NEW INSTRUMENTS AND METHODS OF MEASUREMENT

INTERFERENCE MICROSCOPY

G. V. Rozenberg

The structure of microscopic objects can be revealed only to the extent that the intensities or colors of the individual structural elements of their image prove to be different. For this reason, the problem of contrast has always stood at the center of microscopy. Nevertheless, until comparatively recent times, the possibilities for obtaining contrast images remained very modest. Microscopists were forced to confine themselves to studying only those objects that possessed natural contrast, due to the different capacity of their structural elements to absorb or reflect light. Structures that manifested themselves only in changes of the phase relationships in the light wave passing through them (or reflected from them), but did not affect the distribution of amplitudes over the wavefront, remained inaccessible to observation. To make visible the details of transparent bodies, as well as of bodies with transparency or reflectivity constant over their extent, was possible only by resorting to their artificial staining and by making use of the circumstance that different structural elements, generally speaking, possess unequal capacity to take up the stain.

However, such a method of differential staining of the object left outside the possibilities of microscopy a still very extensive class of objects, including, in particular, all living organisms. Naturally, the problem of obtaining images of transparent objects—reducing, in essence, to the problem of revealing the phase structure of their images—continued to attract serious attention.

Over the last twenty years, as a result of the persistent work of numerous investigators, a number of radical methods for its solution have been found. Among them belongs, for example, the method developed by E. M. Brumberg¹ for obtaining color contrast in ultraviolet rays, which proved highly effective, since an enormous number of organic substances that are completely transparent to visible light possess sharply expressed absorption bands in the ultraviolet region of the spectrum. However, while considerably expanding the range of objects accessible to investigation, this method does not...

solved the problem as a whole, for the determination of the phase structure here is replaced by the determination of the amplitude structure, but in another region of the spectrum. The problem was not solved by the use of the polarization microscope either, since the latter makes it possible to reveal only the dependence of the phase structure of the image of an object on the character of the polarization of the light, without detecting the phase structure itself.

Some other techniques, also used in the microscopy of transparent objects, although they do make it possible successfully to solve a number of problems, have a narrowly limited field of application and, most importantly, usually have the substantial drawback that the production of contrast is due to a considerable distortion of the form of the image of the object.\(^{2,3}\)

The true solution of the problem of revealing the phase structure of an image, and thereby of the problem of the microscopy of transparent objects, was found by using the phenomena of interference, namely, the so-called phase-contrast method and the related interference microscopy.

The phase-contrast method, which in recent years has found wide application, has already been covered quite fully in our literature (see, for example,\(^{2,3,4}\)). By contrast, interference microscopy, despite a number of noteworthy successes, remains chiefly the property of a narrow circle of specialists, possibly because until recently no generalized approach had been found to the consideration of the operation of those devices that are grouped under this name and that were usually considered simply as interference attachments to the microscope.

I. THE INTERFERENCE METHOD OF REVEALING PHASE CONTRAST

A light wave, passing through an inhomogeneous object or being reflected from it, undergoes changes which, generally speaking, amount to the appearance of a dependence of the amplitude and phase of the wave on the coordinates. In other words, as a result of interaction with the object the light wave becomes modulated in space in amplitude and phase.

Let us suppose that initially a plane light wave propagates along the \(z\) axis, i.e. the complex strength of the electric field of the wave is described by the relation

\[ E_0(z)=A_0 e^{i\varphi_0} e^{i(\omega t-kz)}, \tag{1} \]

where \(A_0\) and \(\varphi_0\) are the real amplitude and phase of the wave.

After interaction with the object, the strength of the electric field of the wave will already be a function of all three coordinates:

\[ E(x,y,z)=A(x,y,z)e^{i\varphi(x,y,z)}e^{i\omega t}. \tag{2} \]

This light field, deformed by the object, may be regarded (Fig. 1) as the result of interference of the initial wave incident on the object

of the wave (1), and of a certain diffracted (modulating) field \(E_{\mathrm{d}}(x,y,z)\):

\[ E(x,y,z)=E_0(z)+E_{\mathrm{d}}(x,y,z). \tag{3} \]

Suppose that an optical instrument, for example a microscope, forms in the plane \(z'\) an image of the plane \(z\). If, in this case, the aperture of the instrument is sufficiently large for practically complete preservation of the diffracted field \(E_{\mathrm{d}}\), and if the scale for measuring \(x\) and \(y\) in the image plane \(z'\) is chosen in the appropriate way, then the light field in the plane \(z'\) will be described by the same relations (2) and (3). It is this light field in the plane \(z'\) that is the object of observation.

Existing receivers of light, both photochemical (the eye, photographic emulsion) and photoelectric (including quantum counters), are quadratic detectors, i.e. the magnitude of the effect produced in a light receiver by the action of a light wave upon it is determined exclusively by the intensity of the latter:

\[ I=EE^{*}. \tag{4} \]

According to (2) and (4), the intensity distribution in the image plane \(z'\) has the form

\[ I(x,y)=A^2(x,y). \tag{5} \]

Fig. 1.

Consequently, a light receiver registering the intensity of the light field in the plane \(z'\) will not detect phase inhomogeneities in the image of the object. In order to detect the phase changes introduced into the wave as a result of its interaction with the object, it is necessary first to convert the phase modulation of the wave into amplitude modulation. The performance of this operation is precisely the task of devices intended for revealing phase modulation. The methods used for solving this problem, and the technical realization of the corresponding devices, are very diverse. Some of them (phase contrast, the dark-field method, the shadow method) have been examined in detail by S. M. Rytov\(^4\), who at the same time clarified the close connection of these methods with the methods used for detecting phase modulation in radiophysics. The method underlying interference microscopy also has an analogue in radiophysics. We have in mind the conversion of phase modulation into amplitude modulation by superposing on the oscillation being analyzed a coherent oscillation of the same frequency, produced by a special generator—a heterodyne—with subsequent quadratic detection\(^5\). Exactly the same procedure is followed in optics. To reveal the phase structure of a wave front, the phenomenon of interference of the wave under investigation with a coherent homogeneous light field is used.

\(E' = A'e^{i\varphi'}\), and the resulting interference pattern is perceived by a quadratic detector—the light receiver. This principle lies, in essence, at the basis of all interferometric methods of investigation. It is not difficult to see that the intensity of the resulting interference light field in the image plane is equal to

\[ I = A^2(x,y) + A'^2 + 2A(x,y)A'\cos[\varphi(x,y)-\varphi'], \tag{6} \]

i.e., it depends both on the amplitudes and on the phases of the interfering waves.

Consequently, the interference method, just like heterodyning, makes it possible, to one degree or another (depending on the values of \(A'\) and \(\varphi'\)), to reveal both phase and amplitude modulations.

Before proceeding to a more detailed discussion of relation (6), let us compare the described procedure with the phase-contrast method. In the phase-contrast method\(^{2-4}\), detection of the phase structure is achieved by changing the relative phase shift between the undiffracted and diffracted light fields.

