Path of rays in a thin film. Interference of light in thin films. Stripes of equal slope and equal thickness. Newton's rings. Practical application of interference. Interference of light waves

Stripes of equal slope. The interference fringes are called stripes of equal slope, if they arise when light falls on a plane-parallel plate (film) at a fixed angle as a result of the interference of rays reflected from both surfaces of the plate (film) and emerging parallel to each other.

Stripes of equal inclination are localized at infinity, therefore, to observe the interference pattern, the screen is placed in the focal plane of a collecting lens (as for obtaining images of objects at infinity) (Fig. 22.3).

Rice. 22.3.

The radial symmetry of the lens leads to the fact that the interference pattern on the screen will have the form of concentric rings centered at the focal point of the lens.

Let from air (i, ~ 1) onto a plane-parallel transparent plate with refractive index i 2 and thickness d a plane monochromatic light wave with wavelength is incident at an angle O X(Fig. 22.3).

At the point A light beam S.A. partly reflected and partly refracted.

Reflected beam 1 and reflected at the point IN Ray 2 coherent and parallel. If you bring them to a point with a collecting lens R, then they will interfere in the reflected light.

We will take into account reflection feature electromagnetic waves and, in particular, light waves when they fall from a medium with a lower dielectric constant (and lower refractive index) to the interface between two media: when the wave is reflected from an optically denser medium ( n 2> i,) its phase changes by l, which is equivalent to the so-called “half-wave loss” (±A/2) upon reflection, i.e. the optical path difference A changes to X/2.

Therefore, the optical path difference of the interfering rays is defined as

Using the law of refraction (sin 0 = "2 sind"), and also the fact that i = 1, AB-BC = d/cos O" and AD - AC sin fs-2d tgO" sin O, you can get

Consequently, the optical difference in the wave path A is determined by the angle O, which is uniquely related to the position of the point R in the focal plane of the lens.

According to formulas (22.6) and (22.7), the position of light and dark stripes is determined by the following conditions:

So for the data X,d And n 2 Each inclination of 0 rays relative to the plate corresponds to its own interference fringe.

Stripes of equal thickness. Let a flat monochromatic light wave fall on a transparent thin plate (film) of variable thickness - a wedge with a small angle a between the side faces - in the direction of parallel rays 1 And 2 (Fig. 22.4). The intensity of the interference pattern formed by coherent rays reflected from the top

on the thickness of the wedge at a given point (d And d" for rays 1 And 2 respectively).

Rice. 22.4. Observation of stripes on the equal and lower surfaces of the wedge depends

Coherent pairs of rays (G And G, 2 And 2") intersect near the surface of the wedge (points O and O, respectively") and are collected by a lens on the screen (respectively, at points R And R").

Thus, a system of interference fringes appears on the screen - strips of equal thickness, each of which occurs when reflected from sections of the wedge with the same thickness. Stripes of equal thickness are localized near the surface of the wedge (in the plane 00", marked with a dotted line).

When light beams from an extended light source fall almost normally on a transparent wedge, then the optical path difference

and depends only on the thickness of the wedge d at the point of incidence of the rays. This explains the fact that the interference fringes on the surface of the wedge have the same illumination at all points on the surface where the thickness of the wedge is the same.

If T is the number of light (or dark) interference fringes per wedge segment of length /, then the angle at the top of the wedge (sina ~ a), expressed in radians, is calculated as

Where d] And d 2- thickness of the wedge on which they are located, respectively To-Me and (k + t)-th interference fringes; Oh- the distance between these stripes.

Newton's rings. Newton's rings are a classic example ring strips of equal thickness, which are observed when monochromatic light with wavelength X is reflected from an air gap formed by a plane-parallel plate and a plano-convex lens with a large radius of curvature in contact with it.

Rice. 22.5.

A parallel beam of light is incident normally on the flat surface of a lens (Fig. 22.5). Strips of equal thickness have the form of concentric circles with the center of contact of the lens with the plate.

