Early Telescopes & Geometrical Optics C. A. Griffith, Class Notes, PTYS 521, 2016 Not for distribution. 1
1.2. Image Formation Fig. 1. Snell s law indicates the bending of light at the interface of two materials, shown here for two substances such that n 2 > n 1 and v 2 < v 1. 1. A Few Things about Light Here we discuss some concepts and definitions of basic geometrical optics and the construction of early telescopes. 1.1. Refraction and Reflection The field of classical optics is dominated by two laws. The Law of Reflection states that when light hits a reflecting surface, the ray will be reflected from thesurfaceatananglethatisequaltotheangleofthe incident ray. The Law of Refraction concerns the fact that light moves slower in material with higher density. As a result when light passes from one medium to another, the angle that the light beam makes with the normal to the interface between the two media is less in the medium where light travels slowest, i.e. θ 2 < θ 1 in Fig. 1. The index of refraction is defined as the ratio of the speed of light in a vacuum, c, to the speed of light in a particular medium, v p : n = c/v p Snell s law relates the apparent bending of light caused by the change in the velocity of light as it crosses the boundary between two mediums of indices n 1 and n 2 : n 1 sinθ 1 = n 2 sinθ 2, where θ 1 and θ 2 are the angles from the normal of the interface for the incident and refracted waves(fig. 1). The index of refraction, or the speed of light in a medium, depends on the wavelength of the light. Prisms take advantage of this property and cause light to be dispersed as a function of wavelength. Parallel rays that pass through a lens with convex parabolic surfaces or that reflect from a parabolic concave mirror are brought to a focus. The distance from the lens (or mirror) to the focus is called the focal length, f (Fig. 3). Conversely, concave lenses and convex mirrors cause light to diverge as if emanating from a point. Converging lenses produce real images of distant objects, i.e. that which can be captured on film. When these are combined with an eyepiece lens, they produce a virtual image, that is one that is produced at an infinite distance away (Fig. 4). When combined with other lenses, such as the lens in our eye a real image is formed on our retinas. Note that a series of lenses can bring light to a focus; in that case the focal length is defined as that of a single lens that would bring light to a focus at the same angle of convergence. The focal ratio, or f-number (written as f/# is: F = f/d, where f is the focal length and D is the diameter of the objective lens. A smaller focal ratio indicates a stronger curvature of the lens and is called a faster lens. 1.3. Human Eye Consideryoureye. It consists ofalens that focuses light at the retina, or the back of the eye. The retina is basically a detector array of 125 million photoreceptor cells over a curved focal plane. The iris dilates and contracts to adjust the effective diameter of the lens, which then changes the focal ratio from about f/3 to about f/8. In additional it squeezes the lens into different shapes to adjust its focal length. These adjustments are made to adapt to the distance of the object that your eye is intent upon so that the retina (where the photons are read) lies in the focal plane of the lens. 2. Telescopes 2.1. Refracting Telescopes In 1608 a Dutch optician noticed that distant objects appear larger when viewed through a combination of two lenses, a weak (long focal length) converging lens and a strong (short focal length) diverging lens. 2
Fig. 3. Galilean Telescope intercepts the converging rays from the primary, rendering them parallel again. These rays however subtend a larger angle from the optical axis, thereby increasing the angular size. The image is magnified. Fig. 4. Kepler s telescope is similar to the Galilean telescope, yet of course the image is upside down. Fig. 2. The focal lengths of convex and concave mirrors and lenses. Note that real images are produced only by convex lenses and concave mirrors. This spyglass was subsequently used by Galileo Galilei and is known as the Galileo telescope. Galileo developed this telescope further so that it reached a magnification of 30. With this he investigated the planets and stars. He detected mountains on the moon and the presence of Jupiter s moons. His observations indicated unequivically that Venus orbited the Sun instead of the Earth and that stars extend into the distance, since more of them can be seen with a telescope. Thus he eradicated the Ptolomy and Plato universes where perfect spheres orbit the Earth (only), and all stars sit on a perfect orb that surrounds us(an excellent demonstration of how technology revolutionizes astronomy). In 1630, Kepler replaced the diverging lens with a converging lens, to make a Kepler Telescope. One of the largest of such telescopes is the Lick telescope, which has a 1 meter objective lens. Note how long the telescope is. A long focal length allows for high magnification, while a short focal length allows for a greater field of view (Fig. 6). 2.2. Lenses vs Mirrors Lenses are difficult to make such that they are homogeneous and have a perfect curvature. In addition, large objectives sag under their own weight and absorb light preferentially in the blue and violet. It is easiest to make lenses of spherical curvature, as opposed to parabolic curvature; however, the former of does not focus perfectly as does the latter. The resulting spherical aberration occurs because parallel rays falling on that fall on the lens at different distances from the optical axis are focused at different focal points. Lens also focus light of different wavelengths at different focal lengths; this effect is called a chromatic aberration. Other problems with lenses include 3
Fig. 6. A Cassegrain configuration. Note the position of the prime focus, F o, which has a shorter focal length that the combined primary and secondary mirrors, which collaborate to put the focal plane at F ext. Fig. 5. The Lick 1 meter telescope first saw light in 1888. The telescope is long because it needs to accommodate 17.37 m focal length of the primary lens. the occurrences of comas and astigmatism. The drawback of reflectors is that there is often a greater loss of optical intensity because the surface is not 100% reflective. In addition, the surface deteriorates and must be aluminized every several years. 2.3. Reflecting Telescopes These problems point to reflecting telescopes, most typical of most modern telescopes. If we replace the objective lens with a concave mirror, we eliminate the chromatic aberration. If we use a parabolic mirror, we eliminate the spherical aberration. Alternatively we can add a corrective lens in front of objective, called a Schmidt plate, that focuses all the parallel light at one point. Telescopes that use this device are called Schmidt telescopes. Newton designed a telescope with a concave primary mirror and a convex eyepiece. This design was improved by his french contemporary, Guillaume Cassegrain, who placed a convex mirror in the optical axis, thereby extending the effective focal length of the primary, and improving the angular resolution of the telescope Light was reflected back through a hole in the primary, where it met the eyepiece (Fig. 7). Note that the eyepiece plays the role of taking the real image produced by the primary and producing an imaginary image at infinity, which is made into a real image on your retina using the lens of your eye. In professional astronomy, eyepieces are not used. Instead detectors may be placed e.g. at the Cassegrain focal plane or at the prime focus that is produced when the secondary mirror is removed. 3. Angular Magnification The angular magnification of a telescope is the ratio of the angular extent of an object as seen through the telescope, divided by the angular extent of the object without the telescope. Note that, referring to Figure 5, the angular magnification, M is: M = α IM α OB. Thus from figure 5 we can see that we have two triangles that share one side. In the small angle approximation, tan(α) = α, and M f o f e. For a system with a detector array at the focal plane, there is of course no eyepiece. The angular magnification of the telescope is quantified in terms of the image scale or the plate scale. This is the number of arcseconds, θ, that are imaged onto 1 mm of the detector, and, as evident from figure 5 (e.g. taking the image I to be 1 mm): tan(θ) = 1(mm) f o (mm). 4
From the small angle theory: θ (radians) 1 f o (mm) But we need, by definition, the angle in arcseconds θ (arcseconds) = 206,265 f o (mm) Note that the plate scale depends only on the focal length of the telescope. Thus an observation that requires more angular magnification would best be performed with an instrument on the Cassegrain focal plane rather than at the prime focus. The field of view of a telescope depends also on the linear size, L, of the detector. If you refer back to figure 5, and put a detector at the focal plane of the primary lens, this is kind of equivalent to putting it at the primary focus of a Cassegrain. Then you can see that : tan(θ) = L/2 f o, and, from the small angle approximation, the full field of view is: θ(radians) = L f o. The field of view is larger, for a given detector, when the focal length is smaller. Thus an observation that requires a greater field of view would best be performed with an instrument at the prime focus of a Cassegrain telescope. Note that compared to the Cassegrain focal plane, at the primary focus more photons from an extended object are detected at each pixel, for the same integration time. 3.1. Specific Telescopes Let s consider the Subaru Telescope on top of Mauna Kea at 13,580 ft in Hawaii. It s quite versatile. It has an 8.2 m diameter primary, and four foci. At the prime focus (f/2) instruments such as FMOS, the fiber-fed multi-object spectrograph, take advantage of the wide field of view. FMOS has a 30 arcmin field of view and can simultaneously record 0.9 1.8 µm spectra of 400 objects with a resolving power 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 resolution measurements are possible. For example the FOCAS instrument, placed at the Cassegrain focus, has a 6 arcmin field of view, and a 0.104 arcsec pixel scale. Fig. 7. The layout of the Subaru Telescope, with its four foci. 3.2. Diffraction and Interference In 1803, Thomas Young performed his famous experiment that demonstrated interference from light passing through two closely spaced slits (Fig. 2). He deduced that light behaves like waves. In 1961, Clauss Jonsson of the University of Tubingen performed the 2-slit experiment for electrons. He found, oddly, the if one bombards 2 slits with electrons the result is a diffraction pattern. Well, you might say, maybe the electrons interact with each other bouncing off each other to make such a pattern. In 1974 Pier Giorgio Merli from LAMEL-CNR Bologna, sent electrons through the slits one at a time. Still there was a diffraction pattern. OK, so perhaps the electron interacts with itself. Well, if you put a device that measures which part of the electron passes through which slit, you will see only one electron pass through the slit. And.. and this is really bizarre, you don t see the diffraction pattern. Your observation of the electron made it act like a particle! Any wave (or fraction of an unseen electron) will bend when it passes through an aperture that is nar- 5
distance of sin θ = 1.22 λ D. Fig. 8. Young s sketch of two slit diffraction, which he presented to the Royal Society in 1803. row relative the light s wavelength. This effect of diffraction causes interference of light that crosses the aperture at different places. That is light from one side of the aperture combines with light from another side 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 the light is perfectly out of phase. This superposition of light gives rise to a central peak and a series of fringes of light and dark bands on the other side of the aperture. One can see this diffraction pattern more clearly if one studies the pattern produced by the superposition of light emerging from two narrow slits, as did Thomas Young. Bright bands occur when the path difference between two waves is a whole number of wavelengths. Dark bands occur with the path difference is a odd number of half wavelengths. Thus the grating disperses light according to its wavelength, as does a prism. The diffraction pattern would appear sharper if a large number of equally spaced slits are used. We will find that these effects are important in the understanding of how spectrographs work, and what ultimately limits the spatial resolution of a telescope. 3.3. Diffraction Limit Because of diffraction, the image of a point source on a detector is never a point. Even a telescope in space with a perfect and parabolic primary mirror will record a point source as an extended source, called a point spread function, which consists of a central peak surrounded by concentric ripples (Fig. 9). This airy disk pattern resemble the bands formed from light passing through a single slit, but for two dimensions. Roughly 97.75% of the light is contained in the central disk, whichextendstothefirstdarkringatanangular The second and third minima, containing a fraction of 0.0175 and 0.0042 of the light, encircle the disk at an angular distance of 2.233 λ D and 3.238 λ D. Note that the PSF is extended, even in the absence of terrestrial atmospheric distortions and aberrations in the primary mirror or lens. Ultimately, the diffraction of light by the telescope s aperture establishes the limiting point spread function (PFS). The theoretical limit of the angular resolution of a telescope is set by diffraction can be defined as the angular distance between two sources when the minimum of one source s PSF coincides with the peak of the other source s PSF (Fig. 10). The diffraction limit, or limiting angular separation of two point sources in the sky, was thus estimated by Rayleigh to be: sin θ = 1.22 λ D where λ is the wavelength and D is the telescope diameter. It is easy to calculate this value analytically, especially if you like to play with Bessel functions, but let s move on instead. The calculation of the Airy Disk pattern is stems from the Fraunhofer diffraction equation with the use of Bessel functions. In general the blocakge of light can be modelled using the HuygensFresnel principle. Huygens postulated that every point on a primary wavefront acts as a source of spherical secondary wavelets and the sum of these secondary waves determines the form of the wave at any subsequent time. Fresnel developed an equation using the Huygens wavelets together with the principle of superposition of waves, which models these diffraction effects quite well. It is not a straightforward matter to calculate the displacement given by the sum of the secondary wavelets, each of which has its own amplitude and phase, since this involves addition of many waves of varying phase and amplitude. When two waves are added together, the total displacement depends on both the amplitude and the phase of the individual waves: two waves of equal amplitude which are in phase give a displacement whose amplitude is double the individual wave amplitudes, while two waves which are in opposite phases give a zero displacement. Generally, a two-dimensional integral over complex 6
variables has to be solved and in many cases, an analytic solution is not available. The Fraunhofer diffraction equation simplifies the problem by modeling the light diffracted when both the light source and the viewing plane are effectively at infinity with respect to the diffracting aperture. In this case, the incident light is a plane waveso that the phase of the light at each point in the aperture is the same. The phase of the contributions of the individual wavelets in the aperture varies linearly with position in the aperture, making the calculation of the sum of the contributions relatively straightforward in many cases. Thus for yellow light (λ=0.5 µm) a 1 meter telescope has diffraction limit of Fig. 9. The image of a point source on the focal plane, the point spread function, or PSF, of a telescope operating in the diffraction limit, i.e. with no spherical of otherwise aberration, and no atmospheric blurring. θ radians sin θ = 1.22 5 10 7, or θ arcsec 206,265 1.22 5 10 7 = 0.126 arcsec, while a 10 meter telescope has a diffraction limit of 0.0126 in yellow. Now we can ask what is the size of the pixels that we need to resolve a diffraction limited image on the detector which sits at the focal plane. We can approximate the small angle θ as: θ radians x f, where f is the focal length and x is the separation of two images at the focal plane such that the objects are just resolved. Then, and x f 1.22λ D x 1.22 λ f/#. So let s consider our new nifty camera. At a typical setting of f/8, the pixels spacing that corresponds to the diffraction limited resolution of blue light (0.42 µm) of the camera is x = 1.22 * 8 * 0.42 µm = 4.1 µm. So it would be a bit silly to make the pixels of the CCD smaller than this value, for it would not increase the image resolution (at this focal ratio). It turns out that the pixel size on a Nikon 300 is 5.4 µm. Our eyes are, likewise, well designed. With an average focal length of f/5.7, our eyes have a the diffraction limited resolution on the focal plane (our retina) of 2.85 µm. The maximum density of cones in the Fig. 10. Two points can be considered just resolved ifthepeakofthepointspreadfunctionliesonthefirst trough of the other point spread function, as shown in the middle figure. This is known as Raleigh s criterion. fovea turns out to be 170,000 cones/mm 2, indicating a spacing of 2.5 µm. Well done! The angular resolution is degraded substantially by the turbulence of the Earth s atmosphere. Until recently angular resolutions of only 1 arcsec or at best 0.5 arcsec, during a night of exceptional seeing, were possible. However, a technique called adaptive optics, AO, makes it possible to attain angular resolutions close to the diffraction limit of the telescope. This process involves the adjustment of the shape of the primary or secondary mirror such that the PSF of a bright starin the field of view is minimized. Adjustments are made on timescales of a few hundredths of a second. If no bright stars are available, an artificial laser guide star, LGS, is produced by shining a laser beam into the atmosphere. 7