Optics — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Optics (Multi-step)

• Silvered lens = equivalent mirror: power $P=2P_{lens}+P_{mirror}$, so $\dfrac{1}{F}=\dfrac{2}{f_{lens}}+\dfrac{1}{f_{mirror}}$; then use the mirror formula.
• Cut lenses: cut along the axis $\Rightarrow f$ unchanged; cut across the axis (into two plano-convex) $\Rightarrow f$ doubles for each half.
• Lenses a distance $d$ apart: $P=P_1+P_2-dP_1P_2$. Achromatic doublet: $\dfrac{\omega_1}{f_1}+\dfrac{\omega_2}{f_2}=0$.
• YDSE slab shift: $\Delta y=\dfrac{(n-1)tD}{d}$. Layered apparent depth: $\sum\dfrac{t_i}{n_i}$.
• Two-source contrast: with amplitude ratio $r=\sqrt{I_1/I_2}$, $\dfrac{I_{max}}{I_{min}}=\left(\dfrac{r+1}{r-1}\right)^2$; fringe visibility $=\dfrac{2r}{1+r^2}$.

Worked mini-example. A plano-convex lens ($n=1.5$, $R=20$ cm, so $f_{lens}=\tfrac{R}{n-1}=40$ cm) silvered on its plane face has $\dfrac{1}{F}=\dfrac{2}{40}=\dfrac{1}{20}$, i.e. it acts as a concave mirror of focal length 20 cm. An object 30 cm away then images at $\dfrac{1}{v}=\dfrac{1}{-20}-\dfrac{1}{-30}=-\dfrac{1}{60}$, $v=-60$ cm (real, $m=-2$).