We obtain the condition for the formation of dark rings. They arise where the optical path difference D of waves reflected from both surfaces of the gap is equal to an odd number of half-waves:

where X/2 is associated with the “loss” of a half-wave upon reflection from the plate.

We use both last equations. Therefore, in reflected light the radii of the dark rings are

Meaning T= 0 corresponds to the minimum of the dark spot in the center of the picture.

Similarly, we find that the radii of the light rings are defined as

These formulas for the radii of the rings are valid only in the case of ideal (point) contact of the spherical surface of the lens with the plate.

Interference can also be observed in transmitted light, and in transmitted light the interference maxima correspond to the interference minima in reflected light and vice versa.

Enlightening optics. Lenses of optical instruments contain a large number of lenses. Even a slight reflection of the light of each

Rice. 22.6.

from the surfaces of the lenses (about 4% of the incident light) leads to the fact that the intensity of the transmitted light beam is significantly reduced. In addition, lens flare and background scattered light occur, which reduces the efficiency of optical systems. In prismatic binoculars, for example, the total loss of light flux reaches -50%, but at the boundaries of media it is possible to create conditions when the intensity of light passing through the optical system is maximum. For example, thin transparent films are applied to the surface of lenses. dielectric thickness d with refractive index p ъ (Fig. 22.6). At d - NX/4 (N- odd number) interference of rays G And 2, reflected from the top and bottom surfaces of the film will give a minimum intensity of reflected light.

Typically, optics are cleared for the middle (yellow-green) region of the visible spectrum. As a result, lenses appear purple in reflected light due to a mixture of red and purple. Modern technologies The synthesis of oxide films (for example, by the sol-gel method) makes it possible to create new antireflective protective coatings in optoelectronics based on elements of the metal-oxide-semiconductor structure.

When a light wave is incident on a thin transparent film or plate, reflection occurs from both surfaces of the film.

As a result, coherent light waves arise, which cause light interference.

Let a plane monochromatic wave fall on a transparent plane-parallel film with refractive index n and thickness d at an angle. The incident wave is partially reflected from the upper surface of the film (beam 1). The refracted wave, having partially reflected from the lower surface of the film, is again partially reflected on the upper surface, and the refracted wave (ray 2) is superimposed on the first reflected wave (ray 1). Parallel beams 1 and 2 are coherent with each other, they give an interference pattern localized at infinity, which is determined by the optical path difference. The optical path difference for transmitted light is different from the optical path difference for reflected light, so transmitted light is not reflected from an optically dense medium. Thus, the interference maxima in reflected light correspond to the interference minima in transmitted light, and vice versa.

The interference of monochromatic light on a plane-parallel plate is determined by the quantities ?0, d, n, and u. They correspond to different angles of incidence different points interference pattern (fringe). Interference fringes resulting from the superposition of waves incident on a plane-parallel plate at the same angles are called fringes of equal inclination. Parallel rays 1 and 2 converge at infinity, so the bands of equal inclination are said to be localized at infinity. To observe them, a collecting lens and a screen located in the focal plane of the lens are used.

6.4.2. Let us consider the interference of light on a wedge-shaped film of variable thickness. Let it be on a wedge with an angle? A plane wave falls between the side faces (rays 1, 2 in Fig. 6.10). It is obvious that the reflected rays are 1? and 1 ? ? from the upper and lower surfaces of the wedge (as well as 2 ? and 2 ? ?) coherent with each other. They may interfere. If the angle? is small, then the optical path difference of the rays is 1? and 1.

where dm is the average thickness of the wedge in the AC section. From Fig. 6.10 it is clear that the interference pattern is localized at the surface of the wedge. The system of interference fringes arises due to reflection from places of the film that have the same thickness. These strips are called uniform thickness strips. Using (6.21), we can determine the distance?y between two adjacent maxima for the case of monochromatic light, normal incidence of rays and a small angle?:

A special case of strips of equal thickness are Newton's rings, which appear in the air gap between a plano-convex lens of large radius of curvature R and a flat glass plate, which are in contact at point P. When reflected waves are superimposed, interference fringes of equal thickness appear, which have the appearance of concentric rings at normal incidence of light. In the center of the picture there is an interference minimum of zero order. This is due to the fact that at point P the path difference between the coherent rays is determined only by the loss of a half-wave upon reflection from the surface of the plate. The geometric location of points of equal thickness of the air gap between the lens and the plate is a circle, therefore the interference pattern is observed in the form of concentric dark and light rings. In transmitted light, a complementary pattern is observed - central circle light, the next ring dark, etc.
Let's find the radii of the light and dark rings. Let d be the thickness of the air layer at a distance r from point P. Optical path difference? between the beam that bounced off the plate and the beam that was reflected at the interface between the convex surface of the lens and the air. It is obvious that in transmitted light formulas (6.22) and (6.23) change places. Experimental measurements The radii of Newton's rings make it possible to calculate the radius of a planoconvex lens R using these formulas. When studying Newton's rings as a whole, it is impossible to assess the quality of processing of the surfaces of the lens and plate. It should be noted that when observing interference in white light, the interference pattern takes on a rainbow coloring.

6.4.3. The phenomenon of light interference underlies the operation of numerous optical instruments - interferometers, which are used to accurately measure the length of light waves, the linear dimensions of bodies and their changes, and also measure the refractive indices of substances.
In particular, in Fig. Figure 6.12 shows a diagram of a Michelson interferometer. Light from source S falls at an angle of 450 onto the translucent plate P1. Half of the incident light beam is reflected in the direction of beam 1, half passes through the plate in the direction of beam 2. Beam 1 is reflected by mirror M1 and, returning back, passes through plate P1 () again. Beam of light 2 goes to mirror M2, is reflected from it and, having been reflected from plate P1, goes in the direction of beam 2?. Since beam 1 passes through plate P1 three times, and beam 2 only once, plate P2 (the same as P1, but without a translucent coating) is placed on the path of beam 2 to compensate for the path difference.

The interference pattern depends on the position of the mirrors and the geometry of the light beam incident on the device. If the incident beam is parallel, and the planes of the mirrors M1 and M2 are almost perpendicular, then interference fringes of equal thickness are observed in the field of view. The displacement of the picture by one stripe corresponds to the displacement of one of the mirrors by a distance. Thus, the Michelson interferometer is used for precise measurements of length. Absolute error with such measurements is? 10-11 (m). A Michelson interferometer can be used to measure small changes in the refractive indices of transparent bodies depending on pressure, temperature, and impurities.

A. Smakula developed a method for coating optical devices to reduce light loss caused by its reflection from Zalomny surfaces. In complex lenses, the number of reflections is large, so the loss of light flux is quite significant. To make the elements of optical systems coated, their surfaces are covered with transparent films, the refractive index of which is lower than that of glass. When light is reflected at the air-film and film-glass interface, interference of reflected waves occurs. The film thickness d and the refractive indices of glass nc and film n are selected so that the reflected waves cancel each other. To do this, their amplitudes must be equal, and the optical path difference must correspond to the minimum condition.

When a light wave falls on a thin transparent plate (or film), reflection occurs from both surfaces of the plate. As a result, two light waves arise, which under certain conditions can interfere.

Let a plane light wave fall on a transparent plane-parallel plate, which can be considered as a parallel beam of rays (Fig. 122.1). The plate throws upward two parallel beams of light, one of which was formed due to reflection from the upper surface of the plate, the second due to reflection from the lower surface (in Fig. 122.1 each of these beams is represented by only one beam). When entering and exiting the plate, the second beam undergoes refraction. In addition to these two beams, the plate will throw upward beams resulting from three-, five-, etc. multiple reflection from the surfaces of the plate. However, due to their low intensity, we will not take these beams into account. We will also not be interested in the beams passing through the plate.

