# The Mode Expansion Monitor - Two Interesting Use Cases

Introduction
I’ve noticed a lot of confusion among users regarding the mode expansion monitor (MEM) and I would like to take an opportunity to clarify things. The article on using the mode expansion monitor covers the mathematics required to get started, and interested user are invited to read Snyder and Love “Optical Waveguide Theory” for a more thorough treatment of eigenmode expansion and overlap calculations in regards to finding power coupling coefficients.

Although those resources are very informative, there are enough subtle mechanics involved that the MEM set-up is most clearly demonstrated through a few examples. In the following I will demonstrate the correct usage of MEMs through two interesting use cases.

Note for applications that involve strictly guided modes, ports may be more appropriate and user friendly; since ports objects contain MEMs, mode sources, and frequency monitors. MEMs, however, are more flexible allowing you to look at power coupling from a broader range of sources. Both objects will allow you to extract S-parameters for analysis in INTERCONNECT.

Steps
In summary a mode expansion monitor is used to find the power coupling coefficient into the forward and backward guided modes.

1. Add the Mode Expansion Monitor and choose the mode

The MEM finds the guided modes by invoking a finite difference eigenmode (FDE solver) over the span of the MEM. Settings for the mode calculation are available under the mode expansion tab.

From this menu you can choose the mode that we are interested in: fundamental, fundamental TE or fundamental TM, or ‘user select’. In the above image we have chosen the fundamental mode, and visualized the mode data.

For more information on the settings see FDE Solver - simulation object.. At this point it is important to make sure that the span is wide enough such that the boundary conditions do not effect the results.

1. Add a DFT monitor with the same cross section.

Once the FDTD simulation has run the MEM performs overlap calculations on the fields of a specified frequency-domain field and power monitor (DFT monitor).

Whereas a DFT monitor will simply integrate the Poynting vector over it’s cross section to calculate the power through that monitor. Mode monitors use the field profile in the DFT monitor to determine how much of that power is coupled into the specified mode. The expansion calculation requires that the DFT monitor be aligned with MEM, so that the overlap calculation is performed on the same grid points. This means that the two objects should have the same cross sectional mesh profile and cross sectional span. If their are variations between the DFT and MEM monitor this will certainly cause problems.

1. Choose monitor for expansion

With the DFT monitor set-up correctly you may now add a monitor for expansion. Next specify which monitor to use and give it a name if you like.

1. Run simulation and perform expansion

Expansion allows us to ignore scattered radiation and higher order modes. It is often essential for matching experimental results; furthermore, the ratio of light coupled into different modes provides important insight into the device performance.

QD Single Photon Source to Fiber Mode

QD_Source.fsp (853.6 KB)
QD_Source_analysis.lsf (1.3 KB)

Although FDTD is a classical solver it can be a valuable tool for optimizing the optical set-up and components in quantum photonics experiments. One important design challenge in the field is optimizing the collection of light from solid state Quantum Dots (QD). These serve as single photon sources (SPS), and are usually realized via nano-pillar cavities. The problem with these nano-pillars is that they are composed of III-V semiconductor stacks with high refractive indices. Designing for light extraction efficiency (LEE) is difficult prospects; although, numerous methods have been explored - and could be modeled in FDTD- such as using dielectric antennas, solid-immersion lenses, photonic crystals or nanowires.

Since we are mostly interested in demonstrating how to use the mode expansion monitor, we will employ a simple set-up using a fiber placed directly over the nano-cavity. This is an idealized and simplified case. In reality one would not be able to align a fiber with such accuracy relative to a nano-pillar. Interested users could readily sweep the fiber position to develop a measure of the sensitivity to misalignment.

