Diffraction Grating

Light does not stop dead at the edge of an obstacle. It leans round it, and if the obstacle is about as wide as the wavelength the leaning is large enough to see. A grating takes that behaviour and puts thousands of edges in a row, so that the leaning from every one of them adds up in a few sharp directions and cancels everywhere else. Those directions depend on wavelength, which is what turns a grating into a measuring instrument.

What diffraction is

Diffraction is the bending of light waves round the edges of an obstacle or an aperture when the size of that obstacle is comparable with the wavelength. Instead of travelling in strictly straight lines the light spreads into the region of the geometrical shadow, and a pattern of bright and dark fringes appears where a sharp edge was expected. It is a direct consequence of the wave nature of light.

Huygens' principle explains it. Every point on a wavefront acts as a source of secondary wavelets, and the wavefront a moment later is the envelope of all of them. At an aperture the wavelets near the edges have no neighbours to cancel their sideways spread, so light arrives in directions that geometry alone would forbid.

How a plane grating is made

A plane diffraction grating is a large number of equally spaced parallel slits, all lying in one plane. It is made by ruling fine equidistant lines on an optically flat glass plate with a diamond point, and a laboratory grating carries several thousand of them to the centimetre. The grating used here is ruled with 15000 lines to the inch.

The ruled strips scatter light and act as obstacles. The untouched glass between them transmits, so each unruled strip behaves as a slit. Call the width of a transparent slit a and the width of an opaque ruling b. Then (a + b) is the distance from the centre of one slit to the centre of the next, and it is called the grating element. It is the one dimension of the grating that the measurement depends on, and it follows straight from the number of lines ruled per unit length.

e = (a + b) = 2.54 / N cm

Where the light goes

Monochromatic light falling on the grating makes every slit a source of secondary wavelets. Those wavelets interfere after they leave. In most directions the contributions from the many slits arrive out of step and wipe each other out. In a few directions they all arrive in step, and there the light piles up into a sharp principal maximum.

Light leaving adjacent slits at an angle θ to the normal travels paths that differ by (a + b) sin θ. The wavelets are in step whenever that difference is a whole number of wavelengths.

(a + b) sin θ = nλ

Setting n = 0 gives θ = 0. Every wavelength satisfies it, so all of them go straight through together and the central image is white and undispersed. That is the direct image, and it carries no information about wavelength at all. The useful orders are the ones on either side of it, where θ depends on λ.

The equation does not stop at n = 1. Every whole number gives another direction, each further from the direct image than the last, so the whole spectrum repeats outward as a second order, a third, and so on. It stops when the equation asks for sin θ greater than one, which is no direction at all: the highest order a wavelength can reach is the whole number of times λ divides into the grating element. With the chart's grating the violet reaches a fourth order, the blue and the green a third, and the yellow pair and the red only a second — so sweeping outward the spectrum loses a colour off its long-wavelength end at every turn, and the last thing left is the violet. The chart takes its readings in the first order, and that is where the bench starts, but sweeping the telescope further out finds the rest of them, and each one gives the same wavelength back once its own n is divided out.

Higher orders are fainter. The grating equation says where they are but not how bright, and the single-slit envelope of one ruling hands out less light the further off axis you look. A second order is dimmer than the first, and a ninth is dim indeed, which is one practical reason measurements are usually taken in the first or second.

λ = (a + b) sin θ / n

Why the angle is measured on both sides

Each spectral line appears twice, once to the left of the direct image and once to the right, at equal angles. The spectrometer's circle has an arbitrary zero, so no single reading is an angle of diffraction by itself. Taking the difference between the left-hand and right-hand readings for the same line gives 2θ, and the arbitrary zero drops out.

Measuring both sides buys something else. If the grating is a little off normal, the incident beam meets it at an angle and the two sides no longer come out equal. Half the sum of the two deviations is still very close to the true angle, so the error largely cancels in the mean while a single-sided measurement would carry all of it.

The mercury spectrum

A mercury discharge emits at a set of definite wavelengths rather than across a continuous band, so the grating hands each one its own direction and the spectrum arrives as separated images of the collimator slit. Six of them are bright enough to set the cross-wires on, and the chart takes its readings on four: the violet, the green and the two yellows.

The two yellow lines are the reason the spectrometer is read to a minute of arc. They differ by about 21 Å, which in the first order puts them roughly five minutes apart on the circle. A least count of one minute separates them. A coarser instrument would show one thick yellow line and the observation table would have a row it could not fill.

Resolving power and dispersion

Two things decide whether a grating can separate a pair of close lines. Angular dispersion is how far apart in angle it puts two given wavelengths, and it grows as the grating element shrinks and as the order rises. Resolving power is how narrow it makes each image, and for a grating it is the product of the order and the total number of rulings the beam covers. Ruling more lines to the inch improves both, which is why a fine grating is worth the trouble of making.

The trade runs the other way for orders. A coarse grating has a large element, so λ divides into it many times and a great many orders fit inside ninety degrees, but they are crowded up near the direct image and each is poorly separated. A fine grating throws its first order well out and may have room for only two or three orders altogether. Putting a different grating in the holder and watching what happens to the strip shows both halves of that at once.

How a reading is taken

1. The spectrometer is adjusted for parallel light by Schuster's method or on a distant object, so the collimator delivers a parallel beam and the telescope is focused for it.
2. The grating is mounted on its table and set with its plane perpendicular to the incident beam.
3. The telescope is brought in line with the collimator, the direct image of the slit is set on the cross-wires, and that reading is noted.
4. The telescope is swung to the left-hand first-order spectrum and the cross-wires are set on the violet line, then the green, then each yellow line in turn, both verniers being read each time.
5. The telescope is swung across to the right-hand spectrum and the same four lines are read again.
6. The difference between the readings on the two sides gives 2θ for each line, the two verniers are averaged, and λ follows from (a + b) sin θ = nλ.