Download Review of Basic Principles in Optics, Wavefront

Review of Basic Principles in
Optics, Wavefront and
Wavefront Error
Austin Roorda, Ph.D.
University of California, Berkeley
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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
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 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.
Computation of Geometrical Blur Size
blur[mrad][]blur[minutes]3.44[]DpupilsizemmDpupilsizemm
where D is the defocus in diopters
Application of Blur Equation
• 1 D defocus, 8 mm pupil produces
27.52 minute blur size ~ 0.5 degrees
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: Defocus
Wavefront Aberration
mm (superior-inferior)
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
mm (right-left)
2
3
Wave Aberration: Coma
Wavefront Aberration
mm (superior-inferior)
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
mm (right-left)
2
3
Wave Aberration: All Terms
Wavefront Aberration
mm (superior-inferior)
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
mm (right-left)
2
3
Zernike Polynomials
Wave Aberration Contour Map
2
mm (superior-inferior)
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1
0
1
mm (right-left)
2
Breakdown of Zernike Terms
Zernike term
Coefficient value (microns)
-0.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
0.5
1
1.5
2
astig.
defocus
astig.
trefoil
coma
coma
trefoil
2nd order
spherical aberration
4th order
3rd order
5th order
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
1.22aλθ⋅=
Airy Disk
angle subtended at the nodal point wavelength of the light pupil diameteraθλ≡≡≡
θ
As the pupil size gets larger, the Airy
disc gets smaller.
PSF Airy Disk radius (minutes)
angle subtended at the nodal point wavelength of the
2.5
2
1.5
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
Point Spread Function vs. Pupil Size
1 mm
2 mm
3 mm
4 mm
Perfect Eye
Typical Eye
5 mm
6 mm
7 mm
Larger pupil
Resolution
Unresolved
point sources
Rayleigh
resolution
limit
Resolved
As the pupil size gets larger, the Airy
disc gets smaller.
PSF Airy Disk radius (minutes)
minmin
angle subtended at the nodal point wavelength o
2.5
2
1.5
1
0.5
0
1
2
3
4
5
pupil diameter (mm)
6
7
8
Keck telescope:
(10 m reflector)
About 4500 times
better than the eye!
Convolution
Convolution
(,)
(,) (,)PSFxyOxyIxy⊗=
Simulated Images
20/20 letters
20/40 letters
MTF
Modulation Transfer
Function
low
medium
object:
100%
contrast
contrast
image
1
0
spatial frequency
high
MTF: Cutoff Frequency
1 mm
2 mm
4 mm
6 mm
8 mm
modulation transfer
1
0.5
cut-off frequency
57.3
cutoffafλ=⋅
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
Modulation Transfer Function
0.8
0.6
0.4
0.2
vertical spatial
frequency (c/d)
horizontal spatial
frequency (c/d)
-100
0
c/deg
100
PTF
Phase Transfer
Function
low
medium
object
phase shift
image
180
0
-180
spatial frequency
high
Phase Transfer Function
• Contains information about asymmetry
in the PSF
• Contains information about contrast
reversals (spurious resolution)
Relationships Between
Wave Aberration,
PSF and MTF
The PSF is the Fourier Transform (FT) of the pupil function
()2(,),(,)iWxyiiPSFxyFTPxyeπλ−=
The MTF is the amplitude component of the FT of the PSF
(){},(,)
MTFffAmplitudeFTPSFxy=
xyii
The PTF is the phase component of the FT of the PSF
(){},(,)
PTFffPhaseFTPSFxy=
xyii
The OTF (MTF and PTF) can also be computed as
the autocorrelation of the pupil function
Point Spread Function
Wavefront Aberration
0.5
0
-0.5
-2
-1
0
1
mm (right-left)
Modulation Transfer Function
2
-200 -100
Phase Transfer Function
0.8
0.5
0.6
0
0.4
-0.5
0.2
-100
0
c/deg
100
-100
0
c/deg
100
0
100
arcsec
200
Point Spread Function
Wavefront Aberration
0.5
0
-0.5
-2
-1
0
1
mm (right-left)
Modulation Transfer Function
2
-200 -100
Phase Transfer Function
150
0.8
100
50
0.6
0
0.4
-50
-100
0.2
-150
-100
0
c/deg
100
-100
0
c/deg
100
0
100
arcsec
200
Point Spread Function
Wavefront Aberration
1.5
1
0.5
0
-0.5
-2
-1
0
1
mm (right-left)
Modulation Transfer Function
2
-1000 -500
Phase Transfer Function
150
0.8
100
50
0.6
0
0.4
-50
-100
0.2
-150
-100
0
c/deg
100
-100
0
c/deg
100
0
500 1000
arcsec
Conventional Metrics to
Define Imagine Quality
()()()
Root Mean Square
()()21,, pupil area, wave aberration, avera
Root Mean Square:
Advantage of Using Zernikes to
Represent the Wavefront
te
rm
as
tig
m
at
is
m
.......RMSZZZZ−−=+++
222220212223
de
fo
cu
s
as
te
tig
rm
m
at
is
m
te
rm
tre
fo
il
te
rm
()()()()
……
Strehl Ratio
diffraction-limited PSF
Strehl Ratio = eyedlHH
Hdl
actual PSF
Heye
contrast
Modulation Transfer Function
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20/20
20/10
Area under the MTF
0
50
100
spatial frequency (c/deg)
150
Metrics to Define Image Quality
Other Metrics
Campbell,C.E. (2004). Improving visual function diagnostic metrics with the use of
higher-order aberration information from the eye. J.Refract.Surg. 20, S495-S503
Cheng,X., Bradley,A., Hong,X., & Thibos,L. (2003). Relationship between refractive error
and monochromatic aberrations of the eye. Optom.Vis.Sci. 80, 43-49.
Cheng,X., Bradley,A., & Thibos,L.N. (2004). Predicting subjective judgment of best focus
with objective image quality metrics. J.Vis. 4, 310-321.
Guirao,A. & Williams,D.R. (2003). A method to predict refractive errors from wave
aberration data. Optom.Vis.Sci. 80, 36-42.
Marsack,J.D., Thibos,L.N., & Applegate,R.A. (2003). Scalar metrics of optical quality
derived from wave aberrations predict visual performanc. J.Vis. 4, 322-328.
Sarver,E.J. & Applegate,R.A. (2004). The importance of the phase transfer function to
visual function and visual quality metrics. J.Refract.Surg. 20, S504-S507
Typical Values for Wave Aberration
Strehl Ratio
• Strehl ratios are about 5% for a 5 mm pupil that has
been corrected for defocus and astigmatism.
• Strehl ratios for small (~ 1 mm) pupils approach 1,
but the image quality is poor due to diffraction.
Typical Values for Wave Aberration
Population Statistics
trefoil
coma
coma
trefoil
spherical aberration
Typical Values for Wave Aberration
Change in aberrations with pupil size
rms wave aberration (microns)
1.2
Shack Hartmann Methods
Other Methods
1
Iglesias et al, 1998
Navarro et al, 1998
Liang et al, 1994
Liang and Williams, 1997
Liang et al, 1997
Walsh et al, 1984
He et al, 1999
Calver et al, 1999
Calver et al, 1999
Porter et al., 2001
He et al, 2002
He et al, 2002
Xu et al, 2003
Paquin et al, 2002
Paquin et al, 2002
Carkeet et al, 2002
Cheng et al, 2004
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
pupil size (mm)
8
9
Typical Values for Wave Aberration
Change in aberrations with age
Monochromatic Aberrations as a Function of Age, from Childhood to Advanced Age
Isabelle Brunette,1 Juan M. Bueno,2 Mireille Parent,1,3 Habib Hamam,3 and Pierre Simonet3
Other Optical Factors that
Degrade Image Quality
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 _ of the sampling
frequency.
foveal cone spacing ~ 120 samples/deg
maximum spatial frequency:
60 cycles/deg (20/10 or 6/3 acuity)
MTF: Cutoff Frequency
cut-off frequency
57.3
cutoffafλ=⋅
Nyquist limit
1 mm
2 mm
4 mm
6 mm
8 mm
modulation transfer
1
0.5
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
Thankyou!