1. Austin Roorda, Ph.D. University of Houston College of Optometry
A Review of Optics
Austin Roorda, Ph.D.
University of Houston College of Optometry

2. These slides were prepared by Austin Roorda, except where otherwise noted.
Full permission is granted to anyone who would like to use any or all of these slides for educational purposes.

3. Relationships between pupil size, refractive error and blur
Geometrical Optics
Relationships between pupil size, refractive error and blur
Start with some of the most basic, but profound effects on image quality, which can be explained entirely with geometrical optics. These have to do with the relationships between pupil size, refractive error and blur.
This is the simplest thing to learn, but you could argue that it’s the most serious cause for image quality degradation. After all, how many people are accurately refracted?

4. Optics of the eye: Depth of Focus
When refractive errors are present, then large pupils experience more blur.
2 mm
4 mm
6 mm

5. Optics of the eye: Depth of Focus
Focused behind retina
In focus
Focused in front of retina
2 mm
4 mm
6 mm

6. Courtesy of RA Applegate
7 mm pupil
Bigger blur
circle
Courtesy of RA Applegate

7. Courtesy of RA Applegate
2 mm pupil
Smaller blur
circle
Courtesy of RA Applegate

8. Role of Pupil Size and Defocus on Retinal Blur
Demonstration
Role of Pupil Size and Defocus on Retinal Blur
When do you experience this? You discover this when you are giving a lecture, and you dim the lights. The people in the of the class suddenly have difficulty reading the board. Not because they developed a refractive error. But because their pupils have opened and the refractive error that they can deal with when the light are on produces an intolerable blur with large pupil.
We should never forget these fundamental and simple relationships for image quality. We should not think that aberrations, PSFs, MTFs are so much more important. But we are at a point where the next levels in understanding image quality and how to improve it requires a different way of thinking about how light works. We have to think of light as a wave, and how it interferes and diffracts.
Draw a cross like this one on a page, hold it so close that is it completely out of focus, then squint. You should see the horizontal line become clear. The line becomes clear because you have made you have used your eyelids to make your effective pupil size smaller, thereby reducing the blur due to defocus on the retina image. Only the horizontal line appears clear because you have only reduced the blur in the horizontal direction.

9. Physical Optics The Wavefront
To start with, we will describe the wavefront. This is the one of the most fundamental and useful description of the optical properties of the eye, from which most of the image quality metrics can be derived.

10. What is the Wavefront? parallel beam = plane wavefront converging beam
spherical wavefront
So, what is the wavefront.
Line that is perpendicular to all the rays.
While it is a bit more abstract in the sense of understanding the light paths, it is simpler than rays because many rays can be represented by a single wavefront surface.
Parallel beam = plane or flat wavefront
Converging beam = spherical wavefront

11. What is the Wavefront? parallel beam ideal wavefront = plane wavefront
defocused wavefront
Now, consider if the light is converging to point in front of the image plane, then the wavefront takes a new shape. It is more curved, compared to the ideal wavefront that would be required to focus the light onto the image plane.

12. What is the Wavefront? parallel beam ideal wavefront = plane wavefront
aberrated beam =
irregular wavefront
If the lens has aberrations ….

13. What is the Wavefront? diverging beam = spherical wavefront
aberrated beam =
irregular wavefront
In reverse, a similar ray distortion take place except now that wave aberration is a distortion of an otherwise plane wave.
ideal wavefront

14. The Wave Aberration

15. What is the Wave Aberration?
diverging beam =
spherical wavefront
wave aberration
The wave aberration is a measure of the difference between the ideal wavefront and the actual wavefront. You are able to choose whatever ideal wavefront you want, but you commonly choose the ideal wavefront as one that would focus the light to the image plane, or a plane.
In this example, ideally the light will emerge as a perfect collimated beam, or parallel rays, so the ideal surface is a plane.
Over a pupil, the wave aberration defines a surface, whose height indicates the difference form the ideal surface.

16. Wave Aberration of a Surface-3 -2 -1 1 2 3
Wavefront Aberration
mm (right-left)
mm (superior-inferior)
Here, the different map is plotted across the whole pupil, and looks like a surface, or a contour. They can be plotted in a number of different ways.

17. Diffraction

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

19. 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 geometrical 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

20. Diffraction and Interference
diffraction causes light to bend perpendicular to the direction of the diffracting edge
interference due to the size of the aperture causes the diffracted light to have peaks and valleys

21. rectangular aperture square aperture
Remember one important thing. Smaller apertures generate more diffraction. The closer the edges of the aperture are to each other, the more the perpendicular spread of light. This is counterintuitive but it is true.

22. circular aperture Airy Disc
Because the circular aperture is rotationally symmetric, so is the diffraction pattern. At the focal point of a lens with a circular aperture, you do not get a point, you get an Airy disk pattern.

23. The Point Spread Function

24. The PSF is analogous to the Impulse Response Function in electronics.
The Point Spread Function, or PSF, is the image that an optical system forms of a point source.
The point source is the most fundamental object, and forms the basis for any complex object.
The PSF is analogous to the Impulse Response Function in electronics.

25. The Point Spread Function
The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil.
Airy Disc

26. Airy Disk q

27. As the pupil size gets larger, the Airy disc gets smaller.
2.5 2
1.5
separatrion between Airy disk peak and 1st min
(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 1
0.5 1 2 3 4 5 6 7 8
pupil diameter (mm)

28. Point Spread Function vs. Pupil Size
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
7 mm

29. Small Pupil

30. Larger pupil

31. Point Spread Function vs. Pupil Size Perfect Eye
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
7 mm

32. Point Spread Function vs. Pupil Size Typical Eye
1 mm
2 mm
3 mm
4 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.

33. Observe Your Own Point Spread Function
Demonstration
Observe Your Own Point Spread Function

34. Resolution

35. 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 is used for the size of the Airy disk.
Resolved

36. uncorrected corrected
AO image of binary star k-Peg on the 3.5-m telescope at the Starfire Optical Range
About 1000 times better than the eye!

37. About 4500 times better than the eye!
Keck telescope:
(10 m reflector)
About 4500 times better than the eye!
Wainscott

38. Convolution

39. Convolution

40. Simulated Images
20/20 letters
20/40 letters

41. MTF
Modulation Transfer
Function

42. low medium high object: 100% contrast image contrast spatial frequency
spatial frequency

43. The modulation transfer function (MTF) indicates the ability of an optical system to reproduce (transfer) various levels of detail (spatial frequencies) from the object to the image.
Its units are the ratio of image contrast over the object contrast as a function of spatial frequency.
It is the optical contribution to the contrast sensitivity function (CSF).

44. MTF: Cutoff Frequency 1 mm 1 2 mm 4 mm 6 mm 8 mm 0.5 50 100 150 200
Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size
8 mm
modulation transfer
0.5 50
100
150
200
250
300
spatial frequency (c/deg)

45. Effect of Defocus on the MTF
450 nm
650 nm
Charman and Jennings, 1976

46. PTF
Phase Transfer
Function

47. low
medium
high
object
image
180
phase shift
-180
spatial frequency

48. Relationships Between
Wave Aberration,
PSF and MTF

49. The PSF is the Fourier Transform (FT) of the pupil function
The MTF is the real part of the FT of the PSF
The PTF is the imaginary part of the FT of the PSF

53. Adaptive Optics Flattens the Wave Aberration
AO OFF
AO ON

54. Other Metrics to Define
Imagine Quality

55. Strehl Ratio
diffraction-limited PSF
Hdl
actual PSF
Heye

56. Retinal Sampling

57. Sampling by Foveal Cones
Projected Image
Sampled Image
5 arc minutes
20/20 letter

58. Sampling by Foveal Cones
Projected Image
Sampled Image
5 arc minutes
20/5 letter

59. Nyquist Sampling Theorem

60. Photoreceptor Sampling >> Spatial Frequency1 I 1 I
nearly 100% transmitted

61. Photoreceptor Sampling = 2 x Spatial Frequency1 I 1 I
nearly 100% transmitted

62. Photoreceptor Sampling = Spatial Frequency1 I 1 I
nothing transmitted

63. Nyquist theorem:
The maximum spatial frequency that can be detected is equal to ½ of the sampling frequency.
foveal cone spacing ~ 120 samples/deg
maximum spatial frequency:
60 cycles/deg (20/10 or 6/3 acuity)