JOURNAL OF MODERN OPTICS, 1988, voi,. 35, NO. 1, 145-154 The extended-focus, auto-focus and surface-profiling techniques of confocal microscopy C. J. R. SHEPPARD and H. J. MATTHEWS University of Oxford, Department of Engineering Science, Parks Road, Oxford OX1 3PJ, England (Received 6 January 1987 ; revision receivedf 28 July 1987) Abstract. The extended focus and auto-focus techniques of confocal microscopy, which can be used to give greatly increased depth of field, are compared. These methods rely on the use of high numerical apertures, so that the paraxial approximation is found to give inaccurate theoretical image predictions. An improved approximation is discussed and compared with exact high-angle calculations. The confocal surface-profiling technique, which allows noninvasive investigation surface topography is also considered. 1. Introduction The confocal imaging system [1] results in an improved resolution compared with conventional optical systems. It also has a unique optical sectioning property resulting in the rejection of defocused information from the image [2]. This allows the use of three further imaging techniques [3]. The extended focus [4-7] and autofocus techniques [8, 9] result in a vastly increased depth of field whilst retaining high resolution, whereas the surface-profiling technique [8, 11-14] allows non-invasive investigation of surface topography. This paper-compares the imaging properties of the extended focus and the autofocus techniques. In particular the effects of using these techniques to obtain threedimensional images of a phase edge object, as results from a surface step, are considered. Images of a phase edge in the surface-profiling mode are also discussed, demonstrating the properties and limitations of the method. 2. Automatic focus and surface-profiling techniques In a confocal microscope the object is illuminated with a diffraction-limited focused laser spot and the reflected (or transmitted) light detected with a point-like detector (figure 1). The image is built up point by point by scanning the object in a three-dimensional raster. The intensity recorded in a confocal system for an infinite surface of arbitrary reflection coefficient reaches a maximum when the surface coincides with the focal plane of the system. If the object is thus scanned axially, the distance moved from some datum in order to record the maximum intensity is a measure of the surface height of the specimen at that point. This is the basis of the surface profiling technique [8, 11-14], which can be used to produce one- and two-dimensional plots of surface topography with a sensitivity of better than 20 nm. The results can be presented in the form of an isometric plot [11], or by brightness modulating the display so that bright represents regions of the specimen closest to the objective [8]. t Received in final form 19 October 1987.
1 4 6 C. J. R. Sheppard and H. J. Matthews Beam splitter I Focal i l plane l Figure 1. Schematic diagram of the scanning optical microscope. The maximum intensity may be determined by using either analogue [10, 11] or digital [8] techniques. As well as recording the surface height in this way, the maximum intensity also gives a measure of the local surface reflectivity. The depth of focus of this auto-focus technique is in principle unlimited. If the specimen surface is assumed flat over the illuminated region then if this elemental surface is inclined at an angle y it is found [8] that the maximum signal still occurs when the surface intersects the focal point, but the value of this maximum signal depends on y, as shown in figure 2. Here sin a is the numerical aperture of the system. We would thus expect that the surface-profiling method would exactly follow the surface height as long as it is slowly varying with the resolution element. The image of a single point object in an auto-focus system is shown in figure 3. Here the optical coordinate v is defined as v = kr sin a. (1) Conventional (full illumination) Extended focus Automatic focus 0.8 Figure 2. The variation in the signal from a perfect reflector with the angle y its normal makes to the optical axis.
