Spherical Aberration
Coma
Optical Sine Theorem
Astigmatism
Field Curvature
Distortion
Chromatic Aberration

Aberrations

The theory that we have used to describe the propagation of a ray through an optical system does not describe the true path of the ray. This is not unexpected, since we know that the theory that we have used is only valid to first order. The deviations from geometric optics are known as aberrations. Aberrations are due to inherent shortcomings of a lens, even a lens made of the best glass, and free from manufacturing and other defects. Some aberrations occur even with monochromatic light. These are known as monochromatic aberrations, and include spherical aberration, coma, astigmatism, field curvature and distortion. Other aberrations occur only with light that contains multiple wavelengths. These are known as chromatic aberrations.

Recall that in deriving the formula for the refraction of light, instead of using Snell's law,

we assumed that the angles were small, which allowed us to replace the sine terms with the angle itself. Thus, the law of refraction was stated as

Now let's assume that the angles are large. Using the power series expansion

(12.1)

we can classify our theory according to how many terms we keep from (12.1). If we retain only the first term, so that , then we have first order, or Gaussian, optics. Including the next term (so that ) yields third order optics.

Spherical Aberration

Spherical aberration is the dependence of the focal length on the aperture for nonparaxial rays. To see this, recall that the equation which describes refraction at a spherical surface was given by

formula . (12.2)

This was simplified in the paraxial region by approximating , so that , and thus

. (12.3)

If we extend this to third order, we find that

which implies

. (12.4)

Expanding ₀ and ₁ to third order, we get

formula (12.5)

and

formula . (12.6)

Substituting these into (12.2),

formula . (12.7)

The additional term shows the deviation from the first order theory, and represents the spherical aberration of the lens. It shows us that for a converging lens, the rays nearer the edge are bent more than those near the center, and so come to a focus earlier.

We can split spherical aberration into two parts. The distance between the axial intersection of a ray and the first order focus, f₁, is known as the longitudinal spherical aberration. If we are considering an extended image, the height of a specific ray above the axis at f₁ is called the transverse spherical aberration.

Coma

Coma is an image degrading aberration associated with an point even a short distance from the axis. It comes from the fact that the principal planes can actually only be treated as planes only in the paraxial region. In reality, they are principal curved surfaces.

Consider a bundle of rays passing through a lens. In the absence of spherical aberration, the rays will focus at the focal point behind the lens. However, the effective focal lengths will differ for rays traversing off-axis regions of the lens. This in turn causes the transverse magnification to differ in the offaxis regions. This is not a problem if the object is on the axis, but when the ray bundle comes into the lens at an oblique angle so that the image point is offaxis, coma becomes apparent.

Now consider the effect that an off-axis point has when the image is passed through a lens and projected onto a screen. Assume that the object passes through the lens in such a way that it forms a set of rings on the lens

For comparison with the image on the screen, I have labeled some of the points on the rings. The equivalent points on the screen would look like the figure below

The resulting image is called a comatic circle. The distance from 0 to 1 is called the tangential coma, while the length from 0 to 3 is called the sagittal coma. A little more than half of the energy in the image appears in the roughly triangular region between 0 and 3. The coma flare, which owes its name to its cometlike tail, is often considered the worst of all aberrations, primarily because of its asymmetric configuration.

Optical Sine Theorem

An important result can be derived from a theorem known as the optical sine theorem. Without presenting a formal proof, the theorem states that

, (12.8)

where n_o, y_o, _o, and n_i, y_i, _i are the index, height, and slope angle of a ray in object and image space, respectively, at any aperture size. If the coma is to be zero,

(12.9)

must be constant for all rays. Now consider the effect of a paraxial ray and a ray entering the edge of the lens. The latter ray is called a marginal ray. Since M_T is to be constant over the entire lens, we must have that

which reduces to

. (12.10)

This is known as the sine condition. A necessary criterion for the absence of coma is that the system meet the sine condition.

Astigmatism

When an object lies an appreciable distance from the optical axis the incident cone of rays will strike the lens asymmetrically, giving rise to the aberration known as astigmatism. In order to describe it, picture the plane which contains both the chief ray, which is the ray which passes through the center of the lens, and the optical axis. This plane is knows as the meridianal, or tangential, plane. The sagittal plane is defined as the plane containing the chief ray which is also perpendicular to the tangential plane.

When the object is on the optical axis, the cone of rays is symmetrical with respect to the spherical surfaces of the lens. In this case the meridianal and the sagittal planes are the same, and the ray configurations in all the planes containing the optical axis are identical. In the absence of any spherical aberration, all of the focal lengths are the same and all of the rays arrive at a single focus.

When the object is located off axis, the rays come into the lens at an oblique angle. Now the configuration of the ray bundle will be different in the meridianal and sagittal planes. Because of this, the focal lengths in these planes will be different as well. Basically, the meridianal rays are tilted more with respect to the lens than the sagittal rays, and thus have a shorter focal length. Using Fermat's principal, we find that the focal length difference depends effectively on the power of the lens and the angle at which the rays are inclined. This is known as the astigmatic difference, and it increases rapidly as the rays become more oblique.

Since there are two distinct focal lengths, the incident conical bundle of rays changes after being refracted. The cross section of the beam as it leaves the lens is initially circular, but it gradually becomes elliptical with the major axis in the sagittal plane, until at the tangential focus, F_T, the ellipse degenerates into a line (at third order). All the rays from the object traverse this line, which is known as the primary image. Beyond this point the beam's cross section rapidly opens out until it is again circular. At that location the image is a circular blur known as the circle of least confusion. Moving further from the lens, the beam's cross section again deforms into a line, called the secondary image. This time it is in the meridianal plane at the sagittal focus, F_S.

Field Curvature

Suppose that an optical system was free of all of the other aberrations considered so far. There would then be a one-to-one correspondence between points on the object and image surfaces. Since a planar object normal to the axis will be imaged as a plane only in the paraxial region, we find that for finite apertures the resulting image will usually be a curved surface. This surface is caused by Petzval field curvature.

Consider a spherical object segment, _o. It is imaged by a lens as another spherical image segment, _I. If the object segment is flattened out into the plane '_o, each object point will move towards the lens along its chief ray, thus forming a paraboloidal surface known as the Petzval surface _P. In particular, the displacement x of an image point at height y_i on the Petzval surface from the paraxial image plane is given by

formula (12.11)

where n_j and f_j are the indices and focal lengths of the m thin lenses forming the system. Notice that for the simple case of two thin lenses having any spacing, x can be made zero provided that

or equivalently,

. (12.12)

This is the Petzval condition.

Distortion

Distortion comes from the fact that the transverse magnification, M_T, may be a function of the off-axis image distance, y_i. Thus, that distance may differ from the one predicted by paraxial theory in which M_T is constant. In other words, distortion arises because different areas of the lens have different focal lengths and different magnifications. In the absence of any other aberration, distortion is seen in the misshaping of the image as a whole, even though each point is sharply focused.

Chromatic Aberration

In addition to the previous aberrations, there are chromatic aberrations that arise specifically from multi-wavelength light. Remember that the refraction equation

is a function of the indices of refraction, which in turn vary with wavelength. Thus, rays with different wavelengths will traverse a system along different paths, and this is the basic feature of chromatic aberration.

Since the thin lens equation

is wavelength dependent, the focal length must also vary with the wavelength. The axial distance between two focal points spanning a giving frequency range is termed the axial (or longitudinal) chromatic aberration.

The image of an off-axis point will be formed of the constituent frequency components, each arriving at a different height above the axis. The frequency dependence of f causes a frequency dependence of the transverse magnification as well. The vertical distance between two such image point is a measure of the lateral chromatic aberration.

Last updated: July 24, 1997

Comments to: D-Suson@tamuk.edu