Review of Basic Principles inReview of Basic Principles inOptics, Optics, WavefrontWavefront and and

WavefrontWavefront Error Error

Austin Roorda, Ph.D.University of California, Berkeley

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Geometrical Optics

Relationships between

pupil size, refractiveerror and blur

Optics of the eye: Depth of Focus Optics of the eye: Depth of Focus

2 mm 4 mm 6 mm

2 mm 4 mm 6 mm

In focus

Focusedin frontof retina

Focusedbehindretina

Optics of the eye: Depth of Focus Optics of the eye: Depth of Focus

DemonstrationRole 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 usedyour eyelids to make your effective pupil size smaller, thereby reducing the blur due to defocus onthe retina image. Only the horizontal line appears clear because you have only reduced the blur inthe horizontal direction.

Computation of Geometrical Blur SizeComputation of Geometrical Blur Size

blur[mrad][]blur[minutes]3.44[]DpupilsizemmDpupilsizemm=×=××

where D is the defocus in diopters

Application of Blur EquationApplication of Blur Equation

• 1 D defocus, 8 mm pupil produces27.52 minute blur size ~ 0.5 degrees

Physical Optics

The Wavefront

What is the Wavefront?What is the Wavefront?

converging beam=

spherical wavefront

parallel beam=

plane wavefront

What is the Wavefront?What is the Wavefront?ideal wavefrontparallel beam

=plane wavefront

defocused wavefront

What is the Wavefront?What is the Wavefront?parallel beam

=plane wavefront aberrated beam

=irregular wavefront

ideal wavefront

What is the Wavefront?What is the Wavefront?

aberrated beam=

irregular wavefront

diverging beam=

spherical wavefront

ideal wavefront

The Wave Aberration

What is the What is the Wave AberrationWave Aberration??diverging beam

=spherical wavefront wave aberration

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Wavefront Aberration

mm (right-left)

mm

(sup

erio

r-inf

erio

r)

Wave Aberration: DefocusWave Aberration: Defocus

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Wavefront Aberration

mm (right-left)

mm

(sup

erio

r-inf

erio

r)

Wave Aberration: ComaWave Aberration: Coma

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Wavefront Aberration

mm (right-left)m

m (s

uper

ior-i

nfer

ior)

Wave Aberration: All TermsWave Aberration: All Terms

Zernike Polynomials

-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)

Breakdown of Zernike TermsBreakdown 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

The Point Spread Function

The Point Spread Function, or PSF, isthe image that an optical system

forms of a point source.

The point source is the mostfundamental object, and forms the

basis for any complex object.

The PSF is analogous to the ImpulseResponse Function in electronics.

Airy Disc

The Point Spread FunctionThe Point Spread Function

The PSF for a perfect optical system isthe Airy disc, which is the Fraunhoferdiffraction pattern for a circular pupil.

Airy DiskAiry Disk

θ

1.22aλθ⋅=

angle subtended at the nodal point wavelength of the light pupil diameteraθλ≡≡≡

angle subtended at the nodal point wavelength of the light pupil diameter1.22aaθλλθ≡≡≡⋅=

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8

pupil diameter (mm)

PS

F A

iry D

isk

radi

us (m

inut

es)

As the pupil size gets larger, the Airydisc gets smaller.

Point Spread Function vs. Pupil SizePoint Spread Function vs. Pupil Size

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm

Small PupilSmall Pupil

Point Spread Function vs. Pupil SizePoint Spread Function vs. Pupil Size

1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm

Perfect Eye

Typical Eye

Larger pupilLarger pupil

Resolution

Rayleigh resolution

limit

Unresolved point sources

Resolved

minmin angle subtended at the nodal point wavelength of the light pupil diameter1.22aaθλλθ≡≡≡⋅=

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8

pupil diameter (mm)

PS

F A

iry D

isk

radi

us (m

inut

es)

As the pupil size gets larger, the Airydisc gets smaller.

Keck telescope:(10 m reflector)About 4500 timesbetter than the eye!

Convolution

(,) (,) (,)PSFxyOxyIxy⊗=Convolution

Simulated ImagesSimulated Images

20/40 letters

20/20 letters

MTFModulation Transfer

Function

low medium high

object:100% contrast

image

spatial frequency

cont

rast

1

0

MTF: Cutoff FrequencyMTF: Cutoff Frequency

0

0.5

1

0 50 100 150 200 250 300

1 mm2 mm4 mm6 mm8 mm

mod

ulat

ion

tran

sfer

spatial frequency (c/deg)

cut-off frequency57.3cutoffafλ=⋅

Rule of thumb: cutofffrequency increases by~30 c/d for each mmincrease in pupil size

Modulation Transfer FunctionModulation Transfer Function

vertical spatial frequency (c/d) horizontal spatial

frequency (c/d) c/deg-100 0 100

0.2

0.4

0.6

0.8

PTFPhase Transfer

Function

object

image

spatial frequency

phas

e sh

ift 180

0

-180

low medium high

Phase Transfer FunctionPhase Transfer Function

• Contains information about asymmetryin the PSF

• Contains information about contrastreversals (spurious resolution)

