Period diffraction lattice formula. Diffraction grating

When analyzing the actions of the zone plates, we found out that periodic structures are most effectively working in diffraction. And it is not surprising. After all, the diffraction is a wave effect, and the waves themselves are a periodic structure. Therefore, it can be expected that a set of equidistant cracks must in some cases give a more spectacular and useful diffraction pattern for practical applications.

In this regard, consider the exact optical device - the diffraction grid. Simplest diffraction lattice They call the set of a large number of narrow, parallel, the same, equal to each other of the gaps. Such a lattice works in the transmitted light. Sometimes a diffraction grid in the reflected light is used, which is made by applying a large number of narrow, parallel, the same, equal to each other obstacles to the mirror. Often the grille is made by applying opaque strokes to transparent glass or mirror. Therefore, it is not characterized by the number of slots, but the number of strokes separating the gaps. The first working diffraction lattice made in the XVII century. Scottish scientist James Gregory, who used bird feathers for this. In modern lattices, the number of strokes reaches a million on the surface to several tens of centimeters.

Description of diffraction on the diffraction grid is performed similar to the description of the diffraction in the parallel rays on the slit (Fig. 27.4). The amount of the width of the slit butand the gap between the slits (stroke) B Call lattice period.

Suppose to the lattice perpendicular to its plane falls a bundle of parallel rays, which is further in Fig. 27.4. Responsibility with the principle of Guygens - Fresnel gives secondary interfering waves. We choose some direction of passage of these secondary waves, determined by an angle a. If the movement of the waves between the middle of the adjacent slits is equal to an integer number of waves, then their mutual gain takes place:

Obviously, the same movement of the course will be for the left edges of the slots, and for the right edges, and for any other market points removed from each other d. Moreover, if the gaps are not adjacent and the distance between their centers is not d, but 2D, 3D, ID,..., then from the geometric considerations it is obvious that the difference in the course will increase by an integer once and will remain equal to an integer number of waves. This means a multiple mutual strengthening of waves from all lattice slots and leads to the appearance on the screen of bright highs, called main. The position of the main maxima in accordance with formula (27.21) is set the main formula of the diffraction lattice:

where t \u003d. 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 when the bundles from one slots increase each other, and from others - quench. These additional maxima are usually weak and do not represent interest.

We now turn to the definition of the position of the lows. Obviously, in those directions where the light did not go from one gap, he will not go there and from several. Therefore, condition (27.16) determines the position the main minima of the diffraction lattice:

At the same time, if the position of the main minimum falls on the position of the main maximum, the main maximum disappears.

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

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

Analysis of the formula shows that there is between each two main maxima N - 1 additional lows. At the same time, the more gaps, the more lows between the main maxima and the sharper and brighter the main maxima relative to the dim background between the maxima. If the diffraction lattice is lit by two beams of light with a close wavelength, the lattice with a large number of slots will allow in the diffraction pattern to be clearly divided and determine these wavelengths. And if you light the grid with white light, then each main maximum, except the central, will be unfolded in the spectrum, called diffraction spectrum.

The quality of the diffraction lattice as an optical device is determined by its angular dispersion and resolution. Corner dispersion D. It characterizes the angular width of the spectrum and shows which angles interval accounts for a single wavelength interval:

Taking differential from the relation (27.22), we get

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

The ability to distinguish close spectral lines depends not only on the distance between the lines centers, but also from the lines width. Therefore, another characteristic is introduced in optics - the resolution ability of the optical instrument, which shows how well the device distinguishes small items. For the diffraction lattice under leaving ability Understand the ratio of wavelengths to the difference in close wavelengths, which the lattice is still able to distinguish:

Fig. 27.5.

