One of the known effects that confirm the wave nature of light is diffraction and interference. The main field of their application is spectroscopy, in which diffraction gratings are used to analyze the spectral composition of electromagnetic radiation. The formula that describes the position of the main maxima given by this lattice is considered in this article.
What are the phenomena of diffraction and interference?
Before considering the derivation of the diffraction grating formula, one should get acquainted with the phenomena due to which this grating is useful, that is, with diffraction and interference.
Diffraction is the process of changing the motion of the wavefront when it encounters an opaque obstacle in its path, the dimensions of which are comparable to the wavelength. For example, if sunlight is passed through a small hole, then on the wall you can observe not a small luminous point (what would happen if the light spread in a straight line), but a luminous spot of some sizes. This fact indicates the wave nature of light.
Interference is another phenomenon that is unique to waves. Its essence lies in the imposition of waves on each other. If the wave oscillations from several sources are consistent (they are coherent), then you can observe a stable picture of alternating light and dark areas on the screen. The minima in this picture are explained by the arrival of waves at a given point in antiphase (pi and -pi), and the maxima are the result of waves entering the considered point in one phase (pi and pi).
Both of these phenomena were first explained by the Englishman Thomas Jung when he studied the diffraction of monochromatic light on two thin slits in 1801.
Huygens-Fresnel principle and near and near field approximation
A mathematical description of the phenomena of diffraction and interference is a non-trivial task. Finding its exact solution requires complex calculations involving the Maxwell theory of electromagnetic waves. Nevertheless, in the 20s of the 19th century, the Frenchman Augustin Fresnel showed that, using Huygens's ideas about secondary wave sources, these phenomena can be successfully described. This idea led to the formulation of the Huygens-Fresnel principle, which currently underlies the derivation of all formulas for diffraction by obstacles of arbitrary shape.
Nevertheless, even with the help of the Huygens-Fresnel principle, it is not possible to solve the diffraction problem in general form, therefore, when obtaining the formulas, they resort to some approximations. The main one is a plane wave front. It is such a waveform that must fall on an obstacle in order to simplify a number of mathematical calculations.
The next approximation is the position of the screen where the diffraction pattern is projected relative to the obstacle. This position is described by the Fresnel number. It is calculated as follows:
F = a 2 / (D * λ).
Where a is the geometric dimensions of the obstacle (for example, a slit or a round hole), λ is the wavelength, D is the distance between the screen and the obstacle. If for a particular experiment F << 1 (<0.001), then we speak of the approximation of the far field. The corresponding diffraction bears the name of Fraunhofer. If F> 1, then the near field approximation or Fresnel diffraction takes place.
The difference between the Fraunhofer and Fresnel diffraction lies in different conditions for the phenomenon of interference at small and large distances from the obstacle.
The derivation of the formula for the main maxima of the diffraction grating, which will be given later in the article, involves consideration of the Fraunhofer diffraction.
Diffraction grating and its types
This lattice is a plate of glass or transparent plastic several centimeters in size, on which opaque strokes of the same thickness are applied. The dashes are located at a constant distance d from each other. This distance is called the lattice period. Two other important characteristics of the device are the lattice constant a and the number of transparent slits N. The value of a determines the number of slits per 1 mm of length, therefore, it is inversely proportional to the period d.
There are two types of diffraction gratings:
- Transparent, which is described above. The diffraction pattern from such a grating arises as a result of the passage of a wave front through it.
- Reflective. It is made by applying small grooves on a smooth surface. Diffraction and interference from such a plate arise due to the reflection of light from the vertices of each groove.
Whatever the type of lattice, the idea of its effect on the wave front is to create a periodic disturbance in it. This leads to the formation of a large number of coherent sources, the interference of which is the diffraction pattern on the screen.
The basic formula of the diffraction grating
The conclusion of this formula involves considering the dependence of the radiation intensity on the angle of incidence on the screen. In the far field approximation, the following formula for the intensity I (θ) is obtained:
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 )).
In the formula, the slit width of the diffraction grating is indicated by the symbol a. Therefore, the factor in parentheses is responsible for diffraction by a single slit. The quantity d is the period of the diffraction grating. The formula shows that the factor in square brackets where this period appears describes the interference from the totality of the lattice gaps.
Using the above formula, you can calculate the intensity value for any angle of incidence of light.
If we find the value of the intensity maxima I (θ), then we can conclude that they appear under the condition that α = m * pi, where m is any integer. For the condition of the maxima, we obtain:
m * pi = pi * d / λ * (sin (θ m ) - sin (θ 0 )) =>
sin (θ m ) - sin (θ 0 ) = m * λ / d.
The resulting expression is called the formula for the maxima of the diffraction grating. The numbers m are the diffraction order.
Other ways to write the basic formula for the lattice
Note that the term sin (θ 0 ) is present in the formula given in the previous paragraph. Here, the angle θ 0 reflects the direction of incidence of the front of the light wave relative to the plane of the grating. When the front falls parallel to this plane, then θ 0 = 0 o . Then we get the expression for the maxima:
sin (θ m ) = m * λ / d.
Since the lattice constant a (not to be confused with the slit width) is inversely proportional to d, then through the diffraction grating constant the formula above is rewritten in the form:
sin (θ m ) = m * λ * a.
In order to avoid errors when substituting specific numbers λ, a and d into these formulas, you should always use the appropriate SI units.
The concept of angular dispersion of a lattice
We will denote this value by the letter D. According to the mathematical definition, it is written as follows:
D = dθ m / dλ.
The physical meaning of the angular dispersion D is that it shows at what angle dθ m the maximum will shift for the diffraction order m if the incident wavelength is changed by dλ.
If we apply this expression to the lattice equation, then we get the formula:
D = m / (d * cos (θ m )).
The angular dispersion grating dispersion is determined by the formula above. It can be seen that the quantity D depends on the order m and on the period d.
The larger the dispersion D, the higher the resolution of this grating.
Grating resolution
By resolving power is meant a physical quantity that shows by what minimum two wavelengths can differ so that their maxima appear separately in the diffraction pattern.
Resolution is determined by the Rayleigh criterion. He says: two maxima can be divided in the diffraction pattern if the distance between them is greater than the half-width of each of them. The angular half-width of the maximum for the lattice is determined by the formula:
Δθ 1/2 = λ / (N * d * cos (θ m )).
The resolution of the grating in accordance with the Rayleigh criterion is equal to:
Δθ m > Δθ 1/2 or D * Δλ> Δθ 1/2 .
Substituting the values of D and Δθ 1/2 , we obtain:
Δλ * m / (d * cos (θ m ))> λ / (N * d * cos (θ m ) =>
Δλ> λ / (m * N).
This is the formula for the resolution of the diffraction grating. The larger the number of strokes N on the plate and the higher the diffraction order, the greater the resolution for a given wavelength λ.
Diffraction grating in spectroscopy
Let us write again the main equation of the maxima for the lattice:
sin (θ m ) = m * λ / d.
It can be seen here that the longer the wavelength falls on the plate with strokes, the higher the angles will appear maxima on the screen. In other words, if non-monochromatic light (for example, white) is passed through the plate, then the appearance of color maxima can be seen on the screen. Starting from the central white maximum (zero-order diffraction), further maxima will appear for shorter waves (violet, blue), and then for longer ones (orange, red).
Another important conclusion from this formula is the dependence of the angle θ m on the diffraction order. The larger m, the greater the value of θ m . This means that the colored lines will be more strongly divided among themselves at the maxima for a high diffraction order. This fact was already consecrated when the resolving power of the grating was considered (see the previous paragraph).
The described abilities of the diffraction grating allow it to be used to analyze the emission spectra of various luminous objects, including distant stars and galaxies.
Problem solving example
We show how to use the diffraction grating formula. The wavelength of light that falls on the grating is 550 nm. It is necessary to determine the angle at which first-order diffraction appears if the period d is 4 μm.
The angle θ 1 is easily calculated by the formula:
θ 1 = arcsin (λ / d).
We translate all the data into SI units and substitute into this equality:
θ 1 = arcsin (550 * 10 -9 / (4 * 10 -6 )) = 7.9 o .
If the screen is located at a distance of 1 meter from the grating, then from the middle of the central maximum a first-order diffraction line for a wave of 550 nm will appear at a distance of 13.8 cm, which corresponds to an angle of 7.9 o .