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2012-10-11 Tutorial Zemax 8: Correction II
8 Correction II 1 8.1 High-NA Collimator
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1 8.2 Zoom-System
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6 8.3 New Achromate and wide field system
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8 Correction II
8.1 High-NA Collimator
An achromatic system is typically diffraction limited for a
numerical aperture not larger then NA = 0.1. For a point on axis,
by adding an aplanatic-concentric meniscus-shaped lens, the
numerical aperture can be increased by a factor correponding to the
refractive index of the lens. This principle is shown in the
exercise 6.1 a) Load an achromate with focal length f = 100 mm from
a vendors catalogue with a diameter of at least 30
mm. Take the wavelength to = 632.8 nm. Reduce the entrance pupil
diameter to achieve a diffraction
limited system according to the Marechal criterion of W rms <
/14. Add one aplanatic-concentric meniscus lens of a high-index
medium of appropriate distance and thickness. b) Now add more
meniscus lenses, until the image space NA is larger than 0.6. The
thickness of these lenses should again be chosen appropriate to get
manufacturable components (ratio of diameter to thickness not
larger then 10). Show, that the spherical aberration contribution
of the meniscus is zero for all orders of spherical aberration.
What is the resulting free working distance of the final system ?
c) Now try to further improve the system by numerical optimization.
Is it possible to get better performance ? Think on the definition
of proper constraints to get a reasonable and comparable
system.
Solution: a) In the CVI Melles&Griot catalog, the lens
LAO-100.0-30.0 is loaded as one possible f=100 achromate. The size
is reduced with the help of the menue Tools/Misc/Scale lens with a
factor of 0.1 and the wavelength is set to 632.8 nm. Furthermore
for a more comfortable layout, an additional surface is added in
front of the lens to see the incoming rays.
It is seen in the menue RMS vs defocus, that the numerical
aperture of 0.15 in the image space (for this achromate, may be
different for other choices).
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By try and error, the entrance pupil size is reduced to 23.66 mm
with an image sided numerical aperture of 0.117 to get exactly a
diffraction limited system. The choice of this menue options makes
sense, because be changing the aperture, a small change of the best
final plane occurs. If a refocussing is made with the Quick focus
option, the criterium must be the same in the defocus diagram.
There are small differences between a spot and a wavefront
optimization.
As a material for the aplanatic concentric meniscus lens, the
glass LaSF35 is used, which has a refractive index of 2.015 for the
actual wavelength. The lens is placed in a distance of 1 mm and the
thickness should be 3 mm. To get the desired shape of the bended
meniscus, the front surface is forced to be aplanatic by the
corresponding solve at the surface. For the rear surface, the solve
option Marginal Ray Normal forces the marginal ray to pass this
surface concentric.
The resulting system now has a numerical aperture of
NA=0.232.
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b) If now additional components are added, a factor of 0.6/0.232
= 2.59 is needed. Therefore a refractive index of at least 1.61 is
necessary and must than be applied twice. For example the glass F6
with an index of 1.63 is feasible.
If the numerical aperture is evaluated, we see, that the
paraxial numerical aperture angle is only NA = 0.536. If a ray
trace of the marginal ray is calculated we see, that the paraxial
and the real ray angle is quite different and that the real ray
fulfilles the requirement of NA > 0.6 with a value of NA =
0.635. The reason for this strange deviation is the fact, that the
aplanatic surfaces cause no spherical aberration. This means, that
the intersection length in the image plane is the same as in the
paraxial case. To get this goal, the real ray refraction must have
a change in the angle of the ray, which has this large value
here.
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If a higher index glass is used, for example SF6, we get a
higher numerical aperture also for the paraxial case. If the
Zernike coefficients are calculated for the complete system and
tfor the achromate alone (surface 4), we get the same spherical
coefficients (c9,c16,c25). To achive this result, the focussing
must be made identical. Due to the large difference in th
enumerical aperture, the defocus coefficient c4 is not identical,
if the Quick focus is used as a criterion. If the last distance is
change by 0.00m mm, the c4 coefficient is nearly equal and then
also the the spherical aberrations agree.
