Thermodynamic theory of highly multimoded nonlinear optical systems

Methods

• Power packets are distributed among distinct optical modes with energy levels ε_i and degeneracies g_i, yielding a combinatorial multiplicity W_i.The number of arrangements is W_i = (n_i + g_i − 1)!/[n_i!(g_i − 1)!].Source: Methods · p. 18
• Entropy is defined as S_N = ln W and maximized subject to fixed total internal energy and total packet number.Lagrange-multiplier extremization produces the Bose-Einstein distribution.Source: Mode occupancy distribution · p. 18
• When packet occupancy greatly exceeds degeneracy, n_i ≫ g_i, the Bose-Einstein distribution reduces to a Rayleigh-Jeans distribution.The same high-occupancy condition supports a Taylor expansion of the entropy expression.Source: Mode occupancy distribution · p. 18, Optical Entropy · p. 19
• For highly populated modes, the entropy simplifies to S_N ≃ ∑g_i ln n_i and, after setting g_i = 1, is expressed using the total number of modes.Mode power |c_i|^2 is proportional to packet number n_i, linking the statistical description to optical observables.Source: Optical Entropy · p. 19
• The entropy contains a constant reference floor and a component that responds to nonlinear mode mixing.Temperature and chemical potential are introduced through α = μ(Tn_c)^−1 and β = (Tn_c)^−1.Source: Optical Entropy · pp. 19–20
• The resulting expressions connect modal populations to total optical power and internal energy.These observables are obtained after substituting n_i = n_c|c_i|^2.Source: Optical Entropy · p. 20

Derivation of the first equation of state

• The optical internal-energy and power expressions are manipulated to obtain the thermodynamic relation used as the first equation of state.This relation underlies the subsequent determination of temperature and chemical potential.Source: Derivation of the first equation of… · p. 20
• Chemical potential μ is then obtained from the first equation of state after solving for temperature.The procedure uses μ = 𝒫^−1(U − MμT) as stated in the derivation.Source: Derivation of the first equation of… · p. 20
• For given input power 𝒫 and internal energy U, the known eigenspectrum ε_i allows temperature T to be uniquely determined.The admissible solution must keep every modal occupancy |c_i|^2 positive.Source: Derivation of the first equation of… · p. 20

Extensivity of the optical entropy

• The entropy and thermodynamic variables are derived from the equilibrium entropy expression and the first equation of state.The derivation rewrites μ in terms of U, M, T, and 𝒫 before differentiating the entropy.Source: Extensivity of the optical entropy · p. 20, Thermodynamic driving forces:… · p. 23
• Doubling the number of modes, optical power, and internal energy while preserving the system profile leaves temperature and chemical potential unchanged.The construction doubles M, 𝒫, and U while maintaining the density-of-states structure.Source: Extensivity of the optical entropy · pp. 20–21
• The entropy scales linearly under simultaneous scaling of internal energy, mode count, and optical power: S(λU, λM, λ𝒫) = λS(U, M, 𝒫).This scaling relation expresses the entropy’s extensivity.Source: Extensivity of the optical entropy · p. 21
• The formal chemical potential and optical pressure are introduced consistently with the thermodynamic relations.Optical pressure is defined as the third intensive variable associated with changes in mode number.Source: Thermodynamic driving forces:… · pp. 21–22
• Mode expansion is treated as a self-similar change that preserves the density-of-states profile, with M playing the role of thermodynamic volume.The rescaled density of states is written as D̃(ε) = VD(ε), where V is the scale factor.Source: Thermodynamic driving forces:… · p. 22

Euler equation: second equation of state

• Entropy extensivity enables differentiation with respect to a common scaling factor to obtain the Euler equation.The resulting expression relates entropy to internal energy, mode number, optical power, temperature, pressure, and chemical potential.Source: Euler equation: second equation of state · p. 23
• The Euler equation is consistent with the first equation of state and makes entropy’s extensivity in U, M, and 𝒫 explicit.The derivation uses the previously established thermodynamic relations.Source: Euler equation: second equation of state · p. 23

• Energy flows from higher to lower temperature when the subsystem temperatures have the same sign.Source: Direction of energy flow between two… · p. 24
• When temperatures have opposite signs, energy flows from the negative-temperature subsystem to the positive-temperature subsystem.Source: Direction of energy flow between two… · p. 24
• In a grand canonical-like ensemble, energy and optical power are exchanged subject to conservation of total energy and power.Source: Direction of power and energy flow… · p. 24
• At equal temperature, power flows from higher to lower chemical potential for positive temperature, with the direction reversed for negative temperature.Source: Direction of power and energy flow… · p. 24
• Power exchange ceases once the chemical potentials become equal at equilibrium.Source: Direction of power and energy flow… · p. 24

