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The position of the image formed by a thick lens can be found by the matrix method. This involves multiplying a vector form of the incident vergence successively by matrices representing the refraction by the first surface, the translation to the second surface, and the refraction by the second surface. In this way the exit vergence is determined and from that vergence, the image distance.

A thick lens can be characterized by its surface powers, its index of refraction, and its thickness.
Surface power =
Surface power =
Index =
Thickness d = m

The vergences can be calculated from the lens parameters and the object distance.The change in vergence when the light encounters a refracting surface is equal to the power of the surface:

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 thick lens.


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

Image distance i = m

The equivalent power for the thick lens can be calculated from Gullstrand's equation:


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

Vergence ExampleMatrix definitionsExample for two thin lenses

Lens concepts

Thick lens concepts
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