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Multilayer optical calculations
Steven J. Byrnes
TL;DR
Multilayer optical calculations involve reflection, refraction, and absorption, but some formulas and edge cases are difficult to find consistently documented. The paper derives transfer-matrix formulas for coherent and incoherent planar stacks, including profiles and implementation issues, and explains counter-intuitive absorptive-medium results.
Problem
Formulas for multilayer reflection, refraction, absorption, and related edge cases are difficult to find explicitly and consistently worked out in the literature.
Method
The paper derives transfer-matrix formulas for planar multilayer stacks, including oblique incidence, position-dependent absorption, branch-cut handling, and incoherent layers.
Results
The derivations provide formulas for calculating multilayer reflection and transmission amplitudes, power absorption profiles, and optical properties implemented by tmm.
Takeaways & Limitations
The treatment explains how interference, absorption, and incoherence enter multilayer calculations, including why absorptive starting media can yield T > 1 without stimulated emission.
Takeaways & Limitations
The incoherent analysis treats phase information as completely discarded and is therefore limited to an all-or-nothing coherence model.
Abstract
from arXiv · showhide
When light hits a multilayer planar stack, it is reflected, refracted, and absorbed in a way that can be derived from the Fresnel equations. The analysis is treated in many textbooks, and implemented in many software programs, but certain aspects of it are difficult to find explicitly and consistently worked out in the literature. Here, we derive the formulas underlying the transfer-matrix method of calculating the optical properties of these stacks, including oblique-angle incidence, absorption-vs-position profiles, and ellipsometry parameters. We discuss and explain some strange consequences of the formulas in the situation where the incident and/or final (semi-infinite) medium are absorptive, such as calculating $T>1$ in the absence of gain. We also discuss some implementation details like complex-plane branch cuts. Finally, we derive modified formulas for including one or more "incoherent" layers, i.e. very thick layers in which interference can be neglected. This document was written in conjunction with the "tmm" Python software package, which implements these calculations.
1 Introduction
The document records and explains calculations implemented by the tmm transfer-matrix Python package for multilayer optical stacks. It builds on established derivations while addressing consistency issues, especially for absorptive semi-infinite media.
- The notes explain calculations implemented by the tmm, short for transfer matrix method, Python software package.
- Many textbooks and references contain the derivations, although they are not always easy to find worked out consistently.
- Other programs calculate some or all of the same formulas and generally agree with tmm except for power calculations involving complex-index semi-infinite media.
- The document assumes materials are non-magnetic and isotropic rather than birefringent.
2 Wave propagation
Wave propagation is modeled in a uniform planar stack as forward and backward complex waves whose shared in-plane wavevector component obeys Snell’s law. A complex refractive index can make the normal wavevector component complex, representing absorption.
- The structure consists of smooth planar layers with interfaces normal to z, uniform x-y directions, and light propagating in the x-z plane.
- The electric field is a superposition of forward-moving and backward-moving electromagnetic waves represented with complex sinusoidal quantities.
- The z component of the wavevector may be complex, producing attenuation through absorption while the in-plane components remain real.
- The refractive index determines the wavevector, with n sin θ real and shared across layers according to Snell’s law.
- A complex refractive index has an extinction coefficient whose positive imaginary part indicates absorption and whose negative value corresponds to stimulated emission.
- S- and p-polarization differ in whether the E-field or H-field points in the y-direction, although they coincide for normally incident light.
3 Single-interface reflection and transmission amplitudes
Single-interface reflection and transmission amplitudes are obtained by applying electromagnetic boundary conditions to incident, reflected, and transmitted waves. Superposition then extends these relations to waves incident from both sides.
- At an interface, the relevant amplitudes are incident and reflected waves in layer 1 and a transmitted wave in layer 2.
- The reflection coefficient r and transmission coefficient t are defined as reflected-to-incident and transmitted-to-incident amplitude ratios.
- The Fresnel equations follow by imposing continuity of the appropriate total electric and magnetic field components across the interface.
- With waves incoming from both sides, each outgoing amplitude is a superposition of reflection from one incoming wave and transmission from the other.
- The two one-sided boundary-condition solutions combine linearly to represent all possible combinations of incoming waves.
4 Multilayer thin films
The transfer-matrix method derives reflection, transmission, power-flow, absorption, and ellipsometry formulas for arbitrary multilayer thin films. It also explains how interference and absorptive incident media affect power calculations.
- Complex amplitudes for reflection and transmission: The stack contains semi-infinite first and last layers surrounding finite layers, with forward and backward wave amplitudes assigned at interfaces.For N materials, the incident amplitude in layer 0 is set to 1, while the final layer has transmission amplitude t and no backward wave.
- Complex amplitudes for reflection and transmission: The phase thickness δ_n combines layer thickness with the forward-wave kz, capturing propagation phase and, for complex kz, absorption.The layer relations use e^{iδ_n} and e^{-iδ_n} to connect amplitudes across interfaces.
