Photonic Crystals and Photonic Band Gaps

A photonic crystal does to light what a silicon crystal does to electrons. Electrons in a semiconductor move through a periodic Coulomb potential and their energies split into bands separated by an electronic band gap, and the whole of solid-state electronics lives around that gap. Replace the periodic potential with a periodic dielectric — a lattice of holes in a slab, a woodpile of rods — and photons inherit the same story. This article consumes one electromagnetic result: the source-free Maxwell equations in a linear, lossless, non-magnetic medium collapse into a single Hermitian eigenproblem for the field, and Hermiticity is the whole engine here, exactly as the reality of energy eigenvalues is the engine of quantum band theory. Because the operator is Hermitian, its eigenfrequencies are real and organize into continuous bands ω_n(k) labelled by a band index n and a crystal wavevector k in the first Brillouin zone. A range of ω that no band reaches for any k or polarization is a photonic band gap: light there cannot propagate — an incident beam is totally reflected and its field decays evanescently into the bulk. The gap is coherent Bragg scattering seen through a variational lens: at the zone boundary the medium offers two ways to stand a wave, one hoarding its energy in the high-index material and one forced into the low-index voids, and the frequency split between them is the gap. The reward is control. Break the periodicity on purpose — remove one rod, widen one row of holes — and you carve a state into the forbidden gap that has nowhere to leak: an ultra-high-Q cavity, a waveguide that turns a right-angle corner without radiating, a hollow fiber core guiding light in air. These devices underpin aq-optics-photonic-crystal-fibers, on-chip aq-optics-microring-resonators, and aq-optics-integrated-photonic-circuits.

Key takeaways

A photonic crystal is a dielectric whose permittivity ε(r) is periodic on the scale of the optical wavelength; its modes organize into bands ω_n(k), and gaps between them are photonic band gaps where no propagating mode exists.
The master equation ∇×(ε⁻¹∇×H) = (ω/c)²H is a Hermitian, positive-semidefinite eigenproblem; Bloch's theorem writes each mode as a plane wave times a lattice-periodic envelope, H = e^{ik·r}u(r).
A complete gap is forbidden for every wavevector and polarization; in practice, robust complete gaps are typically obtained in 2D or 3D structures with suitable geometry and sufficiently strong index contrast, whereas a 1D Bragg stack generally provides a stop band whose angular and polarization dependence must be considered.
Gaps open at zone boundaries because the lower (dielectric) band pools field energy in high-ε regions while the upper (air) band, forced orthogonal, sits in low-ε regions at higher frequency — wider gaps need higher contrast.
Maxwell's equations have no built-in length scale: rescaling by s rescales every frequency by 1/s, so band diagrams are universal in ωa/2πc = a/λ.
Key idea: because Θ = ∇×(ε⁻¹∇×) is Hermitian and positive-semidefinite, its eigenvalues (ω/c)² are real and obey a min-max principle — low-frequency modes pool their energy in high-ε material, orthogonality forces the next band into low-ε at higher ω, and periodicity plus enough index contrast turns that split into a complete gap.

The core insight

Because Θ ≡ ∇×(ε⁻¹∇×) is a Hermitian, positive-semidefinite operator, its eigenvalues (ω/c)² are real and obey a variational min-max principle: the lowest modes concentrate displacement-field energy in high-ε regions to minimize frequency, orthogonality pushes the next band into the low-ε regions at higher frequency, and periodicity plus sufficient index contrast opens a complete photonic band gap between them.

Where this lives today

The most visible modern payoff is the silicon photonic-crystal slab: a thin membrane perforated with a triangular lattice of air holes, patterned by lithography, that confines light in-plane through the photonic-crystal band structure and out-of-plane primarily by total internal reflection. Omit one hole and you get an L3 or H1 nanocavity with mode volume near (λ/2n)³ and Q now exceeding a million; couple it to a quantum dot and you have on-chip cavity QED — a Purcell-enhanced single-photon source or a nanowatt-threshold laser. Remove a row of holes and you get a waveguide whose band can be flattened for slow light. In fiber form the same physics gives hollow-core photonic-bandgap fiber, whose microstructured cladding traps light in an air core, carrying high peak powers and gas-filled nonlinear media a solid core could never support. These are commercial components, feeding integrated photonic circuits, frequency combs, and precision spectroscopy.