Usually, the threshold of distinction of lines is determined by the Rayleigh criterion: the optical device allows two adjacent spectrum lines., if the maximum of one of them falls into the nearest minimum of another line (Fig. 27.5). In this case, in the middle between the intensities of lines centers / there is still a commonly distinguishable eye or an intensity device with intensity

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

The position of the nearest email is close to the second wave X 2 Taking into account equations (27.22) and (27.25) is determined by the amount

On the verge of resolution, these provisions (and surveillance angles) coincide:

Thus, the resolution ability of the lattice is the greater, the more strokes in it and the greater the order of the spectrum.

Some of the known effects that confirm the wave nature of light are diffraction and interference. The main area of \u200b\u200btheir use is spectroscopy, in which diffraction gratings are used to analyze the spectral composition of electromagnetic radiation. The formula that describes the position of the main maxima given by this lattice is considered in this article.

What are the phenomena of diffraction and interference?

Before considering the output of the formula of the diffraction lattice, it should be found with the phenomena, thanks to which this lattice is useful, that is, with diffraction and interference.

Guiggens-Fresnel principle and approximation of far and near fields

The mathematical description of the phenomena of diffraction and interference is a nontrivial task. Finding an exact solution requires the fulfillment of complex calculations with the involvement of Maxwell's theory of electromagnetic waves. Nevertheless, in the 20s of the XIX century, the Frenchman Friestel showed that using the Guigens representation of the secondary springs of the waves, one can successfully describe these phenomena. This idea led to the formulation of the Guigens-Fresnel principle, which is currently based on the output of all formulas for diffraction on the obstacles of arbitrary form.

Nevertheless, even with the help of the principle of Guigens-Fresnel, the problem of diffraction is generally not possible, so when obtaining the formulas resort to some approximations. The main one is the flat wave front. It is this form of a wave form should fall on an obstacle that a number of mathematical calculations can be simplified.

The following approximation is the position of the screen, where the diffraction pattern is projected, relative to the obstacle. This provision is described by the number of Fresnel. It is calculated as:

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

The difference between the diffraction of Fraunhofer and Fresnel lies in various conditions for the phenomenon of interference in small and large distances from the obstacle.

The output of the formula of the main maxima of the diffraction lattice, which will be submitted further in the article, involves considering the diffraction of the Frangofer.

Diffraction lattice and its types

This grill is a plate of glass or transparent plastic in size of several centimeters, which causes opaque strokes of the same thickness. 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 a constant lattice A and the number of transparent slots N. The value A determines the number of slots per 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 lattice occurs as a result of passing the wave front through it.
Reflective. It is manufactured by applying small grooves on a smooth surface. Diffraction and interference from such a plate arise due to the reflection of light from the vertices of each groove.

Whatever the type of lattice, the idea of \u200b\u200bits impact on the wave front is to create a periodic perturbation in it. This leads to the formation of a large number of coherent sources, the result of the interference of which is a diffraction picture on the screen.

The main formula of the diffraction lattice

The output of this formula implies consideration of the dependence of the radiation intensity from the angle of its fall on the screen. In the approximation of the far field, the following formula is obtained for intensity I (θ):

I (θ) \u003d i0 * (sin (β) / β) 2 * 2, where

α \u003d pi * d / λ * (sin (θ) - sin (θ0));

β \u003d pi * a / λ * (sin (θ) - sin (θ0)).

In the formula, the width of the diffraction lattice is indicated by the symbol a. Therefore, the multiplier in parentheses is responsible for the diffraction on one gap. The value d is the period of the diffraction lattice. The formula shows that the multiplier in square brackets, where this period appears, describes the interference from the totality of the grid slit.

Taking this formula, you can calculate the value of the intensity for any angle of falling light.

If you find the value of the maxima of the intensity I (θ), then it can be concluded that they appear under the condition that α \u003d m * pi, where M is any integer. For the conditions of the maxima we get:

m * pi \u003d pi * d / λ * (sin (θm) - sin (θ0)) \u003d\u003e

sin (θm) - sin (θ0) \u003d m * λ / d.

The resulting expression is called the formula of the maxima diffraction lattice. The numbers M are the order of diffraction.

