Optics 087 · Waveguides, structured light, and modern optics

Photonic Crystal Band Structure

How can a periodic dielectric block some frequencies of light, yet trap or guide light at defects? Over certain frequency ranges, reflections from the repeating structure add up, which is Bragg reflection, and no wave can travel through: those are the band gaps. The waves that do travel take the Bloch form. A point defect traps light at a frequency inside a gap, acting as a cavity, and a line of defects becomes a channel that guides light along it.

A three-dimensional cutaway shows a 2D TM eigenproblem solved with plane waves. Explore the bands of a square lattice, contours in reciprocal space, supercells with point and line defects, complex fields and checks on the plane-wave cutoff.

Selected normalized eigenfrequency

\nu

$0.50$

Electric-energy fraction in selected region

\eta

$50\%$

Relative algebraic eigen-residual

r_{\rm eig}

$0.00\pi$

Model regime $\text{valid model regime}$

The rods are infinitely long and lossless, only one polarization is solved, the plane-wave basis is cut off at a finite size, and each defect repeats periodically. The gap is found from sampled points and is not a complete 3D band gap. Finite-height slabs, driven transmission and measured devices are not modeled.

Physics tutorial

Audit the physics of Photonic Crystal Band Structure

BackgroundLight in a periodic dielectric is described by a two-dimensional wave eigenproblem: $-\nabla_\perp^2 E_z=(\omega/c)^2\varepsilon_r E_z$ . A Bloch wave is a periodic envelope times a phase that depends on the wavevector. The Lab solves for the electric field along rods that are infinitely long and uniform; it does not model a photonic slab of finite height.

Why it mattersHow can a periodic dielectric block some frequencies of light, yet trap or guide light at defects?

Start with the essentials

Focus question: How can a periodic dielectric block some frequencies of light, yet trap or guide light at defects?
One-sentence intuition: A frequency inside the sampled bulk gap is only the first check. Next, look at the eigenfield and the share of its electric energy inside the region: $\eta_D=\frac{\int_D\varepsilon_r|E_z|^2\,dA}{\int_{\rm cell}\varepsilon_r|E_z|^2\,dA}$ . The confined defect field comes from the solved mode, not a hand-painted Gaussian, and this fraction is not a cavity quality factor.

Core mathematical model

Scalar TM field and the Bloch boundary condition

E_z(\mathbf r+\mathbf R)=e^{i\mathbf k\cdot\mathbf R}E_z(\mathbf r),\quad\nu=\frac{\omega a}{2\pi c}=\frac a{\lambda_0}

Coordinates lie in the plane perpendicular to the rod axis, Z. Time dependence uses a negative-frequency exponential. The relative permeability and the permittivity of air are both one. Only this polarization, with the electric field along the rods, is solved, so the Lab makes no claim about TE gaps or a gap for all polarizations.

A generalized eigenproblem, not a Bragg formula

|\mathbf q+\mathbf g|^2 e_{\mathbf g}=\nu^2\sum_{\mathbf g'}\varepsilon_{\mathbf g-\mathbf g'}e_{\mathbf g'},\quad\mathbf q=\frac{a\mathbf k}{2\pi}

The Fourier coefficients of the circular rods come from integrating over their area. The positive-definite dielectric mass matrix is whitened with a Cholesky factorization, and a real symmetric Jacobi solver then diagonalizes the result. Inversion symmetry makes the Fourier matrices real, but the Bloch fields in space are still complex. The bulk calculation keeps seven plane waves along each reciprocal direction.

Defect supercells and folded bands

\mathbf R_{\rm point}=(5a,5a),\quad\mathbf R_{\rm line}=(a,5a)

A missing rod repeats every five periods in both directions, and its Bloch vector sits at the corner of that supercell's zone. A missing row repeats every five periods across the row. The Lab picks the mode closest to the middle of the sampled bulk gap, choosing only among in-gap modes when any exist. Each channel dot repeats this choice on its own, so the dots are not one continuously tracked band.

Scale invariance versus numerical accuracy

\lambda_0=\frac a\nu,\quad r_{\rm eig}=\frac{\|D\mathbf e-\nu^2 B\mathbf e\|}{\sqrt{\|D\mathbf e\|^2+\|\nu^2 B\mathbf e\|^2}}

With a fixed radius ratio and a nondispersive permittivity, changing the period leaves the normalized frequency unchanged and scales the physical wavelength in proportion. A matrix can be solved accurately and still be a coarse approximation in space. Comparing with a higher cutoff checks one source of error; periodic copies of the defect, sampling in reciprocal space and the finite grid for field integrals are separate sources.

Common difficulties

A missing rod automatically makes a measured resonance?

Typical misconceptionAny bright spot inside a drawn lattice proves a high-quality cavity.

Better mental modelTo call something a cavity, you need a consistent operator, a state inside the gap, spatial confinement and the right boundary for radiation or loss. This Lab solves only the eigenmodes of a periodic, lossless supercell, so it infers no linewidth, source excitation or quality factor.

Run the experiment

01
Scene 1: Bulk Bloch bands
Start with the bulk bands. Slide the wavevector along the boundary path and switch the band index. Compare the field signs, the energy density and the isofrequency contours.
What to observe: Set the rod permittivity to one. The dielectric contrast disappears and the first sampled gap closes. Only the folded plane waves of a uniform medium remain, and a missing rod then makes no optical difference.
02
Scene 2: Point-defect cavity
Remove the central rod, then a whole row. The structure control changes the permittivity matrix itself. Check the selected frequency, how localized the field is and the periodic supercell boundary.
What to observe: Changing the radius changes both the cylinders themselves and their Fourier integrals. Changing only the period stretches the whole cross-section, while the normalized bands stay the same.
03
Scene 3: Line-defect waveguide
Freeze a complex field, change the physical period or the dielectric geometry, then solve again at a higher cutoff. Keep the numerical residual, convergence with cutoff and the finite-supercell error apart.
What to observe: The default point-defect frequency is only an approximation with a finite basis and a periodic supercell. Use the cutoff button to check it. An eigenvalue or a bright central field is not a measured resonance, a transmission or the lifetime of an isolated cavity.