The introduction of such an additional phase shift, by whatever means it is carried out, is, from the standpoint of revealing contrast, equivalent to replacing \(E_0\) in equation (3) by \(E_0 e^{i\psi}\). For \(\psi=0\), i.e., when the natural phase relation between the diffracted and undiffracted waves is preserved, the phase modulation, as we have seen, remains undetected because \(|E_0+E_d|=|E_0|\) (Fig. 2). If, however, \(\psi\) is different from zero, this equality is violated and, along with it, phase modulation is revealed to one degree or another (depending on the value of \(\psi\)). In particular, if the modulation is exclusively of phase character (Fig. 2), then

\[ I = E_0^2\{1 + 2\sin(\varphi-\varphi_0)\sin\psi + \]
\[ +2[1+\cos(\varphi-\varphi_0)](1-\cos\psi)\} \tag{7} \]

and the maximum sensitivity to the detection of the relative phase shift at neighboring points of the light field will occur under the condition

\[ E_0 e^{i\psi} = \pm i(E_0+E_d), \tag{8} \]

i.e., in the case when the modified undiffracted field is shifted in phase by \(\pm \frac{\pi}{2}\) relative to the resulting light field of the unmodified wave (Fig. 2).

Fig. 2.

Thus, the optimum value of \(\psi\) depends essentially on \(E_d\), and when observing objects of various kinds, generally speaking, it is desirable

INTERFERENCE MICROSCOPY

...it is desirable to have the possibility of arbitrarily controlling the magnitude of the relative phase shift of the undiffracted and diffracted waves.

If the phase modulation is small, i.e. the modulating field

$E_{\mathrm{d}} = i(\varphi - \varphi_0)E$ (Fig. 3), then the optimal conditions for its detection correspond to the values $\psi = \pm \dfrac{\pi}{2}$. In this case $E_0 e^{i\psi} = \pm iE_0$, i.e. the undiffracted and diffracted fields prove to be in phase or

Fig. 3.

counterphase (Fig. 4, a), and the intensity of the resultant field is equal to:

\[ I = E_0[1 \pm 2(\varphi - \varphi_0)]. \tag{9} \]

Fig. 4.

Accordingly, for the contrast $\dfrac{I - I_0}{I_0}$ we obtain:

\[ \frac{I - I_0}{I_0} = \pm 2(\varphi - \varphi_0). \tag{10} \]

Thus, for \(\psi=\pm \dfrac{\pi}{2}\), the small phase modulation is entirely converted into amplitude modulation, and the intensity (as well as the contrast) of the light field proves to be a linear function of the phase shift \((\varphi-\varphi_0)\). If, simultaneously with changing the phase of the undiffracted field, one also changes its amplitude, then the contrast of the image also changes. Namely, if the amplitude of the undiffracted field is reduced by a factor of \(p\) and the amplitude of the diffracted field is kept unchanged, then the depth of the resulting amplitude modulation under the condition \(|E_{\mathrm{d}}|\ll |E_0|\) increases by a factor of \(p\) (Fig. 4, b), and consequently the contrast of the image also increases by a factor of \(p\):

\[ \frac{I-I_0}{I_0}=\pm 2p(\varphi-\varphi_0). \tag{11} \]

In some cases it is desirable to obtain not the maximum revelation of phase contrast, but the maximum sensitivity to changes in the absolute magnitude of the modulating field \(|E_{\mathrm{d}}|\). This is achieved under the condition of in-phase or antiphase relation of \(E_0 e^{i\psi}\) and \(E_{\mathrm{d}}\), and in this case (Fig. 4, c)

\[ I=E_0^2+E_{\mathrm{d}}^2\pm 2|E_0|\,|E_{\mathrm{d}}|. \tag{12} \]

Let us now suppose that, instead of changing the phase and amplitude of the undiffracted field \(E_0\), i.e. instead of replacing \(E_0\) by \(\dfrac{1}{p}E_0 e^{i\psi}\), a homogeneous light field \(E'=A'e^{i\varphi'}\), coherent with it, is superposed on the light field \(E_0+E_{\mathrm{d}}(x,y)\). It is not difficult to see that, whatever the amplitude \(A'\) and phase \(\varphi'\) of this field may be, the homogeneous field \(E_0+E'\), formed as a result of the interference of the undiffracted wave with the auxiliary field \(E'\), can be represented in the form

\[ E_0+E'=\frac{1}{p}E_0 e^{i\psi}. \tag{13} \]

Thus, the result of superposing a homogeneous coherent field proves to be completely equivalent to changing the amplitude and phase of the undiffracted wave; i.e., the result of the interference will be identical with the result achieved by the phase-contrast method, provided that in the latter it is possible to control freely not only the phase but also the amplitude relations between the diffracted and undiffracted waves. Therefore, from a fundamental point of view, the interference method of revealing phase modulation gives nothing new in comparison with the phase-contrast method. However, as we shall see below, the ranges of applicability of the methods named are somewhat different; as a result, they are successfully developing in parallel, not competing but complementing one another.

Let us consider in somewhat more detail the requirements imposed on the field \(E'\). In the study of microscopic objects, generally speaking,

INTERFERENCE MICROSCOPY

problems of various kinds arise. In some cases it is necessary to obtain the maximum contrast \(d\ln I\) (for example, in visual observations), whereas in others it is desirable to obtain the greatest difference of intensities \(dI\) (with a photoelectric light receiver). Likewise, along with the need to reveal the contrast of individual structural elements, it may prove expedient to separate the phase and amplitude contrasts, and sometimes also the polarization contrast (for example, for determining refractive indices). Finally, in some cases it is important to determine the sign of the phase contrast, for which purpose the resulting amplitude modulation must be a linear function of the phase shift.

The solution of these problems requires the realization of various phase and amplitude relations between the light field \(E_0+E'\), which forms the image background, and the diffracted wave \(E_d\). Moreover, these relations must depend in an essential way also on the magnitude of the phase shift to be measured. To determine the optimal conditions corresponding to the various cases, let us return to relation (6).

Differentiating (6), we find:

\[ dI=2\left[A+A'\cos(\varphi-\varphi')\right]\,dA +2AA'\sin(\varphi-\varphi')\,d\varphi . \tag{14} \]

If \(A'=0\) or \(\varphi'=\varphi_0\pm mn\) (\(m\) an integer), then only amplitude contrast is revealed, which is evident from relation (5), since in this case the natural phase relations between the diffracted (modulating) wave \(E_d\) and the background \(E_0+E'\) remain unchanged.

If it is necessary to reveal only the phase contrast, leaving the amplitude contrast unrevealed, this is achieved (Fig. 5) under the condition

\[ A'\cos(\varphi-\varphi')=-A. \tag{15} \]

When this condition is satisfied,

\[ dI=-2A^2\tg(\varphi-\varphi')\,d\varphi, \tag{16} \]

and the differential contrast is

\[ \frac{dI}{I}=-2\ctg(\varphi-\varphi')\,d\varphi . \tag{17} \]

Fig. 5.

