Principal Planes: Two Thin Lenses

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The thin lens equation can be used with thick lenses if the principal planes are found.

A thick lens can be characterized by its radii of curvature, its index of refraction, and its thickness.
Radius = m
Radius = m
Index =
Thickness d = m

Using the Cartesian sign convention, the object distance is typically a negative number since it points opposite to the direction of light travel.

Object distance o = m

Performing the indicated matrix multiplications gives the exit vergence from the pair of lenses.


The exit vergence from the final lens determines the image distance with respect to that lens.

Image distance i = m

The equivalent power for the pair of lenses can be calculated from Gullstrand's equation:


Note that the calculation does not take into account the change in lens separation with the angle of the incoming ray. It is typical to do the calculation only for the paraxial rays where the departure from full separation is negligible.

Principal planesSingle thick lensComparison to thin lens

Lens concepts

Thick lens concepts

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