经典Mie理论中的散射系数的另外一种计算

H&B书中经典Mie散射和吸收系数有两种表示方式,一种是29号整理的表示,还有一种常见的表示方式:

式1

两种表示的中间关系式为:

式2

与Bessel函数的关系为:

式3

递推关系式为:

式4

1951年,Aden引入对数微分算法

将Mie散射系数表示为:

式5

式5中Dn的迭代算法表示为:

在matlab中的函数代码可表示为:

function result = Mie_ab(m,x)

% Computes a matrix of Mie Coefficients, an, bn,

% of orders n=1 to nmax, for given complex refractive-index

% ratio m=m'+im' and size parameter x=k0*a where k0= wave number in ambient

% medium for spheres of radius a;

% Eq. (4.88) of Bohren and Huffman (1983), BEWI:TDD122

% using the recurrence relation (4.89) for Dn on p. 127 and

% starting conditions as described in Appendix A.

z=m.*x;

nmax=round(2+x+4*x.^(1/3));

nmx=round(max(nmax,abs(z))+16);

n=(1:nmax); nu = (n+0.5);

sx=sqrt(0.5*pi*x);

px=sx.*besselj(nu,x);

p1x=[sin(x), px(1:nmax-1)];

chx=-sx.*bessely(nu,x);

ch1x=[cos(x), chx(1:nmax-1)];%fai n-1阶

gsx=px-i*chx;

gs1x=p1x-i*ch1x;%y n-1阶

dnx(nmx)=0+0i;

for j=nmx:-1:2      % Computation of Dn(z) according to (4.89) of B+H (1983)

dnx(j-1)=j./z-1/(dnx(j)+j./z);

end;

dn=dnx(n);          % Dn(z), n=1 to nmax

da=dn./m+n./x;

db=m.*dn+n./x;

an=(da.*px-p1x)./(da.*gsx-gs1x);

bn=(db.*px-p1x)./(db.*gsx-gs1x);

result=[an; bn];

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