How it is calculated
For a plane reflection grating with groove spacing \(d\), light incident at angle \(\alpha\) is diffracted into angles \(\beta\) that satisfy the grating equation
Sign convention: both angles are measured from the grating normal and are positive on the same side of it. The zero order is the specular reflection, \(\beta = -\alpha\), and orders \(\pm m\) fall on opposite sides of it. An order exists only if \(|\sin\beta| \le 1\). Otherwise it is evanescent and carries no power.
Mountings
- Littrow: the light returns along its incoming path, \(\alpha = \beta\), so \(\sin\alpha = m\lambda/2d\).
- Constant deviation (Czerny–Turner): the angle between the incident and diffracted beams, \(D_V = \alpha - \beta\), is fixed by the instrument, and the grating rotates to scan. With scan angle \(\phi = (\alpha+\beta)/2\), the equation becomes \(m\lambda = 2d\cos(D_V/2)\sin\phi\), so \(\alpha = \phi + D_V/2\) and \(\beta = \phi - D_V/2\).
Dispersion, resolution, and bandpass
The reciprocal linear dispersion \(d\lambda/dx\) (nm per mm at the focal plane) sets the bandpass: \(\text{BP} \approx w\cdot d\lambda/dx\) for a slit or pixel of width \(w\). This assumes the imaged entrance-slit width matches the exit slit or pixel. The resolving power \(R = |m|N\) is the theoretical limit set by the number of illuminated grooves. Practical spectrometers are usually slit- or pixel-limited well before reaching it. The free spectral range \(\lambda/|m|\) is the band before the next order overlaps, so order-sorting filters are needed when \(\lambda_\text{max} > 2\lambda_\text{min}\) in first order.
Worked example
Take a 1200 l/mm grating (\(d = 833.3\) nm) at \(\alpha = 30^\circ\), first order, 500 nm. Then \(\sin\beta = 0.6 - 0.5 = 0.1\), so \(\beta = 5.74^\circ\). The angular dispersion is 1.206 mrad/nm. With a 300 mm focusing mirror the reciprocal linear dispersion is 2.76 nm/mm, so a 50 µm slit passes about 0.14 nm. Fifty millimetres of illuminated grating gives \(R = 60\,000\), or 8.3 pm at 500 nm. In a Czerny–Turner with \(D_V = 30^\circ\), the same wavelength needs \(\alpha = 33.09^\circ\) and \(\beta = 3.09^\circ\).
Assumptions and limits
- Plane grating in the principal plane. No conical (off-plane) diffraction. Concave and holographic aberration-corrected gratings follow the same equation at their center, but their focal-plane properties differ.
- Geometric bandpass. BP ≈ w·dλ/dx ignores aberrations, diffraction from the optics, and the anamorphic magnification \(\cos\alpha/\cos\beta\). That magnification makes the entrance-slit image differ in width from the slit itself when \(\alpha \ne \beta\).
- Efficiency is not modeled. Which orders are bright depends on blaze angle, groove profile, polarization, and coating. Use the manufacturer's efficiency curves.
References
- C. Palmer, Diffraction Grating Handbook, 8th ed., MKS Instruments / Newport (2020).
- E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Marcel Dekker (1997).