A chopper wheel, a bicell photodetector, and an electronic phase detector are used in a mechanization of theFoucault optical test to obtain a linear focus error voltage for an isolated point image. Geometrical and dif-fraction analyses of the technique are included which show the sensitivity of the technique to aberrationsof higher order than focus.
1. Introduction
This paper describes a technique for sensing the axialposition of the focus of the image of a point source or asmall spot source. The technique works with eithercoherent or incoherent light and is insensitive to wavefront tilt (boresight) as well as certain higher-orderaberrations. It is a mechanization of the Foucaultknife-edge test and requires a chopping wheel, placedin a focal plane of the point image, plus some associateddetectors and signal processing electronics for imple-mentation. A voltage proportional to the focus erroris generated, making the technique useful in servo focuscontrol applications. In addition to a description of thetechnique, geometrical and wave optics analyses (ap-plicable to the extended and point source cases, re-spectively) are given which show the sensitivity of thetechnique to aberrations of higher order than focus.Some experimental focus sensing data are also pre-sented.
II. Description
The Foucault optical test, on which this focus sensordesign is based, is illustrated in Fig. 1. As the knife-edge is raised to intercept the focused beam in Figs. 1(a)and (b), the direction of the knife-edge shadow motionis seen to depend on the axial position of the edge rela-tive to the focus. When cutting the beam at focus, theknife-edge darkens the upper and lower halves of thebeam simultaneously. This paper describes how theFoucault test can be mechanized for sensing focus byreplacing the single knife-edge with a repetitive knife-edge in the form of a rotary chopping wheel and sensing
The author is with MIT Lincoln Laboratory, P.O. Box 73, Lex-
the intensity of the two halves of the chopped beam withseparate photodetectors (a bicell) as shown in Fig. 1(c).The relative phase of the bicell signals depends on theaxial position of the chopper with respect to the beamfocus. This phase is detected electronically to obtaina signal voltage which is proportional to the relativedifference between the positions of the focus and thechopper along the optical axis. The implementationalso includes a field lens to focus the optical systempupil on the bicell so the bicell illumination is inde-pendent of the lateral image position in the focalplane.
Generally the focused spot is small compared with thesize of the working field, which means the detectoroutput waveform phase to be detected is small com-pared with the chopping period. Several methods todetect the phase can be used. In one method (algorithm1) the difference of the ac-coupled bicell signals issampled (and held) at the instant their sum passesthrough zero in the chopping cycle. This is the instantthe chopping wheel tooth has half-covered the image.Zero difference results if both half-pupils darken si-multaneously during the chopping cycle, a conditionidentified as at focus when one performs the Foucaulttest manually. However, with this algorithm, the dif-ference signal depends on the image intensity as well asthe focus position, unless the signal is normalized insome manner. In another method of phase detection(algorithm 2) the ac-coupled bicell signals are convertedto binary levels by a threshold circuit and compared ina binary signal phase detector. With this algorithm, theoutput is independent of image intensity. Phase dif-ference is measured at both the leading and trailingedges of the waveform to achieve electronic and opticalsymmetry in the measurement.
It is of interest to determine by analysis the sensitivityof this focus sensor not only to the focus component ofthe wave front phase error O(x,y) in the system pupilbut also to any other aberrations that may be presentas well. Both geometrical and physical optics concepts
Fig. 1. Foucault knife-edge test with the knife-edge (a) in front and(b) behind focus, (c) a mechanization of the Foucault test for focus
sensing.
can be used for analysis, and a geometrical analysis isdescribed first. A geometrical analysis is appropriatewhen the optical source is resolvable.
