Source

engines/atom_fishbone_coupler/notes/2026-07-27_unit_cell_geometry.md · assembled 2026-07-29 15:57 UTC.

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2026-07-27 — Unit cell geometry schematic (pre-simulation)

Latest simulation status — Stage 2 3D MPB

Completed: first 3D MPB Γ–X band-edge baseline for the requested fishbone (a = 0.23 µm, t = 0.30 µm, W = 1.20 µm, w_slot = 1.00 µm, A = 0.50 µm). The target-near branch is at a/λ = 0.293477 (783.7 nm) at Γ and 0.322780 (712.6 nm) at X. This is a coarse one-branch baseline, not a converged full bandgap calculation. Full details and raw artifacts appear in the Stage 2 entry later on this page.

Latest 2D material check: an SiO₂/air slot guide with a 1.0 µm air gap does support ordinary guided modes, but none is air-gap dominant; the strongest sampled gap electric-intensity fraction is only 8.2%. Details appear in the SiO₂ entry later on this page.

Why

New engine Atom Fishbone Coupler (working name) created from scratch — not a fork. Before any MPB run, lock a shared picture of the unit cell.

Materials

• Core: Si₃N₄
• Cladding: air

Parameters (symbols)

Symbol	Meaning
a	Bragg / unit-cell period along x
t	Film thickness along z
W	Total slot-waveguide width (outer rail edge to outer rail edge), not including fishbone amplitude
w_slot	Air slot width (centered at y = 0)
A	Fishbone amplitude — how far each tooth protrudes beyond the rail in ±y (both sides)
w_fb	Fishbone tooth width along x; default a/2

Derived: each rail width = (W − w_slot) / 2.

Construction (my understanding)

One period 0 ≤ x < a, structure centered on y = 0, midplane z = 0:

1. Left rail (Si₃N₄): full period in x, y ∈ [−W/2, −w_slot/2], z ∈ [−t/2, t/2]
2. Right rail (Si₃N₄): full period in x, y ∈ [+w_slot/2, +W/2], z ∈ [−t/2, t/2]
3. Fishbone +y (Si₃N₄): rectangle, width w_fb (default centered → [a/4, 3a/4] if w_fb = a/2), y ∈ [W/2, W/2+A], same z
4. Fishbone −y (Si₃N₄): mirror, y ∈ [−W/2−A, −W/2]
5. Everything else in the cell: air (including the continuous slot)

Assumptions flagged for review

1. Teeth are rectangular and same thickness as the rails.
2. One tooth pair per period, centered in x.
3. Rails are continuous in x; only outer ±y edges grow teeth.
4. Slot is continuous air (no Si₃N₄ bridge across the slot).
5. W does not include amplitude A (outer extent is W/2 + A).
6. Illustrative numerical defaults in the figure are for readability only.

Schematic

Artifacts

File	Role
schematics/unit_cell.yaml	Parameter + construction definition
schematics/draw_unit_cell.py	Schematic generator
figures/unit_cell_schematic.png	Multi-view figure (top + two yz cuts)
figures/unit_cell_schematic.svg	Vector copy

Regenerate:

python engines/atom_fishbone_coupler/schematics/draw_unit_cell.py

2D slot-waveguide modes @ 780 nm (no fishbone)

Geometry confirmed schematically; first solver step is the plain slot only (no fishbone teeth), as a 2D y–z cross-section eigenmode problem with propagation along x.

Why Tidy3D (not MPB yet)

• MPB is not installed on this host yet, and geometry/units need extra care (overlapping shapes, scale free units).
• Tidy3D’s local ModeSolver is available (tidy3d 2.11.2) and does not need a cloud API key for this cross-section solve.
• Full Bragg unit cell / band diagram still planned for MPB after this baseline.

Geometry (µm) — same defaults as the schematic rails

Parameter	Value
λ	780 nm (0.78 µm)
W	0.80
w_slot	0.10
t	0.40
n_Si₃N₄	2.00 (constant; dispersion model not chosen yet)
cladding	air (n = 1)
fishbone	omitted

Cross-section used by the solver

Mode classification

Tidy3D reports pol_fraction.te / .tm on the mode plane (y–z). We keep the first 2 TE-like and 2 TM-like modes with highest fraction of each polarization among the bound-ish spectrum (num_modes=12, min_steps_per_wvl=30).

