Review of Basic Principles in Optics, Wavefront and Wavefront Error

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

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Geometrical Optics

Relationships between pupil size, refractive error and

blur

Optics of the eye: Depth of Focus

2 mm 4 mm 6 mm

2 mm 4 mm 6 mm

Optics of the eye: Depth of Focus

Focused behind retina

In focus

Focused in front of retina

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

Computation of Geometrical Blur Size

blur[mrad] [ ]blur[minutes] 3.44 [ ]

D pupilsize mmD pupilsize mm

= ×= × ×

where D is the defocus in diopters

Application of Blur Equation

• 1 D defocus, 8 mm pupil produces 27.52 minute blur size ~ 0.5 degrees

Physical Optics

The Wavefront

What is the Wavefront?parallel beam

=plane wavefront

converging beam=

spherical wavefront

What is the Wavefront?parallel beam

=plane wavefront

ideal wavefront

defocused wavefront

What is the Wavefront?parallel beam

=plane wavefront

ideal wavefront

aberrated beam=

irregular wavefront

What is the Wavefront?diverging beam

=spherical wavefront

aberrated beam=

irregular wavefront

ideal wavefront

The Wave Aberration

What is the Wave 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: Defocus

Wave Aberration: Coma

-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: All Terms

-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)

Zernike Polynomials

Wave Aberration Contour Map

0-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

mm (right-left)

mm

(sup

erio

r-inf

erio

r)

-2 -1 1 2

Breakdown of Zernike TermsCoefficient value (microns)-0.5 0 0.5 1 1.5 2

123456789

1011121314151617181920

Zern

ike

term

astig.defocus

astig.trefoilcomacomatrefoil

spherical aberration

2nd order

3rd order

4th order

5th order

The Reason we Measure the Wave Aberration

PTF (phase)

PSF(point spread function)

OTF (optical transfer function)

MTF (contrast)

Image Quality Metrics

The Point Spread Function

The Point Spread Function, or PSF, is the image that an optical system forms of a point

source.

The point source is the most fundamental object, and forms the basis for any complex

object.

The PSF is analogous to the Impulse Response Function in electronics.

The Point Spread Function

The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction

pattern for a circular pupil.

Airy Disc

Airy Disk

θ

1.22aλθ ⋅

=

angle subtended at the nodal point

wavelength of the light

pupil diametera

θ

λ

≡

≡

≡

As the pupil size gets larger, the Airy disc gets smaller.

angle subtended at the nodal point

wavelength of the light

pupil diameter

1.22

a

aθ

λ

λθ

≡

≡

≡

⋅=

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)

Point Spread Function vs. Pupil Size

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

Perfect Eye

Typical Eye

Resolution

Rayleighresolution

limit

Unresolved point sources

Resolved

As the pupil size gets larger, the Airy disc gets smaller.

min

min

angle subtended at the nodal point

wavelength of the light

pupil diameter

1.22

a

aθ

λ

λθ

≡

≡

≡

⋅=

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)

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

Convolution

Convolution

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

Simulated Images

20/40 letters

20/20 letters

MTFModulation Transfer

Function

low medium high

object:100% contrast

image

cont

rast

1

0spatial frequency

MTF: Cutoff Frequency

0

0.5

0

2 mm4 mm6 mm8 mm

mod

ulat

ion

tran

sfer

cut-off frequency

57.3cutoffafλ

=⋅

Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size

1 mm1

50 100 150 200 250 300spatial frequency (c/deg)

Modulation Transfer Function

vertical spatial frequency (c/d) horizontal spatial

frequency (c/d)

0.2

0.4

0.6

0.8

-100 0 100c/deg

PTFPhase Transfer

Function

low medium high

object

image

phas

e sh

ift

0

-180

180

spatial frequency

Phase Transfer Function

• Contains information about asymmetry in the PSF

• Contains information about contrast reversals (spurious resolution)

The Importance of Phase

Relationships Between Wave Aberration,

PSF and MTF

The Reason we Measure the Wave Aberration

PTF (phase)

PSF(point spread function)

OTF (optical transfer function)

MTF (contrast)

Image Quality Metrics

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

( )2 ( , )

, ( , )i W x y

i iPSF x y FT P x y eπλ

− =

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

( ) { }, ( , )x y i iMTF f f Amplitude FT PSF x y=

The PTF is the phase component of the FT of the PSF

( ) { }, ( , )x y i iPTF f f Phase FT PSF x y=

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

Point Spread FunctionWavefront Aberration

-0.5

0

0.5

-200 -100 0 100 200-2 -1 0 1 2arcsecmm (right-left)

c/deg-100 0 100

0.2

0.4

0.6

0.8

c/deg-100 0 100

-0.5

0

0.5

Modulation Transfer Function Phase Transfer Function

Point Spread FunctionWavefront Aberration

-0.5

0

0.5

-2 -1 0 1 2 -200 -100 0 100 200mm (right-left) arcsec

Modulation Transfer Function Phase Transfer Function

0.2

0.4

0.6

0.8

-150

-100

-50

0

50

100

150

-100 0 100 -100 0 100c/deg c/deg

Point Spread FunctionWavefront Aberration

-0.5

0

0.5

1

1.5

-2 -1 0 1 2 -1000 -500 0 500 1000mm (right-left) arcsec

Modulation Transfer Function Phase Transfer Function

0.2

0.4

0.6

0.8

-150

-100

-50

0

50

100

150

-100 0 100 -100 0 100c/deg c/deg

Conventional Metrics to Define Imagine Quality

Root Mean Square

( ) ( )( )

( )( )

21 , ,

pupil area, wave aberration

, average wave aberration

RMS W x y W x y dxdyA

AW x y

W x y

= −

−

−

−

∫∫

Root Mean Square: Advantage of Using Zernikes to

Represent the Wavefront

( ) ( ) ( ) ( )2 2 2 22 0 2 12 2 2 3 .......RMS Z Z Z Z− −= + + +

astig

matism

term

defoc

us te

rm

term

trefoi

l term ……

astig

matism

Strehl Ratio

diffraction-limited PSF

Strehl Ratio = eye

dl

HHHdl

actual PSF

Heye

Modulation Transfer Function

0.30.40.50.6

cont

rast Area under the MTF

20/20 20/10

10.90.80.7

0.20.1

00 50 100 150

spatial frequency (c/deg)

Metrics to Define Image Quality

Other Metrics

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

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

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

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

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

Typical Values for Wave Aberration

Strehl Ratio

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

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

Typical Values for Wave AberrationPopulation Statistics

spherical aberration

comacomatrefoil

trefoil

Typical Values for Wave AberrationChange in aberrations with pupil size

rms

wav

e ab

erra

tion

(mic

rons

) Shack Hartmann MethodsOther Methods

1.2Iglesias 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

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5 6 7 8 9

pupil size (mm)

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

Other Optical Factors that Degrade Image Quality

Retinal Sampling

Sampling by Foveal Cones

Projected Image Sampled Image

5 arc minutes20/20 letter

Sampling by Foveal Cones

5 arc minutes20/5 letter

Projected Image Sampled Image

Nyquist Sampling Theorem

Photoreceptor Sampling >> Spatial Frequency1

I

0

I

0

1

nearly 100% transmitted

Photoreceptor Sampling = 2 x Spatial Frequency1

I

0

I

0

1

nearly 100% transmitted

Photoreceptor Sampling = Spatial Frequency1

I

0

I

0

1

nothing transmitted

Nyquist theorem:The maximum spatial frequency that can be detected is equal to ½ of the sampling frequency.

foveal cone spacing ~ 120 samples/deg

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

MTF: Cutoff Frequency

0

0.5

0

1 mm2 mm4 mm6 mm8 mm

mod

ulat

ion

tran

sfer

cut-off frequency

57.3cutoffafλ

=⋅

Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size

Nyquist limit

1

50 100 150 200 250 300spatial frequency (c/deg)

Thankyou!