The physical picture

Start with the simplest periodic dielectric: a distributed Bragg reflector, a stack of alternating high- and low-index films. A wave satisfying the Bragg condition undergoes strong coherent backscattering, producing counter-propagating waves that form standing-wave Bloch states near the zone boundary. But a standing wave at the zone boundary can be pinned in two distinct ways, and that is the crux. One parks its field antinodes in the high-ε slabs; the other, forced orthogonal, is pushed into the low-ε gaps. Pooling energy in high-ε material is cheap (low frequency), pooling it in air is expensive (high frequency), so the single Bragg frequency splits into two band edges. The frequency interval between the two band edges is a stop band for the corresponding Bloch modes; in a structure with a complete gap, no propagating Bloch mode of any polarization exists in that interval.

Figure

A one-dimensional stack with ε_hi = 12 and ε_lo = 1 offers two standing waves at the zone edge. The lower-frequency dielectric band (blue) pools its field-intensity in the high-ε slabs, the higher-frequency air band (red) is forced into the low-ε voids, and the split is the photonic band gap

Now stack this into two or three dimensions. To obtain a complete band gap in a given dimensionality, the structure must provide a common frequency interval with no propagating Bloch modes for all relevant wavevectors and polarizations. Three-dimensional complete gaps generally require periodic structure and suitable index contrast and geometry in all three dimensions, while two-dimensional photonic crystals can have complete gaps for in-plane propagation. The band diagram below is how this is read: dimensionless frequency ωa/2πc against wavevector across the Brillouin zone, with the forbidden window shaded.

Figure

A one-dimensional photonic band structure in dimensionless frequency ωa/2πc versus wavevector from Γ to the zone edge k = π/a. The dielectric band flattens at ωa/2πc ≈ 0.245 and the air band starts at ≈ 0.315, leaving a photonic band gap of about 25 percent

A defect, finally, is a trap: adding or removing dielectric pulls a state into the gap where the crystal is opaque, so the mode is evanescent in every direction and localizes to a cavity of mode volume near (λ/2n)³ — the point of engineering the band structure.

The core equations

The starting point is a rearrangement of Maxwell's equations: eliminating E between Faraday's and Ampère's laws leaves a single equation for H. Writing it for H rather than E is deliberate — it gives a plain eigenvalue on the right and a Hermitian operator on the left.

\nabla\times\!\left(\frac{1}{\varepsilon(\mathbf{r})}\,\nabla\times\mathbf{H}(\mathbf{r})\right)=\left(\frac{\omega}{c}\right)^{2}\mathbf{H}(\mathbf{r}) \qquad (1)

where: H(r) — the divergence-free magnetic-field mode (the eigenvector); ε(r) — relative permittivity, periodic so that ε(r) = ε(r + R) for every lattice vector R; ω — angular frequency; c — vacuum speed of light. Reading it: the geometry is the input, and the equation returns the allowed frequencies with their field patterns — the eigenvalue being (ω/c)², not ω. The operator Θ ≡ ∇×(ε⁻¹∇×) is Hermitian and, because ε is real and positive, positive-semidefinite. Together these properties make (ω/c)² real and non-negative, so the physical frequencies can be chosen real and non-negative, with orthogonal modes for distinct eigenvalues.

\langle\mathbf{H}_1,\Theta\mathbf{H}_2\rangle=\langle\Theta\mathbf{H}_1,\mathbf{H}_2\rangle,\qquad \langle\mathbf{H},\Theta\mathbf{H}\rangle=\int \varepsilon^{-1}\,|\nabla\times\mathbf{H}|^{2}\,\mathrm{d}^3r\ \ge\ 0 \qquad (2)

where: ⟨·,·⟩ — the inner product ∫H₁*·H₂ d³r; the right integral — the electromagnetic energy functional. Reading it: this is the exact analogue of a Hermitian Hamiltonian — Hermiticity forces real ω, positivity forces (ω/c)² ≥ 0, so no runaway (imaginary-ω) modes exist in a lossless crystal. Periodicity enters through Bloch's theorem. Because Θ commutes with every lattice translation, its eigenmodes are a plane wave times a lattice-periodic envelope, and the frequencies become periodic in the wavevector — so all distinct physics lives in the first Brillouin zone.