Plot convention: TE panels show Re(Ey) (in-plane, across the slot); TM panels show Re(Ez) (out-of-plane). Bottom row is normalized |E|².

Results (n_eff @ 780 nm)

Label	mode index	n_eff	TE frac	TM frac	character (from fields)
TM-1	0	1.6477	0.01	0.99	even Ez on both rails (fundamental)
TE-1	1	1.6144	0.99	0.01	Ey enhanced in the slot (even-ish)
TM-2	2	1.5864	0.02	0.98	odd Ez (left/right opposite)
TE-2	3	1.5666	0.99	0.01	odd Ey (left/right opposite)

Note: for these dimensions the fundamental is TM-like (highest n_eff), with the first TE-like slot mode second. Higher modes with n_eff ≲ 1.2 look weakly confined / continuum-adjacent; modes 9–11 warned about boundary decay.

Mode shapes (first 2 TE + 2 TM)

Separate panels:

Artifacts (this step)

File	Role
package/atom_fishbone/slot_waveguide_modes_2d.py	local ModeSolver driver + plots
runs/slot_waveguide_modes_2d_780nm/summary.json	n_eff / pol fractions for all modes
figures/slot_waveguide_cross_section_2d.png	ε_r cross-section
figures/slot_waveguide_modes_2d_780nm.png	combined TE+TM field panel
figures/slot_waveguide_TE_modes_2d_780nm.png	TE only
figures/slot_waveguide_TM_modes_2d_780nm.png	TM only

Regenerate:

python engines/atom_fishbone_coupler/package/atom_fishbone/slot_waveguide_modes_2d.py
python tools/build_docs_site.py

Caveats

1. n_Si₃N₄ = 2.0 is a flat placeholder at 780 nm (real Si₃N₄ is slightly dispersive / process-dependent).
2. Local ModeSolver warned that remote/subpixel averaging is more accurate; numbers are good for shapes and ordering, not final design tolerances.
3. TE/TM labels follow Tidy3D plane polarization fractions, not every paper’s “quasi-TE” naming for slots — fields are the source of truth.
4. Fishbone is not in this solve. Next physics step: MPB band diagram of the full periodic unit cell (after MPB install + careful non-overlapping geometry).

Still open / next

1. Confirm n_Si₃N₄ model (constant vs dispersive).
2. Install MPB and build the fishbone unit cell carefully (no ambiguous overlaps; consistent scale to µm or a).
3. Band diagram of the Bragg unit cell; compare gap / guided bands to this slot baseline.

2D slot modes — t = 150 nm, W = 1.5 µm, w_slot = 1.0 µm @ 780 nm

Same solver as above (Tidy3D local ModeSolver, y–z cross-section, no fishbone). New geometry requested:

Parameter	Value
λ	780 nm
t	150 nm (0.15 µm)
W	1.5 µm
w_slot	1.0 µm
rail width each	(W − w_slot)/2 = 0.25 µm
n_Si₃N₄	2.00 (constant)
cladding	air

Cross-section

Results (n_eff ≥ 1 preferred; first 2 TE + 2 TM by n_eff within pol)

Label	mode index	n_eff	TE frac	TM frac
TE-1	0	1.1299	0.96	0.04
TE-2	1	1.1171	0.96	0.04
TM-1	2	1.0372	0.02	0.98
TM-2	3	1.0276	0.02	0.98

Compared with the earlier thicker / narrower-slot baseline (t=400 nm, W=0.8, w_slot=0.1), n_eff values are much lower here: thin film + wide 1 µm air gap + only 0.25 µm rails → weakly confined modes sitting just above the air light line. Modes with n_eff ≲ 1 (indices ≥ 4) do not decay at the mode-plane boundaries (continuum / leaky).

For this geometry the two highest-n_eff modes are TE-like, then two TM-like (opposite of the previous baseline, where TM was fundamental).

Mode shapes

Artifacts

File	Role
runs/slot_waveguide_modes_2d_t150nm_W1p5_slot1p0_780nm/summary.json	full spectrum + picks
figures/slot_waveguide_*_t150nm_W1p5_slot1p0_780nm.png	cross-section + field panels

2D slot modes — t = 150 nm, W = 1.3 µm, w_slot = 1.0 µm @ 780 nm

Same solver; thinner rails than the previous W=1.5 case.

