CNR-INOA

Polinomi di Zernike

Bibliografia: roorda e Maeda

CNR-INOA

Sistema di coordinate

Optical

System

Optical

Axis

y

z

xy

x

Object

Plane

Image

Plane

y

x

!

"

Object

Height

h’

h

Image

Height

Optical

System

Optical

Axis

y

z

xy

x

Object

Plane

Image

Plane

y

x

!

"

Object

Height

h’

h

Image

Height

_

y

x

r

a

x = r cos(_)

y = r sin( _)

_ = tan -1(x/y)

r = (x 2+y2)1/2

_

y

x

_

1

x = _ cos(_)

y = _ sin( _)

_ = tan -1(x/y)

_ = r/a = (x 2+y2 )1/2

Normalized Pupil Coordinate SystemPupil Coordinate System

_

y

x

r

a

x = r cos(_)

y = r sin( _)

_ = tan -1(x/y)

r = (x 2+y2)1/2

_

y

x

_

1

x = _ cos(_)

y = _ sin( _)

_ = tan -1(x/y)

_ = r/a = (x 2+y2 )1/2

Normalized Pupil Coordinate SystemPupil Coordinate System

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Sistema di coordinate perl’occhio

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Aberrazione d’onda

L’aberrazione del fronte d’onda, W(x,y), è la distanza, in termini di camminoottico OPD (prodotto tra indice di rifrazione e cammino fisico), tra la sfera diriferimento e il fronte d’onda “reale”.

y

zx

Pupillad’uscita

PianoImmagine

Fronted’onda

aberratoFronte d’onda

sferico diriferimento

Aberrazioned’ondaW(x,y)

{ }

{ } 0,0

2

,

),(2

22

),(

),(1

),(

==

==

!

=

"#$

%&'

(=

yx

yx

ss

yx

d

yf

d

xf

yxWi

p

PSFFT

PSFFTssMTF

eyxpFTAd

yxPSF))

)

*

)

CNR-INOA

I passaggi matematici

CNR-INOA

Sviluppo polinomiale

• L’aberrazione del fronte d’onda W(x,y)può essere sviluppata in termini dipolinomi di Zernike

• Il contributo di ogni termine èindipendente dall’altro

CNR-INOA

Formule matematiche

[ ] [ ]sn

mn

s

s

m

n

m

n

mm

m

m

n

m

n

m

n

m

n

m

n

m

n

m

n

smnsmns

snR

R

mmn

N

N

nnnnmn

mmRN

mmRNZ

2

2)(

0

00

0

3

!)(5.0!)(5.0!

)!()1()(

polynomial radial theis )(

0for 0 , 0for 11

)1(2

factorion normalizat theis

,,4,2, of on values only takecan :given afor

20 , 10 , 0for )sin()(

20 , 10 , 0for )cos()(),(

:as defined are spolynomial ZernikeThe

!!

=

"!!!+

!!=

#===+

+=

+!+!!

$$$$<!=

$$$$%=

&&

&

'''

()&)&

()&)&)&

K

factorial.m

zernike.m

CNR-INOA

Lista dei polinomi di Zernike( )

( )

( )( )

( )( )( )

MMMM

oo

o

)4(c10 4 4 14

mAstigmatisSecondary )2cos(3410 2 4 13

Defocus ,Aberration Spherical 1665 0 4 12

mAstigmatisSecondary )2sin(3410 2- 4 11

)4(sin10 4- 4 10

)3(c8 3 3 9

axis- xalong Coma )cos(238 1 3 8

axis-y along Coma )sin(238 1- 3 7

)3(sin8 3- 3 6

90or 0at axis with mAstigmatis )2(c6 2 2 5

Defocus curvature, Field 123 0 2 4

45at axis with mAstigmatis )2(sin6 2- 2 3

Distortion direction,-in xTilt )cos(2 1 1 2

Distortion direction,-yin Tilt )(sin2 1- 1 1

Pistonor erm,Constant t 1 0 0 0

Meaning ,

frequencyorder mode

4

24

24

24

4

3

3

3

3

2

2

2

!"

