Every simplification the articles and the simulator make, what each one leaves out, and where it stops being true.
A model is useful because it leaves things out. Each entry here names one thing left out, what is real instead, why leaving it out is fine for the purpose, and where that stops being true. In the articles, an id such as APX-004 links to its entry here.
APX-001 lossless components by default
Waveguides, couplers and phase shifters lose no light unless a loss is set.
What we assume
Waveguides, couplers and phase shifters conserve power unless a loss parameter is set.
Reality
Silicon waveguides have propagation loss (order of dB/cm), couplers have excess loss, bends radiate.
Why it's OK here
It isolates interference, which is the concept being taught. Loss is a parameter, and "real-world mode" turns it on.
Where it breaks
Long circuits, high-Q resonators (Q is loss-limited), power budgets.
The effective index changes linearly with wavelength around the design wavelength.
What we assume
n_eff(λ) ≈ n_eff(λ₀) − (n_g − n_eff)(λ − λ₀)/λ₀.
Reality
Higher-order dispersion exists.
Why it's OK here
Over the tens of nm spanned by our sweeps, the first-order term sets the free spectral range correctly.
Where it breaks
Very broadband sweeps, dispersion-engineering topics.
Sources
source still to be added
Used in
The simulator and the lab only.
APX-004 wavelength-independent coupling ratio
A coupler splits the light in the same ratio at every wavelength.
What we assume
A coupler's power coupling κ² does not depend on wavelength.
Reality
Directional couplers are strongly wavelength-dependent; MMIs less so.
Why it's OK here
Teaching the function of a splitter; a dispersive coupler model can be added later.
Where it breaks
Broadband circuits, spectra of MZIs over wide ranges.
Sources
For MMIs: Soldano & Pennings 1995 §VI-C, p. 621 (their outputs stay balanced around the design length) and Table III, p. 625 (reported bandwidths of 80–100 nm); directional couplers: source still to be added
Light from one laser interferes with itself; separate lasers do not interfere, and their powers add at a detector.
What we assume
Light from one laser is coherent: where its paths meet, amplitudes add (interference). One laser may feed several circuit ports, split off-chip. Separate lasers are mutually incoherent, even at the same wavelength: at a detector their powers add. Detectors average over times much longer than the lasers' coherence times and any beat period between them.
Reality
Two superposed waves give I = I₁ + I₂ + 2√(I₁I₂)|g₁₂| cos φ; for uncorrelated waves (g₁₂ = 0) this is I₁ + I₂, with no interference (Saleh & Teich 2019 §12.2A, Eq. (12.2-3)). Sources whose relative phase varies randomly and rapidly add in irradiance (Hecht 2017 §7.1, p. 294). Lasers at different frequencies beat at |ν₂ − ν₁| (Saleh & Teich 2019 §2.6B, Eq. (2.6-12)); a slow detector averages the beat away. But a laser keeps its phase for its coherence time, which can be long, and interference between independent lasers has been observed (Hecht 2017 §9.2.1, p. 404).
Why it's OK here
Continuous-wave circuits read by slow, ideal power detectors. Circuits that need coherent light at several inputs get it from one laser.
Where it breaks
Fast detectors (heterodyne and coherent receivers, beat notes), measurements shorter than the lasers' coherence time, frequency combs, lasers locked to each other.
Sources
Saleh2019 §12.2A Eq. (12.2-3), §2.6B Eq. (2.6-12); Hecht 2017 §7.1 p. 294, §9.2.1 p. 404. Related: Chrostowski & Hochberg 2015 §4.2, p. 111 (a Y-branch cannot combine two incoherent beams to increase power).
Every waveguide carries one mode of one polarisation (TE0).
What we assume
Each waveguide carries one mode (TE0).
Reality
Waveguides can be multimode; polarisation conversion and crosstalk happen.
Why it's OK here
Standard PDK waveguides are designed to be single-mode for one polarisation.
Where it breaks
Y-junction combiners (the "missing" light couples to a higher-order/radiating mode, which must be modelled as a loss port: yBranch.ts does, Chrostowski & Hochberg 2015 §4.2, p. 111), polarisation-diverse circuits.
Sources
source still to be added (single-mode design of PDK waveguides); Chrostowski & Hochberg 2015 §4.2, p. 111 (the second mode of a Y-branch)
A phase shifter adds the same phase at every wavelength.