Ill. Geometrical Analysis
In this analysis the source is assumed to have uniformintensity and to be large enough so each elemental sub-aperture in the system pupil contributes its own imageof the source at the knife-edge, with a lateral displace-ment proportional to the tilt of the wave front aberra-tion in the subaperture. In this case, for wave front tiltsand accompanying image displacements which areperpendicular to the knife-edge, the fraction of thesource image passing the knife-edge determines theintensity of the image of that subaperture in the bicellplane. If the aberration-free source image is centeredon the axis, for small wavefront aberrations one canwrite for the intensity I(xy) at a point (xy) in the bicellplane
I(Xy) = 1 r(x,y) - bIr(Xy) d0(x'y)2 ax
and X- are functions of time having dc or average val-ues equal to the first terms in Eqs. (2a) and (2b), re-spectively. The desired signal information is containedin the last terms of Eqs. (2a) and (2b), and this can beobtained by ac coupling the signals X+(t) and X-(t) toremove the dc component. For the centered knife-edgeposition, the difference of the ac-coupled bicell signalsis
x+-- - f I(X~Y) O(xy) dxdy.'>0 aOx
+ fOkIx(xy y) dxdy,jrX Yi ax (3)
which gives the response of the sensor to focus as wellas any other aberrations that may be present. In ad-dition, since the source size is finite, the difference in Eq.(3) remains constant for other knife-edge positions inthe vicinity of the central position. This is useful fortwo reasons. First, the zero crossing of the sum of theac-coupled bicell signals can be used as a practical in-dicator of the central knife-edge position for datasampling purposes. Second, if the difference in Eq. (3)is constant in the vicinity of the central position, for arepetitive knife-edge moving with constant velocity therelative phase of the zero crossings of the ac-coupledbicell signals also is proportional to X+ - X- as givenin Eq. (3). Consequently we can expect both processingalgorithms to respond similarly.
For circular pupils with uniform illumination, theintegrals in Eq. (3) can be evaluated conveniently whenthe phase function (x,y) is expressed in terms of theZernike polynomials,1 2 U = R (p) cosmO and U-' =Rn (p) sinmO, defined in the unit circle where x = p sinOand y = p cosO. For this case, Eq. (3) becomes simplythe phase gradient components normal to the knife-edge integrated over the half-apertures x > 0 and x <0. These integrations have been done (see AppendixA) for X+ - X- as well as for the analogous differencesignal Y - Y for a knife-edge aligned with the x axis,
(1)
where the knife-edge is aligned with the y axis and lo-cated at x = 0 as shown in Fig. 2, Ir(x ,y) is the intensityon the bicell with the knife-edge removed, and b is aconstant, depending on system parameters, which isassumed to be unity in the following discussion.
The bicell signals X and X- for this knife-edgeposition are simply I(x,y) integrated over the bicellareas x > 0 and x < 0, respectively,
cutting the focused spot in the y direction. In Fig. 3 thevalues of X+ - X- and Y+ - Y- have been tabulatedfor each of the Zernike aberrations up to n = m = 8.The size of each aberration is such that it has a maxi-mum value of unity on the unit circle p = 1. Figure 3shows the largest sensitivity is to focus (m = 0, n = 2).However, significant sensitivity also exists for primaryspherical aberration (m = 0, n = 4) and the higher-orderspherical aberrations. On the other hand, there is nosensitivity to the comas (m = 1, n > 3), and the sensi-tivity to the astigmatisms (m = 2) can be canceled bysplitting the beam into two and chopping one in Y andthe other in X and summing the X and Y responses.Normally for a wave front that is well-corrected thehigher-order aberrations are small and the sensor re-sponse will be given principally by the focus error, to anaccuracy comparable to the level of the remaining ab-errations. Note that, although Fig. 3 shows the theo-retical response of the algorithm for very high orders,the range of validity is limited by the assumption ofgeometrical optics and to aberrations having wave frontslopes less than half of the angular size of the opticalsource.
