# Introduction

This is the note of dipole radiation pattern calculations in a multilayer structure.

The numerical realizations can be found in my Github.

The derivations mainly follow the work of Olivier J. F et .al ^{ref1}

## Extra words

My previous calculations are not helpful since we can not get the emission pattern. In previous cylindrical expressions, the integral in $\mathrm{d}k_{\phi}$ has been transformed into Bessel functions via

If we extract the information of $\mathrm{d}k_{\phi}$, our previous expressions can also be used to get the emission pattern. However, to make this easier to understand, we use plane waves to expand Green’s tensor.

# Green tensor expression

We will use Green tensor to get the dipole’s emission!

I will follow the derivations of Paulus et. al . The derivations will begin from the expression of the Green’s tensor. We still consider only the nonmagnetic material so $\mu=1$ and we use $\mu=\mu_0$ to express the permeability in the vacuum. The equation of the Green’s tensor is as follows

and for a homogeneous media the Green’s tensor can be expressed as

where $R=|\boldsymbol{R}|=|\boldsymbol{r-r_0}|$ is the relative distance and $k_{B}^2=\omega^2\varepsilon \mu$ corresponds to the wave number in the background medium. We can do Fourier transform of Green’s tensor as

and we can express the Green’s tensor as a Fourier transform of each wavevector,

Since we assume that the layers, which will be added later, are perpendicular to the z axis, we first perform the integration over $k_z$ using calculus of residues. Hence, we muse ensure that the integrand vanishes for $k_z\rightarrow\infty$ and rewrite as

where $k_{Bz}=\sqrt{k_B^2-k_x^2-k_y^2}$ is the z component of the wave vector and $\boldsymbol{z}$

Now that we have the plane-wave expansion of the Green’s tensor for an infinite homogeneous background. It is a simple matter to include additional layers. Indeed, the effect of these layers will be to add two plane waves, one propagating upward and one downward, to each Fourier component, The amplitudes of these additional components are determined by the boundary conditions at the different interfaces. Since the Green’s tensor represents the electric field radiated at $r$ by three orthogonal point sources at $r’$, the boundary conditions depend on the polarization of the corresponding Fourier component. It is therefore advantageous to introduce a new orthonormal system

$\boldsymbol{\mathrm{\hat{k}}}(k_{Bz}),\boldsymbol{\mathrm{\hat{I}}}(k_{Bz}),\boldsymbol{\mathrm{\hat{m}}}(k_{Bz})$

Equivalently, another orthogonal system is formed by $\boldsymbol{\mathrm{\hat{k}}}(-k_{Bz}),\boldsymbol{\mathrm{\hat{I}}}(-k_{Bz}),\boldsymbol{\mathrm{\hat{m}}}(-k_{Bz})$. Remark that $\boldsymbol{\mathrm{\hat{I}}}$ is perpendicular to the plane

defined by $\boldsymbol{\mathrm{\hat{k}}}$ and $\boldsymbol{\mathrm{\hat{z}}}$, whereas $\boldsymbol{\mathrm{\hat{m}}}$ lies within this plane. For a given $k_B$, the electric field component parallel to $\boldsymbol{\mathrm{\hat{I}}}$ corresponds

to s polarization and that parallel to $\boldsymbol{\mathrm{\hat{m}}}$ corresponds to p polarization, using the fact that

we can rewrite the Green’s tensor as

To obtain the Green’s tensor $\mathbf{G}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ for a stratified medium, we can now superpose to the free-space Green’s tensor of a homogeneous medium $\varepsilon_l$ the additional terms by formally writing

where $k_{l}^2=\omega^2\varepsilon_l\mu_l$ and $k_{lz}=\sqrt{k_l^2-k_x^2-k_y^2}$. The tensors $\mathbf{R}^{s \uparrow},\mathbf{R}^{s \downarrow},\mathbf{R}^{p \uparrow},\mathbf{R}^{p \downarrow}$ can obviously be interpreted as generalized reflection coefficients that take into account reflections from all existing surfaces. For the explicit calculation of $\mathbf{G}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ it is necessary to consider separately the two cases $z>z_0$ and $z<z_0$. each component of the Green’s tensor is expressed in terms of s- and p-polarized upgoing and down going plane waves with amplitude coefficients $A_{l,\alpha\beta}^{s/p},B_{l,\alpha\beta}^{s/p}$

We need express each component of the Green’s tensor and then we can write the field distribution for an arbitrary orientate dipole. And the coefficients can be obtained from the outer layer to the emitting layer. It’s necessary for us to write the explict form of the Green tensor. We write the Green tensor as