The difference in path acquired by rays 1 and 2 before they converge at point C is equal to

where is the length of the segment BC, is the total length of the segments AO and OS, is the refractive index of the plate.

The refractive index of the medium surrounding the plate is assumed to be equal to unity. From Fig. 122.1 shows that the thickness of the plate). Substituting these values ​​into expression (122.1) gives that

Having made the replacement and taking into account that

it is easy to bring the formula for to form

When calculating the phase difference between oscillations in beams 1 and 2, it is necessary, in addition to the optical path difference, to take into account the possibility of changing the phase of the wave upon reflection (see § 112). At point C (see Fig. 122.1), reflection occurs from the interface between a medium that is optically less dense and a medium that is optically more dense. Therefore, the phase of the wave undergoes a change by . At point O, reflection occurs from the interface between a medium that is optically denser and a medium that is optically less dense, so that no phase jump occurs. As a result, an additional phase difference arises between beams 1 and 2, equal to It can be taken into account by adding to (or subtracting from it) half the wavelength in vacuum. As a result we get

So, when a plane wave falls on a plate, two reflected waves are formed, the difference in their paths is determined by formula (122.3). Let us find out the conditions under which these waves will be coherent and can interfere. Let's consider two cases.

1. Plane-parallel plate. Both plane reflected waves propagate in the same direction, forming an angle with the normal to the plate, equal to angle falls

These waves will be able to interfere if both temporal and spatial coherence conditions are met.

For temporal coherence to occur, the path difference (122.3) must not exceed the coherence length; equal (see formula (120.9)). Therefore, the condition must be met

In the resulting relationship, half can be neglected compared to The expression has a magnitude of the order of unity. Therefore we can write

(twice the thickness of the plate must be less than the coherence length).

Thus, the reflected waves will be coherent only if the thickness of the plate does not exceed the value determined by relation (122.4). Putting , we get limit value thickness equal

Now consider the conditions for maintaining spatial coherence. Let's place a screen E in the path of the reflected beams (Fig. 122.2). The rays arriving at point P are separated in the incident beam by a distance . If this distance does not exceed the radius of coherence pcoh of the incident wave, rays 1 and 2 will be coherent and will create illumination at point P, determined by the value of the path difference corresponding to the angle of incidence. Other pairs of rays coming at the same angle will create the same illumination at other points of the screen. Thus, the screen will be evenly illuminated (in the particular case when the screen is dark). When the beam tilt changes (i.e., when the angle changes), the screen illumination will change.

From Fig. 122.1 it is clear that the distance between the incident rays 1 and 2 is equal to

If we accept that for it turns out and for

For a normal fall at any .

Coherence radius sunlight has a value of the order of 0.05 mm (see (120.15)). At an angle of incidence of 45°, we can put Therefore, for interference to occur under these conditions, the relation must be satisfied

(122.7)

(cf. (122.5)). For an incidence angle of about 10°, spatial coherence will be maintained for a plate thickness not exceeding 0.5 mm. Thus, we come to the conclusion that, due to the limitations imposed by temporal and spatial coherence, interference when the plate is illuminated by sunlight is observed only if the thickness of the plate does not exceed a few hundredths of a millimeter. When illuminated with light with a greater degree of coherence, interference is also observed upon reflection from thicker plates or films.

In practice, interference from a plane-parallel plate is observed by placing a lens in the path of the reflected beams, which collects the rays at one of the points of the screen located in the focal plane of the lens (Fig. 122.3). The illumination at this point depends on the value of quantity (122.3). At , we get maxima, at , we get minima of intensity ( is an integer). The condition for maximum intensity has the form

Let a thin plane-parallel plate be illuminated with diffused monochromatic light (see Fig. 122.3). Let us place a lens parallel to the plate, in the focal plane of which we place the screen. Scattered light contains rays from a wide variety of directions.

