JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Spectroscopy from the Point of View of the Communication Theory. II. Line Widths*A. G. EMSLIE AND GILBERT W. KINGt

A rt/r D. Little, Inc., Cambridge, Massachiusetts(Received November 21, 1952)

A spectrum can be considered as a signal composed of a superposition of randomly spaced Cauchy linesof the form 6/( 2+x 2 ). This signal is passed through a spectroscope which acts as a noisy channel causingdistortion due to finite aperture, slits, prism absorption, and electrical band width. The loss of information inthe Hartley sense can be described by the increase in width of lines in the output signal. Several definitions ofline widths are compared for ease in experimental measurement and calculation as functions of a and theinstrument characteristics. The different widths do not vary in the same manner over the range of experi-mental conditions. In particular the resolution width departs markedly from the others. Because the intensitydistribution is very complicated, the width at half-height and the median width are difficult to calculate. Thelatter is approximately the same as the root-mean-square width which can be more readily calculated. Themost satisfactory width for both experimental and theoretical determination seems to be that of the equiva-lent Cauchy line shape. This is easy to measure from the observed area, which is merely a product of ele-mentary functions and the filter characteristic.

INTRODUCTION

IN Part I the information in a spectrum was con-sidered from the classical point of view of resolving

power. In Part III,2 we shall discuss the information interms of the more fundamental theory of Shannon andshow how the information depends on the separatefactors of signal-to-noise ratio and "effective line width."In the present paper we consider only the line-widthfactor and take the point of view of Hartley that theinformation is given by the number of line widths thatcan be fitted into the total width of the spectrum,neglecting the effect of noise. Clearly, for wide slits, thisis determined by the width of the triangular slit func-tion. For narrow slits and lines, the amount of informa-tion depends on the number of diffraction patternscontained in the spectrum, giving the maximum in-formation from this standpoint. When the lines arenaturally broad, the amount of information is limitedby their own width 3.

In the following, we investigate the width when allthree factors are operating. These results are a quanti-tative measure of the increase in line width due to theexperimental requirements of finite aperture and finiteslits. The method will be extended to include filters andfinite rate of scanning.

Conversely, the results show to what extent thenatural width can be estimated from the observed.

OUTPUT SIGNAL

The original signal is assumed to be composed ofCauchy lines of the form: 3/ir(32+x2). The outputsignal, the intensity distribution in the plane of the

* This work was done in part under a grant-in-aid from theAmerican Cancer Society upon the recommendation of theCommittee on Growth of the National Research Council.

tPresent address: International Telemeter Corporation, LosAngeles 25, California.

I G. W. Kig atd A. G. Enisle, J. Opt. Soc. Am. 41, 405 (1951).2 G. W. King and A. G. Emslie, J. Opt. Soc. Am. 43, 664 (1953).t The effect of Beer's and Lambert's law will be introduced in

another paper.

exit slit, was given by Eq. 4 in Part I. The nontrivialpart, rewritten in units characteristic of the diffractionpattern§, fX/wD, is

1 X o (, B d, x)= I dwf dvf du

4d2J_ O __ , 7r[5I+ (x-u)2]

sinh 2B+sin 2 wXSI(u-V)S 2(V-W) (1)

where is the natural width of the line, B is the prismabsorption, and 2d is the slit width. The function S(x)describes the shape of the slits, which usually is

S(x) =1, -d<x<d, zero elsewhere. (2)

The evaluation of this expression is difficult; but sinceit is a multiple convolution, by Borel's theorem, theFourier transform of F(x) is simply the product of thetransforms of the individual factors:

1 X+,()=-f de-7\B(X)

27r

sin 2 rd sinhB (2- I I )

(erd)i 4B1T1<2 (3)

The variable is proportional to the angular frequencyco of the output signal.

It will be noted that, in the case of no prism absorp-tion, B = 0, the transform of the diffraction patternbecomes (2- |j).

DEFINITIONS OF LINE WIDTHS

Various kinds of line widths are used to describesignals. It is desirable to choose one which adequately

§The wavelength of the radiation is , the aperture of theinstrument is D, and its focal length, f.

[The relationship is = (7rDv'/Xf)r, where v' is the linearvelocity of scanning the optical spectrum in centimeters per secondat the exit slit.

