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(Diffraction on the grating)

- NOTICE:
- Derivations of expressions for diffraction maxima (or minima) on the slit, double slit, and lattice are given here at the secondary level. Who wants to learn more about the derivation of these formulas and see the "correct" derivation of the formulas, can study the following material: Studium difrakčních jevů – TEORIE – doplněk
*(only in Czech)*.

The bending of light (or diffraction) was observed and described as early as around 1660 by the Italian mathematics teacher Francesco Maria Grimaldi. In a darkened room, he let sunlight fall on a small opening and placed various objects in the path of the resulting light to study their shadows. In doing so, he discovered that the shadows were blurred and that they were also bordered by coloured stripes.

The basic assumption of ray optics, that light propagates in a straight line according to the law of straight line propagation, is only valid to a limited extent. In fact, the propagation of light, like that of any wave (e.g., the propagation of sound waves), is affected by its wave properties. We can see this precisely in obstacles that are comparable to its wavelength. In this case, light is bent at the obstacle – so-called **diffraction**. This is manifested by the light spreading partly into the space behind the obstacle where it should never spread according to ray optics, i.e. the light spreads also into the geometric shadow region. The boundary between light and shadow is then not sharp and a so-called **bending (also diffraction) pattern** is formed on the shadow behind the obstacle.

Bending phenomena are generally categorized into two basic types:

If light is incident on an obstacle from a point source at a finite distance from the obstacle, these are

**Fresnel phenomena**.If the source (and the detector screen) is at infinity, then the waves coming from the source are plane and the phase of the oscillations is the same at all points of the obstacle (assuming the waves are incident perpendicular to the obstacle). The condition of an infinitely distant source (or shield) is achieved in practice by using lenses placed in front of and behind the obstacle. In this case, the phenomena are

**Fraunhofer phenomena**. Fraunhofer placed a bending obstacle close to the lens used to image a point or slit light source and studied bending phenomena in the geometric image plane. By decomposing light using an optical lattice, he laid the foundations of lattice spectroscopy.

**We will continue to study only Fraunhofer diffraction.**

Suppose that a plane light wave of wavelength λ is incident on a slit of width *a*. Each point of the slit becomes, according to Huygens' principle, a source of elementary waves, which propagate from it in elementary wavelets and into the space behind the obstacle. Light is then incident on each point on the screen from each point of the slit. Since the slit has been illuminated by a plane monochromatic wave, these elementary waves can be considered coherent and a bending pattern is formed on the screen (see Figure 1).

Fig. 1 – Light intensity in a bending pattern on a slit

(top: graph of intensity waveform; bottom: real pattern)

The plane wave is incident on the slit where the wave bending occurs. At the detection screen, the beam waves have different phases depending on the path they have travelled to that location. The mutual stacking of these coherent wave beams produces the diffraction pattern mentioned above.

We now derive the condition for the appearance of interference minima of the diffraction pattern on the slit. A plane wave of wavelength λ is incident perpendicularly on a slit of width *a*. Rays from the whole slit space are incident on the given spot. For the sake of clarity, only the rays coming from the edges of the slit (points A and B) are shown in the figure. The intensity of light at point P (distant *y* from the axis of the slit) is shown by the red curve. The situation is illustrated in Figure 2

Fig. 2 – derivation of the interference condition of the minimum at the slit.

The difference in trajectory between the wave propagating from point A (the uppermost point of the slit) and the wave propagating from point B (the lower edge of the slit) is Δℓ (indicated in Figure 2; see Note 1). Thus, the difference in trajectory between the wave propagating from the centre of the slit S and the wave propagating from point A is Δℓ/2 due to the similarity of the triangles. Due to the axisymmetry of the bundle of rays pointing from the whole slit surface to point P (the axis in the figure is shown in black), for every ray propagating from any point between A and S, a ray propagating from the points between B and S can be found that is path-delayed by Δℓ/2. The interference minimum condition states that all waves will cancel each other out if this path difference Δℓ/2 is equal to an odd multiple of half waves. If we modify this condition for the entire path difference Δℓ it must therefore be equal to an even multiple of half waves (twice the odd multiple is an even number).

**Note 1:**

To simplify the calculation of the path difference of the waves ℓ_{A} and ℓ_{B}, we assumed that the shield distance *d* is much larger than the slit width *a*. Then the angles α_{A} and α_{B} are approximately equal (denoted α) and the directions of the two waves can be considered parallel, and the path difference of the ℓ_{A} and ℓ_{B} paths will be given by the distance Δℓ. (In the figure, due to the failure to satisfy the condition *a* << *d*, it does not look like this, of course).

We can also see from figure 2 that the path difference of waves Δℓ can be expressed from the right triangle ABD, whose hypotenuse is the width of the slit *a*, the path difference Δℓ is the subtended opposite to the angle α.

Thus, for the path difference, the geometry implies:

Δℓ = *a* ⋅ sinα

We know from wave optics (or from the reasoning derived above) that the path difference for the minimum condition must be equal to twice the odd multiples of half the wavelength.

