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

Hecht 5.7

Monday October 7, 2002

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2

Human Eye, Relaxed 20 mm 15 mm n’ = 1.33

F H H’ F’ 3.6 mm 7.2 mm 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 Sightedness

Far point of the eye is much less than

∞, e.g.

l f Must move object closer to eye to obtain a clear image Normal N.P.

Myopic F.P.

Myopic N.P.

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Myopia

e.g. l f = 2m

l

1 

n

'

s

'  1

f f

How will the near point be affected?

0.5 + 66.7 = 67.2 D is relaxed power of eye – too large!

To move far point to

∞, must decrease power to 66.7

Use negative lens with P = -0.5 D 7

Laser Eye surgery

Radial Keratotomy – Introduce radial cuts to the cornea of the elongated, myopic eyeball Usually use the 10.6 µm line of a CO 2 laser for almost 100% absorption by the corneal tissue Blurred vision Front view 8

Laser Eye surgery

Radial Keratotomy – Introduce radial cuts to the cornea of the elongated, myopic eyeball Usually use the 10.6 µm line of a CO 2 laser for almost 100% absorption by the corneal tissue Distinct vision Front view Flattening 9

Hyperopia – Far Sightedness

Eyeball too small – or lens of eye can’t fully accommodate Image of close objects formed behind retina 10

Hyperopia – Far Sightedness

Suppose near point = 1m

1 1 

n

'

s

'  1  66 .

7  67 .

7

D

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

Use a positive lens with 3 D power to correct this person’s vision (e.g. to enable them to read) Usually means they can no longer see distant objects Need bifocals 11

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 eye θ 13

Magnifier

To maximize the image, bring object as close to the eye as possible, i.e. bring object to near point s = l n = 25 cm h’ θ max

 max 

h l n



h

25

mm

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Hand Magnifier

With thin lens What is the angular magnification of the image for an object at the near point or at

∞?

H,H’ h” θ’ F s’ f L s o l 15 h M ’

Compound Microscope

x’ L F e F o F o x Wish to have intermediate image (h’) just inside the focus of the eyepiece s’ ≈

f

o + L s = x +

f

o

M o



h

'  

s

'

h s

 

L



x



f f o o

F e Recall xx’ =

f

o 2 x’ ≈ L x =

f

o 2 /L

M o

 

L f o

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Compound Microscope

h h” F o L F o h’ F e F e Recall: The magnification of an image formed (a) at the near point is

M e

 25

cm

 1

f e

(b)at infinity

M e

 25

cm f e

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Compound microscope

Total magnification

M



M o M e

 

L f o

  25

cm f e

 

M



M o M e

 

L f o

  25

cm f e

 1  

(Image at infinity) (Image at near point) 18

Compound Microscope

In most microscopes, L ~ 16 - 17 cm Objective Eyepiece

M o

 

L f o

10 X, 20 X, 40 X etc 10X



L f o

 10 

f o

 1 .

7

cm

40X



f o = 0.4 cm

M e

 25

cm f e

(image at

∞)

f

e = 2.5 cm M e = 10 Overall magnification M = 40X10 = 400 19

Compound Microscope

L

A.S.

F e F o F o F e

E n P E x P

Where should the eye be located to view the image?

Optimum viewing –



Place eye near E x P (moving eye away decreases illumination and F.O.V.)



Ensure that exit pupil ~ same size as eye pupil!

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Compound Microscope

L

A.S.

F e F o F o F e

E n P E x P

Chief Ray Marginal ray 21

Numerical Aperture

Measure of light gathering power N. A. = n sin α Cover Glass Oil α g ’ Air α g α o Lens α a O n g 22

Numerical Aperture

If cover glass in air

N

.

A

.



n g

sin 

g

 sin 

a

If cover glass immersed in oil (no = 1.516) – between glass and oil there is essentially no refraction since n g = 1.5

N

.

A

.



n g

sin 

g

' 

n o

sin 

o

 1 .

5 sin 

o

Increases the light gathering power by about 1.5

(N.A. roughly analogous to f# of a lens) 23

Numerical Aperture

In optical fibres Cladding n 2 θ θ c n o α max Core n 1 Cladding n 2 N.A. = n o sin α max = n 1 sin θ = n 1 sin (90 o θ c ) = n 1 cos θ c



n

1 1 

n

2 2

n

1 2 

n

1 2 

n

2 2

This is a measure of the maximum cone of light accepted 24

Viewing distant objects, e.g. stars

star θ Image size on retina h =

f

eye θ

f

eye h 25

θ h”

Telescope

Objective

f

o

f

e Eyepiece θ’ h’ s’ θ’ h T =f eye θ’ 26

Telescope

M

 

f o f e D exit



D o M

Show (magnification of the telescope) (diameter of the exit pupil) 27

The Hubble Space Telescope 0.3 m secondary mirror 2.4 m primary mirror 28

Binoculars

Two telescopes side-by-side Prisms used to erect images Objective Eyepiece 29

Binoculars

“6 X 30” Angular Magnification (M) Diameter of objective lens, D o (mm)

D exit



D o M

Exit 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 mm 30