Techniques of confocal microscopy 147 V)C N C Figure 3. The image of a single point in automatic and extended focus modes, compared with that in a conventional microscope. with k the wavenumber 2tt/1 and r the radial distance from the optical axis. It is seen that the response is sharper than in a conventional microscope. Figure 4 shows contours of the intensity in the three-dimensional image of a single point in a confocal system. The axial optical coordinate u is defined as V Conventional Extended focus /Automatic focus u=4kzsin 2 a/2. (2) The shadow edge is also shown. The region of image space for which the intensity is greater than one per cent of the maximum is indicated. It should be noticed that this is much smaller than the corresponding region for a conventional microscope, which is large and irregularly shaped. This accounts for the superior imaging of confocal microscopy. The behaviour in the auto-focus mode is indicated in figure 4. Near v = 0 the system is focused on the focal plane, but around v=7c the position for maximum signal departs from the focal plane and for large v tends towards the shadow edge. The auto-focus technique is highly nonlinear and hence imaging cannot be expressed simply in terms of a transfer function. Defocused images of a straight edge in a confocal microscope have been presented elsewhere [14]. It is found that the intensity for some parts of the dark region of the image reaches a maximum slightly away from the focal plane [15], so that the auto-focus image of a straight edge is not identical in shape to the confocal image, but is nevertheless sharper than in a conventional microscope. Theoretical images have also been previously given [14] for a phase edge such as is formed by a surface step when examined in a reflection microscope. If the numerical aperture of the system is appreciable the image profile depends both on the numerical aperture and the position of focus. These images are calculated as the superposition of two defocused straight-edge images. Multiple scattering and shadowing effects are ignored and paraxial imaging assumed. From calculation of a series of images for 10
1 48 C. J. R. Sheppard and H. J. Matthews Figure 4. The intensity in the confocal image of a single point. The locus of the auto-focus scan of the image is also shown. The cross-hatched region is that in which the intensity is greater than 0. 01. The corresponding region for a conventional system is shown shaded. different positions of the focal plane relative to the step the auto-focus image and also the surface profile image may be determined [16]. Such images cam be calculated for the paraxial approximation using Fourier optics and defocused transfer functions as presented elsewhere [5]. However for large numerical apertures, which are usually used in practice and are indeed necessary to give substantial resolution in the axial direction, the paraxial approximation breaks down. The transfer function does not reduce any longer to a single integral, but must be evaluated as a double integral, thus greatly extending image computation time. However the intensity at the edge itself as a function of defocus can still be evaluated easily, and can explain some properties of the various imaging methods. For an object consisting of an infinite flat and level perfect reflector the image amplitude U(qb') is found to he [17] U(O') = exp (2io' cos t a/2) where the coordinate 4' is defined sin (2~' sm 2 a /2) 2(psin a/2 U + i sin (24'sin 2 a/2) ] -cos (2(a'sin 2 a/2)~ j, (3) 20' cos t a/2 24' sin e a/2 O'= kz. (4) For the phase step of phase change 0, the intensity at the edge is given by 1(0, 0')=41 U(q'+0/2)+ U(O'-0/2)l2. (5) Introducing a focus parameter n given by n=20'/0 (6)
Techniques of confocal microscopy 149 the intensity variation is shown in figure 5 for an aperture a=60. For the paraxial approximation the expression for the amplitude for a plane reduces to U(tp')=exp [2icb'(1-4 sin e a)] Sin [(O' s a)j2/2]' whereas for the pseudoparaxial approximation, in which the second term in the square brackets of (3) is neglected, we have sin ~casin 2 2) (8) U(cp')=exp [2ic)' cos t a/2] The intensities at the edge according to these two approximations are also shown in figure 5. It should be noticed that the pseudoparaxial approximation gives quite close agreement with the exact high-angle case, but the paraxial approximation gives a very poor prediction. The pseudoparaxial approximation thus gives a good agreement for the intensity at the edge and also far away from the edge, and this is our justification for assuming it to give reasonable agreement everywhere for numerical apertures up to about V3/2 [18]. The pseudoparaxial approximation can thus be used to calculate image profiles at high apertures whilst avoiding the computational problems of the full high-angle theory. Referring to figure 5 (a) we see that the behaviour breaks into two distinct regimes according to whether or not the intensity reaches a maximum when the system is focused mid-way between the two sides of the step (n=0). If it is a maximum, the profile measured by selecting the maximum intensity is shown schematically in figure 6 (a). This behaviour occurs in particular for values of 0 less than about 7t/2 and those lying between about 0. 8671 and 1-717r. If on the other hand it is not a maximum, the measured profile will not be single-valued at the edge, and the profile exhibits a discontinuity (figure 6 (b)). For values of 0 such as 1.717t, the intensity is approximately independent of focus position and the profile has the form shown in figure 6 (c). For values of 0 greater than about 2. 77tt there is always a discontinuity. The discontinuity means that a very sensitive measurement may be made of the step position. For very large step heights the maximum is reached when the system is focused on one half-plane. The overshoot in the profile has been observed in practice [10], but for a range of values of 0 around 2.771 the maximum is predicted to be reached closer to the mid-way position. Figure 6 is of similar form to images calculated for a wide range of step heights and numerical apertures, and these are indeed the only topological forms consistent with the asymptotic behaviour far from the edge. The overall appearance of an auto-focus image is very similar to that in the confocal mode, except that the absolute height information is suppressed. Thus high-resolution diffraction-limited imaging may be achieved with a depth of field vastly greater than in a conventional microscope. 2 (7) 3. Extended-focus technique As an alternative to the auto-focus technique for producing a large depth of focus, the confocal imaging may be used in the extended-focus mode. In this method, the object is again scanned axially but, instead of recording the maximum intensity reached, an integral over all axial positions is performed. As the object is scanned axially, the surface at some time travels through the focal point, and as out-of-focus
1 5 0 C. J. R. Sheppard and H. J. Matthews (a) 1 N C N C (b)
Techniques of confocal microscopy 151 T CC O1 C Figure 5. Intensity at the edge : (a) high angle, (b) paraxial and (c) pseudoparaxial. Aperture a=60. information is detected only weakly in a confocal system this does not greatly affect the image. If on the other hand the axial scanning and integration are performed with a conventional optical system the out-of-focus information produces a blurred final image [4]. The integration may be performed in an analogue manner by photographic recording or by digital techniques. The resultant images are broadly similar to those produced by the auto-focus method. Various properties of the imaging in the extended-focus technique have been discussed elsewhere [5]. The most important difference between the two techniques is that for objects with slowly varying height the extended-focus method results in partially coherent imaging, for which the transfer function has been derived [5]. This has some advantage for numerical calculations and processing. The image of a single point object is shown in figure 3, where it is apparent that the image is very similar to that in the automatic focus mode, but with slightly stronger outer rings (though still weaker than in conventional microscopy). The signal intensity from a plane surface inclined at an angle y is shown in figure 2, where it is seen that the intensity falls off with angle more slowly than with the automatic-focus mode. One result of this is that the contrast at a sloping surface is weaker, but it does also mean that the signal intensity is a closer measure of the surface reflectivity. The image of a straight edge may be calculated by integrating over defocused confocal images [14]. Images of surface steps may be calculated by integrating over all possible focal positions of a surface step in a confocal system [16]. Confocal images fall off in intensity with axial distance so quickly that they converge when integrated from plus to minus infinity. As with the
1 5 2 C. J. R. Sheppard and H. J. Matthews 1 (a) (b) z 0- z (c) z Figure 6. Schematic diagram of surface profile behaviour : (a) maximum at edge occurs midway between half-planes; (b) maximum at edge does not occur mid-way ; (c) intensity at edge is approximately constant. auto-focus images at high-apertures non-paraxial effects become important. The extended focus intensity at the edge for a high-aperture system may be evaluated by integrating (3) to give I(0, cp)= 3 3 ~3(1 -cos3 a)+ sin 20 -cos 3 a sin (2t) cos a) + 1 2(1 -cos a) 2~ 20 cos a 20 2 sin 24 sin (20 cos a) X cos 2cb - + cos a -cos (24 cos a) (9) 20 20 cos a The paraxial approximation gives and the pseudoparaxial approximation sin 2 I(0, 0)=2 1 +cos [ta(2-z sin e a)]c s i ([(si n e a)/2/2]]}, (10) 1 (0,4))= 1 +cos (2tp cos t a/2) si nin 2/2)~ 2 2 ~ (11)
Techniques of confocal microscopy 153 Figure 7. Intensity at the edge in the extended-focus mode. The intensity at the edge according to the high-angle theory is shown in figure 7 for various apertures. For large steps the intensity at the edge tends to a value of onehalf, compared with one-quarter for the auto-focus mode, so that contrast is usually greater for the auto-focus mode. As with the inclined surface greater contrast may be an advantage for visualizing and locating features, but it is in fact an optical artefact in that a true image of surface reflectivity should exhibit no variation in intensity at a phase step. 4. Discussion The use of the pseudoparaxial approximation for calculating images of phase edges is justified. Examination of intensities at the edge shows that there are two distinct forms of behaviour in the auto-focus mode according to the magnitude of the step. The discontinuity in the measured profile can provide a sensitive indication of edge position. Contrast and resolution are in general higher for the auto-focus rather than the extended-focus mode. The intensity at the edge is also identical to the intensity for two reflecting planes as results from a thin film if multiple reflections can be ignored. An optical thickness of about 1'4A can be resolved with a numerical aperture of.j3/2, measurement of thinner layers being complicated by phase effects. In general the methods described in this paper perform much better at very high apertures. References [1] SHEPPARD, C. J. R., and CHOUDHURY, A., 1977, Optica Acta, 24, 1051. [2] SHEPPARD, C. J. R., and WILSON, T., 1978, Optics Lett., 3, 115. [3] SHEPPARD, C. J. R., 1984, 13th Congress of the International Commission for Optics, Sapporo, Japan, pp. 502-503.
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