Resolution in near-field optical microscopy

Ultramlcroscopy 47 (1992) 15-22 North-Holland

Resolution in near-field optical microscopy M Isaacson, J Chne and H Barshatzky Cornell Umverslty, School of Apphed and Engineering Physics, Ithaca, NY 14853-2501, USA Recewed 15 January 1992, at Editorial Office 11 May 1992

A discussion of the relatlonsMp between the spatml resolution in near-field and far-field microscopy is presented m the context of linear optical transfer function theory The results of a s|mphfied theory of near-field optical beam formation are compared with experimental measurements of the near-field probe profile using visible hght and subwavelength probes The radml mtenslty profile of a near-field hght beam is shown to be describable to a first approximation by a simple Gaussmn function

1. Introduction It has been only w~thm the last half century that the concept of super-resolution microscopy m the near field has been vigorously pursued and experimentally demonstrated (e g , refs [1-9]) However, the idea of optical resolution unhindered by far-field diffraction limitations was concewed more than a half century ago in a p a p e r by E H Synge [10] entitled " A Suggested Method for Extending Microscopy Resolution into the Ultra-Microscopic R e g i m e " This "suggestion" was rediscovered in the fifties by O ' K e e f e [11] but only experimentally verified using 3 cm wavelength microwaves in the early seventies [12] The purpose of this article is not to dwell upon the history of near-field imaging and why it has taken so long to come to practical fruition, but rather to look critically at the concept of spatial resolution in the near field We would like to look at the defimtlon of resolution In the same context as it is used in other forms of microscopy The basic principles of near-field optical microscopy have been described before in the literature (e g , a good rewew is gwen in ref [13]) and although there are many ways to implement near-field tmagmg, the basic concept can be best understood if we look at the schematic shown in fig 1 We tmaglne radmtlon incident upon an

opaque screen contammg an aperture much smaller m diameter than the wavelength If we look at the lntenslty dtstrtbutton of the transmitted radiation at some large distance from the aperture screen (large compared to the wavelength), we find that the radtatlon pattern Is dwergmg and that the spatial distribution of the radtatlon intensity m a plane parallel to the screen lS proportional to the modulus squared of the Fourier transform and not the geometric proJection of the aperture

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- Fig 1 Schematic diagram of the near-field diffraction from a sub-wavelength-diameter aperture in a perfectly opaque

0304-3991/92/$05 00 © 1992 - Elsevier Science Publishers B V All rights reserved

screen

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M lsaac~on et a l /

Resolutzon m near-field optical microscopy

On the other hand, if we examine the radiation intensity m the proximity of the aperture exit (a distance from the aperture of the order of its sxze), we find that in this "near-field" region, the radiation is collimated to the aperture size and, m fact, is essentially the geometric projection of the aperture Thus, by placing a sample in this "near-field" region, we can illuminate it by a light probe whose dimension is determined by the aperture size and not by the illuminating wavelength By scanning this probe across the sample such that the sample is always in the near field with respect to the aperture, we can construct a scanning optical probe microscope whose spatial resolution is defined not by the wavelength of the illumination light, but rather by the size of the aperture probe Since such structures have been fabricated with dmmeters below 10 nm (e g , ref [14]), this technique therefore offers the tantahzlng potential of nanometer-scale resolution using visible light Note that although we have discussed the case where we illuminate an object with a collimated beam, this colhmatlon effect works in reverse That is, ff the object is self-luminous (or if light is being transmitted through the sample), then only the light emitted from the surface of the object that is in the near-field region with respect to the aperture will be collected through the aperture And the further away that this aperture "collector" is positioned, the less will be this colhmatlon effect Previous calculations seem to indicate that qualitatively the near-field collimation extends to a distance of the order of D from an aperture of diameter D [15-17] Furthermore, it should be apparent that the general discussion so far holds also to any optical probe or detector which accepts light over a finite extent (such as a pointed optical fiber, for example)

2. Far-field optics Since the time of Robert Hooke [18], optical microscopes have used lenses as the critical imaging elements In fact, the basic elements of the optical microscope have not changed much since Hooke's time - essentially they consist of an

i

x

L

2O . . . . . .

-- . . . . .

