MICHAELSON INTERFEROMETER consists of two mirrors M 1 and M 2 and a semi-permeable reflecting partition S, inclined at an angle of 45° (Fig. 1). This partition transmits 50% of the light incident on it and reflects the remaining 50%. Distances to mirrors L 1 and L 2 are the same: L 1 = L 2 = L. Monochromatic light from the source passes halfway through the partition S, is reflected from M 1 and then hits the detector, half reflected from S (ray 1). This is the path light travels in the direction of the Earth's speed as it moves along its orbit and in reverse side, which corresponds to the swimmer’s movement with and against the current. The other part of the light beam is reflected by the partition S to the mirror M 2, and on way back passes through the partition, hitting the detector (beam 2). This corresponds to the swimmer moving across the current.

If the interferometer is at rest relative to the ether, then the time spent by the first and second rays of light on their path is the same, and two coherent rays enter the detector in the same phase ( cm. COHERENCE). Consequently, interference occurs and a central bright spot can be observed in the interference pattern ( cm. OSCILLATIONS AND WAVES; OPTICS). If the interferometer moves relative to the ether, then the time spent by the rays on their path turns out to be different. Indeed, let c the speed of light relative to the ether, and v speed of the interferometer relative to the ether. Then the time spent on the first path (downstream and back) is equal to

To calculate time t 2 it should be taken into account that while the light travels from the semi-permeable partition to the mirror M 2, the mirror itself moves with the Earth relative to the ether. Therefore, the path traveled by light to the mirror M 2 is equal to the hypotenuse of the triangle. The speed of light does not change, since light moves perpendicular to the direction of the Earth's speed. From simple geometric considerations

Using approximate formulas:

This time delay corresponds to the difference in paths of two rays of light

Consequently, such a difference in the path of the rays will correspond to the total number of wavelengths of light that fit within this path difference, equal to

Interference maxima and minima alternate when the path difference changes by p/2. Thus, having calculated the value n for specific installation parameters and knowing the speed of the Earth, you can find out how the interference fringes should move. Of course, the effect is very small. To strengthen it, Michelson maximized the base of the interferometer L, causing the light to be reflected repeatedly from additional mirrors. In addition, the experiment was carried out a second time with the device rotated 90°, due to which the rays swapped places and the effect of shifting the interference fringes was doubled.

For monochromatic light corresponding to the sodium line with wavelength l = 590 nm, and at L= 11 m, v/c= 10 8, it turns out that the total shift is approximately 0.37 stripes. However, Michelson and Morley claimed, based on preliminary tests of the instrument, that they were able to clearly detect a shift of 0.01 fringes.

Alexander Berkov

IN Michelson interferometer The phenomenon of interference in thin films is used. The phenomenon of interference in this device is carried out by dividing the amplitude of the wave.

What is this device? On a massive pedestal there is a plane-parallel plate ($A$) lightly coated with silver, located at an angle of $45^0$ to the direction of propagation of the rays and two mutually perpendicular flat mirrors $C$ and $D$ (Fig. 1).

Figure 1.

Plate B (Fig. 1) serves as an auxiliary plate; it compensates for the difference in the path of the rays. Light waves extend from ($S$). Some of them are reflected from the silver surface of plate $A$, and some pass through this plate. This is how the process of splitting a wave of light into two coherent waves occurs. The waves that pass through the plate are reflected from the mirrors $C$ and $D$. The reflected waves are again partially reflected and partially transmitted through the silver plate $A$. These waves can interfere in the $AK$ region. This interference pattern is observed through a telescope. Thus, on the plate $A$ the amplitude is divided; the wave front on it is preserved; only the direction of its movement changes.

If, hypothetically, the arm $DA$ is rotated by $90^0$, then the mirror $D$ will end up in position $D"$. A gap appears between $D"$ and $C$, which may be similar to a thin film. If the mirrors $C$ and $D$ are strictly perpendicular, then stripes of equal inclination are observed, which look like circles. The telescope in this case should be adjusted to infinity. If the mirrors $C$ and $D$ are not completely perpendicular, then the gap between us becomes like a wedge, then stripes of equal thickness appear in the form of straight stripes. In this case, the telescope is focused on the silvered edge of the plate $A$.

Interference of monochromatic waves that propagate along the axis of the interferometer

In the case of wave propagation strictly along the axis of the interferometer, the optical difference in the path of the rays ($\triangle $) appears due to the difference in the lengths of the arms ($l_1\ and\l_2\\$) of the interferometer:

The resulting phase difference is:

In a rigorous calculation, one should take into account the change in wave phases upon reflection from mirrors and refraction in the plate $A$; here we will not do this, since this is not of fundamental importance for the interference pattern in our case.

where $E_0$ is the amplitude of the wave before it hits the plate $A$. $\delta =(\varphi )_2-(\varphi )_1$. Therefore, for the intensity observed as a result we obtain:

where $I_0=\frac(1)(2)(E_0)^2$ is the intensity of the wave incoming from the light source.

