JOURNAL OF TilE OPTICAL SOCIETY OF AMERICA

Measurement of the Total Illuminance in a Diffraction Image. Part II. Line Sources

RICHARD BARAKAT, ALFRED NEWMAN, AND RONALD HumPHREYs

Optics Department, Research Division, Itek Corporation, Lexington, Massachlmsetts 02173

(Received 6 February 1964)

A method of measuring the total illuminance of the diffracted image of an incoherent line source is de-scribed. Measurements on a 6-in. focal length objective are reported. Experimental measurements of thiskind give a quantitative measure of lens system performance in terms of total illuminance of the diffractedimage of a line source. The results of theoretical diffraction calculations based upon lens design data werecompared with experimental values and showved good correlation.

1. INTRODUCTION

MEASUREMENTS of the total illuminance for apoint source image were recently reported.' The

purpose of the present paper is to consider the corre-sponding measurements in which the point source isreplaced by an incoherent line source. We restrict our-selves, as in the previous paper, to the on-axis case.

The total illuminance for a line source, also termedcumulative line spread function, 2 Li(vo), is defined asthe amount of energy contained in the diffractionpattern in an infinite slit of half-width vo; thus

Li (tfo) =i N T(v_) d vz, (I. 1)

where T(V.) is the line spread function in the x direction.Since we are only concerned with rotationally sym-metric systems, it does not matter in which directionthe line spread function is taken. The x direction istaken for convenience only. N is a normalizing constantsuch that Li (x) = 1. The dimensionless parameter vo isgiven by

vo (r/XF)z(, (1.2)

where X is measured in microns, F is the f/number ofthe system, and zx is the lateral distance from the originof the line spread function and is also measured inmicrons.

Barakat and Houston 2 have shown that Li(vo), forrotationally symmetric systems can be computeddirectly from a knowledge of the transfer function T(w)via the relation

2 2 sine~owLi(vo)=- T(co) dw, (1.3)

where x¢= 2 is the cutoff frequency of the lens in dinien-sionless units. The cutoff frequency in dimensional unitsis directly related to the f/number of the system by

92, = 111\F, (1-4)

where 1l, is measured in lines/iniii and X\ iii 111111. F'orexample, an f/5 system operating in helium d-light

R. Barakat and A. Newman, J. Opt. Soc. Am. 53, 1365 (1963).2 R. Barakat and A. Houston, J. Opt. Soc. Am, 54, 768 (1964).

(5875.5 A) has a cutoff frequency of approximately 340lines/mm. The transfer function required in (1.3) canbe obtained directly from the lens design data 3' 4 ; con-sequently, we can compare the experimental resultswith theoretical diffraction computations. This topicis treated in Sec. 3.

2. DESCRIPTION OF THE APPARATUS

The apparatus used in this experiment is a modifi-cation of the apparatus used to measure the totalilluminance in the diffracted image of a point source.'A drum carrying circular apertures of successively in-creasing diameters was used to intercept the image ofa point source. The photometered radiation flux of theseapertures then yielded the relative total illuminance ofthe diffracted image. For the present study, we areinterested in the measurement of the total illuminanceof the diffracted image of a line source. Since the axialimage of a point source is circularly symmetric, itmakes no difference in which direction the image isscanned. On the other hand, the axial image of a linesource is symmetric in only one direction: perpendicularto the line spread function. Thus, scanning can only bedone parallel to the image with apertures of equallength and successively greater width, where the widthincreases perpendicular to the line-source image. Thephotometered flux of these apertures then yields therelative total illuminance of the diffracted line source.

In order to perform the experiment without anycostly modification of the existing apparatus, a slitaperture 0.240X6.000 mm was used to mask thecircular apertures. Since the holes were already alignedlaterally to the nearest thousandth of a millimeter, nofurther alignment was necessary. When the center of ahole coincided with that of a slit, a good approximationto a rectangular aperture was produced. The differencein area between a rectangular aperture and that formedby the combination of a slit and hole is about 7% forthe smallest aperture used, less than 2% for the second,and is negligible for holes of larger radii. For this reason,the data taken from the first few holes and slit aperturecombinations were neglected. This neglect is not serious,since Barakat and Houston2 have shown that the total

3R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).R. Barakat and M. Morello, J. Opt. Soc. Am. 52, 992 (1962).