We need write the explicit form of the tensor,

and

Since the definition in the [] let the sign attached to $A_l,B_{l}$, then

Here $\odot$ means the multiply of the corresponding elements in each matrix. We now need calculate the accurate expression of the $A_{\alpha,\beta}^{s/p},A_{\alpha,\beta}^{s/p}$, in the emitting layer $A_{\alpha,\beta}^{s/p},A_{\alpha,\beta}^{s/p}$ should include the dipole’s background part and the reflecting part from other layers. The total reflection coefficients can be calculated iteratively from the outer layer. For the p polarization, for $\beta \neq z$

with $\beta=z$

For the s polarization

In above expressions, $R^{s/p,\uparrow\downarrow}$ are the total reflections in the emitting layer, we define the reflection coefficients in the upper space for each interface as

the same with the lower space

the reflections can be calculated iteratively from the outer layer

where

and

So we can calculate the coefficients in the emitting layer, and the coefficients in other can be calculated via the transmission matrix.

## In the upper space

where

## In the lower space

where

# The field expression and emission pattern

The Green’s tensor can be obtained in previous section, the electric field and magnetic field can be obtained from the Green’s tensor via

And we can write the electric field as

The far field $E_{\infty}$ observed in the direction of the dimensionless unit vector

is determined by the Fourier spectrum $\boldsymbol{E}(k_x,k_y,k_z)$ as $z=0$ as

In the far field , the magnetic field vector is transverse to the electric field vector and the time-averaged Poynting vectors is

calculated as

Since the s and p polarized field are orthogonal, then we can express the far field power as

where the emission pattern is

# Numerical Implementation

Our previous expressions are not suitable for numerical implementation since there are maximum numbers in the exponent. To let is more applicable we rewrite the expressions. We define in the upper space

and in the lower space

If we use these expression, The green tensor should be

where $z_{l}=d_{l+1}$ for $z>z_0$ and $z_{l}=d_{l}$ for $z<z_0$. The coefficients relation between different layers can be rewritten as

## In the upper space

where

## In the lower space

where

# Relations of amplitudes in different layers

In this section , I will complete the derivations of the relations of $G_{\alpha,\beta}$ in different layers. The boundary conditions in each interface can be written as

For convenience we need write the Green’s tensor into a more simple from which can be used to express the boundary conditions. We can express the total green tensor as

where $F_{A/B}$ is the coefficients before A,B. Since only the component $k_{lz},k_{l}$ is related to material properties, we could write $F_{A/B}$ in a more simple form as

where $U_{A/B}$ contains only the coefficients without $k_{lz},\varepsilon$. We write the explicit form for each as follows.

## S polarized light upper/lower layer

## P polarized light upper/lower layer

Then in deriving the relations of coefficients between different layers, we can only use

which is more convenient to write. For convenience, we could also write the Green’s tensor as

so that the boundary conditions can be written as

Then

These equations are all we have. To get the relations, we need decouple the relations for different rows.

### S polarized light

We first use the s polarized light as an example. For S polarized light, we have

Then Eq. $\eqref{Eq:add70}$,$\eqref{Eq:add71}$,$\eqref{Eq:add72}$,$\eqref{Eq:add73}$,$\eqref{Eq:add74}$ would be

Then

So the relation between different layers has been derived and agree well with the reference.

### P polarized light.

For P polarized light, the relation are more complex. We still substitute the detailed expression into Eq. $\eqref{Eq:add70}$,$\eqref{Eq:add71}$,$\eqref{Eq:add72}$,$\eqref{Eq:add73}$,$\eqref{Eq:add74}$. We still have the following relations

Still for the relation

## Appendix

### Dyadic Analysis

In this section a brief review of dyadic analysis is presented. Dyadic operations and theorems provide an effective tool for manipulation of field quantities. Dyadic notation was first introduced by Gibbs in 1884 which later appeared in literature. Consider a vector function $\vec{F}$ having three scalar component $f_{i}$ with ($i = 1,2,3$) in a Cartesian system, that is

Now consider three different vector functions $\vec{F}_{j}$, given by

Which constitute a dyad $\overset\leftrightarrow{F}$ given by

It should be emphasized that

In general a dyad can be formed from the product of two arbitrary vectors $\vec{a}$ and $\vec{b}$ to form $\overset\leftrightarrow{F}=\vec{a}\vec{b}$ . Components of $\overset\leftrightarrow{F}$ can be obtained from a matrix product of a denoted by a 3 × 1 matrix with 1 × 3 matrix. Note that the converse may not be necessarily true. That is, a general dyad may not be expressiable in terms of product of two vector.

## Reference

^{ref1}. Accurate and efficient computation of the Green’s tensor for stratified media ↩