Prof. Jose Sasian
Overview of Aberrations
Lens Design OPTI 517
Prof. Jose Sasian
AberrationFrom the Latin, aberrare, to wander from; Latin, ab, away, errare, to wander.
Symmetry properties
Prof. Jose Sasian
Overview of Aberrations(Departures from ideal behavior)
• Basic reasoning• Wave aberration function• Aberration coefficients• Aspheric contributions• Stop shifting• Structural aberration coefficients
Prof. Jose Sasian
Wavefront
Prof. Jose Sasian
Basic reasoning
• Ideally wavefronts and rays converge to Gaussian image points. This implies that ideally wavefronts must be spherical and rays must be homocentric.
Prof. Jose Sasian
Basic reasoning• Actual image degradation by an optical system implies that the
collinear transformation can not model accurately imaging. In the wave picture for light propagation we notice that wavefronts must be deformed from the ideal spherical shape.
• Wavefront deformation is determined by the use of a reference sphere with center at the Gaussian image point and passing by the exit pupil on-axis point.
Exit pupil
Image plane
Reference sphereis centered at ideal image point
Referencesphere
Deformed wavefront
Prof. Jose Sasian
Basic reasoning• An axially symmetric system can only have an axially symmetric
wavefront deformation for an object point on-axis. In its simplest form this deformation can be quadratic or quartic with respect to the aperture. If the reference sphere is centered in the Gaussian image point then the quadratic deformation can not be present for the design wavelength.
Prof. Jose Sasian
Basic reasoning• For an object point that is off-axis the axial symmetry of the beam is
lost and is reduced to plane symmetry. Therefore for that off-axis beam the wavefront deformation can have axial, plane, or double plane symmetry.
Prof. Jose Sasian
Basic reasoning• The simplest plane symmetric wavefront deformation shapes
represent the primary aberrations. These are:
• Spherical aberration Axially symmetric• Coma Plane symmetric• Astigmatism Double plane symmetric• Field curvature Axially symmetric• Distortion Plane symmetric• Longitudinal Axially symmetric
chromatic• Lateral Plane symmetric
chromatic
Prof. Jose Sasian
Aberration forms:symmetry considerations
Focus
Spherical aberration
Distortion
Field curvatureFocus
Astigmatism
Coma
Sphericalaberration
On-axis
Off-axis
Prof. Jose Sasian
Wave aberration function
• The wave aberration function is a function of the field H and aperture ρ vectors. Because this function represents a scalar, which is the wavefront deformation at the exit pupil, it depends on the dot product of the field and aperture vectors. The assumed axial symmetry leads to a select set of terms.
( ) 2 2000 200 020 111
4 3 2 2 2040 131 222
2 2 3 4220 311 400
, , cos
cos coscos
...
W H W W H W W H
W W H W HW H W H W H
ρ θ ρ ρ θ
ρ ρ θ ρ θ
ρ ρ θ
= + + + +
+ + + +
+ + + ++
H
ρ{ } ∑=
nmj
mlkmlk HWHW
,,,, cos, θρρ
Prof. Jose Sasian
Wave aberration function
• The field vector has its foot at the center of the object plane and the aperture vector has its foot at the center of the exit pupil plane. Both are normalized and so their maximum magnitude is unity. For convenience we draw the first order image of the field vector.
Aperture and field vectors
Exit pupil
Aperture vector Field vector
Image plane
Optical axis
H
ρ
Prof. Jose Sasian
Wave aberration function
• Note that defocus W020 and the change of scale W111 terms are not needed because Gaussian optics accurately predict the location and size of the image. The piston terms W000, W200 and W400 represent a constant phase change that does not degrade the image. These piston terms do not depend on the aperture vector and so they do not produce transverse ray errors.
• Piston terms represent and advance or delay on the propagation of a wavefront.
Prof. Jose Sasian
Summary of primary aberrations
Prof. Jose Sasian
Aberration coefficients for a system of q surfaces
ISW81
040 =2
1
q
Ij
uS A yn=
= − ∆
∑
IISW21
131 = 1
q
IIj
uS AAyn=
= − ∆
∑
IIISW21
222 =2
1
q
IIIj
uS A yn=
= − ∆
∑
( )22014 IV IIIW S S= +
2
1
q
IVj
S Ж P=
= −∑
VSW21
311 =
2 2
1
q
Vj
A uS Ж P A yA n=
= − + ∆ ∑
( )22
1
1q
Vj
S A A y Ж Ay yPn=
= − ∆ − + ∑
Prof. Jose Sasian
Aberration coefficients for chromatic aberrations
LCW21
020 =δλ
1
q
Lj
nC Aynδ
=
= ∆
∑
TCW =111δλ
1
q
Tj
nC Aynδ
=
= ∆
∑
Prof. Jose Sasian
Aberration coefficient parameters
ninycnuA =+=
incynunA =+=
( )ncP /1∆=r
c 1=Ж nuy nuy= −
νδ
nn
nn 1−
=
∆
CF
d
nnn−−
=1
ν
Prof. Jose Sasian
Aberration coefficients
• c is the surface curvature, ν is the ν number or reciprocal dispersive power.
• All with the marginal and chief ray first-order ray traces !!!
