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Aberration Theory

Optical systems convertthe shapes of wavefronts

Lens

AberrationsA perfectly spherical wave willconverge to a point. Any deviationfrom the ideal spherical shape issaid to be an aberration.

Wavefront Error

W(x,y) / n’ > 0

The wavefront error W(x,y)is the difference between aperfect spherical wave and theactual wavefront. It is usuallymeasured in the exit pupil andW(0,0) = 0.

The axis for this system is the Line of Sight.

In general, there is no symmetryin the eye, so W(x,y) can takeon any complex shape.

Wavefront ErrorA reference sphere of radius Ris taken as the ideal sphericalwavefront. R is just the distancefrom the exit pupil to the imageplane.

δη’ is the transverse ray errorand represents the distance between where the ideal and theactual rays strike the image plane.

dl’ is the longitudinal ray error andrepresents the distance the actualray crosses the line of sight fromthe image plane.

W(x,y) / n’

R

δη’

δl’

Image Planer’

Power ErrorIn Visual Optics, longitudinal aberrations are usually given in terms ofa “power error” as opposed to a distance.

P P’

Φ

F’ P P’

Φ+dφ

F’

δl’

Power ErrorThe “power error” is related to the slope of the wavefront

r)r(W

r1d

∂∂

=φ

Defocus Example: 2λs of defocus (λ = 0.5 μm) at edge of 6 mm pupil

222

3

r0001111.0rmm) 3(

mm) 105.0(2)r(W ×=×

=−

Diopters 2222.0mm 0002222.0d -1 ==φ

Power ErrorThe preceding defocus example shows that dφ is constant, whichmeans that dl’ is the same regardless of where the ray entered the eye.

P P’

Φ+dφ

F’

δl’

Measuring Defocus in the Eye

φtest

Ideally, put a test lens atthe principal plane.

Power changes slightlyas lens is moved fromthe principal plane.

Myopia and hyperopiaare defocus, where the prescription is the powerof the test lens thatcancels dφ.

Differential Geometry

• Every point on a continuous surface has two Principal Curvatures.

• These curvatures represent the maximum and minimum curvature through this point and

• The principal curvatures are always along orthogonal axes.

• Calculated from the Fundamental Forms)

Fundamental FormsFirst Fundamental Form

2

xf1E ⎟⎠⎞

⎜⎝⎛∂∂

+= ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛∂∂

=yf

xfF

2

yf1G ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+=

Second Fundamental Form

[ ] 2/12

22

FEGx/fL

−

∂∂= [ ] 2/12

2

FEGyx/fM

−

∂∂∂=

[ ] 2/12

22

FEGy/fN

−

∂∂=

Curvatures

( ) ( )2121

FEG2FM2GLENH 2 κ+κ=

−++

=

21FEGMLNK 2

2

κκ=−−

=

KHH2

KHH12

2

−−=κ

−+=κ

Mean Curvature

Gaussian Curvature

Principal Curvatures

Axial AstigmatismSince the eye is not rotationally symmetric, astigmatism can appearon-axis. This astigmatism is primarily due to the ocular surfaceshaving a toric or biconic shape.

( ) 22

22yyxx xyRRRRz −−+−−=

Principal Curvatures are 1/Rx and 1/Ryat the origin

Biconic SurfacesComputationally similar to the toric surface, but more versatilesince the biconic surface can add asphericity.

( ) ( ) 2y

2

y2x

2

x

y2

x2

RyK1

RxK111

R/yR/xz

+−+−+

+=

0 180 360

Rx

Ry

x

y

Principal Curvatures are 1/Rx and 1/Ry at the origin

Astigmatic Surfaces

• The axes with the maximum and minimum radii of curvature are called the Principal Meridia (any axis is called a meridian).

• There is a steep meridian corresponding to the minimum radius of curvature.

• There is a flat meridian corresponding to the maximum radius of curvature.

• The principal meridia are always perpendicular to each other.