658

VOLUME 43, NUMBER 8 AUGUST, I1953

SPECTROSCOPY AND COMMUNICATION THEORY

characterizes' the shapes of spectrographic lines, isreadily evaluated theoretically, and is easily measuredexperimentally.

Resolution Width a,

The width, 2a,, of a line between the points of in-flection ('F(x)/Ox 2=0) is useful as a criterion of re-solving power,3 as it is the smallest separation at whichtwo lines can be seen as separate peaks. Values of thisquantity4 were given in Part I. In general, a graphicalmethod had to be used.

The caluclation is no easier in transform space sincea, is a solution of the equation

2f dr cosajT sin2rd e-FI sinhB(2- T ) =O. (4)-2

The width a, is the easiest of all to measure experi-mentally, but is one of the most difficult theoretically.Unlike any other, it is independent of the "base line"which is usually difficult to establish. As we shall seebelow, it is not an absolute criterion of the informationcontained in the spectrum.

Width at Half-Height, a,

The width at the point where the intensity (or outputvoltage) is 2 (or I/e or 1/v2) of the central ray is com-monly used. It is subject to uncertainties in the locationof the base line but, like a,, it depends on only one pointof the curve. Moreover, this width is not altogethersuitable for patterns in spectroscopy caused by diffrac-tion in which there are several maxima. It is alsodifficult to evaluate theoretically as it is a solution of asimilar transcendental equation

f,dr(2 cosa,7 - 1)1' (T) = 0-

Median Width a.In order to average over the whole line, it is some-

times useful to consider the width out to the half-areaof the absolute¶ value. It is not difficult to evaluate amfrom experimental data, but the theoretical calculationsare difficult in both x space and r space:

f dxI*(x) I = 2 dxF(x) | (6)0

orr2fdTQl (T) (sinamr)/ (7) = i.. (7)

The Root-Mean-Square Width a,

Gabor5 has suggested a definition of signal width foruse in communication theory, analogous to that of

3 C. M. Sparrow, Astrophys. J. 44, 76 (1916).Referred to as a in Part I., J. Opt. Soc. Am. 41, 405 (1951).¶ As long as the signal is in the form of radiation, *(x), it is

always positive; but when the signal is in electrical form, after thedetector and a filter, it might be negative.

6 D. Gabor, J. Inst. Elec. Eng. 93 Part III, 429 (1946).

quantum mechanics, which in our example would bethe square root of

+00 +80

a 2 (6, B, d)=J' dxT2(x)x2jf dxTI'(x).9~~~~~~0 -00

(8)

Experimentally, this is more tedious to measure, but onthe other hand, it is more characteristic as it involvesthe shape of the whole line rather than one point.Theoretically, this definition is very attractive for, asGabor pointed out, it can be calculated from theFourier spectrum of the signal and its derivative whichare elementary functions in our case.

ag2= f d (d8 / dP2(r) (9)

or

d (r,) (d2 dr2)jf dT 2'(T). (10)

These integrals can be expressed in closed form. It issimpler however, to integrate numerically. In thespectroscopic case there is a further advantage that thelimits of integration of Eq. (9) are finite, (I | <2), whichsimplifies the numerical methods required.

A difficulty which arises with this definition is that ifthe Fourier spectrum has discontinuities, as for example,if the signal is passed through a filter with finite cutoff(as can be done with correlation functions), a, becomesinfinite.

Width of the Equivalent Cauchy Line a,

The area of the line divided by the intensity of thecentral ray has the dimensions of width. It is the widthof the rectangle of unit intensity with the same area asthe pattern. This is not particularly significant. Bymultiplying this ratio by a suitable factor, one canobtain a width of an equivalent curve of the same area,e.g., a Gaussian curve. A most appropriate definition is

+80

dx't (x)2 JO\(0)

=2-7r (0) (°)

(0) 1= = (11)

I . I~~~~~ d4 (,r) 2 f d4QT

the half-width a, of the Cauchy line of the same areaand same central ray. This is the easiest of all widths tocalculate, and one of the easiest to measure experi-mentally. The numerator in the theoretical case isnormalized to sin2B/4B which, when B=0, is 2j underall conditions.

The difficult part of the calculations is the intensity ofthe central ray (the denominator). This can be obtained

659August 1953

A. G. EISLIE AND G. W. KING

in closed form when the spectroscope alone is considered,and simple numerical integration used when variousfilters are included. In contrast to a, the width a,remains finite if the Fourier spectrum has discontinuities(as, for example, after filtering). Experimentally, itinvolves measuring the maximum height and the areaunder the curve, which is simpler than the determina-tion of am or a. Table I and the calculations for actualline shapes show that a, varies in the same way as a,.

