A frequency range where |Tr(M_period)/2| > 1 so no real Bloch wavevector k exists in the infinite periodic limit — waves cannot propagate through the crystal as extended states.

Why does a λ/4 stack reflect strongly?

When n₁d₁ ≈ n₂d₂ ≈ λ/4, reflections from successive interfaces add in phase, producing a wide high-reflectance band centered near λ — the basis of distributed Bragg reflectors.

Why do more periods sharpen the features?

Each period contributes coherent partial reflections; increasing N sharpens transmission dips and raises peak reflectance inside stop bands for a finite stack.

Oblique incidence, TM polarization, absorption, dispersion n(λ), and defect modes inside the gap are not modeled.

1D Photonic Crystal Bandgap

A one-dimensional photonic crystal is a periodic stack of dielectric layers with alternating refractive indices n₁ and n₂ and thicknesses d₁, d₂. This simulator builds the TE transfer matrix for one bilayer period, raises it to the N-th power for a finite coherent stack in air, and computes wavelength-dependent transmission T(λ) and reflection R(λ). For an infinite periodic structure the Bloch condition cos(kΛ) = (T₁₁ + T₂₂)/2 separates propagating bands (|cos| ≤ 1) from stop bands or photonic band gaps (|cos| > 1, no real Bloch wavevector). The canvas shows the layer stack, spectra with gap shading, and the trace of the period matrix versus wavelength; a quarter-wave preset illustrates a high-reflectance band near the design wavelength. Assumptions: normal incidence, scalar TE matrices, lossless dielectrics, and semi-infinite air cladding on both sides.

Who it's for: Undergraduate photonics and solid-state optics students after thin films and before full 2D/3D photonic crystal band-structure solvers.

Key terms

Photonic crystal
Photonic band gap
Transfer matrix
Bloch theorem
Stop band
Bragg stack
Quarter-wave mirror
Multilayer interference

Live graphs

How it works

1D photonic crystal: alternating n₁/n₂ layers, transfer-matrix transmission spectrum, and stop bands where the Bloch trace exceeds unity.

Key equations

cos(kΛ) = (T₁₁ + T₂₂)/2 · |·|>1 → stop band

TE transfer matrix per layer; stack M^N, air | stack | air

Frequently asked questions

What is a stop band?: A frequency range where |Tr(M_period)/2| > 1 so no real Bloch wavevector k exists in the infinite periodic limit — waves cannot propagate through the crystal as extended states.
Why does a λ/4 stack reflect strongly?: When n₁d₁ ≈ n₂d₂ ≈ λ/4, reflections from successive interfaces add in phase, producing a wide high-reflectance band centered near λ — the basis of distributed Bragg reflectors.
Why do more periods sharpen the features?: Each period contributes coherent partial reflections; increasing N sharpens transmission dips and raises peak reflectance inside stop bands for a finite stack.
What is left out?: Oblique incidence, TM polarization, absorption, dispersion n(λ), and defect modes inside the gap are not modeled.

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1D Photonic Crystal Bandgap

Live graphs

How it works

Key equations

Frequently asked questions

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Live graphs

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Key equations

Frequently asked questions