Parameter	Value
λ	780 nm
t	150 nm
W	1.3 µm
w_slot	1.0 µm
rail width each	(W − w_slot)/2 = 0.15 µm
n_Si₃N₄	2.00
cladding	air

Cross-section

Results (first 2 TE + 2 TM by n_eff within pol)

Label	mode index	n_eff	TE frac	TM frac	note
TM-1	0	1.0124	0.01	0.99	only TM above air light line
TE-1	1	1.0070	0.99	0.01	only TE above air light line
TM-2	2	0.9978	0.01	0.99	just below n = 1
TE-2	3	0.9934	0.99	0.01	just below n = 1

Only two modes sit at n_eff ≥ 1 (TM-1, TE-1). Compared with W=1.5 µm (rails 0.25 µm), shrinking total width to 1.3 µm (rails 0.15 µm) drops n_eff further toward cutoff. Higher modes again fail to decay at the mode-plane boundaries.

Mode shapes

Artifacts

File	Role
runs/slot_waveguide_modes_2d_t150nm_W1p3_slot1p0_780nm/summary.json	spectrum + picks
figures/slot_waveguide_*_t150nm_W1p3_slot1p0_780nm.png	plots

2D slot modes — t = 150 nm, W = 1.3 µm, w_slot = 0.5 µm @ 780 nm

Parameter	Value
λ	780 nm
t	150 nm
W	1.3 µm
w_slot	0.5 µm
rail width each	0.40 µm
n_Si₃N₄	2.00

n_eff

Label	mode	n_eff	TE frac	TM frac
TE1	0	1.3479	0.98	0.02
TE2	1	1.3458	0.98	0.02
TM1	2	1.0828	0.02	0.98
TM2	3	1.0649	0.03	0.97

Halving the gap vs 1.0 µm raises TE n_eff substantially (~1.35) and pulls both TE and TM clearly above the air light line. TE is fundamental (pair of nearly degenerate even/odd-like TE supermodes).

E-field only (TE1, TE2, TM1, TM2)

Re(Ey) for TE, Re(Ez) for TM — no intensity plots.

Artifacts

• runs/slot_waveguide_modes_2d_t150nm_W1p3_slot0p5_780nm/summary.json
• figures/slot_waveguide_Efields_TE_TM_t150nm_W1p3_slot0p5_780nm.png

TE1–4 and TM1–4 E-fields (same geometry)

Extended request: also show TE3, TE4, TM3, TM4. Still Re(E) only (Ey for TE, Ez for TM).

Label	mode	n_eff	note
TE1	0	1.3479	guided
TE2	1	1.3458	guided
TE3	5	0.9843	n_eff < 1 (leaky / continuum-adjacent)
TE4	6	0.9775	n_eff < 1
TM1	2	1.0828	guided
TM2	3	1.0649	guided
TM3	4	0.9856	n_eff < 1
TM4	7	0.9761	n_eff < 1

Only four modes sit above the air light line (TE1–2, TM1–2). TE3–4 and TM3–4 are the next TE/TM-classified modes by n_eff; fields do not fully decay at the solver window edges.

Artifact: figures/slot_waveguide_Efields_TE1-4_TM1-4_t150nm_W1p3_slot0p5_780nm.png

2D slot modes — t = 150 nm, W = 1.0 µm, w_slot = 0.5 µm @ 780 nm

Parameter	Value
λ	780 nm
t	150 nm
W	1.0 µm
w_slot	0.5 µm
rail width each	0.25 µm
n_Si₃N₄	2.00

TE1–4 and TM1–4 (E-field only)

Label	mode	n_eff	note
TE1	0	1.1336	guided
TE2	1	1.0998	guided
TE3	5	0.9762	n_eff < 1
TE4	6	0.9749	n_eff < 1
TM1	2	1.0442	guided
TM2	3	1.0108	guided
TM3	4	0.9811	n_eff < 1
TM4	7	0.9710	n_eff < 1

Compared with W=1.3 µm / same gap: TE n_eff drops (1.35 → ~1.13) as rails shrink from 0.40 → 0.25 µm. Still four modes above air light line (TE1–2, TM1–2).

