Aberration (optics)

 
A departure of an optical image-forming system from ideal behavior. Ideally, such a system will produce a unique image point corresponding to each object point. In addition, every straight line in the object space will have as its corresponding image a unique straight line. A similar one-to-one correspondence will exist between planes in the two spaces.
This type of mapping of object space into image space is called a collinear transformation. A paraxial ray trace is used to determine the parameters of the transformation, as well as to locate the ideal image points in the ideal image plane for the system. See also: Geometrical optics; Optical image
When the conditions for a collinear transformation are not met, the departures from that ideal behavior are termed aberrations. They are classified into two general types, monochromatic aberrations and chromatic aberrations. The monochromatic aberrations apply to a single color, or wavelength, of light. The chromatic aberrations are simply the chromatic variation, or variation with wavelength, of the monochromatic aberrations. See also: Chromatic aberration

Aberration measures
The monochromatic aberrations can be described in several ways. Wave aberrations are departures of the geometrical wavefront from a reference sphere with its vertex at the center of the exit pupil and its center of curvature located at the ideal image point. The wave aberration is measured along the ray and is a function of the field height and the pupil coordinates of the reference sphere (Fig. 1).
Transverse ray aberrations are measured by the transverse displacement from the ideal image point to the ray intersection with the ideal image plane.
Some aberrations are also measured by the longitudinal aberration, which is the displacement along the chief ray of the ray intersection with it. The use of this measure will become clear in the discussion of aberration types.


 

Fig. 1  Diagram of the image space of an optical system, showing aberration measures: the wave aberration and the transverse ray aberration. 
 
 
Caustics


Another aberrational feature which occurs when the wavefront in the exit pupil is not spherical is the development of a caustic. For a spherical wavefront, the curvature of the wavefront is constant everywhere, and the center of curvature along each ray is at the center of curvature of the wavefront. If the wavefront is not spherical, the curvature is not constant everywhere, and at each point on the wavefront the curvature will be a function of orientation on the surface as well as position of the point. As a function of orientation, the curvature will fluctuate between two extreme values. These two extreme values are called the principal curvatures of the wavefront at that point. The principal centers of curvature lie on the ray, which is normal to the wavefront, and will be at different locations on the ray.
The caustic refers to the surfaces which contain the principal centers of curvature for the entire wavefront. It consists of two sheets, one for each of the two principal centers of curvature. For a given type of aberration, one or both sheets may degenerate to a line segment. Otherwise they will be surfaces.
The feature which is of greatest interest to the user of the optical system is usually the appearance of the image. If a relatively large number of rays from the same object point and with uniform distribution over the pupil are traced through an optical system, a plot of their intersections with the image plane represents the geometrical image. Such a plot is called a spot diagram, and a set of these is often plotted in planes through focus as well as across the field.
 
Orders of Aberrations


The monochromatic aberrations can be decomposed into a series of aberration terms which are ordered according to the degree of dependence they have on the variables, namely, the field height and the pupil coordinates. Each order is determined by the sum of the powers to which the variables are raised in describing the aberration terms. Because of axial symmetry, alternate orders of aberrations are missing from the set. For wave aberrations, odd orders are missing, whereas for transverse ray aberrations, even orders are missing. It is customary in the United States to specify the orders according to the transverse ray designation.
Monochromatically, first-order aberrations are identically zero, because they would represent errors in image plane location and magnification, and if the first-order calculation has been properly carried out, these errors do not exist. In any case, they do not destroy the collinear relationship between the object and the image. The lowest order of significance in describing monochromatic aberrations is the third order.
Chromatic variations of the first-order properties of the system, however, are not identically zero. They can be determined by first-order ray tracing for the different colors (wavelengths), where the refractive indices of the media change with color. For a given object plane, the different colors may have conjugate image planes which are separated axially. This is called longitudinal chromatic aberration (Fig. 2a). Moreover, for a given point off axis, the conjugate image points may be separated transversely for the different colors. This is called transverse chromatic aberration (Fig. 2b), or sometimes chromatic difference of magnification.
These first-order chromatic aberrations are usually associated with the third-order monochromatic aberrations because they are each the lowest order of aberration of their type requiring correction.
The third-order monochromatic aberrations can be divided into two types, those in which the image of a point source remains a point image but the location of the image is in error, and those in which the point image itself is aberrated. Both can coexist, of course.

Aberrations of geometry


The first type, the aberrations of geometry, consist of field curvature and distortion.

Field curvature


Field curvature is an aberration in which there is a focal shift which varies as a quadratic function of field height, resulting in the in-focus images lying on a curved surface. If this aberration alone were present, the images would be of good quality on this curved surface, but the collinear condition of plane-to-plane correspondence would not be satisfied.


 

Fig. 2  Chromatic aberration. (a) Longitudinal chromatic aberration. (b) Transverse chromatic aberration.
 
