Evaluating the inequivalence and a computationalsimplification for the NBS laserenergy standards

Eric G. Johnson, Jr.

A model with two time constants is used to estimate the inequivalence in response between a laser energypulse and an electrical energy pulse put into a calorimeter of the C series type. The results are as follows:the calorimeter labeled C41 showed a 0.15% inequivalence and the calorimeter labeled C46 showed none.We also find that the complicated model currently used to get the corrected temperature rise of a measure-ment can be replaced by a simpler four-data-point method with no significant loss in accuracy. This simpli-

fication means we can substitute a microprocessor for a large computer to get the corrected temperature risein an electrical calibration or laser energy measurement.

1. Introduction

The National Bureau of Standards uses calorimetersto measure energy in a laser beam. This measurementis traceable to the International System of Units. Weuse electrical energy to calibrate the response of thesecalorimeters and then make corrections to their re-sponse to infer the proper laser energy response. Onecorrection arises because absorbed electrical and laserenergy pulses of exactly the same shape and number ofjoules do not cause the same time response in the outputsignal. This change in the output signal arises from thedifferent locations where the electrical and the laserenergy are absorbed. To simulate the cause for thischange and thereby to infer the inequivalence betweenlaser and electrical energy in a calorimeter, we use anappropriate mathematical model. Briefly, in responseto a change of input power a calorimeter shows heatmodes which are identifiable in a time series signal thatis generated out of a weighted average of the internaltemperature inside the calorimeter. These heat modesshow decaying exponentials with unique time constantsthat are characteristic of the calorimeter, and theygenerate an output signal which is a superpositon of anumber of them. With proper design, as discussed inmore detail in the next section, we can reduce sub-stantially the strength of heat modes actually measuredduring our use of the calorimeter. In the NBS calo-rimeters, most of the energy drives the output into asingle time constant behavior, and the remaining energy

The author is with U.S. National Bureau of Standards, Electro-magnetics Division, Boulder, Colorado 80302.

Received 24 January 1977.

only weakly drives a time behavior that is described inthis paper by a second exponential with a much fastertime constant.

By applying the more detailed model with two timeconstants (labeled henceforth as the two-time-constantmodel) to reduce time series data from the NBS laserenergy meters of the C4 series type,1 2 we establish themagnitude of the inequivalence in response betweenelectrical calibration energy and laser measurementenergy for two of those meters.3

We made no attempt to correct for the inequivalenceof each instrument or to understand why a particularinstrument has a given value. Our purpose is to docu-ment rigorously the sources in the uncertainty in theNBS Laser Energy Standard 1. This Standard is de-fined by three C4 series calorimeters labeled C41, C44,and C46. In our present program for documentationof this Standard, we conclude that the present basis forthe inequivalence estimation needs to be improved 4 5

and that there is no information in the literature whichcan be used6 to deduce size of the inequivalence. Toestablish the inequivalence, we first discuss in Sec. I.Athe essential details of the model with one time constant(labeled henceforth as the one-time-constant model).7

This one-time-constant model is currently used to re-duce data obtained in the NBS laser energy measure-ments and electrical calibrations. To complete theunderstanding of inequivalence, we describe the moredetailed two-time-constant model in Sec. II.B and ex-plain how this model allows us to estimate the magni-tude of the inequivalence. To be sure our conclusionsare correct, we examine the limits of the two-time-constant model in Sec. II.C by discussing a data sum-mary which is used to deduce the inequivalence for C4series calorimeters.

August 1977 / Vol. 16, No. 8 / APPLIED OPTICS 2315

As a secondary benefit of this inequivalence docu-mentation effort, we discuss the application of thefour-point method7 in Sec. III, because it reduces theneed for nonlinear least-squares fit techniques in thedata analysis and because use of such a simplificationhas not been reported in present literature.6 Here weestablish that this four-point method with a quality offit test is as accurate as the complex nonlinear least-squares fit method.7

II. Inequivalence in the Calorimeter of the C SeriesType

Inequivalence is operationally defined in this paperas the difference in response as seen by the temperaturedetectors in a calorimeter due to the differences inspatial and temporal deposition of heat by differentsources at the conversion surface of the calorimeter.Inequivalence as defined here is only one of a numberof causes for errors when correlating the difference be-tween the temperature response of a calorimeter to laserand electrical energy. Although our prime emphasisis to evaluate the inequivalence, we also briefly describethe other causes for error such as window transmissioncorrection, changes in the radiative heat loss out of thewindow, and the various absorption of electrical andlaser energy to delineate clearly between the inequiv-alence as defined in this paper and those other causesfor differences in measurement of laser and electricalenergy.

In the Laser Standards work we use high precisioncalorimeters which are designed to have little inequiv-alence. Specifically, the C series type2 4 is constructedso that the spatial positions for heat deposition by laseror electrical energy are made as nearly the same asreasonably possible. To further reduce the effects ofthe inequivalence, the temperature detectors are de-liberately placed so that there is significant distancebetween them and the heat sources. As a consequenceof this construction, there is a tendency for the strengthof faster decay-rate heat modes3 in the calorimeter tobe significantly attenuated by the time the modereaches the detectors. The relative strength betweenthe various heat modes in a given laser energy mea-surement or an electrical calibration is critically de-pendent on the positions of the heat source and of thedetectors. Because the location of the temperaturedetectors is permanently fixed, we can quantify theinequivalence by measuring the relative strength of thefaster decay-rate heat modes for various heat sources.Ideally we should evaluate all heat modes relative to theslowest decay-rate heat mode. Because the tempera-ture measurements are discretely sampled in time, thusprecluding measurements of heat modes which havedecay rates exceeding that sampling rate, and becausethe analysis of such multiple exponential time depen-dences is extremely complex, we make the approxima-tion that only two different time constants (the slowesttime-decay heat mode and the next slower time-decayheat mode) are sufficient to quantify the magnitude ofthe inequivalence between two heat sources. The re-sults in Sec. II.C show that this approximation must becorrected by secondary information, namely, the known

time shape of the energy pulses, to quantify accuratelythe inequivalence. Sections II.A and II.B develop thenecessary detail to understand the quantification. (TheAppendix derives some of the details for these mod-els.)

A conceptual picture illustrating the essence of theinequivalence is as follows:

(1) We compute from the time series signal out of thecalorimeter a quantity called the corrected temperaturerise AT to which each time constant term in this two-time-constant model makes a contribution which welabel as AT, and ATb. In this model we relate AT toAT, linearly as ATb = YATa; thus the corrected tem-perature rise is AT = (1 + Y)ATa, where Y is the rela-tive strength of the faster time-constant term ATb tothe slower time-constant term ATa

(2) If we put into the calorimeter various knownelectrical energies E, we can construct the relation be-tween the energies and the computed corrected tem-perature rises as E = KAT = K(1 + Y)ATa, where K isthe calibration response for the electrical energy.

(3) If we ignore in this discussion the various correc-tions not relevant to inequivalence, we can assume anideal measurement where a series of known laser ener-gies EL go into the same calorimeter to get the corre-sponding corrected temperature rises ATL. We findthat the relationship between the energies and thecorrected rises is

EL = KLATL = KL(1 + YL)ATa,L,

where KL, YL, and ATa,L are the corresponding valuesfor the laser measurements which can be different fromthe electrical calibrations because the energy source isdifferent.

(4) We now specialize to a case where both the elec-trical and laser input energies are the same. If there isno significant amount of the faster time-constant modein our output signal and hence in our corrected tem-perature rise, Y and YL are zero, and we can now expectthe corrected temperature rises to be equal whichimplies K = KL. Under this circumstance there is noinequivalence. Unfortunately, in the NBS calorimetersthere is a small nonzero value of Y and YL for both theelectrical and laser energy injections into the calorim-eter. This means that even if the laser and electricalenergies are the same and in addition the contributionsto the corrected temperature rise by the slowest time-constant mode are equal, the K and KL are not neces-sarily equal. They are related as K(1 + Y) = KL (1 +YL). In our measurements we get K, Y and YL andthereby infer KL using the above relation. Note thatif Y = YL, then K = KL, and again there is no inequiv-alence although we have two-time-constant behaviorand use a single-time-constant model to reduce thedata.

As clearly mentioned, in addition to the above ine-quivalence, there are other causes for differences intemperature response. These are due to losses in boththe electrical and laser energy.

In the electrical energy case, there are two possiblelosses. One is discussed in Ref. 1. It corresponds to the

2316 APPLIED OPTICS / Vol. 16, No. 8 / August 1977

FRONT HEATER

PRIMARY LASERLASER ABSORPTION SURFACEBEAM- - - - _ -

REARHEATER

THE I4TER-NAL STRUCTURE OF C4 SERIES CALORIMETER

Fig. 1. The internal structure of C4 series calorimeter.

I (a,) I (b)

I II IIII I

ti

tX - t2

I II II II I

I It2 tX t3

TIME

TRUE "POWER ON" TIME

MEASUREMENT SEQUENCE (FOUR TIME

(a) INITIAL RATING PERIOD

(b) POWER ON PERIOD

(c) RELAXATION PERIOD

(d) FINAL RATING PERIOD

( c) I

I

I

(C

I

t4

PERIODS)

ti < t < t2

t2< t < t3

t3 < t < t4

t4 < t < t5

Fig. 2. Measurement sequence (four time periods).

flow of heat from the heater directly to the thermaljacket of the calorimeter and bypassing the temperaturedetectors. This loss is insignificant for the C4 seriescalorimeters. The second loss occurs by the direct ra-diation of heat from the heaters or from the conversionsurface of the calorimeter without a significant fractionbeing recaptured by the conversion surface; therefore,this heat does not affect the detector. This loss is re-duced by making the main heat absorption surface havea small solid angle for direct radiation loss. Thissmall-angle constraint is the case for the C series calo-rimeters and is shown in Fig. 1 by the location for therear heater and laser absorption point (see Refs. 2 and4 for discussion of this loss). The effect of radiation lossdifferences between the front and rear heaters is dis-cussed in Sec. 2.C.

There are three laser energy losses. One is due to thewindow on the C4 series calorimeters. This loss isevaluated by direct transmission measurements on thewindow and is discussed in Refs. 1, 2, and 5. The sec-ond laser loss, the absorptance, results from the failureof the conversion surface to capture all the laser radia-tion. This loss is evaluated in Refs. 1 and 2 and is foundto be insignificant for the C series calorimeters. Thefinal laser loss is similar to the electrical direct radiationloss and is handled similarly, namely, by the choice ofstructure for the absorption surface (see Fig. 1).

In a typical laser energy measurement or electricalcalibration, the temperature detector of a C4 seriescalorimeter has a time response as shown in Fig. 2. Wehave four distinct time periods. We use these four timeperiods in both the one-and two-time-constant mod-els.

The initial and final rating periods are used to deducethe parameters of the calorimeter for a given energymeasurement. These two periods are chosen becauseonly the slowest decay-rate mode is present and becausethere are no unknown sources of heat into the calorim-eter.

The peak response time t3 occurs after the actual timet,, at which the laser or electrical source is turned offbecause it takes time for the heat to flow between theheat conversion point and the temperature detectors.In a true one-time-constant system there would be nosuch delay. In a true two-time-constant system therecan be delay because the sign of the response caused bythe faster decay-rate mode can be opposite to the re-sponse caused by the slower decay-rate mode.

The relaxation period has no unknown heat sources.We expect the strength of various faster decay-rate heatmodes to become insignificant compared to the strengthof the slowest time-decay term during this period.

A. One-Time-Constant ModelTable I summarizes the essential details for the

one-time-constant model as defined by the differentialequation in part (a), where G is the inverse time con-stant of the instrument; the data V(t) are the measuredresponse of the instrument; and P(t) is the appliedpower to the instrument. To simplify this discussion,we assume the effect of the heat source which maintainsthe equilibrium temperature of the instrument is al-ready removed from V(t). In analysis of real data, weuse the values from the two rating periods to deduce theconstant and linear time drift background terms as wellas the unknown parameters defined in part (b), namely,V, V4, and G. These parameters are determinedthrough a nonlinear least-squares procedure with theappropriate orthogonalization to separate the linear-drift dependence from the exponential dependence.Knowing these parameters and assuming that the ap-plied power is constant, we compute the quantities asdefined in part (c), namely Va and P'. Part (d) showsa summarizing response called the corrected tempera-ture rise which does not require us to assume constantapplied power for measurement of energy into a calo-rimeter. Note that the integral in the AT,(t) uses theactual measured data sampled during the power-on and

August 1977 / Vol. 16, No. 8 / APPLIED OPTICS 2317

relaxation periods. We pretend that the various in-duced faster time-decay heat modes act as a source forthe slowest time-decay heat mode and thus set t = t4 inthe one-time-constant corrected temperature rise,AT 1(t4 ), where we know all other heat modes are in-significant. If the applied power is constant and the"one-time-constant is adequate we can use part (e) totest the consistency in the results within the one-time-constant model framework. As indicated in part (f), afurther consistency check arises by varying time t toshow the time dependence of the corrected temperaturerise for the period t < t < t4. If the corrected rise isa constant, the one-time-constant model is sufficientfor real data.

B. Two-Time-Constant ModelTable II summarizes the essential features of the

two-time-constant model. We define the faster time-decay term by two parameters, namely, V5 and G1. We

Table 1. Equations Governing the Single-Time-Constant Modea

(a) Differential equation:

[d + G] V(t) = P(t).

(b) Solution for each time period in listing below:

V(t)

IV1 exp(-G(t - t)j

IV4 exp(-G(t - t4)J1VI exp[-G(t - t)]

+ Va1 - exp[-G(t - t2)J}/(c) Definitions to note:

Time period P(t)

t 1 t t 2 0

tx S tSt5 0

t 2 St <t. PIK--P'

P'M Va G

Va V4 exp[-G(t. - t4 )] - VI exp[-G(tx - tl)]IIX1

X1-1 - exp[-G(tx - t2)](d) Corrected temperature-rise definition (integrate left side of above

differential equation):

AT 1(t) = V(t) - V(t 2) + G 4' V(t)dT.

(e) If measured power is constant, consistency requires

ATi(t) = P'(t. - t2 ) where t > t.(f) Note

ATIt4) = AXT(t.,) if single time-constant model is truefor all time.

a The drift and constant background term have been removed tosimplify the discussion.

Table II. Two-Time-Constant Modela

(a) Differential equation deduced from diffusion theory retainingonly first two heat modes; A is a coupling term:

(+ G I [+ (I> ] V(t) =P(t) + A d P(t).(dt ) [ (GI dt] I d

(b) Solution for each time period in listing below:

V(t) Time period P(t)

IV, exp[-G(t - tl)]}

V4 exp[-G(t - t4)]1+ V5 exp[-GI(t - t3 )J

V, exp[-G(t - t)]

+ Va 1 - exp(-G(t - t2)It+ VbI1 - exp[-G(t -t2)

tI t<t 2 0

t t t 0

t2 t tx P'.

(c) Definitions to note:

Y- Vb/Va,

P'GVa(I + Y),

R P'(t. -t2VAT244),

Va- see one time-constant model,

Vb V5 exp[-Gi(t. - t3)]/1 - exp[-GI(t. - t2)]1.

(d) Corrected temperature-rise definition (here t < t t4) (integrateleft side of above differential equation):

AT2(t) = ATi(t) + G[V(t) - V(t2)] + [V(t) - V2)]11GI.

