Three Proofs — Question 9

Question 9

Opposite lens powers can cancel a color-dependent focus shift

Two thin lenses in contact in air have shape factors G1 and G2 and powers $(n_1-1)G_1$ and $(n_2-1)G_2$. At a reference color, n1=1.50 and n2=1.60. At a second color the index increments are 0.010 and 0.020. Require total power 5 inverse meters at both colors.

1. Derive the two linear equations for the shape factors.

2. Solve for the shape factors and the individual reference-color powers, then verify cancellation of the color shift.

Question 9 - Solution

Opposite lens powers can cancel a color-dependent focus shift

(a) Method and calculation

$$\boxed{0.50G_1+0.60G_2=5\,\mathrm{m^{-1}}}.$$$$\boxed{\Delta\mathcal P=0.010G_1+0.020G_2=0}\Rightarrow G_1=-2G_2.$$

(b) Interpretation and further calculation

$$\boxed{G_1=25\,\mathrm{m^{-1}},\quad G_2=-12.5\,\mathrm{m^{-1}}};\quad\boxed{\mathcal P_1=12.5,\ \mathcal P_2=-7.5\,\mathrm{m^{-1}}}.$$$$\boxed{\Delta\mathcal P=0.010(25)+0.020(-12.5)=0,\quad f_{\mathrm{total}}=0.20\,\mathrm m}.$$

Physical check

The lenses must have opposite powers in this model. Matching these two colors does not establish perfect correction over an entire spectrum.