calibration of incandescent lamps for spectral irradiance by means of absolute radiometers

Calibration of incandescent lamps for spectral irradiance bymeans of absolute radiometers

L. P. Boivin

A method for calibrating incandescent lamps for spectral irradiance by means of absolute radiometers is de-scribed in which a secondary radiometer is calibrated spectrally against absolute radiometers and then usedin conjunction with a series of filters to calibrate the lamps. Considering both narrowband and widebandfilters, an extensive mathematical error analysis is performed. The use of narrowband filters (20-25-nmhalfwidth) is found to be advantageous because very little information is required on the spectral distribu-tion of the lamp being measured. The most serious source of error is a wavelength shift in the measuredspectral transmittances of the filters, especially at shorter wavelengths; for example, at 400 nm, a wavelengthshift error of 1 nm can cause an error approaching 3%. It is estimated that the accuracy of spectral irra-diance measurements made using the method described here will vary between +1 and +0.5% from -350to 800 nm. Measurements on 500-W quartz-bromine spectral irradiance standards are described. Withsuch lamps, only four or five narrowband filters are required to cover the spectral range from the near UVto the near IR. The measured and calibration values agreed to - +0.5% on average with a maximum differ-ence of -1%.

1. IntroductionSince about 1975, the working standards of spectral

irradiance at the National Research Council of Canadahave been a group of 500-W quartz-bromine incandes-cent lamps. These lamps were originally calibratedagainst a spectral radiance standard traceable to theNBS. The spectral irradiance scale was later read-justed after an international intercomparison.1 Be-cause of aging effects in the lamps, it is desirable tocheck their calibration periodically. This paper de-scribes a simple method to calibrate incandescent lampsfor spectral irradiance using absolute radiometers as thebasis of the calibration. This paper also analyzes indetail various sources of error and gives the results ofsome measurements done to verify the method.

A. Principle of the MethodDenote the spectral irradiance to be measured as E(X)

W/cm2 nm-1 in the test plane. The radiation is fil-tered by means of a narrowband interference filterhaving a transmittance T(X) peaking at X = Xm, and the

radiation transmitted by the filter is measured in thetest plane by a detector having an absolute responsivityS(X) V/W* cm-2 ; the detector signal will be given by

V = X T(X)S(X)E(X)dX,-1i

(1)

where (X1 ,X2) is the total bandwidth of the filter. Theabove assumes that the thickness of the filter is negli-gible. In practice a small correction must be appliedto take into account the finite thickness of the filter. Inthe case of an incandescent lamp, E(X) will be close toa Planckian distribution. We assume that in thespectral range (X1,X2) we can write'

E(X) - EI() =E(Xm)X5n exp(C2/XmT)5 exp(c2 /XT)

where c2 = 1.4380 cm * K and T is the color temperatureof the lamp. Substituting the above in Eq. (1), we ob-tain

E(Xm) =V

(2)

X exp(C2/X,,T) X2 S(X)T(X)dXM f ~J)i X exp(C2/XT)

The author is with National Research Council of Canada, Divisionof Physics, Ottawa K1A OR6.

Received 22 April 1980.0003-6935/80/162771-10$00.50/0.

Thus by means of a series of interference filters, whosespectral transmittances are known, and a detector,whose absolute spectral responsivity is known, we canin principle determine the spectral irradiance of a testlamp at a number of wavelengths. At intermediatewavelengths, the spectral irradiances can be obtainedby an interpolation technique, which will be describedlater.

15 August 1980 / Vol. 19, No. 16 / APPLIED OPTICS 2771

LiINTERFERENCE

FILTER

FILTERASSEMBLY

RAD I OMETERAPERTURE

INFRASILDIFFUSER

CAVITYDIFFUSER

N Z L-1 T::

_J .LDIODE

RADIOMETER

SI LICONJDIODEFig. 1. Schematic diagram of radiometer head.

The following questions immediately arise:(1) How closely must the spectral distribution of the

test radiation match the assumed quasi-Planckiandistribution?

(2) How accurately must color temperature T of thelamp be known?

(3) What is the effect of a wavelength shift error inthe measurement of the spectral transmittances of thefilters? (The effect of an error in the magnitude of thespectral transmittance is fairly obvious and conse-quently is not considered here.)

(4) Could one use relatively wideband (e.g., nonin-terference type) filters instead of narrowband ones?This would permit the use of fewer filters to span thedesired spectral range and consequently necessitatefewer measurements. Such filters are also easier tomeasure and yield a greater signal.