P

Fig 2 Schematic representation of light probe formation using a lens Here, we illuminate the back focal plane of a lens, L, with plane wave radiation which the lens focuses to a diffraction limited spot at the sample plane, P, located at a distance F from the lens, where F = LP IS the lens focal length

illumination lens and an objective lens/eyepiece The principal resolution limitation has been due to the hmits set by far-field diffraction, since in conventional microscopes the sample under investigation is always in the far field with respect to the aperture and Imaging lens In the context of linear optical transfer theory (e g , refs [19,20]), the spatial resolution achievable is best described by the modulation transfer function (MTF) The MTF is defined as the VlSlblhty or contrast of the image for slnusoldal intensity distribution of a special spatial frequency in the object plane Then, the different "resolution" criteria appear as various limits of the detectable contrast In this context, we can now look at the resolution attainable in conventional optical microscope systems (1 e , ones which use lenses as imaging elements) Consider the schematic shown i n f i g 2, where we see that plane wave radiation (a parallel beam of hght) illuminates an aperture of radius R in an opaque screen which is in the back plane of a lens of focal length F Therefore, at the sample plane, P, we have a focused beam of hght We will discuss the case of a probe beam forming system (as is used in scanning) since then we can make direct comparisons with scanning near-field microscopy However, one should note that the resolution discussion would be equivalent if we were imaging light coming from the sample Furthermore, we only consider the case of lncoher-

M lsaacsonet a l / Resoluttonm near-fieMopttcalmtcroscopy e n t tmaglng since t h e s c a n n e d p r o b e case m t h e context of th~s dlscusston is an i n c o h e r e n t p r o cess M o r e o v e r , we will a s s u m e a p e r f e c t lens, since a b e r r a t t o n s have no c o u n t e r p a r t in t h e n e a r - f i e l d case If A(p) is t h e a p e r t u r e t r a n s m t s s l o n function, wtth A(p) = 1, p < R a n d A(p) = O, p > R, w h e r e p ts the r a d i a l c o o r d i n a t e m the a p e r t u r e p l a n e , t h e n t h e p r o b e intensity profile at t h e s a m p l e p l a n e , P is just

PSF(r) 10

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(la)

where J denotes the Fourier transform and r d e n o t e s the r a d t a l c o o r d i n a t e at t h e s a m p l e p l a n e , P (e g , ref [19]) I(r) Is s o m e t t m e s called the p o m t s p r e a d function, P S F ( r ) , o f the system It ts the i m a g e of a p o i n t object In t h e a b s e n c e of any lens a b e r r a t i o n , eq ( l a ) just ytelds t h e A i r y disc (e g , ref [20]) with

I(r) = PSF(r) =

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Fig 3 (a) The far-field point spread funcUon, PSF(r), of an

=J[PSF(r)]

=J[,..,C[A(p)]J[A(p)]*]

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w h e r e n is the index of r e f r a c t i o n of t h e m e d i u m b e t w e e n t h e lens a n d the s a m p l e p l a n e , 0 is the c o n v e r g e n c e half angle o f t h e o p t i c a l p r o b e at t h e s a m p l e p l a n e , a n d Jl is a Bessel f u n c t i o n o f first o r d e r This P S F IS shown m fig 3a T h e first m i n i m u m of the P S F in eq ( l b ) c o m e s w h e n r = 0 6 1 A / n s l n 0 S o m e t i m e s this value IS r e f e r r e d to as t h e " s p a t i a l r e s o l u t i o n " of t h e system [21] H o w e v e r , it s h o u l d be e m p h a sized t h a t thts is only o n e spectal d e f i n i t i o n A b e t t e r d e s c r i p t i o n of t h e r e s o l u t i o n can b e obt a i n e d by t h e use of t h e m o d u l a t i o n t r a n s f e r functton, M T F , which is just t h e F o u r i e r transf o r m o f t h e P S F ( n o r m a l i z e d to b e unity at z e r o spatial f r e q u e n c y ) (e g r e f [19]) T h e M T F is given as MTF(f)

17

(lc)

