the human eye has the same 23mm focal length

Phys 531
Lecture 11
7 October 2004
Survey of Optical Systems
Last time:
Developed tools to analyze optical systems
- ray matrix technique
- thick lens picture
Today, look at several common systems
Won’t use matrix methods explicitly
but many lenses thick:
implicitly use thick lens picture
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Outline:
• the eye
• eye glasses
• magnifying glass
• microscope
These and more examples: Hecht 5.7
Next time:
advice for designing systems of your own
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The Eye (Hecht 5.7.1)
Most basic optical system
Components:
- cornea: n = 1.376
- vitreous humor ≈ water: n = 1.33
- lens: n = 1.39 − 1.41
index varies: high in center, low at edges
- iris: variable aperture stop
diameter 2-8 mm
- retina: detector
Again, use thick lens picture
lens system has well defined focal length
3
4
Note : detector is in medium ni = 1.33
Lens equation becomes
fo
fi
n
n
1
1
+ i=
= i
so
si
fo
fi
= object focal length
= image focal length
Irrelevant for ray matrix
Modifies thick lens picture:
fo applies to front focal point
fi applies to back focal point
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System focal length variable
max object distance = ∞ (“relaxed”)
min object distance ≈ 25 cm (varies)
What focal lengths?
Distance from lens to retina ≈ 24 mm ≈ si
Relaxed eye: so = ∞ ⇒ fi = si
For so = 25 cm:
ni
n
1
+ i ⇒ fi = 22 mm
=
fi
so
si
So fi = 22 − 24 mm
Called “accommodation”
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Most focusing power from cornea-air interface
n
fi =
R
n−1
R ≈ 9 mm ⇒ fi ≈ 33 mm
Remaining surfaces:
1
ftotal
≈
1
fcornea
+
1
frest
So
1
1
1
−
=
frest
24 33
88 mm
for relaxed eye
frest = 66 mm for accommodated eye
1
=
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frest adjusted by squeezing lens
muscles relaxed: f long
muscles tense: f short
Minimum achievable so = near point
depends on flexibility of lens
varies with age
Question: Why can’t you see well under water?
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Capabilities of the eye
Resolution:
angular resolution ∆θ ≈ 0.017◦ = 0.3 mrad
Just adequate to resolve crescent of Venus
Correpsonds to about 5 µm on retina
At so = 25 cm,
spatial resolution = so∆θ = 75 µm
Also, wide field of view:
corresponds to 100 Mpixels!
Resolution best in center
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Sensitivity:
Fully expanded pupil, can see I ≤ 10−10 W/m2
from point source
Power = IA
Area = π(4 mm)2 ⇒ P ≈ 10−14 W
Maximum irradiance:
sunlight I ≈ 250 W/m2
pupil area π(1 mm)2
Max power = 10−3 W
But: sun is not point source
power spread out on retina
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Sun subtends angle 10 mrad ≈ 30 × ∆θ
Same intensity from point source:
illuminate area 302 × smaller on retina
≈ 1000× higher image irradiance
Max power from point source ≈ 10−6 W
(≈ damage threshold for laser)
Dynamic range of eye: 10−14 to 10−6 W
eight orders of magnitude
Instantaneous range lower:
∼ five orders of magnitude
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Best artificial detectors:
photographic film
high-end CCDs
dynamic range ≈ four orders of magnitude
10× worse than eye
Upshot:
Can’t build a detector nearly as good as the eye
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Eyeglasses (Hecht 5.7.2)
Common problem: focal length of eye isn’t right
Too strong = near sighted = myopic:
relaxed eye has fi < 24 mm
1
n
n
− >0
somax
f i si
so can’t focus at ∞
=
Maximum distance of focus = far point
Easy to measure
For me, far point ≈ 25 cm
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Also moves near point closer:
for me somin ≈ 7 cm
What is my range of f ? (assuming si = 24 mm)
1
fmin
=
1
1
+
1.33 · 70 mm
24 mm
⇒ fmin = 19 mm
and
1
fmax
=
1
1
+
1.33 · 250 mm
24 mm
⇒ fmax = 22 mm
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Fix with eye glasses
Relaxed eye:
25 cm
Add lens to put image of ∞ at 25 cm:
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What focal length required?
want so = ∞, si = −25 cm
So f = −25 cm
This is my prescription:
D=
1
= −4 diopters
f
How close is near point with glasses on?
so such that si = -7 cm for f = −25 cm
1
1
1
1
=−
+ =
so
25
7
10 cm
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Other vision problems:
Far sighted = hyperopia:
eye’s lens too weak
Correct with positive lens
Astigmatism:
asymmetry in lens
f ’s different along x, y
Correct with cylindrical lens
Question: If you want to start a fire with your glasses,
should you be near-sighted or far-sighted?
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Magnifying glass (Hecht 5.7.3)
At 25 cm, typical eye can resolve 75 µm
Use a lens to see something smaller. . .
what kind?
Want erect, magnified image of real object
• Real object: so > 0
• Erect: m = −si/so > 0 so si < 0
• Magnified: m > 1 so |si| > |so|
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Have
1
1
1
+ =
so
si
f
Want 1/so positive and large
1/si negative and small
Means f should be positive
Recall: get virtual image with positive lens
when so < f
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Picture:
si
so
See image is magnified
But also further away. . .
Resolution improved if image on retina is bigger
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Note size of image on retina proportional
to angular size of object
α
Size on retina = αf
Don’t really know f , just consider α
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Define magnifying power of system (MP)
= angular magnification
α with lens
α without lens
Write as MP = 5×, etc.
=
Could make α without lens very big:
hold object right up to eye
But can’t focus if so < near point
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For magnifying glass, microscope, etc
(not telescope)
Define α for standard distance so = 25 cm
Example:
If I take off my glasses, near point is 7 cm
Object at 7 cm subtends α = y/7 cm
Object at standard 25 cm subtends α0 = y/25 cm
Magnifying power =
α
25
= 3.6
=
α0
7
My bare eyes have MP = 3.6×
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What is MP of glass?
si
so
d
L
Express in terms of practical parameters
d = distance from eye to glass
L = distance from eye to image
f = focal length of glass
Have object size yo
image size yi
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myo
y
Angular size of image α = i =
L
L
magnification m = −si/so:
s yo
α=− i
so L
Without glass α0 =
So MP =
yo
do
α
s do
=− i
α0
so L
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Eliminate so, si:
Have si = d − L
and
So
1
1
1
+ =
so
si
f
d−L
s
si
−1
= i −1=
so
f
f
!
L − d d0
Gives M P = 1 +
f
L
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Two reasonable ways to use:
• Make L → ∞
Achieve by making so → f (so si → ∞)
View image with relaxed eye, d doesn’t matter
d
Get MP → 0
f
Large MP for small f
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Other method:
• Lens close to eye d → 0
1
1
MP → d0
+
L
f
!
To get large MP, want L small
minimum L = near point = d0
d
Then MP → 1 + 0
f
Large if f is small
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Where do we need to put object?
1
1
1
= −
so
f
si
1
1
= +
f
d0
!
1
d
=
1+ 0
d0
f
d0
MP
Recall eye example: MP = d0 /so . . . same
So so =
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Object looks as it would if you could focus at so
Lens makes eye stronger
like being near-sighted
Works well if you can hold object up to eye
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Either method works up to about 4×
f down to 6 cm
For higher MP, not paraxial:
- lens aberrations important
- requires more complex lens
Still works:
Method 2: Jeweler’s loupe
Get MP up to 30×
Impractical if object position fixed
or MP so high you can’t hold steady
Already a problem at 10×
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Method 1: put lens very close to object
since so ≈ f and f is small
Problem: exit pupil is small and far away
- can’t see very much
Question: Where is the exit pupil in this case?
Solution: compound microscope
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Microscope (Hecht 5.7.5)
Use two lenses:
objective: short f , close to object
eyepiece: collects light, match to eye pupil
Typical system:
objective
eyepiece
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Typically don’t care about inversion:
- object creates real inverted image
Objective collects rays at steep angles:
- important to control aberrations
Eyepiece:
• puts final image at ∞
- view with relaxed eye
• provides additional magnification
• matches exit pupil to eye
Aberrations less important than in objective
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What is magnifying power?
Objective: magnification = −si/so
Angular magnification not appropriate:
intermediate image not viewed by eye
Even so, there is a standard length scale
Want objectives interchangeable:
standard position for object, image
Set by tube length
= distance from back focal point to image
= 160 mm
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Then si = 160 mm + f
1
1
1
= −
so
f
si
s
s
MP = i = i − 1
so
f
160 mm + f
160 mm
−1=
f
f
This is magnification written on objective
=
So 20× objective has f = 8 mm
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Total MP = MPobj · MPeyepiece
=
160 mm 250 mm
·
fobj
feyepiece
where MP for eyepiece follows standard convention
Typical objective: 5× to 60×
Typical eyepiece: 5× or 10×
Warning:
Fancy microscopes don’t follow these conventions
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Summary:
• Eyes are impressive instruments, both for
sensitivity and resolution
• Eyeglasses/contacts work by adjusting
location of near and far points
• Effect of magnifying glass is really angular magnification
- Measure with magnifying power
• Microscope uses two lenses to provide
more MP than glass
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