In this paper we will answer these questions by per-forming an error analysis based on the use of mathe-matical models that duplicate realistically spectralresponsivities, spectral transmittances, and spectraldistributions of detectors, filters, and sources used inpractice.

The obvious choice for a detector to be used in theabove appears to be an absolute radiometer. The latterhas the advantages of having a nearly flat spectral re-sponsivity and of measuring irradiance directly andaccurately. However, these radiometers have a rela-tively low responsivity and are somewhat difficult touse. We found it more practical to calibrate spectrallya secondary radiometer against an absolute radiometerand use it for the spectral irradiance measurements.

The type of secondary radiometer we chose incorporatesa silicon diode. The design and method of calibrationof this diode radiometer are described in the next sec-tion.

II. Calibration of the Diode RadiometersFigure 1 gives a schematic diagram of the radiometer

head. The silicon diode is an EG&G UV 444 diode.The diffuser is a sandwich of three infrasil quartz disksground on both surfaces. The outer disk is ground with9-,um grit, whereas the inner ones are ground with27-Aum grit. This type of diffuser was selected to givegood transmission (especially in the UV) and uniformresponsivity (>1%) over the aperture. The inner wallsof the diffuser cavity are painted with Eastman WhiteReflectance Coating 6080. The inner walls of the filterholder are mat black. We verified that negligible in-terreflection errors are caused by the close proximityof the diffuser and the filter. This is important becausethe diode radiometer is calibrated without the filterattachment.

Figure 2 gives the circuit diagram of the radiometeramplifier. The latter has a range switch that spans fourdecades in a precise factor of 10 steps. This is madepossible by using high precision resistors (±0.005%) andstable low-leakage capacitors in the feedback circuit.This precise range switch, combined with the inherenthigh linearity of the diode,2 gives the diode radiometeran important property: it can be calibrated with arelatively large signal and then used throughout itsdynamic range without loss of accuracy.

Figure 3 shows the relative spectral responsivity ofa typical diode radiometer. It will be seen that in the975-800-nm spectral range the curve is nearly linear.This behavior is ideally suited to the calibration tech-nique, which we describe next.

The diode radiometer is calibrated against an abso-lute radiometer at nine discrete wavelengths obtainedfrom a krypton laser: 351,406,476,531,568,647,676,753, and 799 nm. With such a laser, a sufficiently in-tense beam can be obtained at each of the nine wave-lengths to provide a large signal for the absolute radi-ometer: the latter is an accurate but relatively insen-sitive instrument.3 In view of the quasi-linear spectralresponsivity of the diode radiometer in the spectralrange of interest (Fig. 3), a calibration at the above ninewavelengths is sufficient to be able to obtain accuratelya complete spectral calibration for the 350-800-nm in-terval. The usual procedure is to use least squares fit-ting techniques to fit a low-order polynomial throughthe nine calibration points.

Figure 4 gives a schematic diagram of the experi-mental setup used in the calibration. The laser beamgoes through a spatial filter and mirror collimator, re-sulting in a clean collimated Gaussian beam having an1/e 2 diam of -75 mm. This beam is then directed viatwo plane mirrors to a hermetically sealed enclosurecontaining the absolute radiometer and the diode ra-diometers. The radiometers are mounted on a motor-driven rotary table, which allows each radiometer to bebrought in turn into the beam. The radiometer aper-

2772 APPLIED OPTICS / Vol. 19, No. 16 / 15 August 1980

103n

BNC

Light

UV -

Fig. 2. Circuit diagram of

1.0

.8

.6

SCM)

.4

.2

X, nm

Fig. 3. Relative spectral responsivity of a typical dioderadiometer.

tures are placed in the same transverse position in thebeam to an accuracy of ±0.5 mm. The plane of eachaperture is normal to the beam to within ±0.50. Theradiometers can be moved back and forth radially on thetable to make all the aperture planes coincide. Thisadjustment is good to ±0.3 mm. The infrasil windowof the hermetically sealed enclosure is tilted at 200 toavoid interreflection errors and has a 0.60 wedge to

T -t7 O/P

C+15V -15V

radiometer amplifier.

avoid intensity variations in the transmitted beam dueto interference effects.

The absolute radiometer aperture diameters areslightly smaller than those of the diode radiometers.This necessitates a correction that must take into ac-count the Gaussian profile of the incoming beam. Thelatter has a relative intensity profile given by I =exp(-'yr2) in the radiometer aperture plane. It can beeasily shown that the flux received by the diode radi-ometer PD is related to PABS received by the absoluteradiometer by

[1 - exp(-2yR ]= AS[I - exp(-yRABS)I (3)

where RD and RABS are the accurately known apertureradii of the diode and absolute radiometers, respec-tively. In the limit -y - 0, i.e., of a uniform radiationfield,

2PD = PABS2 (4)

RABS

For the beams used in the calibration, y 0.0015, andthe correction factor obtained from Eq. (3) differs fromthat given by Eq. (4) by only -0.1%. Thus -y has to beknown with only modest accuracy to apply this correc-tion.

