Nodal Points

Nodal points are defined as a pair of conjugate points on the axis having unit positive angular magnification. This simply means that a ray of light directed towards one of these points, after refraction through the optical system, appears to proceed from the second point in a parallel direction.

Let H₁P₁ and H₂P₂ be the first and the second principal planes of an optical system and let AF₁ and BF₂ be its first and second focal planes respectively. Consider a point A situated on the first focal plane. From A draw a ray AH₁ parallel to the axis. The conjugate ray will proceed from H₂, a point in the second principal plane such that H₂P₂ = H₁P₁ and will pass through the second focus F₂.

Take another ray AT₁ parallel to the emergent ray H₂F₂ and striking the first principal plane at T₁. It will emerge out from T₂, a point on the second principal plane such that T₂P₂ = T₁P₁ and will proceed parallel to the ray H₂P₂ as the two rays originate at A, a point on first plane. Then the points of intersection of the incident ray AT₁ and the conjugate emergent ray T₂R with the axis give the positions of the two nodal points. It is clear that the two points N₁ and N₂ are a pair of conjugate points and the incident ray AN₁ is parallel to the conjugate emergent ray T₁R. Moreover, tan



In the right angled Δs T₁P₁N₁ and T₂P₂N₂, T₁P₁ = T₂P₂

∠ T₁N₁P₁ = ∠ T₂N₂P₂ =

Here the two Δs are congruent.

∴ P₁N₁ = P₂N₂

Adding N₁P₂ to both sides

P₁N₁ + N₁P₂ + P₂N₂ + N₁P₂

P₁P₂ = N₁N₂

The distance between the principal points N₁ and N₂ is equal to the distance between the principal points P₁ and P₂.

Now consider the two right angled Δs AF₁N₁ and H₂P₂F₂

AF₁ = H₂P₂ and ∠ AN₁F₁ = ∠ H₂F₂P₂

Hence the two Δs are congruent.

∴ F₁N₁ = P₂F₂

Or, F₁P₁ + P₁N₁ = P₂F₂

Or, P₁N₁ = P₂F₂ – F₁P₁

Also, P₁N₁ = P₂N₂,

P₂F₂ = +ƒ₂ and P₁F₁ = –ƒ₁

P₁N₁ = P₂N₂ = (ƒ₁ + ƒ₂)

If the medium on both the sides of the system is optically similar,

ƒ₁ = –ƒ₁

∴ P₁N₁ = P₂N₂ = 0

It means the principal points coincide with the nodal points when the optical system is situated in the same medium (say air).

The planes passing through the nodal points and perpendicular to the axis are termed as nodal planes.

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