One of the well-known effects that confirm the wave nature of light are diffraction and interference. Their main field of application 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 discussed in this article.
What are the phenomena of diffraction and interference?
Before considering the derivation of the diffraction grating formula, one should become familiar with the phenomena due to which this grating is useful, that is, with diffraction and interference.
Diffraction is the process of changing the motion of the wave front when it encounters an opaque obstacle on its way, the dimensions of which are comparable to the wavelength. For example, if sunlight is passed through a small hole, then on the wall one can observe not a small luminous point (which should happen if the light propagated in a straight line), but a luminous spot of some size. This fact testifies to the wave nature of light.
Interference is another phenomenon that is unique to waves. Its essence lies in the imposition of waves on each other. If the waveforms from multiple sources are matched (coherent), then a stable pattern of alternating bright and dark areas on the screen can be observed. The minima in such a picture are explained by the arrival of waves at a given point in antiphase (pi and -pi), and the maxima are the result of waves hitting the point under consideration in one phase (pi and pi).
Both phenomena described were first explained by an Englishman when he investigated the diffraction of monochromatic light by two thin slits in 1801.
The 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 performing complex calculations involving the Maxwellian theory of electromagnetic waves. Nevertheless, in the 1920s, the Frenchman Augustin Fresnel showed that, using Huygens' ideas about secondary sources of waves, one can successfully describe these phenomena. 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 with the help of the Huygens-Fresnel principle, it is not possible to solve the problem of diffraction in a general form, therefore, when obtaining formulas, some approximations are resorted to. The main one is a flat wave front. It is this waveform that must fall on the obstacle so that a number of mathematical calculations can be simplified.
The next approximation is 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<<1 (<0,001), тогда говорят о приближении дальнего поля. Соответствующая ему дифракция носит фамилию Фраунгофера. Если же F>1, then near field approximation or Fresnel diffraction takes place.
The difference between Fraunhofer and Fresnel diffraction lies in the different conditions for the phenomenon of interference at small and large distances from the obstacle.
The derivation of the formula for the main maxima of the diffraction grating, which will be given later in the article, involves the consideration of Fraunhofer diffraction.
Diffraction grating and its types
This grating is a plate of glass or transparent plastic a few 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, as described above. The diffraction pattern from such a grating results from 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 of its effect 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 pattern on the screen.
The 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 the intensity I(θ) is obtained:
I(θ) = I 0 *(sin(β)/β) 2 * 2 , where
α = pi*d/λ*(sin(θ) - sin(θ 0));
β = pi*a/λ*(sin(θ) - sin(θ 0)).
In the formula, the width of the slit of the diffraction grating is denoted by the symbol a. Therefore, the factor in parentheses is responsible for the diffraction by one slit. The value of 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 the set of grating slots.
Using the above formula, you can calculate the intensity value for any angle of incidence of light.
If we find the value of the intensity maxima I(θ), then we can conclude that they appear under the condition that α = m*pi, where m is any integer. For the maximum condition, we get:
m*pi = pi*d/λ*(sin(θ m) - sin(θ 0)) =>
sin (θ m) - sin (θ 0) \u003d m * λ / d.
The resulting expression is called the formula for the maxima of the diffraction grating. The m numbers are the order of diffraction.
Other ways to write the basic formula for the lattice
Note that the formula given in the previous paragraph contains the term sin(θ 0). Here, the angle θ 0 reflects the direction of incidence of the front of the light wave relative to the grating plane. When the front falls parallel to this plane, then θ 0 = 0 o . Then we get the expression for the maxima:
Since the grating constant a (not to be confused with the slit width) is inversely proportional to the value of d, the formula above can be rewritten in terms of the diffraction grating constant as:
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 the angular dispersion of the grating
We will denote this value by the letter D. According to the mathematical definition, it is written as follows:
The physical meaning of the angular dispersion D is that it shows by what angle dθ m the maximum will shift for the diffraction order m if the incident wavelength is changed by dλ.
If we apply this expression to the lattice equation, then we get the formula:
The dispersion of the angular 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 is understood as a physical quantity that shows by what minimum value two wavelengths can differ so that their maxima appear separately in the diffraction pattern.
