# Coupling of scattered TE mode to TM mode using vector Helmholtz equation

I solve a scattered field computation problem using the frequency domain Maxwell's/Helmholtz PDEs. Particularly, I'm studying the behavior of light, which is essentially a plane wave propagating in a 3D domain, as it interacts with a scatterer characterized by a complex refractive index. This interaction can result in diffraction, scattering, absorption, or reflection of the incident light. In ideal conditions for a TE mode, we consider only the $$E_y$$ and there should not be any electric field component ($$E_x=0$$ and $$E_z=0$$) along the direction of propagation (z-direction). However, in practical situations, there may be imperfections/irregularities/resonances on the scatterer boundaries that can cause some field components to scatter in unintended directions. As a result, I have a leakage of the electric field $$E_y$$ and scattering of it into other modes or directions. So for TE mode, $$E_x$$ and $$E_z$$ are not zero anymore, especially on the material interfaces (see Figure).

Before solving the scattered field computation problem, expanding Maxwell's equations

$$\begin{bmatrix} \frac{\partial}{\partial y} E_z - \frac{\partial}{\partial z} E_y\\ \frac{\partial}{\partial z} E_x - \frac{\partial}{\partial x} E_z\\ \frac{\partial}{\partial x} E_y - \frac{\partial}{\partial y} E_x\\ \end{bmatrix} = j \omega \mu \begin{bmatrix} H_x\\ H_y\\ H_z\\ \end{bmatrix}$$

$$\begin{bmatrix} \frac{\partial}{\partial y} H_z - \frac{\partial}{\partial z} H_y\\ \frac{\partial}{\partial z} H_x - \frac{\partial}{\partial x} H_z\\ \frac{\partial}{\partial x} H_y - \frac{\partial}{\partial y} H_x\\ \end{bmatrix} = -j \omega \epsilon(x,y,z) \begin{bmatrix} E_x\\ E_y\\ E_z\\ \end{bmatrix}$$

I derive the set of equations for E-field:

$$\frac{\partial^2}{\partial y^2} E_x + \frac{\partial^2}{\partial z^2} E_x - \frac{\partial}{\partial y}\frac{\partial}{\partial x} E_y - \frac{\partial}{\partial z}\frac{\partial}{\partial x} E_z + k_{0}^2 \varepsilon E_x= 0$$

$$\frac{\partial^2}{\partial z^2} E_y + \frac{\partial^2}{\partial x^2} E_y - \frac{\partial}{\partial z}\frac{\partial}{\partial y} E_z - \frac{\partial}{\partial x}\frac{\partial}{\partial y} E_x + k_{0}^2 \varepsilon E_y= 0$$

$$\frac{\partial^2}{\partial x^2} E_z + \frac{\partial^2}{\partial y^2} E_z - \frac{\partial}{\partial x}\frac{\partial}{\partial z} E_x - \frac{\partial}{\partial y}\frac{\partial}{\partial z} E_y + k_{0}^2 \varepsilon E_z = 0$$

The residual is the result of the summation of all terms comprising the PDE. For the problem, specified by initial/boundary conditions, the residuals are to be zero at the points where the approximate solution is obtained. By minimizing the residual using solvers, I solve the PDE numerically and I could easily compute the light at different modes: $$E_y$$ at TE, and $$E_x$$, $$E_z$$ at TM. My current objective is to compute how much $$E_y$$ light was rescattered to $$E_x$$, $$E_z$$.

Problem: by setting $$E_x$$, $$E_z$$ initially to zero for TE mode, I exclude these variables from the PDEs and cannot compute them later. To address how much of the scattered TE mode light is converted into TM mode light, I should consider the vector Helmholtz equation and somehow determine the coupling of scattered TE mode to TM mode.

Your assistance in providing a proper set of coupled equations would be greatly appreciated. I hope this clarifies the problem, but if you have any further questions, please don't hesitate to inquire.

• I find your description very confusing. What do your equations have got to with that of Helmholtz, how did you get those cross derivatives? What is a "residual"? Is $\epsilon$ a complex quantity? How large are the "imperfections/irregularities" relative to the wavelength and what is their shape? Can you provide a drawing? Oct 22, 2023 at 22:04
• I apologize for any confusion. I'll do my best to provide a clearer explanation. A1: Helmholtz equation is used to model the wave propagation in the presence of the scattering object. It can be derived from Maxwell's equations if we consider the material to be isotropic and non-dispersive and if we omit the electric and magnetic current densities. The cross derivatives are derived from Maxwell's equation using a simple calculus (please find updated equations). A2: The minimization of residuals (see explanation) is a common approach to solving PDEs numerically. Oct 24, 2023 at 7:19
• A3: Yes, ϵ is a complex quantity("a scatterer characterized by a complex refractive index"). A4: Nevertheless, I hardly believe that I have any imperfections/irregularities on the geometry boundaries since I use perfect rectangles to describe the geometries. But they might be one of the reasons for polarization conversion, and it was my first assumption. From my point of view, the most likely reason is that the waves with polarization oriented parallel ($E_y$ in my case) to the permittivity gradient will act as a forcing term and excite other polarization components. A5: I will add a drawing. Oct 24, 2023 at 7:26

## 1 Answer

This is not an answer to your question, which I do not really understand, but it is too long for a comment.

Permittivity is position dependent: $$\epsilon = \epsilon(x,y,z)$$, and can be complex for lossy medium. You seem to have assumed no free charges $$\nabla \cdot \mathbf D = 0.$$

$$\nabla \times (\nabla \times \mathbf {E}) =\nabla \times (-j\mu \omega \mathbf H)=\omega ^2\mu\epsilon \mathbf E \tag{1}$$ For any $$\mathbf U$$ vector field:$$\nabla \times (\nabla \times \mathbf {U}) = \nabla(\nabla \cdot \mathbf U)-\nabla^2 \mathbf U\tag{2}$$ and from $$0=\nabla \cdot\mathbf D =\nabla (\epsilon \mathbf E) =\mathbf E\cdot \nabla \epsilon +\epsilon \nabla \cdot\mathbf E,$$ and then $$\nabla \cdot \mathbf E = -\mathbf E\cdot \tfrac{1}{\epsilon}\nabla\epsilon ,\tag{3}$$ also $$\nabla(\nabla \cdot \mathbf E) = - \nabla(\mathbf E\cdot \tfrac{1}{\epsilon}\nabla \epsilon)\tag{4}.$$ You get $$\nabla^2 \mathbf E+\nabla(\mathbf E\cdot \tfrac{1}{\epsilon} \nabla \epsilon)+\omega ^2\mu\epsilon \mathbf E=0 \tag{5}.$$ It seems to me that the 2nd term in Eq (5) is missing from your setup and instead you have cross derivatives. Note too that if your medium is piecewise continuous then in every "piece" $$\nabla \epsilon = 0$$ and (5) simplifies to the standard Helmholtz equation $$\nabla^2 \mathbf E+\omega ^2\mu\epsilon \mathbf E=0$$ as it should. This is easier to solve by matching boundary conditions directly at the interface between two such homogeneous pieces.

• Thank you for the helpful answer. To provide additional context to my initial question, I start with TE polarization and I aim to compute the amount of light scattered into other modes. For this calculation, I could use the Eq. (5) you provided. I need to solve this PDE for three variables, corresponding to three E-field components. However, I should take into account that there is only a scattered field and no incident field for $E_x$ and $E_z$ for TE: $$\vec{E} = \begin{bmatrix} \vec{i}E_x^{scat}\\ \vec{j}E_y^{tot}\\ \vec{k}E_z^{scat}\\ \end{bmatrix}$$ Please correct me if I am wrong. Oct 26, 2023 at 13:34
• As I mentioned above, I cannot correct you because I do not really understand what your question is, but I caution you that you cannot treat the scattered field as one that only goes "backwards" unless the object is convex seen from the direction of illumination and the medium is homogeneous. Think of, say, three parallel homogeneous layers having two refractive surfaces illuminated with a plane wave. The middle layer will have a standing wave going back and forth between the two surfaces. If the permittivity is continuously varying then you will have all kinds waves going in many directions. Oct 26, 2023 at 14:17