Diffraction grating– an optical device that is a collection large number parallel, usually equally spaced slits. A diffraction grating can be obtained by applying opaque scratches (striations) to a glass plate. Unscratched places - cracks - will let light through, while strokes will scatter and not let light through (Fig. 3).
Rice. 3. Section diffraction grating(a) and its graphic representation (b)
To derive the formula, consider a diffraction grating under the condition of perpendicular incidence of light (Fig. 4). Let us choose two parallel rays that pass through two slits and are directed at an angle φ to the normal.
With the help of a collecting lens (eye), these two rays will fall into one point of the focal plane P and the result of their interference will depend on the phase difference or on their path difference. If the lens is perpendicular to the rays, then the path difference will be determined by the segment BC, where AC is perpendicular to rays A and B. In the triangle ABC we have: AB = a + b = d - the period of the grating, BAC = φ, as angles with mutually perpendicular parties.
From formulas (8) and (9) we obtain diffraction grating formula:
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Rice. 4. Diffraction of light by a diffraction grating
Those. the position of the light line in the diffraction spectrum does not depend on the grating material, but is determined by the grating period, which is equal to the sum of the slit width and the gap between the slits.
Resolution of the diffraction grating.
If the light incident on the diffraction grating is polychromatic, i.e. consists of several wavelengths, then in the spectrum the maxima of individual will be at different angles. The resolution can be characterized angular dispersion:
Consequently, the greater the spectral order k, the greater the angular dispersion.
II. Students' work during a practical lesson.
Exercise 1.
Get permission to take classes. To do this you need:
- have notes in workbook, containing the title of the work, the basic theoretical concepts of the topic being studied, the objectives of the experiment, a table based on the sample for entering experimental results;
– successfully pass control according to the experimental methodology;
– obtain permission from the teacher to perform the experimental part of the work.
Task 2.
Carrying out laboratory work, discussing the results obtained, writing notes.
Devices and accessories
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Rice. 5 Installation diagram |
1. Diffraction grating.
2. Light source.
4. Ruler.
In this laboratory work It is proposed to determine the wavelengths for red and green colors, which are obtained when light passes through a diffraction grating. In this case, a diffraction spectrum is observed on the screen. A diffraction grating consists of a large number of parallel slits, very small compared to the wavelength. The slits allow light to pass through, while the space between the slits is opaque. Total slits – N, with a distance between their centers – d. Diffraction grating formula:
where d is the grating period; sin φ – sine of the angle of deviation from the rectilinear propagation of light; k – maximum order; λ – wavelength of light.
The experimental setup consists of a diffraction grating, a light source and a movable screen with a ruler. The diffraction spectrum is observed on the screen (Fig. 5).
The distance L from the diffraction grating to the screen can be changed by moving the screen. Distance from the central ray of light to a separate line of the spectrum l. At small angles φ.
DEFINITION
Diffraction grating- This is the simplest spectral device. It contains a system of slits that separate opaque spaces.
Diffraction gratings are divided into one-dimensional and multidimensional. A one-dimensional diffraction grating consists of parallel light-transparent sections of the same width, which are located in the same plane. Transparent areas are separated by opaque spaces. Using these gratings, observations are carried out in transmitted light.
There are reflective diffraction gratings. Such a grating is, for example, a polished (mirror) metal plate onto which strokes are applied using a cutter. The result is areas that reflect light and areas that scatter light. Observation using such a grating is carried out in reflected light.
The diffraction pattern on the grating is the result of mutual interference of waves that come from all the slits. Consequently, with the help of a diffraction grating, multi-beam interference of coherent beams of light that have undergone diffraction and coming from all slits is realized.
Diffraction grating period
If we denote the width of the slit in the grating as a, the width of the opaque section as b, then the sum of these two parameters is the grating period (d):
The diffraction grating period is sometimes also called the diffraction grating constant. The period of a diffraction grating can be defined as the distance through which the lines on the grating are repeated.
The diffraction grating constant can be found if the number of lines (N) that the grating has per 1 mm of its length is known:
The period of the diffraction grating is included in the formulas that describe the diffraction pattern on it. Thus, if a monochromatic wave is incident on a one-dimensional diffraction grating perpendicular to its plane, then the main intensity minima are observed in the directions determined by the condition:
where is the angle between the normal to the grating and the direction of propagation of diffracted rays.
