Solution: Path of Ray through a Stack of Lenses
1. The Physical Model
We consider a stack of thin converging lenses, each with focal length $f$, separated by a small distance $l$. Since $l \ll f$, we approximate the discrete system as a continuous medium.
Let the optical axis be the $x$-axis and the transverse displacement be $y$.
Fig 1: The ray enters parallel to the axis at height $h$ and undergoes simple harmonic motion.
2. Derivation
A ray at height $y$ passing through a lens of focal length $f$ deviates by an angle $\Delta \theta = -y/f$. The rate of change of the angle per unit length is:
$$ \frac{d\theta}{dx} \approx \frac{\Delta \theta}{l} = -\frac{y}{fl} $$Since $\theta \approx dy/dx$ (paraxial approximation), we differentiate again:
$$ \frac{d^2 y}{dx^2} = -\frac{1}{fl} y $$3. The Solution
This is the differential equation for Simple Harmonic Motion: $y” + \omega^2 y = 0$, where $\omega = \frac{1}{\sqrt{fl}}$. The general solution is $y(x) = A \cos(\omega x + \phi)$.
Applying Boundary Conditions:
- At $x=0$, position is $y = h$.
- At $x=0$, the ray is parallel, so slope $y’ = 0$.
These conditions yield $A=h$ and $\phi=0$. Thus, the equation of the trajectory is: