1 EE 543 Theory and Principles of Remote Sensing Scattering and Absorption of Waves by a Single Particle

O. Kilic EE Theory of Radiative Transfer We will be considering techniques to derive expressions for the apparent temperature, T AP of different scenes as shown below. Atmosphere Terrain TATA T UP TATA Terrain could be smooth, irregular, slab (such as layer of snow) over a surface. STEP 1: Derive equation of radiative transfer STEP 2: Apply to different scenes

O. Kilic EE Radiation and Matter Interaction between radiation and matter is described by two processes: –Extinction –Emission Usually we have both phenomenon simultaneously. Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption) Emission: medium adds energy of its own (through scattering and self emission)

O. Kilic EE Mediums of Interest The mediums of interest will typically consist of numerous types of single scatterers (rain, vegetation, atmosphere,etc.) First we will consider a single particle and examine its scattering and absorption characteristics. Then we will derive the transport equation for a collection of particles in a given volume.

O. Kilic EE Scattering from a Single Particle When a single particle is illuminated by a wave, a part of the incident power is scattered out and another is absorbed by the particle. The characteristics of these two phenomena can be expressed assuming an incident plane wave.

O. Kilic EE Particle Scattering Phenomena Particle scattering is a complex topic—but we can simplify the view of particle scattering by visualizing the scattered radiation as composed of contributions of many waves generated by oscillating dipoles that make up the particle.

O. Kilic EE Sub-Particle Scattering Effects The radiation scattered by a particle and observed at P results from the superposition of all wavelets scattered by the subparticle regions (dipoles)—from Bohren and Huffman (1983).

O. Kilic EE Scattering by a Single Dipole Recall the dipole radiation pattern Same amount of energy in the forward and backward direction For particles that are very small compared to, forward and backscattering values are same. The incident energy will be redistributed uniformly in azimuth plane  less energy reaching the forward direction.

O. Kilic EE Complex Phenomena Each dipole affects the other such that the resulting field emitted by an oscillating dipole has a contribution that is due to stimulation by the incident field and a contribution due to stimulation by the fields of neighboring dipoles. The charge distribution in the particle forms higher order poles than dipoles and these multipoles also contribute to the scattered radiation.

O. Kilic EE Radiation from Multiple Dipoles – Understanding Larger Particles Recall the N element array: Directivity increases with N 2. Therefore the larger the particle, the more energy is scattered forward. As N increases, there are more sidelobes, thus more nulls. Therefore the larger the particle, the more involved the scattering characteristic gets (more maxima and minima as  varies) 0123N-1 ddd dcos   r

O. Kilic EE Scattering from Large Particles Although the simple discussion of multiple dipoles given above ignores the complicating effects of dipole-dipole interactions, the broad behavior predicted carries over to the more complete calculation of particle scattering.

O. Kilic EE Two Dipoles d P(x,y,z) x z (1) (2)  r r1r1 r2r2 dcos  EiEi Two isolated dipoles emit waves in all directions. At some point P far from the 'particle', these waves superimpose to create the scattered wave along the direction θ. These waves either constructively or destructively interfere depending on their relative phase difference ΔΦ, which is defined in terms of the extra distance the incident wave travels first to the second dipole (r) less the extra distance from the first dipole to P(r cos θ).

O. Kilic EE Excited by an incident wave, two dipoles scattering all directions. In the forward direction (  = 0), the two waves are always exactly in phase regardless of the separation of dipoles. Two Dipole Case

O. Kilic EE Multiple Dipoles The greater the number of dipoles in the particle array, the more they collectively scatter toward the forward direction, For the example shown here, all dipoles lie on the same line, are separated by one wavelength, and interact with each other. The scattered intensity is obtained as an average over all orientations of the line of dipoles (from Bohren, 1988) Forward scattering

O. Kilic EE Particle Extinction Depletion of forward scattering radiation is known as extinction. This decrease occurs due to two main effects: scattering and absorption Total extinction = scattering + absorption Measure of total extinction: Extinction Coefficient or Cross Section Measure of extinction due to scattering: Particle Albedo (ratio)

O. Kilic EE Scattering Cross Section Concept Consider radiation as a stream of photons that flow into a volume containing the scattering particles. Each particle within the volume blocks a certain amount of radiation resulting in a reduction of the amount of radiation directly transmitted through the volume. The reduction in the radiation as it passes through a volume of scatterers can be expressed in terms of a cross-sectional area  ext, which is generally different from the geometric cross- sectional area of the particle.

