Diffraction grating in life. Period of the diffraction grating. Lattice period and principle of its operation

Continuing the reasoning for five, six slits, etc., we can establish next rule: in the presence of gaps between two adjacent maxima, minima are formed; the difference in the path of rays from two adjacent slits for maxima should be equal to the integer X, and for minima - The diffraction spectrum from the slits has the form shown in Fig. Additional maxima located between two adjacent minima create very low illumination (background) on the screen.

The main part of the energy of the light wave passing through the diffraction grating is redistributed between the main maxima formed in the directions where 3 is called the “order” of the maximum.

Obviously than larger number slits, the more light energy will pass through the grating, the more minima are formed between adjacent main maxima, and, therefore, the more intense and sharper the maxima will be.

If the light incident on a diffraction grating consists of two monochromatic radiations with wavelengths and their main maxima will be located in different places on the screen. For wavelengths very close to each other (single-color radiation), the maxima on the screen can turn out to be so close to each other that they merge into one common light strip (Fig. IV.27, b). If the top of one maximum coincides with or is located further than (a) the nearest minimum of the second wave, then by the distribution of illumination on the screen one can confidently establish the presence of two waves (or, as they say, “resolve” these waves).

Let us derive the condition for the solvability of two waves: the maximum (i.e., the maximum of order) of the wave will be obtained, according to formula (1.21), at an angle satisfying the condition. The limiting condition of solvability requires that at the same angle it will turn out

the minimum of the wave closest to its maximum (Fig. IV.27, c). According to what was said above, to obtain the nearest minimum, an additional addition should be made to the path difference. Thus, the condition for the coincidence of the angles at which the maximum and minimum are obtained leads to the relation

If greater than the product of the number of slits and the order of the spectrum, then the maxima will not be resolved. Obviously, if two maxima are not resolved in the order spectrum, then they can be resolved in the spectrum of higher orders. According to expression (1.22), the greater the number of beams interfering with each other and the greater the path difference A between them, the closer the waves can be resolved.

In a diffraction grating, that is, the number of slits is large, but the order of the spectrum that can be used for measurement purposes is small; in the Michelson interferometer, on the contrary, the number of interfering beams is equal to two, but the path difference between them, depending on the distances to the mirrors (see Fig. IV. 14), is large, therefore the order of the observed spectrum is measured in very large numbers.

The angular distance between two adjacent maxima of two close waves depends on the order of the spectrum and the grating period

The grating period can be replaced by the number of slits per unit grating length:

It was assumed above that the rays incident on the diffraction grating are perpendicular to its plane. With an oblique incidence of rays (see Fig. IV.22, b), the zero maximum will be shifted and will appear in the direction. Let us assume that the maximum of order is obtained in the direction, i.e., the difference in the path of the rays is equal to Then Since at small angles

Close to each other in size, therefore,

where is the angular deviation of the maximum from zero. Let's compare this formula with expression (1.21), which we write in the form since then the angular deviation for oblique incidence turns out to be greater than for perpendicular incidence of rays. This corresponds to a decrease in the grating period by a factor. Consequently, at large angles of incidence a, it is possible to obtain diffraction spectra from short-wave (for example, X-ray) radiation and measure their wavelengths.

If flat light wave passes not through slits, but through round holes of small diameter (Fig. IV.28), then the diffraction spectrum (on a flat screen located in the focal plane of the lens) is a system of alternating dark and light rings. The first dark ring is obtained at an angle satisfying the condition

The second dark ring The central light circle, called the Airy spot, accounts for about 85% of the total radiation power passing through the hole and lens; the remaining 15% is distributed among the light rings surrounding this spot. The size of the Airy spot depends on the focal length of the lens.

The diffraction gratings discussed above consisted of alternating “slits” that completely transmit the light wave, and “opaque stripes” that completely absorb or reflect the radiation incident on them. We can say that in such gratings the transmittance of a light wave has only two values: along the slit it is equal to unity, and along the opaque strip it is zero. Therefore, at the boundary between the slot and the strip, the transmittance changes abruptly from unity to zero.

However, it is possible to produce diffraction gratings with a different transmittance distribution. For example, if an absorbing layer with periodically varying thickness is applied to a transparent plate (or film), then instead of alternating completely

Using transparent slits and completely opaque strips, you can obtain a diffraction grating with a smooth change in transmittance (in the direction perpendicular to the slits or strips). Of particular interest are gratings in which the transmittance varies sinusoidally. The diffraction spectrum of such gratings does not consist of many maxima (as shown for conventional gratings in Fig. IV.26), but only of a central maximum and two symmetrically located first-order maxima

For a spherical wave, diffraction gratings can be made consisting of many concentric annular slits separated by opaque rings. You can, for example, apply concentric rings with ink to a glass plate (or transparent film); at the same time central circle, covering the center of these rings, can be either transparent or shaded. Such diffraction gratings are called "zone plates" or gratings. For diffraction gratings consisting of straight slits and strips, in order to obtain a clear interference pattern, it was necessary to maintain constant slit width and grating period; For zone plates, the required radii and thickness of the rings must be calculated for this purpose. Zone gratings can also be manufactured with a smooth, for example sinusoidal, change in transmittance along the radius.

DEFINITION

Diffraction grating- this is the simplest spectral device, consisting of a system of slits (areas transparent to light) and opaque gaps that are comparable to the wavelength.

A one-dimensional diffraction grating consists of parallel slits of the same width, which lie in the same plane, separated by equal-width gaps that are opaque to light. Reflective diffraction gratings are considered the best. They consist of a set of areas that reflect light and areas that scatter light. These gratings are polished metal plates on which light-scattering strokes are applied with a cutter.

The diffraction pattern on a grating is the result of mutual interference of waves coming from all slits. Using a diffraction grating, multi-beam interference of coherent beams of light that have undergone diffraction and coming from all slits is realized.

A characteristic of a diffraction grating is its period. The period of the diffraction grating (d) (its constant) is a value equal to:

where a is the slot width; b is the width of the opaque area.

Diffraction by a one-dimensional diffraction grating

Let us assume that a light wave with a length of . Since the slits of the grating are located at equal distances from each other, the differences in the path of the rays () coming from two adjacent slits for direction will be the same for the entire diffraction grating under consideration:

The main intensity minima are observed in the directions determined by the condition:

In addition to the main minima, as a result of mutual interference of light rays that come from two slits, the rays cancel each other out in some directions. As a result, additional intensity minima arise. They appear in those directions where the difference in the path of the rays is odd number half-wave The condition for additional minima is the formula:

where N is the number of slits of the diffraction grating; — integer values other than 0. If the grating has N slits, then between the two main maxima there is an additional minimum that separates the secondary maxima.

The condition for the main maxima for a diffraction grating is:

The value of the sine cannot be greater than one, then the number of main maxima is:

Examples of solving problems on the topic “Diffraction grating”

EXAMPLE 1

Exercise

A monochromatic beam of light with wavelength θ is incident on a diffraction grating, perpendicular to its surface. The diffraction pattern is projected onto a flat screen using a lens. The distance between two first-order intensity maxima is l. What is the diffraction grating constant if the lens is placed in close proximity to the grating and the distance from it to the screen is L. Consider that

Solution

As a basis for solving the problem, we use a formula that relates the constant of the diffraction grating, the wavelength of light and the angle of deflection of the rays, which corresponds diffraction maximum number m:

According to the conditions of the problem, since the angle of deflection of the rays can be considered small (), we assume that:

From Fig. 1 it follows that:

Let's substitute expression (1.3) into formula (1.1) and take into account that , we get:

From (1.4) we express the lattice period:

Answer

EXAMPLE 2

Exercise	Using the conditions of Example 1 and the result of the solution, find the number of maxima that the lattice in question will give.
Solution	In order to determine the maximum angle of deflection of light rays in our problem, we will find the number of maxima that our diffraction grating can give. To do this we use the formula: where we assume that for . Then we get:

When analyzing the action of zone plates, we found that periodic structures work most effectively in diffraction. And this is not surprising. After all, diffraction is a wave effect, and waves themselves are a periodic structure. Therefore, it can be expected that a set of equally spaced slits should in some cases provide a more effective and useful practical applications diffraction pattern.

In this regard, let's consider a precision optical device - a diffraction grating. The simplest diffraction grating called the totality large quantity narrow, parallel, identical, equally spaced slits. This grating operates in transmitted light. Sometimes a diffraction grating is used in reflected light, which is made by applying a large number of narrow, parallel, identical, equally spaced obstacles to the mirror. Often a lattice is made by applying opaque strokes to clear glass or mirror. Therefore, it is characterized not by the number of slits, but by the number of strokes separating the slits. He made the first working diffraction grating in the 17th century. Scottish scientist James Gregory, who used bird feathers for this. In modern gratings, the number of lines reaches a million on a surface of up to several tens of centimeters.

The description of diffraction on a diffraction grating is similar to the description of diffraction in parallel rays on a slit (Fig. 27.4). The sum of the slot width A and the space between the slots (stroke) b called period of the lattice."