 
Distortion


Distortion, on the other hand, is an aberration in which the images lie in a plane, but they are displaced radially from their ideal positions in the image plane, and this displacement is a cubic function of the field height. This means that any straight line in the object plane not passing through the center of the field will have an image which is a curved line, thereby violating the condition of straight-line to straight-line correspondence. For example, if the object is a square centered in the field, the points at the corners of the image are disproportionally displaced from their ideal positions in comparison with the midpoints of the sides. If the displacements are toward the center of the field, the sides of the figure are convex; this is called barrel distortion (Fig. 3a). If the displacements are away from the center, the sides of the figure are concave; this is called pincushion distortion (Fig. 3b).


 

 

Fig. 3  Distortion. (a) Barrel distortion. (b) Pincushion distortion.


 
Aberrations of point images


There are three third-order aberrations in which the point images themselves are aberrated: spherical aberration, coma, and astigmatism.
Spherical aberration


Spherical aberration is constant over the field. It is the only monochromatic aberration which does not vanish on axis, and it is the axial case which is easiest to understand.


 

 

Fig. 4  System with spherical aberration. (a) Wave aberration function. (b) Rays. (c) Spot diagrams through foci showing transverse ray aberration patterns for a square grid of rays in the exit pupil.

 
 
 
The wave aberration function (Fig. 4a) is a figure of revolution which varies as the fourth power of the radius in the pupil. The wavefront itself has this wave aberration function added to the reference sphere centered on the ideal image. The rays (Fig. 4b) from any circular zone in the wavefront come to a common zonal focus on the axis, but the position of this focus shifts axially as the zone radius increases. This zonal focal shift increases quadratically with the zone radius to a maximum from the rays from the marginal zone. This axial shift is longitudinal spherical aberration. The magnitude of the spherical aberration can be measured by the distance from the paraxial focus to the marginal focus.


 

 

Fig. 5  Caustics in a system with spherical aberration. (a) Diagram of caustics and other major features. (b) Rays whose envelope forms an external caustic.

 
 
The principal curvatures of any point on the wavefront are oriented tangential and perpendicular to the zone containing the point. The curvatures which are oriented tangential to the zone have their centers on the axis, so the caustic sheet for these consists of the straight line segment extending from the paraxial focus, and is degenerate. The other set of principal centers of curvature lie on a trumpet-shaped surface concentric with the axis (Fig. 5a). This second sheet is the envelope of the rays. They are all tangent to it, and the point of tangency for each ray is at the second principal center of curvature for the ray (Fig. 5b).
It is clear that the image formed in the presence of spherical aberration (Fig. 4c) does not have a well-defined focus, although the concentration of light outside the caustic region is everywhere worse than it is inside. Moreover, the light distribution in the image is asymmetric with respect to focal position, so the precise selection of the best focus depends on the criterion chosen. The smallest circle which can contain all the rays occurs one-quarter of the distance from the marginal focus to the paraxial focus, the image which has the smallest second moment lies one-third of the way from the marginal focus to the paraxial focus, and the image for which the variance of the wave aberration function is a minimum lies halfway between the marginal and paraxial foci.
 
Coma


Coma is an aberration which varies as a linear function of field height. It can exist only for off-axis field heights, and as is true for all the aberrations, it is symmetrical with respect to the meridional plane containing the ideal image point. Each zone in its wave aberration function (Fig. 6a) is a circle, but each circle is tilted about an axis perpendicular to the meridional plane, the magnitude of the tilt increasing with the cube of the radius of the zone.


 

 

Fig. 6  System with coma. (a) Wave aberration function. (b) Rays. (c) Spot diagrams through foci, showing transverse ray aberration patterns for a square grid of rays in the exit pupil.
 
 
 
The chief ray (Fig. 6b) passes through the ideal image point, but the rays from any zone intersect the image plane in a circle, the center of which is displaced from the ideal image point by an amount equal to the diameter of the circle. The diameter increases as the cube of the radius of the corresponding zone in the pupil. The circles for the various zones are all tangent to two straight lines intersecting at the ideal image point and making a 60° angle with each other. The resulting figure (Fig. 6c) resembles an arrowhead which points toward or away from the center of the field, depending on the sign of the aberration.
The upper and lower marginal rays in the meridional plane intersect each other at one point in the circle for the marginal zone. This point is the one most distant from the chief ray intersection with the image plane. The transverse distance between these points is a measure of the magnitude of the coma.


 

 

Fig. 7  System with astigmatism. (a) Wave aberration function. (b) Rays. (c) Spot diagrams through foci, showing transverse aberration patterns for a square grid of rays in the exit pupil.
 
