Last Topic - Spherical Optics Lecture Index Next Topic - Telescopes

ASTR 5110, Majewski [FALL 2017]. Lecture Notes

ASTR 5110 (Majewski) Lecture Notes

Telescope Optics II: Aberrations, Diffraction Effects and Image Quality

Some References:

  • Chapters 4, 5, 7, 9, 10 of Schroeder, Astronomical Optics.

  • Chapter 6 of Hecht, Optics.

  • Chapter 7 of Geary, Introduction to Lens Design.

  • Chapter 4 of Bely, The Design and Construction of Large Optical Telescopes.

Edvard Munch's The Scream

A. Spot Diagrams, Ray Fans and Other Definitions

To understand optics beyond the simple spherical and paraxial regime, need to become familiar with 3-D representations and analyses of inclined and off-axis rays.

There are various ways of doing this.

  • In the figure below imagine the entrance pupil of an optical system (think of it as the objective of a telescope) and the rays from an off-axis object point going through it.

    From Geary, Introduction to Lens Design.

  • The chief ray is the ray coming from the object that passes through the center of the pupil (sometimes called the central ray in Schroeder).

  • The plane defined by the optical axis and the object is the tangential plane (or the meridianal plane), and the rays in this plane from the source define the tangential ray fan.

    Rays in the perpendicular plane are in the sagittal plane.

  • One can then map where different rays intercept the entrance pupil:

    (distance from the optical axis normalized against the radius of the aperture stop)

    ... against where the rays intercept the image plane:

    (relative to the position of the chief ray).

    Such a plot is shown schematically in panel (b) above.

  • Were the system unaberrated, all rays from one point on the object would land at the same place as the chief ray and the ray fan plot would be just the x-axis of the plot.

    These plots help to demonstrate how the aberrations work.

    A separate plot could be made for the sagittal plane.

The spot diagram is another way to simulate the size, shape and light distribution in the image surface (plane).

  • Instead of one-dimensional ray fan diagram, this is a two-dimensional representation.

  • Impose a grid of points over the entrance pupil and launch rays from source to each point on grid.

    From Rutten & van Venrooij, Telescope Optics.

  • Plot where the rays pierce a plane perpendicular to the optical axis (typically the image plane is of interest).

    From Geary, Introduction to Lens Design.

The following plot shows both of the above representations for a system without aberrations.

From Geary, Introduction to Lens Design.

Note that these need not be hypothetical constructs, but can be realized with real optical surfaces and a Hartmann mask or Hartmann Screen.

Hartmann screen for the WIYN 3.5-m telescope. Image from National Optical Astronomy Observatory/Association of Universities for Research in Astronomy/National Science Foundation and Copyright WIYN Consortium, Inc., all rights reserved.
  • For a large optic -- like a telescope mirror -- one needs a collimated beam just as large.

  • An alternative method is to feed the telescope with a diverging beam from the center of curvature and view result (e.g., by diverting the outgoing beam with a half-silvered, flat mirror -- like a Netwonian focus).

    From Bely, The Design and Construction of Large Telescopes.

    Note, this methodology can usefully be implemented in a Gregorian reflector for the secondary mirror (often denoted as "M2") in place on the telescope, which is one strong motivator for the Gregorian configuration.

      For the simple Cassegrain, M2 is convex -- doesn't help.

      For both Cassegrain and Gregorian, can't get to center of curvature for the primary ("M1") and have to pull M2 out.

Yet one other way to think about aberrations is what they do to the wavefront.

  • Imagine a wavefront from a source being focused to an image point P.

  • In a perfect system, after passing through the optics, the wavefront should be spherically converging on P and creating an Airy pattern.

  • If there are aberrations in the surface, the wavefront is deformed, and rays are deviated away from their path toward P.

  • Lord Rayleigh determined that a surface produces a noticeably degraded image when the wavefront aberration exceeds λ / 4 (usually thought of in terms of the RMS).

    The problem with this "Rayleigh rule" is that the effect of the wavefront error on the point spread function strongly depends on the type of aberration.

From Hecht, Optics.

There are various ways to describe the perfection of an optical system based on their effect on the image quality.

  • The Full Width at Half Maximum (FWHM) of the point spread function (PSF).

    From The New Astronomy,

    • Most commonly used image descriptor used by optical astronomers (often as a descriptor of the "seeing").

    • Good measure of image size but doesn't give much info on the wings of the PSF.

  • The 80% Encircled Energy (EE) is the angular diameter containing 80% of the energy of the PSF.

    • In a perfect system (no aberrations, no atmosphere, no central obscuration) with a circular aperture (i.e., a perfect Airy disk) of diameter D, 80% of the energy is in diameter ~ 1.8 λ / D.

    • Obviously, depends on λ.

    • Can obviously talk about the EE for any level of energy or any particular radius, and plot the EE as a function of radius, as below.