The correction given by Eq. (3) was derived assumingthat the radiometers have uniform responsivities andthat both are accurately centered on the Gaussianprofile of the beam. The first assumption is certainlyvalid for the type of radiometers we are using, and wewill examine next the errors introduced by decentering


KRYPTON LASER-SL11

Fig. 4. Schematic diagram of the experimental setup used for cali-brating the diode radiometers against the absolute radiometers.

effects. Two questions arise here: (1) What is the errorintroduced by a relative decentering of the radiometerapertures? (2) What is the error, if any, introduced ifthe apertures are centered with respect to each other butdecentered with respect to the Gaussian beam? Tocalculate both of these effects, we must have an ex-pression for the ratio of the responsivity of a radiometerwhen it is decentered by a distance X with respect to aGaussian beam to that when it is perfectly centered onthe beam. Calling this ratio K, it can be shown that

K exp(-.yX 2 ) (1+ y2X2R2) (5)

where the incoming Gaussian beam has a relative in-tensity profile given by exp(-yr2 ), and the radiometeraperture has a radius R. The above expression againassumes that the radiometer has uniform surface re-sponsivity and that R is small compared to the beamwaist radius. These assumptions are perfectly validhere. To answer the first question, a relative decen-tering of the two radiometer apertures by ±0.5 mm, fora nominal radiometer aperture radius of 3 mm and a yof 0.0015, causes an error of only 0.04%. In the secondcase, where both radiometer apertures are perfectlycentered with respect to each other but are decentered

with respect to the Gaussian beam, we put R = 3.0 mmfor the diode radiometer and R = 2.8 mm for the abso-lute radiometer and find that for a decentering of asmuch as 15 mm, the ratio K for the two radiometers willdiffer by only 0.03%, so that the correction given by Eq.(3) will still be valid. Thus, centering of the Gaussianbeam on the radiometer apertures is not critical, pro-vided that these apertures are well centered with respectto each other. The centering tolerance of ±0.5 mmprovided by the rotary table is completely satisfactory,and the correction for nonequal aperture diameters canbe accurately made with only approximate knowledgeof the beam waist radius.

The accuracy of the absolute radiometers is ±0.2%.Using the above calibration technique, the accuracy ofthe diode radiometer calibration is ±0.5% or better atthe nine calibration wavelengths. Using curve fittingtechniques, a low-order polynomial can be fittedthrough the nine calibration points so that the absolutespectral responsivity of the radiometer can be obtainedat any wavelength between 400 and 800 nm with anaccuracy of ±0.5%. From 350 to 400 nm, the accuracyof the interpolation is +1%.

Ill. Error Analysis Using Mathematical ModelsHaving shown how the absolute spectral responsivity

of the diode radiometers is determined, we now returnto the questions in Sec. I regarding the accuracy of theproposed method for measuring spectral irradiance.We will concern ourselves with errors arising from theassumptions made rather than from such obvious causesas calibration errors in the radiometers or inaccuratetransmittance measurements on the filters used. Inparticular, we will study the following:

(1) We assumed that the test source is a quasi-Planckian radiator at the temperature of the nominalcolor temperature of the lamp. We will consider thetest radiation to be either a tungsten radiator or ablackbody at a fixed temperature of 3150 and study theerror introduced when the color temperature estimaterequired in Eq. (2) is in error by various amounts.

(2) The error introduced by a wavelength shift errorin the spectral transmittance curve of the filter.

(3) The error, if any, introduced by an asymmetry inthe filter transmittance curve resulting in an ambiguityin the choice of Xm in Eq. (2).

The above analysis will be done for both narrowbandand wideband filters. If we denote by E'(Xm) the cal-culated irradiance at X = Xm [i.e., the value given by Eq.(2)], the ratio of calculated to true irradiance is givenby

*'X2 S(X)T,(X)E(X)dXE'(Xm)

(6)E(Xm) E(Xm)X5 exp(c 2 /XT) JX2 S(X)Tm(X)dX

A 5 exp(c2 /XT)

where E(X) is the true spectral irradiance, T (X) andTm (X) are the true and measured spectral transmit-tances of the filter, respectively, and S(X) is the relativespectral responsivity of the radiometer. Here we as-


sume in all cases that S(X) is known without error.Finally, T is the assumed color temperature of thelamp.

