2016-12-21 Prof. Herbert Gross Yi Zhong, Norman G. Worku Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str 15 07745 Jena Lens Design II – Seminar 5 (Solutions) 5.1. Interferometer lens If optical spherical surfaces are tested in an interferometer, an additional objective lens is needed to generate a convergent beam. In particular special ratios of diameter and surface radius are necessary to be able to test many surfaces. In reality, mostly a collimated incoming monochromatic beam is used. In this exercise a system should be developed. To avoid problems with imperfect centering of the test surface, mostly a very small field angle is assumed and also corrected to get an insensitivity in the application. Especially in Fizeau-type interferometers one surface of the system is used as a reference. It is preferred to use the last surface for this, this should be coated with a larger reflectivity and must be concentric to the spherical wave. a) Consider a wavelength of 632.8 nm, an incoming collimated beam with diameter 100 mm and first look only on axis. The F-number of the lens should be F# = 1.5. The free working distance is required to be 150 mm. The performance of the focusing lens on axis should be Strehl 99% on axis. The last surface is the Fizeau surface, all lenses are made of SFL6. Determine a system with the appropriate number of lenses to meet this specification. b) Now introduce a field of 10' and reoptimize the system according to the specifications with 95 % Strehl for the 10' field position. Solution: a) First only the axis location is corrected. The F-number of 1.5 corresponds to the numerical aperture of 0.33. The ratio of the semi-diameter and thew free working distance corresponds to this value. But if the last surface is concentric to the image point, it is concave and the free working distance is governed by the marginal ray intersection point at the rim. Therefore the system is of a slightly refocus type. The Fizeau surface is fixed by a solve with a marginal ray normal at the last surface. In the optimization, we first optimize only the radii and set the distances by hand to get appropriate edge thickness values. If we increase the number of lenses step by step, we get the following result.
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2016-12-21
Prof. Herbert Gross
Yi Zhong, Norman G. Worku
Friedrich Schiller University Jena
Institute of Applied Physics
Albert-Einstein-Str 15
07745 Jena
Lens Design II – Seminar 5 (Solutions)
5.1. Interferometer lens
If optical spherical surfaces are tested in an interferometer, an additional objective lens is needed
to generate a convergent beam. In particular special ratios of diameter and surface radius are
necessary to be able to test many surfaces. In reality, mostly a collimated incoming
monochromatic beam is used. In this exercise a system should be developed. To avoid problems
with imperfect centering of the test surface, mostly a very small field angle is assumed and also
corrected to get an insensitivity in the application. Especially in Fizeau-type interferometers one
surface of the system is used as a reference. It is preferred to use the last surface for this, this
should be coated with a larger reflectivity and must be concentric to the spherical wave.
a) Consider a wavelength of 632.8 nm, an incoming collimated beam with diameter 100 mm
and first look only on axis. The F-number of the lens should be F# = 1.5. The free working
distance is required to be 150 mm. The performance of the focusing lens on axis should
be Strehl 99% on axis. The last surface is the Fizeau surface, all lenses are made of SFL6.
Determine a system with the appropriate number of lenses to meet this specification.
b) Now introduce a field of 10' and reoptimize the system according to the specifications with
95 % Strehl for the 10' field position.
Solution:
a) First only the axis location is corrected. The F-number of 1.5 corresponds to the numerical
aperture of 0.33. The ratio of the semi-diameter and thew free working distance corresponds to
this value. But if the last surface is concentric to the image point, it is concave and the free working
distance is governed by the marginal ray intersection point at the rim. Therefore the system is of
a slightly refocus type. The Fizeau surface is fixed by a solve with a marginal ray normal at the
last surface. In the optimization, we first optimize only the radii and set the distances by hand to
get appropriate edge thickness values.
If we increase the number of lenses step by step, we get the following result.
In the merit function, the spot is optimized. As required the z-value of the marginal ray intersection
point relative to the image plane (this is defined to be the global coordinate reference), is forced
to be 150 mm.
The performance on axis is quite good in this case, the Strehl ratio is near to 100 %.
b) Now we introduce the desired field angle, which is 0.16°. It is seen from the results of the
optimization, that there is no chance to meet the specification with only 4 lenses. Therefore we
add one more lens.
To get this result, the weighting of the field locations must be adapted.
If the thickness optimization is allowed, we get a solution with a little better performance still for
the field 0.16667°. The first lenses tend to become thicker in this case.
The 3rd and the 5th surface are compensating for the other positive spherical aberrations. The
first two lenses are of negative power.
5.2. Anamorphic diode collimator
A semiconductor diode with wavelength 650 nm and the divergence / aperture values 0.4 / 0.1 in
the fast and slow axis respectively should be collimated in a circular beam with a diameter of
approximately 8 mm. The collimated beam is now focused into a fiber with numerical aperture of
NA = 0.1.
semiconductor
diode
NAy = 0.4
NAx = 0.1
= 650 nm fiber
NA = 0.1
L1
aspherical
collimator
fast axis
L2
cylindrical lens
L3
L4
focussing lens
circular beamD = 8 mm
cylindrical lens
Find a solution for this problem with only available catalog lenses.
