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Thermodynamic theory of highly multimoded nonlinear optical systems
Fan O. Wu, Absar U. Hassan, Demetrios N. Christodoulides
TL;DR
Nonlinear highly multimoded optical systems require a thermodynamic framework for describing their collective equilibrium behavior. The paper introduces extensive variables and derives equations of state, showing that energy and optical power exchanges obey the second law while also characterizing isentropic processes and Carnot-like cycles.
Problem
The collective dynamics of many nonlinearly interacting optical modes pose theoretical challenges beyond single-mode systems.
Method
The paper constructs a thermodynamic framework using optical entropy, extensive variables, equations of state, and equilibrium mode statistics.
Results
Energy flows from higher to lower temperature for same-sign temperatures, while opposite-sign systems transfer energy from negative to positive temperature; optical power follows chemical-potential gradients at common temperature.
Takeaways & Limitations
The framework describes thermal-equilibrium behavior in highly multimoded nonlinear optical systems and supports analysis of isentropic processes and Carnot-like cycles.
Takeaways & Limitations
The mode-occupancy reduction assumes n_i ≫ g_i, and the eigenenergy ordering depends on whether the system is a waveguide or nonlinear multimode cavity.
Abstract
from arXiv · showhide
The quest for ever higher information capacities has brought about a renaissance in multimode optical waveguide systems. This resurgence of interest has recently initiated a flurry of activities in nonlinear multimode fiber optics. The sheer complexity emerging from the presence of a multitude of nonlinearly interacting modes has led not only to new opportunities in observing a host of novel optical effects that are otherwise impossible in single-mode settings, but also to new theoretical challenges in understanding their collective dynamics. In this Article, we present a consistent thermodynamical framework capable of describing in a universal fashion the exceedingly intricate behavior of such nonlinear highly multimoded photonic configurations at thermal equilibrium. By introducing pertinent extensive variables, we derive new equations of state and show that both the 'internal energy' and optical power in many-mode arrangements always flow in such a way so as to satisfy the second law of thermodynamics. The laws governing isentropic processes are derived and the prospect for realizing Carnot-like cycles is also presented. In addition to shedding light on fundamental issues, our work may pave the way towards a new generation of high power multimode optical structures and could have ramifications in other many-state nonlinear systems, ranging from Bose-Einstein condensates to optomechanics.
Methods
The methods derive optical entropy from the number of ways power packets occupy multimode energy levels, then obtain modal distributions and observables under thermodynamic constraints.
- 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)!].
- Methods: 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.
- Methods: 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.
- Methods: 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.
- Methods: 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.
- Methods: 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.
Derivation of the first equation of state
The first equation of state is used to determine equilibrium temperature and chemical potential from optical power, internal energy, mode count, and the known eigenspectrum.
- 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.
- Derivation of the first equation of state: 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.
- Derivation of the first equation of state: 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.
Extensivity of the optical entropy
The optical entropy is extensive in internal energy, mode count, and optical power, with mode expansion defined self-similarly so the density-of-states profile remains invariant.
- 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.
- Extensivity of the optical entropy: 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.
- Extensivity of the optical entropy: 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.
- Extensivity of the optical entropy: 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.
- Extensivity of the optical entropy: 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.
Euler equation: second equation of state
Using entropy extensivity, the paper derives an Euler equation that provides a second equation of state involving the multimode system’s thermodynamic variables.
- 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.
- Euler equation: second equation of state: 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.
Direction of energy flow between two multimoded systems in thermal contact
The second law determines the direction of energy and optical-power exchange between multimoded subsystems. Energy follows temperature differences, while power follows chemical-potential differences when both quantities can be exchanged.
- Energy flows from higher to lower temperature when the subsystem temperatures have the same sign.
- When temperatures have opposite signs, energy flows from the negative-temperature subsystem to the positive-temperature subsystem.
- In a grand canonical-like ensemble, energy and optical power are exchanged subject to conservation of total energy and power.
- At equal temperature, power flows from higher to lower chemical potential for positive temperature, with the direction reversed for negative temperature.
- Power exchange ceases once the chemical potentials become equal at equilibrium.
