Telescopes

1. Ray diagram to show the image formation in normal adjustment:

⇒ Converging lens:

• A converging lens, also known as a convex lens, is a type of lens that converges light rays It is thicker in the middle than at the edges, causing light rays to bend towards the optic axis (figure 1).
• Properties of a converging lens:
  – Convex shape: Thicker in the middle than at the edges
  – Converges light rays: Light rays bend towards the optic axis
  – Positive focal length: The focal length is positive, indicating that the lens converges light rays
  – Forms real images: A converging lens forms real, inverted images
  – Magnifies objects: A converging lens can magnify objects, making them appear larger than they are.
• Figure 1 Converging lens

⇒ Focal length:

• The focal length of a lens or mirror is the distance between the optical element and the point where parallel light rays converge (or appear to converge). It is a measure of how strongly the lens or mirror converges (figure 2).

  
  Figure 2 Parameters of converging lens

• Focal length is typically denoted by the symbol “f” and is measured in millimeters (mm) or meters (m).
• Types of focal lengths:
  – Positive focal length: Converging lens or mirror (e.g., convex lens)
  – Negative focal length: Diverging lens or mirror (e.g., concave lens)
  – Zero focal length: No convergence or divergence (e.g., flat mirror)

⇒Principle axes:

• The principal axes of a converging lens are the optical axis and the two principal planes.
  – Optical axis (z-axis): The line passing through the center of the lens, perpendicular to the lens surface, and intersecting the lens at its center.
  – Principal plane (x-axis): The plane perpendicular to the optical axis, passing through the lens’s front focal point.
• Principal plane (y-axis): The plane perpendicular to the optical axis, passing through the lens’s back focal point.

⇒ Focal point:

• The focal point of a lens is the point at which rays parallel to the principal axis of the lens are brought to a focus.

⇒ Construction of ray’s diagrams:

• There are three classes of light ray that are used to predict the position of an image formed by a converging lens. These are illustrated in Figure 3.
• Figure 3 Image formation by a converging lens
• when we draw a ray diagram for a lens, we simplify the process of refraction by assuming that it happens in just one part of the lens.
• So, the lens is drawn as a thin vertical line. The arrows pointing out from the centre of the lens, at the top and bottom, indicate that this lens is a converging lens.
  1. A ray parallel to the principal axis (on the left side of the lens) is refracted so that it passes through the focal point on the right side of the lens.
  2. A ray that passes through the optical centre of the lens is undeviated.
  3. A ray that passes through the focal point on the left side of the lens is refracted so that it travels on a line parallel to the principal axis on the right side of the lens.

⇒ Projecting an image:

• Use two of the construction rays to predict where an image will be formed by a converging lens.
• Provided the object lies outside the focal length of the lens, a real image will be formed.
• The image is real when the rays converge at a point. This image can be focused on to a screen.
• Figures 4 (a) and (b) show how two different converging lenses can be used to project an image of a distant object.
• Light rays from the same point on a distant object arrive at the lens very nearly parallel to each other.
• So, for example, rays from the top of a distant object arrive at the lens parallel to each other and rays from the bottom of the same object also arrive parallel to each other.
• Lens B produces a larger image than lens A, because it has a longer focal length. This idea will be used later when we consider the design of an astronomical telescope.
• Figure 4 A lens with longer focal length projects a larger image of a distant object; the image projected by lens B is larger than the image projected by lens A.

2. The magnifying glass:

• Figure 5(a) shows what happens when an object is placed inside the local length of a converging lens.

  
  Figure 5 (a) An object viewed inside the focal length of a lens produces a virtual magnified image. (b) Without a lens, you can only focus on an object at your near point of vision.

• Rays from the top of the object now diverge, and do not come to a focus.
• If your eye is placed behind the lens, the object appears to be bigger and further behind the lens. This is a virtual image.
• It cannot be projected on to a screen and it appears only to the eye on the other side of the lens.
• When the lens is used like this, it is called a magnifying glass.
• The object appears bigger because the lens produces a magnified image at your near point.
• Without the lens, you can only focus on the object at your near point of vision perhaps 25cm away, as shown in Figure 5(b).
• The lens causes magnification because the angle θ in Figure 5(a) is bigger than the angle in Figure 5(b).

3. The Astronomical telescope:

• Figure 6 shows the principle behind the astronomical refracting telescope.
• The objective lens projects a real image of a distant object such as the moon.
• This image is larger for a longer focal length of the objective lens, [math] f_o [/math].
• The eyepiece is now used to magnify this image.
• A short focal length eyepiece produces a larger magnification of the telescope.
• Figure 6  Principle of the telescope
• Using trigonometry
• [math] \tan \alpha = \frac{h}{f_0} \\ \tan \beta = \frac{h}{f_e} [/math]

• Where h is the height of the real image, [math] f_o [/math] is the focal length of the objective lens and [math] f_e [/math] is the focal length of the eyepiece lens. But for the small angle

• [math] tan⁡ α = α \\ tan β = β [/math]
• So
• [math] \alpha = \frac{h}{f_0} \\
  \beta = \frac{h}{f_e} [/math]
  [math]\text{The angular magnification,} M, \text{ of the telescope is defined as} \\
  M = \frac{\text{angle subtended by image at eye}}{\text{angle subtended by object at unaided eye}} = \frac{\beta}{\alpha} [/math]
  [math] \frac{\beta}{\alpha} = \frac{h}{f_e} \times \frac{f_o}{h} \\
  \frac{\beta}{\alpha} = \frac{f_0}{f_e} [/math]
• A telescope is described as being in normal adjustment when the real image produced by the objective lens, is viewed at the focal point of the eyepiece. Under these circumstances, a magnified virtual image is viewed at infinity.

4. Lens aberrations:

• Although refracting astronomical telescopes are very useful instruments, their effectiveness is reduced to some extent by the limitations of their lenses.
• Glass lenses have two main types of aberration, which limit the sharpness of the image that we see.

⇒ Spherical aberration

• Most lenses are ground into a spherical shape, but this is not quite the ideal shape for a lens.
• Figure 7 shows two rays, parallel to the principal axis of a lens, which come from the same distant object.
• Figure 7 Lens with spherical aberration
• The two rays refract at different angles, but they do not pass through the same focal point the ray at the top of the lens,
• A, comes to a focal point closer to the lens than the lower ray, B. As a result of this there is a slight blurring of the image.
• Spherical aberration can be demonstrated easily in the laboratory. A lens is used to project an image of a lamp filament on to a screen.
• If a card with a small hole is placed in front of the lens, you will see that the image becomes sharper. This is because rays pass through only a small part of the lens.
• It is possible to reduce spherical aberration by using a lens with a parabolic shape.
• However, such lenses are very expensive, and they produce some distortion of the image, except for light exactly parallel to the principal axis.

⇒Chromatic aberration

• Figure 8 shows two rays of white light being refracted by a lens.
• The speed of light through glass depends on its wavelength.
• Blue light has a shorter wavelength than red light, and it travels more slowly than red light through glass.
• Consequently, blue light is refracted more than red light, and there are different points of focus for the two colors. This is called chromatic aberration.
• It is possible to reduce the effects of chromatic aberration, but not to remove it entirely, by constructing a lens using two different types of glass.
• Figure 8 Chromatic aberration

7. Reflecting telescope:

• Figure 9 shows the principle behind the Cassegrain reflecting telescope.
• Light from a distant object strikes the primary concave mirror, where the light is reflected towards the focal point at E.
• However, a secondary convex mirror reflects the light again, so that it is focused at F’, where a real image is formed.
• The observer can then see a magnified image through the eyepiece, which is placed behind a hole in the primary mirror.
• Figure 9 Principle of the Cassegrain reflecting telescope.
• A reflecting telescope has several advantages over a refracting telescope.
  – A good astronomical telescope requires a diameter of about 15cm or more, so that sufficient light is gathered. It is very difficult to make a high-quality lens of diameter 15cm, but much easier to make a concave mirror of that size.
  – A reflecting mirror has no chromatic aberration, because light is reflected over a metal surface without passing through glass.
  – Spherical aberration can be reduced more easily in a reflecting telescope by making the concave mirror parabolic in shape. A parabolic mirror focuses light that is parallel to the principal axis accurately at the focal point.
  – It is possible to make reflecting telescopes with larger diameters than refracting telescopes. The world’s largest refracting telescope, at the Yerkes Observatory, has a diameter of 1.0m. There are several reflecting telescopes that have diameters over 8m-for example, the Subaru Telescope in Hawaii has a mirror of diameter 8.2 m. A glass lens with a diameter of over 1 m begins to sag under its own weight, whereas a mirror can be supported by a strong structure behind it.
• The collecting power of a telescope is proportional to its area. Since the area of the telescope mirror is where d is its diameter, the collecting power is 4 proportional to the diameter squared, .
• A refracting telescope does have some advantages over a reflecting telescope.
  – The lenses in a refractor are held in place by a metal tube. So little maintenance is required. The mirror in a reflecting telescope is exposed to the air, and might need recoating.
  – The mirrors in a small reflector can get out of alignment if the telescope gets knocked. So sometimes the mirrors need adjustment. The strong construction of the refracting telescope makes such misalignment less likely.
  – The secondary mirror in a reflecting telescope has the disadvantage of blocking some light from entering the primary mirror.
  – The secondary mirror and its supports will cause some diffraction which will degrade the image