    The Point Spread Function (PSF), whose peak intensity determines the value of Strehl ratio (see below), and encircled energy (EE) of a perfect (aberration-free) and aberrated aperture (0.25 wave peak-to-valley of spherical aberration), as a function of diffraction pattern radius, given in units of λF. In the presence of aberrations, the energy is spread wider, thus the energy encircled within a given pattern radius diminishes. An encircled energy figure like the above can be used to map where the power of the light is distributed in the PSF, and can indicate possible problems with light being taken out of one part of the PSF and scattered to elsewhere (e.g., energy lost from the central Airy disc and being put to larger radius). Figure taken from and caption adopted from

  • The Strehl ratio is the ratio of the peak intensity of the delivered PSF to that expected for a perfect system (i.e., the expected peak of the theoretical Airy disk center).

    • In the figure above, the black PSF has a Strehl ratio of 0.8, and the lost power in the central Airy disk can be seen to be disbursed into the wings of the profile. Still, this PSF is considered relatively good (by the rule of Marechal, below).

    • It can be shown that the Strehl ratio of a system is given approximately by (for ratios > 0.5):

      where ΔΦ is the RMS wavefront error in terms of wavelengths.

    • The rule of Marechal says that an essentially aberration-free system should have Strehl ratio > 0.8.

    • Rule of Marechal is equivalent to a λ / 14 wavefront error.

    • You can go here to find John Graham's calculator to convert from Strehl ratio to a wavefront error.

    • You can go here to find John Graham's calculator to convert from a wavefront error to a Strehl ratio.

  • Our friend the Modulation Transfer Function (MTF) of course gives information on the image quality since the MTF is the amplitude of the OTF, which is the Fourier transform of the PSF.

    • The advantage of working in terms of the MTF is that a system's MTF is the product of the MTFs of its components.

    • Can be used in a two-dimensional form, but often simply use an azimuthal average.

    • MTF becomes zero at the "cutoff spatial frequency", which is the ultimate resolution of the system ( λ / D).

      Recall the MTF of the eye from before:

      A comparison of the MTF of the eye versus an optical CCD camera. The CCD MTF has been normalized to the eye at MTF=0.5. From

    • Since the PSF of a perfect system with a circular aperture can be predicted (Airy function), so too can its Fourier transform.

      MTF(ν) = (2/π) ( Φ - cos Φ sin Φ ), with Φ = arccos (λ ν / D),

      where ν is the spatial frequency (cycles per radian).

      The MTF of the present Hubble Space Telescope optics (solid line), compared to that for an ideal optical system. The abscissa is normalized to the cutoff frequency. From Bely, The Design and Construction of Large Optical Telescopes.

B. Third Order Theory and the Seidel Aberrations

We know that the first-order, spherical optic, paraxial treatment from before was only an approximation.

  • Departures from idealized Gaussian optics are known as aberrations.

Aberrations are of two main types:

  • Chromatic aberrations -- due to wavelength variation of index of refraction.

  • Monochromatic aberrations -- independent of wavelength, and fall into two types.

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

    • Aberrations that deform the image -- stretching it (generally without loss of information/resolution).

These aberrations can be inherent to one optical element or 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 use of additional elements.

The paraxial approximations assumed that for all angles, sin θ ~ θ, so that, for example, Snell's Law can be given by ni θi = nt θt .

Breaks down with rays that either:

  • are incident to the aperture at large distance, y, from optical axis (e.g., marginal rays around the edge).

  • are incident at large angle, θ, to optical axis.

For more accurate treatment of angles, Taylor-expand the sin θ :

If we take the first two terms of this expansion, we have third order theory.

  • The resulting aberrations were studied by mathematician Philipp Ludwig von Seidel in the 1850s and so are known as the Seidel aberrations. (A lunar crater is named in his honor.)

  • Higher order aberrations are still present, and can be mapped with ray-tracing, but the primary aberrations can be understood with third order theory.

  • In third order theory, there are five primary aberrations.

    As might be expected, these aberrations involve polynomial terms of third-order involving y (the distance from optical axis at the entrance pupil -- which is most often the objective -- of radius R ) and θ.

    (Here for clarity we write the f - ratio as N .)

    Primary Monochromatic Aberrations and Scaling Relations

    name type scaling law for given ray scaling law for full telescope
    spherical deteriorate ( y / R ) 3 N - 3
    coma deteriorate θ ( y / R ) 2 θ N - 2
    astigmatism deteriorate θ2 ( y / R ) θ2 N - 1
    field curvature (sag) deform θ2 ( y / R ) θ2 N - 1
    distortion deform θ3 θ3

  • The last four aberrations are off-axis aberrations produced by non-zero θ.

  • Spherical aberration is not dependent on θ and so occurs even for on-axis sources (from rays from the on-axis source hitting the aperture off of the vertex).

  • The following image summarizes the nature of the aberrations.

    From Bely, The Design and Construction of Large Telescopes.

Spherical Aberration

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

  • Parallel rays from a source at infinity hitting different radii do not converge to same point (unless reflected from a parabolic mirror), and the image gets blurred.

  • In two-mirror systems, can get spherical aberration from mismatch of conic constants of primary and secondary (which are rarely tested as a system when figured -- i.e., when polished into final shape).

  • As in our discussion of chromatic aberration in the eye, we can define transverse spherical aberration (TSA) and longitudinal spherical aberration (LSA).

    From Hecht, Optics.

  • The envelope of the rays is called the caustic.

  • The TSA (defined on the paraxial focal plane) is given by:

    where R is the radius of curvature, K is the conic constant, and ρ is the height of the ray from the optical axis (defined as y above).

