Dispersive correlation spectroscopy: a study of mask optimization procedures

Dispersive correlation spectroscopy: a study of maskoptimization procedures

MiIIan M. Mi&Ian and Raymond M. Hoff

A procedure has been established to obtain the design parameters of the masks used in dispersive correlationspectrometers. The SNR equations are used to derive the mask equations. These are later solved for somesimplified band models, and the obtained parameters are used as the initial values in an iterative programwhich determines the final design parameters. The calculations can be performed for various backgroundsand instrumental line profiles in order to achieve the best compromise for the detection of pollutants, andother trace gases, in the atmosphere.

1. Introduction

Dispersive correlation spectrometers, or mask spec-trometers, are a family of instruments which use ab-sorption spectroscopy to detect gases in planetary at-mospheres.1-3 In these instruments, the dispersed in-coming radiation is transmitted selectively through oneor more sampling elements, or masks, before reachingthe photodetector. The various types of instrumentsgrouped under these names differ basically with regardto how the spectrum is sampled: continuously bymeans of one oscillating mask or discretely by cyclicallyalternating two, or more, masks in front of the photo-detector. In either case, the fluctuating photodetectorQutput is suitably processed to yield the desired sig-nal.

Two purposes of the multiwavelength samplingachieved with the masks are to increase the SNR of themeasurement and to enhance, in the passive applica-tions, the instrument's ability to detect the chosen gasunder the presence of other competing absorbers and/orfluctuating backgrounds.

The descriptions of the various instrument types, andsome results obtained with them, have been presentedby a number of authors.4-10 The masks employedconsist of high contrast transparencies of the spectrumof the gas sought 4 or of arrays of exit slits.5' 0 However,other than the fact that those slits are supposed tocorrelate with, or be matched to, some features of thespectrum (the transmission minima, usually), no in-formation seems to be available about their accuratepositioning, and the choice of their number and theirwidths seems to be left to the discretion or ingenuity ofthe researcher.

The authors are with Environment Canada, Atmospheric Envi-ronment Service, Downsview, Ontario M3H 5T4.

Received 19 November 1976.

In this work, the SNR equation of one discrete typeinstrument, a bistable one using two masks, will be usedfor the calculation of the masks. The equations gov-erning the mask parameters are derived from the con-ditions that give a maximum of the SNR. The solutionsof those equations are obtained for simple models andentered in a computer program which in turn yields themask's parameters, i.e., number of slits, their positions,their widths, and the optimum entrance slit of thespectrometer. These give the maximum of the SNR forany given set of backgrounds and concentrations of thegas to be detected.

In Sec. II a brief introduction to the passive spectro-scopic remote sensing problem is given, which indicatesthat multiwavelength sampling may cancel or averageout undesired interferences in the measurement.Section III examines the concept of correlation in adispersive system and shows how a mask can be corre-lated to (similar, or derived from) a given spectrum inorder to achieve a maximum of the cross-correlationfunction between them. In Sec. IV the equation for theslits of the masks are derived, and their solutions forsimplified cases are presented. Finally, the iterativeprocedure used to determine the final mask parametersand the maximum of the SNR is described in Sec. V.

11. Problem in Passive Spectroscopic RemoteSensing

The use of absorption spectroscopy for the mea-surement of gas concentrations is based on the Boug-uer-Beer-Lambert law of absorption:

Nx' (L) = Nx'(O)Iexp[-_ ki(X)cjL]I, (1)

where

June 1977 / Vol. 16, No. 6 / APPLIED OPTICS 1609

NX'(O),NA'(L) are, respectively, the spectral radi-ances at wavelength X before and aftertraversing the path L (W m- 2 nm-sr-1);

ki (X) is the absorption coefficient of speciesi at wavelength X (M2);

ci is the concentration of species i as-sumed homogeneous over the radia-tion beam (m-3);

L is the pathlength along the line ofpropagation (m).

The summation is present over all species. The lawis only exact for truly monochromatic radiation and tothe extent that average absorption coefficients are used,it can be considered an approximate law only.11

The following approach is used to determine theconcentration of one species in the presence of others.In general, use of Eq. (1) at wavelength one will give

1 N )'(L)L N'(0) (2)

The same could be obtained for a total of n selectedwavelengths for which the absorption coefficients of thespecies are known. Then, n equations can be solved forn unknowns.12

A special case of this approach is to measure NA (L)at two neighboring wavelengths, viz,

N1 '(L) = N1'(Q) exp[-kj(1)ciL] exp(-TI),

N2'(L) = N 2'(0) exp[-k(2)cjL] exp(-T2 ), (3)

where c is the concentration of the species of interestand

Tj = k 2W)c2L+... kn(i)cnL, ( = 1,2)

is an optical depth at wavelength j, including the ab-sorption of all other species.

