Anthracene vertical slot-waveguide coupling

Emitter coupling into a vertically-oriented Si3N4 slot waveguide's fundamental mode, on a PEC mirror.

Tidy3D FDTD study of a single-photon emitter (linear dipole in a 200 nm anthracene crystal, n=1.8, center wavelength 780 nm) coupling into the fundamental mode of a vertically-oriented Si3N4 slot waveguide cladded in SiO2. The crystal sits on a perfect-electric-conductor (PEC) mirror, with an optional SiO2 spacer layer in between. The waveguide's propagation direction is vertical (+z): this represents an ordinary slot waveguide cleaved perpendicular to its length, stood on its cleaved facet, with the crystal stamped onto the exposed cross-section.

Both dipole orientations relative to the waveguide are studied:

• TE (Ex): dipole polarized across the slot walls -- the slot-enhanced, design-intent case.
• TM (Ey): dipole polarized parallel to the slot walls -- the orthogonal case, useful for gauging sensitivity to crystal/dipole alignment.

See reports/dry_run_report.html for the full geometry writeup, mode analysis, and FlexCredit cost projections.

Workflow

1. Generate the dry-run report: geometry plots, local (free) mode-solver analysis identifying the TE/TM mode index per cross-section, and projected FlexCredit costs for the full sweep.

python -m anthracene_vertical_slot_coupler_v1.dry_run --estimate-costs

1. Review reports/dry_run_report.html. No FlexCredits are spent by this step -- estimate_cost is Tidy3D's non-billed pre-flight quote.

2. Only after explicitly approving the projected FlexCredit total, run the billed batch:

python -m anthracene_vertical_slot_coupler_v1.run_batch --run

run_batch.py refuses to submit anything if any planned job has an unknown estimated cost, or a per-job cost above the configured approval threshold (StudyConfig.flexcredit_approval_threshold).

1. Analyze downloaded simulation data and render the final report:

python -m anthracene_vertical_slot_coupler_v1.analyze_results

Planned sweep

• Slot geometry x polarization: rail thickness in {150, 300} nm x slot width in {10, 25, 50, 100, 150} nm x {TE, TM}, fixed 450 nm total footprint, zero spacer.
• Spacer thickness x polarization: SiO2 spacer in {50, 100, 150, 200} nm (0 nm is already covered above) x {TE, TM}, at a fixed nominal 150 nm rail / 50 nm slot geometry -- investigates whether standing off the crystal from the PEC mirror improves coupling.
• Reference: bare crystal on the PEC, no waveguide (one polarization; the bare structure is rotationally symmetric about z).

Each simulation is a single broadband job spanning 750-850 nm; per-wavelength results come from multi-frequency monitors, not repeated runs.

Results

Session report (main results)

Anthracene / Vertical Waveguide Coupling: Session Report

First major report covering everything simulated in this session: the baseline slot-waveguide design from the dry-run, a chain of ad hoc single-point and small-sweep variants exploring spacer thickness, crystal thickness, slot width, mirror type, and waveguide core material, culminating in the best design found so far. All coupling-efficiency numbers are at the emitter's 780 nm line unless noted otherwise. All simulations described here have actually been run on Tidy3D's cloud solver (not estimates).

Device concept

A 200 nm anthracene crystal (n=1.8) hosts a linear quantum-emitter dipole (780 nm) at its mid-plane. It sits on an optional SiO2 spacer, on top of a bottom boundary (a PEC mirror in most runs; also tested as a real DBR, and as no reflector at all). Directly above the crystal is a vertically-oriented waveguide -- either a Si3N4/TiO2 slot waveguide (two rails split by a narrow gap) or a solid rod, cladded in SiO2 -- propagating in +z. A dipole polarized across the waveguide's x-axis is the TE case; polarized along y is TM. Coupling efficiency is guided fundamental-mode power (mode monitor, +z) divided by total radiated power (flux box, all sides).

Device schematic (baseline slot-waveguide geometry)

Vertical slot waveguide transverse cross-section

The whole session at a glance

Coupling efficiency at every major milestone this session

Two findings dominate everything else: (1) a plain solid waveguide beats the slot design for this buried-emitter geometry, because the emitter sits below the whole footprint rather than inside the slot gap; and (2) core material index matters a lot -- swapping Si3N4 (n=2.0) for TiO2 (n=2.8) at the same dimensions took the best result from 58%/54% to 75%/74%. The PEC mirror is doing a lot of work too: replacing it with a real 8-pair DBR drops coupling by about a third, and removing the mirror entirely drops it by about two-thirds.

