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15 NBS TECHNICAL NOTE 685

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and

technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to pro- mote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Institute for Computer Sciences and Technology, the Office for Information Programs, and the

Office of Experimental Technology Incentives Program.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consist- ent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

and commerce. The Institute consists of the Office of Measurement Services, and the following center and divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center for Radiation Research — Laboratory Astrophysics 2 — Cryogenics 2 — Electromagnetics 2 — Time and Frequency 2

. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized materials needed by industry, commerce, educational insti- tutions, and Government; provides advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard Reference Materials, the Office of Air and Water Measurement, and the following divisions:

Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services developing and promoting the use of avail- able technology; cooperates with public and private organizations in developing technological standards, codes, and test methods; and provides technical advice services, and information to Government agencies and the public. The Institute consists of

the following divisions and centers: Standards Application and Analysis — Electronic Technology — Center for Consumer Product Technology: Product Systems Analysis; Product Engineering — Center for Building Technology: Structures, Materials, and Safety; Building

Environment; Technical Evaluation and Application — Center for Fire Research: Fire Science; Fire Safety Engineering.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effectiveness in the conduct of their programs through the selection, acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus wthin the executive branch for the development of Federal standards for automatic data processing equipment, techniques, and computer

languages. The Institute consist of the following divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE OF EXPERIMENTAL TECHNOLOGY INCENTIVES PROGRAM seeks to affect public policy and process

to facilitate technological change in the private sector by examining and experimenting with Government policies and practices in order to identify and remove Government-related barriers and to correct inherent market imperfections that impede

the innovation process. THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific information generated within NBS; promotes the development of the National Standard Reference Data System and a system of in- formation analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services

to ensure that the NBS staff has optimum accessibility to the scientific information of the world. The Office consists of the following organizational units:

Office of Standard Reference Data — Office of Information Activities — Office of Technical Publications — Library —

Office of International Standards — Office of International Relations.

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

2 Located at Boulder, Colorado 80302.
Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

**i or / w \

«fATIONAi

OF STANDARD,

UBRAJRY

A SYSTEM FOR CALIBRATING "&*.

LASER POWER METERS ?*o

FOR THE RANGE 5-1000 W fits

E.D. West

L.B. Schmidt

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

& NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued May 1977
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 685

Nat. Bur. Stand. (U.S.), Tech Note 685, 27 pages (May 1977)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC. 20402
CONTENTS

Page

1. GENERAL REQUIREMENTS FOR A CALIBRATION SYSTEM 1

2. THE CALIBRATING SYSTEM 3

3. THE Kl CALORIMETERS 5

4. USE OF THE Kl CALORIMETERS 8

5. ELECTRICAL CALIBRATIONS 9

6. ERRORS DUE TO NON-EQUIVALENCE OF SOURCES 13

7. ABSORPTANCE OF THE CAVITY 14

8. COMPARISON OF Kl-1 AND Kl-2 19

9. ERROR OF A LASER ENERGY MEASUREMENT 21

10. REFERENCES 2 3

ill

A SYSTEM FOR CALIBRATING LASER

POWER METERS FOR THE RANGE 5-1000 W

E. D. West* and L. B. Schmidt

This technical note describes the National Bureau of Standards measurement system for calibration and test of laser power

and energy devices at power levels up to 1 kW and total energies

of several kJ. The main parts of the system are two electrically

calibrated calorimeters and several beam splitters of different

materials. The estimated limits of systematic error in measurements with the two calorimeters are ± 1.9% for one and ± 2.0% for

the other. The two calorimeters have the same precision, and the

standard deviation of an electrical calibration is the same as the standard deviation of a laser energy measurement, indicating

that the laser beam does not affect the precision of the measurement. The standard deviation of an energy measurement depends on the total energy to the calorimeter and is about 0.8% at 1 kJ.

Key words: Calorimeter; laser calorimeter; laser energy;

laser power

1. GENERAL REQUIREMENT FOR A CALIBRATION SYSTEM

The main elements of our laser power, or energy, calibration system are

a laser source, a shutter for interrupting the beam, a beam splitter, a monitor calorimeter, and a calorimeter whose response coefficient has been

evaluated. The devices to be calibrated are compared to one of these calorimeters by substituting the devices to be calibrated, one at a time, for one

of the calorimeters in the calibration system. A typical experimental

arrangement for calibrations is shown in the diagram in figure 1.

The basic requirement of the elements of a calibration system is

stability of their intrinsic characteristics. In a laser power meter calibration the output power of the laser source must be reasonably constant

during the injection time to eliminate serious error due to averaging. The

response of both of the calorimeters to specified energy inputs must repeat,

at the predicted accuracy, to maintain the accuracy of the calibration

system. Equipping the calorimeters so that known amounts of electrical

energy can be substituted for laser energy provides a non-laser check on the

stability of their response to the electrical energy. The stability of the

beam splitter ratio — the ratio of the energies in the transmitted and the

reflected beams -- is the first consideration in selecting a beam splitter.

Long term stability of the ratio of beam splitter is assured when the re- flecting surfaces are of high quality, and the ratio is determined by the

index of the refraction of the material.

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After the calorimeters of a system are electrically calibrated and

their absorptances are determined, the absolute beam splitter ratio is determined as an energy ratio, which is a property of the beam splitter [1]*. At

a particular wavelength the beam splitter ratio should be constant for all

combinations of calorimeters placed in the transmitted and reflected beams.

