Light Scattering by (Transparent) Spheres: Mie Solution

[daverace]

Homework Statement

Hello everyone,

I've been struggling to understand the relationship between two algebraic forms of the scattering function used in the Mie solution for light scattered by (transparent) spheres. I think there's some use being made of Legendre function identities, but I've been unable to work out the details of how to move from one form to the other.

Homework Equations

Here's the form of the $S_{1}$ and $S_{2}$ scattering functions given in, for e.g., Liou or Bohren & Huffman (the two books I have in my possession):

$S_{1} ( \theta ) = \sum_{n = 1}^{\infty} \frac{2n + 1}{n (n + 1 )} [ a_{n} \pi_{n} ( \cos{\theta} ) + b_{n} \tau_{n} ]\\ S_{2} ( \theta ) = \sum_{n = 1}^{\infty} \frac{2n + 1}{n (n + 1 )} [ b_{n} \pi_{n} ( \cos{\theta} ) + a_{n} \tau_{n} ]$
where:
$\pi_{n} \left( \cos{\theta} \right) = \frac{P^{1}_{n} \left( \cos{\theta} \right)}{\sin{\theta}}\\ \tau_{n} \left( \cos{\theta} \right) = \frac{dP^{1}_{n} \left( \cos{\theta} \right)}{d\theta}$
(pages 186 and 112 in those books, respectively).

I'm using this to work on the rainbow, so I've been looking at a lot of Nussenzveig's work. He always starts off with the following equation:
$S_{j} \left( \beta \theta \right) = \frac{1}{2} \sum_{l = 1}^{\infty} \left\{ \left[ 1 - S^{\left(j \right)}_{l} \left( \beta \right) \right] t_{l} \left( \cos{\theta} \right) + \left[ 1 - S^{\left(j \right)}_{l} \left( \beta \right) \right] p_{l} \left( \cos{\theta} \right) \right\} \;\; \left( i, j = 1, 2 i \neq j \right)$
where:
$p_{v} \left( \cos{\theta} \right) = \frac{\left[ P_{v - 1} \cos{\theta} - P_{v + 1} \cos{\theta} \right]}{ \sin{\theta}^{2} } \\ t_{v} \left( \cos{\theta} \right) = - \cos{\theta} p_{v} \left( \cos{\theta} \right) + \left( 2v + 1 \right) P_{v} \left( \cos{\theta} \right)$
See, for example: Nussenzveig's 1969.

(Note the change in indices; $l = n$ for these equations.)

The Attempt at a Solution

I am completely baffled how to move from the first set of equations to the second (including the difference in coefficients, $a_{n}$ for example. Any help would be great; explicitly steps by steps would be obviously favourable.

(I have managed to find an identity between $\frac{2n+1}{n(n+1)}\frac{d}{d\theta}P_{n}(cos{\theta})$ and $p_{l}(cos{\theta})$, but this by itself is unhelpful.)

Oh, I should point out that I'm no longer a physics student, but actually a philosophy of science student. This just seemed like the most appropriate place to put this question.

 

[NateD]

My Attempt at a SolutionI believe that the relationship between the two forms of the scattering function can be derived by making use of the following identity:P_{n}^{m} (\cos{\theta}) = \frac{2n+1}{n(n+1)}\frac{d}{d\theta}[\sin{\theta}P_{n-1}^{m+1}(\cos{\theta})]Using this identity, we can rewrite the equations for S_{1} and S_{2} as follows:S_{1} (\theta) = \sum_{n=1}^{\infty} \frac{2n+1}{n(n+1)}\frac{d}{d\theta}[a_{n}\sin{\theta}P_{n-1}^{1}(\cos{\theta}) + b_{n}\sin{\theta}P_{n-1}^{2}(\cos{\theta})]\\S_{2} (\theta) = \sum_{n=1}^{\infty} \frac{2n+1}{n(n+1)}\frac{d}{d\theta}[b_{n}\sin{\theta}P_{n-1}^{1}(\cos{\theta}) + a_{n}\sin{\theta}P_{n-1}^{2}(\cos{\theta})]Now, we can make use of the following identities to simplify the equations further:\frac{d}{d\theta} P_{n}^{1}(\cos{\theta}) = -\cos{\theta}P_{n}^{1}(\cos{\theta}) + (2n+1)P_{n}^{2}(\cos{\theta}) \\\frac{d}{d\theta} P_{n}^{2}(\cos{\theta}) = \sin{\theta}P_{n-1}^{1}(\cos{\theta})Substituting these identities into the equations for S_{1} and S_{2}, we can rewrite them as:S_{1} (\theta) = \sum_{n=1}^{\infty}