The Calibration of Scanning Monochromators for the Measurement of Absolute Irradiance

Transcription

1 The Calibration o Scanning Monochromators or the Measurement o Absolute Irradiance John J. Merrill R. Gary Layton Certain approximations commonly used in the absolute calibration o a scanning monochromator are examined in terms o the response o the instrument to a monochromatic input. The absolute irradiance due to any spectral eature in the neighborhood o Xo is commonly computed rom the expression W = AB(Xo)/H(Xo), where A is the area o the spectral eature as recorded by the monochromator output trace, B(X) is the spectral irradiance o a stard source, H(X) is the response o the monochromator to B(X) when the monochromator corresponds to wavelength X. As an example, approximations used in justiying such calculations are examined applied to an Ebert 0.5-m monochromator. For the case chosen, the approximation is shown to be valid to an accuracy o 1.5% to 2%, depending upon assumptions made in the calculation. It is ound that the most serious error or this example is introduced by changes in the sensitivity o the monochromator over a wavelength interval comparable with that o the spectral eature under investigation. A second source o error is ound to be the change in the irradiance o the stard source over a wavelength interval comparable to the instrument resolving power. Introduction A common spectroscopic problem is the determination o the relative or absolute irradiance o various spectral components o a radiating source. The spectral components may be a set o relatively monochromatic atomic emission lines, more polychromatic molecular emission bs, continuous emissions rom hot bodies, or combinations o these three types. The spectral irradiance may be determined by separating the incoming radiation into its spectral components by the use o a spectrometer or monochromator. The intensity due to each component is then measured by the system detector. This simple solution is complicated by the act that the response o the spectrometer or monochromator is not the same or all wavelengths o incident radiation. The detector output, whether it is ilm density, phototube current, or any other measured quantity, will be dierent or equally intense monochromatic incident radiations o dierent wavelengths. To overcome this diiculty, it is common practice to calibrate the spectrometer by recording the detector response to a known irradiance, such as the blackbody stard source operating at a known tem- Both authors were with the Physics Department, Utah State University, Logan, Utah, when this work was done; R. G. Layton is now at the State University o New York College in Fredonia, New York. Received 15 April This work was supported by Air Force Cambridge Research Laboratories. perature. The stard technique is to measure the area o the spectral eature as recorded by the monochromator in scanning the relevant wavelengths. This area is multiplied by a calibration actor obtained as the ratio o the stard source irradiance at some wavelength, representative o the spectral eature, the monochromator response to the stard source when the monochromator is set to the same representative wavelength. There are certain approximations involved in such a computation, most o which arise because o the inite resolution o the instrument. The monochromator will respond to a monochromatic input at a range o instrument settings around the nominal wavelength. This leads, o course, to the inite recorded area o a sharp spectral line. During the calibration procedure, however, the instrument setting is constant; but the response is due to radiations rom the continuous source with a range o wavelengths centered around the nominal instrument setting. The purpose o this article is to examine the validity o using the second o these measurements to calibrate the irst. This may be done by considering the monochromator as a linear ilter, an approach that has been used proitably in many calculations involving resolution, absorption, line shape.. 2 The theory will be developed or the general case then 0 applied, as an example, to the calibration o a 4000-A line using an Ebert 0.5-m monochromator. The General Case Let T(X, A) be deined as the response o an instru APPLIED OPTICS / Vol. 5, No. 11 / November 1966

2 - ment to a monochromatic input, 3(X' - X), when the instrument is set at A. Further, let Xo be the wavelength or which T(X, A) is a maximum. For some instruments it may be more useul to deine X 0 as the centroid o T(X, A 0 ), or as the midpoint o the chord connecting the two points o T(X, A 0 ) which are hal the maximum, or some other way o deining the center o T(X, A 0 ). The general development that ollows is valid or any such choice. The instrument response to an input G(X) (representing the spectral irradiance o the spectral eature under investigation) as a unction o the instrument setting is F(A) = G(X) T(X, A)dX. (1) In these terms, the total irradiance o the spectral eature is = G()dX; (2) the area o the eature as recorded by the instrument is where A = F(A)dA = G(X) T(X, A)dXdA 'ao (3) = J G() I(N)dx, I(X) = J T(X, A)dA. (4) Note that I(X) is the area under the curve o the monochromator output vs monochromator wavelength setting with a monochromatic input o unit intensity wavelength X. With the instrument set at A 0 (representing a characteristic wavelength o the eature being considered), the instrument response to the stard source is H(Ao) = J B(X) T(X, Ao)dX, (5) where B(X) is the spectral irradiance due to the stard. A H(Ao) are the experimentally measured quantities, B (X) may be assumed to be known, either rom theoretical calculations or rom prior calibration. From these, it is hoped that W, the irradiance o the spectral eature, may be estimated. To this end, it may be noted that I(X) B(X) are slowly varying unctions o X. That is, the changes in I(X) B(X) will be small over the wavelength interval or which G(X) T(X, A 0 ) have signiicant value. I we set 1(X) = N(Xo) + [I() - I() (6) = I(N) + AI(X), B(X) = B(Xo) + [B(X) - B(Xo)] = B(X,) + AB(X), (7) the integrals representing the experimental measurements become A = G() I(X 0 )dx + G(X) AI(X)dx = M(X 0 ) G(X)dX + G(X) AI(X)dX = IN(Xo) + G() AI(X)dX, H(Ao) = B(X 0 ) T(XN Ao)dx + AB(X) T(X, Ao)dx = B(X 0 ) T(X, A,)dx + AB(X) T(x, Ao)dx. Equation (8) may be solved or W, the desired irradiance, using the deinition o 1(X) rom Eq. (4), with the result that A - G() AI(X)dx We T(X0, A)dA A - J G(X) Al(x)dx] [T(X T(X, Ao)dx L T(o0 Ao)dx] A)dAj The numerator in the last term on the right-h side o Eq. (10) is the instrument response, when set at A 0, to white light o unit intensity. The demoninator is the area o the curve o monochromator output vs monochromator setting with a monochromatic input o unit intensity wavelength X 0. For many situations, these terms are nearly equal, so their ratio is almost unity. The integral in the denominator o the irst term on the right-h side o Eq. (10) may be eliminated by using Eq. (9), with the result that A - G(X) AI(x)dx W = B(X 0 ) 0 LH(Ao) - JAB(X) T(X, Ao)dX X T(x, AO)dX X o I T(>A, A)dA (8) (9).A~dX (11) Exping the expression to irst order in the correction terms AI(X) AB(X) yields. T(X, Ao)dx _ G(X) AI(X)dx W=A H(Ao) \0] Li Ji 0 T(Xo, A)dA L' I AB(X) T(X, Ao)dX + 10 H(Ao) 1. (12) November 1966 / Vol. 5, No. 11 / APPLIED OPTICS 1819

3 Thus, W A[B(Xo)/H(Ao)] (13) is a reasonable estimate or the total irradiance o the spectral eature i, only i, the obvious approximations to the right-h side o Eq. (12) can be justiied. This is the expression commonly used or such computations. The approximations used in justiying the use o Eq. (13) instead o Eq. (11) are not generally valid, should be evaluated or each experimental situation, as is done in the next section or a typical case. Example As a typical case we consider an Ebert 0.5-m monochromator with a distance between entrance exit slits o about 8 cm a grating ruled with 11,800 lines/ cm. In this type o instrument, the light rom the entrance slit is collimated, dispersed, ocused on the exit slit. The light rom a monochromatic source orms a single image o the entrance slit (or each grating order) which is moved past the exit slit by either a change in grating position or wavelength. I we assume that the optical system detector have a lat response over a wavelength interval large enough to include T(X, A) either with X = constant or with A constant, the transmission o the instrument will be proportional to the overlap o the image o the entrance slit (image width 2a) the exit slit (width = 2a). This simple expression should then be multiplied by a slowly varying unction o