JOURNAL OF THE OPTICAL SOCIEITY OF AMERICA

Superresolution in Microscopy and the Abbe Resolution Limit

CHARLES W. MCCUTCHEN

Section on Rheumatic Diseases, Laboratory of Experimental Pathology,National Institute of A rthritis and Metabolic Diseases, National Institutes of Health, Bethesda, Maryland, 20014

(Received 27 March 1967)

It is now wvell established in principle that superresolving optical systems can be made. Such systems,when viewing objects of finite extent, can resolve detail finer than the normal diffraction limit. This hasobvious attractions for microscopy, but we may wonder whether the ultimate diffraction limit for a lens oflarge numerical aperture, the limit which counts in microscopy, can be beaten. Can detail smaller than onehalf the wavelength of light really be made visible? It can be, but only in specialized and probably limitedapplications.INDEX HEADINGs: Resolving power, Diffraction; Microscopy; Optical systems; Fluorescence; Fouriertransforms; Modulation transfer.

IT has been shown by Toraldo di Francial and othersthat superresolution, that is, resolution of detail

finer than the Abbe resolution limit, is possible providedthat one has some prior information about the objectbeing observed. Wolter' and Harris3 have shown thatthis prior information need only be that the object isof finite, and in practice limited, extent. With an ex-tended object this effect can be achieved by confiningthe illumination to a limited area.

So we naturally think of applying superresolution inmicroscopy, for there the illumination normally is con-trolled, and the alternative of using a bigger lens isnot available. But can superresolution really beat theultimate Abbe resolution limit for a lens with an ac-ceptance solid angle of 27r steradians?

ARBITRARY OBJECTS

The answer depends, at least in the scalar approxi-mation, upon what method is used to limit the area ofillumination. Consider a thin, plane object illuminatedby a single monochromatic train of plane waves ofwavelength X travelling perpendicularly to its plane.On the exit side the emerging light will in general varyfrom place to place in both amplitude and phase.

At the object surface a two-dimensional Fourieranalysis of this disturbance would reveal componentsin all azimuthal directions and with space frequencies1/d (d is the spatial period) from zero to infinity. Thosewith frequencies <1/X give rise to propagating wavetrains each travelling at an angle sin-1(X/d) to thenormal to the object plane. Higher space frequenciesin the object cause disturbances that do not propagate.

So, the light departing from the object is unaffectedby space frequencies > 1/X, the Abbe resolution limitfor a grating with sinusoidal transmission illuminatedby coherent light.

Suppose now that we put right against the object astop which intercepts all light outside a certain region.The disturbance immediately at the exit side of the

'G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).2 H. Wolter, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-

Holland Publishing Co., Amsterdam, 1961), p. 203, et seq.3 J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).

object is now the product of the original object dis-turbance and a function which is unity within the stopand zero outside it.

The Fourier transform of the disturbance is the con-volution of the original object spectrum with the Fouriertransform of the stop function. This latter transform isof infinite extent, hence even at space frequencies< 1/X the convolution contains contributions from allfrequencies in the original unconfined object; so infor-mation about them is carried away by the departinglight, albeit in an involved manner, and can, in principle,be decoded. The smaller the stop, the greater will be thesecond moment of its Fourier transform, the greaterthe influence of space frequencies > 1/X upon the propa-gated wave, and hence the lower the precision necessaryin this decoding.

The effect of the stop is similar to that of the localoscillator in a superheterodyne radio receiver. Bothserve to shift signal frequencies into the passband,spatial or temporal, of a subsequent filter.

In a microscope, however, it is not very practical toconfine the illumination by a stop right next to theobject. Instead, the stop is placed behind the condenserlens and the object is illuminated by the image of thisstop. The smallest region of illumination is produced ifthe source is an ideal point, hence again the illuminationis coherent. The function with which we must multiplythe infinite object disturbance is now the diffractionimage of a point, i.e., the Fourier transform of thepupil function, and the Fourier transform of this diffrac-tion image is again the pupil function. Because thepupil function is necessarily limited to N.A.< 1, corre-sponding to d<X, convolving it with the infinite objectspectrum adds at most a spatial bandwidth 1/x to therange, 0 to 1/N, of spatial frequencies that contributeto travelling waves in the case of coherent illuminationof an infinite object by plane waves. There is in thetravelling waves no information about space frequenciesin the object greater than 2/N, and thus no informationfor superresolving techniques to retrieve. The spacefrequency 2/N is just the Abbe resolution limit forso-called incoherent illumination, i.e., illumination asincoherent as we can make it, which is just what we

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SUPERRESOLUTION IN MICROSCOPY

would have if we increased our area of illumination byreplacing the point source by an extended incoherentsource.

Superresolution thus cannot be achieved with thissystem, because the patch of illumination is the wrongshape. Whereas a stop placed right next to the objectgives a sharp-edged region of illumination, the condenserlens can produce only the diffraction image of a sourceand, as the previous paragraph shows, the shape of thisimage is just such as to render superresolutionimpossible.

Apodizing the condenser is of no help, because theresulting illumination pattern still Fourier transformsback into the pupil function of the condenser, which isstill zero beyond N.A.= 1. Nor does it help to replacethe point source by a coherent source of finite extent.The Fourier transform of the illumination is no longersimply the pupil function, but it is still zero beyondN.A.=1.

The immovable upper limit on the condenser apertureis analogous to limiting the range of tuning of the localoscillator in a radio set to a band equal to the bandwidthof the subsequent amplifier stages. However we tunethe oscillator, or however many oscillators we haverunning at once within this limited frequency band, nosignal outside of twice this band will generate in themixer a signal of frequency within the bandwidth of theamplifier.

The stop can, of course, be put near the objectwithout being right against it, in which case it and themap of illumination of the object are Fresnel ratherthan Fourier transforms of each other. However, itwould be difficult to put the stop much nearer than100 wavelengths from the object in practical trans-mission microscopy; in such a case, the Fresnel trans-forms approximate so closely to Fourier transforms thatthe Fourier transform of the illumination would bealmost completely confined to spatial frequencies < 1/Xand superresolution, though possible in principle, wouldin practice require extraordinary precision in the de-coding process.

So there seems to be little hope that superresolutioncan be applied to the observation of arbitrary objectsby transmitted light-which is disappointing, becausethis is the most-used form of microscopy. The path-ologist, for example, can expect no help in his examina-tion of tissue sections.

The trouble lies in the object-limiting stop, whichmust be far closer to the object than it is practical toput it.

On the other hand, for examination of surfaces byreflected light the stop can be as close as desired, andthe problem becomes a technical, and perhaps also aphysical one of making a stop which is sufficiently thinand sufficiently absorbing. In an extreme form, thetechnique would consist of scanning the specimen pasta minute aperture, much smaller than the Abbe limit.Because the stop will inevitably be imperfectly black,

it would be centered in the smallest possible diffractionimage of a coherent source in order to cut down in-formationless light. The light which bounces back wouldthen be measured and displayed, T.V. style, so as tobuild up a picture of the specimen. For this method,no superresolving apodization or spatial processing ofthe received image is required, if the stop is small enough.

Obviously it would be hard to use this method onany but the flattest of surfaces, and I wonder if thereare many jobs it could do that reflection electronmicroscopy would not do better.

With a fluorescent object, illumination by a smallprojected spot does increase the range of spatial fre-quencies in the object which can influence the image.In this case, the object, however illuminated, radiatesincoherently (unless laser action occurs). It is thuscharacterized as an intensity distribution: Relativephases between the amplitudes at different pointschange from moment to moment and convey noinformation.

The distribution of intensity in the image is theconvolution of the intensity distribution in the object(adjusted for instrumental magnification) with thesquare of the amplitude diffraction image of a point.Hence it is the Fourier transform of the product of theFourier transform of the intensity distribution in theobject and the autocorrelation function of the pupilfunction.

The autocorrelation function is just twice as broadas the pupil function itself, so the highest space fre-quency in the object which influences the image is 2/X,just as it was with the most incoherent possible illumi-nation of a nonfluorescent object.

If we now illuminate the object with the smallestpossible diffraction image of a point source the spectrumof the resulting modulated object is the convolution ofthe spatial spectrum of the infinite object with theautocorrelation function of the illuminating pupil. Thisraises by a further 2/X, (X, is the wavelength of thestimulating light) the largest spatial frequency in theinfinite-object spectrum which can influence the image.

We might not call this effect superresolution, becausethe improvement is, even in principle, limited; never-theless a gain of resolution of more than a factor of 2 isnot negligible. A scanning system would be requiredand the final image would be passed through a pinholein the image plane, a pinhole much smaller than thediffraction disk that the observing lens produces. Ineffect, this system scans the object with a spot whoseshape is the product of the intensity diffraction imageproduced by the illuminating lens and the intensitydiffraction image which the pinhole would produce inthe object plane. The Fourier transform of this spot isthe convolution of the autocorrelation function of theilluminating and observing lenses, taking into accountthe different wavelength of the stimulating light, andit extends up to 2(l/X+l/X3,). Eliminating the pinholereduces this bandwidth to 2/XA.

October 1967

CHARLES W. McCUTCHEN

The greatest uncertainty in this procedure would bethe effect of the thickness of the object. How much theout-of-focus light and the refractive-index variation, ifany, will upset the image processing is not obvious, nordoes it seem easy to calculate.

OBJECTS ABOUT WHICH WE HAVEPRIOR INFORMATION

So far we have discussed only those cases in whichthe prior information about the object is created by thestop. Only in this way can superresolution be achievedfor an object of unknown size and shape.

Suppose, however, that we know that the specimenis a discrete object of small size-say a bacterium. Ifthe bacteria are sparsely distributed over the objectplane then even crudely limited dark-field illuminationwill make it very likely that only one bacterium isilluminated, and thus that the object is limited.

To think about this case, it is easiest to imagine that

the bacterium is a larger object modulated by a functionwhich is one within the outline of bacterium and zerooutside it. Then by the previous convolution argumentsuperresolution is possible.

This seems to be the most hopeful of the possibleapplications of superresolution to microscopy, but as inthe case of the fluorescent object we may wonder aboutthe possible deleterious effect of specimen thickness.In both cases we could, if necessary, use specimensthinner than the depth of focus of the instrument-which would be very thin indeed if the Abbe limit weresurpassed by a large factor. However, this would de-mand techniques of viewing and specimen preparationwhich would introduce many of the difficulties whichplague electron microscopy.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge helpful discussionswith W. P. Raney, E. H. Linfoot, and 0. R. Frisch.

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