What we assume
A phase shifter adds the same Δφ at every wavelength.
Reality
A phase shifter changes the effective index over a length L, so with our propagation convention Δφ = 2π Δn_eff L / λ scales roughly as 1/λ, and Δn_eff itself varies slightly with λ.
Why it's OK here
Simulations are at one wavelength for now. Across a sweep of width Δλ the shift changes by about Δλ/λ (1.3 % for 20 nm around 1550 nm).
Where it breaks
Broadband spectra of circuits with large phase shifts.
Sources
Follows from the propagation convention (docs/physics/conventions.md); a source for the size of Δn_eff's dispersion is source still to be added.
The light inside a directional coupler is drawn as a gradual exchange between the two waveguides.
What we assume
The lab draws the light inside a directional coupler's coupling section as the supermode picture behind the coupler model: along the section the field is [[cos ψ, i sin ψ], [i sin ψ, cos ψ]] applied to the inputs, with ψ growing linearly from 0 to asin κ. Before and after the section, the light keeps its input and output values.
Reality
Coupling also happens in the S-bends before and after the straight section (Chrostowski & Hochberg 2015 pp. 101–102 measures their contribution as 2.3–2.8 µm of extra coupling length for rib couplers), and a real coupler's κ depends on wavelength (APX-004).
Why it's OK here
It is a picture of what happens inside a component the simulator treats as lumped. At the coupler's ports it equals the simulator exactly (tested in apps/lab/test/extract.test.ts). It shows students that a coupler moves light gradually.
Where it breaks
The power along the S-bends, and the exact position where the light crosses.
Sources
Chrostowski2015 Eqs. (4.1)–(4.3), pp. 92–93 (κ² = sin²(CL), supermodes); Fig. 4.2, p. 94 (the supermodes' shapes)
APX-009 a bend is a straight waveguide of the same length
A bend has the same effective index and loss as a straight waveguide of equal length.
What we assume
A bent waveguide has the same effective index and no more loss than a straight one; its length is its arc length.
Reality
In a bend the mode shifts outward and changes shape; loss comes mainly from the mode mismatch at the start and end of a bend, and from radiation at small radii (Chrostowski & Hochberg 2015 p. 70). 500 × 220 nm strip bends lose about 0.01 dB per 90° at a 5 µm radius (imec measurement, Chrostowski & Hochberg 2015 p. 70).
Why it's OK here
The lab routes with a 10 µm radius, near where the simulated loss of a conventional 90° strip bend is lowest (Chrostowski & Hochberg 2015 p. 73). Symmetric layouts stay symmetric: equal drawn lengths give equal phases, which is what the circuits taught here rely on. How much a real bend's phase differs from a straight waveguide of the same length is source still to be added.
APX-010 overlapping waveguides are simulated as separate
Waveguides drawn on top of each other are simulated as separate, and marked as a layout problem.
What we assume
Waveguides the student draws on top of each other, across each other or touching are simulated as if they were separate: no coupling, no scattering, no loss where they meet. The lab marks such places as layout problems (magenta marks, "Check the layout") instead of refusing the drawing, so the light keeps flowing while the student edits.
Reality
Drawn shapes on one layer merge into one piece of silicon, so an overlap is not the circuit the drawing shows (source for this and for what light does at a crossing: source still to be added). Even without touching, parallel strip waveguides closer than about 1.6 µm couple noticeably over 1 cm (Chrostowski & Hochberg 2015 §4.1.7, pp. 107–108); the router keeps 2 µm from other cells, but the layout check does not yet flag close waveguides.
Why it's OK here
It is only ever shown together with a problem mark; a correct layout has no overlaps, and then nothing is assumed. Switching the light off instead would make it blink while parts are dragged across each other.
Where it breaks
Any layout with problem marks: its results do not describe a real chip.
Sources
Chrostowski2015 §4.1.7, pp. 107–108 (spacing)
Used in
The simulator and the lab only.
APX-011 radiated light drawn as a fan
The light a Y-branch combiner radiates away is drawn as a faint fan; its power is exact, its shape illustrative.
What we assume
Light a Y-branch combiner loses (its loss port, the second mode) is drawn as a faint fan of two lobes that starts at the junction and spreads forward around the single waveguide, fading with distance. Its brightness follows the radiated power, mapped like a waveguide's (P^(1/2.2), scaled down); the power itself is exact, from the simulator. The fan's shape, reach and angles are illustrative.
