Chapter 27 Lenses and Optical Instruments

Lenses Converging lens Diverging lens

Thin Lenses A thin lens consists of a piece of glass or plastic, ground so that each of its two refracting surfaces is a segment of either a sphere or a plane A thin lens consists of a piece of glass or plastic, ground so that each of its two refracting surfaces is a segment of either a sphere or a plane Lenses are commonly used to form images by refraction in optical instruments Lenses are commonly used to form images by refraction in optical instruments

Thin Lens Shapes These are examples of converging lenses These are examples of converging lenses They have positive focal lengths They have positive focal lengths They are thickest in the middle They are thickest in the middle

More Thin Lens Shapes These are examples of diverging lenses These are examples of diverging lenses They have negative focal lengths They have negative focal lengths They are thickest at the edges They are thickest at the edges

Glass lens (n G = 1.52)

The focal length of a lens is determined by the shape and material of the lens. Same shape lenses: the higher n, the shorter f Lenses with same n: the shorter radius of curvature, the shorter f Typical glass, n = 1.52 Polycarbonate, n = 1.59 (high index lens) Higher density plastic, n ≈ 1.7 (ultra-high index lens)

Q. A parallel beam of light is sent through an aquarium. If a convex lens is held in the water, it focuses the beam (…….. ……………………. ) than outside the water (a) closer to the lens (b) at the same position as (c) farther from the lens n air = 1, n water = 1.33

Rules for Images Trace principle beams considering one end of an object off the optical axis as a point light source. A beam passing through the focal point runs parallel to the optical axis after a lens. A beam coming through a lens in parallel to the optical axis passes through the focal point. A beam running on the optical axis remains on the optical axis. A beam that pass through the geometrical center of a lens will not be bent. Find a point where the principle beams or their imaginary extensions converge. That’s where the image of the point source.

two focal points: f 1 and f 2 Parallel beams: image at infinite!!

Virtual image Magnifying glass

1/p + 1/q = 1/f Magnification, M = -q/p Negative M means that the image is upside-down. For real images, q > 0 and M < 0 (upside-down).

Lens equation and magnification 1/p + 1/q = 1/f M = -q/p This eq. is exactly the same as the mirror eq. Now let’s think about the sign. positivenegative p real object imaginary object (multiple lenses) q real image (opposite side of object) imaginary image (same side of object) f for converging lens for diverging lens M erect image inverted image

two focal points: f 1 and f 2 Parallel beams: image at infinite!! 1/p + 1/q = 1/f 1/2f + 1/q = 1/f 1/q = 1/2f M = -q/p = -1 1/p + 1/q = 1/f 1/f + 1/q = 1/f 1/q = 0  q = infinite

Virtual image Magnifying glass 1/p + 1/q = 1/f 2/f + 1/q = 1/f 1/q = -1/f M = -(-f)/(f/2) = 2

Ex A thin converging lens has a focal length of 20 cm. An object is placed 30 cm from the lens. Find the image Distance, the character of image, and magnification. positive f f = 20, p = 30 1/q = 1/f – 1/p = 1/20 – 1/30 = 1/60 q = 60 M = -q/p = -60/30 = -2 real image (opposite side) < 0 inverted

Magnifier Consider small object held in front of eye Consider small object held in front of eye Height yHeight y Makes an angle  at given distance from the eyeMakes an angle  at given distance from the eye Goal is to make object “appear bigger”: ' >  Goal is to make object “appear bigger”: ' >  y 

Magnifier Single converging lens Single converging lens Simple analysis: put eye right behind lensSimple analysis: put eye right behind lens Put object at focal point and image at infinityPut object at focal point and image at infinity Angular size of object is , bigger!Angular size of object is , bigger! Outgoing rays Rays seen coming from here  f f Image at Infinity  y

Angular Magnification (Standard) Without magnifier: 25 cm is closest distance to view Without magnifier: 25 cm is closest distance to view Defined by average near point. Younger people do betterDefined by average near point. Younger people do better   tan  = y / 25  tan  = y / 25 With magnifier: put object at distance p = f With magnifier: put object at distance p = f '  tan ' = y / f'  tan ' = y / f Define “angular magnification” m  = ' /  Define “angular magnification” m  = ' /  Note that magnifiers work better for older people because near point is actually > 25cm Note that magnifiers work better for older people because near point is actually > 25cm ~y/25 ’~y/f M  = ’/  = 25/f

Example Find angular magnification of lens with f = 5 cm Find angular magnification of lens with f = 5 cm

Eye Glasses Perfect Eye Nearsighted Nearsighted can be corrected with a diverging lens.  A far object can be focused on retina. Optical Instruments

A Farsighted Power of lens: diopter = 1/f (in m) (+) diopter  converging lens (-) diopter  diverging lens Larger diopter  Stronger lens (shorter f)

MaterialnCornea1.38 Aqueous Humor Lens Vitreous Humor 1.34 Air1.00 Water1.33

Combinations of Thin Lenses The image produced by the first lens is calculated as though the second lens were not present The image produced by the first lens is calculated as though the second lens were not present The light then approaches the second lens as if it had come from the image of the first lens The light then approaches the second lens as if it had come from the image of the first lens The image of the first lens is treated as the object of the second lens The image of the first lens is treated as the object of the second lens The image formed by the second lens is the final image of the system The image formed by the second lens is the final image of the system

Combination of Thin Lenses, 2 If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens If the image formed by the first lens lies on the back side of the second lens, then the image is treated at a virtual object for the second lens p will be negativep will be negative The overall magnification is the product of the magnification of the separate lenses The overall magnification is the product of the magnification of the separate lenses