One of the characteristic properties of any wave is its ability to diffract on obstacles whose size is comparable to the length of this wave. The noted property is used in the so-called diffraction gratings. What this is and how they can be used to analyze the radiation and absorption spectra of different materials is considered in the article.
Diffraction phenomenon
This phenomenon consists in changing the path of the rectilinear wave propagation when an obstacle appears in its path. Unlike refraction and reflection, diffraction is noticeable only on very small obstacles whose geometric dimensions are of the order of the wavelength. There are two types of diffraction:
- bending around the wave of an object when the wavelength is much larger than the size of this object;
- wave scattering when passing through holes of different geometric shapes, when the dimensions of the holes are smaller than the wavelength.
The diffraction phenomenon is characteristic of sound, sea and electromagnetic waves. Further in the article we will consider a diffraction grating only for light.
The phenomenon of interference
Diffraction patterns arising at various obstacles (round holes, slits 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 emit waves, preserving the phase difference between them (the property of coherence), then a stable interference pattern can be observed in time.
The positions of the maxima (bright areas) and minima (dark zones) are 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 the point is observed minimum. On the contrary, if two waves come to the point in the same phase, then they will strengthen each other (maximum).
Both phenomena were first described by the Englishman Thomas Jung in 1801, when he studied diffraction by two slits. However, the Italian Grimaldi observed this phenomenon for the first time in 1648, when he examined the diffraction pattern produced by sunlight passing through a small hole. Grimaldi could not explain the results of his experiments.
The mathematical method used to study diffraction
This method is called the Huygens-Fresnel principle. It consists in the statement that in the process of wavefront propagation, 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 Augustine Fresnel in the first half of the XIX century. In this case, 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.
Near and far field diffraction
Diffraction is a rather complex phenomenon, the exact mathematical solution for which requires consideration of Maxwell's theory of electromagnetism. Therefore, in practice, only particular 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" refers to the distance to the screen on which the diffraction pattern is observed.
The transition between the Fraunhofer and Fresnel diffraction patterns can be estimated by calculating the Fresnel number for a particular case. This number is defined as follows:
F = a 2 / (D * λ).
Here λ is the wavelength of light, D is the distance to the screen, a is the size of the object at which diffraction takes place.
If F << 1 (for example, 0.001), then we are talking about Fraunhofer diffraction, if F> 1, then we should consider the near-field approximations.
Many practical cases, including the use of a diffraction grating, are considered in the far field approximation.
The concept of a lattice on which waves diffract
This lattice is a small flat object on which a periodic structure, for example, strips or grooves, is applied in any way. An important parameter of such a lattice 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 bands. Denote it by the letter d, then:
d = 1 / N.
When a plane wave falls on such a lattice, then it experiences periodic disturbances. The latter are displayed on the screen in the form of a picture resulting from the interference of waves.
Types of Gratings
There are two types of diffraction gratings:
- passing, or transparent;
- reflective.
The first are made by applying opaque touches to the glass. It is with such plates that they work in laboratories, they are used in spectroscopes.
The second type, that is, reflective lattices, are made by applying periodic grooves on polished material. A striking everyday example of such a grill is a plastic CD or DVD.
Lattice equation
Considering the Fraunhofer diffraction on a grating, for the light intensity in the diffraction pattern, we can write the following expression:
I (θ) = I 0 * (sin (β) / β) 2 * [sin (N * α) / sin (α)] 2 , where
α = pi * d / λ * (sin (θ) -sin (θ 0 ));
β = pi * a / λ * (sin (θ) -sin (θ 0 )).
Parameter a is the width of one slit, 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 plane of the lattice 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 on the condition of interference maxima, we can come to the following formula:
sin (θ m ) -sin (θ 0 ) = m * λ / d.
The angle θ 0 characterizes the incident wave on the grating. If the wave front is parallel to it, then θ 0 = 0, and the last expression takes the form:
sin (θ m ) = m * λ / d.
This formula is called the diffraction grating equation. The value of m takes on any integers, including negative and zero, it is called the diffraction order.
Analysis of the lattice equation
In the previous paragraph, we found out that the position of the main maxima is described by the equation:
sin (θ m ) = m * λ / 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 the angular distance to the maximum that corresponds to it. Measurement of the corresponding θ m for each wave makes it possible to calculate its length and, therefore, determine the entire spectrum of the emitting object. Comparing this spectrum with data from a known base, we can say by the radiation of which chemical elements it is emitted.
