How Does Diffraction-Grating Spectrometry Work?

Diffraction-grating spectrometry uses a grating’s closely spaced grooves to spread light into its component wavelengths. Each wavelength forms a bright line, or spectral peak, at a different angle. If you know the grating spacing, the diffraction order, and the angle of a peak, you can calculate its wavelength.

The question title does not specify a particular instrument, dataset, or numerical problem, so this article explains the standard calculation for a grating at normal incidence and notes the main variations.

The grating equation

For light arriving perpendicular to the grating surface, bright maxima occur when light from neighboring grooves travels a whole-number multiple of a wavelength farther than the adjacent ray. The condition is

$$m\lambda=d\sin\theta,$$

where:

• $m$ is the signed diffraction order: an integer such as $0, \pm1, \pm2,\ldots$;
• $\lambda$ is the light’s wavelength;
• $d$ is the distance between adjacent grooves;
• $\theta$ is the signed angle between the diffracted beam and the grating’s normal, with its sign indicating which side of the normal the beam lies on.

The central, undispersed beam is the zeroth order, $m=0$. It does not give a wavelength from this equation. The first-order spectrum has $|m|=1$, the second-order spectrum has $|m|=2$, and so on. A grating can produce spectra on both sides of the central beam; the two sides are commonly labelled with positive and negative orders.

Convert line density to groove spacing

Gratings are often specified by their line density, such as 600 lines per millimetre. The spacing is the reciprocal of that density. In this example,

$$d=\frac{1\ \mathrm{mm}}{600}=0.001667\ \mathrm{mm}=1.667\times10^{-6}\ \mathrm{m}.$$

Keep units consistent when using the equation. If $d$ is in metres, the calculated wavelength is in metres. Convert it to nanometres using $1\ \mathrm{nm}=10^{-9}\ \mathrm{m}$ if that is more convenient.

Worked example: calculate a wavelength

Suppose a 600-lines-per-millimetre grating is illuminated normally, and a bright first-order spectral line appears at $20.0^\circ$ from the normal. Then $m=1$, $d=1.667\times10^{-6}\ \mathrm{m}$, and $\theta=20.0^\circ$:

$$\begin{aligned} \lambda&=\frac{d\sin\theta}{m}\\ &=\frac{(1.667\times10^{-6}\ \mathrm{m})\sin(20.0^\circ)}{1}\\ &\approx5.70\times10^{-7}\ \mathrm{m}\\ &\approx570\ \mathrm{nm}. \end{aligned}$$

So the measured line is at approximately 570 nm, in the visible region. For the same grating and wavelength, the second-order angle would satisfy $\sin\theta=2\lambda/d$, giving $\theta\approx43.2^\circ$. Higher orders are possible only while $|m\lambda/d|\leq1$, since the sine of a real angle cannot exceed 1 in magnitude.

Measuring the angle in a spectrometer

A spectrometer may report a reading on a rotating scale rather than the angle directly. First identify the zero-order beam or the instrument’s defined reference direction. The angle in the equation is measured from the grating normal, not automatically from the tabletop, a telescope arm, or an arbitrary scale zero.

If the same spectral line is measured symmetrically on both sides of the central beam, let the separation between its two readings be $\Delta$. For a well-aligned instrument, the angle from the center is approximately $\theta=\Delta/2$. This paired measurement can reduce errors from an uncertain zero reading. Use the instrument’s calibration and alignment procedure if it specifies a different method.

When the light is not incident normally

The compact equation $m\lambda=d\sin\theta$ assumes normal incidence. For oblique incidence, angles and signs depend on the convention used to measure them. With incidence and diffraction angles measured from the normal on opposite sides of it, one common form is

$$d(\sin\alpha+\sin\beta)=m\lambda,$$

where $\alpha$ is the incident angle and $\beta$ the diffracted angle. Other sign conventions give an equivalent equation with a minus sign. Check the instrument or textbook’s angle convention before substituting values; do not use the normal-incidence formula unchanged for an oblique setup.

Common mistakes

• Using line density as spacing. A value in lines per millimetre must be inverted to obtain millimetres per line, then converted to compatible units.
• Using the wrong order. The order is an integer identifying a particular maximum, not a quantity to infer from the angle alone. Check which spectral copy you measured.
• Measuring from the wrong reference. The angle is measured from the grating normal under the stated convention.
• Mixing degrees and radians in software. Many calculators distinguish angle modes. Set the mode to degrees when entering a measured angle in degrees.
• Ignoring overlapping orders. Different wavelengths can appear at the same angle in different orders: for example, a wavelength in second order can overlap a wavelength twice as long in first order. A filter or other instrument information may be needed to identify the correct order.

For a wavelength measurement, the result is only as reliable as the groove spacing, angle reading, alignment, and order assignment. In a real instrument, compare against known spectral lines or follow its calibration procedure before interpreting unknown peaks.