In the event that:

intensity (3) is zero. If:

the intensity is equal to $I_0$, which means: all the energy from the source falls on the “screen”, there is no flow of energy that returns in the direction of the light source.

Comment

The Michelson interferometer is used to measure small distances and small changes in refractive indices. Michelson himself used his interferometer for an experiment to test the connection between the speed of light and the direction of motion of the beam in relation to the Earth.

Example 1

Exercise: In order to calculate the refractive index of ammonia, a glass tube with a vacuum inside is placed in one arm of the Michelson interferometer. Its length is $l=15\cm=15\cdot 10^(-2)m$. If this tube is filled with ammonia, the interference pattern for a wavelength equal to $\lambda =589\nm=589\cdot (10)^(-9)m$ shifts by $192$ band. What is the refractive index of ammonia?

Solution:

The difference in the optical path of the wave ($\triangle $) in vacuum and ammonia can be found as:

\[\triangle =ln-ln_v\left(1.1\right),\]

where $n_v$=1 refractive index for vacuum. Let us write down the condition for interference minima:

\[\triangle =m\frac(\lambda )(2)\ \left(m=0,\pm 1,\pm 2,\dots \right)\left(1.2\right).\]

Let's equate the right sides of expressions (1.1) and (1.2), we get:

Let us express the refractive index from (1.3):

Let's carry out the calculations:

Answer:$n=1.000377.$

Example 2

Exercise: In a Michelson interferometer, when one of the mirrors moves forward, the interference pattern either disappears or appears. What is the displacement ($\triangle l$) of the mirror between two successive occurrences of a clear interference pattern if the waves $(\lambda )_1$ and $(\lambda )_2$ are used?

Solution:

The reason for the disappearance of the interference pattern can be considered to be that the maxima and minima of the interference pattern of waves of different lengths are shifted relative to each other. With a sufficient difference in wavelength, the maxima in the interference of one wave can fall on the minima of another, then the interference pattern completely disappears.

Let us write down the condition for the transition from one clear picture to another:

\[\left(z+1\right)(\lambda )_1=z(\lambda )_2\left(2.1\right),\]

where $z$ is an integer. The required mirror displacement ($\triangle l$) can be defined as:

Using the system of equations (2.1) and (2.2) we express $\triangle l$:

\[\left(z(\lambda )_1+(\lambda )_1\right)=z(\lambda )_2\to z((\lambda )_2-(\lambda )_1)=(\lambda )_1\to z=\frac((\lambda )_1)(((\lambda )_2-(\lambda )_1)),\] \[\triangle l=\frac((\lambda )_1(\lambda )_2)( 2((\lambda )_2-(\lambda )_1)).\]

Answer:$\triangle l=\frac((\lambda )_1(\lambda )_2)(2((\lambda )_2-(\lambda )_1)).$

> Michelson interferometer

Consider the principle of operation Michelson interferometer. Learn what the interference pattern looks like in a Michelson interferometer, circuit design and application.

The Michelson interferometer is the most common configuration in the field of optical interferometry.

Learning Objective

• Understand the principle of operation of the Michelson interferometer.

Main points

• Interferometry uses superimposed waves to obtain information about them.
• A particular drive splits a beam of light into two paths, bouncing back and recombining them to form an interference pattern.
• The most famous application is the Michelson-Morley experiment, where the null result inspired the special theory of relativity.

Terms

• Special relativity: The speed of light remains stable in all frames of reference.
• Superimposed – placed on top of something else.
• Interference is an effect created by superposition due to distortion due to atmospheric or other influences.

Interferometry

Simply put, interferometry is the use of interference in superimposed waves to measure their characteristics. The interferometry method is used in many scientific fields, such as astronomy, engineering, physics, fiber optics and oceanography.

IN industrially It is used to measure small rooms, refractive index and surface irregularities. When two waves of the same frequency are combined, the resulting pattern is based on the difference in their phases. Constructive interference is formed if the waves are in phase, while destructive interference is not. This principle is used in interferometry to obtain information about the initial state of waves.

Michelson interferometer

The Michelson interferometer is the most widely used interferometer, created by A. A. Michelson. The operating principle is to split the light beam into two paths. After this, it recombines them and forms an interference pattern. To create stripes on the detector, the paths must have different lengths and compositions.