A. Mathematical Models UsedFigure 5 shows the model used for S(X) (solid curve)

compared to the actual spectral responsivity. Thecurve for S(X) is cubic, which fits closely the spectral

S(X)

600

X, nm800

Fig. 5. Mathematical model (solid line) for the spectral responsivityof a diode radiometer compared to the actual spectral responsivity

of a typical radiometer (dotted line).

BLACKBOoY, 3150 KTUNGSTEN, 3150 K --- 500 WATT 0.B.

10 LAMP... /

8 /

E /C

Li.J '

4

2

0400 500 600 700 B00

X, nm

Fig. 6. Comparison of the spectral distribution of a 500-W quartz-bromine lamp (dotted line) having a color temperature of 3150 K withthe spectral distributions of a blackbody radiator at 3150 K (solid line)

and a tungsten radiator at 3150 K (dashed line).

responsivity of an earlier type of radiometer. However,for modeling purposes, the curve shown is completelysatisfactory.

For the actual spectral distribution of the source, weused the expression

E(A) E(X)cl (7)

X) 6 X[exp(C2IXT.) - 1] 7

where c and c2 are the familiar blackbody radiationconstants, and (X) is the emissivity. We consideredonly radiators at T = 3150, since this is the approximatecolor temperature of the lamps we are using. For ablackbody radiator, E(X) 1. For tungsten radiators,we derived the following expression for the emissivityby extrapolation and least squares fit to the data of DeVos

4:

E3150(X) = 0.965(0.4141 + 0.4070A - 0.8656X2 + 0.4480X3). (8)

This expression will be accurate to -±0.5% on averagein the 350-800-nm spectral range. For illustrationpurposes, Fig. 6 gives a plot of the spectral distributionof a typical lamp we wish to calibrate (dotted line)compared with the spectral distributions of a blackbody(solid line) and a tungsten radiator (dashed line) at thesame temperature (3150 K).

To represent mathematically the type of filters wehave been using, the following expression was used:

T(X) = exp[-yo(X - Xo)2 ] + b exp[-yi(X - \1)2 ]. (9)

This type of model was used for both narrowband andwideband filters. The shapes of the model filters areshown in Fig. 7 together with the actual shapes of someof the filters we have been using.

B. Results of Calculations

1. Narrowband FiltersFor the measurements, we have been using interfer-

ence filters with a 20-25-nm bandwidth (50% points).This is a compromise bandwidth: a very narrow filter(<10-nm bandwidth) is difficult to measure accuratelyand gives a small signal; on the other hand, a wide filter(>75-nm bandwidth) requires too much prior knowl-edge of the spectral distribution of the lamp beingmeasured (as we show in the next section). The threefilter shapes given in Fig. 7 are very typical of availableinterference filters having a 20-25-nm bandwidth. Themodel curves we used represent these types of filtersrealistically.

Figures 8 and 9 show the variation of E'/E [Eq. (6)]for the case where the test lamp is a tungsten radiatorat 3150 and where there is no wavelength shift error inthe spectral transmittance data [i.e., Tt Tm in Eq. (6)].Figure 8 gives the results corresponding to symmetricalfilter 1 (Fig. 7), and Fig. 9 gives the results correspond-ing to asymmetrical filter 3. The results for filter 2 arealmost identical to those for filter 3 and so are omitted.In Fig. 9, curves a and b denote the two possible sym-metries for filter 3: b denotes the symmetry shown inFig. 7, whereas a denotes a filter of the same shape butreflected in a vertical axis.


1.001

1.000

.999

.998

Fig. 7. Mathematical models for the spectral transmittance curvesof three types of interference filters. For comparison purposes, theactual transmittance curves of typical filters are also shown (left).

Figures 8(A) and 9(A) show the spectral variation ofE'/E when the correct value of T is used in Eq. (6).Even though a blackbody radiator is assumed in thecalculation of E', and the source is actually a tungstenradiator, very little error is introduced by this as-sumption: 0.05% max for the symmetric filter and 0.1%max for the asymmetric filters. If the test lamp is ablackbody at 3150, E'IE =_ 1 essentially. Actual in-candescent lamps will have a spectral distribution in-termediate between the two cases considered here (i.e.,blackbody and tungsten). Thus results calculated fora tungsten source represent a pessimistic error esti-mate.

