Early Telescopes & Geometrical OpticsC. A. Griffith, Class Notes, PTYS 521, 2016

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Fig. 1.— Snell’s law indicates the bending of lightat the interface of two materials, shown here for twosubstances such that n2 > n1 and v2 < v1.

1. A Few Things about Light

Here we discuss some concepts and definitions ofbasic geometrical optics and the construction of earlytelescopes.

1.1. Refraction and Reflection

The field of classical optics is dominated by twolaws. The Law of Reflection states that when lighthits a reflecting surface, the ray will be reflected fromthe surface at an angle that is equal to the angle of theincident ray. The Law of Refraction concerns the factthat light moves slower in material with higher den-sity. As a result when light passes from one mediumto another, the angle that the light beam makes withthe normal to the interface between the two mediais less in the medium where light travels slowest, i.e.θ2 < θ1 in Fig. 1. The index of refraction is definedas the ratio of the speed of light in a vacuum, c, tothe speed of light in a particular medium, vp:

n = c/vp

Snell’s law relates the apparent bending of lightcaused by the change in the velocity of light as itcrosses the boundary between two mediums of indicesn1 and n2:

n1 sinθ1 = n2 sinθ2,

where θ1 and θ2 are the angles from the normal of theinterface for the incident and refracted waves (Fig. 1).

The index of refraction, or the speed of light ina medium, depends on the wavelength of the light.Prisms take advantage of this property and cause lightto be dispersed as a function of wavelength.

1.2. Image Formation

Parallel rays that pass through a lens with con-vex parabolic surfaces or that reflect from a parabolicconcave mirror are brought to a focus. The distancefrom the lens (or mirror) to the focus is called the fo-cal length, f (Fig. 3). Conversely, concave lenses andconvex mirrors cause light to diverge as if emanatingfrom a point. Converging lenses produce real images

of distant objects, i.e. that which can be capturedon film. When these are combined with an eyepiecelens, they produce a virtual image, that is one that isproduced at an infinite distance away (Fig. 4). Whencombined with other lenses, such as the lens in oureye a real image is formed on our retinas. Note thata series of lenses can bring light to a focus; in thatcase the focal length is defined as that of a single lensthat would bring light to a focus at the same angle ofconvergence.

The focal ratio, or f-number (written as f/# is:

F = f/D,

where f is the focal length and D is the diameter ofthe objective lens. A smaller focal ratio indicates astronger curvature of the lens and is called a “faster”lens.

1.3. Human Eye

Consider your eye. It consists of a lens that focuseslight at the retina, or the back of the eye. The retinais basically a detector array of 125 million photore-ceptor cells over a curved focal plane. The iris dilatesand contracts to adjust the effective diameter of thelens, which then changes the focal ratio from aboutf/3 to about f/8. In additional it squeezes the lensinto different shapes to adjust its focal length. Theseadjustments are made to adapt to the distance of theobject that your eye is intent upon so that the retina(where the photons are read) lies in the focal plane ofthe lens.

2. Telescopes

2.1. Refracting Telescopes

In 1608 a Dutch optician noticed that distant ob-jects appear larger when viewed through a combina-tion of two lenses, a weak (long focal length) converg-ing lens and a strong (short focal length) diverginglens.

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Fig. 2.— The focal lengths of convex and concavemirrors and lenses. Note that real images are pro-duced only by convex lenses and concave mirrors.

This “spyglass” was subsequently used by GalileoGalilei and is known as the Galileo telescope. Galileodeveloped this telescope further so that it reached amagnification of 30. With this he investigated theplanets and stars. He detected mountains on themoon and the presence of Jupiter’s moons. His ob-servations indicated unequivically that Venus orbitedthe Sun instead of the Earth and that stars extendinto the distance, since more of them can be seenwith a telescope. Thus he eradicated the Ptolomyand Plato universes where perfect spheres orbit theEarth (only), and all stars sit on a perfect orb thatsurrounds us (an excellent demonstration of how tech-nology revolutionizes astronomy).

In 1630, Kepler replaced the diverging lens with

Fig. 3.— Galilean Telescope intercepts the convergingrays from the primary, rendering them parallel again.These rays however subtend a larger angle from theoptical axis, thereby increasing the angular size. Theimage is magnified.

Fig. 4.— Kepler’s telescope is similar to the Galileantelescope, yet of course the image is upside down.

a converging lens, to make a Kepler Telescope. Oneof the largest of such telescopes is the Lick telescope,which has a 1 meter objective lens. Note how longthe telescope is. A long focal length allows for highmagnification, while a short focal length allows for agreater field of view (Fig. 6).

2.2. Lenses vs Mirrors

Lenses are difficult to make such that they are ho-mogeneous and have a perfect curvature. In addition,large objectives sag under their own weight and ab-sorb light preferentially in the blue and violet. It iseasiest to make lenses of spherical curvature, as op-posed to parabolic curvature; however, the former ofdoes not focus perfectly as does the latter. The result-ing spherical aberration occurs because parallel raysfalling on that fall on the lens at different distancesfrom the optical axis are focused at different focalpoints. Lens also focus light of different wavelengthsat different focal lengths; this effect is called a chro-

matic aberration. Other problems with lenses include

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Fig. 5.— The Lick 1 meter telescope first saw lightin 1888. The telescope is long because it needs to ac-commodate 17.37 m focal length of the primary lens.

the occurrences of comas and astigmatism.

The drawback of reflectors is that there is often agreater loss of optical intensity because the surface isnot 100% reflective. In addition, the surface deterio-rates and must be aluminized every several years.

2.3. Reflecting Telescopes

These problems point to reflecting telescopes, mosttypical of most modern telescopes. If we replace theobjective lens with a concave mirror, we eliminate thechromatic aberration. If we use a parabolic mirror, weeliminate the spherical aberration. Alternatively wecan add a corrective lens in front of objective, calleda Schmidt plate, that focuses all the parallel light atone point. Telescopes that use this device are calledSchmidt telescopes.

Newton designed a telescope with a concave pri-mary mirror and a convex eyepiece. This designwas improved by his french contemporary, GuillaumeCassegrain, who placed a convex mirror in the opti-cal axis, thereby extending the effective focal lengthof the primary, and improving the angular resolution

Fig. 6.— A Cassegrain configuration. Note the posi-tion of the prime focus, Fo, which has a shorter focallength that the combined primary and secondary mir-rors, which collaborate to put the focal plane at Fext.

of the telescope Light was reflected back through ahole in the primary, where it met the eyepiece (Fig.7).

Note that the eyepiece plays the role of taking thereal image produced by the primary and producingan imaginary image at infinity, which is made into areal image on your retina using the lens of your eye.In professional astronomy, eyepieces are not used. In-stead detectors may be placed e.g. at the Cassegrainfocal plane or at the prime focus that is producedwhen the secondary mirror is removed.

3. Angular Magnification

The angular magnification of a telescope is the ra-tio of the angular extent of an object as seen throughthe telescope, divided by the angular extent of theobject without the telescope. Note that, referring toFigure 5, the angular magnification, M is:

M =αIM

αOB

.

Thus from figure 5 we can see that we have two tri-angles that share one side. In the small angle approx-imation, tan(α) = α, and

M ∼

fofe

.

For a system with a detector array at the focalplane, there is of course no eyepiece. The angularmagnification of the telescope is quantified in terms ofthe image scale or the plate scale. This is the numberof arcseconds, θ, that are imaged onto 1 mm of thedetector, and, as evident from figure 5 (e.g. takingthe image I’ to be 1 mm):

tan(θ) =1(mm)

fo (mm).

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From the small angle theory:

θ (radians) ∼1

fo(mm)

But we need, by definition, the angle in arcseconds

θ (arcseconds) =206, 265

fo(mm)

Note that the plate scale depends only on the focallength of the telescope. Thus an observation thatrequires more angular magnification would best beperformed with an instrument on the Cassegrain focalplane rather than at the prime focus.

The field of view of a telescope depends also onthe linear size, L, of the detector. If you refer back tofigure 5, and put a detector at the focal plane of theprimary lens, this is kind of equivalent to putting itat the primary focus of a Cassegrain. Then you cansee that :

tan(θ) =L/2

fo,

and, from the small angle approximation, the full fieldof view is:

θ(radians) =L

fo.

The field of view is larger, for a given detector, whenthe focal length is smaller. Thus an observation thatrequires a greater field of view would best be per-formed with an instrument at the prime focus of aCassegrain telescope. Note that compared to theCassegrain focal plane, at the primary focus morephotons from an extended object are detected at eachpixel, for the same integration time.

3.1. Specific Telescopes

Let’s consider the Subaru Telescope on top ofMauna Kea at 13,580 ft in Hawaii. It’s quite ver-satile. It has an 8.2 m diameter primary, and fourfoci. At the prime focus (f/2) instruments such asFMOS, the fiber-fed multi-object spectrograph, takeadvantage of the wide field of view. FMOS has a 30arcmin field of view and can simultaneously record0.9–1.8 µm spectra of 400 objects with a resolvingpower of R=600. At the Cassegrain focus (f/12.0),and the Nasmyth Focus (Optical) and Nasmyth Focus(IR), with f/12.6 and f/13.6, higher spatial resolutionmeasurements are possible. For example the FOCASinstrument, placed at the Cassegrain focus, has a 6arcmin field of view, and a 0.104 arcsec pixel scale.

Fig. 7.— The layout of the Subaru Telescope, withits four foci.

3.2. Diffraction and Interference

In 1803, Thomas Young performed his famous ex-periment that demonstrated interference from lightpassing through two closely spaced slits (Fig. 2). Hededuced that light behaves like waves.

In 1961, Clauss Jonsson of the University of Tubin-gen performed the 2-slit experiment for electrons. Hefound, oddly, the if one bombards 2 slits with elec-trons the result is a diffraction pattern. Well, youmight say, maybe the electrons interact with eachother bouncing off each other to make such a pat-tern. In 1974 Pier Giorgio Merli from LAMEL-CNRBologna, sent electrons through the slits “one at atime”. Still there was a diffraction pattern. OK, soperhaps the electron interacts with itself. Well, if youput a device that measures which part of the electronpasses through which slit, you will see only one elec-tron pass through the slit. And.. and this is reallybizarre, you don’t see the diffraction pattern. Yourobservation of the electron made it act like a particle!

Any wave (or fraction of an unseen electron) willbend when it passes through an aperture that is nar-

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Fig. 8.— Young’s sketch of two slit diffraction, whichhe presented to the Royal Society in 1803.

row relative the light’s wavelength. This effect ofdiffraction causes interference of light that crosses theaperture at different places. That is light from oneside of the aperture combines with light from anotherside of the aperture leading to a greater light signal,i.e. constructive interference, if the light is in phase,or, no light signal, i.e. destructive interference if thelight is perfectly out of phase. This superposition of

light gives rise to a central peak and a series of fringesof light and dark bands on the other side of the aper-ture. One can see this diffraction pattern more clearlyif one studies the pattern produced by the superpo-sition of light emerging from two narrow slits, as didThomas Young. Bright bands occur when the pathdifference between two waves is a whole number ofwavelengths. Dark bands occur with the path differ-ence is a odd number of half wavelengths. Thus thegrating disperses light according to its wavelength, asdoes a prism. The diffraction pattern would appearsharper if a large number of equally spaced slits areused. We will find that these effects are importantin the understanding of how spectrographs work, andwhat ultimately limits the spatial resolution of a tele-scope.

3.3. Diffraction Limit

Because of diffraction, the image of a point sourceon a detector is never a point. Even a telescope inspace with a perfect and parabolic primary mirror willrecord a point source as an extended source, called apoint spread function, which consists of a central peaksurrounded by concentric ripples (Fig. 9). This “airydisk” pattern resemble the bands formed from lightpassing through a single slit, but for two dimensions.Roughly 97.75% of the light is contained in the centraldisk, which extends to the first dark ring at an angular

distance of

sin θ = 1.22λ

D.

The second and third minima, containing a fraction of0.0175 and 0.0042 of the light, encircle the disk at anangular distance of 2.233 λ

Dand 3.238 λ

D. Note that

the PSF is extended, even in the absence of terres-trial atmospheric distortions and aberrations in theprimary mirror or lens.

Ultimately, the diffraction of light by the tele-scope’s aperture establishes the limiting point spreadfunction (PFS). The theoretical limit of the angularresolution of a telescope is set by diffraction can bedefined as the angular distance between two sourceswhen the minimum of one source’s PSF coincides withthe peak of the other source’s PSF (Fig. 10).

The diffraction limit, or limiting angular separa-tion of two point sources in the sky, was thus esti-mated by Rayleigh to be:

sin θ = 1.22λ

D

where λ is the wavelength and D is the telescope di-ameter. It is easy to calculate this value analytically,especially if you like to play with Bessel functions, butlet’s move on instead.

The calculation of the Airy Disk pattern is stemsfrom the Fraunhofer diffraction equation with theuse of Bessel functions. In general the blocakge oflight can be modelled using the HuygensFresnel prin-ciple. Huygens postulated that every point on a pri-mary wavefront acts as a source of spherical secondarywavelets and the sum of these secondary waves de-termines the form of the wave at any subsequenttime. Fresnel developed an equation using the Huy-gens wavelets together with the principle of superpo-sition of waves, which models these diffraction effectsquite well.

It is not a straightforward matter to calculatethe displacement given by the sum of the secondarywavelets, each of which has its own amplitude andphase, since this involves addition of many waves ofvarying phase and amplitude. When two waves areadded together, the total displacement depends onboth the amplitude and the phase of the individualwaves: two waves of equal amplitude which are inphase give a displacement whose amplitude is dou-ble the individual wave amplitudes, while two waveswhich are in opposite phases give a zero displacement.Generally, a two-dimensional integral over complex

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variables has to be solved and in many cases, an an-alytic solution is not available.

The Fraunhofer diffraction equation simplifies theproblem by modeling the light diffracted when boththe light source and the viewing plane are effectivelyat infinity with respect to the diffracting aperture. Inthis case, the incident light is a plane wave so that thephase of the light at each point in the aperture is thesame. The phase of the contributions of the individualwavelets in the aperture varies linearly with positionin the aperture, making the calculation of the sum ofthe contributions relatively straightforward in manycases.

Thus for yellow light (λ=0.5 µm) a 1 meter tele-scope has diffraction limit of

θradians ≈ sin θ = 1.22 · 5× 10−7, or

θarcsec ≈ 206, 265 · 1.22 · 5× 10−7 = 0.126 arcsec,

while a 10 meter telescope has a diffraction limit of0.0126” in yellow.

Now we can ask what is the size of the pixels thatwe need to resolve a diffraction limited image on thedetector which sits at the focal plane. We can ap-proximate the small angle θ as:

θradians ≈x

f,

where f is the focal length and x is the separation oftwo images at the focal plane such that the objectsare just resolved. Then,

x

f≈ 1.22

λ

D

andx ≈ 1.22 · λ · f/#.

So let’s consider our new nifty camera. At a typicalsetting of f/8, the pixels spacing that corresponds tothe diffraction limited resolution of blue light (0.42µm) of the camera is x = 1.22 * 8 * 0.42 µm = 4.1µm. So it would be a bit silly to make the pixelsof the CCD smaller than this value, for it would notincrease the image resolution (at this focal ratio). Itturns out that the pixel size on a Nikon 300 is 5.4 µm.

Our eyes are, likewise, well designed. With an av-erage focal length of f/5.7, our eyes have a the diffrac-tion limited resolution on the focal plane (our retina)of 2.85 µm. The maximum density of cones in the

Fig. 9.— The image of a point source on the focalplane, the point spread function, or PSF, of a tele-scope operating in the diffraction limit, i.e. with nospherical of otherwise aberration, and no atmosphericblurring.

Fig. 10.— Two points can be considered just resolvedif the peak of the point spread function lies on the firsttrough of the other point spread function, as shownin the middle figure. This is known as Raleigh’s cri-terion.

fovea turns out to be 170,000 cones/mm2, indicatinga spacing of 2.5 µm. Well done!

The angular resolution is degraded substantiallyby the turbulence of the Earth’s atmosphere. Untilrecently angular resolutions of only ∼1 arcsec or atbest 0.5 arcsec, during a night of exceptional seeing,were possible. However, a technique called adaptive

optics, AO, makes it possible to attain angular reso-lutions close to the diffraction limit of the telescope.This process involves the adjustment of the shape ofthe primary or secondary mirror such that the PSF ofa bright star in the field of view is minimized. Adjust-ments are made on timescales of a few hundredths ofa second. If no bright stars are available, an artificiallaser guide star, LGS, is produced by shining a laserbeam into the atmosphere.

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