Interchanging calorimeters in this manner provides a check on the stability

of the beam splitter, and a cross-check on systematic and random errors of

the calibrating system.

The selection of the beam splitter is also influenced by the energy

ranges of the calorimeters and the power or energy output of the laser. The

energies provided in the transmitted and reflected beams must be values that can be measured by the calorimeters to their estimated accuracy. The power,

or energy in the transmitted beam should be as large as possible if one desires to perform calibrations near the maximum output level of the laser.

The injection time of the laser beam must be measured when power meters

are to be calibrated by comparison to a calorimeter. The method of injecting the laser beam into the system must be designed to include the

measurement of this injection time and this measurement must be traceable to

a time standard.

2. THE CALIBRATING SYSTEM

NBS has developed a laser power and energy measurements system with a 5-to-1000-watt range for the calibration of devices for this power range.

The basic measurement problem is the determination of calibration factors

and their associated uncertainties in terms of power or energy as defined by

the international system of units.

The main elements of the NBS calibrating system conform to the requirements discussed previously. One form of an experimental arrangement for

calibrations is shown in the diagram in figure 1. The experimental arrangement is the same principle as the system for lower power [2].

The laser used in the experimental arrangement of figure 1 required

that the laser beam injection time be controlled and measured external to

the laser. This was done by using a water cooled shutter to admit the laser

beam into the system for a predetermined time interval. A red helium neon

laser beam is applied to a light sensor whose output voltage triggers a time

interval counter used to measure the time. The time interval counter is

equipped with a crystal-controlled oscillator that is checked against a 100 kHz standard frequency supplied by the Time and Frequency Division of

*Figures in brackets indicate the literature references

at the end of this paper.
NBS. Another laser used with the calibration system is equipped to allow

the laser beam to be turned on for a specified time interval by an electronic gate. The time interval of the gate is measured by the above time

interval counter.

This system has been evaluated only at 10.6 urn. The beam splitters were selected primarily for the stability of the ratio of the energies in

the transmitted and reflected beams. The power range of this system adds

further requirements in the selection of beam splitters. Beam splitters for

calibration work with 10.6 ym laser beams have been studied by Franzen [1]

For general use he recommends zinc selenide beam splitters, which transmit

70% of the incident beam. When greater transmission is desired, he suggests

potassium chloride beam splitters which transmit 92% of the incident power.

To extend the system capability to a new wavelength, it is necessary to

acquire the laser and measure the absorptance of the calorimeters at the

laser wavelength.

The electrically calibrated calorimeter is the element in the measurement system that provides the link to the international system of units.

The standard volt and ohm have been determined in terms of the base units to

accuracies of parts per million and time measurements are accurate to parts

in 10 13 . These standards are therefore much more accurate than calorimetry

requires, and the methods of using them can be relaxed considerably in the

interest of efficiency and convenience.

The electrically calibrated calorimeter is not itself a standard.

Its relationship to energy resembles that of a potentiometer to voltage

or of a bridge to resistance. It is the means of comparing electrical

energy to laser energy.

The theory of the calorimetric measurement is summarized by two

equations [2] . The first relates the electrical or laser work W done on the calorimeter to the energy equivalent or calibration factor E

multiplying a corrected temperature rise

W = E[T„ - T T + e f{T - T(»)}dt] , (1)

where T is the temperature observed at time t, T and T are temperatures

characterizing respectively the final and initial rating periods [3,4], the

limits of the integral are taken between the times corresponding to these

temperatures, T(<*>) is the steady-state or convergence temperature, and e is the cooling constant. The last two quantities are obtained from the

equation

dT/dt = - e [T - T(«)] . (2)
This equation is integrated to give

T - T(») = [Tj - T(»)] exp [- e (t - t

].) ] (3)

for the initial rating period and a similar equation for the final rating

period with Tj and t

]; replaced by T

p

and t . The rating period data are

fit by least squares to obtain the quantities T

p

, T , e and T(«) required

in eq. (1). More detailed descriptions of the measurement theory and calculations are given in [3,4].

3. THE Kl CALORIMETERS

The two electrically calibrated calorimeters in the NBS system are of

the same design, with laboratory designations Kl-1 and Kl-2. The construction of the calorimeters is illustrated by the cross-sectional diagram in

figure 2. The apparatus is 36 cm long over-all and 15 cm in diameter. The

calorimeter proper consists of a copper absorbing cavity 25 cm long by 4 cm

in diameter attached to and enclosed by a copper shield to enhance the

equivalence of the electrical and laser sources. The main thermal contact

between absorbing cavity and shield is through a copper ring soldered about

half way between the ends of the calorimeter proper. The significance of the

location of the copper ring is discussed in section 6. The calorimeter is

calibrated by means of an electrical heater wound on the outside of the

absorber and the temperature is measured with a resistance bridge, as described by Maier [5], wound on the shield. The calorimeter proper is supported on three stainless steel tubes. These tubes are soldered to threaded

brass caps which are threaded into the temperature controlled jacket that is

1.3 cm thick. These supports are equally spaced on the perimeter of the

temperature controlled jacket. Spaces between the calorimeter proper and

the jacket and between the absorber and the shield are about 1 cm, a space

small enough to avoid convection and the associated non-linearities but large

enough to hold the thermal conductance of the air to a reasonable value.

The temperature controlled jacket is supported on three stainless steel tubes

mounted and spaced equally on the perimeter of the supporting ring. The

construction of these support tubes is similar to those mentioned above.

The temperature controlled jacket has two parallel spiral grooves cut along

its length. Another resistance-thermometer bridge and a heater are wound in

the grooves which allows temperature control of the jacket. A tube leads

into the calorimeter proper for cooling it by an air current to shorten the

cool-down time between experiments. The calorimeter proper and the jacket

are enclosed in an aluminum cover anodized black to absorb stray laser

radiation.
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The right end of the absorber is closed by a polished gold-plated

conical mirror, which reflects the incident laser power to the black-painted

wall. The mirror has an apex angle of 166° and is replaceable. It serves to

spread the incident power so that the power density will be below the damage

threshold of the 3M Nextel* black paint, about 200 W/cm2

. Care in using the

laser is necessary to keep the power density below the damage threshold of

these surfaces.

The calibrating electrical heater consists of 100 ohms of 0.25 mm

diameter manganin wire in a bifilar winding covering about 6 cm of the

absorber between the copper ring and the mirror mount. The heater is operated at power levels between 5 W and 1 kW for a total energy input of 300 J

to 3 kJ (but 1.5 kJ at 1 kW) . The copper heater leads are 0.4 mm in diameter

with a length of about 3.8 cm between the shield and the jacket. For thermal

contact the leads are attached to the shield and the surroundings with Epon

Resin 815 manufactured by Shell Chemical Co. The connection to the calorimeter heater is made inside the shield at the surface of the absorber.

Potential leads are attached midway between the shield and the jacket.

The resistance bridge used for temperature measurement is wound on the

shield near the copper ring. The bridge consists of two copper arms and

two manganin arms each of 100 ohms resistance at the balance temperature,

which is chosen to be just below the steady-state temperature T(°°) so that

the temperature reading is always positive. The bridge is supplied by a 20 mA current source constant to 0.005%. If the four arms are equal, the

bridge output E is given by E = 1/2 IR3AT, where I is the constant current,

R is the resistance per arm, 3 is the temperature coefficient of the copper,

and AT is the difference between the actual temperature and the temperature

at balance. The formula implies a bridge output linear with the temperature

to the extent that 3 is constant. The real question of the linearity of

the calorimeter based on the actual thermometer is discussed with the

electrical calibration data.

The temperature of the temperature-controlled jacket must be brought

to the control point before the calorimeter can be operated. The theory

requires this temperature to be invariant with time during an experiment.

If this temperature varies appreciably, the heat exchange will be incorrectly

evaluated.

The controlled temperature is sensed by a resistance bridge that is the same type as the one used for the calorimeter-proper temperature measurement.

The bridge is fastened in a groove with Epon 815 epoxy resin. It balances

at about 43° C and is powered with 20 mA dc. The output is amplified by an operational amplifier having a low offset voltage and gain stabilized with

*Certain materials are identified in this paper in order to specify adequately the experimental procedure. In no case does such identification

imply recommendation or endorsement by the National Bureau of Standards, nor does it imply that the material or equipment identified is necessarily the

best available for the purpose.

7
a feedback resistor. The amplified signal is fed to intermediate operational

amplifiers which provide proportional and integral (reset) control to an output transistor. This transistor regulates the current in a 30-ft control

heater wound in a groove near the control thermometer.

The temperature is controlled to ± 0.1 mK as calculated from the gain

of the operational amplifier and the bridge parameters. Experience indicates

that temperature control is not the limiting factor in the precision of the

measurement.

The temperature-controlled jacket does not form a complete thermal

enclosure for the calorimeter proper. Some heat is transferred to the room through the opening for the laser beam. To this extent the room is a part

of the thermodynamic surroundings. In principle, the variation in room temperature would set a lower limit on the laser energy to be measured

accurately in isoperibol calorimeters, but we have not yet established that

temperature variation does indeed set the limit.

The output of the thermometer that measures the temperature of the

calorimeter proper is amplified by a dc amplifier linear to 0.01%. This

amplifier is epoxied to the rear cover of the temperature controlled jacket.

The output of the amplifier is read by a digital voltmeter. At 10 sec intervals, the voltmeter reading is transferred to a data coupler and then

punched on a paper tape.

4. USE OF THE Kl CALORIMETERS

In preparing for an experiment, the dc amplifier is allowed to warm up

while the temperature control is established. About two hours after the

temperature of the temperature controlled jacket is under control the rate

of change of the temperature of the calorimeter proper in eq. (2) becomes

small enough to make a measurement. We keep the rate of change in the

initial rating period less than about ten percent of the rate of change in

the final rating period, because we believe that the increase in internal

energy can be evaluated more accurately than the heat exchange. If the

calorimeter proper is cooling rapidly in the initial rating period, the

heat exchange term for the experiment will be large.

The experiment cannot be started until the rate of change of temperature

of the calorimeter proper is small as defined above, but one hour after the

preceding experiment will usually prove more than adequate for this purpose.

In any case, this point is checked in the computer data analysis.

When one is satisfied that the temperature difference is small enough

so that eq. (2) holds, a number of data points of the thermometer response

are logged (20 to 30) and then either the electrical, or the laser input

is made. We frequently log a minimum of 31 points at 10 sec intervals.
After the input, enough points are taken to make sure that 2 to 30 points

are in the final rating period after higher order exponentials have become

negligible and eq. (2) again can be applied. The final rating period can usually be started about 4 00 sec after the input is stopped. The computer

program [3] prints deviations of individual points from the integral form of

the eq. (2) to show that the higher order exponentials have become negligible

before the final rating period is started. The initial testing of the

calorimeters includes the determination of the data point at which the final

rating period can be started.

5. ELECTRICAL CALIBRATIONS

The electrical calibrating circuit diagram in figure 3 is based on principles long in use in calorimetry. The diagram is included to facilitate

discussion of the errors in the measurement. Electrical calibrations are carried out using the calorimeter-proper heater as a four-terminal resistor,

measuring the dc current in the heater, the voltage across it, and the time

the power is on. The heater current is determined from the voltage across a standard resistor. Different standard resistors are used to accommodate

different heater currents. The voltages required for the power computation

are measured with digital voltmeters.

The power source for calibration is a dc supply operating up to 317

volts and 3.15 amps. The current is regulated to 0.1%. This regulation is

better than necessary and is a matter of convenience.

The time is read by a counter-timer with a built-in crystal-controlled

oscillator. The oscillator is checked against a 100 kHz standard frequency

supplied by the Time and Frequency Division of NBS. The timer is connected

in parallel with the calibrating heater and is triggered by the voltage

across the heater. Any leakage current in the timer is by-passed around

the standard resistor and calorimeter heater to avoid a systematic error in

the heater power.

Electrical calibrations have been carried out for two calorimeters Kl-1

and Kl-2 of the type shown in figure 2. The calibration factor E is obtained

by a least squares fit of the corrected temperature rise AT. and the energy

W. according to the equation AT. = W./E. For Kl-1, the electrical calibration is 2.262 J/mV, based on 36 electrical calibrations taken over a period

-3

of two years. The standard deviation of the mean of this factor is 1.5 x 10 or 0.066%. For Kl-2, the calibration factor is 2.276 with the standard

deviation of the mean of 0.058% based on 35 measurements over the same period. The difference in the standard deviation is not significant

(F = 1.3 < 1.9 = F QC {35,34}. See Natrella [6], Chapter 4.)

The residuals from the straight-line fit must be examined for systematic

deviations from a straight line, indicating non-linearity. The maximum

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temperature rise for Kl-1 and Kl-2 is about 3°K and in this range we find

no systematic deviation from linearity.

The straight-line fit takes the energy input to be without significant

random error and calculates a standard deviation s for the corrected

temperature rise. For Kl-1, s AT

= 3.65 mV and for Kl-2 s = 3.58 mV.

From this standard deviation by propagation of error formulas we obtain the

standard deviation of the calibration factor and the standard deviation s E

of individual determinations of the calibration factor. The standard

deviation s depends on the energy input W according to the formula

s £

= (E

2 /W) s AT . (4)

This formula is derived using Ku's propagation-of-error formulas [7]. The

quantity s„ may be looked upon as the standard deviation we would obtain if

Hi we made a large number of measurements at the same energy input W. The

standard deviation s increases as the energy W becomes smaller. This

behavior is consistent with the physics of the measurement from which we would expect very low energy experiments to get lost in the random error. The standard deviation s„ is important because it is an index of the

random error of an electrical energy measurement and therefore bears on a laser energy measurement with the calorimeter. It is taken into consideration later in estimating the error in such a measurement.

Systematic errors in the electrical measurements come from the instrumentation of figure 3. Three different standard resistors have been used,

a nominal one ohm and two nominal one-tenth ohm. The estimated limits of

systematic error are + 0.005% for the one ohm resistor and one of one-tenth

ohm resistors and ± 0.01% for the other one-tenth ohm resistor. The voltages

are read by two digital voltmeters calibrated against a standard cell with

a guarded laboratory potentiometer. The estimated limits of systematic

error in the calibration of the voltmeters are + 0.02%. The standard cell

and the standard resistors are calibrated by the Electricity Division of NBS.

The allocation of power generated in the heater leads constitutes the

largest systematic error in the electrical calibration. The length of each

lead is 1.9 cm between the outer surface of the calorimeter proper and the

temperature-controlled jacket with voltage taps at the midpoints, but the

leads are brought through a radial hole in the 1.6 mm wall of the shield of

the calorimeter proper. The exact size of this hole is not known and the

1.6 mm distance may be too short to bypass to the shield the heat generated

in the lead between the calibrating heater and the shield. The length of

each lead from the heater to the voltage tap at the midpoint above is

6.9 cm. If this heat is equally divided between the calorimeter proper

11
and the temperature-controlled jacket, the heater voltage taps are positioned

wrong by (6.9/2) - (1.9/2) = 2.5 cm. The heater leads are 0.4 mm diameter

copper and the error corresponding to 2 x 2.5 cm is 0.34 ohm or 0.34% of an energy input. The error is in the direction of making the observed number

of joules greater than actual. If we were confident that no heat is bypassed

to the outer calorimeter cylinder, we would correct the calibration factor

in joules per millivolt by multiplying by .9983 and estimating limits of

systematic error of ± 0.17%. If the hole in the shield is small enough,

the heat will be bypassed to the shield and the estimated systematic error depends only on the inaccuracy of locating the voltage taps at the midpoints and the conduction of the voltage taps themselves.

The time interval counters used to measure the time of the electrical

inputs are compared to a time standard accurate to one part in 10 with

an error of one least count or one part per million of a full-scale reading.

The timing error is therefore negligible.

The estimated limits of systematic error in an electrical calibration

are summarized in table I.

Table I. Estimated limits of systematic errors in electrical quantities.

Source of Error . Estimated Limits

Standard cell 0.005%

Calibration of digital voltmeters .04

Standard resistor .01

Heater leads .34

Time .000

Sum (rounded) .40%

Simply summing the systematic errors is a very conservative way of combining

tnem. (Propagation-of-error formulas allow summing percentages in this case .

)

The resistance of the electrical heater provides a convenient means of

monitoring the electrical circuit, figure 3, including the precision of the

two voltmeters and the heater itself. A significant change in the heater

resistance serves as an alert to trouble, possibly in grounding of the

neater or its leads. For calorimeter Kl-1, the heater resistance is 101.12

ohms with a standard deviation of the mean of 0.07 (32 measurements). For

Kl-2, the resistance is 101.42 and the standard deviation of the mean of

0.06.

12
6. ERRORS DUE TO NON-EQUIVALENCE OF SOURCES

The energy supplied to a calorimeter is evaluated by means of the

First Law of Thermodynamics. This law equates the electrical or electromagnetic work done on the calorimeter proper to the change in the

internal energy plus the heat transferred to the surroundings. The change

in internal energy is proportional to T

p

- T as in eq. (1) and is properly

evaluated by the techniques of section 1 independent of the location of the source. The heat exchange term e / (T - >)dt in eq. (1) fails to take

into account the heat exchange due to higher order exponentials [4] . The error is proportional to the input W and the constant of proportionality

depends on the geometric location of the source in the calorimeter proper.

Laser and electrical sources are therefore not exactly equivalent in the errors they cause in accounting for heat exchanged with the surroundings.

To the extent that heat from both sources can be made to flow to the surface

of the calorimeter proper by way of the same isothermal region, this region

acts as a secondary source and makes the constant of proportionality the same for both sources. In figure 2, the copper ring is designed to serve

as the secondary source. To check the design and determine if the copper ring is effective,

experiments were carried out with two different heaters. A small heater was wound on the mirror and electrical calibrations made with it were compared

to the regular electrical calibrations. The mirror location is thermally

quite different from that of the calibration heater, which is located along

the wall where the laser beam is absorbed. Heat from the mirror heater

travels farther to reach the copper ring and allows greater opportunity for

heat to bypass the ring by direct flow to the outer shield of the calorimeter

proper. Comparison of the calibrations using the two different heaters

should therefore give a worse-case estimate of the systematic error from

this source. The wire size of the mirror heater limited the maximum power level of

the electrical calibrations to 6 W, which is at the low end of the range

for the calorimeter. As a consequence, the random errors of the data from

the mirror heater calibrations were somewhat greater than the random errors

of the data from the regular calibrations.

For calorimeter Kl-1, the comparison is based on twenty-five regular

calibration experiments, which give a mean calibration factor of 2.2609

J/mV, and a standard deviation of the mean of 0.002, and eight calibrations

with the mirror heater, which give a mean of 2.2610 and a standard deviation

of the mean of 0.007. The difference between the two calibration factors

is obviously not statistically significant. (The 99% confidence interval

is ± 0.0241.)

13
For calorimeter Kl-2, twenty-one regular calibrations give a mean of

2.2793 J/mV and a standard deviation of 0.0017; eight calibrations with the

mirror heater give 2.2736 and a standard deviation of 0.0059. The difference

between the averages is again not statistically significant (99% confidence

limits ± 0.0204)

The mean calibration factors used in this comparison differ slightly

from the current values given earlier. Although the conclusions reached

would be the same, we feel that the data presented better represent the

state of the measurement system at the time the mirror heater calibrations

were made.

From the preceding calculations we estimate the limits of systematic

error due to the non-equivalence of sources to be 0.0001 ± 0.0241 for Kl-1

and 0.0057 ± 0.0204 for Kl-2, where the stated errors are the 99% confidence

intervals.

7. ABSORPTANCE OF THE CAVITY

The absorptance was measured by the method of West and Schmidt [8] . In

essence, the method consists of placing an auxiliary calorimeter over the

opening in the absorbing cavity. The auxiliary calorimeter is a shallow

cup with a small hole in the center. A laser beam enters the absorbing

cavity through this hole. Laser power not absorbed by the cavity is measured

by the auxiliary calorimeter.

The reflectance 1 - a for the cavity is given by their equation

P

2 /a

2 F

2 1 "

a =

P + P /a F ' (5)

1 2

/ 2 2 where a is the absorptance factor of the cavity, P, is the power to the

calorimeter proper, P_ is the power to the auxiliary calorimeter, a_ is

the absorptance factor of the auxiliary calorimeter and F„ is the fraction

of radiation from the cavity that strikes the auxiliary calorimeter. F_

allows for the hole for the laser beam. The powers P, and P„ are obtained

by simultaneous solution of eqs. (6) below relating the temperature T. of

the two calorimeters to the powers through a set of four experimentallydetermined transfer functions I .

