One of the characteristic properties of any wave is its ability to diffract on obstacles, the size of which is comparable to the wavelength of this wave. This property is used in the so-called diffraction gratings. What they are, and how they can be used to analyze the emission and absorption spectra of different materials, is discussed in the article.
Diffraction phenomenon
This phenomenon consists in changing the trajectory of the rectilinear propagation of a wave when an obstacle appears on its path. Unlike refraction and reflection, diffraction is noticeable only at very small obstacles, the geometric dimensions of which are of the order of a wavelength. There are two types of diffraction:
- wave bending around an object when the wavelength is much larger than the size of this object;
- scattering of a wave when passing through holes of different geometric shapes, when the dimensions of the holes are smaller than the wavelength.
The phenomenon of diffraction is characteristic of sound, sea and electromagnetic waves. Further in the article, we will consider a diffraction grating for light only.
Interference phenomenon
Diffraction patterns appearing on various obstacles (round holes, slots and gratings) are the result of not only diffraction, but also interference. The essence of the latter is the superposition of waves on each other, which are emitted by different sources. If these sources radiate waves while maintaining a phase difference between them (the property of coherence), then a stable interference pattern can be observed in time.
The position of the maxima (bright areas) and minima (dark zones) is explained as follows: if two waves arrive at a given point in antiphase (one with a maximum and the other with a minimum absolute amplitude), then they "destroy" each other, and a minimum is observed at the point. On the contrary, if two waves come in the same phase to a point, then they will reinforce each other (maximum).
Both phenomena were first described by the Englishman Thomas Young in 1801, when he studied diffraction by two slits. However, the Italian Grimaldi first observed this phenomenon in 1648, when he studied the diffraction pattern given by sunlight passing through a small hole. Grimaldi was unable to explain the results of his experiments.
Mathematical method used to study diffraction
This method is called the Huygens-Fresnel principle. It consists in the assertion that in the processpropagation of the wave front, each of its points is a source of secondary waves, the interference of which determines the resulting oscillation at an arbitrary point under consideration.
The described principle was developed by Augustin Fresnel in the first half of the 19th century. At the same time, Fresnel proceeded from the ideas of the wave theory of Christian Huygens.
Although the Huygens-Fresnel principle is not theoretically rigorous, it has been successfully used to mathematically describe experiments with diffraction and interference.
Diffraction in the near and far fields
Diffraction is a fairly complex phenomenon, the exact mathematical solution for which requires consideration of Maxwell's theory of electromagnetism. Therefore, in practice, only special cases of this phenomenon are considered, using various approximations. If the wavefront incident on the obstacle is flat, then two types of diffraction are distinguished:
- in the near field, or Fresnel diffraction;
- in the far field, or Fraunhofer diffraction.
The words "far and near field" mean the distance to the screen on which the diffraction pattern is observed.
The transition between Fraunhofer and Fresnel diffraction can be estimated by calculating the Fresnel number for a specific case. This number is defined as follows:
F=a2/(Dλ).
Here λ is the wavelength of light, D is the distance to the screen, a is the size of the object on which diffraction occurs.
If F<1, then consideralready near-field approximations.
Many practical cases, including the use of a diffraction grating, are considered in the far field approximation.
The concept of a grating on which waves diffract
This lattice is a small flat object, on which a periodic structure, such as stripes or grooves, is applied in some way. An important parameter of such a grating is the number of strips per unit length (usually 1 mm). This parameter is called the lattice constant. Further, we will denote it by the symbol N. The reciprocal of N determines the distance between adjacent strips. Let's denote it with the letter d, then:
d=1/N.
When a plane wave falls on such a grating, it experiences periodic perturbations. The latter are displayed on the screen in the form of a certain picture, which is the result of wave interference.
Types of gratings
There are two types of diffraction gratings:
- passing, or transparent;
- reflective.
The first are made by applying opaque strokes to glass. It is with such plates that they work in laboratories, they are used in spectroscopes.
The second type, that is, reflective gratings, are made by applying periodic grooves to the polished material. A striking everyday example of such a lattice is a plastic CD or DVD disc.
Lattice equation
Considering the Fraunhofer diffraction on a grating, the following expression can be written for the light intensity in the diffraction pattern:
I(θ)=I0(sin(β)/β)2[sin(Nα) /sin(α)]2, where
α=pid/λ(sin(θ)-sin(θ0));
β=pia/λ(sin(θ)-sin(θ0)).
Parameter a is the width of one slot, and parameter d is the distance between them. An important characteristic in the expression for I(θ) is the angle θ. This is the angle between the central perpendicular to the grating plane and a specific point in the diffraction pattern. In experiments, it is measured using a goniometer.
In the presented formula, the expression in parentheses determines the diffraction from one slit, and the expression in square brackets is the result of wave interference. Analyzing it for the condition of interference maxima, we can come to the following formula:
sin(θm)-sin(θ0)=mλ/d.