Thus, when exclusively phase contrast is revealed, the depth of modulation of the resulting light field in intensity and the contrast of the image are proportional to the depth of phase modulation (if the latter is small, and \(\varphi-\varphi'\gg 0\)).

From (16) and (17) it is clear that, in order to obtain the greatest intensity gradients, one should impose the most intense possible

field \(E'\) with a phase shift close to \(\pm \frac{\pi}{2}\). Conversely, in order to obtain the greatest possible contrast it is necessary that the intensity of the background \(E' + E_0\) either vanish (Fig. 6, a), or \(E' + E_0 = -E_{\mathrm{д}}\) (Fig. 6, b).

Fig. 6.

Obviously, these cases reproduce the conditions attained in the method of the darkened field, namely

\[ I = E_{\mathrm{д}}^{2}, \qquad dI = |E_{\mathrm{д}}|\, d|E_{\mathrm{д}}|, \tag{18a} \]

or, respectively,

\[ I = 0, \qquad dI = (dE_{\mathrm{д}})^2, \tag{18б} \]

i.e., there is a quadratic detection of the modulating field independently of its phase. Linear detection of the amplitude of the modulating field, not depending on its phase, will be achieved under the condition of being in phase or in antiphase of \(E' + E_0\) and \(E_{\mathrm{д}}\), if \(|E_{\mathrm{д}}| \ll |E' + E_0|\) (Fig. 7).

Fig. 7.

III. TWO-BEAM INTERFERENCE MICROSCOPES

Above we have seen that, from the fundamental point of view, the possibilities of the phase-contrast and interference methods are identical. However, in practice there are substantial differences between the named methods. First of all, the realization of phase-contrast

...of the method is associated with the necessity of intervening in the very process of image formation by the optical system of the microscope. The conversion of phase modulation into amplitude modulation is achieved here by preparing the intermediate diffraction pattern in the focal plane of the objective lens. \(^{2-4}\) In real optical systems such operative intervention cannot remain painless. The obtaining of a good-quality image proves to be conditioned \(^{2-4}\) by a sufficiently complete spatial separation of the undiffracted and diffracted light fields in the focal plane of the objective. This leads to very substantial limitations with respect to the dimensions and character of the objects examined in the microscope, and also does not allow full use to be made of the capabilities of the optical system of the microscope.

The interference method is free from these limitations. The conversion of phase contrast into amplitude contrast here takes place without intervention in the process of image formation. It is carried out in two ways: either a homogeneous coherent light field is superposed on the image of the object obtained in the usual way with the aid of a microscope, or, with the aid of a special interferometer, an interference image of the object is first obtained at natural size (in this case the phase contrast is converted into amplitude contrast), and then this image is examined with the aid of an ordinary microscope. Below we shall consider the devices used in both variants of the interference method.

a) Interference microscopes with interference in the image plane

Fig. 8. Optical scheme of A. A. Lebedev’s polarization interferometer.

For the first time the interference method of revealing phase contrast as applied to microscopy was carried out by A. A. Lebedev. \(^{6,7}\) The device developed by him was based on a polarization interferometer (Fig. 8). A light beam, polarized on passing through the polarizer \(P\), was split by the birefringent plate (quartz or calcite) \(Kv_1\); then both components passed through a half-wave plate and were again united, passing through the second birefringent plate \(Kv_2\), identical with the first.

The light field obtained as a result of interference of the two polarized components was viewed through the compensator \(K\) and the analyzer \(A\). In practice, the device described was achieved by placing, in front of the objective of an ordinary polarizing microscope, an interference attachment consisting of two quartz plates (each about \(1\ \text{cm}\) thick) and a half-wave plate. With the chosen thickness, the quartz plates did not separate the polarized components, but only displaced them slightly (by approximately \(0.06\ \text{mm}\)) relative to one another. The object was placed between the quartz plates, and in the image plane two coherent images of the object were obtained, slightly displaced relative to one another and interfering with each other.

If the object introduced no phase changes into the wave passing through it, then with crossed polarizer and analyzer the field remained dark. If, however, the phase shift caused by the object differed from zero, then on the dark background there appeared two images of the object, slightly separated from one another. Depending on the phase shifts experienced by each of the components (a birefringent object), the brightnesses of the two images also changed. The magnitude of the phase shift was measured with the aid of the compensator \(K\). As the author notes, the accuracy in determining the refractive index of individual microscopic particles amounted to several units in the fourth decimal place.

Fig. 9. Optical scheme of V. P. Linnik’s microinterferometer.

The next step was taken by V. P. Linnik\(^8\), who constructed an interference microscope for studying the profile of reflecting surfaces, which found wide application under the name of the “Linnik microinterferometer.” The optical scheme of this instrument, based on the operating principle of the Michelson interferometer, is shown in Fig. 9. It consists of a combination of two reflecting microscopes with a common eyepiece, producing superposed coherent images. One of them gives the image of the object under investigation, and the other—the image of a plane mirror \(3\), i.e. it creates a homogeneous coherent field that “reveals” the phase structure of the image of the surface under consideration. If the mirror \(3\) is slightly tilted, i.e. its image forms not a homogeneous field but a field periodically varying along one of the coordinates

phase, interference fringes arise in the image plane, whose form reproduces the phase structure of the image of the object, i.e., the profile of its surface. The interference device is usually made in the form of a small attachment to an ordinary microscope, screwed into the latter’s tube in place of the objective. If both objectives \(O_1\) and \(O_2\), which form part of the arms of the interferometer, are identical, then not only monochromatic light but also white light can be used for observing the surface relief.

Fig. 10. A modification of V. P. Linnik’s microinterferometer for observing opaque objects at low magnification.

In the case of small magnifications, which allow large distances between the object and the objective, the interferometer scheme can be simplified; namely, a miniature Michelson interferometer may be placed in front of the single objective of an ordinary microscope, in which one of the mirrors is replaced by the object under investigation\(^8\). A very substantial advantage of this method is that it eliminates the need for two separate objectives. One scheme of this kind was subsequently implemented by Krug and Lau\(^9\). Its operation is clear from Fig. 10.

Fig. 11. A modification of V. P. Linnik’s microinterferometer for observing transparent objects.

Linnik’s microinterferometer, like its later modifications\(^9\), is intended exclusively for studying the relief of the surfaces of opaque reflecting objects. Krug and Lau\(^9,10\) proposed a somewhat modified version of Linnik’s scheme, reconstructing it as an analogue of the Mach–Zehnder interferometer, which made it possible to apply it also to the study of transparent objects (Fig. 11).

