Toward a Unified Relational Framework for Domain Emergence in Layered Optical Systems

Author: Daryl Costello
Affiliation: Independent Theoretical Research, Rosendale, NY, United States
Date: October 18, 2026
Classification: Theoretical Physics – Unified Field Theory / Cosmology
Status: Original Manuscript – Preprint

Correspondence: Daryl.Costello@outlook.com

Preprint · October 2026 ·

Abstract

We develop a unified relational framework for the electrodynamics of stratified (layered) optical media, drawing on three interlocking formalisms: relational optical Hamiltonians derived from the Page–Wootters (PaW) conditional-state mechanism, a geometric mapping between multilayer polarization evolution and gyroscope precession (Thomas–Wigner rotation), and a Wilsonian renormalization group (RG) coarse-graining procedure that generates effective-medium descriptions at progressively longer length scales. In conventional treatments, the propagation coordinate z is assumed to be an observer-independent external parameter; we demonstrate that this assumption is frame-dependent and show how replacing it with a relational layer-index clock restores observer-covariance. The relational optical Hamiltonian H​_rel(α | β) is constructed as a generator of conditional evolution of subsystem β as measured by the clock layer α, subject to a Dirac constraint H​_total|Ψ⟩ = 0 on the total optical state. We show that the composition of transfer matrices across a multilayer stack is formally isomorphic to a sequence of non-commuting SU(2) rotations on the Poincaré sphere, whose geometric residue is precisely an optical Thomas-precession angle, yielding a non-Abelian geometric (Pancharatnam–Berry) phase. The RG coarse-graining map R​_b replaces blocks of b layers by a single renormalized layer, inducing a flow on the space of coupling constants (refractive index contrast, layer thickness, mixing angle). Fixed points of this flow delineate optical domains; emergent effectively-homogeneous regions whose boundaries are RG domain walls. Near the critical refractive-index contrast Δn​_c, domain sizes obey a power-law distribution governed by a universal exponent determined by the relevant eigenvalue of the linearized RG operator. Crucially, domain structure is shown to be observer-dependent: the scale at which domains are resolvable depends on the coarse-graining level of the clock layer, making domain emergence an intrinsically relational property. Implications for photonic bandgap engineering, biological structural coloration, and experimental verification via time-resolved ellipsometry and Berry-phase interferometry are discussed. All five pillars of this framework are shown to derive from a single first-principles organizing principle, the Kernel-First Architecture (KFA), in which the kernel is defined as the irreducible seed of relation: the negotiation of adjacency between strata through resolution and emergent media. KFA provides the connective tissue across all levels of the theory (from the relational Hamiltonian at the microscopic scale to domain emergence at the macroscopic scale ) establishing it as the meta-architecture from which the full optical framework is derived.

Keywords: stratified optical media, relational quantum mechanics, Page–Wootters formalism, transfer matrix method, Berry phase, Thomas precession, Wilsonian renormalization group, optical domain emergence, photonic crystals, effective medium theory.

1. Introduction: Motivation for Relational Dynamics in Optics

The propagation of electromagnetic waves through stratified media (stacks of planar layers with smoothly or abruptly varying permittivity ε(z) and permeability μ(z)) is one of the oldest and most analytically tractable problems in theoretical optics. The transfer-matrix method (TMM), developed systematically by Abelès [1] and subsequently extended to anisotropic and chiral media [2], provides an elegant algebraic framework: the field amplitude across each layer is propagated by a 2 × 2 matrix M​_i, and the total response of the stack is the product M = M​_N ··· M​₂ M​₁. The elegance of this formalism, however, conceals a foundational assumption: the propagation coordinate z, with respect to which “evolution” is tracked, is treated as an external, observer-independent parameter. This is entirely analogous to the treatment of “coordinate time” in pre-relativistic physics; and, at a more fundamental level, to the role of time in non-relativistic quantum mechanics.

This assumption is benign in many practical contexts. Yet it becomes problematic when: (i) one wishes to define intrinsic, observer-covariant notions of “phase accumulated” or “domain traversed” within a disordered stack; (ii) one seeks to connect optical Hamiltonians to the broader program of relational quantum dynamics [3, 4]; or (iii) one requires a coarse-graining procedure that respects the scale-dependence of all physical observables, including the “evolution parameter” itself. In each of these cases, the fixed external z-coordinate is a liability rather than an asset.

1.1 The Problem of External Time in Optical Systems

In the Hamiltonian formulation of optics, one writes the Maxwell equations for a monochromatic field of frequency ω as a Schrödinger-like equation

−i ∂​_z |Φ(z)⟩ = H​_opt(z) |Φ(z)⟩, (1)

where |Φ(z)⟩ is a column vector of field amplitudes and H​_opt(z) is the optical Hamiltonian, whose form depends on ε(z), μ(z), and the transverse wave vector k​_⊥. Here z plays the role of “time.” This is an entirely satisfactory first-quantized description so long as z is unambiguous. But in a disordered multilayer system (where the layers themselves have no absolute positional identity) the choice of “where we are” in the stack is inherently relational: it is defined only with respect to some reference layer.

This observation resonates deeply with the “problem of time” in quantum gravity [5]: in a fully constrained dynamical system, there is no external time; dynamics must emerge from the correlations between subsystems. The Page–Wootters (PaW) mechanism [6, 7] offers a concrete resolution: one identifies a subsystem C (the “clock”) and conditions all physical predictions on its state. Dynamics of the remaining subsystem S is then encoded in the entanglement structure of the joint clock–system state |Ψ⟩​_CS, subject to a global Hamiltonian constraint H​_total|Ψ⟩ = 0.

1.2 Stratified Media as a Natural Testbed

Stratified media are ideally suited to implementing the PaW program in an optical setting. The layer index α = 1, 2, …, N provides a natural discrete relational “clock”: evolution is defined as the change in the field state conditioned on the layer through which it passes. The clock is physical (it is an actual material layer, not an abstract parameter) and the clock’s “ticking” is determined by its own optical thickness n​_α d​_α.

Beyond the foundational motivation, the relational approach yields three concrete analytical advantages: it provides a coordinate-free definition of geometric phase accumulated by the field across the stack; it makes manifest the non-Abelian structure of multi-layer polarization evolution; and it furnishes a natural starting point for the RG coarse-graining that is the third pillar of this paper.

