line and reporting

deviations are:
MEASURED VALUES OF OPTICAL FLAT PROFILE

1

1 ^ 1 y 1

1

1 .Ub + U. lU

2 1.00 1.81

3 2.00 2.79

4 2.50 3.20

5 3.00 3.05

6 4.00 2.89

7 4.94 2.90

Figure 2.1

PLOT OF PROFILE VALUES CONNECTED BY A SMOOTH CURVE

Figure 2.2

3
(1) If the profile appears to be simply convex or concave let the

reference line pass through the end values, and if the deviation of the

center value from the line is "d", then report the profile to be "d"

microinches convex or "d" microinches concave, depending on the direction

of the curvature (fig. 2.3).

(2) Let the reference line pass through the end values and report

the deviations from this line (fig. 2.4).

(3) Let the reference line pass through the end values, translate

it so that it passes through the center point, and report deviations from

this line (fig. 2.5).

(4) Fit a "least squares" line through all values (with the possible

exception of the end values) and report deviations from this line (fig. 2.6).

The first method is no longer used because it has the overtones of

being a certification rather than a calibration. The last three methods

report the measured profile relative to different lines, but they all

provide the same information. One cannot say that one reference line is

more correct than another. They are all arbitrary and interchangeable.

The fourth method, however, is now being used because all measured values

have some weight in determining the reference line, and therefore an outlying measurement at some point would have a minimal effect on the deviations from the line at other points.

When a given diameter of the test flat is compared to the known master

diameter one flat is placed horizontally on top of the other so that the

two diameters are adjacent and lie in the same vertical plane as shown in

figure 3.1. The lower flat is supported on three thin pads which are equidistant from each other. Two of the pads lie on a line parallel to the

test diameter. Let the distance from each pad to the center be denoted

by where r^ is the radius of the lower flat and k^^ is a constant

between zero and one. Similarly, the upper flat is supported on the lower

flat by three thin pads, each at a distance of k^r^ from the center of the

upper flat. These three pads are also equidistant and have the same orientation as the pads supporting the lower flat. The pad farthest from

the diameter (indicated by +) is slightly thicker than the other two so that the air space between the flats is wedge-shaped. The constants k^

and k^ are chosen in a way that will maximize the effectiveness of

the measurement system. Guidelines for determining these parameters are discussed later.

3. Geometrical Model for 2-Flat Comparison

3.1. Method of Support

4
II , II d MICROINCHES CONVEX

o

o

e

o

yi = y? = 0

Q 9

o

J Q_

Figure 2.3 Figure 2.4

y4 = 0 yi = y7

Least squores line does

not include end points

Q ? o

6

6 - O

Figure 2.5 Figure 2.6

5
2- FLAT SETUP
3.2. Coordinate System

It is necessary to assign a coordinate system to each diameter of

the test flat for the purpose of identifying points along the diameter

at which measurements of the surface profile are to be made. Readings

are taken at n positions along each diameter where n is usually odd.

These positions are denoted by (x^^, . . . >x^) and are usually symmetrically spaced about the center point at inch or half-inch intervals. If

the end positions fall at the extreme edge of the flat they are moved

one-sixteenth of an inch toward the center. The positions are usually

chosen to coincide with those used in previous calibrations. If the

flat has not previously been calibrated the positions are chosen as deemed appropriate. The x^ position is at the A end of the A-B diameter

and the C end of the C-D diameter, and the x position is at the B and

D ends respectively.

3.3. Gravitational Bending

The freeform profile of an optical flat is defined as that form

which the flat would take in a zero-gravity environment. Since no such

environment yet exists in earthbound laboratories the distortion of optical flats by gravitational bending must be considered.

When a flat is supported on three pads, the test diameter will have

an apparent profile (a) which is the sum of its freeform profile (f) and

its bending profile (b) . The bending profile is dependent on the support

radius of the pads and whether the diameter is on the top or bottom surface

of the flat. Many years ago Emerson [4]* showed experimentally that when

the support radius of the pads is set so that minimum bending occurs, then

the bending is small enough to be neglected. This result is discussed in

section 4.2. The current calibration process calls for all test flats

and most master flats to be supported at the minimum bending radius. The

exception is discussed in section 5.2.

4. Some Results of 3-Flat Comparisons

Although the 3-flat method has not been described in detail, it has

been used to give some important results which are relevant to the functioning of the 2-flat method. These results fall into two groups.

4.1. Calibration of NBS Master Flats

The 3-flat method is ideal for calibrating master flats because it

Figures in brackets indicate literature references listed at the end of

this paper.

7
RING SUPPORT AT EDGE

i

Ta»(0

Figure 4.1 Figure 4.2

THREE POINT SUPPORT AT EDGE

B'

B

t:(0)

Figure 4.4
gives an "absolute" calibration. No master values are needed and the

process is relatively free of systematic errors. The measurement scheme

is such that the bending profiles are removed leaving only the freeform

profiles. The more commonly used masters are recalibrated by this method

occasionally to check on their stability.