!""

""

!""

!"

!"

!""

!""

!"

!"

"

!"

!"

!"

!"

os

os

os

Zmnjm

n

#

+#

#

#

#

#

±

CNR-INOA

Aberrazione del fronte d’onda

),(),(

))cos()(())sin()((

),(),(

:spolynomial Zernikeof sum weighteda as expressed is aberration waveThe

max

0

0

1

7

!

! !!

! !

=

=

"

"=

"=

=

#$%

&'(

+"=

=

j

j

jj

k

n

n

m

m

n

m

n

m

n

nm

m

n

m

n

m

n

k

n

n

nm

m

n

m

n

yxZWyxW

mRNWmRNW

ZWW

)*)*

)*)*

Calcola_aberrazione_onda.m

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Utilizzo di un solo indice

Talvolta si ricorre all’utilizzo di un solo indice j. Inquesto caso questa tabella indica l’equivalenza tra ledue notazioni

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Polinomi di Zernike a doppio indiceAzimuthal Frequency, m

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Radial

Order, n

0

1

2

3

4

5

6

Common Names7

Piston

Tilt

Astigmatism (m=-2,2),Defocus(m=0)

Coma (m=-1,1),Trefoil(m=-3,3)

Spherical Aberration(m=0)

Secondary Coma(m=-1,1)

Secondary SphericalAberration (m=0)

( ) ,!"m

nZ

ZernikePolynomial.m

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PSF per i vari termini di ZernikeAzimuthal Frequency, m

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Radial

Order, n

0

1

2

3

4

5

6

Common Names7

Piston

Tilt

Astigmatism (m=-2,2),Defocus(m=0)

Coma (m=-1,1),Trefoil(m=-3,3)

Spherical Aberration(m=0)

Secondary Coma(m=-1,1)

Secondary SphericalAberration (m=0)

( ) ,!"m

nZ

ZernikePolynomialPSF.m

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Zernike Polynomials

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Double-Index Zernike Polynomial MTFs

( ) ,!"m

nZ

Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

RadialOrder, n

0

1

2

3

4

5

6

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MTFyMTFx

Common Names7

Piston

Tilt

Astigmatism (m=-2,2),Defocus(m=0)

Coma (m=-1,1),Trefoil(m=-3,3)

Spherical Aberration(m=0)

Secondary Coma(m=-1,1)

Secondary SphericalAberration (m=0)

Pupil Diameter = 4 mm0 to 50 cycles/degreeλ = 570 nmRMS wavefront error = 0.2λ

ZernikePolynomialMTF.m

CNR-INOA

Double-Index ZernikePolynomial MTFs

Azimuthal Frequency, m-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

RadialOrder, n

0

1

2

3

4

5

6

MTFyMTFx

Common Names7

Piston

Tilt

Astigmatism (m=-2,2),Defocus(m=0)

Coma (m=-1,1),Trefoil(m=-3,3)

Spherical Aberration(m=0)

Secondary Coma(m=-1,1)

Secondary SphericalAberration (m=0)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pupil Diameter = 7.3 mm0 to 50 cycles/degreeλ = 570 nmRMS wavefront error = 0.2λ( ) ,!"m

nZ

ZernikePolynomialMTF.m

CNR-INOA

Simulation based on HumanEye Data

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

sx (cycle/deg)

MTF of Zero Aberration System, 5.4mm pupil

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

sy (cycle/deg)

MTF of Zero Aberration System, 5.4mm pupil

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

sx (cycle/deg)

MTF of Aberrated System, Wrms = 0.85012!

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

sy (cycle/deg)

MTF of Aberrated System, Wrms = 0.85012!