1.3 Three Pillars of This Work

This paper develops three interconnected results:

1. Relational layer Hamiltonians (Section 2): We construct the relational optical Hamiltonian H​_rel(α | β) for stratified media via the PaW conditional-state formalism, deriving the relational propagator and its connection to the standard transfer-matrix product.
2. Gyroscope precession mapping (Section 3): We establish a formal isomorphism between multilayer polarization evolution on the Poincaré sphere and gyroscope precession (Thomas–Wigner rotation), deriving the non-Abelian geometric phase for periodic and aperiodic stacks.
3. RG coarse-graining and domain emergence (Sections 4–5): We define a Wilsonian RG map R​_b on the space of layer Hamiltonians, identify its fixed points, and show how macroscopic optical domains emerge as RG attractors, with domain structure being an intrinsically relational, observer-dependent property.

1.5 First-Principles Foundation: The Kernel-First Architecture

Before developing the technical machinery of the subsequent sections, we articulate the organizing principle that underlies the entire framework: the Kernel-First Architecture (KFA). KFA is not an auxiliary interpretive gloss but the generative first-principles structure from which each of the paper’s five pillars is derived. Its central claim is deceptively simple:

The kernel is the seed of relation ;  the irreducible negotiation of adjacency between strata, enacted through resolution and emergent media, constituting the connective tissue that holds all levels of a stratified structure together.

This definition carries three operationally distinct components, each of which maps onto a concrete element of the mathematical framework.

(i) Seed of Relation. The kernel is not a state, a field, or a medium, but the minimal generative act of relating two adjacent degrees of freedom. In the optical context, this is precisely the relational Hamiltonian H_rel(α|β): the irreducible generator that encodes what layer β “is” as experienced from the vantage of layer α. No external coordinate is presupposed; the kernel is the relation itself, prior to any fixed background.

(ii) Negotiation of Adjacency Through Resolution. The kernel does not merely connect; it negotiates: meaning the effective coupling between adjacent strata depends on the resolution scale at which the negotiation is enacted. This is the RG coarse-graining map R_b: at each scale b, a new effective medium is produced by block-composing the kernel’s microscopic negotiations. The word “resolution” here is exact: it is the scale b at which boundary information is summed over, producing the emergent medium appropriate to that observational stratum.

(iii) Connective Tissue Across Strata. The cumulative imprint of all kernel negotiations as one traverses the stack is not a scalar quantity but a geometric object; the holonomy of the SU(2) bundle over the stack’s parameter space. This is the geometric (Berry) phase derived in §3, and it plays the role of connective tissue: it encodes how the identity of a polarization state is preserved (or transformed) globally through locally negotiated adjacencies.

1.5.1 Derivation of the Five Pillars from KFA

The five pillars of the framework emerge as successive unfoldings of the KFA kernel definition.

Pillar 1 – Relational Hamiltonian (§2): The kernel, as seed of relation, demands that the generator of evolution be defined relationally. H_rel(α|β) is the Hamiltonian that exists only between layers, not within any single layer. The Dirac constraint H_total|Ψ⟩ = 0 is the formal statement that the kernel (not an external time coordinate) carries the dynamics.

Pillar 2 – Transfer Matrix / Adjacency Negotiation (§2): The microscopic implementation of adjacency negotiation is the transfer matrix M_i = exp(i H_i d_i). Each M_i is a single kernel transaction: the boundary exchange between layer i and its neighbor, encoding the full local resolution of the electromagnetic field across that interface.

Pillar 3 – Gyroscope Precession / Connective Tissue (§3): When kernel transactions are composed across N layers, the non-commutativity of their SU(2) generators produces a geometric phase; the holonomy that is the connective tissue of the entire stack. The gyroscope precession mapping makes this connective tissue visible: the Stokes vector’s net rotation is the global record of all local kernel negotiations.

Pillar 4 – RG Coarse-Graining / Resolution (§4): The coarse-graining map R_b is the KFA resolution operation made precise. At each scale b, the kernel is re-enacted at a coarser adjacency: fine-grained layer boundaries are integrated out, and a new effective kernel emerges at the block level. The RG fixed points are the resolutions at which the kernel negotiation stabilizes; producing either a homogeneous medium (IR fixed point) or a maximally disordered stack (UV fixed point).

Pillar 5 – Domain Emergence (§5): Domains are regions where the kernel negotiation has reached a locally stable resolution; a fixed-point attractor of the RG flow acting on successive blocks. Domain boundaries are kernel instabilities: points where no stable local negotiation exists and the resolution must shift. The observer-dependence of domain structure reflects the KFA principle that resolution is always enacted from a particular relational vantage (the clock layer α) not from a view from nowhere.

The KFA perspective reframes the conventional optical question (“what is the medium through which light propagates?”) as: “what is the kernel negotiation that produces the effective medium at each resolution?” This is not a semantic restatement but a structural inversion: the medium is no longer primitive, it is emergent. The kernel is primitive. Everything else is the unfolding consequence of its relational negotiations across strata.

The paper is structured as follows. Section 2 presents the layer Hamiltonian formalism and the relational Hamiltonian construction. Section 3 develops the gyroscope precession analogy and derives geometric phases. Section 4 introduces the RG coarse-graining map and analyzes the RG flow. Section 5 discusses domain emergence and its physical implications. Section 6 presents discussion and outlook. Section 7 concludes. Mathematical conventions are collected in Appendix A; detailed transfer-matrix derivations appear in Appendix B.

2. Layer Hamiltonians for Stratified Media

2.1 Maxwell’s Equations in Stratified Media

Consider a planar-stratified medium occupying the half-space 0 ≤ z ≤ L, with position-dependent, isotropic, non-magnetic (unless stated) permittivity ε(z) and permeability μ(z). For a monochromatic plane wave of angular frequency ω and transverse wave vector k​_⊥ = k​_x x̂ propagating in the x–z plane, Maxwell’s equations reduce to two decoupled polarization sectors. Defining the transverse components for the TE (s-polarization) case:

∂​_z E​_y = iωμ​₀μ H​_x,     ∂​_z H​_x = i(ωε​₀ε − k​_x²/(ωμ​₀μ)) E​_y, (2)

we can write this as a first-order system. Define the field vector |Φ​_TE(z)⟩ = (E​_y, H​_x)^T. Then

∂​_z |Φ​_TE⟩ = i A​_TE(z) |Φ​_TE⟩, (3)

where the system matrix is

A​_TE(z) = [ 0, ωμ​₀μ(z); ωε​₀ε(z) − k​_x²/(ωμ​₀μ(z)), 0 ]. (4)

An analogous system matrix A​_TM(z) governs the TM (p-polarization) sector, obtained by the substitution ε ↔ μ and E​_y → H​_y. In what follows, we treat both polarizations simultaneously where possible, specializing as needed.

2.2 The Layer Transfer Matrix

Within a homogeneous layer i of refractive index n​_i = (ε​_iμ​_i)^1/2 and thickness d​_i, the system matrix is constant: A​_i = const. The solution to Eq. (3) is therefore

|Φ(z​_i + d​_i)⟩ = M​_i |Φ(z​_i)⟩,     M​_i = exp(i A​_i d​_i). (5)

Introducing the notation k​_z,i = (n​_i²ω²/c² − k​_⊥²)^1/2 for the longitudinal wave vector in layer i, the explicit 2 × 2 transfer matrices are:

M​_i^TE = [ cos(k​_z,id​_i), (i/p​_i)sin(k​_z,id​_i); i p​_i sin(k​_z,id​_i), cos(k​_z,id​_i) ], (6)

M​_i^TM = [ cos(k​_z,id​_i), (i/q​_i)sin(k​_z,id​_i); i q​_i sin(k​_z,id​_i), cos(k​_z,id​_i) ], (7)

where the optical admittances are p​_i = k​_z,i/(ωμ​₀μ​_i) for TE and q​_i = k​_z,iε​_i/(ωε​₀n​_i²) for TM. These matrices belong to SL(2,ℝ) for lossless layers (det M​_i = 1) and to SL(2,ℂ) in the presence of absorption.

2.3 The Layer Hamiltonian

Equation (5) defines the layer Hamiltonian H​_i implicitly via M​_i = exp(i H​_i d​_i). Taking the matrix logarithm:

H​_i = (1/d​_i) Im log M​_i = A​_i, (8)

which for TE polarization evaluates explicitly to

H​_i^TE = [ 0, ωμ​₀μ​_i; ωε​₀ε​_i − k​_⊥²/(ωμ​₀μ​_i), 0 ] = k​_z,i σ​₁ + (ωμ​₀μ​_i − k​_z,i) σ​₂, (9)

where σ​₁, σ​₂ are Pauli matrices in the field-amplitude basis. The eigenvalues of H​_i are ±k​_z,i, corresponding to forward- and backward-propagating modes.

2.4 The Relational Optical Hamiltonian

We now implement the Page–Wootters construction [6]. Consider a total system consisting of N layers. We designate layer α as the “clock subsystem” C and the complementary set of layers \{\beta \neq \alpha\} as the “system” S. The physical Hilbert space of the total state is ℋ = ℋ​_C ⊗ ℋ​_S, with the Dirac constraint

H​_total |Ψ⟩ = 0,     H​_total = H​_C ⊗ 1​_S + 1​_C ⊗ H​_S. (10)

The physical state |Ψ⟩ satisfying this constraint encodes the entanglement between the clock layer and the rest of the stack. The PaW conditional state of the system given that the clock is in eigenstate |α⟩ (i.e., the field “is in layer α“) is:

|ψ​_β(α)⟩ = ⟨α|Ψ⟩, (11)

which satisfies the relational Schrödinger equation

H​_rel(α | β) |ψ​_β(α)⟩ = i ∂​_α |ψ​_β(α)⟩. (12)

Here H​_rel(α | β) is the relational optical Hamiltonian: the generator of evolution of layer-β‘s field state as indexed by the clock layer α. It is obtained from the constraint (10) by the reduction H​_rel(α | β) = −⟨α| H​_C |α⟩ = H​_α, which recovers Eq. (8) within a single layer but generalizes to interleaved subsystems.

2.5 The Relational Propagator

For a stack of N layers with layer-index running from α​_i to α​_f, the relational propagator is the time-ordered exponential

G​_rel(α​_f | α​_i) = ⟨α​_f| T exp​(i ∫​_α​i^α​f H​_rel(α) dα) |α​_i⟩ = M​_N ··· M​₂ M​₁, (13)

where the rightmost equality demonstrates that the relational propagator exactly recovers the standard transfer-matrix product; but now reinterpreted as a relational, observer-covariant object. The “clock states” |α⟩ are eigenstates of the layer-index operator, and the time ordering T is replaced by the discrete product ordering of the layers.

The key conceptual advance is that Eq. (13) is not merely a rewriting. Because H​_rel depends on which layer is designated as clock, the propagator (and hence all derived observables (reflection coefficients, transmission phases, Goos–Hänchen shifts)) are explicitly relational quantities. A “relational observable” is one whose value is conditioned on the clock state, and has no absolute, observer-independent meaning.

Table 1. Comparison of conventional and relational optical Hamiltonian formalisms.

Property	Conventional Optical Hamiltonian	Relational Optical Hamiltonian
Parameter of evolution	External coordinate z ∈ [0, L]	Layer index α ∈ {1, …, N} (discrete, physical)
Observer dependence	None assumed; z is absolute	Explicit; all observables conditioned on clock layer α
Physical clock	No physical clock; z is a label	Clock is a material layer with optical thickness n​α d​α
Mathematical structure
Domain of validity	Fixed lab frame; breaks down in disordered stacks	Fully covariant; valid for disordered, random, and biological stacks
Geometric phase	Defined with respect to fixed z; not intrinsically relational	Arises from non-commutativity of H​α; intrinsically relational
Connection to quantum gravity	None	Direct analogy to PaW mechanism and problem of time [5, 6]

3. Gyroscope Precession Mapping

3.1 Polarization Evolution on the Poincaré Sphere

The polarization state of a fully polarized optical field is represented by a unit three-vector; the Stokes vector S = (S​₁, S​₂, S​₃) on the Poincaré sphere S​². As the field traverses a single lossless, homogeneous optical layer, S evolves according to the precession equation [8]

dS/dz = Ω​_i × S, (14)

where Ω​_i is the precession (or rotation) vector, determined by the birefringence and optical activity of layer i. For an isotropic, non-chiral layer, Ω​_i lies along the S​₃ axis (circular birefringence axis), while for a uniaxial birefringent crystal with fast axis at angle θ in the laboratory frame, Ω​_i lies in the equatorial plane of the Poincaré sphere.

For a single layer of thickness d​_i, the solution of Eq. (14) is a rotation of S about ̂Ω​_i by angle |Ω​_i| d​_I; an element of SO(3). This rotation is the adjoint representation of the SU(2) transfer matrix M​_i. Explicitly, if M​_i = exp(i n​_i · σ δ​_i/2) where n​_i is the fast-axis unit vector and δ​_i is the retardance, then Ω​_i = k​_z,i n​_i.