Figures 8(B) and 9(B) show the effect of an improperestimate of the color temperature T to be used in Eq.(2). In Fig. 8(B), results are given for 400 and 550 nm.At 700 nm, the variation becomes negligible (<0.1%from 2500 to 3800 K). In Fig. 9(B), for each symmetry,we show the variation at only one wavelength, that forwhich the variation is maximum. The results for ablackbody radiator are virtually identical to the onesshown for a tungsten radiator. It can be seen that themagnitude of the error is affected by the shape andsymmetry of the filter used. However, in all cases, anerror in T of +100 K results in an error of at most 0.5%in the calculated irradiance. Thus the nominal colortemperature of the test lamp can be used in Eq. (2) withnegligible error.

1.03

1.02

1.01

1.00

.99

.98

.97

2400 2500 3200 3600 4000

TEMPERATURE, K

Fig. 8. Ratio of the calculated irradiance E' to the true irradianceE, when using a narrowband filter of type 1, and where the source isa tungsten radiator at 3150 K: (A) spectral variation for T = 3150

[Eq. (6)]; (B) temperature variation at two wavelengths.

2400 2800 3200

TEMPERATURE, K

Fig. 9. Ratio of calculated irradiance E' to true irradiance E whenusing a narrowband filter of type 3 and where the source is a tungstenradiator at 3150 K. Curves b correspond to filters having the sym-metry shown in Fig. 7, whereas curves a correspond to filters havinga reversed symmetry (i.e., reflected in a vertical axis): (A) spectralvariation for T = 3150 [Eq. (6)]; (B) temperature variation at the

wavelength for which the errors are greatest.

Figure 10 shows the effect on the calculated irra-diance of a wavelength shift error in the spectraltransmittance of the filter; that is, the measured spectraltransmittance curve is identical to the actual one butis shifted up or down in wavelength. This type of erroris very common (unfortunately) due to the nature ofspectrophotometers and is also inherent in interferencefilters, where changes in temperature, angle of inci-dence, or angular width of transmitted beams result incorresponding wavelength shifts in the spectral trans-mittance curves. We give the results for filter 3, but the


zI-

Cl)z

I-

WAVELENGTH

-® b2

a, . . . .

400 500 600 700

X, nm

400 nm

a

Lb 550 nm -

3600 4000. .

- .l.00-E WO O7 nm

.96 50nm

.92-

400 nm.88-

-6 -4 -2 0 2 4 6

WAVELENGTH SHIFT ERROR, nm

Fig.10. Ratio of calculated irradiance to true irradiance as a functionof the wavelength shift error in the spectral transmittance of the filter.The curves were calculated for a tungsten radiator at 3150 K and fora narrowband filter of type 3 having the symmetry shown in Fig. 7.

1.002 b

1.001 a

E'lOO-0-E.999

.998

.997 _

544 546 548 550 552 554 556Xm, nm

Fig. 11. Ratio of calculated irradiance to true irradiance as a functionof Xm in Eq. (6). Results are given for a tungsten radiator at 3150 K,for a narrowband filter of type 3 (both symmetries a and b), whose

peak transmission is at a 550-nm wavelength.

results are almost identical for filter 1 or 2. The resultsare the same for symmetries a and b for filters 1 and 2.Also calculations were made for a tungsten radiator, but,again, results for a blackbody radiator are very similar.It can be seen that this source of error is serious, espe-cially in the blue end of the spectrum, where a shift of±0.35 nm causes a ±1% error in the calculated irra-

diance. A wavelength accuracy of ±0.3 nm is typicalfor a commercial spectrophotometer. On the otherhand, if an interference filter has been measured witha normal incidence parallel beam, then in order to causea wavelength shift of 0.3 nm, the interference filter hasto be tilted by ±30, or a convergent beam through thefilter has to have a half-angular width of 4.5°, or thetemperature of the filter must change by ± 100C.Preventing wavelength shifts due to angular effects isreasonably easy in practice. However, preventingwavelength shifts due to thermal effects may be moredifficult when making measurements on high wattageincandescent lamps.

In Eq. (2), we chose to calculate the irradiance at

wavelength Am corresponding to the peak of the filtertransmission: what happens if one chooses a value Amdifferent from that corresponding to the peak? Thiscould be accidental, for example, resulting from anambiguity in peak X due td a flat top or asymmetry onthe filter transmittance curve. Or it could be deliberateto obtain the irradiance at a more convenient wave-length: for example, the filter peaks at 602.7 nm, butthe spectral irradiance at 600.0 nm is required (e.g., fortabulation purposes). The variation of E'(Am)/E(Am)with Am is of the form

E'(Xm) 1

E(Xm) E(X1m)X5 exp(c 2/XmT)(10)

Figure 11 illustrates this variation for filter 3 for atungsten radiator. The values shown correspond to acenter wavelength of 550 nm, which represents the worstcase in the 400-700-nm spectral range. For the case ofa blackbody, E'/E does not vary by more than 0.005%for a variation in Am of ±5 nm about peak A. Thus thevariation shown in Fig. 11 is proportional to 1/E(X),where E(X) is the emissivity given by Eq. (8). The bestchoice of Am is not necessarily the one corresponding tothe peak of the filter transmission: a shift away fromthis peak X may result in a smaller or greater error de-pending on the exact filter shape. However, we can saythat an offset of ±2 nm in Am from that correspondingto the peak in the filter transmittance will result in anerror of <0.2%. In the case of a blackbody radiator, theerror introduced by such an offset is negligible.