\mathbf{H}_{n,\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}\,\mathbf{u}_{n,\mathbf{k}}(\mathbf{r}),\quad \mathbf{u}_{n,\mathbf{k}}(\mathbf{r}+\mathbf{R})=\mathbf{u}_{n,\mathbf{k}}(\mathbf{r}),\qquad \omega_n(\mathbf{k})=\omega_n(\mathbf{k}+\mathbf{G}) \qquad (3)

where: k — crystal wavevector in the first Brillouin zone; n — band index; u — lattice-periodic envelope; R — Bravais lattice vector; G — reciprocal lattice vector with G·R = 2πm. Reading it: since e^{iG·R} = 1, k and k + G label the same mode, so gaps are read off along the zone boundary. The variational principle supplies the intuition. The lowest band frequency is the minimum of the Rayleigh quotient over all fields, each higher band the minimum over fields orthogonal to those below. Because energy is cheap in high-ε material, the winning mode piles its field there.

\left(\frac{\omega_n}{c}\right)^{2}=\min_{\mathbf{H}\perp\mathbf{H}_{1..n-1}}\frac{\displaystyle\int \varepsilon^{-1}\,|\nabla\times\mathbf{H}|^{2}\,\mathrm{d}^3r}{\displaystyle\int |\mathbf{H}|^{2}\,\mathrm{d}^3r} \qquad (4)

where: the constraint H ⟂ H₁..ₙ₋₁ enforces orthogonality to all lower bands. Reading it: in terms of E, minimizing the quotient concentrates the displacement D = εE in high-ε regions. The lowest (dielectric) band does this; the next, forced orthogonal, sits in the low-ε regions at higher ω. The frequency jump is the gap. Scale invariance is what makes photonic-crystal engineering practical. Rescale the structure by s and the master equation is unchanged provided every frequency is divided by s — there is no absolute frequency scale, only ratios to the lattice constant.

\varepsilon'(\mathbf{r})=\varepsilon(\mathbf{r}/s)\ \Rightarrow\ \omega'=\frac{\omega}{s},\qquad u\equiv\frac{\omega a}{2\pi c}=\frac{a}{\lambda} \qquad (5)

where: s — linear scale factor; a — lattice constant; u — dimensionless (gap-normalized) frequency; λ — vacuum wavelength. Reading it: a gap computed once at u = ωa/2πc is placed at any wavelength by choosing a = uλ; a 5 percent oversize red-shifts the whole diagram by 5 percent. For the 1D stack two useful statements are worth carrying: the Bragg condition sets the central stop-band wavelength, and the transfer-matrix solution gives the stop-band edges and, for a quarter-wave stack, a closed-form expression for the bandwidth.

2\,n_\text{eff}\,\Lambda=\lambda_\text{Bragg}\ \Longleftrightarrow\ k=\frac{\pi}{\Lambda}\ (\text{zone edge}),\qquad \frac{\Delta\omega}{\omega_\text{mid}}=\frac{4}{\pi}\arcsin\!\left(\frac{n_2-n_1}{n_2+n_1}\right) \qquad (6)

where: Λ — the 1D period (lattice constant a); n_eff — averaged index; λ_Bragg — free-space Bragg wavelength; n₁, n₂ — layer indices; Δω/ω_mid — fractional (gap-to-midgap) stop-band width. Reading it: the gap grows with contrast n₂ − n₁ and closes for equal-index layers — n₂/n₁ = 1.5 gives about 26 percent, a silicon/silica pair over 50 percent.

Derivation AFrom Maxwell's equations to the Hermitian master eigenproblem

The goal of this derivation is to show that the whole apparatus of photonic band theory rests on one algebraic move: eliminating E between the two curl equations and solving for H. We reach the curl-curl equation (1) from the source-free Maxwell equations and prove in the same breath that the operator is Hermitian and positive-semidefinite — guaranteeing real band frequencies and an orthogonal mode basis.

Step 1. With time dependence e^{−iωt} and no free charge or current, Faraday's and Ampère's laws read ∇×E = iωμ₀H and ∇×H = −iωε₀ε E. Why: these couple E and H; we keep one field and make the other derived, giving a single second-order eigenproblem.

Step 2. Solve Ampère's law for E = (i/ωε₀ε) ∇×H and substitute into Faraday's law; with c² = 1/μ₀ε₀ the constants collect into (ω/c)², giving equation (1). Why: dividing by ε before the outer curl places ε⁻¹ inside the operator; eliminating H instead would leave ε as a weight on the right, a less convenient generalized eigenproblem.

