## Abstract

We have derived the formula for the Debye-series decomposition for light scattering by a multilayered sphere. This formulism permits the mechanism of light scattering to be studied.
An efficient algorithm is introduced that permits stable calculation for a large sphere with many layers. The formation of triple first-order rainbows by a three-layered sphere and single-order rainbows and the interference of different-order rainbows by a sphere with a gradient refractive index, are then studied by use of the Debye model and Mie calculation. The possibility of taking only one single mode or several modes for each layer is shown to be useful in the study of the scattering characteristics of a multilayered sphere and in the measurement of the sizes and refractive indices of particles.

© 2006 Optical Society of America

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### Equations (31)

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(1)
$${I}_{1}\left(\theta \right)={\left|{S}_{1}^{1}\left(\theta \right)\right|}^{2},\text{\hspace{1em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}{I}_{2}\left(\theta \right)={\left|{S}_{2}^{1}\left(\theta \right)\right|}^{2},$$
(2)
$${S}_{1}^{1}\left(\theta \right)={\displaystyle \sum _{n=1}^{\infty}\text{\hspace{0.17em}}\frac{2n+1}{n\left(n+1\right)}\text{\hspace{0.17em}}}\left[{a}_{n}^{1}{\pi}_{n}\left(\theta \right)+{b}_{n}^{1}{\tau}_{n}\left(\theta \right)\right],$$
(3)
$${S}_{2}^{1}\left(\theta \right)={\displaystyle \sum _{n=1}^{\infty}\text{\hspace{0.17em}}\frac{2n+1}{n\left(n+1\right)}\text{\hspace{0.17em}}}\left[{a}_{n}^{1}{\tau}_{n}\left(\theta \right)+{b}_{n}^{1}{\pi}_{n}\left(\theta \right)\right]\mathrm{.}$$
(4)
$${\pi}_{n}\left(\theta \right)=\frac{1}{\mathrm{sin}\text{\hspace{0.17em}}\theta}{P}_{n}^{1}\left(\theta \right),\text{\hspace{1em} \hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}{\tau}_{n}\left(\theta \right)=\frac{d}{d\theta}{P}_{n}^{1}\left(\theta \right),$$
(5)
$$\begin{array}{l}{a}_{n}^{1}\\ {b}_{n}^{1}\end{array}\}=\frac{1}{2}\left(1-{R}_{n}^{212}-\frac{{T}_{n}^{21}{T}_{n}^{12}}{1-{R}_{n}^{121}}\right)=\frac{1}{2}[1-{R}_{n}^{212}-{\displaystyle \sum _{p=1}^{\infty}{T}_{n}^{21}{\left({R}_{n}^{121}\right)}^{p-1}{T}_{n}^{12}}],$$
(6)
$${T}_{n}^{21}=-\frac{{m}_{1}}{{m}_{2}}\text{\hspace{0.17em}}\frac{2i}{{D}_{n}^{1}},{R}_{n}^{212}=\frac{\alpha {\xi}_{n}^{(2)\prime}\left({m}_{2}ka\right){\xi}_{n}^{(2)}\left({m}_{1}ka\right)-\beta {\xi}_{n}^{(2)}\left({m}_{2}ka\right){\xi}_{n}^{(2)\prime}\left({m}_{1}ka\right)}{{D}_{n}^{1}},$$
