Calculator
Example Data Table
| Mode | m | Lines/mm | Lambda (nm) | Alpha (deg) | Beta (deg) | Notes |
|---|---|---|---|---|---|---|
| General | 1 | 1200 | 532 | 20 | (solve) | Typical visible laser line. |
| General | 2 | 600 | 633 | 15 | (solve) | Higher order increases dispersion. |
| Littrow | 1 | 1800 | 500 | — | — | Use theta; alpha and beta coincide. |
Formula Used
General reflection grating equation: m*lambda = d (sin(alpha) + sin(beta))
- m = diffraction order (integer, can be negative with different sign conventions).
- lambda = wavelength in the propagation medium.
- d = groove spacing (meters), where d = 1 / (lines per length).
- alpha and beta = angles from the grating normal (degrees).
Littrow configuration: when alpha = beta = theta, the equation becomes m*lambda = 2d sin(theta).
Useful optics metrics include angular dispersion d(beta)/d(lambda) = m / (d cos(beta)) and resolving power R = m*N, where N is illuminated grooves.
How to Use
- Select Geometry (General or Littrow).
- Choose what you want to Solve for.
- Enter order m, wavelength, and either lines/mm or spacing.
- Fill angles (alpha, beta) for General, or theta for Littrow.
- Click Calculate to see results above the form.
- Use Download CSV or Download PDF from the results card.
1) What a reflection grating solves
A reflection grating separates light by wavelength using many evenly spaced grooves. The calculator links wavelength, groove spacing, and angles through the grating equation. For visible work, common wavelengths range from 400–700 nm, while groove densities often sit between 300 and 2400 lines/mm.
2) Groove density versus spacing d
Groove density (lines/mm) is the inverse of spacing: d = 1/(lines per mm) in millimeters. For example, 1200 lines/mm gives d = 1/1200 mm = 0.000833 mm = 0.833 um. Entering spacing directly is useful when your grating specification is given in micrometers.
3) Orders and realistic solutions
The order m is an integer: m = 1, 2, 3 and so on, or negative if you flip the sign convention. Higher orders increase dispersion but may become impossible if the required sine argument exceeds 1. If you see “no real angle,” try reducing |m| or increasing d.
4) General geometry and angle meaning
In general mode, the calculator uses m*lambda = d(sin(alpha) + sin(beta)). Angles are measured from the grating normal, which is a common optics convention. Keep alpha and beta consistent on the same side definition; changing sides effectively changes the sign of one angle.
5) Littrow configuration for compact layouts
Littrow sets alpha = beta = theta, so m*lambda = 2d sin(theta). This is popular in monochromators and tunable lasers because the incident and diffracted beams overlap. As theta increases, you can reach a target wavelength with lower groove density, but grazing angles can be harder to align.
6) Dispersion numbers you can interpret
Angular dispersion scales roughly with m/(d cos(beta)), so it rises with higher order and finer spacing. The calculator converts this to deg per nm, and can estimate linear dispersion (mm per nm) if you provide camera focal length. A 200 mm focal length makes small angular changes easier to measure at the detector.
7) Resolving power with a worked example
Theoretical resolving power is R = |m|N, where N is illuminated grooves. If a 10 mm beam hits a 1200 lines/mm grating, N is about 1200*10 = 12000 grooves. With m = 1 at 532 nm, delta-lambda is roughly 532/12000 = 0.044 nm under ideal conditions.
8) Medium index and practical checks
If your light propagates in a medium with refractive index n, the wavelength in that medium is lambda/n. This tool accounts for that to keep geometry consistent. After computing angles, confirm that beta stays within physical limits, and re-check units (nm, lines/mm, um) before concluding.
FAQs
1) Lines/mm or spacing um: which should I enter?
Either works. Spacing (um) overrides lines/mm when provided. Use spacing when your grating datasheet lists d directly, otherwise use lines/mm. The calculator converts both into the same spacing used by the equations.
2) Why do I get “no real beta” or “no real alpha”?
The equation requires a sine argument between -1 and 1. If the geometry demands a value outside that range, no real angle exists. Reduce the order magnitude, choose a larger spacing, or adjust the incidence angle.
3) Are angles measured from the surface or the normal?
This calculator assumes angles from the grating normal, which is standard in many optics texts. If your measurement is from the surface, convert using: angle_from_normal = 90 - angle_from_surface.
4) What is Littrow mode used for?
Littrow means the diffracted beam retraces the incident path: alpha equals beta. It is common for compact spectrometers and tunable cavities because alignment can be simpler and efficiency can improve near blaze.
5) What does resolving power R tell me?
R estimates how well close wavelengths can be separated under ideal conditions. Larger R usually means smaller resolvable delta-lambda. It increases with illuminated grooves and with diffraction order, but real instruments also depend on aberrations and slit widths.
6) Why is refractive index n included?
Wavelength shortens inside a medium: lambda_medium = lambda_vacuum/n. If you work in liquids, prisms, or immersion setups, n changes the effective wavelength in the grating equation, shifting computed angles or required groove density.
7) How is linear dispersion (mm/nm) estimated?
The tool multiplies angular dispersion by the camera focal length: dx/dlambda is approximately f times d(beta)/d(lambda). This gives how many millimeters the spectrum moves per nanometer at the focal plane, helping you size sensors and pixels.