Refractometer for Gases and Liquids Using Phase Contrast*)

The successful use of phase contrast in microscopy naturally led to the idea of using this method for other purposes as well, in particular for the quantitative determination of the magnitudes of phase shifts upon reflection of light or upon the passage of light through a substance. One example of such an application of phase contrast is the apparatus described below—a refractometer for gases and liquids—which has already attained a certain degree of perfection, making it possible to speak not of preliminary experiments, but of serious use of the measuring principle embodied in it.

A liquid or gas whose refractive index is to be measured fills the central part \(A\) of a plane-parallel cuvette, which is a comparatively narrow (\(2p = 1.4\ \mathrm{mm}\)) interlayer separated from the remaining volume \(B\) in the cuvette by opaque walls \(D\) of thickness \(q = 1.25\ \mathrm{mm}\) (Fig. 1). The total height of the cuvette is \(2p + 2q + 2r = 38\ \mathrm{mm}\), its width is \(17.5\ \mathrm{mm}\), and its thickness is \(t = 1 \div 100\ \mathrm{mm}\). The walls of the cuvette are formed by plano-convex lenses with focal length \(1 \div 2\ \mathrm{m}\). Light sources in the form of a slit \(18\text{–}25\ \mu\) wide and \(4.5\ \mathrm{mm}\) long (illuminated by a mercury arc through a light filter or a monochromator selecting the green line \(\lambda = 5461\ \text{Å}\), and a polaroid) are placed in the focal plane of one of the lenses, as a result of which the interior of the cuvette is in a parallel beam of monochromatic rays. The volume \(B\) of the cuvette is filled with liquid

Fig. 1. Diagram of the cuvette.

) E. Ingelstam, Ark. för Fys. 6*, No. 4 (29), 287 (1953).

FROM CURRENT LITERATURE

or with a gas, the refractive index of which, \(n_B\), is known. If the substance filling volume \(A\) has refractive index \(n_A\), then between the light beams passing through volumes \(A\) and \(B\) there arises a phase shift

\[ \varphi=\frac{2\pi}{\lambda}\,t(n_B-n_A). \tag{1} \]

Owing to the smallness of the thicknesses \(2p\) and \(q\), the front of the light wave passing through the cuvette will be noticeably distorted by diffraction phenomena. The author’s calculations showed that, for the described geometry of the cuvette, the diffraction pattern in the focal plane of the exit lens of the cuvette is described by the expression

\[ g(u)=\rho\,\frac{p}{kup}\left[\sin kup\cdot e^{i\varphi}-\sin ku(p+q)\right]+ \]

\[ +\rho\,\frac{p+r}{ku(p+r)}\,\sin ku(p+q+r), \tag{2} \]

where \(g\) is the complex amplitude of the wave in the focal plane, \(u\) is the diffraction angle, \(2\rho\) is the amplitude of the incident wave (there is no amplitude contrast), and \(k=\dfrac{2\pi}{\lambda}\).

Thus (since \(p\ll r\) and \(q\ll r\)) the diffraction pattern in the focal plane of the exit lens of the cuvette is, in its main features, similar to that obtained from a slit of width \(2p\), superposed on a “background,” which is a much finer diffraction pattern from the entire cuvette. Special measurements carried out by the author showed that this pattern corresponds to reality; moreover, the distances between the bands corresponding to diffraction from the slit are equal to \(740\,\mu\), while between the bands corresponding to the “background” they are \(25\,\mu\); the intensity of the “background” rapidly decreases with increasing diffraction angle.

Fig. 2.

As is known, the phase-contrast method, which makes it possible to transform phase modulation of a wavefront into amplitude modulation, consists in changing by \(\pm \dfrac{\pi}{2}\) the relative phase shift between the diffracted and undiffracted light fields (at the same time the amplitude ratio between the fields may also change). For this purpose a “phase plate,” carrying out this operation, is introduced into the focal plane. In the present case, as the “phase plate” the author used a mirror consisting of a metallized glass plate (the phase shift upon reflection from the metallic layer \(\psi\)) of size \(21\times14\) mm or \(18\times10\) mm; moreover, in the central part of the mirror there remained an unmetallized region \(50\,\mu\) wide, extending over the entire length of the mirror (the phase shift upon reflection from this region is equal to zero, and the reflection coefficient is lower than from the metallized one).

area). The mirror was placed in such a way that the center of the diffraction pattern, and only the spectrum of the zero order, fell on this central unmetallized zone. Experimental verification showed that the adjustment must be carried out with an accuracy of the order of 1–3 μ. As a result, a system was obtained corresponding to the case of a positive phase contrast with an exaggerated intensity of the diffracted field. The corresponding vector diagram is shown

Fig. 3. Dependence of \(I_A\) on \(\varphi\) for different \(\psi\).

in Fig. 2. A change in the phase and amplitude relations between fields \(A\) and \(B\) is equivalent to transferring the pole \(O\) to some other point \(O'\). In order to ensure sufficient accuracy in determining \(\varphi\) for any of its values in the interval \(0—2\pi\), it is necessary to have the possibility of changing \(\psi\), namely, of assigning to \(\psi\) at least three different values (poles \(O_1'\), \(O_2'\), \(O_3'\) in Fig. 2). This is achieved by changing the “phase plates”—mirrors with different thicknesses of the metallic coating.

