spectral effects in interferometry

6
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA Spectral Effects in Interferonietry G. D. KAHL AND D. B. SLEATOR Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland (Received March 3, 1958) The influence on the fringe intensities of sources with known spectral energy density functions is analyzed both theoretically and experimentally for double beam interferometers of the Mach-Zehnder type. Simple relations between the spectrum and the fringe intensity and extent of the fringe field are presented. Special attention is given energy sources consisting of a primary source used with a monochromator. Experimental confirmation is demonstrated. I. INTRODUCTION HE quality of interference fringes resulting from T superposing "coherent" wave fields is known to be strongly influenced by the energy spectrum of the wave fields, as well as the source size and type of interferometer employed. A familiar example of the effect of the spectrum is the "white" fringe group, a relatively small number of fringes centered about a locus of zero path difference in an interferometer; the number of observable fringes decreases as the source spectrum is broadened. Truly monochromatic sources are nonexistent; however, they are approximated very well by spectral lines. But for some purposes, extreme intensity is necessary, and a finite frequency band source must be used to augment the intensity. The effects of source size and spectral purity on the interference in the Mach-Zehnder interferometer have been analyzed for several examples of particular symmetric spectral functions and source geometry. 2 The results indicated an equivalence between the influences of spectral width and geometrical extent of the source. A reduction of either one increases both the contrast and number of usable fringes, although the intensity is correspondingly decreased. In many applications, especially those involving short duration photography, the source geometry is limited for various reasons, such as the optical aperture, and the spectral distribution predominates in fixing the quality and intensity of the fringe field. It is with this condition that we are presently concerned. Although a total of six different forms of symmetric spectral distributions are treated in the above references, it is still difficult to predict the effect of a given source spectrum different from those already analyzed. Because of the sensitivity of the fringe field to the spectral distribution, a separate analysis is usually necessary for each type spectral function employed. In what follows, convenient expressions for the intensity of the fringe field in terms of the spectral density of sources of limited geometrical extent are exhibited by means of the Fourier transform. The simplicity of the method permits ready evaluation of a 1 E. H. Winkler, Naval Ordnance Lab. Report 1099 (1950). 2 F. D. Bennett, J. Appl. Phys. 22, 776 (1951). large variety of spectral functions, whether symmetric or not. Examples illustrate the strong dependence of fringe quality on the form of the spectral distribution. An indication is given of the limiting geometrical source size for which this simplified analysis applies, and a means of transition to the more complicated approxi- mate methods for larger sources used in the previous works is outlined. Experimental confirmation is presented. II. SUPERPOSED SCALAR WAVE FIELDS The observable quantity per unit time of two superposed fields W 1 (x,t) and W 2 (x,t) is the absolute square of the resulting amplitude W 1 +W 2 averaged over a time long compared to characteristic periodicities.' Calling this quantity S, we have S= lim(T-Vcoo) Tllf (W' 1 +W 2 ) (W 1 +W 2 )*dt, where the (*) indicates complex conjugate and the upper and lower limits are understood to be + T /2 and - T12, respectively. Thus, S=lim(T->oo )T1{ f Wl*dt+fW2w 2 *dt + (W 1 W 2 *+W 2 W 1 *)dt |, (1) or S= S 1 +S 2 +S 12 . The first two terms are the contribu- tions from each separate field, and the remaining term is the contribution resulting from the interaction, or correlation, between the two fields. If the last term is zero, the fields are "incoherent" with respect to each other. If a relation exists between the fields such that W 2 (,t)=constW 1 (x, t+T), where T is some fixed time interval, then S 12 is the "correlation" function,' 3The methods in this paragraph are similar to those in the first section of an article by D. Gabor, Revs. Modern Phys. 28, 260 (1956). 4 S. 0. Rice, Bell System Tech. J. 23, 282 (1944). TIn Mdf.tion, see Appendix A. 525 VOLUME 48, NUMBER 8 AUGUST, 1958

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Page 1: Spectral Effects in Interferometry

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Spectral Effects in Interferonietry

G. D. KAHL AND D. B. SLEATORBallistic Research Laboratory, Aberdeen Proving Ground, Maryland

(Received March 3, 1958)

The influence on the fringe intensities of sources with known spectral energy density functions is analyzedboth theoretically and experimentally for double beam interferometers of the Mach-Zehnder type. Simplerelations between the spectrum and the fringe intensity and extent of the fringe field are presented. Specialattention is given energy sources consisting of a primary source used with a monochromator. Experimentalconfirmation is demonstrated.