The 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:
Δθ 1/2 = λ/(N*d*cos(θ m)).
The resolution of the grating in accordance with the Rayleigh criterion is:
Δθ m >Δθ 1/2 or D*Δλ>Δθ 1/2 .
Substituting the values of D and Δθ 1/2 , we get:
Δλ*m/(d*cos(θ m))>λ/(N*d*cos(θ m) =>
Δλ > λ/(m*N).
This is the formula for the resolution of a diffraction grating. The greater the number of strokes N on the plate and the higher the order of diffraction, the greater the resolution for a given wavelength λ.
Diffraction grating in spectroscopy
Let us write out once again the basic equation of maxima for the lattice:
It can be seen here that the more the wavelength falls on the plate with strokes, the greater the values of the angles will appear on the screen maxima. In other words, if non-monochromatic light (for example, white) is passed through the plate, 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 (violet, blue), and then for longer ones (orange, red).
Another important conclusion from this formula is the dependence of the angle θ m on the order of diffraction. The larger m, the larger the value of θ m . This means that the colored lines will be more separated from each other at the maxima for a high diffraction order. This fact was already consecrated when the grating resolution was considered (see the 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.
Problem solution example
Let's show how to use the diffraction grating formula. The wavelength of light that falls on the grating is 550 nm. It is necessary to determine the angle at which first-order diffraction appears if the period d is 4 µm.
Convert all data to SI units and substitute into this equality:
θ 1 \u003d arcsin (550 * 10 -9 / (4 * 10 -6)) \u003d 7.9 o.
If the screen is 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 7.9 o .
DEFINITION
Diffraction grating is the simplest spectral instrument. 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 separate opaque gaps. With these gratings, observations are made in transmitted light.
There are reflective diffraction gratings. Such a grating is, for example, a polished (mirror) metal plate, on which strokes are applied with a cutter. The result is areas that reflect light and areas that scatter light. Observation with such a grating is carried out in reflected light.
The grating diffraction pattern is the result of the mutual interference of waves that come from all the slits. Therefore, with the help of a diffraction grating, multipath interference of coherent light beams that have undergone diffraction and that come from all slits is realized.
Grating period
If we denote the width of the slot on the gratings as a, the width of the opaque section - b, then the sum of these two parameters is the grating period (d):
The period of a diffraction grating is sometimes also called the diffraction grating constant. The period of a diffraction grating can be defined as the distance over which the lines on the grating are repeated.
The diffraction grating constant can be found if the number of grooves (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. So, 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 the diffracted rays.
In addition to the main minima, as a result of mutual interference of light rays sent by a pair of slits, they cancel each other out in some directions, 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 additional minima condition is written as:
where N is the number of slits of the diffraction grating; takes any integer value except 0. If the lattice has N slots, then between the two main maxima there is an additional minimum that separates the secondary maxima.
The condition for the main maxima for the 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 passes through a diffraction grating with a wavelength of . At a distance L from the grating, a screen is placed on which a diffraction pattern is formed using a lens. It is obtained that the first diffraction maximum is located at a distance x from the central one (Fig. 1). What is the grating period (d)? |
| Solution | Let's make a drawing. The solution of the problem is based on the condition for the main maxima of the diffraction pattern: By the condition 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:
We express the desired period of the lattice, we get:
|
| Answer |
1. Diffraction of light. Huygens-Fresnel principle.
2. Diffraction of light by a slit in parallel beams.
3. Diffraction grating.
4. Diffraction spectrum.
5. Characteristics of a diffraction grating as a spectral device.
6. X-ray diffraction analysis.
7. Diffraction of light by a round hole. aperture resolution.
8. Basic concepts and formulas.
9. Tasks.
In a narrow, but most commonly used sense, the diffraction of light is the rounding of the borders of opaque bodies by the rays of light, the penetration of light into the region of a geometric shadow. In phenomena associated with diffraction, there is a significant deviation of the behavior of light from the laws of geometric optics. (Diffraction does not only show up for light.)