= A ( AFf) ®A*( AFf) by the c o n v o l u t i o n t h e o r e m (e g , r e f [22]), w h e r e

aberrauon free lens of focal length, F, assuming incoherent Imaging It ~s assumed that the lens is focused onto the sample plane, P, indicated m fig 2 Note that the horizontal axis is given m units of r(R/AF) rather than rnsm(O)/A m order to more easdy show the relationship between the far-field and near-field cases In the hmlt of small 0 and air between the lens plane, L, and sample plane, P, the two expressions are equwalent (b) The modulatton transfer function (MTF) of the imaging system gwmg the point spread function shown m (a) f ts t h e spatial f r e q u e n c y in the s a m p l e p l a n e d e f i n e d as f = 1/r, F ts the lens focal l e n g t h a n d ® r e p r e s e n t s a c o n v o l u t i o n This is shown m fig 3b for t h e P S F shown in fig 3a N o t e that t h e M T F IS just t h e a u t o - c o r r e l a t i o n of the a p e r t u r e t r a n s m i s s i o n funcUon (e g , ref [19]) In a d d i t i o n , it s h o u l d also b e p o i n t e d o u t t h a t t h e spattal r e s o l u t i o n a t t a m a b l e IS tied to t h e allowable contrast So for e x a m p l e , t h e R a y l e l g h c r i t e r i o n for r e s o l u t i o n (in which t h e m a x t m u m intensity o f the p o i n t s p r e a d function f r o m o n e p o i n t object falls on the m i n i m u m for t h e o t h e r ) [21] gives a resolution of r = 0 61A/nsln 0

(2a)

M Isaacson et a l /

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Resolution in near-fieM optical microscopy

with a contrast of the MTF of 9% On the other hand, the Sparrow criterion (in which the second derivative at the center of the intensity sum from two point objects is zero) [23] gives a resolution of r = 0 47A/nsin 0

(2b) y/a

with an MTF contrast of zero One should note that the resolution criteria given above are somewhat arbitrary and the practical resolution limit ts related to a minimum detectable contrast, which is related to the signal to noise ratio of the imaging system In addition, it should be stressed that in all cases, the MTF of the far-field incoherent imaging case depends upon the wavelength of the radiation used All we can do to improve resolution is somewhat reduce the proportionality constant m eq (2), reduce A or increase the numerical aperture n sin 0

3. Near-field optics If we relax the far-field condition, and allow ourselves to investigate the spatial distribution of the radiation in the near field, we can use the same treatment as before to obtain a point spread function and a modulation transfer function for near-field imaging It should be pointed out that to properly deal with the near-field case we must solve the 3D vector diffraction problem Here we will consider a simplified case assuming the Klrchoff formulation [24] and will consider the more rigorous case in a later paper Consider the schematic shown in fig 4 Here, plane wave radiation again dluminates an aper-

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Fig 5 Calculation of the power transmitted through an aperture tn an infinitely thin, perfectly conducting screen The incident radiation is assumed to be a polarized plane wave incident normal to the aperture plane The electric field is parallel to the x-axis and the magnetic field is parallel to the y-ax~s The intensity plot shows the power at a distance of R / I O from an aperture of radius R where R = 3,/50, A = 500 nm (from ref [16])

ture in an opaque screen Only now we consider the spatial distribution of radiation in the immediate proximity of the exit side of the aperture The intensity of radiation here is just given as INF(r) =

IA(p)A(p)*I

= PSFNF(r )

(3a)

where r is again the coordinate in the sample plane (which here is at the aperture exit and is the same as p) and has the same meaning as before Thus, the intensity distribution in the near field IS essentially the geometric projection of the aperture (or more accurately the modulus squared of the aperture function) Note that this is not exactly true, since a more rigorous calculation indicates an increase in the intensity at the aperture edge This has been shown before for the case of apertures m thin perfectly conducting screens and the result of such a calculation by Harootunian [16] is shown in fig 5 In fig 6a we plot a representation of eq (3a) where R is the aperture radius The modulation transfer function for the point spread function gwen by eq (3a) is again the Fourier transform of the PSF

A(p)

MTF(f)NF Fig 4 Schematic representation of hght probe formation without a lens

s

=J[ Iyv(r)] =J[A(p)A(p) *] =~[A(p)]

®J[A(p)]*

(3b)