However, this scheme, like the preceding one, is suitable only in the case of small magnifications. A scheme of an interference microscope for studying transparent objects, applicable also at higher magnifications, was developed earlier by V. A. Savin\(^ {11}\).

The principle of operation of Savin’s interferometer (shown in Fig. 12) is essentially the same as that of Linnik’s microinterferometer.

The optical system creates in the image plane of the microscope, alongside the image of the object, a uniform light field coherent with this image. The only difference of Savin’s interferometer consists in the fact that here, instead of a Michelson interferometer, another interference scheme is taken as the basis, namely the Sagnac interferometer scheme.

Fig. 12. Optical scheme of V. A. Savin’s interference microscope for observing transparent objects.

A light beam from the light source \(I\), after passing through the condenser \(K\) and being reflected from mirror \(Z_0\), is divided by the semitransparent mirror \(P\). Then both components pass in opposite directions through a symmetrical system consisting of two mirrors \(Z_1\) and \(Z_2\) and two identical microscope objectives \(O_1\) and \(O_2\), and give in the common image plane of both objectives \((Iz)\) superposed coherent images of the object planes of objectives \(O_1\) and \(O_2\), interfering with one another and viewed through the eyepiece \(Ok\). The objectives \(O_1\) and \(O_2\) are arranged in such a way that their object pla-

interferometers that form an interference image of the object into the microscope image plane. The object planes \(\Pi p_1\) and \(\Pi p_2\) are displaced relative to one another. The object is placed in one of these object planes (for example, in \(\Pi p_1\)). Then objective \(O_1\) gives, in the image plane \(Iz\), a sharp image of the object, while objective \(O_2\) gives an image of the diffuse light field, produced in the plane \(\Pi p_2\) by the light beam that has passed through the object in the opposite direction.

If the planes \(\Pi p_1\) and \(\Pi p_2\) are separated sufficiently far, then the light field in the plane \(\Pi p_2\) will be comparatively uniform and all the arguments given above concerning the manifestation of phase modulation will be applicable to the image produced by the microscope described.

The same type of instruments also includes the two-objective interference microscope\(^{9}\), whose scheme is shown in Fig. 13. A ray of light from the source \(И\) is split by the semitransparent mirror \(AD\) into two components, each of which twice undergoes total internal reflection at the faces \(AB, A'B'\) and \(AC, A'C'\) of identical glass prisms, and is again combined by the semitransparent mirror \(A'D'\). On the path between the prisms the beams pass through identical condensers \(K_1, K_2\) and microscope objectives \(O_1, O_2\), and in the microscope image plane two superposed images of the object planes of objectives \(O_1\) and \(O_2\) are obtained. In the object plane of objective \(O_2\) the transparent object under examination is placed, and in the path of the second beam a phase shifter \(\Phi\) is arranged, allowing the path difference of the beams to be varied. The interference image of the object can be viewed through the eyepiece and simultaneously photographed by the camera \(\Phi. K.\).

Fig. 13. Two-objective interference microscope for observing transparent objects.

b) Interferometers that form an interference image of the object at natural size, viewed through a microscope

At the basis of the rather similar interference devices described below lies the idea of completely separating the process of “developing” phase contrast from the process of obtaining a magnified image of the object. Thus, the discussion concerns devices intended

for forming an interference image of the object at natural size, suitable for viewing with an ordinary microscope. One such device, shown in Fig. 14, was proposed by Dyson^12.

The illuminating cone of rays from a condenser (not shown in the drawing), which images the light source on the object, falls on a plane-parallel plate, the upper surface of which is coated with a semitransparent layer of silver. On the lower surface of the plate there is deposited a completely reflecting opaque silver “spot,” whose dimensions are somewhat larger than the dimensions of the field of view of the microscope.

Fig. 14. Diagram of Dyson’s interference attachment for microscopy of transparent objects.

Labels in the figure: “window”; microscope objective; spherical reflecting surface; semitransparent surface; object; reflecting “spot”; immersion; specimen slide; cover glass.

A ray of light falling on the upper semitransparent face partly passes through it, reaching the object, and partly is reflected onto the “spot.” Then both parts of the ray fall on a second plane-parallel plate (identical with the first), both faces of which are coated with semitransparent layers of silver; the ray reflected from the “spot” partly passes through it, while the ray that has passed through the object partly passes through it after first undergoing two reflections. As a result, the rays are reunited and proceed further in the common direction. Thus this part of the device acts analogously to a Jamin interferometer. Emerging from the upper plane-parallel plate, the combined rays fall on a spherical reflecting surface (the center of the sphere coincides with the object; the upper

INTERFERENCE MICROSCOPY

face of the upper plane-parallel plate divides the radius of the sphere in half).

Having undergone reflection at the spherical surface, the rays return, undergo partial reflection at the upper face of the upper plane-parallel plate, and are collected in a small “window” made in the reflecting silver layer covering the sphere.

Thus, in the “window” two coherent real images of the light source are obtained: one produced by the rays that have passed through the object, the other by the rays that have bypassed it and have been reflected from the “spot.” As a result of interference in the plane of the “window,” an interference image of the object is obtained, which is then viewed with an ordinary microscope.

Since the cone of the “comparison rays” reflected from the “spot” and forming the “developing” image—a uniform background—is devoid of the central part intercepted by the “spot” during the initial incidence on the system, and has, in the plane of the object, a diameter of several millimeters, the presence of the object does not introduce any appreciable distortions into it.

In the absence of an object, the phase difference of the interfering rays arises only as a result of phase shifts upon reflection. The presence of an object introduces an additional phase difference, which leads to the formation of an interference image of the object: the phase modulation of the light beam passing through the object is thereby converted into amplitude modulation.

Since the dimensions of the “window” only slightly exceed the dimensions of the microscope field of view, while all the numerous secondary reflections give secondary images at distances of several millimeters from the center of the “window,” i.e., they do not enter the microscope objective, the presence of secondary reflections does not lead to distortion of the image and is manifested only in a decrease of its intensity. The author points out that a distinct interference image of the object is obtained not only in monochromatic light, but also in white light.

If the plates are not exactly plane-parallel, but are wedges with a small angle oriented toward each other, then a slight displacement of them relative to one another makes it possible to change the path difference of the interfering rays and thereby to vary both the character of the field on which the image is obtained and the character of the image itself (cf. relation (14)).

The analysis carried out by the author^12 shows that, in the case of exclusively phase contrast (transparent objects), the best results are achieved with a darkened field and equality of the intensities of both interfering rays (cf. Fig. 6, a). In the case of amplitude contrast, the image contrast is then half as great as when an ordinary microscope is used.

It is highly significant that, in order to obtain an interference image of the object, it is necessary that both images of the source

light are precisely superposed on one another, i.e., so that the corresponding points of the two interfering images coincide, for different

Fig. 15. Interference microphotograph of epithelial cells. Magnification 300×; a — bright field; b — dark field.

parts of the image of the light source are mutually incoherent. The relative displacement of the interfering images of the source obtained in the “window” of the interferometer must be small in comparison with the resolution limit determined by the numerical aperture of the illuminating cone of rays.

Fig. 16. Interference microphotograph of frog blood in immersion (oil); magnification 300×.

The visibility of the interference image

\[ V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} \tag{19} \]

(\(I_{\max}\) and \(I_{\min}\) are the greatest and least intensities in the image field) is described in this case by the relation

\[ V=\frac{J_1\left(2\pi d n \sin \frac{\alpha}{\lambda}\right)} {\pi d n \sin \frac{\alpha}{\lambda}}, \tag{20} \]

where \(d\) is the distance between corresponding points of the images, \(\alpha\) is half the aperture angle of the illuminating cone of rays, \(n\) is the refractive index in the object space, and \(J_1\) is a Bessel function.

of first order. This condition imposes stringent requirements on the arrangement of the interferometer plates.

Referring for details of the constructive design of the interferometer to the original paper1, we shall indicate only that the lower plate of the interferometer is mounted on the microscope stage by means of a simple device that makes it possible easily to adjust its position. The upper part of the interferometer, together with the cover glass, is rigidly attached directly to the microscope objective. The slide with the object is placed on the lower plate; the gaps between the slide and the two interferometer plates are filled with immersion liquid. With rough adjustment of the interferometer, fringes of equal thickness are observed, and the adjustment consists essentially in eliminating them. The author notes that handling the interferometer does not require great skill.

a              b

Fig. 17. Interference microphotograph of snail amoebocytes in tissue culture: a — maximum contrast for revealing the outer layers of the cytoplasm; b — the contrast value has been selected for the best revelation of the structure of the nucleus.

It might have been expected that an interferometer of this kind would be suitable only for the study of very thin objects. However, a special calculation carried out by the author shows that these limitations are not so severe and do not lead to serious difficulties in microscopic investigations.

Figures 15, 16, and 17 show examples of microphotographs obtained with the aid of the described device.

An interference microscope of this design, like others, can also be used for measurements of the refractive index of microscopic objects. In particular, measurements of the refractive index

refraction of the cytoplasm of epithelial cells of the tongue led the author to the value \(n = 1.358 \pm 0.010\), which attests to the comparatively high accuracy of the measurements.

Fig. 18. Dyson interference attachment for microscopy of opaque objects.

Later the same author described\(^{13}\) a modified construction of this interferometer, intended for the investigation of opaque objects (Fig. 18). The interferometer is placed in front of the objective of an ordinary reflecting microscope. The radius of the reflecting—

Fig. 19. Wolter interference attachment.

sphere slightly greater than \(2h\). The rays \(o'abcd o'\) and \(o'abO_2do'\), and the rays opposite to them, interfere. As a result, an interference image of the object \(O\), viewed in the microscope, is obtained at \(o'\).

An interferometer design somewhat different in form, but analogous in its principle of operation, was proposed by Wolter\({}^{14}\). The principle of operation of the device is clear from Fig. 19, and we shall not dwell on it. A condenser with an annular diaphragm is used here as the illuminator. The design permits the addition of polarization devices, converting it into an interference-polarization microscope.

III. MULTIBEAM INTERFERENCE MICROSCOPY

Multibeam interference microscopy, which has already found wide application for the study of both transparent and opaque objects (see, for example,\({}^{15—20}\)), represents one variety of the method described above for revealing phase modulation by forming an interference image of the object at natural size and viewing this image through an ordinary microscope. The distinctive feature of this method is that, instead of a two-beam interferometer, a multibeam interferometer—namely, a Fabry–Perot interferometer—is placed before the microscope objective. However, the peculiarities of multibeam interference impose so strong an imprint on the character of the images obtained that the theoretical considerations given above require serious revision. At present there is no complete theory of the multibeam interference microscope, and for the time being only certain general considerations can be stated\({}^{15,16,17,20}\), to which we now turn.

First of all let us consider the case when the transverse extent of the objects considerably exceeds the wavelength of light and when their properties change little over distances comparable with the wavelength of light, i.e. the case of objects quasi-homogeneous from the optical point of view. Then one may confine oneself to the framework of ray optics and apply the usual theory of the Fabry–Perot interferometer\({}^{16,17,20}\).

The structures to be revealed may be either transparent or opaque reflecting ones. In the first case they are objects whose transparency \(\tau(x,y)e^{i\beta(xy)}\) varies from point to point owing to changes either in refractive index (for example, inclusions) or in the thickness of the object. Such objects, when placed between the plates of a Fabry–Perot interferometer, change its optical thickness and, consequently, the phase shift \(\delta\) between the interfering rays of neighboring orders:

\[ \delta = \delta_0 + 2\beta(x,y), \tag{21} \]

where $\delta_0$ is the phase shift in the absence of the object. At the same time, if $\tau(x,y) \ne 1$, the effective transparency of the semitransparent mirror coatings of the interferometer also changes:

$$ T = T_0 \tau^2(x,y), \tag{22} $$

where $T_0$ is the effective transparency of the coatings in the absence of the object. Thus the intensity of the light field produced by the interferometer (see, for example, $^{20}$) in the presence of the object is

$$ I=\frac{I_0 \tau^4(x,y)}{1+F\sin^2[m\pi+\beta(x,y)]}, \tag{23} $$

where $m$ is the order of interference in the absence of the object, $I_0$ is the intensity of the light field in the absence of the object under the condition that $m$ is an integer, and $F$ is the so-called sharpness factor, depending on the effective reflectance $R$ of the semitransparent coatings of the interferometer:

$$ F=\frac{4R}{(1-R)^2}. \tag{24} $$

It follows from (23) that, in a parallel beam of monochromatic rays, weak amplitude contrast

$$ A(x,y)=1-\tau^2(x,y) $$

is revealed without substantial distortions:

$$ d\ln I=-2(1-A)\,dA \simeq -2\,dA. \tag{25} $$

For revealing phase contrast, the optimal conditions are $^{20}$ when

$$ m+\frac{\beta}{\pi}=m_0 \pm \frac{1}{\pi\sqrt{F}}, \tag{26} $$

where $m_0$ is an integer, and moreover

$$ d\ln I \simeq \pm \sqrt{F}\,d\beta. \tag{27} $$

Since usually $F \sim 10^3$, changes in the magnitude $\beta$ by several thousandths are readily detectable*).