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OCTOBER 1964VOLUJME 54, NUMBER 1()

October 1964 I LLU MI NAN CE IN DIFFRACTrION I MAGE, OF LINE 1257

illuminance for a line source is linearly related to line-width for small widths.

The image of the line source, formed by the lenstested, is picked up by a microscope objective and pre-sented to the rectangular apertures. The rectangularapertures are in different planes from the holes in therotating drum. This may be a source of error. The sepa-ration of mask and holes in the back focal plane of the

FIG. 1. Schematic diagram of the total illuminance apparatus:(A) ribbon tungsten filament lamp, (B) condenser, (C) filter,(D) slit, (E) microscope objective, (F) lens system, (G) drum,(H) diffuser, (I) photomultiplier tube.

microscope objective was approximately 0.2 mm; sincethe total power of the objective was approximately 100,the separation of their images in the front focal plane ofthe objective was approximately 0.02 A. For higherf/number, this separation may be negligible because ofthe large depth of focus. However, for low f/numbersystems, this separation can be sufficient to influencethe measurements.

A schematic diagram of the apparatus is shown inFig. 1. The light source is a well-regulated tungstenribbon filament lamp. A 40-mm microscope objectiveserves as a condenser. Since this experiment was per-

14

ZI 12

o 6

-J2n

-5 -4 -3 -2 -I 0 1 2 3 4 5

DEFOCUS IN 0.001 INCH UNITS

FIG. 2. Experimentally determined contour lines of constant totalilluminance for lens operating at J/6.7 in blue light (X-4300A).

formed at finite conjugate distances with an f/6 photo-graphic objective, the condenser imaged the filament ofthe lamp directly on to the first element of the f16objective such that its full aperture was filled withlight. Thus, the objective, a 9-A slit placed in front ofthe condenser, was uniformly illuminated so that everypoint along the slit emitted a cone of light equal to or

C)

0

I-.

4F.3

2 .

-5 -4 -3 -2 -I 0 1 2 3 4 5

DEFOCUS IN 0.001 INCH UNITS

FIG. 3. Experimentally determined contour lines of constant totalilluminance for lens operating at f/6.7 in green light (X-5460 X).

greater than the acceptance cone of the f16 objective.It would have been advantageous to have the accept-ance cone angle just equal to the emitting cone angleof the self-luminous object. In that way, all of theavailable light would have been used to form the image,which would have maximized the signal-to-noise ratioof the photomultiplier tube.

The spectral content of the illumination was con-trolled by using suitable filters located between thecondenser and slit object.

The f16 photographic objective was used at 10:1conjugate with a focal length of 6 in. The geometricalwidth of the slit object in the image plane was alwaysless than half the diameter of the Airy disc formed bythe lens.

The image was picked up by an 8-mm, 5OX Bausch &Lomb metallurgical microscope objective. The fluxpassing through the final rectangular apertures wasmeasured by a 1P21 RCA photomultiplier tube. Theoutput of the photomultiplier tube was presented to aservo potentiometer pen recorder.

The range of apertures used adequately covered theregion of most interest, namely, that immediately sur-rounding the central maximum of the image. The largestaperture was several times wider than that necessary todetect at least 90% of the energy. Thus, the illuminance

U)z0R0

Z

_J(n

I0)

CL

' 5 -4 -3 -2 -I 0 1 2 3

DEFOCUS IN 0.001 INCH UNITS4 5

FIG. 4. Experimentally determined contour lines of constant totalilluminance for lens operating at f/6.7 in red light (X -7000) .

1 257

B AX R A 1 .\T , NEW\MAN, \AND HUMWPI-1IREYS V 54

wd

-J_

-J

0

2 4 6 8 10 12

DIMENSIONLESS HALF-WIDTH V0

FIG. 5. Experimentally determined total illuminance curves forlens operating at f/6.7 in the plane of "best focus" for each coloras compared to the corresponding aberration-free system. Thedotted line corresponds to the aberration-free system; * X-7000A, 0 v-5460 A, A X-4300 A.

contained in this aperture at best focus was chosenas the normalization constant for each series ofmeasurements.

3. DESCRIPTION OF THE MEASUREMENTS

The measurements described in this section weretaken with an f/6.0, 6-in. focal length lens operating at10:1 conjugate. This was the same lens studied in ourprevious paper.' Three spectral distributions were con-sidered: (a) Wratten 29 filter (peak wavelength,X. 7000 A; (b) Wratten 58 filter (peak wavelength,X-5480 A; (c) Wratten 47B filter (peak wavelength,X-~4300 A).

