April 1972

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 62, NUMBER 4

Sensing Small Displacements by Chromatic Asymmetry*

JOSEPH P. KIRKSystems Development Division, International Business Machines Corporation, P. 0. Box 6, Endicott, New York 13760

(Received 2 August 1971)

A method of sensing small displacements between two images of complementary chromaticity is de-scribed. An autocollimator that uses this method of detection has a pointing sensitivity that exceeds visualinterferometric techniques. Two images of complementary chromaticity are superimposed and made tomove in opposite directions in response to a test-mirror rotation. The resulting chromatic asymmetryenables experimental detection of setting errors of 1.5 s of arc for a 0.08-mm-wide test mirror. Analysis ofimage chromaticity when the source is a krypton-ion laser predicts setting errors one-sixteenth of thoseobtained by interferometry.INDEX HEADINGS: Interferometry; Color; Mirrors.

High-precision pointing' is usually done interferometri-cally and a common visual problem is to establish anapparent equal luminance between adjacent regions. Apointing technique is described that requires the detec-tion of chromatic asymmetry in the image formed bysuperposition of two images of different chromaticity. Amodified cyclic interferometer 2 causes the two images tomove in opposite directions in response to a test-mirrorrotation. The mirror system has a dichroic mirror thatcauses light traveling in the clockwise (CW) andcounterclockwise (CCW) direction to be of comple-mentary chromaticity.

A least noticeable rotation (LNR) is defined as theproduct of the setting-error standard deviation in unitsof radians and the ratio of test-mirror width and wave-length. Using the optical system shown in Fig. 1 resultsin a LNR of 1.2X 10-3 and corresponds to a setting errorof 0.12 s of arc for a 1-mm-wide test mirror and awavelength of 0.5,4m. This compares favorably with aLNR of 2X1O-s that is expected from optimum half-shadow interferometric techniques,3 and a LNR of10X 10- that is achieved in practical precision pointinginterferometry.4 It is shown that appropriate choice ofbeam splitters and multicolored lasers will reduce theLNR to 0.63X 10-3.

THEORY

Pointing by chromatic asymmetry is analyzed byconsidering the mirror system shown in Fig. 1. Lightenters through the object slit (2) and travels the route

[2-3-5-6-7-8] in either a CW or a CCW direction. Theoptical path from the object slit (2), to the telescopeobjective (6), along either the CW or CCW path is equalto the focal length of the telescope objective so theobject slit is imaged by autocollimation at (8). Noncyclicimages are removed by using crossed input and outputpolarizers (1), (9), and a half-wave plate (4), oriented at450 with respect to the plane of polarization of the inputpolarizer in one arm. (The noncyclic images can also beremoved by use of multilayer dielectric mirrors.)

Light traveling in the CW direction is reflected twicefrom the dichroic beam splitter (3), and is reflected onceand transmitted once at the neutral beam splitter (5).Light traveling in the CCW direction is transmittedrather than reflected at the dichroic beam splitter so theCW and CCW images at (8) are of complementary

FIG. 1. Autocollimator optical system for pointing bychromatic asymmetry.

APRIL 1972

579

JOSEPH P. KIRK

0.6

0.4

0.2

0.4 0.5 0.6 0.7

FIG. 2. Luminous output of the CW and CCW paths.

chromaticity. The images are viewed through an aux-iliary optical system so they subtend the same visualangle regardless of the image size at (8).

An adjustable slit limits the effective test-mirroraperture (7) and the incoherently illuminated object slit(2) is small compared to the line spread function of thetelescope objective. By application of the VanCittert-Zernike theorem,' it is found that the telescope ob-jective is coherently illuminated. Therefore, the mono-chromatic image of the object slit has an irradianceproportional to a one-dimensional diffraction patternformed by a coherently illuminated lens with the testmirror acting as a slit aperture stop. The image of theobject slit when the source is polychromatic is asuperposition of monochromatic images. The total wave-front aberration is assumed to be less than X/4.

The image formed by light traveling through themirror system in the CW direction has an illuminancedistribution given by

J(sin(ir oAo/X) 2 >,YcW(j3) = -) Y(X)s(X)Tcw(X)dX,

x 7rXo/X XoX

where2 sina

/3= xX0

tral transmittance of the CW path. Yccw(0) is found byreplacing Tcw with Tccw in Eq. (1).

