Quantitative phase imagingin confocal microscopy by optical differentiation
Andrew W. Kulawiec and Duncan T. Moore
The technique of optical differentiation is applied to confocal microscopy for the purpose of quantitativephase imaging. One-dimensional absorptive filters are placed in the pupil of the microscope objective toproduce images related to the local phase slope in the object. With suitable signal processing andintegration, a quantitative phase profile is obtained. This method is demonstrated in a reflection-basedsurface-profiling instrument.
1. IntroductionOptical differentiation through coherent optical pro-cessing has been shown to be an effective method forphase visualization." 2 With this method a one-dimensional absorptive filter, known as a differentia-tion filter, is placed in the Fourier plane of theobjective lens. The spatial-frequency spectrum ofthe object is effectively multiplied by this filter func-tion. If the amplitude transmittance of the filtervaries linearly with position, the image intensity isrelated to the first derivative of the object transmit-tance. In this manner, phase gradients within theobject are revealed.
The advantages of confocal microscopy have beenwell documented. 3 The most notable are the im-proved resolution, the reduction of stray and scat-tered light, and the improved optical-sectioning capa-bilities. In spite of the instrument's remarkableimaging capabilities, its usefulness for quantitativephase measurements is somewhat limited. In thearea of surface profiling, several methods have beenexplored. The simplest method involves scanningan object through focus and storing the position ofthe maximum signal intensity.4 This method uti-lizes the depth-discrimination property of confocalmicroscopes and provides quantitative height mea-
The authors are with The Institute of Optics, University ofRochester, Rochester, New York 14627.
Received 15 November 1993; revised manuscript received 18March 1994.
surements, but the accuracy of this method is limited.Two other techniques, known as the autofocus andextended-focus methods, have been used in confocalmicroscopy to enhance the depth of field of theinstrument.5 These methods produce images thatremain in focus over a large extent but yield littlequantitative information regarding the phase profile.
A method commonly used in scanning microscopyfor producing images related to the object phase is thedifferential phase contrast (DPC) or split-detectormethod.67 With this technique a large-area splitdetector is placed in the objective pupil of the micro-scope. Phase gradients in the object produce a non-symmetric intensity distribution in the pupil. Thesignals from the two halves of the detector aresubtracted to yield an image related to the local phasegradient. In general this method produces only quali-tative phase measurements and lacks the depth-discrimination property of confocal microscopes.Recent research has produced confocal differentialphase contrast (CDPC) images,8' 0 which include theoptical-sectioning capability of confocal microscopes.In addition, research has been done to quantify thephase profile obtained with the DPC microscope."This method is discussed in detail in Subsection 2.D.
A method for performing quantitative phase imag-ing based on optical differentiation and confocalmicroscopy is presented. Images produced by thistechnique are directly related to the slope of theobject's phase distribution along a single direction.By means of a simple calibration and integration ofthe image, a phase profile is obtained. Distinctadvantages of this system are its insensitivity toamplitude variations in the object and the indepen-
dence to slopes in a direction orthogonal to the filteraxis. It is also shown that the maximum allowableslope is twice that of the DPC methods. This methodis demonstrated in a reflection-based confocal scan-ning laser microscope for use in surface profilometry.The experimental setup is described, and preliminaryresults are presented. Limitations of this techniqueare also explored.
2. Theory
A. Optical Differentiation in the CoherentOptical ProcessorTo facilitate a physical understanding of the newsystem described in this paper, it is useful to examinethe simple coherent optical processor currently avail-able for optical differentiation. The basic system isshown in Fig. 1. It can be shown with scalar diffrac-tion theory12 that the image intensity for an inputobject given by t(xl, Yl) and illuminated by a coherentplane wave is given by
I(x3, 3) = K ffP(kFm, Fn)T(m, n)
2x exp[i2rr(mx 3 + ny3)]dmdn . (1)
In Eq. (1) T(m, n) is the spatial-frequency spectrum ofthe object, given by
T(m, n) = f f t(xl, yl)exp[-i2,r(mxl + nyl)]dxldy1.