2. Wideband FilterWe now consider an alternative to the above ap-

proach; that is, instead of using one narrowband inter-ference filter for each test wavelength, we wish to usea small number of relatively wide (i.e., noninterference)filters to cover the spectral range of interest. For eachfilter, one determines the spectral irradiance at anumber of wavelengths by using these wavelengths asAm in Eq. (2). We consider as an example filters havingthe same shape as filter 1 but having a bandwidth (50%points) of 100 nm. The center wavelengths are takento be 400, 550, and 700 nm, respectively. Each filter isused to calculate the spectral irradiances in a ±80-nmwavelength range about the wavelength correspondingto the peak in the filter transmission. The advantagesof the wide filter approach are as follows: (1) fewerfilters, hence fewer measurements, are required to covera given spectral range; (2) the total transmitted flux,hence the radiometer signal, is greater; (3) the filtersneed not be of the interference type, hence are perhapsmore durable, and not so sensitive to angular orientationeffects; (4) the filters are easier to measure. In theanalysis below, we consider the same test sources asbefore and assume the same radiometer spectral re-sponsivity as before.

In the case of a blackbody test source at 3150 K andwith the proper choice of T = 3150 in Eq. (6), spectralirradiances calculated for the three filters (i.e., 400, 550,and 700 nm) have a negligible error (<0.1%); however,


I I l I I I I I I I

when the test source is a tungsten radiator at 3150, thecalculated irradiances have errors as shown in Fig. 12.As in the previous section, E'/E here is proportional to1/E(Am), but here the variation in Am is much greater.It can be seen that errors can be as large as 3%, and evenfor the center wavelength of the filter the error can beclose to 1%. Comparison to the narrowband filter ap-proach shows that the error in the case of a tungstenradiator was never >0.2% even for complex filter shapesand allowing for a +2-nm offset in Am relative to thecenter wavelength.

Going back to a blackbody test source at 3150, wenext study the effect of an improper estimate of T in thecalculation of E' [Eq. (2)]. Figure 13 gives the errorratio E'/E corresponding to various choices of T and forvarious wavelengths. Thus, even if the test source is ablackbody, an error of 100 K in the estimate of tem-perature T can cause an error of as much as 3% in thecalculated irradiance. On the other hand, in the nar-rowband filter approach, the same temperature errorof ±100 K causes an error of <0.5% for blackbody ortungsten test sources.

The errors caused by wavelength shift errors in themeasured filter transmittances are given in Fig. 14 forthe case of a blackbody source. Here the situation isqualitatively very similar to the narrowband filter case,except that the errors caused by a wavelength shift arenot so great for a wideband filter. For example, at 400nm a 1-nm wavelength shift causes an error of 2.3% witha wideband filter and 3% with a narrowband filter.

The wideband filter approach requires more preciseknowledge of the spectral distribution of the test source:one must know if the latter is closer to a tungsten radi-ator or a blackbody radiator. In the former case, oneshould modify Eq. (2) to incorporate an empirical

1.03

-60 -40 -20 0 20 40 60 80

Xm RELATIVE TO THE CENTERWAVELENGTH OF THE FILTER,nm

Fig. 12. Ratio of calculated irradiance to true irradiance as a functionof Xm in Eq. (6). Results are given for a tungsten radiator at 3150 Kand for wideband filters having peak transmittance at 400, 550, and700 nm, respectively. Abscissae give the wavelength Xm. relative to

the center wavelengths of the filters.

equation for the variation of the emissivity with wave-length and temperature. Also the operating tempera-ture of the test lamp must be known to ±20 K to ensureresults accurate to >0.5%. With the narrowband filterapproach, on the other hand, the only information re-quired of the test lamp is its nominal color temperature.Provided this value is within ±100 K of the true valueand whether the lamp at that temperature is behavinglike a blackbody radiator or a tungsten radiator, resultsshould be accurate to >0.5% (provided there are noother sources of error, of course).