(e) If applied power is constant, we find

R = (1 + Y)/(1 + Y'), where Y' Y + Y.

If two time model is correct Y = 0 and R = 1.(f) Can correct Y to Y' by assuming source of inconsistency, R 1,

is due to neglected terms like Vb. Thus we have

Y' = (1 + Y)/R - 1.

a The drift and constant background have been removed to simplifythe discussion.

use the results of the two rating periods from the one-time-constant model analysis to fix V1, V4 , and G; andwe extrapolate that time dependence into the relaxationperiod and use an appropriate nonlinear least-squaresfit to fix V5 and G1. Table III shows that the fasterdecay rate G is about 100 times greater than the slowerdecay rate G for the C4 series calorimeters. Thequantity that measures the inequivalence between laserand electrical energy is indicated by the Y in part (c).This parameter Y is correct only if the applied poweris a strict constant and if the real system drives only twotime constants. It is reasonable to treat the appliedpower as constant for our measurements and to assume

Table Ill. Summary of the Mean Inverse-Time Constants and Estimates of the Standard Deviations of the Meanfor two C4 Instruments

First-inverse- Second-inverse-time constant time constant

G*100 S(G)*100 G1 S(G1 ) NumberofInstrument Source 1/sec 1/sec 1/sec 1/sec measurements

C41 Laser 0.125 0.001 0.15 0.02 7C41 Electrical 0.128 0.002 0.14 0.01 11C46 Laser 0.097 0.003 0.08 0.01 8C46 Electrical 0.095 0.005 0.08 0.01 7

2318 APPLIED OPTICS / Vol. 16, No. 8 / August 1977

that all errors in the two-time-constant model can beincluded in Y' by using concepts of part (e) and part (f).We then use Y' to document accurately the inequiv-alence between various heat sources.

To show how Y is modified to Y' by the addition offaster time-decay processes is tedious. Essentially, theYc term becomes a sum of comparable terms just likethe Y term in part (c), but with many different valuesof Vb and G1 type terms for each additional heat mode.Our experience with the C4 series calorimeter shows thenext faster time-decay-rate modes to be at least fourtimes faster than the second decay-rate mode and thatthe latter has comparable strength to the second heatmode. In brief, we account operationally for the effectof the higher order heat modes by simply changing thevalue of Y to Y'. Thus we now use Y' to characterizethe inequivalence between laser energy measurementsand the electrical calibrations. This correction of Y toY' is not possible if the shape of the energy pulse is un-known and is, therefore, only useful to estimate themagnitude of the inequivalence.

Note that the corrected temperature rise AT2 (t) forthe two-time-constant model in part (d) becomes equalto the corrected temperature-rise AT 1 (t) for the onetime model when the term proportional to V5 which isdecaying exponentially with a G 1 inverse time constantbecomes insignificant. Also note that for the con-stant-applied-power case AT, = VaG(tx t2 ), ATb =YAT 0, and that AT, is the same regardless of whichmodel is used to reduce the data generated during acalibration or measurement.

C. Results for the C Series Calorimeters Labeled C41and C46

Table IV shows the mean of the computed values forY' and its surprisingly small standard deviation S(Y').By way of comparison we also give the mean of Y andits standard deviation which is much larger than thestandard deviation of Y'. We next have the average forthe quality-of-fit ratio during the relaxation period.Each ratio is computed by dividing the standard de-viation of fit during an extended relaxation period(namely, t < t S t) by the corresponding standarddeviation of the rating periods. This large ratio and anexamination of the time-dependent structure of theresiduals show that the two-time model fails to accountfor all nonstatistical behavior during that period. Forconvenience we include the standard deviation S(relax)for the relaxation period. We conclude from these re-sults that the fit during the extended relaxation period