1. Baseline: slot waveguide

150 nm Si3N4 rails, 50 nm slot, 450 nm total footprint, 200 nm crystal, no spacer, ideal PEC mirror -- the nominal geometry from the original dry-run sweep design.

Polarization	Coupling @ 780nm	Spectral range (750-850nm)
TE (dipole across slot)	38.5%	38.4-38.6%
TM (dipole along slot)	23.4%	19.2-32.1%

TE is essentially flat across the band; TM rises steadily from 750 to 850 nm. This is the reference point for every comparison below.

2. SiO2 spacer sweep (0-200 nm)

Same geometry as the baseline, with a SiO2 layer of varying thickness inserted between the crystal and the PEC.

Coupling efficiency vs SiO2 spacer thickness

Spacer (nm)	TE	TM
0	38.5%	23.4%
50	27.3%	18.1%
100	13.9%	11.7%
150	3.4%	4.8%
200	6.6%	1.9%

Finding: 0 nm (crystal directly on the PEC) is optimal. Coupling falls monotonically to a minimum near 100-150 nm, consistent with the crystal moving out of the constructive-interference zone of the dipole's image in the mirror. The interference period (~270 nm half-wavelength in SiO2) suggests a second, weaker maximum may exist further out (~250-300 nm), untested here.

3. Crystal thickness (100 nm vs 200 nm)

Same slot geometry, no spacer, PEC -- crystal thickness reduced from 200 nm to 100 nm, halving the dipole's distance from both the PEC and the waveguide (50 nm each way instead of 100 nm).

Crystal thickness	TE	TM
200 nm (baseline)	38.5%	23.4%
100 nm	57.7%	31.6%

Finding: thinner is better. Roughly +19 points TE, +8 points TM -- moving the emitter closer to both the mirror and the waveguide's near field helps substantially.

4. Slot width (25 nm vs 50 nm)

200 nm crystal, no spacer, PEC, 150 nm rails, 450 nm footprint -- slot narrowed from 50 nm to 25 nm (rails widen correspondingly to keep the footprint fixed).

Slot width	TE	TM
50 nm (baseline)	38.5%	23.4%
25 nm	43.0%	21.4%

Finding: narrower slot helps TE (tighter mode confinement, better near-field overlap: +4.5 points) but slightly hurts TM (-2 points, since TM doesn't benefit from slot confinement and just sees wider rails).

5. Isolating the PEC's contribution (slot waveguide)

Same slot geometry (200 nm crystal, no spacer) with the PEC replaced by an open (PML) boundary and the domain extended below the crystal with SiO2, so downward-radiated power is lost instead of reflected.

Configuration	TE	TM
With PEC (baseline)	38.5%	23.4%
No PEC (open boundary)	12.6%	7.5%

Finding: the PEC accounts for roughly two-thirds of the coupling efficiency. Without it, ~20% of total power radiates straight down and is lost (verified via the flux monitor on that face), and the dipole's radiation-pattern asymmetry (weak along its own polarization axis) is clearly visible in the lateral-flux split.

No-PEC field intensity: downward glow into the substrate that the PEC normally reflects back up

6. Discovery: a regular (non-slot) waveguide beats the slot design

Same crystal/spacer/PEC as the baseline, but the two-rail slot waveguide is replaced by a single solid Si3N4 rod (450 x 300 nm).

Design	TE	TM
Slot waveguide (50 nm slot)	38.5%	23.4%
Regular (solid) waveguide	58.0%	53.9%

Why: the emitter sits below the entire waveguide footprint, not inside the slot gap. A slot removes high-index material from directly above the strongest part of the dipole's near field; a solid rod keeps it there. The usual slot-waveguide advantage (field enhancement for emitters embedded in the gap) doesn't apply to this buried-emitter geometry.

A methodological note carried forward from here: each dipole polarization's simulation uses mirror symmetry (Ex: odd across x, even across y; Ey: the reverse), which restricts that job's own mode solve to one symmetry class -- so mode_index=0 is automatically the correct fundamental mode for that polarization, without needing the Ex/Ey energy-fraction classification the slot waveguide required.

7. Regular-waveguide dimension sweep (W x H)

Width (x, 300/450/600 nm) x thickness (y, 200/300/400 nm), Si3N4 core, otherwise the section-6 baseline.