Colored and monochromatic stripes: (a) – white stripes, where two beams differ in the number of phase inversions; (b) – white stripes, where two beams are characterized by the same number of phase inversions; (c) – pattern of stripes with monochromatic light

The figure below shows how the device works. M 1 and M 2 are two highly polished mirrors, S is the light source, M is a half-silver mirror functioning as a beam splitter, and C is a point on M that is partially reflective. When ray S hits a point on M, it splits into two beams. One ray is reflected towards A, and the second is transmitted through the surface M to point B. A and B are points on highly polished mirrors M 1 and M 2. When the rays hit these points, they are reflected back to point C, where they recombine to create an interference pattern. At point E it comes into view to the observer.

Michelson interferometer diagram showing the path of light waves

Applications

The Michelson interferometer is used to search for gravitational waves. He also played main role in the study of the upper atmospheric layer, determining temperatures and winds through measuring Doppler width and shifts in the spectra of luminescence and aurora.

But still, many people remember the most famous application - the Michelson-Morley experiment. This was an unsuccessful attempt to demonstrate the influence of a hypothetical ethereal wind on the speed of ordinary wind. This inspired the creation special theory relativity.

Michelson interferometer

Animation

Description

The Michelson interferometer is one of the most common skeletal interferometer designs, designed for various applications in cases where spatial alignment of objects generating interfering waves is impossible or for some reason undesirable.

A schematic representation of the Michelson interferometer design is shown in Fig. 1.

Schematic illustration of the Michelson interferometer design

Rice. 1

A beam of light from an almost point source S, located at the focus of the lens, is converted by this lens into a parallel beam (often in modern applications this beam is simply laser radiation, not collimated by an additional lens). Next, this beam is divided into two by a translucent flat mirror SM, each of which is reflected back by mirrors M 1.2, respectively. These two reflected beams form an interference pattern on the SC screen, the nature of which is determined by the ratio of the shapes of the wave fronts of both beams (see Fig. 2).

Wavefronts of beams forming an interference pattern

Rice. 2

Namely, these two beams at the point where the screen is located can have different radii of curvature of the wave fronts R 1,2, as well as the mutual inclination of the latter a. In particular, it is easy to understand that both indicated radii will be the same, and a =0, ​​if and only if the mirrors M 1,2 are both flat (or generally the same shape), and the position of the mirror M 1 in space coincides with mirror image M 2 in the divisor SM, that is, M 2 "(see Fig. 1).

In this case, the illumination on the screen will be uniform, which means ideal alignment of the interferometer.

In the case of a№ 0, R 1 = R 2 (the distances from the divider to the mirrors are adjusted correctly, but the angles of inclination are not), a picture of equidistant direct interference fringes will appear on the screen, as in the interference of waves reflected from two faces of a thin wedge.

In the case of a =0, ​​R 1 No. R 2 (correct angular adjustment, but incorrect distances of the mirrors to the divider), the interference pattern is concentric rings caused by the intersection of two spherical wave fronts of different curvature.

Finally, in the case of a =0, ​​R 1 =R 2, but the non-ideal flatness of one of the mirrors, the picture will be irregular shape“Newton's rings” around the irregularities of the corresponding mirror surface.

All of these changes in the observed pattern occur with very small (tenths of a wavelength in spatial positioning and height of mirror irregularities, and tens of microradians in angular adjustment) deviations of the adjustment parameters from the ideal. If we take this into account, it becomes clear that the Michelson interferometer is a very precise device for monitoring the positioning of an object in space, its angular adjustment and flatness. Special methods Precise measurement of the intensity distribution in the screen plane makes it possible to increase the positioning accuracy to several nanometers.

Timing characteristics

Initiation time (log to -8 to -5);

Lifetime (log tc from -5 to 15);

Degradation time (log td from -8 to -5);

Time of optimal development (log tk from -5 to -4).

Diagram:

Technical implementations of the effect

Technical implementation of the effect

Technical implementation is carried out in full accordance with Fig. 1 content part. The laser beam of a helium-neon laser (for clarity, it is better to expand it with a telescope to a diameter of 10-15 millimeters) is divided into two by a translucent mirror, reflected from two flat mirrors, and a certain interference pattern is obtained on the screen. Then, by carefully adjusting the lengths of the arms and the angular position of the mirrors, the interference pattern in the area of ​​beam overlap on the screen disappears.

Applying an effect

The applications of the Michelson interferometer in technology are very diverse. For example, it can be used for remote monitoring of small deformations (deviations from flatness) of an object (replacing one of the mirrors in Fig. 1). This approach is very convenient when, for one reason or another, close proximity of the object and the reference surface (the second mirror in Fig. 1) is undesirable. For example, the object is very hot, chemically aggressive, and the like.