The background spectral radiance and the extinctiondue to absorption by other species give an effectivebackground spectral radiance

Nj(0) = Nj'(0) exp(-rj). (4)

Solving for c, or hereafter c, yields

1 1 In N 1 (L)-in N 1() (5)

L kj(2)-kj(1)L N 2(L) N 2(0)

The meaning of Eq. (5) is rather interesting. Itimplies that the concentration of the species of interestc can be measured provided that two wavelengths arefound for which (1) the absorption coefficients of thespecies of interest are different, and (2) the effectivebackground radiances are very nearly equal. The sec-ond occurs if both NA'(O) and the absorption coeffi-cients of the other species are equal at both wavelengths.It also occurs if they are not, but their combined effectmakes the effective background radiance equal at bothwavelengths. The measurement of c, of course, is de-pendent on their ratio being close to unity. The errorassociated with its measurement, if the ratio is not unity,could be estimated providing that the maximum andminimum expected concentrations of the other ab-sorbers and the background fluctuations are known.

Using two neighboring wavelengths, then, a measure-ment of c could be made quickly at the expense of atolerable error.

For an optically thin case, this second approach hasbrought us to the heart of a simplified passive remotesensing problem. In passive spectroscopic remotesensing, the background radiances and the presence ofother competing absorbers are not amenable to controlor are even known. In this situation, the idea of sam-pling at two wavelengths, fulfilling (1) and (2) is mostattractive, and can be further explored by extension totwo wavebands or sets of wavelengths with the followingproposition.

Determine if two sampling functions Ml and M2 canbe found for which

f N,\(O) exp[-k(X)cL] M1 (A) dX

= exp(-aicL) f Nx(O) M1(X)dX,

f N,\(O) exp[-k(X)cL]M2(X)dX= exp(-a 2 cL) f N\(O)M2(X)dX, (6)

such that l Ia, - a21 be large, condition (1),f Nx(O) [M1(X) -M2(X)]dX be small, condition (2),

where a, a2 are average absorption coefficients of thegas of interest, as defined by Eq. (6), and Ml(X), M2(X)are the sampling functions (to be defined later).

The answer in principle is affirmative, with the pos-sibility of determining the sampling function(s), M(X),in a variety of ways to suit the measurement and ex-perimental conditions.

Ill. Spatial Correlation Function in DispersiveSpectroscopy

The space correlation function13 is the value of theintegral of the product of two (generating) functions asone of them is displaced with respect to the other, thecorrelation variable being the distance between them.If the two functions are identical, there will be a valueof the correlation variable for which they exactly overlapwith the result that every ordinate, positive or negative,yields a positive product. The final sum, or integral,of all products is therefore large. If, however, thefunctions are dissimilar, or displaced, some productswould be positive and some negative, some cancellationstake place, and the final sum is smaller. The correlationfunction is, then, a measure of their similarity or coin-cidence. Specifically, the autocorrelation functionmeasures the integral of the product of a function withitself, and some of its characteristics are as follows:

(a) It has a positive maximum at the value of thecorrelation variable for which the functions overlap.This value is usually taken as the zero of the correlationcoordinate.

(b) It is an even symmetrical function with respectto this value of the correlation coordinate.

(c) If the generating function is periodic, the auto-correlation function is also periodic and has the sameperiod.

(d) It depends on the frequency content of thegenerating function. The broader the spectrum of the

1610 APPLIED OPTICS / Vol. 16, No. 6 / June 1977

generating function, the more sharply defined is themaximum and the quicker the function falls off fromthat value.

In the case of the dispersed spectrum of the incomingradiation, the cross-correlation function between it anda sampling function can be written

R(t) = .-M(X - )dX (W), (8)

where Pit is the spectral radiant power in the exit planeof the dispersive system (polychromator) (W nm-1);M(X) is the sampling function, or mask, defined as

M(X) 1, for Xi - Ai < X < Xi + Ai,l0, for any other wavelength,

and Xi, Ai are, respectively, the center wavelength andhalf-width of slit i of the mask (nm).

This cross-correlation function is the spectral radiantpower passing through the slits of the mask and has thedimensions of radiant power (W).

On writing Eq. (8), the spectrum is assumed to befully resolved by the polychromator. The applicationto spectrometers with a finite entrance slit width andoptical aberrations will be presented in Sec. V.

Substituting (9) into Eq. (8) and rearranging, thecross-correlation function becomes

R( + Xi,Ai) = Z_ XPQ + Xi + X)dX. (10)i -Ai

Considering now the vibrational bands of a molecularabsorption spectrum on an ideally flat spectral back-ground, it can be shown that for any given slit width 2Ai,each term in this function has a maximum or minimumwhen

Pt + i - i) = PQt + i + i), (11)

i.e., for each value of , the correlation coordinate,placing slit i in a position intercepting equal values ofPA at both ends of the slit. This will be denominatedcondition (a), and, in principle, no restrictions are im-posed on the slit width.

The width of the slit can now be constrained to benarrower than the distance between successive trans-mission troughs, which shall be noted as condition (b).With these two conditions, a maximum is obtainedwhen t + Xi, slit center position, is near the wavelengthof maximum transmission between the troughs. Co-incidence with the peak occurs in symmetrical bands.