Artifacts

• runs/slot_waveguide_modes_2d_t150nm_W1p0_slot0p5_780nm/summary.json
• figures/slot_waveguide_Efields_TE1-4_TM1-4_t150nm_W1p0_slot0p5_780nm.png

2D slot modes — t = 0.30 µm, W = 1.0 µm, w_slot = 0.5 µm @ 780 nm

Same lateral geometry as the previous W=1.0 / gap=0.5 case; thickness raised to 300 nm.

Parameter	Value
λ	780 nm
t	0.30 µm
W	1.0 µm
w_slot	0.5 µm
rail width each	0.25 µm

First truly guided mode (n_eff > 1, air-clad)

Criterion used here: n_eff > n_air = 1 (bound vs continuum for air cladding).

First guided	TM1 (solver mode 0)
n_eff	1.3964
pol	TM fraction 0.98

All guided (n_eff > 1), ordered by n_eff:

rank	label	mode	pol	n_eff
1	TM1	0	TM	1.3964
2	TM2	1	TM	1.3625
3	TE1	2	TE	1.3297
4	TE2	3	TE	1.2809

So with t = 0.30 µm the fundamental is TM, not TE (unlike the thinner t = 0.15 µm case at the same W/gap, where TE led). TE3–4 and TM3–4 remain below the light line.

E-fields TE1–4 & TM1–4

Green border on the panel marks the first guided mode. Re(Ey) for TE, Re(Ez) for TM.

Artifacts

• runs/slot_waveguide_modes_2d_t300nm_W1p0_slot0p5_780nm/summary.json
• figures/slot_waveguide_Efields_TE1-4_TM1-4_t300nm_W1p0_slot0p5_780nm.png

Wide-gap air-mode check — t = 300 nm, W = 1.5 µm, slot = 1.0 µm @ 780 nm

Question

Can two Si₃N₄ rails support a guided mode mostly in a 1 µm air gap, and in particular are the next higher modes (TE3/TE4/TM3/TM4) guided air modes?

Geometry and numerical check

Plain, unpatterned slot cross-section (no fishbone), air cladding:

Parameter	Value
wavelength	780 nm
Si₃N₄ index	2.00 (constant model)
rail thickness	0.30 µm
total rail-to-rail outer width W	1.50 µm
air gap	1.00 µm
individual rail width	0.25 µm
primary solver domain (y × z)	7.5 × 5.3 µm²
confirmation domain (y × z)	9.5 × 6.3 µm²
boundary	PML in y and z

The primary domain leaves 3.0 µm air beyond each outer rail in y and 2.5 µm above/below the film. It is deliberately much wider than the guide. Tidy3D's local ModeSolver was asked for 24 modes. A mode is treated as bound only when n_eff > 1 and its field decays at the mode-plane boundary. The final two columns are sampled electric-field-intensity fractions: in the 1 µm gap, and in the outermost 10% of the computational window, respectively.

Family	order	n_eff (wide)	n_eff (confirmation)	gap	outer 10%	classification
TE	1	1.2834	1.2949	17.7%	~0%	bound, rail-dominated
TE	2	1.2832	1.2946	17.9%	~0%	bound, rail-dominated
TM	1	1.3443	1.3625	6.8%	~0%	bound, rail-dominated
TM	2	1.3441	1.3624	6.5%	~0%	bound, rail-dominated
TE	3	0.9959	0.9973	1.4%	19.0%	radiation / box state
TE	4	0.9956	0.9971	2.1%	21.0%	radiation / box state
TM	3	0.9968	0.9981	18.6%	15.7%	radiation / box state
TM	4	0.9967	0.9977	23.2%	15.3%	radiation / box state

The solver explicitly reported non-decay at the boundary for modes starting at mode index 4 (which contains TM3 and the corresponding higher-order states). Increasing the domain shifts these near-light-line values toward 1 and leaves a large outer-window intensity fraction: the expected signature of the air continuum, not a guided mode. By contrast, the four bound rail modes have negligible outer-window intensity.

