A grating-order scan varies the diffraction order m across three rows at fixed wavelength and spacing, showing how the order angle moves through d * sin(theta) = m * lambda row by row. The arithmetic is exact only for the stated orders, spacings, wavelengths, and rows in the ideal coherent-grating model. Real observations add alignment, finite slit/aperture size, coherence/bandwidth, detector calibration, grating imperfections, peak fitting, finite precision, and measurement uncertainty.
Order raises the required sine marker until the toy sine limit rejects the row. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
First order closes at two fifths
Spacing 5 m times sine marker 2/5 gives 2 m. Order 1 times wavelength 2 m gives 2 m, so this row is accepted.
5m⋅52=2mmλ=1⋅2m=2mA=1
Second order needs four fifths
Spacing 5 m times sine marker 4/5 gives 4 m. Order 2 times wavelength 2 m gives 4 m, so this row is accepted.
5m⋅54=4mmλ=2⋅2m=4mA=1
Third order cannot fit inside the sine limit
Spacing 5 m times sine marker 1 gives 5 m. Order 3 times wavelength 2 m gives 6 m, so this row is rejected.