A diffraction grating, consisting of many narrow slits arranged in a row, is a precision optical element that separates light according to wavelength. Dispersion by a prism can also be explained by the wavelength dependence of the refractive index.
Basic Knowledge
Diffraction grating: When light shines on a periodic structure with slit spacing (grating constant) d, diffracted light appears strongly in directions where the path difference from adjacent slits is an integer multiple of the wavelength.
Diffraction-grating condition: d sin θ = mλ（m = 0, ±1, ±2, …: diffraction order）
m = 0: zeroth-order light (straight transmission), m = ±1: first-order light, and so on.
Dispersion: The property that the refractive index of a medium differs according to wavelength (color). In a prism, shorter wavelengths (violet・blue) have larger refractive indices and bend more strongly (normal dispersion).
Rainbow: The result of refraction+total internal reflection+refraction by raindrops (spherical water droplets). Red is on the outside and violet is on the inside.
Diffraction grating（An element that separates light by wavelength using many parallel slits）
Grating constant d [m]（The spacing between adjacent slits）
Diffraction order m（The integer m in d sin θ = mλ. m = 0 is zeroth-order light）
Dispersion（The wavelength dependence of refractive index. The reason a prism separates white light into colors）
Spectrum（A pattern of light colors separated according to wavelength by dispersion or diffraction）
Further Study (Background・Significance)
A diffraction grating is a core component of a spectroscope. It has hundreds to thousands of grooves per 1 mm, producing much sharper bright lines than Young’s experiment (double slit) (the more slits there are, the sharper the interference becomes).
In astronomy, separating starlight into a spectrum reveals elemental composition, temperature, and radial velocity (Doppler shift) (astronomical spectroscopy). The dark lines in the solar spectrum (Fraunhofer lines) mark the beginning of this spectroscopic analysis.
The rainbow colors of CD/DVD surfaces appear because the pattern of recording pits engraved on the surface functions as a diffraction grating. The grating constant is ~1.5 μm (CD), close to the wavelength range of visible light.
- Diffraction-grating condition: d sin θ = mλ
- The longer λ is (red), the larger θ is (diffracted farther outward)
- Dispersion: shorter wavelengths have larger refractive indices and bend more strongly
- Rainbow: red is outside (the reflection angle in raindrops is large), violet is inside
- The rainbow colors of CDs/DVDs are an application of diffraction gratings
Points to Note (Easily Confused)
① In sin θ = mλ/d for a diffraction grating, θ is larger for longer λ (red), so red appears farther outside (opposite to a prism). ② The maximum value of m is limited by sin θ ≤ 1, giving m ≤ d/λ. ③ In dispersion by a prism, violet bends more strongly and moves inward (toward the base). ④ The grating constant d is the “width per line” (if the problem states “N lines per 1 mm,” then d = 1 mm / N).
Practice
- Light with a wavelength of 500 nm is directed at a diffraction grating with grating constant d = 2.0×10⁻⁶ m. Find the angle of the first-order diffracted light.
- Find the grating constant of a diffraction grating with 400 grooves per 1 mm, and find the first- and second-order diffraction angles for light with a wavelength of 600 nm.
- When white light is dispersed by a prism, which bends more strongly, red or violet? State the reason.