NATIONAL RADIO ASTRONOMY OBSERVATORYGreen Bank, West Virginia

Electronics Division Internal Report No. 119

A LASER DISTANCE MEASURING INSTRUMENT

John M. Payne

JUNE 1972

NUMBER OF COPIES: 150

A LASER DISTANCE MEASURING INSTRUMENT

John M. Payne

CONTENTS Egge

1. 0 Summary ..................................................... ....................................................... 12. 0 Introduction ......................................................... .................................................. 13, 0 Principle of Operation4. 0 Existing Optical Distance Measuring Instruments ....................................... 35, 0 Description of Instrument ............. ......................................................................4

5. 1 General ................................................ ................................................... 45. 2 Choice of Frequencies .......................... ..................................................... 45.3 Optics .....................................................................................55.4 Electronics .................... ..........................................................................5

6. 0 Theoretical Performance of Instrument 97. 0 Tests with Instrument .................... ............................................................... 118. 0 Telescope Installation ............................... ..................................................... 139. 0 Future Development .................................................... ................................. 14

10, 0 Acknowledgments ......... ................................................................................. 14

LIST OF FIGURES

1. Simplified Block Diagram ............................................................................. 152. Photograph of Instrument and Display Unit .......................................... ....... 163. Photograph of Instrument .................... ....................................................... 174. Block Diagram ................................................................................................ 185. Modulating Cavity ................................................................................. ...... 196. RF Drive for Cavity ....................................................................................... 207. Digital Divide Circuits ..................................................... ............................ 218. Local Oscillator Circuits ............................................................. ............... 229. Receiving Circuits 23

10. Integrating Phase Detector ..................................... ........ ............................. 2411. Photograph of Test Range ............................................................................ 2512. Photograph of Instrument in Reber Dish Building ............................................ 2613. Photo of Micrometer Bench .................................................. ....................... 2714. Graph of Results 2815. Photograph of Corner Cube Reflector ......................................................... 2916. Possible Telescope Installation .................................................................... 30

TABLES

1. Summary of Different Forms of Modulated Light Beam Instruments .......... 312. Test Over 60 m Path ...................................................................................... ..........32

A LASER J NCE MEASURING INSTRUMENT

1.0 Summary

This report describes a prototype distance measuring instrument that may be

used to survey the surface of a radio telescope in a fast and accurate manner. The

principle used is simply to measure the transit time of a beam of light out and back over

the path to be measured.

The instrument will permit a survey using a triangulation method by making ac-

curate range measurements over two paths to small reflectors on the surface of the

telescope.

The instrument in its present form will measure ranges up to 60 meters with an

RMS (1a) accuracy of better than ± 0. 1 mm and in a measurement time of 2. 5 seconds.

2. 0 Introduction

The birth of this instrument has been precipitated by the completion of the design

of the 65 meter telescope. A critical part of the construction of the telescope will be the

setting and subsequent checking of the surface panels, a task usually accomplished by

measurements using a tape and theodolite. The design of the telescope requires setting

about 3, 000 points on the surface to an accuracy of ± 0. 1 mm RMS in a few hours, an

almost impossible task using conventional methods.

Although the instrument described in this report was developed primarily to prove

the feasibility of a performance adequate for the setting of the 65 meter telescope, it may

well prove useful in surveying existing instruments. Interest has been expressed in using

the instrument for setting the new surface at Arecibo and the instrument, when repackaged,

would be ideal for this application. No problems are anticipated by operation at a range

of 150 meters.

The instrument uses a simple principle that has been used for many years in

surveying instruments. Commercial instruments using this principle are available but

the best have an accuracy of only ± 1 mm which is not adequate for our needs.

3. 0 Principle of Operation

A high-intensity light beam (in our case a helium neon laser beam) is amplitude

modulated, transmitted over the path to be measured, and reflected back to the instrument.

