Calculate Prism Deviation

What is Prism Deviation?

A prism is an optical element that deflects light through refraction at two or more plane surfaces. The deviation (δ) is the angle by which a light ray is bent from its original path. Prisms are fundamental optical components in spectroscopy, dispersion, telescopes, and various optical instruments. The deviation depends on the prism angle, refractive index, and incident angle.

Basic Concept: Light Refraction in a Prism

When light passes through a prism, it is refracted twice:

• At the entrance surface: The ray is refracted from incident angle i₁ to refraction angle r₁
• At the exit surface: The ray is refracted from refraction angle r₂ to exit angle i₂

The result is a total deviation of the ray by angle δ (delta).

The Deviation Formula

The general deviation formula for a prism is:

• δ – Deviation angle (in degrees)
• i₁ – Incident angle at the entrance surface
• i₂ – Exit angle at the exit surface
• A – Prism angle (angle between entrance and exit surfaces)

Alternatively, if the incident angle i₁ and refraction angles r₁ and r₂ are known:

Snell's Law in Prisms

At each refracting surface, Snell's law of refraction applies:

• i – Incident angle (in air)
• r – Refraction angle (in prism)
• n – Refractive index of prism material

Entrance surface: sin i₁ = n sin r₁ Exit surface: n sin r₂ = sin i₂

Symmetry Condition

The two refraction angles are related by prism geometry:

This follows from the fact that the two refracting surfaces enclose angle A.

Minimum Deviation (δ_min)

An important special case is minimum deviation, which occurs when the light ray passes symmetrically through the prism (i₁ = i₂ and r₁ = r₂ = A/2).

At minimum deviation:

• The ray travels parallel to the prism base
• The deviation is the smallest of all possible deviations
• This condition is often used in spectrometers because it provides the highest resolution

Dependence on Prism Angle and Refractive Index

Dispersion and Color Resolution

The refractive index varies slightly with wavelength – this is called dispersion. This is why prisms can split light into spectra:

• Red light (λ ≈ 650 nm): Smallest deviation (n_red ≈ 1.51)
• Green light (λ ≈ 550 nm): Medium deviation (n_green ≈ 1.516)
• Blue light (λ ≈ 450 nm): Larger deviation (n_blue ≈ 1.531)
• Violet light (λ ≈ 400 nm): Largest deviation (n_violet ≈ 1.545)

These different deviations create the classical prism spectrum (rainbow).

Common Prism Materials and Refractive Indices

Different Prism Types and Angles

Critical Angle and Total Internal Reflection

If the refraction angle r becomes too large, total internal reflection can occur. The critical angle is:

or

Example: For glass with n = 1.5, θ_c = arcsin(1/1.5) ≈ 41.8°

This means that at certain incident angles, light is completely reflected and does not exit the prism. This is desirable in reflection prisms but undesirable in dispersive prisms.

Practical Applications of Prisms

• Spectroscopy: Prisms split light into spectra to analyze wavelength composition
• Telescopes and binoculars: Prisms deflect light without chromatic aberration (reflection prisms)
• Dispersion and color separation: White light is split into rainbow colors
• Polarizers: Certain prism types can polarize light
• Laser applications: Prisms in laser systems for wavelength tuning
• Spectrometers: High-precision dispersion in analytical instruments
• Photography: Viewfinders in SLR cameras use pentaprisms

Numerical Examples

Example 1: General Deviation

Given: Prism angle A = 35°, refractive index n = 1.5, incident angle i₁ = 45°

• Using Snell's law: sin 45° = 1.5 sin r₁ → r₁ ≈ 28.1°
• Symmetry condition: r₂ = A - r₁ = 35° - 28.1° = 6.9°
• Using Snell's law at exit: 1.5 sin 6.9° = sin i₂ → i₂ ≈ 10.3°
• Deviation: δ = i₁ + i₂ - A = 45° + 10.3° - 35° = 20.3°

Example 2: Minimum Deviation

Given: Prism angle A = 60°, refractive index n = 1.52 (Crown glass)

• δ_min = 2·arcsin(1.52·sin(30°)) - 60°
• δ_min = 2·arcsin(0.76) - 60°
• δ_min = 2·49.46° - 60°
• δ_min ≈ 38.9°

Frequently Asked Questions

Q: Why do different colors have different deviation angles?

A: The refractive index varies with wavelength (dispersion). Blue light has a higher n value than red light, so it is deviated more strongly. This is the basis of prism spectroscopy.

Q: How does minimum deviation differ from general deviation?

A: In minimum deviation, the ray passes symmetrically through the prism (i₁ = i₂). This is a special case where deviation reaches its minimum value. General deviation can be calculated for any incident angle.

Q: Can deviation be greater than 90°?

A: Yes, especially with high refractive index and large prism angle. In extreme cases or with totally reflecting prisms, deviations can be significantly larger.

Q: What is the difference between dispersive and reflection prisms?

A: Dispersive prisms deviate different wavelengths differently (for spectral analysis). Reflection prisms deflect all wavelengths equally without chromatic effects, using total internal reflection.

Summary