4.2. Empirical and Theoretical Bending

In 1950 and 1954 three 10.6 inch master flats (designated #1-1, #1-2,

and #1-3) were involved in a thorough study headed by Emerson [4] where the

profiles and bending characteristics of the flats were measured. Bending

profiles of flat #1-3 were determined for six different support radii.

Currently this is the only NBS master flat for which experimentally derived bending corrections exist.

One result of the bending research was the discovery that minimum

bending along the test diameter occurs when the support pads are located

at .70r from the center. This result is not surprising since the circle

of radius

^0

= 72

= .7071r

divides the mass of the flat into equal parts. The bending corrections

for the .70r support were found to be negligible for the 10.6 inch flat,

so it is usually assumed that when flats of this size or smaller are supported at .70r the bending is negligible and the apparent profile is

the same as the freeform profile.

Some theoretical bending values have been developed by Timoshenko [5]

for circular flats using the following two types of support:

I. Ring support at the edge

Figures 4.1 and 4.2 show a flat supported at the edge by a narrow

ring and the corresponding profile of diameter AA' . On the diameter AA',

a distance r from the center, the bending deflection or sag w(r) is given

by the formula

(l+v)

W(r) = 3pg(l-v ) (a -r ) 16Eh^

2 * a - + II (3 +v)(a^ - r^) (4-1)

where

;

a = radius (in.)

h = thickness (in.)

V = Poisson's ratio (dimensionless)

Pg = density (lb. /in.

E = Young's modulus (lb. /in. 2).

9
Note that if the ratio a/h is large, then the first term greatly exceeds

the second, so w(r) can then be considered inversely proportional to the

square of the thickness.

Example . For NBS master flat //1-3, a = 5.3, h = 2.5, v = 0.14,

E = 10.15 X 10 , and pg = .079876. By letting r = 0, the sag at the

center relative to the ring support is computed to be

w(0) = .823 + .174 = .997 microinches.

Note that w(a) = 0.

II. Three point support at the edge

Figures 4.3 and 4.4 show a flat supported at the edge on three points

and the corresponding profile of diameter BB'. The formula for the bending

deflection is given only for r = 0 as Pa^

w(0) = .0362

^

(4-2)

2 3 2 where P = ira q , D = Eh /12(l-v ), and q = pgh. Combining terms then

^^Q^ ^ 1.3647pg(l-v^)a^ . (4-3)

Eh^

Example . For NBS master flat //1-3, using the same values as before,

w(0) = 1.329 microinches.

The value w(a) for the second example is obviously not equal to zero but

since the formula given is only for r = 0 it is unclear how to determine

the deflection for values of r different from zero. Consequently, it is

impossible to directly compare this theoretical bending value with the

observed bending value obtained by Emerson.

These two theoretical formulae, though interesting, have little or no application to the current measuring process. Only by making certain

gross assumptions could they be applied and at the present time it is felt

that such assumptions cannot be justified.

The conclusion of this section then is that when an optical flat is

being used as a master it should be supported at .70r where bending is

assumed to be negligible. It should be supported at another radius only

when experimentally obtained bending values have been established. In

10
that case the apparent profile of the master is computed by adding or subtracting (whichever is appropriate) the bending profile to (or from)

the freeform profile.

5. Measurement Process

5.1. Calibration Facilities

The current optical flat calibration facilities are located in a laboratory which is maintained at a constant temperature of 20° Celsius

and at a near constant humidity. In order to keep the temperature stable

during the measurement process, only the person who is making the measurements is allowed in the room. It is certain that thermal gradients do

exist within the optical flats, but it is assumed that they are small

enough in magnitude to be neglected; therefore, no attempt is made to

correct for temperature. It is also known that the daily variation in

atmospheric pressure affects the wavelength of light, but this effect is

negligible because only small differences in profile are measured.

The master and test flat assembly is encased in a cardboard and aluminum foil insulating shell and is set on ways which allow it to pass back

and forth under a Pulfrich viewer as shown in figures 5.1 and 5.2. The

maximum travel on the ways is slightly more than 16 inches

.

In determining the best setup to use for calibrating a given test

flat, the following facts must be considered:

(1) The test flat must be supported at seven-tenths of its own radius.

(2) The master flat must be supported at seven-tenths of its own radius unless its bending corrections are known.

(3) Due to bending considerations it is best to have the heavier

flat on the bottom.

(4) If the weight of the upper flat will significantly bend the lower

flat then the support pads for the upper flat must be directly above the

support pads for the lower flat.

(5) NBS master flat #1-3 may be supported anywhere from •25r^ to

l.OOr^ if bending corrections are properly applied.

The guidelines in table A have evolved from these considerations.

5.2. Choice of Master Flat and Support Radii

11
TOP VIEW OF CALIBRATION SETUP

OPTICAL

FLAT SHELL

OPENING —

'

WAYS

PULFRICH VIEWER

Figure 5 .