Using the two conditions, a peak correlating mask canbe obtained with one slit for each maximum of thetransmission spectrum. Similarly, a trough mask canbe obtained with one slit per minimum of transmission.In the latter case, condition (b) is implied with respectto the band peaks.

Either one of the two masks can be cross correlatedwith the spectrum by locating it in the exit plane of apolychromator and scanning the spectrum past it. Theoutput of a photodetector, placed behind the mask forinstance, can be plotted as a function of the spectralscan or correlation coordinate to obtain a correlogram.A series of relative maxima and minima can be expectedwith a strong but not necessarily an absolute extremalat the scan position that brings each slit into register

with its respective peak or trough. This is illustratedin Figs. 1 and 2.

Figure 1(a) shows a transmission spectrum of SO2 inthe near uv and three trough masks of eighteen slitseach. Some slit positions are seen to vary when pro-gressively narrow slits are forced to meet condition (a)with respect to asymmetrical bands. Figure 1(b) showsthe corresponding correlograms. Figure 1(c) shows thecorrelograms for three peak masks having the samenumber of slits, approximately the same individual slitwidths as those of the trough mask, and designed for theconsecutive peaks. A similar example for NO2 is pre-sented in Fig. 2, using two sets of two six slit masks.

These graphs are pseudoautocorrelograms of bothspectra, one set emphasizing the minima of transmis-sion, the other the maxima. Because the spectrum ispositive, no correlogram reaches negative values. Someeven symmetry is apparent with respect to, and in theneighborhood of, the mask designed positions whichgive the cross-correlation extremals. This is more ap-parent in the NO2 correlograms. The SO2 presents aclear case of quasi-periodic spectrum in this region, withthe corresponding periodicity in its correlograms. Thecurvature of the broadband, or continuum, over whichthe vibrational bands ride, is responsible for the extre-mals of the pseudoautocorrelograms not being abso-lute.

The continuous type instruments4 6 7"14 use thecross-correlation function directly. In these instru-ments an oscillatory scan is performed with a carefullycontrolled amplitude, and the photodetector output isprocessed to detect the presence and amplitude of someselected frequency components or harmonics of thescanning frequency.

The advantages of using a trough mask over a peakone are not particularly clear in view of Figs. 1 and 2,and either type of mask could be used. The maximumpeak-to-peak amplitude of the correlograms, however,will be usually smaller than the difference between theextiemals of a pair of peak and trough masks withidentical widths for corresponding slits. They will beonly equal if the bands are perfectly regular.

Other factors to consider are that the presence of thegas of interest may consist of a weak absorption on anintense and structured background whose influence inthe signal one often wishes to remove and that thecontinuous type scan may be somewhat inefficient,having a low value of the duty cycle. This arises sincesome of the generated frequencies are not used for thedetection. This is particularly serious with irregularspectra.

In this situation, an alternative is to return to the twosets of wavelengths, the two wavebands concept of Sec.II, and examine the idea of subtracting directly a peakand trough correlogram with a common scan coordinate.For example, this can be achieved by engraving the twomasks on a disk which rotates in front of the photode-tector, and time shares the output.

Two other conditions are required in order to selecta specific mask pair among all the possible ones. Thefirst, condition (c), is that the width for each peak slit,and following (or previous) trough slit, be equal, i.e.,


H-IIIIILLUJJIIWJW LLJ

I I I I I I I I I I I I I I

(a)

where b are the distances between centers of the pairedslits.

The conditions for maxima or minima of this functionare now

P( + Xi - Ai) = P( + Xi + Ai),

P( + i + - Ai) = P( + Xi + i + Ai), (13)i.e., condition (a) for both slits of the pair is returnedand

P( + i + Ai) = P( + X + i - Ai), (14)

i.e., the values of Pi must be equal at the ends of thepaired slits. This becomes condition (d). As soon asa peak and trough of a band are selected, application of

nm280 290 300 310 320 A

300 ppm-m NO2m)

(a)

1111 H

400 410 420 430 440 450 460 470 1\ (nm)

(c)

1.0

peak mask

-50

Cross-correlationmaxima

0 50 (nm)

Fig. 1. Sulfur dioxide correlograms: (a) transmission spectrum of300 ppm-M of SO2 and three masks of eighteen slits used for thecorrelograms; (b) trough mask correlograms obtained from masks with31.8-nm, 15.6-nm, and 3.6-nm total width, respectively. *indicates

largest correlation maximum or minimum.

(b)

+ 118

24

- troughmask

-30 0

Cross-correlationminima

40 (nm)

paired. This is a condition that achieves a zero differ-ence from a spectrally flat background when no gasabsorption is present. The second is obtained from therequirement that the difference between the cross-correlation extremals of the two paired masks be amaximum. This type of instrument, for obvious rea-sons, is expected to operate at this (fixed) position be-tween the spectrum and the paired masks, althoughsome slow spectral scanning may be performed toeliminate residual effects from real life, nonflat back-grounds.

The difference function becomes

G( + 1,3_,As) = P\dX - ('+5 i+i+iPdXi t+ i-Ai + (i+12i-Ai

f E i | p( + A + A)P( + i + bi + ) d| (12)

1.0 (c)

+_ 118

*24-0.3

peakmask

-30

Cross-correlationmaxima

0 40 t (nm)

Fig. 2. Nitrogen dioxide correlograms: (a) transmission spectrumof 300 ppm-m of NO2 and two masks of six slits each used for thecorrelograms; (b) trough mask correlograms obtained from masks with11.8-nm and 2.4-nm total width, respectively; (c) peak mask corre-lograms obtained from masks of the same total width. * indicates

largest correlation maximum or minimum.


'OF 300ppm-mSO2 (pmSO2)

z

I- 0.5

1.0

1.0

50 t nm)

* 318

z0

0.5M .

. 168

36

. . . . . . . .

lo . . . . . . . . . . . I

11 .. .

o A... - -

n I

X1d Xld +-d X1SNR X1 +SNRMAXIMUM DIFFERENCE MAXIMUM SIGNAL TO NOISE

2Ad 0.5D RATIO 2NRrn0.371D

Fig. 3. Optimum slit parameters for maximizing either the differencesignal or the SNR.

-z 0 -

uF

. Difference maxima

-50 0 50 diff (em)

Fig. 4. SO2 difference correlogram obtained with two masks ofeighteen slits, each with 15.6-nm total width.

the four conditions yields a unique value for the widthof the paired slits and the positions of their centers.This is illustrated in Fig. 3.

In a band, or line, spectrum two sets of paired masksare possible depending on whether the peak slits arepaired with the preceding or successive trough. One ofthe two sets yields the maximum difference for a givenabsorption of the gas. The number of slits in eachmask, considering that each pair contributes to thedifference, should equal the number of troughs in thetransmission spectrum.

Figure 4 shows the SO2 difference correlogram fromtwo paired masks of eighteen slits each. The slits usethe same peaks and troughs of the intermediate masksof Fig. 1, and each slit pair fulfills the four conditionswith respect to its trough and successive (longer wave-length) peak. The difference function is more oddsymmetrical in the neighborhood of the point of maxi-mum difference, as compared to the even symmetry ofthe generating cross correlograms. This can be moreeasily noticed in the difference correlogram of the NO 2spectrum of Fig. 5.

Up to now, the calculation of the masks has beendiscussed with respect to the vibrational structure of thespectrum. Relaxation of condition (b) means that amaximization of the difference by applying the re-maining conditions can take place at three scales, eachwith a corresponding solution and each having certainadvantages in some types of gas detection. The first iswith respect to the over-all transmission spectrum, i.e.,continuum, bands and lines. The second solution iswith respect to the bands, as discussed, and the third iswith respect to the rotational lines, if resolved.

In the first case, the solution consists of two slits, eachas wide as the total spectrum, one over it, and the otherjust outside. This pair gives, ideally, the maximumattainable difference for any amount of absorption ofthe gas of interest. Figure 6 shows two less than opti-mum difference correlograms for SO2, one for two slits31.8 nm wide positioned at the centers of the optimum(spectrum wide) slit pairs, the other for two masks ex-tending over an equal spectral span and optimized to

>

S-

z 0

-40

Difference maxima

0

e

I-.

zr

40 diff (nm)

Fig. 5. NO2 difference correlogram obtained with two masks of sixslits, each with 11.8-nm total width.

0 50 100 150 diff (nm)

Fig. 6. SO2 difference correlograms showing the relative magnitudesof the first and second scale solutions.


. . . I . . . . . . . . . . I . . . . . . . .

get the maximum difference from the eighteen strongestvibrational bands (the same as for Fig. 4). The figureillustrates the magnitude of the maxima for the firstscale solution, as compared to the maxima for the sec-ond scale-vibrational bands solution. When using thespectrum-wide slits, the first scale solution would havebeen proportionally larger than for the second scale,using all the peaks and troughs of the bands. This scalesolution would provide a gas sensor most suited to insitu monitoring where the interferences from competingabsorbers are negligible. Because of the simplicity ofthe mask design and the relatively large signal, this typeof sensor appears quite attractive for process controlapplications.

If the rotational lines are resolved, a third scale so-lution exists with a pair of masks having a large numberof very narrow slits. The third scale maximum would,in turn, be smaller than the solution for the vibrationalbands. An increased number of slits, however, producesdifference correlograms with more sharply defined ex-tremals extending over an increasingly narrow range ofthe correlation coordinate, as can be expected from theincreased frequency (or structure) content of the masks.This is also noticeable in Fig. 6, where the effect of theband continuum, which dominates the first scale solu-tion, has almost disappeared in the second scale dif-ference correlogram. A sensor produced using the thirdscale of solution would provide the highest level of re-jection of competing absorbers. Because the rotationalstructure must be resolved, this scale technique wouldbe easier to implement in the ir wavelengths.

