Tutorial : spherical aberration - plane parallel plate

Let consider an optical system with spherical aberration (coefficients B₁ for the third order spherical aberration, C₁ for the fifth order spherical aberration,...) Rays from an on-axis object and making the same angle θ with the optical axis are focusing at the same point (image) because of the cylindrical symetry of the system. This appears in the Taylor expansion by the fact that the spherical aberration terms are not depending on α (see section related to the "geometric aberrations"). However, the position of this image depends on θ (or s in the Taylor expansion). At the limit angle θ = 0, the image coincides with the paraxial image. The spherical aberration represent the variation of image position depending on θ. The maximum distance between the image and the paraxial image is the longitudinal spherical aberration. The transverse spherical aberration is equal to the aberration spot diameter in the paraxial image plane. Note that the spot is circular because of the cylindrical symetry of the system. The spherical aberration depends on the position of the object plane, on the lens shape and obviously on the aperture size. It does not depend on the aperture position and on the distance of the object to the optical axis as the related coefficients in the Taylor expansion are proportionnal to s^2.n+1. It means that the spherical aberration is the same everywhere in a given object plane.

The two figures on the left show the ray tracing of a single spherical lens with spherical aberration (leftmost picture) and the associated spot diagram in the paraxial image plane (rightmost picture).

The left side scheme illustrates the case of a converging spherical lens. The larger the angle θ, the more the image moves away backwards from the paraxial image. The longitunal spherical aberration noted LAS on the picture is then negative, which means that the lens is undercorrected. In this particular case, the transverse spherical aberration is .

The longitunal spherical aberration is positive for a diverging lens which is then overcorrected.

It is possible to minimize the spherical aberration of a single lens in certain conditions. That is for instance the case with an object at infinity and when the radius of curvature of the first facet is around 6 time smaller than the radius of curvature of the second facet (see left figure on the left side). Note that if the lens is reversed (right figure), the spherical aberration is no more minimized. Note also that the lenses displayed on the two picture seems different just because the pictures have different ratio vertical scale/horizontal scale. In general, when considering a lens with convex facets, the configuration with less spherical aberration is the following one : the facet with the smallest radius is oriented towards the "more parallel" rays.

In the general case, an optical system with spherical aberration can be either undercorrected or overcorrected. LAS is not necessary the distance d from the paraxial image to the marginal rays image : in many cases, LAS > d and it is the distance from the paraxial image to the image corresponding to an intermedate angle .

The spherical aberration depends on the position of the object plane and on the aperture size. It does not depend on the aperture position and on the distance of the object to the optical axis as the related terms in the Taylor expansion are proportionnal to s^2.n+1. It means that the spherical aberration is the same everywhere in a given object plane.


Most components as spherical lenses, spherical mirrors, plates with flat and parallel faces,...have spherical aberration whatever the object position. Some others like ellipsoidal or parabolic mirrors and dioptres have in general spherical aberration but may be rigorously stigmatic for particular object positions (see rigorous stigmatism tutorial).

An optical system can be corrected for the spherical aberration (generally partially up to a certain order) by using aberrated components that combined together finally minimize or cancel its effects. Also, a certain type of lenses with aspherial shapes can almost be totally corrected for spherical aberration, even for large apertures. These kind of lenses are for instance used for on-axis objects, for example to inject the beam from a VCSEL (Vertical Cavity Surface Emitting Laser) which emitting area is small (in the range of the micron²) into a single mode fiber optic. A group of two lenses, the first one collimating the beam and the second one focusing the beam at the fiber entrance can be used for an optimal coupling. There are other solutions for minimizing spherical aberration like using a Schmidt plate before a spherical mirror when the object is at infinity...

The leftmost picture shows the ray tracing of a lens which is flat on one side and aspherical on the other side. The aspherical side is optimized to correct almost perfectly the spherical aberration for an object at infinity. Its spot diagram (right picture) shows a spot much smaller than the Airy disk (black circle) defining the diffraction limit of the system (see the "diffraction" tutorial).

Rays refracted through a plane parallel plate are not deviated but only translated. The longitudinal shift of a ray is:

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i is the incidence angle, n is the refraction index and e is the thickness. The transversal shift is then :

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The shift in a plane perpendicular to the ray is :

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Therefore, all incident rays focused at a point A are shifted to the front after transmission through the plate at a distance dx from A depending only on the incidence angle i. This means that the plate generates spherical aberration ( see "spherical aberration" tutorial ).

For sufficiently small incidence angles ( in Gauss conditions ), the longitudinal shift dx calculated above can be approximated as follows :

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dx is no more depending on the incidence angle. Therefore, the conjugate of a point by a plane parallel plate is a point shifted by dx. Note that the magnification is +1.