Rays, parallel planes pattern and falling onto the plate at an angle after reflection from both surfaces of the plate will be collected by the lens at point P and create illumination at this point, determined by the value of the optical path difference. Rays traveling in other planes, but incident on the plate at the same angle, will be collected by the lens at other points located at the same distance from the center of the screen O as point P. The illumination at all these points will be the same. Thus, rays incident on the plate at the same angle will create on the screen a collection of equally illuminated points located in a circle with a center at O. Similarly, rays incident at a different angle Ф" will create a collection on the screen in the same way (but differently, since D is different) illuminated points located along a circle of a different radius. As a result, a system of alternating light and dark circular stripes with a common center at point O will appear on the screen. Each strip is formed by rays incident on the plate at the same angle. Therefore, the interference fringes obtained under the described conditions are called equal fringes. tilt. If the lens is positioned differently relative to the plate (the screen in all cases must coincide with the focal plane of the lens), the shape of the stripes of equal tilt will be different.

Each point of the interference pattern is caused by rays that form a parallel beam before passing through the lens. Therefore, when observing stripes of equal inclination, the screen must be located in the focal plane of the lens, i.e., the way it is positioned to obtain images of objects at infinity. In accordance with this, they say that stripes of different inclinations are localized at infinity. The role of the lens can be played by the lens, and the role of the screen can be played by the retina. In this case, in order to observe bands of equal inclination, the eyes must be accommodated in the same way as when viewing very distant objects.

According to formula (122.8), the position of the maxima depends on the wavelength. Therefore, in white light, a set of stripes shifted relative to each other, formed by rays of different colors, is obtained, and the interference pattern acquires a rainbow color. The ability to observe an interference pattern in white light is determined by the ability of the eye to distinguish shades of light of close wavelengths. Rays that differ in wavelength by less than 20 A are perceived by the average eye as having the same color. Therefore, to assess the conditions under which interference from plates can be observed in white light, it should be set equal to 20 A. This is the value we took when estimating the thickness of the plate (see (122.5)).

2. A plate of variable thickness. Let's take a plate in the form of a wedge with an angle at the apex (Fig. 122.4).

Let a parallel beam of rays fall on it. Now the rays reflected from different surfaces of the plate will not be parallel. Two practically merging rays before falling on the plate (in Fig. 122.4 they are depicted as one straight line, designated by the number) intersect after reflection at point Q. Two almost merging rays 1" intersect at point It can be shown that points Q, Q" and other points similar to them lie in the same plane passing through the vertex of the wedge O. Ray V reflected from the lower surface of the wedge and ray 2 reflected from the upper surface will intersect at point R, located closer to the wedge than Q. Similar rays Г and 3 will intersect at point P, located further from the wedge surface than

The directions of propagation of waves reflected from the upper and lower surfaces of the crystal do not coincide. Temporal coherence will be observed only for parts of the waves reflected from places of the wedge for which the thickness satisfies condition (122.4). Let us assume that this condition is satisfied for the entire wedge. In addition, assume that the radius of coherence is much greater than the length of the wedge. Then the reflected waves will be coherent throughout the entire space above the wedge, and at any distance of the screen from the wedge, an interference pattern will be observed in the form of stripes parallel to the top of the wedge O (see the last three paragraphs of § 119). This is the case, in particular, when the wedge is illuminated by light emitted by a laser.

With limited spatial coherence, the localization region of the interference pattern (i.e., the region of space in which the screen can be positioned to observe the interference pattern on it) also turns out to be limited. If the screen is positioned so that it passes through the points (see screen E in Fig. 122.4), an interference pattern will appear on the screen even if the spatial coherence of the incident wave is extremely small (rays that, before falling on the wedge, intersect at points on the screen coincided).

