Spectral Calibration
Wavelength calibration of a Czerny–Turner spectrograph from mercury lines: linear, quadratic and cubic fits against the grating-equation pixel map.
Detector Range (grating equation)
394.0 – 605.2 nm
α = 33.09°, β₀ = 3.09°
Dispersion at Centre (theory)
0.1040 nm/px
d·cos β₀/(m·f) × pixel pitch
Linear Fit λ = a·i + b
a = 0.103557 nm/px
b = 393.8798 nm at pixel 0
Lines on Detector
6
degree k needs k + 1 lines
Fit Quality
| Degree | RMS residual at lines (nm) | Max |fit − true| over detector (nm) | Dispersion at centre (nm/px) |
|---|---|---|---|
| 1 | 0.0983 | 0.625 | 0.103557 |
| 2 | 0.0904 | 0.494 | 0.103526 |
| 3 | 0.00319 | 0.0120 | 0.104005 |
Calibration Error λ_fit − λ_true
Error of each fitted polynomial against the grating-equation truth at every pixel; dots mark the calibration lines. Outside the outermost lines the fit extrapolates.
Model
x = (i − (N − 1)/2)·p, β = β₀ + atan(x/f), λ = d(sin α + sin β)/m
λ_fit(i) = Σ c_k·u^k, u = (i − (N − 1)/2)/((N − 1)/2), least squares over the lines
The true pixel-to-wavelength map is the grating equation for a Czerny–Turner spectrograph with equal focal lengths and a flat detector normal to the central ray, set by the centre wavelength and the fixed deviation α − β₀ (as on the spectral range page). Both sin β and the atan of the flat detector make it non-linear. When β₀ is small the distortion is nearly odd about the centre, so a quadratic barely improves on a line and a cubic is needed. Each line centroid is offset by Gaussian noise of the given σ (seeded, so the page is reproducible). Aberrations, detector tilt and focal-plane curvature are not modelled. Enter air wavelengths: the grating equation holds for the wavelength in the air around the grating.
References: C. Palmer, E. Loewen, Diffraction Grating Handbook, 6th ed. (Newport, 2005), ch. 2; A. Kramida, Yu. Ralchenko, J. Reader and NIST ASD Team, NIST Atomic Spectra Database (Hg I lines); C. J. Sansonetti, M. L. Salit, J. Reader, Appl. Opt. 35, 74 (1996).
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
The true pixel-to-wavelength map is the grating equation on a flat detector normal to the central ray (no aberrations, tilt or focal-plane curvature); line centroids carry seeded Gaussian noise.
- Palmer C., Loewen E. (2005). Diffraction Grating Handbook, 6th ed., ch. 2. Newport.
- Kramida A., Ralchenko Yu., Reader J., NIST ASD Team. NIST Atomic Spectra Database (Hg I air wavelengths). doi:10.18434/T4W30F
- Sansonetti C. J., Salit M. L., Reader J. (1996). Wavelengths of spectral lines in mercury pencil lamps. Appl. Opt. 35, 74–77. Source of the ASD Hg I lines, ±0.1 pm. doi:10.1364/AO.35.000074