NEW DEVELOPMENTS IN DIFFRACTION MICROSCOPY

The essence of the recently proposed new principle of microscopy—the so-called “diffraction microscopy”—and some results achieved with its aid have already been described in our journal\(^{1,2}\). There was also noted a very serious difficulty of a fundamental nature standing in the way of realizing the invaluable advantages inherent in this method and preventing the creation of a full-fledged diffraction microscope. Recently, however, the possibility of effectively overcoming this, as it seemed, organic defect has been clarified\(^{2,3,4,5}\), and thus a prospect has arisen for the practical realization of devices capable of greatly expanding the range of objects accessible to microscopic investigation.

Let us briefly recall the essence of the method\(^{1,2,6}\). As is known, the process of obtaining an image with the aid of an optical system may be conventionally divided into two stages: 1) formation of an intermediate diffraction image of the object (Fourier transformation) and 2) transformation of the intermediate diffraction image into the final image of the object (inverse transformation). The idea underlying diffraction microscopy consists in the actual separation of these two stages. First, an intermediate diffraction image is obtained and photographed. The positive of this image (a “hologram”) is then placed in an apparatus reproducing the conditions under which it was obtained (but in the absence of the object), and the final image of the object is obtained, which is the diffraction image of the hologram. (Of course, both the obtaining of the hologram and the obtaining with its aid of the image of the object must be carried out in monochromatic radiation.) The value of such a procedure consists in the fact that the two operations may be performed with radiation of different wavelengths (and of different nature). This makes it possible to obtain a hologram,

for example, in X-ray, electron, and similar beams, while the image is in visible light. In this case the magnification proves to be proportional to the ratio of the wavelength of the radiation used to obtain the image to the wavelength of the radiation used to obtain the hologram. (In the case of an X-ray-optical microscope this ratio is of the order of \(10^4—10^5\).) At the same time, the resolving power of the device depends only on the wavelength of the radiation used to obtain the hologram. Thus, under known conditions a diffraction microscope can combine an unusually high resolving power with enormous magnification.

However, in order to obtain a good-quality image it is necessary that the hologram reproduce not only the amplitude but also the phase relations on the hologram. In reality, the phase relations in the process of reproducing the image from the hologram are violated—all parts of the intermediate diffraction image (imitated by the hologram) prove to be in phase. This corresponds to obtaining not a real image of the object, but a so-called Paterson synthesis, with all the shortcomings inherent in it. What is the way to overcome this shortcoming? Of course, the artificial techniques that have been applied, for example, in the case of an X-ray-optical microscope, are a palliative. An effective solution of the problem can be obtained only by a detailed consideration of the nature of the image obtained with the aid of a hologram \(^{3,4,5,6}\).

First of all, an in-phase hologram is obtained in the case when there is not one, but two symmetric objects situated on opposite sides of the so-called “focus” of the hologram. In fact, let both the direct radiation \(A_0 e^{i\psi_0}\) (\(A_0\) is the amplitude, \(\psi_0\) the phase) and the radiation diffracted by the objects, \(A_1 e^{i\psi_1}\) and \(A_2 e^{i\psi_2}\), be incident on some point of the hologram. If we put \(A_1=A_2\), then for the resultant wave

\[ A e^{i\psi}=e^{i\psi_0}\left[A_0+A_1\left(e^{i(\psi_1-\psi_0)}+e^{i(\psi_2-\psi_0)}\right)\right] \]

and under the condition \(\psi_1+\psi_2=2(\psi_0+n\pi)\)

\[ A=A_0+2A_1\cos(\psi_1-\psi_0)\qquad \psi=\psi_0 . \]

In the case \(A_1 \ll A_0\), the distribution of amplitudes on the hologram turns out to be practically identical—up to quantities of order

\[ \left(\frac{A_1}{A_0}\right) \]

—to the distribution of amplitudes obtained in the presence of one object. Consequently, if the condition \(A_1 \ll A_0\) is satisfied, then the hologram obtained from one object, when used as an in-phase one, gives an image of two objects (cf. \(^{1}\)). The same conclusion is easy to reach if one takes into account that an in-phase hologram is an analogue of a zone plate \(^{4}\). As is known, a zone plate simultaneously performs the role of both a converging and a diverging lens with the same focal length, i.e., it simultaneously creates two images: a real and a virtual one, situated on opposite sides of the plate.

Thus, the imperfection of the image obtained with the aid of a hologram (Paterson synthesis) is due to the fact that a diffraction pattern from its imaginary image, obtained on the other side of the hologram, is superposed on the real image of the object \(^{5}\). If it were possible to get rid of the second image or, at least, of the diffraction pattern it creates, then the problem of obtaining a complete image with the aid of an in-phase hologram would be solved.