 
 
Astigmatism


Astigmatism is an aberration which varies as the square of the field height. The wave aberration function (Fig. 7a) is a quadratic cylinder which varies only in the direction of the meridional plane and is constant in the direction perpendicular to the meridional plane. When the wave aberration function is added to the reference sphere, it is clear that the principal curvatures for any point in the wavefront are oriented perpendicular and parallel to the meridional plane, and moreover, although they are different from each other, each type is constant over the wavefront. Therefore, the caustic sheets both degenerate to lines perpendicular to and in the meridional plane in the image region, but the two lines are separated along the chief ray.
All of the rays must pass through both of these lines, so they are identified as the astigmatic foci. The astigmatic focus which is in the meridional plane is called the sagittal focus, and the one perpendicular to the meridional plane is called the tangential focus.
For a given zone in the pupil, all of the rays (Fig. 7b) will of course pass through the two astigmatic foci, but in between they will intersect an image plane in an ellipse, and halfway between the foci they will describe a circle (Fig. 7c). Thus, only halfway between the two astigmatic foci will the image be isotropic. It is also here that the second moment is a minimum, and the wave aberration variance is a minimum as well. This image is called the medial image.
Since astigmatism varies as the square of the field height, the separation of the foci varies as the square of the field height as well. Thus, even if one set of foci, say the sagittal, lies in a plane, the medial and tangential foci will lie on curved surfaces. If the field curvature is also present, all three lie on curved surfaces. The longitudinal distance along the chief ray from the sagittal focus to the tangential focus is a measure of the astigmatism.
The above description of the third-order aberrations applies to each in the absence of the other aberrations. In general, more than one aberration will be present, so that the situation is more complicated. The types of symmetry appropriate to each aberration will disclose its presence in the image.
 
 
Higher-order aberrations


The next order of aberration for the chromatic aberrations consists of the chromatic variation of the third-order aberrations. Some of these have been given their own names; for example, the chromatic variation of spherical aberration is called spherochromatism.
Monochromatic aberrations of the next order are called fifth-order aberrations. Most of the terms are similar to the third-order aberrations, but with a higher power dependence on the field or on the aperture. Field curvature, distortion, and astigmatism have a higher power dependence on the field, whereas spherical aberration and coma have a higher power dependence on the aperture. In addition, there are two new aberration types, called oblique spherical aberration and elliptical coma. These are not directly related to the third-order terms.
Expansions beyond the fifth order are seldom used, although in principle they are available. In fact, many optical designers use the third order as a guide in developing the early stages of a design, and then go directly to real ray tracing, using the transverse ray aberrations of real rays without decomposition to orders. However, the insights gained by using the fifth-order aberrations can be very useful.
 
 
Origin of Aberrations


Each surface in an optical system introduces aberrations as the beam passes through the system. The aberrations of the entire system consist of the sum of the surface contributions, some of which may be positive and others negative. The challenge of optical design is to balance these contributions so that the total aberrations of the system are tolerably small. In a well-corrected system the individual surface contributions are many times larger than the tolerance value, so that the balance is rather delicate, and the optical system must be made with a high degree of precision.
Insight as to where the aberrations come from can be gained by considering how the aberrations are generated at a single refracting surface. Although the center of curvature of a spherical surface lies on the optical axis of the system, it does not in itself have an axis. If, for the moment, the spherical surface is assumed to be complete, and the fact that the entrance pupil for the surface will limit the beam incident on it is ignored, then for every object point there is a local axis which is the line connecting the object point with the center of curvature. All possible rays from the object point which can be refracted by the surface will be symmetrically disposed about this local axis, and the image will in general suffer from spherical aberration referred to this local axis.
A small pencil of rays about this axis (locally paraxial) will form a first-order image according to the rules of paraxial ray tracing. If the first-order imagery of all the points lying in the object plane is treated in this manner, it is found that the surface containing the images is a curved surface. The ideal image surface is a plane passing through the image on the optical axis of the system, and thus the refracting surface introduces field curvature. The curvature of this field is called the Petzval curvature.
In addition to this monochromatic behavior of the refracting surface, the variation in the index of refraction with color will introduce changes in both the first-order imagery and the spherical aberration. This is where the chromatic aberrations come from.
Thus there are fundamentally only three processes operating in the creation of aberrations by a spherical refracting surface. These result in spherical aberration, field curvature, and longitudinal chromatic aberration referred to the local axis of each image. In fact, if the entrance pupil for the surface is at the center of curvature, these will be the only aberrations that are contributed to the system by this refracting surface.
In general, the pupil will not be located at the center of curvature of the surface. For any off-axis object point, the ray which passes through the center of the pupil will be the chief ray, and it will not coincide with the local axis. The size of the pupil will limit the beam which can actually pass through the surface, and the beam will therefore be an eccentric portion of the otherwise spherically aberrated beam.
Since an aberration expansion decomposes the wave aberration function about an origin located by the chief ray, the eccentric and asymmetric portion of an otherwise purely spherically aberrated wave gives rise to the field-dependent aberrations, because the eccentricity is proportional to the field height of the object point.
In this manner the aberrations which arise at each surface in the optical system, and therefore the total aberrations of the system, can be accounted for. See also: Lens (optics); Optical surfaces
Roland V. Shack
 
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