Diffraction is a wave phenomenon that is most clearly manifested when the dimensions of the obstacle are commensurate (of the same order) with the wavelength of light. The relatively late discovery of light diffraction (16th-17th centuries) is connected with the smallness of the lengths of visible light.
21.1. Diffraction of light. Huygens-Fresnel principle
Diffraction of light called a complex of phenomena that are due to its wave nature and are observed during the propagation of light in a medium with sharp inhomogeneities.
A qualitative explanation of diffraction is given by Huygens principle, which establishes the method of constructing the wave front at time t + Δt if its position at time t is known.
1. According to Huygens principle, each point of the wave front is the center of coherent secondary waves. The envelope of these waves gives the position of the wave front at the next moment in time.
Let us explain the application of the Huygens principle by the following example. Let a plane wave fall on a barrier with a hole, the front of which is parallel to the barrier (Fig. 21.1).
Rice. 21.1. Explanation of Huygens' principle
Each point of the wave front emitted by the hole serves as the center of secondary spherical waves. The figure shows that the envelope of these waves penetrates into the region of the geometric shadow, the boundaries of which are marked with a dashed line.
Huygens' principle says nothing about the intensity of the secondary waves. This drawback was eliminated by Fresnel, who supplemented the Huygens principle with the concept of the interference of secondary waves and their amplitudes. The Huygens principle supplemented in this way is called the Huygens-Fresnel principle.
2. According to the Huygens-Fresnel principle the magnitude of light oscillations at some point O is the result of interference at this point of coherent secondary waves emitted everyone wave surface elements. The amplitude of each secondary wave is proportional to the area of the element dS, inversely proportional to the distance r to the point O, and decreases with increasing angle α between normal n to the element dS and direction to the point O (Fig. 21.2).
Rice. 21.2. Emission of secondary waves by wave surface elements
21.2. Slit Diffraction in Parallel Beams
Calculations related to the application of the Huygens-Fresnel principle, in the general case, are a complex mathematical problem. However, in a number of cases with a high degree of symmetry, the amplitude of the resulting oscillations can be found by algebraic or geometric summation. Let us demonstrate this by calculating the diffraction of light by a slit.
Let a plane monochromatic light wave fall on a narrow slot (AB) in an opaque barrier, the direction of propagation of which is perpendicular to the surface of the slot (Fig. 21.3, a). Behind the slit (parallel to its plane) we place a converging lens, in focal plane which we place the screen E. All secondary waves emitted from the surface of the slot in the direction parallel optical axis of the lens (α = 0), come into focus of the lens in the same phase. Therefore, in the center of the screen (O) there is maximum interference for waves of any length. It's called the maximum zero order.
In order to find out the nature of the interference of secondary waves emitted in other directions, we divide the surface of the slot into n identical zones (they are called Fresnel zones) and consider the direction for which the condition is satisfied:
where b is the slot width, and λ - the length of the light wave.
Rays of secondary light waves traveling in this direction will intersect at point O.
Rice. 21.3. Diffraction by one slit: a - path of rays; b - distribution of light intensity (f - focal length of the lens)
The product bsina is equal to the path difference (δ) between the rays coming from the edges of the slot. Then the difference in the path of the rays coming from neighboring Fresnel zones is equal to λ/2 (see formula 21.1). Such rays cancel each other out during interference, since they have the same amplitudes and opposite phases. Let's consider two cases.
1) n = 2k is an even number. In this case, pairwise extinction of rays from all Fresnel zones occurs, and at the point O" a minimum of the interference pattern is observed.
Minimum intensity during slit diffraction is observed for the directions of rays of secondary waves that satisfy the condition
An integer k is called minimum order.
2) n = 2k - 1 is an odd number. In this case, the radiation of one Fresnel zone will remain unquenched, and at the point O" the maximum of the interference pattern will be observed.
The intensity maximum during slit diffraction is observed for the directions of rays of secondary waves that satisfy the condition:
An integer k is called maximum order. Recall that for the direction α = 0 we have maximum zero order.
It follows from formula (21.3) that as the light wavelength increases, the angle at which a maximum of order k > 0 is observed increases. This means that for the same k, the purple stripe is closest to the center of the screen, and the red one is farthest away.
