Ray tracing and ABCD matrix Optics, Eugene Hecht, Chpt. 6

Basics of ray tracing Consider 2D projection Ray uniquely defined by position and angle –Make components of vector Paraxial approximation -- express angle as slope = y ’  = y ’ y z

Example: Propagate distance L Angle (slope) unchanged Position depends on initial position and slope (y 0, y 0 ’) (y 1, y 1 ’)

Example: Go through lens Position unchanged Angle (slope) change depends on position & focal length (y 0, y 0 ’) (y 1, y 1 ’)

ABCD matrix Generalize Can cascade to make single matrix for system Example: go through lens and propagate distance L = f (y 0, y 0 ’) (y 1, y 1 ’) (y 2, y 2 ’)

Example: Fourier transform Propagate distance f, go through lens, propagate f Position and angle swap –note scale factors f and -1/f (y 0, y 0 ’) (y 1, y 1 ’) (y 2, y 2 ’) (y 3, y 3 ’) ff

Example -- 4 f imaging Cascade previous example (y 0, y 0 ’) (y 1, y 1 ’) (y 2, y 2 ’) (y 3, y 3 ’) ff (y 4, y 4 ’) (y 5, y 5 ’) (y 6, y 6 ’) ff

Example -- dielectric interface Snell’s law -- angle changes, position fixed n1n1 n2n2

Other examples Easy to generate ABCD matrices GRIN lens

Optical resonator Condition for stable cavity Return to initial state after integer N round trips