Grating Efficiency Calculator
Scalar blaze efficiency of a ruled grating in Littrow versus wavelength, with the order cutoff and Rayleigh anomalies.
Blaze Wavelength (Littrow, order 1)
501.2 nm
λ_B = 2d·sin φ / m
Scalar Efficiency
93.9 %
shared over 4 orders; × coating reflectance
Littrow Angle θ
17.46°
sin θ = mλ / (2d)
Groove Spacing d
0.8333 µm
λ / d
0.600
λ/d ≥ 0.2: real efficiency depends on polarization
Propagating Orders
-1 … 2
4 orders share the light
Scalar Efficiency in Littrow vs Wavelength (dotted: Rayleigh anomalies)
Model
I_m = sinc²(π·d·ρ·[sin(α − φ) + sin(β_m − φ)] / λ), ρ = cos α / cos(α − φ), η_m = I_m / Σ I_m′, Littrow: α = β = θ
Scalar diffraction from a sawtooth grating with a right-angle apex, mounted in Littrow and turned to follow the wavelength. Each order m′ that propagates gets an envelope I_m′, largest where the facets reflect the light straight back (the blaze wavelength), and the envelopes are scaled to share the incident power (energy conservation across orders). Near the cutoff the envelopes flatten, so the light spreads over the orders. The coating reflectance (about 90 % for aluminium in the visible) and polarization are not included. Scalar theory holds for λ/d below about 0.2; at larger λ/d the TE and TM efficiencies split. At a Rayleigh anomaly an order grazes the grating and passes off, and the real TM efficiency jumps there. In first order, above 2d/3 only orders 0 and 1 propagate, the range where gratings reach their highest efficiency. Order m stops at mλ = 2d.
References: R. Casini, P. G. Nelson, J. Opt. Soc. Am. A 31, 2179 (2014), eqs. 18–25; C. Palmer, E. Loewen, Diffraction Grating Handbook, 6th ed. (Newport, 2005), ch. 2; Lord Rayleigh, Proc. R. Soc. Lond. A 79, 399 (1907).
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
Scalar theory for a right-angle sawtooth in Littrow, with the envelopes of all propagating orders sharing the incident power; no coating reflectance or polarization. Fair for λ/d below about 0.2; above that TE and TM split.
- 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
- Palmer C., Loewen E. (2005). Diffraction Grating Handbook, 6th ed., ch. 2. Newport.
- Lord Rayleigh (1907). On the dynamical theory of gratings. Proc. R. Soc. Lond. A 79, 399–416. doi:10.1098/rspa.1907.0051