In figure 21.3, b shows the distribution of light intensity on the screen depending on the distance to its center. The main part of the light energy is concentrated in the central maximum. As the order of the maximum increases, its intensity rapidly decreases. Calculations show that I 0:I 1:I 2 = 1:0.047:0.017.
If the slit is illuminated with white light, then the central maximum will be white on the screen (it is common for all wavelengths). Side maxima will consist of colored bands.
A phenomenon similar to slit diffraction can be observed on a razor blade.
21.3. Diffraction grating
In the case of slit diffraction, the intensities of the maxima of the order k > 0 are so insignificant that they cannot be used to solve practical problems. Therefore, as a spectral instrument is used diffraction grating, which is a system of parallel equally spaced slots. A diffraction grating can be obtained by applying opaque strokes (scratches) to a plane-parallel glass plate (Fig. 21.4). The space between the strokes (slits) transmits light.
Strokes are applied to the surface of the grating with a diamond cutter. Their density reaches 2000 strokes per millimeter. In this case, the width of the grating can be up to 300 mm. The total number of lattice slots is denoted N.
The distance d between the centers or edges of adjacent slots is called constant (period) diffraction grating.
The diffraction pattern on the grating is defined as the result of mutual interference of waves coming from all slits.
The path of the rays in the diffraction grating is shown in Fig. 21.5.
Let a plane monochromatic light wave fall on the grating, the direction of propagation of which is perpendicular to the plane of the grating. Then the slot surfaces belong to the same wave surface and are sources of coherent secondary waves. Consider secondary waves whose propagation direction satisfies the condition
After passing through the lens, the rays of these waves will intersect at point O.
The product dsina is equal to the path difference (δ) between the rays coming from the edges of neighboring slots. When condition (21.4) is satisfied, the secondary waves arrive at the point O" in the same phase and a maximum of the interference pattern appears on the screen. The maxima satisfying condition (21.4) are called principal maxima of the order k. The condition (21.4) itself is called the basic formula of a diffraction grating.
Major Highs during grating diffraction are observed for the directions of rays of secondary waves that satisfy the condition: dsinα = ± κ λ; k = 0,1,2,...
Rice. 21.4. Cross section of the diffraction grating (a) and its symbol (b)
Rice. 21.5. Diffraction of light on a diffraction grating
For a number of reasons that are not considered here, there are (N - 2) additional maxima between the main maxima. With a large number of slits, their intensity is negligible, and the entire space between the main maxima looks dark.
Condition (21.4), which determines the positions of all the main maxima, does not take into account diffraction by a single slit. It may happen that for some direction the condition maximum for the lattice (21.4) and the condition minimum for the slot (21.2). In this case, the corresponding main maximum does not arise (formally, it exists, but its intensity is zero).
The greater the number of slots in the diffraction grating (N), the more light energy passes through the grating, the more intense and sharper the maxima will be. Figure 21.6 shows the intensity distribution graphs obtained from gratings with different numbers of slots (N). Periods (d) and slot widths (b) are the same for all gratings.
Rice. 21.6. Intensity distribution for different values of N
21.4. Diffraction spectrum
It can be seen from the basic formula of the diffraction grating (21.4) that the diffraction angle α, at which the main maxima are formed, depends on the wavelength of the incident light. Therefore, the intensity maxima corresponding to different wavelengths are obtained in different places on the screen. This makes it possible to use the grating as a spectral device.
Diffraction spectrum- spectrum obtained using a diffraction grating.
When white light falls on a diffraction grating, all maxima, except for the central one, decompose into a spectrum. The position of the maximum of order k for light with wavelength λ is given by:
The longer the wavelength (λ), the farther from the center is the kth maximum. Therefore, the purple region of each main maximum will be facing the center of the diffraction pattern, and the red region will be outward. Note that when white light is decomposed by a prism, violet rays are more strongly deflected.
Writing down the basic lattice formula (21.4), we indicated that k is an integer. How big can it be? The answer to this question is given by the inequality |sinα|< 1. Из формулы (21.5) найдем
where L is the lattice width and N is the number of strokes.
For example, for a grating with a density of 500 lines per mm, d = 1/500 mm = 2x10 -6 m. For green light with λ = 520 nm = 520x10 -9 m, we get k< 2х10 -6 /(520 х10 -9) < 3,8. Таким образом, для такой решетки (весьма средней) порядок наблюдаемого максимума не превышает 3.
21.5. Characteristics of a diffraction grating as a spectral instrument
The basic formula of a diffraction grating (21.4) makes it possible to determine the wavelength of light by measuring the angle α corresponding to the position of the k-th maximum. Thus, the diffraction grating makes it possible to obtain and analyze the spectra of complex light.
Spectral characteristics of the grating
Angular dispersion - a value equal to the ratio of the change in the angle at which the diffraction maximum is observed to the change in wavelength:
where k is the order of the maximum, α -
the angle at which it is observed.
The higher the angular dispersion, the higher the order k of the spectrum and the smaller the grating period (d).
Resolution(resolving power) of a diffraction grating - a value that characterizes its ability to give
where k is the order of maximum and N is the number of lattice lines.
It can be seen from the formula that close lines that merge in the spectrum of the first order can be perceived separately in the spectra of the second or third orders.
21.6. X-ray diffraction analysis
The basic formula of a diffraction grating can be used not only to determine the wavelength, but also to solve the inverse problem - finding the diffraction grating constant from a known wavelength.
The structural lattice of a crystal can be taken as a diffraction grating. If a stream of X-rays is directed to a simple crystal lattice at a certain angle θ (Fig. 21.7), then they will diffract, since the distance between the scattering centers (atoms) in the crystal corresponds to
wavelength of x-rays. If a photographic plate is placed at some distance from the crystal, it will register the interference of the reflected rays.
where d is the interplanar distance in the crystal, θ is the angle between the plane
Rice. 21.7. X-ray diffraction on a simple crystal lattice; dots indicate the arrangement of atoms
crystal and the incident x-ray beam (glancing angle), λ is the wavelength of x-ray radiation. Relation (21.11) is called the Bragg-Wulf condition.
If the X-ray wavelength is known and the angle θ corresponding to condition (21.11) is measured, then the interplanar (interatomic) distance d can be determined. This is based on X-ray diffraction analysis.
X-ray diffraction analysis - a method for determining the structure of a substance by studying the patterns of X-ray diffraction on the samples under study.
X-ray diffraction patterns are very complex because a crystal is a three-dimensional object and X-rays can diffract on different planes at different angles. If the substance is a single crystal, then the diffraction pattern is an alternation of dark (exposed) and light (unexposed) spots (Fig. 21.8, a).
In the case when the substance is a mixture of a large number of very small crystals (as in a metal or powder), a series of rings appears (Fig. 21.8, b). Each ring corresponds to a diffraction maximum of a certain order k, while the radiograph is formed in the form of circles (Fig. 21.8, b).
Rice. 21.8. X-ray pattern for a single crystal (a), X-ray pattern for a polycrystal (b)
X-ray diffraction analysis is also used to study the structures of biological systems. For example, the structure of DNA was established by this method.
21.7. Diffraction of light by a circular hole. Aperture resolution
In conclusion, let us consider the question of the diffraction of light by a round hole, which is of great practical interest. Such holes are, for example, the pupil of the eye and the lens of the microscope. Let light from a point source fall on the lens. The lens is a hole that only lets through part light wave. Due to diffraction on the screen located behind the lens, a diffraction pattern will appear, shown in Fig. 21.9, a.
As for the gap, the intensities of side maxima are small. The central maximum in the form of a light circle (diffraction spot) is the image of a luminous point.
The diameter of the diffraction spot is determined by the formula:
where f is the focal length of the lens and d is its diameter.
If light from two point sources falls on the hole (diaphragm), then depending on the angular distance between them (β) their diffraction spots can be perceived separately (Fig. 21.9, b) or merge (Fig. 21.9, c).
We present without derivation a formula that provides a separate image of nearby point sources on the screen (diaphragm resolution):
where λ is the wavelength of the incident light, d is the aperture (diaphragm) diameter, β is the angular distance between the sources.