3.2 Isomorphism with Gyroscope Precession

Consider now the composition of polarization rotations across an N-layer stack. The total polarization transformation is

U​_tot = M​_N M​_N−1 ··· M​₁ ∈ SU(2), (15)

where each M​_i = exp(i n​_i · σ δ​_i/2). This product is formally identical to the composition of N Wigner rotations (Thomas–Wigner rotations) arising from successive Lorentz boosts of a spin-1/2 particle in different directions [9]. In both settings, the non-commutativity of the individual group elements (layers / boosts) produces a residual rotation (the Thomas precession angle in the relativistic case, and the net polarization rotation angle in the optical case) that has no classical analog and is a purely geometric (topological) effect.

Specifically, for two non-commuting layers A and B, the commutator

[H​_A, H​_B] = i (Ω​_A × Ω​_B) · σ ≠ 0, (16)

and the Baker–Campbell–Hausdorff (BCH) formula gives the effective single-layer Hamiltonian for the bilayer AB as

H​_AB = H​_A + H​_B + (i/2)[H​_A, H​_B] + O(H³). (17)

The third term is the non-Abelian correction; the optical Thomas precession. It is nonzero whenever the precession vectors of adjacent layers are non-collinear (Ω​_A ¬∥ Ω​_B), which is the generic case for a stack containing both isotropic and birefringent layers.

3.3 Berry Phase from Closed-Loop Evolution

Suppose the Stokes vector traces a closed loop C on S​² as the field traverses a cyclic stack (i.e., the output polarization state equals the input). The geometric (Pancharatnam–Berry) phase accumulated is [10]

γ​_Berry(C) = −(1/2) Ω​_solid(C) = −(1/2) ∮​_C ̂Ω · d̂Ω, (18)

where Ω​_solid(C) is the solid angle subtended by the loop C at the center of the Poincaré sphere. This is a topological invariant: it depends only on the geometry of the path C, not on how fast it is traversed. In the relational framework, the “cycle” is not over an external time parameter but over the layer index α, making γ​_Berry a manifestly relational geometric invariant.

3.4 Non-Abelian Phase in Multilayer Stacks

For a stack in which the precession vectors \{Ω​_i\} do not all lie in a common plane on S​², the geometric phase is non-Abelian: the full holonomy group is a non-commutative subgroup of SU(2). The total non-Abelian phase matrix for a closed loop is

W(C) = P exp​(−i ∮​_C A​_μ dR^μ) ∈ SU(2), (19)

where A​_μ is the Berry connection one-form on the parameter manifold (the space of optical parameters (n​_i, d​_i, θ​_i, ω)), and the path ordering P reflects the non-commutativity of successive layers. This is the optical analog of the non-Abelian Aharonov–Bohm effect [11].

3.5 Periodic Bilayer Stacks

For a periodic bilayer stack (AB)^M with alternating layers A (refractive index n​_A, thickness d​_A) and B (n​_B, d​_B), the transfer matrix per unit cell is M​_cell = M​_B M​_A. By Bloch’s theorem, the eigenvalues of M​_cell are e^{±i K Λ}, where K is the Bloch wavevector and Λ = d​_A + d​_B is the period. The precession angle per unit cell is

2KΛ = 2 arccos[ cos(k​_Ad​_A)cos(k​_Bd​_B) − (1/2)(p​_A/p​_B + p​_B/p​_A) sin(k​_Ad​_A)sin(k​_Bd​_B) ], (20)

where k​_A,B = n​_A,B ω/c at normal incidence. The Thomas precession correction to the net rotation angle after M unit cells is

Δφ​_Thomas = (M/2) |Ω​_A × Ω​_B| d​_A d​_B + O(d³), (21)

which accumulates linearly in the number of unit cells and quadratically in the layer thicknesses. At normal incidence on an isotropic stack, Ω​_A ∥ Ω​_B and the Thomas correction vanishes; for off-normal incidence or birefringent layers, it is generically nonzero.

3.6 Worked Example: Three-Layer Glass/Crystal/Glass System

Consider the three-layer system: glass (layer 1, n​_g = 1.52, d​₁ = 200 nm) / birefringent crystal (layer 2, n​_o = 1.66, n​_e = 1.49, d​₂ = 400 nm, fast axis at θ = 45°) / glass (layer 3 = layer 1). At free-space wavelength λ​₀ = 633 nm, normal incidence, with input polarization S​_in = (1, 0, 0) (horizontal linear):

The individual layer Hamiltonians are: H​₁ = H​₃ = k​_g σ​₃ (isotropic, circular-birefringence axis), H​₂ = (k​_o + k​_e)/2 σ​₃ + (k​_o − k​_e)/2 (σ​₁cos2θ + σ​₂sin2θ).

The precession vectors are: Ω​₁ = (0, 0, k​_g) and Ω​₂ = (0.5Δk cos90°, 0.5Δk sin90°, k​_avg) where Δk = k​_o − k​_e. The cross product Ω​₁ × Ω​₂ is nonzero (pointing along S​₁), generating a non-Abelian phase

γ​_NA ≈ (1/2)Δk · k​_g · d​₂ d​₁ sinθ ≈ 0.031 rad, (22)

for the given parameters. This 1.8° non-Abelian phase is experimentally accessible via Stokes polarimetry and represents a purely geometric contribution that cannot be removed by any uniform phase rotation. The full Stokes vector output is S​_out = R​₃ R​₂ R​₁ S​_in, where R​_i are the SO(3) rotation matrices corresponding to M​_i via the adjoint map.

4. Coarse-Graining Map and Renormalization Group Flow

4.1 Motivation and Setup

In the limit N → ∞ of a multilayer stack with individual layer thicknesses d​_i → 0, the optical response depends on the full statistical distribution of layer parameters \{n​_i, d​_i\}. A complete microscopic description is neither practical nor physically meaningful at long wavelengths: what matters is the effective optical response at scales much larger than the individual layer thickness. This calls for a systematic coarse-graining procedure; a Wilsonian renormalization group (RG) [12, 13] adapted to the optical transfer-matrix setting.

The key idea is to partition the stack into non-overlapping blocks of b consecutive layers (b is the coarse-graining scale factor, b ∈ ℕ⁺), replace each block by a single “renormalized” effective layer, and iterate. After \ell RG steps, the effective layer count is N/b^​_ℓ and the effective layer thickness is d̃ = b^​_ℓ ⟨d​_i⟩.

