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Aberration Theory
Geunyoung Yoon, Ph.D.
Assistant ProfessorDepartment of Ophthalmology
Center for Visual ScienceUniversity of Rochester
Optics
Paraxial Optics(First Order Optics)(Gaussian Optics)
Geometrical Optics(Aberration theory)
Diffractive Optics(Fourier Optics)
Quantum Optics
Coherent Optics
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
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
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
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
What Is Wavefront?
Huygens’s principleSnell’s law
Fermat’s principleParaxial (first order) approximation
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
Wavefront vs Ray“Rays are lines normal to the wavefronts at every point of intersection.”
Plane wavefront Spherical wavefront
Ray
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
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
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
=
⋅⋅⋅+−+−=!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
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.”
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
What Kinds Of Wavefront AberrationsAre There?
Monochromatic aberration(Seidel and wave aberrations)
Chromatic aberration(Longitudinal, Transverse)
Monochromatic aberrations (Seidel aberrations)
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Paraxialrays
Marginal rays
Lens
TransverseSA
LongitudinalSA
Monochromatic aberrations (Seidel aberrations)
Spot diagram
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Monochromatic aberrations (Seidel aberrations)
Monochromatic aberrations (Seidel aberrations)
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Monochromatic aberrations (Seidel aberrations)
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Object
Image
lens
�Spherical Aberration
�Coma
�Astigmatism
�Field Curvature
�Distortion
Barreldistortion
Pin-cushiondistortion
Object
Image
Monochromatic aberrations (Seidel aberrations)
Monochromatic aberrations (wave aberrations)
Referenceplane wavefront
Referencespherical wavefront
“The optical deviations of the wavefront from a reference plane or spherical wavefront.”
Wave aberration
Aberratedwavefront
Wave aberrations (defocus)Myopic (near sighted) eye
Perfect eye
Defocusedwavefront
Wave aberrations (higher order)
Eye with higher order aberrations
Perfect eye
Aberratedwavefront
-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
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
Zernike polynomials
Second orderaberrations
Higher orderaberrations
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
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
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
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
Longitudinal (axial) chromatic aberration(LCA)
Long wavelengthShort wavelength
White light
LCA
Bennet and Rabbetts (1989)
LCA
(D)~2.2D
LCA of the human eye
Wavelength (nm)
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
Transverse (lateral) chromatic aberration(TCA)
Long wavelengthShort wavelength
White light
TCA
Why Are These Aberrations Important?
Relationship between aberrations and image quality(Pupil function, Rms, PSF, SR, OTF,
MTF, PTF, Image convolution…)
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.
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
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.
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
1 mm 2 mm 3 mm 4 mm
5 mm 6 mm 7 mm
Point Spread Function vs. Pupil SizePerfect Eye
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
Strehl Ratiodiffraction-limited PSF(with no aberrations)
Hdl
Heye
actual PSF (with aberrations)
Strehl Ratio eyedlHH
(,) (,) (,)PSFxyOxyIxy⊗
Image Convolution
FT-1 [ FT( PSF(x,y) ) °ø FT( O(x,y) ) ] = I(x,y)
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.
Modulation Transfer Function (MTF)
Object100% contrast
Imagediffraction only
Imagediffraction + 0.25D defocus
MTF
Spatial Frequency0
1
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
How Can We Measure These AberrationsOf The Eye?
Different types of wavefront sensorsto measure ocular aberrations
Wavefront sensors for the eye
Objective method
Outcoming lightIngoing light
Tcherning Shack-Hartmann
Subjective method
Ingoing light
Spatially ResolvedRefractometer
Laser RayTracing
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
Spatially Resolved Refractometer Webb, Penney and Thompson (1992)
reference
reference�dx, _dy
Subject adjusts the incidentangle of light until retinal spotintersects reference spot.
Laser Ray Tracing Navarro & Losada (1997), Molebny et al. (1997)
reference�dx, _dy
reference�dx, _dy
Scanning differentlocations of the pupil
CCD
CCD
Tcherning Aberroscope Tscherning (1894)
�dx, _dy
Dot patternmask
CCD
Shack-Hartmann wavefront sensor Liang, Grimm, Goelz, and Bille (1994), Liang and Williams (1997)
Laser beacon�dx, _dy
CCD
Perfect eye Real eye
Comparison of wavefront measurementsusing different wavefront sensors
Shack-Hartmann vs Spatially Resolved Refractometer(Objective vs Subjective methods)
Salmon et al., J. Opt. Soc. Am. A, 15 (1998)
S-H
SRR
Shack-Hartmann vs Laser Ray Tracing(Outcoming vs Ingoing light)
Moreno-Barriuso and Navarro, J. Opt. Soc. Am. A, 17 (2000)
Shack-Hartmann vs Laser Ray Tracingvs Spatially Resolved Refractometer
Moreno-Barriuso et al., Optometry and Vision Science, 78 (2001)
Thank you!