The free working distance now is on axis 11.25 mm (last air
thickness. If the cursor is set onto the intersection point of the
last surface we see, that the z-difference from this point until
the image plane is only 8.7 mm. This is the real free working
distance.
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c) If now the radii of the new three lenses are defined as
variables and a default merit function is used, the Strehl
definition can be improved from S = 0.731 to S = 1. To have a fair
comparison, the focal length of f = 18.63 must be set invariant.
The optimization has made only very small changes to the radii.
By inspection of the Seidel diagram it is seen, that now the
spherical contributions of the surfaces have changed to correct the
residual errors of the achormate, which is not changed here.
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8.2 Zoom-System
A zoom system with afocal setup should be implemented with 3
lenses, a plus-minus-plus setup a zoom range of 9. The input
diameter of the collimated beam is 12 mm, the first focal length
should be f1 = 100
mm. The zoom factor should be located symmetrically around the
magnification = 1. a) Determine the second and the third focal
length and the corresponding distances of the lenses. Formulate a
merit function, that forces the corresponding conditions and values
of the different zoom positions. Take in mind, that the overall
length of the system cannot be constant and that the system should
be afocal for every zoom position. Establish the system by ideal
lenses as a multi-configuration system with 3 zoom positions. b)
The paraxial layout of the system can also be calculated by the set
of formulas of the lecture. Prove the numerical results by the
analytical solution. c) Change the different lenses to real single
lenses of standard glasses and real air distances and perform a
multi-configuration optimization and extend the multiconfiguration
to 5 zoom positions. The default
wavelength = 0.55 m is chosen. What can be said about the
performance of the system ? d) Now to improve the result, the
moving components should be build as achromatic doublets. Try to
find appropriate focal lengths in the vendor catalogs. Re-optimize
the system. Can an overall diffraction limited design be obtained ?
What is the limiting reason ?
Solution:
a) The afocal system is symmetrical for the = 1 constellation.
Therefore also the third lens should have the focal length f3 = 100
mm. In the extreme moving positions of the system, one thickness
shrinks to zero. The unknown parameters are the overall lengths of
the system and the focal length f2 of the moving negative lens.
The multi configuration editor looks like the following
picture:
and in the merit function the afocality and the different
magnifications are required by defining the corresponding operands
for the marginal ray direction (PARB) and height (PARY) for every
configuration.
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In principle, we have in the merit function 4 variables and 6
requirements. But due to the symmetry, the problem has a unique
exact solution.
b) The set analytical formulas give the results
mmmmF
tL m 1004
21002
1
122
max
max
1
mmmmF
tt 67.663
2100
11
max
max
1
minmax
mmfmmmmFF 25,04.0401.01 211
max12
which are identical to the numerical values. The following
figure again shows the basic principle of the layout. In particular
it should be taken in mind, that the symmetrical setup has a
significant larger total length of the system. An invariant setup
is physically impossible.
f1
f2
f3
f2
f2
= 1
= 1/3
= 3
L
f1
f1 f3
f1 f3
f2
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c) First, the two additional configurations are established. The
corresponding marginal ray heights are forced to be 4 and 12 mm
corresponding to magnifications of 2/3 and 2. The multi
configurations editor and the layouts now looks like the following
figures.
In the next step, all lenses are generalized to singlets and the
system is optimized numerically. Here due to the asymmetric use in
the extreme zoom positions, the forced symmetry in the shape of the
first and the third lens is given up. The following points should
be taken into account: - the flag afocal image space is set - the
bending of the lenses is realized by setting the focal length to
remain constant with the help of a solve for the rear radii. - as
lens thicknesses, a first guess of 3, 2 and 7 mm are chosen - as
material BK7 is used - to get realistic solutions, now in the first
and the last configuration the zero-thicknesses are set to variable
to avoid non-physical solutions - in addition to the merit function
of the paraxial layout, a default merit function with angle radius
as criterion and air thicknesses larger than 0.5 mm are forced. If
the optimization is run it is seen, that due to the changed
diameters of the ray bundel for the different zoom positions, a
collision of the second and the third lens takes place in the last
configuration. Therefore a minimum air thickness of 3 mm is used in
the merit function. Now the optimization run gives the following
solution.