1

DIAGRAM OF INTERFEROMETER

OHe SOURCE

4m

Figure 5.2

12
5.3. Preparation for Measurement

When two flats are set up for measurement, the alignment of the

assembly should be checked to be sure that the test diameter passes back

and forth directly under the viewer. When the lamp is turned on a series

of interference fringes will be observed running roughly parallel to the

diameter. It is desirable that these fringes be as nearly parallel to

the diameter as possible. A slight angular movement of the fringes can

be obtained by pressing lightly over one of the thin pads. It is then

very important to check the direction of the wedge by pressing lightly

over the thicker pad. This should make the fringe separation wider . If

the fringe separation becomes narrower it indicates that the "thick" pad

is actually thinner than the thin pads, and the resulting sign reversal

will create errors in the computed values of the profile. This method

of increasing the fringe separation also spreads out the dark fringes

and makes them fuzzier. By controlling the thickness of the pads the

dark area of the fringe can be made just the right width to fit between

the crosshairs of the viewer. For helium yellow light and the Pulfrich

viewer the ideal distance between fringes is 200-300 micrometer units or 2.2-3.3 millimeters.

The points along the diameter at which measurements are to be made

are located with the aid of a paper ruler which has notches cut at the

proper x positions. The ruler is laid on top of the upper flat along side

the test diameter so that it appears with the fringes in the viewer as shown in figure 5.3.

After all preparations have been completed the flats are allowed to

sit for several hours, or preferably overnight, so that they come to

thermal equilibrium. It is likely that during this time there will be a slight angular and lateral movement of the fringes. This is allowable

because the effect of these changes is removed by the data reduction process .

5.4. Procedure for Taking Data

The fringe which most nearly coincides with the test diameter is chosen

as the primary fringe (fig. 5.4). The y position of this fringe is measured

at designated points along the diameter. Beginning at the position the

crosshairs are moved by a micrometer dial until they are centered over

the primary fringe (fig. 5.3). The dial reading y^ is recorded. The

assembly is then moved by the hand crank until the position X2 is centered

under the viewer and the crosshairs are again centered over the fringe.

The reading y^ is recorded. This process is repeated until readings at

all n positions have been made. Then, without delay, the measurements

are repeated in reverse order so that a series of 2n measurements of the

13
INTERFERENCE FRINGES

FRINGE

Figure 5.5

14
fringe profile is generated. Depending on the value of n, the measurements may take up to 30 minutes to complete. These measurements can then

be said to constitute one cycle of data. The usual procedure is to take one to four cycles of data at a time which constitutes a run. Directly

before and after each run several measurements of the distance between

the primary fringe and one of the two adjacent fringes are made. This

distance is called the fringe separation and is denoted by "r". Two independent runs are made per diameter. It is desirable for these runs to

be at least two hours apart in order to test the stability of the fringe

pattern

.

5.5. Mathematical Model of Fringe Profile

In setting up a mathematical model for a run it is necessary to consider that the fringes may drift significantly during the course of the run and the observer may need to take a break of variable length between

cycles. The model which follows accounts for these factors. Within each

cycle the measurements should be equally spaced in time and completed as quickly as possible. Over this period the fringe drift is assumed to be

linear. Between cycles the observer may take a break because nothing is

assumed about the behavior of the fringes over that period of time. The

entire run, however, should be completed within a couple of hours if possible in order to reduce the number of long-term variations which might

arise.

Assume that within a cycle the measurements are spaced by two units of

time, and for the sake of symmetry let the time t=0 fall halfway between the

consecutive measurements at the position. At t=0 during the first cycle

let the distance from the position on the test diameter to the fringe be

given by as shown in figure 5.6. (The fringes are shown as straight lines

for simplicity.) The purpose of the model then is to estimate these values

where i=l,n. At times before and after t=0 the fringe position will drift,

as shown by the dotted lines, and the distance to the fringe will change.

This change can be modeled by assuming that at each end point the fringe has

a constant drift throughout the cycle. Let b^ denote the rate of movement

of the fringe in micrometer units per unit time at the position during

th

the j cycle, and let d^ similarly denote the rate of movement of the

fringe at the x position. The movement of the fringe at the x. position

n ° 1

is then a weighted average of the end effects with the weighting function

depending on x^.

Irregular jumps in the fringe position between cycles can easily be

modeled. Let a. be the distance the fringe moves at the x^ position be15
FRINGE DRIFT DURING CYCLE I AND

FRINGE JUMP BETWEEN CYCLES 18 2

16
th

tween t=0 for the first cycle and t=0 for the j cycle as shown in figure 5.6. Let Cj similarly denote the distance the fringe has moved at

the X position, and note that a^=c^=0. Again the movement of the fringe

n i i

at the position is a weighted average of the end effects. Let m be

the number of cycles per run and let y_j^_. be the l*"^ observation in the

j^^ cycle. Then the measurement equations take the form

y. . = e, + [a. + b.(2i-2n-l)]

fx - X . "I K- ^iJ

+ [c^ + dj (2i-2n-l)]

l\- ^1.

+ if l^i^n, and

(5-1)

y.. = e„ + [a. + b.(2i-2n-l)J

•'ij 2n-i+l ' J J Pn ''2n-i+l"

| L ^n- ^1 J

[c . + d . (2i-2n-l) jF^^nzi+lZl] + e . . if n<i<2n,

where j=l,m and the e . . are independent error values from a distribution

2 with mean zero and variance a .