TE coupling @ 780nm:

W H	200nm	300nm	400nm
300	48.4%	54.1%	54.5%
450	53.9%	58.0%	58.1%
600	53.2%	56.7%	57.1%

TM coupling @ 780nm:

W H	200nm	300nm	400nm
300	43.4%	54.1%	57.5%
450	45.3%	53.9%	57.0%
600	43.4%	51.1%	53.8%

Finding: TE peaks near W=450nm and saturates quickly with H; TM keeps climbing with H over this whole range and prefers narrower W. Best balanced point in the tested grid: W=450, H=400 nm (TE=58.1%, TM=57.0%). TM hadn't saturated at H=400nm, so a larger H may do even better.

8. Core material: TiO2 (n=2.8) instead of Si3N4 (n=2.0)

Best regular-waveguide geometry (450 x 400 nm), core material swapped to TiO2.

Core material	TE	TM
Si3N4 (n=2.0)	58.1%	57.0%
TiO2 (n=2.8)	75.1%	73.6%

Best result of the session. Both modes are broadband and essentially lossless in the mode solver (n_eff ~2.57/2.55, negligible imaginary part). Higher index contrast gives tighter confinement and better near-field overlap with the buried dipole.

Methodology fix along the way: the mode monitor's search bias (target_neff) was still set for the crystal index (1.8), appropriate for Si3N4 but not for TiO2's much higher n=2.8 -- the first attempt completely missed the true fundamental mode and gave spuriously low numbers (~2-8%). Corrected to target_neff=2.2 and rerun; verified against a free local mode solve before trusting the result.

TiO2 TE mode field intensity: guided streak up the rod from the dipole

9. Real mirror: 8-pair TiO2/SiO2 DBR instead of ideal PEC

Best TiO2 geometry (450 x 400 nm), PEC replaced by a quarter-wave DBR (8 pairs, designed for 780 nm, ~1.64 um thick) on a SiO2 substrate terminated in PML.

Mirror	TE	TM
Ideal PEC	75.1%	73.6%
8-pair DBR	54.3%	54.0%

Finding: the DBR reflects almost perfectly (downward flux ~0%, matching the PEC) but coupling is still notably lower. The gap isn't about reflected power -- it's about angle and phase: a PEC reflects every angle with the same simple phase shift, while a quarter-wave DBR's reflection phase and magnitude vary with incidence angle. The dipole's near field contains many angular components, and only the near-normal ones see the DBR's designed response, so the constructive interference that boosts coupling with a PEC is only partially reproduced.

DBR field intensity: layer stack visible below the crystal, negligible field penetrates it

10. No mirror at all: bare crystal/air interface

Same TiO2 waveguide, PEC/DBR removed entirely -- just air (n=1) below the crystal, terminated in PML. Isolates plain Fresnel/total-internal reflection at the n=1.8/n=1 step (critical angle ~33.75 deg) with no engineered mirror.

Bottom boundary	TE	TM	Downward loss
Ideal PEC	75.1%	73.6%	0%
8-pair DBR	54.3%	54.0%	~0%
Bare air (no mirror)	20.9%	20.3%	~14.5%

Finding: bare index contrast alone recovers a real but partial fraction of the benefit -- about a third of what the DBR provides, about a quarter of the ideal PEC. An engineered reflector is doing substantial, necessary work; it isn't just "any index step would do."

Cost accounting

40 billed simulations run this session; total real cost: 1.011 FlexCredits. All jobs were validated and cost-estimated (via Tidy3D's non-billed estimate_cost) before submission, per the project's FlexCredit guardrail.

Mid-session, we noticed the pre-run cost estimate was running 4-10x higher than the real (post-run) billed cost, because the configured run_time (3 ps) was oversized relative to how fast the fields actually decay -- solver logs showed the automatic shutoff triggering within the first 1-2% of allotted time steps across every geometry tested (slot, regular, TiO2, DBR, no-PEC alike). run_time was halved to 1.5 ps, bringing typical-size jobs to within about 2x of their real cost (matching the real-cost floor observed for smaller jobs, ~0.025 FC/job) while retaining a 35-50x safety margin over the observed decay times. This is now the default for all runs going forward.

Summary comparison table

Configuration	TE @ 780nm	TM @ 780nm
Baseline slot WG (150/50/450, 200nm crystal, PEC)	38.5%	23.4%
+ thinner crystal (100nm)	57.7%	31.6%
+ narrower slot (25nm) (200nm crystal)	43.0%	21.4%
No PEC (slot WG) isolation check	12.6%	7.5%
Regular WG discovery (450x300 Si3N4)	58.0%	53.9%
Regular WG optimum (450x400 Si3N4)	58.1%	57.0%
TiO2 core (450x400, PEC)	75.1%	73.6%
TiO2 + real DBR (8 pairs)	54.3%	54.0%
TiO2 + no mirror (bare air)	20.9%	20.3%