M lsaacson et al / Resolution m near-field optical microscopy

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Fig 6 (a) The near-field point spread function, PSF(r) due to an aperture of radms R The sohd hne ~s for an aperture m a perfectly conducting screen (neglecting the intensity increase at the aperture edge), the dashed hne ts for an aperture m an aluminum screen with flmte &electric constant assuming A = 488 nm The point spread function Is evaluated at the exit of the aperture (b) The modulation transfer funchons of the near-field imaging systems gwmg the point spread funcnons shown m (a) Again, the dashed hne ts for an aluminum screen of fimte &electric constant

by t h e c o n v o l u t i o n t h e o r e m This result is p l o t t e d in fig 6b N o t e now that t h e M T F is t h e a u t o - c o r r e l a t i o n o f t h e F o u r i e r t r a n s f o r m of the a p e r t u r e function a n d not that o f t h e a p e r t u r e function itself as in t h e far-field case M o r e o v e r , t h e spatial f r e q u e n c y scale now d e p e n d s only o n the a p e r t u r e size R a n d not on the w a v e l e n g t h o f t h e radiation 2J,(2rrRf)

MTF( f)NF

(2~'Rf)

(3C)

w h e r e R a n d f have the s a m e m e a n i n g s as b e f o r e T h e first m l m m u m of this n e a r - f i e l d M T F now c o m e s at f = 0 6 1 / R , thus t h e s m a l l e r we

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can m a k e the a p e r t u r e a n d h e n c e t h e n e a r field p r o b e , t h e h i g h e r s p a t i a l f r e q u e n c i e s can b e t r a n s m i t t e d t h r o u g h this optical system If t h e a p e r t u r e s c r e e n is n o t p e r f e c t l y o p a q u e , o u r above results have to be s o m e w h a t m o d i f i e d , b u t the m a i n f e a t u r e r e m a i n s u n c h a n g e d , a n d t h a t is that t h e M T F does not d e p e n d on the w a v e l e n g t h if we a r e in the i m m e d i a t e proximity of the s c r e e n This is c o n f i r m e d by simplified calculations which t r e a t the a p e r t u r e as a simple wavegulde T h e effect o f n o n - o p a c i t y of the a p e r t u r e screen can be a p p r o x i m a t e l y u n d e r s t o o d as follows If we c o n s i d e r the a p e r t u r e screen to b e a lossy dielectric, t h e n t h e r a d i a t i o n in the a p e r t u r e leaks into the s c r e e n If o n e a s s u m e s an infinitely long cylindrical w a v e g m d e , the Intensity of t h e TEl1 m o d e b e c o m e s the d o m i n a n t m o d e if the w a v e g u l d e r a d i u s is m u c h s m a l l e r t h a n h M o r e over, it falls off f r o m t h e c e n t e r t o w a r d s t h e sides in an almost G a u s s l a n fashion [25] A s s u m i n g that this intensity d i s t r i b u t i o n is n o t drastically a l t e r e d In going f r o m an infinitely long to finite-length w a v e g u l d e , t h e n c o n s i d e r i n g the a p e r t u r e m a lossy d i e l e c t r i c s c r e e n as a cylindrical w a v e g u l d e o f r a d i u s c o m p a r a b l e to t h e o p e n i n g , we get that the intensity profile in the proximity of the exit has an almost G a u s s l a n a p p e a r a n c e with t h e 1 / e intensity p o i n t of the G a u s s l a n occurring at cR w h e r e R is the physical a p e r t u r e r a d i u s a n d c is a c o n s t a n t which m a y be slightly less t h a n unity for l a r g e r R a n d can b e g r e a t e r t h a n u m t y for small R In o t h e r w o r d s cR is t h e " e f f e c t i v e " a p e r t u r e size T h e c o n d i t i o n for c = 1 ts shown as a d a s h e d hne m fig 6a If we use this G a u s s l a n profile as o u r n e a r - f i e l d p r o b e profile at t h e a p e r t u r e exit, t h e n the p o i n t s p r e a d function b e c o m e s I N F ( r ) = e -(r/oR)2

(4a)

w h e r e R is t h e physical a p e r t u r e r a d i u s T h e n we easily can calculate t h e M T F since the F o u r i e r t r a n s f o r m o f a G a u s s l a n is a G a u s s l a n (e g ref [22]) T h e r e f o r e , the M T F for a G a u s s l a n p r o b e is given by MTF(f)

