Hi Marzanna,

In my graph, the two air spaces are the air thicknesses in the design. Air Space 1 is before the central negative element; Air Space 2 is after the negative element. With the three focal lengths fixed, these are the airspaces required to produce a collimated output with the magnification indicated on the x-axis.

Regarding the fixed track length design, I do not believe it will work with the three focal lengths fixed. With that set of constraints, the only remaining variable is the position of the central element, given by a single air thickness. There are still two conditions to satisfy: 1) the output must be collimated (equivalent to an RAID = 0 operand in the merit function) and; 2) a specific magnification (equivalent to an REAY operand).  So the system is over constrained.

Doing computer algebra with the matrix methods, I found that when the three focal lengths and the track length were fixed and the constraint of collimation was imposed, there were two solutions for magnification. One was smaller than 1, the other greater than 1. But any solution for a different magnification was not collimated.

You can also see this in a simple paraxial design in OpticStudio. The merit function attempts to impose both conditions and cannot meet both, except at the two magnifications. There are two magnifications at which this can be done: 2 and .2.  (Attached as a zar.)

It is likely possible to achieve a fixed track length by splitting the central element into two elements which are allowed to move independently. This would produce the equivalent of a variable focal length central element.

Regarding commercial lenses:

Most of my work with beam expanders has been with fixed magnification Galilean designs. There, for a small input beam, the first element needs to be of very short focal length and therefor small diameter. You may find the same with the Keplerian architecture. Lens choices are limited, but fortunately a number of companies make suitable lenses specifically for such work. A good place to start looking is ThorLabs. (Many of their specialty lenses do not appear in the OpticStudio lens catalog.) Another advantage is that a number of these lenses are aspheric designs optimized for collimation.

I recommend working in OpticStudio with simple lenses, even paraxial lenses, to get a good idea of the focal lengths required, and then looking for similar elements available off-the-shelf. Then you may need to compromise design specifications, like magnification range and track length, to make use of the parts you find

Best,

David

Yes. But how were these equations derived? They could be derived using algebra assuming the Lagrange invariant, or matrix methods.

Hi Marzanna,

I looked at the paper you referenced. To avoid confusion, I edited my Mathematica notebook to match their variable names. When I solve for the air spaces d1 and d2,  I get this:

This matches the equations (5) and (6) given in the paper.

However, it does not match the equations or numbers you give above.

I attach a pdf of the notebook.