RESOLVING POWER OF MULTIBEAM INTERFERENCE MICROSCOPES AND THE UNCERTAINTY RELATION

The well-known successes of multibeam interference microscopy, which makes it possible to resolve objects “in depth” whose dimensions exceed only a few angstroms, i.e., are of the same order of magnitude as interatomic distances, have prompted broad circles of opticians to undertake a serious reconsideration of the question of the resolving power of optical instruments in general. Without touching upon all aspects of this extensive and extremely important problem, we shall deal here only with the interpretation of the resolving power of multibeam devices from the point of view of the uncertainty relation.

Some clarity on this question was introduced by Ingelstam’s article¹, certain propositions of which, however, need clarification on the basis of considerations developed by other authors.

It is generally known (see, for example,²) that for any wave process, including light waves, the Fourier theorem implies an uncertainty relation that limits the accuracy of the simultaneous determination of

coordinate \(x\) and the corresponding component \(k_x\) of the wave number (i.e., momentum):

\[ \Delta x \Delta k_x \geq \frac{1}{2}. \tag{1} \]

Turning to the case of multiple-beam interferometry, let us consider a layer having optical thickness \(\frac{1}{2}z(x)\) and bounded by semitransparent mirrors with amplitude transmittance \(\tau\) and amplitude reflection coefficient \(r\) (we neglect absorption). If a plane monochromatic wave of unit intensity is incident on the Fabry–Perot interferometer formed by such a layer, then (as a result of multiple reflections) a set of transverse partial waves with amplitudes will pass through the layer (or be reflected from the layer)

\[ F_s = A_0 r^{2s} e^{-iskz(x)} \tag{2} \]

(in reflected light there will be added to them a wave with amplitude \(r\), independent of \(x\)), where \(A_0=\tau^2\), \(k=\frac{2\pi}{\lambda}\) is the wave number, and \(\lambda\) is the wavelength. In doing this we have not taken into account diffraction by irregularities of the surface relief of the layer, and also its refraction. In reality, if irregularities exist—and it is precisely they that are of interest for microscopy—diffracted partial waves appear, which in turn will undergo multiple reflections and diffraction. Calculation of the phase and amplitude structure of the resulting interference image is extremely complicated and has not been carried out even in the simplest cases. However, it seems possible to state certain general considerations that make it possible to estimate the resolving power of devices of this kind. In view of the great complexity of the relations involved here, the corresponding estimates should expediently be made on the basis of an analysis of the influence of various factors separately. Some of these will be presented below.

The half-width of the interference fringes (i.e., the transmission maxima) in the case of a Fabry–Perot interferometer, expressed in fractions of the wavelength, is equal to:

\[ w \equiv \Delta\left(\frac{2z}{\lambda}\cos\vartheta\right) = \frac{1-R}{2n\sqrt{R}} = \frac{1}{N_{\mathrm{eff}}}, \tag{3} \]

where \(R=r^2\), \(N_{\mathrm{eff}}\) is the effective number of partial waves (i.e., interfering rays), and \(\vartheta\) is the angle of inclination of the rays to the \(z\)-axis. It follows from (3) that only those of the diffracted rays for which

\[ \vartheta^2 \lesssim \frac{w\lambda}{2z}\equiv \frac{w}{m}, \]

will pass through the interferometer with appreciable intensity, where \(m=\frac{2z}{\lambda}\) is the order of interference. Since the angular spread of the beam of diffracted rays is connected with the transverse dimensions \(\Delta x\) of the irregularities at which diffraction occurs by the relation

\[ \vartheta \cong \frac{\lambda}{\Delta x} \tag{4} \]

(which corresponds to \(\Delta x \Delta k_x = 2\pi\)), only such objects will therefore be resolved that satisfy the condition

\[ \Delta x \gtrsim \lambda \sqrt{\frac{m}{w}}. \tag{5} \]

On the other hand, displacements of the interference maximum exceeding approximately \(0.1w\) are distinguishable, which corresponds to a resolution limit “in depth”

\[ \Delta z \gtrless 0.1w\,\frac{\lambda}{2}. \tag{6} \]

Multiplying (3) by (4), we obtain:

\[ \Delta x\,\Delta z \gtrless \frac{\lambda^2}{r_0}\sqrt{mw} = \frac{\lambda^2}{r_0}\sqrt{\frac{m}{N_{\mathrm{eff}}}}. \tag{7} \]

From simple geometrical considerations it is not difficult to see that

\[ \Delta k_x \cong \frac{2\pi}{\lambda}\sqrt{\frac{\Delta z}{m\lambda}} \cong \frac{2\pi}{x}\sqrt{\frac{0.1w}{2m}}\,{}^*), \tag{8} \]

whence \(\Delta x\Delta k_x \gtrless \sqrt{2}\), in agreement with the uncertainty relation (1).

