THE RESOLUTION PROBLEM IN DIFFRACTION MICROSCOPY

The ideas underlying the method of diffraction microscopy, as well as certain questions concerning the practical realization of the corresponding devices, have already been discussed in the pages of our journal[^1]. Recently, however, several works have appeared that make it possible to examine the principle of diffraction microscopy from a somewhat different point of view and to give a sufficiently substantiated assessment of the real possibilities of this method.

Let us recall that the essence of the method consists in dividing the process of obtaining an image into two stages. First, a photograph of an intermediate diffraction pattern is obtained, and then, in the absence of the object itself, an image is reconstructed with the aid of a positive of this photograph, which has received the name “hologram.” In this process, owing to the violation during reconstruction of the natural phase relations, instead of one image of the object two of its images are obtained, usually located on opposite sides of the hologram. A general theoretical consideration was given in due course by Gabor[^2]. However, the complexity of the relations obtained by him made a more detailed analysis of the features of the method difficult. The possibility of such an analysis—although approximate, yet in many respects sufficiently complete and convincing—appeared after Rogers[^3] drew attention to the analogy between a hologram and a zone plate. Indeed, suppose that a point source of light \(O\) illuminates a point object \(R\) that also scatters light. As a result of in-

interference of the light field created by the source \(O\), and of the coherent light field diffracted by the object \(R\). In some plane \(Q\), perpendicular to the direction \(OR\), a light field is formed whose intensity will be a function of the distance \(x\) from the axis \(OR\). It is not difficult to see that the path difference \(\Delta\) of the interfering rays, in the first approximation, is determined by the relation

\[ \Delta=\frac{x^{2}}{2}\left(\frac{1}{b}-\frac{1}{a+b}\right), \tag{1} \]

where \(a\) is the distance \(OR\), and \(b\) is the distance from the object to the plane \(Q\). Assuming

\[ \frac{1}{f}=\frac{1}{b}-\frac{1}{a+b}, \tag{2} \]

we obtain:

\[ \Delta=\frac{x^{2}}{2f}. \tag{3} \]

Thus, the light field in the plane \(Q\) will be an alternation of bright and dark circles, with the positions of the illumination maxima corresponding to the condition

\[ x_p=\sqrt{2pf\lambda}, \tag{4} \]

where \(\lambda\) is the wavelength, and \(p\) is an arbitrary integer. Here it has been assumed that, when the light is scattered by the object, no phase shift occurs. The presence of a phase shift leads only to a corresponding increase or decrease in the radius of the interference rings, but does not change the overall picture.

Thus, in the case of a point object, the hologram will be a zone plate whose transparency varies as a function of \(x\) according to an approximately sinusoidal law. If the object or the light source is not point-like, then the hologram may be regarded as a superposition of zone plates corresponding to each of the pairs of points of the source and of the object (see below).

It is known that if a zone plate is illuminated by monochromatic light from a point source, then it produces (as a result of the interference of light beams, analogously to a diffraction grating) a focusing action and forms two separate images of the light source. In other words, a zone plate acts simultaneously as a converging and a diverging lens with focal lengths \(\pm f\), where \(f\) is determined by relations (2) or (4).

It follows from relation (4) that the optical power \(\frac{1}{f}\) of the zone plate is proportional to the wavelength \(\lambda\) of the light illuminating it. Therefore the optical properties of the plate, as such, should be characterized by the specific optical power \(\frac{1}{f\lambda}\), expressed in diopters per micron.

All these considerations can also be applied to a hologram obtained from an arbitrary object, since the latter is a superposition of holograms of point objects. An experimental verification of this conclusion was carried out by Rogers³. On the basis of direct meas-

…with holograms obtained with a point source of light and a small scale as the object, he established:

1. Each hologram can be assigned a focal length equal to the focal length of a lens which, if placed at the position of the hologram, forms an image of the light source at the place where the object is located. If the object is extended along the axis \(OR\), then the hologram should be assigned a set of focal lengths corresponding to different sections of the object perpendicular to \(OR\).

2. A hologram forms two “images” of the light source (transforming them into images of the object), and the positions of these images correspond to those which would be obtained if the hologram were replaced by lenses with focal lengths \(\pm f\).