T

1

(t) - T

1

(-) = I 11 (t)P

1 + I 12 (t)P

2

T

2 (t) - T

2 (-J = l 21 (t)V

1 + I 22 (t)P2

(6)

where T, (°°) and T„ (°°) are the temperatures in the steady state with P, = P„ =

0. The I. . are obtained by alternate electrical inputs with first P, = then P

? = 0. Equations (6) for a laser input can then be solved for P, and

P_, which are then used in eq. (5) t to obtain the effective absorptance a.

14
Systematic errors arise in this procedure because: (1) The I.

.

determined electrically are not exactly those required in the laser measurement. This is another aspect of the non-equivalence of sources problem.

(2) The quantities F„ and a„ have systematic errors. There will be another

systematic error because the mean of our observations will differ from the

population mean. Equations (6) require the choice of a particular time t. To choose

this time, we consider the random errors in the I. . and a pseudo-reflectance

experiment. In table II are shown the values of the electrically determined

I. . and the standard deviations for calorimeter Kl-2 at 20 sec intervals

ID after the power (P, or P_) is turned on. The values are averages of four

determinations. Evidently, the precision of these data improves with time

up to 100 sec. In the pseudo-reflectance experiment, we use simultaneous inputs of

electrical power to both the auxiliary calorimeter and the calorimeter proper

to check how well eq. (5) reproduces the known ratio P

? /(P, + P„). The

averages of two experiments are shown in table III. The known value of the

pseudo-reflectance P„/(P, + P_) is 1.45%. Evidently, the best values are obtained for times of 100 sec or less.

On the basis of the data in tables II and III we select 100 sec as the

best time to use for calculating the absorptance. The dual electrical

experiment was not carried out for Kl-2, but it is very similar to Kl-1

both in construction and in the value obtained for the absorptance.

At t = 100 sec, the reflectance at 10.6 ym for calorimeter Kl-1 is

1.47% with a standard deviation s = 0.045 based on four measurements. The

corresponding values for Kl-2 are 1.49% and s = 0.030 based on two measurements. Since the measurement is made with the same auxiliary calorimeter

and Kl-1 and Kl-2 are very similar, we pool the standard deviation to

obtain s_ = 0.042 with four degrees of freedom. The 99% confidence intervals

are 1.47 ± .19% and 1.49 ± .19% respectively.

There is another systematic error because the I . . refer to electrical

and not laser inputs. In figure 4 is a plot of I,, from table II. Excepting

the first point, these observations lie on a straight line with an intercept

at 10 sec, which represents the delay between the heater input and the

response of the thermometer. Theoretically, the data represent part of a curve which eventually reaches a constant value for L,, but here we are only concerned with an adequate representation of the first part of that

curve. In figure 2 the heater is located closer to the thermometer than

the absorbing surface for laser inputs. A laser ray parallel to the axis

and striking the point of the conical mirror will be reflected to the wall

8 cm from the mirror. Rays focused through an aperture in front of the

15
Table II. Summary of the I. . values for two sets of four experiments,

calorimeter Kl-2

Time *n s J 21 s X 12 s J 22 s sec mV/W "6 mV/W % mV/W % mV/W %

20 4.6 11.0 .03 72.6 .08 115.5 1.34 .29

40 13.1 0.9 .18 29.9 .29 28.5 2.61 .19

60 21.3 3.7 .58 13.0 .58 28.4 3.77 .12

80 29.6 3.3 1.30 8.3 1.19 17.2 4.82 .10

100 38.0 1.9 2.41 5.5 2.26 12.3 5.76 .08

120 46.2 1.6 3.92 3.6 3.58 12.0 6.60 .09

140 54.8 1.1 5.86 2.7 5.22 8.2 7.35 .07

160 63.0 1.0 7.94 8.8 7.12 7.3 8.02 .08

180 71.3 1.0 11.04 1.8 9.38 6.2 8.61 .07

200 79.4 0.8 14.24 1.6 11.77 5.7 9.13 .08

220 87.8 0.8 17.83 1.4 14.48 4.6 9.59 .05

Table III. Summary of dual electrical input to Kl-1,

Time (sec)

20

60

100

140

180

220

Pseudo-Reflectance

1.46%

1.48

1.46

1.54

1.59

1.63

16
_ \

\

r\ ^

CL \

X \ h-

^

x

N^ 3 Q " Q_ UJ - x ^

1 \ y\. \ ^5 - HXV/\v q:2 - z> X\ UJ p -

CL X\ CD CO

-z. VN < UJ

«~^—' "~" ' _i x\ < ^

o

en x ^ h- X_\Tax O X v UJ x > _l X>

LlI 1,1.1,1, 1 1 1 1 1

ro

^

UJ

CO

UJ

oj c^ oo co ^a- oj

(M/AW) U

I

Figure 4. Plot of the function I,, required to determine the effective absorptance. The solid line represents an electrical

input. The dotted line represents a possible laser

input. The difference between the two causes a systematic

error in the absorptance.

17
calorimeter will strike the mirror at a greater angle of incidence and be

reflected to the wall closer to the mirror. A laser input will likely be more remote from the thermometer than an electrical input with a cor- respondingly greater delay, so that we would expect a line parallel to that

for the electrical input but displaced to longer times. Such a line, marked

Laser Input (estimated), has been sketched in the figure. Of course, the

actual time displacement will depend on the size and divergence of the beam

as discussed above. For example, one of our laser measurement experiments

with a large beam (^ 1 cm diameter) gives an intercept of 40 sec for a plot

of T, - T, (°°)/P,. Taking a line with the same slope as I,, but displaced

by 10 sec, we find a constant difference of 5 mV/W between I,, determined

electrically and what it might be for a small divergent beam. This smaller

value of I,, gives 1.24% for the effective reflectance at t = 100 sec,

compared to 1.46% for the electrically determined I, ,

.

