Visual Optics I, 20072008

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Visual Optics I, 2007-2008

• Chapter 3
• Retinal Image Quality

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Experimental Grating Resolution
OBJ
Rayleigh Criterion
Fig 3.17 Page 3.19
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Point Spread Function vs. Pupil Size

• Another way to view the interaction between
  diffraction and aberrations is to look at the
  retinal point spread function for various pupil
  diameters
• The point spread function shows the net effect of
  all image-degrading influences (diffraction,
  aberrations, defocus, and others) on an object
  point

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Diffraction-Limited Eye
Notice that the best image (least spread) occurs
with the largest pupil
A . Roorda, University of Houston
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Aberrations increasingly degrade the image for
larger pupils
A . Roorda, University of Houston
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The Strehl Ratio Another View of Resolution
NOT IN NOTES

• Strehl Ratio for any optical system compares peak
  intensity of actual PSF vs. that of the same
  system if it were diffraction-limited
• For a typical eye
• 5 mm pupil, SR 0.05
• 1 mm pupil, SR 1.0

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Q1. The Rayleigh Criterion predicts that the eye
should obtain optimum resolution for a pupil
diameter of

• 1.3 mm
• 3 mm
• 4 mm
• 5 mm

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Resolution Limit of the Eye Anatomy vs. Optics
Page 3.20
This is the optical limit for resolution
according to the Rayleigh Criterion
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Page 3.20
Foveal Cone Mosaic
Figure 3.18 Upper foveal cone mosaic, showing
the region of highest density (smallest cones) in
the foveola (slightly below center). Note that
the pattern is not entirely regular (number of
neighbors per cone). An occasional rod can be
seen toward the edges. Lower the pattern in the
central foveolar region (lower) shows a regular
hexagonal (six-neighbor) cone matrix.
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Anatomical Optimization of the Fovea
Page 3.21
Foveolar hexagonal cone matrix

• Avascular
• Maximum cone density
• 11 projection cone ? ganglion cell
• Outer retinal layers displaced away from fovea
• Macular xanthophyll pigment absorbs blue light
  (reduces scatter and chromatic aberration)

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Anatomical (Receptor) Limit of Resolution
Page 3.21
Foveolar hexagonal cone matrix
To resolve two images as separate, there must be
at least one unstimulated receptor in between the
stimulated receptors
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Anatomical (Receptor) Limit of Resolution
Page 3.21
One unstimulated receptor in between two
stimulated receptors
Angular separation for receptor limit
Figure 3.19 Two closely adjacent sources
subtending angle ? at the nodal point of the
(simplified schematic) eye and stimulating two
receptors separated by one unstimulated receptor.
Based on a cone separation of 2 ?m, angle ?
corresponds to a distance of 4 ?m on the fovea?
This is the receptor limit for resolution.
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Anatomical (Receptor) Limit of Resolution
Page 3.21
Rayleigh Criterion
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Anatomical (Receptor) Limit of Resolution
Page 3.21
OBJ
Rayleigh criterion closely matches receptor limit
? foveal cones match the optics of a
diffraction-limited system
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Nyquist Limit for Resolution of Finite Objects

• 2 ?M cones in central foveola means around 140
  cones per degree
• Typical cited value is 120 cones/degree
• To resolve a finite image, cones must be twice as
  dense as the finest image detail
• The smallest resolvable letter detail would have
  a spatial frequency of 60 cycles/degree
• 20/10 VA chart letters have a SF 60
  cycles/degree
• According to the Nyquist Limit, letters smaller
  than 20/10 cannot be resolved

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Combining Interference Diffraction
Page 3.24
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Construction for First Order Interference Maximum
Page 3.24
Figure 3.20 - Double slit interference with a
distant screen. Construction for finding the
path difference between waves traveling from the
two slits to point Q on a distant screen (making
an angle ? with the axis). Since the
construction is identical geometrically to that
shown in Figure 8 for diffraction, points along
the line AC are all equidistant from Q.
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Construction for First Order Interference Maximum
Page 3.26
Figure 3.21 - Double slit interference with a
distant screen. When BC ?, waves from each
slit are in phase at A and C (equidistant from
point Q on the screen). The result is
constructive interference at the screen.
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Construction for First Order Interference Maximum
Page 3.25
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Interference versus Diffraction same Geometry
Page S8
Figure S5 Same constructions for (a)
interference maxima and (b) diffraction minima
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Interference versus Diffraction
Diffraction at each slit? for BC ? (min 1)
must be much ______ ?
greater
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Interference/Diffraction versus Slit Width
Page S9
Narrow slits
Wider slits
Figure S6 - (a) Double slit interference pattern
for narrow slits. Diffraction has little effect
due to the large angle of diffraction, so the
screen pattern is due to interference alone. (b)
Same slit separation, but wider slits. Solid line
shows net effect at screen. Approaching the first
diffraction minimum, diffraction effects (dashed
line) progressively reduce the intensity of
interference maxima (fourth interference maximum
canceled in this case).
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Single vs Double Slit
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Interference/Diffraction versus Slit Width
Page S10
Figure S7 upper
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Screen Intensity Profile
Slit Width ? e.g. 5 ? 10?4 mm Slit Separation
20 ? 10?2 mm
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Interference/Diffraction versus Slit Width
Page 3.29
Figure 3.24 lower
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Screen Intensity Profile
Slit Width 2? 10?3 mm Slit Separation 20
? 10?2 mm
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Page 3.30
Figure 3.25upper
Slit Width 5? 2.5 ? 10?3 mm Slit Separation 20
? 10?2 mm
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Interference/Diffraction versus Slit Width
Page 3.30
Figure 3.25 lower
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Screen Intensity Profile
Slit Width 10? 5 ? 10?3 mm Slit Separation 20
? 10?2 mm
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Slit Width ? e.g. 5 ? 10?4 mm Slit Separation
20 ? 10?2 mm
Slit Width 2? 10?3 mm Slit Separation 20
? 10?2 mm
Slit Width 5? 2.5 ? 10?3 mm Slit Separation 20
? 10?2 mm
Slit Width 10? 5 ? 10?3 mm Slit Separation 20
? 10?2 mm
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Multi-slit Interference/Diffraction
Page 3.32
Figure 3.26 - Interference patterns for same slit
spacing for (a) two slits, (b) three slits, (c)
four slits, (d) many slits (diffraction grating).
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Diffraction Grating
Page 3.33
Figure 3.27 - Pattern obtained from white light
with a diffraction grating. The central maximum
is white, higher order maxima are wavelength
dependent. Red (longer wavelength) diffracts
through a larger angle than blue (shorter
wavelength)
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Many Slits Diffraction Grating
I
q
Fig 3.26 Page 3.32
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Many Slits Diffraction Grating
Showing all wavelengths
I
Green
Yellow
Blue
Orange
Violet
Red
m 1
m 2
m 0
q
Fig 3.27 Page 3.33
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Practice Problem 3
The two figures below show screen intensity
profiles for a double slit. Slit separation is
the same in both cases, but slit width differs.
In which figure are the slits wider?
1st order Diffraction minimum
Interference maxima
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Practice Problem 3
The two figures below show screen intensity
profiles for a double slit. Slit separation is
the same in both cases, but slit width differs.
In which figure are the slits wider?
1st order Diffraction minimum
A
B
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Practice Problem 3
The two figures below show screen intensity
profiles for a double slit. Slit separation is
the same in both cases, but slit width differs.
In which figure are the slits wider?
A
?
B
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Practice Problem 4
All four figures show screen intensity profiles
for a double slit. In which figure is the slit
separation greatest?
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Practice Problem 4
Greatest slit separation?
Interference
Greatest separation ? smallest angle (maxima
closest together)
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Practice Problem 4
Greatest slit separation?
Interference
Greatest separation ? smallest angle (maxima
closest together)
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Practice Problem 5
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Practice Problem 6
All four figures show screen intensity profiles
for a double slit.
In which figure is the slit width greatest?
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Practice Problem 6
Greatest slit width?
Diffraction
Greatest width ? smallest angle of diffraction
(1st order minimum)
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Practice Problem 6
Greatest slit width?
Diffraction
?
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Greatest slit separation
Smallest slit separation
Greatest slit separation
Widest slit?
Widest slit
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Slit Width 0.5 ?m Slit Sepn 10 ?m
Slit Width 0.25 ?m Slit Sepn 2.5 ?m
Slit Width 2.5 ?m Slit Sepn 10 ?m
Slit Width 2.5 ?m Slit Sepn 5 ?m