Step 3. Form ⟨H, ΘH⟩ = ∫H·∇×(ε⁻¹∇×H) d³r and integrate by parts twice, dropping boundary terms (fields periodic or vanishing at infinity), giving ∫ε⁻¹|∇×H|² d³r — real and, since ε greater than 0, non-negative. Why: a real symmetric form makes Θ Hermitian, so eigenvalues are real; non-negativity gives (ω/c)² ≥ 0, so ω is real and no growing modes exist. Hermiticity also makes the modes orthogonal and complete — the basis the defect derivation expands in.*

Go deeper · Derivation B · 1 minWhy a gap opens at the Brillouin-zone boundary

Derivation BWhy a gap opens at the Brillouin-zone boundary

The goal here is to show, with a two-state degenerate-perturbation argument, how a weak periodic modulation can open a gap at the zone edge when its Fourier component couples the degenerate states, with a gap width that is first order in the modulation depth, and that the sign of the split is fixed by where each standing wave stores its energy. Take a nearly homogeneous 1D medium ε(z) = ε̄ + Δε cos(2πz/a) with small Δε, whose unperturbed modes are plane waves e^{ikz} with ω = ck/√ε̄, and ask what the modulation does at the zone boundary. Result (gap-1D):

\omega_\pm=\frac{c}{\sqrt{\bar\varepsilon}}\,\frac{\pi}{a}\left(1\mp\frac{\Delta\varepsilon}{4\bar\varepsilon}\right),\qquad \frac{\Delta\omega}{\omega_0}=\frac{\Delta\varepsilon}{2\bar\varepsilon}

Step 1. At the zone edge the states k = +π/a and k = −π/a are degenerate and differ by one reciprocal-lattice vector $G=2\pi/a$ . Why: for a modulation containing a Fourier component at $G$ , this component couples the two states, while other plane waves are generally off resonance in the weak-modulation limit, so the problem reduces to an effective two-by-two matrix.

Step 2. The effective matrix is [[ω₀, V], [V, ω₀]] with coupling V ∝ (Δε/ε̄)ω₀ set by the G-th Fourier coefficient of ε; its eigenvectors are the standing waves cos(πz/a) and sin(πz/a). Why: the modulation lifts the degeneracy as a periodic potential does for a nearly-free electron — the standing waves are the true zone-edge eigenstates.

Step 3. The eigenfrequencies are ω_± = ω₀ ± V. The cosine antinodes sit in the high-ε half (lower frequency), the sine antinodes in the low-ε half (higher frequency); the gap is Δω = 2V, linear in Δε. Why: this reproduces the variational field-concentration argument from an independent calculation, confirming the lower band hoards energy in high-ε. *Why these two results agree: as Δε → 0 the bands collapse to the empty-lattice (free-photon) dispersion, and in that same limit Δε/2ε̄ is exactly the weak-contrast expansion of the arcsin formula (6) — the perturbation result and the exact transfer-matrix formula are consistent.*

Go deeper · Derivation C · 1 minThe localized defect mode inside the gap

Derivation CThe localized defect mode inside the gap

The goal of this derivation is to explain why breaking the periodicity at a single site produces a state inside the gap that is exponentially localized, and to connect its confinement to the mode volume and quality factor that make photonic-crystal cavities useful. We model the defect as a local perturbation on the perfect-crystal operator Θ₀ and track a state pulled off a band edge into the forbidden window. Result (defect-decay):

\mathbf{H}_\text{def}(\mathbf{r})\sim e^{-\kappa|\mathbf{r}|},\qquad \kappa=\kappa(\omega_\text{def})>0\ \ \text{for}\ \ \omega_\text{def}\in(\omega_-,\omega_+)

Step 1. At any frequency inside the gap the perfect crystal has no Bloch mode with real k; the dispersion relation is satisfied only by a complex wavevector k = k_r + iκ. Why: a real k would name a propagating band state, and none exists in the gap — the analytic continuation into the gap has an imaginary part, the evanescent decay rate.

Step 2. Adding dielectric locally lowers a mode's frequency and drags a state down from the air band into the gap (a donor mode); removing material raises one up from the dielectric band (an acceptor mode). Why: this is the variational principle applied in one cell — more high-ε near the defect lets a mode lower its energy.