The phase of the returned beam is measured with respect to a reference, the phase shift

2

being proportional to the total distance the transmitted light beam has travelled.

This principle is illustrated in Figure 1 in which the light beam is modulated at

a frequency 1m'

the one-way distance to be measured is d, and the phase of the returned

signal is cps

.

The returned light beam will be shifted in phase by -2—d where X is the wave-

length of the modulating frequency. If X < 2d, then the phase will be shifted through

more than one cycle and ambiguities in distance reading will arise.

The instrument we have developed uses a modulating frequency of 550 MHz, so

range ambiguities will occur about every 27 cm. This means, of course, that the dis-

tance to be measured has to be known in advance to better than this value. A dual-

frequency system is often used to resolve such ambiguities, and the present instrument

could be modified to work in this way. However, it seems unnecessary for measuring

radio telescopes.

Referring to Figure 1, the phase detector output will be proportional to

where Or is the reference phase and is the phase of the returned signal.

2d - nA x 360 degrees

where (2d - nX) < X and n is an integer.

For a distance determination of O. 1 mm accuracy using a X of 55 cm, we re-

quire a phase measuring accuracy and phase stability within the instrument of 0. 13°,

The high accuracy of the instrument stems directly from the high modulation

frequency used. The highest frequency used in a commercial instrument is 75 MHz for

which the manufacturer claims an accuracy of ± 1 mm.

The velocity of light through air varies with air temperature, pressure, and

relative humidity. At a range of 60 m we need to know air temperature to about 1. 5 °C

and air pressure to about 4 mm of mercury to achieve the 0. 1 mm accuracy. The

effects of relative humidity over such a distance can be neglected.

3

4. 0 Existing Instruments

There are many ways of implementing the simple principle illustrated in Figure

1 and Table 1 gives a summary of various instruments using this general principle. An

instrument coming very close in performance to our requirements is the Mekometer III

that was developed at the National Physical Laboratory in the U.K. This instrument is

not at present commercially available.

The Mekometer uses an ingeneous phase measuring technique that nulls the phase

of the received light beam by mechanically changing the path length of the received sig-

nal a calibrated amount. By slightly shifting the transmitted signal in frequency and re-

nulling several times the distance may be obtained. This method has the advantage of

eliminating any errors inherent in the measurement of phase electronically but is not

suitable for our application due to the fairly long period required for a measurement.

Some of the instruments listed in Table I overcome any potential instability in

the phase measuring circuits by switching between an internal optical reference path

and the signal path in a manner similar to a load-switched radiometer.

Our previous experience with mixers and phase measuring techniques suggested

that mixing down the 550 MHz signal and reference waveforms to a low frequency and

comparing the phases directly should yield a measurement accuracy of 0. 1° in phase.

Before building the instrument we tested various R. F. components (hybrids,

filters, amplifiers and mixers) and reached the conclusion that provided the components

were stable in temperature to better than ± 1 °C the phase detection to 0. 1° was cer-

tainly feasible.

Before building the prototype instrument we decided that the instrument should

have the following specifications:

Accuracy better than ± 0. 1. mm RMS

on a single reading.

Range: 1 m - 60 m.

Measurement time less than 5 secs.

Digital display automatically updated.

Reflector size 1" diameter or less.

5. 0 agniption_ollp,,Amment

5. 1 General

A photograph of the instrument and its display unit is shown in Figure 2. More

detailed views of the instrument are shown in Figure 3. Cigarette smoke was blown

in to make the laser beam visible. No effort was taken to make the instrument small;

like most prototypes, it just grew. For the final tests of the instrument, it was housed

in a temperature controlled box, the light beam passing through a quartz window. A

block diagram of the instrument is shown in Figure 4.

5. 2 Choice of Frequencies

There are three main frequencies within the instrument:

1) The modulating frequency.

2) The IF frequency.

3) The digital circuitry clock frequency.