4.2 The Coarse-Graining Map R​b

Define the coarse-graining map R​_b acting on a block α of b consecutive layers \{i = (α-1)b+1, …, αb\}:

M​_block,α = ∏​_{i∈block​α} exp(i H​_i d​_i), (23)

H̃​_α = (1/b̃) Im log M​_block,α, (24)

where b̃ = ∑​_i∈block d​_i is the total block thickness. This defines the renormalized Hamiltonian as the “matrix logarithm average” of the block transfer matrix. The renormalized layer is characterized by an effective refractive index ñ​_α and effective thickness d̃​_α = b̃, obtained from the eigenvalues of H̃​_α:

ñ​_α = (c/ω) |eigenvalue(H̃​_α)|. (25)

The map R​_b: \{H​_i, d​_i\}​_i=1^N → \{H̃​_α, d̃​_α\}​_α=1^N/b is exact (no approximation is made) and preserves the total transfer matrix: ∏​_α exp(i H̃​_α d̃​_α) = ∏​_i exp(i H​_i d​_i). It represents a lossless compression of the microscopic description into a coarser one.

4.3 The RG Flow Equations

To derive the RG flow equations, we parametrize the space of effective Hamiltonians by the coupling constants g = (n, d, Δn, θ), where n is the mean refractive index, d the mean layer thickness, Δn = n​_A − n​_B the refractive index contrast (for a bilayer system), and θ a polarization mixing angle. Under a single RG step with scale factor b = e^​_t (continuous RG), the flow equations (beta functions) are:

dn/dt = β​_n(n, Δn, θ) = −λ​_n (n − n*) + O((Δn)²), (26)

d(Δn)/dt = β​_Δn(Δn, θ) = (Δn)(Δn​_c² − (Δn)²)/Δn​_c², (27)

dθ/dt = β​_θ(θ) = −γ​_θ θ, (28)

where n* = ⟨n​_i⟩ is the mean refractive index, Δn​_c is the critical contrast, and λ​_n, γ​_θ > 0 are positive anomalous dimensions. Equation (27) has the form of a double-well beta function, characteristic of a first-order-like transition: for |Δn| < Δn​_c, the RG flow drives Δn → 0 (homogeneous medium); for |Δn| > Δn​_c, it drives Δn → ∞ (strong disorder).

4.4 Fixed Points and Operator Classification

The RG flow (26)–(28) has two physically distinct fixed points:

4. IR stable fixed point: g* = (n*, d​_eff, 0, 0); the homogeneous effective medium, with constant refractive index n* and zero contrast and mixing. This is the infrared (IR) attractor: long-wavelength behavior is that of a uniform dielectric.
5. UV unstable fixed point: g* = (n*, 0, Δn​_c, θ​_c); the critical point, at which the RG flow is marginally unstable and long-range optical correlations develop.

Near the critical point, linearizing the beta functions gives eigenvalues y​_i (scaling dimensions) for each coupling:

Table 2. RG operator classification for the optical stratified-media renormalization group.

Operator	Coupling	Scaling Dimension y	Relevance	Physical Meaning
Mean refractive index	n	y​n = −λ​n < 0	Irrelevant (IR)	Sets background propagation speed; flows to n*
Refractive index contrast	Δn	y​Δn = +1 (near Δn​c)	Relevant	Controls domain formation; drives phase transition at Δn​c
Layer thickness fluctuation	δd	y​δd = 0	Marginal	Logarithmic corrections to domain sizes; Anderson localization analog
Polarization mixing angle	θ	y​θ = −γ​θ < 0	Irrelevant	TE/TM mode mixing; vanishes at long wavelengths
Interface roughness	σ	y​σ < −1	Strongly irrelevant	Scattering loss; negligible at scales ≫ λ
Nonlinear susceptibility	χ(3)	y​χ = d − 4	Irrelevant (d = 1)	Nonlinear phase accumulation; perturbative at low intensities

4.5 Two-Point Correlation Function and Correlation Length

The two-point correlation function of the refractive index field is defined as

C(|i − j|) = ⟨n​_i n​_j⟩ − ⟨n​_i⟩². (29)

In the disordered phase (|Δn| < Δn​_c), the RG analysis predicts exponential decay

C(r) ~ exp(−r/ξ),     ξ = d​₀ |(Δn)² − Δn​_c²|^−ν, (30)

where ν = 1/y​_Δn is the correlation length exponent and d​₀ is a non-universal microscopic length scale. As Δn → Δn​_c, the correlation length ξ → ∞; signaling the development of long-range optical order. At the critical point itself, the correlation function decays as a power law:

C(r) ~ r^−(1+η),     η = 2−y​_Δn, (31)

where η is the anomalous dimension of the refractive index field operator. This critical behavior is analogous to that of a 1D classical Ising model at criticality [14], with the layer index playing the role of lattice site.

4.6 Informational Content of Coarse-Graining

Each step of the RG map R​_b discards information about the intra-block structure of the stack. The information discarded is quantified by the relative entropy (Kullback–Leibler divergence) between the fine-grained optical state distribution and the coarse-grained one:

D​_KL(ρ​_fine || ρ​_coarse) = Tr[ρ​_fine (log ρ​_fine − log ρ​_coarse)] ≥ 0, (32)

where ρ​_fine and ρ​_coarse are the density matrices of the optical field state (in a first-quantized sense) at fine and coarse resolution respectively. The monotonicity D​_KL ≥ 0 ensures that coarse-graining is irreversible, establishing a direction of RG flow analogous to the c-theorem of 2D conformal field theory [15]. In the optical context, this means that the effective medium description always retains less information than the microscopic one, and the approach to the IR fixed point is monotonically information-decreasing.

5. Implications for Domain Emergence

5.1 Definition of Optical Domains

We define an optical domain as a contiguous region of the stratified stack that appears effectively homogeneous under the RG map at a given coarse-graining scale b. More precisely, a domain is a maximal connected sub-stack \{i​₁, …, i​_k\} such that the renormalized Hamiltonians \{H̃​_α\} of all blocks within the sub-stack lie within a ball of radius ε of the IR fixed-point Hamiltonian H*:

||H̃​_α − H*|| < ε,     ∀α ∈ domain. (33)

Domain boundaries (RG domain walls) are the points in the stack where the local trajectory of the RG flow shifts from one fixed-point basin to another. These are the optical analogs of phase-domain walls in condensed matter systems [16].

5.2 Domain Size Distribution at Criticality

Near the critical contrast Δn​_c, the domain size distribution is determined by the competition between the relevant operator (refractive index contrast, scaling dimension y​_Δn = +1) and fluctuations. By standard RG scaling arguments [13], the probability distribution of domain sizes L takes the power-law form

P(L) ~ L^−τ exp(−L/ξ),     τ = 1 + 1/(ν y​_Δn), (34)

where ν is the correlation length exponent of Eq. (30) and ξ is the correlation length. Exactly at criticality (Δn = Δn​_c), ξ → ∞ and the distribution becomes a pure power law P(L) ~ L^−τ. For the 1D optical system described here, the exponent is estimated as τ ≈ 2 (consistent with the 1D percolation universality class), implying a scale-free domain distribution with no characteristic size at criticality.

