Normalized scattering diagram for atmospheric haze

Vadim Raskin Elizabeth City State University, Box 83, Elizabeth City, North Carolina 27909. Received 11 March 1981. 0003-6935/81/193290-02$00.50/0. © 1981 Optical Society of America.

It is well-known that the particles of which atmospheric haze is composed have such dimensions that for the visible region of the electromagnetic spectrum it corresponds to non-Rayleigh scattering. This case must be described ac­cording to the general diffraction formulas. It is important to note, however, that usually in atmospheric haze the parti­cles with a large refractive index or absorption coefficient in the visible region of the spectrum are absent. Therefore it is possible to obtain the scattering diagram for the polydisperse aerosol system in analytical form using the approximate method of calculation of light scattering by the particles proposed by Shifrin.1 He developed an integro-differential equation for scattering by tenuous particles—particles that satisfy two conditions:

(1) The polarizability of the particle relative to the ambient medium must be small compared to 1.

(2) The phase shift of the central ray is <2. By such an approach we avoid numerical tabulation of nu­merous possible cases and analyze the optical properties of some aerosol systems much more easily.

In this Letter we assume the sphericity of the particles of atmospheric aerosol.

The expression for the normalized scattering diagram is presented as a product of the expression for a Rayleigh scat­tering diagram for a monodisperse aerosol with the radius of the particles equal to the mean radius corresponding to the given distribution by some special function. This is just a convenient form of presenting the end result of lengthy inte­grations.

In some atmospheric situations the particle size distribution of atmospheric haze can be approximately described by

where a is the radius of the aerosol particle. The mean radius of particles corresponding to Eq. (1) is

If the number of particles per cubic centimeter is N, using the expression N =∫∞

0Aam exp(-pa)da we obtain

and the normalized distribution Eq. (1) is

We shall now find for this size distribution the normalized scattering diagram using for the scattering diagram of the single particle expression obtained in the first iteration of Shifrin's integro-differential equation for determining the scattered field of tenuous particles.1 If we designate I(β,a) as the scattering diagram of the individual particle, and I(β) as the scattering diagram for polydisperse system, we can' write

3290 APPLIED OPTICS / Vol. 20, No. 19 / 1 October 1981

where $(a) is given by Eq. (1). Therefore,

(see Ref. 2). β = angle of scattering and X = wavelength of incident light set at 0.55 mK. The polarizability given by Lorentz-Lorentz formula is

where n = the relative refractive index and I0 = intensity of the incident light. Therefore,

The result of integration in Eq. (5) can be represented in the following form:

where I(β) is the scattering diagram for atmospheric haze with particle size distribution given by Eq. (1) and IR(a) is Ray-leigh's scattering diagram for monodisperse aerosol with particles having radius a and with concentration N.

where |α |2 is the coefficient whose numerical value is de­pendent on the properties of the substance of the particle. For water this equals 2.304 × 10-3.

where S = (1 + u2)-l/2 and r = arctanu, and

To obtain the normalized diagram we must know the scat­

tering coefficient of the polydisperse aerosol. We define it by

where K(λ) is the scattering coefficient of the polydisperse aerosol, and K(λ,a) is the scattering coefficient of the indi­vidual aerosol particle. The scattering coefficient of the in­dividual particle according to van de Hulst3 can be expressed as

where

The result of integration in Eq. (14) can be expressed as

where

and K∞, is the scattering coefficient for very large particles:

In(β) the normalized scattering diagram is defined by

This characteristic is independent of the number of aerosol particles N. As the end result of our calculations we achieved the following expression for the normalized scattering diagram of atmospheric haze with a particle distribution given by Eq.(l)

References 1. K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow,

1951), NASA Technical Translation TT477,1968. 2. K. S. Shifrin and V. F. Raskin, "Average indicatrice corresponding

gamma-distribution," in Proceedings, Voyekov's Main Geo­physical Observatory (Leningrad, 1961), Vol. 109.

3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1 October 1981 / Vol. 20, No. 19 / APPLIED OPTICS 3291