To achieve the stability required it seemed essential that these frequencies be

derived from a common, stable, crystal oscillator.

Corrections for variation in the speed of light due to changes in atmospheric

conditions may also be applied directly to the instrument by pulling the frequency of this

master oscillator. The frequencies were chosen as follows:

Master Oscillator ....... ....... f........ 60. 97463 MHz

Modulating Frequency ......... ...... f......... 9f........ 54$. 7717 MHzm o

IF Frequency ........................I

= —97 = 3. 72160 kHz21'

Clock Frequency o 20. 32487 MHz3

With these frequencies and the group* velocity of light at 0 °C and 760 mm of

mercury the digital phase meter will have a sensitivity of 20 counts per millimeter of

distance per cycle of IF signal, and the digital display may be arranged to read directly

in cm, the least significant digit being 0. 001 cm.

* The group velocity is correctly used since it is the velocity of the modulation envelopewhich is required.

5

5. 3 Optics

The optics of the instrument were kept very simple and no effort was directed

towards compactness. A more engineered version of the instrument could be made

much smaller by using a shorter focal length receiving mirror.

The light source we used was a Hughes Helium Neon Laser Model 3076H. This

laser has a power output of 3 mW, a beam divergence of 0. 8 milliradian, and is linearily

polarized to better than 1000: 1.

The optical path is made clear from Figure 4. The polarized laser beam first

passes through a cavity that is tuned to the modulating frequency. In the high electric

field region of the cavity is a crystal of potassium dihydrogen phosphate (KDP) through

which the laser beam passes. When an electric field is applied to this crystal the plane

of polarization of the laser beam is rotated. On emerging from the cavity the beam

passes through a polarizing prism that converts the polarization angle modulation to

amplitude modulation.

Two mirrors then direct the beam to the distant target which is a retroreflector.

This has the property of reflecting the light beam back along its path with a high degree

of precision over a wide range of angles of the incident light.

The returned light beam has a diameter of about 3 inches at the receiving mirror

at a range of 60 m. This light is then focused, via a mirror, onto the receiving photo

diode.

For linear operation of the instrument it is essential that no modulated laser light,

other than the directly reflected beam, find its way into the receiving diode. It is also

important that there are no multiple reflections between the diode and the distant reflec-

tor. This is explained more fully later.

5. 4 Electronics

5. 41 Signal Flow

The master oscillator drives a x3 transistor multiplier and power amplifier that

provides 500 mW at 182. 92389 MHz to the x3 transistor multiplier built into the top part

of the cavity. This multiplier provides 1 watt at 548. 7717 MHz to the modulating crystal.

The master oscillator is divided by a 14-bit counter to produce a 3. 72160 kHz

signal which is added to the modulating frequency to produce the local oscillator fre-

quency via the LO circuits.

The received light signal is first detected and amplified in the avalanche photo-

diode which is matched to 50 ohms at 550 MHz. The 550 MHz signal is now filtered,

amplified and mixed to produce an IF signal at 3, 72160 kHz. This signal is squared

using a Schmitt trigger and the leading edge is used to start the clock pulses at

20.324878 MHz into the integrating digital phase detector. The clock pulses are

stopped by the leading edge of the reference signal, the accumulated clock pulses being

a measure of the phase difference between the two waveforms.

The digital circuitry permits the averaging of 100, 1000, or 10, 000 IF signal

periods, the number of periods being selectable by a switch.

As previously mentioned, the frequencies have been chosen to give a scale factor

of 20 clock pulses per millimeter for one IF cycle, At the end of an integration period,

which then includes the sum of many IF cycles, the accumulated count is displayed on a

5-digit L. E. D. display in centimeters, the least significant digit being 0. 001 cm.

5. 42 Modulator

The modulator assembly is shown in Figure 5. The KDP crystal was purchased

from Isomet Corporation sealed in a plastic case with optical windows coated for 6328A.

