Photonic Crystal Bandgap Calculator

This tool models a 1D photonic crystal (alternating layers) and estimates the first stopband near the Bragg condition.

• Snell's law (layer angles): \(n_0\sin\theta_0 = n_i\sin\theta_i\)
• Bragg center wavelength (order m): \( m\,\lambda_0 = 2\,(n_H d_H\cos\theta_H + n_L d_L\cos\theta_L) \)
• Fractional bandwidth (approx.): \(\Delta\lambda/\lambda_0 = \tfrac{4}{\pi}\arcsin\big(\tfrac{|n_H-n_L|}{n_H+n_L}\big)\)
• Band edges: \(\lambda_{\min}=\lambda_0-\tfrac{\Delta\lambda}{2}\), \(\lambda_{\max}=\lambda_0+\tfrac{\Delta\lambda}{2}\)
• Frequency band: \(f=c/\lambda\) and energy: \(E=hc/\lambda\)

These relations are widely used for quick design estimates of multilayer mirrors. Rigorous band diagrams require solving Maxwell's equations with periodic boundary conditions.

1. Enter the refractive indices for the high and low layers.
2. Enter each layer thickness and choose the correct units.
3. Set the ambient index and the external incidence angle.
4. Keep m = 1 unless you need higher-order stopbands.
5. Click Calculate Bandgap to see results above the form.
6. Use Download CSV or Download PDF for reporting.

1) What a photonic bandgap means

A photonic bandgap is a spectral region where a periodic structure strongly suppresses propagation by destructive interference. In 1D multilayers, this appears as a stopband in reflection and transmission. The calculator estimates this stopband around the Bragg condition, which is a practical first-pass design target.

2) Why 1D multilayers are common

Alternating thin films are the simplest photonic crystals to fabricate with repeatable thickness control. They are widely used as dielectric mirrors, edge filters, and resonator stacks. Even when a final design uses 2D or 3D periodicity, 1D estimates help set the correct wavelength scale and material contrast.

3) Role of refractive-index contrast

Higher index contrast increases reflectivity per interface and typically widens the stopband. The bandwidth expression used here depends on the normalized contrast |nH−nL|/(nH+nL). Small contrast yields a narrow bandgap that is sensitive to thickness errors, while large contrast supports a broader gap that is more tolerant.

4) Thickness design and quarter-wave condition

For a strong primary stopband at normal incidence, designers often choose quarter-wave optical thickness: nH dH ≈ nL dL ≈ λ0/4. At oblique incidence, the effective optical thickness includes cos(θi) inside each layer. The ratio check reported by the calculator helps you see how closely the design follows this condition.

5) Angle tuning and bandgap shift

As incidence angle increases, refraction reduces the internal propagation angle in each layer and changes the optical path length. The stopband center generally shifts to shorter wavelengths (higher frequencies). This angle tuning is useful in filters, but it can also cause polarization splitting in real stacks, especially at large angles.

6) Band edges, frequency, and energy views

Band edges can be expressed as wavelengths, frequencies, or photon energies depending on your application. Telecom and spectroscopy often use wavelength; RF photonics and resonators may prefer frequency; detector and emitter studies often use energy in eV. This calculator provides all three, so you can compare designs across disciplines easily.

7) Practical tolerances and material dispersion

Real coatings have thickness tolerances, surface roughness, and wavelength-dependent refractive indices (dispersion). A 1–2% thickness drift can noticeably shift the center wavelength, especially for narrow gaps. If your materials are dispersive, use refractive indices evaluated near the target wavelength and re-check the stopband across your range.

8) When to use rigorous simulation

This tool is intentionally fast and design-oriented. For high accuracy, strong oblique incidence, polarization-specific performance, absorption, chirped stacks, or non-quarter-wave designs, use a transfer-matrix method and, for periodic photonic crystals, solve Maxwell eigenmodes (e.g., plane-wave expansion or FDTD). Treat this calculator as your reliable starting point.

1) Is this a 1D, 2D, or 3D photonic crystal calculator?
It is a 1D multilayer (Bragg stack) approximation. It estimates the stopband center and an approximate bandwidth for alternating layers. 2D and 3D crystals require different numerical band-structure methods.

2) What refractive indices should I enter?
Use the indices at your target wavelength, not a generic catalog value. If your materials are dispersive, choose values near the expected center wavelength and re-run the calculator for nearby wavelengths to assess sensitivity.

3) Why does the bandgap shift with incidence angle?
The effective optical path inside each layer changes with refraction and the cosine factor in the propagation direction. This modifies the Bragg condition, typically shifting the stopband to shorter wavelengths as the angle increases.

4) What does the Bragg order m mean?
The order sets which interference condition you target. m = 1 is the primary stopband and usually the strongest. Higher orders can appear at shorter wavelengths, but they are less commonly used for robust mirror designs.

5) How accurate is the bandwidth formula?
It is a standard quick-estimate for quarter-wave stacks at normal incidence. It does not fully capture polarization effects, strong oblique incidence behavior, absorption, or chirped designs. Use transfer-matrix simulation for precision.

6) What does the quarter-wave ratio indicate?
It compares optical thicknesses of the two layers. A ratio near 1 suggests a well-balanced quarter-wave design for the chosen angle. Larger mismatch can reduce stopband strength and alter the band edges from the estimate.

7) Why might I see total internal reflection warnings?
At large external angles or strong index differences, Snell’s law can predict an evanescent wave in a layer. In that case, the simple stopband estimate is not reliable. Reduce the angle or adjust indices.