aberrations of the eye, measurement
TRANSCRIPT
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Aberration Theory
Geunyoung Yoon, Ph.D.
Assistant ProfessorDepartment of Ophthalmology
Center for Visual ScienceUniversity of Rochester
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Optics
Paraxial Optics(First Order Optics)(Gaussian Optics)
Geometrical Optics(Aberration theory)
Diffractive Optics(Fourier Optics)
Quantum Optics
Coherent Optics
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Outline�What Is Wavefront?
Huygens’s principle, Snell’s law, Fermat’s principleParaxial (first order) approximation
�What Kind Of Wavefront Aberrations Are There?Monochromatic aberration (Seidel and wave aberrations)Chromatic aberration (Longitudinal, Transverse)
�Why Are These Aberrations Important?Relationship between aberrations and image quality(Pupil function, PSF, MTF, Image convolution…)
�How Can We Measure These Aberrations Of The Eye?Different types of wavefront sensors
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Outline�What Is Wavefront?
Huygens’s principle, Snell’s law, Fermat’s principleParaxial (first order) approximation
�What Kind Of Wavefront Aberrations Are There?Monochromatic aberration (Seidel and wave aberrations)Chromatic aberration (Longitudinal, Transverse)
�Why Are These Aberrations Important?Relationship between aberrations and image quality(Pupil function, PSF, MTF, Image convolution…)
�How Can We Measure These Aberrations Of The Eye?Different types of wavefront sensors
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Outline�What Is Wavefront?
Huygens’s principle, Snell’s law, Fermat’s principleParaxial (first order) approximation
�What Kind Of Wavefront Aberrations Are There?Monochromatic aberration (Seidel and wave aberrations)Chromatic aberration (Longitudinal, Transverse)
�Why Are These Aberrations Important?Relationship between aberrations and image quality(Pupil function, PSF, MTF, Image convolution…)
�How Can We Measure These Aberrations Of The Eye?Different types of wavefront sensors
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Outline�What Is Wavefront?
Huygens’s principle, Snell’s law, Fermat’s principleParaxial (first order) approximation
�What Kind Of Wavefront Aberrations Are There?Monochromatic aberration (Seidel and wave aberrations)Chromatic aberration (Longitudinal, Transverse)
�Why Are These Aberrations Important?Relationship between aberrations and image quality(Pupil function, PSF, MTF, Image convolution…)
�How Can We Measure These Aberrations Of The Eye?Different types of wavefront sensors
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What Is Wavefront?
Huygens’s principleSnell’s law
Fermat’s principleParaxial (first order) approximation
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Wavefront vs Ray“A wavefront is a surface over which an optical disturbance has a constant phase.”
Harmonic wave function )sin(),( tkxAtx ωψ −=
Phase
Plane wavefrontSpherical wavefront
x
x
x
x
t=0 t
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Wavefront vs Ray“Rays are lines normal to the wavefronts at every point of intersection.”
Plane wavefront Spherical wavefront
Ray
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Huygens’s Principle“Every point on a primary wavefront serves as the source of spherical secondarywavelets, such that the primary wavefront at some later time is the envelope ofthese wavelets.”
Primaryspherical wavefront
Secondaryspherical wavefront
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Snell’s law
Fast medium(smaller refractive index, ni)
Slow medium(larger refractive index, nt)
θi
θt
θr
normal
)sin(
)sin(
i
t
t
i
n
n
θθ
=
Reflection:
Refraction:
θi = θr
θi > θt when ni < nt
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Fermat’s Principle
OPnSOnOPL ti ⋅+⋅=
2222 )( xabnxhn ti −+⋅++⋅=
OPnSOnOPL ti ⋅+⋅=
“The path actually taken by light in going from some point S to a point Pis the shortest optical path length (OPL).”
ni
θi
θt
nt
P
O
S
θr
P
h
b
x a-x
a
= 0 to minimize OPLdOPLdx
0)(
)(2222=
−+
−−⋅+
+⋅
xab
xan
xh
xn ti
)sin(
)sin(
i
t
t
i
n
n
θθ
=
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⋅⋅⋅+−+−=!7!5!3
sin753 θθθ
θθ
n1
θ
n2
SP
so si
po piR
R
nn
s
n
s
n
io
1221 −=+
i
i
o
o
p
RsRn
p
RsRn θθ sin)(sin)( 21 −=
+
θθ ≈sinApproximation
Paraxial Optics (First order optics)
Lens maker’s formula
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Paraxial Optics (First order optics)
n1 n2
SP
“The emerging wavefront segment corresponding to these paraxial rays isessentially spherical and will form a ”perfect” image at its center P.”