Rice. 21.9. Diffraction by a circular hole from two point sources
21.8. Basic concepts and formulas
End of table
21.9. Tasks
1. The wavelength of light incident on the slit perpendicular to its plane fits into the width of the slit 6 times. At what angle will the 3rd diffraction minimum be seen?
2. Determine the period of a grating with a width L = 2.5 cm and N = 12500 lines. Write your answer in micrometers.
Solution
d = L/N = 25,000 µm/12,500 = 2 µm. Answer: d = 2 µm.
3. What is the diffraction grating constant if the red line (700 nm) in the 2nd order spectrum is visible at an angle of 30°?
4. The diffraction grating contains N = 600 lines per L = 1 mm. Find the largest order of the spectrum for light with a wavelength λ = 600 nm.
5. Orange light at 600 nm and green light at 540 nm pass through a diffraction grating having 4000 lines per centimeter. What is the angular distance between the orange and green maxima: a) first order; b) third order?
Δα \u003d α op - α z \u003d 13.88 ° - 12.47 ° \u003d 1.41 °.
6.
Find the highest order of the spectrum for the yellow sodium line λ = 589 nm if the lattice constant is d = 2 μm.
Solution
Let's bring d and λ to the same units: d = 2 μm = 2000 nm. By formula (21.6) we find k< d/λ = 2000/ 589 = 3,4. Answer: k = 3.
7. A diffraction grating with N = 10,000 slots is used to study the light spectrum in the 600 nm region. Find the minimum wavelength difference that can be detected by such a grating when observing second-order maxima.
Continuing the reasoning for five, six slots, etc., we can establish the following rule: if there are slots between two adjacent maxima, minima are formed; the difference in the path of the rays from two adjacent slits for the maxima should be equal to an integer X, and for the minima - The diffraction spectrum from the slits has the form shown in Fig. Additional maxima located between two adjacent minima create a very weak illumination (background) on the screen.
The main part of the energy of the light wave that has passed 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 greater the amount of light energy that passes through the grating, the more minima are formed between adjacent main maxima, the more intense and sharper the maxima will be.
If the light incident on the diffraction grating consists of two monochromatic radiations with wavelengths and their main maxima are 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 so close to each other that they merge into one common bright band (Fig. IV.27, b). If the top of one maximum coincides with or is located further (a) than the nearest minimum of the second wave, then the presence of two waves can be reliably established by the distribution of illumination on the screen (or, as they say, these waves can be “resolved”).
Let us derive the condition for the solvability of two waves: the maximum (i.e., the maximum order) of the wave will turn out, according to formula (1.21), at an angle that satisfies the condition.
the minimum of the wave closest to its maximum (Fig. IV.27, c). According to the above, in order to obtain the nearest minimum, an additional addition should be added 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 slots 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 waves can be resolved.
In a diffraction grating, i.e., the number of slots is large, but the order of the spectrum that can be used for measuring 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, which depends on the distances to the mirrors (see Fig. IV. 14), is large, so the order of the observed spectrum is measured by very large numbers.
The angular distance between two neighboring maxima of two nearby waves depends on the order of the spectrum and the grating period
The grating period can be replaced by the number of slots per unit length of the grating:
It was assumed above that the rays incident on the diffraction grating are perpendicular to its plane. With oblique incidence of the rays (see Fig. IV.22, b), the zero maximum will be shifted and will turn out in the direction.
are close to each other in size, so
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 the angular deviation with oblique incidence is greater than with 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-wavelength (for example, X-ray) radiation and measure their wavelengths.
If a plane light wave does not pass 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
At the second dark ring The share of the central light circle, called the Airy spot, accounts for about 85% of the total radiation power that has passed through the hole and lens; the remaining 15% is distributed between 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 strips" 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: it is equal to unity along the slit, and zero along the opaque strip. Therefore, at the interface between the slot and the strip, the transmittance changes abruptly from unity to zero.
However, diffraction gratings can also be made with a different transmission coefficient distribution. For example, if an absorbing layer with a periodically changing thickness is applied to a transparent plate (or film), then instead of alternating completely
transparent slits and completely opaque stripes, it is possible to obtain a diffraction grating with a smooth change in the transmittance (in the direction perpendicular to the slits or stripes). Of particular interest are gratings in which the transmittance varies according to a sinusoidal law. The diffraction spectrum of such gratings does not consist of many maxima (as shown for ordinary gratings in Fig. IV.26), but only of a central maximum and two symmetrically located first-order maxima
For a spherical wave, it is possible to make diffraction gratings consisting of a plurality of concentric annular slots separated by opaque rings. It is possible, for example, to ink concentric rings on a glass plate (or on a transparent film); while the 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 rectilinear slits and stripes, in order to obtain a distinct interference pattern, it was necessary that the slit width and grating period be constant; for zone plates, the necessary radii and thicknesses of the rings must be calculated for this purpose. Zone gratings can also be made with a smooth, for example sinusoidal, change in the transmittance along the radius.
One of the important optical devices that have found their application in the analysis of emission and absorption spectra is a diffraction grating. This article provides information that allows you to understand what a diffraction grating is, what is the principle of its operation and how you can independently calculate the position of the maxima in the diffraction pattern that it gives.
At the beginning of the 19th century, the English scientist Thomas Young, studying the behavior of a monochromatic beam of light when it was divided in half by a thin plate, obtained a diffraction pattern. It was a sequence of bright and dark stripes on the screen. Using the concept of light as a wave, Jung correctly explained the results of his experiments. The picture he observed was due to the phenomena of diffraction and interference.
Diffraction is understood as the curvature of the rectilinear trajectory of wave propagation when it hits an opaque obstacle. Diffraction can manifest itself as a result of the wave bending around an obstacle (this is possible if the wavelength is much larger than the obstacle) or as a result of a curvature of the trajectory, when the dimensions of the obstacle are comparable to the wavelength. An example for the latter case is the penetration of light into slots and small round holes.
The phenomenon of interference is the superposition of one wave on another. The result of this overlay is a curvature of the sinusoidal form of the resulting wave. Particular cases of interference are either the maximum amplification of the amplitude, when two waves arrive in the considered zone of space in one phase, or the complete attenuation of the wave process, when both waves meet in the given zone in antiphase.
The described phenomena allow us to understand what a diffraction grating is and how it works.
Diffraction grating
The name itself says what a diffraction grating is. It is an object that consists of periodically alternating transparent and opaque stripes. It can be obtained by gradually increasing the number of slots on which the wave front falls. This concept is generally applicable to any wave, however, it has found use only for the region of visible electromagnetic radiation, that is, for light.
A diffraction grating is usually characterized by three main parameters:
- Period d is the distance between two slits through which light passes. Since the wavelengths of light are in the range of a few tenths of a micrometer, the value of d is of the order of 1 μm.
- The grating constant a is the number of transparent slots that are located on a length of 1 mm of the grating. The lattice constant is the reciprocal of the period d. Its typical values are 300-600 mm-1. As a rule, the value of a is written on the diffraction grating.
- The total number of slots is N. This value is easily obtained by multiplying the length of the diffraction grating by its constant. Since typical lengths are several centimeters, each grating contains about 10-20 thousand slots.
Transparent and reflective grilles
It has been described above what a diffraction grating is. Now let's answer the question of what it really is. There are two types of such optical objects: transparent and reflective.
A transparent grating is a glass thin plate or a transparent plastic plate on which strokes are applied. The grooves of the diffraction grating are an obstacle for light, it cannot pass through them. The stroke width is the aforementioned period d. The transparent gaps remaining between the strokes play the role of slits. When performing laboratory work, this type of lattice is used.
A reflective grating is a polished metal or plastic plate, on which grooves of a certain depth are applied instead of strokes. The period d is the distance between the grooves. Reflective gratings are often used in the analysis of radiation spectra, since their design allows the distribution of the intensity of the diffraction pattern maxima in favor of higher-order maxima. The CD optical disc is a prime example of this kind of grating.
The principle of operation of the lattice
For example, consider a transparent optical device. Let us assume that light having a flat front is incident on a diffraction grating. This is a very important point, since the formulas below take into account that the wavefront is flat and parallel to the plate itself (Fraunhofer diffraction). Strokes distributed according to the periodic law introduce a perturbation into this front, as a result of which a situation is created at the output of the plate, as if many secondary coherent radiation sources are operating (the Huygens-Fresnel principle). These sources lead to the appearance of diffraction.
