FourierTransform¶

Type:	section
Appearance:	simple
Excludes:	CartesianToFourierTransform, ConvertToPowerFlux, CreateReducedBasis, DensityIntegration, DipoleEmission, EvaluateReducedBasis, ExportFields, ExportGrid, FarField, FluxIntegration, GridStatistics, IntegrationWeights, LayeredMediaSolver, ModeOverlap, MultipoleExpansion, OpticalImaging, PupilField, Radiation, ResonanceExpansion, ResonanceOverlap, ScatteringMatrix, Superposition, SurfaceDensityIntegration

Use this post-process to obtain the Fourier transform of a time-harmonic electromagnetic field in a homogeneous half-space.

In a nutshell, the outgoing electromagnetic field is determined and decomposed into plane waves. These enter the half-space and travel towards infinity. In a subsequent post-process, the so computed Fourier spectrum may serve as the input for an optical imaging system, c.f., OpticalImaging.

Computing the Fourier transform is a comprehensive topic. For instance, geometric features such as periodicity have a strong impact on the Fourier transform:

Twofold periodic scatterers yield discrete Fourier spectra (discrete diffraction orders, or Fourier modes).
When the scatterer is isolated, the scattered field has a continuous Fourier spectrum. But, the outgoing field may also contain a discrete spectrum stemming from the reflection and transmission of illuminating plane waves.
The Fourier transform of a onefold periodic structure is a mixture of both: It is discrete in the direction of periodicity, and it is continuous in the other direction.

JCMsuite automatically adapts to these scenarios.

Example: A Fourier transform post-process may be specified as follows:

PostProcess {
  FourierTransform {
    FieldBagPath = "./project_results/fieldbag.jcm"
    OutputFileName = "./project_results/fourier_modes.jcm"
    NormalDirection = Y
    Format = JCM-ASCII
    NumericalAperture = 0.75
  }
}

Theoretical background

It is required that the problem setup has a homogeneous and isotropic material distribution in a half-space above a plane. $\varepsilon_+$ and $\mu_+$ denote the corresponding scalar permittivity and permeability, respectively. The angular wave number is given by $k_+=\omega \sqrt{\varepsilon_+ \mu_+}$ .

The scatterer (optical device) is contained in the lower half-space. For the sake of a simpler notation, it is assumed that the considered half-space is located above a plane with normal vector in $z$ -direction. Hence, the plane is fully fixed by its intersection point $(0, 0, z)$ with the $z$ -axis.

The Fourier transform of the electric field intensity with respect to $x, y$ is given by

$\begin{eqnarray*} \VField{E}^{\wedge}(\pvec{k}_\perp, z) & = & \frac{1}{4\pi^2} \int_{\rnum^2} \VField{E}(\pvec{x}_\perp, z) e^{-i\pvec{k}_\perp \cdot \pvec{x}_\perp} \dd \pvec{x}_\perp, \end{eqnarray*}$

where $\pvec{x}_\perp = (x, y)$ and $\pvec{k}_\perp = (k_x, k_y)$ . The scaling factor in front of the integral is chosen so that the inverse Fourier transform,

$\begin{eqnarray*} \VField{E}(\pvec{x}_\perp, z) & = & (\VField{E}^{\wedge})^{\vee}(\pvec{x}_\perp, z) = \int_{\rnum^2} \VField{E}^{\wedge}(\pvec{k}_\perp, z) e^{i\pvec{k}_\perp \cdot \pvec{x}_\perp} \dd \pvec{k}_\perp, \end{eqnarray*}$

reads as an infinite superposition of plane waves. The integrals are understood in a general (distributional) sense. Hence, the Fourier transform $\VField{E}^{\wedge}$ may split into a continuous spectrum and a discrete spectrum consisting of $\delta$ -functions. This way, the periodic geometries as well as isolated geometries are treated within a common framework.