Is the setup diffraction limited? Explain the shape of the residual spot pattern. What are
the reasons for the residual aberrations in the system? What can be done to further
improve the result?
Solution
If the desired beam diameter after the collimation of the fast axis is 8 mm, the focal length of the
first lens is
mmNADf y 10/2/
Since the numerical aperture of the fast axis is high, it is recommended to use an aspherical
collimator lens, which is corrected for spherical aberration on axis. If such a lens is found in the
lens catalogs, it must be considered:
1. the lens should be used without cover glass plate
2. if a working wavelength near to the 650 nm is found, it is an advantage
Possible solution:
Catalog Asphericon, lens with the No A12-10HPX
Necessary steps to process this lens:
1. load the lens
2. turn around
3. set NA to 0.4 and vignetting factors in field menu to VCX = 0.75.
Alternatively, the front surface of the collimating lens can be established by an elliptical aperture.
If the axes of the ellipse are set in a ratio of 1:4, the desired light cone is obtained in approximation.
In this case exactly the tan(u) values are related and therefore the numerical apertures as sin(u)
values are only roughly obtained.
4. change wavelength to 650 nm
5. optimize first distance to collimate this wavelength (default merit function, with criterion: direction
cosines). Alternatively, the option QUICK ADJUST can be used with the first distance as variable
and the angle spot as an afocal criterion.
A footprint diagram shows the elliptical beam cross section behind the lens.
In the next step, a Galilean telescope with factor = 4 must be found to enlarge the diameter of
the x-section to the same value as in the y-section. First a negative cylindrical lens with a rather
short focal length must be found.
Possible solution: Lens with 25 mm negative focal length in the catalog of Melles Griot:
RCC-25.0-15.0-12.7-C
The lens is inserted behind the collimating asphere and rotated around the x-axis by 90° to work
in the x-section.
The distance to the collimator is not very relevant and is fixed to be 5 mm.
For a Galilean telescope with factor 4, the second lens must have a focal length of 4x25.1 mm =
100.4 mm. In the same lens catalog one can found the following lens:
RCX-30.0-20.0-50.9-C
The lens is inserted, turned around to get a better performance and also tilted by 90° in the
azimuth.
A first guess gives a distance of 100-25=75 mm between the telescope lenses to get a collimated
x-section. But from the spot diagram with direction cosine option it is seen, that the angle
distribution is not equal in both sections. Due to the finite positions of the principal planes of the
lenses, the distance must be optimized with an angle criterion default merit function.
Again as an alternative, the QUICK ADJUST feature can be used to find the optimal lens distance
in the telescope.
Spot diagram before and after this focussing operation with the same scale:
The footprint diagram now shows a rather circular cross section. The residual error can be
neglected and comes from the fact, that for this wavelengths, the catalog focal lengths are not
exact.
The data are now the following:
To focus the beam into a fiber with numerical aperture 0.1, the focal length must be not smaller
than
f = 4.32 mm / 0.1 = 43.2 mm. A lens of approximately this size can be found in the catalog of
Melles Griot as an achromate. This helps in getting a better correction:
LAO-44.0-14.0
This lens is inserted to complete the system. Finally the last distance is optimized to get a minimal
spot size.
It is seen, that the spot is nearly diffraction limited.
5.3. Anamorphic Prism Beam Expander
With two identical prisms, which are arranged in opposite order (1st: tip at top, 2nd tip at bottom)
and both are incident perpendicular, a collimated beam is changed in its diameter in the section
of the incidence plane.
The magnification can be calculated by the formula
sin(a) = sqrt( (1-m) / (n2-m) ),
with the tilt angle a of the second surface.
a) Select a system with two prism made of BK7 at a wavelength of = 550 nm for a
magnification of m = 0.5. The incoming collimated beam diameter is 10 mm, the thickness
of the prisms is 5 mm. Check the elliptical cross section at the output. What is the stretching
factor between the prisms?
b) If the incoming beam is not perfectly collimated but shows a residual divergence of 5 mrad,
determine the residual aberrations of the system. Select an image plane in a distance, that
the c4-Zernike coefficients is zero.
Solution:
a) The refractive index is 1.5185224, therefore from the above formula we find a tilt angle of
31.74792°. The distance between the prisms is chosen to be 10 mm. In the system the
second surfaces of both prisms is tilted as a single surface by -31.74792° and +31.74792°
respectively. Two coordinate breaks are established with opposite tilt angles around the x-
axis of 53.0363665° corresponding to the new axis direction after the first prism.
Note: Here instead of analytical formula, it is also possible to use zemax numerical
optimization to determine the tilts of second surface of each prism and the tilt and decenter
of the second prism (as discussed in the seminar).
The magnification is exactly m = 0.5, this is shown by the footprint diagram. Between the prisms,
the streteching factor is sqrt(2) = 0.707 corresponding to a cross section diameter of 3.5365.
Therefore the stretching is exactly symmetrical.
b) The object distance is fixed to be 1000 for a numerical aperture of NA = 0.005. Optimizing the
final distance for c4 = 0 delivers d = -636.43 mm.
The remaining aberration is dominated by astigmatism of 14 waves. All other aberrations can be