• Each subsystem’s energy-temperature relation can be calculated individually at fixed optical power.Source: Graphical way of predicting the final… · p. 24
• The common equilibrium temperature is obtained from the intersection of U_A(T) and U_0 − U_B(T).Source: Graphical way of predicting the final… · p. 25
• The method predicts final temperatures for the simulated canonical-like ensembles in Figs. 2 and 3.Source: Graphical way of predicting the final… · p. 25
• Adiabatic evolution preserves mode occupancies and entropy while eigenenergies, internal energy, temperature, and chemical potential change.Source: Graphical way of predicting the final… · p. 25
• During an isentropic process, temperature and chemical potential scale by the same factor as the eigenenergies.Source: Isentropic invariants · p. 25

Terminology

• A canonical-like ensemble contains subsystems exchanging internal energy but not optical power through a diathermic wall.Source: Terminology · pp. 25–26
• A grand canonical-like ensemble additionally permits optical-power exchange through a diathermic permeable wall.Source: Terminology · p. 26

Simulations

• Ensemble and propagation-distance averaging are expected to agree when the nonlinear system is ergodic.Source: Simulations · p. 26
• The equilibrium distribution is unaffected by how states are grouped into cells with associated degeneracies.Source: Figure 1 · p. 28
• Doubling the number of sites can preserve the density-of-states profile through self-similar enlargement or alter it through alloy-like enlargement.Source: Figure 2 · p. 29

• The discrete optical field evolves through complex site amplitudes coupled by propagation constants, neighboring-site coupling, and nonlinear terms.Source: Normalizations and conserved… · pp. 33–34
• The Hamiltonian matrix has eigenvalues ε_i and orthonormal eigenvectors, allowing the state to be represented in modal coordinates.Source: Normalizations and conserved… · pp. 33–34
• The Hamiltonian is conserved during propagation because it has no explicit dependence on z.Source: Normalizations and conserved… · p. 34
• Optical power P = Σ|c_i|^2 = Σ|a_m|^2 is also conserved during evolution.Source: Normalizations and conserved… · p. 34
• The linear Hamiltonian equals the modal energy sum Σ ε_i|c_i|^2, and the optical internal energy is defined as U = −H_L.Source: Normalizations and conserved… · p. 34

• The discrete coupled nonlinear Schrödinger equations describe normalized optical-field evolution in the Lieb lattice.Source: Lieb lattice: Governing equations · p. 36
• The indices m and n label unit cells, while κ1 and κ2 are the vertical and horizontal nearest-neighbor coupling coefficients.Source: Lieb lattice: Governing equations · p. 36

• κ1 and κ2 respectively couple cavities along unit-cell edges and diagonals, while the final term represents Kerr nonlinearity.Source: 3D array of optical cavities:… · p. 37
• The three-dimensional lattice uses site indices l, m, and n along the x, y, and z coordinates.Source: 3D array of optical cavities:… · p. 37

• The coupled equations describe normalized optical fields for two orthogonal polarizations, a and b, in an optical lattice.Source: Governing equation for the two-species… · p. 37
• The nonlinear terms represent self-phase modulation, cross-phase modulation, and four-wave mixing with coefficients A, B, and C.Source: Governing equation for the two-species… · p. 37
• The Hamiltonian of the coupled system is conserved and includes the polarization-dependent nonlinear interactions.Source: Governing equation for the two-species… · pp. 37–38
• For circular polarization, A=1, B=2, C=0; for linear polarization, A=1, B=2/3, C=0, so the two optical powers are individually conserved.Source: Governing equation for the two-species… · p. 38

Lattice design for a canonical-like setting

• The canonical-like setting is governed by the coupled equations with A=1, B=2/3, and C=0.Source: Lattice design for a canonical-like… · p. 38
• The canonical-like design uses three groups of polarization-maintaining sites, with x̂ and ŷ polarizations confined to opposite sides and overlapping in the middle layer.Source: Lattice design for a canonical-like… · p. 38
• The left and right groups act as subsystems, while the middle group functions as a thermal interaction layer.Source: Lattice design for a canonical-like… · p. 39
• The grand-canonical-like design uses A=1, B=2/3, and C=1/3, enabling the corresponding ensemble configuration.Source: Lattice design for a grand… · p. 39
• Its geometry contains side groups for the two polarizations and a central group of circular waveguides serving as the interaction layer.Source: Lattice design for a grand… · p. 39