Diffraction grating wavelength theory
Understand diffraction grating wavelength: the equations, assumptions, supporting concepts and measurement method.
01Why bright orders occur
Light from adjacent illuminated grooves reaches a distant direction with path difference d sin(theta) at normal incidence. Reinforcement occurs when that difference is an integer number of wavelengths. The integer is the diffraction order m. The central undeviated image is order zero; matched positive and negative orders lie on either side.
Why is the central image unsuitable for determining wavelength from its angle alone?
For m = 0, the condition gives theta = 0 for every wavelength, so it contains no wavelength-dependent angular displacement.
02Line density is not line spacing
A line density tells how many periods fit in one unit of length. Spacing d is its reciprocal. For 300 lines per millimetre, one millimetre divided by 300 gives 0.003333 mm, or 3.333333 × 10⁻⁶ m. Use metres when calculating wavelength in SI units. Confusing millimetres and metres introduces a factor of one thousand.
Does a larger groove density mean larger spacing?
No. More grooves per millimetre means a smaller distance between neighbouring grooves.
03Use a pair of symmetric readings
Measure the same order and spectral line on the left and right. If signed angles about the central direction are alphaL and alphaR, take theta as half their separation. A constant zero offset common to both readings cancels. Real circular scales may wrap through zero, so first unwrap readings into a consistent interval. This cancellation does not correct an incorrectly aligned grating or an incorrectly identified line.
If the left and right readings are −19.27° and +19.27°, what angle enters the grating equation?
Use 19.27°, half the 38.54° separation; using the full separation would overestimate wavelength.
04A graph extracts wavelength
For fixed spacing and wavelength, sin(theta) increases linearly with positive order. Plot order m horizontally and sin(theta) vertically. Call the fitted gradient s to avoid confusing it with order m. The wavelength is d times s. Both axes are dimensionless, so the gradient is dimensionless and multiplication by d supplies units of length. Retain a fitted intercept to inspect deviations rather than forcing zero.
Which graph makes the normal-incidence relationship linear?
sin(theta) against order, with slope lambda/d. Angle itself is not exactly proportional to order.
05Angles, radians and permitted orders
The displayed angle is in degrees. Convert it to radians before using sine in software or in a calculator set to radians. Do not replace sine by the angle at the larger settings. An order exists geometrically only when |m lambda/d| is at most one. Visibility also depends on grating efficiency, finite slit width, aperture and intensity; the existence condition alone does not promise a bright observable line.
Does the small-angle approximation remain good at approximately 55.6°?
No. Use the full sine function. This lesson deliberately includes angles too large for sin(theta) approximately equal to theta.
06Uncertainty and systematic effects
An angle uncertainty u(theta) in radians contributes approximately cos(theta) times u(theta) to uncertainty in sin(theta). Equal angle precision therefore does not produce identical vertical uncertainties across all orders. A simple unweighted fit omits this difference, while an uncertainty in grating spacing changes every wavelength estimate together. Normal-incidence errors, line width and telescope centring can also dominate rounded angular resolution.
Does a small fitted slope error include uncertainty in groove spacing?
No. Grating calibration is an additional shared uncertainty and must be considered separately.
The generator uses wavelength 550 nm and density 300 lines/mm, giving d = 10⁻³/300 m. For orders 1 to 5, theta = arcsin(m lambda/d) is rounded to 0.01 degrees. The stored y value is sin(theta_deg × pi/180), x is the integer order and the estimated wavelength is fitted gradient times d. The fifth-order sine is 0.825, so it is geometrically permitted. The reference wavelength is a chosen teaching parameter rather than an identification of a particular atomic emission line. The displayed raw angle represents half a symmetric left/right separation; independent telescope readings are not generated.