In addition to the main minima, as a result of the mutual interference of light rays sent by a pair of slits, in some directions they cancel each other, resulting in additional intensity minima. They arise in directions where the difference in the path of the rays is an odd number of half-waves. The condition for additional minima is written as:
where N is the number of slits of the diffraction grating; takes any integer value except 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 expression:
The value of the sine cannot exceed one, therefore, the number of main maxima (m):
Examples of problem solving
EXAMPLE 1
| Exercise | A beam of light having a wavelength .passes through a diffraction grating. A screen is placed at a distance L from the grating, onto which a diffraction pattern is formed using a lens. It is found that the first diffraction maximum is located at a distance x from the central one (Fig. 1). What is the diffraction grating period (d)? |
| Solution | Let's make a drawing. The solution to the problem is based on the condition for the main maxima of the diffraction pattern: According to the conditions of the problem, we are talking about the first main maximum, then . From Fig. 1 we get that:
From expressions (1.2) and (1.1) we have:
Let us express the desired period of the lattice, we obtain:
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| Answer |
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 0 is incident perpendicular to the plane of the diffraction grating. 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 an odd number of half-waves. 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 to the 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: |
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, the greater the number of 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., 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 be
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 be obtained 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 us 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 a plane light wave passes not through slits, but through round holes 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 length of the slit it equal to one, and along the opaque strip - 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, on a glass plate (or on transparent film) apply concentric rings with ink; in this case, the central circle enclosing 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; at zone plates For this purpose, the required radii and thickness of the rings must be calculated. Zone gratings can also be manufactured with a smooth, for example sinusoidal, change in transmittance along the radius.
An important role in applied optics is played by the phenomena of diffraction by openings in the form of a slit with parallel edges. At the same time, the use of light diffraction at a single slit for practical purposes is difficult due to the poor visibility of the diffraction pattern. Diffraction gratings are widely used.
Diffraction grating- a spectral device used to decompose light into a spectrum and measure wavelength. There are transparent and reflective grilles. A diffraction grating is a collection of a large number of parallel lines of the same shape, applied to a flat or concave polished surface at the same distance from each other.
In a transparent flat diffraction grating (Fig. 17.22), the width of the transparent line is equal to A, width of the opaque gap - b. The quantity \(d = a + b = \frac(1)(N)\) is called constant (period) of the diffraction grating, Where N- number of lines per unit length of the grating.
Let a plane monochromatic wave be incident normally to the grating plane (Fig. 17.22). According to the Huygens-Fresnel principle, each slit is a source of secondary waves that can interfere with each other. The resulting diffraction pattern can be observed in the focal plane of the lens onto which the diffracted beam falls.
Let us assume that light diffracts on slits at an angle \(\varphi.\) Since the slits are located at equal distances from each other, then the differences in the paths of rays coming from two adjacent slits for a given direction \(\varphi\) will be the same in within the entire diffraction grating:
\(\Delta = CF = (a+b)\sin \varphi = d \sin \varphi .\)
In those directions for which the path difference is equal to an even number of half-waves, an interference maximum is observed. On the contrary, for those directions where the path difference is equal to an odd number of half-waves, an interference minimum is observed. Thus, in directions for which the angles \(\varphi\) satisfy the condition
\(d \sin \varphi = m \lambda (m = 0,1,2, \ldots),\)
the main maxima of the diffraction pattern are observed. This formula is often called diffraction grating formula. In it, m is called the order of the main maximum. Between the main maxima there are (N - 2) weak side maxima, but against the background of the bright main maxima they are practically invisible. As the number of strokes N (necks) increases, the main maxima, while remaining in the same places, become increasingly sharper.
When observing diffraction in non-monochromatic (white) light, all the main maxima, except the zero central maximum, are colored. This is explained by the fact that, as can be seen from the formula \(\sin \varphi = \frac(m \lambda)(d),\) different wavelengths correspond to different angles at which interference maxima are observed. The rainbow stripe, which generally contains seven colors - from violet to red (counted from the central maximum), is called the diffraction spectrum.
The spectrum width depends on the lattice constant and increases with decreasing d. The maximum order of the spectrum is determined from the condition \(~\sin \varphi \le 1,\) i.e. \(m_(max) = \frac(d)(\lambda) = \frac(1)(N\lambda).\)
Literature
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P. 517-518.