aberration theory - university of arizona · microsoft powerpoint - class03_08bw.ppt author:...
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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