Conclusions and open directions

• Best design found so far: solid (non-slot) TiO2 waveguide, 450x400 nm, 200 nm crystal, no spacer, ideal PEC -- TE=75.1%, TM=73.6%.
• Every geometric lever tried (thinner crystal, no spacer, solid vs. slot, higher-index core) helped in the same direction: get more high-index material as close to the dipole as possible.
• The PEC mirror is not a detail -- it is the single largest lever in the whole study. A real DBR captures roughly 70% of its benefit; no mirror at all captures roughly 25-28%.
• Not yet tried: combining the biggest wins together (TiO2 core + 100nm crystal + no spacer + optimized W/H), a W x H sweep at n=2.8, spacer values beyond 200nm to find the next constructive-interference peak, and DBR pair-count/design-wavelength tuning.
• The run_time reduction and TE/TM symmetry-based mode selection (both discovered mid-session) should carry forward into any future full-sweep run of this project.

Dry-run preflight (baseline)

Anthracene / Vertical Slot Waveguide Coupling: Dry Run

This is a pre-flight report. No billed FDTD solve is launched by this dry run. Cost figures below come from Tidy3D's non-billed estimate_cost pre-flight quote.

Device concept

A 200 nm anthracene crystal (n=1.8) hosts a linear quantum-emitter dipole (780 nm) at its mid-plane, treated as a laterally infinite thin film. It sits on an optional SiO2 spacer, which sits on a perfect electric conductor (PEC) that forms the z-minus boundary condition of the simulation domain (not a finite-thickness structure). Directly above, in contact with the crystal, is a Si3N4 slot waveguide cladded in SiO2 -- but oriented vertically: propagation is +z, not in-plane. Physically this is an ordinary slot waveguide, cleaved perpendicular to its length, stood on its cleaved facet, with the crystal stamped onto the exposed cross-section and a reflector later deposited on top (the reflector is out of scope here; see Limitations). The slot runs along x; a dipole polarized along x sits across the slot walls (slot-enhanced, TE), while a dipole polarized along y sits parallel to the walls (TM). The objective is coupling efficiency: guided fundamental-mode power (mode monitor, +z) divided by total radiated power (flux box: 4 sides + top).

Device schematic: x-z cross-section through the slot, with the SiO2 spacer shown at 200 nm for illustration (0 nm is the baseline)

Vertical slot waveguide: transverse (x-y) cross-section, propagation +z out of the page

Key parameters

Parameter	Value
emitter line / analysis band	780 nm / 750-850 nm (21 pts)
indices (crystal / Si3N4 / SiO2)	1.8 / 2.0 / 1.444
lateral domain / fine mesh	3200 nm sq., PML sides / 20 nm near slot
run time / shutoff / FC threshold	3.0e-12 s / 1e-05 / 0.50 FC per job

Methodology notes

• TE / TM assignment. Rather than assuming mode ordering, each distinct cross-section's fundamental modes are solved locally with Tidy3D's free mode solver plugin (no cloud billing) and classified by whether Ex or Ey carries more energy. This mapping is stored per-geometry so the FDTD mode monitor's mode_index=0,1 results are attributed correctly -- important because, as the plot below shows, the ordering flips for the 300 nm rail geometries.
• Symmetry. x=0 and y=0 are geometric mirror planes; an Ex dipole is odd across x=0 and even across y=0 (Ey is the reverse), so each job simulates only ¼ of the domain, cutting cost roughly 4x.
• Why the flux box omits its bottom face: between the dipole and the PEC there is only lossless dielectric, so the net time-averaged Poynting flux through any horizontal plane there is exactly zero in steady state (a pure standing wave) -- the PEC-adjacent face isn't needed.
• The top-of-waveguide reflector from the device concept is not modeled; this study isolates the coupling step. The base_no_waveguide reference (bare crystal on the PEC) provides total-emitted-power context; it needs only one polarization since it is rotationally symmetric about z.

Planned simulation sweep

29 total planned jobs: 1 no-waveguide reference, 20 slot-geometry x polarization combinations, and 8 spacer-thickness x polarization combinations (all built on top of the same 14 distinct cross-sections below, x2 for TE/TM where applicable). Each job is a single broadband simulation; the full 750-850 nm spectrum comes from one run via multi-frequency monitors, not one run per wavelength.