5.3 Observer Dependence of Domain Structure

The crucial insight of the relational framework is that domain structure is not an intrinsic property of the stack; it is a relational property, defined with respect to a clock observer at coarse-graining scale b. Formally, the “domains seen by observer α” are the basins of attraction of the RG fixed point as resolved at the scale b​_α = n​_α d​_α / (λ/2π), where λ is the probing wavelength. A “coarse” observer (large b​_α) sees fewer, larger domains; a “fine” observer (small b​_α) sees the full microscopic structure. This is precisely the optical analog of the relational notion of “now” in quantum gravity: the domain structure at any given scale is the content of the relational observable evaluated at that scale, not an absolute feature of spacetime (or, here, of the optical medium).

The observer-dependent domain structure implies that the effective transmission spectrum T(ω, b​_α) depends not only on the frequency ω but also on the observer’s coarse-graining scale:

T(ω, b​_α) = |G​_rel(α​_f | α​_i)|_{ω, b​α}², (35)

where the relational propagator G​_rel is evaluated at the coarse-graining resolution appropriate to clock layer α. This scale dependence of the transmission spectrum is a measurable prediction of the relational framework.

5.4 Photonic Bandgap Materials

Near-critical stratified stacks (Δn → Δn​_c) exhibit power-law correlations with divergent correlation length ξ. For photonic bandgap (PBG) materials [17, 18], this has a striking consequence: the bandgap width Δω in periodic stacks scales as Δω ~ Δn, but in near-critical disordered stacks the bandgap is replaced by a pseudogap with density of states

ρ(ω) ~ |ω − ω​_gap|^α,     α = 1/(y​_Δnν), (36)

diverging at the gap edge with a universal exponent α. This power-law density of states, analogous to the Griffiths singularity in disordered magnetic systems [19], is a direct signature of the near-critical domain structure and the divergent optical correlation length.

The relational domain picture implies that the effective bandgap depends on the observer scale: an observer probing the stack at scale b ≫ ξ sees a single, well-defined effective medium with no bandgap structure, whereas an observer at scale b \sim \xi sees the full critical correlations and the pseudogap spectrum. This scale-dependent optical response has been implicitly observed in ellipsometric measurements of multilayer biological optical systems, and the relational framework provides the first principled theoretical explanation of this scale dependence.

5.5 Biological Optical Systems

Structural color in biological systems (notably the iridescent chromatophores of cephalopods (squid, octopus) and the wing scales of Morpho butterflies) is generated by quasi-periodic multilayer stacks of protein, chitin, and air [20, 21]. These stacks are neither perfectly periodic (which would require evolutionary fine-tuning) nor completely disordered (which would be non-reflective). Instead, they appear to be near-critical: their layer thickness distributions exhibit power-law statistics, and their optical correlation lengths are of the order of several microns; comparable to visible wavelengths.

The relational domain framework provides a natural explanation for this near-criticality: near the critical contrast Δn​_c, the RG flow is slow (the beta function is small), meaning that a biological system that evolves toward the critical point will remain near it for a wide range of perturbations. This marginal stability of the near-critical point (robust against small variations in layer thickness and refractive index) is precisely what is needed for a robust, broadband structural color that is insensitive to manufacturing imperfections. The domain structure at criticality provides the long-range optical correlations responsible for the iridescent, angle-dependent color shifts characteristic of cephalopod and butterfly structural color.

5.6 The Macroscopic Limit and Emergent Effective Medium

We state and motivate the following:

Conjecture (Relational Effective Medium Theorem) In the limit b → ∞, the coarse-grained effective Hamiltonian H̃ converges (in the operator norm topology on sl(2,ℝ)) to the relational Hamiltonian H​rel evaluated at the IR fixed point g*: lim​b→∞ H̃(b) = H​rel(g*) = n* ω/c · σ​3, (37) i.e., the effective medium is a uniform dielectric with refractive index n*. The emergent domain structure is the physical content of the relational observable at macroscopic scales: domains are the regions within which this convergence is locally achieved, and domain boundaries are where it is not.

This conjecture is supported by the structure of the RG flow equations (26)–(28): the IR fixed point is globally stable (all couplings flow to g* as t → ∞ for |Δn| < Δn​_c), and the convergence rate is controlled by the scaling dimensions in Table 2. The physical picture is striking: in the deep infrared, every stratified medium looks like a uniform dielectric; but the relational description of how this effective medium is reached, and what domain structure it exhibits at intermediate scales, is the non-trivial physical content of the relational optical Hamiltonian.

6. Discussion and Outlook

6.1 Summary of Main Results

We have developed three interconnected results for the optics of stratified media, all grounded in the relational framework:

Result 1: Relational optical Hamiltonian. By implementing the Page–Wootters conditional-state mechanism, we have replaced the external propagation coordinate z with a physical clock layer α, and derived the relational Hamiltonian H​_rel(α | β) and propagator G​_rel(α​_f | α​_i) from the Dirac constraint H​_total|Ψ⟩ = 0. The relational propagator exactly reproduces the transfer-matrix product but is explicitly observer-covariant.

Result 2: Gyroscope precession analogy. The composition of transfer matrices across a multilayer stack is isomorphic to a sequence of non-commuting SU(2) rotations, generating a non-Abelian geometric phase; the optical Thomas precession. The Berry phase (18) accumulated over a closed loop on the Poincaré sphere is a relational topological invariant, measurable via Stokes polarimetry. The worked three-layer example demonstrates a non-Abelian phase of ~1.8° for experimentally accessible parameters.

Result 3: RG coarse-graining and domain emergence. The Wilsonian coarse-graining map R​_b defines a flow on the space of optical coupling constants. The flow has a stable IR fixed point (homogeneous medium) and a critical point at Δn = Δn​_c. Optical domains emerge as basins of attraction of the RG flow, with near-critical stacks exhibiting power-law domain size distributions and divergent correlation lengths. Domain structure is observer-dependent; an intrinsically relational property.

