# Diffraction Gratings

A diffraction grating is a spatial dispersion element that is commonly used in optics for spectrum analysis of broadband signals [Goodman (2004)]. Let us first consider a complex transmittance function, relating the input and output waveform distributions, of a thin sinusoidal diffraction grating of Fig. 5.3:

where

Now consider a plane wave propagating along the z-axis given by

so that *y(x, y,* 0_{-}; t) = 1, where the explicit time dependence *e ^{jwt}* is dropped for convenience. The wave just after the grating can then be expressed as:

Considering the lens transmittance function t_{1}(x, *y;* w), wave output after the lens is given by

where *d* is the focal length of the lens. The output wave, in the focal plane, is

where *h(x,y*) is the free-space impulse response. Using the paraxial- wave approximations, the output wave can finally be written as:

|/(x,y; z = d)| = |*y(x,*y; l+) * *h(x,*y)|^{2}

where 50 is the Dirac-delta function. This equation consists of three terms where 5(y) corresponds to the 0^{th}-order diffraction and the other two terms, which are symmetric about the origin, are the ±1^{st}- order diffraction terms. It is clear from the equation above that the location of intensity maxima of the ±1^{st} diffraction order depends on the signal wavelength (or the frequency w). Selecting one of these orders, we obtain the frequency-to-space mapping relation of this system where each frequency *w* is mapped onto a specific point [0, y(w)] on the output plane according to the relation:

**Figure 5.3 **Thin sinusoidal grating excited with a normally incident plane wave.

This is the 1D frequency scanning along the у-axis using a thin sinusoidal diffraction grating. While a thin sinusoidal grating is taken here as an example, a large variety of diffraction gratings exist, which are used for spectral analysis using their higher-order diffraction patterns.