Let a beam of parallel rays fall on the grating perpendicular to its plane, which then Rice. 27.4 in accordance with the Huygens-Fresnel principle gives secondary interfering waves. Let us choose a certain direction of passage of these secondary waves, determined by the angle a. If the difference in the wave paths between the centers of adjacent slits is equal to an integer number of waves, then their mutual amplification takes place:

Obviously, the same path difference will be for the left edges of the slits, and for the right edges, and for any other markers located at a distance from each other d. Moreover, if the slits are not adjacent and the distance between their centers is equal to d, A 2d, 3d, id,..., then from geometric considerations it is obvious that the path difference will increase an integer number of times and remain equal to an integer number of waves. This means multiple mutual amplification of waves from all grating slits and leads to the appearance of bright maxima on the screen, called the main ones. The position of the main maxima in accordance with formula (27.21) is given the basic formula of a diffraction grating:

Where t = 0, 1, 2, 3,... - the order of the main maxima. They are located symmetrically relative to the central maximum, for which T = 0.

In addition to the main maxima, there are additional maxima, when beams from some slits reinforce each other, and from others they cancel each other. These additional highs are usually weak and of no interest.

Let us now move on to determining the position of the minima. It is obvious that in those directions where the light did not go from one slit, it will not go there even from several. Therefore, condition (27.16) determines the position main minima of the diffraction grating:

Moreover, if the position of the main minimum falls on the position of the main maximum, then the main maximum disappears.

However, in addition to these minima, additional minima will appear due to the arrival of light in antiphase from different slits. Let us make a simplified estimate of their position, neglecting the role of the strokes. In this approximation, the entire lattice appears to be a single slit, the width of which is equal to Nd, Where N- number of grating slits. By analogy with formula (27.23) we have

It is immediately clear that this estimate includes the positions of more strictly calculated (taking into account the role of primes) main maxima (27.22). It is obvious that these false positions must be eliminated. After this, a fairly accurate formula is obtained to determine the position of a large number additional minima of the diffraction grating:

Analysis of the formula shows that between each two main maxima there is N- 1 additional minimums. Moreover, the more slits, the more minima between the main maxima and the sharper and brighter the main maxima relative to the dim background between the maxima. If a diffraction grating is illuminated by two beams of light with a similar wavelength, then a grating with a large number of slits will allow these wavelengths to be clearly separated and determined in the diffraction pattern. And if you illuminate the grating with white light, then each main maximum, except for the central one, will turn out to be decomposed into a spectrum called diffraction spectrum.

The quality of a diffraction grating as an optical device is determined by its angular dispersion and resolution. Angular dispersion D characterizes the angular width of the spectrum and shows what range of angles falls on a unit wavelength interval:

Taking the differential from relation (27.22), we obtain

When working with a diffraction grating, small angles are usually used, so that cos a ~ 1. Therefore, we finally obtain that the angular dispersion (and the angular distance between the centers of close spectral lines) is greater, the greater the order of the spectrum and the smaller the grating period:

The ability to distinguish close spectral lines depends not only on the distance between the centers of the lines, but also on the width of the lines. Therefore, another characteristic is introduced in optics - the resolution of an optical device, which shows how well the device distinguishes small details of an object. For a diffraction grating under resolution understand the ratio of wavelength to the difference between nearby wavelengths that the grating is still able to distinguish:

Rice. 27.5

Typically, the line discrimination threshold is determined by the Rayleigh criterion: an optical device resolves two adjacent spectral lines, if the maximum of one of them falls into the nearest minimum of the other line(Fig. 27.5). In this case, in the middle between the intensities of the centers of the lines / there is also a minimum with intensity that is usually visible to the eye or instrument

The position of the main high of the first wave is given by equation (27.22):

Position of the nearest additional low of the close second wave X 2 taking into account equations (27.22) and (27.25) is determined by the sum

At the resolution threshold, these positions (and viewing angles) coincide:

Thus, the greater the resolution of the grating, the more lines it contains and the greater the order of the spectrum.

Some of the well-known effects that confirm the wave nature of light are diffraction and interference. Home area their applications are spectroscopy, in which for analysis spectral composition electromagnetic radiation diffraction gratings are used. The formula that describes the position of the main maxima given by this lattice is discussed in this article.

What are the phenomena of diffraction and interference?

Before considering the derivation of the diffraction grating formula, it is worth becoming familiar with the phenomena that make the grating useful, that is, diffraction and interference.

Huygens-Fresnel principle and far- and near-field approximations

The mathematical description of the phenomena of diffraction and interference is a non-trivial task. Finding its exact solution requires complex calculations involving Maxwellian theory electromagnetic waves. Nevertheless, in the 20s of the 19th century, the Frenchman Augustin Fresnel showed that using Huygens' ideas about secondary sources of waves, these phenomena can be successfully described. This idea led to the formulation of the Huygens-Fresnel principle, which currently underlies the derivation of all formulas for diffraction by obstacles of arbitrary shape.

