1.Rayleigh and Mie scattering. 2.Phase functions. 4.Single and multiple scattering

Transcription

1 5 November 2014 Outline 1.Rayleigh and Mie scattering 2.Phase functions 3.Extinction 4.Single and multiple scattering Luca Lelli Room U2080 Phone

2 Scattering fundamentals Scattering can be broadly defined as the redirection of radiation out of the original direction of propagation, usually due to interactions with molecules and particles Reflection, refraction, diffraction etc. are actually all just forms of scattering Matter is composed of discrete electrical charges (atoms and molecules dipoles) Light is an oscillating EM field excites charges, which radiate EM waves These radiated EM waves are scattered waves, excited by a source external to the scatterer The superposition of incident and scattered EM waves is what is observed

3 Scattering geometry

4 Scattering geometry Backward scattering = (= 180 ) Forward scattering =0

5 Types of scattering 1. Elastic scattering the wavelength (frequency) of the scattered light is the same as the incident light (Rayleigh and Mie scattering) 2. Inelastic scattering the emitted radiation has a wavelength different from that of the incident radiation (Raman scattering, fluorescence) 3. Quasi-elastic scattering the wavelength (frequency) of the scattered light shifts (e.g., in moving matter due to Doppler effects)

6 Rayleigh and Mie scattering

7 Rayleigh and Mie scattering example Brighter beam Enhanced forward scattering (Mie) in the direction of observation Laser beam Wavelength 532 nm Beam splitter Shallower beam Side scattering (Rayleigh)

8 More types of scattering 1) Single scattering Photons scattered only once Prevails in optically thin media (τ << 1), since photons have a high probability of exiting the medium (e.g., a thin cloud) before being scattered again Also favored in strongly absorbing media (ω << 1) 2) Multiple scattering a) 1 photon g=0 Prevails in optically thick, strongly scattering and non-absorbing media Photons may be scattered hundreds of times before emerging

9 Parameters governing scattering (1) The wavelength (λ) of the incident radiation (2) The size of the scattering particle, usually expressed as the non-dimensional size parameter, x: x = 2 r r is the radius of a spherical particle, λ is wavelength (3) The particle optical properties relative to the surrounding medium: the complex refractive index Scattering regimes: x << 1 : x ~ 1 : x >>1 : Rayleigh scattering Mie scattering Geometric scattering

10 Atmospheric particles Type Size Number concentration Gas molecule ~10-4 µm < cm -3 Aerosol, Aitken < 0.1µm ~10 4 cm -3 Aerosol, Large µm ~10 2 cm -3 Aerosol, Giant > 1 µm ~10-1 cm -3 Cloud droplet 5-50 µm cm -3 Drizzle drop ~100 µm ~10 3 m -3 Ice crystal µm m -3 Rain drop mm m -3 Graupel mm m -3 Hailstone ~1 cm m -3 Insect ~1 cm <1 m -3 Bird ~10 cm <10-4 m -3 Airplane ~ m <1 km -3

11 Some refractive indices Substance nr ni (n = nr+ i ni) Water Water (ice) NaCl (salt) H2SO (NH4)2SO SiO (λ = 550 nm) Carbon (λ = 550 nm) Mineral dust (λ = 550 nm) The most significant absorbing component of atmospheric particles is elemental carbon (soot); reflected in the large value of the imaginary part of the refractive index. Other common atmospheric particles are purely scattering.

12 Scattering regimes UV Visible Near IR Thermal IR Microwave Only single scattering Only spheres Particle Radius 1 cm 1 mm 100 µm 10 µm 1 µm 0.1 µm 10 nm 1 nm Geometric Optics x=2000 Mie Scattering x=0.2 Rayleigh Scattering x=0.002 Negligible Scattering Hail Raindrops Drizzle Cloud droplets } Dust, Smoke, Haze Aitken Nuclei Air Molecules 0.1 µm 1 µm 10 µm 100 µm 1 mm 1 cm 10 cm Wavelength

13 Scattering phase function x=10 Forward scattering x=3 x=1 Scattering lobes derived from Mie theory for homogeneous spheres x=0.1 The scattering phase function, or phase function, gives the angular distribution of light intensity scattered by a particle at a given wavelength

14 Rayleigh scattering phase function Atmospheric composition: N2 (78%), O2 (21%), Ar (1%) E Size of N2 molecule: 0.31 nm Size of O2 molecule: 0.29 nm Size of Ar molecule: 0.3 nm! Visible wavelengths ~ nm Vertically polarized => Size parameter << 1! E Horizontally polarized Unpolarized!

15 Rayleigh and Mie scattering Figure 1.2: Normalized angular distribution of the scatterd light for 4 di erent size parameters. (a) Rayleigh limit (b) x =0.01 (c) x =0.1 (d) x = 10. The green curve is the parallel incident polarization. The red is the perpendicular one and the blue one for unpolarized light.

16 Mie scattering phase function µm 1.064µm 1.64µm 2.13µm Cloud of poly-dispersed water droplets of mean radius 6 micron 100 phase function scattering angle, degrees

17 8 < (! E int! E ext ) d! S =! 0 : (!! H int H ext ) d! S =! 0 Maxwell equations + boundary conditions at particle surface Rayleigh and Mie scattering

18 8 < (! E int! E ext ) d! S =! 0 : (!! H int H ext ) d! S =! 0 Maxwell equations + boundary conditions at particle surface Rayleigh and Mie scattering "!E # k,e! E?,e = eik(r z) ikr " S2 S 3 S 4 S 1 #"!E # k,i! E?,i Plane wave + 4 complex amplitudes S_i

19 8 < (! E int! E ext ) d! S =! 0 : (!! H int H ext ) d! S =! 0 Maxwell equations + boundary conditions at particle surface Rayleigh and Mie scattering "!E # k,e! E?,e = eik(r z) ikr " S2 S 3 S 4 S 1 #"!E # k,i! E?,i Plane wave + 4 complex amplitudes S_i "!E # k,e! E?,e = eik(r z) ikr " S2 0 0 S 1 #"!E # k,i! E?,i S3 = S4 = 0 sphere not depolarizing

20 8 < (! E int! E ext ) d! S =! 0 : (!! H int H ext ) d! S =! 0 Maxwell equations + boundary conditions at particle surface Rayleigh and Mie scattering "!E # k,e! E?,e = eik(r z) ikr " S2 S 3 S 4 S 1 #"!E # k,i! E?,i Plane wave + 4 complex amplitudes S_i "!E # k,e! E?,e = eik(r z) ikr " S2 0 0 S 1 #"!E # k,i! E?,i S3 = S4 = 0 sphere not depolarizing After a change of coordinates [ (x,y,z) -> (r, phi, theta) ], the observed intensity I_e, result of the illumination of a sphere by I_i can be written as: I e (, )= I i k 2 R 2 ( S 1( ) 2 sin 2 + S 2 ( ) 2 cos 2 ) G. Mie, Beiträge zur optik trüber medien, speziell kolloidaler metallösungen, Ann. der Physik 25 (1908), 377.

21 Rayleigh and Mie scattering Forward peak Refractions and internal reflections Rayleigh Mie Figure 1.3: Plot of A = S 1 ( ) 2 + S 2 ( ) 2 for a spherical water particle as function of size parameter x and four di erent scattering angles for.

22 Bigger particles > more scattering. Mie scattering Large particles > consider the fine-scale scattering from the surface microstructure and then integrate over the larger scale structure. If the surface isn t smooth, the scattering is incoherent. If the surfaces are smooth, then we use Snell s Law and angle-of-incidence-equals-angle-of-reflection. Add up all the waves resulting from all the input waves, taking into account their coherence, too (Mie theory) Incident E size parameter 43 refractive index 1.40 parallel polarization Mie regime Geometrical optics Ray tracing

23 Mie scattering phase function Secondary Rainbow Primary Rainbow x=100 x=10,000 Rainbow: for large particles (x = 10,0000), the forward and backward peaks in the scattering phase function become very narrow (almost non-existent). Light paths are best predicted using geometric optics and ray tracing Primary rainbow: single internal reflection Secondary rainbow: double internal reflection

24 Mie scattering phase function Glory Glory Fogbow x=30 Corona Forward Diffraction Peak dary Rainbow Fogbow Rainbow x=100 Fogbow spikes in scattering phase function present but not sharp as for rainbows. Hence the separation of colors (due to varying refractive index) is not as vivid as a normal rainbow.

25 Mie scattering phase function Glory Glory Fogbow x=30 Corona Forward Diffraction Peak dary Rainbow Fogbow Glory Rainbow x=100 Glory opposite end of the phase function from the corona. Sun at the back. Glories have vivid colors if the range of drop sizes in the fog is relatively narrow, otherwise white

26 Mie scattering phase function Lunar Glory corona Glory Fogbow x=30 Corona Forward Diffraction Peak dary Rainbow Fogbow Glory Rainbow x=100 Corona for intermediate values of the size parameter (x), the forward scattering peak is accompanied by weaker sidelobes. If you were to view the sun through a thin cloud composed of identical spherical droplets (with x = 100 or less), you would see closely spaced rings around the light source. The angular position of the rings depends on wavelength, so the rings would be colored. This is a corona. Because few real clouds have a sufficiently narrow distribution of drop sizes, coronas are usually more diffuse and less brightly colored.

27 Extinction Extinction = removal of light from its travel path due to both absorption and scattering

28 Extinction Extinction = removal of light from its travel path due to both absorption and scattering I I (s) ds b ext ( ) Incident light intensity Outgoing light intensity Differential travel path through a medium of volume dv, section area da and radius r Coefficient (or strength) of attenuation The Beer-Lambert-Bouguer extinction law I (s) =I (0) e R b ext ( )ds = I (0) e ( ) ( ) Optical thickness of the volume (unitless). Depends on the medium: absorption and scattering of both molecules and particles

29 0.6 Clear atmosphere (Rayleigh scattering) Reflection top-of-atmopshere [-] Wavelength [nm]

30 0.6 Clear atmosphere (Rayleigh scattering) Reflection top-of-atmopshere [-] I Wavelength [nm]

31 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer = Wavelength [nm]

32 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer = Wavelength [nm]

33 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Wavelength [nm]

34 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Wavelength [nm]

35 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Wavelength [nm]

36 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Wavelength [nm]

37 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Wavelength [nm]

38 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Wavelength [nm]

39 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Wavelength [nm]

40 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Wavelength [nm]

41 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Cloud =5, ms Wavelength [nm]

42 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Cloud =5, ms Wavelength [nm]

43 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Cloud =5, ms Cloud =10, ms Wavelength [nm]

44 Reflection top-of-atmopshere [-] I Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Cloud =5, ms Cloud =10, ms Wavelength [nm]

45 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Cloud =5, ms Cloud =10, ms Wavelength [nm]

46 Reflection top-of-atmopshere [-] Clear atmosphere (Rayleigh scattering) Aerosol layer =0.05 Cloud =1, ss Cloud =5, ss Cloud =10, ss Cloud =1, ms Cloud =5, ms Blue Green Red Cloud =10, ms Wavelength [nm]

47 Conclusions to be drawn from the analysis of the spectra - For a pure molecular atmosphere (clear sky), Rayleigh scattering follows 4 - Aerosol layer enhances scattering and the signal at top-of-atmosphere increases - Clouds as perfect reflectors (i.e. single-scattering) shield the atmosphere below - Consistently, the spectra are independent on optical thickness (i.e. light is not allowed to penetrate clouds) - Clouds as real objects (i.e. multiple scattering) enhances scattering - Clouds are spectrally neutral - Clouds are Mie scattering objects in this spectral range - The thicker the cloud, the stronger is multiple scattering, the higher is absorption of gases inside the cloud (i.e. oxygen around 760 nm is deeper)