At a small wedge angle, the difference in the path of the rays can be calculated with a sufficient degree of accuracy using formula (122.3), taking as b the thickness of the plate at the point where the rays fall on it. Since the path difference for the rays reflected from different parts of the wedge is now unequal, the illumination of the screen will be uneven - light and dark stripes will appear on the screen (see the dotted curve in Fig. 122.4 showing the illumination of the screen E). Each of these stripes arises as a result of reflection from sections of the wedge with the same thickness, as a result of which they are called stripes of equal thickness.

When the screen is displaced from position E in the direction from the wedge or towards the wedge, the degree of spatial coherence of the incident wave begins to affect. If in the screen position indicated in Fig. 122.4 through E, the distance between incident rays 1 and 2 will become of the order of the coherence radius, the interference pattern on the screen E will not be observed. Likewise, the picture disappears at the screen position indicated by

Thus, the interference pattern resulting from the reflection of a plane wave from a wedge turns out to be localized in a certain region near the surface of the wedge, and this region is narrower, the lower the degree of spatial coherence of the incident wave. From Fig. 122.4 it is clear that as we approach the top of the wedge, they become more favorable conditions both temporal and spatial coherence. Therefore, the clarity of the interference pattern decreases as you move from the top of the wedge to its base. It may happen that the pattern is observed only for the thinner part of the wedge. For the rest of the screen, uniform illumination occurs.

Almost stripes of equal thickness are observed by placing a lens near the wedge and a screen behind it (Fig. 122.5). The role of the lens can be played by the lens, and the role of the screen can be played by the retina. If the screen behind the lens is located in a plane conjugate to the plane indicated in Fig. 122.4 through E (accordingly, the eye is accommodated to this plane), the picture will be the clearest. When you move the screen on which the image is projected (or when you move the lens), the picture will deteriorate and disappear completely when the plane adjacent to the screen goes beyond the localization region of the interference pattern observed without the lens.

When observed in white light, the fringes will be colored, so that the surface of the plate or film appears to have a rainbow coloration. For example, thin films of oil or oil spread on the surface of water, as well as soap films, have this color. The tarnished colors that appear on the surface of steel products during quenching are also caused by interference from a film of transparent oxides.

Let us compare the two cases of interference upon reflection from thin films that we have considered. Stripes of equal inclination are obtained by illuminating a plate of constant thickness with scattered light, which contains rays of different directions, varying within more or less wide limits). Bands of equal inclination are localized at infinity. Stripes of equal thickness are observed when a plate of variable thickness is illuminated with a parallel beam of light). Bands of equal thickness are localized near the plate. In real conditions, for example, when observing rainbow colors on a soap or oil film, both the angle of incidence of the rays and the thickness of the film change. In this case, bands of a mixed type are observed.

Note that interference from thin films can be observed not only in reflected, but also in transmitted light.

Newton's rings. A classic example of strips of equal thickness are Newton's rings. They are observed when light is reflected from a plane-parallel thick glass plate and a plano-convex lens with a large radius of curvature in contact with each other (Fig. 122.6). The role of a thin film, from the surfaces of which coherent waves are reflected, is played by the air gap between the plate and the lens (due to the large thickness of the plate and lens, interference fringes do not arise due to reflections from other surfaces). With normal incidence of light, stripes of equal thickness look like concentric circles, and with oblique incidence - ellipses. Let's find the radii of Newton's rings obtained when light is incident normally to the plate. In this case, the optical path difference is equal to twice the gap thickness (see formula (122.2); it is assumed that in the gap ). From Fig. 122.6 it follows that are the radii of the dark rings. The value corresponds to the point at the point where the plate and lens touch. At this point, a minimum intensity is observed, due to a change in phase when the light wave is reflected from the plate.