I. INTRODUCTION

HE quality of interference fringes resulting fromT superposing "coherent" wave fields is known tobe strongly influenced by the energy spectrum of thewave fields, as well as the source size and type ofinterferometer employed. A familiar example of theeffect of the spectrum is the "white" fringe group, arelatively small number of fringes centered about alocus of zero path difference in an interferometer; thenumber of observable fringes decreases as the sourcespectrum is broadened. Truly monochromatic sourcesare nonexistent; however, they are approximated verywell by spectral lines. But for some purposes, extremeintensity is necessary, and a finite frequency bandsource must be used to augment the intensity.

The effects of source size and spectral purity on theinterference in the Mach-Zehnder interferometer havebeen analyzed for several examples of particularsymmetric spectral functions and source geometry. 2

The results indicated an equivalence between theinfluences of spectral width and geometrical extent ofthe source. A reduction of either one increases both thecontrast and number of usable fringes, although theintensity is correspondingly decreased. In manyapplications, especially those involving short durationphotography, the source geometry is limited for variousreasons, such as the optical aperture, and the spectraldistribution predominates in fixing the quality andintensity of the fringe field. It is with this condition thatwe are presently concerned. Although a total of sixdifferent forms of symmetric spectral distributions aretreated in the above references, it is still difficult topredict the effect of a given source spectrum differentfrom those already analyzed. Because of the sensitivityof the fringe field to the spectral distribution, a separateanalysis is usually necessary for each type spectralfunction employed.

In what follows, convenient expressions for theintensity of the fringe field in terms of the spectraldensity of sources of limited geometrical extent areexhibited by means of the Fourier transform. Thesimplicity of the method permits ready evaluation of a

1 E. H. Winkler, Naval Ordnance Lab. Report 1099 (1950).2 F. D. Bennett, J. Appl. Phys. 22, 776 (1951).

large variety of spectral functions, whether symmetricor not. Examples illustrate the strong dependence offringe quality on the form of the spectral distribution.An indication is given of the limiting geometrical sourcesize for which this simplified analysis applies, and ameans of transition to the more complicated approxi-mate methods for larger sources used in the previousworks is outlined. Experimental confirmation ispresented.

II. SUPERPOSED SCALAR WAVE FIELDS

The observable quantity per unit time of twosuperposed fields W1 (x,t) and W2 (x,t) is the absolutesquare of the resulting amplitude W 1+W2 averagedover a time long compared to characteristic periodicities.'Calling this quantity S, we have

S= lim(T-Vcoo) Tllf (W'1+W2) (W 1+W 2)*dt,

where the (*) indicates complex conjugate and theupper and lower limits are understood to be + T /2 and- T12, respectively. Thus,

S=lim(T->oo )T1{ f Wl*dt+fW2w2*dt

+ (W 1W2*+W2W1*)dt |, (1)

or S= S1+S2 +S1 2 . The first two terms are the contribu-tions from each separate field, and the remaining termis the contribution resulting from the interaction, orcorrelation, between the two fields. If the last term iszero, the fields are "incoherent" with respect to eachother.

If a relation exists between the fields such thatW2(,t)=constW 1 (x, t+T), where T is some fixedtime interval, then S12 is the "correlation" function,'

3The methods in this paragraph are similar to those in thefirst section of an article by D. Gabor, Revs. Modern Phys. 28,260 (1956).

4 S. 0. Rice, Bell System Tech. J. 23, 282 (1944). TIn Mdf.tion,see Appendix A.

525

VOLUME 48, NUMBER 8 AUGUST, 1958

Page 2: Spectral Effects in Interferometry

G. D. KAHL AND D. B. SLEATOR

R(r), of the field Wi, where

R(r) =lim(T-*)o [ WiQ1) Wi* +T)

±W1*(1)Wl(1+)]dt. (2)

Physically R(r) represents the mutual intensity of adivided wave train, one part of which has been delayedby a time r with respect to the other.