From each source (the gap between the strokes) a wave propagates that is coherent to all other N-1 waves. Now suppose that a screen is placed at some distance from the plate (the distance must be sufficient for the Fresnel number to be much less than one). If you look at the screen along a perpendicular drawn to the center of the plate, then as a result of the interference superposition of waves from these N sources, for some angles θ, bright stripes will be observed, between which there will be a shadow.
Since the condition of interference maxima is a function of the wavelength, if the light falling on the plate was white, multi-colored bright stripes would appear on the screen.
Basic Formula
As mentioned, the incident flat wave front on the diffraction grating is displayed on the screen in the form of bright bands separated by a shadow region. Each bright band is called a maximum. If we consider the amplification condition for waves arriving in the region under consideration in the same phase, then we can obtain the formula for the maxima of the diffraction grating. It looks like this:
Where θ m are the angles between the perpendicular to the center of the plate and the direction to the corresponding maximum line on the screen. The value m is called the order of the diffraction grating. It takes integer values and zero, that is, m = 0, ±1, 2, 3, and so on.
Knowing the grating period d and the wavelength λ that falls on it, we can calculate the position of all the maxima. Note that the maxima calculated by the formula above are called principal. In fact, between them there is a whole set of weaker maxima, which are often not observed in the experiment.
You should not think that the picture on the screen does not depend on the width of each slit on the diffraction plate. The slot width does not affect the position of the maxima, but it does affect their intensity and width. Thus, with a decrease in the gap (with an increase in the number of strokes on the plate), the intensity of each maximum decreases, and its width increases.
Diffraction grating in spectroscopy
Having dealt with questions about what a diffraction grating is and how to find the maxima that it gives on the screen, it is curious to analyze what will happen to white light if a plate is irradiated with it.
We write again the formula for the main maxima:
If we consider a specific order of diffraction (for example, m = 1), then it is clear that the larger λ, the farther from the central maximum (m = 0) the corresponding bright line will be. This means that white light is split into a range of rainbow colors that are displayed on the screen. Moreover, starting from the center, purple and blue colors will first appear, and then yellow, green will go, and the farthest maximum of the first order will correspond to red.
A property of the wavelength diffraction grating is used in spectroscopy. When it is necessary to know the chemical composition of a luminous object, for example, a distant star, its light is collected by mirrors and directed to a plate. By measuring the angles θ m, one can determine all the wavelengths of the spectrum, and hence the chemical elements that emit them.
Below is a video that demonstrates the ability of gratings with different N numbers to split the light from the lamp.
The concept of "angular dispersion"
This value is understood as the change in the angle of occurrence of the maximum on the screen. If we change the length of monochromatic light by a small amount, we get:
If the left and right parts of the equality in the formula for the main maxima are differentiated with respect to θ m and λ, respectively, then an expression for the dispersion can be obtained. It will be equal to:
The dispersion must be known when determining the resolution of the plate.
What is resolution?
In simple terms, this is the ability of a diffraction grating to separate two waves with close λ values into two separate maxima on the screen. According to Lord Rayleigh's criterion, two lines can be distinguished if the angular distance between them is greater than half their angular width. The half-width of the line is determined by the formula:
Δθ 1/2 = λ/(N*d*cos(θm))
The difference between the lines according to the Rayleigh criterion is possible if:
Substituting the formula for the variance and half-width, we obtain the final condition:
The resolution of the grating increases with an increase in the number of slots (strokes) on it and with an increase in the order of diffraction.
The solution of the problem
Let's apply the acquired knowledge to solve a simple problem. Let light fall on the diffraction grating. It is known that the wavelength is 450 nm, and the grating period is 3 μm. What is the maximum order of diffraction that can be observed on a crane?
To answer the question, you should substitute the data into the lattice equation. We get:
sin(θ m) = m*λ/d = 0.15*m
Since the sine cannot be greater than one, then we obtain that the maximum order of diffraction for the specified conditions of the problem is 6.
