NIST

PUBLICATIONS

NBSIR 76-993

The Calibration of a Pentaprism

i

Charles P. Reeve

Ralph C. Veale

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

January 1976

Final

U S DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANOAROS

.U55

//76"e93

1975

NBSIR 76-993

THE CALIBRATION OF A PENTAPRISM

Charles P. Reeve

Ralph C. Veale

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

January 1976

Final

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

James A. Baker, III, Under Secretary

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

CONTENTS

Page

1. Introduction 1

2. Description and Uses 1

3. Geometrical Models 3

3.1. Angular Errors between Faces - Transmitted Beam 5 3.2. Angular Errors between Faces - Reflected Beams 6

3.3. Errors in Vertical Alignment with Incident Beam 9

3.4. Maximum Deviation of Incident Beam from Normal 13

4. Method of Calibration 13

4.1. Setup Procedures 15

4.2. Measurement of Parameters 15

4.2.1. Deflection Angle of Transmitted Beam 15

4.2.2. Angle between Reflected Beams 18

4.2.3. Indices of Refraction 21

4.2.4. Interior Angles 21

5. Effects when Indices of Refraction Change 22

6. Example 23

7. Conclusion 26

Tables 1 and 2 27

Tables 3 and 4 28

References 29

THE CALIBRATION OF A PENTAPRISM

by

Charles P. Reeve and Ralph C. Veale

1 . Introduction

The Dimensional Technology Section is currently responsible for

the calibration of pentaprisms at the National Bureau of Standards.

This responsibility requires that we be familiar with

1) a good method of calibrating pentaprisms

2) their common uses

3) effects of errors in their internal geometry, and

4) effects of errors in their alignment with a beam of light.

Information which we have been able to obtain on the above subjects is

currently quite scattered throughout different sources and needs to be

unified. Additionally, through investigation carried out at the Bureau, we have learned some important facts relative to the functioning of a pentaprism that we have been unable to find documented anywhere else.

Therefore, the purpose of this paper is to collect from these sources

all the facts about pentaprisms which we believe are relevant to their

calibration and usage. These facts are discussed individually in the

following sections where material is liberally incorporated from the

references

Although some of the mathematics may appear awesome, the results

are fairly simple. The method of calibration is also simple and the

data reduction can easily be done by hand with the aid of a small calculator ..

2. Description and Uses

The word pentaprism is a commonly used contraction of "pentagonal

prism" which, as its name indicates, is a five-sided prism (fig. 1).

It is one of a class of objects knovm as "constant deviation prisms"

whose purpose is to bend a beam of light exactly 90° regardless of the

1
Figure 2

Figure 3

2
angle of incidence of the light. Each of the five faces is perpendicular to both the base and the top. Faces 1 and 2 serve as entry and

exit planes for a beam of light, and the angle between them is nominally

90°. Faces 3 and 4 are silvered and serve as reflectors, and the angle

between them is nominally 45° . Face 5 serves no purpose other than

truncating the unusable portion of the wedge between faces 3 and 4.

The function of the pentaprism is governed by the law of reflection

which states that if an incident beam of light is reflected off two

planes, then the reflected beam is bent by an angle which is equal to

twice the supplement of the acute angle between the reflectors. Thus

in a pentaprism the beam deviation is 2(180 - 45)° = 270°. It can easily be shown that if the incident beam is not normal to the face

which it enters, it will be refracted on entry and exit but through

the same angle so that the effects cancel each other (fig. 2). Of course this assumes that the entry and exit faces are exactly 90°

apart. The case where that is not true is covered in the next section.

Sometimes a pentaprism is mounted in a square metal housing with

ground locating faces and is called an op tical square (fig. 3), referring to its functional use. The faces of the pentaprism are usually

mounted not quite parallel to the corresponding faces of the mount to

avoid ghost reflections directly off the prism face when the mount is

aligned to the line of sight. This is especially important if the

optical square is being used with an autocollimator because the ghost

image could interfere with the returning beam.

A pentaprism (optical square) is useful in many types of optical

measurements where the line of sight or a beam of light must be turned

at a right angle. For example, it may be used with an autocollimator

to check the parallelism of Internal surfaces, perpendicularity of a translational movement, or relative squareness of two or more machine

surfaces. These and other examples are given in Farago [IJ* and

Hume [2,3]

3. Geometrical Models

In the past it was usually assumed that if a pentaprism was observed to bend a beam of light within a few seconds of a right angle

then the 45° and 90° angles must also be within a few seconds of nominal.

This has been found to be an erroneous assumption. A recent sampling

of five pentaprisms indicated that the 90° angle is likely to deviate

several minutes from nominal. In view of this it is mandatory to ex- amine the effects of such deviations. This is accomplished in the next

two sections. Also considered will be two characteristics of a "per-

*Numbers in brackets indicate references listed at the end of this paper.

3
90° +a

/ \

/ 1 /

/ 1 /90°-\l/i\

190° + ,^ ^02 9 J

Figure 4

4
feet" pentaprism: the effects of tilting it in a vertical plane and

its usable angular range of incident light.

3.1. Angular Errors between Faces - Transmitted Beam

Assume here that the 90° angle between the faces is in error by a small amount 4) and that the internal bending of the beam is in error by

a small amount <^ due to an error of ^ in the 45° wedge angle. The path

of a transmitted beam in a horizontal plane is shown in figure 4. Let

the observed bending angle of the beam outside the pentaprism be given by

90° + a. It is then essential to know a as a function of (J), ^, 6^, y^,

and y2 where 9^ is the deviation of the incident beam from normal,

is the index of refraction of the medium (usually air), and y^ is the

index of refraction of the pentaprism. Assume all angles in figure 4 are positive as shown. Then 90° - i) + - = 90° + 4), or

<f> + ij; = 63 - e^, (3-1)

and 90° - ij; + 6^ - = 90° + a, or

a + i) = - By (3-2)

The refraction law gives

y^ sin 6-|^ = s±n B and

y^ sm = y^ sin

Writing these equations in the form

a = B^ - B^ - ^Ij,

] , = Sin — sin 6- 1 , ^ 3/

= (J) + 4; + e^.

-1/^1

'2 = 4,

(3-3)

C3-4)

5
and substituting from bottom to top gives the expression

a = Sin \ — sm

^^1

-1/^1

(f) + i/; + Sin — sin 9, ^2 ^ \

- ^. (3-5)

Using the power series expansions for sine and arcsine functions and

ignoring all powers of 9^ greater than two and powers of (() + i|j great€

than one, equation (3-5) reduces to the approximation equation

a«—^2 /^2 \ 1 /^2 ^1 \ 2

<!)+ — - lU + ^ — - — (cf) + i|;) 8,. (3-6)

Let a(4), 0^, be the representation of a function of the

five variables in the system, and note that

a(0, 0, 0^, y^, M^) = 0 (3-7)

in both the true (3-5) and approximate (3-6) expressions for a. This

indicates that a well-made pentaprism is a true square for all normal

and off-normal beams.

Let Aa be defined by

Aa = a((f), ^, 9^, y^, - a(((), 0, y^, y^) . (3-8)

Then Aa is the fluctuation of a as 9^ varies from zero to some maximum

value. In equation (3-5) this fluctuation is a maximum when (}>, and

9^ all have the same algebraic sign. In table 1 Aa is evaluated using

equations (3-5) and (3-6) for specific values of (|), 4^, 9^, y^, and

The values from the approximate expression (3-6)*^ are in parentheses.

It is obvious that the approximate expression is very good over this

range of values for the arguments. The effectiveness of the pentaprism,

in terms of the small magnitude of Aa caused by manufacturing angle errors (\>. and ^ and alignment angle error 6^, is impressive.

3.2. Angular Errors between Faces - Reflected Beams

Figure 5 illustrates the paths of the two most intense reflected

"ghost" beams for the pentaprism. The angle between these beams,

2^ = 9-, - 9-, is a function of (j) , 9^, y^ , and y_. Later it will

/ o i i ^ *When evaluating this expression angles must be expressed in radians.