Other ways to write the main formula for the lattice

Note that in the previous paragraph, the formula is present sin (θ0). Here, the angle θ0 reflects the direction of falling the front of the light wave relative to the lattice plane. When the front drops parallel to this plane, then θ0 \u003d 0o. Then we get an expression for the maxima:

sin (θm) \u003d m * λ / d.

Since the permanent lattice A (not to be confused with the width of the slit) is inversely proportional to the value of D, then through a constant diffraction lattice of the formula above rewrite in the form:

sin (θm) \u003d m * λ * a.

In order not to errors in the substitution of specific numbers λ, a and d into these formulas, you should always use the corresponding units of C.

The concept of the angular dispersion of the lattice

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

The physical meaning of the angular dispersion D lies in the fact that it shows how the angle dθm will be shifted to the maximum for the form of the diffraction M, if you change the length of the incident wave to Dλ.

If you apply this expression for the lattice equation, then the formula will turn out:

D \u003d m / (d * cos (θm)).

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

The greater the dispersion D, the higher the resolution of this lattice.

Resolution lattice

Under the resolution understands the physical quantity, which shows which the minimum value there may be two wavelengths so that their highs on the diffraction pattern appeared separately.

The resolution is determined by the Rayleigh criterion. It says: two maxima can be divided into a diffraction pattern if the distance between them turns out more than half-sewn each of them. The angular half-width of the lattice maximum is determined by the formula:

Δθ1 / 2 \u003d λ / (n * d * cos (θm)).

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

Δθm\u003e Δθ1 / 2 or D * Δλ\u003e Δθ1 / 2.

Substituting the values \u200b\u200bof D and Δθ1 / 2, we get:

Δλ * m / (d * cos (θm))\u003e λ / (n * d * cos (θm) \u003d\u003e

Δλ\u003e λ / (m * n).

This is the formula of the resolution of the diffraction lattice. The larger the number of N on the plate and the higher the form of the diffraction, the greater the resolution for this wavelength λ.

Diffraction grating in spectroscopy

We repel once again the main maxima equation for the lattice:

sin (θm) \u003d m * λ / d.

Here it is seen that the larger the wavelength drops on the plate with strokes, the maxima on the screen will appear at large angles. In other words, if there is a non-monochromatic light (for example, white) through the record, then the appearance of color maxima can be seen on the screen. Starting from the central white maximum (zero-order diffraction), maxima will appear further for shorter waves (purple, blue), and then for longer (orange, red).

Another important conclusion from this formula is the dependence of the angle θm on the diffraction procedure. The greater M, the greater the value θm. This means that colored lines will be more separated by each other at high-order diffraction. This fact was already consecrated when the resolution of the lattice was considered (see the previous paragraph).

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

An example of solving the problem

We show how to use the formula of the diffraction lattice. The wavelength of light that falls the lattice is 550 nm. It is necessary to determine the angle at which the first order diffraction appears if the period d is 4 microns.

θ1 \u003d arcsin (λ / d).

Transfer all data into units of SI and substitute in this equality:

θ1 \u003d arcsin (550 * 10-9 / (4 * 10-6)) \u003d 7.9o.

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

Definition

Diffraction grating - This is the simplest spectral device consisting of a system of slots (transparent for light of plots), and opaque gaps that are comparable with a wavelength.

One-dimensional diffraction grating, consists of parallel slits of the same width, which lie in the same plane separated by the same over-transparent width for light by the gaps. Reflective diffraction gratings are considered the best. They consist of a set of areas reflecting light and sections that light scatter. These lattices are polished metal plates on which the scattering light of the touches is applied with a cutter.

The diffraction pattern on the lattice is the result of the mutual interference of the waves coming by all the cracks. With the help of a diffraction lattice, a multipath interference of coherent beams of light, diffraction, and which come from all the cracks are realized.

