Prof. Jose SasianOPTI 518
Introduction to aberrations
OPTI 518Lecture 14
Prof. Jose SasianOPTI 518
Topics
• Structural aberration coefficients• Examples
Prof. Jose SasianOPTI 518
Structural coefficients
Ж
Requires a focal system
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Structural stop shifting parameter
s’ is the distance from the rear principal plane to exit pupil
s is the distance from the front principal plane to entrance pupil
2
2 2P P Py y yS SЖ Ж
'
2 1 2 1 ' 2 'P Py y s sSЖ Y s n Y s n
1 / 2nu Y y
Using ω on we can express:
2P Py ySЖ
Prof. Jose SasianOPTI 518
Field curves
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Review of concepts
• Thin lens as the thickness tends to zero
• Shape of a lens and shape factor• Conjugate factor to quantify how the lens
is used. Related to transverse magnification
• Must know well first-order optics
1 2 1 2tn
Prof. Jose SasianOPTI 518
Shape and Conjugate factors
nu ' 1' 1
mYm
1 2 1 2
1 2 1 2
c c R RXc c R R
Lens bending concept
Prof. Jose SasianOPTI 518
Shape X
X=-1
X=0
X=-1.7
X=-3.5X=3.5
X=1.7
X=1
X=0
Prof. Jose SasianOPTI 518
Shape
X=0
X=0
X=-1X=-2X=-3
X=1X=2X=3
Prof. Jose SasianOPTI 518
Shape or bending factor
• Maintains optical power of thin lens• For a given shape factor and lens
thickness the lens aspect ratio changes
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
2K
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Spherical MirrorA spherical mirror can be treated as a thin lens with n=-1
If a dummy plane unfolding surface is added
2
1
11
000
I
II
III
IV
V
L
T
XYY
14
011401
A
BC
D
EF
n=-1
Prof. Jose SasianOPTI 518
Field curves
Rt~f/3.66Rm~f/2.66Rs~f/1.66Rp~f/0.66
Prof. Jose SasianOPTI 518
Application of CoddingtonEquations
Prof. Jose SasianOPTI 518
Application of CoddingtonEquations
Prof. Jose SasianOPTI 518
Thin lensSpherical aberration and coma
I II
Fourth-order
Prof. Jose SasianOPTI 518
Spherical aberration of a F/4 lens
•Asymmetry•For high index 4th order predicts well the aberration
Prof. Jose SasianOPTI 518
Thin lens spherical aberration
n=1.517
I
Y
X
Prof. Jose SasianOPTI 518
Thin lens coma aberration
n=1.517
II
X
Y
Prof. Jose SasianOPTI 518
This lensAplanatic solutions
X=4
Y=5n=1.5
' 1' 1
mYm
Prof. Jose SasianOPTI 518
Thin lensSpherical and coma @ Y=0
III
N=1.5N=2N=2.5N=3N=3.5N=4
Strong index dependence
Prof. Jose SasianOPTI 518
Thin lens special casesstop at lens
Prof. Jose SasianOPTI 518
Thin lens special cases stop at lens
Prof. Jose SasianOPTI 518
Thin lens special cases stop at lens
Prof. Jose SasianOPTI 518
Thin lens special cases stop at lens
Prof. Jose SasianOPTI 518
Thin lens special cases stop at lens
Prof. Jose SasianOPTI 518
Achromatic doubletTwo thin lenses in contactThe stop is at the doublet
1 2
1 2
11
22
y y
1 2
21
1
12
2
1
YY
YY
Prof. Jose SasianOPTI 518
Achromatic doubletCorrection for chromatic change of focus
1 2
1 2
11 2
1 2
22 1
1 2
1
1
L
L
L
0L
For an achromatic doublet:
2,
,0
kP kk
L L ki P
yy
Prof. Jose SasianOPTI 518
Achromatic doubletCorrection for spherical aberration
3 2 2 3 2 21 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2I A X B X Y CY D A X B X Y C Y D
43,
,0
kP kk
I I ki P
yy
For a given conjugate factor Y, spherical aberration is a functionOf the shape factors X1 and X2 . For a constant value of spherical
aberration we obtain a hyperbola as a function of X1 and X2.
Prof. Jose SasianOPTI 518
Achromatic doubletCorrection for coma aberration
22
,, ,
0
kP kk
II II k k I ki P
yS
y
2 21 1 1 1 1 2 2 2 2 2II E X FY E X F Y
For a given conjugate factor Y and a constant amount of coma the graph of X1 and X2 is a straight line.
Prof. Jose SasianOPTI 518
Achromatic doubletAstigmatism aberration
2, , ,
0
2k
kIII III k k II k k I k
i
S S
1 21 1III
Astigmatism is independent of the relative lens powers, shape factors,or conjugate factors.
Prof. Jose SasianOPTI 518
Achromatic doublet
1 2
1 2IV n n
2 1
2 1
n n
Field curvature aberration
,0
kk
IV IV ki
0IV For an achromatic doublet there is no field curvature if
Prof. Jose SasianOPTI 518
Achromatic doubletDistortion and chromatic change of magnification
, ,0
k
T T k k L ki
S
2
2 3, , , , ,
0 ,
3 3k
PV V k k IV k III k k II k k I k
i P k
y S S Sy
00
V
T
Prof. Jose SasianOPTI 518
Cemented achromatic doublet
1 1 2 212 21
1 2
1 1 2 2
1 2
2 1
1 12 1 2 1
1 11 1
1 1
X Xc c
n n
X Xn n
X X
2 1
1 2
2 1
1 2
11
11
nn
nn
For achromat
For a cemented achromatic lens the graph of X1 and X2 is a straight line.