6
7
be shown how the measurement of 2C in conjunction with the measure- ment^of a can be used to calibrate the interior angles of thrp'nLAssume that all angles in figure 5 are positive as shown. Then

. 90° - + 63 - = 90° + (j), or

* + = 63 - e^, (3_9)

^""^ ' 90° - ^ - 63 + = 90° + <},, or

0 + ^ = - 63. (3_io)

The refraction law gives

sin 0^ = sin 0^ and

j \ sin 0^ = sin 0^, (3_ii)

and the reflection law gives

^5 ~ ^3 ^^'^

^8 " ^1- (3-12)

Writing these equations in the form

2, = 0^ - 63,

^8

=

'7 = Sin-lgsin

0^),

= <^ + + 03, (3_13)

65 = 63,

63 = <J) + !|J + 0^,

-1/^1

8
and substituting from bottom to top gives the expression

25 = Sin — sin 2((}) + ijj) + Sin e 1* (3-14)

When 2E, is measured with an autocollimator the pentaprism is of necessity adjusted so that 9^ is small (usually less than 5'). Therefore

equation (3-14) can be reduced using the first order approximations to

the sine and arcsine functions to give the approximation equation

Now consider the case of a perfect pentaprism where the incident

beam is not perpendicular to the pentaprism face in a vertical plane.

Assume that the incident beam is horizontal but the pentaprism is tilted

toward the beam by an angle x as shown in figures 6 and 8. It is natural to assume that the transmitted beam would show both a vertical and

horizontal component of change from the x = 0 position (fig. 7). An

experiment was conducted where a pentaprism was tilted through a 9' arc with a sine bar and the vertical and horizontal changes in angle were

read by an autocollimator. The observed data are given in table 2.

There is apparently a one-to-one correspondence between the tilt angle

and the vertical deflection of the beam, but no correspondence between

the tilt angle and the horizontal deflection of the beam. The vertical

deflection may be of interest in certain situations, but not in the

calibration process since the autocollimators are set to read horizontal

angle only. Therefore consideration will be given only to deflection in

the horizontal plane.

It is interesting to verify geometrically that the tilt angle does

not significantly affect the angle of deflection horizontally. When the

pentaprism is in a horizontal position with x=0 as shown in figure 7,

consider the points a, b, and c which lie at the intersection of the

plane z=h and the three vertical edges which form the 45° wedge angle.

(Note that the point c lies on the truncated edge.) Let the pentaprism

be tilted by rotating it about the x axis through an angle x, and consider the new points a*, b', and c' which lie at the intersection of

the plane z=h and the same three vertical edges respectively as shown

in figures 6 and 8. By calculating the (x,y) coordinates of the six

points the apparent change in the horizontal wedge angle y can be

computed. The coordinates are as follows:

25 = 2—(4) + ^) . (3-15)

3.3. Errors in Vertical Alignment with Incident Beam

9
Figure 6
point (x,y) point (x,y)

(d,0) a' (d,-=^-^^^)

2

\_ cos T

^ • 2 b (0,d) b' (0, —2— " e_sin_:L) cos T COS T

c (e,e) c' (e , e cos t).

If the origin Is shifted to point c' then the coordinates of a' and b'

become

-e sln^T

(d-e, e cos t) = (d-e, ) and

cos T cos T

/ d e sln^T V , d-e v (-e, e cos t) = (-e, ) cos T cos T cos T

respectively. Consider these two points to be vectors A = (a^, a^) and

B = (b^, 1)2) In the (x,y) plane and recall that the expression for their

dot product can be written

cos y' = (3-16)

V2

2 I I r~2 T

a^ + a^, |B|=-yb^ + h^, and y' Is the

angle between the vectors. Then

/ jN J. e(e-d)

e(e-d) +

Y~

, cos T cos Y = ,^ 2 n r 2n * (3-17)

(d-e)" +

2 cos T e2 ^ id^)2 cos T

Substituting d = e(l - tan 22 1/2°) = e(2 - /2) Into eq. (3-17) gives

2

, cos T +1

' 1 4 1 \COS T + 6 cos T +1

COS y' = / / \

(3-18)

Let Y = Y + Ay = 45° + Ay- Then

11
Ay = Cos

-1 cos T + 1 cos T + 6 COS T + 1

- 45' (3-19)

Now consider the effect of Ay on the deviation a' of the reflected

beam from a right angle inside the prism (fig. 8). The reflection law

gives 270° + a' = 2(180° - y') = 2(180° - 45° - Ay) = 270° - 2Ay, hence

a' = -2Ay. (3-20)

