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Gratings Tutorial

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 when = 0°,

Grating Equation 1

(1)

known as the grating equation. The equation states that a grating with spacing a will deflect light at discrete angles (theta sub m), dependent upon the value λ, where  is the order of principal maxima. The diffracted angle, theta sub m, 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 m, different wavelengths of light will exit the grating at different angles. For white light sources, this corresponds to a continuous, angle-dependent spectrum.

 

Transmission Grating
Figure 1. Transmission Grating

Transmission Gratings

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 narrow-width grooves separated by distance a. The incident light impinges on the grating at an angle theta sub i, as measured from the surface normal. The light of order m exiting the grating leaves at an angle of theta sub m, 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:

Grating Equation 2

(2)

where both  and  are positive if the incident and diffracted beams are on opposite sides of the grating surface normal, as illustrated in the example in Figure 1. If they are on the same side of the grating normal,  must then be considered negative.

 

Reflective Grating
Figure 2. Reflective Grating

Reflective Gratings

Another very common diffractive optic is the reflective grating. A reflective grating is traditionally 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 grating equation for reflective gratings can be found:

Grating Equation 3

(3)

where is positive and is negative if the incident and diffracted beams are on opposite sides of the grating surface normal, as illustrated in the example in Figure 2. If the beams are on the same side of the grating normal, then both angles are considered positive.

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, theta sub i = theta sub m, we find the only solution to be m=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) Gratings

Blazed Grating
Figure 4. Blazed Grating, 0th Order Reflection
Blazed Grating
Figure 3. Blazed Grating Geometry

The blazed grating, also known as the echelette grating, is a specific form of reflective or transmission diffraction grating designed to produce the maximum grating efficiency in a specific diffraction order. This means that the majority of the optical power will be in the designed diffraction order while minimizing power lost to other orders (particularly the zeroth). Due to this design, a blazed grating operates at a specific wavelength, known as the blaze wavelength.

The blaze wavelength is one of the three main characteristics of the blazed grating. The other two, shown in Figure 3, are a, the groove or facet spacing, and gamma, the blaze angle. The blaze angle gamma 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 geometries similar to the transmission and reflection gratings discussed thus far; the incident angle () and th order reflection angles () are determined from the surface normal of the grating. However, the significant difference is the specular reflection geometry is dependent on the blaze angle, gamma, and NOT the grating surface normal. This results in the ability to change the diffraction efficiency 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 theta sub i is reflected at theta sub m for m = 0. From Eq. 3, the only solution is theta sub i = –theta sub m. This is analogous to specular reflection from a flat surface.

Blazed Grating
Figure 6. Blazed Grating, Incident Light Normal to Grating Surface
Blazed Grating
Figure 5. Blazed Grating, Specular Reflection from Facet

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, theta sub r, from a blazed grating occurs at the blaze angle geometry. This angle is defined as being negative if it is on the same side of the grating surface normal as theta sub i. Performing some simple geometric conversions, one finds that

Grating Equation 2

(4)

Figure 6 illustrates the specific case where theta sub i= 0°, hence the incident light beam is perpendicular to the grating surface. In this case, the 0th order reflection also lies at 0°. Utilizing Eqs. 3 and 4, we can find the grating equation at twice the blaze angle:

Grating Equation 2

(5)

Littrow Configuration

The Littrow configuration refers to a specific geometry for blazed gratings and plays an important role in monochromators and spectrometers. It is the angle  at which the grating efficiency is the highest. In this configuration, the angle of incidence of the incoming and diffracted light are the same, theta sub i = theta sub m, and m > 0 so

Grating Equation 2

(6)

Blazed Grating
Figure 7. Littrow Configuration

The Littrow configuration angle, Theta sub L, is dependent on the most intense order (m = 1), the design wavelength, lambda sub D, and the grating spacing a. It is easily shown that the Littrow configuration angle, Theta sub L, is equal to the blaze angle, gamma, at the design wavelength. The Littrow / blaze angles for all Thorlabs' Blazed Gratings can be found in the grating specs tables.

Grating Equation 2

(7)

It is easily observed that the wavelength dependent angular separation increases as the diffracted order increases for light of normal incidence (for theta sub i= 0°, theta sub m increases as m 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, Free Spectral Range, defined as:

Grating Equation 2

(8)

 

where lambda is the central wavelength, and m 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 and relatively low groove density. 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.

 

Holographic Gratings
Figure 8. Holographic Grating

Holographic Surface Gratings

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 8.

Please note that dispersion is based solely on the number of grooves per mm and not the shape of the grooves. Hence, the same grating equation can be used to calculate angles for holographic as well as ruled blazed gratings.

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