= e -(~f~n)2

(4b)

M Isaacson et a l /

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Resolutton m near-fieM opttcal rmcroscopy

and ~s plotted in fig 6b as a dashed hne for the case c = 1 Note that the MTF of the Gausslan probe results In a slightly lower spatial frequency cut-off then the hard-edged opaque aperture transmission function but is not sigmficantly different other than that there are no osctllatlons Further calculations made by propagating a Gaussian near-field probe seem to indicate that the shape remains Gausslan in the near-field regime and the beam width remains relatively constant for a distance from the aperture screen of the order of the aperture diameter [25] Moreover, we can effectively measure this near-field probe profile (or point spread function) by imaging objects smaller than the probe size Using mlcrofabrlcated aluminum structures, we have experimentally determined the beam profile at a distance of ~ 10 nm from the aperture probe plane The results are shown in fig 7 along with the Gausslan fit to the data and seem to indicate the validity of this approximate Gausslan model We have also included in this figure experimental results from Betzlg et al [9] who use a pointed single-mode fiber as a probe With such a model, we hope to be able to develop simulation methods for near-field imaging Although there have been some near-field image simulations done in the past (e g , refs [26-28]), they have been done using simplified structures such as gratings We

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Fig 7 Experimentally determined near-field hght probe profiles at a distance of 10 nm from a sample consisting of small metalhc structures The circles are taken using a metalllzed hollow glass pipette of the type described m ref [30], the squares come from ref [9] using a metalhzed pointed single mode fiber In both cases, the metallic structures would be considered ~mall compared to the beam size

~21

09

07

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1

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5o

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~ too t~ A P E R T U R E RADIUS (am)

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Fig 8 Calculation of the near-field light probe radms (defined as the 1 / e attenuation point of intensity) as a function of the physical radius of the probe aperture assuming a thick aluminum screen waveguide A = 488 nm, and a single TEII mode in the aperture

must now push these methods forward to reahstlcally simulate images of more complex structures before we can use near-field scanning optical microscopy as a quantitative tool The point to be made here is that the effective near-field point spread function can be approximated by a Gausslan profile whose 1 / e intensity value is of the same order as the radius of the aperture or pointed tip probe used Because the effective tip radius in real probes can be larger than the actual radius (due to the fact that in the visible region of the spectrum no material is perfectly opaque or conducting), there is a point of no return whereby making a smaller physical probe actually does not result m a smaller effective probe This is shown in fig 8 where we show ratio of the 1 / e intensity point radius to the physical aperture radius plotted against the physical aperture radius R for hollow aluminum p~pes using radiation of wavelength A = 488 nm and the Bethe approximation for the Poyntlng vector [29] These approximate curves would indicate for this configuration effective aperture radu close to 10 nm are achievable But the reader should keep the approximate nature of these calculations m mind before drawing conclusions

M Isaacson et al / Resolution in near-fteM optzcal rmcroscopy

4. Conclusion It has not been the purpose of this article to review the entire field of near-field microscopy, but rather to just look critically at the issue of spattal resolution We have used concepts developed (and accepted) for far-field optics imaging systems and have tried to present these in the context of near-field Imaging When one is discussing a new imaging or microscopy technique, there ts always the tendency to let enthusiasm get in the way of hard facts when it comes to comparing the new techntque wtth existing or complementary methods Claims of resolution tend sometimes be rather vague and quahtatlve It has been the hope to put near-field imaging resolution on a more quantttatlve footing than in the past and to begin to look more at how we can better simulate near-field images A point to be kept in mind in these discussions ts that we have been assummg lmear optical transfer theory Rigorously, this is not true in near-field optical microscopy, since the probe shape (and hence the point spread functton) is not truly mvarlant of the sample since the nearfield radiation distribution depends upon the boundaries imposed by the sample-probe geometry Furthermore, our calculations have been performed at the aperture exit Clearly, the probe profile spreads out as we get further away from the aperture Moreover, the spatial resolution not only depends upon the contrast (through the MTF), it also depends upon the detected signal strength Thus, the optical power throughput through such sub-wavelength probes must inevitably also be included m the discussion of system resolution much as it is done in photoelectron microscopy where stgnal strength ts also an issue (e g , ref [31]) We are just in the infancy m looking at the initial issues concerning resolution in near field imaging However, the n u m b e r of groups throughout the world working on this new microscopy method ts increasing at a rapid pace Eventually, there will be agreement on suitable criteria for system resolution and our understandmg of the near-field imaging process will increase so that we can routinely use image simulation to

