FDO Cell

Transport matrix for cell

Focussing quad:

mf = {{1, 0}, {-(1/f), 1}}

Full length defocussing quad:

md = mf /. f -> -f

Drift space:

mdrift = {{1, L}, {0, 1}}

Total cell:

MatrixForm[
      mcell = Simplify[
          (mdrift /. L -> L2) . md . (mdrift /. L -> L1) . 
            mf]]

Get Trace for stability condition:

Trmcell = 
      Simplify[mcell[[1, 1]] + 
          mcell[[2, 2]]]

Stability condition is is -1<=1/2 Tr[M] <= 1 or here that L1*L2/(2 f^2)<=2 or that f>Sqrt[L1*L2]./2

If f=Sqrt[L1*L2]/2, then all rays focus at a single spot.  Since L2>L1, that spot has to be in L2.  Use simple optics to find these spots.

Let x be distance from defocussing quad where focus occurs.  Then L2-x = object distance for focussing quad:

eq1 = (1/(L2 - x)) + (1/s) == (1/f)

For defocussing quad, focus is virtual and:

eq2 = -(1/(s - L1)) + (1/x) == -(1/f)

solve1 = 
      Simplify[Solve[{eq1, eq2} /. f -> ((Sqrt[L1 L2])/2), {x, s}]]

solve2 = 
      PowerExpand[
        solve1[[1]] /. L1 -> (L2/4)]

See accompanying figure for picture of this.