O. Kilic EE Cross Section (  ext ) and Efficiency (Q ext ) of Particle Extinction Particles 'block' a certain amount of radiation from penetrating through the slab. The reduced transmission can be described in terms of a cross-sectional area  ext shown as the shaded area around the particle which is different than the geometric cross section,  g (area of the particle projected onto a plane perpendicular to the direction of incident wave) While this view of extinction is a simple one to visualize, it is not entirely correct as extinction occurs through subtle interference effects.

O. Kilic EE Absorption and Scattering When  ext exceeds the value of the geometrical cross-sectional area of the particle, Q ext > 1 and more radiation is attenuated by the particle than is actually intercepted by its physical cross-sectional area. Since this extinction occurs by absorption or by scattering or by a combination of both, it follows that: Q ext depends on the refractive index of the material, the wavelength of radiation and the size of the particle.

O. Kilic EE Q ext vs Size for Spherical Water Droplets Plot of Q ext, calculated from Lorentz-Mie theory, as a function of the size parameter x for a water sphere illuminated by light of a wavelength of 500 nm. Extinction is highly dependent on the size of the particle. x = 2  D max / Q ext blue = 450 nm red = 650 nm D<  nm  Blue is scattered D =

O. Kilic EE Q ext vs Size The rapid rise in extinction as x increases (i.e., toward shorter wavelengths = higher frequencies) is a general characteristic of non-absorbing particles that are smaller than the incident wavelength. Thus, blue light is extinguished (scattered) more than red light, leaving the transmitted light reddened in comparison to the incident light.

O. Kilic EE The Extinction Paradox The tendency for the extinction Q ext either to oscillate around the value of 2 as x → ∞ or to converge to the value of 2. Intuition suggests that if we consider extinction as just the radiation that is blocked by the particle, then the extinction cross section is just the shadow projected by the very large particle. This geometrical view of extinction predicts that the limiting value of Q ext is 1 and not 2.

O. Kilic EE The Extinction Paradox (2) However, no matter how large the particle, in the vicinity of its edges rays do not behave the way simple geometrical arguments say they should. The energy removed from the forward direction can be thought of as being made up of a part that represents the amount blocked by the cross-sectional area of the particle and a part diffracted around the particle's edge.

O. Kilic EE Fundamental Extinction Formula Consider a component of a plane wave (with unit amplitude) incident on a particle Suppose the scattered wave at O’ is a spherical wave (i.e. far field) Ref: Van de Hulst (1957, p.29) Amplitude function; accounts for the fact that the intensity of the scattered radiation varies with scattering angle. z

O. Kilic EE Extinction Formula Extinction is defined at θ = 0 (forward direction) and  ext can be derived in terms of S(θ) This relation is known as the fundamental extinction formula. In deriving this relation, consider points near θ = 0 provided x, y < < z.

O. Kilic EE Forward Direction Approximation For x, y < < z Total field in the forward direction:

O. Kilic EE Intensity in the Forward Direction Total Intensity: O is the integral of the first term and corresponds to the geometric area projected by the particle on a distant screen.  ext is the integral of the second term and can be interpreted as the amount of power reduced by the presence of the particle (due to extinction). O -  ext

O. Kilic EE Extinction Term where Extinction depends on particle size, and amplitude function along forward direction. Using the method of stationary phase the integral of the second term is reduced to:

O. Kilic EE Extinction Factors One may gain the false impression that extinction occurs through blocking an amount of incident em wave of magnitude  ext. In fact, extinction is a subtle interference phenomenon and not one of merely blocking as the extinction paradox reminds us. In discussing this paradox, we learn that the particle not only 'blocks' an amount of incident energy that is defined by the geometric cross section, but also removes some of the energy of the original wave via interference. More to come on how to calculate S(  )…

