From Figure 2 it can be seen that angle β’’ – which is the new diffraction angle measured from the non-tilted grating normal – becomes β’’ = β’ – θ for a transmission grating but β’’ = β’ + θ for a reflection grating. The new diffraction angle (β’) is now measured from the tilted grating normal. When the grating is tilted in the incoming beam by an angle θ, the angle of incidence is reduced to α’ = α-θ. For the transmission grating the angles of transmitted orders are positive below the grating normal. Angles on the left side of the grating are positive when measured counterclockwise. β’’ is measured from the non-tilted grating normal. We will use Figure 2 in order to find the relation between the diffraction angle β’’ as a function of tilt angle θ. Where m is the diffraction order (0, -1, +1, -2, +2, etc.). The general grating equation that determines the relation between angle of incidence (α), diffraction angle (β), the wavelength of light (λ), and the period (Λ) reads: In order to calculate the actual deflection angles one has to consider the grating equation and add a tilt to the grating normal. The plots in Figure 1 were calculated by the grating equation (see next Section for details). Figure 1 compares the total change in 1 st order deflection angle for a transmission and reflection grating compared to the change of the 0 th order for a typical grating groove density of 1200 l/mm. where l is the wavelength of incident light and m is an integral number of diffraction bands. Again, since the higher order diffractions follow the 0 th order diffraction, the m th order diffraction is almost unaffected by tilting the grating. For a transmission grating however, the 0 th order goes straight through the grating and is not affected by tilting the grating. The higher order diffractions basically follow the 0 th order diffraction so, the m th order diffraction also shifts angularly by twice the tilt angle of the grating. For a reflective grating, the 0 th order is obviously reflected back from the surface just as if the grating was a plane mirror so, when the grating is tilted the 0 th order diffraction shifts by twice the angular tilt. For the 0 th order diffraction the diffraction angle (β) equals the angle of incidence (α). First we will consider the 0 th order diffraction order for both types of gratings while tilting the grating slightly. In the following, all angles are measured relative to the normal of the grating. The main difference can be seen on the figures above.
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