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 3D representations and analyses of inclined and offaxis 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 offaxis
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 xaxis 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 onedimensional ray fan diagram, this is a twodimensional 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 3.5m 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 halfsilvered, 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, http://www.newastro.com/newastro/book/C2/chapter2_P5.asp.
 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 (aberrationfree) and aberrated aperture (0.25 wave peaktovalley 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 http://www.telescopeoptics.net/Strehl.htm.
 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 aberrationfree 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 twodimensional 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 http://www.gigapxl.org/technologyresponse.htm.
 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 firstorder, 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 multipleelement system.
 The advantage of multipleelement 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 n_{i} θ_{i} = n_{t} θ_{t} .
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, Taylorexpand 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 raytracing, 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 thirdorder 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  θ^{2} ( y / R )  θ^{2} N ^{ 1} 
field curvature (sag)  deform  θ^{2} ( y / R )  θ^{2} N ^{ 1} 
distortion  deform  θ^{3}  θ^{3} 
 The last four aberrations are offaxis aberrations produced by nonzero θ.
 Spherical aberration is not dependent on θ and so occurs even for onaxis sources (from
rays from the onaxis 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 twomirror 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 (TSA_{min})  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 Σ_{LC} in the Hecht figure above.
 These ideas are summarized in the spot diagrams and ray fan plots below, where Σ_{LC}
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 (aberrationfree) and aberrated aperture (0.25 wave peaktovalley 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 http://www.telescopeoptics.net/Strehl.htm.
Example of spherical aberration. From http://eckop.com/aberrations/.
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 ~ 1520%.
 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 1.3mm error in the placement of the device used to measure the
shape of the primary when being made by PerkinElmer.
 The result was a 38mm 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)
singledish radio telescope in the world, operating at 3cm to 6m 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 offaxis, "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.
Coma
Comatic aberration is an important image defect created by the fact that rays
from an offaxis 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 / (fratio) ^{2}.
 This means that the aberration is field dependent and increases linearly
with offaxis 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
fratio.
 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 "headtail" 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).
From http://astron.berkeley.edu/~jrg/Aberrations/node7.html.
 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 offaxis stars.
From http://eckop.com/aberrations/ .
Astigmatism
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 (socalled "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.
From http://astron.berkeley.edu/~jrg/Aberrations/node8.html.
 The image of an offaxis 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 crosslike pattern becomes stronger with increased degrees of astigmatism.
From Hecht, Optics, Fourth Edition.
 Recall that this aberration goes as θ^{2} y , and for
a full circular aperture, as θ^{2} / (fratio) .
 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 offaxis 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 offaxis image will simply look out of focus when compared to the unaberrated 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 http://eckop.com/aberrations/. 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 Σ_{p} in the above figure).
In the absence of the previous aberrations, the resulting focal surface
(the socalled 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 https://starizona.com/acb/basics/equip_optics101_curvature.aspx.
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 http://eckop.com/aberrations/.
Image of a flowering tree showing the effects of strong field curvature. From
https://www.bhphotovideo.com/explora/photography/tipsandsolutions/opticalanomaliesandlenscorrectionsexplained".
 Note that positive lenses have curvature towards the lens whereas
negative lenses have curvature away from the lens.
From http://foto.hut.fi/opetus/350/k03/luento4/luento4.html.
 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:
n_{1} f_{1} = n_{2} f_{2}
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 http://www.donzoptix.co.uk/ASTRO/CCD/CCD.html. 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 http://www.google.com.gi/patents/US6803988.
 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.
From http://astron.berkeley.edu/~jrg/Aberrations/node9.html.
 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 NOAOHydra multifiber spectrograph focal
plane, which gets warped to the
curved focal plane of the telescope focus. Photo by Meagan Morscher, from http://www.astro.wisc.edu/~morscher/index.html.
(Right) The curved focal plane slitmask for the IMACS spectrograph of the Magellan Telescope. Photo by Steven Majewski.
Distortion
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.
From http://astron.berkeley.edu/~jrg/Aberrations/node10.html.
 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 http://www.digitalphotosecrets.com/tip/1043/doprimelensestakebetterpicturesthanzooms/.
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
telescopeoptics.net
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 a_{n} 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 Z_{n} term, the higher the spatial frequency
and, usually, the lower the amplitude, a_{n} 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 Vshape 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.
From http://ottawa.rasc.ca/astronomy/astro_facts/j_dolland.html.
One can partially counteract chromatic aberration by creating achromatic doublets
from combining positive and negative lenses that counteract each other.
From http://hyperphysics.phyastr.gsu.edu/hbase/geoopt/aber2.html.
 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:
V_{Y} = ( n_{Y}  1 ) / ( n_{B}  n_{R} )
where n_{Y}, n_{B}, and n_{R}
are the index of refraction at some specific yellow, blue and red
wavelengths.
V_{d} 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 offaxis 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 fratio.
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.
RitcheyChretien 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 onaxis (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 offaxis 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 31inch Tinsley Reflector is a classical Cassegrain (but slow,
f/15.5 to increase the comafree 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 twohyperboloidal mirror arrangement, one could correct for both spherical
aberration and coma over a large field.
 Thus, the RitcheyChretien telescope is an aplanatic Cassegrain with
two hyperboloidal mirrors.
 The field of view of an RC Cassegrain is limited by astigmatism.
To first order, this is given (in radians) by:
RC astigmatism ~ θ^{2} / 2N
where N is the fratio.
This results in a larger field of view with good images than the classical
Cassegrain, but again slow optical systems are preferred.
 Though RC 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 200inch 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] worldleading 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 indepth 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 "offaxis" source "sees" the mirror the same
way as an onaxis 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 offaxis 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.
From http://members.shaw.ca/quadibloc/science/opt02.htm.
 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 offaxis 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 offaxis points  there are no offaxis 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:
 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.
 Insert field flattener.
Place a planoconvex 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 48inch 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 200inch
telescope, and a second version of the survey in the 1990s).
 SchmidtCassegrains with spherical primaries
are also now common designs for amateur telescopes,
like the 8inch 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 offaxis sources due to the presence of a stop.
Because the problem is worse as the offaxis 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 http://www.covingtoninnovations.com/dslr/vgnHyadesVignetted.jpg) and an extreme
example from a terrestrial image
(image from http://www.birdnet.co.uk/digital_tips.html).
From Rutten & van Venrooij, Telescope Optics.
 It can be shown (see homework) that to have no vignetting,
D_{M} = D_{C} + 2 D_{F}
 In general, this practice is not followed, and often what is done is
a compromise, tolerating some vignetting:
D_{M} = D_{C} + D_{F}
 Note that manufacturers of amateur telescopes generally make
D_{M} = D_{C} 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.
SchmidtCassegrain 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
SchmidtCassegrains, 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 40inch telescope,
which is of the BakerSchmidt design.
 Systems like this have a relatively large, aberrationcorrected 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.
NearFocal 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, RC and other Cassegrains.
Prime Focus and Cassegrain Correctors:
 The normally severely fieldlimited 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 100inch Newtonian
focus and used later at the prime focus of the Palomar 200inch.
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 RitcheyChretien telescopes,
for example.
 Example of the widefield corrector designed for the
MMT telescope to create very wide fields of view
(1 degree) for spectroscopy or imaging (0.5 degree).
From http://cfawww.harvard.edu/cfa/oir/MMT/MMTI/widefield.html.
 The RC 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 J_{1} is a firstorder 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 noncircular 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 2.5m shows
clearly the crosslike 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, disklike 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 http://www.dtm.ciw.edu/vonbraun/obs_mishaps/images/int_reflection2.html.
For some applications where faint surface brightnesses are sought, and such diffraction
patterns are detrimental, offaxis 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.
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.
