József Cserti Eötvös University, Faculty of Science, Department of Physics of Complex Systems The beautiful rainbow International Conference on Teaching Physics Innovatively New Learning Environments and Methods in Physics Education ELTE University, Faculty of Science, Budapest, Hungary 17-19 August 2015.
Primary and secondary rainbow Photograph: Ákos Horváth
Primary and secondary rainbow Photograph: Balázs Gyüre Photograph: László Grób Photograph: J. Cs.
The house where Newton grew up Woolsthorpe by Colsterworth, Woolsthorpe Manor, UK
More rainbows Triple Rainbows Photograph: Géza Gáspárdy
Supernumerary arcs Atmospheric Optics http://www.atoptics.co.uk/
Short history before Descartes (not complete) Aristotle (384 – 322 BC): an unusual kind of reflection of sunlight from clouds Alexander of Aphrodisias (fl. 200 AD): Alexander's dark band Roger Bacon (1266): the first measurment of the angle, 42 degrees Theodoric of Freiberg (German monk, 1304): first experiment with a spherical flask filled with water Rene Descartes (1637): Geometrical optics
Descartes' 1637 treatise, Discourse on Method Rene Descartes (1596- 1650)
Snell's law (also known as the Snell–Descartes law or the law of refraction) plane of the Willebrord Snellius interface (born Willebrord Snel van Royen) in Leiden (1580- 1626)
Snell's law (also known as the Snell–Descartes law or the law of refraction) plane of the Willebrord Snellius interface (born Willebrord Snel van Royen) in Leiden (1580- 1626)
Snell's law (also known as the Snell–Descartes law or the law of refraction) plane of the Willebrord Snellius interface (born Willebrord Snel van Royen) in Leiden (1580- 1626)
Classification of the rays inside the sphere p is the number of chords the ray makes inside the sphere Incident light p=1 p=0 Incident light Incident light p=2 p=3 Incident light primary rainbow secondary rainbow
Role of the impact parameter The impact parameter is the distance of an incident ray from the central axis of the droplet. incident light p=3 incident light impact parameter p=4 incident light incident p=5 light p=7
Dispersion The refractive index depends on the frequency (color) of the light. incident white light p=2 p=3 incident white light Each color of light has its own rainbow angle. Each rainbows slightly displaced from the next.
Role of the impact parameter and the dispersion incident light incident p=17 light p=17 incident light p=17
Primary rainbow Parallel incident light, one reflection inside the droplet, p=2 incident light p=2 p=2
Primary rainbow Parallel incident light, one reflection inside the droplet, p=2 incident light p=2 p=2 p=2
Secondary and higher order rainbow p=3 or p>3 secondary rainbow p=3 p=3 incident light p=4
Descartes' Theory Geometrical optics α−β P α β ρ β α β R 180 −2β β α
Descartes' Theory Scattering angle as a function of the impact parameter 180 p=2 135 Alexander's dark band 90 p=0 p=3 45 p=1 1 b 0.2 0.4 0.6 0.8 Scattering angles have an extreme value as a function of the impact parameter
Cartesian ray Caustics caustic
Cartesian ray for different colors incident light secondary rainbow incident light primary rainbow
Higher order rainbows incident light Observer
Summary (Geometrical Optics) H. Moyses Nussenzveig: The Theory of the Rainbow , Scientific American, Vol. 236 , p. 116 (1977)
Beyond the Geometrical Optics Rainbow and the wave nature of light Airy's theory, the supernumerary arcs The shape of the initial wavefront AB B changes with time D George Biddell Airy (1801-1892) P A A A A’ v 2 1 A’ P’(u,v) Q O u P 1 R B’ B θ d B’ B 1 2 Observer C 1 C 2 Interference of light starting caustic from the wavefront A' B' D’ G. B. Airy: On the intensity of light in the neighbourhood of a caustics , Transactions of the Cambridge Philosophical Society, 6 379 (1838)
Supernumerary arcs Airy's theory 1 0.5 George Biddell Airy -2.5 -2 -1.5 -1 -0.5 0.5 1 (1801-1892) -0.5 Computer simulation of -1 the supernumerary arcs -1.5 D -2 i s r u e p c t e i r o n n -2.5 u s m o e f r t a h r e y a r c s G. B. Airy: On the intensity of light in the neighbourhood of a caustics , Transactions of the Cambridge Philosophical Society, 6 379 (1838)
Experiment Computer Supernumerary arcs Airy's theory stepper motor water droplet Laser primary Mirror rainbow Intensity Detector 1 int Experiment 0.8 Airy's theory Supernumerary arcs 0.6 0.4 Glass ( n=1.467 ), R = 5.25 mm laser ( = 650 nm), kR = 50749 0.2 0 154 154.2 154.4 Andrásné Hunh, Eötvös University, Budapest 2005.
