Principal points of achromatic lens pair

For two lenses to form an achromatic combination with the same glass, the separation must be x = (f₁ + f₂)/2.

Given: f₁ = 8 cm, f₂ = 4 cm.

Step 1: separation

x = (8 + 4)/2 = 6 cm

Step 2: equivalent focal length

1/F = 1/8 + 1/4 − 6/(8×4) = 0.125 + 0.25 − 0.1875 = 0.1875
F   = 5.33 cm

Step 3: principal points

α = F x / f₂ = 5.33 × 6 / 4 = 8 cm    (H₁ right of L1)
β = F x / f₁ = 5.33 × 6 / 8 = 4 cm    (H₂ left of L2)

 L1     H2          L2   H1
 |------.-----------|----.
 0     2.0          6.0  8.0   (cm from L1)

H₁ lies 8 − 6 = 2 cm beyond the second lens; H₂ lies 6 − 4 = 2 cm after the first lens, so the principal points are crossed.

Answer: x = 6 cm, F = 5.33 cm; the first principal point is 8 cm to the right of the 8 cm lens (2 cm beyond the second lens); the second principal point is 4 cm to the left of the 4 cm lens (2 cm from the first lens).