Multilayer Stack Design Calculator

Inputs

Model a thin-film coating using the transfer-matrix method. Use quarter-wave layers for mirrors, or set custom thickness for filters and antireflection stacks.

Formula used

This calculator uses the thin-film transfer-matrix method. Each layer contributes a characteristic matrix that accounts for phase accumulation and impedance mismatch.

• Snell’s law per layer: n0 sin(θ0) = ni sin(θi)
• Phase thickness: δi = 2π ni di cos(θi) / λ
• Optical admittance (s): qi = ni cos(θi) · (p): qi = ni / cos(θi)
• Layer matrix: Mi = [[cosδ, i sinδ/qi],[i qi sinδ, cosδ]]
• Total matrix: M = Π Mi from top to bottom

From M, the reflection coefficient r and transmission coefficient t are computed using boundary admittances q0 and qs. Reflectance is R = |r|², and transmittance is T = (qs/q0)|t|² for lossless real indices.

How to use this calculator

1. Enter the design wavelength and incident angle.
2. Set the incident medium and substrate refractive indices.
3. Add layers from top to bottom with index and thickness mode.
4. Use quarter-wave layers for mirror and stopband designs.
5. Enable the spectrum sweep to inspect stopbands and passbands.
6. Press Calculate to view results above the form.
7. Use CSV or PDF buttons to export your setup.

Multilayer stack design article

1) Why multilayer stacks matter

Multilayer optical stacks control reflection and transmission using interference. A few nanometers of thickness error can shift a feature by several nanometers in wavelength, so design tools must be consistent and repeatable. This calculator reports reflectance, transmittance, and reflection phase for quick comparisons.

2) Data you enter and what it represents

Each layer is defined by refractive index n and thickness d in nanometers. Common values include SiO2 near 1.45 and TiO2 near 2.35 in the visible. You also set the incident medium index n0, substrate index ns, and incidence angle.

3) Quarter-wave and half-wave building blocks

Quarter-wave layers use d = \u03bb/(4n), creating strong constructive and destructive interference in alternating high/low index pairs. Half-wave layers use d = \u03bb/(2n) and often act as “spacers” that preserve phase while adjusting physical thickness. The calculator converts these modes into real thickness automatically.

4) Bragg mirrors and stopband behavior

Alternating quarter-wave layers form a Bragg mirror. Increasing pair count raises peak reflectance and steepens edges. With high contrast, the stopband widens; a common approximation for normalized stopband width is \u0394\u03bb/\u03bb \u2248 (4/\u03c0) asin((nH-nL)/(nH+nL)). Use the spectrum sweep to see the band shape directly.

5) Antireflection stacks and matching

For antireflection designs, the goal is to reduce Fresnel mismatch between media. A single quarter-wave layer works best when n \u2248 \u221a(n0 ns) at the design wavelength, but broadband performance typically needs multiple layers with gradually stepped index. Sweep plots help reveal residual ripple across 400–700 nm or your chosen range.

6) Angle and polarization effects

As the incident angle increases, the optical path changes by cos(\u03b8i) within each layer, shifting spectral features toward shorter wavelengths. s and p polarizations also diverge because their admittances differ; the calculator can compute each polarization or an unpolarized average. If internal reflection occurs, adjust indices or reduce angle.

7) Practical thickness tolerance and iteration

Real coatings have thickness and index variation. A \u00b11% thickness error at 550 nm can move a quarter-wave condition by several nanometers, and small index drift changes phase as well. Use the layer table to test “what-if” offsets quickly, then compare reflectance and phase to judge sensitivity before fabrication.

8) Using exports for documentation

After calculating, export CSV for records, audits, or spreadsheets, and export PDF for sharing design snapshots. When you iterate, keep wavelength, angle, and substrate constant so changes reflect only the stack. For mirror designs, repeating the sequence is a fast way to explore performance scaling.

FAQs

1) What does “unpolarized” mean here?

It is the average of s and p results at the same wavelength and angle. It is useful for unpolarized sources or when you want a single summary value for reflectance and transmittance.

2) Why can absorbance be near zero?

The model assumes real refractive indices, so layers are treated as lossless. In that case, energy is conserved and A = 1 - R - T should be close to zero, aside from rounding.

3) How do quarter-wave layers help mirrors?

Alternating quarter-wave layers create repeated impedance mismatches that add reflections in phase at the design wavelength. More pairs generally increase reflectance and deepen the stopband, especially when index contrast is high.

4) What thickness should I enter for “custom” mode?

Enter the physical thickness in nanometers. The calculator will use that value directly in the transfer-matrix calculation. Use the spectrum sweep to verify whether your custom thickness achieves the intended passband or notch.

5) Why do results change with incident angle?

The phase thickness depends on cos(\u03b8i) inside each layer. Increasing angle reduces the effective optical thickness and shifts spectral features. Polarization also splits at higher angles, which is why s and p can differ.

6) What causes the internal reflection warning?

If n0 sin(\u03b80)/ni exceeds 1 for a layer or the substrate, Snell’s law implies no propagating solution. Lower the angle, increase the layer index, or change the substrate index to avoid that condition.

7) How many wavelength points should I use in the sweep?

Use 101–301 points for fast design checks, and 501+ points for sharper features like high-pair Bragg mirrors. More points improve curve resolution but increase computation time on slower hosting.