To explore this problem further, we compared data taken with a different

electrical heater on calorimeter 2, figure 1 [8]. The heater used to obtain

most of the data was wound on the cylindrical surface. A second heater was fastened to the flat surface, which would be closer to the laser input. The

values of I^p obtained with this heater were then used with the original I-,, to obtain the effective absorptance (a = 1.38%) and with the decreased values

of I-,, to obtain a = 1.19%. Considering all cases, the calculated effective

absorptance ranges from 1.19 to 1.46%. Although both of these values are

less than 1.46%, we consider that the error might also be of the other sign

and allow a systematic error of 0.2% for the bias in the I. .. We must also allow for the fact that the laser beams used were small

and covered only the central part of the mirror. A larger beam would be

reflected to the walls nearer the mirror and might give a different value

for the absorptance. We estimate this error to be less than 0.2%. A

better procedure for obtaining a random sample would give a better evaluation of this error.

To summarize the systematic error in the absorptance measurement for

each calorimeter, we estimate ± 0.19% from the random error of the

absorptance measurement and + 0.4% for the sum of the error in the I. . and

in the sampling.

18
8. COMPARISON OF Kl-1 AND Kl-2

When two calorimeters are used to measure a beam ratio as in figure 1,

the ratio is biased because of the systematic errors (6,,6

?

) in energy

measurement by each calorimeter. The true beam ratio can be obtained by

interchanging the calorimeter sub-systems so the bias between calorimeter

sub-systems can then be calculated. In this instance each calorimeter subsystem includes a K-l calorimeter, a temperature controller, a constant current supply, a signal amplifier, and a digital voltmeter that reads the

response of the temperature bridge. In one position the beam ratio R, is

the ratio of energies to the two calorimeters

E

l( l + 6 1 )AT1 K

l E

2 (l + 6 2 )AT2 (')

where the E's are the calibration factors and the AT's are the corrected

temperature rises. Reversing the calorimeters, the second ratio R„ is

E,(1+6,)At'

R. = — f (8) E

1 (1+61 )AT1 where the primes are used to indicate that the temperature rises will be

different in the two experiments. The true ratio R is equal to R, or R,

corrected for the errors, that is

E AT E AT' R = -± ±- = -^ f . (9)

E

2 AT2 E

1 AT1 Multiplying expressions from eq. (7) and (8) we obtain

E, (1+6,) AT, E, (1+6,) AT,

R,R, = -± ± ±- -£ = f (10)

E

2 (1+62 )AT2 E

1 (1+61 )AT1 which simplifies to

AT, AT

R,R, = —- —4 (ID

AT

2 AT

X that is, the product R-iR, I s independent of the calibration factor and its

associated error. Multiplying together the two expressions for R in eq. (9) we obtain

AT. AT,

2 —i —4 . (12)

AT

2 AT1

19
By inspection of eqs . (11) and (12)

R = /ItjRj . (13)

Substitution in eqs. (7) from (9) gives

R

1 = R(l+6

1 )/(l+62

) , (14)

and similarly from eqs. (8) and (9)

R

2 = R(l+52 )/(l+61

) . (15)

From eqs. (14) and (15)

(•1 + 6, 1 + 6_

R

i - R2 = R rrx ~ T-nr> > (16)

which simplifies to

R

l - R

2 26

1 + & ±

2 - 26

2 - 6 2 2

R (1 + 6 X

) (1 + 6 2) (17)

For Kl-1 and Kl-2, 6 < 0.02 to 0.03, so that we can use the approximations

2

6 << 6 and 6 << 1 to obtain

R, - R

2 R

= 2

(6-l - 6 2

) . (18)

This equation is the basis for deducing the difference in the systematic

errors of the two calorimeter sub-systems from intercomparisons using a beam

splitter.

The calorimeter sub-systems of Kl-1 and Kl-2 have been intercompared

using a gallium arsenide beam splitter. Twenty-five measurements with Kl-2

as monitor give an average ratio of 1.7919 and a standard deviation of

0.0143. Seventeen measurements with Kl-1 as monitor give an average of

1.8154 and a standard deviation of 0,0142. From eq. (18) we calculate a difference in systematic errors of 0.65%. This difference is large enough

that we accept the statistical hypothesis that there is a real systematic

difference between the two calorimeters at the 99% confidence level. The

accuracy of the digital voltmeters are regularly monitored at NBS by equipment that is traceable to the international system of units. The scale errors of these digital voltmeters are below .01%; therefore their contributions to the systematic difference of the sub-systems are insignificant in

comparison to the total systematic difference.

20
9. ERROR OF A LASER ENERGY MEASUREMENT

The over-all error is obtained by the methods of combining errors discussed in [6]

.