So we write the condition for the position of the bending minima:

a ⋅ sinα = k ⋅ 2⋅ |
λ |

2 |

which can be modified to form:

*a* ⋅ sinα = *k* ⋅ λ

(1)

where: | a |
[m] | – width of the slit |

λ | [m] | – wavelength | |

α | [°; rad] | – the angle (direction) for which the given minimum occurs | |

k |
– an integer except zero – read Note 2(for k = 0 the main bending maximum occurs!) |

The maxima then occur between the given minima – marked in white in Figure 1 and with the given values of *k* (so-called orders of maxima).

**Note 2:**

In some texts dealing with diffraction at the slit, relation (1) is presented as a condition for diffraction maxima on the basis of an erroneous derivation of the ray path difference from points A and B. But this is not true! It is probably due to the similarity of the derivation (and also the similarity of the relation) for the condition of the maximum on the diffraction grating. The validity of the erroneous conclusion that relation (1) is a condition on the maximum is erroneously illustrated in some texts by showing that the relation correctly determines a zero-order maximum. This may be true, but unfortunately zero order is the only maximum that the relation correctly determines in this case! Condition (1) is indeed (and is experimentally supported) a condition for first and higher order minima (*k* = 1, 2, …). Setting *k* = 0 is physical nonsense because there is no zero minima in the diffraction pattern!

The most pronounced bending patterns are created when light is allowed to fall on multiple narrow slits. A system of a large number of slits is called an optical lattice (see below). The parameters of the array of slits are the width of the slit *a* and the distance between the centres of adjacent slits *b* – called the **grating constant** (or *grating period*) for a grating.

To derive the condition of the interference maximum on the lattice, we first derive a relation for the bending of light on a system of two slits (called a *double slit*).

Let the mono-frequency light from the distant source fall on the double slit. Again, we will assume that the width of the slit *a* and the distance between the centres of adjacent slits *b* are much smaller than the distance of the shield d from the pair of slits. Two phenomena occur when a plane wave is incident – diffraction at each slit (see earlier) and then also interference of two coherent waves from the two slits.

The light spreads again in all directions beyond the slits. In the same way as at one slit, we consider rays that have diverged from their original direction by an angle α, striking at point P and emanating from corresponding points – as indicated in Figure 3. The corresponding points are the upper edges of the first and second slits (their mutual distance is *b*).

Fig. 3 – derivation of the interference maximum on the double slit

The figure then shows for the path difference of these rays ℓ_{1} and ℓ_{2}:

Δℓ = *b* ⋅ sinα

where: | Δℓ | [m] | – path difference |

b |
[m] | – distance between slots | |

α | [°; rad] | – deviation of the wave from the original direction of the plane wave |

There will be amplification in directions where the wave path difference is equal to an even multiple of half waves.

The condition of the interference maximum in the case of a double slit is:

*b* ⋅ sinα = *k* ⋅ λ

(2)

where: | b |
[m] | – distance between slots |

λ | [m] | – wavelength of the light used | |

α | [°; rad] | – the angle (direction) for which the maximum occurs | |

k |
– integer (so-called order of maximum) |

The interference minimum occurs in directions where the path difference is equal to an odd multiple of half waves.

The resulting bending pattern on the double slit will still be affected by diffraction on the slits themselves, as mentioned. Thus, the resulting pattern will be a composite of broader maxima and minima that correspond to the bending at one slit (the envelope curve) and a series of light and dark bands that result from the interference of light waves from the two slits ("modulating" the pattern – see Figure 4; right part). Thus, the narrow equidistant bands can be said to result from interference from the individual slits, while the broad regions of light and dark are the result of diffraction across the width of the slit.

Fig. 4 – comparison of the diffraction pattern for a single slit (left) and a pair of equally wide slits (right)

The grating can be thought of as a system composed of a large number of slits placed at equal distances *b*. Each of the slit points can again be thought of as a point source of light from which light propagates according to Huygens' principle. From the analogy with the double slit (see earlier), then, for the path difference of the corresponding slit points of the lattice, the same relation for the interference maximum as for the double slit follows. The following figure illustrates the nature of the diffraction pattern for different numbers of identical slits. Note that for two or more slits the positions of the maxima (or minima) do not change, only the area of the minima widens and the width of the maxima narrows ("focusing" of the maxima).

Fig. 5 – comparison of diffraction patterns for different numbers of slits (1, 2, 3 … 7) and grating

The condition for the interference maximum on the grating corresponds to the condition for the double slits, only in the case of the grating the maxima are "sharper" (see Figure 5):

*b* ⋅ sinα = *k* ⋅ λ

(3)

where again: | b |
[m] | – grating constant |

λ | [m] | – wavelength of the light used | |

α | [°; rad] | – the angle (direction) for which the maximum occurs | |

k |
– integer (order of the maximum) |