IV. Mask Studies

A. Preliminary ConsiderationsThe optimization of the mask, for any operational

instrument, is not usually done for the maximum of thedifference, which may be directly proportional to thesignal, but rather for the maximum of the SNR whenthe instrument is photon noise limited. The discussionregarding the continuous and discrete system stillapplies for the SNR, and the simpler solutions for themaximum of the difference have served to indicate thethree possible scales of solution in the most general caseof mask design. In the passive remote sensing of gasesin the atmosphere, naturally occurring phenomena willlimit or influence the choice of solutions. Unavailableor limited backgrounds, their structure and slope in thespectral region of interest, their variability during theday, season, and unresolved spectral lines in the spec-trum of the gas to be detected are some of these. In anycase, the appropriate instrumental equations, signal orSNR, must be used to obtain the mask parameters.

The approach that follows is based on the work of oneof the authors.'5 Simplifying assumptions are initiallyused to obtain analytical solutions for the parametersof each slit pair. These parameters are subsequentlyutilized as the initial values in an iterative computerprogram which determines the final mask parametersfor the maximum SNR. In the program, however, thereal expected background(s) and the physical charac-teristics of the actual instrument are used. For this

work the signal to noise ration equation of the Corre-lation Spectrometer' 6 (COSPEC I) will be used to il-lustrate the calculation process.

B. Instrumental Response and Slit Pair EquationsThe output equation of the COSPEC I type of in-

struments is given by either of the following expres-sions:

V(t) = o I- - or Vo ,() R2J, (15)RIQJ IR Q + R2(0)1

The SNR, on the basis of shot-photon noise limita-tions,' 5 is given by

SNR() = k R1 () - R2Q)[R1() + R2(Q)J112 (16)

where V(t) is the output in volts,Vo is an instrument reference voltage,k, is an instrument constant (W-1/2),

RiQt),R2(Q) are the cross-correlation functions be-tween the dispersed spectrum and thepaired masks MX), M2(X) (W), and,

(, the cross-correlation coordinate, is relat-ed to the angular position of the dispers-ing element (grating) (nm).

The two expressions for the output are but variationson the same theme, i.e., a difference correlogram ob-tained by time sharing a photomultiplier output whosebias is controlled by either one or both of the sampledradiant powers0 '' ,,5 or by other means involving insome manner either the total or some fraction of theavailable radiant power. The form of the SNR equa-tions is not affected by the photomultiplier biasing if itsgain remains unchanged during the signal detectioncycle.

Writing the cross correlation in the form of Eq. (12)and substituting gives

SNR(4)

2

i, | [P(Q + Ai + A) - P( + Ai + bi + X)]dA}

(z { Jf4 [P(Q + Ai + A) + P(Q + Ai + bi + A)]dX 1/2

(17)

This is a function of n, the number of slits, Xi theirinitial center positions in mask 1, Xi + i, their initialcenter positions in the paired mask 2, and Ai, the slitwidths.

Equation (17) can be used to solve for the parametersof a single slit pair for n = 1. Taking the partial deriv-atives with respect to X1, 61, and A1, equating to zero, andsimplifying, the three following equations are ob-tained:

PQ + Xi -Al) = P + X + Al1), (18)

PQ + X + 61 - AD = P + Al + 61 + A, (19)

and


= K

[P(Q + A + Al) - 3P(Q + A + 61 + Al)]

Sj; P(Q + A1 + A)dA + [3P(Q + A + A1 )-Al~~~ A-P~ +Al+ 1 A/1)I.*Jn P( + A + 61 +A)dA= 0 (20)

Equations (18) and (19) are similar to Eqs. (11) and(13) and have the same implications, i.e., condition (a),requiring equal values of PA at both ends of the slit, isreturned for both slits. Notice that on writing Eqs.(15)-(17), the only condition retained from the previousfour has been condition (c) requiring that paired slitshave equal widths.

p X+Ae /A

PA = Q *h fJAe NAdA = Q-h I N(A + )dA, (22)

where * denotes convolution,Q is the acceptance solid angle of the system

(sr),h is the entrance slit height (m),

Nx is the incident spectral radiance (Wm-2 sr-nm ), and

Ae is the entrance slit half-width ().The nominal etendue of the system is then 2Ae h Q (M2sr).

For a single exit slit pair, substitution of Eq. (22) inthe SNR gives

fJ -Al[N(A + t+ AXI+ A) -N(A + t+ AXi+ 61 + A ]dXld

A f [ [N(A + +Xl+ A) +N(A + +Xl+ 6+ )]dA)

SNR1 () = k2 - (23)

With the SNR as given in Eq. (17) it can be shownthat the solutions for each slit pair are uncoupled fromthe rest of the slits.'5 The parameters for each slit paircan then be obtained by solving Eqs. (18), (19), and (20)for each peak and successive (or previous) trough. Thisleaves undetermined n, the number of slits in eachmask. The slit equations, however, are not uncoupledwhen the entrance slit width enters in the SNR equa-tions, and the procedure to determine both n and theoptimum entrance slit width will be indicated in Sec.V.