In the above, the electric field $\VField{E}$ is the total field. It is shown in the appendix to this section, that the total field allows a splitting into an upward directed part and a downward directed part, $\VField{E}_{\uparrow}$ and $\VField{E}_{\downarrow}$ , respectively:

$\begin{eqnarray*} \VField{E}(\pvec{x}_\perp, z) & = & \VField{E}_{\uparrow}(\pvec{x}_\perp, z) +\VField{E}_{\downarrow}(\pvec{x}_\perp, z) \\ {} & = & \int_{\rnum^2} \VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\uparrow \cdot \pvec{x}} + \VField{E}_{\downarrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\downarrow \cdot \pvec{x}}\dd \pvec{k}_\perp, \end{eqnarray*}$

where $\pvec{k}_{\uparrow}=(\pvec{k}_\perp, k_z)^{\mathrm{T}}$ , $\pvec{k}_{\downarrow}=(\pvec{k}_\perp, -k_z)^{\mathrm{T}}$ , and with $k_z = \sqrt{k_+^2-|\pvec{k_\perp}|^2}$ .

To discuss this splitting in greater detail, it is assumed that the material in the upper half-space is non-dissipative, so that $k_+$ is a real positive number, and hence,

$\begin{eqnarray*} \Re(k_z)>0, \Im(k_z)=0, & \mbox{for} & |\pvec{k_\perp}|<k_+, \\ \Re(k_z)=0, \Im(k_z)>0, & \mbox{for} & |\pvec{k_\perp}|>k_+. \end{eqnarray*}$

When $|\pvec{k_\perp}|<k_+$ , $\VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\uparrow \cdot \pvec{x}}$ is an upward propagating plane wave, whereas $\VField{E}_{\downarrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\downarrow \cdot \pvec{x}}$ travels downward. For $|\pvec{k_\perp}|>k_+$ , $\VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\uparrow \cdot \pvec{x}}$ is evanescent in $z$ -direction, since $\Im(k_z)>0$ and hence $\lim_{z\rightarrow \infty} e^{i k_z z} = 0$ . Again, $\VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\downarrow \cdot \pvec{x}}$ has the opposite behavior: it increases exponentially in the positive $z$ -direction.

Note

JCMsolve computes the upward directed Fourier spectrum $\VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp)$ . Its definition is independent of the $z$ -position of the plane.

From the energy conservation principle, it follows that the downward directed part $\VField{E}_{\downarrow}$ only stems from the illumination. However, the illumination may also contain upward directed waves due to reflections or transmissions. Generally, the illuminating field may allow for a splitting into upward and downward directed parts:

$\begin{eqnarray*} \VField{E}^{(\mathrm{illum})}(\pvec{x}_\perp, z) & = & \VField{E}^{(\mathrm{illum})}_{\uparrow}(\pvec{x}_\perp, z)+\VField{E}^{(\mathrm{illum})}_{\downarrow}(\pvec{x}_\perp, z), \end{eqnarray*}$

with corresponding splitting of its Fourier transform

$\begin{eqnarray*} \VField{E}^{(\mathrm{illum}), \wedge}(\pvec{k}_\perp, z) & = & \VField{E}^{(\mathrm{illum}), \wedge}_{\uparrow}(\pvec{k}_\perp)e^{ik_z z} +\VField{E}^{(\mathrm{illum}), \wedge}_{\downarrow}(\pvec{k}_\perp)e^{-ik_z z}. \end{eqnarray*}$

For example, when the scatterer is embedded into a homogeneous medium or into a layered media stack, and when illuminated by a plane wave from below, $\VField{E}^{(\mathrm{illum})}_{\downarrow}$ is zero and $\VField{E}^{(\mathrm{illum})}_{\uparrow}$ is the prescribed plane wave as it leaves the multilayer stack. Then, the Fourier transform of the upward directed field $\VField{E}_{\uparrow}$ can be decomposed into the scattered part and the illumination part:

$\begin{eqnarray*} \VField{E}^{\wedge}_{\uparrow}(\pvec{k}_\perp) & = & \VField{E}^{(\mathrm{scat}), \wedge}_{\uparrow}(\pvec{k}_\perp) +\VField{E}^{(\mathrm{illum}), \wedge}_{\uparrow}(\pvec{k}_\perp). \end{eqnarray*}$