This linear model can be solved by the usual method of least squares

2n m

2 (i.e., minimizing E Z e. .). Let the 2nm observations be given by the

i=l j=l ^

vector

y = (Yii ••• y2n,l W2m y2n,m^''

^^"^^

and let the n+4m-2 unknowns be given by the vector

f^) = (e, . . . 9 b, d, a. b_ c„ d„ ... a b c d )' (5-3) 12 nll2222 mmmm

where 9 is nxl and Q is (4m-2)xl. Since a^=c^=0, they are not included

17
in the model. Let p, q, u, and v be vectors of order n whose components

are given by

X - X . n 1 1 X - Xt n 1

q . = 1-p . , 1 1

(5-4)

u. = (2n-2i+l)p. , and

v^ = (2n-2i+l)q_j^

where i=i,n. Let I be the identity matrix of order n and let K be the nxn matrix with I's on the minor diagonal and O's elsewhere, i.e.,

K =

0 0

0 0

0 1

1 0

0 1

1 0

0 0

0 0

(5-5)

(Note that premultiplication of an nxl vector by K flips the vector.) Let

the 2nm residuals be given by the vector

z = (^11 ^21 ^2n,l '*' ^Im ^2m "** ^2n,m^ ' (5-6)

and let the (2nm)x(n+4m-2) matrix of coefficients be given by

X =

-I -u -V 0 0 0 0 0 0 0 0'

K Ku Kv 0 0 0 0 0 0 0 0

I 0 0 P -u q -V 0 0 0 0

K 0 0 Kp Ku Kq Kv 0 0 0 0

I 0 0 0 0 0 0 P -u q -V

_K 0 0 0 0 0 0 Kp Ku Kq KvThe measurement equations are then given by

y = ^ U , (5-8)

18
The least squares estimation takes the form

E(y) = x(^^^ where

(5-9)

Var(y) = a^I .

The normal equations are given by

X'xQ = X'y (5-10)

and the estimates are given by

(^?^ = (X'X)-^X'y =

(j;, JJ)

X'y (5-11)

where V is nxn, U is nx(4m-2), and W is (4m-2)x(4m-2) . The predicted

values of the observations are given by

(5-12)

and the deviations by

d = y - y . (5-13)

The estimate of a is given by

s = /d'd/(2nm-n-4m+2) . (5-14)

(At this point the above parameters are expressed in micrometer units.)

In summary, the results of this fitting procedure which are of primary

interest are the set of n estimated fringe positions, 0^, and the estimated

standard deviation of a single measurement, s.

5.6. Computation of Test Flat Profile

Each estimated fringe profile is now converted to an estimate of the

air space between the flats along the given diameter. Figure 5.5 shows a cross section of the setup which is cut perpendicular to the x axis at

the position x,. The separation d. at the position x. is related to the

fringe profile 6,^^ by

19
xe.

where is the separation of the flats along the primary fringe, and X is the wavelength of the monochromatic light. (The d_j^ values are now

expressed in microinches . ) The separation d_j^ is also related to the profiles of the two flats. Figure 5.7 shows a cross-section of the setup

along the test diameter as viewed horizontally with the upper flat being

designated U and the lower flat L. For the purposes here it does not

matter which is the test flat and which is the master. Let r^ and r'^ be

parallel auxiliary lines separated by an arbitrary distance h which pass

through the upper and lower flats respectively, and for each line let the

direction toward the other line be the positive direction. The adjacent

profiles of the upper and lower flats can then be taken as deviations

a^ and a^ from the lines r^ and r''^ respectively. At every position x_j^ the relation between the parameters takes the form

h = a^ + a^' + d. . (5-16)

1 1 X

Substitution of the expression for d_^ in equation 5-15 into equation 5-16

gives the following expression for the sum of the apparent profiles of the

two flats:

The value of h was arbitrary so it may be set equal to d^. Then

TT T

Let the vector of profile values aV + be denoted by a^ for the first ^ 11 I run and a^^ for the second run. Intuitively, the difference a^ - a^^ should

be linear in x because the flats are rigid bodies. A statistical test is

performed under this assumption in order to determine if there is a significant between-run error term. If such a term is found to be significant

then the measurement process is said to be "not in control" and the measurements are repeated.

The test is implemented as follows. Let s^ and s^^ be the respective

standard deviations of a single measurement of runs 1 and 2, given in eq20
RELATIONSHIP OF PROFILES
uation 5-14, which have been converted to micro inches , and let

(5-19)

A constant term and a linear term are removed from the vector of z values as follows: Let

X =

1

1 X.

1 X

n_

and a = (5-20)

The least squares estimation of a takes the form

E(z) = Xa

and

Var(z) = Va' (5-21)

where V is as given in equation 5-11. The total variance can be

written as

2 2,2

t w b

(5-22)

2 2 where a is the within-run variance and a, is the between-run variance. w ^ b

The estimates a are of no importance. The vector of residuals is given

by

d=z-z=[I- X(X'V '•X) ""-X'V ^]z, (5-23)

and the estimate of the total variance is given by

2 ^ d'V d

't n-2

(5-24)

The within-run variance is estimated by

s

2 = s 2^2+ s w I II

(5-25)