In other discussions of the performance of a spectro-scope it is convenient to approximate the output lineshape by the Cauchy curve a/7r(ac +x2 ), with the samearea and central ray; in which case 4+ ()= 2 exp (- a,.r)and a =-a.

Since the original signal is represented by a Cauchycurve of half-width , the relation of a to a is a simplemeasure of the distortion due to the spectroscope.

COMPARISON OF WIDTHS

Simple Patterns

Table I gives the ratios of the above line widths for avariety of simple functions, the last three of which arethe limiting cases of spectroscopy; the diffraction

TABLE I. Relationships among the various line widthsfor a number of simple "line shapes."

ac/a. ac/ah ac/a. ac/aa,

Rectangle 0.637 0.637 1.274 1.104Exponential 0.919 0.919 0.919 0.901Gaussian 0.798 0.677 1.182 1.128Diffraction 0.767 0.718 0.635 1.155Triangle 0.550 0.637 1.087 1.008Cauchy 1.732 1.000 1.000 1.000

graphical method and is replotted in Fig. -1, multipliedby Vl in order to show how it converges to the otherhalf-widthsatlarge a. The width at half-height ah(6, B, 0)was obtained from examination of the numerical valuesof H(x); the median width am(a, B, 0) from numericalintegration. The root-mean-square width, a,, was ob-tained by integration of the transform of H(x) to give

a(a, 0, )>-=a(-e-4 5 +1+4+86 2 )i/

(-e- 4 6 +1-46+86 2)i. (12)The Cauchy width is obtained by evaluating** H(O):

(13)

Figure 1 shows the relationship between the variouswidths as a function of , d being zero in this case. Atlarge a (5), it is possible to show analytically that all thewidths approach asymptotically the straight line

Ala=a=am=ao=a =(1+ 6+ )- (15)Only when 26 is negligible does this reduce to thelimiting value of a for the Cauchy line. The curves havea finite slope at = 0. For these line shapes the widthsa, and am are almost identical.

Narrow Slits and Finite Prism AbsorptionThe various widths, as functions of , have been ob-

tained for finite prism absorption in a similar fashion.The widths a and a, can be given in closed form

a,2(6, B, 0)= ( 2-B 2 )

X{e- -

2(cosh4B- 1)/B2-a (sinh4B)/B- 1}

{e-4 6 - 2(cosh4B- 1)/B2+a (sinh4B)/B- 1}'(16)

pattern, sin2x/x2 , for an infinitely narrow line with in-finitely narrow slits; the Cauchy curve for a finite lineand narrow slits; and the triangle for purely geometricoptics where the slits are wide.

It can be noted that the more manageable width a, iswithin 16 percent of a. There is a serious lack of corre-lation of a, with the other widths, and of all of themwith the resolution a, so that line widths must be usedwith care in estimating resolving power.

Any of them can be used in principle to determine thenatural width . In the following sections we shallcalculate and compare the different widths for variousexperimental conditions.

Narrow Slits

Intermediate between the limiting case of a diffractionpattern (a=0, d=O) and the Cauchy line (= o, d=O)is the intensity pattern at narrow slit (d=O), H(x), forfor which an explicit expression was given in Part I. Thesecond factor in the last expression in Eq. (3) for thetransform is unity, and some of the widths can be givenin closed form.

The resolution width a vas obtained in Part I by a

a, (6, B, 0)= ( 2-B 2)/

For a=0(a-B coth2B+Be-2 csch2B).

a, (0, B, 0) = cothB/B.

(17)

No Diffraction

Another immediate case lies between the Cauchy andtriangular form. Since a and d are measured in units ofXf/wD, large a or d corresponds to the case whendiffraction is negligible. The transform, Eq. (3), becomesnegligible before the finite cutoff at = 2, so that form-ally the limit of the various integrals can be extended toinfinity, making them more tractable.

In this case the line shape itself can be given in closedform so that a can be obtained by numerical evalua-tion. The limiting expression for large d is

a,=d+ (1/r)(log2d-2 log2d/ah+ah2/4d2) (18)= d+ (0.31830 log2d/a-004335)6. (19)

** The transform of 11(x), Eq. (I=4O) for BmO can be invertedto give

H(i , 0; x)=Re(1+2iz-ei)2z, z=x+iS (14)from wvhich Eq. (13) followvs directly by substitution of x=0.

