Hologram Diffraction Calculator
Find the diffraction angle of light leaving a holographic grating for any order, wavelength, and incident angle. Useful for designing holographic optical elements, spectrometers, and display systems.
About this calculator
The grating equation governs how light is redirected by a periodic structure: sin(θ_m) = sin(θ_i) + (m × λ) / d, where θ_m is the diffraction angle for order m, θ_i is the incident angle, λ is the wavelength, d is the grating period, and m is the diffraction order (0, ±1, ±2, …). Solving for θ_m requires an arcsine, which only has a real solution when the argument stays within [−1, 1]; orders that violate this are said to be evanescent and do not propagate. The formula used here implements exactly that check with clamping. The zeroth order (m = 0) always exits at the specular reflection angle. Higher orders fan out at larger angles; the spread depends on the ratio of wavelength to grating period.
How to use
Example: grating period d = 1 μm, wavelength λ = 0.532 μm (green laser), incident angle θ_i = 0°, diffraction order m = 1. Step 1 — compute sin(0°) = 0. Step 2 — add (1 × 0.532) / 1 = 0.532. Step 3 — θ_1 = arcsin(0.532) ≈ 32.1°. So the first-order diffracted beam exits at about 32° from the grating normal. For m = −1: 0 + (−1 × 0.532)/1 = −0.532, giving θ = −32.1°, symmetric about the normal. Orders with |sin value| > 1 are evanescent and the calculator returns 0.
Frequently asked questions
What is the difference between diffraction order 0 and order 1 in a holographic grating?
The zeroth diffraction order (m = 0) behaves like ordinary specular reflection or transmission — the grating has no dispersive effect on it and the exit angle equals the incident angle. The first order (m = ±1) is the primary diffracted beam and is the one most commonly exploited in spectrometers, holographic displays, and wavelength-selective filters. Higher orders (m = ±2, ±3, …) exist but carry progressively less energy for most practical gratings. In holographic optical elements, design effort typically focuses on maximizing first-order efficiency while suppressing the zeroth order.
How does grating period affect the diffraction angle of a holographic element?
A smaller grating period d causes larger diffraction angles for the same wavelength and order, because the ratio λ/d grows. When d approaches λ, the first-order beam diffracts at very steep angles, and higher orders become evanescent. Conversely, a very large d relative to λ produces small deflection angles and brings all orders close together. Holographic grating designers choose d to place the desired diffraction order at a specific angle, which is why gratings for visible light typically have periods in the 0.3–2 μm range.
Why does the diffraction calculator clamp the arcsine argument to the range minus one to one?
The arcsine function is only defined for real inputs between −1 and 1; values outside this range would produce a complex (imaginary) angle, which physically means the diffracted order cannot propagate in the medium. This condition is called evanescent diffraction. The clamping ensures the calculator returns a real number rather than an error, and the result of 0 signals that the requested order does not exist for those parameters. In practice, you should note when clamping occurs and reduce the diffraction order or increase the grating period to produce a propagating beam.