Can you draw a lens diagram for two convex lenses in series? Lenses and Telescopes Can you draw a lens diagram for two convex lenses in series?

Rays parallel to the principle axis If the distance between the two lenses is adjusted so that it is equal to f1 + f2 then you get the following diagram: f

Rays not parallel to the principle axis Things to remember: The incident rays are all parallel The centre ray does not refract All the rays meet at the focal point of BOTH lenses The rays continue in a straight line to the second lens The central ray refracts as if it came parallel to the focal point The other rays exit the second lens parallel to the central ray Extending the previous exercise, a telescope is unlikely to be perfectly aligned with the object to view in the exact centre. Therefore, how would the rays appear if there is a small angle subtending from the unaided eye to the object? f

Telescope lenses and angles The two main angles you can label on this diagram are the angle subtended by the eye to the object (α) and the angle subtended from the eye to the image (β) f β α Eyepiece Objective lens

Magnifying power Recall that the power of a lens (in Dioptres) is the reciprocal of the focal length (in metres): The magnifying power is then the ratio of the objective lens focal length to the eyepiece focal length: Another useful formula is that the magnifying power is also the ratio of the angles subtending to the image and object:

Lens formula The focal length of a lens can be calculated using the following formula, if the image and object distances are known: How does this formula change if the distance to the object approximates to infinity? (i.e. a star)

Chromatic aberrations The angle at which a light ray refracts at is slightly dependant on the wavelength of the light (You know this from Newton’s prism experiments to split light into a spectrum) If light of various colours is refracted through one or more lenses then the different colours form images at slightly different positions. This causes a blurring of the colours

A picture of a star showing chromatic aberration

Minimising chromatic aberrations You can reduce the effect of chromatic aberrations by: Using a larger lens and using only the centre of the lens Using a coloured filter to allow only one wavelength of light through the lens The problem with these solutions is that they both reduce the amount of light getting through the lens and hence will reduce the sensitivity of the optical device

Summary A telescope can be created with two lenses, an objective lens and an eyepiece The two lenses are arranged so that the sum of their focal lengths is equal to their separation The ray diagrams for this must be created so that the rays all pass through the focal point and all refract parallel at the eyepiece Chromatic aberrations occur due to different wavelengths of light refracting different amount; this can be reduced by using only the centre of a lens or a coloured filter

How can a mirror be used to focus light? Reflecting telescope How can a mirror be used to focus light?

Reflecting telescope Using parabolic mirrors to focus light gets rid of the problem of chromatic aberrations; This is because the angle of reflection is independent of wavelength. Light that travels parallel to the normal will reflect through the principle focus Light that travels along the principal axis will reflect straight back along that line. Any ray that strikes the principal axis will reflect at the same angle as the angle of incidence

Focusing light Light from a distant object can therefore be focused onto a point from a concave mirror without chromatic aberration

Newtonian Telescope Light is reflected off a primary mirror before being sent, via a plane mirror, to the eyepiece at the side. Note that the rays will be parallel before entering your eye. The lens in your eye brings them to focus on your retina

Cassegrain Telescope This works in a similar way to a Newtonian telescope except that there is a convex mirror that straightens out the rays after they have been focused from the primary mirror. Spherical aberrations still can exist in mirrors but large diameter mirrors with large focal lengths and using light only from closer to their centres avoids this.

Mirrors versus lenses The main advantages of a mirror over a lens are: Lighter Only the surface needs to be perfect as opposed to the body of the lens Thin so can be adjusted as it is moved using pressure pads Cheaper to manufacture Able to be made in segments and fitted together No chromatic aberrations

Spherical aberrations Spherical aberrations occur in both lenses and mirrors and it is because light hitting different sections do not always meet at the principle focus These cannot be overcome simply with design changes in shape as it varies with the angle of the incident light The effect can be minimised by using oversized lenses / mirrors and closing the aperture to only use the centre when pin-sharp images are needed

Spherical aberrations

Summary The use of mirror in telescopes means that they can be built cheaper, bigger and removes the chromatic aberrations Newtonian and Cassegrain telescopes differ slightly in the positions of the eyepiece and in the use of plane or convex mirrors in the centre You must be able to draw ray diagrams for each of these telescopes Remember that the rays that enter the eye must be parallel so a final eyepiece is used to achieve this Spherical aberrations can be removed by only using the middle of a large lens or mirror

Charge Coupled Devices How can a digital camera take a picture if there is no photographic film?

CCDs (Charge Coupled Devices) A CCD is the device that is used by digital cameras and sensors in optical telescopes to receive and store an image. These work by having light photons strike their surface, releasing electrons in a manner not dissimilar to the photo-electric effect. The number of electrons released is proportional to the brightness / intensity of the incoming source.

How a CCD works As the photons strike the pixels then an electron-hole pair is created and the array holds a charge. Periodically the array is made to release it's charge in a set sequence. This creates a linear number sequence which is the encoded picture. A modern day CCD digital camera has 16 million pixels or more! As the picture is taken these are filled with charge. As the picture is saved these all drain out one at a time to be stored on the chip or SD memory card!

Animation of a CCD draining This animation shows how the pixels are "drained" and the data is stored. This is only a 20 pixel array! Colour images are obtained using filters of RGB over the pixels. Click for animation; links to an external internet page

Advantages of CCD's over Photographic film CCD's operate over a large range of wavelengths from 400nm - 1000nm which means that they can see into the near Infra-Red Using phosphorous you can get incoming UV and X-rays to create visible light by fluorescence therefore extending the range that CCD's can detect. Hold a mobile phone camera in front of a TV remote and see the white flashes that your eye cannot detect

Quantum Efficiency This is a measure of how many of the incident photons are detected by the device. The human eye is only 1% efficient, photographic film is 4% efficient and CCD's are up to 70% efficient This means that objects need to be viewed for less time to get an image and more faint objects can be photographed when CCDs are used instead of photographic film

Example calculations The light from a distant star is incident on a 1m diameter objective lens. The average wavelength of the light is 550nm. Only 172 photons are detected per second on the CCD 1) Using a quantum efficiency of 70% how many photons arrive at the objective lens? 2) How much average energy does each photon carry? 3) Given the surface area of the objective lens what is the number of W/m2? 4) Assuming the star is similar to that of the sun with an output of 1026 Watts, how far away is it? Solar radius is 6.955 X 108 m

1) Using a quantum efficiency of 70% how many photons arrive at the objective lens? 2) How much average energy does each photon carry?

3) Given the surface area of the objective lens what is the number of W/m2?

4) Assuming the star is similar to that of the sun with an output of 1026 Watts, how far away is it? Solar radius is 6.955 X 108 m Converting to light years this gives 26000 light years distance which is within the Milky Way Galaxy (about 40000 light years across)

Summary Electronic devices use CCDs which collect charge when light photons strike them CCDs are better than photographic film because: They have a wider range of wavelengths that they are sensitive to They have a higher quantum efficiency (70%) If you know the number of photons received per second then the quantum efficiency is the percentage that are detected

Non-optical Telescopes What are the benefits in not using light to probe the universe?

Resolving Power When an electromagnetic wave passes through an aperture then it diffracts creating a diffraction pattern. From a point source (i.e. a distant star) going through a circular aperture this looks like this:

Resolving two objects If two objects are close together and far away (i.e. their angular separation is very small) then their diffraction patterns will begin to overlap This means that if the angular separation is too small then they will appear as a single object; they cannot be resolved

Rayleigh Criterion If the maximum of one lines up with the minimum of another one then this is the limit of the sources being resolvable. This is the Rayleigh Criterion.

Resolving binary stars Resolved objects Rayleigh criterion Unresolved objects

Calculating resolving power The resolving power needed to resolve two point sources can be calculated using: Example A car is 5km away and has a headlights 1.5m apart. What resolving power is needed to resolve the headlights?

Calculating the resolving power for telescopes Given the diameter of the telescope aperture d and the wavelength of the incident waves then the Rayleigh criterion can be calculated for any telescope. This is it's maximum resolving power. If the resolving power for objects exceeds this then the telescope cannot resolve them.

Keplar 22-b An exoplanet has been detected orbiting Keplar 22. The radial distance of the planets orbit is 127 million km. Keplar 22 is 600 ly from the Earth. How big does a telescope need to be on Earth to resolve the Star / planet pair?

Summary When light goes through an aperture it diffracts If the diffraction patterns overlap from two objects then the two objects cannot be resolved The Rayleigh criterion is where two objects can just be resolved The further away and the closer together two objects are then the smaller the angular separation is; hence the more difficult they are to resolve