The crystal needs a DC bias of about 500 volts to ensure operation in the linear region

and an RF voltage of 200 V p. p. to give a modulation depth of around 10 percent. We

decided the best solution to the problem of driving the modulator would be to incorporate

it in a cavity.

We found that the case the crystal was encapsulated in was rather lossy and re-

duced the Q of the cavity, resulting in an RF drive level of about 1 watt.

The cavity was fine tuned by an adjustable vane giving variable capacitance be-

tween the center conductor and the floor of the cavity.

The final x3 multiplier was situated in the top of the cavity and was coupled into

the cavity via a loop to the center conductor. This loop was arranged to give an input

impedance of 50 ohms to simplify testing of the multiplier and cavity.

Another loop in the top of the cavity served to couple out about 10 mW of power

at 550 MHz for the local oscillator circuits.

7

A crucial part of the design of the instrument is the prevention of any of the RF

modulating signal leaking into the received signal path. This isolation was achieved by

putting the RF power circuits in the top of the cavity, having a good fitting lid on the

cavity and also by double shielding the receiver.

5. 43 lylaltiplier Chain

The RF multiplier chain is shown in Figure 6 and is conventional in design.

We used an RCA "overlay" transistor for the final multiplier and found that the voltage

variable collector to base capacitance permitted very efficient multiplication.

5. 44 Di stal

The digital divide circuits are shown in Figure 7. The master oscillator signal

passes through a buffer amplifier and logic gates and is then divided by three to produce

the clock frequency for the phase detector and display circuitry and also drives a 14-bit

counter to give the 3. 72160 kHz signal to the local oscillator circuits.

5. 45 Local Oscillator Circuits

The local oscillator circuits are shown in Figure 8. A voltage-controlled crystal

oscillator running at approximately 60. 947 MHz is multiplied by nine in a transistor

multiplier and filtered through a narrow band cavity filter. The output from the multi-

plier is then split with a hybrid, one output providing the system local oscillator, the

other output feeding a mixer via a circulator and a 20 dB pad. The LO port of this

mixer is fed with +7 dBm at the modulating frequency from the cavity. The IF output

of this mixer is then amplified and mixed with the 3. 72 160 kHz signal from the digital

divide circuits. The output of this mixer is amplified and passed through a low-pass

filter and applied to the voltage-controlled crystal oscillator. With this phase-locked

loop in a "locked" condition, the LO output will always be exactly 3. 72 160 kHz away

from the modulating frequency.

8

5. 46 Receiving Circuits

The receiving circuits are shown in Figure 9. The photodiode, bandpass filter,

RF amplifier, mixer, and IF amplifier are all mounted inside a double-shielded box

attached to the side of the receiving telescope. The diode and matching circuits are

contained in a separate shielded box. Care was taken to screen and decouple all supply

leads into the receiver box.

The Schmitt trigger was used rather than a straight limiter to avoid false trigger-

ing of the digital readout circuits through noise on the zero crossing of the IF signal.

The main reason for the filter in the signal line was to reject a strong signal at 500 MHz

that results from a longitudinal mode of oscillation in the laser. This signal was rejected

because of the possibility of intermodulation products in the mixer.

5. 47 Integrating Phase Detector

The 3. 7 kHz start (reference) and stop (signal) square waves enter the phase

detector through D-type flip-flops and both are delayed by one clock pulse. (The clock

is about 20 MHz.)

The output of gate A, pin 14, will be a burst of clock pulses at the 3. 7 kHz IF,

the number of pulses being proportional to the phase difference between the signal and

reference IF waveforms. The clock pulses are then gated into the display counter (F,

etc. ) via the counters B, C, D and the flip-flop E.

The circuitry permits the averaging of phase differences for 100, 1000, or

10, 000 cycles of the IF. For example, for averaging 100 cycles of the intermediate

frequency, the counter B divides the clock pulses representing phase shift by 100.

After 100 IF cycles, determined by the "enable" counters, the clock pulses accumulated

in the display counter are gated into the display.

Operation is similar in the averaging of 1000, or 10,000, cycles and the display

counter always counts up in increments of 0. 001 cm.

9

6. 0 Theoretical Performance of Instrument

6. 1 Si nag atio

The transmitted light beam is attenuated somewhat by the various optical com-

ponents and the actual transmitted power is about 1 mW. With a 1" target retroreflec-

tor, a beam divergence of 0. 8 milliradian and a 3" diameter receiving mirror, the

power at the photodiode is about 0.2 mW. This results in such a high signal-to-noise

ratio that negligible phase noise results.

There are other sources of noise, however, that can only be evaluated by mea-

surement. The other noise contributions come from A. M. noise on the laser and intensity

fluctuations of the returned light due to atmospheric irregularities and instabilities, The

noise contribution from the laser was measured in the lab and was found to be negligible.

6.2 Signal Leakage

Any small signal at the modulating frequency finding its way into the receiver will

add to the desired signal and produce phase errors which will in turn produce cyclic

errors in the measured distance with a spatial periodicity of half a wavelength (27. 3 cm).

The isolation required between the inside of the cavity and the input to the receiver is

about 140 dB to keep phase errors as low as 0. Or.

6. 3 Oscillator Stability

The required stability for the crystal oscillator for an accuracy of ± 0. 1 mm

in 60 m is 1 part in 6 x 10 5 which is easy to achieve with a crystal oscillator in a tem-

perature enclosed environment.

6. 4 Atmospheric Effects

The atmosphere affects the instrument in several different ways.

1) Short term effects lar fluctuations in the beam. When a

beam of light is projected over a path close to the ground it is subject to turbulence

that can be severe at times. During our tests we found angular deviations as much as

0.2 milliradian peak to peak. This caused the returned signal to vary greatly in strength

and at times drop to zero. As would be expected, under these conditions the accuracy of

the instrument was decreased. This condition occurred only rarely, generally when the

dndp per mm Hgt °C

-10 4. 1005 x 10-7

0 3.9504 x 10-7

10 3.8109 x 10-7

20 3.6809 x 10-7

30 3.5595 x 10-7

- 10 -

weather was sunny and calm. Our experiments with the optical reference platform on

the 140-foot telescope suggest that these effects are greatly reduced a few feet above

the surface of the ground.

2) Short term effects giving ges in refractive index. The effects

mentioned above are due to "blobs" of air at slightly different temperatures through

the beam. These blobs will have the effect of continually changing the effective refrac-

tive index of the light path. No calculations have been made on this effect: experiments

show it to be very small.

3) Variation in refractive index with atmos eheric conditions. The group

refractive index of air is given by the equation

(n -1)_go .._.P...._ 5. 5 e x 10-8 n = 1 + x —g t 760 t 1 + 273. 15

1 +

273. 15

where ngo

= group refractive index of air at 0 °C, 760 mm Hg, water vapor pres-

sure = 0, 0. 3% CO 2 = 1. 000300 for the helium neon laser frequency.

p = pressure in mm Hg.

e = partial pressure of water vapor in mm Hg.

t = temperature in °C.

From this equation we obtain the following:

dna) dp for constant t

pmmHg t °C

dnat constant pressuredt

- 11. 8426 x 10-7

-10.9914 x 10-7

-10.2288 x 10-7

-9. 5428 x 10-7

-8.9237 x 10-7

760

760

760

760

760

-10

0

10

20

30

---- per °Cdt

dnde = O. 55 x 10

-7 per mm Hg and changes negligibly with tem-

perature and pressure. The effects of varying e from a dry to a saturated atmosphere

change the range of 60 meters by less than 0. 1 mm.