Reality
The out-of-phase part of the light excites the second-order mode or radiation modes of the single waveguide and leaves into the cladding and substrate, as an interference pattern that depends on the exact junction (Chrostowski & Hochberg 2015 Fig. 4.25 simulates one). Where it ends up (absorbed, scattered, or coupling into nearby waveguides) is not modelled.
Why it's OK here
Without it, a combiner looks as if it destroyed light. The fan shows that the light leaves the circuit, and how much; its two lobes echo the second mode's null on the axis. The inspector gives the exact number ("Radiated").
Where it breaks
The fan's size and direction are not a field calculation; radiated light never reaches other waveguides or detectors in the lab.
Sources
Chrostowski2015 §4.2, p. 111 (the combiner's second mode), Figs. 4.23 and 4.25, pp. 112–113 (simulated field of a combiner, single and out-of-phase inputs)
MMI couplers are ideal and lossless, and their sizes are estimates.
What we assume
The 1×2 MMI splits into two equal images in phase; the 2×2 MMI is an exact 50:50 coupler with its cross path 90° behind its bar path. Both are lossless, perfectly balanced and the same at every wavelength. The multimode section's propagation phase is counted as the strip waveguide's over the section's length (half before, half after the lumped model), as for the directional coupler. Sizes follow the paper's approximate formulas: section width 6 µm (estimate), effective width and beat length from its Eqs. (4) and (6) with the 220 nm slab's index, length 3L_π/8 (1×2, symmetric interference) or 3L_π/2 (2×2, general interference), about 35 µm and 141 µm. Access waveguides are straight 500 nm strips at ±W_e/4, without tapers.
Reality
Real MMIs have excess loss (0.1–3 dB reported) and imbalance, depend on wavelength (L_π ∝ 1/λ, Eq. (40)), reflect part of the light a combiner can't image back (§VI-D), and are designed with a mode solver and tapered access waveguides. The section's own phase comes from its fundamental mode (Eq. (5)), not the strip's.
Why it's OK here
The ideal S-matrices follow from the self-imaging analysis exactly (Eqs. (21), (37)), and they show what MMIs are for: in-phase splitting, and a coupler whose split doesn't depend on a gap. In an interferometer both arms pass the same MMIs, so the section's common phase cancels.
Where it breaks
Absolute phases through an MMI, spectra, loss budgets, footprints of real devices.
Sources
Soldano1995 Eqs. (4)–(7), p. 616; Eq. (21), p. 618; Eqs. (34)–(37) and Table I, p. 620; Table III, p. 625; Chrostowski & Hochberg 2015 §3.2.2, p. 53 (slab index 2.845, SiO₂ 1.444)
Used in
The simulator and the lab only.
APX-013 the light inside an MMI is drawn from the mode propagation analysis
The light inside an MMI is drawn from the paper's mode analysis; the numbers come from the simulator.
What we assume
The lab draws the light inside an MMI's multimode section as the paper's guided-mode propagation analysis gives it: sine-like modes over the effective width (Eq. (23)), phases from the quadratic approximation (Eq. (7)), the light arriving at each face from the simulator, each access waveguide's field taken as a Gaussian (1/e half-width 0.4 µm, estimate). Light going both ways adds as power.
Reality
The real field follows the section's exact modes (Eq. (5) is approximate, §VI-B), includes radiation modes, and the access waveguides' true mode shape.
Why it's OK here
It shows self-imaging as the paper explains it, with the student's own input light. At the output face the images carry the simulator's split to within a few percent (tested in apps/lab/test/examples.test.ts); the numbers students read come from the simulator, not from the drawing.
Where it breaks
Fine details of the pattern, its brightness away from the images.
Sources
Soldano1995 Eqs. (7)–(12), (23), pp. 616–618; Figs. 5 and 7, pp. 619–620 (such patterns)
Used in
The simulator and the lab only.
APX-014 ideal waveguide crossing
A waveguide crossing passes the light straight through, without loss or crosstalk.
What we assume
A crossing passes each arm straight through with transmission 1; nothing leaks into the other arm or comes back. Its phase is its drawn length's, as a strip waveguide. The lab draws the double-etch design of Bogaerts et al. 2007 (Fig. 2(b)): the parabolic expanders as a silicon area, 500 nm arms, 8 × 8 µm in all.