660 Vol. 43

a,=611(e-11 126).

SPECTROSCOPY AND COMMUNICATION THEORY

This is only in error by 4 percent at d = 5. For small d,

ahL2=82 +2d2 . (20)

The closed expression for the line shape can also beintegrated, in this case, to give the following formulafor a,,, which was solved numerically:

G[ (am+ 2d)/] - 2G (am/6)+G[

where'(am- 2d)/8]= 7r(d/6)2

G(x)= fdxfdxfdx/(1+x2)

= 2 (X2 - 1) tan-2x-x log(1+X2 ).

For d large the limiting law is

a,,,= (2 -V2)d+ (2/7) (v2 - 1)X (1- am2 /8d2- loga /2d)b,

= 0.585d+0.5768

(21)

limiting laws with those for the other widths:

a,= (2/7r)d +(2/7r2) log2d/5, d>>and

a. 2 -=62+4/3d 2 , d<<.

(32)

(33)

In Part I, it was shown that the resolution width,a,, is

a.2= 152+4/3d2 (34)

for all values of d. In Fig. 2, V3a is plotted to show theagreement at d=0 (a Cauchy line) with all the otherhalf-widths, as in Fig. 1. It is seen that V3-a, rapidlydiverges at large d. In fact, even a, alone crosses a, atd= 106. This clearly means that Sparrow's undulation

(22)

d>> (23)

(24)

which is in error by only 5 percent at d = 8. (The resultfor a triangle is am= (2-'2)dj.) For d small

a.2= 52+ 2 'd 2 d<<. (25)

Equation (8) for the root-mean-square width, a, inx space is difficult to evaluate, but Eq. (9) in r spacecan be integrated in the case of no diffraction since it isnot necessary to cut off the integrand at T=2. Theresult is

a, 2= 62 +d2fl (a/d)/f2(/d),where

f'(x)= (1/15)[42+ (32- 120x2+40x4 )x-l tan-12/x+ (-124+120X 2-80x4)x-l tan-ll/x+ -40+5x 2 - 3x4) log(x2+4)+ (40 - 60x2+ 12x

4) log (X2+ 1)

+ (10X2 -9x 4 ) logX2 ],

and

(26)

(27)

f 2 (x) = [(16- 12x2)x 1l tan-12/x+ (- 8+24x2 )

Xx-l tan'l/x+ (- 12+x2) log (x2+4)+ (12-4x2 ) log(x2 +1)+3x2 logx2 ]. (28)

a

FIG. 1. The various line widths of a line with natural width aafter passing through the spectroscope, with narrow slits.

When d is largea,2= 62 + 2d

2 d>>6, (29)

and when d is small

a 2=

2 + d2, d<<K. (30)

It is seen that the formula for a, is the same as foram when d is small. In fact, the values of a, and ace arevery close for all values of d/6.

The equivalent Cauchy width is easily obtained:

a,= d( tan-12d/8- [log(1 +4d2 /82)]/ (4d/8) }-l. (31)

Although this is a simple, explicit form from which a,can be obtained exactly, it is of interest to compare the

criterion is unsuitable for shapes of this type, since adoublet must always be resolved at a separation of 2a,.Further investigation shows that, in the case of nodiffraction, the above equation (derived from Sparrow'scriterion) is valid only up to d=-\r283. Beyond that theseparation 2a8, at which the presence of two lines isrevealed by two maxima, is given by

ar 2 (2d22

2 )+1 (4d2±82)>, d>26. (35)

The two incipient maxima now appear, not at the mid-point, but at a separation 2x, where

x2 - 2 (2d2-82 ) - ' (4d2 82 ) l, d>v25. (36)

The quantity a, is the proper measure of resolving

661August 1953

AND G. W. KING Vol.43

3

2

AND SLITS

0 1 2 3 4 5

FIG. 2. The ratio of the various line widths to the natural widtha after passing through the spectroscope, when the diffractionpattern is narrow.

power. It is shown in conjunction with a in Fig. 2. Thisbreakdown of Sparrow's criterion also occurs withdiffraction and slits (discussed in the next section) andseems to be caused by the discontinuous derivatives ofthe triangular slit functions.

Sharp Lines

If the natural width is small, then 8--O, and the firstfactor in Eq. (3) becomes unity. Again, it is possible togive a closed expression for the contour of the line,allowing ah and am to be obtained by numerical methods.The width a was computed by numerical integrationof Eq. (9). The Cauchy width is the only one expressiblein closed form:

a,.=4d2/[4d Si(4d)-2 sin22d-('y+log4d-Ci(4d)], (37)

in which y is Euler's constant. For small d

a,.2 1+ (4/9)d 2, (38)and for large d

a,.= (2/r)d+ (l+ y)/7r2+ (1/70) log4d. (39)

This is good to 3 percent at d=4.The resolution width a for =0, as calculated by

Sparrow's criterion, was given as a function of d inPart I (Fig. 6) and oscillated violently. Further investi-

gation shows the curve has a vertical tangent at d r.Beyond this, the undulation condition is inadequate forthe reason mentioned above in the case of no diffraction(d>-). The true resolution, a, obtained by super-imposing contours of two equal lines is plotted in Fig. 3,and is found to be within a few percent of the ultimatevalue for the triangle (d large), a,=d.

GENERAL CASE

The line shape for the general case of finite lines,diffraction, and slits can be given in a closed form; but itinvolves the exponential integral of a complex variablewhich has not yet been tabulated. Thus, it is impractical

7.

6

4

a

d

FIG. 3. The various line widths of a very narrow line afterpassing through a spectroscope with finite aperture and slits ofwidth d diffraction units.

to obtain a and a,, directly. We have seen above thatin the three special cases, neglecting one effect at a time,a,,, is essentially equal to a,, so we can conclude that thisidentity will hold for spectroscopic lines in general. It isrelatively easy to compute a, in the general case, andthe results are shown in Fig. 4. They can be comparedwith a which is easier to obtain experimentally. Theresolution width, a, was given in Part I. For reasonablylarge and not too large d, the true resolution width, a,will be very close to the values given for a.

The most practical width is, however, a, which iseasy to obtain both theoretically and experimentally,and results for the general case are given in Fig. 5. The

662 A . G . EIS LI E

SPECTROSCOPY AND COMMUNICATION THEORY

theoretical expression is

ac= 8d2 /[F(-5+2id)-2F(-b)+F(--2id)]with

F(u) = eu+E(u)-uE(u)+u lnu-lnu,where

E (u) = f due-u/u

(40) 7

6(41)

(42)

u=2(ix-b).

This width has the further advantage that it can beeasily calculated for more general situations of other

a5

esetal the same.0.,

.6 5 6 .7u/ ,AUGHY Ll NE~~~.6

O~~~~~~~~~~~~~~~~7 1 A 4~~~~~~

natural line shapes, modification by Beer's law, and byelectrical filtering, by numerical integration in r space.

To facilitate the use of the graphs for estimating thenatural width, we have marked on the curves fora0(am) and a6 the relative intensity at which the widthin question cuts the contour. (These were obtained bycarrying out the Fourier transform integral for theparticular values x= a.)

a

11 | ' d

FIG. 5. The equivalent Cauchy line width as a function ofnatural width 8, slit width d, and finite aperture.

ILLUSTRATION

The general question of how much information is ina spectrum often in practice devolves to the questionof what the natural width of the lines is, or how much ofthe apparent width is due to the distortion of theinstrument.

The figures allow one to make this estimate by meas-uring one or more of the widths discussed. Since the slitwidth is known, and the scale factor to d is ?rD/Xf, onecan read from the graphs the natural width in theseunits. If several of the widths are measured, addedinformation can be obtained, For example, an isolatedvibrational band of benzene at 40K was measured, withslits of 195 microns. Since the aperture of the instrumentwas D=5.6 cm, its focal length 27 cm, and the wave-number is 1015 cm-1, the equivalent value of d is 6.3diffraction units. The observed widths were ah=6.0,

am=3.9, a, =4.0, and a,=4.5. Reading from the graphsall these values confirm an estimate of a very smallnatural width (less than 3=0.5). The method of con-verting the value of a in diffraction units to wavenumbers was given in Part .1, tt

ft The following typographical errors should be noted: Equa-tion (I-36) should read dv/dx'= (8kifvP 2 )-' and vo for NaCl is125 cm-.

663August 1953