To give a 0. 1 mm accuracy in a 60 meter range, we need to know:

Temperature to approximately 1. 5 °C

Pressure to approximately 4 mm Hg

We anticipate that in use on a telescope a reference target would be used to

calibrate out instrumental and atmospheric instabilities so even the simple corrections

outlined above may be unnecessary.

7. 0 Tests with Instrument

The first tests were performed over short path lengths in the laboratory. The

target was moved on a micrometer stage and the calibration of the instrument checked

and found to be correct.

As usual with this kind of project, there were many different configurations

tried and discarded. Perhaps the most significant of these was the way in which the

reference waveform was derived. Initially we felt that the "safest" way of deriving

the reference waveform was to split the outgoing light and use a second photodiode and

mixer to obtain the IF signal. After careful testing we found this to be unnecessary and

used the more direct approach shown in the block diagram.

- 12 -

The instrument in its early days had the signal photodiode, mixer and IF ampli-

fier in separate boxes. We found this to be far from ideal so far as leakage was con-

cerned: much better performance was obtained when all these components were packaged

together in one doubly-shielded box.

The outdoor tests were performed over a 60 m range (total path = 120 meters) with

the instrument in the Reber dish building and the reflector on a micrometer bench in a

small hut. The photographs in Figures 11, 12 and 13 show the arrangement.

The instrument stability over the 60 m range was first checked and gave the re-

sults shown in Figure 14, curve A.

The target was then moved in increments of one inch over a total of 32 inches and

the digital readout recorded for each inch. A typical result is shown in Table 2. For a

360° phase change the display will read 27.307 cm, so this number is added to the dis-

played number at each crossover.

An analysis of the results given in Table 2 gives a slope of 2. 540180 cm/in and

a standard deviation on a single measurement of 0. 00757 cm. The errors for each inch

are plotted in curve B, Figure 14, and can be seen to be not noise-like but follow a cyclic

pattern. This cyclic error is made clearer in the results of Figure 14, curve C. We

subtracted out the repetitive features of curve C with the results shown in curve E which

yielded a standard deviation of ± 0. 043 mm on a single reading. The cyclic errors are,

we think, due to leakage of the modulating signal into the receiver.

These results suggest that the accuracy of an engineered instrument may well be

better than 0. 05 mm.

- 13 -

8. 0 Telescope Install ation

A possible way of installing the instrument on a telescope is shown in Figure 16.

B is regarded as a fixed point near the vertex of the dish and C is a fixed point near the

focus. The distance from the instrument reference plane to the reflector at D via the

mirror at B is first measured. Mirror B is then tilted out of the beam and the distance

from the instrument to D via mirror C is measured.

Provided the angle CDB is less than 45° a single corner cube may be used: for

angles of incidence separated by more than 45° the reflector assembly would have to use

two corner cubes.

One possible way of steering the mirrors would be use a coarse positioning sys-

tem (several arc minutes) to direct the beam onto the corner cube and then servo control

the mirror via a quadrant detector in the returned beam. With the mirror servo con-

trolled in this manner the time taken to go from target to target would be very short,

certainly less than one second.

A system such as this would be necessary when attempting to survey the 140-foot

at different telescope positions as we know the focal point moves by about 1/2". With an

open loop system this movement would result in misdirecting the beam onto the reflector.

A good first installation on the 140-foot would be to incorporate the mirror at B

with the instrument so making a complete unit and just measure the path ABD for about

ten reflectors. The reflectors could be made to use existing target holes. This installa-

tion would require virtually no telescope time and the structure at the vertex is ideal for

mounting the instrument. If these first tests were successful it would be a simple matter

to add another mirror at the focal point.

- 14 -

9. 0 Future Development

If we continue work on this instrument the next step will almost certainly be tele-

scope installation of a more engineered version of the present instrument.

There is another development that may be worth pursuing and that is using the

surface of the dish as the reflector. Argon lasers (or CO 2) are now readily available with

output powers of several watts. The output of such a laser, when focused to a spot on the

surface of the dish, would provide adequate power to the receiver just by scattering.