Exact description: the Mie's theory Light = Electromagnetic field as a solution of the Maxwell's equations James Clerk Maxwell: On the Physical Lines of Force (1862) Rainbow = Scattering of an electromagnetic plane wave by a homogeneous dielectric sphere Helmholtz's equation: E or the B field Solution: Infinite series, can be treated efficiently only numerically Quantum mechanics: Scattering by potential well Gustav Mie: Beitrage zur Optik trüber Medien, speziell kolloidaler Metallösungen , Ann. Phys., Leipzig 25 , 377-445 (1908). M. Born and E. Wolf: Principles of Optics , Pergamon Press, New York, 1989 (6th eds.). Ludwig V. Lorenz, 1890 Peter J. W. Debey, 1909
Experiment and the Mie's theory primary Water ( n=1.33 ), R = 1.82 mm rainbow Intensity laser ( = 650 nm), kR = 17583 1 int. Experiment 0.8 Mie's theory Supernumerary arcs 0.6 0.4 0.2 0 137 137.5 138 138.5 θ Descartes' theory Andrásné Hunh, Eötvös University, Budapest 2005.
Rainbow related optical phenomena: Corona Ring around the Sun due to volcanic ash The Eruption of Krakatoa 1883 Corona around the Moon Rings around the Sun (black spot: shield the sun not to damage eyesight Imaged by Richard Fleet (Glows, Bows & Haloes) in Wiltshire, England during the summer of 2003 Corona = diffraction of light by small particles http://www.atoptics.co.uk/droplets/corona.htm
Rainbow related optical phenomena: Glory Glories can be seen on mountains and hillsides, from aircraft, and is directly opposite the Sun Glory = a large angle (close to 180º ) scattering of light by a water droplet surface wave Photograph: Géza Király Photograph: Péter Vankó incident light back scattered light Photograph: Imre Derényi http://www.atoptics.co.uk/droplets/glory.htm
Some references Atmospheric Optics: http://www.atoptics.co.uk/ http://www.phy.ntnu.edu.tw/java/Rainbow/rainbow.html Raymond L. Lee, Jr., and Alistair B. Fraser: The Rainbow Bridge, Rainbows in Art, Myth, and Science, Penn State University Press; 1st edition (July 2001) http://www.usna.edu/Users/oceano/raylee/RainbowBridge/Chapter_8.html H. Moyses Nussenzveig: The Theory of the Rainbow , Scientific American, Vol. 236, p. 116 (1977) John A. Adam: The mathematical physics of rainbows and glories, Physics Reports 356 , 229–365 (2002) Philip Laven, Geneva, Switzerland: Freely available MiePlot computer program (Microsoft Windows) for Mie theory, http://www.philiplaven.com/ My works (Hungarian): A szivárvány fizikája, Fizikai Szemle 2005. év szeptemberi, októberi és decemberi számában Fizikus szemmel a szivárványról, Fizikai Szemle 2006. év szeptemberi számában a Mindentudás az Iskolában rovatában A szivárvány fizikai alapjai, T ermészet Világa 2007. év májusi és júniusi számában, 202. és 258. oldal
Acknowledgement Gyula Dávid, T amás Geszti, Péter Gnädig, Ottó Haiman, Gábor Horváth, Andrásné Huhn, Krisztián Kis-Szabó, András Pályi, Péter Pollner, Géza Tichy, T amás Weidinger, and Philip Laven Waterfall at Jajce by Tivadar Kosztka Csontváry
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