19.26477
1.667
μm
599.88
lines/mm
0.329934
1
3
600
0.916667
nm
0.000635484
rad/nm
0.036410535
deg/nm
19.26477
1.667
μm
599.88
lines/mm
0.329934
1
3
600
0.916667
nm
0.000635484
rad/nm
0.036410535
deg/nm
The Diffraction Grating Calculator solves the grating equation mλ = d sin θ for any unknown — the diffraction angle, wavelength, or slit spacing. It also computes the resolving power R = mN, angular dispersion, and the maximum observable diffraction order, giving you a complete characterization of the grating's spectral performance.
Diffraction gratings are among the most important optical components in spectroscopy. By splitting light into its constituent wavelengths with far greater resolution than a prism, they enable precise measurement of atomic emission lines, laser wavelengths, and stellar spectra. This calculator handles both transmission and reflection gratings — the fundamental equation is the same.
When a plane wave of wavelength λ strikes a grating with slit spacing d, constructive interference occurs at angles satisfying the grating equation:
$$m\lambda = d\sin\theta$$
where m = 0, ±1, ±2, ... is the diffraction order. The zeroth order (m = 0) passes straight through at θ = 0 regardless of wavelength; higher orders separate wavelengths angularly.
The resolving power of a grating determines the smallest wavelength difference it can distinguish:
$$R = mN = \frac{\lambda}{\Delta\lambda_{\min}}$$
where N is the total number of illuminated slits. Higher order and more slits both improve resolution.
The angular dispersion measures how rapidly the diffraction angle changes with wavelength:
$$\frac{d\theta}{d\lambda} = \frac{m}{d\cos\theta}$$
This determines the physical separation of spectral lines at the detector plane. The maximum observable order is limited by the condition sin θ ≤ 1, giving $$m_{\max} = \lfloor d/\lambda \rfloor$$.
Gratings are specified by their line density (lines per mm). A 600 lines/mm grating has d = 1/600 mm = 1.667 μm. Typical scientific gratings range from 300 to 2400 lines/mm, with higher line densities providing greater dispersion but lower maximum order.
If the result shows θ approaching 90°, the order is near the physical limit — the diffracted beam grazes the grating surface and intensity drops sharply. If mλ/d exceeds 1, no real angle exists for that order. The resolving power R tells you that wavelengths differing by less than λ/R cannot be distinguished. For example, R = 600 at λ = 550 nm means spectral features closer than about 0.92 nm will overlap.
Inputs
Results
Green light (550 nm) diffracts at 19.27° in first order through a standard 600 lines/mm grating. The resolving power of 600 can separate the sodium D lines (589.0 and 589.6 nm, Δλ = 0.6 nm) only marginally — you would need m = 2.
Inputs
Results
A 1200 lines/mm grating (d = 0.833 μm) shows first-order diffraction at 41.3°, corresponding to λ ≈ 549.5 nm. Only first order exists for visible wavelengths at this line density.
A diffraction grating is an optical element with many equally spaced parallel slits (or grooves). When light passes through, each slit acts as a source of secondary wavelets that interfere constructively at specific angles determined by the grating equation mλ = d sin θ, separating white light into its spectral components.
Resolving power R = mN depends on the diffraction order m and the total number of illuminated slits N. More slits and higher orders both improve resolution. A grating with 1000 slits in second order has R = 2000, meaning it can resolve wavelengths differing by λ/2000.
The maximum order is m_max = floor(d/λ). Since sin θ cannot exceed 1, the grating equation requires mλ ≤ d. For a 600 lines/mm grating with 550 nm light, m_max = floor(1667/550) = 3.
Gratings offer higher resolving power, linear dispersion (nearly equal angular spacing per nm), and work across a wider wavelength range. Prisms have higher throughput (no light lost to multiple orders) and no overlapping orders but provide nonlinear dispersion that depends on the glass material.
Angular dispersion dθ/dλ = m/(d cos θ) measures how fast the diffraction angle changes with wavelength in rad/nm. Higher angular dispersion means greater physical separation of spectral lines at the detector, improving the ability to resolve closely spaced features.
Yes. The second-order diffraction of wavelength λ appears at the same angle as first-order diffraction of 2λ. For example, 400 nm in second order overlaps with 800 nm in first order. Filters or cross-dispersers are used to separate overlapping orders in spectrometers.
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