Relationships Between Wave Aberration,

PSF and MTF

()2(,),(,)iWxyiiPSFxyFTPxyeπλ−=

(){},(,)xyiiMTFffAmplitudeFTPSFxy=

The PSF is the Fourier Transform (FT) of the pupil function

The MTF is the amplitude component of the FT of the PSF

(){},(,)xyiiPTFffPhaseFTPSFxy=The PTF is the phase component of the FT of the PSF

The OTF (MTF and PTF) can also be computed asthe autocorrelation of the pupil function

arcsec-200 -100 0 100 200

mm (right-left)-2 -1 0 1 2

-0.5

0

0.5

c/deg-100 0 100

0.2

0.4

0.6

0.8

c/deg-100 0 100

-0.5

0

0.5

Point Spread FunctionWavefront Aberration

Modulation Transfer Function Phase Transfer Function

Point Spread Function

arcsec-200 -100 0 100 200

mm (right-left)

Wavefront Aberration

-2 -1 0 1 2-0.5

0

0.5

c/deg

Modulation Transfer Function

-100 0 100

0.2

0.4

0.6

0.8

c/deg

Phase Transfer Function

-100 0 100

-150

-100

-50

0

50

100

150

c/deg-100 0 100

-150

-100

-50

0

50

100

150

c/deg-100 0 100

0.2

0.4

0.6

0.8

mm (right-left)-2 -1 0 1 2

-0.5

0

0.5

1

1.5

arcsec-1000 -500 0 500 1000

Point Spread FunctionWavefront Aberration

Modulation Transfer Function Phase Transfer Function

Conventional Metrics to Define Imagine Quality

Root Mean SquareRoot Mean Square()()()()()21,, pupil area, wave aberration, average wave aberrationRMSWxyWxydxdyAAWxyWxy=−−−−∫∫

Root Mean Square:Root Mean Square:Advantage of Using Advantage of Using ZernikesZernikes to to

Represent the WavefrontRepresent the Wavefront

()()()()222220212223.......RMSZZZZ−−=+++

astig

matism

term

defoc

us te

rm

astig

matism

term

trefoi

l term ……

StrehlStrehl Ratio Ratio

diffraction-limited PSF

Hdl

Heye

actual PSF

Strehl Ratio = eyedlHH

Modulation Transfer FunctionModulation Transfer Function

00.10.20.30.40.50.60.70.80.9

1

0 50 100 150

spatial frequency (c/deg)

cont

rast Area under the MTF

20/20 20/10

Metrics to Define Image QualityMetrics to Define Image Quality

Other Metrics

Campbell,C.E. (2004). Improving visual function diagnostic metrics with the use ofhigher-order aberration information from the eye. J.Refract.Surg. 20, S495-S503

Cheng,X., Bradley,A., Hong,X., & Thibos,L. (2003). Relationship between refractive errorand monochromatic aberrations of the eye. Optom.Vis.Sci. 80, 43-49.

Cheng,X., Bradley,A., & Thibos,L.N. (2004). Predicting subjective judgment of best focuswith objective image quality metrics. J.Vis. 4, 310-321.

Guirao,A. & Williams,D.R. (2003). A method to predict refractive errors from waveaberration data. Optom.Vis.Sci. 80, 36-42.

Marsack,J.D., Thibos,L.N., & Applegate,R.A. (2003). Scalar metrics of optical qualityderived 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 tovisual function and visual quality metrics. J.Refract.Surg. 20, S504-S507

Typical Values for Wave AberrationTypical 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

Typical Values for Wave AberrationTypical Values for Wave AberrationPopulation Statistics

spherical aberration

comacomatrefoil

trefoil

Typical Values for Wave AberrationTypical 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

Typical Values for Wave AberrationTypical 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

Other Optical Factors that Degrade Image Quality

Retinal Sampling

Projected Image Sampled Image

5 arc minutes20/20 letter

Sampling by Foveal ConesSampling by Foveal Cones

5 arc minutes20/5 letter

Projected Image Sampled Image

Sampling by Foveal ConesSampling by Foveal Cones

NyquistNyquist Sampling Theorem Sampling Theorem

Photoreceptor Sampling >> Spatial Frequency

I

0

1

I

0

1

nearly 100% transmitted

I

0

1

I

0

1

nearly 100% transmitted

Photoreceptor Sampling = 2 x Spatial Frequency

I

0

1

I

0

1

nothing transmitted

Photoreceptor Sampling = Spatial Frequency

Nyquist theorem:The maximum spatial frequency that canbe detected is equal to _ of the samplingfrequency.

foveal cone spacing ~ 120 samples/deg

maximum spatial frequency:60 cycles/deg (20/10 or 6/3 acuity)

MTF: Cutoff FrequencyMTF: Cutoff Frequency

0

0.5

1

0 50 100 150 200 250 300

1 mm2 mm4 mm6 mm8 mm

mod

ulat

ion

tran

sfer

spatial frequency (c/deg)

cut-off frequency57.3cutoffafλ=⋅

Rule of thumb: cutofffrequency increases by~30 c/d for each mmincrease in pupil size

Nyquist limit

Thankyou!