Image formation in scanning microscopes with partially coherent source and detector

Transcription

1 OPTICA ACTA, 1978, VOL. 25, NO. 4, Image formation in scanning microscopes with partially coherent source and detector C. J. R. SHEPPARD and T. WILSON Oxford University, Depart of Engineering Science, Parks Road, Oxford, U.K. (Received 13 October 1977) Abstract The effect of partial coherence of both the source and the detector in a scanning microscope is investigated. The transfer function for the system is derived and various special cases discussed. If the effective source and effective detector are coherent and incoherent, respectively, the microscope (Type 1) is of the form of the scanning transmission electron microscope (STEM). If the effective source and the effective detector are both coherent, the microscope (Type 2) is of the form of the scanning acoustic microscope. Scanning optical microscopes of both these types may be constructed. The effect of using a Type 2 scanning microscope in dark-field is discussed. This arrangement has the advantage over the Type 1 dark-field microscope that imaging for weak contrast specimens may be made linear. 1. Introduction Since the publication of the theory of Fourier imaging in the conventional transmission microscope by Hopkins [1] there has been some development which has been concerned with the theory of imaging in the scanning transmission microscope [2-5]. Although the imaging process is substantially similar in electron microscopes and light microscopes there are differences between microscopes employing these two forms of radiation. In the electron microscope spherical aberration is very important and phase objects are usually examined by defocusing ; in the light microscope the optical system is usually diffractionlimited and absorption in the specimen may produce contrast although phase objects are also studied using special techniques. It should be remembered that if the imaging process is analysed, taking into account the complex nature of the lens pupil functions and object transmittance, the results may be applied to cases where either light or electrons are used. It has been shown that a scanning microscope which employs a point source and an incoherent detector behaves identically to a conventional microscope in which the objective and condenser lenses are the same as the objective and collector lenses respectively of the scanning microscope. One result of this is that the aberrations of the collector lens are unimportant-it is solely a light-gathering device. A second result is that the resolution of the microscope is primarily determined by the properties of the objective lens, whereas the relative numerical aperture of the collector lens affects the coherence of the detection system. Coherence effects in both electron and light microscopes have been considered by Barnett [6]. The detection system is in general partially coherent, even though the detector is incoherent. This is analogous to the fact that the object illumination in a conventional microscope is partially coherent, although the source is incoherent. The analogy may be taken a step further.

2 316 C. J. R. Sheppard and T. Wilson Figure 1 shows three forms of scanning microscope which may be shown to behave identically to one another [7]. In 1 (a) we have an effective detector which is analogous to the effective source of the conventional microscope. In 1 (c) the radiation from the object is focused onto the detector, which is analogous to the critical illumination system, and may be termed `critical detection'. In 1 (b) the detector is placed in the back focal plane of the collector lens, and we have `Kohler detection'. The size of the detector also affects the imaging. In the limiting case as the detector becomes point-like (figure 2 (a)) imaging becomes coherent and both lenses contribute to the resolution [7]. This microscope has been termed ` confocal ' or Type 2 to distinguish it from the previously described Type 1 scanning microscope. It exhibits an overall improvement in imaging performance. Figure 2 (b) shows a further form of this type of microscope which employs a coherent detector. This is an equivalent way of regarding the Type 2 microscope, and has been realized in practice in the acoustic microscope [8] using an amplitude-sensitive detector. It may be achieved in an optical microscope by using superheterodyne detection [9] j -. In the present paper we generalize to the case of a finite detector, which is equivalent to a partially coherent effective detector. OBJECTIVE LENS COLLECTOR LENS 4, 1 (a) ( b) I'Mr 41000, -1 (c) Figure 1. Scanning microscopes of Type 1. (a) As in the STEM. (b)' Kohler detection'. (c) ' Critical detection ' POINT DETECTOR COHERENT DETECTOR Figure 2. Scanning microscopes of Type 2. (a) Using a point detector. scanning acoustic microscope. (b) As in the t The acoustic microscope and the superheterodyne microscope in fact give the square root of the intensity signal from the Type 2 microscope discussed here.

