Chapter 2.  Primer of Image Aberrations and their Graphical Representation.

The five "3rd order" or "Seidel" aberrations which afflict optical systems are: spherical aberration, coma, astigmatism, field curvature, and distortion [cf. for a fuller treatment:  H. Rutten and M. van Venrooij, Telescope Optics, Evaluation and Design (Willmann-Bell, 1988), pp. 21-35; G.H. Smith, Practical Computer-Aided Lens Design (Willmann-Bell, 1998), pp. 55-97; or for a more advanced treatment, W. Smith, Modern Optical Engineering , 3rd ed. (McGraw-Hill, 2000), pp. 61-89].  In order to evaluate how well a lens functions, one must gain a basic understanding of how these aberrations are represented in graphical form.  In addition, several "1st order" aberrations also occur:  defocus, longitudinal chromatic aberration (chromatic variation of focal length), and lateral color (chromatic variation of magnification).  Chromatic variations of the 3rd order aberrations also occur.  But the only one of importance is chromatic variation of spherical aberration, also called "spherochromatism" or "Gauss Error."  This is an insidious problem for apochromatic lenses and must be carefully watched.  The following primer is meant to show the reader how these aberrations appear in two different graphical forms:  the transverse ray fan plots and the spot diagrams.

The easiest way to understand how well a telescope functions is by means of the spot diagram.  Rutten and van Venrooij use spot diagrams almost exclusively in their book for image evaluation; and thus, the reader can consult their work for extensive examples.  Essentially a spot diagram shows how tightly rays of light passing through the telescope converge to a focus.  Typically the standard against which the convergence is measured for visual systems is the Airy disk, the small circle of light which a perfectly functioning telescope achieves at focus.  This circle has a finite size, and no smaller image can be formed in any telescope than the Airy disk because of the physical phenomenon of diffraction.  What follows is an example of a spot diagram:

Coma in a 200mm f/6 Newtonian  
Figure 1:  Spot Diagram Showing Coma

The small black circle seen at the center of the figure represents the Airy disk for an 200 f/6 Newtonian reflector.  The large comet-shaped scatter of blue points inside and above the circle represents the physical scatter of light rays caused by the aberration called "coma."  Here we have a spot diagram showing approximately what you would see at the edge of a 1.3 degree field of view in an 200mm f/6 Newtonian.  Instead of a sharply focused, small round image (the Airy disk), bright stars in such a telescope are blurred into fan-shaped objects several times larger than the minimum size possible.  

In fact, however, only the bright pointed end of a comatic blur and its concentrated wings can actually be seen in a real telescope.  The smeared out "tail of the comet" is generally too faint to be visible.  Nevertheless, one can easily grasp from this single representation the essence of a spot diagram and why it is so useful for quickly evaluating the image quality of a high-resolution visual telescope.  We want all the light rays from a given star to be concentrated well within the Airy disk.  But instead, at the field edge of our 200mm f/6 Newtonian the rays are badly smeared out.  So obviously this instrument cannot form a sharp image there, however excellent it is on-axis.

As we saw in Chapter 1, the other main graphical representation of telescope performance is the transverse ray fan plots.  The comatic blur of the above Newtonian looks as follows in a transverse ray fan plot::

Coma Ray Fan
Figure 2:  Ray Fan Plots Showing Coma

The axes labeled "PY" and "PX," as we saw in Chapter 1, represent radial distances ("pupil zones") at which rays of light enter the entrance pupil of the telescope in the y/z- (meridional) and x/z- (sagittal) planes.  "EY" and "EX" represent the "error distances" for rays of a given pupil zone by which they miss the optical axis as measured in a transverse direction away from the selected focus.  We would like in all cases for the ray fan plots to appear as exactly straight, horizontal lines, coinciding with the PY- and PX- axes.  That would indicate perfect imagery.  The degree to which they deviate from straight and horizontal, and do not coincide with the abscissae of the graphs, is a measure of the image aberration produced by the optical system.  In Figure 2 above, we see that the sagittal ray fan plot looks perfect.  But the meridional plot is quite bad, just as the spot in Figure 1 is also quite bad.

Now, at this point an attentive reader will easily have noticed that the ray fan plots for spherical aberration (shown in Chapter 1), and those above for coma differ from one another markedly.  The plots for spherical looked like polynomials of degree 3, while the tangential plot for coma looks like a polynomial of degree 2 and the sagittal plot is a straight line, making the graphs for the two aberrations easy to distinguish from one another.  This is not an accident.  Most of the aberrations also leave unique signatures in ray fan plots.  And thus, by learning the individual signatures--even without understanding why those signatures are formed- -the reader can quickly diagnose many of the aberrations which afflict a telescope.  The spot diagrams, on the other hand, can present tangles of dots in which individual aberrations are hard to discern.  But the spots too are useful, because no matter how complicated they are, we can still judge the severity of a telescope's collective aberrations by comparing the spot sizes to the Airy disk, allowing us quickly to ascertain whether the mass of aberrations is large enough to ruin the high-resolution performance or not.  So the spot diagrams give us a quick tool to judge image quality, and the ray fan plots give us a useful tool to understand what aberrations dominate and need correcting.

So let us go systematically through the image aberrations and see their characteristic representations in ray fan plots and spot diagrams.  The first to consider is defocus.  Ordinarily, one would not consider defocus an aberration because at the telescope one would simply move the eyepiece forward or back to correct the problem.  Even so, it is useful to see how defocus is represented in the plots.  For an object inside focus we have the following ray fan plots:

Inside Focus
Figure 3:  Ray Fan Plots Showing Defocus (Inside)

For the same object outside focus, we have:

Outside Focus
Figure 4:  Ray Fan Plots for Defocus (Outside)

Thus, simple defocus manifests itself in ray fan plots as straight but tilted lines.  If the lines slope in an upward (positive) direction, then we are examining the image from inside focus; if the lines slope in a downward (negative) direction, then we are outside focus.  Contrast this with the spot diagram, which looks exactly the same for equal amounts of defocus inside and out:

Defocus Spot Diagram  
Figure 5:  Spot Diagram Showing Defocus (Inside or Outside)

As in the previous spot diagram, the Airy disk once again appears as a small black ring, whereas the much larger bull's-eye pattern of blue points represents the out-of-focus star image.  If, on the other hand, we were examining the image exactly at focus, the ray fan plots for this perfect telescopic system would look like straight horizontal lines and the spot diagram would show one tiny point of light at the center of the Airy disk.  Of course, I must again emphasize that because of the physics of light, in a real telescope--no matter how small the image blur may appear in a spot diagram--the smallest actual image is defined by the Airy disk.

In the last chapter we saw the ray fan plots for one type of spherical aberration.  For reference, it is presented again:

Undercorrected Ray Fan
Figure 6:  Ray Fan Plots Showing Undercorrection (Paraxial Focus)

This type is called "undercorrection," because it represents the natural error of a concave spherical mirror or convex singlet lens constructed with spherical surfaces.  When imaging a star at infinity, such a mirror or lens tends to redirect the rays impinging on its outer pupil zones to a shorter focus than rays impinging closer to its middle zones.  We saw this graphically illustrated in Figure 3 of the last chapter.  Unless corrective steps are taken to give the mirror or lens an aspheric shape (or unless we make a doublet lens), either optic will exhibit undercorrected spherical aberration when imaging a star at focus.   The corresponding spot diagram  for Figure 6 above is as follows:

Undercorrected Spot
Figure 7:  Spot Diagram Showing Undercorrection (Paraxial Focus)

Similarly, one may also have "overcorrected" spherical aberration, when rays in the outer pupil zones come to a longer focus than rays in the inner zones.  An example of overcorrected spherical appears as follows in a ray fan plots:

Overcorrected Ray Fan
Figure 8:  Ray Fan Plots Showing Overcorrection (Paraxial Focus)