For isolated structures, the Fourier transform $\VField{E}^{\wedge}_{\uparrow}(\pvec{k}_\perp)$ of the scattered field is a smooth function of $\pvec{k}_\perp$ (except a branch cut singularity at $|\pvec{k}_\perp|=k_+$ ). In contrast to this, the Fourier transform $\VField{E}^{(\mathrm{illum})}_{\uparrow}$ of the illumination is often discrete. For example, when the illumination has a transmitted or reflected plane wave in the upper half-space with k-vector $\pvec{k}^{(\mathrm{illum})}$ and amplitude $\pvec{A}$ , then

$\begin{eqnarray*} \VField{E}^{(\mathrm{illum}), \wedge}_{\uparrow}(\pvec{k}_\perp) & = & \VField{A} \delta(\pvec{k}_\perp-\pvec{k}^{(\mathrm{illum})}_{\perp}). \end{eqnarray*}$

The presence of a continuous and a discrete spectrum in $\VField{E}^{\wedge}_{\uparrow}$ necessitates a proper book-keeping in the output file. This is discussed below, when detailing the output format.

Periodic geometries

See Electromagnetics for the discussion of periodic boundary conditions in the context of electromagnetic field computation. There, it is claimed that the electromagnetic field satisfies Bloch-periodic boundary conditions

$\begin{eqnarray*} \VField{E}(\pvec{x}+\pvec{a_i}) & = & \VField{E}(\pvec{x})e^{i\pvec{k}_{\mathrm{B}} \cdot \pvec{a_i}}, \end{eqnarray*}$

with a given Bloch vector $\pvec{k}_{\mathrm{B}}$ , and for one or more lattice vectors $\pvec{a_i}$ . To fit with the representation above, it is assumed that the lattice vectors are located in the $xy$ -coordinate plane. This way, one gets for the upward propagating field:

$\begin{eqnarray*} \VField{E}_{\uparrow}(\pvec{x}_\perp+\pvec{a}_{\perp, i}, z) & = & \VField{E}_{\uparrow}(\pvec{x}_\perp, z)e^{i\pvec{k}_{\perp, \mathrm{B}} \cdot \pvec{a}_{\perp, i}}. \end{eqnarray*}$

I) Twofold periodic case:

There exist two lattice vectors $\pvec{a}_{\perp, 1}$ and $\pvec{a}_{\perp, 2}$ with reciprocal grid vectors $\pvec{b}_{\perp, 1}$ , $\pvec{b}_{\perp, 2}$ , such that

$\begin{eqnarray*} \pvec{b}_{\perp, i} \cdot \pvec{a}_{\perp, j} & = & 2\pi \delta_{ij}. \end{eqnarray*}$

The function $\VField{E}_{\uparrow}e^{-k_{\perp, \mathrm{B}} \cdot \pvec{x}_\perp}$ is lattice periodic, and therefore, it allows for an expansion into discrete Fourier modes:

Warning

As a convention the lattice vectors $\pvec{a}_{\perp, 1}$ and $\pvec{a}_{\perp, 2}$ are ordered, so that $\pvec{a}_{\perp, 1}(1)$ ( $x$ -direction of the first lattice vector) is maximized, and $\pvec{a}_{\perp, 1}$ , $\pvec{a}_{\perp, 2}$ together with $\pvec{n}$ form an orthonormal coordinate system. Here, $\pvec{n}$ is the normal propagation direction of the scattered field, that is $\pvec{n}=[0\,0\,1]$ for NormalDirection=Z and $\pvec{n}=[0\,0\,-1]$ for NormalDirection=-Z.