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Third Order Optics
!3sin
3θθθ −=
n1 n2
SP
“The paraxial approximation, sin_ ≈ _, is somewhat unsatisfactoryif rays from the periphery of a lens are considered.”
Paraxial rays
Perpheral raysAberrations
−+
++
−=+
2
2
2
121221 11
2
11
2 iiooio sRs
n
Rss
nh
R
nn
s
n
s
n
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What Kinds Of Wavefront AberrationsAre There?
Monochromatic aberration(Seidel and wave aberrations)
Chromatic aberration(Longitudinal, Transverse)
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Monochromatic aberrations (Seidel aberrations)
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
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�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Paraxialrays
Marginal rays
Lens
TransverseSA
LongitudinalSA
Monochromatic aberrations (Seidel aberrations)
Spot diagram
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�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Monochromatic aberrations (Seidel aberrations)
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Monochromatic aberrations (Seidel aberrations)
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
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Monochromatic aberrations (Seidel aberrations)
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Object
Image
lens
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�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Barreldistortion
Pin-cushiondistortion
Object
Image
Monochromatic aberrations (Seidel aberrations)
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Monochromatic aberrations (wave aberrations)
Referenceplane wavefront
Referencespherical wavefront
“The optical deviations of the wavefront from a reference plane or spherical wavefront.”
Wave aberration
Aberratedwavefront
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Wave aberrations (defocus)Myopic (near sighted) eye
Perfect eye
Defocusedwavefront
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Wave aberrations (higher order)
Eye with higher order aberrations
Perfect eye
Aberratedwavefront
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-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Wavefront Aberration
mm (right-left)
mm
(su
perio
r-in
ferio
r)
Wave Aberration of a Surface
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Mathematical description of the aberration.Zernike circle polynomials
Cn
mW(_,θ) = Z n
m(_,θ)∑
Wavefrontaberration
Zernikecoefficient
Zernike polynomials(wavefront mode)
Z 2
2(_,θ) = _2 cos2θastigmatism
m: angular frequency
n: radial order
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Zernike polynomials
Second orderaberrations
Higher orderaberrations
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Wavefront mode for each Zernike polynomial-5 -4 -3 -2 -1 0 1 2 3 4 5
2
3
4
5
nm
astigmatism defocus astigmatism
coma coma trefoiltrefoil
spherical 2nd astigmatism2nd astigmatismquadrafoil quadrafoil
2nd coma2nd comapentafoil pentafoil2nd trefoil 2nd trefoil
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Wavefront aberration and Zernike coefficients
= 0.3°ø - 0.2°ø + 0.4°ø
0.3°ø - 0.5°ø - 0.2°ø
0.2°ø + ……………
+
+
Wave aberration Zernikecoefficients
Zernike polynomial(mode)
Waveaberration
Zernikecoefficients
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0
0.2
0.4
0.6
0.8
1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Wavefront rms error
( )∑=2m
nCrms 22
21 rmsStrehl
−≈λπ
when rms is small.
Diffraction-limited(Strehl ≥ 0.8)
4
λ≤−vpW
Rayleigh’s _/4 rule
rms (_)
Stre
hl ra
tio
14
λ≤rms
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Chromatic Aberration
( )
−⋅−=
21
111
1
RRn
fLensmaker’s formula:
1.51
1.515
1.52
1.525
1.53
1.535
1.54
93
94
95
96
97
98
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Wavelength (mm)
Refra
ctiv
e in
dex
Focal length (mm
)
BK7 Schott glassR1, R2 = 50mm
n = n(_) � f = f (_) for polychromatic light
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Longitudinal (axial) chromatic aberration(LCA)
Long wavelengthShort wavelength
White light
LCA
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Bennet and Rabbetts (1989)
LCA
(D)~2.2D
LCA of the human eye
Wavelength (nm)
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0
0.2
0.4
0.6
0.8
1
400 450 500 550 600 650 700
Spectral sensitivity of the human eye
Wavelength (nm)
Sens
itivi
ty
0D
-0.8D
-0.3D
0.4D
0.2D
Optical effect of eye’s LCA on image quality ≈ 0.2D defocus for monochromatic light
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Transverse (lateral) chromatic aberration(TCA)
Long wavelengthShort wavelength
White light
TCA
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Why Are These Aberrations Important?
