Geometrical Optics

Geometrical Optics Refraction, reflection at a spherical/planar interface Hecht, Chapter 5 Wednesday Sept. 11, 2002

Same for all rays Independence of path For any rays traveling from point S to another point P in an optical system the optical path lengths are identical!!

z y x y x Reflection by plane surfaces r1 = (x,y,z) r2= (-x,y,z) r1 = (x,y,z) r3=(-x,-y,z) r4=(-x-y,-z) r2 = (x,-y,z) Law of Reflection r1 = (x,y,z) → r2 = (x,-y,z) Reflecting through (x,z) plane

n2 θ2 θ2 n1 θ1 θ1 θ1 θC θ1 Refraction by plane interface& Total internal reflection n1 > n2 P Snell’s law n1sinθ1=n2sinθ2

Examples of prisms and total internal reflection 45o 45o 45o Totally reflecting prism 45o Porro Prism

Fermat’s principle • Light, in going from point S to P, traverses the route having the smallest optical path length • More generally, there may be many paths with the same minimum transit time, e.g. locus of a cartesian surface

Optical System Imaging by an optical system O and I are conjugate points – any pair of object-image points - which by the principle of reversibility can be interchanged I O Fermat’s principle – optical path length of every ray passing through I must be the same

Cartesian Surfaces • A Cartesian surface – those which form perfect images of a point object • E.g. ellipsoid and hyperboloid O I

Surface ƒ(x,y) Cartesian refracting surface n’>n n P(x,y) n’ y x I O s s’

Cartesian refracting surface • Surface ƒ(x,y) will be cartesian for points points O and I if… ___________________________________ • The equation defines an ovoid of revolution for a given s, s’ • Equality means all paths are equal (i.e. for all x,y) • We then have perfect imaging by Fermat’s principle • But we can see that the surface will be cartesian for one set of s, s’ (no too useful)

Paraxial ray approximation • We would like a single surface to provide imaging for all s, s’. • This will be true if we place certain restrictions on the bundle of rays collected by the optical system • Make the PARAXIAL RAY APPROXIMATION • Assume y << s,s’ (i.e. all angles are small) • x << s, s’ (of course)

Paraxial ray approximation • All distances measured from V (i.e. assume x=0) • All angles are small sinα ≈ tan α ≈ α ; cos α = 1 • Snell’s law nθ = n’θ’

θ1 θ2 α Ф α’ R s s’ n n’ V O C I

Refraction at spherical interfaces • Light travels left to right • V = origin – measure all distances from here • R = positive to the right of V, negative to the left • S = positive for real objects (i.e. one to the left of V), negative for virtual • S’ = positive for real image (to right of V), negative for virtual images • Heights – y,y’ – positive up, negative down

θ1 θ2 α Ф s’ s Refraction at a spherical interface: Paraxial ray approximation y C Note: small angles means that s + x ≈ s α + Ф = θ1

θ2 Ф α’ Refraction at a spherical interface: Paraxial ray approximation I C α’ + θ2 = Ф

Refraction at a spherical interface: Paraxial ray approximation • Snell’s law ____________________________ • Leads to…

Geometrical Optics