# Dipole's Emission In Multi-Layered Structure

# 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

If we extract the information of

# 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

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

where $R=|\boldsymbol{R}|=|\boldsymbol{r-r*0}|k*{B}^2=\omega^2\varepsilon \mu$ 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

where

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

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

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})

we can rewrite the Green’s tensor as

To obtain the Green’s tensor

where $k*{l}^2=\omega^2\varepsilon_l\mu_lk*{lz}=\sqrt{k

*l^2-k_x^2-k_y^2}* \mathbf{R}^{s \uparrow},\mathbf{R}^{s \downarrow},\mathbf{R}^{p \uparrow},\mathbf{R}^{p \downarrow} \mathbf{G}\left(\mathbf{r}, \mathbf{r}^{\prime}\right) z>z_0 z<z_0 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 *{\alpha,\beta}^{s/p},A*{\alpha,\beta}^{s/p}*{\alpha,\beta}^{s/p},A*{\alpha,\beta}^{s/p}

with

For the s polarization

In above expressions,

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

is determined by the Fourier spectrum

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}*0z*{l}=d_{l}

## 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

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}k*{lz},k

*{l}* F{A/B}$ in a more simple form as

where $U*{A/B}k*{lz},\varepsilon$. 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.

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.

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

Now consider three different vector functions

Which constitute a dyad

It should be emphasized that

In general a dyad can be formed from the product of two arbitrary vectors

## Reference

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