Previous Topic - Human Eye Lecture Index Next Topic - Aberrations

ASTR 5110, Majewski [FALL 2019]. Lecture Notes

ASTR 5110 (Majewski) Lecture Notes

Spherical Optics, the Paraxial Regime and Introduction to Telescopes

Some References:

  • Chapters 2-5 of Schroeder.

  • Chapter 5 of Hecht.

M.C. Escher, Hand With Reflecting Sphere.

A. Introductory Statements About Approaches to Optical Theory

A common function of an optical device is to collect and reshape a portion of an incident wavefront with the purpose of forming an image of the object.

Various approaches are taken to understand imaging optical systems.
  • In dealing with light as a wave, must account for both interference and diffraction effects. This can become quite complicated.

    E.g., Schroeder doesn't begin discussing this type of analysis in earnest until Chapter 10.

    When the wave character of light cannot be ignored, we call on the field of physical optics.

  • In limit where the wavelength of light approaches zero, we can use the techniques of geometrical optics -- as we have done in our analysis of the eye.

    In this simplified type of analysis, light is treated as moving along rays and the paths of these rays are manipulated by the positioning of reflecting/refracting surfaces following the laws of reflection/refraction.

    Ray tracing is the basis of many modern, computerized optical analysis programs, like, for example, ZEMAX.

    Geometrical optics is often a more straightforward means to understand many optical problems; e.g., geometrical optics is covered as early as Schroeder's Chapter 2.

  • A different perspective is offered by the variational calculus of Fermat's Principle.

    This is the subject of Schroeder's Chapter 3.

    Fermat's Principle states that the actual path that a ray follows from one point to another through an optical train is the one that minimizes the time of travel between the two points.

    Said another way, the time taken along a path of a ray between two points has no more than an infinitesimal difference from the time taken in other closely adjacent paths.

    A way to think about Fermat's Principle is by considering wave interference of adjacent paths and realizing that the highest transmission of light occurs where there is the least destructive interference (see discussion in Schroeder Section 3.3).

    • An example of the application of Fermat's Principle is the ray path shown in the Figure below (see also Schroeder Section 3.2):

      Figure and associated text from Hecht.

      Because the index of refraction for a medium, n, is defined as the speed of light in the medium compared to the speed in a vacuum, n = c / v, then the last equation is simply a statement of Snell's Law of Refraction:

      ni sin θi = nt sin θt

      Another way to describe the transit time is by the sum of the s / v in each medium, where s is the spatial path length in that medium and v is the speed of light in the medium.

      This sum of the s / v is called the optical path length (OPL) of the ray, and another way of stating Fermat's Principle is that light rays follow a direction whereby the OPL is minimized.

    • The By the way, the Law of Reflection follows if we take ni = -nt .

      Then θi = - θt .

B. Introductory Comments on Aspheres

In many optical applications we want to take diverging rays and make them parallel (i.e., collimate them) -- e.g., flashlights, projectors, searchlights.

While, e.g., with a telescope, we want to take collimated rays and bring them to a focus.

Let's look at what these types of situations imply for refracting and reflecting surfaces:

  • Refracting surface:

    • In the image below we have light emanating from a source S impinging on a glass surface.

      Because the index of refraction of glass is higher than air, the speed of the wavefront slows down in the glass and the wavefront flattens out (panel a).

      If we want the wavefront perfectly flat, we want the light rays to be collimated (panel b).

    • This is the same as saying all rays intersecting a vertical line (DD' ) are in phase at these intersections (panel c).

      Thus, for any point A on the surface of the glass, we want the rays leaving S at the same time and refracted at A to arrive at DD' at the same time (in phase).

      This is the same as saying that all OPLs through all possible A must all be equal:

    • This implies that:

      which is the definition of/equation for a hyperbola of eccentricity e = (nt / ni ) .

    • Other examples of refracting surfaces for similar transformations between diverging, converging and collimated wavefronts yield hyperboloidal or ellipsoidal surfaces:

  • Reflecting surface:

    • Similar arguments hold for reflecting surfaces.

      The telescope mirror takes collimated light from a distant source and brings it to a focus (F).

    • For this to happen, the OPLs for all rays coming from the wavefront must be equal:

    • Imagine the flat surface Σ, which is parallel to the wavefront, to the right of the surface.

      Obviously, we also have that:

    • This means that the mirror surface must actually have a shape so that for any point A we have length AD = FA , which is the definition of a parabola.

      The ideal mirror surface for a telescope to bring collimated, on-axis rays to a common focus is a parabola.

Clearly, commonly required optical surfaces are actually rotated conic sections, as was first suggested by Johannes Kepler.

