Diffraction Gratings Tutorial
Diffraction gratings, either transmissive or reflective, can separate different wavelengths of light using a repetitive structure embedded within the grating. The structure affects the amplitude and/or phase of the incident wave, causing interference in the output wave. In the transmissive case, the repetitive structure can be thought of as many tightly spaced, thin slits. Solving for the irradiance as a function wavelength and position of this multi-slit situation, we get a general expression that can be applied to all diffractive gratings,
known as the grating equation. The equation states that a grating with spacing , of th order, will diffract light at a wavelength of at an angle of . The diffracted angle, , is the output angle as measured from the surface normal of the grating. It is easily observed from Eq. 1 that for a given order , different wavelengths of light will exit the grating at different angles. For white light sources, this corresponds to a continuous, angle-dependent spectrum.
One popular style of grating is the transmission grating. This type of grating is created by scratching or etching a transparent substrate with a repetitive, parallel structure. This structure creates areas where light can scatter. A sample transmission grating is shown in Figure 1.
The transmission grating, shown in Figure 1, is comprised of a repetitive series of grooves of narrow width and separation . The incident light impinges on the grating at an angle , as measured from the surface normal. The light of order exiting the grating leaves at an angle of , relative to the surface normal. Utilizing some geometric conversions and the general grating expression (Eq. 1) an expression for the transmissive diffraction grating can be found:
Another very common diffractive optic is the reflective grating. A reflective grating is made by depositing a metallic coating on an optic and ruling parallel grooves in the surface. Reflective gratings can also be made of epoxy and/or plastic imprints from a master copy. In all cases, light is reflected off of the ruled surface at different angles corresponding to different orders and wavelengths. An example of a reflective grating is shown in Figure 2. Using a similar geometric setup as above, the general expression for the reflective grating is identical to the tranmission grating equation (see Eq. 2).
Both the reflective and transmission gratings suffer from the fact that the zeroth order mode contains no diffraction pattern and appears as a surface reflection or transmission, respectively. Solving Eq. 2 for this condition, = , we find the only solution to be =0, independent of wavelength or gratings spacing. At this condition, no wavelength-dependent information can be obtained, and all the light is lost to surface reflection or transmission.
This issue can be resolved by creating a repeating surface pattern, which produces a different surface reflection geometry. Gratings of this type are commonly referred to as blazed (or ruled) gratings. An example of this repeating surface structure is shown in Figure 3.
Blazed (Ruled) Grating
Blazed Grating, 0th
Blazed Grating Geometry
The blazed grating shown in Figure 3 is characterized by two main variables: , the groove or facet spacing, and , the blaze angle. The blaze angle is the angle between the surface structure and the surface parallel. It is also the angle between the surface normal and the facet normal.
The blazed grating features the same geometries as the transmission and reflection gratings discussed thus far; the incident and exit angles are determined from the surface normal of the grating. However, the significant difference is the surface reflection geometries are determined based on the blaze angle, , and NOT the surface normal. This results in the ability to change the diffraction geometries by only changing the blaze angle of the grating.
The 0th order reflection from a blazed grating is shown in Figure 4. The incident light at angle is reflected at for = 0. From Eq. 2, the only solution is = . This is analogous to specular reflection from a flat surface.
Blazed Grating, Specular Reflection
The specular reflection from the blazed grating is different from the flat surface due to the surface structure, as shown in Figure 5. The specular reflection, , from a blazed grating occurs at the blaze angle geometry. Performing some simple geometric conversions, one finds that
Utilizing Eqs. 2 and 3, we can find the grating equation for a blazed grating at twice the blaze angle:
There is one final case that plays an important role in monochromators and spectrometers, the Littrow configuration. In this configuration, the angle of incidence of the incoming and diffracted light are the same, = , and > 0 so
The Littrow configuration angle, , is dependent on the most intense order ( = 1), the design wavelength, , and the grating spacing . It is easily shown that the Littrow configuration angle, , is equal to the blaze angle, , at the design wavelength. The Littrow / Blaze angles for all Thorlabs' Blazed Gratings can be found in the grating specs tables.
It is easily observed that the wavelength dependent angular separation increases as the diffracted order increases for light of normal incidence (for =0, increases as increases). There are two main drawbacks for using a higher order diffraction pattern over a low order one: (1) a decrease in efficiency at higher orders and (2) a decrease in the free spectral range, , defined as:
where is the central wavelength, and is the order.
The first issue with using higher order diffraction patterns is solved by using an Echelle grating, which is a special type of ruled diffraction grating with an extremely high blaze angle. The high blaze angle is well suited for concentrating the energy in the higher order diffraction modes. The second issue is solved by using another optical element: grating, dispersive prism, or other dispersive optic, to sort the wavelengths/orders after the Echelle grating.
While blazed gratings offer extremely high efficiencies at the design wavelength, they suffer from periodic errors, such as ghosting, and relatively high amounts of scattered light, which could negatively affect sensitive measurements. Holographic gratings are designed specially to reduce or eliminate these errors. The drawback of holographic gratings compared to blazed gratings is reduced efficiency.
Holographic gratings are made from master gratings by similar processes to the ruled grating. The master holographic gratings are typically made by exposing photosensitive material to two interfering laser beams. The interference pattern is exposed in a periodic pattern on the surface, which can then be physically or chemically treated to expose a sinusoidal surface pattern. An example of a holographic grating is shown in Figure 6.