Interference patterns. Maximum and minimum conditions

Interference patterns are light or dark bands that are caused by rays that are in phase or in phase with each other. Light and similar waves are added when superimposed, if their phases coincide (either in the direction of increasing or decreasing), or they compensate each other if they are in antiphase. These phenomena are called constructive and destructive interference, respectively. If a beam of monochromatic radiation, all waves of the same length, passes through two narrow slits (the experiment was first carried out in 1801 by Thomas Young, an English scientist who thanks to him came to the conclusion about the wave nature of light), two resulting beams can be directed On a flat screen, on which instead of two overlapping spots formed interference fringes - a pattern of uniformly alternating light and dark areas. This phenomenon is used, for example, in all optical interferometers.

Superposition

The defining characteristic of all waves is a superposition that describes the behavior of superimposed waves. Its principle is that when more than two waves are superimposed in space, the resultant perturbation is equal to the algebraic sum of the individual perturbations. Sometimes this rule is violated for large perturbations. Such simple behavior leads to a number of effects, which are called interference phenomena.

The phenomenon of interference is characterized by two extreme cases. In the constructive maxima of the two waves coincide, and they are in phase with each other. The result of their superposition is the amplification of the disturbing effect. The amplitude of the resulting mixed wave is equal to the sum of the individual amplitudes. And, conversely, in destructive interference, the maximum of one wave coincides with the minimum of the second - they are in antiphase. The amplitude of a combined wave is equal to the difference between the amplitudes of its constituent parts. In the case where they are equal, the destructive interference is complete, and the total disturbance of the medium is zero.

Young's experiment

The interference pattern from two sources unequivocally indicates the presence of overlapping waves. Thomas Jung suggested that light is a wave that obeys the principle of superposition. His famous experimental achievement was the demonstration of constructive and destructive interference of light in 1801. The modern version of Jung's experiment is inherently different only in that it uses coherent light sources. The laser uniformly illuminates two parallel slits in an opaque surface. The light passing through them is observed on the remote screen. When the width between the slots is much longer than the wavelength, the rules of geometrical optics are observed - two illuminated areas are visible on the screen. However, when the cracks approach, the light diffracts, and the waves on the screen overlap. Diffraction itself is a consequence of the wave nature of light and another example of this effect.

Interference pattern

The principle of superposition determines the resulting intensity distribution on the illuminated screen. An interference pattern occurs when the path difference from a slot to a screen is equal to an integer number of wavelengths (0, λ, 2λ, ...). This difference ensures that the highs come at the same time. Destructive interference occurs when the path difference is equal to an integer number of wavelengths shifted by half (λ / 2, 3λ / 2, ...). Jung used geometric arguments to show that superposition leads to a series of equidistant bands or high-intensity areas corresponding to areas of constructive interference separated by dark areas of complete destructive.

Hole spacing

An important parameter of the geometry with two slits is the ratio of the wavelength of the light wave λ to the distance between the holes d. If λ / d is much less than 1, the distance between the bands will be small, and the overlapping effects will not be observed. Using closely spaced slots, Jung was able to separate the dark and light areas. Thus, he determined the wavelengths of the colors of visible light. Their extremely small value explains why these effects are observed only under certain conditions. In order to separate the parts of constructive and destructive interference, the distances between the sources of light waves must be very small.

Wavelength

Observation of interference effects is a difficult task for two other reasons. Most light sources emit a continuous spectrum of wavelengths, resulting in the formation of multiple interference patterns superimposed on each other, each with its own interval between the bands. This eliminates the most pronounced effects, such as areas of total darkness.

Coherence

In order to be able to observe interference for an extended period of time, it is necessary to use coherent light sources. This means that the radiation sources must maintain a constant phase relationship. For example, two harmonic waves of the same frequency always have a fixed phase relationship at each point of space - either in phase, or in antiphase, or in some intermediate state. However, most light sources do not radiate true harmonic waves. Instead, they emit light, in which random phase changes occur millions of times per second. Such radiation is called incoherent.

