Diffraction grating. Operating principle of a diffraction grating. Diffraction grating period
The grille looks like this from the side.
Applications are also found reflective grilles, which are obtained by applying fine strokes to a polished metal surface with a diamond cutter. Imprints on gelatin or plastic after such engraving are called replicas, but such diffraction gratings are usually of low quality, so their use is limited. Good reflective gratings are those whose total length is about 150 mm, with a total number of lines of 600 pcs/mm.
Main characteristics diffraction grating- This total number of strokes N, shading density n (number of strokes per 1 mm) and period(lattice constant) d, which can be found as d = 1/n.
The grating is illuminated by one wave front and its N transparent lines are usually considered as N coherent sources.
If we recall the phenomenon interference from many identical light sources, then light intensity is expressed according to the pattern:
where i 0 is the intensity of the light wave that passed through one slit
Based on the concept maximum wave intensity, obtained from the condition:
β = mπ for m = 0, 1, 2... etc.
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Let's move on from auxiliary angleβ to the spatial observation angle Θ, and then:
(π d sinΘ)/ λ = m π,
Major maxima appear under the following conditions:
sinΘ m = m λ/ d, with m = 0, 1, 2... etc.
Light intensity in major highs can be found according to the formula:
I m = N 2 i 0.
Therefore, it is necessary to produce gratings with a small period d, then it is possible to obtain large ray scattering angles and a wide diffraction pattern.
For example:
Continuing from the previous one example Let us consider the case when, at the first maximum, red rays (λ cr = 760 nm) deviate by an angle Θ k = 27 °, and violet rays (λ f = 400 nm) deviate by an angle Θ f = 14 °.
It can be seen that using a diffraction grating it is possible to measure wavelength one color or another. To do this, you just need to know the period of the grating and measure the angle at which the beam deviated, corresponding to the required light.
Diffraction grating– an optical device that is a collection of a large number of parallel slits, usually equally spaced from each other. 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. Cross section of a diffraction grating (a) and its graphical 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.
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.
Continuing the reasoning for five, six slits, etc., we can establish the following rule: if there are gaps between two adjacent maxima, a minima is 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 large quantity light energy will pass through the lattice, the more minima are formed between adjacent main maxima, and, consequently, 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 of small diameter (Fig. IV.28), then the diffraction spectrum (on a flat screen located in the focal plane of the lens) is a system of alternating dark and light rings. The first dark ring is obtained at an angle satisfying the condition
The second dark ring The central light circle, called the Airy spot, accounts for about 85% of the total radiation power passing through the hole and lens; the remaining 15% is distributed among the light rings surrounding this spot. The size of the Airy spot depends on the focal length of the lens.
The diffraction gratings discussed above consisted of alternating “slits” that completely transmit the light wave, and “opaque stripes” that completely absorb or reflect the radiation incident on them. We can say that in such gratings the transmittance of a light wave has only two values: along the slit it is equal to unity, and along the opaque strip it is zero. Therefore, at the boundary between the slot and the strip, the transmittance changes abruptly from unity to zero.
However, it is possible to produce diffraction gratings with a different transmittance distribution. For example, if an absorbing layer with periodically varying thickness is applied to a transparent plate (or film), then instead of alternating completely
Using transparent slits and completely opaque strips, you can obtain a diffraction grating with a smooth change in transmittance (in the direction perpendicular to the slits or strips). Of particular interest are gratings in which the transmittance varies sinusoidally. The diffraction spectrum of such gratings does not consist of many maxima (as shown for conventional gratings in Fig. IV.26), but only of a central maximum and two symmetrically located first-order maxima
For a spherical wave, diffraction gratings can be made consisting of many concentric annular slits separated by opaque rings. You can, for example, apply concentric rings with ink to a glass plate (or transparent film); 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; For zone plates, the required radii and thickness of the rings must be calculated for this purpose. Zone gratings can also be manufactured with a smooth, for example sinusoidal, change in transmittance along the radius.
1. Diffraction of light. Huygens-Fresnel principle.
2. Diffraction of light by slits in parallel rays.
3. Diffraction grating.
4. Diffraction spectrum.
5. Characteristics of a diffraction grating as a spectral device.
6. X-ray structural 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, light diffraction is the bending of light rays around the boundaries of opaque bodies, the penetration of light into the region of a geometric shadow. In phenomena associated with diffraction, there is a significant deviation in the behavior of light from the laws of geometric optics. (Diffraction is not limited to light.)
Diffraction is a wave phenomenon that manifests itself most clearly in the case when the dimensions of the obstacle are commensurate (of the same order) with the wavelength of light. With few lengths visible light associated with the rather late discovery of light diffraction (16-17 centuries).
21.1. Diffraction of light. Huygens-Fresnel principle
Diffraction of light is a complex of phenomena that are caused by 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 for constructing the wave front at time t + Δt if its position at time t is known.
1.According to Huygens' principle each point on 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 Huygens' principle using the following example. Let a plane wave fall on an obstacle with a hole, the front of which is parallel to the obstacle (Fig. 21.1).
Rice. 21.1. Explanation of Huygens' principle
Each point of the wave front isolated by the hole serves as the center of secondary spherical waves. The figure shows that the envelope of these waves penetrates the region of the geometric shadow, the boundaries of which are marked with a dashed line.
Huygens' principle says nothing about the intensity of secondary waves. This drawback was eliminated by Fresnel, who supplemented Huygens' principle with the idea 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 Huygens-Fresnel principle the magnitude of light vibrations at a certain point O is the result of the interference at this point of coherent secondary waves emitted everyone elements of the wave surface. The amplitude of each secondary wave is proportional to the area of the element dS, inversely proportional to the distance r to point O and decreases with increasing angle α between normal n to element dS and direction to point O (Fig. 21.2).
Rice. 21.2. Emission of secondary waves by wave surface elements
21.2. Slit diffraction in parallel beams
Calculations associated with the application of the Huygens-Fresnel principle are, in general, complex. math 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 flat monochromatic light wave fall on a narrow slit (AB) in an opaque barrier, the direction of propagation of which is perpendicular to the surface of the slit (Fig. 21.3, a). We place a collecting lens behind the slit (parallel to its plane), in focal plane which we will place the screen E. All secondary waves emitted from the surface of the slit in the direction parallel optical axis of the lens (α = 0), the lens comes into focus in the same phase. Therefore, at 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 slit surface 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 - light wavelength.
Rays of secondary light waves traveling in this direction will intersect at point O."
Rice. 21.3. Diffraction at one slit: a - ray path; 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 slit. 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 suppression of rays from all Fresnel zones occurs and at point O" a minimum of the interference pattern is observed.
Minimum intensity during diffraction by a slit is observed for the directions of rays of secondary waves satisfying the condition
The integer k is called on the order of the minimum.
2) n = 2k - 1 - odd number. In this case, the radiation of one Fresnel zone will remain unquenched and at point O" the maximum interference pattern will be observed.
The maximum intensity during diffraction by a slit is observed for the directions of rays of secondary waves satisfying the condition:
The integer k is called order of maximum. Recall that for the direction α = 0 we have maximum of zero order.
From formula (21.3) it follows 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 stripe is furthest 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 quickly decreases. Calculations show that I 0:I 1:I 2 = 1:0.047:0.017.
If the slit is illuminated by white light, then the central maximum on the screen will be white (it is common to all wavelengths). Side highs will consist of colored bands.
A phenomenon similar to slit diffraction can be observed on a razor blade.
21.3. Diffraction grating
In slit diffraction, the intensities of maxima of order k > 0 are so insignificant that they cannot be used to solve practical problems. Therefore, it is used as a spectral device diffraction grating, which is a system of parallel, equally spaced slits. A diffraction grating can be obtained by applying opaque streaks (scratches) to a plane-parallel glass plate (Fig. 21.4). The space between the strokes (slots) allows light to pass through.
The strokes are applied to the surface of the grating with a diamond cutter. Their density reaches 2000 lines per millimeter. In this case, the width of the grille can be up to 300 mm. The total number of grating slits is denoted N.
The distance d between the centers or edges of adjacent slits is called constant (period) diffraction grating.
The diffraction pattern on a grating is determined as the result of mutual interference of waves coming from all slits.
The path of rays in a 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 surfaces of the slots belong to the same wave surface and are sources of coherent secondary waves. Let us consider secondary waves whose direction of propagation 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 adjacent slits. When condition (21.4) is satisfied, secondary waves arrive at point O" in the same phase and a maximum interference pattern appears on the screen. Maxima that satisfy condition (21.4) are called main maxima of order k. Condition (21.4) itself is called the basic formula of a diffraction grating.
Major Highs during diffraction by a grating are observed for the directions of rays of secondary waves satisfying the condition: dsinα = ± κ λ; k = 0,1,2,...
Rice. 21.4. Cross section of a diffraction grating (a) and its symbol (b)
Rice. 21.5. Diffraction of light by a diffraction grating
For a number of reasons that are not discussed here, between the main maxima there are (N - 2) additional maxima. With a large number of slits, their intensity is negligible and the entire space between the main maxima appears dark.
Condition (21.4), which determines the positions of all main maxima, does not take into account diffraction at a separate slit. It may happen that for some direction the condition will be simultaneously satisfied 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 slits 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 intensity distribution graphs obtained from gratings with different numbers of slits (N). The periods (d) and slot widths (b) are the same for all gratings.
Rice. 21.6. Intensity distribution at different meanings N
21.4. Diffraction spectrum
From the basic formula of a diffraction grating (21.4) it is clear that the diffraction angle α, at which the main maxima are formed, depends on the wavelength of the incident light. Therefore, intensity maxima corresponding to different wavelengths are obtained in different places on the screen. This allows the grating to be used as a spectral device.
Diffraction spectrum- spectrum obtained using a diffraction grating.
When white light falls on a diffraction grating, all maxima except the central one will be decomposed into a spectrum. The position of the maximum of order k for light with wavelength λ is determined by the formula:
The longer the wavelength (λ), the farther the kth maximum is from the center. Therefore, the violet region of each main maximum will face the center of the diffraction pattern, and the red region will face outward. Note that when white light is decomposed by a prism, violet rays are more strongly deflected.
When writing 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 width of the grating, and N is the number of lines.
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 device
The basic formula of a diffraction grating (21.4) allows you to determine the wavelength of light by measuring the angle α corresponding to the position of the kth maximum. Thus, a diffraction grating makes it possible to obtain and analyze 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 maximum, α -
the angle at which it is observed.
The higher the order k of the spectrum and the smaller the grating period (d), the higher the angular dispersion.
Resolution(resolving power) of a diffraction grating - a quantity characterizing its ability to produce
where k is the order of the maximum, and N is the number of grating lines.
It is clear from the formula that close lines that merge in a first-order spectrum can be perceived separately in second- or third-order spectra.
21.6. X-ray diffraction analysis
The basic diffraction grating formula 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 onto 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
x-ray wavelength. If a photographic plate is placed at some distance from the crystal, it will register the interference of reflected rays.
where d is the interplanar distance in the crystal, θ is the angle between the plane
Rice. 21.7. X-ray diffraction by a simple crystal lattice; the dots indicate the arrangement of atoms
crystal and the incident X-ray beam (grazing angle), λ is the wavelength of the X-ray radiation. Relationship (21.11) is called Bragg-Wolfe condition.
If the wavelength of X-ray radiation is known and the angle θ corresponding to condition (21.11) is measured, then the interplanar (interatomic) distance d can be determined. X-ray diffraction analysis is based on this.
X-ray structural analysis - a method for determining the structure of a substance by studying the patterns of X-ray diffraction on the samples being studied.
X-ray diffraction patterns are very complex because the crystal is a three-dimensional object and the X-rays can diffract into different planes under 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, and the x-ray pattern 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 using this method.
21.7. Diffraction of light by a circular hole. Aperture resolution
In conclusion, let us consider the issue of light diffraction by a round hole, which is of great practical interest. Such openings are, for example, the pupil of the eye and the lens of a microscope. Let light from a point source fall on the lens. A lens is an opening that allows only Part light wave. Due to diffraction on the screen located behind the lens, a diffraction pattern will appear as shown in Fig. 21.9, a.
As for the gap, the intensities of the side maxima are low. 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 a 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).
Let us present without derivation a formula that provides a separate image of close point sources on the screen (aperture resolution):
where λ is the wavelength of the incident light, d is the diameter of the hole (diaphragm), β is the angular distance between the sources.
Rice. 21.9. Diffraction at a circular hole from two point sources
21.8. Basic concepts and formulas
End of the table
21.9. Tasks
1. The wavelength of light incident on the slit perpendicular to its plane is 6 times the width of the slit. At what angle will the 3rd diffraction minimum be visible?
2. Determine the period of a grating with width L = 2.5 cm and having 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 constant of the diffraction grating if in the 2nd order spectrum the red line (700 nm) is visible at an angle of 30°?
4. The diffraction grating contains N = 600 lines at L = 1 mm. Find highest order spectrum for light with wavelength λ = 600 nm.
5. Orange light with a wavelength of 600 nm and green light with a wavelength of 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 = α or - α z = 13.88° - 12.47° = 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 us reduce d and λ to the same units: d = 2 µm = 2000 nm. Using formula (21.6) we find k< d/λ = 2000/ 589 = 3,4. Answer: k = 3.
7. A diffraction grating with a number of slits N = 10,000 is used to study the light spectrum in the region of 600 nm. Find the minimum wavelength difference that can be detected by such a grating when observing second-order maxima.