3 Image formation in scanning microscopes 317 We also consider the effects of a finite source. This has been done for the Type 1 scanning microscope [10] where the effect is to degrade the image. In the present paper we derive the transmission function for the generalized scanning microscope, that is, one with finite source and finite detector, and examine various special cases. 2. Analysis of imaging in the generalized microscope We consider the optical system as shown in figure 3, and restrict ourselves to the one-dimensional case for simplicity ; the two-dimensional result being an obvious extension. The source is assumed to have a mutual intensity [11] given by J(xl ; x1 '). The pupil function of the objective (probe-forming) lens is P 1 (6 1 ) and that of the collector lens is P2(62). x l x 0 x \~ d f f ikk S(x t ) PP (t; I ) t(x s x0 ) P2 %) D(x 2 ) \\ SOURCE EFFECTIVE OBJECT EFFECTIVE DETECTOR SOURCE DETECTOR Figure 3. Optical system of the generalized scanning microscope. The mutual intensity in the object plane is given by J(x0 ;x0')= f f 00 J(xl ;xl')hlcx +Xf/M\hl*C x +X1/M'dxldxl', where h1 is the impulse response of the first lens, that is the Fourier transform of P1, M is the spot demagnification, d/f, and we have ignored a constant factor. Here f, the distance of the lens from the object plane, is approximately equal to the focal length of the lens. If the object when scanned has an amplitude transmittance t(x6 -x0 ), we find the measured intensity at the detector of intensity sensitivity D(x2 ) by propagating the mutual intensity to the detector plane, putting X2'= x 2 and integrating over the detector to give I(xa)= f f f fjj(xl ; x1 )h1cx0+~ /Ml hl*c x0+~ /M l t(x" -x 0 )t*(xa-x )' 00 ( 1 ) xh2cx0+x2/m\h2*c x0+xf /M )D(x 2 )dxi dxi 'dxo dxo 'dx2. (2) If the source is incoherent, the mutual intensity in the x 1 plane is J(x1 ;x1')=8(x1-x1')s(x1), (3) where 8 is the Dirac delta function and S(x l ) is the intensity distribution of the

4 318 C. J. R. Sheppard and T. Wilson source. This gives an image intensity I(x8)=ffffS(xl)h1 -~ x 0 +x1/m\ hl* Cx +~IM I t(x8 -x )t*(x 8 -x ') xh2cx0 +X21M \h2*(xo' +X 1M)D(x2)dx l dx dx 'dx 2. (4) This expression may be used to calculate the image intensity of a non-periodic object. Let us consider the case of a point-like object 8(x 8 -x ). Then we have I(xe)= f S(xl)Ihl( xs+x1/m)} f 2dx1f D(x2) h2( x g +~f21m)i 2 dx2i ( 5 ) = {S(Mx8) 1 h1(x81af)1 2 }{D(Mxe) Ih2(x81ltf)1 2 }, where denotes the convolution operation. The image is sharpest when both source and detector are points and is degraded when either or both have finite size. As both become large, imaging becomes very poor. We now consider imaging in terms of the Fourier transforms of the object transmittance. We have +'0 *8-x )= f T(m)exp2irjm(xe -x )dm, (6) (7) t*(xe-x ')= f T * (p)exp -2 lrjp(xe - xo')dp, and substituting in equation ( 2), the intensity in the image may be written in the form where I(x8)= f f C(m ; p)t(m)t*(p) exp 2irj(m-p)xe dmdp, (8) C(m ;p)=fffffj(xi X X {hl*(xo, +f1m ( xo +xl/m)h(xo+x2/m\ Af ;x t ')~h 1 2 exp - 2lrjmx ~ h2* C x0 +~f1m\ exp 2irjpx ' } D(x 2 ) dxl dx l ' dx dx ' dx 2. (9) The expressions in curly brackets are Fourier transforms, and using a combination of the shift and similarity theorems [12, p. 276] we obtain +~ \ C(m ;p)=fffffj(xl ;x1')expmf(e1x1-61'x1')d(x2)exp M (m-~f-p+of lx 2 xp1(ei)p1*(e1')p2(afm-s1)p2*(afp-61 ')dx 1 dx l'dx2 del d61 '. (10) Considering the case of an incoherent source for which equation (3) holds,

