A Gaussian beam of 1 mm diameter was .... Rc.3 The variation in the electric field component Ex in the plane of .... sinO(l -sin2 Ocos' a)dadO. 2. 0 0. 3. 1. 4. 4.
SCANNING VOI. 16,327-332 (1994) 0 FAMS, Inc.
Received February 2, 1994
The Scanning Optical Microscope (Extracts on Near-Field Microscopy)From a 1975Report R. KOMPFNER, C. J. R. SHEPPARD, D. WALSH, A. CHOUDHURY, J. N. GANNAWAY, P. G. HALE
University of Oxford, Department of Engineering Science Oxford, U.K.
Summary:The followingconsists of extractsconcerningnear-field microscopy from a Progress Report prepared in April 1975 as Oxford University Engineering Labomtory Report No. 1883/77, describing work initiated by the late Professor R. Kompfner. Preliminary experimental work on near-field microscopy is outlined, and a theory for contrast formation is presented. Key words: diffraction theory, evanescent waves, image formation, near-field microscopy
Introduction The approach to super-resolutionwe have chosen to follow relies on the radiation which can penetratethrough a small hole in an opaque screen, a hole a small fraction of a wavelength in diameter. If the object is very close to the screen, preferably touching it, and is moved in a television-likescan, the amount of light reaching a detector beyond the object will depend on the features of a layer of the object closest to the screen,namely, its refractive index, its absorption, and its scattering distribution; the resolution will be of the orderof the hole size. We shall report on some theoretical calculations, on the effect of variations in the dielectric constant, and on an experiment with a hole and various dielectrics scaled up to X-band.
Resolution and Super-Resolution Our approach to super-resolutionrests on the principle of illuminating an object through a hole substantially smallerthan the waist of a Gaussian beam and recording the transmitted or scattered 1ight amplitude. ~
C. J. R. Sheppard currently at Department of Physical Optics, School of Physics, University of Sydney, Sydney, NSW, Australia Address for reprints:
It is intuitively clear that the object has to be very close to the hole for the method to be effective. In other words, one will expect that the “depth of focus” will now be of the same order of magnitude as the hole diameter,and the smallerthe hole, the closer the object has to be, but the better the resolution will be within the smaller depth of focus. A theoretical estimate of the contrast in the super-resolution microscope will be given in the Appendix. The experimental approach has included attempts at making small holes in thin metal layers on glass plates, so far unsuccessful. Because the object has to slide close up against the metal layer during the scanning process, the metal layer has to be tough and smooth, because the amount of light transmitted through the hole depends on the thickness of the layer, it has to be thin. And because substantial improvementin resolution can only be obtained by holes that are small compared with a wavelength, most of the usual methods of making smd1holes have turned out to be impractical. R. Lemons (of Stanford University) has suggested a novel method of making a hole smaller than the smallest cross section of a laser beam depending on the existence of substances which have resonant absorptions in the wavelength range of interest. Such absorption becomes nonlinear at light intensity high enough that it “saturates.”That means that the higher the intensity the lower is the loss so that a light beam with, say, a Gaussian intensity distribution will experience less loss near the axis than away from it. The result is that the light that penetrates through such a layer (assumed to be thin with respect to a wavelength) has a distribution which is n m w e r than the incident intensity distribution.A computercalculation of a system of this kind showed that a beam could be narrowed by a factor approaching one half. In practice, such a saturable absorber screen, scanned by a deflected light beam, would permit, in principle, super-resolution to be obtained with a stationary object and hence might enable one to scan at high speed,something which does not seem possible with a real hole in a metal layer as discussed earlier. A preliminary experiment with a saturable absorber (a dye called DQTCI) will be described further on.
