Tutorial : focal lengths - principal planes - real lens interfacing different media

Using the Snell-Descarte law with the first order approximation applying in Gauss conditions, the different parameters paraxial parameters are given by the formulas belows. For simplicity, lets define the parameter k :

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R₁ and R₂ are the respective radii of the first and the second surfaces, n₀ and n₂ are the respective refraction indexes of the object and image spaces, n₁ is the refraction index of the lens and e is the center thickness of the lens.

The positions of the primary and secondary principal planes are given by :

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The focal lengths are given by the formulas : .

Note that which means that f = - f_i if the object space and the image space have the same refraction indexes and in particular if the lens is used in air or vacuum.

There are lens shapes that give particular results. For example, if the second surface of a lens is plane, the primary principal plane is tangent to the first surface. If the lens is reversed so that the plane surface is the first one, the secondary principal plane is tangent to the second surface.