O. Kilic EE Terminology for Radiation/Scattering from a Particle Scattering Amplitude and its relation to amplitude function, S(  ) Differential Scattering Cross Section: –Bistatic radar cross section –Backscattering cross section Scattering Cross Section Absorption Cross Section Total Cross Section (Extinction Cross Section) Albedo Phase Function

O. Kilic EE Scattering Amplitude Consider an arbitrary scatterer, and a linearly polarized incident plane wave with unit amplitude: Imaginary, smallest sphere D i EiEi o EsEs R The scatterer redistributes the incident electric field in space: rr

O. Kilic EE Scattering Amplitude (2) f(o,i) is a vector which represents the amplitude, phase, and polarization of the scattered field in the far field in the direction of o when the particle is illuminated in the direction i with unit amplitude  f(o,i) depends on four angles. Note that even if the incident wave is linearly polarized, the scattered wave in general is elliptically polarized.

O. Kilic EE Scattering Amplitude(3) Note: Comparison to slide #22 suggests: (ik)f  (o,i) = S  (  ) E s behaves like a spherical wave in the far field.  : polarization of the incident wave Forward scattering Theorem:

O. Kilic EE Scattering Cross Section Definitions: Power Relations Differential Scattering Cross-section Scattering Cross-section Absorption Cross-section Total Cross-section

O. Kilic EE Differential Scattering Cross Section (m 2 /St) Consider the scattered power flux density S s at a distance R from the particle along the direction, o caused by an incident power flux density S i. where

O. Kilic EE Differential Scattering Cross Section ss i o R AsAs Units: m 2 /St Suppose the observed scattered power flux density in the direction o is extended uniformly over 1 Steradian of solid angle about o. Then the cross section of a particle which would cause just this amount of scattering would be  d. The differential cross section varies with o and i. Total power if S s was uniformly distributed over the sphere Or, equivalently:

O. Kilic EE Bistatic Radar Cross Section Suppose the observed power flux density in the direction o is extended uniformly in all directions from the particle over the entire 4  steradians of solid angle. Then the cross section which would cause this would be 4  times  d for the direction o.

O. Kilic EE Backscattering Cross Section (Radar Cross Section) Suppose the observed power flux density in the direction opposite to the incident illumination (o = -i) is extended uniformly in all directions from the particle over the entire 4  steradians of solid angle. Then the cross section which would cause this would be 4  times  d for the backscatter direction.

O. Kilic EE Scattering Cross Section (m 2 ) Consider the total observed scattered power at all angles surrounding the particle. The cross section of a particle which would produce this amount of scattering is called the scattering cross section,  s.

O. Kilic EE Scattering Cross Section (2) Unit: m 2 Integrate over all scattered angles over a sphere surrounding the particle

O. Kilic EE Absorption Cross Section (m2) Consider the total power absorbed by the particle. The cross section of a particle which would correspond to this much power is called the absorption cross section,  a.

O. Kilic EE Absorption Cross Section (2)

O. Kilic EE Total Cross Section (Extinction Cross Section) The sum of the scattering and the absorption cross sections is called the total cross section,  t (or the extinction cross section) Recall that these all depend on the incident energy direction.

O. Kilic EE Size Dependence If the particle size is much larger than : total cross section is twice the geometric cross section. (Van de Hulst: Extinction paradox discussed earlier). Use GO approximation. If the size is much smaller than  the scattering cross section is inversely proportional to 4 and proportional to the square of the particle volume. (Kerker). Use Rayleigh approximation.

O. Kilic EE Albedo The ratio of the scattering cross section to the total cross section is called the albedo of a particle. Low Albedo: If  a >>  s  a<<1 Incident energy is mostly absorbed than scattered Usually occurs for moderately lossy particles at large compared to D. e.g. rain at microwave frequencies High Albedo: If  s >>  a  a>>1 Incident energy is mostly scattered than absorbed Usually occurs for large particle sized compared to e.g. tree trunks at microwave frequencies

O. Kilic EE Phase Functions A dimensionless quantity that relates the differential cross section to the total cross section. It represents the amount of scattered power and has no relation to the phase of the wave. It is commonly used in radiative transfer theory. The name “phase function” has its origins in astronomy, where it refers to lunar phases.

O. Kilic EE Phase Functions: p and  Note that phase functions are dimensionless.

O. Kilic EE Phase Function (2) i Analogy: directivity p: The ratio of the differential scattering cross-section of a particle to an equivalent particle that would scatter the total extinction power uniformly.

O. Kilic EE Phase Function Relations

O. Kilic EE Phase Function Relations (2)

O. Kilic EE Phase Function Relations (3)

O. Kilic EE Integral Representations of Scattering Amplitude and Absorption Cross Section If the shape of a particle is simple (e.g. sphere, cylinder), one can obtain exact expressions for cross sections and scattering amplitude. For instance, Mie solution is used for spheres. However, in many practical cases, the shape of a particle is arbitrary. In such cases, general integral representations of scattering amplitude can be used.

O. Kilic EE Arbitrary Dielectric Body Consider a dielectric body whose relative dielectric constant is a function of position: Assume that the dielectric body occupies a volume V, and is surrounded by free space,  0. Also assume that the permeability of the body is  0. 00 V

O. Kilic EE Maxwell’s Equations Maxwell’s equations for this medium are: where J is the source for the incident field and is the position dependent permittivity given as follows: (1) (2) (3)

O. Kilic EE Maxwell’s Equations for the Incident and Total Fields The total field which satisfies eqn 1 and 2 is the sum of the incident and scattered fields for the region outside the particle. (4) The incident field satisfies the Maxwell’s equations in the following form: (5) and exists independent of the particle.

O. Kilic EE Maxwell’s Equations for the Scattered Field Using the linearity of Maxwell’s equations, the scattered field expressions can be derived as follows: (6) (7) (4) in (1) (4) in (2)

O. Kilic EE Equivalent Current for Scattered Field From (5) From (4) (8) (9)

O. Kilic EE Equivalent Current Source Equivalent source inside the particle that generates the scattered fields. Note that: V J EsEs (10)

O. Kilic EE Solution Recall the use of vector potential A and scalar potential  to solve for the radiated fields given a current source distribution in space. (11)

O. Kilic EE Solution (2)

O. Kilic EE Solution for Scattered Fields J eq (12) (13) (14)

O. Kilic EE Far Field Solution for the E s Field In order to obtain the scattering amplitude, we consider E s in the far field. (15)

O. Kilic EE Far Field Approximations In the far field, (16)

O. Kilic EE Scattering Amplitude – Integral Form This is an exact expression for the scattering amplitude f(o,i) in terms of the total electric field E(r) inside the particle. E(r) is not known in general, therefore this eqn is not a complete description in terms of known quantities. Using (16) in (15), (17) Dependent on incidence direction

O. Kilic EE Approximate Calculations of f(o,i) Rayleigh Scattering: When the particle size is much smaller than the incident wavelength (kD <<1), the internal field can be calculated approximately for simple geometries. Because of the small size, the impinging field within or near the object can be treated as a static field; i.e. no time variation.

O. Kilic EE Small Dielectric Sphere rr Solve for Laplace’s equation: (Note that there are no charges) a

O. Kilic EE Solution of Laplace’s Equation for Dielectric Sphere

O. Kilic EE Electrostatic Solution for Dielectric Sphere Illuminated by a Plane Wave (18)

O. Kilic EE Scattering Amplitudes of a Dielectric Sphere (Rayleigh) Using (18) in (17),

O. Kilic EE General Form of Scattering Amplitudes for a Dielectric Sphere (Rayleigh) 

O. Kilic EE Differential Cross Section of a Dielectric Sphere (Rayleigh)

O. Kilic EE Scattering Cross Section

O. Kilic EE Out of the blue… Why is the sky blue? The blue of the sky (shorter wavelength) scatters more than the red portion because of the -4 dependence; This was a great puzzle in the last century, and was finally explained by Rayleigh. Rayleigh noted that the scatterers need not be water or ice as wa commonly believed at that time, but that the molecules of air itself can contribute to this scattering.

O. Kilic EE Efficiency, Q s Note: This is valid for small ka (Rayleigh).

O. Kilic EE Absorption Cross Section The absorption cross section for a dielectric body is the volume integral of the loss inside the particle.

O. Kilic EE Efficiency, Q a

O. Kilic EE Total Cross Section The total cross section is the sum of scattering and absorption cross sections. Note that we can not apply the forward scattering theorem, since this would give a total cross section of zero when. Real if  r is real.

O. Kilic EE Total Cross Section Rayleigh approximation does not hold for forward scattering theorem. In general summing the absorption and scattering cross sections for a particle for a given approximate value of the internal field gives a much better approximation to the total scattering cross section.

O. Kilic EE Example Consider a plane wave with unit magnitude, polarized in the x direction and propagating in the z direction. This wave is incident upon a small dielectric sphere of radius a. Calculate the scattered field in the far field.

O. Kilic EE Solution In the far zone of the sphere, the electric field components are given by: where

O. Kilic EE E  Component

O. Kilic EE E  Component

O. Kilic EE Solution

O. Kilic EE Rayleigh-Debye Scattering (Born Approximation) Now consider the scattering characteristics of a particle whose relative dielectric constant  r is close to unity. In this case, the field inside the particle may be approximated by the incident field.

O. Kilic EE Scattering Amplitude – Born Approximation Using the integral equation for scattering amplitude Define Thus i o  i-oi-o Largest dimension of the particle; i.e. diameter Fourier Transform of

O. Kilic EE Observations The scattering amplitude is proportional to the Fourier transform of evaluated at the wave number k s. In general if is concentrated in a region small compared with the wavelength, the cross section is spread out in k s and thus in angle , and the scattering is almost isotropic. If the particle size is large compared to a wavelength, the scattering is mostly concentrated in a small forward angular region,  ~0.

O. Kilic EE Rayleigh-Debye Scattering Cross Section

O. Kilic EE Rayleigh-Debye Absorption Cross Section Where we use:

O. Kilic EE Example: Homogenous Sphere Let k s be along a unit vector direction denoted by i s. Let i s be along the z’ direction. Because of spherical symmetry we can assign this arbitrarily. x’ y’ z’ r’ isis

O. Kilic EE Limitations of Rayleigh and Born The forward scattering theorem is applicable to any scatterer of arbitrary shape and size. However, for Rayleigh and Born approximations, the forward scattering theorem does not yield the total cross section. In fact, the scattering amplitude f(o,i) is real for a lossless particle in these approximations  no loss as the way passes through the particle!

O. Kilic EE Limitations of Rayleigh and Born The reason for this shortcoming is that the scattering amplitude f(o,i) is predominantly real, and this real part describes the general angular characteristics of scattering. The imaginary part is small, but this represents the total loss, which includes both the part scattered out and that absorbed by the object  related to the extinction cross section. If we have a solution which is better than Rayleigh and Born, forward scattering theorem can be used to calculate extinction cross section.

O. Kilic EE WKB Technique This better approximation may be provided by the WKB interior wave number technique. Note that Rayleigh is applicable to small particles; i.e. ka <<1 Born is applicable when (  r -1)kD << 1, where kD itself can be large, as long as the dielectric value of the particle is close to free space.

O. Kilic EE WKB Technique WKB technique is valid when (  r -1)kD >> 1 and  r -1 < 1 Therefore, it provides a solution for which Rayleigh and Born approximations are not applicable.

O. Kilic EE WKB Technique In WKB approximation, the field inside the particle is approximated by a field which propagates with the propagation constant of the medium of the particle. It is also assumed that since  r -1 < 1, the refraction angle is equal to the incident angle, And the wave propagates in the direcion of the incident wave. The transmission coefficient T at the surface of the particle is approximated by that for normal incidence.

O. Kilic EE WKB Technique For an incident field: The internal field at point B is approximated by: z1z1 z2z2 A B C

O. Kilic EE From where We obtain

O. Kilic EE Mie Theory The exact solution of scattering of a plane wave by an isotropic, homogenous sphere was obtained by Mie in See Kerker, 1969, Chapter 3 for a detailed derivation or any other standard textbook on radiation and scattering.

O. Kilic EE Mie Theory

O. Kilic EE References Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley otes/at622_sp06_sec14.pdfhttp://langley.atmos.colostate.edu/at622/n otes/at622_sp06_sec14.pdf Van deHulst, Light Scattering by Small Particles (1957, p.29) Bohren and Huffman, Absorption and Scattering of Light by Small Particles. Ishimaru