Physics · Ch 6 — Optics
Diffraction in Grating
Diffraction in Grating
A diffraction grating is a plane sheet of transparent material ruled with a very large number of closely and equally spaced opaque lines (a modern commercial grating has around 6000 lines per centimetre); the rulings act as opaque obstacles of width , and the transparent gaps between them act as slits of width . The combined width of one ruling and one slit, , is the grating element; points on successive slits separated by exactly this grating element are corresponding points. A plane wavefront of monochromatic light incident normally on the grating diffracts at every slit; the path difference between diffracted light from one pair of corresponding points, at angle , is , and this same path difference applies for every pair across the whole grating. A bright maximum forms wherever , (the order of diffraction), giving the grating equation ; writing for the number of grating elements (lines) per unit width -- the value usually printed on the grating -- this becomes . Despite resembling the single-slit minimum formula , this grating formula is instead a condition for a maximum, with order ; the two must not be confused. The grating equation, used with a spectrometer (measuring the angle between two symmetric first-order i …
What this figure shows. A plane transmission grating AB, made of alternating transparent slits of width a and opaque rulings of width b (grating element a+b), is struck by a plane wavefront of monochromatic light. Diffracted light travelling at angle theta from a pair of corresponding points on adjacent slits is shown converging (via a lens, or effectively at a distant screen) at a point P; the path difference between this pair of corresponding points, (a+b) sin(theta), is exactly the quantity the grating maxima c …
What this figure shows. A sodium lamp illuminates the slit of a spectrometer's collimator, which produces a parallel beam striking a diffraction grating mounted on the turntable/prism table; the telescope, shown rotated to two symmetric angular positions on either side of the straight-through (n=0, undiffracted) direction, picks up the two first-order (n=1) diffracted images. The angle between these two symmetric telescope readings is 2 theta, so half that measured angle gives theta, the diffraction angle used directly in the grating wavelength for …
What this figure shows. A diffraction grating illuminated with white light produces, on either side of a central (m=0) white maximum, a series of spread-out spectra for each successive diffraction order (m=1, m=2, ...); within each order, red appears at the largest diffraction angle and violet at the smallest, since red's longer wavelength satisfies the grating equation sin(theta)=Nm(lambda) at a larger angle theta for the same order m -- illustrating both that the central maximum is colourless (zero path difference is achromatic) and that each higher diffraction order spre …
Worked out. A grating with 4000 slits per centimetre produces a second-order (m=2) diffraction maximum at 30 degrees. The number of lines per metre is N = 4000 times 100 = 4 times 10^5 per metre. Substituting into sin(theta) = Nm(lambda), rewritten as lambda = sin(theta)/(Nm) = sin(30 degrees)/(4e5 times 2) = 0.5/(8e5) = 6.25e-7 m = 6250 angstrom -- a wavelength squarely in the red part of the visible spectrum, recovered purely from the grating's line density and the measured diffraction a …
Worked out. Light of wavelength 500 nm strikes a grating and produces a fourth-order (m=4) bright line at 30 degrees. Rearranging sin(theta) = Nm(lambda) for N gives N = sin(theta)/(m lambda) = 0.5/(4 times 500e-9) = 2.5e5 lines per metre, which converts to 2500 lines per centimetre -- showing how the same grating equation can be solved either for wavelength (given a known grating) or, as here, for the grating's own line density (given a known wavelength), depending on which quantity the experiment is desig …