wavelength g(x) representing the wavelength dependence o detector sensitivity relection coeicients o the various optical elements. Letting x 0 be the distance between the center o the image the center o the exit slit, 7'(X, A) = 9(X) a D(x - xo)dx, (14) where D(x - x 0 ) is the intensity distribution o the image o the entrance slit. To ind an explicit expression or xo(x, A) we must consider the geometry o the monochromator. The angles used are deined in Fig. 1. The angle 0 is ixed as the angle between a ray passing through the center o the entrance slit the ray passing through the center o the exit slit. The ollowing relations obtain a + /2 = i, (15) a - 0 (16) d(sini + sino) = nx, (17) where d is the grating constant, n is the order o diraction. For the Ebert geometry, i y = -/2, xo = tany y, (18) where is the ocal length o the spectrometer mirror, y is assumed to be small. By substituting Eqs. (15) (16) into Eq. (17), the ollowing relation is obtained (or irst order): A, Entrance Ray C Exit Ray Grating Normal Fig. 1. Monochromator geometry. The vector AO represents the direction o the entrance ray incident upon the grating at 0. OC represents the direction o exit rays which are ocused on the exit slit. OB is the bisector o the angle AOC. Incident radiation with wavelength will be diracted in the direction OD, which will coincide with OC when the monochromator setting, represented by the angle a, corresponds to X. or sin(ce + 0/2) + sin(a - ) = X/d, (19) since cos(0/2) + cosa sin(0/2) + sina cos3 - /2 (which implies X = A), this expression re- For 3 = duces to cosa sing = X/d. (20) Exping Eq. (19) in terms o y, 2 cos(0/2) sina = A/d. (21) sina cos(0/2) + cosa sin(0/2) + sin(a - 0/2)cosy - cos(a -.0/2) sin-y = X/d. (22) By making the approximations cos-y = 1 siny =, substituting A/d or 2 cos(0/2) sin(a) rom Eq. (21), Eq. (22) may be reduced to which gives (A - X)/d = y cos(a - /2), (23) y = (A - )/[d cos(a - /2) (24) Using this result, the integrals in the irst term on the right-h side o Eq. (12) may now be written explicity as Jo\(+ ( A Vid. T(X, Ao)dx -X = JUg\ J- DX- d cos(ao -/2) dx dx (25) T(Xo, A)dA = Wg() -+ D(x _ (A - X) )dx da. (26).10 -a d cos(a -/2) 1820 APPLIED OPTICS / Vol. 5, No. 11 / November 1966

4 The integrals may be put into similar orm by substituting into Eq. (25) Xo)/d cos(adz = da/d (A, - X) d cos(ao - 4/2) Z = [(A - [cos(c- /2)]-1 d - -dx d cos(ao -<>/2) (27) + (A - Xo) sin(a - /2) (28) 2d cos(a - -p/2) cos(k/2) cosa ( into Eq. (26). The limits are set by observing that, in both integrations, the entire range or which T has nonzero values is covered, the only dierence being in the direction. Reversing the sign o Eq. (25) exchanging the limits, the integrals o Eqs. (25) (26) may be written as T(x, Ao)dX -d g(y) D(x - y) cos(ao - 0/2)dx dy (29) _. a d +. T(A0, A)dA = - (Y0) +ad(x - z) [cos(a - /2)]-1 _ a (A - Xo) sin(a - /2) -1 2d cos(a - /2) coso/2 cosa d Numerical evaluation o the two terms in brackets in Eq. (30) at a wavelength Xo o 4000 A over a wavelength interval o 20 A gives, or the monochromator described (a ;-; 14, 0/2 ::: 4 30', y < 3%) [cos(ao - /2)V- 1 = 1.01 (31) (A - Xo) sin(a - /2) < 2.03 X (32) 2d cos 2 (a - / 2 ) cos(a/2)cosa - Thus, Eq. (32) may be neglected in comparison with Eq. (31). The remaining problems in comparing the integrals are the variation o cos(a-0/2), which changes with A but not with X, the variation o g(x), which varies with X but not with A. The evaluation can be made by noting that Ymin (x)dx < y(x) (x)dx < yxnax (x)dx, (33) where y (x) (x) are positive in the range o integration. To scan the 20-A wavelength interval in the neighborhood o 4000 A, a changes by less than 3 min o arc. This corresponds to a change in cos(a - q/2) o 0.02% a change in g(x) o 1%, as estimated or our system by measuring the system response to the blackbody stard as a unction o spectrometer setting. Thus, I T(X, Ao)dX < T(Xo, A)dX (34) The second approximation requiring justiication involves the value o the quantity G(X) AI(X)dx A 0(X) [T(X, A) - T(Xo, A)]dAdA G(X) I(X)dX (35) Using Eq. (26), the A integral in the numerator may be written as where AI(X) = [T(X, A) - T(Xo, A)]dA = i - (X)D(x - ( - X)) gs(xo)d~x - (A - X) dadx d cos(a - /2)) d = 12 -a [9(X) h(u, x) D(u)du ugx = al - g(x0) h(uo, x)d(uo)duo] dx,,io = uo u = x - {(A - X)l/ [d cos(a -/2)] h(u, x) = d