1.16/00nm

1.08 -

1.04 550 nm

E .96-

.92

.88

.84-

.80-

.76 L L ,2400 2800 3200 3600

TEMPERATURE, K

Fig. 13. Ratio of calculated irradiance to true irradiance as a functionof temperature T in Eq. (6); results are given for a blackbody at 3150K and for wideband filters having peak transmittances at 400, 550,

and 700 nm, respectively.

1.04

1.02 -

1.00 _-

.98 -

.96

-2 -I 0 I 2

WAVELENGTH SHIFT ERROR, nm

Fig. 14. Ratio of calculated irradiance to true irradiance as a functionof the wavelength shift error in the spectral transmittance of the filter.The curves were calculated for a blackbody at 3150 K and for wide-band filters having peak transmittances at 400, 550, and 700 nm,

respectively.


I I I I I

00 nm

550nm

00 nm_

l l I I I-

IV. Experimental ResultsWe now describe measurements made on two test

lamps. We chose the narrowband filter approach sinceit requires the least information on the test lamp.Furthermore, for these lamps, relatively few filters arerequired to cover a wide spectral range. The 500-Wquartz-bromine lamps used have a nominal color tem-perature of 3150 K and are part of a series of spectralirradiance standards used in our laboratory. The rel-ative spectral distribution of such a lamp is shown inFig. 6 (dotted curve). Five interference filters were usedfor the measurements, the center wavelengths being400.0, 450.0, 546.6, 600.0, and 702.5 nm, respectively.The bandwidth of the filters is between 20 and 25 nm.The transmittances of the filters were measured on aCary spectrophotometer at 1-nm wavelength intervals.For the lamp measurements, the filters are mounted ina cell attached to the radiometer head (Fig. 1). Thedistance between the radiometer aperture and the lampis set at 50.0 cm. However, the presence of the filtersbetween the lamp and the detector effectively shortensthis distance by -t/3 (where t is the filter thickness),so that the measured irradiance must be corrected forthis effect. Wavelength shifts of the filter transmissioncurves due to angular effects or thermal effects are es-timated to be <0.2 nm.

The following interpolation technique was used toobtain spectral irradiances at other than the testwavelengths. Assuming that the spectral irradianceE(X) varies as E(X) = K/A 5 exp(C/X), then for two ad-jacent test wavelengths A1 and A2, with correspondingmeasured spectral irradiances E1 and E2, we have E1 =K/A5 exp(C/Ai) and E2 = K/X exp(C/X2) from which

c ln(El/E2) -5 ln(X2/Xl)1/X2 -1/X1

Using the above values of C and K, E(X) for A1 < X < A2can be calculated. How accurate is this interpolationtechnique? If the test lamp were a tungsten radiator

at a temperature of 3150 K, using test wavelengths of400,500,600, and 700 nm would result in the followinginterpolation errors (assuming no error in E1 or E2):0.1% max for 400 < X < 500; 0.7% max for 500 < X < 600;and 0.3% max for 600 < X < 700. If measurements weremade every 50 nm instead of every 100 nm, the maxi-mum interpolation error is decreased to -0.15%. If thetest source is a blackbody at 3150 K, on the other hand,the maximum interpolation error is <<0.1% even ifmeasurements are made only every 100 nm. The testlamps being measured here behave very much likeblackbody radiators at 3150 K. Therefore, the aboveinterpolation technique can be used with negligibleinterpolation error per se to obtain spectral irradiancesat other wavelengths. The interpolation technique alsoserves as a useful check on the choice of T used in Eq.(2). The constant C in Eq. (11) should very nearlyequal 1.438/T for a blackbody. The value of T so de-rived should be compatible with the value of T assumedin deriving the values of E1 and E2. For example, forthe two lamps we measured, we assumed T = 3150 K.Now, if for ( 1,El) and ( 2,E2) in Eq. (11), we take thevalues corresponding to 400 and 702.5 nm, we find thatthe value of T derived from the interpolation formulais 3130 K for both lamps, which agrees well with theassumed value.

Table I compares the spectral irradiances derivedusing the method described in this paper with the cali-bration values. For each lamp, the average differencebetween the two sets of values is -0.5%, and the maxi-mum difference is -1%.