Conclusion

For an ordinary two-rail Si₃N₄/air slot guide, this requested 1 µm gap does not produce a bound, gap-dominant air mode. The slot effect remains visible in the guided TE pair, but only about 18% of sampled electric intensity is in the gap; TM is even less gap-localized. The apparently air-like TM3/TM4 states are not guided.

More generally, a conventional index-guided mode in an all-air cladding must remain above the air light line (n_eff > 1). A very wide low-index gap loses the evanescent overlap that makes a slot mode work. A truly air-localized guided mode therefore needs an additional confinement mechanism, e.g. a 2D photonic bandgap / hollow-core design, antiresonant guidance, or a periodic structure that opens a stop band around the air defect. The fishbone's eventual periodic design could be explored for this purpose; the plain slot alone cannot provide it here.

Artifacts

File	Role
runs/slot_waveguide_modes_2d_t300nm_W1p5_slot1p0_wide_780nm/summary.json	24-mode wide-domain spectrum and localization metrics
runs/slot_waveguide_modes_2d_t300nm_W1p5_slot1p0_xwide_780nm/summary.json	larger-domain confirmation spectrum
figures/slot_waveguide_cross_section_2d_t300nm_W1p5_slot1p0_wide_780nm.png	cross-section
figures/slot_waveguide_modes_2d_t300nm_W1p5_slot1p0_wide_780nm.png	TE1–TE4 and TM1–TM4 field panels

Regenerate:

.venv/bin/python engines/atom_fishbone_coupler/package/atom_fishbone/slot_waveguide_modes_2d.py \
  --W 1.5 --slot 1.0 --thickness 0.3 --pad-y 3.0 --pad-z 2.5 \
  --num-modes 24 --per-pol 4 --tag t300nm_W1p5_slot1p0_wide_780nm

Interrupted Si₃N₄ slab — 0.5 µm air gap, t = 300 nm @ 780 nm

Geometry

Two Si₃N₄ slabs, each extending arbitrarily far in ±y, are separated by a 0.5 µm-wide air gap. The film is 0.30 µm thick, air-clad above and below.

This is materially different from a finite-width slot guide: in the infinite-width limit each side is a laterally unbounded slab waveguide. To check the limit numerically, I used a wide finite proxy with two 2.0 µm wide Si₃N₄ regions (W = 4.5 µm including the gap), with 3.0 µm lateral air padding on both exterior sides and 2.5 µm above/below the film.

Result

It does not support a discrete, air-guided gap mode.

The six highest-n_eff modes of each polarization are conventional slab modes:

Family	n_eff range (first 4)	gap electric-intensity fraction
TE-like	1.6965 → 1.6182	0.48% → 1.26%
TM-like	1.6251 → 1.5939	0.04% → 0.20%

All have negligible field at the exterior computational boundary, so they are vertically guided slab modes. Their field is overwhelmingly in the Si₃N₄, not the gap. No computed bound state is localized to the 0.5 µm air defect.

Why the infinite-width limit rules it out

The air gap is a lower-index interruption inside the high-index slab. It is a lateral anti-guide: a wave can lower its effective index barrier by spreading indefinitely into either Si₃N₄ half-slab. Thus the finite-proxy slab modes become a continuous set parameterized by lateral wavevector as rail width goes to infinity; none becomes a normalizable gap-localized eigenmode.

The familiar slot enhancement can raise the normal electric field in a low-index gap between finite rails, but it does not turn a low-index trench in an unbounded high-index slab into a bound air-core waveguide. To make this geometry air-guiding, add lateral reflection/confinement, for example a 2D photonic bandgap surrounding the gap, an antiresonant reflector, or finite width rails.

Artifact

• runs/slot_waveguide_modes_2d_t300nm_W4p5_slot0p5_wide_slab_780nm/summary.json

Stage 2 — 3D MPB fishbone unit cell, first band-edge baseline

Requested physical geometry

Parameter	Value
period a	0.23 µm
Si₃N₄ thickness t	0.30 µm (carried forward from the preceding stage)
rail outer-to-outer width W	1.20 µm
continuous air slot w_slot	1.00 µm
rail width (W − w_slot)/2	0.10 µm
fishbone amplitude A	0.50 µm
tooth length along x	a/2 = 0.115 µm
core index model	n_Si3N4 = 2.00, hence ε = 4.0
cladding	air (ε = 1)

MPB construction and unit handling

MPB uses dimensionless units with a = 1. The driver therefore divides every physical length by 0.23 µm before creating geometry:

Quantity	MPB units
x period	1.0000
t	1.30435
W	5.21739
slot	4.34783
rail width	0.43478
A	2.17391
tooth length	0.50000

The isolated guide is represented by an air supercell of size 1 × 18.2609 × 10.0000 (x, y, z), i.e. 1.0 µm air padding on each exterior y and z side. The rails are separate full-period blocks. Each tooth occupies the y interval [W/2, W/2 + A] (or its negative mirror), while the rail ends at ±W/2; they touch only on a face and have zero overlapping volume. This matters because MPB does not require, or benefit from, overlapping dielectrics to make a connected shape.

First completed 3D MPB result — targeted Γ–X band edge

MPB was newly installed for this stage. A complete first-pass run used:

Solver setting	Value
spatial resolution	8 pixels / a (grid 8 × 147 × 80)
mesh size	5
eigensolver residual	1e-4
target frequency	a/λ = 0.30
computed branch	one full-vector 3D eigenbranch nearest target
k path	Γ (0,0,0) → X (0.5,0,0)

k point	a/λ	equivalent wavelength for a = 230 nm
Γ	0.293477	783.7 nm
X	0.322780	712.6 nm

The selected branch therefore rises by 0.029303 a/λ from Γ to X. The result is plotted below.

Interpretation and limitation

This establishes that the exact non-overlapping 3D geometry runs in MPB and places a target-near branch around the 780 nm design neighborhood at Γ. It is not yet a converged full band diagram or a photonic bandgap claim: the first pass traces one target-selected branch at only the two band edges, with a coarse spatial resolution. In this large air supercell, an untargeted multi-band solve is dominated by nearly-degenerate air-box states; it should next be split into mirror-symmetry sectors and converged in resolution, supercell padding, k-point density, and number of branches before interpreting guided-band gaps or radiation loss.

Artifacts

File	Role
package/atom_fishbone/fishbone_band_3d.ctl	reproducible MPB unit-cell construction and Γ–X solve
package/atom_fishbone/plot_fishbone_band_3d.py	parses MPB frequencies and plots bands
runs/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_GammaX/mpb_stdout.txt	raw MPB log
runs/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_GammaX/mpb_stdout.csv	Γ/X frequency data
runs/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_GammaX/fishbone_band_3d-epsilon.h5	3D dielectric grid
figures/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_GammaX.png	plotted result

Regenerate:

RUN=engines/atom_fishbone_coupler/runs/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_GammaX
mkdir -p "$RUN"
(cd "$RUN" && mpb ../../package/atom_fishbone/fishbone_band_3d.ctl > mpb_stdout.txt)
.venv/bin/python engines/atom_fishbone_coupler/package/atom_fishbone/plot_fishbone_band_3d.py \
  "$RUN/mpb_stdout.txt" \
  engines/atom_fishbone_coupler/figures/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_GammaX.png

Stage 2 update — MPB band-edge axis in π/a units

The band-diagram x-axis now uses k_x/(π/a), appropriate for the 1D Bloch path. The displayed edge segment is 0.75 π/a to 1.00 π/a (X), which MPB expresses as k_x = 0.375 to 0.5 in its native 2π/a units.

k_x/(π/a)	MPB k_x	a/λ	λ for a = 230 nm
0.75	0.375	0.302415	760.5 nm
1.00 (X)	0.500	0.322687	712.9 nm

Only one line is shown because this preliminary MPB calculation deliberately targeted one full-vector eigenbranch near a/λ = 0.30; it was not a multi-band calculation. The next full diagram must use symmetry-sector filtering / guided-mode identification to avoid the large air-supercell's near-degenerate radiation-box bands.

Artifacts:

• runs/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_edge/
• figures/fishbone_band_3d_W1p2_gap1p0_A0p5_a0p23_r8_target0p30_edge.png

SiO₂/air slot waveguide — 1.0 µm gap @ 780 nm

Aim and chosen geometry

This is a finite, air-clad SiO₂ slot waveguide without fishbone teeth. To give the 1.0 µm gap a favorable test, each rail is 1.0 µm wide and the film is 0.50 µm thick. The wide rails avoid simple rail cutoff; the 3.0 µm lateral and 2.5 µm vertical air padding make the outer boundaries non-participating.