Nevertheless, even using the Huygens-Fresnel principle to solve the problem of diffraction in general view fails, therefore, when obtaining formulas, they resort to some approximations. The main one is the plane wave front. It is precisely this waveform that must fall on the obstacle in order to simplify a number of mathematical calculations.

The next approximation lies in the position of the screen where the diffraction pattern is projected relative to the obstacle. This position is described by the Fresnel number. It is calculated like this:

Where a is the geometric dimensions of the obstacle (for example, a slot or a round hole), λ is the wavelength, D is the distance between the screen and the obstacle. If for a particular experiment F

The difference between Fraunhofer and Fresnel diffractions lies in the different conditions for the interference phenomenon at small and large distances from the obstacle.

The derivation of the formula for the main maxima of a diffraction grating, which will be given later in the article, assumes consideration of Fraunhofer diffraction.

Diffraction grating and its types

This lattice is a plate of glass or transparent plastic several centimeters in size, on which opaque strokes of the same thickness are applied. The strokes are located at a constant distance d from each other. This distance is called the lattice period. Two other important characteristics of the device are the lattice constant a and the number of transparent slits N. The value of a determines the number of slits per 1 mm of length, so it is inversely proportional to the period d.

There are two types of diffraction gratings:

Transparent, which is described above. The diffraction pattern from such a grating arises as a result of the passage of a wave front through it.
Reflective. It is made by applying small grooves to a smooth surface. Diffraction and interference from such a plate arise due to the reflection of light from the tops of each groove.

Whatever the type of grating, the idea behind its effect on the wavefront is to create a periodic disturbance in it. This leads to the formation of a large number of coherent sources, the result of the interference of which is a diffraction pattern on the screen.

Basic formula of a diffraction grating

The derivation of this formula involves considering the dependence of the radiation intensity on the angle of its incidence on the screen. In the far-field approximation, the following formula for intensity I(θ) is obtained:

I(θ) = I0*(sin(β)/β)2*2, where

α = pi*d/λ*(sin(θ) - sin(θ0));

β = pi*a/λ*(sin(θ) - sin(θ0)).

In the formula, the width of the diffraction grating slit is denoted by the symbol a. Therefore, the multiplier in parentheses is responsible for diffraction at a single slit. The value d is the period of the diffraction grating. The formula shows that the factor in square brackets, where this period appears, describes the interference from a set of grating slits.

Using the above formula, you can calculate the intensity value for any angle of incidence of light.

If we find the value of intensity maxima I(θ), we can come to the conclusion that they appear provided that α = m*pi, where m is any integer. For the condition of maxima we obtain:

m*pi = pi*d/λ*(sin(θm) - sin(θ0)) =>

sin(θm) - sin(θ0) = m*λ/d.

The resulting expression is called the diffraction grating maxima formula. The m numbers are the order of diffraction.

Other ways to write the basic formula for a lattice

Note that the formula given in the previous paragraph contains the term sin(θ0). Here the angle θ0 reflects the direction of incidence of the light wave front relative to the grating plane. When the front falls parallel to this plane, then θ0 = 0o. Then we get the expression for the maxima:

sin(θm) = m*λ/d.

Since the grating constant a (not to be confused with the slit width) is inversely proportional to d, the formula above can be rewritten in terms of the diffraction grating constant as:

sin(θm) = m*λ*a.

To avoid errors when substituting specific numbers λ, a and d into these formulas, you should always use the appropriate SI units.

The concept of grating angular dispersion

We will denote this quantity by the letter D. According to mathematical definition, it is written by the following equality:

The physical meaning of angular dispersion D is that it shows by what angle dθm the maximum for diffraction order m will shift if the incident wavelength is changed by dλ.

If we apply this expression to the lattice equation, then we get the formula:

D = m/(d*cos(θm)).

The angular dispersion of a diffraction grating is determined by the formula above. It can be seen that the value of D depends on the order m and the period d.

The greater the dispersion D, the higher the resolution of a given grating.

Grating resolution

Resolution means physical quantity, which shows by what minimum value two wavelengths can differ so that their maxima appear separately in the diffraction pattern.

Resolution is determined by the Rayleigh criterion. It says: two maxima can be separated in a diffraction pattern if the distance between them is greater than the half-width of each of them. The angular half-width of the maximum for the grating is determined by the formula:

Δθ1/2 = λ/(N*d*cos(θm)).

The resolution of the grating in accordance with the Rayleigh criterion is equal to:

Δθm>Δθ1/2 or D*Δλ>Δθ1/2.

Substituting the values of D and Δθ1/2, we get:

Δλ*m/(d*cos(θm))>λ/(N*d*cos(θm) =>

Δλ > λ/(m*N).

This is the formula for the resolution of a diffraction grating. The greater the number of lines N on the plate and the higher the diffraction order, the greater the resolution for a given wavelength λ.

Diffraction grating in spectroscopy

Let us write out again the basic equation of maxima for the lattice:

sin(θm) = m*λ/d.

Here you can see that the longer the wavelength falls on the plate with the streaks, the larger the angles, the maxima will appear on the screen. In other words, if non-monochromatic light (for example, white) is passed through the plate, then you can see the appearance of color maxima on the screen. Starting from the central white maximum (zero-order diffraction), further maxima will appear for more short waves(purple, blue) and then for longer ones (orange, red).

Another important conclusion from this formula is the dependence of the angle θm on the diffraction order. The larger m, the larger the value of θm. This means that the colored lines will be more separated from each other at the maxima for high order diffraction. This fact was already highlighted when the resolution of the grating was considered (see previous paragraph).

The described capabilities of a diffraction grating make it possible to use it to analyze the emission spectra of various luminous objects, including distant stars and galaxies.

Example of problem solution

Let's show you how to use the diffraction grating formula. The wavelength of the light that falls on the grating is 550 nm. It is necessary to determine the angle at which first-order diffraction occurs if the period d is 4 µm.

θ1 = arcsin(λ/d).

We convert all the data into SI units and substitute this equation:

θ1 = arcsin(550*10-9/(4*10-6)) = 7.9o.

If the screen is located at a distance of 1 meter from the grating, then from the middle of the central maximum the line of the first order of diffraction for a wave of 550 nm will appear at a distance of 13.8 cm, which corresponds to an angle of 7.9o.

Diffraction grating

Very large reflective diffraction grating.

Diffraction grating- an optical device operating on the principle of light diffraction is a combination of large number regularly spaced strokes (slots, protrusions) applied to a certain surface. The first description of the phenomenon was made by James Gregory, who used bird feathers as a lattice.

Types of gratings

Reflective: Strokes are applied to a mirror (metal) surface, and observation is carried out in reflected light
Transparent: Strokes are applied to a transparent surface (or cut out in the form of slits on an opaque screen), observation is carried out in transmitted light.

Description of the phenomenon

This is what the light from an incandescent flashlight looks like when it passes through a transparent diffraction grating. Zero maximum ( m=0) corresponds to light passing through the grating without deviation. Due to lattice dispersion in the first ( m=±1) at the maximum, one can observe the decomposition of light into a spectrum. The deflection angle increases with wavelength (from violet to red)

The front of the light wave is divided by the grating bars into separate beams of coherent light. These beams undergo diffraction by the streaks and interfere with each other. Since each wavelength has its own diffraction angle, white light is decomposed into a spectrum.

Formulas

The distance through which the lines on the grating are repeated is called the period of the diffraction grating. Designated by letter d.

If the number of strokes is known ( N), per 1 mm of grating, then the grating period is found using the formula: 0.001 / N

Diffraction grating formula:

d- grating period, α - maximum angle of a given color, k is the order of the maximum, λ is the wavelength.

Characteristics

One of the characteristics of a diffraction grating is angular dispersion. Let us assume that a maximum of some order is observed at an angle φ for wavelength λ and at an angle φ+Δφ for wavelength λ+Δλ. The angular dispersion of the grating is called the ratio D=Δφ/Δλ. The expression for D can be obtained by differentiating the diffraction grating formula

Thus, angular dispersion increases with decreasing grating period d and increasing spectrum order k.

Manufacturing

Good gratings require very high manufacturing precision. If at least one of the many slots is made with an error, the grating will be defective. The machine for making gratings is firmly and deeply built into a special foundation. Before starting the actual production of gratings, the machine runs for 5-20 hours at idle speed to stabilize all its components. Cutting the grating lasts up to 7 days, although the time of applying the stroke is 2-3 seconds.

Application

Diffraction gratings are used in spectral instruments, also as optical sensors of linear and angular displacements (measuring diffraction gratings), polarizers and filters of infrared radiation, beam splitters in interferometers and so-called “anti-glare” glasses.

Literature

Sivukhin D.V. General course physics. - 3rd edition, stereotypical. - M.: Fizmatlit, MIPT, 2002. - T. IV. Optics. - 792 p. - ISBN 5-9221-0228-1
Tarasov K.I., Spectral devices, 1968