Refraction through a prism

A prism is a wedge of transparent material — glass, typically — bounded by two plane faces that meet at the angle of the prism (or refracting angle), $A$. Light entering one face refracts toward the normal (glass is denser than air), crosses the glass, and refracts again away from the normal as it exits the second face. Because both bends happen in the same rotational sense, they add rather than cancel, and the emergent ray leaves at an angle $\delta$, the angle of deviation, from the direction it would have kept had the prism not been there.

Naming the angles. Let $i_1$ be the angle of incidence at the first face and $r_1$ the angle of refraction there; let $r_2$ be the angle of incidence at the second face (measured inside the glass) and $i_2$ the angle of emergence. Simple geometry of the quadrilateral formed by the two normals and the ray path inside the prism gives two relations that hold for every ray through the prism, not just a special one:

where: $A$ is the fixed refracting angle of the prism (a property of the glass block, not of the ray), $r_1,r_2$ are the internal refraction angles at the two faces, and $i_1,i_2$ are the incidence and emergence angles measured outside the glass, in air.

How deviation changes as you rotate the prism. If you keep the prism fixed and slowly change the angle of incidence $i_1$ (by rotating the incoming beam, or the prism, on a spectrometer table), the deviation $\delta$ does not change monotonically. It falls steeply from a large value at grazing incidence, flattens into a broad, shallow dip, and then rises steeply again as the ray approaches grazing emergence at the far face. That dip has one lowest point, the angle of minimum deviation, $\delta_m$. Because the curve is so flat near the bottom, a spectrometer reading taken anywhere close to the true minimum still gives a good estimate of $\delta_m$ — which is exactly why this method, rather than an arbitrary angle of incidence, is the standard laboratory technique for measuring a glass's refractive index. By the principle of reversibility of light, a ray traced backward through the prism must trace the same geometric path forward; the only configuration consistent with the deviation curve having a single turning point is the symmetric path, where the ray inside the glass runs parallel to the base and

This symmetric condition is the key that unlocks the prism formula, derived below.

Thin-prism shortcut. For a thin prism (small $A$) with small angles of incidence throughout, all the sines can be replaced by the angles themselves, and the geometry collapses to $\delta\approx(n-1)A$ — independent of $i_1$. This is useful for quick estimates and for understanding why thin prisms are used to produce small, controlled deviations in optical instruments, but it is not a substitute for the exact minimum-deviation formula in a general problem.

Start from the two general geometric relations $A=r_1+r_2$ and $\delta=i_1+i_2-A$, apply the symmetric condition that holds exactly at minimum deviation, and combine with Snell's law at the first face to arrive at the formula relating the glass's refractive index $n$ to the measurable angles $A$ and $\delta_m$.

Step 1. At the angle of minimum deviation, the ray path through the prism is symmetric: $i_1=i_2=i$ and $r_1=r_2=r$. Substituting $r_1=r_2=r$ into $A=r_1+r_2$ gives $A=2r$, so $r=\dfrac{A}{2}$.

Step 2. Substituting $i_1=i_2=i$ into $\delta=i_1+i_2-A$ at the minimum gives $\delta_m=2i-A$, so $i=\dfrac{A+\delta_m}{2}$.

Step 3. Apply Snell's law at the first face, where the ray travels from air (refractive index 1) into the glass (refractive index $n$): $n=\dfrac{\sin i}{\sin r}$.

Step 4. Substitute the expressions for $i$ and $r$ found in Steps 1 and 2 into Snell's law to obtain the prism formula:

where: $n$ is the refractive index of the prism material relative to the surrounding medium (usually air), $A$ is the prism's refracting angle, and $\delta_m$ is the angle of minimum deviation. Every quantity on the right is directly measurable with a spectrometer, which is why this single equation is the working basis of the standard laboratory determination of $n$.

A prism has a refracting angle of 60° and the angle of minimum deviation for a certain light is found to be 30°. Find the refractive index of the prism material.

Step 1. Write down the prism formula and identify the given values: $A=60°$, $\delta_m=30°$.

Step 2. Substitute into the formula:

Step 3. Evaluate the sines and divide: $\sin 45°=0.7071$, $\sin 30°=0.5$, so $n=\dfrac{0.7071}{0.5}=1.414$.

The refractive index of the prism material is approximately 1.414, which is $\sqrt2$.

Competitive stretch

Suppose instead the refractive index is given as $n=1.5$ and the refracting angle is $A=40°$; find $\delta_m$. Rearranging the prism formula: $\sin\left(\dfrac{A+\delta_m}{2}\right)=n\sin\left(\dfrac{A}{2}\right)=1.5\times\sin(20°)=1.5\times0.342=0.513$. Taking the inverse sine, $\dfrac{A+\delta_m}{2}=\sin^{-1}(0.513)\approx30.9°$, so $A+\delta_m\approx61.8°$, giving $\delta_m\approx61.8°-40°=21.8°$. The angle of minimum deviation is approximately 21.8°.

• Assuming $i_1=i_2$ always holds. This symmetric condition is true only at the one specific angle of incidence that produces minimum deviation, not for every ray passing through the prism.
• Confusing $A$ with a base angle. $A$ is the refracting (apex) angle between the two refracting surfaces of the prism, not a base angle of the triangular cross-section.
• Sign errors in $\delta=i_1+i_2-A$. Students sometimes add $A$ instead of subtracting it, or mix up $r_1+r_2=A$ with $i_1+i_2=A$.
• Overusing the thin-prism approximation. $\delta\approx(n-1)A$ only applies for small $A$ and small angles of incidence; it is not a substitute for the exact minimum-deviation formula in a general problem.

Ask the tutor: want a walkthrough of why the deviation is minimum exactly when $i_1=i_2$, or a practice problem that gives you $\delta_m$ and asks you to find the prism's apex angle instead?