In the case of reflecting structures, whose reflection coefficient $\rho(x,y)e^{ia(x,y)}$ changes from point to point either because of a change in the reflectance of the object, or because of unevenness of the surface relief $\left(\rho=1 \text{ and } a(x,y)=4\pi \frac{h(x,y)}{\lambda}, \text{ where } h \text{ is the depth}\right.$

*) Further increase of the sensitivity of the method to small phase shifts is possible by using a doubled interferometer. See, for example, $^{20,25}$.

depressions), the expression for the intensity of the light field produced by the interferometer assumes a considerably more complicated form^20. The amplitude contrast for reflecting structures proves to be considerably distorted. However, if only phase modulation takes place (for example, due to irregularities of the relief), equivalent to a variation of the optical thickness of the interferometer, and if the surface under investigation is coated, in order to increase the reflecting power, with a sufficiently thick layer of silver^16,17,20*), then (neglecting absorption of light in the layers of silver and in the object) the intensity of the light field is given by the relation

\[ I=\frac{I_0 F\sin^2\eta}{1+F\sin^2\eta}, \tag{28} \]

where

\[ \eta=\left(m+\frac{2h}{\lambda}\right)\pi. \tag{29} \]

Assuming, as before, \(\eta=m_0\pi\pm\frac{1}{\sqrt{F}}\), we obtain the optimal conditions for revealing the relief:

\[ d\ln I\simeq \frac{2\sqrt{F}}{\lambda}\,dh, \tag{30} \]

which corresponds to the possibility of revealing surface-relief irregularities of the order of several angstroms^16,20.

Examples of the images obtained in this way^16,19 are given in Figs. 20 and 21.

Along with the direct revelation of phase contrast by changes in the intensity of the interference field, another method is also widely used (employed also in the Linnik microinterferometer). If the plates of the Fabry–Perot interferometer are inclined relative to one another, then the image field is crossed by a system of sharp interference fringes, whose contours reproduce lines of equal phase shifts, i.e., lines of equal level of the surface of the object. The theory of this method has been considered in detail in^16,20, and we shall not discuss it.

*) Only in the case where the semitransparent coatings of the interferometer are fairly thick (250–300 Å) silver films do the phase relationships between the interfering beams not give rise to additional interference patterns in transmitted and reflected light (as is also the case for an unmetallized plate of transparent dielectric).

In the case of thinner silver films, or films of other metals, the phase relationships prove to be different, and expression (28) for the intensity of the light reflected from the interferometer assumes a considerably more complicated form^20.

Fig. 20. Microrelief of the surface of a diamond crystal, revealed by the immersion reflection method of the Fabry–Perot interferometer; one of the plates of the interferometer is the surface under investigation. The conditions correspond to relation (30).

Fig. 21. Multiple-beam interference microphotograph of a portion of the surface of a diamond crystal. Magnification \(74\times\). Since the groove depth on the surface is of the order of \(50\,\text{Å}\), they cannot be observed by other methods of revealing phase contrast.

Examples of microphotographs of this kind16, 19 are shown in Figs. 22 and 23.

In view of the fact that the magnitude of the phase shift \(\delta_0\) between the interfering rays of neighboring orders, which determines the intensity of the interference pattern, is inversely proportional to the wavelength of the light, in multiple-beam interferometry there is a very sharp dependence of the intensity (and contrast) on the wavelength (compare formulas (23) and (26)). Therefore, to obtain contrast it is necessary to use well-monochromatized light20. If it is desired to obtain color contrast, one may use radiation consisting of a set of suitably selected (two or three) monochromatic components15. The use of white light is possible only under the condition that the technique for obtaining the so-called bands of equal chromatic order16, 20, i.e., microspectroscopy, is employed.

Fig. 22. Detail of the surface shown in Fig. 20; magnification \(1260\times\). The plates of the interferometer are mutually inclined.

Fig. 23. Interference microphotograph of the surface of rolled steel, obtained with mutual inclination of the interferometer plates.

The relations given above ignored the phenomenon of diffraction. If the dimensions of the object or of its structural elements are comparable with the wavelength of light, then allowance for diffraction phenomena, which sharply alter the character of the interference image, becomes necessary. To clarify the essence of the question, let us suppose that between the plates of the interferometer there is a single inhomogeneity whose dimensions are smaller than or of the order of the wavelength of light. The light field formed by the interferometer will then represent the result of the superposition of the homogeneous field that existed in the absence of the inhomogeneity and the coherent field diffracted by the inhomogeneity. Therefore the inhomogeneity may—

will be considered as an additional coherent light source with the corresponding distribution of brightness (and phases) over the angles. It is not difficult to see that the action of such a point light source, located between the plates of the interferometer, is equivalent to the superposition, on the homogeneous light field created by the interferometer in the absence of inhomogeneity, of the light field created by two

Fig. 24. Diagram of the formation of two systems of equidistant virtual sources \(I_1, I_2, I_3,\ldots\) and \(I'_1, I'_2, I'_3,\ldots\) of coherent diffracted waves.

systems of equidistant coherent sources (Fig. 24), whose intensities decrease in a geometric progression, analogously to the case of the formation of nonlocalized interference fringes of equal monochromatic order \(^{16,20}\). The interference pattern formed in this way has not yet been subjected to theoretical analysis. However, without touching on the question of the degree of correspondence of the image to the object, it is clear that only the inhomogeneity itself will lie in the object plane of the microscope and give a sharp image. The remaining virtual sources will create diffuse images that distort the pattern (cf. Figs. 20 and 22).

In the case of a complex structure, the distortions introduced by multiple-beam interference into the diffracted field will have an even more intricate character as a consequence of the repeated diffraction of the diffracted rays during their multiple passage through the structure under investigation.

Here we shall subject to theoretical consideration only the simplest case of a strictly periodic phase grating \(^{17}\). When a parallel beam of rays passes through such a grating, both in reflected and in transmitted light, in addition to the undiffracted beam \(E_0\), there is obtained a system of diffracted beams \(E_m\), emerging at angles \(\vartheta_m\), and the relative phase shift between any of the diffracted beams and the undiffracted beam is equal (in the absence of amplitude modulation) to \(\psi \pm \frac{\pi}{2}\), where

\[ \tg \psi = \frac{s_1 - s_2}{s_1 + s_2}\tg \frac{\Delta \varphi}{2}, \]

\(s_1\) and \(s_2\) are the widths of two adjacent alternating strokes of the grating, and \(\Delta \varphi\) is the phase difference introduced by the grating between undiffracted rays passing through adjacent strokes of the grating. By virtue of the assumed symmetry of the grating, \(E_m = E_{-m}\) and \(\vartheta_m = -\vartheta_{-m}\).

