rainbows

1. rainbows Khor Wei Sean 2O218

2. Part I Rainbow How does a rainbow form? Special characteristics of rainbow Mathematical approximation The second and higher order rainbow The differences with the single rainbow The formation principle Experimental results using a pendant drop

3. How does a rainbow form? Basic theory: light refraction

4. Special characteristics of rainbow Specific arc shape The deflective angle 40 - 42 o o

5. Special characteristics of rainbow Color pattern: • outer red to inner blue • Different refraction index: • Blue light 1.340 • Red light 1.331 • deflective angles are slightly different

6. supernumerary bows Supernumerary bows • Consisting of several faint rainbows • Mostly on the inner side of the primary rainbow.

7. How does supernumerary bows form? • Formed by the interference between rays of light following slightly different paths with slightly varying lengths within the raindrops. • Supernumerary rainbows are clearest when raindrops are small and of similar size.

8. Airy theory is good for explaining the specific arc angle. Airy theory assumes that the light waves are non polarized - not suitable for polarized light. Airy theory has a large error when size of water drop is small It is good for primary bow, but less accurate for the supernumerary arcs Airy theory

9. Mie theory An analytical solution of Maxwell’s equation for the scattering of electromagnetic radiation by spherical particles A long polynomial chain, require computation. Its validity has been proved by experiments. Unfortunately, it gives us little physical insight into the process that produces the rings

10. Comparison of Airy and Mie theory • The ripples on the Mie theory calculations are due to interference between the primary rainbow rays and the reflected rays from the surface of the water drop. • Airy and Mie theory are consistent at some level. For primary rainbow.            r=103.4057µm, λ=0.65µm, x=2πr/λ=1000, n=1.333, perpendicular polarization.

11. Comparison of airy and Mie theory with natural light • When we sum up the result of each theory over all visible light. The difference between the two theory reduces. • Dark gaps between the supernumerary arcs have some subtle differences

12. The secondary rainbow Reversed color Darker and broader Below the primary rainbow deflective angle 50 - 53 o o

13. Formation principle of secondary rainbow The light experience 2 internal reflection in water drops. As a result, the order of light reversed. The light peak broadened, and less luminance.

14. The light experience 2 internal reflection in water drops. • As a result, the order of light reversed. • The light peak broadened, and less luminance. Formation principle of secondary rainbow

15. Higher order rainbow

16. Too dim to be observed in natural. • Can be observed in laboratory. • Most of them are monochromatic experiments. Higher order rainbows

17. Higher order rainbows Rose of rainbows surrounding a drop J Walker, “How to create and observe a dozen rainbows in a single drop of water”, July 1977

18. Reference • R Greenler, “Rainbows, Halos and Glories”, Cambridge University Press, 1989. • Philip Laven, “How are glories formed?” • Philip Laven, “Atmospheric glories: simulations and observations” • Wikipedia • Raymond L. Lee, Jr., “Mie theory, Airy theory, and the natural rainbow” • C W Chan and W K Lee, “measurement of Liquid Refractive Index By Using High-Order Rainbows” • P H Ng, M Y Tse and W K Lee, “Observation of High Order Rainbows Formed By a Pendant Drop” • Waves in composites and metamaterials/Airy theory http://en.wikiversity.org/wiki/Waves_in_composites_and_metamaterials/Airy_theory • Supernumerary Rainbows http://www.atoptics.co.uk/bows.htm