Parameter	Value
λ	780 nm
core	SiO₂, n = 1.45 (constant model)
t	0.50 µm
total width W	3.00 µm
air slot	1.00 µm
rail width	1.00 µm each
domain, y × z	9.0 × 5.5 µm²
solver	local Tidy3D ModeSolver, 20 modes

Results

All eight shown modes lie above the air light line and decay at the outer boundaries, so they are guided. However, their electric intensity is localized in the oxide rails, not in the 1 µm air gap.

Label	mode	n_eff	gap electric-intensity fraction	classification
TE1	0	1.3081	1.5%	guided, rail-dominated
TE2	1	1.3074	1.8%	guided, rail-dominated
TE3	4	1.1748	7.0%	guided, rail-dominated
TE4	5	1.1726	8.2%	guided, rail-dominated
TM1	2	1.2762	1.0%	guided, rail-dominated
TM2	3	1.2752	1.0%	guided, rail-dominated
TM3	6	1.1648	4.4%	guided, rail-dominated
TM4	7	1.1607	4.6%	guided, rail-dominated

Conclusion

The 1 µm SiO₂ slot is a conventional finite-rail waveguide, not an air-core guide. Lower core contrast makes the modes broader, but it does not create a mode that lives primarily in the gap. Of the tested modes, TE4 puts the most sampled electric intensity in the gap (8.2%)—still far from air dominated.

Artifacts

File	Role
runs/slot_waveguide_modes_2d_sio2_t500nm_W3p0_slot1p0_wide_780nm/summary.json	20-mode spectrum and localization metrics
figures/slot_waveguide_cross_section_2d_sio2_t500nm_W3p0_slot1p0_wide_780nm.png	SiO₂/air cross-section
figures/slot_waveguide_modes_2d_sio2_t500nm_W3p0_slot1p0_wide_780nm.png	TE1–4 and TM1–4 fields/intensities

SiO₂/air slot waveguide — tiny 0.15 × 0.15 µm rails, 1.0 µm gap @ 780 nm

Geometry

Requested geometry: two SiO₂ rails, each 0.15 µm wide × 0.15 µm thick, separated by a 1.0 µm air gap. Thus W = 1.30 µm. The computation used the same generous 3.0 µm lateral and 2.5 µm vertical air padding as the other wide domain tests.

Results

Label	mode	n_eff	gap electric-intensity fraction	interpretation
TE1	1	0.99927	28.8%	below air light line; radiation-box state
TE2	2	0.99592	1.1%	radiation-box state
TE3	5	0.99316	14.8%	radiation-box state
TE4	8	0.98847	13.8%	radiation-box state
TM1	0	1.00036	31.5%	marginally bound / near-cutoff
TM2	3	0.99571	26.4%	radiation-box state
TM3	4	0.99467	2.4%	radiation-box state
TM4	6	0.99139	1.8%	radiation-box state

Only TM1 sits infinitesimally above the air light line, and even it has 5.6% of sampled electric intensity in the outermost 10% of the domain. It should be treated as near-cutoff rather than a robust guided air mode. The other modes' tempting gap fractions arise because they are extended air-continuum states; the field plots visibly reach the computational boundary.

TE and TM on separate rows

For readability, the plot puts TE field/intensity rows first and TM field/intensity rows second.

Conclusion

Making the SiO₂ rails tiny does not create the desired 1 µm air-gap guide. It removes robust index guidance instead: all TE modes and all but one TM mode are below n_eff = 1; the remaining TM1 is only 0.00036 above cutoff. This is not a usable air-guided mode.

Artifacts

File	Role
runs/slot_waveguide_modes_2d_sio2_t150nm_rails150nm_slot1p0_wide_780nm/summary.json	20-mode spectrum and localization metrics
figures/slot_waveguide_cross_section_2d_sio2_t150nm_rails150nm_slot1p0_wide_780nm.png	tiny-rail cross-section
figures/slot_waveguide_TE_TM_rows_2d_sio2_t150nm_rails150nm_slot1p0_wide_780nm.png	requested TE/TM row-separated field view