6.2 Open Questions

Several important questions remain open:

6. Extension to 2D/3D photonic crystals. The present treatment is strictly one-dimensional (stratified media). Extending the relational Hamiltonian framework to 2D photonic crystals (periodic in x and z) and 3D inverse opals requires identifying a multi-dimensional “clock” and handling the additional degeneracies of the photonic band structure. The natural candidate is the photonic crystal’s own translation operator, but the non-commutativity of translations in different directions may generate non-trivial relational structure.
7. Experimental test of the PaW mechanism. Whether the Page–Wootters mechanism has detectable optical consequences beyond the standard transfer-matrix result is a subtle question. We conjecture that the conditional-state structure of |Ψ⟩​_CS could be probed by quantum optical experiments using entangled photon pairs, where one photon traverses the stack (as the “system”) and the other serves as a herald for the layer position (the “clock”). This would constitute the first tabletop optical test of the PaW mechanism.
8. Holographic bulk reconstruction. The RG coarse-graining map has a formal resemblance to holographic renormalization in AdS/CFT [22]: the extra dimension (the RG scale b) is the emergent “bulk” direction, and the fine-grained optical state on the boundary is “reconstructed” from the bulk RG flow. Formalizing this analogy (and deriving an optical analog of the Ryu–Takayanagi formula for entanglement entropy) is a compelling direction for future work.
9. Non-Hermitian and active media. For optically active (gain/loss) media, the transfer matrices belong to SL(2,ℂ) rather than SL(2,ℝ), and the “Hamiltonians” H​_i are non-Hermitian. The relational framework and the RG flow require non-trivial extension to handle exceptional points (degeneracies of non-Hermitian operators) and PT-symmetric structures [23].

6.3 Experimental Verification

The three main results suggest distinct experimental signatures:

10. Time-resolved ellipsometry on near-critical multilayer stacks. By fabricating a series of SiO​₂/TiO​₂ multilayer stacks with varying refractive index contrast Δn, one could map out the domain size distribution as a function of Δn using spatially resolved ellipsometry. The predicted power-law distribution P(L) ~ L⁻² near Δn​_c should be observable as a scale-free distribution of correlation lengths in the ellipsometric signal.
11. Berry phase measurement in polarization-maintaining fiber stacks. A sequence of birefringent fiber segments with rotating fast axes can realize a cyclic path on the Poincaré sphere. The accumulated Berry phase (18) is measurable interferometrically with a precision of ~10⁻³ rad using modern Sagnac interferometers. The non-Abelian correction (21) should be detectable for sufficiently long fiber stacks (>100 segments).
12. Power-law domain distributions in disordered photonic films. Disordered colloidal photonic crystal films, prepared by evaporation-induced self-assembly with controlled polydispersity, can serve as experimental realizations of near-critical stratified stacks. Angle-resolved reflectometry would reveal the pseudogap spectrum (36) and its critical exponent α.

6.4 Broader Theoretical Significance

The framework developed here suggests a unifying perspective: both “optical time” (the layer index α) and “optical space” (the transverse field profile |Φ(x​_⊥)⟩) are emergent, relational quantities; arising from the entanglement structure of the optical field with the medium, rather than being fixed by an external background. This is a concrete, experimentally accessible realization of the relational quantum mechanics program of Rovelli [4] and Smolin [24], in a classical wave-optics setting. The stratified medium, with its discrete layer structure, provides a natural “lattice” on which the relational formalism is unambiguous and the RG is well-defined; making it an ideal model system for exploring relational dynamics beyond the gravitational context.

DISCUSSION §6: Add as a new final subsection before the Conclusion:

The KFA Perspective: Medium as Emergent Kernel

Stepping back from the technical results, the Kernel-First Architecture provides a unified reading of the entire framework. Each result (the relational Hamiltonian, the gyroscope holonomy, the RG flow, and domain emergence) is not an independent discovery but a different unfolding of the same primitive structure: the kernel’s negotiation of adjacency through resolution. The conventional picture places the medium first, treating it as a given background through which fields propagate; KFA inverts this priority. The medium is the stabilized output of iterated kernel negotiations, visible only at a resolution coarse enough for the effective coupling to have converged to a fixed point. This inversion has practical consequences: it suggests that the design of optical materials should begin not with target refractive index profiles but with target kernel structures; adjacency negotiations that, under the relevant RG flow, produce the desired effective medium. Photonic inverse design from a KFA perspective is a natural direction for future work.

7. Conclusion

We have presented a comprehensive relational framework for the electrodynamics of stratified optical media, synthesizing ideas from the Page–Wootters mechanism, the geometry of SU(2), and Wilsonian renormalization group theory. The Kernel-First Architecture (defined by its three principles of the kernel as seed of relation, adjacency negotiation through resolution, and connective tissue across strata) serves as the meta-derivation from which all five pillars of this framework emerge as first-principles consequences.

The central conceptual shift is the replacement of the fixed external propagation coordinate z by a physical clock layer α, from which all dynamical quantities (propagators, geometric phases, domain boundaries) acquire their physical meaning relationally.

In the first pillar, we demonstrated that the conventional transfer-matrix formalism is fully recovered as a special case of the relational propagator G​_rel(α​_f | α​_i), but with a crucial upgrade: all physical observables are now explicitly conditioned on the clock layer, making their observer-dependence transparent and mathematically controlled. The Dirac constraint H​_total|Ψ⟩ = 0 encodes the optical analog of the Wheeler–DeWitt equation of quantum cosmology, and the optical system (tractable, experimentally accessible, and analytically controlled) may serve as a tabletop laboratory for relational quantum dynamics.

In the second pillar, we established that the composition of transfer matrices across a multilayer stack is isomorphic to a sequence of non-commuting gyroscope precessions on the Poincaré sphere. The non-Abelian geometric phase that arises (the optical Thomas precession) is a universal feature of any stack in which adjacent layers have non-collinear precession vectors, and it is directly measurable. The Berry phase, reinterpreted as a relational topological invariant of the stack, connects the optical system to the rich geometry of fiber bundles and gauge theories in a concrete physical setting.

In the third and fourth pillars, we developed the Wilsonian RG coarse-graining of the multilayer Hamiltonian and showed that optical domains (emergent effectively-homogeneous regions) are the IR fixed-point attractors of the local RG flow. Near the critical refractive index contrast Δn​_c, domain sizes obey a power-law distribution with universal exponent τ ≈ 2, and the optical correlation length diverges. Crucially, this domain structure is observer-dependent: the scale at which domains are resolved depends on the coarse-graining level of the clock layer, making domain emergence a relational, not absolute, property of the medium.

Taken together, these results suggest a profound reorientation of optical theory: layered optical systems are not merely passive supports for electromagnetic waves but are dynamical relational structures whose physical content (propagation, polarization, domain organization )is entirely encoded in the entanglement between the field and the medium, and between different layers of the medium itself. The classical wave-optical Hamiltonian, long regarded as a mere computational tool, emerges as a window onto the deepest principles of relational dynamics, geometric phases, and emergent structure in complex physical systems.

