From f1d35fbf000a4608950352ee27bf11dfe3c0e7bf Mon Sep 17 00:00:00 2001 From: DorisReiter <132465655+DorisReiter@users.noreply.github.com> Date: Wed, 3 Apr 2024 16:21:50 +0200 Subject: [PATCH 1/4] Update maxwell.md New description of 2D calculations --- docs/math/maxwell.md | 132 +++++++++++++++++++------------------------ 1 file changed, 57 insertions(+), 75 deletions(-) diff --git a/docs/math/maxwell.md b/docs/math/maxwell.md index 12b22897..55c43c33 100644 --- a/docs/math/maxwell.md +++ b/docs/math/maxwell.md @@ -173,140 +173,122 @@ In these equations we have used $\mathbf{D}=\varepsilon_0\varepsilon\mathbf{E}$ These equations sometimes referd to as boundary conditions. It should be noted, that here boundary refers to the behaviour of the fields at the interface between two materials in contrast to boundary conditions at the edge of a simulation box.While the calculations can be done with keeping $\varepsilon(\mathbf{r})$, the boundary conditions can give a useful sanity check. They also indicate, that at interface a fine grid is required, while at areas of homogeneous materials larger grid can be chosen. -## Eigenvectors propagating in $x_3$-direction +# Reduction two 2-Dimensions -Assuming no sources and currents present, Maxwell's simplifies to +Often structures are constructed in such a way, that they are strongly patterned in a plane (e.g. the $xy$-plane) and are uniform in the direction perpendicular to the plane. This already holds for the simple +example of a waveguide, which has interface within the plane, but is (almost) infinitely extended in the perpendicular direction. -$$ \begin{aligned} &\nabla\cdot \left(\varepsilon\vec{\mathcal{E}}\right) = 0 \\ - & \nabla\cdot \left(\mu\vec{\mathcal{H}}\right) = 0 \\ - & \nabla\times\vec{\mathcal{E}} = - \mu \frac{\partial \vec{\mathcal{H}}}{\partial t} \\ - & \nabla\times\vec{\mathcal{H}} = \varepsilon \frac{\partial \vec{\mathcal{E}}}{\partial t} \end{aligned} -$$(maxwell_no_sources) +For simulations, just solving the equation within the plane can reduce the problem greatly, also from the computational point of view. The equation also reduce to a much simpler version, which we derive here. -By combining the latter two equations of {eq}`maxwell_no_sources` -we get for the $\mathcal{E}$ +We will discuss only the electric field here. Equations for the magnetic field can be derived in a similar way. We will make a plane wave ansatz assuming a monochromatic wave with frequency $\omega$ via -$$ - & - \nabla\cdot \left(\varepsilon\vec{\mathcal{E}}\right) - = - 0 - - & - \nabla\times \left( \frac{1}{\mu}\nabla\times\vec{\mathcal{E}} \right) - = - \varepsilon \frac{\partial^2 \vec{\mathcal{E}}}{\partial t^2} -$$ (maxwell_telegraph) +$$\mathbf{E}(\mathbf{r},t) = \mathbf{E}(\mathbf{r}) e^{-i(\omega t- \mathbf{k} \cdot \mathbf{r})}$$ -If we restrict the problem to a 2D-plane $\Omega \in \mathbb{R}^2$ like done in -{cite}`Vardapetyan2003,Vardapetyan2002,Vardapetyan2002_2`, -i.e. a plane with $\vec{x}=(x_1,x_2)$ and -assuming propagation only in $x_3$-direction with a propagation constant $\beta$, -the equations simplify for the harmonic case with a frequency of $\omega$ to: +Now we assume that the wave propagates along the $z$-direction. The electric field is only structured in the $x,y$-plane and thus depends only on $\mathbf{r}_{\perp }=(x,y) $. Along the propagation direction we assume it to be homogeneous. Then the fields are $$ - \mathcal{E}(\vec{x},x_3,t) - = - (\vec{E}(\vec{x}),E_3(\vec{x}))\mathrm{e}^{i(\beta x_3 - \omega t)} + mathbf{E}(\mathbf{r}) \to \mathbf{E} ( \mathbf{r}_{ \perp }) = \left(\mathbf{E}_{ \perp }(\mathbf{r}_{ \perp } ),E_z( \mathbf{r}_{\perp})\right) g +$$ - \mathcal{H}(\vec{x},x_3,t) - = - (\vec{H}(\vec{x}),H_3(\vec{x}))\mathrm{e}^{i(\beta x_3 - \omega t)} $$ + \mathbf{k} \to k_z +$$ -Using these, the curl can be written as +We further separate the amplitude into the amplitude within the $xy$-plane $\mathbf{E}_{ \perp }=(E_x,E_y,0)$ and along the propagation direction $E_z$. Note that $\vE$ is a 2D vector in the xy plane with , while $E_z$ is a scalar. Both amplitudes depend only on $\mathbf{r}_{ \perp }$, while in the $z$- direction we assume a homogeneous system described by a plane wave +ansatz. $$ - \nabla \times - = - \begin{pmatrix} - 0 & -i \beta & \partial_y \\ - i \beta & 0 & -\partial_x \\ - -\partial_y & \partial_x & 0 - \end{pmatrix} +\mathbf{E}(\mathbf{r},t) \to \left(\mathbf{E}_{ \perp }(\mathbf{r}_{ \perp }),E_z(\mathbf{r}_{ \perp })\right) e^{-i(\omega t- k_z z)} $$ -and the derivative with respect to time becomes +With these assumption we can reduce the Maxwell and wave equation respectively. For the derivations in these equations appearing we have + +$$ \begin{aligned} +\partial_z \mathbf{E}(\mathbf{r},t) = i k_z \mathbf{E}(\mathbf{r},t) \\ + \partial_t \mathbf{E}(\mathbf{r},t) = -i \omega \mathbf{E}(\mathbf{r},t) +\end{aligned}$$ + +where $\partial_i = \partial/\partial i$ is the partial derivative after the quantity $i$. + +Now we use these assumptions to rewrite the wave equation $$ - \frac{\partial}{\partial t} - = - i \omega +\mathbf{\nabla}\times \left( \frac{1}{\mu(\mathbf{r})} \mathbf{\nabla} \times \mathbf{E}(\mathbf{r},t)\right) - \partial_t^2 \mathbf{E}(\mathbf{r},t) = 0 $$ -This leads to the equations +We remove the time dependence of the fields, such that only the field +amplitudes enter and obtain two equations +$$\begin{aligned} + & \mathbf{\nabla} \times \frac{1}{\mu(\mathbf{r}_{ \perp } ) } \mathbf{\nabla} \times \mathbf{E}_{ \perp }(\mathbf{r}_{ \perp } ) + - \left[ \omega^2 \varepsilon(\mathbf{r}_{ \perp }) +\frac{k_z^2}{\varepsilon(\mathbf{r}_{ \perp })} \right] + \mathbf{E}_{\perp}(\mathbf{r}_{ \perp }) + i \frac{k_z}{\mu(\mathbf{r}_{ \perp })} \grad E_z(\mathbf{r}_{ \perp })= 0 {#eq:2D_wave1 label="eq:2D_wave1"}}\\ + \div \left( \varepsilon(\mathbf{r}_{\perp}) \grad E_z (\mathbf{r}_{\perp}) \right) + \omega^2 \varepsilon(\mathbf{r}_{\perp}) E_z (\mathbf{r}_{\perp}) - i k_z \div \left( \frac{1}{\mu(\mathbf{r}_{\perp})} \mathbf{E}_{\perp}(\mathbf{r}_{\perp}) \right) = 0 {#eq:2D_wave2 label="eq:2D_wave2"} + \end{aligned} $$ - & - \nabla \times \left(\frac{1}{\mu} \nabla \times \vec{E}\right) - - \omega^2 \epsilon \vec{E} - + \frac{\beta^2}{\mu}\vec{E} - + \frac{i \beta}{\mu} \nabla E_3 - = 0 - & - \nabla \cdot \left(\frac{1}{\mu} \nabla E_3\right) - + \omega^2 \epsilon E_3 - - i \beta \nabla \cdot \left( \frac{1}{\mu} \vec{E} \right) - = 0 +Note that the top equation is a 2D equation, in the sense that it acts only in the $(x,y)$ plane, while the bottom equation is just a scalar equation. - & - \nabla \cdot \left( \epsilon \vec{E} \right) - + i \beta \epsilon E_3 - = 0 +We can proceed analogously for the Maxwell's equation + +$$ + \mathbf{\nabla} \left( \varepsilon(\mathbf{r})\mathbf{E}(\mathbf{r},t) \right) =0 $$ -and the boundary conditions at $\partial\Omega$, -where $\vec{n}$ is the unit vector orthogonal to the boundary: +and remove the time dependence from the equations simplifying it to $$ - &\vec{E} \times \vec{n} = 0 - - &E_3 = 0 + \mathbf{\nabla} \left( \varepsilon(\mathbf{r}_{ \perp }) \mathbf{E}_{\perp}(\mathbf{r}_{ \perp })\right) + + i k_z \varepsilon(\mathbf{r}_{ \perp }) E_z(\mathbf{r}_{ \perp })=0 $$ -Defining +Again this is a scalar equation. Now we have four equations for three free parameters $E_x,E_y,E_Z$, hence our system is overestimated. Hence, we can skip one equation, which we choose to be +Eq. [\[eq:2D_wave2\]](#eq:2D_wave2){reference-type="ref" reference="eq:2D_wave2"}. + +We use the definition $$ - E_3^{\text{new}} = i \beta E_3 + E_e^{\text{new}} = i \beta E_z $$ -converts the problem to a eigenvalue problem with the eigenvalue $\beta^2$ +to rewrite -$$ +$$\begin{aligned} & \nabla \times \left(\frac{1}{\mu} \nabla \times \vec{E}\right) - \omega^2 \epsilon \vec{E} + \frac{\beta^2}{\mu}\vec{E} + \frac{1}{\mu} \nabla E_3^{\text{new}} - = 0 - + = 0\\ & \nabla \cdot \left(\frac{1}{\mu} \nabla E_3^{\text{new}}\right) + \omega^2 \epsilon E_3^{\text{new}} + \beta^2 \nabla \cdot \left( \frac{1}{\mu} \vec{E} \right) - = 0 - + = 0\\ & \nabla \cdot \left( \epsilon \vec{E} \right) + \epsilon E_3^{\text{new}} - = 0 + = 0 \end{aligned} $$ +and convert the problem to an eigenvalue problem with the eigenvalue $\beta^2$ + Variational problem: -$$ +$$\begin{aligned} & \left( \frac{1}{\mu} \nabla \times \vec{E}, \nabla \times \vec{F} \right) - \omega^2 \left( \epsilon \vec{E}, \vec{F} \right) + \left( \frac{1}{\mu} \nabla E_3^{\text{new}}, \vec{F} \right) = - - \beta^2 \left( \frac{1}{\mu} \vec{E}, \vec{F} \right) - + - \beta^2 \left( \frac{1}{\mu} \vec{E}, \vec{F} \right)\\ & \left( \epsilon \vec{E}, \nabla q \right) - \left( \epsilon E_3^{\text{new}}, q \right) = - 0 + 0 \end{aligned} +$$ $$ ## PML From 9d4232e5f30c064281f8326be3a97ec606d20ab5 Mon Sep 17 00:00:00 2001 From: Doris Reiter Date: Wed, 24 Apr 2024 17:36:11 +0200 Subject: [PATCH 2/4] Added documentary for 2D calculations --- docs/math/maxwell.md | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/docs/math/maxwell.md b/docs/math/maxwell.md index 55c43c33..cbf695bf 100644 --- a/docs/math/maxwell.md +++ b/docs/math/maxwell.md @@ -173,7 +173,7 @@ In these equations we have used $\mathbf{D}=\varepsilon_0\varepsilon\mathbf{E}$ These equations sometimes referd to as boundary conditions. It should be noted, that here boundary refers to the behaviour of the fields at the interface between two materials in contrast to boundary conditions at the edge of a simulation box.While the calculations can be done with keeping $\varepsilon(\mathbf{r})$, the boundary conditions can give a useful sanity check. They also indicate, that at interface a fine grid is required, while at areas of homogeneous materials larger grid can be chosen. -# Reduction two 2-Dimensions +# Reduction to two dimensions Often structures are constructed in such a way, that they are strongly patterned in a plane (e.g. the $xy$-plane) and are uniform in the direction perpendicular to the plane. This already holds for the simple example of a waveguide, which has interface within the plane, but is (almost) infinitely extended in the perpendicular direction. @@ -187,14 +187,14 @@ $$\mathbf{E}(\mathbf{r},t) = \mathbf{E}(\mathbf{r}) e^{-i(\omega t- \mathbf{k} \ Now we assume that the wave propagates along the $z$-direction. The electric field is only structured in the $x,y$-plane and thus depends only on $\mathbf{r}_{\perp }=(x,y) $. Along the propagation direction we assume it to be homogeneous. Then the fields are $$ - mathbf{E}(\mathbf{r}) \to \mathbf{E} ( \mathbf{r}_{ \perp }) = \left(\mathbf{E}_{ \perp }(\mathbf{r}_{ \perp } ),E_z( \mathbf{r}_{\perp})\right) g + \mathbf{E}(\mathbf{r}) \to \mathbf{E} ( \mathbf{r}_{ \perp }) = \left(\mathbf{E}_{ \perp }(\mathbf{r}_{ \perp } ),E_z( \mathbf{r}_{\perp})\right) g $$ $$ \mathbf{k} \to k_z $$ -We further separate the amplitude into the amplitude within the $xy$-plane $\mathbf{E}_{ \perp }=(E_x,E_y,0)$ and along the propagation direction $E_z$. Note that $\vE$ is a 2D vector in the xy plane with , while $E_z$ is a scalar. Both amplitudes depend only on $\mathbf{r}_{ \perp }$, while in the $z$- direction we assume a homogeneous system described by a plane wave +We further separate the amplitude into the amplitude within the $xy$-plane $\mathbf{E}_{ \perp }=(E_x,E_y,0)$ and along the propagation direction $E_z$. Note that $\vec{E}$ is a 2D vector in the xy plane with , while $E_z$ is a scalar. Both amplitudes depend only on $\mathbf{r}_{ \perp }$, while in the $z$- direction we assume a homogeneous system described by a plane wave ansatz. $$ @@ -222,8 +222,8 @@ amplitudes enter and obtain two equations $$\begin{aligned} & \mathbf{\nabla} \times \frac{1}{\mu(\mathbf{r}_{ \perp } ) } \mathbf{\nabla} \times \mathbf{E}_{ \perp }(\mathbf{r}_{ \perp } ) - \left[ \omega^2 \varepsilon(\mathbf{r}_{ \perp }) +\frac{k_z^2}{\varepsilon(\mathbf{r}_{ \perp })} \right] - \mathbf{E}_{\perp}(\mathbf{r}_{ \perp }) + i \frac{k_z}{\mu(\mathbf{r}_{ \perp })} \grad