    The marginal rays have the largest TSA (as shown in the equation and in figure below) and the largest LSA (as shown in the figure).

    From Schroeder, Telescope Optics.

    Clearly, one can reduce the TSA by stopping down the aperture (i.e., bring the available optical surface to a more nearly paraxial regime), at a loss of total light through the system.

  • The place where the blur of the image will be minimum (TSAmin) -- i.e., where the caustic is narrowest -- is known as the circle of least confusion.

    Obviously, it lies between the paraxial and the marginal focus.

    It is shown as the plane ΣLC in the Hecht figure above.

  • These ideas are summarized in the spot diagrams and ray fan plots below, where ΣLC lies at position (c):

  • The net effect of spherical aberration is to take light out of the central Airy disk and put it into the rings (reducing the Strehl Ratio).

    • Imagine where the rays from the aberrated parts of the wavefront shown below will end up in the focal plane.

    From Hecht, Optics.

    Even λ / 4 optics reduces the irradiance of the central image by ~20%, as shown in this figure that we already saw above, for an optical system aberrated to λ / 4 by spherical aberration:

    The Point Spread Function (PSF), whose peak intensity determines the value of Strehl ratio (see below), and encircled energy (EE) of a perfect (aberration-free) and aberrated aperture (0.25 wave peak-to-valley of spherical aberration), as a function of diffraction pattern radius, given in units of λF. In the presence of aberrations, the energy is spread wider, thus the energy encircled within a given pattern radius diminishes. An encircled energy figure like the above can be used to map where the power of the light is distributed in the PSF, and can indicate possible problems with light being taken out of one part of the PSF and scattered to elsewhere (e.g., energy lost from the central Airy disc and being put to larger radius). Figure taken from and caption adopted from

    Example of spherical aberration. From

Example: Hubble Space Telescope

  • Soon after HST was put into orbit (1990), astronomers discovered that they could not find a good focus of its images (the circle of least confusion was very ugly).

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

    Strehl ratio ~ 15-20%.

  • Too much energy was in the Airy rings and in a diffuse light halo.

    Classical situation of spherical aberration.

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

  • Later it was determined that the HST primary was polished to exquisite precision, but to an incorrect shape!

    Too flat at margins by ~ λ / 2.

    The 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 (removing the high speed photometer).

  • COSTAR inserted a pair of small mirrors (10 mm and 30 mm) into beam, one of which is a flat and the other an aspheric surface that:

    • added an inverse λ / 2 spherical aberration to the marginal ray paths, and

    • redirected the corrected beam to the other remaining HST instruments (the Faint Object Camera, the Faint Object Spectrograph, and the Goddard High Resolution Spectrograph).

    (Left) The COSTAR instrument being readied. (Middle) How COSTAR removes the residual spherical aberration before feeding the HST beam to other instruments. (Right) UVa professor Kathy Thornton as part of the 1993 Hubble Servicing Mission 1 using the Endeavour Space Shuttle places COSTAR into HST axial instrument bay.

  • COSTAR restored >70% of the energy in the central disk, which increased the faintness limit of HST by several magnitudes and with much cleaner images.

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

  • NASA also replaced WF/PC1 with WF/PC2 (a spare instrument built on the ground), which had its own corrective optics within its relay mirror system:

  • COSTAR was removed in the fifth HST servicing mission in 2009 because eventually all HST instruments incorporated their own corrective optics.

Example: Arecibo Telescope

Arecibo (in Puerto Rico) is the largest (1000 ft. primary) single-dish radio telescope in the world, operating at 3-cm to 6-m wavelengths.

  • Because it is not steerable, the primary is spherical, so that a wide range of directions can be accessed by just moving the "prime focus" receiver to point at different pieces of its spherical face from the same fixed position above.

  • To quote Hecht: The telescope surface is then " `omnidirectional' but also equally imperfect in all directions".

  • The electronic schemes for removing spherical aberration were inefficient, so in 1997 the telescope was upgraded with an all off-axis, "Gregorian" concave secondary/tertiary system that compensates for the spherical aberration using a chain of aspherical surfaces.

    Shapes of the two added "mirrors" are configured to make the OPLs of all rays equal at the receiver.

From Hecht, Optics, Fourth Edition.


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

From Hecht, Optics, Fourth Edition.

  • Note that, like spherical aberration, coma is worse for rays traversing the objective farther from the optical center, but what the rays do is different:

    • For spherical aberration rays on opposite sides of the center cross the focal plane on opposite sides of the chief ray.

    • For coma, rays on opposite sides of the optical center land on the same side of the chief ray.

  • The reason why coma is a more insidious problem is that it asymmetrically affects the image, as seen in the spot diagram and ray fan diagrams below (the ray fan plot functions shown are all negative, for example).

  • Recall that this aberration goes as θ y 2, and for a full circular aperture, as θ / (f-ratio) 2.

    • This means that the aberration is field dependent and increases linearly with off-axis angle, θ.

    • Because of the y 2 dependence, rays from opposite sides of the optical center land on same side of chief ray.

    • The aberration depends strongly (as the inverse square) with the f-ratio.

    • Of course, one can limit coma by stopping down the aperture, but at the expense of less light through the system.

  • Coma has a very characteristic "head-tail" appearance that gives rise to its name (i.e., like a comet).

    • The coma shape comes from a differential transverse magnification for different ray cones coming from the optical surface.

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

    • In addition, the larger the ring, the more distant is the center of the comatic circle from the position of the chief ray.

    • In the figure below, the outer ring (unprimed numbers) is the ring from the marginal rays.

      Slightly more than half of the energy goes into that part of the image in the triangular region between the chief ray (0) and the point 3.

      From Hecht, Optics, Fourth Edition.

    • The image below defines the tangential coma and the sagittal coma.

    • The tangential coma is three times larger than the sagittal coma (see Schroeder).


  • Coma can also appear when optical elements in a telescope are misaligned (tilted or not coaxial with one another).

    • In this case the additional coma is "field independent", having the same amplitude throughout.

    • This type of coma is easily identified (how?).

    Image demomstrating images with and without the effect of coma on off-axis stars. From .


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 (so-called "refractive astigmatism" -- a poor name since the astigmatism we are discussing here can be caused by refractive elements of course).

The astigmatism here happens for symmetric lenses, and is sometimes referred to as oblique astigmatism or marginal 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 (sagittal to tengential) 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 (i.e., nominally an Airy pattern, caused by the diffraction of the entrance pupil) will look differently at different points along the optical axis.


  • The image of an off-axis point source near the circle of least confusion looks something like an Airy disk, but with an increasing biaxial asymmetry of spokes (corresponding to the meridianal and sagittal planes) as the level of astigmatism grows.

    This cross-like pattern becomes stronger with increased degrees of astigmatism.

    From Hecht, Optics, Fourth Edition.

  • Recall that this aberration goes as θ2 y , and for a full circular aperture, as θ2 / (f-ratio) .

    • This aberration is thus strongly dependent on θ.

    "If stars are imaged through a lens with astigmatism, as shown in Figure 1.14, the image shape will depend on which plane of focus was selected as the image plane. If the image plane is positioned on the tangential focus, the off-axis image of the stars would resemble lines tangent to an imaginary circle centered on the optical axis. When the image plane is positioned at the medial focus the off-axis image will simply look out of focus when compared to the un-aberrated image. As for the case when the image plane is positioned at the sagittal focus, the image of the stars will resemble short line segments that are positioned on lines that go through the optical axis." Caption and image taken directly from Here medial focus means "at the circle of least confusion", as shown in the Geary image above.

Field Curvature

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

  • The figure shows that outside of the paraxial region the mapping from object to image conjugate points actually follows spherical surfaces (we assumed planes before).

    From Hecht, Optics, Fourth Edition.

  • If one flattens the object surface to a plane, then the focal surface must curve even more (shown as Σp in the above figure).

    In the absence of the previous aberrations, the resulting focal surface (the so-called Petzval surface) is paraboloidal.

    The sag of the surface scales in the same way as astigmatism.

  • If one is using a flat detector and the field curvature is significant compared to the desired area of the focal plane, one cannot have the entire field of view in focus.

    Significnt field curvature will cause problems when imaging onto a flat detector. From

    Stars imaged through a system with Petzval field curvature showing images in focus in the center of the field, and with the focus incresinging blurry near the edges. "This is very similar to the image at the medial focus of a system suffering from astigmatism." Image and modified caption from

    Image of a flowering tree showing the effects of strong field curvature. From".
  • Note that positive lenses have curvature towards the lens whereas negative lenses have curvature away from the lens.


  • One can find suitable combinations of positive and negative mirrors or lenses that exactly counteract one another and make the net sag = 0.

    You can show that this happens when the optical elements satisfy the Petzval condition:

    n1 f1 = n2 f2

    This is the philosophy behind field flattener lenses.

    Typically used with "air space" field flatteners, which are placed in the beam with a gap to the focal plane:

    From The theory behind making such devices was worked out by Joseph Petzval, a Slovakian/Hungarian mathematician, physicist, and inventor who was is a founding father of geometrical optics and the study of optical aberrations and who was trying to solve the problem of field curvature in opera glasses, portrait cameras.

  • A simple kind of single element field flattener, useful right near the focal plane, might look something like the following (see also examples shown below in Schmidt telescope discussion).

    A simple field flattner that lies close to th focal plane. Note the following key difference of this image ant the previous one. In the previous one (from donzoptix) the two lenses shown are part of the field flattener; note the incoming light on the left in that image is already converging from the telescope objective. In the image just above the left lens is the objective itself. From

  • If astigmatism is present there will be separate surfaces for the sagittal and tangential foci.

    • The tangential focal surface always lies 3 times farther from the Petzval surface as the sagittal focus (and on the same side).

    • With no astigmatism, the two planes merge at the Petzval surface.


  • If one cannot remove the field curvature, an alternative is to have a curved detector in the focal plane.

    • For imaging, this is very hard to do with CCDs of course, but photographic glass plates or gelatin films can be curved like this, either by pressing them against a properly shaped hard surface (for backward curving field) or by sucking them with a vacuum against this surface (for inward curving surface).

    • For spectroscopy, the focal plane is the point where one places slit masks or fibers, and these usually involve metal plates that can be curved to the Petzal surface shape.