IV. Wave Optics Analysis
If the optical source is small compared to the Airydisk, the effects of diffraction are significant, and a waveoptics analysis is required to find the sensitivity of thesensor to focus and the various other aberrations.Linfoot 3 has given a diffraction analysis of the Foucaultoptical test which permits the computation of the pupilimage intensity when the knife-edge is at the cutoffmidpoint for the case of uniform pupil illumination andsmall wave front errors. Writing the wave front phaseerror 0(x,y) as the expansion
O(xy) = aijxiyjij20
(4)
where P(xy) is small enough that
(1r )2l 2r aij2 << 1
Linfoot shows (Ref. 3, Chap. 2, Article 5) that the pupilimage plane intensity I(x,y) at the knife-edge cutoffmidpoint for a vertical knife-edge as in Fig. 2 is givenby
I(xy) = Io(x~y) + {42r2/X} IL ajxi-yj(-l)r (
X(X + y)r - (X -y/)r
(5a)
= I(xy) + AI(xy), (5b)
where Io(xy) is the image intensity at the cutoff mid-point in the absence of aberration, AI(xy) is the changein intensity due to the aberration, and y' = V(1 - y 2 ).
Equation (5a) is valid for the strip -1 < y < 1; I(xy) iszero outside that strip. The upper value in the bracketsis to be taken for (x,y) inside the geometrical image ofthe pupil, and the lower, outside. Because of diffrac-tion, Io(xy) has significant values outside the pupilimage (the halo, see Ref. 3), but as Linfoot notes, only
Fig. 3. Relative values of the signals X+ - X and Y+ - Y for theZernike aberrations up to n = m = 8 as given by the geometrical
analysis.
En\0
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 80 0 3 _3 1 _1 _1 1 11 11
2 2 8 a 4 4 6W
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
+2 -a1 1 14 4 6 6 1919 50 0 0 0 0 0 0 0
00 0 0 0 0
0 0 0 0 0 0
4848 12 12 640 640
0 0 0 0 0 00 0 0 0
0 0 0 07 _7 1 1
64 64 20 20
0 0 0 00 0
163 163
o o o o o o o o~~ss sso o o o o o~~0
+ m 1LEGEND I
Fig. 4. Relative values of the signals X+ - X and Y+ - Y for theZernike aberrations up to n = m = 8 as given by the diffraction
analysis.
the image intensity inside the pupil image is affected bythe presence of small wave front aberrations. In ad-dition, note that Io(xy) is symmetrical about the y axis.Consequently, to compute the difference of the ac-coupled bicell signals (X+ - X-) at the cutoff midpointfor the present case it is sufficient to integrate onlyAI(x ,y) over the same semicircles as for the geometricalanalysis:
- = SJo S(x3y)dxdy o AI(xy)dxdy.fftay2)S5lft (P+ y 2)Sl5
(6)
An example of this computation for focus is given inAppendix B. The relative values of X+ - X- and Y+- Y- computed in this manner are shown in Fig. 4 forthe Zernike aberrations up to m = n = 8.
Comparison of Figs. 3 and 4 shows many similaritiesin the response to different aberrations for the resolvedand unresolved source cases. However, the responseto spherical aberration relative to that for focus in thepoint source case is considerably smaller than for theextended source case. Note that Figs. 3 and 4 show onlythe relative responses for the different aberrations forthe two cases. To compare bicell difference signals onan absolute basis, the equations given above can be usedto compute the absolute signal levels for cases of inter-est. Also, note in the point source case that, because asignificant amount of diffracted light falls outside thegeometrical image of the pupil, it is necessary for thebicell detector size to be several times the pupil imagesize if the midpoint of the bicell signal cutoff is to beused for the determination of the knife-edge cutoffmidpoint.
V. Experimental Results
The technique has been implemented in the labora-tory to sense the focus of an f/30 He-Ne beam. Wide-band detectors and preamplifiers were used to preservethe fidelity of the light cutoff waveforms. A voltage vsfocus discriminant obtained by chopping at 80 Hz andsampling the bicell difference signal at the cutoff mid-point (algorithm 1 described above) is shown in Fig. 5.The slope of this discriminant depends on the size of thefocused spot, being greater for smaller spot diameters.The rms noise level in the 10-100-Hz band when thisdiscriminant was measured corresponded to 0.005X p-pfocus, although no special precautions were taken tominimize noise.
