Lecture 13 Light scattering and absorption by atmospheric particulates. Part 1: Principles of scattering. Main concepts: elementary wave, polarization, Stokes matrix, and scattering phase function. Rayleigh scattering. Objectives: 1. Principles of scattering. Main concepts: elementary wave, polarization, Stoke matrix, and scattering phase function. 2. Rayleigh scattering. Required Reading : L02: 1.1.4; 3.3.1 Additional/advanced Reading : Bohren, G.F., and D.R. Huffmn, Absorption and scattering of light by small particles. John Wiley&Sons, 1983. 1. Principles of scattering. Main concepts: elementary wave and light beam, polarization, Stoke matrix, and scattering phase function. Figure 13.1 Simplified visualization of scattering of an incident wave by a particle. 1 How scattering works : Consider a single arbitrary particle composed of many dipoles. The incident electromagnetic field induces dipole oscillations. The dipoles oscillate at the frequency of the incident field and therefore scatter radiation in all directions. In a given direction of observation, the total scattered field is a superposition of the scattered wavelets of these dipoles. Scattering of the electromagnetic radiation is described by the classical electromagnetic theory , considering the propagation of a light beam as a transverse wave motion (collection of electromagnetic individual waves) . r Electromagnetic field is characterized by the electric vector E and magnetic r vector H , which are orthogonal to each other and to the direction of the r r propagation. E and H obey the Maxwell equations . Poynting vector gives the flow of radiant energy and the direction of propagation as (in cgs system) r c r r S = E × H [13.1] 4 π r S is in units of energy per unit time per unit area (i.e. flux); r r NOTE : E × H means a vector product of two vectors. Thus 1 c 2 I = E ∆Ω 4 π Since electromagnetic field has the wave-like nature, the classical theory of wave motion is used to characterize the propagation of radiation. Consider a plane wave propagating in z-direction (i.e., E oscillates in the x-y plane). r The electric vector E may be decomposed into the parallel El and perpendicular Er components, so that E = a exp( − iδ ) exp( − ikz + iω t ) [13.2a] l l l E r = a r exp( −iδ r ) exp( −ikz + iω t ) [13.2b] 2 where al and ar are the amplitude of the parallel El and perpendicular Er components, respectively; δδδl and δδδr are the phases of the parallel El and perpendicular Er components, respectively; k is the propagation (or wave) constant, k = 2 πππ/λ,λ,λ, and ω is the circular frequency, ωωω = kc=2 πππc/ λλλ Eq.[13.2] can be written in cosine representation as E l = a l cos( ζ + δ l ) E r = a r cos( ζ + δ r ) where ζ = kz − ω t and ζ+δ is called the phase . Then we have E l / a l = cos( ζ ) cos( δ l ) − sin( ζ ) sin( δ l ) [13.3] E r / a r = cos( ζ ) cos( δ r ) − sin( ζ ) sin( δ r ) and thus 2 2 2 (El / al ) + (E r / ar ) − (2 El / al )(E r / ar ) cos(δ ) = sin (δ ) [13.4] where δδδ = δδδr - δ l is the phase difference (or phase shift ). Eq.[13.4] represents an ellipse => elliptically polarized wave If δδδ = m π (m =0, +/1; +/-2…), then sin (δδδ) =0 and Eq.[13.4] becomes 2 El Er El Er ± = 0 or = ± [13.5] al ar al ar Eq.[13.5] represents two perpendicular lines => linearly polarized wave If δδδ = m π/2 (m = +/-1; +/-3,…) and al = ar = a, Eq.[13.4] becomes 2 2 2 El + Er = a [13.6] Eq.[13.6] represents a circle => circularly polarized wave In general, light is a superposition of many waves of different frequencies, phases, and amplitudes. Polarization is determined by the relative size and correlations between two electrical field components. Radiation may be unpolarized, partially polarized, or completely polarized. 3 Figure 13.2 Example of vertically polarized light. Natural sunlight is unpolarized . If there is a definite relation of phases between different scatterers => radiation is called coherent. If there is no relations in phase shift => light is called incoherent Natural sunlight is incoherent. The property of incoherent radiation: The intensity due to all scattering centers is the sum of individual intensities. NOTE: In our course, we study the incoherent scattering of the atmospheric radiation. NOTE: The assumption of independent scatterers is violated if the particles are too closely packed (spacing between particles should be several times their diameters to prevent intermolecular forces from causing correlation between scattering centers). • Eq.