Photonic Crystal Bandgap Engineering

Lattice Geometry

Rod radius r/a 0.20

Rod index n_rod 3.40

Background index n_bg 1.00

Presets

Band Gap (TM, bands 1–2)

Gap–midgap ratio Δω/ω_m

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Midgap ωa/2πc

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Fill fraction f

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Plane waves used

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How it works

For TM-polarized light (E field along the rod axis) in a 2D photonic crystal, Maxwell's equations reduce to a scalar eigenproblem in reciprocal space:

Σ_G' κ(G−G') |k+G| |k+G'| E(G') = (ωa/2πc)² E(G)

κ(G) = Fourier coefficient of 1/ε(r)
     = f/ε_rod + (1−f)/ε_bg           (G = 0)
     = 2f (1/ε_rod − 1/ε_bg) J₁(|G|r)/(|G|r)   (G ≠ 0)

This page diagonalizes that matrix exactly (a real 2D plane-wave expansion, ~5×5 = 25 reciprocal-lattice vectors truncated at |G| ≤ 2·2π/a) at every k-point along the irreducible Brillouin-zone path Γ→X→M→Γ, then plots the resulting bands. A gap between band 1 and band 2 means: for every direction of propagation in the plane, no TM mode exists in that frequency window — light in that range simply cannot propagate through the crystal, regardless of angle.

Rod radius r/a — sets the fill fraction f = πr²/a² and hence the size of the Fourier perturbation κ(G).
n_rod / n_bg — the index contrast ε_rod/ε_bg is what drives band splitting; low contrast (n_rod ≈ n_bg) gives no gap at all, exactly like the nearly-free-electron limit in solids.
The 3D view renders the actual rod lattice the solver used — instanced cylinders on a square grid with the chosen radius and material tint.

Real devices: this is the physics behind photonic-crystal fibers, distributed-feedback lasers and on-chip waveguide filters — a periodic dielectric with no gap-forming defect blocks certain wavelengths from propagating in-plane, which is exactly how a line-defect "wire" or point-defect "cavity" is later carved into the same lattice to guide or trap light.

Dielectric rods (square lattice) Band 1 Band 2

solving bands…

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