The ratio

S = sl/s^

t w

(5-26)

22
has an F distribution with (n-2 ,4nm-2n-8m+4) degrees of freedom. Taking

the level of significance of the test to be .01, the value

F = F gg(n-2,4nm-2n-8in+4) (5-27)

2 is obtained from table B. If S > F then the between-run error, a, , is

b determined to be significant and the system is "not in control". If

S < F then the between-run error is determined to be not significant and

the mean of the two runs,

^I ^11

a =

^ (5-28)

is computed and separated into components,

T M

a = a + a (5-29)

T M

where a and a are the respective apparent profiles of the test flat

and master flat. The master flat profile is subtracted to give

a^ = a - a^ . (5-30)

M

Recall from section 3.3 that if a bending correction b is used in

M M

determining a , then the sign of b depends on whether the master flat

is on the top or bottom. If the bending profile of the master flat is

unknown then it is usually assumed to be zero provided the master flat

is supported at 0.70r„.M

T T

The computed profile a also contains an unknown bending value b . If the test flat is used in the orientation opposite the one in which it was calibrated, there will be a difference in the respective apparent

T profiles of +2b . This difference can usually be assumed negligible

when the flat is supported at 0.70r^.

The exact degree of bending in a given flat is highly dependent on

its radius and thickness (see section 4.2). There is evidence that flats

which are 10" or less in diameter have negligible bending when supported

at 0.70r. For example, when the NBS 10.6" master flat is supported at

0.70r the maximum bending along its diameter has been shown experimentally to be 0.02 microinches.

T The values a given in equation 5-30 are referenced to some arbitrary line. The final profile values a which are referenced to the

least squares line through the profile must be determined. Frequently

the end values of the profile will fall off sharply if measurements

23
were made near the edge of the flat where there is dubbing. If these

values were equally weighted they might distort the appearance of flat- ness along the diameter. To account for this possibility a diagonal

weighting matrix,

W =

w^ 0

0 1

0 0

0 0

0 0

0 0

1 0

0 w

n _

(5-31)

is introduced where w^=w^=0 if the end points are in the dubbing and w-=w =1 if not. Let

1 n X = WX (5-32) w

where X is as defined in equation 5-20. The final vector of profile

values is then given by

= [1 - X(X'www X )~"^X'] a^ = Qa^ (5-33)

5.7. Error Analysis

The uncertainty of the computed profile values can be expressed as the sum of random and systematic components of error. The random com- ponent is taken to be the three standard deviation limit for random

error. The systematic component is taken to be the sum of all other

known effects which contribute error to the system.

Let Sp, the pooled standard deviation of a single measurement (in

microinches) , be estimated from all runs made on the diameters of the

test flat. From equations 5-18 and 5-11,

Var(a^) = Var(a^^) = Vs

and

Var(a) = Var

II'

+ Var

'II = V

2 s

(5-34)

(5-35)

From equation 5-30,

Var(a ) = Var(a) + Var(a ) = Var(a)=V (2 (5-36)

24
M

since the values a are constants. From equation 5-33

^2 ^2

Var(a^) = Q[Var(a^)] Q' = Q[v(2^)]Q' = QVQ' (^^j, (5-37)

t ll therefore the standard deviation of the i profile value of the test

flat is

KQVQ'),,

a.

1

Two non-negligible sources of systematic error are the uncertainty

of the master flat values and the non-linearity of the ways which move

the flats back and forth under the viewer. Let E^^ be the largest uncertainty of the master flat values (including bending uncertainties) , and

let be the maximum error caused by the non-linearity of the ways. For

the current setup the ways are believed to be straight to within .0002

inch which corresponds to .47 micrometer units. This error is converted

to test flat profile error by

Ew = ^ (5-39) max

where r is the maximum fringe separation for all runs on the test flat

max ° ^ diameters. All other sources of systematic error are assumed to be negligible; therefore, the limit for total systematic error is

E = + , (5-40)

and the total uncertainty of the computed profile values is

= 3s^^ + E . (5-41)

One shortcoming of this calibration process is its lack of strict

statistical controls. The test for between-run variance given in section 5.6 is not broad enough to detect all possible errors which might

occur. A test of the "computed" standard deviation against an "accepted"

standard deviation is not possible since each calibration involves certain

factors which make it unique. Two such factors are the fringe sharpness,

which varies depending on the fringe separation and the reflective properties of the flats, and the skill of the particular observer in centering

the fringe in the crosshairs of the viewer. Over many years these factors

25
have combined to give standard deviations of a single measurement between

.05 and .25 microinches. In the absence of a strict statistical test,

the standard deviation is accepted if it lies within this range.

Since there is no check standard in the process, a change in the

master values could go undetected until the master itself is recalibrated.

To guard against the possibility of such an error the computed profile of

the test flat is checked against a previous calibration if one exists.

There is strong evidence that no significant long-term change in the profile of master flats occurs when used at a temperature of 20°C. The NBS

10.6 inch master flat, for example, has been calibrated four times over

a period of 18 years and the maximum disagreement in values is .13 microinch. This difference is easily within the limits of random measurement

error. To assure their stability, all NBS master flats are periodically

calibrated by the 3-flat method.