Prof. Jose SasianOPTI 518
Crown in front: BK7 and F8
Prof. Jose SasianOPTI 518
Flint in front: BK7 and F8
Prof. Jose SasianOPTI 518
Cemented achromatic doublet
Prof. Jose SasianOPTI 518
Cemented doublet solutions
Crown in front
Flint in front
Prof. Jose SasianOPTI 518
Two mirror afocal system
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Prof. Jose SasianOPTI 518
Merssene afocal systemAnastigmatic
Confocal paraboloids
Prof. Jose SasianOPTI 518
Paul-Baker systemAnastigmatic-Flat field
AnastigmaticParabolic primarySpherical secondary and tertiaryCurved fieldTertiary CC at secondary
Anastigmatic, Flat fieldParabolic primaryElliptical secondary Spherical tertiaryTertiary CC at secondary
Prof. Jose SasianOPTI 518
Meinel’s two stage optics concept (1985)
Large DeployableReflector
Second stage correctsfor errors of first stage;fourth mirror is at the
exit pupil.
Prof. Jose SasianOPTI 518
Aplanatic, Anastigmatic, Flat-field, Orthoscopic (free from distortion, rectilinear, JS 1987)
Spherical primary telescope.The quaternary mirror is near the exit pupil. Spherical aberration and
Coma are then corrected with a single aspheric surface. The Petzval sum is zero.If more aspheric surfaces are allowed then more aberrations
can be corrected.
Prof. Jose SasianOPTI 518
Lister objective
Prof. Jose SasianOPTI 518
Lister objective• Two achromatic doublets that are spaced• Telecentric in image space• Normalized system
11100
A
A
IA
IB
Жyu
Prof. Jose SasianOPTI 518
Lister objective
The aperture stop is at the first lens.The system is telecentric
111
B
A B
B
yy
Prof. Jose SasianOPTI 518
Lister objective
Prof. Jose SasianOPTI 518
Condition for zero coma
22,
, ,0
2 2 2 2
2 2
2
2
0
1 0
1
kP kk
II II k k I ki P
II A A IIA B B IIB
IB
II B IIA B IIB
BIIA IIB
B
yS
y
y y
y y
yy
Prof. Jose SasianOPTI 518
Condition for zero astigmatism
2, , ,
0
2
2
1 0
1 1 0
2 02
12
2
1
kk
III III k k II k k I ki
III A B B IIB
III B B IIB
B B IIB
BIIB
BIIB
B
B BIIA
B
S S
y
y y
y y
y
yy
y y
y
, ,
2k P k P k
k
y yS
Ж
Prof. Jose SasianOPTI 518
Lister Objective
2
2 2
2 2
221
1
1 212123
3
IIB IIA
B BB
B B
B B
B B B
B
A
IIA
IIB
y yyy y
y y
y y y
y
Choose:
Prof. Jose SasianOPTI 518
Petzval objective
R +52.9 -41.4 +436.2 +104.8 +36.8 45.5 -149.5N-crown=1.517; N-flint=1.576
T 5.8 1.5 23.3/23.3 2.2 0.7 3.6F=100 mm
Prof. Jose SasianOPTI 518
Petzval objectiveFlattening of the tangential field curve
Prof. Jose SasianOPTI 518
Variance criterion
2 22 2 3
020 040 220 222 111 131 311
2 22 2040 131 222
1 1 1 212 2 4 3
1 1 1180 72 24
W W W W W H W W H W H
W W H W H
220 222
222 220
122
W W
W W
Zero sagittal field curvature
Prof. Jose SasianOPTI 518
Zero tangential field curve curvature
,0
2
2
1
1 1
3
1 1 23
2 13
23
1
2 12 3 23
2
1
1
kk
IV IV k B IVA IVBi
III B B IIB
IV III
IVB B IIB B B IIB
IVB IIB
IV
BIIB
IVIV B
B BIIB
B B
B BIIA
B
IIA IIB
B
y
y y
y y y y
y
y
yy yy y
y y
yq
y
2 13 2
IV
BB
qyy
Prof. Jose SasianOPTI 518
Diffractive opticshigh-index model
1/d
Prof. Jose SasianOPTI 518
Diffractive opticshigh-index model
1/d
Prof. Jose SasianOPTI 518
Ray diffractive law (1D)
Prof. Jose SasianOPTI 518
Diffractive lens (n very large)
A=0B=0C=3D=1E=0F=2
23 1I Y
2II Y 1III
0IV
0V 1
Ldiffractive
0T
Prof. Jose SasianOPTI 518
MIRRORS
Prof. Jose SasianOPTI 518
Spherical Mirror
Prof. Jose SasianOPTI 518
Aspheric mirrors
Prof. Jose SasianOPTI 518
Aspheric mirrors
Prof. Jose SasianOPTI 518
Spherical mirror stop at mirror
Prof. Jose SasianOPTI 518
Spherical mirror with stop shift
Prof. Jose SasianOPTI 518
Stop shifting
'
2 1 2 1 ' 2 'P Py y s sSЖ Y s n Y s n
Prof. Jose SasianOPTI 518
Spherical MirrorObject at infinity Y=1
Prof. Jose SasianOPTI 518
Aspheric mirror
Prof. Jose SasianOPTI 518
With stop shift
Prof. Jose SasianOPTI 518
Paraboloid with object at infinity
Prof. Jose SasianOPTI 518
Aspheric mirror with corrector plateShift stop from mirror and insert corrector at stop
Prof. Jose SasianOPTI 518
Special aplanatic cases
Prof. Jose SasianOPTI 518
Wright and Schmidt systems
Prof. Jose SasianOPTI 518
Wright and Schmidt systems
Prof. Jose SasianOPTI 518
Summary
• Structural coefficients• Basic treatment• There is more to it• Use of Y-Ybar diagram• Analysis of simple systems• See R. Shack notes