 Solution

Represent a paraxial ray by its height and forward slope $(x,\theta {)}^{\mathsf{T}}$, with $\theta =dx/dz$. Free-space propagation leaves the slope unchanged and increases height by $L\theta$, givingA thin lens leaves the ray height unchanged. A parallel incident ray at height $x$ must emerge toward the focal point, so its slope changes by $-x/f$. By linearity, the same increment applies to any paraxial incident slope:These ray transfer matrices have determinant one in air and compose in reverse matrix order to the physical order of traversal.