C. Entrance Slit Equations

With the assumptions considered, the spectral radi-ant power in the dispersed spectrum PA is the convo-lution of the incoming spectrum with an instrumentaltransfer function or instrument profile which is a deltafunction. In a real spectrometer, the instrumentalprofile is a combination of imaging, aberrations, anddiffraction effects which also change with wavelength.Following the successive approximation approach, wewill now investigate the effect of a finite entrance slitconsidering the simplest instrumental profile, i.e., theentrance slit image with no aberration or diffractioneffects. The complementary assumption is that theimage does not vary with wavelength over the spectralregion of interest. With these assumptions PA is givenby

P = UNA*hA(A - A). (21)

The entrance slit width is defined as

A(A) 1 for Al <Ae,10 for AI >Ae

The spectral radiant power becomes

On deriving the equations for the maximum of the SNR,the equations governing the solution for A, and Ae areidentical on account of symmetry. This gives the so-lution

A, = A1. (24)

Thus, for a maximum of the SNR, the half-width ofthe entrance and exit slit must be equal. This width,however, must satisfy Eq. (20) with respect to the highlyresolved brands of the spectrum.

Equations (18), (19), and (20) hold for any Px re-gardless of the entrance slit width by which it has beenproduced. The difference in the SNR, however, is thatthe maximum obtained is only relative. The absolutemaximum is obtained with the common solution for theentrance-exit slit widths, and this also applies to thethree scales of solution previously discussed.

When considering all the bands of the spectrum, thedistances between peaks vary and so do the Ai's for theexit slit pairs. Similarly each exit slit pair, optimizedfor one band, would also require its matched entranceslit. In this situation the solution for the compromise,or final, entrance slit Aef, and the number of slits of themasks n are coupled. The determination of both Aefand n can be done as indicated in Sec. V.

D. Slit Pair Solutions for Simple Band ModelsAs will be shown in the following section, a first ap-

proximation in the mask design can be obtained by as-suming a simple band model for the gas of interest.Two models are of particular interest. The first is asinusoidal band model of the form

PA = A + (B/2) (sinX - 1),

which would be applicable to a periodic band such as


3f6>7r+A

Fig. 7. Simplified band models: (a) sinusoidal of the form PA = A+ (B/2) (sinA - 1); (b) isolated cosine of the form PA = A - (B/2)

(cos + 1) for IAI < r and PA = A for IAI > 7r.

Table 1. Initial Slit Parameters from Band Models

MaximumBand model Maximum SNR difference

Sinusoidal X, = 7r/2 X /= r2

PA = A + BsinX -1) 5, = r 7 TP 2=A +~(sinX~ 1) A\ = 1.166 rad A\ = 1.571, (/2) rad

Isolated cosine X= 0 = 0

PX = A-2-(cosX + 1) >I 127r +A l I>2r

1X < T

PX = A 1X > r A < 1.803 rad A, = 3.142(7Tr)rad

(Varies with depth (Independent ofof band) depths)

the vibrational bands of SO2. The second is an isolatedcosine of the form

Pl = A -(cos + 1) for Al < r,2

P = A for A > r,

which would be applicable to an isolated absorption linesuch as the entire electronic transition of SO2 in the uv(the first scale solution discussed previously). Thesetwo models are shown in Fig. 7.

Applying these two models to Eqs. (18), (19), and (20)yields the results shown in Table I. The calculationshows that for a maximum of SNR for a sinusoidal band,the width of the slits (2Ai) should be 0.371 times theperiodicity of the bands rather than the 0.5D for themaximum difference signal. This is shown in Fig. 3 asthe maximum SNR solution. A second result is that foran isolated band the optimum width varies with thedepth of the band in the following fashion:

B/A Ai0 1.803 rad

10-6 1.80310-2 1.80110-1 1.7830.25 1.750.50 1.691.00 1.60.

Thus differences in the optimum slit width will occurfor varying gas concentrations.

V. Mask Optimization Procedure

A. Preliminary ParametersTo initiate the mask calculation process the following

data are required: (1) several high resolution trans-mission spectra of the gas to be detected, covering therange of concentrations expected to be measured and(2) a high resolution spectrum of the background ra-diance (or a suitable weighted average). The obtain-ment of the last of these may require a lengthy mea-surement program in the case of applications to passivesensing. Finally (3) the instrumental transfer functionis needed which includes the instrumental line profileF(X) and the transmittance of all its optical componentsA(X).

To calculate the initial band parameters, the mostlikely bands to be used are selected on the basis of theavailable background. For example, for the passiveremote sensing of SO2 using sunlight in the near uv,bands below 300 nm are not considered.

The initial slit widths Ai (0) are obtained by applyingthe result for A (Table I) to the distances betweensuccessive peaks:

Ai = 1.166 D- = 0.186Di.