(7)
$${T}_{n}^{12}=-\frac{2i}{{D}_{n}^{1}},{R}_{n}^{121}=\frac{\alpha {\xi}_{n}^{(1)\prime}\left({m}_{2}ka\right){\xi}_{n}^{(1)}\left({m}_{1}ka\right)-\beta {\xi}_{n}^{(1)}\left({m}_{2}ka\right){\xi}_{n}^{(1)\prime}\left({m}_{1}ka\right)}{{D}_{n}^{1}},$$
(8)
$$\alpha =\{\begin{array}{cc}1& \text{\hspace{0.17em} \hspace{0.17em}TE \hspace{0.17em}wave}\\ \frac{{m}_{1}}{{m}_{2}}& \text{\hspace{0.17em} \hspace{0.17em}TM \hspace{0.17em}wave}\end{array},\text{\hspace{1em} \hspace{0.17em} \hspace{0.17em}}\beta =\{\begin{array}{cc}\frac{{m}_{1}}{{m}_{2}}& \text{\hspace{0.17em} \hspace{0.17em}TE \hspace{0.17em}wave}\\ 1& \text{\hspace{0.17em} \hspace{0.17em}TM \hspace{0.17em}wave}\end{array},$$
(9)
$${D}_{n}^{1}=-\alpha {\xi}_{n}^{(1)\prime}\left({m}_{2}ka\right){\xi}_{n}^{(2)}\left({m}_{1}ka\right)+\beta {\xi}_{n}^{(1)}\left({m}_{2}ka\right){\xi}_{n}^{(2)\prime}\left({m}_{1}ka\right).$$
(10)
$${\xi}_{n}^{(1)}\left(mkr\right)=mkr{h}_{n}^{(1)}\left(mkr\right),$$
(11)
$${\xi}_{n}^{(2)}\left(mkr\right)=mkr{h}_{n}^{(2)}\left(mkr\right)\mathrm{.}$$
(12)
$${Q}_{n}^{1}={R}_{n}^{212}+\frac{{T}_{n}^{21}{T}_{n}^{12}}{1-{R}_{n}^{121}}={R}_{n}^{212}+{\displaystyle \sum _{p=1}^{\infty}\text{\hspace{0.17em}}{T}_{n}^{21}{\left({R}_{n}^{121}\right)}^{p-1}{T}_{n}^{12}},$$
(13)
$$\begin{array}{l}{a}_{n}^{2}\\ {b}_{n}^{2}\end{array}\}=\frac{1}{2}\left(1-{Q}_{n}^{2}\right),$$
(14)
$${Q}_{n}^{2}={R}_{n}^{323}+\frac{{T}_{n}^{32}{Q}_{n}^{1}{T}_{n}^{23}}{1-{R}_{n}^{232}{Q}_{n}^{1}}={R}_{n}^{323}+{\displaystyle \sum _{p=1}^{\infty}\text{\hspace{0.17em}}{T}_{n}^{32}{Q}_{n}^{1}{T}_{n}^{23}{\left({R}_{n}^{232}{Q}_{n}^{1}\right)}^{p-1}}.$$
(15)
$$\begin{array}{l}{a}_{n}^{3}\\ {b}_{n}^{3}\end{array}\}=\frac{1}{2}\left(1-{Q}_{n}^{3}\right),$$
(16)
$${Q}_{n}^{3}={R}_{n}^{434}+\frac{{T}_{n}^{43}{Q}_{n}^{2}{T}_{n}^{34}}{1-{R}_{n}^{343}{Q}_{n}^{2}}={R}_{n}^{434}+{\displaystyle \sum _{p=1}^{\infty}\text{\hspace{0.17em}}{T}_{n}^{43}{Q}_{n}^{2}{T}_{n}^{34}{\left({R}_{n}^{343}{Q}_{n}^{2}\right)}^{p-1}},$$
(17)
$$\begin{array}{l}{a}_{n}^{l}\\ {b}_{n}^{l}\end{array}\}=\frac{1}{2}\left(1-{Q}_{n}^{l}\right),$$
(18)
$${Q}_{n}^{j}={R}_{n}^{j+1,j,j+1}+\frac{{T}_{n}^{j+1,j}{Q}_{n}^{j-1}{T}_{n}^{j,j+1}}{1-{R}_{n}^{j,j-1}{Q}_{n}^{j-1}},$$
(19)