• Lens optimization started!!!
is the Lagrange invariant.Ж
Prof. Jose Sasian
Comments on aberrations• Third-order or fourth-order ?• A well corrected system has its third-order aberrations
almost zero• Aberration cancellation is the main mechanism for
image correction• Presenting to the optimization routine a system with its
third-order aberrations corrected is a good starting point
• Some simple systems are designed by formulas that relate third-order aberrations
• Note symmetry in third-order aberration coefficients• Wave aberrations seem to be simpler to understand
than transverse ray aberrations
Prof. Jose Sasian
Example
Prof. Jose Sasian
Example
Prof. Jose Sasian
Summary of aberrations
Aspheric surfaces(non-spherical)
• Conic (or conicoids)• Cartesian ovals• Polynomials on x/y, r-theta, Zernikes• Bernstein polynomials, Bezier curves• Splines• NURBS• Freeform surfaces• User definedProf. Jose Sasian
Prof. Jose Sasian
Conic plus polynomial(much used in lens design)
( ) [ ] ...)1(11
1010
88
66
4422
2
++++++−+
= SASASASAScK
cSSZ
S x y2 2 2= +
C is 1/r where r is the radius of curvature; K is the conic constant; A’s are aspheric coefficients
Aspheric contribution can be thought of as a cap to the spherical part
Sag= sphere + aspheric cap
Prof. Jose Sasian
Aspheric contributions to the Seidel sums
IS aδ =
0IVSδ =
0=LCδ
IIyS ay
δ =
3
VyS ay
δ
=
0=TCδ
2
IIIyS ay
δ
= nyca ∆−= 432ε
nyAa ∆= 448
ISW81
040 =
IISW21
131 =
IIISW21
222 =
( )22014 III IVW S S= +
VSW21
311 =
Prof. Jose Sasian
Aspheric contributions explanation
Prof. Jose Sasian
Aspheric contributions depend on chief ray height at the surface
•When the stop is at the aspheric surface only spherical aberration is contributed given that all the beams see the same portion of the surface .•When the stop is away from the surface, different field beams pass through different parts of the aspheric surface and other aberrations are contributed
Prof. Jose Sasian
Stop Shifting
Exit pupil Image plane
Optical axis
•Stop shifting is a change in the location of the aperture stop along the optical axis•Stop shifting does not change the f/#•Stop shifting does not change the optical throughput•Stop shifting selects a different portion of the wavefront for off-axis beams
Prof. Jose Sasian
Stop shifting may produce light vignetting
Prof. Jose Sasian
Stop shifting
Prof. Jose Sasian
Change of Seidel sums with stop shifting
0=ISδ
III SyyS δδ =
IIIIII SyyS
yyS
2
2
+
∂=
δδ
0=IVSδ
{ } IIIIIIIVV SyyS
yySS
yyS
32
33
+
++=
δδδδ
0=LCδ
LT CyyC ∂
=δ
ISW81
040 =
IISW21
131 =
IIISW21
222 =
( )22014 III IVW S S= +
VSW21
311 =
Prof. Jose Sasian
Wave coefficients in terms of Seidel sums
ISW81
040 =
IISW21
131 =
IIISW21
222 =
( )22014 III IVW S S= +
VSW21
311 =
Prof. Jose Sasian
The ratio yyδ
Can be calculated at any plane in the optical system
new old new old new oldu u y y A ASu y A− − −
= = =
is the stop shifting parameterS
Prof. Jose Sasian
Structural coefficients σ
IpI yS σφ 34
41
=
3
2
2V V
p
ЖSy
σ=
2 212II p IIS Жy φ σ=
LpL yC φσ2=
2III IIIS Ж φσ=
2T TC Жσ=
2IV IVS Ж φσ=
Marginal ray height atthe principal planes
py
Prof. Jose Sasian
Structural coefficients:Thin lens (stop at lens)
[ ]DCYBXYAXySI ++−= 2234
41 φ
[ ]2 212IIS Жy EX FYφ= −
2IIIS Ж φ=
2 1IVS Ж
nφ=
0=VS
2 1LC y φ
ν=
0=TC
( )212
−+
=nnnA
( )2
2
1−=
nnD
( )( )1
14−+
=nnnB
( )11−+
=nn
nE
nnC 23 +
=
nnF 12 +
=
12
12
21
21
rrrr
ccccX
−+
=−+
=
uuuu
mmY
−+
=−+
=''
11
))(1( 1 xccncn −−=∆∆=φ
Surface optical power φ
Prof. Jose Sasian
Bending of a lens• Maintains the
optical power• Changes the
optical shape• Meniscus, plano-
convex, double-convex, etc.
• Shape factor 12
12
21
21
rrrr
ccccX
−+
=−+
=
Prof. Jose Sasian
Using a lens design program• Must be able to interpret correctly the information
displayed by the program• In some instances the program is right but we think it is
wrong. So we must carefully review our assumptions• When there is disagreement between you an the
program, there is an opportunity to learn• Verify that the program is modeling what you want• Check and double check• You must feel comfortable when using a program. • Read the manual• Play with the program to verify that it does what you
think it does• Must reach the point when it is actually fun to use the
program
Prof. Jose Sasian
Summary
• Review of aberrations• Aspheric surfaces• Stop shifting• Aberration coefficients• Structural aberration coefficients
• Next class derivation of coefficients
Prof. Jose Sasian
Mode matching concept
• Same mode diameter• Same amplitude distribution• Same phase distribution• Same polarization• Same x,y,z position• Same angular position
Prof. Jose Sasian
Fiber coupling efficiency
( )2 2
20 0
0 0 0 0
4
z z
εω ω λω ω πω ω
=
+ + +
( ) ( )m mx x dx+∞
∗
−∞
Ψ Ψ∫( ) ( ) 1m mx x dx+∞
∗
−∞
Ψ Ψ =∫
“overlap integral”
z
Herwig Kogelnik
Top Related