The ideal source is a laser

Interference is still observed when two incoherent sources are superimposed in space, but interference patterns change randomly, together with a random phase shift. Sensors of light, including the eyes, can not register a rapidly changing image, but only a time-averaged intensity. The laser beam is almost monochromatic (that is, it consists of one wavelength) and highly coherent. It is an ideal light source for observing interference effects.

Frequency determination

After 1802, the wavelengths of visible light measured by Jung could be correlated with the insufficiently accurate velocity of light available at a time to approximately calculate its frequency. For example, in green light it is about 6 × 10 ¹⁴ Hz. This is many orders of magnitude greater than the frequency of mechanical oscillations. For comparison, a person can hear a sound with frequencies up to 2 × 10 ⁴ Hz. What exactly fluctuates with such speed, remained a mystery for the next 60 years.

Interference in thin films

The observed effects are not limited to the double slit geometry used by Thomas Young. When reflection and refraction of rays from two surfaces separated by a distance comparable to the wavelength occurs, interference occurs in thin films. The role of film between surfaces can be played by vacuum, air, any clear liquids or solids. In visible light interference effects are limited to the size of several micrometers. A well-known example of a film is a soap bubble. The light reflected from it is a superposition of two waves - one reflected from the front surface, and the other - from the back. They are imposed in space and stacked with each other. Depending on the thickness of the soap film, two waves can interact constructively or destructively. A complete calculation of the interference pattern shows that for a light with one wavelength λ constructive interference is observed for a film of thickness λ / 4, 3λ / 4, 5λ / 4, etc., and destructive for λ / 2, λ, 3λ / 2, ...

Formulas for calculation

The phenomenon of interference has found many applications, so it is important to understand the basic equations that apply to it. The following formulas allow us to calculate the different quantities associated with interference for the two most common cases.

The arrangement of the bright bands in Young's experiment, i.e., areas with constructive interference, can be calculated using the expression: y _light. = (ΛL / d) m, where λ is the wavelength; M = 1, 2, 3, ...; D is the distance between the slits; L is the distance to the target.

The location of dark bands, that is, areas of destructive interaction, is determined by the formula: y _dark. = (ΛL / d) (m + 1/2).

For another kind of interference - in thin films - the presence of a constructive or destructive overlay determines the phase shift of the reflected waves, which depends on the thickness of the film and the index of its refraction. The first equation describes the case when there is no such displacement, and the second describes a shift of half the wavelength:

2nt = mλ;

2nt = (m + 1/2) λ.

Here λ is the wavelength; M = 1, 2, 3, ...; T is the path traversed in the film; N is the refractive index.

Observation in nature

When the sun illuminates a soap bubble, you can see bright colored bands, because different wavelengths are subjected to destructive interference and are removed from the reflection. The remaining reflected light appears to complement the removed colors. For example, if as a result of destructive interference there is no red component, then the reflection will be blue. Thin oil films on water produce a similar effect. In the nature of feathers of some birds, including peacocks and hummingbirds, and the shells of some beetles look rosy, while changing color when changing the viewing angle. The physics of optics here consists in the interference of reflected light waves from thin layered structures or arrays of reflecting rods. Similarly, pearls and shells have an iris due to the imposition of reflections from several layers of mother-of-pearl. Precious stones, such as opal, exhibit beautiful interference patterns due to the scattering of light from regular structures formed by microscopic spherical particles.

Application

There are many technological applications of light interference phenomena in everyday life. The physics of camera optics is based on them. The usual antireflective coating of lenses is a thin film. Its thickness and refraction of the rays are chosen in such a way as to produce destructive interference of the reflected visible light. More specialized coatings consisting of several layers of thin films are designed to transmit radiation only in a narrow wavelength range and, therefore, are used as light filters. Multilayer coatings are also used to enhance the reflectivity of mirrors of astronomical telescopes, as well as optical resonators of lasers. Interferometry - the exact measurement methods used to record small changes in relative distances - is based on observing the shifts of dark and light bands produced by reflected light. For example, measuring how an interference pattern will change allows one to establish the curvature of the surfaces of optical components in fractions of the optical wavelength.