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Revision problem

Chapter 18 problem 37 page 612

Suppose you point a pinhole camera at a 15m tall tree that is 75m away….

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Optical Instruments

• Thin lens equation

• Refractive power

• Cameras

• The human eye

• Combining lenses

• Resolution

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Optical Instruments - continued

Optical imaging and color in medicine

Integral part of diagnosis

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Thin lens equation

Instead of using ray tracing, we can use similar triangles to find the relationship between f, s and s’

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Thin lens equation

Magnification triangles:

s

s

h

hm

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Thin lens equation

Focusing triangles:

f

fs

h

h

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Thin lens equation

Combining

sffs

fs

s

s

s

f

fs

h

h

111

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Thin lens equation

• Focal length, f

• Distance from object to lens, s

• Distance from image to lens, s’

ssf

111

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Sign conventions

• Object distance, s

• is always positive for this course.

• Focal length, f

• is positive for converging lens, or concave mirror

• Is negative for diverging lens or convex mirror

• Magnification, M, and image height, h’

• are positive when image is upright

• are negative when image is inverted

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Sign conventions

• Image distance s’

• Is positive for real images

• Is negative for virtual images

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Sign Conventions for Lenses and Mirrors

Slide 19-11

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Magnification

• Now use a sign convention, to indicate whether image is upright (positive) or inverted (negative)

s

s

h

hM

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Refractive power

A thicker lens will refract light at a larger angle and have a shorter focal length, f.

We define the refractive power, P, as

Measured in diopters, 1D=1m-1

fP

1

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Refractive power of lenses in contact

If two lenses are touching (or at least, very close), their refractive powers add.

Useful for lenses which are close together – such as corrective eye lenses

Measured in diopters, 1D=1m-1

21 PPPtotal

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Camera

• Simple single lens camera.

• Image is focused by a convex lens

• Shutter used to allow the light into the camera

• Recorded on CCD (used to be photosensitive paper, 35mm in width)

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Camera

CCD (Charge Coupled Device) is a 2D array of 1to >20 million pixels – each of which is a photosensitive semiconductor with color filter

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Camera

• Focusing achieved by moving the lens towards or away from the image.

• Exposure is controlled by changing the diameter of an iris behind the lens and the shutter time

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Camera exposure

• Exposure is related to the amount of light which is recorded.

• Controlled by shutter speed and iris size

• Shutter speed is the time the shutter is open.

• Needs to be shorter for fast moving images

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Camera exposure

• Shutter speed is the time the shutter is open.

• Needs to be shorter for fast moving images

• Expressed as fractions of a second – 1/500s to 1/30s

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Camera exposure

•Iris size controls the effective diameter of the lens

•Measured as the f-number, the ratio of the diameter of the lens, d, and the focal length

Focal length, f is fixed, and light intensity goes as area, (d2 ), or 1/(f-number)2

Labeled as f-stops on a camera

d

fnumberf

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Human Eye

• Focusing by the fixed cornea, and the variable lens

• Exposure controlled by the iris

• Recorded by the retina which contains photosensitive cells

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Human Eye Focusing

• The cornea acts as a fixed lens.

• Corrections to the focusing applied by stretching the ciliary muscles to curve the lens, called accommodation

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Human Eye Focusing

• Far point – lens muscles relaxed – longest focal length

• Near point – lens muscles fully contracted, shortest focal length

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Corrective lenses

Two common types of conditions require corrective lenses

• Myopia or near sightedness rays converge in front of the retina when the lens muscles are relaxed

• Hyperopia or far sightedness rays converge behind the retina when the lens muscles are relaxed

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Correcting Myopia

Add a concave lens to diverge the light rays (negative focal length)

This increases the far point

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Correcting Hyperopia

Add a convex lens

Occurs when the eye is about 50 years old, and the lens becomes less elastic, and cannot curve.

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Simple Magnifying lens

Increases the apparent size of an object.

Angular size for the magnified object is now

f

htan

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Simple Magnifying lens

• Increases the apparent size of an object.

• Compare the angular size at near point and for the magnified object

• Magnifies up to 20f

cmM

cm

h

f

h

near

magnified

near

magnified

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25

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Compound Microscope

Simplest form contains two lenses

• Objective lens to create real image

• Eyepiece lens to magnify real image

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Microscope

Magnification from the objective lens

obj

objf

L

s

sM

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Microscope

Magnification from the eyepiece lens

eye

eyef

cmM

25

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Microscope

Total magnification is the product of the two

eyeobj

eyeobjtotalf

cm

f

LMMM

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Telescope

Two stage magnification, but with weaker objective lens

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Telescope

We want the angular magnification

obj

eyeM

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Telescope

Objective lens angle

obj

objf

h

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Telescope

Eyepiece lens angle

eye

eyef

h

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Telescope magnification

Total magnification

eye

obj

obj

eye

f

fM

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Reflecting Telescope

Need large aperture to capture more light –large objective lens.

Easier to make a mirror than a lens, Newton invented a reflecting telescope.

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Resolution of optical instruments

Imperfections in the lens are called aberrations

Two main types

• Spherical aberration – poor focusing

• Chromatic aberration – color dispersion n(λ)

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Correcting aberrations

• Spherical aberration – remove the edges of the lens, using a smaller iris, but reduces image intensity

• Chromatic aberration – use 2 lenses

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Resolution from the wave model

• Telescopes, microscopes and lenses all have dimensions >> λ

• Images do not, however, when the instruments are used at their limits of resolution

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Resolution from the wave model

• To separate two circular images, we would get 2 circular diffraction patterns

• Airy disk – with ring fringes.

• The central disk has a radius

D

22.1

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Telescope Resolution

• Called Rayleigh’s criterion, relates the angular resolution α, wavelength, λ, and object lens diameter

D

22.1

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Resolution of a Microscope

At the object end of a microscope, the angular separation, θmin, and minimum resolvable distance, dmin will be

D

ffd

D

22.1

22.1

minmin

min

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Resolution of a Microscope

We replace D with 2f tanΦ, which is nearly 2f sinΦ.

sin

61.0min d

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Resolution of a Microscope

Some microscopes use a transparent oil which decreases the λ, and decreases the minimum resolution

sin

61.0min

nd o

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Resolving power of a Microscope

The resolving power of a microscope is defined by

Where NA is the numerical aperture

NARPd o61.0

min

sinnNA

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Resolving power of a Microscope

Values of the numerical aperture are around 1 for an immersion microscope, so the resolving power of a microscope can be as small as 0.5λ, half the wavelength of light.

Smaller wavelengths can be obtained by using electron microscopes, where the object is irradiated with beams of electrons, to get from 2000x magnification to x1,000,000x

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Summary

• Thin lens equation

• Refractive power

• Cameras

• The human eye

• Combining lenses

• Resolution