A discriminant obtained by chopping at 400 Hz andusing the binary signal phase detector (algorithm 2)with the same f/30 He-Ne beam is shown in Fig. 6. Theslope of this discriminant does not vary with the size ofthe focused spot, and the extent of its linear portion isgreater than algorithm 1. The rms noise measured inthe 10-100-Hz band using algorithm 2 corresponded to0.015X p-p focus. In both algorithms described herea 50% duty factor chopper was used, and the average ofdata taken at both the leading and trailing edges of thedetector signal waveform is shown. The technique alsohas been demonstrated using a resolved source withwhite light.
VI. Discussion
In the preceding, the performance of this focussensing technique has been discussed for wave frontswith small phase aberrations and uniform time-in-variant intensity distributions. In many practical sit-uations, the higher-order aberrations may be large, thebeam intensity may vary with time, or the bicells maynot be centered on the pupil image, and it is of interestto consider the operation of the sensing technique inthese conditions.
For the case of large aberrations producing badlyaberrated far-field distributions at the chopper blade,geometrical analysis indicates that half-aperture beamcutoff midpoints occur simultaneously at the axial po-sition where the medians of the far-field distributions
wCD
-0
a.
0 -2
-4
-6
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
RELATIVE WAVEFRONT FOCUS (m P-p)
Fig. 5. Output voltage vs focus discriminant obtained for an f30He-Ne beam using algorithm 1 described in the text.
6
4
2 4W
-
00
-4
-6
-12 - -4 0 4 8 12
RELATIVE WAVEFRONT FOCUS, ({Lm p-p)
Fig. 6. Output voltage vs focus discriminant obtained for an f/30He-Ne beam using algorithm 2 described in the text.
from the half-apertures are coincident. This positionis not necessarily the point of best focus. An autofocussystem using this sensor would adjust the focus to makethese medians coincident.
The case of time varying aperture illumination, par-ticularly when the variation occurs at rates greater thana small fraction of the chopping frequency, makes theelectronic detection of the beam cutoff midpoint diffi-cult. To achieve some independence from the effect ofintensity fluctuations, a separate, unchopped sampleof the focused beam, obtained with a beam splitter, canbe detected independently, and these detector signalscan be used to normalize the chopped beam signals.Two methods of normalization have been used for thispurpose, and they both require dc-restored detector
signals for operation. The first method involves elec-tronic division of the chopped beam signal by the ref-erence beam signal. Writing the modulating effect ofthe chopper on the beam as C(F,t), where F is focus andt is time, and the light intensity before chopping as L(t),the signal from one bicell detector is the product L(t)C(F,t). Dividing this signal by L(t), as detected by the
unchopped reference channel, produces the desiredwaveform C(F,t) independent of the half-aperture in-tensity. This signal can be ac-coupled to the phasedetector as before. The second method involves elec-tronic subtraction in which half of the reference channelsignal is subtracted from the bicell detector signal:
L(t) C(F,t) - 1/2L(t) = L(t)[C(F,t) - 1/2]. (7)
Since C (F,t) is a 50% duty factor signal alternating be-tween 0 and 1, [C(F,t) - 1/2] is a symmetrical signal al-ternating between -1/2 and +1/2. Thus, variations inL(t) modulate the amplitude of [C(F,t) - 1/21 but do notaffect the times of the zero crossings which correspondto the beam cutoff midpoints. Hence, phase detectionbased on the zero crossings of the dc-coupled waveformL(t) [C(F,t) - 1/2] is independent of L (t). Generally,wideband electronic subtraction is easier and cheaperto implement than electronic division.