Obviously, this diagram looks very much like Figure 6 reflected across the horizontal axis.  Moreover, comparing Figures 6 and 8 to Figures 3 and 4, we can see that overcorrected spherical aberration bears some resemblance to negative defocus (i.e. being inside focus), while undercorrected spherical shows a resemblance to positive defocus (i.e. being beyond focus).  That is to say, if we have an optic suffering from no spherical aberration and we are examining it inside focus (negative defocus, cf. Figure 3), then light from progressively larger annuli of the entrance pupil falls progressively further away from the optical axis--while never yet having reached it--from our vantage point along the optical axis.  What that means can be seen in the following diagram:

Negative Defocus
Figure 9:  Negative Defocus in a Perfect Lens

The above lens achieves a perfect focus (i.e. shows no spherical aberration) at the vertical black line on the extreme right side of the layout.  Somewhat to the left of that focus there appears another black line, which cuts across the ray fan marking a position inside focus (i.e. showing negative defocus).  It is easy to see that rays from progressively larger annuli of the entrance pupil fall further and further from the optical axis, when considered from the vantage point of that second black line.

Now examine what happens to rays at the paraxial focus of an optic exhibiting overcorrection:

Overcorrected Layout
Figure 10:  Lens Showing Overcorrected Spherical Aberration

Here the paraxial focus is marked by a small black vertical line [Remember that the "paraxial" region of a lens is that which immediately surrounds the optical axis].  And again we see a tendency for rays coming from progressively larger annuli in the entrance pupil to fall away from the paraxial focus rather as they did in Figure 9 for negative defocus.  This illustrates the justification for saying that the global positive tilt of overcorrected spherical aberration shown in Figure 8 resembles the global tilt of negative defocus shown in Figure 3.

Likewise, in the case of positive defocus (i.e. when we are beyond focus) for an optical system suffering from no spherical aberration, light from the entrance pupil has already crossed the optical axis and is now diverging from our vantage point, falling further and further away from the axis as we consider progressively larger annuli of the entrance pupil.  Undercorrected spherical aberration presents a rather similar effect, the outer zones coming to a focus before the paraxial zones and already diverging by the time we reach paraxial focus (as we saw in Figure 3 of the last chapter).  Notice that in Figures 6 and 4 above, the global tilt of the lines in both cases is positive.

Indeed, we could say that in Figures 6 and 8 we are not at the best focus, because clearly there is an average non-zero tilt to the lines.  By shifting focus we should be able to reach a better balance.  That is true and for the undercorrected kind of spherical aberration it will look as follows:

Undercorrected 2 Ray Fan
Figure 11:  Ray Fan Plots Showing Undercorrection (Best Focus)

Here there is no systematic positive or negative slope to the lines as wholes.  Notice that the scale of the EY- and EX- axes is now much smaller than before (+/-10 microns instead of +/-50 microns in Figures 6 and 8), confirming that the average focus error is now less.  The spot diagram corresponding to Figure 11 is also clearly better than that shown above for undercorrected spherical aberration at the paraxial focus (Figure 7):

Undercorrected 2 Spot
Figure 12:  Spot Diagram Showing Undercorrection (Best Focus)

It was said earlier that the ray fan plots for spherical aberration look like the graphs of a 3rd order polynomial.  Actually, that was a simplification and true only for the fundamental form of spherical aberration, also called "3rd order," or "Seidel" spherical.  But when 3rd order spherical is eliminated or minimized, higher orders of the aberration may then stand out.  Moreover, there may also be mixtures of various orders of spherical aberration.  5th order will appear in some of our future ray fan plots, and show the appearance of a polynomial of degree 5.  But high order spherical (7th, 9th, etc). will not appear and need not detain us.  Still it is useful to see the graph for a residual of 5th order spherical aberration, when 3rd order has been corrected:

5th Order Spherical
Figure 13:  Ray Fan Plots for 5th Order Spherical Aberration

For the case of coma, which only affects off-axis images in well-built optical systems, I gave a spot diagram and ray fan plot at the beginning of this chapter (Figures 1 and 2).  Astigmatism, which also only affects off-axis images in well-built telescopes, is caused by the existence of two different foci, one for the tangential plane and another for the sagittal plane.  When we stop along the optical axis between the two foci, we see equal and opposite amounts of defocus in both the tangential and sagittal ray fan plots:

Figure 14:  Ray Fan Plots for Astigmatism (Medal Focus)

Here we are past the focus for the tangential plane, but before that for the sagittal plane.  The corresponding spot diagram shows a disk of light which resembles a defocused star.   But because we have reached a medium defocus in the two optical planes, we cannot shrink the disk any further:

Astigmatism Spot
Figure 15:  Spot Diagram for Astigmatism (Medial Focus)

If we move away from this focal position, the image sharpness may become better in one plane, but it will be worse in the other.  Thus, the star disk will assume a continuously varying elliptical shape depending on our z-axis position.  The slopes seen in the ray fan plots will also change.  But under no circumstances can they become the same as one another (contrast this with Figures 3 and 4 for defocus without astigmatism).  What follows are the ray fan plots and spot diagram for the same optic as in Figures 14 and 15, but for a different, randomly selected focal position:

Astigmatism 2 Ray Fan Plot
Figure 16:  Ray Fan Plots for Astigmatism (Outside Sagittal Focus)

Astigmatism Spot 2
Figure 17:  Spot Diagram for Astigmatism (Outside Sagittal Focus)

Thus, both the spot diagram and the ray fan plot give us ways of diagnosing astigmatism.  If we see elliptically shaped spots, or if we see tangential and sagittal ray fan plots with differing slopes, then our optic suffers from astigmatism.  Lenses for visual refractors normally do show a small off-axis astigmatism.

Field curvature is also a normal feature of visual telescopes and is rarely objectionable.  This aberration causes on-axis and off-axis stars to focus not in a plane perpendicular to the optical axis, but on a curved focal surface normal to that axis:

Field Curvature
Figure 18:  Field Curvature

The above figure shows a singlet lens bent to a shape which minimizes coma, and given an aspheric front surface in order to minimize spherical aberration.  Stars in two different field positions are being imaged.  The blue rays emanate from an on-axis star, and the green rays from one 10 degrees off-axis.  The curved image surface is clearly visible as a black arc on the right side connecting the two foci.

For a photographic lens this much field curvature would be intolerable, since film and CCD detector usually comprise flat surfaces.  In that case, the star off-axis would look far out of focus, if we placed our film or CCD chip at the axial focus.  But the human eye can automatically refocus on different parts of the telescope's field of view.  Thus, field curvature is seldom a problem for visual telescopes.

Distortion is the tendency for an optical system to image straight lines as curves.  The problem grows with larger field angles and can cause severe difficulties for photographic objectives which may image fields 50 degrees or more in angular extent.  But visual telescope objectives rarely show more than 2-degree fields.  And so image distortion caused by the objective is usually not an issue.  Only certain highly tilted, unobstructed reflectors show noticeable distortion.  Eyepieces suffer much more, but analyzing them lies beyond the scope of this web site.  So we need not worry about distortion.

Among the so-called "1st order" aberrations, we have already discussed defocus for a single (monochromatic) wavelength of light.  Unfortunately, because lenses operate by refraction, and because the amount of refraction produced by a lens varies with the wavelength of light due to "Snell's Law," we must consider what happens to several of the aberrations when more than one wavelength of light passes through a lens.  The first aberration to consider is axial chromatic aberration, also called "longitudinal chromatic aberration," or "primary spectrum."  This error occurs because a single converging lens acts as a kind of focusing prism, spreading the variously colored foci out along the optical axis, with the violet focus falling shortest, then the blue, green and so on to the deep red, which falls longest.  Axial chromatic aberration can be seen graphically in a polychromatic ray fan plot for the lens shown in Figure 18:

Axial Color
Figure 19:  Ray Fan Plots for Primary Longitudinal Chromatic Aberration

Here the variously colored lines represent various wavelengths of light passing through our lens.  The wavelengths chosen (listed in microns) can be seen at the bottom of the caption.  They are typical wavelengths used in optical analysis.  The violet line represents the g- spectral line at 0.436 micron, and marks the approximate limit of human vision at shorter wavelengths; the blue line represents the blue F- spectral line at 0.486 micron, which along with the red C- spectral line at 0.656 micron is brought to common focus in order to achromatize a lens (see Chapter 3a).  The green line at 0.546 micron represents the green e- spectral line, close to the brightest portion of the spectrum for the human eye.  The e-line is often used in lens design for analyzing the 3rd order monochromatic aberrations.  And the gold colored line seen in Figure 19 represents the r-line at 0.707 micron in the deep red, near the limit of human vision at longer wavelengths.  

Primary spectrum can be seen even more graphically in the following polychromatic layout of a very fast singlet lens with extended spectral coverage:

Longitudinal Chromatic Aberration
Figure 20:  Polychromatic Layout of a Singlet Lens

In order for a lens to perform its best, all these different colors of light ought to be brought to an identical focus.  But in practice this is never possible with lenses and can only be achieved using optical systems composed entirely of mirrors.  Still, it is possible to come close to the ideal for lenses by employing special types of glass.  Such lenses are called "apochromatic" and will be discussed in detail in Chapters 4a  and 4b.

In Figures 19 and 20 above, it is easy to see that this singlet lens falls woefully short of ideal.  The color spread is huge.  At the focus for e-light (notice that the green line is horizontal in Figure 19), we are well beyond the focus for g- and F- (notice the negative slope of their graphs in Figure 19), and well in front of the focus for C- and r-.  It is utterly impossible to find a good polychromatic focus for this lens, and in practice the human eye looking through it would see a kaleidoscope of colors and vague, ill-defined objects.  This is despite the fact that the eye is only very slightly sensitive to g- and r-, and moderately sensitive to F- and C-.  The spot blur which corresponds to Figure 19 is the following:

Chromatic Spot
Figure 21:  Spot Diagram for Primary Longitudinal Chromatic Aberration

The blur is nearly 10 mm in diameter for violet and the Airy disk is barely visible as a black dot at the center of the blur.  Clearly chromatic aberration can be a very serious problem for lenses.  And that is true not only for axial chromatic, but also for another form called "lateral chromatic aberration" or "lateral color."  It can often happen that because some colors of light come to a longer focus than the main design color (usually e- or the nearby d-line), and others to a shorter focus, a lens will not have a uniform focal length and therefore magnification for all colors.  In which case a star off-axis may be smeared out into short spectrum pointing toward the optical axis, since the bluer rays experience the lens as a shorter focus optic imparting less magnification than the redder rays.  Lateral color appears in ray fan plots as a series of colored lines separated from one another as follows:

Lateral Color
Figure 22:  Ray Fan Plots for Lateral Color

The corresponding spot diagram shows a similar effect, where one can see the Airy disk as a tiny circle at the center of the diagram:

Lateral Color Spots
Figure 23:  Axial Spot Diagram for Lateral Color

Any good lens images all colors of light directly on top of one another.  Two smaller effects which can also be seen here are chromatic variation of astigmatism and defocus.

The last polychromatic aberration to consider is "spherochromatism," or "Gauss Error."  Spherochromatism is the variation of spherical aberration with wavelength.  A well-behaved example of this error is the following:

Figure 24:  Spherochromatism

Violet and blue rays show overcorrected spherical aberration somewhat past their best focus for this optic (compare Figure 8), while deep red and full red show undercorrected spherical somewhat inside best focus (compare Figure 6).  Green shows excellent correction and is at its focus.  Generally in real lens systems, the spherochromatism does not look so neat as the above, and frequently the violet rays diverge quit far from the other colors.  Spherochromatism is one of the most important residual problems with apochromatic lenses and can indeed ruin otherwise interesting designs, if it is bad enough  For now, though, it is sufficient for the reader merely to get an impression of how spherochromatism manifests itself in ray fan plots.

The above examples of aberrations do not exhaust the list of possibilities.  But they are the main ones seen in on-axis and only slightly off-axis images, such as we encounter in high-resolution visual objectives.

Chapter 3a

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