Table IV. Inequivalence Data Summary

Qual-In- ity of S(re-

stru- Y' S(Y') Y S(Y) fit lax)ment % % % % ratio ,uV N Source

C41 -1.45% 0.02% -1.82% 0.39% 19 17 7 LaserC41 -1.30 0.03 -1.04 0.25 43 32 11 ElectricalC46 -1.06 0.03 -1.05 0.15 39 33 8 LaserC46 -1.07 0.03 -1.03 0.23 95 70 7 Electrical

Table V. Electrical-Calibration Data Summary

K S(K)Instrument (J/mV) (J/mV) N % Range

C41 (front heater) 4.32171 0.0010 2 -C41 (rear heater) 4.2529 0.0013 11 0.16C46 4.8888 0.0017 7 0.2

Table VI. Summary of a Comparison Between Front Heater and RearHeater Data at the Same Energy for C41 Inequivalence.

S(Y') S(relax)Position Y' (%) Relaxation (,uV) N K/(4.2529)

Front -1.52% 0.01 131 3 2 1.0162Rear -1.31 0.03 71 16 2 1.0

is quite variable and that significant higher order heatmodes are present. The number N of measurementsare indicated for each type of source and instrument.We note that applied energy and power values for eachsource were quite variable, and there is no indicationthat the Y' has any dependence on these values.

We see that the difference between the Y' for eachsource into the C41 instrument is 0.15% and that thedifference for the C46 instrument is not significant.Briefly, for the C41 calorimeter, the corrected temper-ature-rise response to a given laser energy and powerpulse is 0.15% less than the corresponding corrected riseto an electrical calibration at the same energy andpower. Because the standard deviation S(Y') is nomore than 0.03%, we could correct for the inequivalencein the C41 calorimeter if the laser beam shape used herewas unchanged during various calibrations. Becausethe laser beam changes shape we do not correct for theinequivalence, rather we just note its potential size inthe error budget of the NBS laser Standard 1.

To demonstrate that the Y' correction cannot nec-essarily account for all differences in laser and electricalenergies into the C4 series calorimeters, we show someelectrical calibration measurements in Tables V and VI.Figure 1 shows the geometric position for the twoheaters and clearly indicates that the front heater cansignificantly induce a much larger direct radiation heatloss with the consequence that we must increase theelectrical calibration constant to account for situation.Since the Y' for the front heater is only 0.21% largerthan for the back heater, we find that there remains a1.4% loss to be attributed to this loss. Because analysisof heat radiation needs data we do not have, it is notpossible to account for how much is heat-radiation loss.We can only note that the solid angle for direct radiationfrom the region due to the front heater is at least fourto five times that due to the back heater where mostlaser energy is absorbed. We have inferred in our useof the C4 series calorimeters as laser standards that theback heater's direct-radiation loss and the absorbedloser-energy direct-radiation loss are equal because theyhave the same solid angle for direct radiation. Thelaser-absorption measurements1 2 suggest that the in-ference is adequate because the remaining measurement

August 1977 / Vol. 16, No. 8 / APPLIED OPTICS 2319

uncertainties in the window transmission coefficientand in the electrical-calibration constant swamp anyinformation on the possible differences in radiation lossbetween laser measurements and electrical calibrations.We operationally avoid significant change in the coef-ficients in Eq. (A7) by being sure the laser radiationalways strikes the back of the can.

III. Four-Point MethodWhile we were evaluating the inequivalence, the

computer program computed the corrected temperaturerise for the four-point method using the definitions al-ready discussed in Sec. II.A. Instead of using thenonlinear least-squares fit to fix V4, V1, and G, we as-sumed a constant background term and used four datapoints to compute algebraically these four quantities.The data points used are the beginning and end valuesof each rating period (see Fig. 2). To confirm that thereis no significant error in using only the four-pointmethod, we compute a quality of fit parameter using theresiduals of the data measured during the rating periodsrelative to the appropriate four-point solution shownin Table I, part (b) for these periods. (Remember thisdiscussion has no background term.) The quality of fitparameter is three times "the root-mean-squared de-viation of these residuals" divided by the computedcorrected temperature rise AT1(t4). If all measurementwhere the quality of fit parameter is greater than 0.05%is deleted, we find that the four-point method impliesan error in its corrected temperature rise relative to thetwo-time-constant model's value not greater than 0.1%for thirty-three out of thirty-five times, (see Table VII).Since our practical precision of the corrected temper-ature rise in most laser measurements is around 0.1%for the energies at which we use the C4 series to calibratepower meters, we conclude there is no need for thenonlinear least-squares fit technique. To use thefour-point method, it is absolutely necessary to performthe quality of fit test and to confirm by other means thatit is unlikely for the background term to have a signifi-cant drift. We note that this analysis assumes thatduring the power-on period the only applied power isfrom the laser beam or from the electrical calibrationsource and that the fluctuations in the backgroundpower during the power-on and relaxation periods arecomparable to those implied by the quality of fit duringthe rating periods. These latter conditions are alsoassumed for the nonlinear least-squares fit model cur-rently under use.2 4,7

IV. Inferred ResultsBecause past guesses of the inequivalence error are

0.5% and we now have quantitative estimates of thiserror as less than 0.2%, we conclude that it is possibleto reduce the net delivered laser-beam uncertainty fromthe currently accepted worst-case measurement of1.4-1.1%.

Table VII. Four-Point-Method Summarya

% Error relative to AT 2(t4) Number of measurements b

0.0 160.1 170.2 2

a In this table all data are mixed together regardless of source orinstrument. b Selection criterion of the good measurements is: Allcases with less than a 0.05% quality of fit parameter.

As a consequence of the results from the four-pointmethod, it is possible to reduce significantly the amountof paper tape and connect time to a large computer byusing a small-memory microprocessor to do real-timereduction of the temperature data.

I thank Martin Reilly, Aaron Sanders, Stephen Jar-vis, Anne Rumfelt, and Wilbur Anson for useful dis-cussions on various parts of this paper. The data forthis analysis were obtained by Joe Skudler.

Appendix: Derivation from the Diffusion EquationFor the convenience of the reader, I show some detail

on the derivation of selected notions defined in the mainbody of this paper such as the one-time-constant model,the corrected temperature rise, the inequivalence ratio,and the correction to that ratio.

Assume that the 1-D diffusion equation with a sourcedescribes the essential physics for linear calorimtry.(We completely neglect nonlinear radiative heat losseffects.) The equation is

a 1 a /a Sat c ax ax C

where q(x,t)C(x)k (x)

(Al)

is the temperature distribution in K,is the heat capacity in J/(Km3),is the heat conductivity in W/(Km2),and

S(x,t) is the heat source density in W/(m3 ).To get the equation form used in the main text, we as-sume there exists a complete set of orthonormal modesdefined by Eq. (A2) and by appropriate boundaryconditions which are purposely undefined here.Thus

G01ix) = a (k 01)'

where GI is the inverse-time constant for the th modeand where

f dx0j(x)0,(x) = a61 .

Next, we assume that the heat source and the tem-perature distribution can be described as a linear su-perposition of these modes, namely,

0(x,t) = E T(t)0j(x) and S(x,t) = E P(t)0i(x).I I

Finally we assume the temperature detectors whichgenerate the voltage signal are represented by

V(t) = E d f 0(x,t)0j(x)dx, (A3)

2320 APPLIED OPTICS / Vol. 16, No. 8 / August 1977

(A2)

where dl are the coefficients for the physical placementof the detectors in the calorimeter.

Given the above definitions, we find that the voltagesignal is described as

V(t) = E dTI(t),

where TI(t) is given by the differential equation

[d/(dt)]TI = GIT1 + P1.