Conic sections which are sections of rotation for optical surfaces. The numerical values shown are the conic constants, K = -e2 (see Schroeder Section 3.5). Note the size of an f/4 aperture corresponding to the vertex radius of curvature, R, for all of the conic sections shown. Faster mirrors would have a larger relative aperture for this same R. From Rutten & van Venrooij, Telescope Optics.

In applications that we are typically interested in, the conic section of choice is not a circle (rotated to create a spherical surface), which is just one case from a large family of possibilities.

The asphere (i.e., non-spherical surface) is described by a conic constant as follows:

  • The equation of a circle is, of course, given by:

    y2 + z2 = R2

    If we translate the circle so the origin of our coordinate system is coincident with the vertex of the optical system the equation of the circle is:

    y2 - 2zR + z2 = 0

    From Geary, Introduction to Lens Design.

    The darkened region symmetric about the optical vertex is the optical surface of interest.

  • Now, generalizing to the section of a conic asphere, we simply introduce the ellipticity into the equation:

    y2 - 2zR + (1 - e2) z2 = 0

    Note that e2 = (a2 - b2) / a2 where a is the semi-major axis and b the semi-minor axis of the conic section.

  • The vast array of possible conic sections are encapsulated in the conic constant (also known as the Schwarzschild constant ), K = - e2.

    The following table (Schroeder, p.41) describes the conic constants:

    Note there are two types of ellipses:

    • prolate ellipse -- the two focal points that define the shape of the ellipse lie on the optical axis (i.e., the horizontal in the figure above).

    • oblate ellipse -- the two focal points that define the shape of the ellipse lie on opposite sides of optical axis.

  • In the above equation, the constant R is the vertex radius of curvature (i.e., the radius of curvature at the optical axis of the optical element).

    • In the above figure, all of the conic sections are shown with the same R.

    • Note the similarity of the shapes of these different surfaces near the vertex.

      For a relatively small diameter optic, the surfaces would be indistinguishable (note the f/4 aperture shown).

C. Spherical Versus Aspherical Surfaces

We will spend much of our initial discussion of optics in the regime of spherical optics in order to derive some general themes.

But first: If the general shape of desired optical surfaces is aspherical, why bother spending considerable time on such a specific case as a sphere?

  1. Aspherical surfaces are typically very hard to make and to measure.

    • Thus, they are very much more expensive.

    • Use of aspherical optics is only relied upon when there is no other way to get what you want from spherical optics or when, in the long run, it is found to be cheaper to use aspheres over simpler surfaces.

      Often it is preferable to use multiple spherical elements when a smaller number of aspheres could accomplish the same optical effect.

    • The figure below shows why spherical surfaces are so much simpler to build: The glass grinding and polishing tool or lap, or (based on the substrate typically used) a pitch lap (here the bottom piece) need only be a sphere and the spherical surface desired (in the mirror blank in the person's hand) can be obtained by any random rotational, meridianal or azimuthal motion on this tool (when the two pieces are only separated by abrasive).

      From Hecht, Optics.


      As grinding proceeds, both surfaces become more spherical.

      As an aside, polishing is a long process, involving the use of successively finer grit (typically cerium oxide or carborundum = silicon carbide, which is almost as hard as diamond) and griding surfaces to achieve the required surface smoothness.


    • But the symmetry of the spherical lap is lost for aspheres:

      • Although there is azimuthal symmetry, each radial zone requires a different lap shape, conforming to the increasing local radius of curvature (for the parabola or hyperbola; see Schroeder Eqn. 3.5.6) from vertex to edge.

      • The radial curvature is no longer the same as the azimuthal curvature.

    • The first quality, mass-produced aspheres were for the Kodak disk camera in 1982 (and these glass lens elements were made in injection molds and were one part of high quality, but mass produced, multielement lenses that were corrected for spherical aberation).

  2. In many cases spherical surfaces are sufficient.

    • Note the convergence of the conic surfaces at the vertex in the above figure.

      The equivalent size of an f/4 aperture surface (corresponding to this particular vertex radius of curvature) is shown against this family of aspheres, and it may be seen how close all of their shapes are across this surface when they have the same vertex radii of curvature (as in the figure).

    • Faster optics (i.e., those with a larger diameter) show increasingly more deviation from the spherical approximation near the vertex.

  3. Our derivation of the optimal shapes above (e.g., the parabola) were for collimated rays coming in parallel to the optical axis.

    • But rays coming in significantly off-axis are no longer necessarily best served by parabolic surface (they have significant coma, as we shall see).

    • Spherical surfaces, while having significant spherical aberration, are free of other aberrations.

  4. To first order, we can describe many basic optical properties in the limit of a spherical surface.

    • In the paraxial approximation (i.e., when we focus on rays very near the optical axis -- see below) all of the surfaces have the same radius of curvature, R.

Making Astronomical Aspheres

Despite the difficulties in making aspheres, a number of advances have been made to make and polish them for astronomical applications.

This has been necessarily, in large part, because of the modern interest in very fast (low f/ratio) telescope objectives (which make telescopes more compact and inexpensive). As stated above, faster optics are less well approximated by spheres.

  • Computer-Controlled Optical Surfacing:

    • A computer controls a non-rotating but orbiting grinding tool.

    • The computer controls 6 motions of the tool: three positional, two tilt and one orientation (which is a fixed angle to the optical axis).

    • The lap actually wears itself to the correct shape as it moves over different radial zones.

    • Generally one needs limited-size laps to work well -- smaller laps imply longer time to polish a surface and these small grinding tools can create high spatial frequency irregularities if not controlled very well.

  • Membrane and strip polishing:

    • A flexible membrane carrying the polishing pitch moves back and forth under a strip of actuators that place the correct pressure for appropriate material removal at each radius.

      Meanwhile, the mirror blank spins under the moving membrane.

      From Wilson, Reflecting Telescope Optics II
  • Rotating furnace:

    • As it turns out, a liquid held in a rotating cylinder will, due to centrifugal force, flow outward, ultimately creating a parabolic surface (see homework problem.)

    • Applying this idea to molten glass, the Steward Observatory Mirror Lab, under Roger Angel, pioneered the casting of large, fast, paraboloidal mirrors in rotating furnaces.

      The huge facilities are located under the football stands at the UofA.

      The mirror casting furnace with one of the LBT 8.4-m mirrors inside. The furnace is 22 feet high and 39 feet across. From

      The mirror here is rotating at 7-8 RPM. From

      The upper half of the oven consists of the furnace floor (hearth) and a 28 ton, removable dome. The hearth and dome are lined with refractory tiles embedded with over 270 heating elements rated at 8,000 watts each making the oven the equivalent of 2,160 household space heaters! Over 600 thermocouples inserted into the furnace sense temperature during casting. From

      The bottom half of the furnace consists of a turntable that supports the hearth, and a 16 foot wide, instrumented, rotating pedestal. The furnace is turned by two 40 horsepower, DC servo motors at a rate particular to the mirror being cast. For the 6.5 meter mirrors, the rate was 7.4 rpm. Power is supplied by two connections (500 kW and 750 kW) to the Tucson Electric Power Company. There are also two 500 kW diesel generators for backup. All systems that control furnace rotation, heating, and monitoring are backed up by redundant systems. Roughly 1 megawatt of power is required at peak temperature. From

    • One advantage of this technique is that the mirror can be cast not only as a paraboloid, but as a meniscus, by using an appropriately shaped mold in the furnace.

      (Consider the glass disk shape that one would need to start with -- and then grind out -- with a normal, non-rotating furnace.)

      Other advantages of a thin, meniscus mirror will become obvious later.

    • The telescopes that UVa has access to that have mirrors fabricated in this way include the LBT 2x8.4-m, the MMT 6.5-m, the Magellan 2x6.5-m and the ARC 3.5-m.

  • Stressed-lap polishing:

    • Uses a large, circular grinding surface (for high rate of glass removal and natural smoothing over large spatial frequencies).

    • The lap is constantly stressed and deformed to proper shape by bending and twisting.

    • Method used by the Steward Observatory Mirror Lab to polish aspheres.

      Stressed-lap polishing the MMT f/5 secondary mirror at the Steward Mirror Lab. From

      Polishing of a 6.5-m mirror (the same ones used both in the Magellan and MMT telescopes) in the Steward Observatory Mirror Lab. The polishing machine here, called the "Large Optical Generator (LOG)" uses stress-lap polishing to make mirrors finished to a precision of 15-20 nanometers (λ/25 for green light).

    • The lap shape is driven by cross-connected actuators around periphery of polisher.

      Top view of 60 cm stressed lap, showing 12 actuators around plate periphery. Twisting and bending moments are produced by arranging the tension bands in sets of equilateral triangles. From

      Side view of the stress lap with one actuator shown. The electrical motor torque is converted to a linear force on the tension bands. The result is a curvature on the pitch surface. From

  • Stressed-workpiece polishing:

    • Deform the workpiece, e.g., against another surface like a steel plate (e.g., with a vacuum).

    • Grind the opposite surface as a flat or a sphere.

    • Release the stresses.

    • Used, e.g., in creating the Schmidt corrector plates for the Meade or Celestron telescopes we use for ASTR 1230 Night Lab, as shown below.

      From Wilson, Reflecting Telescope Optics II
  • Ion beam figuring:

    • Sputters material from the workpiece at the atomic level, by momentum of a directed ion beam bombarding the surface.

      From Wilson, Reflecting Telescope Optics II
  • Liquid mirror telescopes (LMTs):

    • Newton originally proposed using a rotating liquid (e.g., mercury) itself as a perfect paraboloid, but first done for 35 cm telescope in 1872.

    • Revived in last few decades (primarily by Canadian collaborations) as technical problems overcome.

    • Primary limitation is that liquid mirror telescopes can only look at zenith.

      Limits science to survey type projects, with drift-scan CCD imaging.

      A 2.7-m liquid mirror telescope in Canada. From
  • Primary technical challenge is suppression of ripples on surface from:

    • wind

    • vibrations

    • misalignment of rotational axis

    Air bearings are one modern solution to smooth, accurate rotation.

  • Primary practical problem is that mercury vapors and oxides are very toxic.

  • But a big advantage is cost: The Large Zenith Telescope, a 6-m LMT, was built at a cost of about $500,000 (which is about 1% the cost of a conventional telescope of similar aperture).

D. Some Basic Results of Paraxial, Spherical Optics

Paraxial optics, or Gaussian optics relates to the simplification of analyzing optical systems based only on rays near the optical axis of the system.

  • By focusing on paraxial rays, we can get reasonable results by first order approximations.

  • Normal discussions of optics start with this simplification because it allows us to understand some basic properties of optical systems, but ignores higher order effects that complicate the true images created by an optical system.

Aberrations are higher order effects introduced into the wavefront and are a critical aspect to understanding real optical systems.

Refraction/Reflection at a Spherical Surface and Derivation of the Mirror Equation

(See Schroeder Section 2.2-2.3.)

In the figure below we have a cuved surface and a horizontal line as the optical axis of that surface, intersecting it at point O.

The long-dashed line represents the normal to the (spherical) optical surface, and so, by definition, intersects the optical axis at C, the center of curvature. The radius of curvature is given by R.

Note the refracted (solid line) ray in this generalized figure showing refraction. This ray, nominally headed toward the focus point B at a distance s from the optical element while on the left side of the surface, is -- due to refraction -- actually brought to a second focus or image focus at the point B', which is at a distance s' from the optical element:

From Schroeder.

The point B is the conjugate point of B' and is the first focus or object focus of the ray (and is at distance s, as we normally have for the distance of the object from the optical surface). In this case it happens to be a virtual focus, because the incoming rays are converging to the optical axis in the image space, on the right, rather than diverging from the object space, on the left (see "human eye" webpage where the normal case of object focus is shown).

(Also note that in Schroeder's usual economy of description, one can also think of this as reflection from a spherical surface, if one thinks of the ray starting from the point B.)

From Snell's Law:

n sini = n' sini'

The paraxial approximation is defined as that regime when we can assume sini = i and sini' = i' .

This means that Snell's Law becomes:

n i = n' i'

Then, from simple geometry we have two triangles where:

i + u = φ, and i' + u' = φ

Then, inserting into the paraxial version of Snell's Law:

n ( φ - u ) = n' ( φ - u' )


n'u' - nu = (n' - n) φ

From Schroeder.

Then, applying the paraxial approximation again, we have approximately right triangles and:

φ ~ y / R , u ~ y / s , and u' ~ y / s'

and inserting these relations to the previous equation, we have:

n' y / s' - n y / s = ( n' - n ) y / R

n' / s' - n / s = ( n' - n ) / R

(Note that because of differences in sign definitions for s and for the angles here, some books -- like Hecht -- derive the above equation with a positive sign on the left side. Obviously this is fine as long as you adopt the same definitions for variables when you use those equations.)

Now, if the first focus (B) happens to be at infinity (i.e., we have an incoming collimated beam), then the distance along the optical axis s' of the conjugate focus at B' is the focal length of the surface, i.e., s' = f. The same holds for the other conjugate distance, s, when the second focus is at infinity.

From Schroeder.

Note that in the case of the mirror, once again we can take that n = - n' . This yields:

n' / s' + n' / s = ( n' + n' ) / R

and we derive the so-called Mirror Equation :

1 / s' + 1 / s = 2 / R

... and, assuming s = infinity, we learn that the focal length of a spherical mirror is given by f = s' = R / 2 .

This means that (collimated) light coming from infinity near the optical axis is brought to a focus at a point halfway to the radius of curvature of a spherical mirror.


Note that in the right side of the equation of refraction for a spherical surface (or the mirror equation) there is no reference to the object or image, only to the vertex radius of curvature and the index of refraction.

The invariance of the right side of the equation to the direction of the light makes it useful, and we define it as the power:

P = ( n' - n ) / R

If we recall the definitions of the first and second focal lengths (i.e., setting the focus points s or s' to infinity), we then find:

P = ( n' - n ) / R = n' / f ' = n / f

Defining power in this way (in a manner already familiar in our study of eyeglasses) is the starting point for analyzing multi-surface systems -- just as we found for the addition of lenses in our discussion of eyeglasses.

Two Surfaces

  • When we use multiple, widely spaced surfaces (as in a reflecting telescope) the geometry mimics that one would derive for the "thick lens" (as opposed to the "thin lens" approximation we have seen previously).

  • Thick lens (surface separation large compared to lens diameter) in air..

    Situation as shown, but with n1 = n2' = 1 and n1' = n2 = n :

    From Schroeder.

    From n' / s' - n / s = ( n' - n ) / R = P we have for each surface:

    n / s1' - 1 / s1 = ( n - 1 ) / R1 = P1

    1 / s2' - n / s2 = ( 1 - n ) / R2 = P2

    The incoming ray in the figure has s1 = infinity intersecting the first surface at y1 and the second at y2 .

    From two sets of similar right triangles, both sets including either y2 or y1 as one side,

    y2 / y1 = s2 / s1' = ( s1' - d ) / s1' = s2' / f '

    where f ' is the effective focal length of the whole system, and, as for our discussion of eyes, the net power is P = 1 / f ' .

    We can find the effective focal length by setting s1 = infinity and s2 = s1' - d in the equations for P1 and P2 and combining the results with the previous equation to solve for P = 1 / f ' :

    P = 1 / f ' = (1 / s2' ) [( s1' - d ) / s1' ]

    then one finds through substitution and algebra (see Schroeder Section 2.4.a) that the net power of the system is:

    P = 1 / f ' = P1 + P2 - ( d / n ) P1 P2

    This is the equation for a thick lens, or for a two-mirror system (like a reflecting telescope), or, as it turns out, two thin lenses in series.

  • Thin lens (surface separation small compared to lens diameter) in air.

    In this case, d is small, so s2 = s1' and we can directly coadd the equations for each surface given above (the two just below the last figure shown) to get the lens-makers formula :

    1 / s' - 1 / s = ( n - 1 ) ( 1 / R1 - 1 / R2 ) = P1 + P2

    and as might be expected, the net power of the thin lens is given by the combined power of its two curved surfaces as:

    P = P1 + P2 = 1 / f '

    And, as we discussed in the case of eyeglasses, this holds for the combination of multiple powered surfaces close to one another.

Objects and Images -- A Primer

For your benefit, here is a nice summary on how to do geometric optic ray tracing.

Transverse Magnification

Linear magnification or transverse magnification is simply the ratio of the image size to the object size, and we can sshow that it is proportional to the image focus and the object focus.

The ray in the following figure connects the conjugate points Q and Q' corresponding to object h and image h' .

Because this ray goes through the center of curvature, C, it is perpendicular to surface, and undeviated by it.

From Schroeder.

We describe the transverse magnification as the ratio of heights of the image to object:

m = h' / h

But h' = ( s' - R ) tan φ and h = ( s - R ) tan φ and this can be used to show (see Schroeder Section 2.2.b):

m = h' / h = ( s' - R ) / ( s - R ) = n s' / n' s

where the last bit comes from the application of the refraction at a spherical surface equation (the first red equation in Section D above) -- you should prove this to yourself (fun algebra!).

  • Since h and h' are in opposite z directions, m < 0 and the image is inverted .

  • For the thin lens case we have two surfaces and we have m1 = s1' / n s1 and m2 = n s2' / s2 .

    The net magnification is m = m1 m2 = s' / s .

  • For a mirror in air, m = - s ' / s .

Plate Scale

For astronomical sources at infinity (i.e., s = infinity), the previous definition of magnification is not meaningful.

When we use a telescopes to image onto a detector (or onto some other instrument, like the slit of a spectrograph), it is desirable to know how the angular extent of the sky maps onto the linear dimensions of our focal plane.

The conversion from angular extent to linear scale on the focal plane is called the plate scale (because of the traditional association with imaging the focal plane onto a photographic plate), or the focal plane scale.
  • The figure below shows the relevant ray trace diagram for rays coming from an astronomical source on the optical axis and for another at an angle θ from the optical axis going through a lens.

    Modified from Hecht.

  • Since the rays from the two sources going through the optical center vertex are undeviated, they preserve θ upon exit from the lens.

  • In the paraxial regime tan θ ~ θ ~ x / f .

  • Thus, the plate scale is given by θ / x = 1 / f .

  • Note that that it is common to quote plate scales in arcsec/mm, which is then given by scaling the above equation by 206265 arcsec / radian, so that

    θ / x = 206265 arcsec / f (mm).

Angular Magnification

The magnification most people are familiar with is angular magnification, i.e., enlarging the image of a source on the retina of the eye.

This type of magnification only makes sense with eyepiece viewing, since, as we have seen above, for a single optical element the incoming and outgoing angle for rays between two sources at infinity are the same.

  • Magnifying the appearance of terrestrial sources is the original function for which a telescope was invented.

  • Below is the ray diagram for a telescope used to view a (terrestrial) object at finite (but far) distance.

    From Hecht.

  • The first lens (the objective) forms a virtual (intermediate) image of the object just beyond its focal point.

  • The second lens (the eyepiece) uses its power to make a new, larger image of the intermediate image on the eye.

    The net appearance of the final image on the retina is larger than would have been possible than with an unaided eye.

  • Focus is usually achieved by moving the eyepiece in and out.

For astronomical sources, the sources are at infinity so the intermediate image forms at the focal length of the objective, fo.

From Hecht.

  • If the eyepiece is situated so that the virtual image is at its object focus, parallel rays will exit to be viewed by the normal eye.

    (If you are near or farsighted you can take off your glasses and adjust the eyepiece distance to have the rays diverge or converge to compensate as needed -- but if you have astigmatism, you should leave your glasses on.)

  • The telescope in this configuration (for the normal eye) actually has no net focal length since parallel rays in are turned into parallel rays out (it is afocal).

  • Now suppose that you want to look at a field of view subtending a half-angle α.

    From Hecht.

    Rays from the source extended through the optical center are undeviated, as usual.

    Rays that pass through the first focus of the objective (e.g., Fo1C in the figure) are passed parallel to the optical axis (CE) which then go through the second focus of the eyepiece (EFe2).

    Fo1 and Fe2 are conjugate points.

  • As may be seen, the apparent half angle at this point is αa , which is magnified:

    M = αa / α

  • Note that in the paraxial region

    α ~ tan α = BC / fo

    αa ~ tan αa = DE / fe = BC / fe

    So that

    M = αa / α = - fo / fe

    The magnification delivered by an eyepiece of focal length fe on a telescope with a focal length fo is given by the ratio of these two numbers.

But again, in normal astronomical usage we do not use an eyepiece, but, rather, record images in the focal plane of the objective.

Our primary interest in telescopes come in their:

  • light gathering power, defined by the area of the objective. We can intercept more light from a source than, for example, our eye. The larger the area, the more light collected.

  • resolution (abiility to see fine detail), where the larger the aperture, the smaller is the diffraction-limited image (ignoring atmospheric effects).

"One-Element" Telescope Systems

One element telescopes can use lenses (refractors) or mirrors (reflectors) for the single optical element (the "objective").
  • Refractors: Once the dominant telescope type, but rarely made today for professional use.

    Several disadvantages of refractors compared to reflectors explains their decline in use for large professional telescopes:

    • A lens must be transparent and free of internal flaws; a mirror doesn't (only the reflecting surface needs to be "clean").

    • A lens can only be supported around the rim, a mirror can be supported all along the back. The potential for sagging increases as the lens diameter increases.

    • A lens has chromatic aberration, a mirror doesn't.

    The world's largest refractor is the 40-inch Yerkes refractor in Williams Bay, WI (a place I was very lucky to have done my graduate work!).

  • Reflectors: The figure below shows the "prime focus" configuration in a single mirror telescope with focal length f = R / 2 in the paraxial limit.

    From Hecht.

    In large telescopes, the astronomical instrument receiving the light is positioned inside the telescope for "prime focus" work.

    In the days when photography was the dominant light detecting technology, the astronomer had to work the camera and ride along with it in a prime focus cage.

    (Left) Observer sitting in Hale 200-inch PF cage. (Right) Elevator to the Hale 200-inch PF cage.

  • Click here to see how the prime focus cage is used (photos from the last ever photographic run at the 200-inch prime focus by myself in 1995).

  • For smaller telescopes, just the detector is mounted at the prime focus, as, e.g., in the case of the Schmidt telescope (which is technically not a simple one-optic system...)

    From Optic/Optic/node9.html and

  • The Newtonian (invented by ole' Isaac) is a modified version of the prime focus, using a flat (i.e., unpowered) secondary mirror (often called the diagonal) to bring the beam more conveniently outside of the telescope.

    From Hecht.

  • Note that there is some loss of incoming light in the prime focus configuration and its variants by the presence of the prime focus cage, focal plane detector or Newtonian flat.

Two-Mirror Telescopes

By "two-mirror" here, I mean two mirrors with power (it is convenient to think of the Newtonian, with only one powered optic, as a modified prime focus configuration).

The thick lens formula can be applied to the two-mirror telescope.

Two primary types (both bring focus to behind the primary [i.e. first] mirror [the objective ]):

  • Cassegrain = convex secondary --> inverted image.

    From Schroeder; but note that the distance, d, is labeled wrong in this figure and should be the distance between the two mirrors, as defined earlier.

    A convex secondary effectively slows down the telescope, increasing its focal length by m = f / f1 ...

    ... making the system as if the primary has the same aperture but longer focal length.

    For example, note the net focal length, f and primary focal length f1 as shown in this figure.

    From Rutten & van Venrooij.

    Recall the power of such a system is given by:

    P = P1 + P2 - ( d / n ) P1 P2

    ... so that the final focal length is given by

    P = 1 / f = 1 / f1 + 1 / f2 - ( d / n ) / f2 f1

    and, realizing that (in the Schroeder convention) n = -1 for a mirror

    and d < 0, we have ( d / n ) > 0, or
    f = f1 f2 / ( f1 + f2 - |d | )

    • The net focal length of the system is given by the focal lengths of the two mirrors and their separation.

    • In the Schroeder convention, convex mirrors have P = 1 / f < 0 and concave mirrors have P = 1 / f > 0.

      Thus, the negative power of the convex Cassegrain secondary makes the net system power smaller.

    • Do not confuse the net focal length of the system with the back focal length (b.f.l.), or secondary-focal surface distance, which is the distance between the last optical element in the system (e.g., the secondary mirror) and the focal plane of the telescope, shown by f1 - s2 + f1 β in the Schroeder figure below and by bfl in the Rutten figure above.

    • It can be shown (see Schroeder Section 2.5.b) that b.f.l. = m k f1, where k = y2 / y1 is the ratio of the ray heights at the margins of the two mirrors (i.e., the relative mirror radii).

    • Note that the power of each mirror is given by the spherical mirror equation, P = 1 / f = 2 / R .

    From Schroeder.

  • Gregorian = concave secondary --> erect image.

    See lower panel (b) in Schroeder image above.

    Notice that the concave secondary "catches" diverging rays that have already passed f1, with the net result of producing a non-inverted (erect) image.

    With two concave mirrors, Gregorians were originally easier to make, but once convex secondaries became practical to make, Gregorians fell out of favor because they require longer overall telescope tubes (because of the greater d needed to let beam off primary get to divergence).

    But Gregorians are becoming popular again:

    • Concave mirrors still easier to make.

    • Gives access to a point in telescope where one can put a test source for alignment and figuring.

    • Easier to baffle against stray light.

    The LBT has Gregorian optical paths:

    The f/15 Gregorian optical path of the Large Binocular Telescope.

(Read Section 2.5.a of Schroeder carefully to understand the sign conventions of the magnification, focal length and power in two-mirror calculations.)

Telescope Ports and Multiple-Port Telescopes

Telescopes can have a variety of different optical configurations, and sometimes the same telescope can be reconfigured to send the beam to different ports (i.e., focal planes):

  • Multiple reflections shorten telescope length for long focal lengths.

    • Keeps telescope and telescope enclosure smaller, which....

    • ... saves money for telescope construction.

    • ... makes it easier to deal with thermal and mechanical stability.

    • Allows introduction of additional optical surfaces that can be used to modify the beam.

  • Same telescope (with fixed tube) can be reconfigured to different focal lengths.

    • As we shall see, different focal lengths lead to different "plate scales" (image sizes) and allows different fields of view or resolution for the same focal plane area.

  • Can have different instruments mounted at different "ports".

Traditional reflectors (e.g., Palomar 200", Kitt Peak 4-m) were often designed with at least three configurations possible (prime, cassegrain, coude), while the Palomar 200" also has a Newtonian.

This allows a variety of instruments to be mounted having different demands depending on:

  • desired entrance f/ratio or plate scale

  • instrument mass/size -- mechanical flexure concerns

  • desire for multiple instruments mounted at one time

    (possible quick changes by different types of secondary/tertiery mirrors put in beam).

The Mayall 4-m telescope at Kitt Peak. The telescope switches from prime focus to Cassegrain by flipping around the top end (the black section), which has a Cassegrain secondary on one side and a prime focus cage on the other. Note location and f/ratios of Prime Focus (f/2.7), Cassegrain Focus (f/8.0) and Coude Focus (f/160) in the righthand figure.

Note that one can even have different secondaries in Cassegrain mode for different desired f-ratios.

Typical configurations would be:

  • Prime focus:

    • Minimal number of surfaces (maximizes reflectivity and minimizes thermal emissivity).

    • Usually between f/1.5 - f/3.

    • Plate scale is large -- useful for wide field of view camera.

    • In fact, usually need a corrector for off-axis rays.

    • Because of placement, generally limited to smaller instruments.

  • Cassegrain focus:

    • Preferred focus for most equatorially-mounted telescopes.

    • Easily accessible and small number of surfaces.

    • Can handle larger, but not massive instruments.

    • Usually between f/7 and f/15.

    • Spectrograph, higher resolution camera, IR camera.

  • Newtonian focus:

    • Side port, must be counterweighted on opposite side if up the tube.

    • Can also send through declination axis (as shown below in Hale 200-inch).

    • Spectrograph or wide field of view camera, or something with a design that precludes fiting it at prime focus.

    • Not as popular today.

  • Coude focus:

    • Very long focal length Cassegrain focus with about 4 to 7 fold mirrors (i.e., powerless, flat mirrors that help to "fold" the beam) located in tube and polar axis in equatorially-mounted telescope.

    • Sends focus to a fixed place (the Coude room).

    • Can mount very large/massive instruments, especially those where stability desirable, like a VERY high resolution spectrograph.

    • Typically f/30 to f/200.

    • Field of view very small -- few arcseconds -- and field rotates. No imaging.

    • Number of optical surfaces means lower throughput.

    • Falling out of use because:

      • fiber optics can now deliver light to bench-mounted spectrographs.

      • more alt-az mounted telescopes being built.

  • Nasmyth focus:

    • A Cassegrain focus ported out to the elevation axis of an altitude-azimuth mounted telescope by a flat mirror (like the Palomar Newtonian "through the declination trunnion focus" shown below, but in this case projected through the altitude bearing).

    • Advantage of Nasmyth focus over Newtonian is that it is a stable one (doesn't change angle with respect to gravity field/floor -- moves only with azimuth).

      Allows placement of heavy instruments on well-supported platforms.

      Length of telescope tube can be less of an issue at Nasmyth focus.

      (Top Left) ARC 3.5-m telescope with view of Nasmyth platform. (Top Right) Mike Skrutskie's CorMass spectrograph mounted to ARC 3.5-m Nasmyth focus. (Bottom Left) CorMass mounted to Nasmyth of Magellan 6.5-m Clay telescope. More of the stable Nasmyth platform here is visible. (Bottom Right) Schematic view of the Nasmyth focus (from
    • Do not have to rebalance telescope with different instruments.

    • As we shall see at APO 3.5-m, facilitates quick changes of instruments.

    • Disadvantage of Nasmyth focus over Newtonian is that focal plane rotates, and at variable rate -- must have instrument rotator in the beam or ignore rotation.

      Latter option possible for stellar spectroscopy.

Schematic view of Keck Telescope with Nasmyth platforms shown. From

EXAMPLE: Palomar 200-inch telescope

  • Multiple focus ports:

    Click on the above image to see a cut-away view of the Palomar 200-inch telescope. Four separate optical paths are illustrated (and this does not include that going to the Newtonian focus, which could not be shown in this representation). Note that two possible Coude configurations are shown, depending on the declination of the source. The various f/ratios are given on the lower left, just above the word "TWO" in "THE TWO HUNDRED INCH TELESCOPE".

    The fifth focus possibility on the 200-inch is a Newtonian focus through the declination axis bearing. The observer can mount a large instrument in the huge yoke arm of the telescope.

  • Note the problem of the Prime Focus: It is in the beam of the telescope and therefore more difficult to access.

    The prime focus was traditionally used with the astronomer riding inside the telescope through the night. Now the astronomers can use cameras that are remotely operable.

    Note on the images: These are part a series of classic drawings of the 200-inch made by Russell Porter, considered a genius of mechanical drawing. These drawings, which are deadly accurate on what the 200-inch looks like, were drawn from the blueprints before the telescope was even built! Click here for the complete set of these drawings, which are considered "masterpieces" of the art. Permission of California Institute of Technology.

Previous Topic - Human Eye Lecture Index Next Topic - Aberrations

Unless otherwise attributed, material copyright © 2005, 2007, 2009, 2011, 2013, 2015, 2017, 2019 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 511 and Astronomy 5110 at the University of Virginia.