(2)
P(x2, Y2) is the pupil function of the processor, and xjand yj are illustrated in Fig. 1. Equation (1) statesthat the intensity is the inverse Fourier transform ofthe product of the object spectrum and the scaledpupil function, modulus squared. The functionP(x2, Y2) is typically a circular aperture multiplied bythe appropriate amplitude and phase function. It isassumed in this expression that the physical diameterof the lenses is much larger than the diameter of pupilP. If the pupil function is an absorptive filter thatvaries linearly with position in one direction, byFourier theory' 3 the intensity is given by the firstderivative of the object's transmittance, modulussquared. For a pure phase object the image is thederivative of the phase distribution squared. In thecase in which the object consists of both phase and
ObjectPlane
t x,
0Pupil ImagePlane Plane
P(X2,Y2) I(x3,y3)
1I A
F F F
Fig. 1. Diagram of the coherent optical processor.
transmittance or reflectance variations, the image is amixture of these variations.
One can determine the response of this system to aphase slope by using an object transmission functiongiven by
t(xl, Yi) = exp(ikxl tan 0). (3)
This object is simply a phase wedge in the xl directionwith a slope given by tan 0. Inserting this object intoEq. (1), one finds that the intensity as a function ofslope is simply a scaled version of the pupil functionitself:
I(x 3 , y3) = K IP(F tan 0, 0) 12. (4)
For a clear circular aperture of radius po the image isindependent of the wedge angle 0, provided F tan 0 isless than p. In this expression and in others tofollow, K is a normalization constant and F is the focallength, as shown in Fig. 1. For larger phase slopesthe image is zero. If the pupil function is a circularaperture multiplied by a linear absorptive filter, theoutput intensity is directly related to the slope of thephase wedge. This result can be understood withthe following physical explanation. The plane waveilluminating the wedge is refracted by an angledetermined by the slope of the wedge. The refractedbeam is then focused in the pupil plane, where it isattenuated by the filter. Light from different wedgeangles focuses to different locations in the pupil,resulting in a slope-dependent attenuation. Any com-plicated object phase distribution can be reduced to aseries of wedges of different slopes. Hence the result-ant intensity distribution is a mapping of local phaseslopes. This image can then be integrated to pro-duce the object phase function. The only limitationof this method is the maximum slope of the phasefunction. The maximum slope transmitted is givenby pa divided by the focal length F.
B. Confocal MicroscopeThe imaging properties of a confocal microscope arequite different from those of the coherent opticalprocessor. Figure 2 shows a general system layout.This figure depicts a general microscope configura-tion, which can be adapted to different systems. Weobtain confocal operation by illuminating the sourceplane with coherent collimated light, placing a pin-hole centered in the image plane, and by scanning theobject t(xl, yl). It can be shown3 that the imageintensity as a function of scan coordinates (x5, Ys) is
Source Object Pupil ImagePlane Plane Plane Plane
S(xo,y,) t(x,,y) P(x,y) I(xa,Ys)
I1 1F F F F F F
Fig.2. Diagram of the generalized confocal optical system.
where T(m, n) is the spatial-frequency spectrum ofthe object as given in Eq. (2) and c(m, n) is thecoherent transfer function given by
c(m, n) = J J S(xo, yo)P(-xo - XFm, -yo - XFn)
x dxodyo.
varies with the slope of the wedge. The maximumslope transmitted is given by tan mna,, = ±2po/F for aphase wedge in transmission. For a reflection sys-tem the maximum phase slope is given by tan Omnai =+pO/F because of the doubling of the angle onreflection. In either case the transfer function issymmetric about zero, resulting in no informationabout the sign of the phase wedge. Without knowl-edge of the sign of a phase slope, any quantitativereconstruction is impossible. To obtain more infor-mation, an asymmetry must be introduced into thetransfer function. This can be accomplished by plac-ing a spatially varying absorptive filter in the pupilplane, such as,
(6) x2 + y2 < p2
This transfer function is simply the convolution ofthe apertures located in the source and pupil planes.For the case of clear circular apertures the transferfunction is the familiar inverse cosine function,
I p[ p2/2)
2 cos- P P 1- P p< 2poc~m, n) Ir \ 2po 2po [ k2po /
0 p>2po
(7)
where p2 = (XF) 2(m2 + n2) and is the radius ofpupils S and P. This function is plotted in Fig. 3.It is important to note that the spatial-frequencycutoff is twice that of a typical coherent imagingsystem.
C. Optical Differentiation in Confocal MicroscopyConsider the response of the confocal microscope tothe phase wedge described in Eq. (3). A straightfor-ward calculation shows that the image intensity is aconstant given by
I(x 8,YS) = c 2 (8)
The image is now a scaled version of the transferfunction. For clear circular apertures the intensity
1.0
0.8 -
0.6-
1.0
Fig. 3. Plot of c(m, 0) for clear circular apertures.
b b
P(X' A = 2p+ -2O
* (9)x2 + y > p2
This particular pupil function is linear in absorptionalong the x direction. The slope is given by theconstant b, where b may assume values between zeroand one. When this function is inserted into Eq. (6),along with a clear circular aperture for S(xo, yo), theresulting transfer function is no longer symmetric.Figure 4 shows several plots of c(m, 0) for differentvalues of the absorption slope b. As the slope ap-proaches a minimum of zero, the transfer functionapproaches that of the conventional confocal micro-scope.
The asymmetry introduced in the transfer functionproduces different outputs for phase slopes that areequal in magnitude but of opposite sign. For a givenimage intensity, however, two different slopes arepossible, still resulting in ambiguity. The proposedmethod for resolving this ambiguity is to use twoidentical absorptive filters, oriented 1800 to eachother. By simply inserting a beam splitter after thefirst lens on the image side of the object, the light canbe directed to two separate filters. The transferfunction for the reversed filter is identical to theprevious function but flipped about the zero fre-quency axis. Each filter path has its own pinholeand detector, resulting in two signals to be processed.The result when the two signals are subtracted is
1.0
b=O0.8 b=0.2
b=0.5
0.6 b=0.75b=1
Fig. 4. Plots of c(m, 0) with linear absorptive pupil filters.
Fig. 5. (a) Plot of the difference signal for the linear absorptivefilter (b = 1). (b) Plot of the sum signal for the linear absorptivefilter(b = 1).
shown in Fig. 5(a) for the case b = 1. Note that thisresult is the difference between image intensities,which are proportional to the transfer functionsmodulus squared. The difference signal is an oddfunction of the phase slope. For small slopes aboutzero the signal is nearly linear; but the result isambiguous for larger slopes. To obtain a completelyunambiguous response to slope, the difference isdivided by the sum of the two signals. The sum is aneven function of slope, as shown in Fig. 5(b), anddecreases monotonically as the phase slope ap-proaches the maximum transmitted slope. The ef-fect of calculating this ratio is to create a resultantsignal that can vary between -1 and 1 and is aone-to-one mapping between the output signal andphase slope. This relationship is plotted in Fig. 6.
The complete method for determining a phaseprofile consists of scanning the object while recordingthe signals from both detectors, calculating the sumand differences of the two signals, computing thedifference divided by the sum, converting the data toslope with the inverse of Fig. 6, and finally integratingthe signal. The effect of changing the slope of thelinear absorption filter is simply to change the slope ofthe difference/sum (diff/sum) ratio or the calibrationcurve. This is illustrated in Fig. 7 where the ratio isplotted for several values of slope b.
In addition to providing an unambiguous slopemeasurement, this technique is insensitive to trans-
0
1.0
0.5
0.0
-1.0 II-1.0 -0.5 0.0
Normalized Slope
Fig. 6. Plot of the diff/sum ratio for linear(b= 1).
