1. 生活的需求？
身體的需求？
生命的需求？

2. Special topic: the human eyes

3. The human eyes Eye Components and optical properties
Image quality analysis
Defocus
Diffraction limit
Aberration
Scattering
Adapted from original slide courtesy of Austin Roorda
(vision.berkeley.edu/roordalab)

4. Anatomy of the human eye

6. On-axis image formation
All rays emanating at yi0 arrive at yt2 irrespective of departure angle ai0
Diopter equal to the reciprocal of the focal length measured in metres
Power of the spherical surface
[Unit: diopter D, 1D = 1 m-1]

7. Ambient refractive index of image side
Object at infinity
Image focal length
Ambient refractive index of image side
1/Power
Look in this way:

8. The reduced eye nt=4/3 ni=1 Dioptric power = 60 D
To simplify image formation in the eye we use the reduced eye. The reduced eye has a single refracting surface
nt=4/3
ni=1
Dioptric power = 60 D
The reduced eye is sufficient to explain most aspects of image formation in the eye.
It is ‘reduced’ because there is only one refracting surface, while the real eye has a two surface, aspheric cornea, a pupil and a lens with a gradient refractive index.

9. Components of the human optical system – cornea
Cornea (first surface)
transition from air (n =1)
to front surface of
cornea (n = 1.376)
radius of curvature = 7.7 mm
power:
The first surface of the cornea is responsible for most of the refraction of the eye.

10. Components of the human optical system – cornea
Cornea (second surface)
Transition from back surface
of cornea (n = 1.376) to the
aqueous humor (n = 1.336)
radius of curvature = 6.8 mm
power:
The second surface of the cornea is actually a lens, and it reduces the overall power of the cornea.
total power of cornea ~ +43 D

11. Components of the human optical system – pupil
Function:
Govern image quality
Depth of focus
Control light level?
Size affected by:
Light conditions
Attention
Emotion
Age
The functions of the pupil.

12. Pupil as an aperture stop
The pupil is perfectly located to maximize
the field of view of the eye
Extremely wide field of view
Entrance pupil
Aperture stop .
Cornea
Field of view: the angle subtended from the center of the entrance pupil to the edges of the field stop.
So you don’t feel limited field of view when you bath in sunshine.

13. Pupil vs. light intensity
The range of light intensities in the environment is enormous!
clear
blue
sky
snow in
sunlight
solar
disc
rod
threshhold
room light
10-6
10-5
10-4
10-3
10-2
10-1 1 10
102
103
104
105
106
107
108
109
1010
Lumen
(cd/sqm2)
Rodieck, B. The First Steps in Seeing
Why can you perceive bright and dark environment?
Although pupil do equalize light intensity, its effect is only ~ 16 times.
The eye perceives light on a logarithmic scale (Weber–Fechner law)

14. Components of the human optical system - crystalline lens
Gradient index of refraction
n = at surfaces
n = at the equator
n ~ 1.41 at the center
Little refraction takes place at the surface but instead the light curves as it passes through.
For a homogenous lens to have same power, the overall index would have to be greater than the peak index in the gradient
Total power of lens ~ 21 D

15. Components of the human optical system - ciliary body
Accommodation
The relaxed eye is under tension
at the equator from the ciliary body.
This keeps the surfaces flat
enough so that for a typical eye
distant objects focus on
the retina.
睫狀肌

16. Ciliary body and accommodation
In the accommodated eye, the ciliary muscle contracts and relaxes the tension on the equator of the lens.
Surface curvature increases.
Power of the lens increases.
Power of the accommodated lens ~ D

17. Components of the human optical system - retina
Images are sampled by
millions of rods and cones.
We often discuss vision in terms of visual angle. This is so we can easily go back and forth from object to image space. Large objects outside the eye form small images but the subtended angle is the same.
1 degree of visual angle = 60 minutes or arc = 292 microns on the retina
1 minute of arc = 4.87 microns
Foveal cone is 2.5 microns = 0.5 minutes of arc

18. Retina Fovea: 5 degrees from optical axis
Optic disc: 15 deg from fovea, 10 deg from optical axis.
studftp.stut.edu.tw/~492f0059/Weekly-Eye.htm

19. Fundus image optic disc posterior pole fovea 10 deg 5 deg
Left eye image

20. Optic nerve & blind spot
Close the left eye, focus the right eye on a single point. Keeping your head motionless, with the right eye about 3 or 4 times as far from the page as the length of the red line, look at each character in turn, until the black circle vanishes.
Same for left eye

21. Visual angle
It is the angle subtended at the second nodal point by the image
Equal to the angle subtended at the first nodal point by the object
The nodal points are points in the optical system where the light passing through emerges at the same angle
The second nodal point in the eye is about 16.5 mm from the retina
Consider a 1 mm image on the retina…
We often discuss vision in terms of visual angle. This is so we can easily go back and forth from object to image space. Large objects outside the eye form small images but the subtended angle is the same.
1 degree of visual angle = 60 minutes or arc = 288 microns on the retina
1 minute of arc = 4.87 microns
Foveal cone is 2.5 microns = 0.5 minutes of arc N’ q N q

22. Visual angle Rule of finger Typically expressed in radian
Your fingertip occupies 1o with a straightened arm
Moon subtends about 0.5o
Typically expressed in radian
1 radian = degrees
Do you know why radian is preferred?
1 minute ~ 4.8 mm on retina
1 foveal cone ~ 2.5 mm

23. Image of lunar eclipse on retina at 1 deg from fovea
Moon subtends about 0.5 degrees
Cones at 1 degree from fovea are about 5 microns in diameter.
Moon spans about 144 microns
Image is sampled by about 29 cones across (~650 in total)
Cones at fovea are smaller (~ 2.5um), and thus moon image is sampled by ~60 cones across.
Resolution limited by cone size and density

24. Spatial distribution of rods and cones

25. Spectral response of photoreceptors
wavelength (nm)
normalized spectral absorptance
0.00
0.25
0.50
0.75
1.00
400
450
500
550
600
650
700
L cones
M cones
S cones
rods

26. S, M, and L cone arrangement
JW 1 deg nasal
JW 1 deg temporal
macaque 1.4 deg nasal
AN 1 deg nasal
Roorda, Nature 397, 520 (1999)

27. Transmission of the ocular media
Invisible, useful for imaging purpose
Boettner and Wolter, 1962

28. Effect of pupil size Remember this simple test?
Have you ever heard people say that “I don’t need glass except at night”?
Can you explain it?

29. Depth of focus is a function of pupil size
Hyperopia
只有老花眼鏡可以生火
Now we move to how the optics affect image quality.
First effect is geometrical. How does the pupil size change the depth of focus of the eye?
Myopia
2 mm
4 mm
6 mm

30. Depth of focus is a function of pupil size
Focused behind retina
In focus
The same defocus error causes more blur for larger pupils
Focused in front of retina
2 mm
4 mm
6 mm

31. Computation of geometrical blur
q: visual angle W b q x l
D is defocus in diopters

32. Computation of geometrical blur
where D is the defocus in diopters
1 D defocus, 8 mm pupil produces minute blur size ~ 0.5 degrees
A distant star will become the size of the Moon!
More introduction to other aberrations will be given next week

33. Depth of focus is a function of pupil size
瞇眼看物也是一樣的原理

34. Diffraction
“Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction”
Sommerfeld, ~ 1894

35. Fraunhofer Diffraction
Also called far-field diffraction
Occurs when the screen is held far from the aperture.
Occurs at the focal point of a lens
When a parallel beam passes though an aperture, the light distribution does not simply take the shape of the aperture, like diffraction theory would predict.
Because light interferes with itself, diffraction occurs and the light forms what is called a diffraction pattern. When the aperture is far from the screen, then one type of pattern, called a Fraunhofer pattern is formed.
A Fraunhofer diffraction pattern also forms at the focal point of a lens

36. Fraunhofer Diffraction
rectangular aperture
square aperture
Remember one important thing. Smaller apertures generate more diffraction.

37. The Airy Disc circular aperture Airy Disc
Sir George Biddel Airy: Inventor of spectacles for astigmatism

38. The Airy Disk q

39. Point spread function vs. pupil size perfect eye
1 mm
2 mm
3 mm
4 mm
How bad is the wavefront aberration?
Here is an example from a typical human eye.
5 mm
6 mm
7 mm

40. Resolution Unresolved point sources Rayleigh resolution limit Resolved
Two points are resolved at the Rayleigh resolution limit when the peak of the Airy disc from one point is above the first minimum of the other.
Therefore, the equation for the Rayleigh resolution limit is the same as in the previous slide.

41. Resolution Larger pupil  better resolution
0.5 1
1.5 2
2.5 3 4 5 6 7 8
pupil diameter (mm)
minimum angle of resolution
(minutes of arc 500 nm light)
This shows the inverse relationship between pupil size and potential image quality. Larger pupils can resolve smaller objects. Recall that the human eye can only resolve about 60 c/deg
Larger pupil  better resolution
Recall the pinhole glass: smaller pupil  better resolution
Trade off?

42. Point spread function vs. pupil size typical eye
2 mm
3 mm
4 mm
1 mm
pupil images
followed by
psfs for changing pupil size
5 mm
6 mm
7 mm
How bad is the wavefront aberration?
Here is an example from a typical human eye.
More introduction to other aberrations will be given next week

43. Observe your own point spread function
Or test with a LED

44. Retinal straylight in the human eye
slides courtesy of
Tom van den Berg
Thomas J. T. P. van den Berg, Michiel P. J. Hagenouw, and Joris E. Coppens
The Ciliary Corona: Physical Model and Simulation of the Fine Needles Radiating from Point Light Sources IOVS, 46: (2005).

45. Tom van den Berg

46. Ciliary corona Actual subjective appearance of straylight: a pattern of very fine streaks, not at all like the circularly uniform (Airy disc-like) scattering pattern of particles of approximate wavelength size
Tom van den Berg

47. Central diffraction pattern from 2, 3, 4, 50 randomly placed particles
Tom van den Berg

48. Diffraction pattern for 1000 particles, as a function of wavelength, including spectral luminosity effect.
Tom van den Berg

49. Diffraction pattern for 1000 particles for broad band light
Diffraction pattern for 1000 particles for broad band light. wavelength band = nm
Tom van den Berg

50. Tom van den Berg

51. Optical illusions
Our vision is not solely determined by the ocular imaging system
Brain processing is the dominator

52. Intelligent brain
Play with your right eye again. What happened to the line?

53. Other imaging instruments

54. Other imaging instruments
Single-lens magnifier
Photographic camera
Microscope
Telescope

55. A single-lens magnifier
Can we take a photo of this virtual image?
What is the magnified physical quantity?
Viewing angle

56. Magnifying power The closest point on which the eye can focus
Normal viewing distance ~ 254 mm

57. Magnifying power What is the largest MP? Most common case
Try with a lens very close to your eye
Most common case

58. Single lens magnifier
A 2.5x lens
P = 10 D
f = 0.1 m
MP = 2.5 (L =∞)

59. The camera Pinhole camera
Images of a solar eclipse through a leaf canopy

60. Pinhole size vs. clarity
Pinhole too small: diffraction takes over

61. Single lens reflex camera
Iris diaphragm
(aperture stop)

62. Depth of focus vs. f/# Focal length divided by the aperture diameter.

63. Retina vs. digital camera

64. Retina vs. digital camera

65. The compound microscope
Objective magnification
Eyepiece magnification
Combined magnification

66. The compound microscope
Typical value:
Normal viewing distance: 254 mm
Tube length: L1 = 160 mm
For a 5x objective: fo = 32 mm
For a 10x eyepiece: fe = 25.4 mm
Combined MP: 50x

67. The telescope

68. Astronomical telescope
Infinity conjugates

69. HW 2-1
Ball lens: We intend to use a spherical ball lens of radius R and refractive index n as magnifier in an imaging system, as shown in the figure. The refractive index satisfies the relationship 1 < n < 4/3, and the medium surrounding the ball lens is air (n = 1).
a) Calculate the effective focal length (EFL) of the ball lens. Use the thick lens model with appropriate parameters.
b) Locate the back focal length (BFL), the front focal length (FFL) and the principal planes of the ball lens.
c) An object located at distance d to the left of the back surface of the ball lens, where Show that the object is one half EFL behind the principal plane, and use this fact to find the location of the image plane.
d) Is the image real or virtual? Is it erect or inverted? What is the magnification?

70. HW 2-2
Work out the system matrix for the composite element and use it to answer the following questions.
a) What is the optical power of this composite element?
b) If a plane wave is incident from the left, where will it focus?
c) This system is used to image an object at infinity. Is the image real or virtual?

71. HW 2-3
Mirror-in-a-pool: Consider a perfectly focusing paraboloid mirror filled with a fluid of refractive index n. The mirror surface is described by the equation s(x)= x2/4f, where f is the focal length of the mirror. The fluid is present up to a height of f. Light is incident from the top as shown in the figure. You may neglect the slight reflection that occurs when the light rays go from the air into the fluid.
a) Calculate the portion of the incoming ray bundle which will exit from the fluid as a divergent ray bundle after focusing.
b) Show that the remaining rays will exit as a parallel ray bundle.

72. HW 2-4
It is determined that a patient has a near point at 50 cm. If the eye is approximately 2 cm long.
a) How much power does the refracting system have when focused on an object at infinity? When focused at 50 cm?
b) What power must the eye have to see clearly an object at the standard near-point of 25 cm?
c) How much power should be added to the patient’s vision system by a correcting lens?
d) The preceding calculation overlooks the separation between the correction lens and the eye. Can you find out the best position-power combination of the correction lens?