In essence, the method of solving the problem under consideration is based on the generally known fact that the character of a diffraction pattern in any one plane completely determines its character in any other planes. Therefore, knowledge of the amplitudes in two different planes is equivalent to knowledge of the amplitudes and phases in one plane. Thus, the forced in-phase nature of the hologram, when an image is obtained with its aid, can be compensated by a second hologram. One of the possible methods of carrying out such compensation is described below.^3 Let us suppose that the production of the hologram, as well as its transformation into an image, is carried out in parallel rays,^7 and that the hologram \(H_1\) is obtained at a distance \(f\) from the object \(O\) (Fig. 1, \(a\)). Then, after removing the object and placing the hologram in its former position, we shall obtain two images of the object: a real one (\(I_1\)) and a virtual one (\(I_2\)), situated at equal distances \(f\) on opposite sides of the hologram (Fig. 1, \(c\)). In this case, upon the real image \(I_1\) there will be superposed the diffraction pattern produced by the virtual image \(I_2\). Suppose now that we have independently obtained an image of this diffraction pattern; for this it is sufficient to photograph the diffraction pattern \(H_2\), produced by the object \(O\) at a distance \(2f\) (Fig. 1, \(b\)). If now, at the place of the image \(I_1\) obtained with the aid of the hologram \(H_1\) (Fig. 1, \(c\)), we place the negative (the “antihologram”) of the diffraction pattern \(H_2\), then the diffraction pattern produced by the image \(I_2\) will thereby be completely covered by the antihologram, and photographing the image \(I_1\) behind the antihologram \(H_2\) should yield a correct image of the object.

Verification of the method described showed the validity of the considerations on which it is based.

Fig. 1. Diagram of the arrangement of holograms and images.

As the light source there was used the image (3 \(\mu\) in diameter) of a circular aperture (20 \(\mu\) in diameter), illuminated by monochromatic light (mercury line \(\lambda = 4358\) Å). This image was placed at the focus of a lens (focal length 175 mm), which produced a parallel beam of rays. The object was part of a transparent scale (1 inch long), the divisions of which corresponded to 0.01 inch. The distance \(f\) from the object to the hologram \(H_1\) was 11.0 cm. The antihologram \(H_2\) was obtained at a distance of 22.0 cm from the object. Since the image was obtained in parallel rays, the linear magnification was equal to unity. The final image was obtained on the same setup, at a distance of 11.0 cm from the hologram \(H_1\). The results obtained are shown in Fig. 2.

Photograph \(a\) is an ordinary photograph of the object (scale); \(b\)—a reproduction of the principal hologram \(H_1\); \(c\)—the image of the object ob-

obtained with the aid of this hologram alone, i.e., without eliminating the effect of the virtual image; \(d\)—the antihologram \(H_2\); \(e\)—the image of the object obtained with the aid of both holograms, i.e., after eliminating the effect of the virtual image \(I_2\).

From a comparison of the figures it is not difficult to see that, although complete elimination of the diffraction background has not been achieved, it is nevertheless greatly weakened—the image is incomparably cleaner than that obtained with the aid of a single hologram, and the diffraction rings around the numerals have almost disappeared. One may expect that further development of the technique for obtaining and reproducing holograms will lead to radical improvements. The principal difficulties here lie in the proper choice of the conditions for exposure and development of the photographic plates (the straightness and steepness of the characteristic curve), as well as in the possible reduction of phase distortions introduced by the holograms into the wave front passing through them. The authors point out that the most conspicuous image defects—spots and rings, chaotically scattered over the field of the picture—are an instrumental effect, by no means connected with defects of the method. They are caused by deficiencies of the objective lens and by dust particles that had settled on its surface and on the surfaces of the holograms. Naturally, as in every interference method, the fabrication of holograms and their reproduction require a high degree of care, and a well-developed technique is needed in order to obtain good-quality images.

Fig. 2. Images obtained by the method of diffraction microscopy according to the scheme of Fig. 1.

Summing up, one may assert that an effective method has been found for overcoming the principal shortcoming of diffraction microscopy—the in-phase character of the hologram—which leads to distortion of the image. Thus, the realization of the diffraction microscope is becoming merely a matter of its technical improvement, which, given the present state of experimental technique, appears relatively uncomplicated. It is unlikely that its use in electron microscopy will present difficulties. The situation is considerably more complicated with the X-ray optical microscope, for it is not yet entirely clear by what route an antihologram can be obtained. In any case, a broad future is opening up before diffraction microscopy, and one may expect that in the near future it will occupy an important place in microscopic investigations.

G. R.

CITED LITERATURE

1. UFN 35, 595 (1948).
2. UFN 43, 144 (1951).
3. W. L. Bragg and G. L. Rogers, Nature 167, 190 (1951).
4. G. L. Rogers, Nature 166, 237 (1950).
5. W. L. Bragg, Nature 166, 399 (1950).
6. D. Gabor, Proc. Roy. Soc. A197, 454 (1949).
7. M. F. Hain and J. Dyson, Nature 166, 315 (1950).
8. M. J. Buerger, J. Appl. Phys. 21, 909 (1950). See also².