Estimate dispersion quickly for prism and grating. Use angles, focal length, and index slope measured. Generate exports and compare designs before building hardware today.
| Mode | Key Inputs | Typical Output | Use Case |
|---|---|---|---|
| Prism | A = 60°, λ₀ = 589.3 nm, (n₁,λ₁)=(1.5168,486.1), (n₂,λ₂)=(1.5143,656.3) | dδ/dλ ≈ few × 10⁻⁴ deg/nm | Estimate spectral spread for glass prism setups. |
| Grating | g = 1200 lines/mm, m = 1, λ = 632.8 nm, f = 300 mm, W = 25 mm | dx/dλ ≈ 0.2–1 mm/nm (depends on β) | Design camera length and pixel sampling. |
A common prism relation is
n = sin((A+δ)/2) / sin(A/2).
Differentiating gives the angular dispersion:
dδ/dλ = [2 sin(A/2) / cos((A+δ)/2)] · (dn/dλ).
In minimum deviation, δ is computed from δ = 2 asin(n sin(A/2)) − A.
The derivative dn/dλ may be estimated from two index points or a Cauchy model.
The grating equation is mλ = d (sinα + sinβ).
The angular dispersion is
dβ/dλ = m / (d cosβ).
Linear dispersion at the focal plane is approximated by dx/dλ ≈ f · dβ/dλ
for small angles, where f is the imaging focal length.
Dispersion links wavelength change to a measurable shift in angle or position. Higher dispersion spreads nearby wavelengths farther apart, easing separation on a detector. This calculator reports angular dispersion (deg/nm or rad/nm) and, for gratings, linear dispersion (mm/nm) at the focal plane.
For common optical glasses in the visible band, the refractive index usually decreases with wavelength, giving a negative dn/dλ. Typical slopes can fall around 10−5 to 10−4 per nm, depending on glass type and wavelength range. The prism formula scales the dispersion by geometry through sin(A/2) and cos((A+δ)/2).
Many prism spectrometers are aligned near minimum deviation because the beam path is symmetric and sensitivity to small angle errors is reduced. In this mode, δ is computed from n and the apex angle A. If n·sin(A/2) approaches 1, the minimum-deviation solution becomes impractical.
If you have two index values (n₁ at λ₁ and n₂ at λ₂), the calculator uses a two-point slope (n₂−n₁)/(λ₂−λ₁). This is convenient when you only have sparse data. For smoother behavior, the Cauchy model n(λ)=A+B/λ²+C/λ⁴ (λ in µm) provides a continuous derivative.
Ruled and holographic gratings commonly range from about 300 to 2400 lines/mm. A 1200 lines/mm grating has spacing d ≈ 0.833 µm. Angular dispersion increases with diffraction order |m| and with groove density, and it grows rapidly as β approaches 90°, where cosβ becomes small.
In a spectrograph, the camera or imaging lens converts angles to positions. Using dx/dλ ≈ f·dβ/dλ, a 300 mm focal length can turn modest angular changes into measurable millimeter shifts. Larger f increases separation but can demand a larger detector or tighter alignment.
For gratings, resolving power is approximated by R = mN, where N is the number of illuminated grooves. With g = 1200 lines/mm and W = 25 mm, N ≈ 30,000, so R ≈ 30,000 in first order, giving Δλ ≈ 632.8/30000 ≈ 0.021 nm. Real instruments may be limited by slit width, aberrations, and pixel sampling.
Use dispersion outputs to match detector pixel size and desired spectral span. If linear dispersion is too small, consider higher groove density, higher order (with care), or a longer focal length. If dispersion is too large, you may reduce order, reduce f, or choose a prism for broader, smoother coverage.
Angular dispersion is the change in output angle per wavelength change. Linear dispersion converts that angle to a position shift at the focal plane, typically in mm per nm, using the imaging focal length.
Most transparent optical materials show normal dispersion, where refractive index decreases as wavelength increases. That makes dn/dλ negative across much of the visible spectrum.
Use it when you have coefficients for your material or want a smooth derivative at λ₀. It is helpful when two-point data are noisy, far apart, or not centered near your operating wavelength.
Littrow means the incidence and diffraction angles are equal, which often maximizes efficiency near the blaze condition and simplifies alignment. The calculator then solves β from mλ = 2d sinβ.
Illuminated width increases the number of grooves N that contribute coherently. Since R = mN, doubling W approximately doubles resolving power, reducing the estimated minimum resolvable wavelength interval.
Slit width, aberrations, focus errors, detector pixel size, stray light, and finite source linewidth can dominate. The displayed Δλ is an idealized estimate based on resolving power, not a full system model.
Prisms often give smoother, lower-order dispersion across a wide band with fewer overlapping orders. Gratings can provide higher dispersion and resolution but may require order-sorting filters and careful geometry.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.