Determine lens system for known magnification and available space

Question

I'm trying to create a system of two lenses with a set magnification in a limited space for the setup, i.e. the distance from the object to the first lens $g_1$ has a minimum ($152\,$cm) and the whole lens setup is limited ($40\,$cm). The defined length of the setup $L$ is defined below. For a sketch see image:

In total there are the following important parameters:

• object size $G=4.136\,$mm, is known
• magnification $V\simeq0.2579$
• focal length $f_i$ of each lens (should be reasonable so one can buy it)
• distance to objects $g_i$ (where $g_1$ has a minimum of $g_{min}=152\,$cm)
• distance to image $b_i$
• image size $B_i$ ($B_2$ is set due to the magnification)
• distance between lenses $d=b_1+g_2$

where $i\in\{1,2\}$ and the setup distance $L=(g_1-g_{min})+d+b_2\overset{!}{<}40\,$cm. Since the image is captured by a camera in position $B_2$ the image distance of the second lens is required as $b_2>0$.

My question is, whether there is an elegant way of finding the best possible fit of lenses and positions that I don't know of or if is this only solvable by calculating the whole system for varying $f_i,g_i,d$?

Answer 1

You can start with a single lens to achieve the magnification you want. This will give you a range of values for the focal length. If the orientation of $B_2$ does not matter, then you would get two intervals for the focal length: $f = \frac{g_1}{1 \pm \frac{1}{V}}$

This is the effective focal length of your two-lens system. Optimize your setup with available lenses and their separation using Gullstrand's equation. Then you will have to get the front and back working distances to see if they match your requirements

If not possible, then you could try adding a third lens. There are online calculators for this purpose or use Zemax