Distribution of illuminance in the image of a slit

## Diffraction grating

A diffraction grating in spectral instruments is used to disperse light into a spectrum. The diffraction grating is an optical element consisting of a series of regularly packed grooves that have been placed onto planar or concave grating surface. Gratings can be either transmissive or reflective. In addition, there are Amplitude and Phase Diffraction Gratings. The periodic refractive index modulation of the first type of gratings leads to changes in the incident wave amplitude.

In phase diffraction gratings the grooves have a special shape which periodically changes the phase of a light wave. Planar reflective phase diffraction gratings with a triangular grooves shape, also called echelette gratings, are the most widely used.

## Grating equation

The wave front of the incident light that falls on a diffraction grating is split into several coherent beams by its grooves. The coherent beams, diffracted by each groove, interfere and form the resulting spatial intensity distribution of light. The intensity distribution is proportional to the product of two terms, the interference function  and the intensity function of a single slit  The interference function  is determined by the interference of  coherent beams coming from grating grooves. The slit intensity function  is related to the light diffracted from one groove.

Two coherent parallel rays, diffracted on one groove spacing of the grating, have the difference in their path lengths  or else , where is the angle of incidence, is the angle of diffraction. The corresponding phase difference is   The function of  – is a periodical one with different intensive principal maxima. The positions of principal maxima correspond to . It can be rewritten as , where  is the diffraction order. Eq. (2) can be used to replace  in Eq. (1):  . Combining this result and Eq. (3) gives , substituting into Eq. (1) gives:  Eq. (4) is called the grating equation. It shows that the principal maxima occur at angles which are represented by non-zero integers  .

Between the adjacent principal maxima  secondary maxima are located whose intensity decreases in proportion to , and  minima, where the intensity is zero. The grating equation for monochromators can be written in a more convenient form. The difference between  and  at any grating rotation is constant. This value is equal to , and it is determined by monochromator optical design.

Here we have defined   and   , where  is the scan angle which is measured from the grating normal to the bisector of the beams,  is the deviation angle between the incident and diffraction directions. Thereafter, the grating equation may be simply expressed as: , or otherwise:

A lens can be used to collect the diffracted light coming from the grating, and to form spectra for each value of  in its focal plane. For  (zeroth-order maximum), the spectrum is not formed, because  for all wavelengths. In addition, the zero-order maximum direction corresponds to the specular reflection, i.e.  .

## Blaze wavelength

How to calculate the blaze wavelength for the tested spectral range to select a grating.

The reflectivity of a grating depends on the groove angle. Changing the groove angle, the diffraction maximum of  function may be combined with the principal maximum of  function of any order. The diffraction maximum direction is determined by the specular reflection of the incident beam on the groove facet (not on the grating plane). Thus, it takes place when the angles  and  meet the next conditions:

Then, the spectrum in this diffraction order will have the highest intensity. The angle  is called the blaze angle, and the corresponding wavelength is the blaze wavelength  . If your spectral range is defined, the blaze wavelength  is given by: , where  and  – are the boundaries of the spectral range. Equation (8) helps to choose the grating.

Example 1. The tested spectral range is 400…1200 nm, i.e.  400 nm,  1200 nm. From Eq. (8):  600 nm. Select a grating with the blazed wavelength of 600 nm.

Example 2. The tested spectral range is 600…1100 nm. Calculations using Eq. (8) give the value of 776 nm. The grating with such blaze wavelength is not listed. A grating with the closest blaze wavelength should be selected, i.е. 750 nm.

## Energy-efficient range of gratings

The energy-efficient range of grating is a spectral range in which the grating has a reflectivity coefficient of more than 0.405. It can be estimated as:  Value of  depends on the diffraction order. It is maximal in the first order, and then decreases rapidly in the higher orders. For the first diffraction order:  . The wavelengths that define this spectral range are:  and  .

## Free spectral range

How to overlap the Free spectral and Energy-efficient ranges.