But the most significant technical application of the Michelson interferometer is the use of this circuit in optical gyroscopes based on the Sagnac effect to control the shift of the interference fringe generated by rotation.

Literature

1. Physics. Big encyclopedic dictionary. - M.: Big Russian Encyclopedia, 1999.

2. Sivukhin D.V. General course physics. Optics. - M.: Nauka, 1985.

3. Landsberg G.S. Optics. - M.: Nauka, 1976.

Keywords

• interference
• monochromatic
• ray path difference
• refractive index
• zero interference band

Sections of natural sciences:

There are many types of interference devices called interferometers. In Fig. Figure 123.1 shows a diagram of the Michelson interferometer. A beam of light from source 5 falls on a translucent plate coated with a thin layer of silver (this layer is shown in the figure by dots). Half fallen luminous flux is reflected by the plate in the direction of beam 1, half passes through the plate and propagates in the direction of beam 2. Beam 1 is reflected from the mirror and returns to where it is divided into two beams of equal intensity. One of them passes through the plate and forms beam 1, the second is reflected in the direction of S; this bundle will no longer interest us. Beam 2, reflected from the mirror, also returns to the plate where it is divided into two parts: beam 2 reflected from the translucent layer and beam passed through the layer, which we will also no longer be interested in. Light beams 1 and 2 have the same intensity.

If the conditions of temporal and spatial coherence are met, beams 1 and 2 will interfere. The result of interference depends on the optical path difference from the plate to the mirrors and back. Beam 2 passes through the thickness of the plate three times, beam 1 only once. In order to compensate for the different optical path differences that arise due to this (due to dispersion) for different wavelengths, a plate exactly like but not a silver-plated plate is placed on the path of beam 1. This equalizes the paths of beams and 2 in the glass. The interference pattern is observed using a telescope T.

Let us mentally replace the mirror with its virtual image in a translucent plate. Then rays 1 and 2 can be considered as arising due to reflection from a transparent plate limited by planes. Using adjustment screws, you can change the angle between these planes; in particular, they can be installed strictly parallel to each other. By rotating the micrometer screw, you can smoothly move the mirror without changing its tilt.

Thus, you can change the thickness of the “plate”; in particular, you can make the planes intersect each other (Fig. 123.1,6).

The nature of the interference pattern depends on the alignment of the mirrors and on the divergence of the light beam incident on the device. If the beam is parallel and the planes form an angle that is not equal to zero, then in the field of view of the pipe rectilinear stripes of equal thickness are observed, located parallel to the line of intersection of the planes. In white light, all stripes, except the zero-order stripe coinciding with the line of intersection, will be colored. The zero band turns out to be black, since the beam is reflected from the plate from the outside, and beam 2 from the inside, as a result of which a phase difference arises between them, equal to white light. The bands are observed only when the thickness of the “plate” is small (see (122.5)). In monochromatic light corresponding to the red line of cadmium, Michelson observed a clear interference pattern with a path difference of the order of 500,000 wavelengths (the distance between them is approximately 150 mm in this case).

With a slightly diverging beam of light and a strictly parallel arrangement of the planes and Mb. stripes of equal inclination are obtained, having the form of concentric rings. As the micrometer screw rotates, the rings increase or decrease in diameter. In this case, either new rings appear in the center of the picture, or decreasing rings contract to a point and then disappear. Shifting the pattern by one stripe corresponds to moving the mirror to the floorboard of the wavelength.

Using the device described above, Michelson carried out several experiments that went down in the history of physics. The most famous of them, carried out jointly with Morley in 1887, had the goal of detecting the motion of the Earth relative to a hypothetical ether (we will talk about this experiment in § 150). In 1890-1895 Using the interferometer he invented, Michelson made the first comparison of the wavelength of the red line of cadmium with the length of a normal meter.

In 1920, Michelson built a stellar interferometer, with which he measured the angular sizes of some stars. This device was mounted on a telescope. A screen with two slits was installed in front of the telescope lens (Fig. 123.2).

The light from the star was reflected from a symmetrical system of mirrors mounted on a rigid frame mounted on a cart. The internal mirrors were stationary, but the external ones could move symmetrically, moving away from the mirrors or approaching them. The path of the rays is clear from the figure. Interference fringes appeared in the focal plane of the telescope lens, the visibility of which depended on the distance between the external mirrors. By moving these mirrors, Michelson determined the distance between them at which the visibility of the stripes became zero. This distance must be on the order of the coherence radius of the light wave coming from the star. According to (120.14), the coherence radius is equal. From the condition, the angular diameter of the star is obtained