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If now the spot diagram is observed it is seen, that the
performance is significantly far off from beeing diffraction
limited.
d) In a last step, the three lenses are substituted by
achromates to generate a larger number of degree of freedom to
correct the system. As an example, the lenses AAP - 100.0 – 25.4 of
CVI Melles Griot and 62486 from the Edmund Optics are possible
candidates. To guarantee that the focal lengths of the doublet
components remain constant, three lines with the EFLY are defined
to specify the partial focal lengths to the lens-groups.
In a first step, only the air distances are allowed to be
varied. It is seen, that the diameter of the third lens is too
small and the thicknesses must be enlarged. The results looks fine
only for the configurations 2-4.
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In a third step, also the radii are allowed to be changed and
the thicknesses are appropriate. Then the results looks better, but
is still not diffraction limited.
The reason for this bad result is, that the overall length of
the system is quite short for the ambitious zoom factor and
numerical aperture. In reality, in addition the broadband behavior
for several wavelength and the correction for finite field points
must be considered. This shows, that the correction of zoom system
is quite complicated.
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8.3 New Achromate and wide field system
A new achromate is a cemented component, that is optimized for
large field applications. In this case, the Petzval condition is
the basis for the refractive power distribution and the
achromatization is no longer a correction goal. In this case, a
large difference in the refractive indices is more important than a
large spreading of the Abbe number. a) Establish a classical
achromate made of BK7 and SF6 with focal length f = 100 mm and an
object at infinity from the scratch. Insert a stop 20 mm before the
lens and select the wavelength 546.07 nm and an entrance pupil
diameter of 4 mm. The system should be used on axis andfor a
maximum field angle of 20°. Optimize the radii for the default
merit function and evaluate the system performance. b) Extend the
system to a finite imaging system with magnification m = -1 by
doubling the system. The first group of lenses should by reversed
and the stop is in the middle of the system to get a complete
symmetric layout. What is the dominating aberration. Insert a
second field point in the zone of the field at 14°. c) Now the
second part of the symmetric system is scaled down by a factor of
2. This delivers a system with a magnification of m= -0.5.
Therefore we no longer have a symmetric layout. What happens with
the system quality ? d) To improve the system add two meniscus
lenses made of SK12 near to the cemented components towards the
stop with the exact 1:2 scaling. Optimize the radii to improve the
system while preserving the mirror-symmetry. Can the astigmatism be
removed ? e) Now optimize the radii without any anti-symmetric
constraints. Show, that the performance can be improved
significantly. Finally enlarge the numerical aperture of the system
to a value, that is just below the diffraction limit. To have some
more degrees of freedom, the distances around the meniscus lenses
are also set as variable. To guarantee a useful solution, the
magnification (PMAG -0.5) and the overall length of the system
(TOTR 190.5) must be forced in the merit function to be constant.
Show, that the field curvature is corrected by the mensicus shaped
lenses to a very good flattened field.
Solution: a) The approach delivers the follwing system:
The performance is nearly diffraction limited, a residual
astigmatism is seen for the field point.
b) the extension of the systems gives the following layout.
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The astigmatism is enlarged in the field, this is seen at the
line-shaped focus at 20° field. c)
The scaling can not be performed by a simple single command in
Zemax. Therefore for the values of the distances and thicknesses
are set by hand to half the values, the radii are calculated by
setting a pick up of the corresponding surface with a scaling
factor of -2 (not -0.5 !). It is seen, that the astigmatism for the
outer field is still the limiting aberration. d) The system now
looks as follows:
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It is seen, that the performance is much better, there is nearly
no longer astigmatism inside. e) A complete arbitrary optimization
of the radii gives the following result:
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In the second step, the entrance pupil diameter is enlarged to a
value of 7 mm. In the merit function the two requirements are
added:
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The rms-plot against the focus shows nearly coincident image
locations over the field. This means, that the field curvature is
well corrected.
