Contents 1 Small size parameter approximation 2 From molecules 3 Cause of the blue color of the sky 4 In optical fibers 5 In porous materials 6 See also 7 References 8 Further reading 9 External links Small size parameter approximation[edit] The size of a scattering particle is often parameterized by the ratio x = 2 π r λ displaystyle x= frac 2pi r lambda where r is its characteristic length (radius) and λ is the wavelength of the light. The amplitude of light scattered from within any transparent dielectric is proportional to the inverse square of its wavelength and to the volume of material, that is to the cube of its characteristic length. The wavelength dependence is characteristic of dipole scattering[3] and the volume dependence will apply to any scattering mechanism. Objects with x ≫ 1 act as geometric shapes, scattering light according to their projected area. At the intermediate x ≃ 1 of Mie scattering, interference effects develop through phase variations over the object's surface. Rayleigh scattering applies to the case when the scattering particle is very small (x ≪ 1, with a particle size < 1 /10 wavelength[4]) and the whole surface re-radiates with the same phase. Because the particles are randomly positioned, the scattered light arrives at a particular point with a random collection of phases; it is incoherent and the resulting intensity is just the sum of the squares of the amplitudes from each particle and therefore proportional to the inverse fourth power of the wavelength and the sixth power of its size.[3][5] In detail, the intensity I of light scattered by any one of the small spheres of diameter d and refractive index n from a beam of unpolarized light of wavelength λ and intensity I0 is given by I = I 0 1 + cos 2 θ 2 R 2 ( 2 π λ ) 4 ( n 2 − 1 n 2 + 2 ) 2 ( d 2 ) 6 displaystyle I=I_ 0 frac 1+cos ^ 2 theta 2R^ 2 left( frac 2pi lambda right)^ 4 left( frac n^ 2 -1 n^ 2 +2 right)^ 2 left( frac d 2 right)^ 6 [6] where R is the distance to the particle and θ is the scattering angle. Averaging this over all angles gives the Rayleigh scattering cross-section[7] σ s = 2 π 5 3 d 6 λ 4 ( n 2 − 1 n 2 + 2 ) 2 displaystyle sigma _ text s = frac 2pi ^ 5 3 frac d^ 6 lambda ^ 4 left( frac n^ 2 -1 n^ 2 +2 right)^ 2 [8] The fraction of light scattered by a group of scattering particles is the number of particles per unit volume N times the cross-section. For example, the major constituent of the atmosphere, nitrogen, has a Rayleigh cross section of 6969509999999999999♠5.1×10−31 m2 at a wavelength of 532 nm (green light).[9] This means that at atmospheric pressure, where there are about 7025200000000000000♠2×1025 molecules per cubic meter, about a fraction 10−5 of the light will be scattered for every meter of travel. The strong wavelength dependence of the scattering (~λ−4) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths. From molecules[edit] Figure showing the greater proportion of blue light scattered by the atmosphere relative to red light. The expression above can also be written in terms of individual molecules by expressing the dependence on refractive index in terms of the molecular polarizability α, proportional to the dipole moment induced by the electric field of the light. In this case, the Rayleigh scattering intensity for a single particle is given in CGS-units by[10] I = I 0 8 π 4 α 2 λ 4 R 2 ( 1 + cos 2 θ ) . displaystyle I=I_ 0 frac 8pi ^ 4 alpha ^ 2 lambda ^ 4 R^ 2 (1+cos ^ 2 theta ). Cause of the blue color of the sky[edit] Main article: Diffuse sky radiation Scattered blue light is polarized. The picture on the right is shot through a polarizing filter: the polarizer transmits light that is linearly polarized in a specific direction. The strong wavelength dependence of the scattering (~λ−4) means
that shorter (blue) wavelengths are scattered more strongly than
longer (red) wavelengths. This results in the indirect blue light
coming from all regions of the sky.
α scat = 8 π 3 3 λ 4 n 8 p 2 k T f β displaystyle alpha _ text scat = frac 8pi ^ 3 3lambda ^ 4 n^ 8 p^ 2 kT_ text f beta where n is the refraction index, p is the photoelastic coefficient of the glass, k is the Boltzmann constant, and β is the isothermal compressibility. Tf is a fictive temperature, representing the temperature at which the density fluctuations are "frozen" in the material. In porous materials[edit]
λ−4 Rayleigh-type scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials.[14] The strong contrast in refractive index between pores and solid parts of sintered alumina results in very strong scattering, with light completely changing direction each 5 micrometers on average. The λ−4-type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70 nm) obtained by sintering monodispersive alumina powder. See also[edit] Rayleigh sky model Ricean fading Optical phenomenon Dynamic light scattering Tyndall effect Critical opalescence Marian Smoluchowski Rayleigh Criterion Aerial perspective Parametric Process Bragg's law References[edit] ^
John Strutt (1871) "On the light from the sky, its polarization and colour," Philosophical Magazine, series 4, vol.41, pages 107–120, 274–279. John Strutt (1871) "On the scattering of light by small particles," Philosophical Magazine, series 4, vol. 41, pages 447–454. John Strutt (1881) "On the electromagnetic theory of light," Philosophical Magazine, series 5, vol. 12, pages 81–101. John Strutt (1899) "On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky," Philosophical Magazine, series 5, vol. 47, pages 375–394. ^ A. T. Young, "Rayleigh scattering," Appl. Opt. 20, 522–535, 1981
^ a b "Cornell lectures" (PDF). Retrieved 2 April 2014.
^
Further reading[edit] C.F. Bohren, D. Huffman, Absorption and scattering of light by small
particles, John Wiley, New York 1983. Contains a good description of
the asymptotic behavior of
External links[edit] HyperPhysics description of Rayleigh scattering Full physical explanation of sky colo |