Inequality (7) shows that the uncertainty relation imposes restrictions on the product of the resolving limits of a multiple-beam interference microscope in the image plane \((\Delta x)\) and in depth \((\Delta z)\), without restricting either of them separately. In accordance with this, in principle any resolving power in depth can be achieved, but only at the expense of a corresponding decrease in resolving power in the image plane. Thus, if one requires \(\Delta z = 1\) Å, then for \(\lambda = 5000\) Å, \(z = 1\) cm \((m = 4\cdot 10^4)\) and \(N_{\mathrm{eff}} = 100\) \((R = 0.97)\), according to (7) we find \(\Delta x = 2.5\) mm.

Fig. 1. Schematic image of a multiple-beam interference fringe, illustrating the relation between \(\Delta x\) and \(\Delta z\).

The obtained relation finds its practical expression in the sharp difference of the scales of the interference pattern in the directions \(z\) and \(x\) (both for fringes of equal thickness and for fringes of equal chromatic order). This is illustrated schematically in Fig. 1 for \(m = 400\) and \(N_{\mathrm{eff}} = 100\).

\[ \Delta z \sim \frac{\lambda}{2000} \]

\[ \Delta x \sim 200\lambda \]

\[ {}^*)\ \text{Ingelstam}^{1}\ \text{erroneously assumes}\ \Delta k_x=\frac{2\pi}{\lambda}\frac{\Delta z}{m\lambda} \ \text{and, using the uncertainty relation, obtains}\ \Delta x\Delta z \gtrless m\lambda^2. \]

Let us now turn to another aspect of the question. Suppose that the inhomogeneity has width \(\Delta x\). In order that the rays diffracted by it at an angle \(\vartheta\), after undergoing \(N_{\rm eff}\) reflections, retain an unchanged path difference, it is necessary that they not pass beyond the limits of the inhomogeneity. Otherwise the path difference will undergo a jump, which will lead to a broadening of the interference fringes and to a decrease in the accuracy of the determination of \(z\). This condition may be written in the form

\[ \vartheta<\frac{\Delta x}{zN_{\rm eff}} =\frac{\Delta x\cdot w}{m\lambda} <\frac{20}{m\lambda^{2}}\Delta z\,\Delta x, \]

whence, taking into account the relation between \(\vartheta\) and \(\Delta x\), we find:

\[ \Delta z(\Delta x)^2>\frac{m}{20}\lambda^3 \]

and, taking (5) into account, we again arrive at relation (7).

Consideration of the resolving power of phase-contrast and multiple-beam interference microscopes also leads to relations of the type (9). In this case relation (7) takes the form

\[ \Delta x\,\Delta z \gg \frac{\lambda^2}{2\pi z}, \tag{10} \]

i.e., in the case of multiple-beam devices the conditions are improved by \(\sqrt{\dfrac{m}{N_{\rm eff}}}\) times.

Let us note that our consideration applied to the case of a single inhomogeneity with constant \(\Delta z\). In essence, in this case we have (Fig. 2) a superposition of a set of coherent images of the inhomogeneity, of which only one \((s=1)\) is in the focus of the microscope.^5 The remaining images, corresponding to virtual equidistant objects \(a'\), \(a''\), etc., are produced by diffracted beams that cover an ever larger and larger area as \(s\) increases. This circumstance is discussed in detail by Ingelshtam. The presence of neighboring inhomogeneities will obviously introduce substantial changes into the behavior of the diffracted rays.

Fig. 2. Diagram of the course of rays diffracted by an inhomogeneity \(a\) in a Fabry–Perot interferometer, illustrating the formation of virtual objects and the increase of \(\Delta x\) owing to their coexistence. Increasing the number of reflections leads to a decrease in \(\Delta z\).

Foster^6 analyzed the case of a strictly periodic phase grating. It is essential that multiple reflections introduce changes into the phase and amplitude relations not only between the undiffracted and diffracted fields, but also between the individual components of the diffracted field. The resulting phase shift for a ray diffracted at an angle \(\vartheta\) is approximately equal to

\[ \Delta\varphi \simeq 2\pi \frac{z}{\lambda}\left(1-\sqrt{1-\frac{n^2\lambda^2}{d^2}}\right), \tag{11} \]

where \(n\) is the order of the diffraction spectrum and \(d\) is the grating constant. In the case \(n=1\) and \(\lambda \ll d\), \(\Delta\varphi \simeq \dfrac{\pi z\lambda}{d^2}\). Since \(\Delta\varphi\) corresponds to the error in determining \(z\), \(\Delta z = \dfrac{\lambda}{2\pi}\Delta\varphi\), and \(d=\Delta x\), we therefore find:

\[ \Delta z\cdot(\Delta x)^2 \simeq \frac{m}{2}\lambda^3. \tag{12} \]

The value of the right-hand side obtained here, exaggerated in comparison with (9), has resulted from the fact that, instead of the phase error for the resulting field, we have taken the phase error for a single diffracted field.