Since the exiting beam is not perpendicular to the prism face it is

refracted. The resulting deviation a of the reflected beam from a right

angle outside the prism is related to a ' by

hence

-l/^2

a = Sin — sin a ' 1 , ^1

-1P2 . , ,1 a = Sxn — sin(-2Ay)

L^i J

(3-21)

(3-22)

Substituting eq. (3-19) into eq. (3-22) gives

-1

a = Sin

~y2(cos^T - 2 cos^T +1)'

yj^(cos'^T + 6 cos^T +1)

If T is small then eq. (3-23) can be approximated by

^2\

(3-23)

(3-24)

Values of a are computed for selected values of x in table 3 using,

the exact equation (3-23) and the approximate equation (3-24) . It is

very clear from table 3 that any tilt angle less than 3° has no measurable effect on the reflected beam in the horizontal plane. Since the

tilt during calibration is never more than a few minutes, the tilt angle

can be dismissed as a cause for concern.

It should be noted that a tilt along the base of the face which is

perpendicular to the one just considered gives the same effect since the

pentaprism functions in a symmetric manner.

12
3.4. Maximum Deviation of Incident Beam from Normal

Occasionally it may be of interest to know the range of incident

beam angles for which there is transmission through the pentaprism.

The two extreme cases (for a perfect pentaprism) are shown in figure 9

where 9. is the deviation of the incident beam from normal and 6 is the

1 r deviation of the refracted beam from normal. Some mathematical manipulation gives the result

= Tan"^(

^ I^

) = 16° 19' 29.8", (3-25)

hence

e, = Sinl— sin 6 ^ = Sin"""" /

, ] . (3-26)

^ V^l V Vy^VTTTTf/

If = 1.0 and = 1.5, then 9^ = 24° 56' 14.4".

For incident beams entering at the center of the face (fig. 10),

the maximum deviation is

9 = Tan'V , = 8° 19' 53.6", (3-27)

.4 +.2/2

hence

1 = Sin sin 9 ^ = Sin V ] . (3-28)

^1 V ^25 + 16/2/

( )

\y, \l25 + 16/2/

If = 1.0 and y^ = 1.5, then 9^ = 12° 33' 12.7".

4. Method of Calibration

The method of calibrating a has been in use at the National Bureau

of Standards for several years and has similarities to a method described

by Hume [3] . The method of calibrating the remaining parameters was recently developed at NBS and has been implemented as part of the calibration service.

13
14
4.1. Setup Procedures

The measurement setup calls for a surface plate, two autocollimators, and two high quality front surface type mirrors. The autocollimators and mirrors are set on the surface plate as shown in fig. 11.

The mirrors should be mounted so they are perpendicular to the surface

plate. The angles between the axes of the autocollimators and the perpendiculars to the mirrors should be close enough to 90° that either

autocollimator will read on scale with any of the four possible orientations of the pentaprism. This can be accomplished by placing the

pentaprism so that it reflects the beam from autocollimator A into

mirror 1. Mirror 1 is adjusted until autocollimator A reads on scale,

then the pentaprism is rotated 90° so that it reflects the beam from

autocollimator B into mirror 1. Autocollimator B is adjusted so that

it reads on scale, then the pentaprism is rotated another 90° so that

it reflects the beam from autocollimator B into mirror 2. Mirror 2 is

adjusted so that autocollimator B reads on scale. At this point everything should be adequately aligned, but as a check the pentaprism should

be rotated 90° again to be sure that autocollimator A and mirror 2 are properly aligned.

Each autocollimator should be checked for reading vertical angle

by tilting the pentaprism slightly toward it and observing a change in

reading. If there is no change then the autocollimator is properly

reading horizontal angle only. After all adjustments are completed

there should follow a period of waiting time for temperature stabilization.

4.2. Measurement of Parameters

In the normal usage of a pentaprism the angle of deflection of the

transmitted beam, a, is considered to be the most important parameter.

However, in a high precision calibration of a pentaprism it is necessary

to know the magnitude of other parameters which may affect a in certain

situations. The methods of measuring these parameters are outlined in

the following sections.

4.2.1. Deflection Angle of Transmitted Beam

Let the angles between the autocollimator axes and the perpendiculars to the mirrors be given by 90° + 3^, 90° + 90° + 63, and 90° + g as shown in figure 11. (The autocollimator axis is taken to be the line

along which the collimated light exits.) Readings are made with the pent

prism in each of its four positions. Care should be taken that with each

reading the front surface reflection of the pentaprism is outside the

field of view of the autocollimator. The readings take the form

15
AUTOCOLLIMATOR

B

AUTOCOLLIMATOR

A

Figure 11

16
(4-1)

^Al = \ - + ^1 ^1'

^A2 = \ + ^ - ^2 ^2'

^B2 = - " ^3 ^ "3'

^Bl = ^B °' - ^4 ^4

where A. and A are the respective autocollimator readings for light A fi which returns parallel to the exit beam, and the e's are independent

2 error values from a distribution with mean zero and variance a^. The

autocollimators are assumed to be of the usual type where a clockwise

rotation of a mirror gives a larger reading. The subscripts of r denote which autocollimator is being read and which mirror is being used.

The above system of four equations and seven unknowns can be

solved for a by applying the restraint

6^ + B2 + 33 + = 0 (4-2)

which follows naturally since the sum of the four angles involving 6 must be exactly 360°. If the error terms are Ignored then

= i^-^Al ^A2 - ^B2 ^Bl^ • ^""-^^

The standard deviation of a cannot be computed from equations (4-1) because there is no redundancy in the system, but by taking several sets

of four readings the mean value, a, and its standard deviation s , can

a be computed. Each set of four readings can be made in a couple of

minutes so it is reasonable to take several sets of readings with the

pentaprism in both the top up and bottom up positions. Suppose that n

sets of readings are taken in each position. Let a^, a^, ... denote

the top up values and aj^, denote the bottom up values. Then

OL = — /] a. and a' = — a I where n is the number of sets of readings, n ,-^1 1 n /-< 1 1=1 i=l

In order to test whether the top up values differ from the bottom up values, a test similar to one given by Natrella [4] is employed as follows:

17
1) Choose the level of significance of the test. (Assume .01)

2) Look up t for 2n - 2 degrees of freedom in table 4.

. y yJ

3) Compute a and s, a' and s' for the two sets of n measurements,

(s and s' are the sample standard deviations of the two sets.)

V2

^ ,2

S "T" s

. (s is the pooled standard deviation.)

2 P

5) Compute U = t r>nc s %j— . .995 p^n

6) If |a - a'l > u, decide that a and a' do_ differ. Check the

measuring system and repeat the measurements.

II ~ Ct "I" ct

a - a' I < u, compute a = —^ , the grand mean of the

measured values of the pentaprism deviation from 90". Coms _ pute s_ =

"^^^ » the standard deviation of a . a The uncertainty of a can be expressed as the sum of random and

systematic components of error. The random component is taken to be

the three standard deviation limit 3s_. The process is thought to be

a free of systematic errors with the possible exception of the Aa values

given in table 1. The estimated limit, E, for this error can be added

to the random component to give a total uncertainty of the a estimate of

'] - ' " ' U = 3s_ + E. a

4.2.2. Angle between Reflected Beams

When a manual autocollimator is set to read the two reflected

beams shown in figure 5, the field of view contains a weak and a strong

image as shown in figure 12. (Note the position of the non-reflector.)

Although the angle between the two beams is 2^, the autocollimator

divides the angle in half and the separation is read as ? . (The vertical separation shown in figure 12 is typical but of no consequence.)

The readings take the form

r = A +

s 1

r = A + C +

w 2

(4-4)

18
WEAK

IMAGE

Figure 12

19
where the subscripts of r denote which image is being read. The e's are independent error values from a distribution with mean zero and

2 variance o^. Ignoring the error terms as in the previous section, then

5 - r„ - r^. (4-5)

1 TM

Several pairs of measurements are made and the mean value, E, = — / C • J=l ^ and its standard deviation, s_, are computed where m is the

number of pairs of readings. (Again the sign of is correct if the

autocollimator is of the usual type.)

