Resolving Power of Grating

A grating hands every wavelength its own direction, so two different wavelengths arrive at two different places. That much is the grating equation and nothing more. It says where the two images stand but says nothing about how wide they are, and a pair of images can be in different places and still be impossible to tell apart. What decides the width is the number of rulings the light covers, and that is what this experiment measures.

What resolving power is

Resolving power is the ability of a grating to show two very close wavelengths as two separate spectral lines instead of one merged blur. It is written as a pure number: the wavelength being worked at, divided by the smallest difference in wavelength the instrument can still tell apart there.

R = λ / dλ

Because it is a ratio of two wavelengths it has no units, and because dλ sits underneath it a large number means a good instrument. An R of 274 means the grating can separate wavelengths that differ by one part in 274. At the yellow end of the mercury spectrum that is about 21 ångström.

Rayleigh's criterion

The two nearby wavelengths λ and λ + dλ each produce their own diffraction pattern, a principal maximum with minima on either side of it. When the two wavelengths are close the two patterns overlap, and the eye sees their sum, because light of two different wavelengths does not interfere. Whether that sum reads as one line or two is a question about the shape of a curve, and somebody has to say where the boundary is.

Lord Rayleigh put it where the pattern itself provides a landmark. Two spectral lines are said to be just resolved when the principal maximum of one wavelength falls exactly on the first minimum of the other. At that exact overlap the resultant still shows a shallow trough, about 81 % of the peak intensity, which is enough for the eye or a detector to register two lines rather than one.

The 81 % is not a coincidence and not a fitted number. Two equal maxima that far apart put 8/π² of one peak's intensity at the midpoint between them, and 8/π² is 0.8106. It falls out of the shape of the pattern.

The three conditions

The figure below shows all three cases side by side, and the simulation moves between them as the stop on the grating is opened and closed.

• Unresolved. The two wavelengths are too close together for the instrument. Their patterns overlap so heavily that the resultant is a single smooth hump with no trough at all, and the lines cannot be told apart.
• Just resolved. The principal maximum of λ + dλ falls exactly on the first minimum of λ. The resultant shows a trough at about 81 % of the peak. This is the borderline case, and it is the one the formula is derived from.
• Well resolved. The wavelengths are further apart than the limit. The resultant drops close to zero between the two peaks and the lines are unambiguous.

Where R = nN comes from

Let the grating have element e = a + b, with N rulings covered by the beam, observed in the n-th order. The derivation is four short steps and one substitution.

The principal maximum for λ. Adjacent slits give a path difference of e sin θ, and the n-th order maximum stands where that difference is a whole number of wavelengths.

e sin θ = nλ

The principal maximum for λ + dλ. The slightly longer wavelength has its own n-th order maximum, a little further out at θ + dθ.

e sin(θ + dθ) = n(λ + dλ) = nλ + n dλ

The first minimum next to the n-th maximum of λ. Here the number of rulings enters for the first time. A principal maximum is sharp because all N slits agree, and the first direction in which they stop agreeing is the one where the path difference across the whole grating is one extra wavelength beyond nNλ.

N e sin(θ + dθ) = (nN + 1) λ

Rayleigh's criterion. The lines are just resolved when that minimum coincides with the principal maximum of λ + dλ, which is exactly what writing the equation above at the angle θ + dθ says. Dividing it through by N gives the same quantity as the second equation.

e sin(θ + dθ) = nλ + λ/N

Both expressions are e sin(θ + dθ), so they are equal. The nλ cancels from each side, leaving n dλ = λ/N, and rearranging that gives the result.

R = λ / dλ = nN

Two things and only two things are in it. The wavelength has gone, the grating element has gone, and the angle has gone. Everything the instrument can do to a close pair is decided by how many rulings are lit and which order is being looked at.

What that means at the bench

The mercury lamp gives a yellow pair that is close enough to be a real test.

Yellow I

5769.6 Å

Yellow II

5790.7 Å

Separation, dλ

21.1 Å

Mean wavelength, λ

5780.2 Å

R needed, λ/dλ

274

A grating ruled with 15000 lines to the inch has a grating element of 1.6933 × 10⁻⁴ cm, so a centimetre of ruled width carries 5905 rulings. In the first order that is a resolving power of 5905, more than twenty times what this pair demands, and the two lines come apart easily. Stop the beam down to half a millimetre of ruled width and only about 274 rulings are left lit. That is the moment the pair reaches Rayleigh's limit, and it is the same 274 that λ/dλ gives.

This is worth dwelling on, because it is the whole experiment. The number measured from the two wavelengths and the number counted off the rulings are the same number, arrived at from opposite ends. Neither depends on which grating is in the holder. A coarser grating needs a wider strip lit to reach the same figure, and a higher order reaches it on a narrower one, but the figure itself is fixed by the pair being looked at.

Resolving power is not dispersion

The two are easy to confuse and they are not the same thing. Angular dispersion, dθ/dλ, is how far apart in angle the grating puts two given wavelengths. Resolving power is how narrow it makes each image. A grating can have plenty of one and little of the other.

The simulation makes the difference visible in a way a formula cannot. Closing the stop leaves every line standing at exactly the angle it stood at before, because the grating equation knows nothing about how much of the plate is lit. The dispersion is untouched. What changes is the width of each maximum, and that is enough to turn two lines into one.

Both improve with order, which is the one thing they have in common. Going to the second order doubles R and also spreads the spectrum wider. The price is brightness, since the single-slit envelope hands out less light the further off axis you look, and a faint pair is harder to set a cross-wire on however well separated it is.

How the measurement is made

1. The spectrometer is adjusted for parallel light by Schuster's method or on a distant object, and the grating is set normal to the beam.
2. 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.
3. The telescope is swung into the first-order spectrum on one side, and the cross-wires are set first on Yellow I and then on Yellow II, both verniers being read each time.
4. The telescope is swung across to the other side and the same two lines are read again.
5. The difference between the two sides gives 2θ for each line, the two verniers are averaged, and λ follows from e sin θ = nλ.
6. The mean of the two wavelengths and the difference between them give R, which is the result.