The precision of a laser energy measurement is taken to be the same as that of an electrical calibration as calculated by eq. (4) . This procedure

is justified by comparing the precision of electrical and beam splitter

experiments. Franzen [1] has made beam ratio measurements using Kl-1 and

Kl-2 with a gallium arsenide beam splitter. For a set of twenty-five mea- surements with Kl-1 as calorimeter 1, figure 4, and Kl-2 as calorimeter 2,

Franzen observed the standard deviation of the beam ratio to be 0.80%. The

average energy to Kl-1 was 867 J and to Kl-2 481 J. Comparing the standard

deviations in per cent from these energies according to eq. (4) , we get

0.42% and 0.74% respectively for Kl-1 and Kl-2. Combining these we get

0.85% as the estimated standard deviation of a beam splitter measurement.

The observed standard deviation (0.80%) and that calculated from the electrical calibrations (0.85%) do not differ significantly (F test [6]) and we reject the hypothesis that the precision of a laser energy measurement is

different from the precision of an electrical measurement.

The systematic errors are combined in two different ways, depending on how they were estimated. Those determined by a sampling process are combined

by root-mean-square; those estimated from calculations or otherwise are

summed. Although summing is a pessimistic method of combining errors, the

evidence is that root-mean-square combination of systematic errors of this

kind leads to over-optimistic error statements [6]

.

For calorimeter Kl-1, the error due to the non-equivalence of sources

is estimated from the difference between the calibration factors with the

regular and the mirror heater to be .0001 + .0241 at the 99% confidence

level. Taking the sum as a conservative estimate, we get .0242 t 2.2609 = 1.07%. The random component of error in the absorptance a is 0,19%. These

two are combined by root-mean-square to give 1.08%. The estimated limits of

systematic error in the electrical calibrations is ± 0.4% and the estimates

for the absorptance error are 0.2% in the I. . and 0.2% for insufficient

variation of the incident beam location. The combined estimated limits of

systematic error are ± (1.08 + 0.4 + 0.2 + 0.2) = ± 1.9% for calorimeter

Kl-1.

For calorimeter Kl-2, the non-equivalence error is 0.0057 ± .0204 or

1.21%. Combining with the 0.19% from the absorptance measurement, we get

1.22%. The other estimates are the same as for Kl-1, leading to a combined

estimate of ± 2.0%.

21
We obtain the limits of error of a single measurement of energy by

adding to the estimated limits of systematic error the product ts, where the

t statistic is chosen for the 99% confidence level and the number of electrical calibration experiments less one and the standard deviation s is

calculated by eq. (4) from the standard deviation sY„, also obtained from the

electrical calibration data. For a 1 kJ measurement with calorimeter Kl-1,

fcs. /W is 2.262 x 3.65 v 1000 x 100 = 0.82% and t = 2.7 for 35 degrees of

freedom and 99% confidence interval. The estimated limits of error are

± (2.7 x 0.82 + 1.9) = ± 4.1%. The corresponding numbers for Kl-2 are 0.81%

and 4.2%.

22
10. REFERENCES

[1] Franzen, D.L., Precision beam splitters for C0

2 lasers, Applied Optics,

14, 647-652 (March 1975) .

[2] West, E.D., Case, W.E., Rasmussen, A.L. , and Schmidt, L.B., A reference

calorimeter for laser energy measurements, J. Res. Nat. Bur Stand.

(U.S.), 7 6A (Phys. and Chem.), No. 1, 13 (Jan. -Feb. 1972).

[3] West, E.D., Data analysis for isoperibol laser calorimetry, Nat. Bur.

Stand. (U.S.), Tech. Note 396, 34 pages (Feb. 1971).

[4] West, E.D. and Churney, K.L., Theory of isoperibol calorimetry for laser

and power energy measurements, J. Appl. Phys., 41, 2705-2712 (May 1970).

[5] Maier, C.G., Resistance thermometers for chemists, J. Phys. Chem.,

Vol. 34, 2860-2868 (1930).

[6] Natrella, M.G., Experimental statistics, Nat. Bur. Stand. (U.S.),

Handbook 91 (1966) .

[7] Ku, H.H., Precision measurement and calibration, statistical concepts

and procedures, Nat. Bur. Stand. (U.S.), Spec. Publ. 300, .1, 436 pages

(Feb. 1969) .

[8] West, E.D. and Schmidt, L.B., Spectral-absorptance measurements for

laser calorimetry, J. Opt. Soc . Am., 65, 573-578 (May 1975).

23
NBS-T14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS-TN-685

_l_

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A SYSTEM FOR CALIBRATING LASER POWER METERS

FOR THE RANGE 5-1000 W

5. Publication Date May 1977

6. Performing Organization Code 276.08

7. AUTHOR(S)

E.D. West and L.B. Schmidt

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. 2768120

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as block 9

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 2 00- word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) This technical note describes the National Bureau of Standards measurement

system for calibration and test of laser power and energy devices at power levels up

to 1 kW and total energies of several kJ. The main parts of the system are two

electrically calibrated calorimeters and several beam splitters of different materials.

The estimated limits of systematic error in measurements with the two calorimeters

are ± 1.9% for one and ± 2.0% for the other. The two calorimeters have the same precision, and the standard deviation of an electrical calibration is the same as the

standard deviation of a laser energy measurement, indicating that the laser beam does

not affect the precision of the measurement. The standard deviation of an energy

measurement depends on the total energy to the calorimeter and is about 0.8% at 1 kJ.

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)

Calorimeter; laser calorimeter; laser energy; laser power.

18. AVAILABILITY [^Unlimited

^2 For Official Distribution. Do Not Release to NTIS H Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. C13

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