      (Left) The circular metal plate with the fibers across it shown here is the NOAO-Hydra multifiber spectrograph focal plane, which gets warped to the curved focal plane of the telescope focus. Photo by Meagan Morscher, from (Right) The curved focal plane slitmask for the IMACS spectrograph of the Magellan Telescope. Photo by Steven Majewski.


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.

  • Varies as θ 3 and does not depend on where rays intercept optic.

  • 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.


  • In astronomy, not usually not that much of a problem, because this can usually be calibrated out.

  • But can be a problem in terrestrial photography:

    Barrel (left) and pincushion (right) distortion in photos of buildings. Barrel distortion always happens in wide angle lenses, but the extreme ends of the focus of a variable zoom lens can introduce either pincushion or barrel distortion (the latter happening at the wide end of the zoom). Photos from

C. Wavefront Representation of Aberrations

(See Schroeder Section 10.3.)

We used geometrical optics to describe the Seidel aberrations.

But, as described above, we can also think of aberrations in terms of their effects on the wavefront.

    From Bely, The Design and Construction of Large Optical Telescopes.

  • Each of the above aberrations can be described in terms of a characteristic phase difference pattern imposed on the idealized, spherical, converging wavefront.

    From Bely, The Design and Construction of Large Optical Telescopes.

  • These (and other) wavefront error patterns can be described in polar coordinates, r (the radius normalized to the outer radius of the converging cone) and φ (the angle of the meridian):

    These common wavefront error patterns, Z(r, φ) are known as the Zernike polynomials.

    Here is an image from a very nice description of Zernike polynomials at that shows the definition of the polar coordinates (though using ρ as the name of the radial variable) in (1), a demonstration of the shape of the wavefront along an optic diameter for several aberrations in (2), Zernike representaions of the first three Seidel aberrations in (3) and a map of the wavefront deviations around the edge of the optic for several Seidel aberrations in (4):

  • It is common to then describe an aberrated wavefront, W(r,φ) by means of this set of orthogonal Zernike polynomials, as follows:

    where the an are the coefficients of the different wavefront error components making up the total aberrated wavefront.

  • The following table gives the Zernike polynomials up to 5th order for a circular aperture with no central obscuration, with their associated aberration type.

    (Schroeder Table 10.6 shows the same up to 3rd order for an annular aperture.)

    (Note that there are other ways of writing these polynomials -- e.g., the Seidel polynomials -- that are also used.) One uses linear algebra to solve for these terms in matrix form.

  • The higher the Zn term, the higher the spatial frequency and, usually, the lower the amplitude, an of the wavefront error.

    The first 20 terms are generally sufficient to describe the major, classical wavefront errors imposed by the optics.

    They cannot practically describe irregular aberrations such as those imposed by atmospheric turbulence or microroughness in mirror surfaces, which would require a very large number of terms.

    This website gives more detailed information on the formulation, usefulness and dangers of relying on Zernike polynomials. (2011: This website is now password protected, unfortunately.)

  • By taking images of the pupil of the telescope, one can see wavefront differences and diagnose and correct problems in the mirror shape.

    For example, the Large Binocular Camera (LBC) on the Large Binocular Telescope uses analysis of the pupil images derived from slightly out of focus stars to assess how to reshape the mirror to account for basic wavefront problems.

    A gallery of LBC images showing a variety of pupil images with differently aberrated wavefronts can be found here. (The V-shape notch in the images is the shadow of the secondary mirror truss for the LBT.)

    Note the use of Zernike polynomials to describe the nature of the aberration and to send information to the mirror support system on how to adjust the mirror actuators to adjust the mirror shape.

    And here is a handy cartoon cribsheet explaining how to correct the telescope mirror shape using "by eye active optics". Note how the aberrated image corrections correspond to the Table 4.2 shown above.

D. Chromatic Aberration and Achromats

We discussed chromatic aberration in reference to the human eye, but let's formally introduce the subject here for completeness, and to preface additional applications in the creation of telescope optics.

  • Cause is the variable index of refraction as a function of wavelength.

  • For a positive lens, more power for bluer light, which focuses before redder wavelengths.


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 longitudinal chromatic aberration to join together, for example a blue and a red wavelength.

    From Rutten & van Venrooij, Telescope Optics.

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

    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...).

    The chart below shows the variety of possible types of glass offered by one manufacturer (Schott), categorized by:

    • the index of refraction (at 588 nm -- the wavelength of a Na line), shown on the ordinate.

    • the dispersion, which is the difference in refractive indices between two wavelengths (in this case 486 and 656 nm -- the Hβ and Hα lines), on the abscissa.

      The latter property gives some idea of the prismatic dispersive power of the glass.

      A common way of defining this dispersive power is, for example:

      VY = ( nY - 1 ) / ( nB - nR )

      where nY, nB, and nR are the index of refraction at some specific yellow, blue and red wavelengths.

      Vd involves the three specific wavelengths named just above.

    From Rutten & van Venrooij, Telescope Optics.

    Note that because crown glass is more resistant to wear, it is usually used in the outermost element of a telescope objective.

  • With additional elements, can bring more wavelengths parfocal -- one additional wavelength with each element.

