1 Huygens’ and Fermat’s principles (Hecht 4.4, 4.5) Application to reflection & refraction at an interface Monday Sept. 9, 2002

2 Huygens’ wave front construction Given wavefront at t Allow wavelets to evolve for time Δt r = c Δt ≈ λ New wavefront What about –r direction? See Bruno Rossi Optics. Reading, Mass: Addison-Wesley Publishing Company, 1957, Ch. 1,2 for mathematical explanation Construct the wave front tangent to the wavelets

3 Plane wave propagation New wave front is still a plane as long as dimensions of wave front are >> λ If not, edge effects become important Note: no such thing as a perfect plane wave, or collimated beam

4 Geometric Optics As long as apertures are much larger than a wavelength of light (and thus wave fronts are much larger than λ) the light wave front propagates without distortion (or with a negligible amount) i.e. light travels in straight lines

5 Physical Optics If, however, apertures, obstacles etc have dimensions comparable to λ (e.g. < 10 3 λ) then wave front becomes distorted

6 Let’s reflect for a moment

7 Hero’s principle Hero (150BC-250AD) asserted that the path taken by light in going from some point A to a point B via a reflecting surface is the shortest possible one

8 Hero’s principle and reflection AB A’ O R O’

9 Let’s refract for a moment

10 Speed of light in a medium Light slows on entering a medium – Huygens Also, if n → ∞ = 0 i.e. light stops in its track !!!!! See: P. Ball, Nature, January 8, 2002 D. Philips et al. Nature 409, (2001) C. Liu et al. Physical Review Letters 88, (2002)

11 Snel’s law Willebrord Snel ( ) discovers the law of refraction Descartes ( ) publish the, now familiar, form of the law (viewed light as pressure transmitted by an elastic medium) n 1 sin 1 = n 2 sin 2

12 Huygens’ ( ) Principle: Reflection and Refraction of light Light slows on entering a medium Reflection and Refraction of Waves Reflection and Refraction of Waves Click on the link above

13 Total internal reflection n1n1 n2n2 θCθC n 1 > n – Discovered by Kepler

14 Pierre de Fermat’s principle 1657 – Fermat ( ) proposed a Principle of Least Time encompassing both reflection and refraction “The actual path between two points taken by a beam of light is the one that is traversed in the least time”

15 n1n1 n2n2 Fermat’s principle n 1 < n 2 A O B θiθi θrθr x a h b What geometry gives the shortest time between the points A and B?

16 Optical path length n1n1 n4n4 n2n2 n5n5 nmnm n3n3 S P

17 Optical path length Transit time from S to P Same for all rays

18 Fermat’s principle t = OPL/c Light, in going from point S to P, traverses the route having the smallest optical path length

19 Optical effects LoomingMirages

20 Reflection by plane surfaces r 1 = (x,y,z) x y r 2 = (x,-y,z) Law of Reflection r 1 = (x,y,z) → r 2 = (x,-y,z) Reflecting through ((x,z) plane x y z r 2 = (-x,y,z) r 3 =(-x,-y,z) r 4 =(-x-y,-z) r 1 = (x,y,z)

21 n2n2 Refraction by plane interface & Total internal reflection n1n1 n 1 > n 2 θCθC P θ1θ1 θ1θ1 θ1θ1 θ1θ1 θ2θ2 θ2θ2 Snell’s law n 1 sinθ 1 =n 2 sinθ 2

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

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

24 Cartesian Surfaces Cartesian surfaces – those surfaces which form perfect images of a point object E.g. ellipsoid and hyperboloid