This yields also their matching entrance slit widths [Eq.(24)].

The initial X (0) and (0) for each slit pair are de-termined by moving about the respective peak andtrough a length equivalent to their respective Ai (0),until condition (a) (Fig. 3) is met.

Each exit slit pair is now ranked according to theirindividual SNR. The sinusoidal band approximationis used, which gives for each exit slit pair

SNRi(0) = 0445Bi (2A- B) 1/2 (25)

The Ai, B, Di are the same as in Fig. 7 for each of theapproximated peaks and troughs. The width of thematching entrance slits are ranked in the same order astheir exit slit pairs.

The SNR of the mask, for the initial parametersSNRT(0), is obtained. This is done by starting with thehighest ranking slit pair and adding, one at a time, thenext ranked slits. The SNR is expected to rise, flattenoff, and decrease as the number of slit pairs is increased.The slit pair that starts the decrease is discarded as wellas the remaining slit pairs of lower rank and theirmatching entrance slits. The criterion for inclusion ofslit pair k in the mask is


9 JULY 197610:00 EST

5 DEC. 197611-00 EST

TORONTO, ONTARIO43.8N, 79.3W

EO O_ M O FIO n MASK 1O CD E1lC MASK #2

o 0 E 0 0 E MASK# 3[ 0 0 0 f 1 E MASK#41 2, 3 4 5 6 7

300 310 320WAVELENGTH (nm)

and decreased by small steps and the corresponding X1and 31 obtained to meet condition (a) until the maxi-mum attainable SNR1 is obtained. This becomesSNRi(1). The final Ai(1), Xi(1) and 61(1) are stored forthe next iteration. The same process is repeated foreach successively ranked exit slit. As each SNRi (1) ismaximized, the SNRtotal for those slits is determined.This value is also expected to follow the same type ofrise as before, with increasing number of slits, and aftera few iterations with progressively wide entrance slits(see below) some of the initially ranked slits may bediscarded.

Fig. 8. Ultraviolet spectral radiance at Toronto, Ontario with finaldesigned mask positions superimposed. The dotted line indicatesSNR cutoff chosen by the computer program (using the summer noonspectrum). The two extra slits in each mask are those used in the

present commercial instrument (see text).

( E BkDk r1Z_ B 5D/f

. 1/2

+ (2k- - B)Ak 1 Ž 1.k-1E_ (2A- Bi)hs

This process is repeated for increasing entrance slitwidths until the maximum of all the SNRtotal is ob-tained. The corresponding entrance slit width, thenumber of exit slit pairs left, their widths, and positionsare the ones which yield the absolute maximum of theSNR for the background and gas concentration con-sidered. In a further step, the process may be repeatedwith different gas concentrations and backgrounds.

(26)

This process determines the initial number of exit slitsn(0), their widths Ai (0), and their center positions Xi (0)and Xi(0) + bi (0). It also gives the range of entrance slitwidths which may be considered. These parametersare obtained for a piecewise sinusoidal approximationto a highly resolved spectrum. In this approximation,they are expected to-hold for increasingly smeared-outspectra, as produced by increased entrance slit widths,providing that those slits remain narrower than thenarrowest of the entrance slits found.

B.. Program Outline

The computer program calculates

(lNx -exp[-k(X)cL] +. . . [(A) -y(A)])*F(A),

I

(27)

The result of the computation for skylight and theSO2 vibrational bands is shown in Fig. 8. Shown aretwo spectral radiances (digitized on a somewhat largergrid than that used for the program input) taken inwinter and summer at Toronto, Ontario. The programchose five slits for the mask design corresponding to slits1 through 5 shown in Fig. 8. Shown in Fig. 8 are the foursets of slits used in the COSPEC IV16 family of instru-ments which use seven slits. The last two slits, whilenot necessary for a strong SNR in summer, were foundexperimentally to be necessary to obtain a large SNRin cases of low sun elevation angle (winter, early morn-ing). This points out that a general mask to be designedmust be a compromise for varying background condi-tions.

IX

where y(X) is the spectral response of the photodetectorused in the instrument. The equation is purposelybroken up into components I and II to indicate that afull radiation transfer equation could be included, in (I),rather than the Bouguer-Beer-Lambert, optically thinsolution.

The convolution of Eq. (27) with the entrance slit, toobtain the dispersed spectrum, and posterior crosscorrelation with the masks is substituted by a sum ofcross correlations of Eq. (27) with the individual crosscorrelation between the selected entrance slit and eachof the exit slits. This can be done on account of thesymmetry of the entrance and exit slits, which makestheir convolution and cross correlation the same.

The iterative process starts by using the narrowestof the ranked entrance slit widths left. It is cross cor-related, with the highest ranking exit slit pair, this inturn with Eq. (27), and the SNR is calculated. Thevalue of the initial slit width, A1(0), is then increased

C. Mask Selection

As we have seen, there are, in principle, as many op-timum masks as the original number of ranked entranceslits. Of these, only one yields the absolute maximumof the SNR for the given conditions. The others may,however, give better rejection to the effects of back-ground changes, interferences by other absorbers,etc.