0.5 1.0
absorptive filters
mittance or reflectance variations in the object. Foramplitude variations that vary slowly when comparedwith the size of the illuminating spot, the reduction inlight occurs in both filter paths equally. By dividingthe difference signal by the sum, we can cancel thesevariations.
The technique outlined above provides only onecomponent of the phase slope, namely, the compo-nent along the direction of the absorptive filter.Although phase slopes in the orthogonal direction arenot measured with this technique, it is importantthat they not interfere with the measurement. Todetermine the response to a phase slope along anarbitrary direction, we select the following inputobject:
t(x1, yl) = exp[ik(xl tan O, + Yi tan 0)], (10)
where tan , and tan Oy correspond to slopes in the xland yi directions, respectively. Inserting this objectinto Eq. (5), we find the image intensity:
| Y tan O, tan 0 2,(Xs,, s) = K C A A/ (11)
This result simply states that both axes of thetwo-dimensional (2-D) coherent transfer function canbe scaled in terms of phase slope. We obtained the
1.0
0.5
0
E<-0.5
b=.75
.1.0-1.0 -0.5 0.0 0.5 1.0
Normalized Slope
Fig. 7. Plots of the diff/sum ratio for several values of filterslope b.
transfer function plots shown in Figs. 3-7 by takingthe O, = 0 line from the 2-D transfer functions. Thefull 2-D coherent transfer function for a single linearabsorptive filter [b = 1 in Eq. (9)] is plotted in Fig. 8.As before, we obtained the signal by taking the ratioof the difference and the sum for two filters. Thisratio is plotted in Fig. 9. Clearly slopes orthogonalto the filter do not affect the measurement of slopesalong the filter. This plot also shows that the limita-tion on the magnitude of the slope must include bothorthogonal components of the slope. This limitationcan be written as
[(tan 0,,)2 + (tan OY)2]1/2 < pF (12)
for a reflection system.
D. Comparison to the Split-Detector Method
The system described above is similar to the split-detector method, also known as DPC or CDPC. Inboth methods two detectors are used to measure amodulation of intensity in the pupil because of localphase slopes in the sample. Each detector measuresthe light in one half of the pupil. One obtains thedifferential phase-contrast images by subtracting theintensities measured by the detectors. As describedin Section 1, a large-area split detector may be used toproduce nonconfocal DPC images, or pinholes may beused in a suitable configuration to produce CDPCimages. A direct comparison can be made betweenCDPC and the use of linear absorptive filters bychoosing as a pupil filter the function
P(x, ) = ox
2+ y
2< p2 , X > 0
x2
+ y2
> p2 , x < 0
and performing the same analysis as for the linearabsorptive filter. The coherent transfer functionc(m, 0) is shown in Fig. 10 for this half-pupil functionalong with c(m, 0) for the linear absorptive filter(b = 1). As before the response to a phase wedge isgiven by the coherent transfer function. The differ-ence in intensities that is due to two such filters isplotted in Fig. 11(a) as a function of the normalizedphase slope along with the difference signal for the
c(m,n)
n
Fig. 8. 2-D surface plot of c(m, n) for the linear absorptive filter(b = 1); m and n correspond to spatial frequencies in the x and ydirections.
diff/sum
y-slope
x-slope
Fig. 9. 2-D surface plot of the diff/sum ratio for the linearabsorptive filter (b = 1).
linear absorptive filter (b = 1). This curve shows theresponse typically obtained with a CDPC microscope.For small phase slopes, less than approximately onethird of the cutoff, the response is unambiguous.Proper calibration and integration provide the correctprofile. For larger phase slopes, however, the resultbecomes ambiguous. If the difference signal is di-vided by the sum signal, shown in Fig. 11(b), theoperating range can be increased to one half of thecutoff. The resulting response curve is shown inFig. 12. For slopes greater than ± po/2F, the outputsignal is independent of slope. These regions corre-spond to the case in which there is no light on one ofthe detectors. With the absorptive filters, both detec-tors are illuminated until the phase slope exceeds± po/F. The absorptive filter system images twicethe range of slopes of the CDPC system.
In addition the signal obtained with the CDPCmethod is affected by phase slopes orthogonal to thedirection of the filter. Figures 13 and 14 show the2-D coherent transfer function c(m, n) for the splitfilter in Eq. (13) and the signal obtained by dividingthe difference signal by the sum signal. As with Fig.9, Fig. 14 gives the response to a phase wedge with aslope in an arbitrary direction, as described by Eq.(10). The variations in signal along lines of theconstant x slope indicate that the measurement of aslope along the filter direction is influenced by slopesorthogonal to the filter direction. This result se-verely limits the method for accurate quantitative
0.o
0.4-
0.3 -
0.2 -l
0.1
0.0 -
-1.0 -0.5 0.0 0.5 1.0Normalized Slope
Fig. 10. Plot of c(m, 0) for a half-pupil. The dashed curvecorresponds to c(m, 0) for a linear absorptive filter (b = 1).
Fig. 11. (a) Plot of the difference signal for half-pupils. (b) Plotof the sum signal for half-pupils. The dashed curves correspondto linear absorptive filters (b = 1).
phase measurements and gives the absorptive-filtermethod a distinct advantage.
3. Experimental SetupThe method described above has been implementedin a reflection-based surface-profiling instrument.Figure 15 illustrates the main components and layoutof this system. The light from a linearly polarized10-mW He-Ne laser is spatially filtered and collimated.The 40x, 0.65-numerical aperture (N.A.) microscopeobjective focuses the beam to a small spot on the
.2a:
U)
a
1.0Normalized Slope
Fig. 12. Plot of the diff/sum ratio for half-pupils. The dashedcurve corresponds to the diff/sum ratio for linear absorptive filters(b = 1).
0.5
c(m,n)
Fig. 13. 2-D surface plot of c(m, n) for half-pupil; m and ncorrespond to spatial frequencies in the x andy directions.
sample and collects the reflected light. The combina-tion of polarizing beam splitter and quarter-waveplate is used to eliminate ghost reflections and tomaximize throughput. A nonpolarizing beam split-ter then directs the reflected light along two separatepaths through the absorptive filters to a lens andpinhole combination. Two silicon photodiodes oper-ating photovoltaically then detect the light behind thepinholes. The currents generated are the input sig-nals to a sum and difference amplifier, and the outputsignals are digitized in the computer. We build upan image by mechanically scanning the object in one,two, or three dimensions. Lateral scanning is accom-plished with crossed roller-bearing stages driven byinchworm piezoelectric motors. These devices havea range of 25 mm with a minimum step size of 4 nm.Axial or through-focus scanning is done by movingthe microscope objective with a piezoelectric device.This device has a range of 100 im with a resolution of10nm.
The one-dimensional (1-D) absorptive filters usedare simple variable-density beam splitters. In thesebeam splitters the optical density varies linearly withposition along one direction. The amplitude trans-mittance varies exponentially in the x direction.Although the system has been modeled with linearabsorptive filters, exponentially varying filters pro-duce similar results. In particular the only differ-ence is a small change in the shape of the diff/sumratio versus slope curve. The range of slopes is notaffected.
pinhole f=200mm BS PBS objective30 lgm 40X, 0.65 NA
Fig. 15. Diagram of the experimental surface-profiling instru-ment: f, focal length; BS, beam splitter; PBS, polarizing beamsplitter.
Objects with height variations greater than thedepth of field must be scanned axially to obtain thesignal at best focus. The sum and difference signalscorresponding to best focus are found when the sumsignal is at its maximum. This is equivalent to theautofocus method commonly used in confocal micros-copy to extend the depth of field arbitrarily. Alterna-tively a servo system incorporating an autofocussensor and feedback could be implemented to main-tain the focus of the objective. The ratio of the twosignals is then calculated and converted to the objectslope with the calibration curve.