$${Q}_{n}^{j}={R}_{n}^{j+1,j,j+1}+{T}_{n}^{j+1,j}{Q}_{n}^{j-1}{T}_{n}^{j,j+1}\times {\displaystyle \sum _{p=1}^{\infty}{\left({R}_{n}^{j,j+1,j}{Q}_{n}^{j-1}\right)}^{p-1}},$$
(20)
$${A}_{n}^{3}\left(mkr\right)=\frac{{\xi}_{n}^{(1)\prime}\left(mkr\right)}{{\xi}_{n}^{(1)}\left(mkr\right)},$$
(21)
$${A}_{n}^{4}\left(mkr\right)=\frac{{\xi}_{n}^{(2)\prime}\left(mkr\right)}{{\xi}_{n}^{(2)}\left(mkr\right)}\mathrm{.}$$
(22)
$${A}_{n}=1/\left[n/z-{A}_{n-1}\right]-n/z,$$
(23)
$${B}_{n}\left(z\right)=\frac{{\xi}_{n}^{1}\left(z\right)}{{\xi}_{n}^{2}\left(z\right)}=\frac{{\xi}_{n-1}^{1}\left(z\right)}{{\xi}_{n-1}^{2}\left(z\right)}\frac{\left[n/z-{A}_{n-1}^{3}\left(z\right)\right]}{\left[n/z-{A}_{n-1}^{4}\left(z\right)\right]}.$$
(24)
$${T}_{n}^{l}={T}_{n}^{l+1,l}{T}_{n}^{l,l+1}\mathrm{.}$$
(25)
$${Q}_{n}^{j}={R}_{n}^{j+1,j,j+1}+{T}_{n}^{\text{\hspace{0.17em} \hspace{0.17em}}j}{Q}_{n}^{j-1}\text{\hspace{0.17em}}{\displaystyle \sum _{p=1}^{\infty}{\left({R}_{n}^{j,j+1,j}{Q}_{n}^{j-1}\right)}^{p-1}},$$
(26)
$${R}_{n}^{j+1,j,j+1}=\frac{1}{{B}_{n}\left({m}_{j+1}k{r}_{j}\right)}\times \frac{\alpha {A}_{n}^{4}\left({m}_{j+1}k{r}_{j}\right)-\beta {A}_{n}^{4}\left({m}_{j}k{r}_{j}\right)}{-\alpha {A}_{n}^{3}\left({m}_{j+1}k{r}_{j}\right)+\beta {A}_{n}^{4}\left({m}_{j}k{r}_{j}\right)},$$
(27)
$${R}_{n}^{j,j+1,j}={B}_{n}\left({m}_{j}k{r}_{j}\right)\times \frac{\alpha {A}_{n}^{3}\left({m}_{j+1}k{r}_{j}\right)-\beta {A}_{n}^{3}\left({m}_{j}k{r}_{j}\right)}{-\alpha {A}_{n}^{3}\left({m}_{j+1}k{r}_{j}\right)+\beta {A}_{n}^{4}\left({m}_{j}k{r}_{j}\right)},$$
(28)
$${T}_{n}^{\text{\hspace{0.17em} \hspace{0.17em} \hspace{0.17em}}j}=\frac{{m}_{j}}{{m}_{j+1}}\frac{{B}_{n}\left({m}_{j}k{r}_{j}\right)}{{B}_{n}\left({m}_{j+1}k{r}_{j}\right)}\times \frac{{A}_{n}^{4}\left({m}_{j}k{r}_{j}\right)-{A}_{n}^{3}\left({m}_{j}k{r}_{j}\right)}{-\alpha {A}_{n}^{3}\left({m}_{j+1}k{r}_{j}\right)+\beta {A}_{n}^{4}\left({m}_{j}k{r}_{j}\right)}\times \frac{{A}_{n}^{4}\left({m}_{j+1}k{r}_{j}\right)-{A}_{n}^{3}\left({m}_{j+1}k{r}_{j}\right)}{-\alpha {A}_{n}^{3}\left({m}_{j+1}k{r}_{j}\right)+\beta {A}_{n}^{4}\left({m}_{j}k{r}_{j}\right)},$$
(29)
$$\begin{array}{l}{a}_{n}\\ {b}_{n}\end{array}\}={T}_{n}^{32}{R}_{n}^{212}{R}_{n}^{232}{R}_{n}^{212}{T}_{n}^{23}$$
(30)
$$\begin{array}{l}{a}_{n}\\ {b}_{n}\end{array}\}={T}_{n}^{32}{T}_{n}^{21}{R}_{n}^{11}{T}_{n}^{12}{T}_{n}^{23}$$
(31)
$$m\left(\rho \right)={m}_{c}+\left({m}_{s}-{m}_{c}\right)\left({e}^{b\rho}-1\right)/\left({e}^{b}-1\right),$$