E_z(\mathbf{r}_{ \perp })= 0 {#eq:2D_wave1 label="eq:2D_wave1"}}\\ - \div \left( \varepsilon(\mathbf{r}_{\perp}) \grad E_z (\mathbf{r}_{\perp}) \right) + \omega^2 \varepsilon(\mathbf{r}_{\perp}) E_z (\mathbf{r}_{\perp}) - i k_z \div \left( \frac{1}{\mu(\mathbf{r}_{\perp})} \mathbf{E}_{\perp}(\mathbf{r}_{\perp}) \right) = 0 {#eq:2D_wave2 label="eq:2D_wave2"} + \mathbf{E}_{\perp}(\mathbf{r}_{ \perp }) + i \frac{k_z}{\mu(\mathbf{r}_{ \perp })} \nabla E_z(\mathbf{r}_{ \perp })= 0 \\ + &\nabla \cdot \left( \varepsilon(\mathbf{r}_{\perp}) \nabla \cdot E_z (\mathbf{r}_{\perp}) \right) + \omega^2 \varepsilon(\mathbf{r}_{\perp}) E_z (\mathbf{r}_{\perp}) - i k_z \nabla \cdot \left( \frac{1}{\mu(\mathbf{r}_{\perp})} \mathbf{E}_{\perp}(\mathbf{r}_{\perp}) \right) = 0 \end{aligned} $$ @@ -242,10 +242,9 @@ $$ + i k_z \varepsilon(\mathbf{r}_{ \perp }) E_z(\mathbf{r}_{ \perp })=0 $$ -Again this is a scalar equation. Now we have four equations for three free parameters $E_x,E_y,E_Z$, hence our system is overestimated. Hence, we can skip one equation, which we choose to be -Eq. [\[eq:2D_wave2\]](#eq:2D_wave2){reference-type="ref" reference="eq:2D_wave2"}. +Again this is a scalar equation. Now we have four equations for three free parameters $E_x,E_y,E_Z$, hence our system is overestimated. Hence, we can skip one equation, as discussed later -We use the definition +We $$ E_e^{\text{new}} = i \beta E_z @@ -289,7 +288,7 @@ $$\begin{aligned} = 0 \end{aligned} $$ -$$ + ## PML @@ -299,6 +298,7 @@ $$ The mode profiles of bent waveguides can be calculated using the previously derived math with an transformed effective refractive index defined as + {cite}`AzizurRahman2013,shyroki2006exact,Jedidi2007,Xiao2012,Dehghannasiri2017` $$ From 298c90705ee4c398088e26a0d3ea2ae238599a85 Mon Sep 17 00:00:00 2001 From: Doris Reiter Date: Wed, 24 Apr 2024 17:46:52 +0200 Subject: [PATCH 3/4] sentence corrected --- docs/math/maxwell.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/math/maxwell.md b/docs/math/maxwell.md index cbf695bf..660c5489 100644 --- a/docs/math/maxwell.md +++ b/docs/math/maxwell.md @@ -244,13 +244,13 @@ $$ Again this is a scalar equation. Now we have four equations for three free parameters $E_x,E_y,E_Z$, hence our system is overestimated. Hence, we can skip one equation, as discussed later -We +We use the definiton $$ E_e^{\text{new}} = i \beta E_z $$ -to rewrite +to rewrite the equations to $$\begin{aligned} & From e1ec263e0f5a7676a39ead7c1dbafe0c25df7c47 Mon Sep 17 00:00:00 2001 From: Doris Reiter Date: Wed, 24 Apr 2024 18:17:31 +0200 Subject: [PATCH 4/4] e ->3 --- docs/math/maxwell.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/math/maxwell.md b/docs/math/maxwell.md index 660c5489..0b7013c2 100644 --- a/docs/math/maxwell.md +++ b/docs/math/maxwell.md @@ -247,7 +247,7 @@ Again this is a scalar equation. Now we have four equations for three free param We use the definiton $$ - E_e^{\text{new}} = i \beta E_z + E_3^{\text{new}} = i \beta E_z $$ to rewrite the equations to