A diffraction grating is often used to determine the spectrum of the light incident on it, since it allows it to be split into individual colors. In this article, we consider what a diffraction grating is, a constant and its period, and we give an example of solving the problem using this optical device.
Diffraction phenomenon
Its essence is to change the direction of wave propagation when it encounters an obstacle in its path. The diffraction result is clearly distinguishable if the size of the obstacle is comparable to or less than the wavelength. The diffracted wave is able to penetrate in the area behind the obstacle, where it could not get if it moved along a straight line.
The figure below shows an example of diffraction of a sea wave.
It is seen how the direct wave front after passing through the obstacle takes on the shape of a circle.
The mathematical description of diffraction is carried out using the Huygens-Fresnel principle, which states that each point of the wavefront is a source of a secondary wave of some intensity.
Diffraction is often accompanied by interference. Thanks to these two phenomena, one can observe the so-called diffraction patterns.
Diffraction grating
This lattice is a transparent plate on which opaque strokes with a certain period are applied. When light passes through such a plate, it introduces a periodic perturbation into its wavefront. As a result, a number of secondary sources arise that emit coherent waves. As a result of interference, coherent waves form on the screen a set of maxima and minima, that is, a diffraction pattern.
An important characteristic of the lattice is the number of strokes N deposited per 1 mm of its length. This value of N is called the diffraction grating constant, d is the grating period, which is equal to the reciprocal of N. The period is usually expressed in microns. The geometric meaning of d is simple - it is the distance between two adjacent strokes on the lattice. Period d is a parameter of the lattice equation, which will be given in the next section.
The grating described above is called translucent or transparent. There is also a reflected diffraction grating, which is a collection of periodic grooves deposited on a smooth surface of the material. An example of this type of grill is a DVD.
Lattice equation
In the far field approximation (Fraunhofer diffraction), the equation for the lattice is as follows:
sin (θ m ) = m * λ / d.
This expression shows at what angles θ m the maxima (bright stripes) will appear on the screen. Here m = 0, ± 1, 2, 3, ... These numbers are called the diffraction order. The quantity λ is the wavelength that diffracts on the grating, d is the period, that is, the reciprocal of the diffraction grating constant. The length of the lattice itself, as can be seen from the formula, plays no role. The angles θ m are measured between two straight lines: a perpendicular restored to the center of the lattice, and a straight line directed from the center to a maximum of the order of m.
The above formula directly follows from the condition of the interference maximum. In laboratory work, it is used to determine either the constant diffraction grating when λ is known, or the wavelength when d is known.
The use of a diffraction grating in spectroscopy
The above lattice equation allows us to conclude that the angles θ m at which the maxima appear depend on the wavelength. The larger it is, the larger these angles (long waves are better diffracted than short ones). This means that if you direct white light to the grating, it will decompose it into a number of colors like a dispersion prism. Moreover, the sequence of colors, starting from the center (m = 0), will go from purple to red.
Each maximum for the corresponding diffraction order and white light will represent a “rainbow”. The only maximum that will always be white is the center or zero (m = 0).
The phenomenon of the decomposition of white light into its individual components allows the use of a diffraction grating in spectroscopy. For example, passing light from a distant galaxy through a lattice, and then analyzing the spectrum obtained, it is possible to tell with certainty which elements are present in the galaxy, what is their temperature, at what speed does this galaxy move relative to us (in the latter case, the Doppler effect is taken into account).
Problem solving example
We show how to use the lattice equation, using the example of solving a simple problem. Let the diffraction grating constant be 300 strokes per 1 mm. It is necessary to determine at what angle the first order maximum will be observed for the violet (400 nm) and red (700 nm) waves.
Given that the number of strokes N is inversely proportional to the period d, we rewrite the lattice equation in the form:
sin (θ m ) = m * λ * N.
The angle for the first maximum is:
θ 1 = arcsin (λ * N).
We substitute the data in SI units into this expression, we get:
For violet: θ 1 = arcsin (400 * 10 -9 * 300 * 10 3 ) = 6.89 o .
For red: θ 1 = arcsin (700 * 10 -9 * 300 * 10 3 ) = 12.12 o .
If the screen is placed at a distance of 1 meter from the grating, then the red and violet stripes for the first diffraction order will be at a distance of about 9 cm from each other.