Although most of the above controls are not rigid, they have proven

to be adequate for protection against gross errors.

6. Graphical Display of Profiles

Ideally the calibration of an optical flat along a given diameter

would yield a continuous curve which represents the profile of the flat

along the diameter. Such a calibration could be accomplished under the

current measurement system by letting n take on a very large value, but

the data-taking process would be quite laborious for the observer. In

general, high quality optical flats have smooth profiles which can be

adequately represented by measuring the profile at a few points and con- necting the points with a smooth curve. In pre-computer days a hand

drawn graph of each diameter was sometimes submitted as a supplement to

the Report of Calibration. Such a graph was useful in giving the user a good idea of the profile at a glance. The present computer facilities

allow a graph of each diameter to be plotted as part of the data reduction process.

One good method of fitting a smooth curve to a discrete set of

points involves the use of cubic spline functions. Let the set of points

{ (x_j^,a_j, ) , i=l,n} represent the profile of a diameter. In each of the

intervals {(x^,x^_^^), i=l,n-l} a cubic polynominal is fit to the points

subject to the condition that the cubic between points x, , and x. must ^ 1-1 1

agree with the cubic between points x_j^ and at the point x^ in their

first and second derivatives. Also, either the first or second derivatives, but not both, must be specified at the end points x^ and x^. For

this problem it seems most natural to set the second derivatives equal to

26
zero. The resulting equations form a tridlagonal system of 3n-3 equations

In 3n-3 unknowns. Methods for formulating and solving this system are given In Ahlberg [1] and UNIVAC [6] . Once the coefficients of the n-1

cublcs are obtained the Interpolated value at a given point Is determined by plugging the value of Into the appropriate cubic.

Interpolated values were computed at 201 equally spaced points for

each of the diameters In the example which follows (fig. 7.3). The stars

indicate the actual measured values. The Interpolated values are connected

by straight lines, but due to their closeness they give the appearance of

a smooth, continuous curve.

7 . Example

The following example Is presented In order to Illustrate the calibration process. An eight inch single surface test flat identified as L9210 was calibrated along its A-B and and C-D diameters. As given in

section 5.2, the flat was supported on top of the NBS 10.6 inch master

with both sets of support pads being placed at a radius of 0.70 (4 lnches)=

2.8 inches. This necessitated using the empirically derived bending cor- rections of the master flat for this support radius. Three cycles of measurements were made per run at 11 positions along each diameter. A sample

data sheet for one run is given in figure 7.1. The data reduction process was then carried out as shown below where the step numbers correspond to

those given in figure 7.2.

Step Operation Reference

(1) Choose positions along x axis sec. 3.2.

(2) Compute sum of upper and lower flat sec. 5.5., 5.6.

profiles (in microlnches) for runs 1 and (eq . 5-11 & 5-18)

II from mathematical model

(3) Compute s^ and s^^ for the two runs (eq. 5-14)

(4) Compute (run I prof lie) -(run II profile) (eq . 5-19)

(5) Compute residuals from straight line fit (eq. 5-23)

(6) Compute s^ for straight line fit (eq. 5-24)

2 2 2

(7) Compute the ratio 3^/(3^+3^^) (eq. 5-26)

(8) Obtain F value from table B (eq. 5-27)

(9) If good F test, compute mean of profiles (eq. 5-28)

from two runs

27
OPTICAL PLAT CALIBRATION - 1 FLAT VS. MASTER

Side

1 T

2 (D

Diameter

C-D

Run

®

2

Test Flat Lot # L ?Z I Q

Observer £> £rhe^

Date 4 / Z / 74-

Comments: 9'/<? /I/*^.

Inch Test Flat #

Inch Master Flat # (-3

Test Flat on: bottom

Support Radius — .7 (H-- 2,$"

Do not pause during a cycle. OK to pause between cycles.

Cycle 1 Cycle 2 Cycle 3 Pos.

w { I t t

A

I XLiW lie O J w \w w t

1

1/ " J53 347 34-a 35c? 347

2

1/ "

Vlr 3^6 344

3

1 ly " 3Z7 32.2 3 la Ji/ 3\S

4

"i h " 'J P c 2jy 2<r3

5 o V

" O/x. 2-/6 JIL 2Gb -til 26 7

6

ji " •f Z6 7 245" 242 244 -I ^ /

7

/I /

" 22 i» 2/4 2/Z —* Zog

8

/— II IS J /4 7 /4f

9

/ h II to A, //2_

10 "a. 53 SI

11 n 15/ " 2-1 2o 2J '7 IS

1213

1415

16

17

18

19

20

21

22

23

Cycle 4

Are the end positions in the dubbing? ^^yes) no

Compute

Fringe

Separation

Fringe 1: 271 277

Fringe 2: ^1

Difference

:

2\2~ 110 Z-l^

Average Difference; 2/2.. o

IISCOMM NBS DC 3/75

Figure 7.1

28
A-B Diameter

CD (2) (2) (4) (5) (9) (10) (11) (12) (14) (16)

