Download Roorda - A Review of Optics

LIGHT AND THE RETINAL IMAGE: KEY POINTS
Light travels in (more or less) straight lines: the pinhole camera’s inverted
image
Enlarging the pinhole leads to BLUR
How a lens prevents blur: refraction reunites light rays by bending them
Point-to-point projection from object to inverted image
Refraction: which way is light bent? Slowing in glass: lifeguard analogy.
The eye: retina, lens and cornea; fovea, periphery and blind spot
Focus errors; distant vision and near vision
Myopia, hypermetropria, emmetropia, accommodation; emmetropization
Visual angle and image size: q (in radians) = size/distance
q (in degrees) = (180/p ) * size/distance
q (minutes of arc) = 60 * (180/p ) * size/distance
Point spread function: width is 1 minute in visual angle, or 5 microns (.005 mm)
Sources of light spread making the image imperfect:
focus error; chromatic aberration; other aberrations; diffraction
Direct observation of the image: Helmholtz’s ophthalmoscope
Quality of the image: spread is about 5 microns (1 minute of arc)
Visual resolution limit: about 1 minute of arc or 30 cpd (for 20/20 vision)
Can vision be perfected?? William’s magic mirror and laser surgery
Aliasing through sampling by the photoreceptor mosaic: Nyquist limit (60cpd)
A Review of Optics
Austin Roorda, Ph.D.
University of Houston
College of Optometry
(Most of) these slides were prepared by Austin Roorda,
(UC Berkeley Optometry School) and used by
permission.
Geometrical Optics
Relationships between
pupil size, refractive
error and blur
Optics of the eye: Depth of Focus
2 mm
4 mm
6 mm
Optics of the eye: Depth of Focus
Focused
behind
retina
In focus
Focused
in front
of retina
2 mm
4 mm
6 mm
7 mm pupil
Bigger blur
circle
Courtesy of RA Applegate
2 mm pupil
Smaller blur
circle
Courtesy of RA Applegate
Demonstration
Role of Pupil Size and Defocus on Retinal Blur
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.
Physical Optics
The Wavefront
What
is
the
Wavefront?
parallel beam
=
plane wavefront
converging beam
=
spherical wavefront
What
is
the
Wavefront?
parallel beam
=
plane wavefront
ideal wavefront
defocused wavefront
What
is
the
Wavefront?
parallel beam
=
plane wavefront
ideal wavefront
aberrated beam
=
irregular wavefront
What
is
the
Wavefront?
diverging beam
=
spherical wavefront
ideal wavefront
aberrated beam
=
irregular wavefront
The Wave Aberration
What is the Wave Aberration?
diverging beam
=
spherical wavefront
wave aberration
Wave Aberration of a Surface
Wavefront Aberration
mm (superior-inferior)
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
mm (right-left)
2
3
Diffraction
Diffraction
“Any deviation of light rays from a
rectilinear path which cannot be
interpreted as reflection or refraction”
Sommerfeld, ~ 1894
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
rectangular aperture
square aperture
circular aperture
Airy Disc
The Point Spread Function
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.
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
Airy Disk
1.22  
q
a
q
separatrion between Airy disk peak and 1st min
(minutes of arc 500 nm light)
As the pupil size gets larger, the Airy
disc gets smaller.
1.22  
q
a
2.5
q  angle subtended at the nodal point
2
  wavelength of the light
1.5
a  pupil diameter
1
0.5
0
1
2
3
4
5
pupil diameter (mm)
6
7
8
Point Spread Function vs. Pupil Size
1 mm
5 mm
2 mm
3 mm
6 mm
4 mm
7 mm
Small Pupil
Little spreading due to defocus or aberrations
So diffraction is limiting
Larger pupil:
Less diffraction (not shown)
But more blur and more aberrations
Aberrations
Size
Perfect Eye (Diffraction
1 mm
2 mm
3 mm
4 mm
Limited)
5 mm
6 mm
7 mm
Point Spread Function vs. Pupil
Size
Typical
Eye
with 3aberrations
1 mm
2 mm
mm
4 mm
pupil images
followed by
psfs for changing pupil size
5 mm
6 mm
7 mm
Demonstration
Observe Your Own Point Spread Function
Resolution
Unresolved
point sources
Rayleigh
resolution
limit
Resolved
Keck telescope:
(10 m reflector)
About 4500 times
better than the eye!
“Pupil” is 10M:
almost no diffraction
Wainscott
• Compound eye:
• Each facet must
be large to fight
diffraction
• Many facets
(pixels) needed to
capture details
Convolution with the PSF
Convolution
PSF ( x, y )  O( x, y)  I ( x, y )
Simulated Images
20/20 letters
20/40 letters
MTF
Modulation Transfer
Function
low
medium
object:
100%
contrast
contrast
image
1
0
spatial frequency
high
• 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).
MTF: Cutoff Frequency
cut-off frequency
1 mm
2 mm
4 mm
6 mm
8 mm
modulation transfer
1
0.5
f cutoff
a

57.3  
Rule of thumb: cutoff
frequency increases by
~30 c/d for each mm
increase in pupil size
0
0
50 100 150 200 250
spatial frequency (c/deg)
300
Effect of Defocus on the MTF
450 nm
650 nm
Charman and Jennings, 1976
Relationships Between
Wave Aberration,
PSF and MTF
Retinal Sampling
Sampling by Foveal Cones
Projected Image
20/20 letter
Sampled Image
5 arc minutes
Sampling by Foveal Cones
Projected Image
20/5 letter
Sampled Image
5 arc minutes
Nyquist Sampling Theorem
Photoreceptor Sampling >> Spatial Frequency
1
I
0
1
I
0
nearly 100% transmitted
Photoreceptor Sampling = 2 x Spatial Frequency
1
I
0
1
I
0
nearly 100% transmitted
Photoreceptor Sampling = Spatial Frequency
1
I
0
1
I
0
nothing transmitted
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)