[13.4] shows that, in the general case, three independent parameters al, ar and δδδ are required to characterize an electromagnetic wave. These parameters are not measured. Therefore, a new set of parameters (which are proportional to the intensity) has been proposed by Stoke. Stokes parameters : so-called intensity I, the degree of polarization Q, the plane of polarization U, and the ellipticity V of the electromagnetic wave 4 * * I = E l E l + E r E r * * Q = El El − E r E r [13.7] * * U = E l E r + E r E l * * V = −i( E l E r − E r E l ) They are related as I 2 = Q 2 + U 2 + V 2 [13.8] Stokes parameter can be also expressed as 2 2 I = a l + a r 2 2 Q = al − ar [13.9] U = 2 a l a r cos( δ ) V = 2a l a r sin( δ ) Actual light consists of many individual waves each having its own amplitude and phase. NOTE: During a second, a detector collects about millions of individual waves. Measurable intensities are associated with the superposition of many millions of simple waves with independent phases. Therefore, for a light beam the Stokes parameters are averaged over a time period and may be represented as 2 2 I = a l + a r = I l + I r 2 2 Q = a l − a r = I l − I r [13.10] U = 2 a l a r cos( δδδ ) V = 2 a l a r sin( δδδ ) where .... denote the time averaging. For a light beam, the time averaged Stokes parameters are related as I 2 ≥ Q 2 + U 2 + V 2 [13.11] 5 The degree of polarization DP of a light beam is defined as DP = (Q 2 + U 2 + V 2 ) 1 / 2 / I [13.12] The degree of linear polarization LP of a light beam is defined by neglecting U and V as Q I − I LP = − = − l r [13.13] I I l + I r Unpolarized light : Q = U = V = 0 Fully polarized light : I 2 = Q 2 + U 2 + V 2 Linear polarized light : V = 0 Circular polarized light : V = I The scattering phase function P(cos ΘΘΘ) is defined as a non-dimensional parameter to describe the angular distribution of the scattered radiation as 1 ∫ P(cos Θ)dΩ = 1 [13.14] 4π Ω where Θ is the scattering angle between the direction of incidence and observation. NOTE : The phase function is expressed as P(cos ΘΘΘ) = P (θθθ', ϕϕϕ', θθθ, ϕϕϕ), where (θθθ', ϕϕϕ') and ( θθθ, ϕϕϕ) are the spherical coordinates of incident beam and direction of observation, and (see L02: Appendix C): cos (Θ)Θ)Θ) = cos (θθθ')))cos (θ) + sin (θθθ')))sin (θ) cos (ϕ(ϕ(ϕ '-ϕϕϕ) [13.15] Isotropic scattering : P(cos ΘΘΘ)=1 Forward scattering refers to the observations directions for which Θ < π/2 Backward scattering refers to the observations directions for which Θ > π/2 6 2. Rayleigh scattering Consider a small homogeneous spherical particle (e.g., molecule) with size smaller than r r the wavelength of incident radiation E0 . Then the induced dipole moment p0 is r r p0 = αE0 [13.16] where ααα is the polarizability of the particle. NOTE : Do not confuse the polarization of the medium with polarization associated with the EM wave! According to the classical electromagnetic theory, the scattered electric field at the large distance r (called far field scattering) from the dipole is given (in cgs units) by r r 1 1 ∂p E = sin( γ ) [13.17] c 2 r ∂t r where γγγ is the angle between the scattered dipole moment p and the direction of observation. In an oscillating periodic field, the dipole moment is given in terms of induced dipole moment by r r p = p 0 exp( −ik (r − ct )) [13.18] and thus the electrical field is r r exp( − ik (r − ct )) E = − E k 2α sin( γ ) [13.19] 0 r E0r pr γγγ1 Dipole Direction of Direction of ΘΘΘ E0l scattering incident radiation γγγ2 pl (out of page) γ1=π/2; γ2=π/2-Θ NOTE: Plane of scattering (or scattering plane ) is defined as a plane containing the incident beam and scattered beam in the direction of observation. 7 Decomposing the electrical vector on two orthogonal components perpendicular and parallel to the plane of scattering, we have exp( − ik (r − ct )) E = − E k 2α sin( γ ) [13.20] r 0 r r 1 exp( − ik ( r − ct )) E = − E k 2α sin( γ ) l 0 l r 2 Using γ1=π/2; γ2=π/2-Θ and that 1 c 2 I = E , [13.21] ∆Ω 4π perpendicular and parallel intensities (or linear polarized intensities) are 4 2 2 I r = I 0 r k α / r [13.22] 4 2 2 2 I l = I 0 l k α cos (Θ /) r Using that the natural light (incident beam) in not polarized ( I0r =I 0l =I 0/2 ) and that k=2 π/λπ/λπ/λ , we have 4 2 I 0 2 2π 1 + cos (Θ ) I = I r + I l = α [13.23] r 2 λ 2 Eq.[13.23] gives the intensity scattered by molecules for unpolarized incident light, called Rayleigh scattering.