Mie Total and Differential Backscattering Cross Sections at Laser Wavelengths for Junge Aerosol Models M. P. McCormick, J. D. Lawrence, Jr., and F. R. Crownfield, Jr.

The first two authors are with NASA Langley Research Station, Hampton, Virginia 23365; the last named author is with The College of William and Mary, Williamsburg, Virginia 23185. Received 17 July 1968.

Scattering functions for polydisperse aerosol distributions cal­culated using the rigorous Mie theory have been reported pre­viously.1,2 The interpretation of laser backscatter experiments in the atmosphere has created a need for the extension of previous calculations to laser wavelengths and parameters characteristic of the atmospheric aerosol.

This note reports calculations of backscattering and total cross sections for a Junge size distribution of spherical aerosol particles with a refractive index of 1.5. These calculations have been per­formed at the laser wavelengths of ruby (0.6943 μ), neodymium (1.06 μ), and their second harmonics (0.3472 μ and 0.5300 μ, re­spectively).

The radiation backscattered by a volume element of the atmo­sphere located a distance Z from the laser, expressed as the power incident on a coaxial receiver, is given by

where E is the transmitted energy, A is the receiver area, q(Z) is the transmissivity of the atmosphere given by

where β(Z) is the scattering coefficient, f(Z) is the scattering func­tion or volume cross section, and c is the speed of light. The scat­tering volume is assumed to contain both molecules, whose scat­tering properties are described by Rayleigh theory, and aerosols, whose scattering properties are described by Mie theory. Only the aerosol component of the scattering is considered here.

For a continuous size distribution of aerosols which obeys the Junge3 size distribution dn(r) = br−(v + 1)dr, the aerosol scatter­ing function as defined by Bullrich1 is

where

i1.2(α,η,θ) are the Mie intensity functions for light with electric vector perpendicular and parallel, respectively, to the plane through the direction of propagation of the incident and scat­tered wave; α = 2πr/λ is the particle size parameter; λ is the wavelength of incident radiation; dn(r) is the number of particles with radii between r and r + dr per unit volume; r is the radius of the spherical aerosol particle; η is the index of refraction of the spherical aerosol particle; θ is the scattering angle measured between the forward direc­tion and scattered wave; b is a constant dependent on the total number of particles per unit volume; v is the size distribution parameter; N = ∫r2

r1dn(r) is the total number of Mie particles with radii be­tween r1 and r2 per unit volume.

2424 APPLIED OPTICS / Vol. 7, No. 12 / December 1968

Similarly, the scattering coefficient, β is given by

where am(α,η) and bm(α,η) are the Mie coefficients (cf. van de Hulst4) .

The function dσ/dΩ, = f/N is the average cross section per unit solid angle per particle and the function σ = β/N is the total cross section per particle. Values of dσ/dΩ, for 180° scattering and σ are listed in Table I for particles having an index of refrac­tion of 1.5. The computations have been performed for the laser wavelengths 1.06 μ, 0.6943 μ, 0.5300 μ, and 0.3472 μ for values of the size distribution parameter representative of the atmospheric aerosol and for four sets of particle radii limits.

The Mie coefficients were calculated using Deirmendjian's5

recursion formulas with sixteen significant digits. The recursion was terminated when the real and imaginary parts of am and bm

were all less than 10 - 8 . The integral expressions were calculated using the trapezoidal rule for incremental Δα = 0.01 for α ≤ 2.0 and incremental Δα = 0.10 for α ≥ 2.0.

References 1. K. Bullrich, Advances in Geophysics (Academic Press, Inc.,

New York, 1964), Vol. 10. 2. D. Deirmendjian, Appl. Opt. 3 , 187 (1964). 3. C. E. Junge, Air Chemistry and Radioactivity (Academic Press,

Inc., New York, 1963). 4. H. C. van de Hulst, Light Scattering by Small Particles (John

Wiley & Sons, Inc., New York, 1957). 5. D. Deirmendjian, R. Clasen, and W. Viezee, J. Opt. Soc. Amer.

51 , 620 (1961); See also D . Deirmendjian and R. J. Clasen, RAND Rep. R-393-PR (1962).

December 1968 / Vol. 7, No. 12 / APPLIED OPTICS 2425