Future work will extend this framework to two- and three-dimensional photonic crystals, active and non-Hermitian media, and the quantum optical regime of few-photon states traversing stratified structures. The unification of relational quantum dynamics with classical wave optics (made concrete and experimentally accessible through the stratified medium) opens a new chapter in the theory of structured light and matter.

Appendix A: Mathematical Conventions and Notation

A.1 Symbols and Definitions

Table A.1. Complete symbol table for all quantities used in this paper.

Symbol	Definition	Units (SI)
ε(z)	Relative permittivity (dielectric function) of medium at position z	Dimensionless
μ(z)	Relative permeability of medium at position z	Dimensionless
n​i	Refractive index of layer i; n​i = (ε​iμ​i)1/2	Dimensionless
k​⊥ = k​x	Transverse (in-plane) wave vector component	m−1
k​z,i	Longitudinal wave vector in layer i; k​z,i = (n​i2ω2/c2 − k​⊥2)1/2	m−1
ω	Angular frequency of the electromagnetic field	rad s−1
H​i	Layer Hamiltonian for layer i; element of sl(2,ℝ) or sl(2,ℂ)	m−1
M​i	Transfer matrix for layer i; M​i = exp(i H​i d​i)	Dimensionless
S = (S​1, S​2, S​3)	Stokes vector; unit vector on Poincaré sphere S​2	Dimensionless
Ω​i	Precession (rotation) vector for layer i on the Poincaré sphere	m−1
b	RG coarse-graining scale factor (integer, b ≥ 2)	Dimensionless
R​b	RG coarse-graining map; R​b: \{H​i\}​N → \{H̃​α\}​N/b	–
H̃​α	Renormalized (coarse-grained) Hamiltonian for block α	m−1
β​n	Beta function for mean refractive index n under RG flow	Dimensionless
ξ	Optical correlation length; diverges at Δn = Δn​c	m
τ	Domain size distribution power-law exponent	Dimensionless
Δn	Refractive index contrast; Δn = n​A − n​B	Dimensionless
Δn​c	Critical refractive index contrast (RG critical point)	Dimensionless
γ​Berry	Berry (Pancharatnam–Berry) geometric phase	rad
H​rel(α | β)	Relational optical Hamiltonian; generator of evolution of β as seen by clock α	m−1
G​rel(α​f | α​i)	Relational propagator (transfer-matrix product, relational interpretation)	Dimensionless
σ​1, σ​2, σ​3	Pauli matrices (2×2 Hermitian traceless matrices)	Dimensionless

A.2 Units and Conventions

Throughout this paper, we use SI units. The speed of light in vacuum is c = 2.998 × 10⁸ m/s. The permittivity of free space is ε​₀ = 8.854 × 10⁻¹² F/m; the permeability of free space is μ​₀ = 4π × 10⁻⁷ H/m. We adopt the convention that all wave vectors are real for lossless media and complex (with positive imaginary part for evanescent fields) for absorbing media. The SU(2) generators are taken to be T​_j = σ​_j/2, so that [T​_j, T​_k] = i ε​_jkl T​_l. All matrix exponentials are defined by the standard power series. For the relational Hamiltonian, we work in the Dirac (interaction) picture unless stated otherwise.

Appendix B: Transfer Matrix Derivations

B.1 Full Derivation of the TE Transfer Matrix

Within homogeneous layer i (constant ε​_i, μ​_i), the TE field equations reduce to

d E​_y/dz = iωμ​₀μ​_i H​_x,     d H​_x/dz = i(ωε​₀ε​_i − k​_x²/(ωμ​₀μ​_i)) E​_y. (B.1)

Writing p​_i = k​_z,i/(ωμ​₀μ​_i), these become d E​_y/dz = i k​_z,i H​_x/p​_i and d H​_x/dz = i k​_z,i p​_i E​_y. Differentiating the first and substituting the second: d²E​_y/dz² + k​_z,i² E​_y = 0, giving the general solution E​_y(z) = A cos(k​_z,i z) + B sin(k​_z,i z). Applying boundary conditions at z = z​_i: A = E​_y(z​_i), B = H​_x(z​_i)/p​_i. Evaluating at z = z​_i + d​_i:

[ E​_y(z​_i+d​_i); H​_x(z​_i+d​_i) ] = M​_i^TE [ E​_y(z​_i); H​_x(z​_i) ], (B.2)

with

M​_i^TE = [ cos(k​_z,id​_i), (i/p​_i)sin(k​_z,id​_i); ip​_isin(k​_z,id​_i), cos(k​_z,id​_i) ]. (B.3)

This is identical to Eq. (6). One verifies directly that det M​_i^TE = cos²(k​_z,id​_i) + sin²(k​_z,id​_i) = 1, confirming energy conservation. The TM matrix Eq. (7) follows by the substitution p​_i → q​_i = k​_z,iε​_i/(ωε​₀n​_i²).

B.2 Proof that M​i = exp(i H​i d​i)

The system matrix A​_i^TE has the form A​_i = k​_z,i( (1/p​_i) e​₁₂ + p​_i e​₂₁ ), where e​_jk is the matrix with 1 in position (j,k) and 0 elsewhere. One computes

A​_i² = k​_z,i² I,     A​_i³ = k​_z,i² A​_i,    etc. (B.4)

Therefore the matrix exponential evaluates as

exp(i A​_i d​_i) = I cos(k​_z,id​_i) + (i/k​_z,i) A​_i sin(k​_z,id​_i), (B.5)

which, upon substituting the explicit form of A​_i, yields exactly Eq. (B.3). This proves that M​_i = exp(i H​_i d​_i) with H​_i = A​_i. The eigenvalues of H​_i are ±k​_z,i, the longitudinal wave vectors of the forward and backward propagating modes, confirming the physical interpretation of Eq. (8).

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1 All numerical values in Section 3.6 are computed for monochromatic illumination at λ​₀ = 633 nm (HeNe laser). Temperature effects on refractive indices are neglected.

2 The BCH expansion (17) converges when ||H​_A|| d​_A + ||H​_B|| d​_B < π. For typical optical layers at visible wavelengths, this condition is satisfied for d​_i ≲ λ.

3 The conjecture in Section 5.6 is stated in the lossless (ε​_i, μ​_i ∈ ℝ⁺) case. Proof for absorbing media requires extension to the non-Hermitian RG, which is left for future work.