05 Aberrations GY

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
















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


�Coma

�Astigmatism

�Field Curvature

�Distortion

Paraxialrays

Marginal rays

Lens

TransverseSA

LongitudinalSA


Spot diagram


�Coma

�Astigmatism

�Field Curvature

�Distortion




�Coma

�Astigmatism

�Field Curvature

�Distortion



�Coma

�Astigmatism

�Field Curvature

�Distortion

Object

Image

lens


�Coma

�Astigmatism

�Field Curvature

�Distortion

Barreldistortion

Pin-cushiondistortion

Object

Image


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!

Documents

05 Aberrations GY