How to use
- Enter values in any two fields: Refractive Index n, Prism Angle A, or Minimum Deviation δ.
- Leave the third field blank (do not type 0 unless you truly mean 0).
- Select Compute Missing Quantity.
- Read the result in the output panel below the button.
Units: angles are in degrees. The refractive index n is unitless.
For physically meaningful results, typical prisms have A between about 10° and 70° and glass indices around 1.4–1.9.
If you are working with a liquid prism or a polymer, your index may be lower; if you are working with dense flint glass, it may be higher.
At minimum deviation, the relationship between refractive index n, apex angle A, and minimum deviation δ is:
n = sin((A + δ)/2) / sin(A/2)
The calculator uses the same equation in three different ways depending on which field is blank:
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Compute δ (when n and A are known):
δ = 2·asin(n·sin(A/2)) − A.
This is a direct evaluation, but it only works when the arcsine input is within the real domain.
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Compute n (when A and δ are known):
n = sin((A + δ)/2) / sin(A/2).
This is also direct and is commonly used in optics labs to estimate the refractive index of a prism from measured angles.
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Compute A (when n and δ are known):
there is no simple closed form in elementary functions because A appears inside and outside sine terms.
The page uses a numerical iteration (Newton’s method) to find an A that satisfies the equation.
Note on notation: many textbooks write the deviation as D or δm (minimum deviation).
This calculator uses δ to match the common “minimum deviation” symbol.
Worked examples
The examples below are intentionally detailed so you can verify your own hand calculations.
They also show what “reasonable” numbers look like, which helps you spot unit mistakes (degrees vs radians) or swapped fields.
Example 1: Find minimum deviation δ from n and A
Suppose you have a crown-glass prism with apex angle A = 60° and refractive index n = 1.50
at a specific wavelength (for example, the sodium D line at 589 nm). To find the minimum deviation δ:
- Compute sin(A/2) = sin(30°) = 0.5.
- Compute n·sin(A/2) = 1.50 × 0.5 = 0.75.
- Compute asin(0.75) ≈ 48.590°.
- Then δ = 2 × 48.590° − 60° ≈ 37.18°.
In the calculator: enter n = 1.5 and A = 60, leave δ blank, and click the button.
Small differences may appear due to rounding and the exact trig evaluation.
If you see a wildly different value (for example, a negative deviation for a typical glass prism), re-check that you left exactly one field blank.
Example 2: Find refractive index n from A and δ
In a teaching lab, you might measure the minimum deviation angle and use it to estimate the prism’s refractive index.
Assume you have a prism with A = 60° and you measure δ = 38.6° at minimum deviation.
To compute n:
- Compute (A + δ)/2 = (60° + 38.6°)/2 = 49.3°.
- Compute sin((A + δ)/2) = sin(49.3°) ≈ 0.757.
- Compute sin(A/2) = sin(30°) = 0.5.
- Divide: n ≈ 0.757 / 0.5 ≈ 1.514.
In the calculator: enter A = 60 and δ = 38.6, leave n blank, and compute.
The result should be close to typical crown glass values.
If you repeat the measurement with a different color filter, you will usually get a slightly different index because of dispersion.
Assumptions and interpretation
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Minimum deviation condition: the formula applies when the ray path is symmetric through the prism.
If your prism is not aligned to minimum deviation, the measured deviation will be larger than the minimum and the formula will not match.
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Monochromatic (single-wavelength) light: refractive index depends on wavelength.
Use an index value specified for the wavelength you care about, or compute n separately for each wavelength.
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Geometric optics model: ignores diffraction, polarization, and detailed Fresnel reflection losses.
Those effects matter in precision metrology, but the minimum-deviation relation is still the standard starting point.
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Angles in degrees: inputs and outputs are in degrees; internally the script converts to radians for JavaScript trig functions.
If you copy values from a source that uses radians, convert them to degrees first.
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Prism in air: the common textbook form assumes the prism is surrounded by air (n≈1).
If the prism is immersed in another medium (like water or oil), the effective index contrast changes and you should use a modified relation.
Limitations and common input pitfalls
This tool is designed for typical lab and design calculations, but there are constraints and edge cases.
Understanding them will help you interpret “no real solution” messages and avoid confusing results.
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Domain limits for arcsin (when solving for δ): the term
n·sin(A/2) must be between −1 and 1.
If it is outside that range, there is no real-valued minimum-deviation solution for those inputs.
Practically, this can happen if you enter a very large index or a very large apex angle.
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Numerical solving for A (when solving for A from n and δ): the page uses Newton’s method.
For non-physical combinations (for example, δ too small for a given n) or values near singularities (where
sin(A/2) is near zero),
the iteration may converge slowly or fail.
If you suspect this, try slightly different starting values or verify that your δ is truly a minimum deviation measurement.
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Exactly one blank field: the calculator requires two known values.
If you leave two fields blank (or none), it will prompt you to correct the input.
This is intentional so the script can unambiguously decide which quantity to compute.
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Rounding and significant figures: refractive index values are often quoted to 3–5 decimal places.
If you enter a rounded n, your computed δ will also be rounded.
For lab reports, keep track of measurement uncertainty in A and δ; small angle errors can shift n.
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Temperature and wavelength dependence: published indices depend on temperature and wavelength.
If you compare your computed n to a datasheet, ensure the conditions match (e.g., 20°C and the same spectral line).
Reference table (A = 60°)
The table below illustrates how minimum deviation changes with refractive index for a prism with apex angle 60°.
Values are approximate and assume the minimum-deviation condition with monochromatic light.
Use it as a quick sanity check: higher refractive index generally produces a larger minimum deviation for the same apex angle.
Minimum deviation is widely used in prism spectrometers because it provides a stable alignment condition.
Near the symmetric path, small rotations of the prism often change the deviation more slowly than they do away from minimum deviation,
which makes it easier to find a repeatable setting.
In many spectrometers, you rotate the prism until the observed spectral line reaches a turning point; that turning point corresponds to minimum deviation.
If you are using this calculator for a lab report, it can help to write down the measurement workflow explicitly:
measure A (often by reflection methods or by direct mechanical measurement), align the prism to minimum deviation for a chosen spectral line,
measure δ, then compute n.
If you repeat the measurement for multiple spectral lines, you can build a small dispersion curve n(λ).
As you rotate a prism in a fixed incoming beam, the outgoing beam direction changes.
There is typically a rotation angle where the deviation reaches a minimum value; at that point the ray path inside the prism is symmetric.
This is not merely a mathematical trick: it is a real, observable turning point used in spectrometer alignment.
In the minimum-deviation relation, A appears in both the numerator and denominator inside sine functions.
That makes it a transcendental equation in A.
The page therefore uses Newton’s method to iteratively find an A that satisfies the equation for your given n and δ.
For typical prism values, convergence is fast.
Not directly.
The common formula on this page assumes the prism is in air.
If the surrounding medium has refractive index n0, you would typically replace n with a relative index (n/n0)
and re-derive the relation accordingly.
For quick estimates, you can sometimes approximate by using the relative index, but for accurate work you should use the correct immersed-prism equations.
For a 60° prism made of common glass (n around 1.5), minimum deviation is often on the order of 35°–40°.
Smaller apex angles produce smaller deviations; higher-index materials produce larger deviations.
If you compute a deviation near 0° for a typical glass prism with A around 60°, that usually indicates an input mistake.