$\begin{eqnarray*} \VField{E}_{\uparrow}(\pvec{x}_\perp, z)e^{-k_{\perp, \mathrm{B}} \cdot \pvec{x}_\perp} & = & \sum_{n_1,n_2=-\infty}^{\infty} \VField{C}_{n_1, n_2} e^{i \pvec{k}_{\perp, n_1, n_2} \cdot \pvec{x}_\perp} e^{ik_z z}, \end{eqnarray*}$

with $\pvec{k}_{\perp, n_1, n_2} = n_1 \pvec{b}_{\perp, 1}+n_2 \pvec{b}_{\perp, 2}$ , and with $k_z$ as defined above.

With this, the upward propagating field $\VField{E}_{\uparrow}$ can be written as an inverse Fourier transform

$\begin{eqnarray*} \VField{E}_{\uparrow}(\pvec{x}_\perp, z) = \int_{\rnum^2} \VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp)e^{i k_z z} \dd \pvec{k}_\perp, \end{eqnarray*}$

with

$\begin{eqnarray*} \VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) & = & \sum_{n_1,n_2=-\infty}^{\infty} \VField{C}_{n_1, n_2} \delta(\pvec{k}_\perp - \pvec{k}_{\perp, n_1, n_2}). \end{eqnarray*}$

Hence the Fourier spectrum is fully discrete.

II) Onefold periodic case:

It is assumed that the single grid vector $\pvec{a}_1$ is aligned with the $x$ -coordinate direction, i.e., $\pvec{a}_{\perp, 1} = (a_x, 0)$ . The reciprocal lattice vector is given as $(b_x, 0) = (2\pi/a_x, 0)$ . Considerations similar to the treatment of the twofold periodic case lead to a Fourier transform of the following shape,

$\begin{eqnarray*} \VField{E}_{\uparrow}^{\wedge}(k_x, k_y) & = & \sum_{n_1=-\infty}^{\infty} \VField{E}_{\uparrow}^{\wedge}(k_y) \delta(k_x - n_1 b_1). \end{eqnarray*}$

Hence, it is discrete in $k_x$ but still a continuous spectrum in $k_y$ .

Storage format

The computed Fourier transform is stored in a JCM table under the path OutputFileName. Each row in the table corresponds to a plane wave with angular wave number $\pvec{k}_j$ stored in the first the columns. Summing up (superimposing) these plane waves gives an approximation of the upward directed field:

$\begin{eqnarray*} \VField{E}_{\uparrow}(\pvec{x}_\perp, z) & = & \int_{\rnum^2} \VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\uparrow \cdot \pvec{x}} \dd \pvec{k}_\perp \approx \sum_{j=1}^{N_\mathrm{modes}} \VField{E}_{j}^{\wedge} e^{i\pvec{k}_j \cdot \pvec{x}}. \end{eqnarray*}$

The output JCM table file has the following columns:

Columns 1-3: KX, KY, KZ

This is $\pvec{k}_j$ as explained above.
Columns 4-5: N1, N2

These columns are only present for geometries with a pure discrete spectrum. The integer numbers refers to the diffraction orders as defined by the reciprocal grid vectors, so that $[k_x, k_y] = \pvec{k}_{\mathrm{B}} + n_1 \pvec{b}_{\perp, 1}+n_2 \pvec{b}_{\perp, 2}$ .
Columns …: <Quantity>X_<iF>, <Quantity>Y_<iF>, <Quantity>Z_<iF>,

Here, <Quantity> is the tag of the field quantity, e.g., ElectricFieldStrength or MagneticFieldStrength. The index <iF> stands for the field index. The $x$ , $y$ , and $z$ components of one field are stored in consecutive columns.

The continuous and discrete spectra are treated in the same way. However, when the user wishes to perform an up-sampling of the computed Fourier spectrum, one must distinguish between the continuous and the discrete spectra: Only the continuous spectrum can be interpolated on a finer mesh in the $\pvec{k}$ -space. The discrete spectrum is fixed. To support for this, the header of the table contains an integer entry NContinuousSpectrumModes, here called $N_{\mathrm{cont}}$ . The first $N_{\mathrm{cont}}$ corresponds to an sampling of the continuous spectrum on a regular grid in the $\pvec{k}$ -space. Note that $\VField{E}_{j}^{\wedge}$ is equal to $\VField{E}_{\uparrow}^{\wedge}$ at $\pvec{k}_j$ up to an scaling factor proportional the the cell volume of the underlying mesh in the $\pvec{k}$ -space.