    In the following example showing the focal length differences as a function of wavelength, the C, d and F are wavelengths corresponding to the lines named above -- Hα, Na D and Hβ, respectively.

E. Monochromatic Aberration Compensation

The last example of correcting chromatic aberrations demonstrates a general rule of thumb for optics:

  • Generalized Schwarzschild theorem:

    "One can generally correct n primary aberrations with n reasonably separated, powered optical elements."

Many modern telescopes employ this principle to correct out some aberrations.

  • In many cases, this involves inserting refracting elements in the beam to correct aberrations introduced by the primary mirror.

  • Such combined reflecting (catoptric) and refracting (dioptric) systems are called catadioptric.

Recall that a paraboloidal mirror fixes spherical aberration.

  • Thus, we can take out the first Seidel aberration, but then are left with the rest -- i.e., all of the aberrations attendant to off-axis sources (coma, astigmatism, etc.).

  • Because of these other problems, uncorrected prime focus reflectors (and Newtonian reflectors -- which are a simple variant of the prime focus with only 1 powered element) tend to have relatively small fields of view within which images of good quality may be seen.

  • Coma is the dominant aberration limiting field of view in a single mirror (or Newtonian) system.

    The length of the coma is given roughly by:

    Coma ~ 3 θ / 16 N 2

    where N is the f-ratio.

    Thus only very slow mirrors will really deliver anything like an acceptable field of view.

  • Several methods have been used to fix this problem in reflecting telescopes.

Ritchey-Chretien Cassegrain Telescopes

With two mirrors, one can create an aplanat -- a system free of both spherical aberration and coma.

  • The principle upon which combining conic sections is based is the theorem of Apollonius: The normal to a conic surface always bisects the angle formed by rays from the two foci joining the intersection of the normal and the surface.

    From Bely, The Design and Construction of Large Optical Telescopes.

    This means that, based on the simple law of reflection, all optical rays from a source at one focus will always reflect to the other focus to form a stigmatic (i.e., perfect) image of the source.

  • When joining two conic sections, if the second conic is placed with one of its foci coincident with one of the foci of the first surface, the second surface will reimage stigmatically to its other focus.

    Recall the typical paraboloid/hyperboloid combinations for a classical Cassegrain and classical Gregorian from before:

    From Bely, The Design and Construction of Large Optical Telescopes.

    But here stigmatic images are only formed on-axis (because one focus of a parabola is at infinity, requiring rays to come in on axis to form a stigmatic image).

    Thus, the classical Cassegrain still has coma and other off-axis aberrations, but less so because the equivalent f/ratio is slower than just the primary due to the magnification of the secondary mirror.

  • The Fan Mountain 31-inch Tinsley Reflector is a classical Cassegrain (but slow, f/15.5 to increase the coma-free field).

  • The Magellan Telescope is a classical Gregorian (f/11 at Nasmyth port and f/15 at Cassegrain port).

  • Each of the sides of The Large Binocular Telescope is a classical Gregorian (f/15 at beam combining port). Also see here.

It was proposed by Schwarzschild (in 1905) and developed by Ritchey and Chretien (in 1910) that if one slightly departed from a paraboloidal/hyperboloidal combination to a two-hyperboloidal mirror arrangement, one could correct for both spherical aberration and coma over a large field.

  • Thus, the Ritchey-Chretien telescope is an aplanatic Cassegrain with two hyperboloidal mirrors.

  • The field of view of an R-C Cassegrain is limited by astigmatism. To first order, this is given (in radians) by:

    R-C astigmatism ~ θ2 / 2N
    where N is the f-ratio.

    This results in a larger field of view with good images than the classical Cassegrain, but again slow optical systems are preferred.

  • Though R-C systems are the most common choice for large Cassegrains these days, typically additional optics are added to correct for astigmatism. Some relevant examples (to UVA students):

Interesting historical aside, from Wikipedia: "Ritchey intended for the 200-inch Hale Telescope to be an RCT. His design would provide sharper images over a larger usable field of view. However, he and Hale had a falling out. Hale refused to adopt the new design, with its complex curvatures, and Ritchey left the project. (Given the large delays in construction, Hale could be forgiven for some amount of risk aversion.) Ritchey would later be vindicated, as the Hale telescope turned out to be the last [sic] world-leading telescope to have a parabolic primary mirror." -- Of course, this is no longer a true statement, given the parabolic mirrors being delivered by the Steward Observatory Mirror Lab.

Schmidt Telescopes/Cameras

(See Schroeder Chapter 7 for much more in-depth coverage of Schmidt telescopes.)

The basic principle of the Schmidt optical system is to make use of certain advantages of a spherical primary and correcting out its primary disadvantage (i.e., spherical aberration).

  • We saw in the example of the Arecibo telescope that an advantage of a spherical surface is that any sky direction to the surface sees the same (spherical) surface.

  • If we adopt the vertical of the Arecibo telescope as the optical axis of the telescope, we see that an "off-axis" source "sees" the mirror the same way as an on-axis source.

  • The same principle can be applied to an optical telescope as shown below.

    • In the normal telescope -- whether with a spherical or paraboloidal mirror -- there are coma and other off-axis aberrations because the entrance pupil is the mirror itself.

      • This can be seen in the left panel below: An oblique beam of parallel rays has no axis of symmetry with respect to the mirror, and forms the image to one side of the incoming beam.

      • The image will not be symmetrical (will have coma and astigmatism).

    • Now imagine placing a pupil stop at the center of curvature of a spherical mirror (right panel in above figure).

      • Every bundle of rays passing through this stop, whether it is parallel to the optical axis or not, has its own axis of symmetry.


      • This axis of symmetry is the chief ray (defined by the center of the aperture stop), which always hits the mirror normally to the surface.

      • All rays from an off-axis source are symmetric about the chief ray, which acts like its own optical axis.

      • Thus, there is (in principle) no coma or astigmatism.

        (In effect, because there are no off-axis points -- there are no off-axis aberrations.)

      • The result is a system with a potentially enormous field of view.

  • Thus, with an appropriate pupil and spherical mirror we have removed two aberrations.

    But we are left with spherical aberration and curvature of field.

    Correcting these will leave us with a system that can take perfect images over an enormous field of view.

    • The important advance formulated in 1929 by Bernhard Schmidt (while sailing across the Indian Ocean on an eclipse expedition) was the introduction of a thin corrector plate to correct out the spherical aberration.

    • The shape of the surface is a toroidal curve.

      • Rays hitting the thinnest part of the plate (the neutral zone) are passed without deviation.

      • Rays hitting the corrector outside the neutral zone see a negative power lens that bends the rays slightly outward.

      • Rays hitting the corrector inside the neutral zone see a positive power lens that bends the rays slightly inward.

      Thus, all rays are brought to the same focus.

    • This leaves curvature of field, which is usually dealt with either by:

      1. Bending the detector.

        Easily done with thin photographic plate glass, when curvature not too large.

        Can only be approximated with a mosaic of flat CCD chips.

        The radius of curvature of the film is half the radius of curvature of the mirror.

      2. Insert field flattener.

        Place a plano-convex lens with plano side in contact with film or near CCDs to flatten field.

      The following series of drawings from Hecht show the progression of design principles for the Schmidt. The first and last panels demonstrate the curvature of field.

      Left figure From Hecht, Optics, Fourth Edition, showing the progression of logic in the creation of a Schmidt telescope.

      Similar to the Hecht figure, but from Optics and Vision, by Pedrotti & Pedrotti.

      The right panels are from Palomar Observatory, showing (top) Hubble using the Palomar Schmidt, and below that, the Russell Porter drawing of the Palomar Schmidt, looking down into the telescope and showing the plate holder.

  • The first Schmidt telescope was built in 1930 and the most famous was built in 1949 at Mt. Palomar (see picture with Edwin Hubble above).

    • The Palomar 48-inch Schmidt can take clear images over 6 degrees across (compared to fractions of a degree from normal Cassegrain telescopes).

    • It is an f/2.5 telescope.

    • It was used to create the two Palomar Observatory Sky Surveys (one in the 1950s to find targets for the new Palomar 200-inch telescope, and a second version of the survey in the 1990s).

  • Schmidt-Cassegrains with spherical primaries are also now common designs for amateur telescopes, like the 8-inch Meade and Celestron telescopes we use in ASTR 130 night lab:

    Here the design consideration is driven by economy:

    • Spherical primary is easy to make in high production.

    • With stressed workpiece technology, easy to make the corrector plates.

  • Of course, since a Schmidt telescope introduces a refracting element, it introduces some chromatic aberration.

    • This is small because the power is low.

    • Can be minimized by appropriate location of the neutral zone in corrector.

  • Note that the diameter of the primary mirror should be larger than that of the corrector, or you will have vignetting.

    • Vignetting is the narrowing of the bundle of rays coming from off-axis sources due to the presence of a stop.

      Because the problem is worse as the off-axis angle increases, the result is a gradual fading of the image at the periphery of the focal plane.

      Two examples of vignetting, shown in an astronomical image of the Hyades (image from and an extreme example from a terrestrial image (image from
      From Rutten & van Venrooij, Telescope Optics.
    • It can be shown (see homework) that to have no vignetting,

      DM = DC + 2 DF

    • In general, this practice is not followed, and often what is done is a compromise, tolerating some vignetting:

      DM = DC + DF

    • Note that manufacturers of amateur telescopes generally make DM = DC to save cost at expense of considerable vignetting.

      From Rutten & van Venrooij, Telescope Optics.
The Schmidt design is also used for cameras, and are often the imaging end of astronomical spectrographs.

Schmidt-Cassegrain Telescopes

The Schmidt revolution in catadioptrics inspired a number of derivatives.

  • Soon afterward, various optical designers sought ways to improve access to the image plane, by using convex secondary mirrors between the primary and corrector.

  • The primary can be an asphere or a sphere, depending on desired field of view and f/ratio.

  • As a class, these catadioptric telescopes are often called Schmidt-Cassegrains, and there are many types, depending on the relative positions and shapes of the two mirrors and Schmidt corrector.

    • By using a corrector and a pupil stop, can use the mirrors to correct other aberrations.

    • A goal is try to obtain an anastigmatic system (one that is free of spherical aberration, coma and astigmatism).

    • One example is represented by the Fan Mountain Observatory 40-inch telescope, which is of the Baker-Schmidt design.