An explanation should be given for the difference in intensity

of the two reflected beams. One beam obviously loses some of its

intensity while bouncing around inside the pentaprism. Let

I = intensity of incident beam from autocollimator,

= reflectivity of entrance and exit faces, and

= reflectivity of silvered faces.

Then the intensity of the strong reflected beam is

I3 = Pel. C4-6)

and the intensity of the weak reflected beam is

Iwees ^ = p^(l - P^)^P^I . (4-7)

Their ratio is

I

J

, = (1 - pXpI . (4-8)

I e s w

Obviously this ratio may vary considerably depending particularly on

the reflectivity of the silvered faces of the given pentaprism. Sometimes the images may be so nearly equal in intensity that one cannot

be definitely identified as the weaker. In that case some grease can

be applied to the pentaprism face which is 90° away from the autocollimator and the weaker image will gradually disappear.

20
4.2.3. Indices of Refraction

A perfect pentaprism functions independently of its index of

refraction, but an imperfect pentaprism does not. To some degree

the index of refraction must be included in the characterization of

its performance. It is not a practical matter to measure the index

of refraction every time a pentaprism is calibrated, so certain as- sumptions have to be made. The manufacturer's value for the index

of refraction, if available, is taken to be the correct value. Otherwise the value = 1.5 is used. This corresponds closely to the

"average" value for crown glass. The index of refraction of the air

is taken to be = 1.0.

4.2.4. Interior Angles

The mean value a which was computed in section 4.2.1. can be

substituted into equation (3-6) to give

(4-9)

where the 6^ term is dropped since 6^ is purposely kept very small.

Similarly the mean value C which was computed in section 4.2.2. can

be substituted into equation (3-15) to give

- ^2 ^2

C= — — i^ . (4-10)

•^1 ^1

From equations (4-9) and (4-10) the estimates <\) and if) are computed to be

and (4-11)

^

\l) = I

- a .

Let Y be the deviation from nominal of the 45° angle of the pentaprism

(a slight change in notation from section 3.3.). By the reflection

properties

(4-12)

21
Recall that 4) is the deviation from nominal of the 90° angle of the

pentaprism. Then the estimated deviations from nominal of the two

interior angles are

l/, ^l\p 1 =

and

'

il"- - rj^ - 2

= ^ - a..

(4-13)

and their respective standard deviations are

and (4-14)

s s + s_

5 a

Note that if C r 0 then y is a function of indices of refraction

y^^

and y^'

5. Effects when Indices of Refraction Change

The index of refraction of most materials is a function of both

temperature (T) and wavelength of light (A) . At high levels of precision errors can arise when a pentaprism is used at a different

wavelength and/or temperature than when it was calibrated. These

errors can be accounted for by applying correction formulae which

take into account the conditions under which the pentaprism was calibrated and the conditions under which it will be used at a given time.

Let y^ and y^ be the indices of refraction during calibration and let

y| and y^ be the indices of refraction during usage at a later date

under different conditions. (For example, the pentaprism could be

used in a liquid medium, with a laser, or at an extreme temperature.)

The deflection angle of the transmitted beam, a', under the new con- ditions is then given by

^-l),+a, (5-1)

22
and its standard deviation is given by

s .= -1/1-^ - ll^s^ + s^ . (5-2)

The upper bound on the error associated with a' due to a non-zero

incident beam angle is given by

Aa' = y — -

' (5-3)

where 9^ is the largest value of the incident beam's deviation from

normal

In order for the user to make these corrections (if he so chooses)

the values T, X, y^, ]x^, a, ^, s^, and s_ are all given in the Report of

a C Calibration. The values y, ij^, s-, and s- are also given.

6. Example

The data sheets for a sample pentaprism calibration are shown in

figures 13 and 14. Four sets of measurements of a were made in both

the top up and bottom up positions. The statistical test shows that

there is no significant difference between the two mean values. Four

sets of measurements of C were made in the top up position. The index

of refraction values were estimated and the remaining parameters com- puted .

Suppose that this pentaprism were later used in the following

two ways.

Case I: Pentaprism used with a laser for which the index of

refraction was 1% higher than during calibration.

Case II: Pentaprism used in an underwater test where = 1.33.

In both cases assume that the maximum incident beam deviation from

normal (9^) was 5°, and that = 1.0 and y^ = 1.5. Using the metl:

of section 5 the following parameters can be computed.

23
Cia^IBRATION OF A PENTAPRISM (OPTICAL SQUARE) sheet 1 of 2

a

Lot # L ^754-

Serial # T.'tlT.ioi (NGS)

Observer

Date S I Z.O I IS

'a1 'a2 - 'b2 + 'b1>

top

bottom

top

bottom

top

bottom

top

bottom

top

bottom

top

bottom

top

bottom

top

bottom

up

up

upup

up

up

up

up

up

up

up

up

up

up

up

up

Al A2 B2 Bl top up («) bottom up (a*)

7.0 7.5" Z, 3 -/. Soo

7.2 if.

2.4- - / . S"Oo

7. o i.L Z. 2. -1.4-7^

7.0 7. 3 2.4 -1.4'Z.S

7.f ^.S Z.3 -1. 375"

7.3 2, 2. - /. S"Oo

I'f 7, 3 2. 6 -1. 35-0

7. 1 1. 5 S.3 Z. 3 - 1 . ^SO

# sets (n) - 4"

n k(n)

2 9.925

3 4.604

4 3.707

5 3.355

6 3.169

7 3.055

8 2.977

mean (o and «') - — / 4-ZS

std. dev. (s and s") - «^734

2 ^ .2

8 + S

'

k(n) - 3. 7q7

./S4- (a)

|S _ i.| . .044- (b)

u . k(n)*Sp*^

If (b) > (a) f check measuring system and repeat measurements.

If (b) < (a) , compute a - * °

' - ~ L^f , and s - -2- - ,OLl

Figure 13

5/75

24
CALIBRATION OF A PENTAPRISM (OPTICAL SQUARE) sheet 2 of 2

opaque|

A/1

Lot # L 95'4'3

Serial # T. 472 3q /

Estimated indices of refraction

- 1,5

Observer C/^/e * C

Date S I Zo I 75"

Temperature (T) - 2.0 1 O °c. Estimated wavelength of

light (X) - A

Image

weak

strong

weak

strong

weak

strong

weak

strong

weak

strong

weak

strong

reading

5

4^.4

fy.4

weak - strong (()

2. . 4o. /

2 • 3^.5-

Z • 4o./

# sets (m) - 4-

(J. - 1)1 +S - - O ' -5f.4 "

J - f - s - _2 • 4f J -

mean (f)

std. dev. (s)

» — -7=-

2 2

2 I

^ • J9.7

•)^ s' +

.112. "

.07 "

Figure 14 7/75

25
Case I Case II

± 1.0

X 1.33

1.515 ^2

= 1.5

a' = 0.2" a' = -41.0"

S 1 =

a 0.02" S 1 =

a 0.10"

Aa' = 0.3" Aa' = 0.1"

7. Conclusion

A discussion of some facts relevant to the calibration of a pentaprism and a description of the calibration process itself have been presented. Three major conclusions can be drawn:

1) Angular errors between the faces of the pentaprism directly

affect the angle of deflection of the transmitted beam. If these errors

are 5" or less then variations in the angle of incidence (up to 15°) have

a negligible effect (less than 0.5") on the angle of deflection of the

beam (sec . 3.1.).

2) Angular errors in the vertical alignment of the pentaprism face

with the incident beam directly affect the angle of deflection of the

incident beam in the vertical plane, but if these errors are less than

3° then they have a negligible effect (less than 0.3") on the deflection

of the incident beam in the horizontal plane (sec. 3.3.).

3) The new calibration process allows a thorough characterization

of the pentaprism which was lacking in the old calibrations.

26
Table 1.

.ues (in seconds) of Aa = a((J), ^ ^2) - a(4>. 0, Mn1

^1 = 1.0 Aa - from (2xact equation (3-5)

^2 = 1.5 (Aa) - from approximate equation (3-6) + i' !i

1" 5" 0 rill 20 1 20'

1° .00 .00 .00 .01 .OA .23

(.00) (.00) (.00) (.01) (.04) (.15)

20 .00 .00 .01 .03 .16 . 76

(.00) (.00) (.01) (.03) (.15) (.61)

3° . 00 .01 .02 .07 .36 1. 60

(.00) (.01) (.02) (.07) (.34) (1.37)

50 .00 .02 . 06 . 19 .98 4.21

(.00) (.02) (.06) (.19) (.95) (3.81)

7° .01 .03 .13 .38 1.92 8.07

(.01) (.03) (.12) (.37) (1.87) (7.46)

10° .01 .07 .26 .78 3.92 16.27

(.01) (.06) (.25) (.76) (3.81) (15.23)

15° .03 .15 .59 1.78 8.97 36.81

(.03) (.14) (.57) (1.71) (8.57) (34.27)

Table 2.

