# 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.

Phase shift of light hitting scribed surface

Typically, light is directed to a grating either from a great distance or by projection through optics that collimate the light. The result is that (at least to a very good approximation), the incoming light appears to be a plane wave. Thus, when a particular phase (say, the crest) of the wave hits one facet or echellette of a grating, the wavefront has not yet reached the corresponding point on the next echellete. For an incidence angle α, measured from the normal to the plane of the grating, the extra distance or phase shift through which the wave front must propagate to reach the next echellette is d sin α. Once the wave hits an echellete, it scatters in every direction, but it can only be seen where the scattering from the set of echelletes gives constructive interference.

Click on the images below to see a bigger image: