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DIFFRACTION GRATINGS  Raman Spectroscopy:
The diffraction gratings, commonly used by spectrographs and monochromators, are characterized by the size, the pitch, how the grooves are obtained, the "blaze", the order for which they are optimized and bythe shape of the surface.
The pitch of a diffraction grating, determined by the number of grooves per millimeter, is the distance between two adjacent grooves.The gratings can be obtained by engraving, by replica of engraved gratings or holographycally.
The "blaze" is the particular inclination of the grating grooves (engraved or of an its replica), aimed to improve the efficiency in a particular spectral range (see our electromagnetic sprectrum poster) and order.The order refers to the diffraction order, for which the grating has been optimized by means the blaze. The shape of the gratings can be flat or concave.
The diffraction gratings consist of a repetitive structure of linear elements (typically reflective) whose width is comparable to λ. Each pair of adjacent rows behaves like a slit: when it is illuminated by a collimated radiation, the light is reflected (transmitted) with a large divergence (in the plane perpendicular to the slit itself), whose angle depends on the ratio between the slit width and the wavelength of light. The adjacent slit transmits or reflects the light in the same way. At a short distance from the slits the two signals overlap. For a viewing angle (with respect to the grating normal direction) equal to the angle of incidence, the phase differences between the two signals are zero: the beam is reflected (diffraction order "zero"). The diffracted radiation from each of the two slits (hence with a different angle with respect to the reflected radiation) travels a different path; if the difference of path is equal to an integer multiple of a certain wavelength, the two beams have the same phase for that wavelength: by overlapping one each other, the amplitudes are summed and the resulting intensity, which is the square of the sum of the single amplitudes, becomes four times the intensity of each of them (if the two amplitude are the same). For different angles, also the phases are different: when the path difference is half a wavelength, the phases are in opposition, and the resulting amplitude (and, consequently, the intensity) vanishes.
This phenomenon is repeated, for different angles, for all the wavelengths contained in the incident radiation. When the path difference is of 1 wavelength, the first order is generated; if the path difference is of 2 wavelengths, the second order is generated; the numbering of subsequent orders is the same of the number of the entire wavelengths that separates the two contribution. The optical path differences are produced by both the sides of the slits. When theangles of the different orders are greater than the reflection angle, then the order numbers are positive. If the angles are smaller than the reflection angle, the orders are negative.So far only two slits were considered. However the diffraction gratings consist of several adjacent slits. The total effect is that the spectral images, formed on the focal plane of the focusing optics, better approximates the image of the slit that has generated them how much larger is the amount of slits contained in the grating. For this reason, in order to obtain the highest resolution,the largest possible number of grooves should be used, hence the maximum possible surface.
If, for a certain angle and a certain wavelength, the difference of the optical path between two orders is an integer number k, for a double wavelength, the difference is k +1. Hence the two wavelengths, even if spatially overlapping, belong to different orders. The set of wavelengths that can be observed without overlapping is limited.
The dispersion is the spectral range contained in a certain angular range (usually indicated as the space, measured in mm, on the focal plane of the optical system that provides the formation of the spectrum image  Angstrom per millimeter  Δβ / Δλ). The dispersion grows with the diffraction order: the second order is dispersed two times the first, the third order is three times the first, etc..
The dispersion is inversely proportional to the grating pitch: more is dense the grating, greater is its dispersion.
The dispersion within a certain order, depends on the inverse cosine of the pitch: for small angles, the dispersion is almost linear (in contrast to the dispersion of a glass prism, which strongly varies between blue and red).The beams which illuminated the slits were obtained by illuminating a "window" parallel to the slits that constitute the grating on the focal plane of a collimator (window = entrance slit of the monochromator). When the light transmitted or reflected by the grating is collected with an optical system, on its focal plane is formed the image of the entrance slit with the light of the wavelengths contained in the incident radiation.
For higher angles, the optical path difference becomes two or more wavelengths. Even in these cases, the amplitudes are summed if the phases are equal. Therefore for the same wavelength, are generated several images of the first slits : as mentioned aboveare formed the various orders.The diffraction gratings can be obtained by mechanical ruling (it was the only method available in the past). The realization of a diffraction grating by means mechanical ruling is a very complex operation. Starting from an engraved grating, many replicas can be obtained by depositing many layers on the engraved grating itself
The shape of the grooves of an engraved grating is similar to that shown in Fig.16.Fig. 16  Shape of the slits of an engraved grating.
The particular inclination of the grating surface is used to maximize the energy sent onto a particular order at a certain wavelength (of "blaze" λ).
The small incision defects produce unexistent lines ("ghost") in the spectrum .Many modern cheaper gratings are holographycally obtained: two beams of coherent radiationcoming from the same source (laser) interfere by forming a certain angle on the surface of a photosensitive material. The pattern that is formed is the radiation interferogram. The profile of the grooves of the grating is sinusoidal: no blaze is possible to obtain.
The holographic gratings are less efficient than the engraved ones, but they are much less delicate (the surface of an engraved gratingcannot be touched and it is very difficult to clean).
Spectrographs and monochromators are constituted by an entrance slit, a collimator, a dispersing element (grating or prism), a focusing system and by the plane on which the spectrum is formed (in monochromators even by an exit slit). In some cases is used a concave grating which works also as a collimator and as afocusing system.
The collimator and the focusing system are usually concave mirrors (Fig. 16  next page). When they are spherical mirrors, their notinaxis functioning, produces aberration phenomena(like astigmatism and coma), which can be avoided by using toric mirrors.
MONOCHROMATORS AND SPECTROGRAPHS
Most of the spectrographs has optical structure similar to that shown in Fig. 17.
Fig.17– Spectrograph optical structure
The spectrum is constituted by many images of the entrance slit, how many are the wavelengths contained in the source. If the optical system is free by aberration effects, the spectrum will be made by perfect images of the entrance slit. However if the mirrors are spherical, the images will be affected by astigmatism: Fig.18 shows the ray behavior close to the focus.Fig.18
The images of the entrance slit on the sagittal and tangential focus are two perpendicular lines. At halfway between these two lines, the beam appear circular. When it is important to solve the spectral content of the points along the entrance slit by avoiding the mixing between the spectral content of the other points , e.g. as when the slit is illuminated by a linear array of optical fibers, is preferable to use the sagittal focus as it offers a better resolution along the height.
Conversely, when we are more interested to the spectral resolution, is preferable to use the tangential focus. Finally, for a visual observation, the best compromise between the sagittal and the tangential focus is the visual focus. Actually the three surfaces, tangential, visual and sagittal, are inclined one with respect to theothers.In order to optimize one of the three planes, it is necessary to properlyrotate the surface where the spectrum is formed.
If the spectrographs use toroidal mirrors, instead of spherical mirrors, the three foci coincide. However this happens only in the centre, in correspondence of the visual focus. Nevertheless the three above planes remain inclined: even in this case the focus optimization requires the rotation of the spectral surface (Fig.19).
Fig. 19