Assuming that the aperture of the objective is sufficiently large for the objective to transmit all the diffracted beams, and that the strokes of the grating are parallel to the \(y\) axis, we obtain for the complex amplitude of the field in the image plane:

\[ E(x) = E_0 + 2 \sum_{m=1}^{\infty} E_m \cos mx . \tag{31} \]

If the grating is placed in a Fabry--Perot interferometer, then the diffracted rays will again and again undergo reflection and diffraction, i.e. (in contrast to two-beam interferometers) changes will occur not only in the relative amplitude and phase ratios between the undiffracted and diffracted fields, but also in the corresponding ratios between the individual components of the diffracted field. Assuming strict parallelism of the interferometer plates, as well as independence of the amplitude and phase shifts from the angle of incidence, we find that the complex amplitude of the light field in the image plane of the microscope is equal to

\[ E'(x) = E'_0 + 2 \sum_{m=1}^{\infty} E'_m \cos mx, \tag{32} \]

where, in the case of observations in transmitted light,

\[ E'_m = E'_{-m} = \tau \left( E_m + \sum_{k=1}^{\infty} \rho^k B_{k,m} \right). \tag{33} \]

Here, \(\tau\) and \(\rho\) denote the moduli of the amplitude coefficients of transmission and reflection of the semitransparent coatings of the interferometer,

\[ \begin{aligned} B_{1,0} &= E^{\mathrm{pr}}_{0} E^{\mathrm{otr}}_{0} \rho e^{-i\delta_{0}} + 2 \sum_{m=1}^{\infty} E^{\mathrm{pr}}_{m} e^{-i\delta_{0}} \cos \vartheta_{m}, \\[6pt] B_{1,m} &= E^{\mathrm{pr}}_{0} E^{\mathrm{otr}}_{m} \rho e^{-i\delta_{0}} + 2 \sum_{m'=1}^{\infty} E^{\mathrm{pr}}_{m'} \left( E^{\mathrm{pr}}_{m'-m} + E^{\mathrm{otr}}_{m'+m} \right) \cdot e^{-i\delta_{0}} \cos \vartheta_{m'} , \end{aligned} \tag{34} \]

and \(B_{k,m}\) are obtained from (34) by replacing \(E^{\mathrm{pr}}_{0}\) and \(E^{\mathrm{pr}}_{m}\) by \(B_{(k-1),0}\) and \(B_{(k-1),m}\), respectively; here \(E^{\mathrm{pr}}_{m}\) and \(E^{\mathrm{otr}}_{m}\) are the complex amplitudes of the diffracted wave of the \(m\)-th order in the transmitted and reflected light, respectively, while \(\delta_{0}\) is the phase difference between interfering rays of neighboring orders in the absence of a grating.

Thus, the changes undergone by each of the diffracted beams depend substantially on the amplitudes and phases of the other diffracted beams. Expression (32) can be transformed to the form:

\[ E'(x) = \tau \left( E^{\mathrm{pr}}_{0} + 2 \sum_{m=1}^{\infty} E^{\mathrm{pr}}_{m} \cos mx \right) \sum_{p=0}^{\infty}(x), \tag{35} \]

where the symbol

\[ \sum_{p=0}^{\infty}(x) \]

denotes the sum

\[ \sum_{p=0}^{\infty}(x) = \sum_{p=0}^{\infty} \rho^{k} \left( E^{\mathrm{otr}}_{0} + 2 \sum_{m'=1}^{\infty} E^{\mathrm{otr}}_{m'} \cos m x \right)^{k} N_{k} e^{-i(k\delta_{0}-\Omega_{k})}. \tag{36} \]

Here the real numbers \(N_{k}\) and \(\Omega_{k}\) are functions of the coordinate \(x\), characterizing the measure of deviation of \(\delta_{0}\cos\vartheta_{m}\) from \(\delta_{0}\).

In the case of observation in reflected light, expression (35) takes the form:

\[ E'(x) = \rho' + \tau^{2} \left( E^{\mathrm{otr}}_{0} + 2 \sum_{m=1}^{\infty} E^{\mathrm{otr}}_{m} \cos mx \right) \sum_{p=0}^{\infty}(x), \tag{37} \]

where \(\rho'\) is the reflectivity of the coating layer of the interferometer when light is incident from outside.

Owing to the presence of the numbers \(N_{k}\) and \(\Omega_{k}\), the calculation of \(E'(x)\) becomes extremely difficult. If, however, \(\delta_{0}\) is small and the diffraction angles \(\vartheta_{m}\) are small, then one may put \(N_{k}=1\) and \(\Omega_{k}=0\), and for the intensity of the image one obtains expressions coinciding with the usual-

INTERFERENCE MICROSCOPY

with the expressions obtained in the approximation of ray optics, i.e., with expressions (23) and (28). In this case the amplitude structure of the image corresponds exactly to the phase structure of the object.

According to \(^{17}\), the error introduced into the calculation of the phase of the diffracted ray by the assumption that \(N_k = 1\) and \(\Omega_k = 0\), i.e. \(\delta_0 \cos \vartheta_m = \delta_0\), is approximately equal to

\[ \delta_0 (1 - \cos \vartheta_m) \simeq \delta_0 \left( 1 - \sqrt{1 - \frac{m^2 \lambda^2}{(s_1 + s_2)^2}} \right), \tag{38} \]

where \(s_1 + s_2\) is the grating period. Thus, the distortions of the image increase as the diffraction order \(m\) increases and as the grating period decreases. Thus, for an interferometer with a plate spacing of \(0.01\) mm and a grating with a constant of \(0.05\) mm, for \(\lambda = 5 \cdot 10^{-5}\) cm the error in the phase of the diffracted ray of the first order is only \(0.004\pi\), but for a grating with a period of \(0.005\) mm it already increases to \(0.4\pi\). As the author’s calculations \(^{17}\) have shown, the deviations of \(N_k\) and \(\Omega_k\) from their ideal values (1 and 0, respectively), and consequently the distortions of the image, are especially large near the boundaries of the grating strokes. Furthermore, under oblique illumination the equality \(\vartheta_m = -\vartheta_{-m}\) is violated, which leads to a sharply pronounced asymmetry of the image. Appreciable distortions are also introduced by the mutual skew of the interferometer plates.

Finally, an increase in the reflection coefficient \(R\) of the semitransparent coatings of the interferometer entails an increase in the role of reflections of higher orders, and consequently also distortions of the image.

Thus the results of the theoretical consideration, carried through to the end only in the case where diffraction is neglected, i.e., in the case where the objects are large and quasi-homogeneous and ray optics is applicable, may be summarized as follows: diffraction phenomena lead to distortions of the image, and the distortions are the stronger the larger the diffraction angles, i.e., the smaller the object.

In conclusion, let us note that even in the absence of diffraction by the object there exists a whole series of factors imposing strict limitations on the dimensions of the objects considered, on the resolving power of the microscope used, and on the degree of precision in manufacturing and adjusting the interferometer \(^{16,20}\). These limitations reduce essentially to the following (for more detail see \(^{20}\)):

1. The dimensions of the inhomogeneities of the semitransparent coatings of the interferometer \(^{20,21}\) reach \(1\,\mu\), and therefore the resolution limit of the optical system of the microscope must not exceed this value.

2. The half-width of the interference fringes must exceed the resolution limit, which restricts the order of the interfering rays, i.e., the reflectivity of the coating.

3. Interfering rays of sufficiently high order must enter the objective, which restricts the angle of mutual inclination of

plates, and consequently the number of interference fringes in the field of view as well.