Just such superposed fields occur in interferometersof the Mach-Zehnder and related types, where a beamWo originating from one source point P is divided intotwo part beams W1 and W2 which are again superposedin the neighborhood of a receptor point Q. If Wois collimated the part beams are plane wave trains;otherwise, they usually may be approximated as such,at least in a small region near Q. If the dispersiveelements are matched and the frequency dependenttransmission is proportional in each arm of theinterferometer, by taking Q to be the point of zerooptical path difference for the part beams, and definingr as

(3)we have

different examples of spectral density functions. Allof these spectral functions can be readily approximatedusing a continuum light source and a conventionalmonochromator. For simplicity, we assume that thedivided beams are of equal intensity (i.e., a=1) andthat the origin of our reference vector x is at the pointwhere the path difference (-S 2) is zero.

1. Monochromatic Spectra (Fig. 1)

The spectral energy is concentrated in one frequency.The spectral density function G(f) is

G 1(f)=A6(f-fo), (8)

where A is a constant and represents the Diracfunction. Therefore, R r) = A cos (27rfor) and R(0) = A;using Eq. (5) we have

S= 2A cos27rfor. (9)

If we substitute Eq. (3) for T and note that v=foXo,our intensity expression is

S= 2A coS2Err(f2-fl) -. /XO]. (10)

Equation (10) is the well-known formula for theintensity of interfering coherent monochromatic planewaves.

where is the unit normal vector to the plane wavefronts of Ws, o is a field vector originating from Q,and v is the wave velocity. The constant a is the ratioof the amplitude of beam 2 to beam 1, and depends onthe coatings of the divider plates.

We may express the average intensity of Eq. (1) as

(5)

2. Triangular Spectral Distribution (Fig. 1)

The spectral energy function is taken to be zero forall frequencies except those in the range fo-Af<f:fo+Af. Requiring that Af is smaller than fo, G(f) is

G (1) S r)

where the dependence of R on is implicit becauseT T(X).

The correlation function R(T) and the spectraldensity G(f) are Fourier cosine transforms of eachother, that is,

and

R(T) = f G(f) cos(27rfr)df

0G(f) = 4, R(-r) cos(2rfr)dT.

fO

ILf- to

(6)

(7)

By knowing G(f), R(T) can be found and hence theresulting field intensity can be predicted.5

III. EXAMPLES

We shall illustrate the dependence of fringe intensitiesformed in plane wave interferometers for several

6 Usually the procedure is reversed; G(f) is deduced fromobservations of R(r) as, for example, in detecting the spectrumof turbulence. G. I. Taylor, Proc. Roy. Soc. (London) 164, 476(1938).

co

f- fo

f- fo

FIG. 1. Theoretical intensity functions, S(r), for typicalspectral density functions. The ratio A f/fo is taken to be 0.1 forthe finite band widths.

526 Vol. 48

1= (f2-fl) ':�/V,

W2 (-t, t+ T) = aWl

S(2) =R(0)12+a1R(0)12+aR(,r),

Page 3: Spectral Effects in Interferometry

SPECTRAL EFFECTS IN INTERFEROMETRY

defined to be

G2(f)=B(Af+f-fo), (fo-Af<Pf o),=B(Af-f+fo), (fo< ffo+Af), (11)= 0, all other f.

Using this, R(T)= 2B(1-cos27rAfr) (cos27rfor)/(27rT)2and R(O) = B(Af) 2 ; the intensity function becomes

S= B(Af) 2[1+ (cos27rfor) (sin2 7rAfr)/ (7rzfT)2]. (12)

The second term resembles the interference diffractionpattern from a double slit. The high-frequency wavecos (27rfor) representing the fringes is modulated by thelow-frequency sin2 (7rAfr) and damped by the factor(7rAfr)-2 . It is symmetric about =0 and has a maxi-mum there; it is zero when either 2foT=n- orAft= , where n, m =±i (1,2,. ). The fringes appearin groups. The primary group occurs in the regionI-rAf[ I<1 and shows the greatest intensity; the other

groups appear symmetrically on either side of theprimary, and are just one half as wide as the primarygroup, diminishing in intensity as grows large. Thedistance from T= 0 to a fringe near the center of anyother group is almost precisely an integral number ofunit fringe widths measured near the center of theprimary groups; thus accurate measurements spanningany number of groups are possible if the fringes arevisible.