C.J.R. Sheppard Department of Physical Optics School of Physics University of Sydney Sydney, NSW Australia
Experiments Relating to Super-Resolution One proposed method of obtaining super-resolutionentails the use of a metal screen containing a hole of subwavelength
328
Scanning Vol. 16,6 (1994)
diameter. There are various techniques which have been employed to produce such a small hole. One method is by electron-beam exposure of a resist followed by etching or deposition of the metal, and efforts have been made to obtain holes made by this method. A second method is the deposition of the metal film onto a glass slide on which latex spheres are placed which are washed off after the deposition. It was thought that natural holes would be present in evaporated metal films, one of which might be selected for the super-resolution experiments, but examination with a scanning electron microscope has shown that the holes which do form are irregularlyshaped and usually do not traverse the thickness of the film. Preliminary experimentshave been made on the beam-narrowing effects of a nonlinear dye. A Gaussian beam of 1 mm diameter was narrowed by 20% by nonlinear dye in solution in a dye cell. Complicationshave arisen caused by beam-heating effects in the dye. Experiments have also been started on a microwave simulation of the electromagnetic effects of a small hole. These experiments are being conducted at X-band in a waveguide. The Appendix discusses the field which is produced behind a perfectly conducting sheet containing a hole when it is illuminated by a plane wave with a wavelength much greater than the diameter of the hole. The field behind the hole consists of a propagating component and an evanescent component. Only the propagating component is measured by a detector. If a dielectric is now placed behind the hole, various processes should be considered. One might expect dielectric contrast from two distinct mechanisms.One, which is discussed in the Appendix, is due to the secondary radiation of a propagating field by the medium which is immersed in the oscillating evanescentfield of the hole. The second is due to a change in the effective size of the hole which is caused by its being partially immersed in a dielectric medium. The variation of each of these components with dielectric constant may be approximated by a power law, and thus the total power is the sum of two power terms. Preliminary results have supported this. However, theory also predicts the variation with the hole size, and at present there are some difficulties in explaining the observed results.
Appendix Contrast in the Super-Resolution Microscope The Theory of Diffraction as Applied to Small Holes
The Kirchhoff theory of diffraction is based on the Kirchhoff integral theorem, which expresses the amplitude at any point in space in terms of the distribution in amplitude and its first normal derivative over any surface.' This may be considered as being equivalent to the assumption of Huygen's principle. Kirchhoff's theory then goes on to assume a given distribution of amplitude and its derivative over this surface, and hence to calculate the resultant. However, this process has some
complications. First, the assumed distributions usually are taken to be those of the undisturbed primary wave over the aperture and zero elsewhere in the screen, and this assumption violates Green's theorem? Second,the method is non-self consistent, that is, it does not produce the assumed variation in amplitude over the surface of the screen and aperture. This is because the assumed distribution in amplitude and its derivative are not independent. The problem may be solved by assuming one or the other, resulting in Rayleigh's formulae which are cases of the Dirichlet and Neumann problems, respectively. These formulae may be expressed as surface integrals over the aperture only, in which case they are applicable to the cases of perfectly hard and perfectly soft screens. The Kirchhoff theory may be seen to be the arithmetic mean of these two solution^.^ Kirchhoff's theory is found to agree with experimental results for the field at large distances from the aperture when the aperture is large compared with the wavelength. Furthermore, the diffracted intensity for microwaves has been m e a s ~ r e dand ~ . ~the intensity on the axis has been shown to agree with that predicted by the Kirchhoff theory if the diameter of the aperture is greater than about a wavelength. Kirchhoffstheory may be modified in various ways to overcome some of the difficulties.5First, the solution to Rayleigh's equation for assumed amplitude or its normal derivativeas for the unperturbed primary wave give consistent results, but these do not give correct values for both the amplitude and its derivative in the aperture.Second, a solutionmay be obtained which gives the unperturbed values for the amplitude and its derivative in the aperture, but then the boundary conditions are not satisfiedon the screen.Braunbek7v8assumed that the derivative of the amplitude at any given point on the edge was given by Sonmefeld's theory for the straight edge. Rayleigh's equations are integral equations, which may be soluble using Fourier transform methods.6The scattered field may be represented as an angular spectrum of plane waves, including evanescent For a small hole, the integral equations may