1. If the plates of the interferometer are inclined relative to one another, then the interfering rays of different orders are displaced somewhat relative to one another in the plane of the object. Obviously, this displacement must be smaller than both the dimensions of the object and the resolving limit of the microscope.

2. The degree of nonparallelism of the illuminating light beam, and consequently also the permissible dimensions of the light source, are substantially limited by the dimensions of the object.

Taking all these requirements into account, we see that obtaining an undistorted image of an object in the case of multiple-beam interferometry is conditioned by severe restrictions on the dimensions of the object observed and on the permissible resolving power of the microscope. As the sensitivity of the device to the detection of phase modulation increases, so too does the severity of these restrictions. In practice, multiple-beam interference microscopy can be applied only to optically quasi-homogeneous objects whose transverse dimensions substantially exceed the wavelength of light and whose thickness does not exceed several microns[^15]. At the same time, the sensitivity of the method of multiple-beam interferometry to the detection of phase contrast is incomparably higher than the sensitivity of other methods. It makes it possible, for example, to reveal surface irregularities whose dimensions are of the order of, or even smaller than, interatomic distances in the crystal lattice of a solid[^20]. This also determines the field of application of this method.

Basically, this is the investigation of the relief of surfaces, the study of ultrasonic waves in crystals, the observation of sufficiently large transparent bacteria, the measurement of small refractive indices (including dispersion of the refractive index and birefringence), and so forth.[^15],[^16],[^20]

IV. COMPARISON OF DIFFERENT METHODS FOR DETECTING PHASE CONTRAST

Let us proceed to a comparison of the possibilities and technical advantages of the various methods of microscopy of transparent objects.

a) The phase-contrast method

The production of contrast is conditioned by interference in the process of image formation by the microscope, namely by the introduction of a phase plate into the rear focal plane of the objective. The possibility of separate action on the diffracted and undiffracted light fields is connected with a sufficiently sharp spatial separation of these fields. This occurs only under the condition that ...

large diffraction angles, small dimensions of the object and the light source, and a small aperture of the light beam illuminating the object.

Thus, the phase-contrast method is applicable only to such objects whose dimensions are small or comparable with the wavelength of light, namely to very thin objects containing small inhomogeneities with sharply varying refractive indices. Coarse structures are transmitted by the phase microscope with considerable distortions.

In addition, the introduction of phase diaphragms into the focal plane of the objective leads, on the one hand, to rather considerable losses of light and, on the other hand, to incomplete use of the possibilities provided by the optical system of the microscope (limitation of the aperture).

At the same time, the phase-contrast method has the important advantage of simplicity in operation, although the production of variable phase and amplitude contrast is achieved by comparatively complicated technical means. Like other methods, it permits investigation both in monochromatic and in white light, with wide use of color contrast.

b) Two-beam interference microscopy

Since the detection of phase modulation here is not associated with intervention in the process of image formation, the restrictions imposed on the thickness of the object, the dimensions of its details, the sharpness of changes in its properties, and the aperture of the condenser practically disappear. Thus not only is the range of transparent objects accessible to microscopic investigation substantially broadened, but it also becomes possible to make full use of the optical system of the microscope. At the same time, all the possibilities for revealing phase contrast that are inherent in the phase microscope are retained.

However, this is achieved at the cost of a very substantial complication of the optical system of the microscope or by equipping an ordinary microscope with a sufficiently complex interference device, which in a number of cases complicates the technique of microscopic investigation.

Among the merits of the method may be counted the comparatively easy variability of the interference conditions, which makes it possible to attain optimal visibility of the object. As in the phase-contrast method, the use of white light and the production of color contrast are possible here.

Among the shortcomings, a significant place is occupied by large losses of light in the interference device, as well as by limitations imposed by a number of designs on the available magnifications owing to the fact that the presence of the interferometer in front of the object prevents a reduction of the distance between the object and the objective.

c) Multiple-beam interference microscopy

The distortions introduced by the interferometer into the phase relations between the components of the diffracted field, as well as the limitations imposed on the resolving power of the microscope, the angle of mutual inclination of the plates, and the aperture of the illuminating light beam, make this method inapplicable to very small objects. The method is applicable exclusively to coarse structures of small thickness with properties that vary slowly in the plane of the object \(^{15—20}\).

At the same time, the sensitivity of the method to small phase shifts is incomparably (approximately by 2–3 orders of magnitude) higher than in the other methods. Therefore, for example, in the investigation of surface relief, the method of multiple-beam interference microscopy makes it possible to obtain magnifications and resolving power in the direction normal to the plane of the object that are tens and hundreds of times greater than in the other methods.

The linear magnification in depth may reach 400,000, and the resolving power \(3 \div 5\) Å, i.e. considerably higher even than that of the electron microscope. However, such large magnifications and resolving power in the direction perpendicular to the plane of the object are obtained at the cost of a corresponding decrease in magnification and resolving power in the plane of the object.

This is especially clearly manifested if the reflectivity \(R\) of the semitransparent coatings of the interferometer is varied: a decrease in \(R\), bringing the interference conditions closer to those of two-beam interferometry, entails a decrease in phase sensitivity and at the same time makes it possible to obtain greater resolution in the plane of the object.

Among the shortcomings of the method we should note the comparatively complex technique of microscopic observations, especially in the case of reflecting objects, when preliminary silvering of their surface is necessary.

d) Combined methods

Thus, the fields of application of the various methods for revealing phase contrast by no means coincide; these methods complement one another and extend the possibilities of microscopy of transparent objects. In addition to improving the techniques for using these methods, as well as the devices intended for them, further development is also proceeding in the direction of creating universal devices that combine various possibilities for revealing contrasts of different types (amplitude, phase, and polarization) \(^{8, 14, 22, 23}\), and also of combining different methods for revealing phase contrast \(^{24}\).

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Fig. 25. Mixed phase-amplitude-polarization contrast. On the right, an amplitude grating; on the left, a phase grating; below, birefringent objects.

Fig. 26. Mixed phase-amplitude-polarization contrast. A piece of glass.

INTERFERENCE MICROSCOPY

Fig. 25 shows a color microphotograph of a phase and amplitude grating covered with birefringent plates, obtained by means of an apparatus with combined amplitude-phase-polarization contrast.

Fig. 26 is a microphotograph of a glass splinter made with the same apparatus.

The combination of the methods of phase and interference microscopy24 apparently makes it possible to achieve a substantial improvement in the conditions for obtaining an image over the entire field of view. An example of such a combined microphotograph is given in Fig. 27. The author points out the particular expressiveness of the color contrast obtained in this case. It should be borne in mind, however, that such “universal” microscopes are already very complex and costly instruments, requiring considerable skill in handling, and that their field of application therefore cannot prove to be very broad.

Fig. 27. Epithelial cells. The microphotograph was obtained by combining interference and phase contrasts.

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1. 12. ↩