Step 3. Because the tails decay over a few lattice constants, the mode volume shrinks toward V_mode ∼ (λ/2n)³ while Q = ω_def/Δω_res is limited only by residual leakage — out-of-plane radiation for a 2D slab, with no radiation loss in an ideal infinite 3D crystal. Why: near-diffraction-limited volume with very high Q gives these nanocavities their enormous Purcell factor, behind single-photon sources and low-threshold lasers. *Why this makes sense: κ vanishes at the band edges — where the gap just opens and the evanescent decay barely begins — and peaks at mid-gap, where the crystal is most opaque, consistent with the tunneling picture of evanescent penetration.*

Limiting cases

Case	Result	Meaning
Long-wavelength (homogenization)	ω → 0, v_ph → c/√ε_eff	When λ ≫ a the crystal looks uniform; the lowest band is linear and the medium acts as an effective dielectric with ε_eff set by the fill fraction.
Weak-contrast (empty-lattice)	$\Delta\omega/\omega \approx \Delta\varepsilon/2\bar{\varepsilon} \propto \Delta\varepsilon$	Bands are nearly free-photon, folded into the zone; gaps open only at zone boundaries, width first order in the modulation depth.
Strong-contrast / complete gap	n_hi/n_lo ≳ 2 (diamond), ≳ 2.8 to 3 (many 2D, woodpile)	A full 3D gap needs the directional stop bands to overlap; diamond does it near contrast 2 (silicon/air suffices). Below threshold only a directional or single-polarization gap survives.
Defect (mid-gap) state	ω_def ∈ (ω₋, ω₊), \	H\	∼ e^{−κr}	A localized state inside the gap decays evanescently; κ is largest at mid-gap, so the deeper the state, the tighter the confinement.
Band edge (slow light)	$v_g = \mathrm{d}\omega/\mathrm{d}k \to 0$	Where a band flattens the group velocity collapses, enhancing light–matter interaction per length — the basis of slow-light devices.

Common trapA photonic band gap is the same thing as a Bragg mirror's reflection band

It is not. A 1D Bragg mirror (distributed Bragg reflector) has only a stop band, and a stop band is directional: it forbids propagation near-normal to the layers, but as the incidence angle grows the effective period changes, the band slides, and eventually closes — no single frequency is reflected from every direction. A complete (omnidirectional) photonic band gap forbids propagation for every wavevector and polarization in the specified frequency interval. Such gaps require multidimensional periodicity and are favored by sufficiently large index contrast, although the required contrast depends on the lattice geometry, filling fraction, and polarization. Most 2D lattices open a gap for only one polarization (TE or TM). Two further traps: the eigenvalue is (ω/c)², not ω, so the operator has no built-in frequency scale; and the lowest band always starts at ω = 0, because photons have no rest mass.

Worked examples

1 · The stop band of a silicon / silica Bragg mirror (confidence builder)

A quarter-wave Bragg reflector for the telecom band uses silicon (n₂ = 3.48) and silica (n₁ = 1.45), centered at λ₀ = 1.55 μm. Find the gap-to-midgap ratio, the layer thicknesses, and the stop-band edges.

Step 1. The fractional width comes from equation (6): Δω/ω_mid = (4/π) arcsin[(n₂ − n₁)/(n₂ + n₁)]. The contrast is (3.48 − 1.45)/(3.48 + 1.45) = 2.03/4.93 = 0.4118, so Δω/ω_mid = (4/π) arcsin(0.4118) = (1.2732)(0.4244) = 0.540 — a 54 percent gap, far beyond the n₂/n₁ = 1.5 case that gives only 26 percent.

Step 2. Each layer is a quarter-wave thick, d = λ₀/(4n): silicon d₂ = 1550/(4 × 3.48) = 111.4 nm, silica d₁ = 1550/(4 × 1.45) = 267.2 nm, period a = 378.6 nm — sub-wavelength, so these mirrors are grown as thin films.

Step 3. A fractional width of 0.540 spans $\omega_{\rm mid}(1\pm0.270)$ . In wavelength ( $\lambda\propto1/\omega$ ) the edges are $1550/1.270=1220$ nm and $1550/0.730=2123$ nm, so at normal incidence the mirror reflects from about 1220 nm to 2123 nm — the whole telecom band and beyond. Δω/ω_mid = 0.54; layers 111 and 267 nm; stop band ≈ 1220 to 2120 nm.

