Paraxial rays and ABCD matrices

Let z increase along light propagation from left to right, let y increase upward, and let the small angle $\theta=dy/dz$ be positive for an upward-going ray. Our column vector is $(y,u)^{\mathsf T}$ with $u=n\theta$, called the reduced angle. Height is in metres and reduced angle is dimensionless. The approximation replaces sine and tangent by their angles in radians and requires small ray angles and small surface slopes throughout the system.

Within a uniform region of index n, propagation through positive axial distance d changes height by $d\theta=du/n$ and leaves direction unchanged. Thus the upper-right matrix entry has units of length. At a curved interface take radius R positive when the center of curvature lies to the right of the vertex. Linearizing Snell’s law about that local surface normal changes reduced angle by minus surface power times height, while height is continuous.

If a ray first meets element one and then element two, the total matrix is $M_2M_1$: the rightmost factor acts first on a column. Interchanging factors usually changes the system because propagation before bending differs from propagation after bending. A converging thin lens in air has positive focal length f and lower-left entry $-1/f$. An initially parallel ray above the axis therefore leaves with a negative slope, as physical convergence requires.

Both elementary reduced-angle matrices have determinant one, so their product does too for ideal first-order transport without ray rejection. That fact expresses preservation of a suitable phase-space area, not preservation of every individual ray height or angle. If a reference uses $(y,\theta)$ instead, refraction can give determinant $n_1/n_2$. Mixing that vector with reduced-angle matrices is a common error that can hide in air-only examples and fail immediately at a dielectric boundary.

All rays from one point in an object plane share its height but can have different launch angles. At an image plane they must return to a common height. The equation $y_2=Ay_1+Bu_1$ shows that first-order imaging therefore requires B to vanish. At those conjugate planes A is the transverse magnification. A negative A means inversion, while the determinant condition gives $D=1/A$ and connects angular spread to magnification.

This condition assumes the considered rays pass through the clear apertures; a matrix does not reject an obstructed ray automatically. It also cannot reveal spherical aberration, polarization, diffraction, or the finite brightness of a source. A good workflow first checks the paraxial map, then traces representative aperture rays, and finally asks whether higher-order or wave effects matter. A beautifully multiplied matrix is still only as valid as the linearization used at every surface.