Dispersive Element Design
Diffraction grating parameters: grating equation, angular/linear dispersion, blaze profile.
Groove Spacing d
0.8333 µm
β at 550.0 nm (centre)
-2.70°
d(sin α + sin β) = mλ
Angular Dispersion (centre)
0.06883 °/nm
1.201 mrad/nm
Reciprocal Linear Dispersion
8.324 nm/mm
d·cos β / (m·f), at the centre
Blaze Angle φ
19.27°
sin φ = λ_B / (2d)
Blaze Wavelength at α (order 1)
495.5 nm
d[sin α + sin(2φ − α)] / m
Scalar Efficiency (centre)
74.7 %
λ/d = 0.66 ≥ 0.2: scalar estimate, real efficiency depends on polarization
Resolving Power per mm Width
1200
R = mN, N = grooves illuminated
Diffraction Angle
Reciprocal Linear Dispersion (f = 100 mm)
Scalar Blaze Efficiency at α = 45°
Model
mλ = d(sin α + sin β), dβ/dλ = m/(d·cos β), dλ/dx = d·cos β/(m·f), R = mN
I_m = sinc²(π·d·ρ·[sin(α − φ) + sin(β_m − φ)] / λ), ρ = cos α / cos(α − φ) (α ≥ φ), cos φ (α < φ)
η_m = k·I_m / Σ I_m′, k = min(1, ρ(β) / ρ(α)), ρ(β) as ρ with α → β
α and β are measured from the grating normal, positive on the same side. The reciprocal linear dispersion d·cos β/(m·f) holds on a focal plane normal to the diffracted ray; a flat detector normal to the central ray sees x = f·tan(β − β₀), which multiplies it by cos²(β − β₀). The blaze wavelength is entered as the catalogue value (first order, Littrow); at another incidence angle the facets reflect into β = 2φ − α, which moves the peak. The efficiency is scalar theory for a sawtooth with a right-angle apex: the envelopes of all propagating orders share the incident power, and when β > α the riser hides part of the lit facet from the exit direction (shadowing), leaving the fraction k. Coating reflectance and polarization are not included; it is fair for λ/d below about 0.2. The free spectral range λ/m does not depend on the groove density.
References: C. Palmer, E. Loewen, Diffraction Grating Handbook, 6th ed. (Newport, 2005), ch. 2; R. Casini, P. G. Nelson, J. Opt. Soc. Am. A 31, 2179 (2014), eqs. 18–25.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Grating equation and dispersion are exact geometry; the blaze efficiency is scalar theory for a right-angle sawtooth, shared over the propagating orders with facet shadowing (no coating reflectance or polarization; fair for λ/d below about 0.2).
- Palmer C., Loewen E. (2005). Diffraction Grating Handbook, 6th ed., ch. 2. Newport.
- Casini R., Nelson P. G. (2014). On the intensity distribution function of blazed reflective diffraction gratings. J. Opt. Soc. Am. A 31, 2179–2184 (eqs. 18–25). doi:10.1364/JOSAA.31.002179