    • Systems like this have a relatively large, aberration-corrected field (the FMO telescope has a flat angular field diameter more than a degree), but the large central obscuration means significant loss of effective light gathering power (and actually lowers the contrast and sharpness of images).

Maksutov telescopes were invented because the Schmidt corrector plate was found to be too hard to make in the 1930s.

  • It was found that a deep, but spherical meniscus lens can correct the spherical aberration of a spherical mirror by introducing the opposite spherical aberration.

Near-Focal Plane Field Correctors

(See Schroeder Chapter 9.)

Field correctors are lenses or combinations of lenses placed in the converging beam of the telescope, typically right in front of the focus.

They enlarge the usable field of the instrument by correcting existing aberrations without introducing other serious ones.

These are almost universally used in all telescope types.

Field Flatteners:

  • We have already seen how field flatteners can remove curvature of field aberration.

    The lens increasingly lengthens the OPL radially outward.

    From Rutten & van Venrooij, Telescope Optics.

    If the original radius of curvature is r, the radius of curvature of the field flattener is given by:

    R = r ( n - 1 ) / n

    where n is the index of refraction, and there is no astigmatism in the system.

    Need to place very close to focal plane to limit chromatic aberrations.

  • Often used for Schmidt, R-C and other Cassegrains.

Prime Focus and Cassegrain Correctors:

  • The normally severely field-limited prime focus and Cassegrain fields can be substantially corrected (e.g., for coma, astigmatism) by using correctors.

  • The Ross corrector was designed in 1935 for the Mt. Wilson 100-inch Newtonian focus and used later at the prime focus of the Palomar 200-inch.

    This corrector has no power but removes coma.

    From Rutten & van Venrooij, Telescope Optics.

  • The Wynne triplet corrector uses lenses with all spherical surfaces to mate to either paraboloidal or hyperboloidal primaries.

    Thus, useful to correct the prime foci of Ritchey-Chretien telescopes, for example.

  • Example of the wide-field corrector designed for the MMT telescope to create very wide fields of view (1 degree) for spectroscopy or imaging (0.5 degree).


  • The R-C Cassegrain is already corrected for spherical aberration and coma, so it is of interest to fix astigmatism and field curvature.

    Can correct astigmatism and some field curvature with an aspheric plate resembling a Schmidt plate placed in the converging Cassegrain beam.

F. Diffraction Effects

We have repeatedly visited the concept of the point spread function (PSF) produced by an optical system.

Up to know we have always discussed the PSF in terms of that produced by a perfect optical system with a circular aperture.

  • In the case of the perfect optical system with circular aperture and monochromatic light, the PSF is given by the Airy function:

    where J1 is a first-order Bessel function, C is a constant and x = π D θ / λ .

  • Rayleigh/Sparrow criteria.

    From Bely.

However, a variety of aspects of the telescope can degrade the telescope PSF.

  • In a Cassegrain telescope, the aperture is obstructed by the secondary.

    • The PSF of the annular entrance pupil is:

      where ε is the ratio of the obscuration diameter to the aperture diameter.

    • The result is a reduction of the Strehl ratio and more energy at larger radii.

      From Bely, The Design and Construction of Large Optical Telescopes.

    • As may be seen in this example from HST, the central obscuration reduces the MTF at lower angular frequencies, meaning that one has reduced contrast for low frequency (large scale) features.

      The MTF of the present Hubble Space Telescope optics (solid line), compared to that for an ideal optical system with no central obscuration and no wavefront errors. The abscissa is normalized to the cutoff frequency. From Bely, The Design and Construction of Large Optical Telescopes.

    • Segmented mirrors may have non-circular shapes, changing the PSF shape in two dimensions.

      From Bely, The Design and Construction of Large Optical Telescopes.

    • Secondary support vanes will create diffraction spikes.

      The following are views of the observed and modeled actual complex pupil functions of HST.

      • Note the diffraction off of the secondary vane supports and the three primary mirror support pads.

      • Note also the annular patterns of microroughness of the mirror, leftover from its polishing.

      From Bely, The Design and Construction of Large Optical Telescopes.

      Typical image of a star from a Cassegrain focus, showing the (Fourier transform) result of the secondary support vanes:

      The diffraction spikes from this image of an overly exposed star from the du Pont 2.5-m shows clearly the cross-like pattern created by the secondary support vanes, slightly rotated counterclockwise. The extra vertical "spikes" are actually a problem caused by the CCD pixels being supersaturated, leading to blooming of charge straight up the CCD column. The faint, disk-like halo around the star is the result of multiple internal reflections between the CCD chip, the dewar window and the filters in the camera. From

      For some applications where faint surface brightnesses are sought, and such diffraction patterns are detrimental, off-axis design telescopes are considered:

      From Bely, The Design and Construction of Large Optical Telescopes.

    • Dust and mirror surface defects scatter light and create faint halos around sources, and basically "fill in" the dark Airy rings (especially at shorter wavelengths).

      From Hasan et al. (1995), PASP, 107, 289.

      Last Topic - Spherical Optics Lecture Index Next Topic - Telescopes

      Unless otherwise attributed, material copyright © 2005, 2007, 2009, 2011, 2013, 2015, 2017 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 511 at the University of Virginia.