# A Brief Review of Mueller Matrix Calculations Associated with Oceanic Particles
## 2 Fundamental Concepts for Mueller Matrix Calculations  
### [PERSON]'s Equations and the Volume/Surface-Integral Equations  
Since all of the rigorous light-scattering computational methods should obey [PERSON]'s equations, we will first give a brief introduction to the role that both the volume/surface methods for mapping the near field to far field play in the final solutions. We will only consider time-harmonic electromagnetic waves and dielectric particles. The dielectric particles are assumed to be isotropic and have a linear response to an applied field. In this case, [PERSON]'s equations in the medium while using SI units are as follows:  
\[\
abla\cdot\dot{\mathbf{E}}\left(\stackrel{{\rightarrow}}{{r}} \right)=0,\,\
abla\times\mathbf{E}(\stackrel{{\rightarrow}}{{r}} )=-\mu\frac{\partial\mathbf{H}(\stackrel{{\rightarrow}}{{r}})}{ \partial t}, \tag{1}\]  
\[\
abla\cdot\dot{\mathbf{H}}\left(\stackrel{{\rightarrow}}{{r}} \right)=0,\,\
abla\times\mathbf{H}(\stackrel{{\rightarrow}}{{r}} )=\varepsilon\frac{\partial\mathbf{E}(\stackrel{{\rightarrow}}{{r}} )}{\partial t}, \tag{2}\]  
where \(\mathbf{E}\) and \(\mathbf{H}\) are the electric and the magnetic fields, respectively; \(\varepsilon\) and \(\mu\) are the permittivity and permeability of the medium. Using the Fourier transformation, an arbitrary incident field in the time-domain can be transformed into the summation of the fields in the frequency-domain, or the time-harmonic fields. Assuming that the time-harmonic field follows \(\exp(-i\omega t)\), where \(\omega\) is the angular frequency of the electromagnetic wave, [PERSON]'s equations in a time-independent form become:  
\[\
abla\cdot\dot{\mathbf{E}}\left(\stackrel{{\rightarrow}}{{r}} \right)=0,\,\
abla\times\mathbf{E}(\stackrel{{\rightarrow}}{{r}} )=i\omega\mu\mathbf{H}(\stackrel{{\rightarrow}}{{r}}), \tag{3}\]  
\[\
abla\cdot\dot{\mathbf{H}}\left(\stackrel{{\rightarrow}}{{r}} \right)=0,\,\
abla\times\mathbf{H}(\stackrel{{\rightarrow}}{{r}} )=-i\omega\varepsilon\mathbf{E}(\stackrel{{\rightarrow}}{{r}}). \tag{4}\]  
Using Equations (3) and (4), the vector Helmholtz equations for the electric and magnetic fields are  
\[\left(\
abla^{2}+k^{2}\,\right)\mathbf{E}(\stackrel{{ \rightarrow}}{{r}})=0, \tag{5}\]  
\[\left(\
abla^{2}+k^{2}\,\right)\mathbf{H}(\stackrel{{ \rightarrow}}{{r}})=0, \tag{6}\]  
where \(k\) is the wave number and \(k^{2}=\omega^{2}\mu\varepsilon\). For oceanic particles, the surrounding medium and the scattering particles are assumed to be nonmagnetic, thus \(\mu=\mu_{0}\), where \(\mu_{0}\) is the vacuum permeability. The light speed \(\mathrm{c}\) in vacuum is equal to \(1/\sqrt{\mu_{0}\varepsilon_{0}}\), where \(\varepsilon_{0}\) is the permittivity in vacuum. Consequently, the refractive index \(\mathrm{m}\) of the medium is \(m=c/v=\sqrt{\varepsilon/\varepsilon_{0}}\), where \(v\) is the light speed in the medium. Since the electric and the magnetic fields are dependent on each other, we will use the electric field to describe the electromagnetic field. The volume integral and surface integral equations of the electric field can be deduced from [PERSON]'s equations and the vector Green function [15].