2 · Placing a 2D gap at 1.55 μm, and a fabrication error

A triangular lattice of air holes in silicon has its lowest TE gap centered at ωa/2πc ≈ 0.30 with a gap-to-midgap ratio of about 20 percent. What lattice constant a puts the mid-gap at λ = 1.55 μm, and if the pattern is fabricated 5 percent too large, where does the gap move?

Step 1. By scale invariance (equation 5) the mid-gap is fixed by a/λ = u. Setting u = 0.30 and λ = 1.55 μm gives a = uλ = 0.30 × 1550 = 465 nm — no band structure is recomputed, the dimensionless diagram is placed by one multiplication. A 20 percent gap then runs 1409 to 1722 nm, bracketing the C-band.

Step 2. A 5 percent oversize is s = 1.05: every frequency scales by 1/s, every wavelength by s, so the mid-gap shifts to λ' = 1.05 × 1550 = 1628 nm — a pure 5 percent red-shift, no new physics, which is why nanometre lithography control matters. a = 465 nm; a 5 percent oversize red-shifts the mid-gap from 1550 nm to 1628 nm.

3 · How weak a modulation still opens a gap

A nearly homogeneous 1D medium has mean permittivity ε̄ = 12 (n̄ ≈ 3.46) with a small sinusoidal modulation Δε = 2. Using the perturbation result, find the fractional gap, and compare it with the strong-contrast Bragg value.

Step 1. The gap-1D result gives Δω/ω₀ = Δε/(2ε̄) = 2/(2 × 12) = 0.083, an 8.3 percent gap — small and linear in the modulation depth, as degenerate perturbation theory predicts. Placed at 1.55 μm (where λ = 2√ε̄ a gives a = 224 nm) the window is about 129 nm wide.

Step 2. Weak modulation always opens some gap at the zone edge, but a wide, robust gap — and any complete 2D or 3D gap — demands strong contrast. Δε = 2 in ε̄ = 12 gives only 8.3 percent, versus 54 percent for the silicon/silica mirror.

Physics of the portrait

Photonic band gaps were not stumbled on in a spectrometer; they were predicted on paper by two theorists in the same 1987 volume of Physical Review Letters, and the answer reorganized a corner of optics around solid-state language.

Origins · 1 minEli Yablonovitch and how this was discovered

Eli Yablonovitch

b. 1946 · inventor of the photonic band gap

Eli Yablonovitch was working on semiconductor-laser efficiency when he asked a deceptively simple question: could a periodic dielectric inhibit spontaneous emission by denying an excited atom any mode to radiate into? His 1987 paper (Physical Review Letters 58, 2059) proposed exactly that — a 3D structure with a full photonic band gap would suppress the spontaneous decay that limits laser and solar-cell performance — and in the same volume Sajeev John (Physical Review Letters 58, 2486) showed disorder in such a structure could Anderson-localize photons. Those two ideas launched the field. To prove a complete gap was achievable, Yablonovitch in 1991 worked at microwave wavelengths, drilling a face-centred-cubic array of intersecting holes into a dielectric block and measuring a genuine omnidirectional gap. That structure became Yablonovite, the first demonstrated 3D photonic band-gap crystal, and by scale invariance the same design shrunk four orders of magnitude works in the infrared. His framing — that one can engineer the density of optical states as freely as a chip designer engineers a band structure — is the thread connecting Bragg mirrors, defect nanocavities, slow-light waveguides, and hollow-core fibers into one subject.

History

1887

— Rayleigh shows a one-dimensional periodic stack has a frequency stop band

1972

— Bykov proposes that a periodic structure can control spontaneous emission

1987

— Yablonovitch and John independently introduce the photonic band gap

1991

— Yablonovitch machines Yablonovite, the first 3D band-gap crystal at microwave frequencies

1996

— Russell, Knight and colleagues demonstrate the first photonic-crystal fiber

2003

— Noda's group reaches photonic-crystal nanocavity Q above 10⁵ in silicon

Going further

Texts. Joannopoulos, Johnson, Winn and Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (2008), chapters 2 to 7 · Saleh and Teich, Fundamentals of Photonics, 3rd ed., chapter 7 · Yeh, Optical Waves in Layered Media · Ashcroft and Mermin, Solid State Physics (Bloch's theorem, Brillouin zones). The zone-edge result Δω/ω₀ = Δε/2ε̄ is standard degenerate perturbation theory; the quarter-wave arcsin formula follows from the transfer-matrix trace. Key numerical results: silicon/silica (n = 3.48, 1.45) gives a 54 percent gap with layers 111 and 267 nm at 1.55 μm; n₂/n₁ = 1.5 gives 26 percent; Δε = 2 in ε̄ = 12 gives 8.3 percent. Complete-gap contrast thresholds are near 2 for the diamond lattice and 2.8 to 3 for woodpile and many 2D lattices. The photonic band-gap concept was introduced by Yablonovitch (Physical Review Letters 58, 2059, 1987) and John (Physical Review Letters 58, 2486, 1987).