Enlightening optics. Interference upon reflection from thin films underlies the antireflection of optics. The passage of light through each refractive surface of the lens is accompanied by reflection of approximately 4% of the incident light. In complex lenses, such reflections occur many times and the total loss of light flux reaches a noticeable value. In addition, reflections from lens surfaces result in glare. In coated optics, to eliminate light reflection, a thin film of a substance with a refractive index different from that of the lens is applied to each free surface of the lens. The thickness of the film is selected so that the waves reflected from both its surfaces cancel each other. A particularly good result is achieved if the refractive index of the film is equal to the square root of the refractive index of the lens. Under this condition, the intensity of both waves reflected from the film surfaces is the same.

Interference in a thin film. Alpha is the angle of incidence, beta is the angle of reflection, the yellow beam lags behind the orange one, they are brought together by the eye and interfere.

Obtaining a stable interference pattern for light from two spatially separated and independent light sources is not as easy as for water wave sources. The atoms emit light in trains of very short duration, and coherence is broken. Such a picture can be obtained relatively simply by making sure that waves of the same train interfere. Thus, interference occurs when an initial beam of light is split into two beams as it passes through a thin film, such as the film applied to the surface of the lenses of coated lenses. A ray of light passing through a film of thickness will be reflected twice - from its inner and outer surfaces. The reflected rays will have a constant phase difference equal to twice the thickness of the film, causing the rays to become coherent and interfere. Complete quenching of the rays will occur at , where is the wavelength. If nm, then the film thickness is 550:4 = 137.5 nm.

The rays of neighboring parts of the spectrum on both sides of the nm do not interfere completely and are only attenuated, causing the film to acquire color. In the approximation of geometric optics, when it makes sense to talk about the optical difference in the path of the rays, for two rays

Maximum condition;

Minimum condition

where k=0,1,2... and is the optical path length of the first and second beams, respectively.

The phenomenon of interference is observed in a thin layer of immiscible liquids (kerosene or oil on the surface of water), in soap bubbles, gasoline, on the wings of butterflies, in tarnished flowers, etc.

As an option:

In nature, one can often observe the rainbow coloring of thin films (oil films on water, soap bubbles, etc.) arising from the interference of light reflected by two film surfaces. Let a plane monochromatic wave fall on a plane-parallel transparent film with refractive index n and thickness d at an angle i (for simplicity, consider one ray).

On the surface of the film at point O the beam

will be divided into two: partially reflected from the upper surface of the film, and partially refracted. The refracted ray, having reached point C, will be partially refracted into the air (n 0 = 1), and partially reflected and go to point B. Here it will again be partially reflected (we will not consider this path of the ray in the future due to low intensity) and refracted, emerging into the air at an angle i. Rays 1 and 2 emerging from the film are coherent if the optical difference in their path is small compared to the coherence length of the incident wave. If a collecting lens is placed on them, they will converge at one of the points P of the focal plane of the lens and give an interference pattern, which is determined by the optical path difference between the interfering rays. Optical path difference arising between two interfering rays from point O to plane AB: where is the refractive index environment taken equal to 1, and is due to the loss of a half-wave upon reflection of light from the interface. If n>n 0 (n

Newton's rings.

Newton's rings. They are a classic example of stripes of equal thickness, observed when light is reflected from an air gap formed by a plane-parallel plate and a plane-convex lens with a large radius of curvature in contact with it.

A parallel beam of light is incident normally on the flat surface of the lens and is partially reflected from the upper and lower surfaces of the air gap between the lens and the plate. When reflected rays overlap, stripes of equal thickness appear, which, under normal light incidence, have the form of concentric circles. In reflected light, the optical path difference (taking into account the loss of half upon reflection), provided that n=1, and I=0, where d is the gap width. r is the radius of curvature of the circle, all points of which correspond to the same gap d. Considering d=r 2 /2R. Hence, .

Equating to the conditions of maximum and minimum, we obtain expressions for the radius of the mth light and dark rings: By measuring the radii of the corresponding rings, we can (knowing the radius of curvature of the lens) determine and vice versa, find the radius of curvature of the lens.