be solved in terms of a power series in ka,where k is the wave number and a is the radius of the hole, to give the diffracted wave at large distances from the hole.6For the near field, spheroidal wave functions must be used"J2 althoughin the limit at ka +Othese become Legendrepolynomialsfor the angular distributionand associated Legendre functions for the radial distribution. The case of finite screen thickness (for a prolate spheroid approximating to a circular disk) has been considered by Spence and Granger.I3 The Kirchhoff theory is a scalar theory of diffraction and hence cannot be expected to give correct results for polarised radiation. If the theory is applied to each of the six rectangular components of the electric and magnetic vectors, then the six wave functions so obtained do not in general satisfy Maxwell's equations. This difficulty can be solved by introducing certain additional contour integrals representing the effect of fictitious line charges along the rim of the aperture. Magnetic surface currents and charges must be introduced to
R. Kompfner et al.: The scanning optical microscope
account for discontinuitiesin the field vectors on crossing the screen.14Then, by the uniqueness theorem, the calculated field which satisfies the boundary conditions is the required solution. An alternative approximation is the assumptionthat the tangential component of H is negligible at the screen?15which is equivalent to the assumption of an electric dipole layer in the aperture. A second alternative approximation is the assumption that the tangential component of E is undisturbed in the apertu~-e,”.’~ which is equivalent to the assumption of a magnetic dipole layer in the aperture. The vector form of Babinet’s theorem for diffraction by complementary screens may be formulated in terms of a complementary incident field.I7In this way the diffraction by a hole may be determinedfrom the problem of a circular disk. The field far from a plane aperture may be calculated by variational techniques.l8 The exact solution of Maxwell’s equations for the case of a very small circular aperture in an infinite, perfectly conducting plane was considered by Bethe.” Bouwkamp20corrected this theory and gave solutions for the near field in terms of Legendre functions in spheroidal coordinates. The near field is a static, oscillating field with magnetic and electric components which are 90” out of phase with each other. Bouwkamp2’has also considered the problem by solving integro-differential equations for the field and obtained values for the far field and the field of the hole in terms of a power series in (ka).The field has been evaluated in terms of spheroidal wave functions by Meixner and Andrejewski22and Andrejewski2’. Young’s theory of diffraction,which considers the resultant intensitiesas being due to interference between the geometric wave and a wave reflected from the boundary of the aperture, has been shown to be equivalent to the Kirchhoff theory in the region where the latter is a~plicable.~~ A n d r e ~ sconsidered *~ a vector form of Young’s theory and showed that this gave good agreement for the field in the plane of an aperture of a few wavelengths in diameter. The measurements were extended to aperturessmaller than one wavelength by Robinson26and H a d l ~ c kIt. ~was ~ found that the Young theory was not now applicable but could be modified to account for multiple reflections. There was thus a peak in the intensity at the centre of the aperture when its diameter was equal to half the wavelength. Derwin28measured the diffraction by a slit in a perfectly absorbing screen and showed that, at a distance of greater than half the wavelength or in the plane of the slit for slit widths greater than one wavelength, the results were the same as for a conducting screen. Measurements of the diffraction by circular apertures with diameters less than the wavelength have been made and compared with the results of Bouwkamp’s theory.2’
329
oidal coordinatesu, v, 0 which are related to Cartesian coordinates by
z = auv x =a
J m c o s $
in the ranges
The field may be expressed in terms of spheroidal wave functions, which for the case of a hole much smallerthan the wavelength degenerate to Legendre polynomials. Figure 1 shows the variation in electric and magnetic field strengths along the axis of the hole. Here E,, H,are the amplitudes of the incidentelectric and magnetic fields.The magnetic and electric fields are 90” out of phase with each other, and the electricfield strength is proportionalto (ka).Figure 2 shows the variation in Ex,the component of the electric field parallel to the incident electricfield, in a plane perpendicular to the plane of polarisation. Further from the aperture the field becomes more spread out, as is to be expected. Figure 3 shows the variation in Exin the plane of polarisation. In the plane of the aperture, Exexperiences a discontinuity at the edge of the aperture. If z >> a, the value of Exbecomes cylindrically symmetric and tends to a value given by
(3)
l h
The Theory of Bouwkamp 0
Bouwkamp” has calculated the diffracted field produced by a small hole in a perfectly conducting sheet. The field near to the hole of radius a may be expressed in terms of oblate spher-
0.5
1
1.5
2
2.5
3
z/a
FIG.1 The variation in electric and magnetic field strengthsalong the axis of a hole radius a.