Prerequisites. Waveguide Modes and the Wave Equation · Total Internal Reflection and Optical Guiding · Numerical Aperture and Acceptance Angle. Leads to. Photonic-Crystal and Hollow-Core Fibers · Microring Resonators and Optical Cavities on Chip · Integrated Photonic Waveguides and Circuits · Coupled-Mode Theory and Directional Couplers.

Ask the tutor: Want to see how an L3 nanocavity carves a mode out of a 2D silicon gap, or how a hollow-core fiber's cladding gap replaces total internal reflection? Just ask.

Is a photonic band gap for light really the same idea as an electronic band gap in silicon?

Structurally, yes — photonic band theory borrows Bloch's theorem, Brillouin zones, and band diagrams directly from solid-state physics, and in both a wave in a periodic medium organizes into bands with gaps of no propagating states. But two differences matter. First, photons obey a Hermitian eigenproblem with eigenvalue (ω/c)², not a Schrödinger equation with eigenvalue E, so the operator is scale-invariant and vector — polarization is a genuine extra label. Second, photons have no rest mass, so the lowest band always starts at ω = 0 with a linear slope; there is no gap below the first band the way there is a floor to electron energies.

Why does the lower band store its energy in the high-index material?

It is forced to by the variational principle. The band frequency is the minimum of the Rayleigh quotient (4), which in terms of E is minimized by concentrating the displacement D = εE where ε is large — piling energy into high-index material is cheap. The lowest mode does exactly this and earns the name dielectric band; the next band must be orthogonal to it, so its energy is forced into the low-ε regions at higher frequency. That separation is the gap, which is why higher contrast makes wider gaps.

What is the difference between a stop band and a complete gap?

A stop band is directional: a 1D Bragg mirror forbids propagation only near its stacking direction, and tilting the incidence slides the band until it closes — it blocks some directions, never all. A complete photonic band gap is forbidden for every wavevector and polarization at once, an omnidirectional mirror. Getting one needs periodicity in 2D or 3D so the directional stop bands overlap, plus high contrast so the overlap is non-empty. Many 2D lattices manage a gap for only one polarization; a full 3D gap is harder still, which is why the diamond and woodpile geometries are prized.

Why are band diagrams always drawn in dimensionless units ωa/2πc?

Because Maxwell's equations carry no intrinsic length or frequency scale. Shrink a crystal by s and the solutions are identical except that every frequency is multiplied by s (equation 5). A gap computed once therefore describes a whole family of structures, and the only meaningful quantity is the wavelength-to-lattice ratio u = ωa/2πc = a/λ — so one diagram serves a centimetre microwave crystal and a 400-nanometre optical one, and to build at a target wavelength you set a = uλ. It also explains why a few percent lithography error walks the gap off your design wavelength by that same few percent.

How does a defect trap light if the surrounding crystal is transparent to nothing at that frequency?

That is exactly why it traps. At a defect frequency inside the gap the surrounding crystal has no propagating Bloch mode — the only solutions have a complex wavevector, so any field there decays exponentially. A mode created at the defect has nowhere to escape to: it is evanescent in every direction and stays pinned, decaying over a few lattice constants. Adding dielectric pulls a state down from the air band, removing it pushes one up from the dielectric band, and either way you get a cavity with mode volume approaching (λ/2n)³ and Q limited only by residual leakage.

Does high index contrast alone guarantee a complete gap?

No — contrast is necessary but not sufficient; geometry matters just as much. High contrast makes the directional stop bands wide, but a complete gap also needs those bands, at different frequencies for different directions, to overlap into one common window. Some lattices arrange this far better than others: the diamond lattice opens a full 3D gap near contrast 2 (silicon and air suffice), whereas many 2D lattices and the woodpile need 2.8 to 3, and some geometries never open one at all. Polarization adds a further hurdle: a 2D lattice can have a fine TE gap and no TM gap, so a complete 2D gap is a stringent target.