Graphical way of predicting the final temperature of canonical-like ensembles consisting of several subsystems
The equilibrium temperature of canonical-like multimoded ensembles can be predicted graphically from subsystem energy-temperature relations. The same framework also describes isentropic changes, in which temperature and chemical potential scale together with the eigenenergies.
- Each subsystem’s energy-temperature relation can be calculated individually at fixed optical power.
- The common equilibrium temperature is obtained from the intersection of U_A(T) and U_0 − U_B(T).
- The method predicts final temperatures for the simulated canonical-like ensembles in Figs. 2 and 3.
- Adiabatic evolution preserves mode occupancies and entropy while eigenenergies, internal energy, temperature, and chemical potential change.
- During an isentropic process, temperature and chemical potential scale by the same factor as the eigenenergies.
Terminology
The paper distinguishes ensemble terminology, simulation-averaging procedures, and ways of enlarging optical systems. These definitions clarify which quantities are exchanged and how numerical and thermodynamic constructions are interpreted.
- Terminology: A canonical-like ensemble contains subsystems exchanging internal energy but not optical power through a diathermic wall.
- Terminology: A grand canonical-like ensemble additionally permits optical-power exchange through a diathermic permeable wall.
- Simulations: Ensemble and propagation-distance averaging are expected to agree when the nonlinear system is ergodic.
- Simulations: The equilibrium distribution is unaffected by how states are grouped into cells with associated degeneracies.
- Simulations: Doubling the number of sites can preserve the density-of-states profile through self-similar enlargement or alter it through alloy-like enlargement.
Normalizations and conserved quantities in a discrete optical system
The discrete optical system is formulated through site amplitudes, modal eigenstates, and a Hamiltonian description. Its evolution conserves both the Hamiltonian-based internal energy and optical power.
- The discrete optical field evolves through complex site amplitudes coupled by propagation constants, neighboring-site coupling, and nonlinear terms.
- The Hamiltonian matrix has eigenvalues ε_i and orthonormal eigenvectors, allowing the state to be represented in modal coordinates.
- The Hamiltonian is conserved during propagation because it has no explicit dependence on z.
- Optical power P = Σ|c_i|^2 = Σ|a_m|^2 is also conserved during evolution.
- 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.
Lieb lattice: Governing equations
The Lieb lattice is modeled by discrete coupled nonlinear Schrödinger equations for the evolution of normalized optical fields across its sites.
- The discrete coupled nonlinear Schrödinger equations describe normalized optical-field evolution in the Lieb lattice.
- The indices m and n label unit cells, while κ1 and κ2 are the vertical and horizontal nearest-neighbor coupling coefficients.
3D array of optical cavities: governing equation
The three-dimensional coupled-cavity array is governed by a discrete field-evolution equation whose couplings connect neighboring cavities and whose final term represents Kerr nonlinearity.
- κ1 and κ2 respectively couple cavities along unit-cell edges and diagonals, while the final term represents Kerr nonlinearity.
- The three-dimensional lattice uses site indices l, m, and n along the x, y, and z coordinates.
Governing equation for the two-species canonical-like and grand canonical-like ensembles
The two-species optical ensembles are modeled with coupled discrete nonlinear Schrödinger equations that include linear coupling and polarization-dependent nonlinear interactions.
- The coupled equations describe normalized optical fields for two orthogonal polarizations, a and b, in an optical lattice.
- The nonlinear terms represent self-phase modulation, cross-phase modulation, and four-wave mixing with coefficients A, B, and C.
- The Hamiltonian of the coupled system is conserved and includes the polarization-dependent nonlinear interactions.
- 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.
Lattice design for a canonical-like setting
The canonical-like optical ensemble uses separated polarization-preserving site groups connected through a thermal interaction layer, while the grand-canonical-like design additionally permits power exchange.
- 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.
- Lattice design for a canonical-like setting: 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.
- Lattice design for a canonical-like setting: The left and right groups act as subsystems, while the middle group functions as a thermal interaction layer.
- Lattice design for a canonical-like setting: The grand-canonical-like design uses A=1, B=2/3, and C=1/3, enabling the corresponding ensemble configuration.
- Lattice design for a canonical-like setting: Its geometry contains side groups for the two polarizations and a central group of circular waveguides serving as the interaction layer.