Pos ^11 z d a a

1 a a ®*T

a a

.06 19.18 8.47 10.71 .02 13.82 .01 13.81 -2.10 .048 .34

.50 18.82 8.74 10.08 .01 13.78 .12 13.66 -1.54 .031 .29

1.50 17.72 9.05 8.68 .04 13.39 .21 13.18 - .41 .034 .30

2.50 15.97 8.70 7.27 .07 12.33 .21 12.12 .15 .036 .31

3.50 14.91 9.16 5.75 -.02 12.04 .35 11.69 1.32 .037 .31

4.00 13.88 8.87 5.00 -.05 11.38 .34 11.04 1.48 .037 .31

4.50 12.20 7.95 4.25 -.09 10.07 .32 9.75 .99 .037 .31

5.50 8.55 5.74 2.81 -.10 7.15 .27 6.88 - .26 .036 .31

6.50 6.20 4.69 1.51 .04 5.45 .32 5.12 - .41 .034 .30

7.50 2.80 2.73 .07 .02 2.77 .16 2.60 -1.32 .031 .29

7.94 1.78 2.30 - .52 .07 2.04 .06 1.98 -1.23 .048 .34

(3)

^11

"

.144 (6)

s = .148

t (13)

P 1

/[(.144)^+C.: + C.132)^+C

L35)^

9 .136)^J/4

(7)

2

s

t .0219

0.56 ' 025 y"

2 ^ 2 .0390

"ll

(15) 174 y"

C8) F_99(9,90) = 2.62 E = . 199 y"

C-D Diameter

Pos ^1 ^11 z d a M

a

i a a

a a

.06 9.15 9.74 -2.59 .07 10.44 .01 10.43 -1.33 .048 .34

.50 9.09 9.89 -2.81 -.07 10.49 .12 10.37 - .96 .031 .29

1.50 8.74 9.56 -2.82 .10 10.15 .21 9.94 - .39 .034 .30

2.50 7.79 8.93 -3.14 -.04 9.36 .21 9.15 - .19 .036 .31

3.50 7.92 9.18 -3.26 .02 9.55 .35 9.20 .84 .037 .31

4.00 7.71 9.12 -3.41 -.04 9.42 .34 9.08 1.22 .037 .31

4.50 6.92 8.36 -3.43 .03 8.64 .32 8.31 .95 .037 .31

5.50 4.67 6.46 -3.79 -.15 6.56 .27 6.30 - .08 .036 .31

6.50 3.40 5.34 -3.94 -.11 5.37 .32 5.05 - .34 .034 .30

7.50 1.44 3.55 -4.11 -.11 3.49 .16 3.33 -1.06 .031 .29

7.94 1.03 2.83 -3.80 .29 2.93 .06 2.87 -1.08 .048 .34

"ll

=

.132

.136

- 321 *The runs were not repeated even

t * though this number exceeds the

maximum F ratio 2.62. There was

2 apparently a bad reading in the

.1030 * 2.87 dubbing area at the 7.94 position

4- 2 ^11

.0359 on the

value

; X axis

of .29 as indicated by the

in the "d" column.

= .137

Figure 7.2

29
8 INCH OPTICAL FLAT NO. L9210 A-B DWMCTER

1.00 2.00 3.00 4.00 5.00 POSmON ON DIAMETER (INCHES)

6X)0 7.00 8.00

8 INCH OPTICAL FLAT NO. L9210 C-D DUMETER

1.00 2i» 3X10 4.00 5.00 POSmON ON DUMETER (INCHES) 6.00 7.00

Figure 7.3

30
(10) Obtain profile values for master flat

}

(11) Subtract master profile from mean profile (eq. 5-30)

(12) Compute final profile values relative to (eq . 5-33)

least squares line

(13) Compute pooled standard deviation of single sec. 5.7.

measurement

(14) Compute standard deviation of final profile (eq. 5-38)

values

(15) Compute limits for systematic error sec. 5.7.

(eq. 5-39 & 5-40)

(16) Compute total uncertainty for final values (eq. 5-41)

A smooth curve was fit to each set of values in step (12) by the method

described in section 6. The graphs are given in figure 7.3.

8. Conclusion

The data reduction part of the calibration process is now fully computerized. The output includes the usual Report of Calibration plus a plot of the profile of each diameter.

There are a couple of pitfalls to be avoided in using the calibrated

flat. First, the values on the Report of Calibration are valid only when

the flat is supported in the exact manner as it was during calibration.

If the flat is supported in a different manner (such as a ring support

around the edge) then it will distort to some extent due to gravitational

bending. In that case the user must determine the bending corrections

for himself unless he chooses to ignore them. Second, if the flat is

properly supported on three pads as described in sections 3.1., 4.2., and

5.2. then minimum bending occurs only along those diameters which are parallel to the lines which connect the pads. This means that in the usual

case where the A-B and C-D diameters are perpendicular the reported values

cannot be legitimately used simultaneously. This is illustrated in figure 8.1. One method of overcoming this difficulty would be to measure

the profile along diameters A-B, C-D, and E-F which are 60° apart as shown

in figure 8.2.

The main weakness of this calibration process is that it is more "onedimensional" than "two-dimensional". (An example of a true two-dimensional

calibration using an entirely different technique is given by Dew [2] .)

