Theoretical background

Lorenz-Mie Theory (LMT)

Note

The Lorenz-Mie Theory or (LMT for short) is a framework that can be used to find exact solution of the scattered field considering a plane wave incident to a scatterer with a certain geometry. The solution is usually written in the form of an infinite summation which, of course, has to be truncated. PyMieSim is a library which does solve the equations in order to retrieve plenty of important informations. It is to be noted that in all the library, the angles \(\theta\) and \(\phi\) are defined as in a spherical coordinate system as shown in the following figure.

Note

Here are few of the most important relations governing the PyMieSim library.

\[ \begin{align}\begin{aligned}&S_1=\sum\limits_{n=1}^{n_{max}}\frac{2n+1}{n(n+1)}(a_n\pi_n+b_n\tau_n)\\&.\\&S_2=\sum\limits_{n=1}^{n_{max}}\frac{2n+1}{n(n+1)}(a_n\tau_n+b_n\pi_n)\\.&\\&\text{Fields} = E_{\theta}(\phi,\theta) \vec{\theta} + E_{\phi}(\phi,\theta) \vec{\phi}\\.&\\&\text{SPF} = \sqrt{ E_{\parallel}(\phi,\theta)^2 + E_{\perp}(\phi,\theta)^2 }\end{aligned}\end{align} \]

Stokes parameters:

\[ \begin{align}\begin{aligned}&I = \big| E_x \big|^2 + \big| E_y \big|^2\\.&\\&Q = \big| E_x \big|^2 - \big| E_y \big|^2\\.&\\&U = 2 \mathcal{Re} \big\{ E_x E_y^* \big\}\\.&\\&V = 2 \mathcal{Im} \big\{ E_x E_y^* \big\}\end{aligned}\end{align} \]

Scattering properties

Note

There are many properties of the scatterer that might be useful to know such as: - scattering efficiency - extinction efficiency - absorption efficiency - back-scattering efficiency - ratio of front and back scattering - optical pressure efficiency - anisotropy factor g

Those parameters can be computed using PyMieSim according to those equations.

\[ \begin{align}\begin{aligned}&Q_{sca} = \frac{2}{x^2}\sum_{n=1}^{n_{max}}(2n+1)(|a_n|^2+|b_n|^2)\\&Q_{ext} = \frac{2}{x^2} \sum_{n=1}^{n_{max}} (2n+1) \mathcal Re \{ a_n+b_n \}\\&Q_{abs} = Q_{ext}-Q_{sca}\\&Q_{back} = \frac{1}{x^2} \Big| \sum\limits_{n=1}^{n_{max}} (2n+1)(-1)^n (a_n - b_n) \Big|^2\\&Q_{ratio} = \frac{Q_{back}}{Q_{sca}}\\&Q_{pr} = Q_{ext} - g * Q_{sca}\\&g = \frac{4}{Q_{sca} x^2} \Big[ \sum\limits_{n=1}^{n_{max}} \frac{n(n+2)}{n+1} \text{Re} \left\{ a_n a_{n+1}^* + b_n b_{n+1}^*\right\} + \sum\limits_{n=1}^{n_{max}} \frac{2n+1}{n(n+1)} \text{Re}\left\{ a_n b_n^* \right\} \Big]\\&A_s = \pi r^2\\&\sigma_{i} = Q_{i} A\\&\mu_{sca} = \sigma_{sca} C\\&\mu_{ext} = \sigma_{ext} C\\&\mu_{abs} = \sigma_{abs} C\end{aligned}\end{align} \]

With:

C: the scatterer concentration in the sample.

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An and Bn coefficients:

From the An and Bn coefficients, we can retrieve many useful properties of the scatterer and scattered far-fields. Those are complementary to the Cn and Dn coefficient (for near-field properties) which we do no compute with PyMieSim at the moment. Depending on the scatterer geometry, all those coefficient may vary. Here we have three example which are available with the PyMieSim library.

Note

Sphere

\[a_n = \frac{ \mu_{sp} \Psi_n(x) \Psi_n^\prime(M x) - \mu M \Psi_n^\prime(x) \Psi_n(M x)} {\mu_{sp} \xi_n(x) \Psi_n^\prime(M x)- \mu M \xi_n^\prime (x) \Psi_n(M x)}\]

\[b_n = \frac{ \mu M \Psi_n(x) \Psi_n^\prime(M x) - \mu_{sp} \Psi_n^\prime(x) \Psi_n(M x)} {\mu M \xi_n(x) \Psi_n^\prime(M x) - \mu_{sp} \xi_n^\prime (x) \Psi_n(M x)}\]

With:

\(\psi_n = x \psi^{(1)}_n (x) = \sqrt{x \pi/2} J_{n+1/2} (x)\).

\(M = k_{sp}/k\) is the relative complex refractive index.

\(x = \pi d / \lambda\).