5 Image formation in scanning microscopes 319 we may express S(xl ) and D(x2 ) in terms of their Fourier transforms to give +`0 Fs (u) = f S(x 1 ) exp { - 2irjuxl } dxl, +'0 FD (v) = f D(x 2 ) exp { - 2irj vx2 } dx 2 (11) C(m ;p) = f0fsi AMfl~FD C(pM)+( AMfl')j i( l)pl * ( l ' ) 00 xp2(afm - ei)p2 * (Afp - e1 ' )deldel '. (12) We have thus expressed the intensity variation in the image of an arbitary specimen by equation (8), in which C(m ;p) is a transfer function for the given microscope system given by equation (12). The transfer function, which is the same as Burge's [10) transmission cross-coefficient, is a property only of the microscope and not of the specimen. For any two spatial frequencies m ;p present in the object, equation (9) gives the magnitude of the output intensity at a spatial frequency of m -p. The results of this section could be produced without using the concept of mutual coherence. All we need to do is to find the intensity distribution at each plane of the system in turn. 3. Special cases of the generalized scanning microscope The transfer function C(m ; p) is determined by the pupil functions of the lenses and the distributions of the source and detector. For given pupil functions we have various special cases, when the source or detector are either pointlike or infinite. By infinite we mean that the demagnified source size is large compared with the point-spread function of the objective lens ; that is, the source is large compared with the point-spread function on the source side of the lens. This is in fact not very large at all ; for the scanning optical microscope or STEM it might be of the order of a few hundred micrometres. On the other hand, a point-like source or detector must be no bigger than a few tens of microns. If the source is point-like but the detector is infinite, we have p-m (el - 61' )\ FFs = 1, FD =B + (13) \ M' ;1Mf and putting these values into equation (12) we obtain +,0 t C (m ; p) = f Pl(61)P1 * ( Sl+(p -m )Af)IP2(Afm - 61)I 2 dd1, (14) which is thus independent of the aberrations of the collector lens. Transforming the variable by putting e2 = Af m - f l, and assuming the pupil function of the collector lens is everywhere either zero or unity, we obtain C(m ;p) = f -OD P2(e2)Pl(Afm - S2)Pl * (Afp - 62)dd2

6 320 C. J. R. Sheppard and T. Wilson which is the same result as was obtained before for the Type 1 scanning microscope [7]. This then is the case of the normal STEM. If the source is infinite but the detector point-like, we have FD= 1, Fs=B ( 1/-S1 l, AMf / and if P1 is now everywhere either zero or unit y (16) +'0 C(m ;p)= I Pl(61)P2(Afm-61)P2*(Afp-e1)del, (17) which is the same as equation (15) with P 1 replaced by P 2 and vice versa. By reciprocity we might have expected such a microscope to behave identically to the Type 1 scanning microscope. Thus a STEM may be constructed by illuminating the object in a conventional electron microscope and scanning a point detector in the image plane. This system might be compared with scanning electron diffraction [13], where the diffraction pattern rather than the image is scanned relative to the detector. If both source and detector are point-like, we have which gives FD =Fs =1, (18) C(m ; p)={p1(afm) P2(Afm)}{Pl * (Afp) P2 * (Afp)}, ( 19 ) which is again the same result as was obtained before [7], this time for the Type 2 scanning microscope. As explained in an earlier paper, a Type 2 microscope exhibits a modest improvement in resolution over a Type 1 microscope, and this may be further improved by using a lens with annular pupil function. This type of microscope is very easily constructed for use with light, but there are problems in doing so with electrons as the object is usually immersed inside the objective lens, although it might be possible to design a symmetrical double lens as both objective and collector. The use of annular lenses in scanning microscopes has also been discussed in a previous paper [14], which was primarily concerned with light as radiation. With electrons, the reduction in spherical aberration given by an annular lens would be of great importance, but the loss of electrons in traversing the annulus would reduce the signal to noise ratio. A further special case is when both source and detector are infinite. Then we have p - m 61-61, Fs =8 1Mf ), FD M + ~ (20) 8C AMf The product of these functions is always zero unless m =p, and then the intensity in the image given by equation (8) is at zero spatial frequency. An image is therefore not formed, and we might term this a Type 0 microscope. 4. The transfer function for the generalized microscope We now consider the form of the transfer function C(m ; p) as a function of two spatial frequencies m and p. Let us assume that P1 and P2 are both unity if 161 < a, and zero otherwise, that is we neglect aberrations. Then for the Type 1

7 Image formation in scanning microscopes 321 microscope (this behaves the same as the conventional microscope), the transfer function is as in figure 4 (b). The cut-off frequency for p = 0 is m = 2a/A f. The transfer function is calculated as the area in the e,, e1 ' plane which is common to the squares of sides P l and P2 (the latter being displaced by Afm, Afp) and which lies also within the strips of widths FS and FD (the latter being displaced a distance (p - m).f / M) which lie parallel to the line 1 = 1 ' (figure 5). For simplicity we have shown FS and FD as simple functions which are zero outside of a certain range and unity otherwise, although, of course, in practice their magnitude will vary with distance, they might in fact be of a gaussian form. This geometrical construction becomes much simpler in the special case of a Type 1 microscope as then FS and FD become a constant and a 8-function (or vice versa). Similarly for a Type 2 microscope both FS and FD are constants. The transfer function is then as shown in figure 4 (a) For a Type 0 microscope the transfer function is only non-zero along the line m =p (figure 4 (c)). For in-between cases we can see that the region of m, p space where the transfer is non zero is reduced continuously from the Type 2 to the Type 0 microscope, Type I being an intermediate case. In a previous paper [7] we discussed the effects the transfer function has on the image of particular objects. In general, for objects with a range of spatial frequencies of approximately equal amplitude the Type 2 microscope exhibits superior resolution to that of the Type 1 or conventional microscope. If the object consists of small amplitude variations on a large background, as often occurs in biological specimens, the cross-frequency terms are negligible, and we need only consider the C(m ; 0) values. For instance, if t(xo ) = 1 + b cos 2lTmx o, (21) (a) (b) (c) Figure 4. Transfer function C(m ; p) for special cases. (a) Type 2 microscope. (b) Type 1 microscope. (c) Type 0 microscope.

8 322 C. J. R. Sheppard and T. Wilson 1 P" t Figure 5. Region of integration for C(m ; p). where b is, in general, complex and of small magnitude, and might be either real (absorption object) or imaginary (phase object), neglecting terms in b 2 we obtain from equation (8) I(x,,) = C(m ; 0)b cos 2rrmxg, (22) t i that is, we are interested only in values of the transfer function with p=0. In this case microscopes intermediate between Types 2 and 1 behave identically to each other, but microscopes between Types 1 and 0 have a degraded performance. The latter microscopes are those considered by Burge and Dainty [10], who discussed imaging of phase objects for which the value of the imaginary part of C(m ; 0) is the important quantity. These authors also consider the dark-field microscope, and show that for a Type 1 microscope an object of the form of equation (21) gives only a spatially constant image. The transfer function is non-linear, and if two spatial frequency components are present, the image is a small difference-frequency component. A Type 2 dark-field microscope may be constructed with one lens of circular pupil and one of annular pupil of sufficient diameter that the pupil functions do not overlap when placed on top of each other. This form of microscope has distinct advantages over conventional dark field microscopes as C(m ; -m) is non-zero except when m=0. Thus single spatial frequencies are imaged. If the square root of the intensity signal is found, the transfer function is linear, and response is enhanced at high spatial frequencies relative to a conventional microscope with an objective of circular pupil function. The Type 1 scanning microscope with point source (or detector) is equivalent to a conventional microscope. There is however no conventional equivalent of the Type 2 scanning microscope known to the authors. In the conventional microscope the source is infinite and this produces an incoherent

9 Image formation in scanning microscopes 323 effective source. The effect of a finite source size on imaging in the conventional microscope has been discussed by Yamamoto et al. [15]. In this case the effective source is partially coherent. Imaging is different from that in the scanning microscope : the main effect is that only a small region of object may be illuminated, resulting in the limiting case to an image of the source rather than the object. We may no longer therefore express the imaging in terms of a transfer function as in equation (8). Equivalence of scanning microscopes and conventional microscopes requires a coherent source and an incoherent detector for the scanning microscope (or an incoherent source and a coherent detector) and an incoherent source for the conventional microscope. This condition is the same as having a coherent effective source and a partially coherent effective detector in the scanning microscope (or vice versa) and a partially coherent effective source in the conventional microscope. There has been some confusion in the literature concerning this point which has mainly resulted from differences in terminology. Our terminology is as shown in figure 3. It should be noted that equivalence is true for partially coherent imaging, that is, it is not restricted to coherent imaging as has been stated elsewhere. 5. Edge-ringing in the scanning microscope Equation (4) may be written in the form I(xg)= J Jh(xo ;x ')t(xe-x )t*(xe-x0')dx dxo, (23) where +00 XI XI h(xo ;x ') = { f S(x l )h1 Cx + Af /M h l )*(' x0+af / M)dx l } 00 / J X + { f D(x 2 )h2 ' x0+x21 h2* X0+x /M\ dx2 } (24) is a spread function for imaging of a pair of points. Equation (23) is a very general expression, not only applicable to imaging in the scanning microscopes considered here. For a symmetric system we can show that h(x ;xo')=h(-x ; -x ') (25) and if S(x l ), D(x 2 ) are real h(xo ;xo')=h*(xo' ;x0). (26 ) We can also see that the function h(x,,7 x 8 ) gives the intensity of the image of a point object. Substituting equations (7) into (23) and comparing with (8) we see that C(m ; - p)= f f h(xo ; x ') exp { - 2-j(mxo+px ')} dxo dxa, (27) that is C(m ; - p) is the Fourier transform of the spread function h(xo ; xo ).

10 324 C. J. R. Sheppard and T. Wilson For the special case when p = 0 C(m ; 0) = f [ f h(xo ; x o') dxo ' exp { - 2irjmx0 } dxo, (28) J that is, the Fourier transform of the expression in square brackets. Kintner and Sillitto (16) have considered the condition that no edge-ringing is observed in partially coherent imaging of a straight edge. Their results may be developed for a more general case by starting from equation (23) and differentiating for the case of a forward step object Using (26) t(x)=0 x<0 i =1 x>0 1 (29) di+ _ + f h(x9 ; x o')dxo ' + h(x 0 ; x e ) f dxo. (30) dx9 - x dixe = ; xo') J f (31) Rl h(xa dxo }. Kintner and Sillitto (16) argue that a necessary and sufficient condition for edgeringing not to occur is that di di _ ` dxo dx e = Rl { f h(x g ; xo ') dxo' } '>0, where I_ is the intensity in the image of a reversed step function. This means that the real part of the inverse Fourier transform of C(m ; 0) must be everywhere non-negative. It is interesting that this same quantity C(m ; 0) is also of importance in the imaging of weak objects. ACKNOWLEDGMENTS The authors would like to thank Professor M. Chodorow for useful suggestions concerning this work, which was supported by the Science Research Council. REFERENCES [1] HOPKINS, H. H., 1953, Proc. R. Soc. A, 217, 408. [2] WELFORD, W. T., 1972, Y. Microscopy, 96, 105. [3] BARNETT, M. E., 1973, Optik, 38, 585. [4] ZEITLER, R., and THOMSON, M. G. R., 1970, Optik, 31, 258, 359. [5] THOMSON, M. G. R., 1973, Optik, 39, 15. [6] BARNETT, M. E., 1974, Y. Microscopy, 102, 1. [7] SHEPPARD, C. J. R., and CHOUDHURY, A., 1977, Optica Acta, 24, 1051.

11 Image formation in scanning microscopes 325 [8] LEMONS, R. A., and QuATE, C. F., 1974, Appl. Phys. Lett., 24,163. [9] SAWATARI, T., 1973, Appl. Optics, 12, [10] BURGE, R. E., and DAINTY, J. C., 1976, Optik, 46, 229. [11] BORN, M., and WOLF, E., 1975, Principles of Optics (Pergamon Press). [12] GOODMAN, J. W., 1968, Introduction to Fourier Optics (McGraw-Hill). [13] SHEPPARD, C. J. R., and AHMED, H., 1976, Corrosion Sci., 16, 819. [14] SHEPPARD, C. J. R., 1977, Optik, 48, 329. [15] YAMAMOTO, K., ICHIOKA, Y., and SuzuKI, T., 1976, Optica Acta, 23, 987. [16] KINTNER, E. C., and SILLITTO, R. M., 1977, Optica Acta, 24, 591.