Principle of amplitude grating

An optical element composed of a large number (thousands or even tens of thousands) of slits of equal width and spacing is called a diffraction grating. However, there are already many types of modern gratings, and some diffraction units of gratings are no longer narrow slits in the usual sense. In order to include the definition of diffraction gratings in these gratings, gratings are defined as optical elements that can produce periodic spatial modulation of the amplitude or phase of incident light, or both simultaneously. Grating can be classified into amplitude grating and phase grating based on the modulation of its incident light.

Overview diagram of amplitude grating

The multi slit Fraunhofer diffraction device includes a diffraction screen. The diffraction screen has many slits of equal width and spacing, with a slit width of and a spacing of d. It can modulate the amplitude of incident light spatially and periodically. This type of diffraction screen is also known as an amplitude type rectangular grating, with d referred to as the grating constant.

An amplitude grating is a device that can modulate the amplitude of incident light waves according to the laws of sine or cosine functions, without affecting the phase distribution. The complex amplitude transmission coefficient is given by the following formula.

This is a 1D grating that is infinitely long in the x direction and has a certain B range. B represents the modulation amplitude of the grating on the amplitude of the light wave, and d is the grating constant. This grating can be obtained by capturing the interference fringes of two plane waves at a certain angle in the propagation direction. As shown in the figure, the complex amplitude transmission coefficients of sine grating and rectangular amplitude grating are displayed.

Intensity distribution of amplitude grating

Assuming a sine grating contains N interference fringes with a spacing of d. So, when a monochromatic plane wave with unit amplitude vertically illuminates a grating, the amplitude distribution on the plane immediately behind the grating can be written as (assuming the grating transmission coefficient varies along the x₁ direction)

Among them, the above equation represents within the grating range, the following equation represents outside the grating range, and B is a constant less than 1.

Calculate the diffraction intensity distribution of a grating with N units (each stripe can be regarded as a diffraction unit). Just need to obtain the diffraction factor of the unit, and then multiply it by the multi beam interference factor to obtain it. For the sinusoidal grating under discussion, the complex amplitude generated by unit diffraction is

PS. Multi beam interference factor is，and β is the wavelength of light waves.

So, the intensity distribution of the sinusoidal grating diffraction pattern is

Note the formula, the above equation can also be written as

In the formula, the diffraction intensity of the unit includes a large number of terms, making it difficult to accurately draw its image. Only their amplitude distribution diagrams are shown in Figures a to c. Figures a and b belong to diffraction factors and interference factors, respectively, and Figure c is their product. It can be seen that the diffraction pattern of the sine grating only contains zero order and ±1st order spectral lines. Similarly, the broadband of spectral lines is inversely proportional to the number of periods N of the grating. When N approaches infinity, the spectral line width decreases to zero, which can be mathematically represented by three delta functions.