# Interference of Light

## Interference of Light

the superposition of light waves, which usually produces a characteristic spatial distribution of light intensity (interference pattern) in the form of alternating light and dark fringes as a result of the violation of the principle of addition of intensities. Some interference phenomena were observed by I. Newton but could not be explained using his corpuscular theory. The correct explanation of interference as a typical wave phenomenon was furnished by T. Young and A. Fresnel in the early 19th century.

Interference of light occurs only when the phase difference is constant over time, that is, when the waves are coherent. Before the invention of lasers, coherent beams of light could be produced only by the division and subsequent combination of rays from the same light source. In this case, the phase difference of the oscillations is constant and is determined only by the rays’ path difference Δ.

There are several methods for generating coherent light beams. For example, in the Fresnel experiment (Figure 1) two plane mirrors I and II, forming a dihedral angle close to 180°, give two virtual images S1 and S2 of the source S. A light fringe is observed on the screen AB when the path difference A of the rays S1M and S2M is equal to an even number of half-waves, whereas a dark fringe results when Δ is equal to an odd number of half-waves. Another method was proposed by Young (Figure 2). Light from an opening S falls onto a screen AB with two openings (or slits) S1 and S2. Interference of light is observed on screen CD. The distance between the neighboring light or dark interference fringes Δτ ≈ λ/α, where a is the angle S1MS2 at which the interfering rays converge. In these experiments interference of light is observed only upon addition of waves emitted by the same point of the source. Interference fringes corresponding to various points of the source are shifted with respect to each other, and the interference patterns are blurred in the case of superposition. The limiting size of the source that still yields a clear interference pattern is determined by the relationship d = λ/β, where β is the angle between the diverging rays from the source (for example, the angle S1SS2 in Figure 2).

Figure 1. Diagram of Fresnel’s experiment

This limitation does not apply in the case of interference of light reflected by two surfaces of a plane or moderately wedge-shaped transparent plate (Figure 3). In this case the path difference Δ ═ 2hn cos i’ + λ/2 is produced, where h is the thickness of the plate, n is its index of refraction, and i ’ is the angle of refraction. The additional path difference λ/2 arises because of the difference in the phase shift upon reflection from the upper and lower surfaces of the plate, respectively. In strictly plane-parallel plates (with a precision of a fraction of λ.), rays that are incident on the plate at the same angle i will have the same path difference, and the resulting interference fringes in this case are called equal-inclination fringes. They are localized at infinity and therefore may be observed in the principal focal plane of the lens. In thin plates of variable thickness, the lines corresponding to the maximums and minimums pass through the points corresponding to identical plate thickness and are called equal-thickness fringes. They are localized in the plane of the plate. In this case, a given interference fringe in monochromatic light describes a line that corresponds to the same plate thickness (Figure 4). If the light is not monochromatic, superposition of the patterns described above occurs for various wavelengths that do not interfere with each other.

Figure 2. Diagram of Young’s experiment

In this case the positions of the maximums and minimums are displaced, and the observer, therefore, sees a sequence of colored fringes. This interference phenomenon in thin films explains the rainbow colors in oil and petroleum spots on water and the iridescent colors on tempered metals. Interference of light in thin films plays a large role in coated optical components, in interference light filters, and in interference microscopy. Interference of light in thin films is studied in thin-film optics.

Figure 3. Interference in a plane-parallel plate

The possibility of observing interference depends on the degree of monochromaticity of the light. Only a few interference fringes may be observed in white light near Δ = 0, and the fringes are colored because the positions of the maximums and minimums depend on the wavelength. When a narrow spectral line is isolated from the source, the maximum path difference Δmax may be as high as several dozen centimeters. Clear interference fringes may still be observed at Δmax ≈ λ2/Δλ, where Δλ is the width of the spectrum. The quantity Δmax may be related to the time τ during which the phase of the wave remains unchanged—that is, the emitted wave is a segment of a sinusoid (“wave train”). In this case, Δmax is found to be equal to the length of the wave train: Δmax = λ2/Δλ = cτ (c is the velocity of light), which explains the impossibility of light interference at Δ > Δmax, since the respective wave trains in two interfering rays cease to overlap.

The limitations on the dimensions of the source in the experiments mentioned above are removed if a laser, which has spatial coherence, is used as the light source; interference may be observed upon superposition of the waves emitted by different points of the source. The high monochromaticity of laser radiation makes possible the observation of interference of light at very large path differences.

Figure 4. Typical instances of equal-thickness fringes

At very low light intensities, when sensitive detectors are used to record individual photons, interference of light is manifested as a statistical phenomenon. The average number of quanta incident on a given area of the screen during a fixed time interval gives the same intensity distribution as does the usual mode of observation. This is in full agreement with quantum theory, according to which interference of light takes place as a result of “interference of the photon with itself rather than as a result of superposition of different photons.

Interference of light is used very widely in measuring the wavelength of radiation, in the study of the fine structure of spectral lines, in determining the density, indexes of refraction, and dispersion properties of substances, in measuring angles and linear dimensions of parts in lengths of a light wave, and in monitoring the quality of optical systems. The operation of interferometers and interference spectroscopes, as well as the method of holography, is based on interference of light.

An important case of interference of light is the interference of polarized rays. In general, when two differently polarized coherent light waves are added, vector addition of their amplitudes takes place, leading to elliptical polarization. This phenomenon is observed, for example, upon passage of linearly polarized light through anisotropic mediums. Upon entering such a medium, a linearly polarized ray is divided into two coherent rays that are polarized in planes that are perpendicular to each other. As a result of the differing states of polarization, the velocity of propagation of the rays in the medium differs, and a phase difference Δ, which depends on the distance traversed in the substance, arises between them. The value of Δ will determine the state of elliptical polarization; in particular, for Δ equal to a whole number of half-waves, the polarization will be linear.

Interference of polarized rays is widely used to observe and study stresses and deformations in solids and to create particularly narrow-band light filters, as well as in crystal optics to determine the structure and orientation of the axes of a crystal and in mineralogy to identify minerals and rocks.

### REFERENCES

Landsberg, G. S. Optika, 4th ed. Moscow, 1957. (Obshchii kurs fiziki, vol. 3.)
Vavilov, S. I. Mikrostruktura sveta, part 2. Moscow, 1950.
Born, M., and E. Wolf. Osnovy optiki. Moscow, 1970. (Translated from English.)

M. D. GALANIN

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