V. ConclusionsIn this paper we described a method for calibrating

incandescent lamps for spectral irradiance by means ofabsolute radiometers; the calibration is not done di-rectly but via a secondary radiometer, which itself iscalibrated spectrally against absolute radiometers. The

Table 1. Comparison of Calculated Spectral Irradiances E'(X) to Actual Values E(X)

E(X),,uW/cm2

-nm-1 E'(X),,uW/cm2 -nm-1 A E(X),iuW/cm

2 * nm-1 E'(X),MW/cm2

-nm-1 A

X,nm Lamp 053 Calculated % Lamp 084 Calculated %

400 1.227 1.229 +0.16 1.224 1.221 -0.25420 1.660 1.671 +0.66 1.660 1.660 0440 2.171 2.185 +0.65 2.171 2.172 +0.05460 2.769 2.757 -0.44 2.766 2.740 -0.95480 3.384 3.373 -0.33 3.380 3.352 -0.83500 4.040 4.027 -0.33 4.030 4.001 -0.73520 4.708 4.706 -0.04 4.701 4.675 -0.56540 5.380 5.398 +0.34 5.379 5.362 -0.32560 6.081 6.099 +0.30 6.079 6.057 -0.36580 6.808 6.794 -0.21 6.795 6.746 -0.73600 7.490 7.472 -0.24 7.471 7.417 -0.73620- 8.056 8.114 +0.72 8.034 8.054 +0.25640 8.627 8.722 +1.10 8.611 8.658 +0.55660 9.184 9.290 +1.15 9.168 9.222 +0.59680 9.737 9.815 +0.80 9.707 9.743 +0.37700 10.241 10.294 +0.52 10.210 10.218 +0.08


secondary radiometer is then used in conjunction witha series of filters to calibrate the lamps. We performeda mathematical analysis of the errors to be expectedusing narrowband or wideband filters. For this anal-ysis, we assumed that the test source is a blackbody ortungsten radiator at 3150 K. However, the results ofthe calculations, and in particular the conclusions de-rived from them, will be valid for other source temper-atures. We found that the use of narrowband filters(e.g., -20-25-nm bandwidth at 50% points) is advan-tageous because very little information is required onthe spectral distribution of the lamp being measured;the nominal color temperature of the lamp, accurateonly to within ±100 K, is all that is required. The mostserious source of error is a wavelength shift error in themeasured spectral transmittance of the filter. Thissource of error is particularly serious in the blue part ofthe spectrum, where a wavelength shift of only 1 nm cancause an error approaching 3%. This source of error isslightly less serious when using wideband filters. Thissource of error may be the limiting factor in the accuracyobtainable with this calibration technique. There areother sources of error that we did not discuss in thispaper because their influence on the measurements canbe immediately determined: for example, errors in thecalibration of the radiometer, or in the filter transmit-tance measurement, will cause errors of the same mag-nitude in the calculated spectral irradiances. However,both of these sources of error should be 0.5% or less inpractice for the 350-800-nm spectral range. Othersources of error, such as alignment errors and distancesettings, can be made negligible. We expect that theaccuracy of spectral irradiance measurements using themethod described here should be ±1% at the blue endof the spectrum and better than 0.5% at the red end.

We have given the results of spectral irradiancemeasurements made on 500-W quartz-bromine spectralirradiance standards. The spectral distribution of theselamps is very similar to that of a blackbody at 3150 K.We have shown that with such lamps only four or fivenarrowband filters are necessary to determine thespectral irradiance curve from the near UV to the nearIR. Our measurements differed from the calibrationby 0.5% on average with a maximum difference of-1%.

OPTICAL SCIENCE &ENGINEERING SHORT

COURSE

The course will be given at the Dou-bletree Inn in Tucson, Arizona, 12-23January 1981. The purpose of the courseis to acquaint both the specialist and thenonspecialist engineer or scientist withthe latest techniques in the design andengineering of optical systems. Thecourse comprises 18 3-h lectures; de-tailed notes will be supplied.

The wide range of topics that will becovered includes geometrical and physi-cal optics, optical system layout and de-sign, Fourier methods, digital image pro-cessing, polarized light, radiometry, imagequality, interferometry and optical testing,thin films, photodetectors, and visible andinfrared systems.

Address inquiries to Philip N. Slater,Optical Systems & Engineering ShortCourses, Inc., P.O. Box 18667, Tucson,Arizona 85731, or telephone 602-885-3798 and leave a message.

The author wishes to acknowledge the technical as-sistance of Frank McNeely in the fabrication of the ra-diometers and in setting up the facilities for the inter-comparison of radiometers.

References1. M. Suzuki and N. Ooba, Metrologia 12,123 (1976).2. W. Budde, Appl. Opt. 18, 1555 (1979).3. L. P. Boivin and T. C. Smith, Appl. Opt. 17, 3067 (1978).4. J. C. De Vos, Physica 20,107 (1954).


calibration of incandescent lamps for spectral irradiance by means of absolute radiometers

Documents