If the integral fo-G2 (f)df remains finite as AF isdecreased to zero, the primary group grows very wideand the monochromatic intensity function is ap-proached. This may be seen by taking the limit of (12)as Af--O while B(Af) 2 remains finite.

3. Rectangular Spectral Distribution (Fig. 1)

Again letting A be a constant we define:

G3 (f)=A, (fo-of~f<fo+Af)-; (13)= 0, all other f.

Proceeding as before, the intensity function becomes

S= 2Az f[l+ (cos27rfor) (sin2rAfr)/(2ArfT)]. (14)

This result is identical with that obtained previouslyin reference (2) by other methods for the limiting caseof a point source with the above spectrum. As in theprevious example, groups symmetric about =O arepresent. The high frequency is again cos(27rfor), butthe modulating frequency term is sin 27rzfr. Thezeros occur where 2

foT=p-4 and 2Afr= q, where p andq are nonzero integers. The primary fringe groupoccupies the region where 2fr I<1, which impliesthat for the same band width Af, this group is onlyone-half as wide as that for the triangular distribution inexample 2. Moreover, the damping factor is less severe,so the secondary groups will be more prominent.

Another distinction should be noted: Alternate (odd)groups are characterized by negative values of

(sin27rAifT), so the maxima of (cos27rfor) correspond tominima of the intensity function (14) in the oddgroups and to maxima in even groups; hence thefringes undergo a change of "phase" in adjacentgroups. Therefore measurements between a fringemaximum centered in an even group and a fringemaximum centered in an odd group can be in error byone-half unit fringe spacing, where unit fringe spacingis again defined as the spacing between adjacent maximanear the center of the primary group.

The extension to other examples is obvious. Therestriction a= 1 is only for convenience of exposition.

IV. SPECTRAL DENSITY FUNCTION

In practice, the effective spectral energy admittedto the interferometer is obtained from some primarysource of energy, which is filtered to limit the frequenciesto a suitably small band width. If P(f) is the spectraldensity function of the primary source and L(f) is thefrequency response of the filter, the resulting spectraldensity function is G(f) = constP(f)XL(f). Themethodof determining G(f) will be carried out for the prismmonochromator used in the experiments.

The monochromator consists of entrance and exitslits, collimating lenses, and a prism dispersing elementwhich is assumed linear over a small frequency range.With the primary source filling the entrance slit withuniform intensity, an image of the entrance slit isformed near the exit slit for each frequency presentin the primary source. Each such image is displacedlaterally from the others by an amount depending onthe angular dispersion of the prism. But only theenergy in that portion of the entrance slit imagesoverlapping the exit slit passes into the optical system.The monochromator is adjusted so that the image atfrequency o is centered on the exit slit; the imageformed by frequency f is displaced laterally a distanceg(f), found from

g(f) e [(f-fo)XoF/foJ/ (dX/dO)X,, (15)

where F is the focal length of the focusing lens, Xo isthe wavelength at frequency o, and d/dO is theangular dispersion of the prism.

If the exit slit and entrance slit images are rectanglesof width 2 and w respectively, the area of overlapof the exit slit and the image at frequency f is propor-tional to the width of overlap, L(f), given by

L(f)= (wI+w2)/2-g(f), Ig(f)J < (wl~w2)/2,

L(f) = O Ig(f)I> (W1+W2)12. (6

L is symmetric about fo. The cutoff frequencies arethose which make I g(f)I equal to (+w 2)/2. Thespectral energy density passing the exit slit is

G(f)= constP(f) XL(f). (17)

If P(f) is constant over the frequency range of L(f),(corresponding to many actual examples of continuous

August 1958 527

Page 4: Spectral Effects in Interferometry

G. D. KAHL AND

-I to)I 1.0)

:tuanu us itatr V

(,)

FIG. 2. Fringe photographs using (a) triangular and (b) rec-tantular spectral functions. The left edge is centered on the primarygroups; secondary groups are bracketed.

sources), then G(f) = KL(f), where K is constant.Under these conditions, by adjusting w and 2,several different shapes of G(f) are possible. These are

(la) Wl=W2, K finite; G(f) is a symmetric triangularfunction with apex centered at fo;

(lb) w 1=w 2-40, K-- o; G(f) approaches the Diraca function;

(2a) W17-W2, (W1, W2 ZO); G(f) is a trapezoid centeredon fo;

(2b) 2 finite, wr--O, K-oo; a rectangle centeredon fo.

The foregoing depends on the area of overlap beingproportional to the width of overlap and P(f) beingconstant between the cutoff frequencies of L. If theentrance slit image exhibits strong curvature (whichsometimes occurs in prism type monochromators) amore detailed treatment is necessary.

V. EXPERIMENTAL RESULTS

The spectral effects on fringe intensity were testedusing the Mach-Zehnder interferometer adjusted for

Q

9

t,

Ati

.S

.S

Is

I

E2

2!

200 ;I~-

15C

I I,,< - ""~~~~~~~~(a

(b) \Z __

'cc~ ~ "-

,I I~~~~~~~~

40 80 120 160Half width of spectral distribution A

FIG. 3. Theoretical and observed number of fringes in theprimary group for (a) triangular, and (b) rectangular spectralfunctions centered on 5460 A'.

fringes perpendicular to the plane of centers of theelements (the "preferred" adjustment.) Film observa-tions were made on the conventional plane passingthrough the "axis of rotation" of the instrument andoriented nearly perpendicular to both emerging beams.The primary source was a frosted tungsten light bulbused with a 60° carbon bisulfide prism monochromator(Fi= 13 in.) adjusted for the 5460 A green mercuryline. The angular dispersion at minimum deviation forthis wavelength was computed to be 25 500 A/radian.

Pictures were obtained for various slit width settings,maintaining equal widths of both entrance and exitslits for each picture, thereby approximating thetriangular spectral function of Eq. (11), correspondingto the intensity function of Eq. (12). A representativepicture is shown in Fig. 2(a). Since the half width(Af) of the frequency band is related to the slit settingsby Eqs. (15) and (16), the predicted number of fringesin the primary group can be compared with the experi-ment. The results are shown in Fig. 3.

.

-0)

0'

00)

.

+2 F

+1

I Secondary Groups

X . 1-1 -L Primary Group _ 1...rIt.."

-2~~~~~~~~~6 , , . . ,~._ . I-6

-60 -40 -20 0 + ro +40 +60Fringe Number

FIG. 4. Measured deviation of fringes from "ideal" position vsfringe number for the rectangular spectral function. The "ideal"position assumes constant fringe spacing.

To approximate the rectangular spectral functionthe entrance slit was made relatively wide (0.135 in.)while the exit slit was kept quite narrow (0.006 in.and 0.003 in.); a representative picture is seen inFigure 2(b). The comparison between measured andpredicted number of fringes in the primary group arealso shown in Fig. 3. To test the predicted phase changeof the fringes in alternate groups for the rectangularspectral function the fringe positions across severalgroups were measured. The difference between themeasured positions and that computed by assumingthe fringes equally spaced (with the spacing found atthe center of the primary group) is shown in Fig. 4.The fringe position at the center of adjacent groupsis seen to be displaced by about one-half fringe spacing.Since the slope of the curve in Fig. 4 measures thechange in fringe spacing, we observe that the largestchanges in spacing occurs between the groups; thespacing is nearly constant over most of the primarygroup and at the center of any given group.

..

.

528 Vol. 48D. B. SLEATOR

Page 5: Spectral Effects in Interferometry

SPECTRAL EFFECTS IN INTERFEROMETRY

VI. SOURCE SIZE

The intensity functions considered apply to pointsources or a small element of a finite source; for afinite size source, the total intensity is the sum ofcontributions from all elements of the source. Finitesource effects for monochromatic spectra in theMichelson or Mach Zehnder interferometer type havebeen discussed previously2 6 7 as well as those forGaussian type distribution in references 1 and 2. There-fore we will merely demonstrate that source size hasnegligible influence on the present experiments, andindicate the method for more detailed study. Anexample of the equivalence of source size and spectrumis given in Appendix B.

The function r depends on the location of the sourceelement. For the aforementioned "preferred" orienta-tion of the interferometer,8 the associated with anarbitrary source point is

r(4) = (0) coso, (18)

where q is the angle between the central and an ar-bitrary source point viewed from the interferometercollimating lens.

The fringes of each family for a given source pointlie between alternate pairs of zero's of the functioncos27rfoT(o). The central (zeroth order) fringes fromeach family are coincident independent of 0; however,fringes of corresponding nonzero order from differentfamilies are not coincident because of the differingvalues of 0.

