The ensemble average of the Stokes parameters .. of the scattered wave is explicitly calculated for the second order. It is found that the polarization of the scattered wave depends on the polarization of the incident wave. The handiness of wave 9 = 0 is the same as that of the incident, while it is.
However, the handedness of the total scattered wave does not change by second order. It is found that the results of the lossless case are insensitive to first order of k. When one calculates the scattered wave, due to the illumination of a collection of particles, and ignores the interaction of the particles, one speaks of single scattering.
The theory presented by Hartel, however, does not include polarization of the scattered wave. His transfer equation is a continuity equation for a 4-dimensional vector with 4-component Stokes parameters of the scattered wave.
FORMULATION OF THE PROBLEM
Em,flo
Loielectric Particle
The first term involves the interaction of the i-th particle's volume elements, which would exist even if all other particles were absent, while the second term describes an interaction between the i-th particle and all others . If we want the intensity pattern of the scattered field, we need to calculate the Poynting vector in the far zone. If we now look at the set of the particles and neglect the self-interaction terms, we can show that (see Appendix D). the different orders add incoherently because of the assumption about. randomness in the position and orientation of the particles.
Multiplying by N is due to - s c -sc. the assumption of randomness that causes the intensities from some particles to add up. This is so because N is approximately equal to N. 15 was derived under the assumption that the particles are. of such a size and so far apart that they see only the far-field field of each particle. m is the refractive index of the medium. If we assume the ratio of two consecutive terms in 2. 12 equal to 10 and the ratio of the intensity of the second-order multiple scattering to the dominant self-interaction term in 2. then D must satisfy the following inequality:.
Later, for the sake of obtaining a simple form for the scattered wave intensity, we will assume that our particles have the same shape, dimensions, and sensitivity. We want to find the distributed field of the far zone in .! characterized by r, 8,cp w.r.t. fixed x,y,z system.
Einc
El ,
We wrote F(9) and not F(9, cp) because the averaging procedure will eliminate the cp dependency regardless of its form. particles, provided there are no losses. This would not be true if the collection of the particles showed a cp dependence on average. The polarization properties of E(i)(r} depend entirely on - s c - the vector e Xe XE which is independent of the material medium, - r - r - o. the shape, size, orientation and sensitivity of the particles .
Considering the correspondence {x,y,z)~(9,cp,r) we understand that the backscattered wave is circularly polarized but opposite to the incident wave, while the forward wave is . scattered wave is circularly polarized and has the same handedness. If the incident wave is left-handed circularly polarized, we can easily see that the backscattered wave is again c.p. is. but from opposite direction, while the forward scattered wave is c. P. and of the same dexterity as the incident wave. We can easily understand the above results if we consider that the observer who decides the direction of rotation of the electric vector always walks behind the wavefront.
If Ei. because each particle is measured from a common origin, say the center of the volume occupied by the particles, then 4. 1. But there are two reasons for using 4. 1. 2. a) Most particles in pairs satisfy 4. When we perform the integration over the volume i -of this particle, e can be replaced by its average value e • The reason is v. Because of the assumption of far-field field interaction between particles, we understand that if a. if the particle is located near the origin, then every other particle is at a distance much greater than the wavelength.
On the other hand, the linear dimensions of the particles are of the order of a wavelength, thus the change in direction of e over the volume of the ith particle. is truly negligible for all the particles but the one located near the origin. As we did in section 3.1, we will again assume that all the particles have the same shape, size and susceptibility. the rules set out in Appendix D we have:. to any point within the volume V occupied by the particles. where ~· is the radius vector from the origin to any point within the volume V occupied by the particles. We will now show that F. Now e - r is a fixed direction in space and L will depend on the orientation of the particle w. r. t. the fixed direction.. averaged over a,~,)' all the possible orientations of the particle are included and F.
1 cannot possibly depend on the fixed direction e (9, SPECIAL EXAMPLES D is proportional to 1/(ka)2. Thus, by making ka very small, i.e. much smaller than /.., we cannot lose the contribution to the self-interaction, unless D becomes large to make the multiple scattering more important than the self-interaction. b}j .6. The end limits of this range depend on ka, the refractive index n of the surrounding .. m. medium, and the size of the reaction occupied by the particles. Also, the ratio between the forward scatter intensity and the backscatter intensity does not depend on any e. THE INTEGRAL EQUATION OF THE SCATTERING PROBLEM We assume that the constitutive parameters of the medium a:re Em,µ0 • We can consider the system of the medium plus the particle as a new medium with constitutive parameters E ,µ. Now the homogeneous solution of A-5 is just the incident. r) whereas the special solution is the scattered -1nc-. The incident wave induces a current inside the volume V p of the particle which is given by I= -iwP where P is the relative (to the surrounding medium) polarization given by. The far-field field is a TEM wave and behaves like a plane wave near a given direction. The sense of rotation must be determined by the observer who sees the wave receding from him. If D-8 represents a scattered field of the nth order, i.e. then it makes sense to calculate the Stokes parameters D- i 3 for this order. This is so because there are several types of independent waves, so the average Stokes parameter of the composite waves says S. theorem, which can be easily demonstrated in our example. Due to the periodicity of the integrand (= f{cos a')), the limits can be replaced by 0 and 2iT. Therefore, I becomes independent of the azimuth angle 1 STI" 4
CONCLUSIONS