The microscope objective chosen for this instru-ment correctly determines surface slopes as great as+40.5 deg. We find this maximum slope by calculat-ing the inverse sine of the N.A. of the objective. InTable 1 we list the maximum slopes obtainable withcommonly used microscope objectives. The surfaceslope is listed in terms of both degrees and absoluteslope.
4. ResultsBecause of variations in the absorption profiles of thefilters and alignment errors, it is essential to performa calibration of the system. The calibration entailsmeasuring the response to a series of known surfaceslopes. For this purpose a small steel ball bearing ofknown diameter has been used. This object providesa continuous range of slopes that can be preciselycalculated because of its excellent sphericity. A ballbearing with a diameter of 790 ± 1 pum (as measuredwith a micrometer) was used for this purpose.
Figure 16(a) shows a plot of the sum and differencesignals obtained by scanning in one dimension alongthe center of the ball bearing. The high-frequency
Table 1. Maximum Surface Slopes Imaged with Common MicroscopeObjectives Using the Absorptive Filter Method
Fig. 16. 1-D line scan of the 790-.m ball bearing: (a) sum anddifference signals, (b) diff/sum ratio signal.
variations in the data are due to surface roughnessand small changes in reflectivity on the surface of thesphere. Figure 16(b) shows the resulting ratio signal.These line scans are plotted against the lateral posi-tion in the scan. We can convert the horizontal axisto slope by calculating the derivative of a sphere (orcircle for the 1-D case), using the known diameter.The curve in Fig. 16(b) can then be inverted to theplot slope versus the diff/sum ratio. Noise causedby roughness in the sample is removed at this stepwith a linear filter smoothing algorithm. The re-sults of these steps yield a calibration curve, as shownin Fig. 17. The calibration curve in this measure-ment is not exactly asymmetric, most likely because
of alignment errors and differences in the profiles ofthe two absorptive filters. It is also evident that theslope range is not as great as theoretically predicted,clearly because of the signal-to-noise ratio. As seenin Fig. 16 both sum and difference signals are rapidlyapproaching zero at the edges of the ball bearing.The process of dividing two small numbers is muchmore sensitive to noise, causing a lack of accuracy atlarge slopes.
As a test of this technique a second ball bearing oflarger diameter was measured as an unknown object.(The diameter of the second ball bearing was latermeasured with a micrometer to be 991 1 jIm.)The results of this measurement are shown in Figs.18 and 19. Figure 18 contains the sum, difference,and ratio signals for the 1-D line scan. The ratiocurve of Fig. 18(b) was converted to a slope with thecalibration curve of Fig. 17 and numerically inte-grated. The result of this integration is shown inFig. 19(a) plotted versus both lateral position andactual slope in degrees. Also plotted in Fig. 19(a) is acircle with a diameter of 991 jim (dotted curve).Figure 19(b) shows a measure of error obtained bysubtracting the circle from the data. The rms heighterror over the entire data was 0.53 jim. The errorover the 300-jim center was only 0.060 im rms.
5. DiscussionThis paper has presented a new method for quantita-tive phase imaging based on optical differentiation
2500
2000 -(a)
1500 -
50
01
-5000 100 200 300 400 500
Lateral Position (m)
1.0
(b)
0.5
.2
0
0 100 200 300 400 SooLateral Position (m)
Fig. 18. 1-D line scan of the 991-pum ball bearing: (a) sum anddifference signals, (b) diff/sum ratio signal.
6
.
CD
A
2
3.
