Find DC1, DC2, ΔL parameters that match a target spectrum (e.g., 980/1550 WDM coupler).
Waveguide
Targets
Each row: at this wavelength, want P_bar = target (weight scales the cost contribution).
λ (μm)
P_bar target
Weight
Bounds
Best parameters
DC1 gap
—
DC1 L
—
DC2 gap
—
DC2 L
—
Path difference ΔL
—
Best cost
—
Spectrum at best parameters
Range auto-set to span all targets. Target points marked.
How it works
This tool designs an asymmetric Mach–Zehnder interferometer (MZI): two directional
couplers (DC1, DC2) joined by two arms whose optical path lengths differ by ΔL. Because the coupling
is wavelength dependent and the arm phase grows with 1/λ, the device acts as a wavelength filter —
for example a 980/1550 nm WDM coupler that routes one band to the bar port and the other to the cross port.
Bar-port transfer. With κ₁L₁ and κ₂L₂ the coupling angles of the two DCs and
Δφ = (2π·n_eff/λ)·ΔL the arm phase,
P_bar = |cos(κ₁L₁)·cos(κ₂L₂) − sin(κ₁L₁)·sin(κ₂L₂)·ejΔφ|²
= cos²(κ₁L₁)·cos²(κ₂L₂) + sin²(κ₁L₁)·sin²(κ₂L₂) − 2·cos(κ₁L₁)cos(κ₂L₂)·sin(κ₁L₁)sin(κ₂L₂)·cos Δφ.
The cross port carries P_cross = 1 − P_bar.
Coupling coefficient κ(λ). From coupled-mode theory for two identical slab
waveguides (effective-index method, TE–TE):
κ = 2·h²·γ·e−γg / [β·(h²+γ²)·(1+γa)], where h and γ are the lateral guided/evanescent
transverse wavenumbers, β = n_eff·k₀ the propagation constant, a = W/2, and g the coupler gap.
κ decays exponentially with gap and shifts with wavelength — the source of the spectral selectivity.
Index model. Cladding index from the Sellmeier equation (fused silica, Malitson 1965);
core n_core = n_clad/(1 − Δ/100). The fundamental slab mode solves u·tan u = w with u² + w² = V²,
V = k₀·(d/2)·√(n_core² − n_clad²), applied vertically (H) then horizontally (W) to give n_eff(λ).
Optimization. Minimize the weighted least-squares cost
C = Σᵢ wᵢ·(P_bar(λᵢ) − targetᵢ)² over the five variables (g₁, L₁, g₂, L₂, ΔL). A reproducible
multi-start Nelder–Mead search (Mulberry32 seeded RNG) explores the bounded parameter box; moves that
leave the bounds are clipped and lightly penalized. Each target row contributes one wavelength
constraint, so two rows (e.g. bar = 1 at 0.98 μm, bar = 0 at 1.55 μm) yield a WDM design.
The response is wavelength selective because both κ(λ) and Δφ(λ) vary with λ. Feed any solution into
the forward MZI Spectrum tool to inspect its full
transmission curve.