Scanning Vol. 16,6 (1994)
330
1.2
'
1
'
m
h
essary to account for the near field. However, the field of the hole is only given by this distribution for large distances, and so the propagating components of the field are only given by this distribution for large distances. Hence, the propagating components of the field of the hole are the same as those for this spherical wave. The evanescent components are those given by the near field, that is, with the electric field proportional to (ku). The values of Bethe's distant fields for E, along the x axis and H,along they axis are given by
z=o
0
0.5
1
1.5
2
,
eJkR
E = s ( k a ) - -(KAj) 3a kR
YIa
FIG.2 The variation in the electric field component E,rin a plane perpendicular to the plane of polarisation.
1.2
1
z=O/
811-4 '
I
where K is a unit vector in the direction of propagation of the wave. If 8is the angle between K and the z direction and the a is the azimuthal angle, then the Poynting vector is given by
0.8
E i R ( I -sin2 8cosz a). S =l 6y 9 a (kR)2 0
0.5
1
1.5
2
x/a Rc.3 The variation in the electric field component Ex in the plane of polarisation.
For large z, Exthus falls off as 1k2.Similarly, H,tends to avalue
H.. 4n' H, 3 a ( r 2 + z2 )3/2 For large z, H,falls off as I/?. Note that although this paragraph applies for the condition where z >> a, the condition zk c< I must still apply. The distant field (zk>> 1) has been calculated by Bethel9and it is found that this falls off as dMMwhere R is the spherical radius coordinate,as does a spherical wave, with both the elecThus, whatever tric and magnetic fields proportional to (k~7)~. the value of ka, we can choose a value of z large enough for the near field to be negligible compared with the distant field. At and have a magnitude values of Wa= 1 the fields fall off as proportional to (~LI)~. Now the total field of the illuminated aperture may be expressed as a sum of plane waves, including evanescent waves which propagate in the xy plane with a reduced wave number, and which are exponentiallydecaying in the zdirection.I0The distant field has the form of a spherical wave with an amplitude which varies with angle and which may be considered as such a sum of plane waves, the evanescent waves being nec-
(7)
Contrast Mechanism Assuming the Field Distribution of Bouwkamp
Let us considerwhat happens when an object is placed in the region close behind the hole. Now the electricfield in the vicinity of the hole consists of an evanescentcomponent which has a magnitudeproportional to (ka)and a propagating component with a magnitude proportionalto (ka)*. In the space behind the hole, the evanescent field has an ass@ ciated displacementcurrent densityjw&J3which radiates,producing an additional propagating component, the energy of which is a factor (ka)*smallerthan the direct propagating component. Let us assume that the screen and hole are immersed in a medium of permittivity E, and that we now place a slab of material permittivity& + & and conductivity oin the space from z = zIto z = 3.The radiated field is given by
neglecting the componentsof field E,,,which is small, and E., which produces a quadrupole radiation. The angle 0 is the angle between the directionof propagation and thex axis. Here we have assumed that E is not changed by the presence of the specimen, which seems reasonable if o i s small, as the tangential component of E is continuous across the boundary of the specimen.
