# Diffraction Gratings

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## Light Interference

Light acts as a wave, so different portions of a light beam appear brighter or dimmer due to the interaction of the electromagnetic field in the waves. Intensity is proportional to the square of the electric field. If

$E = E_o \sin ωt$

then

$I = \dfrac{E_o^2}{8π}.$

If $$E$$ from one wavelet interacts with another wavelet, the phase shift between the two waves modifies the observable intensity,

$I = E_o0^2 \dfrac{|\cos ϕ|}{8π}$

where $$ϕ$$ is the phase shift. When $$ϕ = π/2$$ or $$3π/2$$ radians, the intensity goes to 0. This is illustrated in the figure below. The upper wave is taken as the reference phase. In the left-hand inset, the second wave is in phase, giving an output with the same phase and summed amplitude. In the middle inset, the second wave lags the first by 1/4 wavelength (π/2 radians, giving $$ϕ = π/4$$), so the sum is reduced in amplitude compared to the first case. In the last inset, the two waves are shifted by half a wavelength (π radians, giving ϕ = π/2), and the summed electric field vanishes (see the flat line?).

Click on a thumbnail image below to see a bigger image for the shifts:

Interference_In_Phase

Interference_Quarterwave_Shift

Interference_Halfwave_Shift

## The Grating Construction

A periodically-roughened surface: the diffraction grating

While the drawing doesn't indicate scale, d (the groove spacing) is typically small -- much less than 1 mm. Common groove densities (1/d) are 300, 600, 1200, 2400, 3600, and 4800 grooves per mm, corresponding to d = 3.3333 µm to 208.33 nm respectively.