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

neglected.