Algorithmic Design of Plasmonic Grating Couplers

The goal in this work is to get light into a 1D-MIM slot waveguide cut into an upper sheet of gold from a 3D Guassian mode illuminating a lower sheet of gold. The strange geometry is designed to minimize light spillage in the half-space to minimize the noise on an NSOM probe. To do this, I make use of a sample with a glass substrate, 100nm Au layer, 200nm of air, and another 100nm of gold. By never having holes/slots coincident between the two Au sheets, the top half-space will be very dark when the structure is illuminated from below with a Guassian beam. What is explored here is how to design the holes/slots to get the light from the Guassian beam into a 1D MIM mode. However, the mathematics and techniques involved in this process have become interesting in their own right.

In this work, we

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notebooks:brian:3d_guassian_to_arbitrary_2d_via_numerical_simulation
3D Guassian to Arbitrary 2D via Numerical Simulation

The goal in this work is to get light into a 1D-MIM from a 3D Guassian mode with as little light spillage in the half-space in which the NSOM probe will be working as possible. To do this, I make use of a sample with a glass substrate, 100nm Au layer, 200nm of air, and another 100nm of gold. By never having holes/slots coincident between the two Au sheets, the top half-space will be very dark when the structure is illuminated from below with a Guassian beam. What is explored here is how to design the holes/slots to get the light from the Guassian beam into a 1D MIM mode useful for the TLONC project. However, the mathematics and techniques involved in this process have become interesting in their own right.

as scatterers at these points map one field onto another coherently. For the 2D-MIM analytic mode, I made use of Hankel functions to describe elementary modes of a point radiator in 2D.

The goal is to get light into a 1D MIM mode. Reversing this field, we will likely find that a 1D MIM waveguide which is well matched to a 2D MIM mode does not precisely radiate as an elementary Hankel function. It will have some complicated radiation pattern.

Therefore, here we make use of COMSOL to calculate radiation pattern of the 1D MIM into the 2D MIM mode to accurately generate diffraction gratings for arbitrary geometries. The steps go as follows.

    We simulate the 1D MIM to 2D MIM transition in reverse (i.e. with the 1D MIM as the source) and optimize so that this transition is very efficient.
    We check to see that if this radiation pattern was indeed operated in the forward direction, it would couple light back the MIM waveguide. Of course the system is reciprocal, but not all the radiated modes will be used. For instance, MI SPP's on the top surface and 3D scattered fields will not be used by structure in the bottom sheet of gold.
    The 2D MIM field is exported (E2

).
Mathematica is used define an analytic 3D Gaussian beam (E1
) and find the arg(E1E2)=0 lines in which \abs(E1E2)>A

. The latter condition essentially windows the grating structure to where both beams are intense enough for it to matter. This geometry is fattened from infinitesimally thin lines into 100nm slits to form a grating structure.
This grating is imported back into COMSOL and simulated using a Gaussian beam source. It is verified that the functionality of the grating is similar to the reversed fields in step 2.

The results are presented below.

First we begin with the reversed COMSOL Simulation (herringbone2.mph). The light begins in the 1D MIM and transitions to the 2D MIM with very little reflection. This is due to the design of the coupler. As it expands, the energy “wants” to transition to a similar impedance waveguide. In this case the closest mode is the 2D MIM. The radius at the end of the slit was optimized to minimize the VSWR.

The field between the 2D MIM is exported for use later.(exported_data.txt) Additionally, the geometry is exported for later use. inter_geo.mphbin

The reversed field is used as the boundary source for an approximate “forward” simulation.

It is apparent that if this field were recreated using a diffraction coupler (or other means), it would indeed couple into the 1D MIM.

The exported field is imported into Mathematica (E1
) and an incoming Gaussian is defined (E2

). linegenerator.nb
MIM Reversed Radiated Field (E1
) Gaussian Incident from Below the Sample (E2
).

These two are multiplied to yield their /arg(E1E2)
and /abs(E1E2)

arg(E1E2)
\abs(E1E2)

This is used to design a grating (pattern.dxf):

This grating design is combined with the exported geometry to form a new simulation. sim_with_gc.mph The results of which are shown below in the forward direction.
Field just below the top sheet Cross Section

Note how the field begins in the 3D Guassian mode and is smoothly coupled into the 1D MIM waveguide. Most importantly for our future endeavors in TLONC, very little of the energy makes it into free-space modes on the top-side of the structure. In other words, the top-side is dark and we would expect to be able to perform NSOM measurements with such a device and very cleanly measure the 1D MIM mode.
Field just above the top sheet
Transmission Line Optical Nano Circuits, GratingCoupler
Discussion
Nader Engheta, 2013/11/25 17:28

Dear Brian, Great results. Nader
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In diffraction_grating_to_mim_waveguide_design, I initially presented an idea in which a diffraction grating could be used to map a free space 3D Gaussian beam (E1) into a 2D MIM Gaussian beam (E2). The diffraction grating geometry was calculated semi-analytically by finding the places in which arg(E1E2)=0 as scatterers at these points map one field onto another coherently. For the 2D-MIM analytic mode, I made use of Hankel functions to describe elementary modes of a point radiator in 2D.

Therefore, here we make use of COMSOL to calculate radiation pattern of the 1D MIM into the 2D MIM mode to accurately generate diffraction gratings for arbitrary geometries. The steps go as follows.

We simulate the 1D MIM to 2D MIM transition in reverse (i.e. with the 1D MIM as the source) and optimize so that this transition is very efficient.

We check to see that if this radiation pattern was indeed operated in the forward direction, it would couple light back the MIM waveguide. Of course the system is reciprocal, but not all the radiated modes will be used. For instance, MI SPP's on the top surface and 3D scattered fields will not be used by structure in the bottom sheet of gold. Mathematica is used define an analytic 3D Gaussian beam (E1) and find the arg(E1E2)=0 lines in which \abs(E1E2)>A. The latter condition essentially windows the grating structure to where both beams are intense enough for it to matter. This geometry is fattened from infinitesimally thin lines into 100nm slits to form a grating structure.
This grating is imported back into COMSOL and simulated using a Gaussian beam source. It is verified that the functionality of the grating is similar to the reversed fields in step 2.

The results are presented below.

The field between the 2D MIM is exported for use later.(exported_data.txt) Additionally, the geometry is exported for later use. inter_geo.mphbin

The reversed field is used as the boundary source for an approximate “forward” simulation.

It is apparent that if this field were recreated using a diffraction coupler (or other means), it would indeed couple into the 1D MIM.

The exported field is imported into Mathematica (E1
) and an incoming Gaussian is defined (E2

). linegenerator.nb
MIM Reversed Radiated Field (E1
) Gaussian Incident from Below the Sample (E2
).

These two are multiplied to yield their /arg(E1E2)
and /abs(E1E2)

arg(E1E2)
\abs(E1E2)

This is used to design a grating (pattern.dxf):

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