When the test mirror is aligned with the optical axis,the superimposed CW and CCW images form an imagethat consists of two regions in which the chromaticityvaries. These regions are also separated by a brightneutral region. As the test mirror is rotated, the CW andCCW images move in opposite directions because theCCW path from point (7) to (8) in Fig. 1 has an evennumber of reflections and the CW path an odd number.This causes the image to become asymmetrical in color.Each image moves by

2AA:=-AO,

Xo(2)

where A is the width of the test mirror and AO is therotation of the mirror from its position when the CWand CCW images are superimposed.

The relative tristimulus values of the image areobtained by evaluation of Eq. (1). The integration isapproximated by a summation over 80 values of wave-length for which the tristimulus values of the spectrumare tabulated.' The tristimulus values of the imageformed by the superimposed CW and CCW images thathave been displaced by AO\ are

whereY(0) = cw(O+AO)+ Yccw(O -AO), (3)

Ycw(j4AO)

80 /sin[7rXo0(jhAI)/Xn]\ 2 rX0

= E - y(Xn)s(Xn)Tcw(Xn)nil 7rS0(Ak4AB)/Xn XOXn

and similarly for X(0) and Z(0).It is assumed, for the purpose of calculation, that light

from opposite regions of the image are separatelyintegrated to give average chromaticities, as if the light

(1)

and a is the half-angle subtended by the test mirror, Xois the center of the visible spectrum, x is measured fromthe optical axis, xo is the first zero of the Xo image, y(X)is the tristimulus value of the spectrum, s(N) is thespectral radiance of the source, and Tcw(X) is the spec-

I

Iy

'0.3 1

>

In

2

0 r-,-I.0

8, 0 0 P3 04

I.0

FIG. 3. Relative luminance of the CCW and CW diffractionpatterns showing the sample regions.

580 Vol. 62

April 1972 SENSING DISPLACEMENTS B

from each region had entered separate integratingspheres. It is the difference in average chromaticity ofthe two regions that is calculated as a function of 46.A LNR is then predicted by comparing this chromaticitydifference with the least-noticeable chromaticity dif-ferences.

In order to establish the regions to be sampled, anormalized relative luminance

YN (A) = Y (Q)/ Y (A) max (4)

is calculated. The left- and right-hand regions are thendefined by values of fi satisfying the conditionmin<Y(f)<max and are bounded by 0i1f 2 and /3a34,respectively. The integrated tristimulus values for theleft and right sample regions are given by

Left side:

Jp2 ra2

XL= XQ3)df, YL= fY()df ,611

ra2ZL= Z(O)d#;

Right side:

r:4 f4XR= J X(Q)dr3, YR = Y(8)dg,

)

The C.I.E. 1931 chromaticity diagram coordinatesare then computed for each side by

XX= -X)

X+ Y+Z

yY=X

X+ Y+Z

The comparison between calculated chromaticitydifferences and least-noticeable chromaticity differencesis easily done on a geodesic chromaticity diagram 7 whereleast-noticeable chromaticity differences are unit dis-tances. Following the notation of MacAdam thecoordinates on the geodesic chromaticity diagram are tand q. The distance R between two points representingthe chromaticity of the right and left sample regions inthe geodesic chromaticity diagram is

R= [(aL- R)'+ (nL-r VR)2J1. (7)The chromaticity difference R increases with X# and

observation of this difference indicates that A# is nolonger zero. Because unit distances in the geodesicchromaticity diagram are least-noticeable color differ-ences, it follows that the ratio of Af and R is a leastnoticeable X6 given by

AOAOLN=-. (8)

R

(5)

Y CHROMATIC ASYMMETRY 581

860 000 900 900 940 900 980

'lo

FIG. 4. Geodesic chromaticity locus when a tungsten source isused. A: Chromaticity of the pattern when A4= 0.02. Values of j3are labeled along the curve. B: Chromaticity of the 0.4< •N<0.6sample regions as a function of M3. The tick marks correspond toincrements of AO=0.02. C: Chromaticity of the 0.01 Y•v< 0.99sample regions as a function of A3. The tick marks correspond toincrements of 4A= 0.02.

A least-noticeable rotation of the test mirror, LNR, is

AjLN ALNR=-=-AO

2 X(9)

and can be compared with values obtained fromexperiment.

ANALYSIS AND EXPERIMENT

The autocollimator system used to establish experi-mental limits on the setting errors is shown in Fig. 1.The object slit is illuminated by a tungsten-halogenlamp that has a color temperature8 of 3212 K. Thecalculated spectral luminous output of the CW andCCW paths is plotted in Fig. 2 and will be used tocalculate the theoretical LNR. The test mirror, whoseorientation is being sensed, is 3 mm long in the directionparallel to the axis of inversion and a width that variesfrom 0.03 and 0.3 mm by means of an overlaid adjust-able slit. This slit acts as the limiting aperture of the170-mm focal-length telescope objective (6). This rangeof test-mirror widths is limited by the fact that aper-tures of less than 0.03 mm produce diffraction imagesthat are too large, and apertures greater than 0.3 mmresult in an experimental pointing sensitivity that ex-ceeds the 0.1-s resettability of the rotational device used.The calculated relative illuminance of the CW andCCW diffraction images are shown in Fig. 3.