(A4)

(A5)

If we solve this equation, we can write the formal solu-tion for the voltage signal as

V(t) = E di t P1(t) exp[-G(t - t)]dt.I fi

AT 1(t) = J (d + Go) V(t)dt, (All)

where t > t. Performing the integration gives

ATI(t)

= B hild Go (t. - t2 ) t

G - 1) {exp[GI(t. - t)]- exp[G(t 2 -t)]

(A12)

If you wait until t = t4, when the higher modes are allgone, we obtain

(A6)

To complete the assumptions for our derivation of themodels in the text, we further assume that both theelectrical and laser sources place their energy in a waywithin the calorimeter so that

PI(t) = h1P(t) (A7)

is true. This form assumes that the spacial distributionof the energy from either the laser or electrical sourceis sufficiently constant that we can ignore operationallythe possible changes in coefficients or the possible factthat Eq. (A7) is not strictly true. The structure of theabsorption can, as shown in Fig. 1, make this a reason-able approximation. The case where the electricalheaters were radically moved and where the calibrationcoefficient changed significantly shows that Eq. (A7)is an approximation. With the above discussion ourbasic waveform for the voltage signal is

V(t)= dihi f P(t) Exp[-Gj(t - t)]dt. (A8)

To make our demonstration of the relationships inthe main text as simple as possible, we now assume thatthe pulse shape of the source is an initial condition pulseplus a rectangular power pulse, namely,

P(t) = Ab(t - to) + B[6(t - t2 ) - (t - t,

AT 1(t4 ) = B(t. - t2)Go E d - (A13)

We can now define for this special pulse shape thecomponents to the corrected temperature rise as

ATa - Bhodo(t. - t2 ), ATb - Bhldl(t. - t2 )GO/G,

and have the remaining modes contribution to thecorrected temperature rise grouped as

ATc --Go E- (t. - t2 ).i>1 GI

The ratio ATb/ATa = hjd1 Go/(hodoGj). If you nowidentify the Va and Vb defined in Tables I and II of themain text, you will find that Va = Bhodo/G and that Vb= B1h lg ld1/G1, where G -- Go. The Y is given as Y =h1d1Go/(hodoGj) = Vb/Va = ATb/ATa as we assertedin the main text.

To understand the Y' construction in Table II(e), wenote that Yc = AT/ATa. The rest of the operations inthis table should be obvious. I reemphasize that thisY' correction can only be done if you know the shape ofthe energy pulse of which the rectangular pulse is thesimplest.

(A9)

where to << t2, so that only the lowest I = 0 mode ispresent by time t = t1 and where the applied electricalor laser source is given by the various values of B.Under these assumptions, the time behavior for thevoltage signal V(t), is

V(t) = Adoho exp[-Go(t - to)] for t1 S t S t2

= Adoho exp[-Go(t - to)]

+ B E G dI {1 - exp[G14t2 - t)]} for t2 S t t,

= Adoho exp[-Go(t - to)]

+ B E -. l exp [GI(t 3 - t)] - exp[G(t 2 - 0)]

for

tx, t. (A10)

If we note the definition of corrected temperature risefor the one-time-constant model, we have

References1. E. D. West, W. E. Case, A. L. Rasmussen, and L. B. Schmidt, J. Res.

Nat. Bur. Stand. Sect. A: 76, 13 (Jan.-Feb. 1972).2. W. E. Case, "The C4 series system-equipment, computer pro-

grams, and measurement procedures" (To be NBS Tech. Note).3. E. D. West and K. L. Churney, Appl. Phys. 41, 2705 (May 1970).4. E. D. West and W. E. Case, IEEE Trans. Instrum. Meas. IM-23,

422 (December 1974).5. E. G. Johnson, W. E. Case, and C. Selby, "The 8/31/75 statement

of uncertainty for 'C series' calorimeter intercomparison," NBSInternal Report.

6. S. R. Gunn, "Calorimetric measurements of laser energy and power1975 supplement," Lawrence Livermore Lab Report UCRL-51873(1975). This report is an update on an extensive review of theliterature.

7. E. D. West, Nat. Bur. Stand. (U.S.) Tech. Note 396, (February1971).

August 1977 / Vol. 16, No. 8 / APPLIED OPTICS 2321