In the preceding sections, analyses were given whichshowed the sensitivity of the technique to higher-orderaberrations. These analyses were done for bicellscentered on the pupil image. In practice, however,exact centering is not possible, and the sensitivity tohigher-order aberrations may be different from thatshown in Figs. 3 and 4. In the geometrical analysis, thesensitivity for specific offsets and aberrations can becomputed by modifying the limits of integration in theanalysis given above. It can be shown in this mannerthat small offsets do not significantly change the re-sponse to focus. No corresponding wave optics analysishas been done. Experimentally, it has been found thatthe null focus indication is not sensitive to moderateshifts in the bicell position.
VII. Summary
The mechanized Foucault test for point image focussensing described here has a relatively simple imple-mentation and provides a voltage-vs-focus discriminantwhich makes it useful as a sensor for servo focus controlsystems. The sensor is insensitive to image position(boresight) and coma, as well as a large fraction of otherhigher-order Zernike aberrations. Some sensitivity tospherical aberration and some higher-order aberrationsexists however, the amount depending on the size of thesource. Astigmatism aligned with the chopping axescan be detected by subtracting the x and y focus errorsignals if chopping in both directions is provided, oralternatively the sensor discriminant can be made in-sensitive to astigmatism by summing the x and y focuserror signals. The focus error can be determined atboth the leading and trailing edges of the choppingcycle, providing a data update rate that is twice thechopping frequency. Finally, the effect of beam in-tensity fluctuations on the electronic signal processing
can be reduced by signal normalization, a procedurewhich is practical in this focus sensor because thenumber of subaperture signals which must be normal-ized is only four.
The author wishes to acknowledge the contributionsof Earl P. Holbrook, who assisted in the design andtesting of a focus sensor using the principles describedhere, and of Jan Herrmann for stimulating discussionsduring the development of the technique. The en-couragement of L. C. Marquet and D. P. Greenwoodduring the development is appreciated.
This work was sponsored by the Navy High EnergyLaser Project Office, PMS-405, Naval Sea SystemsCommand.
Appendix A
The integration of the gradients of the Zernike ab-errations for the geometrical analysis is illustrated inthis Appendix. For the Zernike polynomial cosineterms,
O(x,y) = Uln(p,O) = Rn(p) cosmO,
the x gradient is
a a a p a 0-O(xy) =- U(pO) = - U(p,) + U'(p,0)a-dx ax ap ax 0o ax
(Al)
Since p = X2 + y2 and 0 = tan-' (x/y),
-= sin and -= -cosO.ax ax p
Writing /ap Rm(p) as R'm(p), the x gradient is there-fore
- Un(p,0) = R (p) cosmO sin - -R n (p) sinm0 cos0. (A2)ax P
Equation (3) in the text can now be written as an inte-gration in polar coordinates over the semicircles 0 < 0< r and -ir < 0 < 0:
- = - ax Un (p,0)da + X -o- Um(p,O)daLxO ax n x<0 Oxn
=-S pdp 3 dOR'(p) cosmO sino
- Rn(p) sinm0 cosOj
+ go pdp X dO [R'm(P) cosmo sino
m-- Rm (p) sinm0 cos -P nI
(A3)
An analogous expression can be derived for the Zernikepolynomial sine terms Unm (p,0) = Rm(p) sinm0, and theintegrations can be performed for the various R m (p) todetermine the value of X+ - X- for each Zernike ab-erration. An analogous procedure can be carried outfor the orthogonal chopping case giving Y+ -Y-.
For example, to compute the sensitivity to focus (n= 2, m = 0), we note from Ref. 1 that R°(p) = 2p2
- 1,
so that R"2(p) = 4p. Substituting into Eq. (A3), we
The bicell difference signals have been computed in thismanner for a number of Zernike aberrations and theresults are tabulated in Fig. 3.
