Chapter 27Chapter 27

Lenses and Optical Lenses and Optical InstrumentsInstruments

Lenses

Converging lens

Diverging lens

Thin LensesThin Lenses

A thin lens consists of a piece of A thin lens consists of a piece of glass or plastic, ground so that each glass or plastic, ground so that each of its two refracting surfaces is a of its two refracting surfaces is a segment of either a sphere or a segment of either a sphere or a planeplane

Lenses are commonly used to form Lenses are commonly used to form images by refraction in optical images by refraction in optical instrumentsinstruments

Thin Lens ShapesThin Lens Shapes

These are These are examples of examples of convergingconverging lenses lenses

They have positive They have positive focal lengthsfocal lengths

They are thickest They are thickest in the middlein the middle

More Thin Lens ShapesMore Thin Lens Shapes

These are These are examples of examples of divergingdiverging lenses lenses

They have They have negative focal negative focal lengthslengths

They are thickest They are thickest at the edgesat the edges

Glass lens (nG = 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 fLenses 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.Q. A parallel beam of light is sent through an aquarium. A parallel beam of light is sent through an aquarium. If a convex lens is held in the water, it focuses the beam (……..If a convex lens is held in the water, it focuses the beam (……..……………………. ) than outside the water……………………. ) than outside the water

(a) closer to the lens (b) at the same position as

(c) farther from the lens

nair = 1, nwater = 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: f1 and f2

Parallel beams: image at infinite!!

Virtual image

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.

positivepositive negativenegative

pp real object real object imaginary objectimaginary object

(multiple lenses)(multiple lenses)

qq real imagereal image

(opposite side of object)(opposite side of object)imaginary imageimaginary image

(same side of object)(same side of object)

ff for converging lensfor converging lens for diverging lensfor diverging lens

MM erect imageerect image inverted imageinverted image

two focal points: f1 and f2

Parallel beams: image at infinite!!

1/p + 1/q = 1/f1/2f + 1/q = 1/f1/q = 1/2f

M = -q/p = -1

1/p + 1/q = 1/f1/f + 1/q = 1/f1/q = 0 q = infinite

Virtual image

Virtual image

Magnifying glass

1/p + 1/q = 1/f2/f + 1/q = 1/f1/q = -1/f

M = -(-f)/(f/2) = 2

Ex. 27.1 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

MagnifierMagnifier Consider small object held in front of eyeConsider small object held in front of eye

• Height Height yy• Makes an angle Makes an angle at given distance from the at given distance from the

eyeeye Goal is to make object “appear bigger”: Goal is to make object “appear bigger”: '' > >

y

MagnifierMagnifier Single converging lensSingle 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 Angular size of object is , bigger!, bigger!

Outgoingrays Rays seen coming

from here

ff Image atInfinity

1 1 1

q f p

y

Angular Magnification Angular Magnification (Standard)(Standard)

Without magnifier: 25 cm is closest distance to viewWithout magnifier: 25 cm is closest distance to view• Defined by average near point. Younger people do Defined by average near point. Younger people do

betterbetter tan tan = = yy / 25 / 25

With magnifier: put object at distance With magnifier: put object at distance pp = = ff '' tan tan '' = = yy / / ff

Define “angular magnification” Define “angular magnification” mm = = ' / ' / Note that magnifiers work better for older people because Note that magnifiers work better for older people because

near point is actually > 25cmnear point is actually > 25cm

~y/25’~y/fM= ’/ = 25/f

ExampleExample

Find angular magnification of lens Find angular magnification of lens with with ff = 5 cm = 5 cm

255 Standard

525

1 6 Maximum5

m

m

Eye Glasses

Perfect EyeNearsighted

Nearsighted can be corrected with a diverging lens. A far object can be focused on retina.

Optical Instruments

AA

Farsighted

Power of lens: diopter = 1/f (in m) (+) diopter converging lens (-) diopter diverging lens

Larger diopter Stronger lens (shorter f)

MaterialMaterial nn

CorneaCornea 1.381.38

Aqueous Aqueous HumorHumor

1.33-1.33-1.341.34

LensLens 1.41-1.41-1.451.45

Vitreous Vitreous HumorHumor

1.341.34

AirAir 1.001.00

WaterWater 1.331.33

Combinations of Thin LensesCombinations of Thin Lenses

The image produced by the first lens is The image produced by the first lens is calculated as though the second lens calculated as though the second lens were not presentwere not present

The light then approaches the second The light then approaches the second lens as if it had come from the image of lens as if it had come from the image of the first lensthe first lens

The image of the first lens is treated as The image of the first lens is treated as the object of the second lensthe object of the second lens

The image formed by the second lens is The image formed by the second lens is the final image of the systemthe final image of the system

Combination of Thin Lenses, 2Combination of Thin Lenses, 2

If the image formed by the first lens If the image formed by the first lens lies on the back side of the second lies on the back side of the second lens, then the image is treated at a lens, then the image is treated at a virtual objectvirtual object for the second lens for the second lens• p will be negativep will be negative

The overall The overall magnificationmagnification is the is the product of the magnification of the product of the magnification of the separate lensesseparate lenses