Angle θ0 characterizes the incident wave on the grating. If the wave front is parallel to it, then θ0=0, and the last expression becomes:
sin(θm)=mλ/d.
This formula is called the diffraction grating equation. The value of m takes on any integers, including negative ones and zero, it is called the order of diffraction.
Lattice equation analysis
In the previous paragraph, we found outthat the position of the main maxima is described by the equation:
sin(θm)=mλ/d.
How can it be put into practice? It is mainly used when the light incident on a diffraction grating with a period d is decomposed into individual colors. The longer the wavelength λ, the greater will be the angular distance to the maximum that corresponds to it. Measuring the corresponding θm for each wave allows you to calculate its length, and therefore determine the entire spectrum of the radiating object. Comparing this spectrum with the data from a known database, we can say which chemical elements emitted it.
The above process is used in spectrometers.
Grid resolution
Under it is understood such a difference between two wavelengths that appear in the diffraction pattern as separate lines. The fact is that each line has a certain thickness, when two waves with close values of λ and λ + Δλ diffract, then the lines corresponding to them in the picture can merge into one. In the latter case, the grating resolution is said to be less than Δλ.
Omitting the arguments about the derivation of the formula for the grating resolution, we present its final form:
Δλ>λ/(mN).
This small formula allows us to conclude: using a grating, you can separate the closer wavelengths (Δλ), the longer the wavelength of light λ, the greater the number of strokes per unit length(lattice constant N), and the higher the order of diffraction. Let's dwell on the last one.
If you look at the diffraction pattern, then with increasing m, there really is an increase in the distance between adjacent wavelengths. However, to use high diffraction orders, it is necessary that the light intensity on them be sufficient for measurements. On a conventional diffraction grating, it falls off rapidly with increasing m. Therefore, for these purposes, special gratings are used, which are made in such a way as to redistribute the light intensity in favor of large m. As a rule, these are reflective gratings, the diffraction pattern on which is obtained for large θ0.
Next, consider using the lattice equation to solve several problems.
Tasks to determine diffraction angles, diffraction order and lattice constant
Let's give examples of solving several problems:
To determine the period of the diffraction grating, the following experiment is carried out: a monochromatic light source is taken, the wavelength of which is a known value. With the help of lenses, a parallel wave front is formed, that is, conditions for Fraunhofer diffraction are created. Then this front is directed to a diffraction grating, the period of which is unknown. In the resulting picture, the angles for different orders are measured using a goniometer. Then the formula calculates the value of the unknown period. Let's carry out this calculation on a specific example
Let the wavelength of light be 500 nm and the angle for the first order of diffraction be 21o. Based on these data, it is necessary to determine the period of the diffraction grating d.
Using the lattice equation, express d and plug in the data:
d=mλ/sin(θm)=150010-9/sin(21 o) ≈ 1.4 µm.
Then the lattice constant N is:
N=1/d ≈ 714 lines per 1 mm.
Light normally falls on a diffraction grating having a period of 5 microns. Knowing that the wavelength λ=600 nm, it is necessary to find the angles at which the maxima of the first and second orders will appear
For the first maximum we get:
sin(θ1)=λ/d=>θ1=arcsin(λ/d) ≈ 6, 9 o.
The second maximum will appear for the angle θ2:
θ2=arcsin(2λ/d) ≈ 13, 9o.
Monochromatic light falls on a diffraction grating with a period of 2 microns. Its wavelength is 550 nm. It is necessary to find how many diffraction orders will appear in the resulting picture on the screen
This type of problem is solved as follows: first, you should determine the dependence of the angle θm on the diffraction order for the conditions of the problem. After that, it will be necessary to take into account that the sine function cannot take values greater than one. The last fact will allow us to answer this problem. Let's do the described actions:
sin(θm)=mλ/d=0, 275m.
This equality shows that when m=4, the expression on the right side becomes equal to 1,1, and at m=3 it will be equal to 0.825. This means that using a diffraction grating with a period of 2 μm at a wavelength of 550 nm, you can get the maximum 3rd order of diffraction.
The problem of calculating the resolution of the grating
Assume that for the experiment they are going to use a diffraction grating with a period of 10 microns. It is necessary to calculate by what minimum wavelength the waves near λ=580 nm can differ so that they appear as separate maxima on the screen.
The answer to this problem is related to the determination of the resolution of the considered grating for a given wavelength. So, two waves can differ by Δλ>λ/(mN). Since the lattice constant is inversely proportional to the period d, this expression can be written as follows:
Δλ>λd/m.
Now for the wavelength λ=580 nm we write the lattice equation:
sin(θm)=mλ/d=0, 058m.
Where we get that the maximum order of m will be 17. Substituting this number into the formula for Δλ, we have:
Δλ>58010-91010-6/17=3, 410- 13 or 0.00034 nm.
We got a very high resolution when the period of the diffraction grating is 10 microns. In practice, as a rule, it is not achieved due to the low intensities of the maxima of high diffraction orders.