The characteristic of the diffraction lattice serves its period. The period of the diffraction lattice (D) (its constant) is called the value equal to:

where a is the width of the slit; B - the width of an opaque area.

Diffraction on a one-dimensional diffraction lattice

Suppose that the light wave is perpendicular to the plane of the diffraction lattice with a length. Since the slots at the lattice are located at equal distances from each other, the difference in the course of the rays (), coming from two adjacent cracks, will be the same for the entire diffraction lattice under consideration:

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

In addition to the main minima, as a result of the mutual interference of the beams of light, which go from two cracks, in some directions of the rays quit each other. As a result, additional intensity minima arise. They appear in those directions where the path difference make up the odd number of half feet. The condition of additional minima is the formula:

where n is the number of slots of the diffraction lattice; - In addition to 0, in the event that the grille has N slots, then there are additional minimum between the two main maxima, which are separated by secondary maxima.

The condition of the main maxima for the diffraction lattice is:

The size of the sinus can not be more units, then the number of main maxima:

Examples of solving problems on the topic "Diffraction Lattice"

Example 1.

The task

On the diffraction grid, the monochromatic beam of light with a wavelength drops perpendicular to its surface. On the flat screen, the diffraction pattern is projected by lenses. The distance between the two maxima of the first-order intensity is l. What is the constant diffraction lattice, if the lens is placed in the immediate vicinity of the lattice and the distance from it to the screen is L. Think that

Decision

As the basis for solving the problem, we use the formula that binds a diffraction lattice constant, the light wavelength and the beam deflection angle that corresponds to the diffraction maximum number M:

By the condition of the problem, since the angle of deflection of the rays can be considered small (), then we will take that:

Figure 1 it follows that:

Substitut in formula (1.1), the expression (1.3) and take into account that, we obtain:

From (1.4) Express the lattice period:

Answer

Example 2.

The task	Using the terms of Example 1, and the result of the solution, find the number of highs that the lattice will give.
Decision	In order to determine the maximum angle of deflection of the beams of light in our problem, we find the number of highs that our diffraction grating can give. To do this, use the formula: where we put that when. Then, we get:

Definition

Diffraction lattice Called the spectral device, which is a system of a number of slots separated by opaque gaps.

Very often in practice, a one-dimensional diffraction grid is used, consisting of parallel slits of the same width, which are in one plane, which are separated equal to the width of opaque gaps. Such a grid is made using a special dividious machine that makes parallel strokes on a plate of glass. The number of such strokes can be more than one thousand per millimeter.

Reflective diffraction gratings are considered the best. This is a set of areas that reflect light with areas that light reflect. Such lattices are a polished metal plate, on which the scattering light of the strokes are applied with a cutter.

The pattern of diffraction on the lattice is the result of the mutual interference of the waves, which go through all the cracks. Consequently, with the help of a diffraction lattice, a multipath interference of coherent beams of light, which have been diffraction and which come from all the cracks are realized.

Suppose that on the diffraction lattice, the width of the slit will be A, the width of the opaque section - b, then the value:

is called a period of (constant) diffraction lattice.

Diffraction pattern on a one-dimensional diffraction lattice

Imagine that a monochromatic wave drops to the plane of the diffraction lattice. Due to the fact that the gaps are located at equal distances from each other, then the difference between the ray (), which come from the pair of neighboring slots, for the chosen direction will be the same for the entire diffraction lattice:

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

In addition to the main minima, as a result of the mutual interference of the beams of light, which sends a pair of cracks, in some directions they quit each other, it means that additional minima appear. They arise in directions, where the difference between the rays make up the odd number of half-fell. The condition of additional minimums is recorded as:

where n is the number of the slits of the diffraction lattice; k 'takes any integer values \u200b\u200bexcept 0 ,. If the lattice has n slots, then between the two main maxima there are additional minimum that shared the secondary maxima.