The superimposed diffraction images combine to forman image consisting of two regions of differing chro-maticity separated by a bright neutral region, with theentire pattern in a dark surround. As the test mirror isrotated, the pattern is no longer symmetrical in color,and it is this asymmetry that indicates that the mirroris not aligned with the optical axis. The visual problem

JOSEPH P. KIRK Vol. 62

TABLE I. LNR's for different sources and sample regions.

\Region 0.01 < Yv•<0.99 0.4< YN<0.6\sampled

Source \3213K blackbody 0.95X 10-3 0.67 X 10-3Argon-ion laser 1.7 X 10-3 1.0 X 10-3Krypton-ion laser 0.83 X 10-3 0.63 X 10-3

TABLE II. Experimental LNR's..--.

Aperture Visual angularwidth half-width(mm) LNR (deg)

0.03 2.2X1O' 4.1 (5Xeyepiece)0.08 1.2X10-3 2.5 (8Xeyepiece)0.14 2.5X10-3 3.6 (20Xeyepiece)0.3 2.5X10-' 1.7 (2OXeyepiece)

of detecting a mirror misalignment is one of observinga chromatic asymmetry in an image that has a widerange of illuminance. 9

The chromaticity of the pattern, plotted on thegeodesic chromaticity diagram, is shown as curve A inFig. 4 for the case when AOSf= 0.02. Along the curve arevalues of ,3 indicating the chromaticity of particularpoints in the image. The chromatic asymmetry is shownby the fact that opposite sides of the image havedifferent chromaticities caused by the relative dis-placement of the superimposed CW and CCW images.

The chromaticity of the two sample regions, definedby 0.4< YNv(O) < 0.6, are plotted as curve B of Fig. 4.When AB= 0, the chromaticity of both sample regionsare equal and are indicated by the encircled point oncurve B. As the mirror is rotated the chromaticity ofeach sample region moves in opposite directions alongcurve B. The increments correspond to values ofZ,3=0.02, 0.04, 0.06, etc.

A similar curve, C, is plotted for sample regionscorresponding to 0.01< YN<0.99. The calculatedLNR's for curves B and C of Fig. 4 are 0.95X 10-3, and0.67X 10-3 respectively, and are listed in Table I.

The experimental LNR's listed in Table II wereestablished by one observer over the course of two daysusing the optical system shown in Fig. 1. The minimumsetting error, LNR= 1.2X 10-, compares favorablywith what is expected from optimum half-shadowinterferometric techniques thatpredictaLNR=2X 1°-.Using the optical system of Fig. 1 as a wave-front re-versing interferometer, setting errors of LNR= 70X 10-3for 0.03-mm-wide faces are achieved. The more con-ventional precision pointing interferometer using aKoster's prism and a larger test mirror gives pointingerrors 4 of LNR= 1OX 10-. Therefore, pointing bychromatic asymmetry is at least five times more sensi-tive than that achieved interferometrically.

On the other hand, the LNR's predicted by use of thegeodesic chromaticity diagram are smaller than thosefound by experiment. The diffraction image presents the

cw100

T (%)

14 . C.

568 647X (m.)

FIG. 5. Transmission of CW and CCW paths for use withkrypton-ion laser.

observer with the problem of detecting a chromaticasymmetry in an image of varying chromaticity and alow level and varying luminance with the entire patternin a dark surround. MacAdam's color-matching datawere taken using a bipartite-field colorimeter where theregion of differing chromaticity is in a bright surround. 7

In order to have better agreement with color-matchingdata, the experimental situations need to be madesimilar.

