0
°
0
°
0
°
0
°
0
°
0
°
0
°
0
°
0
°
0
°
The Prism Calculator analyzes the passage of light through a triangular prism, computing all refraction angles and the angular deviation of the beam. Prisms are fundamental optical elements used in spectrometers, periscopes, binoculars, and laser systems. Their ability to disperse white light into its constituent colors was famously demonstrated by Isaac Newton in 1666.
When a light ray enters a prism, it refracts at the first surface, travels through the glass, and refracts again at the second surface. Each refraction is governed by Snell’s law. The total angular change between the incident and emergent rays is the deviation angle δ, which depends on the prism’s apex angle A, its refractive index n, and the angle of incidence θi.
A particularly important quantity is the minimum deviation δmin, which occurs when the light ray passes symmetrically through the prism (the entry and exit angles are equal). At minimum deviation, the formula simplifies to $$\delta_{\min} = 2\arcsin\!\left(n\sin\frac{A}{2}\right) - A$$ This relationship is widely used to measure the refractive index of glass samples with high precision by finding the angle of minimum deviation experimentally.
Prisms serve multiple roles in optics. Dispersive prisms separate wavelengths for spectroscopy. Reflecting prisms (like Porro prisms in binoculars) use total internal reflection to redirect beams without metallic mirrors. Polarizing prisms (like Nicol and Wollaston prisms) separate ordinary and extraordinary rays in birefringent crystals.
This calculator handles the complete ray-tracing through a prism: the refraction angle at entry, the internal incidence angle at the exit face, the exit angle, the total deviation, and the minimum deviation. It is an essential tool for anyone designing or analyzing prism-based optical systems.
The deviation angle varies with wavelength because the refractive index is wavelength-dependent (chromatic dispersion). This is why a prism splits white light into a rainbow spectrum—violet light, having a higher refractive index, is deviated more than red light.
Light passing through a prism undergoes two refractions. The geometry is governed by Snell’s law at each surface and the prism’s apex angle A.
At the first surface (entry):
$$n_1 \sin\theta_i = n\sin\theta_{r1} \implies \theta_{r1} = \arcsin\!\left(\frac{\sin\theta_i}{n}\right)$$
Inside the prism, geometry gives the incidence angle at the second surface:
$$\theta_{i2} = A - \theta_{r1}$$
At the second surface (exit):
$$n\sin\theta_{i2} = n_1 \sin\theta_{r2} \implies \theta_{r2} = \arcsin\!\left(n\sin\theta_{i2}\right)$$
Total deviation:
$$\delta = \theta_i + \theta_{r2} - A$$
Minimum deviation occurs when θi = θr2 (symmetric passage):
$$\delta_{\min} = 2\arcsin\!\left(n\sin\frac{A}{2}\right) - A$$
The deviation angle δ tells you how far the emergent beam is deflected from its original direction. Smaller apex angles and lower refractive indices produce less deviation. The minimum deviation δmin is the smallest possible deviation for a given prism—useful for calibrating spectrometers. If the exit angle calculation fails (result is NaN), it means total internal reflection occurs at the second face and the ray does not emerge from the prism.
Inputs
Results
A 60° crown glass prism with a 45° incidence angle produces a deviation of about 37.9°. The minimum deviation for this prism is 37.2°, achieved at symmetric passage.
Inputs
Results
Dense flint glass (n = 1.7) in a 60° prism. At θ<sub>i</sub> ≈ 58.2° the ray passes nearly symmetrically, producing a deviation very close to δ<sub>min</sub> ≈ 56.4°.
Minimum deviation is the smallest angle by which a prism deflects light. It occurs when the light ray passes symmetrically through the prism, meaning the angle of incidence equals the angle of emergence. At minimum deviation, the ray inside the prism is parallel to the base. This condition is used experimentally to measure refractive indices with high accuracy.
The refractive index of glass varies with wavelength (chromatic dispersion). Violet light has a higher refractive index than red light, so it is bent more. When white light passes through a prism, each color is deviated by a different amount, spreading the beam into a spectrum. This is the principle behind prism spectrometers.
If the angle of incidence at the exit face exceeds the critical angle, the light cannot emerge—it undergoes total internal reflection and is redirected inside the prism. This is deliberately exploited in Porro prisms (used in binoculars) and right-angle prisms (used as mirrors) to redirect light beams without silvered surfaces.
By measuring δmin and knowing the apex angle A, you can compute the refractive index using: n = sin((A + δmin)/2) / sin(A/2). This method, performed with a spectrometer, is one of the most accurate techniques for measuring refractive indices of transparent solids.
A dispersive prism is designed to separate wavelengths (colors) using refraction and chromatic dispersion—used in spectrometers and monochromators. A reflecting prism uses total internal reflection to redirect light beams (like Porro prisms in binoculars or pentaprisms in SLR cameras). Some prisms serve both functions.
Yes. A larger apex angle increases the deviation for all wavelengths, and since the dispersion depends on the difference in deviation between colors, a larger apex angle generally produces greater angular separation between spectral lines. However, very large apex angles may cause total internal reflection at the exit face, limiting the usable range.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Lens Calculator
Geometric Optics Calculators
Mirror Equation Calculator
Geometric Optics Calculators
Magnification Calculator
Geometric Optics Calculators
Focal Length Calculator
Geometric Optics Calculators
Lens Maker Equation Calculator
Geometric Optics Calculators
Snell's Law Calculator
Geometric Optics Calculators
