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Transcript

Huygens’ and Fermat’s principles (Hecht 4.4, 4.5) Application to reflection & refraction at an interface Monday Sept. 9, 2002 1 Huygens’ wave front construction Construct the New wavefront wave front tangent to the wavelets r = c Δt ≈ λ Given wavefront at t Allow wavelets to evolve for time Δt What about –r direction? See Bruno Rossi Optics. Reading, Mass: Addison-Wesley Publishing Company, 1957, Ch. 1,2 for mathematical explanation 2 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 3 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 4 Physical Optics If, however, apertures, obstacles etc have dimensions comparable to λ (e.g. < 103 λ) then wave front becomes distorted 5 Let’s reflect for a moment 6 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 7 Hero’s principle and reflection A B R O’ O A’ 8 Let’s refract for a moment 9 Speed of light in a medium c v n 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, 490-493 (2001) C. Liu et al. Physical Review Letters 88, 23602 (2002) 10 Snel’s law 1621 - Willebrord Snel (1591-1626) discovers the law of refraction 1637 - Descartes (1596-1650) publish the, now familiar, form of the law (viewed light as pressure transmitted by an elastic medium) n1sin1 = n2sin2 11 Huygens’ (1629-1695) Principle: Reflection and Refraction of light Light slows on entering a medium Reflection and Refraction of Waves Click on the link above 12 Total internal reflection 1611 – Discovered by Kepler θC n1 n2 n1 > n2 13 Pierre de Fermat’s principle 1657 – Fermat (1601-1665) 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” 14 Fermat’s principle A n1 θi h O n1 < n2 x n2 b What geometry gives the shortest time between the points A and B? θr a B 15 Optical path length S n1 n2 n3 n4 n5 P nm 16 Optical path length Transit time from S to P m 1 t ni si c i 1 m OPL ni si i 1 P OPL n(s)ds S P c OPL ds v S Same for all rays 17 Fermat’s principle t = OPL/c Light, in going from point S to P, traverses the route having the smallest optical path length OPL t c 18 Optical effects Looming Mirages 19 Reflection by plane surfaces y r1 = (x,y,z) z r2= (-x,y,z) r1 = (x,y,z) x r3=(-x,-y,z) x y r4=(-x-y,-z) r2 = (x,-y,z) Law of Reflection r1 = (x,y,z) → r2 = (x,-y,z) Reflecting through ((x,z) plane 20 Refraction by plane interface & Total internal reflection n2 θ2 θ2 n1 > n2 θ1 θ1 θ C θ1 θ1 n1 P Snell’s law n1sinθ1=n2sinθ2 21 Examples of prisms and total internal reflection 45o 45o 45o Totally reflecting prism 45o Porro Prism 22 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 Optical O System Fermat’s principle – optical path length of every ray passing through I must be the same I 23 Cartesian Surfaces Cartesian surfaces – those surfaces which form perfect images of a point object E.g. ellipsoid and hyperboloid 24