Transcript Roorda - A Review of Optics

A Review of Optics
Austin Roorda, Ph.D.
University of Houston
College of Optometry
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.
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
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!
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
Larger pupil
Point Spread Function vs. Pupil Size
Perfect Eye
1 mm
5 mm
2 mm
3 mm
6 mm
4 mm
7 mm
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
Demonstration
Observe Your Own Point Spread Function
Resolution
Unresolved
point sources
Rayleigh
resolution
limit
Resolved
uncorrected
corrected
AO image of binary star k-Peg on the 3.5-m
telescope at the Starfire Optical Range
q min
1.22   1.22  900109


 0.064secondsof arc
a
3.5
About 1000 times better than the eye!
Keck telescope:
(10 m reflector)
About 4500 times
better than the eye!
Wainscott
Convolution
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
PTF
Phase Transfer
Function
low
medium
object
phase shift
image
180
0
-180
spatial frequency
high
Relationships Between
Wave Aberration,
PSF and MTF
The PSF is the Fourier Transform (FT) of the pupil function
2
i W ( x, y ) 

PSF  xi , yi   FT  P( x, y )e 



The MTF is the real part of the FT of the PSF
MTF  f x , f y   Amplitude  FT  PSF ( xi , yi )
The PTF is the imaginary part of the FT of the PSF
PTF  f x , f y   Phase  FT  PSF ( xi , yi )
Adaptive Optics Flattens the Wave Aberration
AO OFF
AO ON
Other Metrics to Define
Imagine Quality
Strehl Ratio
diffraction-limited PSF
Strehl Ratio =
Hdl
H eye
H dl
actual PSF
Heye
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)