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 to understand what a diffraction grating is, what is the principle of its operation, and how to independently calculate the position of the maxima in the diffraction pattern that it gives.
Diffraction and interference
At the beginning of the 19th century, the English scientist Thomas Jung, studying the behavior of a monochromatic light beam when it is divided in half by a thin plate, obtained a diffraction picture. 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 arose due to the phenomena of diffraction and interference.
Diffraction is understood to mean the curvature of a rectilinear wave propagation path when it hits an opaque obstacle. Diffraction can occur as a result of the wave enveloping an obstacle (this is possible if the wavelength is much larger than the obstacle) or as a result of 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 consists in superimposing one wave on another. The result of this overlap is the curvature of the sinusoidal shape 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 this zone in antiphase.
The described phenomena make it possible 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. You can get it if you gradually increase the number of slots on which the wave front falls. This concept is generally applicable to any wave, but it has found use only for the region of visible electromagnetic radiation, that is, for light.
The 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 lie in the range of several tenths of a micrometer, the value of d is of the order of 1 μm.
- The constant lattice a is the number of transparent slits that are located at a length of 1 mm of the lattice. 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 a diffraction grating.
- The total number of slits N. This value is easily obtained by multiplying the length of the diffraction grating by its constant. Since typical lengths are several centimeters, each lattice contains about 10-20 thousand slots.
Transparent and reflective gratings
What was described above is a diffraction grating. Now we will answer the question of what it really is. There are two types of such optical objects: transparent and reflective.
A transparent lattice is a glass thin plate or a plate of transparent plastic, on which strokes are applied. Strokes of the diffraction grating are an obstacle to light, through which it cannot pass. Stroke width - this is the aforementioned period d. The transparent gaps remaining between the strokes play the role of gaps. When performing laboratory work, this type of grating is used.
A reflective grating is a polished metal or plastic plate on which grooves of a certain depth are applied instead of strokes. Period d is the distance between the grooves. Reflective gratings are often used in the analysis of radiation spectra, because their design allows you to distribute the intensity of the maxima of the diffraction pattern in favor of higher order maxima. An optical CD is a prime example of this type of diffraction grating.
The principle of operation of the grill
For example, consider a transparent optical device. Suppose that light having a planar 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). The strokes distributed according to the periodic law introduce perturbation into this front, as a result of which a situation arises at the exit from the plate, as if many secondary coherent radiation sources are working (Huygens-Fresnel principle). These sources lead to diffraction.
A wave propagates from each source (the gap between the dashes), which is coherent with all other N-1 waves. Now suppose that a screen is placed at some distance from the plate (the distance must be sufficient so that the Fresnel number is much less than unity). If you look at the screen along the perpendicular drawn to the center of the plate, then as a result of interference superposition of waves from these N sources for some angles θ, bright bands will be observed, between which there will be a shadow.
Since the condition of interference maxima is a function of wavelength, if the light incident on the plate was white, multi-colored bright stripes will appear on the screen.
Basic formula
As was said, 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 bar is called a maximum. If we consider the amplification condition for waves arriving in the considered region in the same phase, then we can obtain the formula for the maxima of the diffraction grating. It has the following form:
sin (θ m ) = m * λ / d
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 quantity 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 lattice 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 the main ones. In fact, between them there is a whole set of weaker maxima, which are often not observed in the experiment.
Do not think that the picture on the screen does not depend on the width of each slit on the diffraction plate. The width of the gap does not affect the position of the maxima, however, it affects their intensity and width. So, 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 interesting to analyze what will happen to white light if they irradiate the plate.
We again write the formula for the main maxima:
sin (θ m ) = m * λ / d
If we consider a specific diffraction order (for example, m = 1), then it can be seen that the larger λ, the farther from the central maximum (m = 0) there will be a corresponding bright line. This means that white light is split into a series of rainbow colors that are displayed on the screen. Moreover, starting from the center, first violet and blue colors will appear, and then yellow, green and the farthest maximum of the first order will correspond to red.
The wavelength grating property is used in spectroscopy. When it is necessary to find out the chemical composition of a luminous object, for example, a distant star, its light is collected by mirrors and sent to a plate. By measuring the angles θ m, it is possible to 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 numbers N to split the light from a lamp.
The concept of "angular dispersion"
This value is understood as the change in the angle of the maximum on the screen. If you change the length of monochromatic light by a small amount, we get:
D = dθ m / dλ
If the left and right sides of the equality in the formula for the main maxima are differentiated with respect to θ m and λ, respectively, then we can obtain the expression for the variance. It will be equal to:
D = m / (d * cos (θ m ))
The dispersion must be known when determining the resolution of the plate.
What is the resolution?
In simple words, this is the ability of the 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 (θ m ))
The difference between the lines in accordance with the Rayleigh criterion is possible if:
Δθ m > Δθ 1/2
Substituting the formula for the variance and half-width, we obtain the final condition:
Δλ> λ / (N * m)
The resolving power of the grating increases with an increase in the number of slits (strokes) on it and with an increase in the diffraction order.
The solution of the problem
We 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 lattice period is 3 μm. What is the maximum diffraction order that can be observed on a crane?
To answer the question, substitute the data in the lattice equation. We get:
sin (θ m ) = m * λ / d = 0.15 * m
Since the sine cannot be greater than unity, then we find that the maximum diffraction order for the indicated conditions of the problem is 6.