The range of wavelengths in a given spectral order for which superposition of light from adjacent orders does not occur is called the free spectral range. Consequently, there is a well-defined relationship between the angle of diffraction and the wavelength. The free spectral range is determined as:  if the following condition is satisfied:  . For the first order of diffraction,  , and  . That is to say, the free spectral range is one octave region. To superimpose the free spectral range on the Energy-efficient range, the next condition:  should be satisfied. In this case, the grating reflection coefficient for  is not less than 0.68 within the free spectral range.

Example. If  600 nm, then  450 nm, and  900 nm. Thus, the Free spectral range of this grating is superimposed on the Energy-efficient range. This occurs at all wavelengths from 450 nm to 900 nm.

## Dispersion

The angular dispersion is a measure of the angular separation of light rays of different wavelength. The expression for the angular dispersion obtained by differentiating the grating equation is defined as:  For a given wavelength, Eq. (12) shows that the angular dispersion is a function of the angles of incidence  and diffraction . It does not depend on the groove number. The linear dispersion of a spectral instrument  is the product of the angular dispersion and the effective focal length. The reciprocal linear dispersion  is more often considered.

## Resolving power

The theoretical resolving power is given by  , where  is the limit of resolution. The resolution of the diffraction grating is determined by the spectral width of the instrumental function  . For the grating, the width of the instrumental function is the width of principal maxima of the interference function:  . Therefore  The spectral resolution of the diffraction grating is equal to the product of the diffraction order  and the total number of grooves . Using the grating equation, the resolving power can be rewritten as , where the quantity  is simply the ruled width of the grating. As expressed by Eq. (15), for the given  and  the  value can only be increased by increasing the size of the grating. The expression for the resolving power can be represented in another form using Eq. (12) and Eq. (15): , where  is the width of the diffracted beam,  is the angular dispersion. Eq. (16) shows that the resolving power is proportional to the angular dispersion.

## Grating efficiency range

For each grating with the groove spacing  , there is a boundary value of wavelength  . When  and  ,  can be defined from the grating equation as  .

Therefore, gratings with different numbers of grooves should be used to operate in different spectral regions:

• for UV range: 3600 – 1200 l/mm;
• for VIS range: 1200 – 600 l/mm;
• for IR range: less than 300 l/mm.

## Cоncave diffraction grating

A concave grating can act not only as a dispersive system but also as a focusing one. All the equations describing the spectroscopic characteristics (angular dispersion, resolution, and the free spectral range) are the same as for a planar grating. Compared with planar gratings, the concave gratings have astigmatism. Astigmatism may be reduced by using the varied line-space diffraction gratings or gratings with aspherical surfaces.

## Holographic diffraction grating

The scattered light intensity defines the diffraction grating quality. This intensity and the intensity of the “ghosts” (artificial lines) increase due to the presence of small defects on the groove facets or due to periodic errors in the spacing of the grooves. In comparison with the ruled gratings, advantages of holographic gratings are: any “ghost” absence and low intensity of scattered light. However, reflective holographic phase gratings have the sinusoidal shape of the groove (Fig. 2), i.e. it is not echelette gratings. They have a considerably lower efficiency.

Production of holographic gratings with triangular groove profiles, so-called “blazed” holographic gratings, leads to additional defects on the groove facets which increase the stray light intensity. Moreover, the ideal triangular profile is not achievable, and it reduces the energy efficiency of such gratings.

## Distribution of illuminance in a slit image

The illumination distribution over the slit image depends on the optical system aberrations, and on the slit illumination method.

### Aberrations

Spherical aberrations, coma, astigmatism, field curvature, image distortion, and chromatic aberrations.

#### Aberrations

An ideal optical system gives a point of light in the image plane. In practice, the optical system is pretty close to the ideal one. But if the beam width is limited, and the light source is off-axis, the rules of paraxial optics are failed. And so, the optical image should be distorted. Designing of the optical systems, it is necessary to correct aberrations. Aberrations should have been corrected when designing optical systems.

#### Spherical aberrations

The result of this aberration is that the images of objects are often blurry. The effect of spherical aberration manifests itself in two ways: the center of the image remains more in focus than the edges, and the intensity of the edges falls relative to that of the center. This aberration is the only one that remains in the case, if a point object is located at the main optical axis of the system. Particularly, spherical aberration is significant in high-aperture systems.

#### Coma

The point projected image in the presence of comatic aberration takes an elongated shape, like a comet.