12C',z0

0-,

F- .-J

LI

1C4I-

to4'2.

-20 -15 -10 -5 0 5 10 15 20

DEFOCUS IN 0.001 INCH UNITS

Fic. 6. Experimentally determined contour lines of constant totalilluminance for lens operating at f/I 2.5 in red light (Qv-7000 A).

The lens system was mounted on the apparatus andits optical axis was found by the usual methods of ad-justing and rotating the lens until image motion was ata minimum during rotation. The final lateral alignmentof rectangular apertures with the optical axis wascarrie(d out photoelectrically. For the smaller apertures,it was found that the illuminance remained very closeto a maximum over a range of a few thousandths ofinch of longitudinal defocussing. However, this was nottrue for the larger apertures. Thus the receiving plane,

defined as the plane of "best focus," is all but impos-sible to locate by merely maximizing the illuminance ofa particular aperture. Experience showed that the onlyreliable way of defining "best focus" is to examine andcompare the total illuminance curve for many receivingplanes near, but both within and beyond the focus.

The longitudinal defocussing was measured by a dialindicator which could be read to 0.0001 in. Totalilluminance measurements were taken in many receivingplanes. From the total illuminance curves, data wereobtained for constant Li (vo) and replotted as a function

C-)

DEFCU IN 0.0 NC NT

'4-

LI-/

0X

F-

- tO -5 0 5 0o 5 20

DEFOCUS IN 0.001 INCH UNITS

FIG. 7. Experimentally determined contour lines of constanttotal illuminance for lens operating at f/12.5 in green light(X-5460 A).

of half-width of slit in microns and defocussing in inches.In this way, a contour map of the total illuminance at afixed f/number and for a specified spectral content wasgenerated.

Figure 2 shows the contour lines of constant totalilluminance for the objective in blue light (X-4300 A).The f/number becomes f/6.7 instead of f/6.0, becausethe measurements were made at finite conjugates. Theplane of "best focus" was defined as that point on theoptical axis where 50% of the total illuminance wascontained in the small half-width v0. This plane islabeled as zero on the abscissa on all the pertinent

1.0

W8 2,' 6 8 lo

-J . II

o .2I-

00 2 4 6 8 0 2

DIMENSIONLESS HALF-WIDTH V.

FIG. 8. Experimentally determined total illuminance curves forlens operating at f/12.5 in the plane of "best focus" for each coloras compared to the corresponding aberration-free system. Thedotted line corresponds to the aberration-free system; * X-7000A, o X-5460 A.

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October 1964 ILLUMINANCE IN DIFFRACTION IMAGE OF LINE 125

z2

-J

0

0 2 4 6 8 10 12

DIMENSIONLESS HALF-WIDTH Vo

FIG. 9. Comparison of theoretical and experimental valuesof Li(vo) in receiving plane designated as l1=0 for lens operatingat f/6.63 in green light (X-5460 A) - - - theoretical,

experimental.

graphs. Note the symmetry of the curves about theplane of "best focus."

Figures 3 and 4 show the contour lines for the sameobjective in green (A 5460 A) and red light (X'7000A), respectively. Note here that the curves are almostidentical and quite insensitive to defocusing as com-pared to the case shown in Fig. 2. This similarity indi-cates that the spherical aberration is almost equal forgreen and red light.

In Fig. 5 we compare the total illuminance curves inthe plane of "best focus" corresponding to the threecolors taken from Figs. 2-4, with the perfect system

1.0

c--------------

- e 4 /

z -

0.6-2

/-~0.4-

H0.210

0 2 4 6 8 10 2

DIMENSIONLESS HALF-WIDTH Vo

FIG. 10. Comparison of theoretical and experimental values ofLi(ivo) in defocused receiving plane 31=0.001 in. for lens operatingat f/6.63 in green light (X-5460 A) - - - - theoretical,

experimental.

(dashed curve) calculated by Barakat and Houston.2

When vo is small, it has been shown2 that

Li(vo) - (2/71r)vor(0), (3.1)

where r(O) is the Struve criterion. In other words,Li (vo) varies linearly with vo as compared with a quad-ratic dependence on vo for the total illuminance due to apoint source. The slope of Li(vo) in the vicinity of theorigin is a direct measure of the performance of thesystem in that

=(3.2)

where T(wo) is the transfer function. Thus for fixed Vo

(where vo is small), the slope is a direct measure of theintegrated transfer function. For most applications,r(O) should be as large as possible.

The photographic objective was stopped down to aneffective f 12.5 aperture, and the same measurementsrepreated for red and green light. Figures 6 and 7illustrate the contour lines at f112.5 for red and greenlight, respectively. Again, note the similarity in thesetwo contour maps. Here the depth of focus is at leasttwice that shown in Figs. 3 and 4. Also, notice thesymmetry about the plane of best focus. This indicatesthat the objective is almost perfect at f/12.5.

I.c

UJ 0.8(C)2

0.6 28

.-J 0.4-

-J

H0.2-0

00 2 4 6 8 10 12

DIMENSIONLESS HALF-WIDTH V0

FIG. 11. Comparison of theoretical and experimental valuesof Li(vo) in defocused receiving plane a1=0.002 in. for lens operat-ing at f/6.63 in green light (X-5460 A) - - - theoretical,-- experimental.

In Fig. 8 we compare Li(vo) in the plane of best focus(as taken from Figs. 6 and 7) with a perfect system(dashed curve). Unlike Fig. 5, here we see that for redlight the objective is perfect (within measurementcapability), and falls off only slightly for green light.In Fig. 5 the departure from a perfect system is appre-ciable, but again the correction for red light is betterthan that for green.

4. THEORETICAL AND EXPERIMENTALCORRELATION

Since the design data for this objective were known,we availed ourselves of the opportunity to check theo-retically computed values vs experimental values of the

z0H

z

c

0~

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6

NORMALIZED SPATIAL FREQUENCY1.8 2.0

FIG. 12. Theoretical transfer function of lens operating atf/6.63 in green light (X-5460 A): (A) 61=0, (B) 51=0.001 in.,(C) 31=0.002 in.

125 9

3ARAKAAr, NEWTMAN, AND HUMPHREYS

total illuminance for a line source. The transfer functionT(w) of the lens was evaluated from the lens design datausing the methods of Barakat and Morello.3' 4 The wave-length employed in the calculations was X=5460 A.The theoretical transfer function was first evaluated inthe paraxial plane, and then for any other desired re-ceiving planes. In this manner, we generated a largenumber of transfer functions in receiving planes sepa-rated by distances of 0.001 in. These data were used toevaluate Li(vo) in (1.3) for the following values of thedimensionless parameter v0: vo=0(0.50)16.0. To revertback to dimensional coordinates, we used (1.2).

The experimental values of Li(vo) were obtained bythe methods already described. In order to increase theprecision of the experiment, we employed an inter-ference filter with a peak transmittance of 5460 Awith a-417 A half-peak transmittance in place of theWratten filter. This substitution of filters led to a con-siderable loss of light in the experiment, with attendentdifficulties.

The results are shown in Figs. 9-11 for three con-secutive receiving planes in the region of best focus..The corresponding transfer functions are shown inFig. 12. The correlation is quite good, as can be seenfrom the figures. However, the correlation is appreciablyless close when the system is defocused. In essence, thisindicates that theoretically, the focus deterioratesfaster than it does in practice.

Before attempting to account for the discrepanciesbetween the observations and theory, it may be worth-while to explain how the correlation shown in thefigures was obtained. The experimental values in theregion of best focus were measured in receiving planesseparated distances of 0.001 in. The experimental andtheoretical data were plotted separately, but to thesame scales. The two sets were overlaid and displacedhorizontally until they most closely agreed. The corre-lations shown in Figs. 5. 8-11 were obtained from curvesmatched in this manner.

There is much uncertainty concerning the best matchof the theoretical with the experimental curves. Further-more, it should be mentioned that these theoreticalvalues were computed prior to the measurements. Theparameters used in the calculations were for 6: 1 con-jugate planes while the experimental values were ob-tained at 10: 1 conjugate planes. Thus one would indeedexpect a sharper focus in the theoretical calculations.The amount of defocussing in the experiment is con-trolled mechanically. Over long distances the error iscumulative; therefore, the separations of the experi-mental receiving planes measured are uncertain.Another source of error is in the lens data. Although weused the lens design data in our computations, it mustbe realized that the actual lens does not conformexactly to the design data. In spite of these errors, thecorrelation between theory and experiment is good.

Vol. 54