The process described above is used in spectrometers.
Grating resolution
It is understood as such a difference between two wavelengths that appear in the form of separate lines in the diffraction pattern. The fact is that each line has a certain thickness, when two waves with close values of λ and λ + Δλ are diffracted, then the corresponding lines in the picture can merge into one. In the latter case, it is said that the resolution of the grating is less than Δλ.
Omitting the arguments regarding the derivation of the formula for the resolving power of the lattice, we present its final form:
Δλ> λ / (m * N).
This small formula allows us to conclude: using the grating, the closer the wavelengths (Δλ) can be separated, the longer the wavelength of light λ, the greater the number of strokes per unit length (lattice constant N), and the higher the diffraction order. Let us dwell on the latter in more detail.
If you look at the diffraction pattern, then with an increase in m the distance between adjacent wavelengths actually increases. 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 rapidly decreases 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 reflecting gratings, on which a diffraction pattern is obtained for large θ 0 .
Next, we consider the use of the lattice equation to solve several problems.
Tasks for determining diffraction angles, diffraction order and lattice constant
Here are some examples of solving several problems:
- To determine the period of the diffraction grating, the following experiment is carried out: a monochromatic light source whose wavelength is a known quantity is taken. Using lenses, a parallel wave front is formed, that is, conditions are created for Fraunhofer diffraction. Then this front is directed to a diffraction grating, the period of which is unknown. In the resulting picture, angles for different orders are measured using a goniometer. Then, according to the formula, the value of the unknown period is calculated. We carry out this calculation with a specific example.
Let the wavelength of light be 500 nm, and the angle for the first diffraction order is 21 o . From these data, it is necessary to determine the period of the diffraction grating d.
Using the lattice equation, we express d and substitute the data:
d = m * λ / sin (θ m ) = 1 * 500 * 10 -9 / sin (21 o ) ≈ 1.4 μm.
Then the lattice constant N is equal to:
N = 1 / d ≈ 714 strokes per 1 mm.
- Light normally falls on a diffraction grating having a period of 5 μm. Knowing that the wavelength is λ = 600 nm, it is necessary to find the angles at which the first and second order maxima appear.
For the first maximum we get:
sin (θ 1 ) = λ / d => θ 1 = arcsin (λ / d) ≈ 6.9 o .
The second maximum appears for the angle θ 2 :
θ 2 = arcsin (2 * λ / d) ≈ 13.9 o .
- Monochromatic light is incident on a diffraction grating with a period of 2 μm. Its wavelength is 550 nm. It is necessary to find how many diffraction orders appear on the resulting picture on the screen.
This type of problem is solved as follows: first, the dependence of the angle θ m on the diffraction order for the conditions of the problem should be determined. After this, it will be necessary to take into account that the sine function cannot take values greater than one. The latter fact will provide an answer to this problem. We perform the described actions:
sin (θ m ) = m * λ / d = 0.275 * m.
This equality shows that for m = 4 the expression on the right-hand side will become 1.1, and for m = 3 it will be 0.825. This means that using a diffraction grating with a period of 2 μm at a wavelength of 550 nm, a maximum of the 3rd diffraction order can be obtained.
The task of calculating the resolution of the grating
Suppose that for the experiment they are going to use a diffraction grating with a period of 10 μm. It is necessary to calculate at what minimum length 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 determining the resolving power of the lattice under consideration for a given wavelength. So, the two waves can differ by Δλ> λ / (m * N). Since the lattice constant is inversely proportional to the period d, this expression can be written as follows:
Δλ> λ * d / m.
Now for the wavelength λ = 580 nm, we write the lattice equation:
sin (θ m ) = m * λ / d = 0.058 * m.
Whence we get that the maximum order m will be 17. Substituting this number in the formula for Δλ, we have:
Δλ> 580 * 10 -9 * 10 * 10 -6 / 17 = 3.4 * 10 -13 or 0.00034 nm.
We obtained a very high resolution when the period of the diffraction grating is 10 μm. In practice, as a rule, it is not achieved due to the low intensities of the maxima of high diffraction orders.