Thin and thick lens imaging

Continue using light traveling along positive z and reduced ray vector $(y,n\theta)^{\mathsf T}$. For a lens in air, define object distance $s_o=z_{H_1}-z_o$ and image distance $s_i=z_i-z_{H_2}$. A real object on the incoming side has positive object distance, and a real image on the outgoing side has positive image distance. For a thin lens the two principal planes coincide with its plane; a converging element has positive f.

Multiply translation from object to lens, the lens matrix, and translation to the proposed image. Requiring the total B entry to vanish gives the lens equation. The corresponding A entry is transverse magnification. A negative image distance means the outgoing rays diverge as though they originated from an image on the incoming side; extending them backward locates a virtual image. Such an image can be viewed, but cannot be caught on a screen placed at its apparent position.

A homogeneous thick lens with surface powers $\Phi_1$ and $\Phi_2$, axial thickness t, and internal index n has matrix $S_2T(t,n)S_1$ between its surface vertices. Its net power is $\Phi_1+\Phi_2-(t/n)\Phi_1\Phi_2$. The separation term shows why simply adding surface powers is only a thin-element approximation. Each surface power uses its own incoming and outgoing indices and the signed radius defined in the preceding lesson.

When the external medium is air on both sides and net power is nonzero, effective focal length is the negative reciprocal of C. A parallel input ray has outgoing height Ay and slope Cy at the last vertex. It reaches the axis after distance $-A/C$, which is the back focal length measured from that vertex. The effective focal length is instead measured from the second principal plane. These distances agree only for special placements of that plane.

Principal planes are equivalent reference planes that let a centered first-order system obey a thin-lens-style conjugate equation. They need not coincide with a glass surface, remain inside the glass, or lie at the mechanical center. For an air-to-air matrix, the first plane lies $(1-D)/P$ to the right of the first vertex, and the second lies $(A-1)/P$ to the right of the last vertex. These offsets follow by factoring the matrix into two translations and a thin powered element.

For a real design, label the vertex coordinates and principal-plane coordinates before inserting any object distance. This prevents a physically meaningful but numerically wrong detector placement. Principal-plane equivalence remains paraxial: it does not imply that actual rays bend abruptly at those planes or follow the constructed lines inside the element. If C vanishes the system is afocal, effective focal length is not finite, and this particular finite-power factorization must be replaced by the appropriate matrix description.