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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


References: Birney, Gonzalez & Oesper, pp. 108-112; Roy & Clarke, Section 16.5.


Chromatic aberration is a characteristic of lenses, and arises because of the wavelength variation of the index of refraction.

We have already discussed this in reference to the human eye.

  • For a single lens, there is no single place where all wavelengths come into the same focus.

  • But one can generally find that between the extremes of the red and blue focal lengths, there is one place where the smallest possible image from all wavelengths can be found -- the circle of least confusion.

    Here the image is in the sharpest focus, but it is not a point.

From Roy & Clarke, Astronomy: Principles and Practice.

One can partially counteract chromatic aberration by creating achromatic doublets from combining positive and negative lenses that counteract each other.


  • Can only pick two wavelengths in the chromatic aberration to join together, for example a blue and a red wavelength. Other wavelengths will come to a focus elsewhere (though not as badly as if there were no achromat).

    From Rutten & van Venrooij, Telescope Optics.

  • Traditional refractors were designed to be optimized to bring the red and blue parfocal in order to minimize the secondary spectrum across wavelengths over which the eye is most sensitive.

    The focal length of a doublet lens as a function of wavelength. From Rutten & van Venrooij, Telescope Optics.

  • The lenses are usually made of different index of refraction glasses (like crown and flint), in order to suppress other aberrations (e.g., spherical, coma...).

  • With additional elements, can bring 3 wavelengths parfocal, etc.

    In the following example, the C, d and F correspond to specific spectral lines names above -- Hα (6563 A), Na D (about 5890 A) and Hβ (4863 A), respectively.

    Relative focal lengths as a function of wavelength as one adds optical elements.

  • But that there are some differences in the focal length with wavelength is something to keep in mind for later in the semester, when we try to focus the 26-inch refractor for CCD imaging at different wavelengths.


You will eventually be using the McCormick refractor to take images with a CCD camera. Note that the focus of the telescope varies with wavelength, due to the chromatic aberration in the lenses of the objective.

The McCormick refractor has a focus curve much like this.

Note what this implies for filtering the CCD.

  • As may be seen, even though one loses light, it is useful to put a filter in the telescope to limit the wavelength range.

    This will make sure the light that does come through the filter has approximately the same focal length.

  • Taking images with a green filter will give the clearest images because here there will be the smallest range of focal length.

  • Taking images with a blue filter will generally yield poorer quality images because of the much larger range in focal length within the blue wavelength range.


Monochromatic aberrations are those that affect all wavelengths equally.

There are two types of these Seidel aberrations:

  • Aberrations that deteriorate the image -- making it less clear.

    There are three most important aberrations of this type:

    • spherical aberration

    • coma

    • astigmatism

  • Aberrations that deform the image -- stretching it.

    There are two most important aberrations of this kind:

    • field curvature

    • distortion

These aberrations can be inherent to one optical element and can be in the final image from a multiple-element system.

  • The advantage of multiple-element systems is that we hope to correct out aberrations from one element by adding successive elements.

  • Generally one corrects these aberrations in the order they are listed above (this is generally the order considered of decreasing importance).

Spherical Aberration

We've already discussed spherical aberration in the context of the human eye.

  • In this aberration, parallel rays from a source at infinity hitting different radii from the center of the optic do not converge to same point, and the image gets blurred.

  • For a spherical surface, this does not happen, which is the source of the name for this aberration (though any shape but a parabola has this same problem).

  • As with chromatic aberration, there is a circle of least confusion.

  • The spread of the focus along the optical axis is known as longitudinal spherical aberration and the spread of an image in a plane perpendicular to the optical axis is called the transverse spherical aberration
    From Hecht, Optics.

  • As pointed out before, a way to remove spherical aberration is to make the mirror surface a parabola:

    From Hecht, Optics.

    Recall that a parabola is that surface of points for which the distance of each point from a plane is equal to the distance of that point from a single focus point.

    • So, e.g., in the above figure (panel b) the distance between D1 to A1 is the same as the distance from A1 to F.

      More generally, any point on the parabola has a distance from the plane Σ that is equal to its distance from the the focal point F.

    In this configuration all parallel rays come to a single focus, which is not the case for a sphere:

    The two large figures show the difference in foci for a spherical and a parabolic mirror. But the inset on the left focusing on the region near the optical axis of the spherical surface shows that the rays near the optical axis do more or less come to one focus. This shows that for large f/ratios, spherical aberration is less of a problem (think about why that is from this figure). From

    However, even though parabola's do not have spherical aberration, it is much more complicated (i.e. expensive) to make optics with a parabolic than a spherical shape.

Historical Example: Hubble Space Telescope

  • Soon after the Hubble Space Telescope (HST) was put in orbit (1990), it was discovered that it could not be put into good focus (at the circle of least confusion, the images were still very ugly).

  • The Airy diffraction limit desired was being reached (0.1 arcsec), but only 12% of energy there compared to expected 70%.

  • Too much of the light from stars was put into the Airy rings and in a diffuse light halo, and the point-spread-function. has a Strehl ratio of only ~ 15-20%.

    The telescope was showing classical signs of spherical aberration.

    Image from the HST Wide Field/Planetary Camera 1 (WF/PC1). The "tendrils" are from diffraction off the secondary support struts.

  • It was later determined that HST primary was polished to exquisite precision, but to an incorrect shape!

    The mirror was too flat at the margins by ~ λ / 2.

    The error was a result of a 1.3-mm error in the placement of the device used to measure the shape of the primary when being made by Perkin-Elmer.

  • The result was a 38-mm longitudinal spherical aberration:

    From Hecht, Optics, Fourth Edition.

The response of NASA was a dramatic servicing mission with the Space Shuttle to insert "corrective eyewear" -- the Corrective Optics Space Telescope Axial Replacement (COSTAR) -- into the HST instrument bay.

  • COSTAR restored >70% of the energy in the central disk, increasing its magnitude limit by several magnitudes and much cleaner images.

    The galaxy M100 before (left) and after (right) COSTAR.

  • More recently installed HST instruments were designed with optical corrections in their optics that eventually allowed COSTAR to be removed.


Comatic aberration is an important image defect created by the fact that rays from an off-axis source (i.e., rays coming in at an angle) do not all converge at the same point in the focal plane.

From Hecht, Optics, Fourth Edition.

  • Images that are formed with coma have a very characteristic "head-tail", comet-like appearance (see below) that gives rise to the name of this aberration.

  • Each ring of points on the optical surface creates its own comatic circle on the focal plane.

  • The larger the ring, the more distant is the center of the comatic circle from the position of the chief ray (which is the ray coming through the center of the optic).

  • The result is a "comatic blur" with a triangular-like shape pointing radially away from the optical center.


  • Interestingly, spherical optics, while they have spherical aberration, do not have coma (because any angle coming in from the field of view sees the same spherical shape at any angle).

    This fact comes into play later when we discuss the Schmidt telescope.


    This is different than the vision defect we talked about earlier, which was due to differences in the effective radius of curvature along different meridians.

    The astigmatism here happens for symmetric lenses, and is sometimes referred to as oblique astigmatism.

    Imagine a system that has no spherical aberration or coma.

    • In this aberration, one recognizes that the angles that rays in the sagittal and meridianal planes make a different angle with respect to the optical surface.

      This results in different focal lengths for rays in these two planes, with the meridianal plane focus closer.

      From Hecht, Optics, Fourth Edition.

    • The ray bundle in such a situation has a very particular shape, with it first becoming elliptically shrunk in the tangential direction as the tangential focus is approached -- eventually becoming a sagittal line.

      Then the bundle expands tangentially again while shrinking in the sagittal direction as the sagittal focus is reached, at which point a tangential line is formed.

      The difference in the focal lengths is known as the astigmatic difference.

      In between the two foci is a circle of least confusion.

    • In general, the image of a point source will look differently at different points along the optical axis.


    Field Curvature

    This aberration occurs when the image plane (focal plane) is not flat, but curved.


    This can be a problem for large area detectors, which need to follow a curved focal plane to stay in focus.


    This originates from a change in the plate scale (i.e. focal lengths or transverse magnifications) with field angle.

    • Has the effect of shifting image positions on the focal plane.

    • Has the effect of distorting the image.

    • In the absence of other aberrations, individual points are sharply focused, but the image as a whole is deformed.

      • In positive or pincushion distortion, points move radially outward.

      • In negative or barrel distortion, points move radially inward.


    • This aberration is often considered not much a problem, because we can calibrate the distortion out and there is no loss of information.


    • Parabolic mirrors do not have spherical aberration, but they do have coma and produce good images only for light approaching nearly parallel to the optical axis (i.e., only for the center of field of view).
    • Only objects near the optical axis are in focus at the focal plane. Objects at larger angle to the optical axis get increasingly "fuzzier" due to "coma".
    • This is a serious problem when one wants to make a telescope that images large fields of view.

    • To overcome this limitation, Bernhard Schmidt invented a design that went back to a spherical mirror (to eliminate coma) --- but he introduced a thin correcting lens to remove the spherical aberration introduced by the spherical mirror.
    • In this system, one puts the detector at the prime focus (which usually has a curved focal plane -- e.g., film can be curved to match the focal plane).

    • A telescpe using a lens/mirror combination is called a "catadioptric" system.
    • Their importance is that they give good images over wide field.
      • Schmidt telescopes are useful for efficiently mapping the sky.

    • The first Schmidt telescope was built in 1930 and the most famous was built in 1949 at Mt. Palomar, designed to map the sky and search for interesting targets in preparation for their exploration by the famous 200-inch telescope.

      Edwin Hubble guiding the Palomar 48-inch Schmidt telescope.

    • A variant on the Schmidt design, called a Schmidt-Cassegrain, is now a popular telescope design for amateur telescopes (like the 10-inch Meade in the doghouse and the 8-inch Meade telescopes we use in ASTR 130 night lab):

      These telescopes take advantage of the spherical primary mirror with the Schmidt correcting plate, but then introduce a secondary mirror that redirects the light to a Cassegrain focus:

      Here the design consideration is driven by economy:

      • Spherical primary is easy to make in high production.

      • Ironically, it is relatively easy to make the apparently complex corrector lens shape using "stressed workpiece technology".

        How to make a Schmidt corrector plate. First deform the thin flat glass plate against a steel template using a vacuum (second picture). Then grind the opposite surface flat (third picture). Then release the vacuum stress (bottom figure). Voila!

        From Wilson, Reflecting Telescope Optics II.

    • The result is a relatively inexpensive telescope with good image over wide field; long focal lengths are possible in this folded, compact package.

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    All material copyright © 2002,2006,2012,2015 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the University of Virginia.