Requiring that for every family the Nth fringesfrom central shall all concide to within a small phase krestricts the permissible values of k to 0 <•m, implying

fo I T(r )-T(0) < k. (19)

Since cosq051-02/2 and fT(0)cN, using Eq. (18)we have

Nom2/2 = k. (20)

For all experiments, 0t m was smaller than 0.013 radian,corresponding to a maximum slit length of in. anda 29-in. focal length of the interferometer collimator.Allowing k to be 0.1, we find that there should be 1180fringes on either side of the central fringe fulfillingour criterion. Hence for these experiments the influenceof source size (independent of spectral effects) shouldbe negligible.

It should be observed that where a precise integrationover the source is required, the spectral density functionvaries for each transverse element across the exit slitwidth. In fact, for each strip element parallel to thejaws, the function G(f) is rectangular, centered on afrequency which depends linearly on the transversedistance from slit center to the strip element. However,

6 F. D. Bennett, J. Appi. Phys. 22, 184 (1951).7 G. D. Kahl and F. D. Bennett, J. AppI. Phys. 23, 673 (1951).8 F. D. Bennett and G. D. Kahl, J. Opt. Soc. Am. 43, 71 (1953).

performing the indicated integration for equal entranceand exit slit widths with the approximation cosq51yields the same intensity function found by assumingthe entire slit an effective point source with the tri-angular power spectrum.

APPENDIX A

The relation between the correlation function andspectra of real functions is discussed in detail inreference 4. We can include complex functions, anda short derivation is given here.

We assume that W(t) is zero outside the interval-T/2<t<T/2 and can be represented by the Fourierintegral

W(t) = fwA (f)e2"rftdf,_0

where

A (f) = fw (t)e-2riftdt

Then

f W(t)W*(t+,)dt=J fW(t)f-T/2 -. 0 oo

X [A*(f)-2,Tftes21ifT]dfdt.

(Al)

(A2)

(A3)

Interchanging the order of integration and making useof (A2) yields

T/2 21ofT1 W(1)W*(t+r)dt= JA (f)A*(f)e-2 ifTdf; (A4)-T/2 f.

thus, the average value of [W()W(t+r)+conj.] is

-T/2

= (4/T)f A (f)A*(f) cos27rfrdf, (A5)

since A (f)A * (f) and cos27rfr are even functions of f.By letting T approach o, and defining:

G(f) = lim(T-o )E(4/T)A (f)A*(f)], (A6)

the left side of (A5) becomes R(r), corresponding toEq. (2) of the text, and we have

(A7)R(r) =f G(f) cos2'rfrdf.0

It follows immediately that

G(f) =4f R(T) cos2rfTdr.0

(A8)

529August 1958

Page 6: Spectral Effects in Interferometry

G. D. KAHL AND D. B. SLEATOR

APPENDIX B. SPECTRUM VS SOURCE SIZE

The equivalence between a monochromatic, finitesource and a finite spectrum band, point source can beeasily demonstrated with a simple example. We considera monochromatic circular source of frequency fi andradius R centered on the axis of the interferometercollimator of focal length F. Using Eqs. (10) and (18),the field intensity is

R

S(T) = 2r D (r) { +cos[27rflr (0) coso]) rdr, (B 1)

where D(r) is a measure of the luminous intensity per

unit area of the source, and

r=F tanq5. (B2)

By letting D be a simple function of r such that

D(r)=D(0) cos3 0, (B3)

where D(O) is a constant, Eq. (21) integrates directly.We have

S(r) = 2D(0)h{ 1+ (cos[27rfir (0) (1- h)])X (sin[27rfir (0) k])/ (27rfiT (0)) },

where1- 2h= cos[tan'l (RIF) ].

(B4)

(B5)

By choosing f=fo/(1-h), we see that except for amultiplicative constant, Eq. (B4) has the same formas Eq. (14); the two have the same damping when

h/(1-k) = Af/fo. (B6)

Under these conditions the fringe pattern for thecircular monochromatic source is indistinguishablefrom that of the small source with rectangular spectraldistribution, given by Eq. (14). The values of D(r)given by (B3) differ very little from a constant, decreas-ing only 5% for f as large as 10°; usually the instru-ment aperture limits 4 to considerably smaller angles.

530 Vol. 48