0)v
w
Slope (degrees)-30 -20 -10 0 10 20 30
I I I I I I I
(a)60/\
/O \
C e I I I I0 1o00 200 300 400 5C
Lateral Position (m)
0 100 200 300 400 500Lateral Position (m)
Fig. 19. 1-D line scan of 991-.m ball bearing. (a) The surfaceprofile obtained by the calibrating diff/sum ratio then integrating;the dotted curve corresponds to a circle of 991-l.m diameter. (b)The difference in the profile from the circle.
and confocal microscopy. The advantages of thissystem include the large range of phase slopes im-aged, the improved lateral resolution, and the insensi-tivity to amplitude variations in the object. In addi-tion many of the techniques devised for confocalmicroscopy are applicable to this system. Thesetechniques include the variety of methods availablefor scanning the object and detecting the reflected ortransmitted light.
In the area of profilometry this method can provideboth surface figure and roughness data. It can beused to measure profiles of aspheric surfaces as wellas spheres and flats. The limits on height variationsand scan areas are determined only by the mechanicalscanning configuration, not by the principles of theslope-detection method. It is therefore possible tocharacterize large optics with this method. In addi-tion, the amount of time necessary to collect thesurface profile depends on the scanning configuration.The experimental setup described in Section 3 re-quires several minutes for each line profile, but asystem employing beam scanning, rather than objectscanning, would require significantly less time.
The method has been demonstrated in the reflec-tion instrument described above. The same prin-ciples can be applied to a transmission instrument aswell. In this case variations in refractive index or
thickness are determined. Other possible areas ofuse include optical data storage and biomedical imag-ing.
Support for this research was provided by theCenter for Optics Manufacturing.
References1. R. A. Sprague and B. J. Thompson, "Quantitative visualiza-
tion of large variation phase objects," Appl. Opt. 11, 1469-1479 (1972).
2. J. C. Bortz and B. J. Thompson, "Phase retrieval by opticalphase differentiation," in Wavefront Sensing, N. Barakat andC. Koliopoulos, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 351,71-79 (1982).
3. T. Wilson and C. J. R. Sheppard, Theory and Practice ofScanning Optical Microscopy (Academic, London, 1984).
4. D. K. Hamilton and T. Wilson, "Three-dimensional surfacemeasurement using the confocal scanning microscope," Appl.Phys. B 27, 211-213 (1982).
5. C. J. R. Sheppard and H. J. Matthews, "The extended-focus,auto-focus and surface-profiling techniques of confocal micros-copy," J. Mod. Opt. 35, 145-154 (1988).
6. D. Kermisch, "Visualization of large variation phase objects,"
in Image Processing, J. C. Urbach, ed., Proc. Soc. Photo-Opt.Instrum. Eng. 74, 126-129 (1976).
7. D. K. Hamilton and C. J. R. Sheppard, "Differential phasecontrast in scanning optical microscopy," J. Microsc. 133,27-39(1984).
8. J. P. H. Benschop, "Confocal differential phase contrast inscanning optical microscopy," in Scanning Imaging Technol-ogy, T. Wilson and L. Balk, eds., Proc. Soc. Photo-Opt.Instrum. Eng. 809,90-96 (1987).
9. J. P. H. Benschop, "Phase detection using scanning opticalmicroscopy," in Integrated Circuit Metrology, Inspection, andProcess Control II, K. M. Monahan, ed., Proc. Soc. Photo-Opt.Instrum. Eng. 921, 123-130 (1988).
10. A. E. Dixon, S. Damaskinos, M. R. Atkinson, and L. H. Cao, "Anew transmission and double reflection scanning beam confo-cal microscope: applications in transmission," in ScanningMicroscopy Instrumentation, G. S. Kino, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1556, 144-153 (1991).
11. M. R. Atkinson, A. E. Dixon, and S. Damaskinos, "Surface-profile reconstruction using reflection differential phase-contrast microscopy," Appl. Opt. 31, 6765-6771 (1992).
12. J. W. Goodman, Introduction To Fourier Optics (McGraw-Hill,New York, 1968).
13. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics(Wiley, New York, 1978).
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