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gain a more quantitative understanding of nearfield scannmg opttcal microscopy images

Acknowledgements This work was supported by the US Air Force Office of Scientific Research grant No 91-0219 The authors would like to thank Professor Dr E Zeltler for helpful discussions and the referee for useful comments Thts paper was lnttlally presented in August 1991 at the 49th Annual EMSA Meeting m a symposium entitled "Resolution in Microscopy" The authors would like to thank Professor M Sarlkaya, the symposium organizer for giving us the opportunity to reflect on this issue

References [1] U Dung, D W Pohl and F Rohner, I Appl Phys 59 (1986) 3318 [2] M Isaacson, E Betz~g, A Harootuman and A Lewis, Ann NY Acad Scl 483 (1986) 448 [3] A Harootuman, E Betzlg, M Isaacson and A Lewis, Appl Phys Lett 19 (1986) 674 [4[ E Betzlg, M Isaacson and A Lewis, Appl Phys Lett 51 (1987) 2541 [5] U Ch Fischer, U T Durlg and D W Pohl, Appl Phys Lett 52 (1988) 249 [6] S Okazakl, H Sasatlm, H Hatano, T Hayashl and T Nakamura, Mlkrochlm Acta (Wlen) 111 (1988) 87 [7] R C Reddlck, R J Warrnack and T L Ferrell, Phys Rev B 39 (1989) 767 [8] D Courjon, J M Vlgoureaux, M Spajer, K Sarayeddme and S Leblanc, Appl Opt 29 (1990) 3734 [9] E Betzlg, J K Trautman, T D Harris, J S Wemer and R L Kostelak, Science 251 (1991) 1468 [10] E H Synge, Phil Mag 6 (1928) 356 [11] J A O'Keefe, J Opt Soc Am 46 (5) (1956) 510 [12] E A Ash and G Nlcholls, Nature 237 (1972) 510 [13] D Pohl, in Adv in Optical and Electron Microscopy 12, Eds C J R Sheppard and T Mulvey (Academic Press, London, 1991) p 243 [14] A Muray, M Isaacson, I Adesida and B Whitehead, J Vac Sci Technol B 1 (1983) 1901 [15] E Betzlg, A Harootuman, A Lewis and M Isaacson, Appl Opt 25 (1986) 1890 [16] A Harootuman, PhD Dissertation, Cornell University (1987) [17] A Roberts, J Appl Phys 65 (1989)2896 [18] R Hooke, Mlcrographla (Royal Soc, London, 1665)

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M lsaacson et al / Resolutton m near-fteld opttcal mtcroscopy

[19] J W Goodman, Introduchon to Fourier Optics (McGraw-Hall, New York, 1968) [20] E H Lmfoot, Fourier Methods m Optical Image Evaluation (Focal Press, London, 1964) [21] L Raylelgh, Phil Mag 8 (1879) 261 [22] I N Sneddon, Fouraer Transforms, 1st ed (McGraw-Hill, New York, 1951) [23] C M Sparrow, Astrophys J 44 (1916) 76 [24] M Born and E Wolf, Principles of Optics, 6th ed (Pergamon, New York, 1980) p 379 [25] H Barshatsky, PhD Dissertation, Cornell Umverslty (1992)

[26] E Betzlg, A Harootuman, A Lewis and M Isaacson, Appl Opt 25 (1986)1890 [27] E Marx and E C Teague, Appl Phys Lett 51 (1987) 2073 [28] C Girard and M Spajer, Appl Opt 29 (1987) 3726 [29] H A Bethe, Phys Rev 66 (1944)163 [30] M Isaacson, J Chne and H Barshatzky, J Vac Scl Technol B 9 (1991) 3511 [31] O H Grlffith and W Engel, Eds, Specaal Issue on Emission Microscopy and Related Techmques, Ultramlcroscopy 36 (1991) 1-274