If, with the aid of an additional lens (see Fig. 7), an image of the cuvette is obtained, then the intensity \(I_A\) of the image of region \(A\) will depend on \(\varphi\). The corresponding dependences for the values of \(\psi\) corresponding to \(O_1'\), \(O_2'\), and \(O_3'\) in Fig. 2 are shown in Fig. 3. A high accuracy in measuring \(\varphi\) is obtained, naturally, on the rectilinear portions of these curves.

Fig. 4. Phase-contrast image of the cuvette. \(a\)—without taking diffraction into account, \(b\)—taking into account diffraction of the beam passing through region \(B\), \(c\)—taking into account diffraction at the phase plate.

The image of the cuvette obtained by means of the described phase-contrast method is distorted by circumstances of two kinds. First, by a fine diffraction “ripple” due to the second term in expression (2); second, by diffraction at the phase plate (Fig. 4). The second effect is especially significant, as can be seen from Fig. 5, which shows a recording of the intensity of the image of a cuvette filled with air for different \(\varphi\) (different values of the air pressure in volume \(A\)). A rather noticeable dependence of the intensity of the image of region \(B\) (and \(D\)) on the value of \(\varphi\) (i.e., on the pressure in region \(A\)) is observed. This circumstance must be taken into account either by preliminary calibration of the instrument, or by a detailed calculation of this dependence. The approximate character of this dependence is shown in Fig. 6.

Fig. 5. Registration trace of the dependence of the intensity of the cuvette image on the distance to the center of the image for various air pressures in region A. The air pressure in region B is conventionally taken to be equal to zero.

Fig. 6. Dependence of the amplitude of the light wave in the image plane of the cuvette on the magnitude of the phase shift \(\varphi\) for regions A, B, and D.

The author determines \(\varphi\) in the following way. In the plane of the image of the cuvette a mask with two slits (of width \(0.15\ \mathrm{mm}\)) is installed, one of which cuts out a portion of the image of region \(A\), and the other—of region \(B\) or \(D\). The light passing through the slits is directed onto two photomultipliers, and the ratio of the photocurrents is measured. A schematic layout of the arrangement is shown in Fig. 7.

Since by the method described \(\varphi\) is determined only with an accuracy up to an integral multiple of \(2\pi\), additional rough measurements of \(\varphi\) are necessary, carried out either by directly counting the number of complete periods of variation of \(\varphi\) (if a gradual change in \(\varphi\) is being determined), or with the aid of an auxiliary Jamin interferometer, installed immediately after the cuvette and removed after the rough determination of \(\varphi\) in order to refine this value by the method described.

Fig. 7. Schematic layout of the refractometer.

The refractometer constructed by the author made it possible to carry out measurements of \(\varphi\) with an error not exceeding \(2\pi \cdot 10^{-3}\), i.e., with a cuvette thickness of \(10\ \mathrm{mm}\), to determine \(n-1\) with an accuracy up to \(0.04 \cdot 10^{-6}\).

The upper limit of the measurable values of \(\varphi\) is determined by the degree of monochromaticity of the light, both owing to the uncertainty in \(\varphi\) that arises when the light is insufficiently monochromatized (dispersion of the medium) and to the uncertainty in \(\psi\) (dispersion of the phase shift upon reflection from the metal of the mirror). The author notes that, using the green mercury line, it is possible to measure (at \(t = 10\ \mathrm{mm}\)) \(n-1\) within the limits \((0.05 \div 1500)\cdot 10^{-6}\). If, instead of the photoelectric method of comparing the intensities of the images of regions \(A\) and \(B\) (or \(D\)), a visual method is used, the measurement accuracy proves to be an order of magnitude lower.

The author analyzes in detail the possible sources of error, as well as the areas of application of the instrument. In particular, he points out the possibility of continuous monitoring of the pressure or concentration of a gaseous or liquid mixture, and also draws attention to the small working volume \(A\) of the cuvette, which makes it possible to carry out measurements with small quantities of substance.

V. Yur’ev