QD single photon source
It is not possible to model a single photon classically; however, an ideal SPS produces perfectly uncorrelated photons, so we may model this using an incoherent unpolarized source and measure how many photons the fiber collects on average. For this 2D axially symmetric QD set-up we can approximate an incoherent point dipole using two simulations weighting the horizontal in-plane polarization by two. The out of plane dipole in 2D does not capture the correct radiation pattern.
$$< | \mathbf{E} | > = \frac{1}{3} \left( | 2 \cdot \mathbf{E} _{Hor} |^2 + | \mathbf{E} _{Ver} |^2 \right)$$

Mode expansion monitor
After setting up the structure the next thing to do is set up the MEM and DFT monitors. One thing to be concerned about here is the size mismatch between the nanopillar and the fiber. If we look at the modes we find two degenerate modes in polarization due to the fiber symmetry. The higher order modes are unphysical.

Mode profiles of the E-field magnitude for these solutions are seen below. We have set the MEM and FDTD span to be 28.4 $\mu m$, so that the fields extend far enough into the cladding such that they do not interact with the metal BC. See starting with Metal BC FDE.

To do some quick sanity checks we could expand a few DFT monitors along the propagation distance of the fiber. This is possible as long as they have the same cross-section as the MEM.

If the power coupling changes for the further monitor then we should place the monitor even further. The MEM determines how much light would reach our detector set-up and ignores the scattered light, which would certainly be lost before propagating the >10mm required to the detector. Below we seethe expansion for mode 1 at two DFT monitor locations (y = 1.54, 2.77 $\mu m$ ) and we observe convergence so we can be confident in this set-up.

The horizontal and vertical dipole will couple into the first and second mode respectively, which can be seen by running the simulation. Normalized power into the fiber is given in the following plot.

The wighted power coupling is given.

This could be interpreted as the probability that a photon will collected by the fiber into a propogating mode, which seems to be around 1.5%. Compared to the upward power which is about 22%.

In the script we have used a normalization factor to get the output power to 1. This doesn’t tell the entire story since the amount of power output from dipole will depend on it’s local environment. This can seen by looking at the purcell enhancement of the dipole which provides a connection to the QM properties via the local density of optical states. To design this structure it would be best to optimize the cavity for maximum output power at the target wavelength. See this post for some discussion on VCSEL cavities, the same strategy would apply for these structures.

OLED Slab Mode Analysis
OLED_slab_analysis.fsp (7.1 MB)
OLED_slab_analysis.lsf (1.0 KB)

In the Mode Analysis of a OLED structure AppGallery example we look at the mode profiles of a slab and discuss how effectively the active layer will couple into these various modes. For more information on OLED methodology and OLED(2D) for relevant results anThe OLEDS stack is defined as follows. Glass, nitride and ITO layers are provided as lossless, and dispersionless dielectrics with indices given below; whereas the organics and aluminum cathode are given with n at 500nm - the center wavelength.

From the mode expansion monitor we find the first three supported modes which are given below. These were calculated using 0.5nm mesh, but in order to solve this in FDTD within a reasonable time frame we set the mesh size to 2nm.

mode 1: 3.7 dB/micron (essentially an SPR mode)
mode 2: 0.18 dB/micron (waveguide mode with some loss)
mode 3: 1.7 dB/micron (combined dielectric/SPR mode)

With some intuition about the dipole radiation pattern we could probably guess which modes would be important; however, here we will demonstrate how to measure the proportion of radiation into each mode.

Since we are injecting the radiation into a non-homogeneous medium the power injected will be greater than the source power. In this set-up the materials are lossy and dispersive so we need to use the transmission box method to accurately capture dipole power. Here we see enhancement across the bandwidth especially at high frequency, where injected 1.57 times that which would be injected in a homogeneous medium.

It may be helpful to look at the farfield radiation pattern using a far_field_indexchange analysis group to account for the glass air interface if one would like to measure the far-field radiation pattern or angular dependent color shift. Here we measure the transmission that is coupled into the glass, and compare it with the power coupled into the modes above.

If we look at the power ratio coupled into each mode we that the lossy SPP modes dominate.

To look at the polarization specific coupling refer to the simulation file. This results indicates the importance of developing some patterning on the interface to increase extraction efficiency.
For more information see OLED methodology and OLED(2D) for relevant results and workflows.

Note the results will change slightly the depending on the location of the dipole. To accurately model the electro-luminescent layer one should sweep dipole positions, adding the incoherent results.