How small Λ must be depends on the wavelength and the angle of incidence. The most straightforward way to realize a single order is to make the grating period Λ small enough to eliminate all other nonzero orders as solutions to the Grating Equation (1). The condition for a single diffracted order:įor many gratings, especially those designed for use with lasers, it is desirable for all of the incident light to be diffracted into a single order to minimize loss in the overall system. Nevertheless they are real, and can emerge from the edge of the glass substrate. These orders are totally internally reflected inside the substrate, and therefore do not propagate in the air region outside the substrate. Examples are the +2 nd and –3 rd orders shown in Figure 6. However, it is important to recognize that there may be orders that do not exist as solutions to (1), yet they exist and propagate inside the substrate. Since the incident and diffracted orders are all measured in air, the form of the Grating Equation in (1) should be used.
Here the grating interface is shown as the front surface of a glass substrate. For the reflected orders, n m = n i, and the Grating Equation becomesįor the transmitted orders, n m = n t, and the Grating Equation becomesĪ typical transmission grating is illustrated in Figure 6. Where n m is the index of refraction of the region into which the diffracted light travels. To be more complete, if a grating is at an interface between two media with indexes n i and n t, the Grating Equation (1) is written as The graph in Figure 3 also shows a dashed line called the “Littrow line.” When the curve associated with a particular m θ i), whereas when n t is larger than n i the opposite occurs. Try out PGL’s “Grating Calculator” tool to visualize the Grating Equation just as shown in Figure 3. For example, for θ smaller than about 5º only the +1 st and –1 st orders exist, while for θ larger than about 38º only the –1 st, –2 nd, and –3 rd orders exist. From the graph it is apparent that different orders exist for different angles of incidence. Here the specific case of θ = 10º illustrated in Figure 2 is indicated by the solid dots on the graph. The dependence of the angles of diffraction on the angle of incidence can be more completely visualized on the graph shown in Figure 3. For diffracted light traveling left-to-right, θ m ≥ 0, whereas for diffracted light traveling right-to-left, θ m ≤ 0. Incident light is shown traveling left-to-right, for which the angle θ ≥ 0. Note the sign conventions for the angles. This 0 th order is typically not considered a diffracted order since it does not provide any angular dispersion (change in angle with change in wavelength). Referring to Figure 2, there will be three diffracted orders ( m = –2, –1, and +1) along with the specular reflection ( m = 0). As an example, suppose a HeNe laser beam at 633 nm is incident on an 850 lines/mm grating. The larger the period Λ, or the lower the frequency f, the more orders there are. (2) A graphical example of the grating equation: Figure 2 In terms of f the grating equation becomes Often gratings are described by the frequency of grating lines instead of the period, where f (in lines/mm) is equal to 10 6/Λ (for Λ in nm). For a given angle of incidence, θ, it gives the angle of diffraction θ m for each “order” m for which a solution to (1) exists. Since AB = Λsinθ m and A’B’ = Λsinθ, where Λ is the grating period and θ m and θ are the angles of diffraction and incidence, respectively, relative to the surface normal, the condition for constructive interference is Mathematically, the difference between paths AB and A’B’ is a multiple of the wavelength when AB – A’B’ = mλ, where m is an integer and λ is the wavelength of light (typically stated in nm). If the difference between adjacent green-blue ray paths diffracted off of identical locations on adjacent periods is equal to a multiple of the wavelength of light, the two blue rays interfere constructively. The light is diffracted in many directions, only one of which is indicated by the blue rays. Referring to Figure 1, imagine a beam of light represented by the two green rays incident on the binary (rectangular profile) grating shown. Constructive interference leads to the grating equation: Figure 1
HOW TO FIND THE CUT OFF WAVELENGTH FROM A GRAGH SERIES
If the surface irregularity is periodic, such as a series of grooves etched into a surface, light diffracted from many periods in certain special directions constructively interferes, yielding replicas of the incident beam propagating in those directions. When light is incident on a surface with a profile that is irregular at length scales comparable to the wavelength of the light, it is reflected and refracted at a microscopic level in many different directions as described by the laws of diffraction. Gratings are based on diffraction and interference:ĭiffraction gratings can be understood using the optical principles of diffraction and interference.
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