R. Kompfner et al.: The scanning optical microscope
Summing over an infinite slab between z = zI to z = z2, the total E is in a given direction #is given by
(4)
33 1
and therefore the ratio is
where 77 is the collection efficiency,
Here we have neglected refraction effects at the surface of the specimen. Assuming thaty 9 &are constant in the region of interactionof specimen,we obtain
B 2a
77 = 0 0
?sinO(l -sin2 Ocos’ a)dadO 2
3 4
1 4
= 1 - -cosp - -cos3p.
whereX = d a etc. Let us denote the triple integral, the effective radiating power, by F, then E@=
j w p s i n $ ~ ( a +j w ( & +6&))Eoa3((akj)Fe’kR 4 nR
(1 1)
The value of 77 is plotted in Figure 4. It has a value of about 0.6 for practical objectives. To summarize, the specimen produces a radiating field which is dependent upon its dielectric constant, that is, it produces dielectric contrast. Furthermore, if the specimen is very close to the hole, the resolutionof the image will be of the order of magnitude of the size of the hole. Let us now examine the effect of the propagating components of the field produced by the hole. The power collected by an objectivelens which subtends an angle of 213 is (from Eq. 17)
Similarly, the magnetic field is given by p2n
H,
P2
=@,
4
=I
h
2
-sin2 Ocos’ a)sin OdadO
0 0
or
or
where 77 is the collectionefficiency, which is the same as for the component originating from the evanescentpart of the field. Let us neglect absorption in the specimen.Then the total signal power collected by a lens which subtends an angle 28 is
p ZK
nsin 0(1-sin2 Ocos2a)dadO, 11 I1
(14) where we have transformedto sphericalcoordinates about the z axis instead of the x axis. We have also neglected terms in The power incidenton the hole is
P FIG.4 The collection efficiency q as a function of the angular semiaperture of the objective lens p.
Scanning Vol. 16,6 (1994)
332
We have thus calculated two components of power collected at the objective lens, one component arising from the evanescent part of the field of the hole and the other from the propagating part. It may be shown that the power which results from the cross-productterms of the two components of field cancel out because the latter are 90"out of phase with each other (this is in fact true only if we may neglect absorption). To obtain an image of the specimen we must detect the signal power P I in a background of power Pz. If the time taken to record the picture is t, then the energy incident on the hole in the time taken to scan a singlepicture element is
Let us now assume that the specimen is placed between z = a12 and z = 3 ~ 1 2which , gives P = 0.6 and take the following numerical values: P=IW ~v=~xIO-"J
6&I & = 0.1 17 = 0 . 6 . Then the time needed to produce an adequate picture for a given hole size is as follows:
where P is the total incident power, dois the diameterof the diffractionlimited spot, and D is the dimensionof the square area of the specimenwhich is probed. From Eq. (16) the number of photons collected from the evanescent field is N , = 2 P a ' o m k 6 F 2 q ( S E~)/ 3hvdiD2
(22)
Similarly the number of photons collected from the propagating field is
N2 =
P(ak)8tF2q( 6~I E )
96 hvzN2
(24)
and
N2 =
4 P(ak)6tq 27 hvzN2 *
Now in practice N2>>N , , thus we can write the signal-to-noise ratio of the picture S I N = N , I NY2
t/Nz
We note that the minimum hole radius is inversely proportional to the tenth root of the power, and hence a very large power increase is necessary to produce a noticeablechange in the resolution which may be obtained.
References
256 P a s m k 4 q 27 hvdi D2 '
If the number of lines in the picture is N = DDa, and we further assume that the diameter of the diffraction-limitedspot is given by do= 2A = 4 d k , then
N, =
alh 0.25 0.1 0.05 0.025
(26)
neglecting degradation of the signal-tenoise ratio by the amplification system. Now the eye cannot distinguish an area of brightness B from an adjacentarea of brightnessB f AB unless the ratio of signal-to-noise is approximately five times the ratio of B to Taking ( B I B )= 10, an adequate picture will be produced if
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