Derivation of Prism Formula

In Physics, a prism is defined as a transparent, polished flat optical element on which light reflects. In a prism, one surface must be angled because objects with two parallel polished surfaces are not considered as prisms. Prisms are made up of materials that are transparent to the wavelengths for which they are designed. The materials generally used to make a prism are glass, plastic, and fluorite. The traditional shape of a prism is somewhat triangular.

Types of Prism

Dispersive Prisms– Dispersive prisms are used for breaking up the light into its constituent colors because the refractive index is dependent on the frequency. The white light that enters into this prism is a mixture of various wavelengths and frequencies. After passing through the prism, each light frequency bent in a different direction. Some examples of dispersive prisms are the Triangular prism, Abbe prism, Pellin–Broca prism, Amici prism, Compound prism, and Grism.

Reflective Prisms – Reflective prisms are used for reflecting light. To flip, invert, rotate, deviate, or displace the light beam, a reflective prism is used. This type of prism is used in binoculars or single-lens reflex cameras for the purpose of getting an erect image. If it is not used in these devices, then the image will be upside down for the users. Some common examples of reflective prisms are Porro prism, Porro–Abbe prism, Amici roof prism, Pentaprism, Abbe–Koenig prism, Schmidt–Pechan prism, Bauernfeind prism, Dove prism, and Retroreflector.

Polarising Prisms– Polarising prisms are used for splitting a beam of light into components of varying polarisation. Polarising prisms are made up of materials that are known as birefringent crystalline. Some common examples of the polarising prism are Nicol prism, Wollaston prism, Nomarski prism, Rochon prism, Senarmont prism, Glan–Foucault prism, Glan–Taylor prism, and Glan–Thompson prism.

Deflecting Prisms– Deflecting prisms are used for deflecting the beam of light at a fixed angle. Beam steering can be done by using a pair of these prisms. On rotating the prisms, the beam gets deflected into any desired angle within the limit of a conical “Field of Regards”. Some examples of deflecting prisms are Rhomboid prisms and Deck prisms.

Beam-Splitting Prisms– Prisms that are used to splitting a single beam into two or more beams are known as beam-splitting beams. Some of the common examples of this prism are Beam splitter cube and Dichroic prism.

Derive the Prism Formula

The prism formula is derived from Snell’s law. It states the relationship between the angle of incidence and the angle of refraction. 

Let us understand the derivation now. 

\[\mu = \frac{sin i}{sin r}\]    (by Snell’s law)

\[\delta = i_{1} - r_{1} + i_{2} - r_{2}\] . . . . (Eq. 1)

\[\delta = i_{1} + i_{2} - (r_{1} + r_{2})\]

\[\angle ALO + \angle AMO = 2rl \angle s\]  (From Quadrilateral and ㄥALO = ㄥAMO = 90°)

∠LAM + ∠LOM = 2rt ∠s (sum of four ∠s of a quadrilateral = 4rt ∠s)  (Eq.2)

\[\angle r_{1} + \angle r_{2}\] + ㄥLOM = 2rt ∠s (Eq.3)

ㄥLAM = \[\angle r_{1} + \angle r_{2}\] ( Comparing Eq.2 and Eq.3)

\[A = \angle r_{1} + \angle r_{2}\]

\[\delta = i_{1} + i_{2} - A\] ( Substituting A in Eq.1)

\[i_{1} + i_{2} = A + \delta \angle i_{1} = \angle i_{2}\]

\[\angle r_{1} = \angle r_{2} = \angle r\]

ㄥALM = ㄥLMA = 90° - ∠r

Thus, AL = LM and LM॥ BC

ㄥA = \[\angle r_{1} + \angle r_{2}\]

A = 2r ( Since, \[\angle r_{1} = \angle r_{2} = \angle r\] )

\[r = \frac{A}{2} i_{1} + i_{2} = A + \delta\]

\[i_{1} + i_{2} = A + \delta m 2 i_{1} = A + \delta m\]

\[i_{1} = \frac{(A + \delta m)}{2}\]

∴  \[ \mu = [ sin [\frac{(A + \delta m)}{2}]] [Sin(A/2)] \]

The prism formula and the answer to the common question ‘How to prove the prism formula?’ are discussed above.

The above derivation can also be termed as the refractive index of prism formula derivation. Students can write this solution when they are asked to derive an expression for the refractive index of prism in terms of angle of prism and angle of minimum deviation.

Derive an Expression for Deviation Through a Thin Prism

A prism whose refracting angle is small ( within 10°) is known as a thin prism. In this type of prisms, by default, the angle of incidence and the angle of refraction are also small.

Consider a thin prism of angle A and refractive index n. Let i₁ be the angle of incidence, r₁ be the angle of refraction at the first face, and in the second face, r₂ be the angle of incidence, and i₂ be the angle of emergence. Let us consider d as the angle of deviation.

d = (i₁ + i₂) − A……….. (1)

A = r₁ + r₂       ………… (2)

For a thin prism, when the angle of incidence i₁ is small, the angles r₁, r₂ , and i₂ are also small (for small values of θ, sinθ ≈ θ in radians)

∴ n = sin i₁/sin r₁ ≈ i₁/r₁  or    i₁ = nr₁

Also, n = sin i₂/sin r₂ ≈ i₂/r₂  or   i₂ = nr₂

From Eq.1,

d = (nr₁ + nr₂) − A = n(r₁ + r₂) − A = nA − A ( ∵ A = r₁ + r₂)

∴  d = A(n − 1)

Using the above formula, students can derive an expression for deviation through a thin prism.