Appendix B
The diffraction analysis requires finding the coeffi-cients aij in Eq. (4) for the Zernike aberration, substi-tuting these in the expansion for AI(xy) [Eq. (5)], andintegrating AI(xy) over the bicell halves. For example,for the focus aberration (n = 2, m = 0),
Integrating AI(xy) over the X+ half of the bicell, wefind
X+= 7 4X2 (-4xy')dxdy = . _r)
Since AI(xy) is odd in x,
X- +
and
+ 47r3 1 3~ B4
X ( 2)
The value -3/2 has been entered as the X+- X sen-sitivity to focus in Fig. 4, which shows the relative sen-sitivity of the technique to a number of Zernike aber-rations for both knife-edge orientations calculated inthis manner. The factor (47r3)/X, common to all entriesin Fig. 4, has been deleted.
References1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,
1959).2. D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978),
Appendix 2.3. E. H. Linfoot, Recent Advances in Optics (Oxford U.P., London,
1958), pp. 128-175.
Books continued from page 1886
Most physicists and engineers will find the authors's use of mathe-matics almost always helpful, usually comfortable, and occasionallydelightful.
The references to papers, books, and dissertations are generallyappropriate although by no means exhaustive. Indeed the authorsstate in their Preface that ". . .we included the works that we foundhelpful in preparing this manuscript ... (and) ... a few very recentreferences to serve as entry points to the current literature .... Nosystematic attempt has been made to establish historical priori-ties . . . " With this caveat, the bibliography is generally satisfactory,although the references will not lead the reader to important paperson some issues (such as alternative SNR measures), and there are noreferences at all for a few topics (such as Monte Carlo analysis ofscattered radiation). The authors' familiarity with the literature onmost of the subjects covered is evident, however, and they clearly tryto give credit where credit is due, occasionally referring-as a hypo-thetical example-to "the work of Smith and co-workers [Jones etal. (1970)]."
This book-like Vol. 1-has been edited and produced with greatcare. The typography is excellent, the illustrations are helpful andclear, and typographical errors are few. Curious but not serious isa consistent error in the Author Index (common to both volumes)which causes the quoted page numbers (p') to be biased by +6 for p'> 669. The correct page numbers (p) are given by the decoding al-gorithm: p = p' - 6 for 669 < p' • 678.
Together Vols. 1 and 2 constitute the definitive textbook on radi-ological imaging theory. Technical advances in this rapidly evolvingfield may soon provide new material for complementary books, butthe conceptual structure developed so well here should prove usefulto graduate students, physicists, and engineers inside and outsideradiological imaging for years to come.
CHARLES E. METZ
Physical and Biological Processing of Images. Edited by 0.J. BRADDICK and A. C. SLEIGH. Springer-Verlag, Heidelberg,1983. 403 pp. $33.00.
This volume contains the proceedings of a symposium sponsoredby the Rank Prize Funds organization and held at The Royal Society,London, on 27-29 Sept. 1982. Its purpose: to discuss the interdis-ciplinary (that is, the physical, biological, and psychophysical) aspectsof image processing in vision. To do so, a wide range of topics wasselected, comparing wherever possible similar processes in biologicaland artificial systems. A detailed review of so diverse a matter iswell-nigh impossible, especially since points of contact with moreconventional optics, outside of its psychophysical branch, are few.
The areas covered are grouped into six major parts. Part 1 is anOverview of the visual system and its relationship to computer pro-cesses. In Part 2 we read about Local Spatial Operations on theImage. How do elements as unreliable as nerve cells make the brainso reliable? By information processing through parallel channels.The fly's eye with its many light detectors (ommatidia) is a good ex-ample. Other contributions discuss mathematical models, acuitymeasurements by contrast transfer, and 2-D processing and fil-tering.
Part 3 deals with Early Stages of Image Interpretation, which in-cludes visual algorithms, motion measurements, coding, and stere-opsis. Here I find the one article that perhaps comes closest toarousing interest in readers of Applied Optics: How to extract themost information from images changing in time? The applicationis that of cineangiocardiography, taking a sequence of x-ray imagesrecorded on film at the rate of 50 frames/sec after the injection of anx-ray-opaque dye. From these images any temporal, spatial, andstructural changes affecting the heart can be determined by com-puter.