Appendix: Splitting into downward and upward directed waves.

It is shown how the total field $\VField{E}$ can be split into an upward directed part and a downward directed part, $\VField{E}_{\uparrow}$ and $\VField{E}_{\downarrow}$ , respectively:

$\begin{eqnarray*} \VField{E}(\pvec{x}_\perp, z) & = &\VField{E}_{\uparrow}(\pvec{x}_\perp, z) +\VField{E}_{\downarrow}(\pvec{x}_\perp, z) \end{eqnarray*}$

The electric field satisfies Maxwell’s equations in the upper homogeneous half-space with scalar permittivity $\varepsilon_+$ and permeability $\mu_+$ , and with no impressed sources:

$\begin{eqnarray*} \curl \mu_+^{-1} \curl \VField{E}(\pvec{x}_\perp, z) -\omega^2 \varepsilon \VField{E} & = & 0, \\ \divo \varepsilon_+ \VField{E}(\pvec{x}_\perp, z) & = & 0. \end{eqnarray*}$

This second equation is the divergence condition. Applying this on both sides of the above equation defining the inverse Fourier transform above yields

$\begin{eqnarray*} \left ( \begin{array}{c} i\pvec{k}_\perp \\ \partial_{z} \end{array} \right ) \times \mu_+^{-1} \left ( \begin{array}{c} i\pvec{k}_\perp \\ \partial_{z} \end{array} \right ) \times \VField{E}^{\wedge}(\pvec{k}_\perp, z) -k_+^2 \VField{E}^{\wedge}(\pvec{k}_\perp, z) & = & 0, \\ \left ( \begin{array}{c} i\pvec{k}_\perp \\ \partial_{z} \end{array} \right ) \cdot \VField{E}^{\wedge}(\pvec{k}_\perp, z) & = & 0, \end{eqnarray*}$

with the angular wave number $k_+=\omega \sqrt{\mu_+\epsilon_+}$ of the upper half-space.

Expanding these equations shows that each component $E_j$ of the electric field satisfies the scalar equation

$\begin{eqnarray*} (-\partial_{zz}+\pvec{k}_\perp \cdot \pvec{k}_\perp-k_+^2) E_j^{\wedge}(\pvec{k}_\perp, z) & = 0, \end{eqnarray*}$

which permits the general solution

$\begin{eqnarray*} E_j^{\wedge}(\pvec{k}_\perp, z) = E_{j, \uparrow}^{\wedge}(\pvec{k}_\perp)e^{i k_z z} + E_{j,\downarrow} ^{\wedge}(\pvec{k}_\perp)e^{-i k_z z}, \end{eqnarray*}$

with $k_z = \sqrt{k_+^2-|\pvec{k_\perp}|^2}$ .

Switching back to the vectorial Maxwell’s equations and applying the inverse Fourier transform yields the desired representation of the electric field in the upper half-space:

$\begin{eqnarray*} \VField{E}(\pvec{x}_\perp, z) & = & \VField{E}_{\uparrow}(\pvec{x}_\perp, z) +\VField{E}_{\downarrow}(\pvec{x}_\perp, z)=\int_{\rnum^2} \VField{E}_{\uparrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\uparrow \cdot \pvec{x}} + \VField{E}_{\downarrow}^{\wedge}(\pvec{k}_\perp) e^{i\pvec{k}_\downarrow \cdot \pvec{x}}\dd \pvec{k}_\perp, \end{eqnarray*}$

where $\pvec{k}_{\uparrow}=(\pvec{k}_\perp, k_z)^{\mathrm{T}}$ and $\pvec{k}_{\downarrow}=(\pvec{k}_\perp, -k_z)^{\mathrm{T}}$ .