Aberrations in Optical Components (lenses, mirrors) 1)Chromatic Aberrations: Because the index of refraction of the lens material depends slightly on wavelength, the focal length also depends on wavelength, so different wavelengths will form images at slightly different places. (Doesn’t occur for mirrors, since  =  ’ for all wavelengths) 2) Spherical Aberration (both lenses and mirrors): We have made the paraxial approximation, that all rays are near the principal axis. As rays get further from the principal axis, their focus point changes, so if the lens or mirror collects light through a large diameter aperture, the image will be blurry. To sharpen the image, use a smaller diameter aperture in front of or behind the lens (or in front of the mirror) to restrict the light to being close to the axis; in a camera this requires a longer exposure time to compensate for the reduced amount of light entering the camera/second. aperture

Camera A converging lens forms a real image on the film or electronic (photodiode coupled to CCD or CMOS) detector. In more expensive cameras, the distance between the lens and CCD/film is adjustable so it can be made = q (i.e. 1/q = 1/f-1/p) to keep the image sharp. An expensive camera will also have an adjustable aperture so one can limit spherical aberrations, compensating by increasing the exposure time.

Less expensive cameras have a fixed lens position and aperture, with diameter D <<f. The image will be in reasonable focus on the CCD/film as long as the image distance >> f. In the limit, D  0, you don’t even need a lens: the camera is a pin-hole camera: only “one ray” from each point on the object reaches the CCD/film. Cameras in cell phones typically replace the mechanical shutter with an “electronic shutter” which turns the pixels in the CCD detector on/off. Also, better cell phone cameras have an autofocus, that moves the lens slightly to get better focus.

Eye Pupil is aperture which controls how much light enters eye. Real image formed on retina where light sensors (rods and cones) send signals through optic nerve to brain. Refraction occurs in outer surface (tears, cornea), aqueous humor, and lens. Ciliary muscles can cause the lens to contract, decreasing its focal length, to keep image in focus on retina.

For a “normal” eye, parallel rays from  are focused on the retina when the lens is “relaxed” (i.e. not compressed). Therefore, the normal eye can focus on  : normal “far point” =  f For a nearsighted eye, the focal length of the relaxed eye-lens (i.e. its largest value) is less than the length of the eyeball, so the eye cannot focus on objects at infinity (i.e. the eyeball is “too long”. To bring light from infinity to focus on the retina, place a diverging lens in front of the eye.

For rays from a closer object to be focused on the retina, the lens must be compressed to decrease f: 1/f = 1/p + 1/q, to keep q = size of eyeball. The shortest distance that the eye can focus on = near point. For a normal eye, near point  25 cm. If the object is closer than the near point, it will be focused outside the eye (behind the retina); i.e. the image on the retina will be blurry.

A farsighted person’s near point is larger than 25 cm. This can happen because the eyeball is “too short” or because the lens cannot compress (and reduce its focal length) enough. (The latter often happens with old age as the lens becomes less flexible or the ciliary muscle weakens.) an

A farsighted person’s image can be corrected by placing a converging lens in front of the eye, which will bring the focus onto the retina.

Magnifying Glass Recall that a converging lens will form a magnified virtual image if the object is inside the focal point (p 1.

It is more useful to consider the “angular magnification”, which compares how large a viewer perceives the image to be with how large the object could appear without use of the lens; i.e. compares the sizes of the images (with and without the magnifying glass)on the retina. The size of the image of an object on the retina    h/p (tan    ). When viewing a small object, the image on the retina will have its maximum size when the object is placed at the eye’s near point:  0  h/L NP L NP

L L = -q The angle for the image formed by the magnifying glass is  = h’/L = h/p Therefore, the angular magnification m =  /  0 = (h/p) / (h/L NP ) = L NP /p To see the image clearly, one needs L to be between the eye’s near point (L NP ) and its far point (L FP ). Since L = -q = pf/(f-p)  p = Lf /(L+f) so L NP f/(L NP +f) < p < L FP f/(L FP +f) (L NP /L FP ) (L FP /f +1) < m < (L NP /f + 1) image at far point: relaxed eye image at near point: maximum m For normal eye (or if wearing corrective lens): L np = 25 cm, L fp =  : 25 cm/f < m < 25 cm/f + 1

Example: If f = 6 cm, < m < If f = 2.5 cm 10 < m < 11 The smaller f [i.e. the “fatter” the lens], the larger m. relaxed eye maximum m

For more magnification, first use a lens (the objective lens) to form an enlarged real image, and then use a magnifying glass (the eyepiece) to make an enlarged virtual image of that: Compound Microscope

Objective Lens: M O = -q 1 /p 1 1/p 1 = 1/f O – 1/q 1 p 1 = q 1 f O /(q 1 - f O )  M O = - (q 1 -f O ) / f O For large M O, need large q 1 : put object slightly outside the focal point f O. [Note the microscope length L > q 1, so the larger M O, the longer the microscope must be.] Compound Microscope

Telescope Unlike the microscope, which forms a large image of very small objects, the telescope forms images of very large objects (e.g. planets) that are very far away: p object  . Therefore the rays that come from each point in the object are parallel. The telescope is used to increase the angular spread of rays coming from each point on the object, to make it “look larger”. Want to create a virtual image in which  image >  object.

Let  object =  O and  final image = . For an object at p = , the objective lens forms a real, inverted image at its focal point, f O. The size of the image = h’, where h’ = -f O tan  O  -f O  O. i.e.  O  -h’/f O If the image is at the focal point of the eyepiece, f e, it will form a virtual image at , making an angle  tan  = h’ / f e.

Telescope m = - f O / f e L = f O + f e The amount of light/per second that enters the telescope is proportional to the area of the objective lens, so to image faint objects, a large diameter lens is needed. However, it is difficult to support a large lens by its edges (don’t want to block light by building in supports), because gravity will cause it to deform. The largest refracting (i.e. lens) telescope is at the Yerkes Observatory in Wisconsin. It’s objective lens has a diameter of 100 cm and a focal length f O = 19.4 m with a maximum magnification |m| = 194; i.e. it’s “eyepiece” has an “effective” focal length of f e = 10 cm. [“Effective”, because modern research telescopes do not have simple eyepieces for viewing.] Because of atmospheric disturbance, it is actually not useful to have |m| larger than  200 for earth based telescopes. The collecting power (i.e. area of the objective) is a much more important criterion.

Galileo’s telescope (1609), with which he first observed the moons of Jupiter and the phases of Venus, was 92 cm long. Its objective lens had a diameter of 3.7 cm and focal length = 98 cm. (Its eyepiece had a diverging lens with a focal length of 2.2 cm, so its total length was less than f O and it was non-inverting.) It had a magnification |m|  40. Inexpensive home telescope with diameter = 7 cm and f O = 70 cm. It has two eyepieces: f e = 20 mm  |m| = 35. f e = 4 mm  |m| = 175 (  Yerkes!)

To overcome the difficulties in constructing a large objective lens, large telescopes use a reflecting, concave mirror. (Since light does not pass through the mirror, it can be supported from behind). Use of a mirror also avoids chromatic aberration. Also, to remove spherical aberration, a parabolic mirror is used: for a parabolic mirror, all rays from  are brought to focus at the focal point, not just paraxial rays. A disadvantage is that it is necessary to block some of the light in the center, but this fraction of the light can be kept very small.

The objective mirror of the main telescope in the MacAdam Student Observatory has a diameter of 50 cm and a focal length 3.45 m. The maximum magnification |m|  200, corresponding to an “effective” f e  1.7 cm.

Most radio telescopes have parabolic mirrors (good metal surfaces are perfect reflectors of radio waves) that can be directed at the point of the sky of interest. A radio detector is placed at the focus of the parabola ; all the radio waves that hit the parabola are focused there. Imaging is done by changing the direction of the parabola’s axis. The sensitivity (radio power detected) is proportional to the area of the dish. The largest radio telescope is in Aracebo, Puerto Rico, with a diameter of 305 m. It is built into the ground and cannot be moved (or pointed). Therefore, instead of parabolic, it is spherical. Radio waves from different directions in the sky are measured by moving the detector along cables strung above the dish to the point where they are  focused.