31
BENDING PROPERTIES FOR .70 R SUPPORT

Figure 8.1

Figure 8.2

32
Despite this shortcoming, the process does give a high degree of measurement accuracy and it has sufficiently met the needs of a large number of

optical flat users for many years.

I wish to acknowledge three people who were of great assistance to me

in the writing of this paper. I thank Mr. Ralph C. Veale of the Dimensional

Technology Section who spent much time acquainting me with the many facets

of this complex measurement process. I also thank Mr. Joseph M. Cameron of

the Office of Measurement Services for his many helpful suggestions on improving the paper. Finally, I thank Dr. James J. Filliben of the

Statistical Engineering Laboratory for suggesting the mathematical model

and statistical test which appear in sections 5.5. and 5.6.

33
Table A. Guide for setting up flats

Test flat

diameter

Test flat

position

Which

master?

Master support radius

Apply master

bending corr

0" - 7" top 10.6" no

7" - 10.6" top 10.6" .70r^ yes

12" either 12.25" .70r^ *

16" either 16" .70r^

Odd sizes over 10.6" are calibrated either by overlapping or by a 3-flat com parison with two other flats of the same size. The method of overlapping in

volves calibrating the central segment and the two end segments separately

and then fitting them together.

* Test flat diameter may vary slightly from master flat diameter. Since

bending values are not known for these two master flats, an error term

must be added to the master flat uncertainty if its support radius is

not exactly .70r .

34
Table B. Percentiles of the F distribution

Values of F (n-2 , Amn-2n-8m+4)* for certain m and n_

• 99

m

11 1 2 3 4

5 9.78 5.09 4.51 4.29

6 7.01 4.22 3.83 3.67

7 5.64 3.70 3.41 3.29

8 4.82 3.35 3.12 3.03

9 4.28 3.10 2.91 2.82

10 3.89 2.90 2.74 2.67

11 3.60 2.76 2.62 2.56

12 3.37 2.63 2.51 2.46

13 3.18 2.53 2.42 2.37

14 3.03 2.44 2.34 2.30

15 2.91 2.37 2.28 2.23

16 2.80 2.31 2.21 2.18

17 2.70 2.25 2.16 2.13

18 2.62 2.20 2.11 2.09

19 2.55 2.16 2.07 2.06

20 2.49 2 .12 2 . 03 2 .02

ZX Z . Uo z . uu 1 Q Q

22 2.37 2.04 1.97 1.95

23 2.33 2.01 1.94 1.92

24 2.28 1.99 1.92 1.90

25 2.24 1.97 1.90 1.88

from the twelfth edition of C.R.C. Standarc

Tables, with some interpolation.

35
References

[1] Ahlberg, J. H., Nilson, E. N., and Walsh, J. L., The Theory of

Splines and Their Applications, Academic Press, 1967, section 2.1.

[2] Dew, G. D., The Measurement of Optical Flatness, Journal of Scientific Instruments, Vol. 43, pp. 409-415, 1966.

[3] Dew, G. D. Systems of Minimum Deflection Supports for Optical Flats

Journal of Scientific Instruments, Vol. 43, pp. 809-811, 1966.

[4] Emerson, Walter B., Determination of Planeness and Bending in Optical Flats, Journal of Research of the National Bureau of Standards,

Vol. 49, No. 4, pp. 241-247, October 1952.

[5] Timoshenko, S., and Woinowski-Krieger, S., Theory of Plates and

Shells, 2nd Edition, McGraw-Hill Book Co., 1959, sections 20 and 65

[6] UNIVAC, MATH-PACK Programmers Reference, Sperry Rand Corporation,

1970, pp. 2-68 through 2-72.

USCOMM-NBS-DC 36
NBS-114A IREV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RI-I^ORT NO. NBSIR 75-9752. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

XLlC V_>ct J. X UI. d L XUll UL dll VJ^LXL-dX r Xci L by Interf erometric Comparison to a Master Optical Flat

5. Publication Date December 1975

6. Performing Organization f-ode 213.12

7. AUTHOR(S)

Charles P. Reeve

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. 2131120

11. Cont r ac t /Gr an t 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 sigr)ificant information. If document includes a significant

bibliograpfiy or literature survey, mention it here.)

Optical flats are normally calibrated by one of two methods at the National

Bureau of Standards. The most frequently used method involves a direct interf erometric comparison of a test flat to a master flat. The purpose of this paper

is to present a detailed description of this measurement process. Some of the

topics discussed are methods of supporting the flats, effects of gravitational

bending, measurement of interference fringes, formulation and solution of the

linear least squares model, analysis of errors, choic,^^J,^__prof ile reference line,

and graphical display of profiles. An example is worked out in a step-by-step

fashion in order to illustrate the process, and a statistical test is incorporated

to check whether the process is in control.

17. 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)

Calibration; diameter; gravitational bending; interf erometry; least squares line;

linear model; optical flat; profile; standard deviation; uncertainty.

18. AVAILABILITY H Unlimited

1 For Official Distribution. Do Not Release to NTIS

1

' Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402. SD Cat. No. CH

1 y! Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

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