Orders for entire visible spectrum

Grating equation: d sin θ = mλ, where m is the order. Order m exists only if sin θ ≤ 1, i.e. mλ ≤ d.

Given: 250 grooves/mm, so d = 1/250 mm = 4.0×10⁻⁶ m = 4000 nm; λ ranges from 400 nm to 700 nm.

1. Highest order in which the longest wavelength (700 nm) appears: m ≤ d/λ_max = 4000/700 = 5.71. So m = 1 to 5.
2. Highest order for 400 nm: m ≤ 4000/400 = 10 (not the limiting case).
3. The entire spectrum (400 to 700 nm) is therefore present only up to m = 5. The 6th order contains only part of the spectrum, from violet up to 4000/6 ≈ 667 nm.
4. Overlap check: the red end of order m (mλ = 700m) overlaps the violet end of order m+1 (400(m+1)) if 700m > 400(m+1), i.e. m > 1.33.

(The angles follow from sin θ = mλ/d; e.g. first-order red is at sin θ = 0.175, θ = 10.1°.)

Answer: the entire visible spectrum appears in 5 orders (m = 1 to 5) on each side; only the first order is free from overlap with its neighbour.