Reality
The measured double-etch crossing loses 0.16 dB and leaks below −40 dB into the side arms; a direct crossing of two strips loses about a third of the light.
Why it's OK here
A well-designed crossing is close to ideal, and the lab's point is topology: crossings let circuits that aren't planar be drawn at all.
Where it breaks
Circuits with many crossings in series (21 real ones cost 3.4 dB), and cross-talk budgets.
Sources
Bogaerts2007 Figs. 1, 2 and 4, pp. 2801–2803
Used in
The simulator and the lab only.
APX-015 ring resonator cells with lumped couplings
A ring couples to its bus at one point, independent of wavelength, without loss or backscattering.
What we assume
The ring cells (all-pass and add-drop, radius 10 µm) couple ring and bus through a lumped coupler at the point where they are closest, with κ² set directly (one value for both couplings of an add-drop ring), independent of wavelength. The heater's phase shift is lumped into the ring. The ring's length is its drawn circumference, a bend counted as a straight waveguide (APX-009); no loss, no backscattering.
Reality
The coupling builds up along the curved gap and depends on gap and wavelength; bends lose light and change n_eff; sidewall roughness backscatters light into the other direction (Bogaerts et al. 2012 §2.6); a heater heats part of the ring.
Why it's OK here
With these assumptions the cells reproduce the textbook ring formulas exactly (Bogaerts et al. 2012 Eqs. (1), (5), (6), tested), which is what students learn: resonances one free spectral range apart, all-pass phase, add-drop filtering. In the drawn light only, each coupling is spread over 3 µm either side of the point (the directional-coupler picture of APX-008, exchange evenly along it), so brightness and phase do not jump where bus and ring meet; it equals the exact light outside that stretch.
Where it breaks
Real extinction and Q (they need loss), wavelength dependence of κ, split resonances from backscattering.
Sources
Bogaerts2012 §2.1–2.3, Eqs. (1), (5), (6), (9), pp. 48–50
Used in
The simulator and the lab only.
APX-016 a heated waveguide's index changes like bulk silicon
A heated waveguide's index changes like bulk silicon's, with one uniform temperature.
What we assume
The phase change of a heated waveguide section is Δφ = 2π (dn/dT) ΔT L / λ, with the thermo-optic coefficient of bulk silicon (dn/dT = 1.87 × 10⁻⁴ K⁻¹, si-dndt) for the waveguide's effective index, and one uniform temperature rise ΔT over the heated length L.
Reality
Part of the mode's power travels in the oxide, whose dn/dT is 6.3× smaller (Chrostowski & Hochberg 2015 §3.1.2, p. 51), so dn_eff/dT is somewhat smaller than silicon's; the temperature under a heater is not uniform and heat spreads beyond it, to neighbouring waveguides too (Chrostowski & Hochberg 2015 §6.5.2, p. 237); dn/dT itself depends on temperature and wavelength.
Why it's OK here
The articles use it for orders of magnitude (tens of kelvin for half a cycle over 100 µm) and label the result an estimate; most of the power of a 500 × 220 nm strip mode is in the silicon.
Where it breaks
A heater's real efficiency (mW per π), thermal crosstalk, and anything that needs the phase to better than about 20 %.
Sources
Chrostowski2015 §3.1.1, p. 51 (dn/dT); §3.1.2, p. 51 (oxide); §6.5.2, p. 237
APX-017 a slab waveguide as the picture of a strip waveguide
The waveguide article explains modes with an infinitely wide slab instead of a strip.
What we assume
The waveguide article explains modes with a slab waveguide: a silicon layer between oxide, infinitely wide, solved exactly for TE polarisation (packages/sim/src/slab.ts).
Reality
A strip waveguide confines light in both directions of its cross-section. Its effective index is lower than the slab's (2.443 for 500 × 220 nm against 2.845 for the 220 nm slab, Chrostowski & Hochberg 2015 p. 61 and p. 53), and its mode must be computed numerically.
Why it's OK here
The slab shows every idea the article needs (a mode as a pattern that fits, the effective index between core and cladding, the field reaching into the cladding, one mode for a thin core) with an exact solution, and it is the first step of the book's own design procedure (Chrostowski & Hochberg 2015 §3.2, p. 53).
Where it breaks
Numbers for real strip waveguides (use soi-strip-neff), the width dependence, polarisation.