LASER SOURCES

If a multicolored laser is used to illuminate theentrance slit, there will be sufficient luminous flux suchthat the auxiliary optical system can sample localregions of the pattern and display these in a bipartitefield with illuminances similar to that used in obtainingcolor-matching data. Under these conditions reliablepredictions of experimental LNR's are expected byusing the geodesic chromaticity diagram. LNR's arecalculated for the cases where the source illuminatingthe entrance slit is first a krypton and then an argonlaser. In Fig. 1, the beam splitter located at position (3)is chosen in each case so that the CW and CCW imagescontain equal luminous flux. The spectral transmittancesof the CW and CCW paths are shown in Figs. 5 and 6for use with the two lasers. The chromaticities of thediffraction images and the chromatic difference of thesample regions as a function of mirror rotation areplotted in Figs. 7 and 8. The chromaticity plot whenusing the krypton-ion laser as source, Fig. 7, is similarto that obtained when using the tungsten source. CurvesB and C show the locus of chromaticity of the twosample regions as the test mirror is rotated and thecorresponding LNR's are listed in Table I. The chro-maticity of the image when an argon laser is used assource is shown in Fig. 8, where curves B and C havebeen displaced by +5 and -5 units, respectively, alongthe t axis. The corresponding LNR's are larger due tothe limited spectral range of the argon-laser lines. Even

T (%) I -...

X ("W)

FIG. 6. Transmission of CW and CCW paths for use withan argon-ion laser.

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April1972 SENSING DISPLACEMENTS BY CHROMATIC ASYMMETRY

FIG. 7. Geodesic chromaticity locus when a krypton-ion lasersource is used. A: Chromaticity of the pattern when 43=0.02.Values of jl are labeled along the curve. B: Chromaticity of the0.4< YN<0.6 sample regions as a function of AO. The tick markscorrespond to increments of A3 =0.02. C: Chromaticity of the0.01< YN•O.99 sample regions as a function of M3. The tickmarks correspond to increments of 43=0.02.

in this case, the LNR's are less than are expected frominterferometric measurements and the laser will supplysufficient luminous flux for accurate color matching.

CONCLUSION

A method of sensing small displacements has beenpresented that, when applied to an autocollimator, re-sults in a pointing sensitivity exceeding that possiblewith an interferometer. It is a new technique that isparticularly suited to the alignment of small faces butcan be used in any conventional autocollimator applica-tion. A comparison of the chromatic asymmetry of theimage with color-matching data indicates that, withsufficient luminous flux, this technique is 16 times moresensitive than interferometry. Lasers supply sufficientluminous flux and spectral range to realize this previ-ously unobtained sensitivity.

ACKNOWLEDGMENTv

The author wishes to thank Professor H. H. Hopkinsfor suggesting the topic discussed here and for advisingthe author in the investigation that followed. Duringthe course of the work, the author received financialsupport from an IBM Resident Graduate Student

FIG. 8. Geodesic chromaticity locus when an argon-ion lasersource is used. A: Chromaticity of the pattern when A#=0.02.Values of /3 are labeled along the curve. B: Chromaticity of the0.4< YN<0.6 sample regions as a function of AO. The tick markscorrespond to increments of AO=0.02. C: Chromaticity of 0.01< YN<0.99 sample regions as a function of AO. The tick markscorrespond to increments of A#=0.02.

grant. The constant encouragement of Dr. A. J. Lavinduring the course of the author's studies is gratefullyacknowledged.

REFERENCES

* Work done in part at the University of Reading, Reading,Berkshire, England, and was submitted to that university inpartial fulfillment of the degr6e of Doctor of Philosophy.

1 Only precision pointing techniques using the human eye asdetector will be considered here. Much higher sensitivities can beachieved by photoelectric techniques. See, for example, R. V.Jones, J. Sci. Instr. 38, 2 (1961); 38, 37 (1961).

2 H. H. Hopkins and H. J. Tiziani, Brit. J. Appl. Phys. 17, 50(1966).

3R. J. Kennedy, Proc. Nat. Acad. Sci. U. S. 12, 621 (1926).4 G. H. Lovins, Appl. Opt. 3, 883 (1964).5 See, for example, H. H. Hopkins, in Advanced Optical Tech-

niques, edited by A. C. S. vanHeel (North-Holland, Amsterdam,1967), p. 215 or M. Born and E. Wolf, Principles of Optics(Pergamon, New York, 1965), p. 510.

6 Optical Society of America, Committee on Colorimetry, TheScience of Color (Thomas Y. Crowell Co., 1953, available only fromOptical Society, 2100 Pennsylvania Ave., NW, Washington, D. C.20037), Ch. 8.

7 D. L. MacAdam, Appl. Opt. 10, 1 (1971).8 The source used in these experiments was the inside surface of

the filament coil of a Philips 12-V 50-W, 7027/J7 tungsten-halogen lamp operated at 10.6 V.

9 For the particular system considered here, the calculatedretinal illuminance of the central maxima of the superimposeddiffraction images is approximately 75 trolands giving an apparentbrightness of 6 cd/m2 . This is to be compared with a luminance of300 cd/m2 used by MacAdam in establishing small-field chro-maticity discrimination.

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