Cardinal points and equivalent focal length

Cardinal points

Cardinal points are six points on the axis of a coaxial optical system which let us find the image of any object without tracing rays through each surface.

1. First and second focal points (F₁, F₂): an object at F₁ gives parallel rays on emergence; parallel rays entering the system converge to F₂.
2. First and second principal points (H₁, H₂): the points where the principal planes cut the axis. The principal planes are conjugate planes with unit lateral magnification (+1).
3. First and second nodal points (N₁, N₂): a ray directed toward N₁ emerges parallel to its original direction as if from N₂. If the medium on both sides is the same (air), N₁ and N₂ coincide with H₁ and H₂.

Equivalent focal length of two thin lenses

Take lenses of focal lengths f₁, f₂ a distance d apart in air.

 h1 ---|-------|---
       L1      L2      ray parallel to axis
       |<--d-->|       at heights h1 (L1), h2 (L2)

1. A ray parallel to the axis at height h₁ strikes L1. After L1 it converges toward the point at distance f₁ with slope −h₁/f₁.
2. At L2 (distance d away) its height is h₂ = h₁ − d·h₁/f₁ = h₁ (1 − d/f₁).
3. L2 deviates the ray further by h₂/f₂, so the final slope is −h₁/f₁ − h₂/f₂ = −h₁/f₁ − h₁(1 − d/f₁)/f₂.
4. For the equivalent lens of focal length F, the final slope is −h₁/F. Therefore

1/F = 1/f₁ + 1/f₂ − d/(f₁f₂)

F = f₁f₂ / (f₁ + f₂ − d)

Positions of the principal points follow from the same figure:

• H₁ is at α = F d / f₂ from L1 (on the side of L2).
• H₂ is at β = F d / f₁ from L2 (on the side of L1).

For d = 0 (lenses in contact) this reduces to 1/F = 1/f₁ + 1/f₂.