It will be realized that this report deals mainly with observation to the distant end

of the 60 m baseline. The location of the zero-point of the instrument has not been deter-

mined. However, the stability observations (paragraph 7) determine that both these points

remain stable to better than ± 0. 1 mm. In practice, short range measurements would be

made to reference targets close to the instrument as well as long range measurements to

targets of interest.

In conclusion, we may say the instrument in its present form satisfies the re-

quirements of the 65 meter telescope (1 0. 1 mm RMS) and when rebuilt in a form suitable

for use on a telescope there are indications that this accuracy may be doubled.

10.0 Acknowledgments

C. Pace in Charlottesville who designed and built the digital circuits, Dr. J. W.

Findlay who worked on the testing of the instrument and analyzed the results, and Steve

Mayor and Ralph Becker in Green Bank who built the instrument.

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- 26 -

IP

" viikdokAmazoril„,„„:

Figure 12 — Photograph of Instrument in Reber Dish Building

A:A1

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RESULTS

INSTRUMENT STABILITY — OVER RANGE OF 60 METERS

DISTANCE MEASUREMENT — OVER RANGE OF 60 METERS

bag

Figure 14 — Graph of Results

Figure 15 — Photograph of Corner Cube Reflector

SERVO CONTROLLEDMIRROR

ADDITIONAL COMPONENTS REQUREDFOR TRIANGULATION MEASURMENTS

QUADRANTDETECTOR

DISTANCE MEASURINGINSTRUMENT

■SERVO CONTROL

1•* ONE OF MIRROR AXIS

SERVO CONTROL

BEAM SPLITTER

RETURN BEAMOF OTHER AXIS

POSSIBLE TELESCOPEI NS TALLATION

- 30 -

Figure 16 — Possible Telescone Installation

TA

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E 1

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ode.

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cs1

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ital

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play

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and

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ly.

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omet

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ium

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ator

.

± 2

mm

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ith

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th. N

ulle

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ator

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z

_ Dep

ends

on oper

ator

.

2.5

secs

± 0

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Mec

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lled

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ging

pat

hle

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.

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erci

ally

avai

labl

e.

NR

AO

Des

crib

ed in

this

rep

ort.

Hel

ium

neo

nla

ser.

550

MH

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Mix

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m-

pare

d di

rect

ly.

- 32 -

TABLE 2

Test Over 60m Path

Slope ---- -2. 541099 ± 0. 000145 cm

Intercept 82. 2613 ± O. 0027 cm

RMS (10 ± 0. 00757 cm

Slope Corrected for Atmosphere = -2. 541099 x 0. 99964 = -2. 540180 cm

MicrometerPosition

Inch

DigitalReadout

cm

Readout Error

cm

0. 9483. 4926. 0308. 564

11. 10613. 65116. 20018. 73521. 26123. 80926. 351

1. 5944. 1416. 6719. 211

11. 75614. 30516. 85619. 38821. 92224. 45227. 004

2. 2364. 7797. 3149. 841

12. 39814. 94417. 49720. 02922. 56325. 106

0. 9483. 4926. 0308. 564

11. 10613. 65116. 20018. 73521. 26123. 80926. 35128. 90131. 44833. 97836. 51839. 06341. 61244. 16346. 69549. 22951. 75954. 31156. 84959. 39261. 92764. 45467. 01169. 55772. 11074. 64277. 17679. 719

-0. 002-0. 005-0. 002+0. 005+0. 005+0. 001-0. 007- O. 001+0. 014+0. 007+0. 006-0. 003-0. 009+0. 002+0. 004

0. 000-0. 008-0. 018.-0. 009-0. 002+0. 009- 0. 002+0. 001- 0. 001+0. 006+0. 020+0. 004- 0. 001-O. 013- 0. 004+0. 003+0. 001

3231302928272625242322212019181716151413121110

987654321