Up to now we have taken into account only the factors connected with diffraction on the inhomogeneity under consideration. However, there is also a whole series of other factors connecting \(\Delta z\) with \(\Delta x\). Among them are, for example, limitations imposed on the number of interference fringes located in the field of view of the microscope, on the angle of mutual inclination of the interferometer plates (the necessity that \(N_{\mathrm{eff}}\) partial beams fall into the objective), on the displacement of partial beams over the surface of the wedge under successive reflections, on the half-width of the interference fringes (it must be greater than \(\Delta x\)), on the dimensions of the light source and the degree of collimation of the light beam, etc. The most important of these limitations are considered in \({}^{3}\) and \({}^{7}\), and we shall not touch upon them here. Let us note only that they likewise lead to the inevitability of an increase in \(\Delta x\) as \(\Delta z\) decreases, strengthening the relation of uncertainty.

Let us also point out the extremely interesting features of the so-called superposition interference fringes, which have a peculiar zigzag-shaped form and make it possible to achieve a further refinement of measurements of \(z\), at the cost of corresponding losses in the determination of the horizontal coordinate \({}^{8}\).

Thus, we must state that limitations on the resolving power “in depth” are imposed exclusively by circumstances of a technical, but by no means fundamental, nature. However, an increase in resolving power in depth loses its meaning as soon as we are deprived of the possibility of ensuring constancy of \(z\) over ever larger areas, whose dimensions

\[ \Delta x \gg \frac{\lambda^2}{\Delta z}. \]

Thus, it is well known that the granularity of the structure of silver coatings used in multiple-beam interferometry, having a scale of about \(1\mu\), is in no way revealed in multiple-beam interferograms. In this case we obtain with great accuracy not the true value \(z(x)\), but a certain mean value for a comparatively large area, and it is not at all clear how this mean is related to the true relief \(z(x)\). It has only been established experimentally \({}^{3}\) that, when coatings are deposited, the dependence of the mean value of \(z\) on \(x\) is preserved with very high accuracy (in any case of the order of \(5\) Å), which makes it possible to use the methods of multiple-beam interferometry to study the mean microrelief of a surface. In this connection one should note the error of Koehler’s attempt \({}^{9}\) to treat small irregularities of interference fringes of equal chromatic order as directly corresponding to microirregularities of the relief of the interferometer surfaces.

In conclusion, let us note that the main task of microscopy, including interference microscopy, is the reproduction, in the image that is the subject of direct measurement, of the qualities of the imaged object. For this, generally speaking, there is no need for the image to resemble the object (although this is often desirable for reasons of simplicity in processing the experimental data). However, for complete

valuable study of an object require both rationally designed instruments and a full theory of the action of these devices, making it possible to determine in sufficient detail from the image various properties of the object.

A complete theory of the multibeam interference microscope does not yet exist, and the possibility of estimating its resolving power by means of an uncertainty relation is an important practical step. Of course, it also has great significance in principle, since, on the one hand, it makes it possible to evaluate correctly the capabilities and range of application of multibeam devices and, on the other hand, it destroys the firmly rooted myth of an allegedly fundamental limitation on the resolving power of the microscope. As is now clear, suitably designed optical devices can in principle provide any resolving power in depth. Since, however, in many cases it is desirable that the resolving powers in depth and in the image plane be the same, in these cases, according, for example, to (7), we obtain

\[ \Delta x \simeq \Delta z \gg \frac{\lambda}{4} \]

in accordance with the conditions that obtain for ordinary microscopes. At the same time it is obvious that it would be futile to try to increase substantially the resolving power of the microscope in the image plane, since this would prove possible only in one dimension and under the indispensable condition that in the other two dimensions (including in depth) the object retain homogeneity on a scale increasing as \(\Delta x\) decreases—a case that is extremely rare and of no practical interest. Taking the circumstances set forth into account is extremely important not only in designing and estimating the capabilities of multibeam interference microscopes, but also for the correct interpretation of the results obtained with their aid.

G. Rozenberg

CITED LITERATURE

1. E. Ingelstam, Ann. f. Fys. 7, No. 24, 309 (1954).
2. L. I. Mandelstam, Complete Collected Works, vol. V.
3. G. V. Rozenberg, UFN 47, No. 2, 173 (1952).
4. P. M. Duffieux, Rev. d’optique 33, No. 10, 489 (1954).
5. G. V. Rozenberg, UFN 50, No. 2, 271 (1953).
6. R. C. Faust, Proc. Roy. Soc. 211A, No. 1105, 240 (1952).
7. F. A. Korolev, High-Resolution Spectroscopy, Gostekhizdat, 1953.
8. A. M. Slauert, Nature 168, 861 (1951); S. Tolanski, Contraste de phase et contrast par interférences, Paris, 1952, p. 163; H. P. Polster, J. O. S. A. 39, 1038 (1949); S. C. Hunter and F. R. N. Nabarro, Phil. Mag. 43, 538 (1952); P. J. Lindberg, Rep. Optic. Labor. Roy. Inst. Techn. Stockholm, No. 23 (1952).
9. W. F. Koeler, J. O. S. A. 43, No. 9, 743 (1953). See also UFN 52, No. 4, 650 (1954).