A second computer program models the instrument'soutput and is used to produce the output correlogramsfor the different expected backgrounds, interferentconcentrations, etc., and each of the successive optimummasks. Of these, the one giving the derived perfor-mance is selected, and the instrument's output programcan then be used to predict the instrument's behaviorunder any expected range of conditions.


z5-

3 -4

290

References

1. M. Bottema, W. Plummer, and J. Strong, Astrophys. J. 139, 1021(1964).

2. M. Bottema, W. Plummer, J. Strong, and R. Zander, Astrophys.J. 140, 1640 (1964).

3. A. R. Barringer, in Proceedings Third Symposium on RemoteSensing (U. Michigan Press, Ann Arbor, Mich., 1964).

4. A. R. Barringer and J. P. Shock, in Proceedings Fourth Sympo-sium on Remote Sensing (U. Michigan Press, Ann Arbor, Mich.,1966).

5. J. Strong, Appl. Opt. 6, 179 (1967).6. R. B. Kay, Appl. Opt. 6, 776 (1967).7. D. T. Williams and B. L. Kolitz, Appl. Opt. 7, 607 (1968).8. C. B. Ludwig, R. Bartle, and M. Griggs, NASA Contractors Report

CR 1380, U.S. Clearinghouse S.T.1 N69-31961 (1968).

9. M. M. Millan, S. J. Townsend, and J. H. Davies, U. Toronto, In-stitute for Aerospace Studies (Toronto, Ontario, Canada, 1969),Rpt. 146.

10. G. S. Newcomb and M. M. Millin, IEEE. Trans. Geosci. Electron.8, 149 (1970).

11. R. M. Goody, Atmospheric Radiation (Oxford U. P., London,1964).

12. R. P. Bauman, Absorption Spectroscopy (Wiley, New York,1962).

13. F. H. Lange, Correlation Techniques (VanNostrand, Reinhold,New York, 1967).

14. A. J. Moffat, J. R. Robbins, and A. R. Barringer, Atmos. Environ.5,511 (1971).

15. M. M. Millan, Ph.D. Thesis (U. Toronto, Toronto, Ontario,Canada, 1972).

16. A trademark of Barringer Research, Ltd., Toronto, Ontario,Canada.

Meetings ScheduleOPTICAL SOCIETY OF AMERICA

2000 L Street N.W,, Washington D.C. 20036

1-3 June 1977 CONFERENCE ON LASER ENGINEERING ANDAPPLICATIONS, Washington Hilton Hotel Information:J. W. Quinn at OSA or CIRCLE NO.59 ON READER SERVICECARD

6-9 June 1977 LASER RADAR, 8TH INTERNATIONAL CON-FERENCE, Drexel University, Philadelphia Information:J. W. Quinn at OSA or CIRCLE NO. 57 ON READER SERVICECARD

6-9 June 1977 INTERNATIONAL CONFERENCE ON MULTI-PHOTON PROCESSES, Rochester Information: J. W.Quinn at OSA or CIRCLE NO. 56 ON READER SERVICECARD

20-24 June 1977 FOURIER TRANSFORM INFRARED SPEC-TROSCOPY INTERNATIONAL CONFERENCE, University ofSouth Carolina, Columbia Information: J. W. Quinn atOSA or CIRCLE NO. 58 ON READER SERVICE CARD

9-11 August 1977 OPTICAL PROPAGATION THROUGHTURBULENCE, RAIN, AND FOG, OSA TOPICAL MEETING,University of Colorado, Boulder Information: J. W. Quinnat OSA or CIRCLE NO. 60 ON READER SERVICE CARD

9-14 October 1977 ANNUAL MEETING, Royal York, TorontoInformation: J. W. Quinn at OSA or CIRCLE NO. 64 ONREADER SERVICE CARD

16-18 January 1978 INTEGRATED AND GUIDED WAVE OP-TICS, OSA/IEEE MEETING, Salt Lake City Information:J. W. Quinn at OSA or CIRCLE NO. 61 ON READER SERVICECARD

30 October-3 November 1978 ANNUAL MEETING, Jack TarHotel, San Francisco Information: J. W. Quinn at OSAor CIRCLE NO. 53 ON READER SERVICE CARD

23-24 June 1977 OPTICAL FABRICATION & TESTINGWORKSHOP, Ethan Allen Inn, Danbury, Connecticut In-formation: J. W. Quinn at OSA or CIRCLE NO. 55 ONREADER SERVICE CARD

7-12 October 1979 ANNUALcana, Rochester, New Yorkat OSA or CIRCLE NO. 54 ON

MEETING, Holiday Inn/Ameri-Information: J. W. Quinn

READER SERVICE CARD


Documents

Dispersive correlation spectroscopy: a study of mask optimization procedures