For both bands of equal inclination and bands of equal thickness, the position of the maxima depends on the wavelength. Therefore, a system of light and dark stripes is obtained only when illuminated with monochromatic light. When observed in white light, a set of stripes shifted relative to each other is obtained, formed by rays of different wavelengths, and the interference pattern acquires a rainbow color. All arguments were given for reflected light. Interference can also be observed in transmitted light, and in this case there is no loss of a half-wave. Consequently, the optical path difference for transmitted and reflected light differs by /2. those. Interference maxima in reflected light correspond to minima in transmitted light, and vice versa.

As an option:

Another method of obtaining a stable interference pattern for light is the use of air gaps, based on the same difference in the path of two parts of the wave: one immediately reflected from the inner surface of the lens and the other passing through the air gap under it and only then being reflected. It can be obtained by placing a plano-convex lens on a glass plate with the convex side down. When the lens is illuminated from above with monochromatic light, a dark spot is formed at the point of fairly close contact between the lens and the plate, surrounded by alternating dark and light concentric rings of varying intensities. Dark rings correspond to interference minima, and light ones correspond to maxima; at the same time, dark and light rings are isolines of equal thickness of the air gap. By measuring the radius of a light or dark ring and determining its serial number from the center, you can determine the wavelength of monochromatic light. The steeper the surface of the lens, especially closer to the edges, the smaller the distance between adjacent light or dark rings

The rainbow coloring of soap bubbles or gasoline films on water occurs as a result of the interference of sunlight reflected by the two surfaces of the film.

Let a plane-parallel transparent film with a refractive index P and thickness d a plane monochromatic wave with a length of (Fig. 4.8).

Rice. 4.8. Interference of light in thin film

The interference pattern in reflected light occurs due to the superposition of two waves reflected from the top and bottom surfaces of the film. Let us consider the addition of waves emanating from the point WITH. A plane wave can be thought of as a beam of parallel rays. One of the beam rays (2) directly hits the point WITH and is reflected (2") upward in it at an angle equal to the angle of incidence. Another ray (1) hits the point WITH in a more complicated way: first it is refracted at a point A and spreads in the film, then is reflected from its lower surface at point 0 and, finally, comes out, refracted, outward (1") at point WITH at an angle equal to the angle of incidence. Thus, at the point WITH the film casts upward two parallel rays, one of which was formed due to reflection from the lower surface of the film, the second - due to reflection from the upper surface of the film. (Beams resulting from multiple reflections from film surfaces are not considered due to their low intensity.)

Optical path difference acquired by rays 1 and 2 before they converge at a point WITH, is equal

Assuming the refractive index of air and taking into account the relationships

We use the law of light refraction

Thus,

In addition to the optical path difference , the change in wave phase upon reflection should be taken into account. At the point WITH at the air interface – film" is reflected from optically denser medium, that is, media with a high refractive index. At not too large angles of incidence, in this case the phase undergoes a change by . (The same phase jump occurs when a wave traveling along a string is reflected from its fixed end.) At the point 0 At the film-air interface, light is reflected from an optically less dense medium, so that no phase jump occurs.

As a result, an additional phase difference arises between beams 1" and 2", which can be taken into account if the value decrease or increase by half the wavelength in vacuum.

Therefore, when the relation

it turns out maximum interference in reflected light, and in the case

observed in reflected light minimum.

Thus, when light falls on a gasoline film on water, depending on the viewing angle and film thickness, a rainbow coloring of the film is observed, indicating an increase in light waves with certain wavelengths l. Interference in thin films can be observed not only in reflected, but also in transmitted light.

As already noted, for the observed interference pattern to occur, the optical path difference of the interfering waves should not exceed the coherence length, which imposes a limit on the thickness of the film.

Example. On soap film ( n = 1.3), located in the air, a beam of white light falls normally. Let us determine at what minimum thickness d film reflected light with wavelength µm will be maximally amplified as a result of interference.

From the condition of the interference maximum (4.28) we find the expression for the film thickness

(angle of incidence ). Minimum value d it turns out when: