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Optical systems:The eye, magnifier, microscope, telescope & binoculars Hecht 5.7

Monday October 7, 2002

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3

Human Eye, Relaxed

3.6 mm

7.2 mm

20 mm

n’ = 1.33

15 mm

F F’H H’

P = 66.7 D

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Accommodation Refers to changes undergone by lens to

enable imaging of closer objects Power of lens must increase There is a limit to such accommodation

however and objects inside one’s “near point” cannot be imaged clearly

Near point of normal eye = 25 cm Fully accommodated eye P = 70.7 for s =

25 cm, s’ = 2 cm

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Myopia: Near Sightedness

Eyeball too large ( or power of lens too large)

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Myopia – Near SightednessFar point of the eye is much less than Far point of the eye is much less than ∞, e.g. ∞, e.g. llff Must move object closer to eye to obtain a clear imageMust move object closer to eye to obtain a clear image

MyopicMyopic

F.P.F.P.

Normal N.P.Normal N.P.

MyopicMyopic

N.P.N.P.

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Myopiae.g. le.g. lff = 2m = 2m

fs

n

l f

1

'

'1

0.5 + 66.7 = 67.2 D0.5 + 66.7 = 67.2 D

is relaxed power of eye – too large!is relaxed power of eye – too large!

To move far point to To move far point to ∞, must decrease power ∞, must decrease power to 66.7to 66.7

Use negative lens with P = -0.5 DUse negative lens with P = -0.5 D

How will the How will the near point be near point be affected?affected?

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Laser Eye surgeryRadial Keratotomy – Introduce radial cuts to the Radial Keratotomy – Introduce radial cuts to the cornea of the elongated, myopic eyeballcornea of the elongated, myopic eyeball

Usually use the 10.6 Usually use the 10.6 µm line of a COµm line of a CO22 laser for laser for

almost 100% absorption by the corneal tissuealmost 100% absorption by the corneal tissue

Front viewFront view

BlurredBlurredvisionvision

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Laser Eye surgeryRadial Keratotomy – Introduce radial cuts to the Radial Keratotomy – Introduce radial cuts to the cornea of the elongated, myopic eyeballcornea of the elongated, myopic eyeball

Usually use the 10.6 Usually use the 10.6 µm line of a COµm line of a CO22 laser for laser for

almost 100% absorption by the corneal tissuealmost 100% absorption by the corneal tissue

Front viewFront viewFlatteningFlattening

Distinct Distinct visionvision

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Hyperopia – Far Sightedness

Eyeball too small – or lens of eye can’t fully accommodateEyeball too small – or lens of eye can’t fully accommodate

Image of close objects formed behind retinaImage of close objects formed behind retina

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Hyperopia – Far Sightedness

Suppose near point = 1mSuppose near point = 1m

Ds

n7.677.661

'

'

1

1

Recall that for a near point of 25 cm, we need 70.7DRecall that for a near point of 25 cm, we need 70.7D

Use a positive lens with 3 D power to correct this Use a positive lens with 3 D power to correct this person’s vision (e.g. to enable them to read)person’s vision (e.g. to enable them to read)

Usually means they can no longer see distant Usually means they can no longer see distant objects - objects - Need bifocalsNeed bifocals

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Correction lenses for myopia and hyperopia

http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/V/Vision.html

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Hand magnifier

Observation with unaided eyeObservation with unaided eye

θθ

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MagnifierTo maximize the image, bring object as close to the eye To maximize the image, bring object as close to the eye as possible, i.e. bring object to near point s = las possible, i.e. bring object to near point s = ln n = 25 cm= 25 cm

θθmaxmax

mm

h

l

h

n 25max

h’h’

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Hand Magnifier

With thin lensWith thin lens

θθ’’

ff

FF

H,H’H,H’

hhMM’’

LL

s’s’

h”h”

ll

ssoo

What is the angular magnification of the What is the angular magnification of the image for an object at the near point or at image for an object at the near point or at ∞?∞?

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Compound Microscope

FFoo FFoo

FFee

FFee

LL

Wish to have intermediate image (h’) Wish to have intermediate image (h’) just inside the focus of the eyepiecejust inside the focus of the eyepiece

s’ s’ ≈ ≈ ffoo + L + L

xx

s = x + s = x + ffoo o

oo fx

fL

s

s

h

hM

''

Recall xx’ = Recall xx’ = ffoo22

x’x’

x’ x’ ≈ L≈ L

x = x = ffoo22/L/L

oo f

LM

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Recall: The magnification of an image formed Recall: The magnification of an image formed

(a)(a) at the near point isat the near point is

(b)(b)at infinity at infinity

Compound Microscope

ee f

cmM

25

FFoo FFoo

FFee

FFee

LL

hh

h’h’

h”h”

125

e

e f

cmM

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Compound microscope

eoeo f

cm

f

LMMM

25

Total magnificationTotal magnification

(Image at infinity)(Image at infinity)

1

25

eoeo f

cm

f

LMMM (Image at near point)(Image at near point)

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Compound Microscope

oo f

LM

ObjectiveObjective EyepieceEyepiece

10 X, 20 X, 40 X etc10 X, 20 X, 40 X etc

10X 10X

In most microscopes, L ~ 16 - 17 cmIn most microscopes, L ~ 16 - 17 cm

cmff

Lo

o

7.110

40X 40X ffoo = 0.4 cm = 0.4 cm

ee f

cmM

25

ffee = 2.5 cm = 2.5 cm

Overall magnification M = 40X10 = 400Overall magnification M = 40X10 = 400

MMee = 10 = 10

(image at (image at ∞)∞)

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Compound Microscope

FFoo FFoo

FFee

FFee

LLA.S.

EnP

ExP

Where should the eye be located to view the image?Where should the eye be located to view the image?

Optimum viewing – Optimum viewing – Place eye near EPlace eye near ExxP (moving eye away decreases illumination and F.O.V.)P (moving eye away decreases illumination and F.O.V.)Ensure that exit pupil ~ same size as eye pupil!Ensure that exit pupil ~ same size as eye pupil!

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Compound Microscope

FFoo FFoo

FFee

FFee

LLA.S.

EnP

ExP

Chief RayChief Ray Marginal rayMarginal ray

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Numerical ApertureMeasure of light gathering powerMeasure of light gathering power

Cover GlassCover Glass

ααgg

ααaa

AirAirOilOil

ααgg’’ααoo

nngg

N. A. = n sin N. A. = n sin αα

LensLens

OO

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Numerical Aperture

aggnAN sin1sin..

If cover glass in airIf cover glass in air

ooogg nnAN sin5.1sin'sin..

If cover glass immersed in oil (no = 1.516) – between glassIf cover glass immersed in oil (no = 1.516) – between glassand oil there is essentially no refraction since nand oil there is essentially no refraction since ngg = 1.5 = 1.5

Increases the light gathering power by about 1.5Increases the light gathering power by about 1.5

(N.A. roughly analogous to f# of a lens)(N.A. roughly analogous to f# of a lens)

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Numerical Aperture

Core nCore n11

Cladding nCladding n22

Cladding nCladding n22

In optical fibresIn optical fibres

nnooααmaxmax

θθθθcc

N.A. = nN.A. = noosinsinααmaxmax= n= n11sin sin θθ = n = n11sin (90sin (90o o - - θθcc) = n) = n11cos cos θθcc2

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121

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1 1 nnn

nn

This is a measure of the maximum cone of light acceptedThis is a measure of the maximum cone of light accepted

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Viewing distant objects, e.g. stars

ffeyeeyeθθ

Image size on retina h = Image size on retina h = ffeyeeyeθθ

starstar

hh

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Telescope

hhTT=f=feyeeyeθθ’’

ffoo ffee

ObjectiveObjective EyepieceEyepiece

s’s’

h”h”

h’h’

θθ’’

θθ’’θθ

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Telescope

ShowShow

e

o

f

fM (magnification of the telescope)(magnification of the telescope)

M

DD oexit (diameter of the exit pupil)(diameter of the exit pupil)

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The Hubble Space TelescopeThe Hubble Space Telescope

2.4 m primary2.4 m primarymirrormirror

0.3 m secondary0.3 m secondarymirrormirror

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BinocularsTwo telescopes side-by-sideTwo telescopes side-by-side

Prisms used to erect imagesPrisms used to erect images

EyepieceEyepiece

ObjectiveObjective

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Binoculars

““6 X 30”6 X 30”

Angular Magnification (M)Angular Magnification (M) Diameter of objective lens, DDiameter of objective lens, Doo (mm) (mm)

M

DD oexit

Exit pupil = 5 mm, a good match to the normal pupil diameterExit pupil = 5 mm, a good match to the normal pupil diameter

For night viewing, a rating of 7 X 50 is better: i.e. ~ 7 mmFor night viewing, a rating of 7 X 50 is better: i.e. ~ 7 mm