Mode j Coefficient (µm) RMS Coefficient (µm)

0 0 0

1 0 0

2 0 0

3 1.02 0.416413256

4 0 0

5 0.33 0.134721936

6 0.21 0.074246212

7 -0.26 -0.091923882

8 0.03 0.010606602

9 -0.34 -0.120208153

10 -0.12 -0.037947332

11 0.05 0.015811388

12 0.19 0.084970583

13 -0.19 -0.060083276

14 0.15 0.047434165

Total RMS Wavefront Error (µm) 0.484608089

WaveAberration.m WaveAberrationPSF.m

WaveAberrationMTF.m

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What are ZernikePolynomials?

• set of basic shapes that are used to fitthe wavefront

• analogous to the parabolic x2 shape thatcan be used to fit 2D data

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Properties of ZernikePolynomials• orthogonal

– terms are not similar in any way, so the weighting of oneterms does not depend on whether or not other terms arebeing fit also

• normalized– the RMS wave aberration can be simply calculated as the

vector of all or a subset of coefficients

• efficient– Zernike shapes are very similar to typical aberrations found

in the eye

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Measurement SetupPupil

Retina

Iris

RealAberratedWavefront

IdealPlanar

Wavefront

y

zx

IncomingLight Beam

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Shack-Hartmann SensorLayout

CCDPupil Relay OpticsPBS

LightSource

LensletArray

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Shack-Hartmann WavefrontSensor

rr’ r”

f f f f

p p’

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Perfect eye

Wavefront Lens Array CCD Array

Aberrated (typical) eye

Shack-Hartmann WavefrontSensor

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Lenslet Array

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Alcune immagini

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BD KW SMShack-Hartmann Images

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WavefrontWavefront MapsMaps(at best focal plane)(at best focal plane)

BD KW SM

0.33 DS –0.17 DC X 87

6.42 DS –0.6 DC X 126

0 DS –1 DC X 3

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Wavefront sensor imageWavefront sensor image Wavefront aberrationWavefront aberration

Aberrations of an RK patientAberrations of an RK patient

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Aberrations of a LASIK patientAberrations of a LASIK patient

Wavefront sensor imageWavefront sensor image Wavefront aberrationWavefront aberration

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Post - RKPost - RK Post - LASIKPost - LASIK

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Cheratocono

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LAC per cheratoconounaided eye custom contact lens

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PSF (pupilla 5 mm)unaided eye custom contact lens

rms = 4.16strehl ratio = 0.0008

1 degree1 degree

rms = 1.48strehl ratio = 0.004

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Variazione dellamappacorneale alpassare deltempo conocchio aperto

W.CharmanContact Lens& Anterior Eye28 (2005)75-92

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PSF through-focus (5 mmpupil)

0

2

4

6

8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

defocus [D]

rms

wav

e ab

erra

tion

(mic

rons

)

0.00000

0.00100

0.00200

0.00300

0.00400

0.00500

strehl ratio

The highest strehl ratio does not correlate withrms when aberrations are high

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Simmetria tra i due occhi

Junzhong Liang andDavid R. Williams

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Metrics to Define Image Quality

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-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wave Aberration Contour MapWave Aberration Contour Map

mm (right-left)

mm

(sup

erio

r-inf

erio

r)

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Breakdown of Zernike Terms

-0.5 0 0.5 1 1.5 2123456789

1011121314151617181920

Zern

ike

term

Coefficient value (microns)

astig.defocus

astig.trefoilcomacomatrefoil

spherical aberration

2nd order

3rd order

4th order

5th order

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Root Mean Square( ) ( )( )

( )

( )

21, ,

pupil area

, wave aberration

, average wave aberration

RMS W x y W x y dxdyA

A

W x y

W x y

= !

!

!

!

""

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Root Mean Square( ) ( ) ( ) ( )

2 2 2 22 0 2 1

2 2 2 3.......RMS Z Z Z Z

! != + + +

astig

matism

term

defoc

us te

rm

astig

matism

term

trefoi

l term ……

Include the terms for which you want to determine theirimpact (eg defocus and astigmatism only, third orderterms or high order terms etc.)

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Problemi nel RMS

J.S. McLellan et al.Vis. Res. 46 (2006)3009-3016

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Point Spread Function

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Strehl Ratiodiffraction-limited PSF

Hdl

Heye

actual PSF

Strehl Ratio = eye

dl

H

H

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Typical Values for Wave Aberration

• Strehl ratios are about 5% for a 5 mm pupil that hasbeen corrected for defocus and astigmatism.

• Strehl ratios for small (~ 1 mm) pupils approach 1,but the image quality is poor due to diffraction.

Strehl Ratio

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Typical Values for Wave AberrationPopulation Statistics

spherical aberration

comacomatrefoil

trefoil

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Typical Values for Wave Aberration

Iglesias et al, 1998Navarro et al, 1998Liang et al, 1994Liang and Williams, 1997Liang et al, 1997Walsh et al, 1984He et al, 1999Calver et al, 1999Calver et al, 1999Porter et al., 2001He et al, 2002He et al, 2002Xu et al, 2003Paquin et al, 2002Paquin et al, 2002Carkeet et al, 2002Cheng et al, 2004

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9pupil size (mm)

rms

wav

e ab

erra

tion

(mic

rons

) Shack-Hartmann MethodsOther Methods

Change in aberrations with pupil size

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Typical Values for Wave AberrationChange in aberrations with age

Monochromatic Aberrations as a Function of Age, from Childhood to Advanced AgeIsabelle Brunette,1 Juan M. Bueno,2 Mireille Parent,1,3 Habib Hamam,3 and Pierre Simonet3

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Convolution

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( , ) ( , ) ( , )PSF x y O x y I x y! =

Convolution

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Simulated Images

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“I have never experienced any inconvenience from thisimperfection, nor did I ever discover it till I made theseexperiments; and I believe I can examine minute objectswith as much accuracy as most of those whose eyes aredifferently formed”

Thomas Young (1801) on his own aberrations.

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References[1] MacRae, S. M., Krueger, R. R., Applegate, A. A., (2001), Customized Corneal Ablation, The Quest forSuperVision, Slack Incorporated.

[2] Williams, D., Yoon, G. Y., Porter, J., Guirao, A., Hofer, H., Cox, I., (2000), “Visual Benefits of CorrectingHigher Order Aberrations of the Eye,” Journal of Refractive Surgery, Vol. 16, September/October 2000,S554-S559.

[3] Thibos, L., Applegate, R.A., Schweigerling, J.T., Webb, R., VSIA Standards Taskforce Members (2000),"Standards for Reporting the Optical Aberrations of Eyes," OSA Trends in Optics and Photonics Vol. 35,Vision Science and its Applications, Lakshminarayanan,V. (ed) (Optical Society of America, Washington,DC), pp: 232-244.

[4] Goodman, J. W. (1968). Introduction to Fourier Optics. San Francisco: McGraw Hill

[5] Gaskill, J. D. (1978). Linear Systems, Fourier Transforms, Optics. New York: Wiley

[6] Fischer, R. E. (2000). Optical System Design. New York: McGraw Hill

[7] Thibos, L. N.(1999), Handbook of Visual Optics, Draft Chapter on Standards for Reporting Aberrations ofthe Eye. http://research.opt.indiana.edu/Library/HVO/Handbook.html

[8] Bracewell, R. N. (1986). The Fourier Transform and Its Applications. McGraw Hill

[9] Mahajan, V. N. (1998). Optical Imaging and Aberrations, Part I Ray Geometrical Optics, SPIE Press

[10] Liang, L., Grimm, B., Goelz, S., Bille, J., (1994), “Objective Measurement of Wave Aberrations of theHuman Eye with the use of a Hartmann-Shack Wave-front Sensor,” J. Opt. Soc. Am. A, Vol. 11, No. 7,1949-1957.

[11] Liang, L., Williams, D. R., (1997), “Aberration and Retinal Image Quality of the Normal Human Eye,” J. Opt.Soc. Am. A, Vol. 14, No. 11, 2873-2883.