Build layer stacks and inspect wave behavior. See overall matrix, reflectance, and transmittance at once. Adjust layers quickly for coatings, filters, and mirrors easily.
This tool uses the 2×2 characteristic matrix for a stack of planar layers at normal incidence. For a layer with refractive index n, thickness d, and wavelength λ, the phase thickness is:
δ = 2π n d / λ
The layer matrix is:
M = [ [ cosδ, i sinδ / n ], [ i n sinδ, cosδ ] ]
The overall matrix is the ordered product M_total = M1 · M2 · … · MN. Using incident admittance η0 = n0 and substrate admittance ηs = ns, reflection and transmission coefficients are:
r = (η0A + η0ηsB − C − ηsD) / (η0A + η0ηsB + C + ηsD)
t = 2η0 / (η0A + η0ηsB + C + ηsD)
Reflectance is R = |r|² and transmittance is T = (ηs/η0)|t|² for real indices.
| Parameter | Value |
|---|---|
| Wavelength | 550 nm |
| Incident index (n₀) | 1.000 |
| Substrate index (nₛ) | 1.520 |
| Layer 1 | n = 1.38, d = 100 nm |
| Layer 2 | n = 2.10, d = 70 nm |
| Layer 3 | n = 1.38, d = 100 nm |
Layered media appear in anti-reflection coatings, dielectric mirrors, Fabry–Pérot filters, and thin-film sensors. A transfer matrix compactly tracks how forward and backward waves evolve across many interfaces, so the whole stack behaves like one equivalent interface at a chosen wavelength.
The calculator assumes normal incidence and real refractive indices. You enter incident index n₀ (often 1.000 for air), substrate index nₛ (around 1.52 for common glass), plus each layer’s index and thickness. Thickness units are converted internally to nanometers.
The key quantity is the phase thickness δ = 2π n d / λ. For visible optics, λ typically spans 400–700 nm. A quarter‑wave layer at the design wavelength uses d = λ/(4n); for λ = 550 nm, a layer with n = 1.45 is about 94.8 nm thick.
Each layer contributes a 2×2 characteristic matrix. The stack matrix is the ordered product M_total = M1·M2·…·MN, matching the physical order from the incident medium toward the substrate. This approach remains efficient even for dozens of layers, which is common in high‑reflectivity coatings.
With admittance η ≈ n at normal incidence, the matrix elements A, B, C, D yield complex amplitude coefficients r and t. Power reflectance is R = |r|², while T = (nₛ/n₀)|t|². For real indices and ideal math, R + T should be close to 1.
An uncoated glass surface reflects about 4% at normal incidence because R = ((n₀−nₛ)/(n₀+nₛ))². A single quarter‑wave AR layer can reduce that to roughly 1% near its design wavelength when the layer index is near √(n₀nₛ). As a guide near 550 nm, SiO₂ is about 1.45, while TiO₂ and Ta₂O₅ are typically around 2.1–2.3. Multilayer Bragg mirrors (alternating high/low index) can exceed 99% reflectance with enough pairs and strong index contrast.
If you see R + T drift from 1, it is usually rounding, extreme layer counts, or unrealistic inputs. Very thick layers create rapidly varying sin/cos terms, so results become highly wavelength‑sensitive. For thin films, a 1–2 nm thickness change can noticeably shift a narrowband resonance. For absorption or oblique incidence, the model must be extended to complex indices and polarization‑dependent admittances.
Use the CSV export to compare designs across multiple runs (for example, sweeping thicknesses around a quarter‑wave value). The PDF report is useful for sharing a chosen stack with fabrication notes. For broadband performance, repeat the calculation at several wavelengths and look for stable low R or high R bands.
It is a 2×2 matrix that links the electric and magnetic field components (or wave amplitudes) across a layer stack. Multiplying layer matrices gives one overall matrix for the full coating.
This version assumes real refractive indices. Metals and absorbing films require a complex refractive index (n + iκ). The formulas still work, but the implementation must be extended to complex n values.
Small deviations usually come from floating‑point rounding and formatting. Larger deviations can occur with extreme values, many layers, or inconsistent inputs. For real indices and stable math, R + T should stay close to 1.
A layer whose optical thickness equals one quarter of the design wavelength, so n d = λ/4. It introduces a π/2 phase shift and is widely used in AR coatings and Bragg reflectors.
It depends on index contrast and target reflectance. Alternating high/low index pairs increase reflectance rapidly; many practical dielectric mirrors use 6–12 pairs to reach very high reflectance near a design wavelength.
Yes. The product order corresponds to the physical order from incident side to substrate. Swapping layers generally changes the interference condition and can significantly change reflection and transmission.
Yes, conceptually. The method is general for waves in stratified media. Ensure your wavelength and thickness units are consistent, and interpret refractive index as an effective wave impedance parameter appropriate to your RF material model.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.