Astigmatic SurfacesIn general, and in the eye, the principal meridia do not lie along the x and y axis, but are rotated through some angle θo.

( ) ( ) 2y

o22

y2x

o22

x

yo22

xo22

R)(sinrK1

R)(cosrK111

R/)(sinrR/)(cosrz

θ−θ+−

θ−θ+−+

θ−θ+θ−θ=

Astigmatic Power Errorθ+θ=+= 222222 sinBrcosArByAx)r(W

θ+θ=∂

∂=φ 22 sinB2cosA2

r)r(W

r1d

For a given value of θ, dφ is a constant. Axial astigmatism can bethought of as defocus that depends on meridian.

Axial Astigmatism

If a refraction is performed through a series of slits rotated atvarious angles, the refractive error will oscillate between aminimum and maximum value in the presence of axialastigmatism.

Jackson Crossed CylinderCrossed cylinders have a power +D along one meridian and a power -D along the orthogonal meridian.

Spherical Equivalent = Average power

Chromatic Aberration

θ2

θ1

Index n1

Index n2(λ)

Snell’s Lawn1*sin θ1 = n2(λ) *sin θ2

Chromatic AberrationChromatic Aberration is wavelength dependent defocus

( ))(A2)(d

rA)r(W 2

λ=λφλ=

-2

-1.5

-1

-0.5

0

0.5

400 500 600 700LC

A (D

) New Eye ModelNavarro Eye ModelWald & GriffinBedford & Wyszecki

Wavelength (nm)

-2

-1.5

-1

-0.5

0

0.5

400 500 600 700LC

A (D

) New Eye ModelNavarro Eye ModelWald & GriffinBedford & Wyszecki

Wavelength (nm) For a given λ, dφ is constant

Duochrome (Bichrome) Test

Red Sharper Green Sharper Equal

Transverse Chromatic Aberration

Wavefront Tilt or Prism

The wavefront remains the ideal spherical shape, but it is tilted relative to the ideal wavefront.

IdealPrism Error

Ideal Image Point

Real Image Point

Wavefront Tilt

Strabismus or “Lazy Eye” causes wavefront tilt

The brain sees independent images that donot fuse in the brain. The brain does not like this discrepancy, so it will start ignoringone image. If this is not fixed early, one eyewill become “blind”, even though it works fine from an optical standpoint.

Hirschberg Test

E E’

FoveaPupillary AxisκLOS

The subject fixates ona point source. The reflection of the sourceshould be slightly nasalof the pupil center.

Asymmetry betweenthe eyes suggestsocular misalignment

Spherical Aberration

With spherical aberration, the wave-front is rotationally symmetric, butis not spherical. The location of thefocus varies with the distance fromthe centerline.

Ideal Image Point

Spherical Aberration

Spherical Aberration

If a refraction is performed through a series of annular rings,the refractive error will vary with the size of the average ringdiameter, when spherical aberration is present.

Longitudinal Spherical Aberration

012345678

0 2 4 6 8Pupil Diameter (mm)

LSA

(D) Van Meeteren Fit

Navarro ModelKooijman ModelNew Eye ModelGullstrand-LeGrand Eye Model

Night Myopia

There is a myopic shift of the smallest point that is formedon the retina as the pupil dilates.

Oblique Astigmatism

“Off-axis” astigmatism Δ given as the difference between the meridional and sagittal foci

Δ= α1.5 x 10-2 Diopterswhere α is the field anglein degrees

Aberrations From Raytracing Defocus and Axial Astigmatism

P’

n’

M’ F’

'M'P'n

'F'P'nd −=φ

P’

n’

F’λF’λref

λλ

−=φ'F'P'n

'F'P'nd

ref

λref is typically 587.6 nm

Aberrations From Raytracing Chromatic Aberration

P’

n’

M’ F’

'F'P'n

'M'P'nd −=φ

Aberrations From Raytracing Spherical Aberration