Relationship between aberrations and image quality(Pupil function, Rms, PSF, SR, OTF,
MTF, PTF, Image convolution…)
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Image quality
Wave aberrations Image planePSF, SR, MTF…
object
image
?
How well can an optical system form image?
We can improve image quality by correcting the aberrations.
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Aberration vs Image quality
Point Spread Function(PSF)
Optical Transfer Function(OTF)
Strehl Ratio
Modulation Transfer Function(MTF)
Phase Transfer Function(PTF)
FT
FTImage convolution
autocorrelation
Pupil function(aberration) ( ) ( )
= yxWiyxAyxP ,2
exp,),(λπ
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Point Spread Function (PSF)
( )( )2, yxPFTPSF =
The Point Spread Function, or PSF, is the image that anoptical system forms of a point source.
The point source is the most fundamental object, and formsthe basis for any complex object.
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Point Spread Function (PSF)
Airy Disc
The PSF for a perfect optical system is the Airy disc, which isthe Fraunhofer diffraction pattern for a circular pupil.
Geometrical optics Real world
Diffractionperfectoptics
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1 mm 2 mm 3 mm 4 mm
5 mm 6 mm 7 mm
Point Spread Function vs. Pupil SizePerfect Eye
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pupil images
followed by
psfs for changing pupil size
1 mm 2 mm 3 mm 4 mm
5 mm 6 mm 7 mm
Point Spread Function vs. Pupil SizeTypical Eye
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Strehl Ratiodiffraction-limited PSF(with no aberrations)
Hdl
Heye
actual PSF (with aberrations)
Strehl Ratio eyedlHH
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(,) (,) (,)PSFxyOxyIxy⊗
Image Convolution
FT-1 [ FT( PSF(x,y) ) °ø FT( O(x,y) ) ] = I(x,y)
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Modulation Transfer Function (MTF)
( ) ( )( )[ ]yxPSFFTffMTF yx ,Re, =
The Modulation Transfer Function, or MTF, is a measure ofthe reduction in contrast from object to image.
The ratio of the image modulation to the object modulation atall spatial frequencies.
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Modulation Transfer Function (MTF)
Object100% contrast
Imagediffraction only
Imagediffraction + 0.25D defocus
MTF
Spatial Frequency0
1
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Optics Simulations- Have fun!!! -
http://webphysics.ph.msstate.edu/javamirror/java/light.htm
http://www.optics.arizona.edu/jcwyant/math.htm
Fundamental Optics
Wavefront Theory and Fourier Optics
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How Can We Measure These AberrationsOf The Eye?
Different types of wavefront sensorsto measure ocular aberrations
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Wavefront sensors for the eye
Objective method
Outcoming lightIngoing light
Tcherning Shack-Hartmann
Subjective method
Ingoing light
Spatially ResolvedRefractometer
Laser RayTracing
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Measurement principle of wavefrontsensors for the eye
xdkx
yxW∆⋅=
∂
∂ ),(
Measurement of wavefront slope (1st derivative of wavefront)averaged over each subaperture on the pupil
ydky
yxW∆⋅=
∂
∂ ),(
Original wavefrontW(x,y)
Averagedwavefront slope
subaperture
Spot displacement�dx, _dy
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Spatially Resolved Refractometer Webb, Penney and Thompson (1992)
reference
reference�dx, _dy
Subject adjusts the incidentangle of light until retinal spotintersects reference spot.
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Laser Ray Tracing Navarro & Losada (1997), Molebny et al. (1997)
reference�dx, _dy
reference�dx, _dy
Scanning differentlocations of the pupil
CCD
CCD
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Tcherning Aberroscope Tscherning (1894)
�dx, _dy
Dot patternmask
CCD
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Shack-Hartmann wavefront sensor Liang, Grimm, Goelz, and Bille (1994), Liang and Williams (1997)
Laser beacon�dx, _dy
CCD
Perfect eye Real eye
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Comparison of wavefront measurementsusing different wavefront sensors
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Shack-Hartmann vs Spatially Resolved Refractometer(Objective vs Subjective methods)
Salmon et al., J. Opt. Soc. Am. A, 15 (1998)
S-H
SRR
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Shack-Hartmann vs Laser Ray Tracing(Outcoming vs Ingoing light)
Moreno-Barriuso and Navarro, J. Opt. Soc. Am. A, 17 (2000)
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Shack-Hartmann vs Laser Ray Tracingvs Spatially Resolved Refractometer
Moreno-Barriuso et al., Optometry and Vision Science, 78 (2001)
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Thank you!