Main sweep: every (rail thickness, slot width) cross-section, 450 nm footprint, zero spacer

Spacer sweep: crystal and dipole shift up off the PEC as spacer thickness increases

cross-section	waveguide	rail thickness nm	slot nm	footprint nm	spacer nm
no_waveguide	no	-	-	-	0
h150_slot010_w450_spacer000	yes	150	10	450	0
h150_slot025_w450_spacer000	yes	150	25	450	0
h150_slot050_w450_spacer000	yes	150	50	450	0
h150_slot100_w450_spacer000	yes	150	100	450	0
h150_slot150_w450_spacer000	yes	150	150	450	0
h300_slot010_w450_spacer000	yes	300	10	450	0
h300_slot025_w450_spacer000	yes	300	25	450	0
h300_slot050_w450_spacer000	yes	300	50	450	0
h300_slot100_w450_spacer000	yes	300	100	450	0
h300_slot150_w450_spacer000	yes	300	150	450	0
h150_slot050_w450_spacer050	yes	150	50	450	50
h150_slot050_w450_spacer100	yes	150	50	450	100
h150_slot050_w450_spacer150	yes	150	50	450	150
h150_slot050_w450_spacer200	yes	150	50	450	200

Projected FlexCredit costs

Projected FlexCredit cost per job (blue=TE, orange=TM); total 3.001 FC

Known-cost total: 3.0010 FC across 29 planned jobs, all under the 0.50 FC per-job threshold.

No job has been run and no FlexCredits have been spent. Review this total, then explicitly approve before run_batch.py --run is invoked.

Mode analysis (free, local mode solver -- no cloud billing)

Fundamental-mode effective index vs slot width, TE and TM, both rail thicknesses

Both a slot-aligned (TE) and orthogonal (TM) guided mode exist at every geometry tested, with n_eff between SiO2 (1.444) and Si3N4 (2.00). Representative mode profiles (the slot-mode field pileup is visible in the TE panels):

h150_slot010_w450_spacer000: TE and TM mode profiles

h150_slot050_w450_spacer000: TE and TM mode profiles

h150_slot150_w450_spacer000: TE and TM mode profiles

h300_slot050_w450_spacer000: TE and TM mode profiles

Meshing

Mesh refinement near the narrowest (10 nm) slot -- global auto grid plus targeted overrides at the crystal/slot and dipole

Curated geometry cross-sections

Full tidy3d renders for a representative subset (every other geometry appears in the montages above):

no_waveguide

no_waveguide: xz

no_waveguide: yz

no_waveguide: xy_dipole

h150_slot010_w450_spacer000

h150_slot010_w450_spacer000: xz

h150_slot010_w450_spacer000: yz

h150_slot010_w450_spacer000: xy_dipole

h150_slot010_w450_spacer000: xy_waveguide

h150_slot050_w450_spacer000

h150_slot050_w450_spacer000: xz

h150_slot050_w450_spacer000: yz

h150_slot050_w450_spacer000: xy_dipole

h150_slot050_w450_spacer000: xy_waveguide

h150_slot150_w450_spacer000

h150_slot150_w450_spacer000: xz

h150_slot150_w450_spacer000: yz

h150_slot150_w450_spacer000: xy_dipole

h150_slot150_w450_spacer000: xy_waveguide

h300_slot050_w450_spacer000

h300_slot050_w450_spacer000: xz

h300_slot050_w450_spacer000: yz

h300_slot050_w450_spacer000: xy_dipole

h300_slot050_w450_spacer000: xy_waveguide

h150_slot050_w450_spacer200

h150_slot050_w450_spacer200: xz

h150_slot050_w450_spacer200: yz

h150_slot050_w450_spacer200: xy_dipole

h150_slot050_w450_spacer200: xy_waveguide

Expected post-run outputs

• Coupling efficiency (guided power / total power) versus wavelength, TE vs TM, for every slot geometry.
• 2D heatmaps at 780 nm: coupling efficiency vs (rail thickness, slot width), separately for TE and TM.
• Spacer-thickness sweep: coupling efficiency vs SiO2 spacer thickness, TE and TM.
• Field-intensity cross-sections for the best-coupling geometry, to build intuition for the coupling mechanism.

Limitations / scope notes

• The top-of-waveguide reflector is not modeled; this study isolates the coupling step only.
• The crystal is a simple isotropic dielectric (n=1.8) here; real anthracene is birefringent, but the emitter's dipole orientation is the controlling factor for this study.
• Total footprint width is fixed at 450 nm; only rail thickness and slot width are swept. A width sweep is a natural extension if this geometry looks promising.

Downloads

FDTD simulation study only -- no GDS/STL deliverable files exist in this tree.

---

Source: nanophotonic_devices/emitter_coupling/anthracene_vertical_slot_coupler_v1/