- Complex amplitudes for reflection and transmission: Multiplying interface and propagation relations yields a matrix whose entries determine the reflection and transmission amplitudes, r = M̃10/M̃00 and t = 1/M̃00.The method therefore calculates r and t for an arbitrary multilayer thin film, along with internal amplitudes.
- Ellipsometric parameters: Ellipsometry parameters can be calculated from the s- and p-polarization reflection amplitudes, but textbook definitions may differ by sign or phase conventions.The paper notes that complex phase angle is used in these parameters and that definitions may require sign or π/2 adjustments.
- Calculating Poynting vector: The normal Poynting vector is computed as net forward power at a point, normalized to the total incoming power, for both s- and p-polarization.Absorbed energy density is obtained as the negative spatial derivative of this normalized Poynting-vector component.
- Counter-intuitive results when the initial medium is absorptive: Interference between forward and backward waves produces oscillations in absorption and Poynting-vector profiles, including hot-spots and nodes.When the initial medium is absorptive, the resulting power entering the stack need not equal 1 − R, and R + T can differ from 1.
5 Branch cuts
Complex Snell-law angles require branch-cut handling because θ and π − θ represent different wave assignments. The choice is observable in semi-infinite boundary layers but not in finite layers when both waves are solved.
- Branch-cut ambiguity: Snell’s law determines θ_i through a complex arcsine, but the arcsine is ambiguous because it has branch cuts in the complex plane.The two non-equivalent solutions are θ and π − θ, corresponding to different forward- and backward-wave assignments.
- Finite layers: In finite-thickness layers, swapping θ_i with π − θ_i exchanges v_i and w_i without changing observable quantities such as reflectance or absorption.Both forward- and backward-traveling waves are solved in these layers, making the labeling choice immaterial.
- Semi-infinite layers: The θ versus π − θ choice matters in the starting and final semi-infinite layers because their forward or backward amplitudes are fixed by boundary conditions.The starting layer sets the forward amplitude to 1, while the final layer sets the backward amplitude to 0.
- Implementation: Naively using θ_i = arcsin(n_0 sin(θ_0)/n_i) can select the wrong branch, including for the final layer during total internal reflection.The recommended implementation is to check whether θ or π − θ satisfies the appropriate criteria rather than relying on a programming language’s arcsine convention.
6 Thick “incoherent” films
The paper models very thick layers incoherently by averaging away unresolved Fabry–Pérot fringes and retaining intensities rather than phase. This simplifies multilayer calculations but cannot represent partial coherence or depth-resolved absorption without additional coherent averaging.
- 6.1 Introduction: Incoherent layers are appropriate when they greatly exceed the wavelength and their Fabry–Pérot fringes are strongly averaged out.Fringe averaging may result from wavelength, angle, or thickness variations.
- 6.1 Introduction: The incoherent calculation discards phase upon entry and propagates only intensity, making coherence treatment all-or-nothing.More sophisticated incoherence methods exist, while coherent simulations with varied parameters provide a fallback.
- 6.2 Calculation method: The calculation organizes coherent layers into stacks and tracks forward and backward power flows across incoherent layers using transmissivity, reflectivity, and matrix relations.Capitalized power variables distinguish these quantities from coherent-wave amplitudes.
- 6.3 Absorption profile, Coherence length: Total absorption and transmission through an incoherent layer do not require the coherence length when it is long relative to a wavelength but short relative to the layer thickness.The integrated effect of decaying oscillations is approximately independent of their precise decay properties under these conditions.
- 6.3 Absorption profile, Coherence length: Depth-resolved absorption does require the coherence length, so the software omits it for incoherent layers and recommends coherent simulations averaged over varying parameters.Near interfaces, absorption oscillations gradually decay into a smooth exponential profile.
A Appendix: Sign convention for reflection amplitude
The appendix explains why p-polarized reflection amplitudes have two valid sign conventions: the scalar amplitude depends on the chosen polarization unit vector. The paper adopts Convention A, while observable quantities remain unaffected by the naming choice.
- Why a sign convention exists: The p-polarization reflection-amplitude sign differs because both a scalar amplitude and its unit vector can be sign-flipped without changing the electric-field vector.Thus the underlying vector fields are unambiguous even though scalar reflection amplitudes differ.
- Convention A and Convention B: Convention A and Convention B assign opposite signs to the backward p-polarized scalar amplitude while using equivalent physical field vectors.The displayed forward and backward field decompositions make the convention difference explicit.
- Normal incidence: At normal incidence, Convention B makes s- and p-polarized reflection amplitudes equal, matching the equivalence of the polarizations there.This convention is tied to the relative phase of electric fields.