cos(a- _ ~~ /2) 1 +- x-u sin(a -/2) ]1 da = -, 2 cos(o/2) cosa du ul = x + {X/[d cos(a - /2)] (36) (37) with obvious extensions with X = X 0 in the integral over Uo Ṫhe limits o integration in both the inner integrals o Eq. (36) are broad enough to include all values o u or which D(u) has nonzero value. Thus, the integrals are not changed i we set u = ulo = + c. Then the two integrals are identical. Noting that I(X) may be written in these terms as AI(X) is given by I(X) = g(x) AI(X) = {1 - h(u, x)d(u)du, [g(xo)/g(x)l}i(x), Eq. (35) becomes G(X) AI(X)dx [1 - g(xo)/g(x)]g(x)i(x)dx A J I Io G(X)I(X)dx (38) (39) (40) November 1966 / Vol. 5, No. 11 / APPLIED OPTICS 1821

5 Over the 20-A range o wavelengths covered by G(X), g(x) diers rom g(xo) by less than 0.5%. Thus, using the theorem o Eq. (33), J G(X)AI(X)dX/A < (41) The third inal quantity requiring evaluation is AB(X) T(X, Ao)dx E AB(X) T(x, Ao)dA H(Ao) E B(X) T(X, AO)dX 0 (42) For a blackbody stard source the spectral intensity is given by Planck's law: B(X, T) = CX-5 (ec2/xt - 1)1. (43) For T C X = 4000 A, C 2 /XT 28, Eq. (43) may be represented reasonably by the Wien radiation law: B(X, T) CoseC2/XT (44) For constant T = C, AX = 4-1 A [representing the wavelength width o T(X, A 0 ), or the instrument resolving power], X 0 = 4000 A, AB(X)/B(X) = [C,/XT) - 5] AX/Xo 23AX/Xo -z (45) Using the theorem o Eq. (33) once again, AB(X) T(X, Ao)dX/H(Ao) < (45) This term is probably considerably smaller than this value. For example, i T(X, A) were symmetrical around X = X 0, the linear error term represented by Eq. (45) would give no contribution to the integral o Eq. (42), the evaluation would have to be extended to the second term in the Taylor series expansion o AB(X), which is quadratic in AX leads to an upper bound value o 0-1 or the quantity in Eq. (42). A better estimate might be to assume that D(x - x 0 ) is symmetrical about x = x 0, in which case AB(X) T(X, A)dX 11(A 0 ) - L ( o [g(x yb0o) irnax - (0.006) (0.01). (47) FUTURE MEETINGS OPTICAL SOCIETY OF AMERICA To this must be added the value resulting rom the (AX) 2 term in the expansion o AB(X), yielding a limit, or the case o symmetric D(x - x 0 ), o about Combining the results o Eqs. (34), (41), (46), (47), Eq. (12) reduces, in this example, to [- A B(Xo)/H(Xo)l/WI < or 0.02, (48) depending upon which estimate o the last error term is used. Conclusion The results o the previous section indicate that the commonly used method o calibrating spectroscopic measurements must be careully examined or each combination o wavelength, spectroscopic analyzer, calibrating source resolving power, to mention the most obvious o the variables involved. Such justiication must show that the values o the error integrals o Eq. (12) are appropriately bounded. Unless proper precautions are taken appropriate justiications provided, all absolute relative spectral irradiance measurements are subject to unknown systematic errors, the ultimate precision attainable by such measurements is severely limited. The most serious error arises rom changes in detector sensitivity optical transmission o the spectrometer over the wavelength interval included in the spectral eature. I this interval is small, as is the case or a sharp spectral line, the calibration procedure can be justiied to within approximately l%. I the interval is large, as or an extended molecular b system, the problem is more diicult. In this case it may be necessary, or example, to calibrate the instrument output point by point beore measuring the area under the spectral eature. A second source o error is the change in the emission o the blackbody stard over a wavelength interval comparable to the resolving power o the instrument. This will contribute only a small error to measurements made with high resolving power, but may contribute signiicantly i, or example, the resolving power o the instrument is compromised to achieve greater sensitivity. Reerences 1. Gilbert W. King A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951). 2. James E. Stewart, Appl. Opt. 4, 609 (1965) April, Neil House, Columbus, Ohio October, 52nd Annual Meeting, Sheraton Cadillac Hotel, Detroit, Mich March or April, Spring Meeting, Washington, D.C October, 53rd Annual Meeting, Pittsburgh-Hilton Hotel, Pittsburgh, Pa APPLIED OPTICS / Vol. 5, No. 11 / November 1966