3. When the wavelength of the illuminating light is changed, the product \(f\lambda\) remains constant.

4. Upon photographic enlargement of the hologram, its focal length increases in proportion to the square of the linear enlargement, in accordance with (4).

5. The magnification obtained by the method of diffraction microscopy is equal to the product of the magnification in obtaining the hologram (projection), the photographic enlargement of the hologram, and the magnification in reconstruction of the image (projection).

6. The image of a complex object can be obtained by photographic superposition of holograms of its parts and subsequent reconstruction of the complex image.

7. Instead of an amplitude hologram, one may use a phase hologram, making it from transparent gelatin by means of the Carbro process (a relief transparent image).

8. An image obtained by reconstruction with the aid of a hologram may in turn be regarded as a hologram.

Thus Rogers’ experiments\(^3\) confirm with a sufficient degree of accuracy the view of the hologram as the most general case of a zone plate; a hologram may be regarded simply as the superposition of a number of zone plates. As we shall see below, such an interpretation makes it possible by simple methods to examine the problem of the resolving power of a diffraction microscope. In particular, a zone plate consisting of alternating transparent and opaque zones with sharp boundaries should be regarded as a superposition of a series of holograms of point objects corresponding to the harmonic components of the expansion of the transparency of the hologram as a function of \(x\). Accordingly, such a plate will possess a set of multiple focal lengths (the smallest of which, the principal one, \(f_0 = \dfrac{x_1^2}{\lambda}\), where \(x_1\) is the radius of the zero zone) and will form not two, but many pairs of images of different orders. This conclusion, as is known, is also confirmed by experiment\(^7\).

Let us now determine what number of zones can practically be obtained on the hologram of a point object. It is not difficult to see that the width of the \(p\)-th zone is equal to \(k_p x_1\), where \(k_p=\sqrt{2(p+1)}-\sqrt{2p+1}\). Suppose that the hologram is obtained without the use of optics (by direct projection) and that the light source has diameter \(d\). Simple considerations show that the largest number \(n\) of zones obtained under these conditions is determined by the condition

\[ k_n > \frac{b}{a(a+b)} \cdot \frac{d^2}{\lambda}, \tag{5} \]

since zones with \(p>n\), produced by different (incoherent) parts of the light source, will completely overlap.

On the other hand, if the resolving power of the photographic emulsion is equal to \(N\) lines per centimeter, then the greatest number of zones that can be obtained in photographing is determined by the condition

\[ k_n x_1 > \frac{1}{2N}, \tag{6} \]

or

\[ k_n^2 > \frac{a}{b(a+b)} \cdot \frac{1}{4N^2\lambda}. \tag{7} \]

When using a zone plate whose effective diameter is \(D\), the angular distance between the center of the principal maximum and the first minimum is equal to \(1.22\,\frac{\lambda}{D}\). The aperture of the hologram \(D\) is determined by the greatest number \(n\) of zones obtained:

\[ D = 2x_n . \tag{8} \]

Hence, taking into account the dependence of \(f\) on \(\lambda\), the photographic magnification \(M\) of the hologram and the relation of \(x\) to \(a\), \(b\), and \(\lambda\), we find that the limit of resolution in the image of two close points of the object is determined by the condition

\[ y = 0.61 \sqrt{\frac{ab\lambda}{(a+b)2n}} . \tag{9} \]

If the number of zones \(n\) is limited by the resolving power of the emulsion, then, taking

\[ k_n \cong \frac{1}{\sqrt{8(n+1)}}, \]

according to (7) and (9) we have:

\[ y = 0.61 \frac{a}{(a+b)N}. \tag{10} \]

If, however, the number of zones is limited by the extent of the light source, then, according to (5) and (9),

\[ y = 1.22 \frac{b}{a+b}\, d . \tag{11} \]

It is very remarkable that in both cases the limit of resolution depends neither on the wavelength of the light used to obtain the hologram nor on the wavelength of the light used to reconstruct the image. Thus the hopes, persistently emphasized in early works (see [1]), of obtaining a high resolving power by producing the hologram in rays of very short wavelength (X-rays, electron beams, etc.) prove untenable. In particular, for the case of X-rays under conditions optimal for modern technology:

\[ \lambda = 10^{-8}\ \text{cm}, \qquad N = 10^4\ \text{cm}^{-1}, \qquad d = 10^{-3}\ \text{cm}, \qquad a+b = 500\ \text{cm}, \]

one obtains \(y\) of the order of several tenths of a micron, i.e. of the same order as in an ordinary electron (and even optical) microscope. Thus, the use of X-radiation in this case does not lead to any substantial increase in resolving power.