Sources
Osgood2021 §3.3–3.4; Chrostowski & Hochberg 2015 §3.2.2, p. 53 (the 220 nm slab, reproduced in packages/sim/test/wave.test.ts)
Used in
The simulator and the lab only.
APX-018 atoms as driven oscillators with one resonance
The material widget uses identical atoms with one resonance and no damping.
What we assume
The *Light in a material* widget (F2) models a material as identical atoms, each an electron bound to its nucleus like a mass on a spring, with one resonance frequency ω₀ and no damping. Its index follows the dense-medium dispersion equation with one oscillator, (n² − 1)/(n² + 2) = (S/3)/(1 − (ω/ω₀)²), with S ∝ the density of atoms. The wave keeps its amplitude inside: nothing is reflected at the faces and nothing absorbed.
Reality
A material has several resonances with different strengths (oscillator strengths f_j, Eq. (3.73)); the electrons are damped, so light near a resonance is absorbed and n becomes complex; each face reflects part of the light; even in bright sunlight the electrons move by less than 10⁻¹⁷ m (Hecht 2017 p. 79), not the drawn millimetres.
Why it's OK here
The widget shows the mechanism, not a material. Below resonance and far from it, damping hardly matters (ω₀² − ω² ≫ γω, Hecht 2017 p. 81) and the model gives the observed trend, n > 1 rising towards resonance (p. 103). The numbers for silicon and glass in the text come from measurements, not from this model.
Where it breaks
Near or above resonance (absorption, n < 1), any real material's n(λ), reflection and transmission at a surface.
Sources
Hecht2017 §3.5.1, Eqs. (3.63)–(3.66), (3.70), pp. 79–80, Eqs. (3.73)–(3.74), p. 81; §4.2, pp. 101–103, Figs. 4.9–4.11
The 2D mode pictures compute one field component on a grid, which puts n_eff of a silicon strip about 1–2 % too high.
What we assume
The figures that draw a waveguide's mode in its cross-section (F7, the strip; F9, the coupler's supermodes) solve for the dominant field component alone: E_x for TE, E_y for TM. The component obeys the Helmholtz equation, with its normal part n²E continuous at index jumps, on a rectangular grid of 20 nm cells; the field is zero at the edge of the window (packages/sim/src/modes2d.ts).
Reality
A mode of a high-contrast waveguide is hybrid: all six field components are non-zero, and the small components are coupled to the large one at the corners and sidewalls. A full-vector mode solver gives n_eff = 2.443 for the 500 × 220 nm strip (Chrostowski & Hochberg 2015 Fig. 3.14, p. 61); the semi-vectorial solver gives about 2.48. The cross-over length of a 200 nm coupler comes out 10 % shorter than the book's 37.5 µm (§4.1.6, p. 106). The grid adds staircase edges, smoothed by averaging n² over each cell.
Why it's OK here
The figures show what a mode looks like (its shape, its zeros, how far it reaches into the oxide, which modes exist), which the approximation gets right. In the infinitely wide limit it reproduces the slab exactly (2.845 TE, 2.051 TM, Chrostowski & Hochberg 2015 p. 53; packages/sim/test/modes2d.test.ts). The numbers the articles and the lab calculate with are the book's full-vector values (soi-strip-neff, soi-strip-dc-crossover).
Where it breaks
N_eff to better than 2 %, the small field components, modes near cut-off (they reach the window's edge), polarisation conversion.
The fibre figure uses the LP modes, which assume the core's index is only slightly higher than the cladding's.
What we assume
The fibre's modes are linearly polarised (LP) modes with a scalar field u(r) cos lφ, Bessel functions J_l in the core and K_l in the cladding, which holds when n₁ − n₂ ≪ n₁ (packages/sim/src/fibre.ts).
Reality
The exact modes of a round core are hybrid (HE, EH, TE, TM); several of them share each LP mode's pattern and differ slightly in n_eff (Saleh & Teich 2019 §10.2A).
Why it's OK here
A standard glass fibre is weakly guiding (Δ ≈ 0.25 % in the figure's fibre, Saleh & Teich 2019 Example 10.2-2), so the LP modes are accurate there; the cut-offs reproduce Table 10.2-1 (packages/sim/test/fibre.test.ts).
Where it breaks
High-contrast cores (silicon, or glass in air), polarisation effects, splitting between the hybrid modes of one LP group.