- Glancing incidence: At glancing incidence, Convention A gives an intuitive interpretation of r = −1 as destructive interference at the interface.The reflected and incident waves travel in the same direction at grazing angles.
- Choice used in this document: The paper uses Convention A because neither convention is intrinsically right or wrong, although each offers different interpretive advantages.The cited textbooks are divided between the two conventions.
B Appendix: R and T with an absorptive starting medium
The appendix defines reflected and transmitted power by extrapolating measurements away from an absorptive starting medium, beyond the interference region. This normalization preserves R = |r|2 but can make power_entering differ from 1 − R and permit R + T ≠ 1.
- Definitions of R and T: R and T are defined from powers measured after propagation to a large distance, with exponential factors removing absorption in the initial medium.R uses the returning power after a round trip, while T uses power leaving the final interface.
- Definitions of R and T: The normalization is intended to evaluate reflected and transmitted power beyond the region where incoming and outgoing waves coherently interfere.The limiting distance need only be sufficiently large to lie outside that interference region.
- Interpretation: The chosen R and T definitions are useful for applications such as antireflective coatings on tinted glass, although they are not unique.The software's incoherent calculation is based on such situations.
- Absorptive starting medium: R = |r|2 remains valid, but an absorptive starting medium can make the normalized Poynting vector at the first interface differ from 1 − R.Interference changes the local power flow near the interface.
- Absorptive starting medium: When the starting medium is absorptive, R + T can differ from 1 because interference creates an extra or deficient absorption contribution near the first interface.For complex r, the oscillatory absorption does not necessarily cancel against the non-oscillating baseline.
B.1 Accounting for this effect in the incoherent calculation
The incoherent calculation accounts for additional near-interface absorption by treating power transmission between incoherent layers without coherent interference corrections. Its broader optical treatment also requires careful branch selection for complex refractive indices and does not cover unstable gain solutions.
- Accounting for near-interface absorption: Extra absorption near incoherent interfaces is included because absorption associated with oscillation cutoffs contributes to each layer's total absorbed light.This is part of the layer-by-layer absorption calculated by the incoherent method.
- Accounting for near-interface absorption: The net power crossing an interface between incoherent layers is Pf,0T01 − Pb,1T10 because transmitted beams have no coherent partner producing interference corrections.Forward and backward powers on both sides determine the crossing flux.
- Stimulated emission: Stimulated-emission media can yield mathematically finite steady-state solutions that are physically unphysical when the actual system is unstable and fields grow exponentially.The algorithm finds the finite solution, even when gain saturation would be needed for a physical outcome.
- Complex refractive indices: Complex Snell-law angles require choosing between two cosine branches, θ and π − θ, because arcsine has branch cuts.The sine is unambiguous, but the cosine determines the relevant propagation choice.
- Complex refractive indices: The branch choice matters in the starting and ending semi-infinite layers but not in intermediate finite-thickness layers.In finite layers, exchanging forward and backward wave labels leaves observable quantities unchanged.
D.1 Computer implementation
The implementation should avoid relying on a direct complex arcsine result alone, instead checking the selected angle against decay and forward-energy criteria. These criteria are shown to be mutually consistent across ordinary, total-internal-reflection, boundary, and negative-index cases.
- Angle selection: Directly using θi = arcsin(n0 sin(θ0)/ni) can select the wrong angle because programming-language branch-cut implementations differ, especially under total internal reflection.Rounding near a branch cut and software changes can also make this approach non-robust.
- Angle selection: A robust implementation computes the arcsine, checks the resulting angle, and replaces it with π − θ when necessary; only the starting and ending semi-infinite layers require this check.
- Branch criteria: For absorbing media with Im n > 0, selecting Im(n cos θ) > 0 makes the forward-propagating electric field decay rather than amplify.The additional forward-energy conditions are Re[n cos θ] ≥ 0 for s-polarization and Re[n cos θ*] ≥ 0 for p-polarization.
- Branch criteria: The decay and forward-energy requirements are consistent: choosing Im(n cos θ) > 0 also gives positive real-part conditions for the relevant s- and p-polarization expressions.The text supports this through separate theorems for s- and p-polarization and a square-root argument.
- Special cases: The angle analysis distinguishes total internal reflection, ordinary propagation, and the boundary case, with decay determining the sign in total internal reflection and forward-energy conditions determining it in ordinary propagation.For negative-index media, Im n < 0 can still describe absorption when the Poynting-vector direction is opposite the wavevector.
- Special cases: Flipping the sign of ni changes cos θi and ni cos θi only up to possible sign flips, so the consistency proof remains unchanged.
D.5 Everything else
The treatment excludes active media and leaves additional cases such as n = 0 unexamined.
- Active media cannot be analyzed by this approach because refractive index at one wavelength does not determine which solution is physically appropriate.
- Cases such as n = 0 are not analyzed.