#### Astigmatism

An optical system with astigmatism is one where rays that propagate in two perpendicular planes have different focus. The scattering pattern is a family of ellipses with a uniform illuminance distribution. In the analysis of astigmatism, it is most common to consider rays, which propagate in two special planes. The first plane is the meridional plane. This is the plane which includes both the object point being considered and the axis of symmetry. The second special plane is the sagittal plane. This is defined as the plane, orthogonal to the meridional plane. The centers of curvature of both sections are called foci and the distance between the centers is a measure of astigmatism.

#### Field curvature

It is an optical aberration in which a flat object normal to the optical axis cannot be brought into focus on a flat image plane.

#### Image distortion

Distortion is a deviation from rectilinear projection, a projection in which straight lines remain straight in an image. In distortion, image magnification varies with distance from the optical axis.

#### Chromatic aberrations

There are two types of chromatic aberration: axial (longitudinal), and transverse (lateral). Axial aberration occurs when different wavelengths of light are focused at different distances, i.e., different points on the optical axis (focus shift). Transverse aberration occurs when different wavelengths are focused at different positions in the focal plane (because of the magnification). Chromatic aberration occurs in optical systems comprising elements with different refractive indices for different wavelengths of light. It does not occur in mirrors. That makes mirrors particularly efficient in monochromators, spectrographs and other optical systems.

### Illuminance of entrance slit

Coherent and incoherent illumination.

Essential for the intensity distribution across the width of a spectral line has a device entrance slit illumination character, i.e. the degree of coherence. In practice, the entrance slit illumination is not strictly coherent or incoherent. However, you can come very close to one of these two extreme cases. Coherent slit illumination can be realized if the light point source will be located at the focus of a large-diameter condenser. Another way is a lensless illumination, when a small light source is placed at a long distance from the slit. Incoherent illumination can be obtained by using a condenser lens to focus light onto the entrance slit of the instrument. Other illumination methods are intermediate. The importance of their differentiation is associated with the fact that interference phenomena may occur at the coherent light illumination. Interference phenomena are not observed in case of any incoherent light.

If the main requirement is the highest resolution, the aperture of the diffraction grating should be filled with coherent light in a plane which is perpendicular to the slit. To ensure the maximum brightness in spectra, the incoherent illumination should be used. In this case, the aperture is filled also in the plane which is parallel to the slit.

## Filling the objective aperture with light

F/# matcher

One of the main parameters that characterizes the spectral instrument is its light-gathering power (aperture ratio). It is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. Aperture is defined as the ratio of the collimator mirror focal length  to its diameter . In practice, the reciprocal value is often used, called the F-number :  In spectral devices, mirrors are usually rectangular. But  for rectangular mirrors may be estimated on base of the area of the collimator  with the circle diameter of  For example, the area of the collimator is  . Then  or  For high aperture optical systems, such as objective lenses, optical fibers, etc., instead of  another characteristic – numerical aperture is usually used. The numerical aperture is defined by:

If an intermediate source is used, you can get his reduced image on the entrance slit with the help of a collecting lens, and as a result of that you can increase the flow of light entering the device. However, the solid angle of the incoming radiation is increased in the spectral instrument. The collimator is refilled, the part of the radiation is scattered, creating a parasitic background.

The optimal image of the extended incoherent light source on the device entrance slit is achieved when the solid angle of the incident light beam is equal to the device input angle.

###### Fig.4. F/# matcher layout.

The following notation is used:  ;  ;  The flow of light depends on the product  , where  is the area of the entrance slit;  is the input solid angle. If the slit and collimator are filled with the light, no any additional lenses and mirrors can help to increase the total radiation flux passing through the system. The maximal input solid angle for the given spectral device is a constant value which depends on the collimator size and its focal length:

A special device (F/# matcher) is used to match the working apertures of the light source and spectral instrument. The light beam is outgoing from the optical fiber or from the light source directly. F/# matchers are used in conjunction with a spectral instrument providing its maximum light-gathering power.

#### F/# matcher features:

• Use of full aperture of spectral instrument
• Stray light reduction
• Good spectral and spatial image quality
• Possibility to use filters with various thickness without focus distortion