On the Polarization Angle in the Scattering Problem

[摘要] No research has not yet been attempted on the polarization angle on the secondary scattering of the sun's ray in the earth's atmosphere. The author has developed its mathematical characteristics and has given useful evaluation. In Fig. 2, P and Q are two points in the earth's atmosphere. Define any rectangular coordinate system X1Y1Z1 with its centre at Q, X1 being directed to the sun. The direct insolation at Q can be resolved into two plane polarized lights, the one travelling to - X1 and oscillating in Z1 direction, the other to -X1 and in Y1.ω and ω' are the angles of Z1 and Y1 suspended by QP. Define a rectangular coordinate system X2Y2Z2, choosing P to the origin, X2 to PQ, and Z2 to the direction of oscillation when the former polarized light proceeds from Q to P. Define another system X2' Y2'Z2', choosing P to the origin, X2' to PQ, and Z2' to the direction of oscillation when the latter polarized light proceeds from Q to P. Ω and Ω' are the angles of Z2 and Z2' suspended by PO', O' being a point on the earth's surface. Then the intensity of secondary scattering at O' generated by air particles at P and Q is proportional tosin2ωsin2Ω+sin2ω'sin2Ω'.For brevity let us write D=sin2ωsin2Ω, D'=sin2ω'sin2Ω'. The author calls D+D'as the polarization angle. This is naturally a function of the sun's altitude h and the positions of P and Q. Define a system of rectangular coordinates X'Y'Z' with its centre at O', Z' being directed normally upwards, X'lying in the vertical plane passing the sun and towards the sun-side. A is the angle between two planes O'Z'P and Z'X', θ is between O'P line and X'Y' plane. (see Fig. 4). XYZ system is produced by the parallel translation of X'Y'Z' system from the origin O' to the earth's centre O. Let the new system x1y1z1 be produced by rotating XY plane by the angle A around Z axis, then rotating the new XZ plane by the angle ∠POZ around the new axis Y. (see Fig. 5). ∠POZ is denoted by a. Let θ1 be the angle ∠OPQ, and A1 be the angle between two planes OPQ and x1z1. The author has had the next conclusions.D' is always independent to h, and D and D' are always indifferent to the substitutionθ1, A1→π-θ1, π+A1.A). When A=0. D and D' are indifferent to the substitution θ1, A1→π-θ1, π-A1→π-θ1, π+A1. WhenA=π/2 D'is equal to D for A=0, The substitutionθ1, A1→π-θ1, .π-A1 is possible to D'for any h and D for h=0 and 90°. When.A=π. D' is equal to that for A=0. The substitution for .D and D' is the same as that for A=0.B) When h=O. D(A+π/2)=D(A) D'(A=π)=D'(A=0) D+D'is identical A=0, π, and D for A=0, π/2, π. When h=90. D(A+π/2)=D'(A), D'(A+π/2)=D(A) D+D' is identical for A=0, π/2, π, and D for A=0, π.C). When α=0. 1) When θ=0. Representing D, D' by D(A, A'), D'(A, A1), then D (π, π-A1)=D (0, A1) D'(π, π-A1)=D'(0, A1) . and D+D' is equal for both cases. Further D'(0, A1)=D'(0, π-A1) D(π, A1) =D'(π, π-1) . Especially for h=0 and 90°, the following four are each equal: D(0, A1), D(0, π-A1), D (π, A1) and D (π, π-A1). 2) When θ=90°. In this case there exists the same conclusion as for θ=0, and moreover, especially for A=0, h=0, D is independent to A1. 3) For any values of h. D(π/2, A1)=D(0, A1+π/2) D'(π/2, A1)=D'(0, A1+π/2).

[学科分类] 大气科学