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Shape Factor and Aberration
“Bending Lenses”
The shape of a lens is defined by the shape factor, X:
𝑐1 + 𝑐2
𝑋=
𝑐1 − 𝑐2
The following lenses all have the same focal length, but different shape factors:
𝑋 < −1
𝑋 = −1
𝑋=0
𝑋 = +1
𝑋 > +1
Making curvature changes in a lens without changing its power is known as
“bending” the lens. A lens can be bent in Zemax by using a solve on the second
surface to maintain the element power:
For the lens above with shape factor, 𝑋 = −1, the Zemax LDE is:
and the solve on surface 2, Curvature is:
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This solve causes the second surface of the lens to take the value that maintains
lens power as Z= 0.05 mm^-1, or F=1/Z=20mm. The first surface of the lens can
then be made variable without changing the power of the lens.
The lens’s aberrations will change as its shape factor changes, since this
changes the real angles of incidence of the rays impinging on it. We can monitor
the aberrations using the operands available in the Merit Function Editor (MFE):
Clicking on the ‘Help’ menu item in the MFE,
brings up the following operand list:
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The list goes on for several pages. Clicking on an operand brings up the
definition and instructions for use. For example, clicking on ‘COMA’ in the above
list gives this information box:
The terms “Surf” and “Wave” refer to the columns in the MFE which will be
labeled as such when the COMA operand is entered.
Other operands for third-order aberrations of interest are:
 SPHA: Spherical Aberration
 FCUR: Field Curvature
 ASTI: Astigmatism
Install these four operands in the MFE, setting the “Surf” value to “0”, so that the
aberration sum for both surfaces is returned:
(Here, the “Hide Column” feature under the “View” menu has been used to shrink
unused columns in the editor.)
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We set Zemax up with two field points at ∞, one on axis and one at 5 degrees
(since only spherical aberration exists on axis – the others require an off-axis
field):
Likewise, since these are all monochromatic aberrations, we use only one
wavelength:
We can then update the MFE and see what aberrations Zemax reports:
What do these aberrations look like in the analysis windows?
Spherical aberration is present on-axis, and is the change of focal length with ray
distance from the axis. In the ray diagram, it looks like this:
Positive Spherical
aberration
...and miss the
paraxial focal point
by a larger amount.
(TRA)
Rays further
from the axis…
…focus closer
to the lens…
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Spherical aberration looks like this in a Transverse Ray Aberration (TRA) “ray
fan” – Spherical Aberration is characterized by a focus error that changes with
distance from the pupil center:
OBJ: 0.0000 (deg)
ey
ex
Py
Px
Transverse Ray Fan Plot
2/26/2013
Maximum Scale: ± 5000.000 µm.
0.550
Shape.zmx
Configuration 1 of 1
Surface: Image
For comparison, here is what pure focus error in a paraxial system looks like:
OBJ: 0.0000 (deg)
ey
ex
Py
Px
Transverse Ray Fan Plot
2/27/2013
Maximum Scale: ± 1000.000 µm.
0.550
Surface: Image
LENS.ZMX
Configuration 1 of 1
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Coma only affects off-axis fields – it is a change in magnification with ray
distance from the center of the entrance pupil:
Coma, in the ray fan plot, manifests as asymmetry in the Tangential TRA (in the
Meridonal plane):
OBJ: 10.00 (deg)
ey
ex
Py
Px
Transverse Ray Fan Plot
2/27/2013
Maximum Scale: ± 5000.000 µm.
0.550
Surface: Image
Shape.zmx
Configuration 1 of 1
(Note: The y-z plane is always the “meridonal” plane in Zemax, even though that
is not the optical definition; Optically, the meridonal plane is the plane containing
both the ray and the z-axis. The two definitions agree if “meridonal” rays are
confined to the y-z axis.)
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OBJ: 10.00 (deg)
0.5500
20000.00
Ray Pattern with Coma at the Paraxial Focal Plane:
Surface: IMA
IMA: 3.507 mm
Spot Diagram
2/27/2013
Field
RMS radius
GEO radius
Scale bar
Units are µm.
:
2
:
1688.44
:
5179.88
: 2e+004
Reference : Chief Ray
Shape.zmx
Configuration 1 of 1
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Using our Merit Function which reports 3rd order aberrations, we can write a
Zemax macro (see: “Zemax Programming Language”, chapter 25 in the Manual).
Here is a macro in the “ZPL” language that loops our system through 50 different
Shape Factors and prints out the aberrations:
!ThirdOrd.zpl
!
!Print out 3rd order aberrations vs. shape factor
!Run on Shape.zmx
!Editors LDE and MFE must be open for macro to work correctly
!
!Number of points to print:
N = 50
!
!Surfaces in Shape.zmx
frontsurf = 3
backsurf = 4
!Rows in MFE:
Spha = 1
Coma = 2
Asti = 3
Fcur = 4
!
!Get starting curvatures (should already be at one extreme)
c1 = CURV(frontsurf)
c2 = CURV(backsurf)
!
Inc = -(c2+c1)/N
!
!Loop through shape factors:
FOR S1, c1, -c2, Inc
!Set curvature of frontsurf:
SetSurfaceProperty frontsurf, 2, S1
UPDATE all
!Get back surface curvature:
S2 = CURV(backsurf)
!Calculate the shape value:
Shape = (S1+S2)/(S1-S2)
!Extract the aberrations from the MFE:
SA = OPER(Spha,10)
CO = OPER(Coma,10)
AS = OPER(Asti,10)
FC = OPER(Fcur,10)
!
PRINT Shape, " ",SA," ",CO," ",AS," ",FC
NEXT
ZPL is much like early “Basic”, but contains a number of functions that access
Zemax editor values – such as “OPER”, which accesses values in the MFE – and
keywords – like “UPDATE” – which drive Zemax operations. Macros are run
from the “Macros” menu from the top of the Zemax window, and they must be in
the Macro Directory. Not sure where the Macro Directory is, or you would like to
change it? Look under “File/Preferences/Folders”.
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The output of the “ThirdOrd” macro is a text window:
You can save this text to a file and use it in a plotting program (or you can deal
with the primitive plot capabilities of ZPL):
Spherical Aberration & Coma vs. Shape Factor
Spherical & Coma vs. Shapefactor
100
Sph Abb
Coma
80
60
Aberration
in Waves
40
20
0
-20
-40
-1.5
-1
-0.5
0
S
h
a
p
e
Factor
Shape Factor
0.5
1
1.5
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Expanded View:
Spherical & Coma vs. Shapefactor
Sph Abb
Coma
20
15
10
5
0
-5
-10
-0.4
-0.2
0
0.2
0.4
0.6
Shape Factor
0.8
1
1.2
1.4
As can be seen from the close-up, both S.A. and Coma are minimized at a
Shape Factor of ~0.7
Adding weights to the spherical aberration and Coma lines of our MFA;
Plus making the radius of the first lens surface variable and optimizing, we get
this lens (R1=11.7825, R2=-79.00985, Shape Factor = 0.7405):
pg. 10