The beautiful rainbow International Conference on Teaching Physics - - PowerPoint PPT Presentation

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.

József Cserti

Eötvös University, Faculty of Science, Department of Physics of Complex Systems

The beautiful rainbow

Primary and secondary rainbow

Photograph: Ákos Horváth

Primary and secondary rainbow

Photograph: László Grób Photograph: Balázs Gyüre Photograph: J. Cs.

The house where Newton grew up

Woolsthorpe by Colsterworth, Woolsthorpe Manor, UK

More rainbows

Photograph: Géza Gáspárdy Triple Rainbows

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

• r the law of refraction)

Willebrord Snellius (born Willebrord Snel van Royen) in Leiden (1580- 1626) plane of the interface

Snell's law

(also known as the Snell–Descartes law

• r the law of refraction)

Willebrord Snellius (born Willebrord Snel van Royen) in Leiden (1580- 1626) plane of the interface

Snell's law

(also known as the Snell–Descartes law

• r the law of refraction)

Willebrord Snellius (born Willebrord Snel van Royen) in Leiden (1580- 1626) plane of the interface

Classification of the rays inside the sphere

primary rainbow secondary rainbow p is the number of chords the ray makes inside the sphere

Incident light Incident light Incident light Incident light

p=0 p=1 p=2 p=3

Role of the impact parameter

impact parameter The impact parameter is the distance of an incident ray from the central axis of the droplet. incident light incident light incident light incident light

p=3 p=4 p=5 p=7

Dispersion

incident white light

Each color of light has its own rainbow angle. Each rainbows slightly displaced from the next.

The refractive index depends on the frequency (color) of the light.

incident white light

p=2 p=3

Role of the impact parameter and the dispersion

incident light incident light incident light

p=17 p=17 p=17

Primary rainbow

incident light

Parallel incident light, one reflection inside the droplet, p=2 p=2 p=2

Primary rainbow

incident light

Parallel incident light, one reflection inside the droplet, p=2 p=2 p=2 p=2

Secondary and higher order rainbow

p=3 or p>3

incident

light

secondary rainbow p=4 p=3

p=3

Descartes' Theory

Geometrical optics α α α β β β β R ρ P α−β 180 −2β

Descartes' Theory

Scattering angle as a function of the impact parameter Scattering angles have an extreme value as a function of the impact parameter Alexander's dark band

0.2 0.4 0.6 0.8 1 b 45 90 135 180

p=0 p=3 p=1 p=2

Cartesian ray

Caustics caustic

Cartesian ray for different colors

primary rainbow secondary rainbow

incident light incident light

Higher order rainbows

incident light Observer

• H. Moyses Nussenzveig: The Theory of the Rainbow, Scientific American, Vol. 236, p. 116 (1977)

Summary

(Geometrical Optics)

Beyond the Geometrical Optics

Rainbow and the wave nature of light Airy's theory, the supernumerary arcs

B

2

B

1

A

2

A

1

C

2

C

1

P B D D’ B’ A’ A

1

P

P’(u,v)

u v

O A’ B’ R Q

θd

• G. B. Airy: On the intensity of light in the neighbourhood of a caustics,

Transactions of the Cambridge Philosophical Society, 6 379 (1838)

The shape of the initial wavefront AB changes with time caustic

George Biddell Airy (1801-1892) Observer

Interference of light starting from the wavefront A' B'

Supernumerary arcs

Airy's theory

• G. B. Airy: On the intensity of light in the neighbourhood of a caustics,

Transactions of the Cambridge Philosophical Society, 6 379 (1838) George Biddell Airy (1801-1892)

• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1

• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1

D i r e c t i

• n

s

• f

t h e s u p e r n u m e r a r y a r c s

Computer simulation of the supernumerary arcs

Experiment

Supernumerary arcs Airy's theory

Andrásné Hunh, Eötvös University, Budapest 2005.

154 154.2 154.4 0.2 0.4 0.6 0.8 1 int water droplet Laser Computer Mirror Detector stepper motor

Intensity

Experiment Airy's theory

primary rainbow Supernumerary arcs

Glass (n=1.467), R = 5.25 mm laser ( = 650 nm), kR = 50749

Exact description: the Mie's theory

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

James Clerk Maxwell: On the Physical Lines of Force (1862)

Light = Electromagnetic field as a solution of the Maxwell's equations 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

Experiment and the Mie's theory

Andrásné Hunh, Eötvös University, Budapest 2005.

137 137.5 138 138.5 0.2 0.4 0.6 0.8 1

int.

θ

primary rainbow

Experiment Mie's theory

Supernumerary arcs Intensity Descartes' theory

Water (n=1.33), R = 1.82 mm laser ( = 650 nm), kR = 17583

Rainbow related optical phenomena: Corona

Corona = diffraction of light by small particles

http://www.atoptics.co.uk/droplets/corona.htm

Rings around the Sun

(black spot: shield the sun not to damage eyesight

Ring around the Sun due to volcanic ash The Eruption of Krakatoa 1883 Corona around the Moon

Imaged by Richard Fleet (Glows, Bows & Haloes) in Wiltshire, England during the summer of 2003

Rainbow related optical phenomena: Glory

Glory = a large angle (close to 180º ) scattering of light by a water droplet

http://www.atoptics.co.uk/droplets/glory.htm Photograph: Péter Vankó Photograph: Géza Király

Glories can be seen on mountains and hillsides, from aircraft, and is directly opposite the Sun

Photograph: Imre Derényi incident light back scattered light surface wave

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