Observed changes in angle of beam deflection in horizontal and vertical

planes vs. angle of pentaprism tilt toward beam.

= tilt angle of pentaprism on sine bar

h^ = observed angle in horizontal plane

v^ = observed angle in vertical plane

i T, h. h,-h_ V, v.-v„ (v,-v_)-T,

1 1 lO X lO iO i

0 0' 0.0" 4' 3.2" 0' 0.0" 0' 9.6" 0' 0.0" 0.0"

1 2' 3.8" 4' 2.8" 0' -0.4" 2' 13.5" 2' 3.9" 0.1"

2 4' 7.5" 4' 3.4" 0' 0.2" 4' 15.0" 4' 5.4" -2.1"

3 6' 11.3" 4' 3.2" 0' 0.0" 6' 21.9" 6' 12.3" 1.0"

4 8' 15.0" 4' 3.5" 0' 0.3" 8' 26.0" 8' 16.4" 1.4"

27
Table 3.

Predicted changes in angle of beam deflection in horizontal plane vs. angle of pentaprism tilt toward beam, where " 1.0 and ~ 1.5.

T = tilt angle of pentaprism

a = (angle of beam deflection) - 90°

exact a approx. a

T eq. (3-23) eq. (3-24)

10' .0000" .0000

' jU . uuuz

1° .0036" .0036

30 .2909" .2907

5° 2.249" 2.243"

10° 36.25" 35.89"

20° 9' 58.17" 9' 34.19"

30° 53' 9.79" 48' 26.84"

Table 4.

Values of the t distribution

for .01 level of significance

df *^.995 df ^.995 df '^.995

2 9.925 12 3.055 22 2.819

4 4.604 14 2.977 24 2.797

6 3.707 16 2.921 26 2.779

8 3.355 18 2.878 28 2.763

10 3.169 20 2.845 30 2.750

28
References

1. Farago, Francis T. , Handbook of Dimensional Measurement, Industrial'

Press, Inc., New York, 1968, pp. 212-215, 280-282.

2. Hume, K. J., Engineering Metrology, Third Edition, MacDonald

Technical and Scientific, London, 1970, pp. 168, 230-231, 233.

3. Hume, K. J., Metrology with Autocollimators , Hilger & Watts Ltd,

London, 1965, pp. 19, 49, 114-115, 131-133.

4. Natrella, Mary G. , Experimental Statistics, National Bureau of

Standards Handbook 91, U.S. Government Printing Office,

Washington, D.C., 1963, pp. 3-23 to 3-26.

USCOMM-NBS-DC

29
NBS-n4A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 76-9932. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

The Calibration of a Pentaprism

5. Publication Date January 1976

6. Performing Organization Code

7. AUTHOR(S) Charles P. Reeve

Ralph C. Veale

8. Performing Organ. Report No. NBSIR 76-993

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Wotk Unit No. 2131120

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 tess factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

A pentaprism, or optical square, is one of a class of objects known

as "constant deviation prisms" whose purpose is to bend a beam of

light exactly 90° regardless of the angle of incidence of the light.

It is used in many optical measurements where the line of sight or

a beam of light must be turned at a right angle. The calibration

process which is described is fairly simple and requires two mirrors,

two autocollimators, and a surface plate. Consideration is given to

the effects of small angular errors in both the internal geometry of

the pentaprism and the positioning of the pentaprism relative to an autocollimator . An example of the calibration of the important pentaprism parameters is given. A simple statistical test is employed for

testing whether the measurement system 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

)

Autocollimator; calibration; constant deviation prism; geometrical model;

optical square; pentaprism; reflection; refraction; standard deviation;

uncertainty.

18. AVAILABILITY STI Unlimited

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