In a diffraction grating, the relationship between the spacing of the grating (i.e.dem. the distance between adjacent grooves or slits), the angle of incidence of the wave (light) with respect to the grating, and the diffracted wave of the grating is called the grating equation. Like many other optical formulas, the lattice equation can be derived using the Huygens–Fresnel principle, which states that any point on a wavefront of a propagating wave can be considered a point wave source, and a wavefront can be found at any later point by adding the contributions of each of these individual point wave sources on the previous wavefront. d: distance between slits (lattice period) N: number of slits per mm (the groove density, equal to the inverse of the lattice period) m: diffraction order (m = 0, ± 1, ± 2,…) λ: wavelength From this relationship, we can see that all the luminous components corresponding to m = 0 (light of order zero) are emitted in a straight line and that it is therefore not possible to separate the wavelengths with this order. It can also be seen that for m ≠ 0 the diffraction angle ß is different for each wavelength. For this reason, gratings can be used to separate white light into its constituent wavelengths. The diffraction angle ß also varies with the N-groove density and the α angle of incidence. One point to consider is that, depending on the N groove density, it may not be possible to obtain diffracted light. For example, if the angle of incidence is α = 30° and the groove density N = 2400 grooves/mm, applying the equation to first-order light (i.e. m = +1) with a wavelength λ of 700 nm gives sin ß = 1.18, then diffracted light cannot be obtained in this case. Quantum electrodynamics (QED) provides an additional derivation of the properties of a diffraction grating with respect to photons as particles (at a certain level).
QED can be described intuitively with the path-integral formulation of quantum mechanics. As such, it can model photons to potentially track all paths from a source to an endpoint, any path with a certain probability amplitude. These probability amplitudes can be represented as a complex number or equivalent vector – or, as Richard Feynman simply calls them in his book on QED, “arrows.” The diffracted beams, which correspond to successive orders, can overlap, depending on the spectral content of the incident beam and the density of the grating. The higher the spectral order, the greater the overlap in the following order. A diffraction grating is able to scatter a beam of different wavelengths in a spectrum of related lines due to the principle of diffraction: in a certain direction, only waves of a certain wavelength are obtained, while the rest is destroyed by interference with each other. The arrays give exceptionally high resolutions of spectral lines. The resolution (R) of an optical instrument represents the ability to separate closely spaced lines in a spectrum and is equal to the wavelength λ divided by the smallest difference (Δλ) in two wavelengths that can be detected; i.e. R = λ/Δλ.
For a 10-centimeter-wide array with 10,000 lines per centimeter, the resolution in the first order of diffraction would be 100,000. For an emission of wavelength in the ultraviolet, say λ = 300 nanometers (3 × 10-7 meters), a wavelength difference of Δλ = 3 × 10-12 meters (about 1/100 of the diameter of an atom) should theoretically be possible. Lattices are made by two methods, ruler and holography. A high-precision control motor creates a master grid by polishing grooves with a diamond tool against a thin layer of sprayed metal applied to a surface. Masterlet replication enables the production of control arrays, which make up the majority of diffraction gratings used in dispersive spectrometers. These arrays can be triggered for specific wavelengths, typically have high efficiency, and are often used in systems requiring high resolution. Ladder gratings are a type of coarse control grids, that is, they have low groove density, high flame angles and use high diffraction orders. The virtue of a ladder network lies in its ability to provide high dispersion and resolution in a compact system design.
Diffraction order overlap is an important constraint of scale gratings, which require some type of order separation usually provided by a prism or other grating. Holographic arrays are created by a sinusoidal interference pattern etched into the glass. These networks have less dispersion than linear lattices, are designed to minimize aberrations, and can have high efficiency for a single polarization plane. Networks can be reflective or transmissive, and the surface of a grid can be flat or concave. Planar gratings typically offer higher resolution over a wide range of wavelengths, while concave gratings can act as both scattering and focusing elements in a spectrometer. Diffraction gratings are also used to evenly distribute light before e-readers such as the Nook Simple Touch with GlowLight. [27] The iridescent colors of peacock feathers, mother-of-pearl feathers and butterfly wings are most often confused with diffraction grids. Simmering in birds[30], fish[31] and insects[30][32] is often caused by thin-film interference rather than diffraction grating. Diffraction produces the full spectrum of colors when the angle of view changes, while thin-film interference usually creates a much narrower range. Flower surfaces can also produce diffraction, but plant cell structures are usually too irregular to produce the fine slit geometry needed for a diffraction grid.
[33] The iridescent signal of flowers is therefore only very locally perceptible and therefore not visible to humans and insects visiting flowers. [34] [35] However, natural grids occur in some invertebrates, such as peacock spiders,[36] the antennae of seed shrimp, and have even been found in Burgess Shale fossils. [37] [38] The dispersion of a grating is determined by the lattice equation, which is usually written as follows: An idealized diffraction grating consists of a series of spaced slits {displaystyle d} that must be wider than the wavelength of interest to cause diffraction. Assuming a plane wave of monochromatic light of wavelength λ {displaystyle lambda} with a normal incidence on a lattice (i.e. the wavefronts of the incident wave are parallel to the plane of the main lattice), each slit in the lattice acts as a near-point wave source from which light propagates in all directions (although this is usually limited to the anterior hemisphere from the point source). Of course, each point of each slit reaches the incident wave plays as a point wave source for the diffraction wave, and all these contributions to the diffraction wave determine the detailed distribution of the light properties of the diffraction wave, but the diffraction angles (at the grating level) where the intensity of the diffraction wave is highest are determined only by these quasi-point sources, that correspond to the slots in the network.