In the case of obtaining a hologram with the aid of an electron beam, additional restrictions arise.^5 The point is that obtaining a sufficiently small light source here is associated with the use of electron optics (as the “source,” a reduced image of a source of comparatively large size is usually used). In this case, the individual parts of the “source,” first, can no longer be regarded as incoherent and, secondly, all possible aberrations of the electron-optical system become significant.

The authors^5 estimated the influence on the resolving power of spherical and chromatic aberration of the electron lens, as well as that of unstable voltage (wavelength) and current. In addition, they determined the permissible dimensions of the field of view and the exposure times necessary for obtaining the hologram. Referring the reader for details to the original paper, let us note that the use of electron optics makes it possible to obtain source sizes incomparably smaller than in the case of x-rays and, in connection with this, substantially greater resolving power that does not, incidentally, exceed the resolving power of an ordinary electron microscope. Thus, estimates for two different types of optical systems led the authors to values of the resolution limit of several angstroms for exposure times of the order of hundreds of seconds.

The experiments carried out by the authors showed that, indeed, the exposure must have a duration from several minutes to several hours; however, the resolution limit is practically obtained approximately an order of magnitude higher than the theoretical estimate (20–50 Å). At the same time, the method of obtaining the hologram by direct projection proved noticeably worse than the method of obtaining a hologram using a lens (electron) between the object and the hologram. The authors note that one of the most difficult technical problems is the stabilization of the position of the object with an accuracy of up to several angstroms during a rather long exposure.

Thus, in the field of electron microscopy as well, the method of diffraction microscopy offers no substantial advantages.

In this connection, the problem of finding effective methods for obtaining images in x-rays without resorting to an intermediate hologram again becomes of undoubted interest. One such method, apparently, may be the use, instead of lenses, of zone plates.^6 The fabrication of such plates is possible, for example, by making a hologram of a point object and subsequently reducing it.^4 As is easily seen, the focal length \(f'\) of such a hologram reduced by \(M\) times, when it is used in radiation with wavelength \(\lambda'\), is related to the focal length \(f\) of the same hologram at natural size, obtained with radiation whose wavelength is \(\lambda\), by the relation

\[ \frac{f}{f'} = M^2 \frac{\lambda'}{\lambda}. \tag{12} \]

Consequently, in passing from \(\lambda \simeq 5 \cdot 10^{-5}\ \text{cm}\) (visible light) to \(\lambda \simeq 5 \cdot 10^{-8}\ \text{cm}\) (x-rays), the focal length will remain unchanged for \(M = 31.5\), which does not present any serious technical difficulty. The basic question, apparently, consists in developing a technique for obtaining reduced reproductions of such holograms that would ensure the opacity of the corresponding zones for x-ray radiation,^4 while preserving adequate resolving power. It is not excluded that along this path some success will be achieved in the creation of an x-ray microscope.

G. Rozenberg

References

1. Usp. Fiz. Nauk 35, 595 (1948); 43, 144 (1951); 44, 622 (1951).
2. D. Gabor, Proc. Roy. Soc. A197, 454 (1949).
3. G. L. Rogers, Proc. Roy. Soc. Edinb. 63, No. 3, 193 (1950–51); for certain questions concerning image distortion during its reconstruction, see G. L. Rogers, ibid., 63, No. 4, 313 (1951–52).
4. A. V. Baez, J. Opt. Soc. Am. 42, No. 10, 756 (1952).
5. M. E. Haine and T. Mulvey, J. Opt. Soc. Am. 42, No. 10, 763 (1952).
6. O. E. Myers, Am. J. Phys. 19, 359 (1951).
7. S. M. Raiskii, Usp. Fiz. Nauk 47, No. 4, 515 (1952).