\(\lambda\) is the wavelength in the surrounding medium.

References [1] Eq(III.88-91).

important: It is to be noted that PyMieSim assume \(\mu_{sp} = \mu\) at the moment. It might change in a future update.

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Note

Cylinder

\[a_n = \frac{ M J_n(M x) J_n^\prime (m x) - m J_n^\prime (M x) J_n(m x) } { m_t J_n(M x) H_n^\prime (m x) - m J_n^\prime (M x) H_n(m x) }\]

\[b_n = \frac{ m J_n(m_t x) J_n^\prime (m x) - m_t J_n^\prime (m_t x) J_n(m x) } { m J_n(m_t x) H_n^\prime (m x) - m_t J_n^\prime (m_t x) H_n(m x) }\]

With:

\(M\) is the refractive index of the scatterer.

\(m\) is the refractive index of the medium.

\(H_n\) is the Hankel function of first kind of order n.

References [5] Eq(8.30-32).

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Note

Core/Shell sphere

\[a_n = \frac{ \psi_n \left[ \psi_n' (m_2 y) - A_n \chi_n' (m_s) \right] - m_2 \psi_n'(y) \left[ \psi_n (m_2 y) - A_n \chi_n (m_2 y) \right]} {\xi_n (y) \left[ \psi_n' (m_2 y) -A_n \chi_n' (m_2 y) \right] - m_2 \xi_n' (y) \left[ \psi_n(m_2 y) - A_n \chi_n (m_2 y) \right]}\]

\[ \begin{align}\begin{aligned}b_n = \frac{ m_2 \psi_n(y) \left[ \psi_n' (m_2 y) - B_n \chi_n' (m_2 y) \right] - \psi_n' (y) \left[ \psi_n (m_2 y) - B_n \chi_n (m_2 y) \right]} {m_2 \xi_n(y) \left[ \psi_n' (m_2 y) - B_n \xi_n' (m_2 y) \right] - \xi_n' \left[ \psi_n (m_2 y) -A_n \chi_n (m_2 y) \right]\\ }\end{aligned}\end{align} \]

With:

\[A_n = \frac{ m_2 \psi_n (m_2 x) \psi_n' (m_1 x) - m1 \psi_n'(m_2 x) \psi_n(m_1x)} {m_2 \xi_n (m_2x) \psi_n' (m_1 x) - m_1 \xi_n' (m_2 x) \psi_n (m_1 x)}\]

\[B_n = \frac{m_2 \psi_n (m_1 x) \psi_n' (m_2 x) - m_1 \psi_n (m_2 x) \psi_n' (m_1 x)} {m_2 \chi_n' (m_2 x) \psi_n (m_1 x) - m_1 \psi_n' (m_1 x) \chi_n (m_2 x)}\]

and:

\[x = \frac{2 \pi R_{core}}{\lambda}, \: y = \frac{2 \pi R_{shell}}{\lambda}, \: m_1 = \frac{n_{core}}{n_{medium}}, \: m_2 = \frac{n_{shell}}{n_{medium}}.\]

\[\chi_n (x) = -x\sqrt{\frac{\pi}{2x}} N_{n+1/2} (x) \leftarrow \text{Neumann}\]

References [8] Eq(4-5).

Generalized Lorenz-Mie Theory (GLMT)

Note

Coming soon

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Coupling mechanism

Note

There are two main coupling mechanisms, coherent coupling and non-coherent coupling. For instance, photodiode collect light via a non-coherent mechanism. On the other part, fiber optic CoherentMode mode collects light in a coherent way and as such they usually collect a lot less light but they add additional information on the sample studied.

Mathematically they are defined as follows:

\[ \begin{align}\begin{aligned}C_{coh.} &= \Big| \iint_{\Omega} \Phi_{det} \, . \, \Psi_{scat}^* \, d \Omega \Big|^2\\C_{Noncoh.} &= \iint_{\Omega} \Big| \Phi_{det} \Big|^2 \,.\, \Big| \Psi_{scat} \Big|^2 \, d \Omega\end{aligned}\end{align} \]

It is to be noted that the coherent coupling definition is derived from the coupled mode theory which remains true as long as the parallax approximation is also true. Furthermore, this coupling is what we would call centered coupling. It means that the scatterer is perfectly centered with the detector. Even though it doesn’t affect so much the non-coherent coupling coupling, it can largely affect coherent coupling.

To take into account the effect of transversal offset of the scatterer, we define the footprint of the scatterer.

\[\eta_{l,m}(\delta_x, \delta_y) = \Big| \mathcal{F}^{-1} \big\{ \Phi_{det} \, . \, \Psi_{scat} \big\} \Big|^2\]

Thus, we can compute the mean coupling as the mean value of \(\eta_{l,m}\)

\[\widetilde{\eta}_{l,m} = \big< \eta_{l,m}(\delta_x, \delta_y) \big>\]