The condition of the main maxima for the diffraction lattice is the expression:

Since the size of the sinus can not be more units, then the number of main maxima:

If the white light is passed through the grille, then all the maxima (except the central m \u003d 0) will be decomposed in the spectrum. In this case, the violet area of \u200b\u200bthis spectrum will be addressed to the center of the diffraction pattern. This property of the diffraction lattice is used to study the composition of the spectrum of light. If the lattice period is known, the calculation of the wavelength of light can be reduced to finding an angle that corresponds to a maximum direction.

Examples of solving problems

Example 1.

The task

What is the maximum order of the spectrum, which can be obtained using a diffraction lattice with a constant m, if a monochromatic beam of light with a wavelength m is perpendicular to it?

Decision

As a basis for solving the problem, we use a formula that is a condition for observing the main maxima for the diffraction pattern obtained by passing light through the diffraction grid:

The maximum value is the unit, so:

From (1.2) we will express, we get:

Cut out:

Answer

Example 2.

The task

Through the diffraction grid, monochromatic light with a wavelength is passed. At the distance L from the grid set the screen. With the help of a lens located near the lattice, create a projection of the diffraction pattern. In this case, the first maximum diffraction is at a distance of L from the central one. What is the number of strokes per unit of the length of the diffraction lattice (N) if the light falls on it normally?

Decision

Make a drawing.

Wide distribution in the scientific experiment and technique received diffraction latticeswhich are a set of parallel, located at equal distances of the same slots, separated equal to the width of opaque gaps. Diffraction grilles are made using a dividing machine that applies a stroke (scratch) on glass or other transparent material. Where scratch was carried out, the material becomes opaque, and the gaps between them remain transparent and actually play the role of gaps.

Consider first the diffraction of light from the grille on the example of two cracks. (With an increase in the number of cracks, diffraction maxima becomes only narrower, brighter and distinct.)

Let but -gap width, a b. - The width of the opaque gap (Fig. 5.6).

Fig. 5.6. Diffraction from two slots

Period diffraction lattice - This is the distance between the middle of the adjacent slots:

The movement of two extreme rays is equal to

If the path difference is equal to the odd number of half-breed

the light sent by the two slots due to the interference of the waves will be mutually quenched. The minimum condition is

These minima are called additional.

If the movement difference is equal to the annual number of half-breed

that waves sent by each slit will mutually strengthen each other. The condition of interference maxima taking into account (5.36) has the form

This is a formula for main Maxims of Diffraction Lattice.

In addition, in those directions in which none of the gaps does not distribute light, it will not be distributed and with two cracks, that is main minima lattice will be observed in the directions defined by condition (5.21) for one slot:

If the diffraction lattice consists of N.gaps (modern lattices used in the appliances for spectral analysis have up to 200 000 Strokes and period d \u003d 0.8 microns, that is, order 12 000 Strochors. 1 cm), then the condition of the main minima is, as in the case of two slots, the ratio (5.41), the condition of the main maxima - the relation (5.40), and the condition of additional minimahas appearance

Here k "can take all integer values \u200b\u200bexcept 0, n, 2n, ....Therefore, in the case of N.the gaps between the two main maxima is located ( N-1) Additional minimums separated by secondary maxima creating a relatively weak background.

The position of the main maxima depends on the wavelength l.. Therefore, when the white light is passing through the grill, all maxima, except the central, decompose into the spectrum, the purple end of which is addressed to the center of the diffraction pattern, and the red - outwards. Thus, the diffraction grille is a spectral device. Note that while the spectral prism deflects the purple rays, the diffraction grille, on the contrary, deflects the red rays stronger.

An important characteristic of any spectral device is resolution.

The resolution of the spectral instrument is a dimensionless value.

where is the minimum difference of wavelengths of two spectral lines, at which these lines are perceived separately.

Determine the resolution of the diffraction lattice. Position of the middle komaximum wavelength

determined by the condition

The edges k.- gO maximum (i.e. nearest additional minima) for wavelength l. Located at angles that satisfy the ratio: