Abstract
The convoluted nonlinear behaviors of heavily multimode photonic structures have been recently the focus of considerable attention. The sheer complexity associated with such multimode systems, allows them to display a host of phenomena that are otherwise impossible in fewmode settings. At the same time, however, it introduces a set of fundamental challenges in terms of comprehending and harnessing their response. Here, we develop an optical thermodynamic approach capable of describing the thermalization dynamics in large scale nonlinear photonic tightbinding networks. For this specific system, an optical SackurTetrode equation is obtained that explicitly provides the optical temperature and chemical potential of the photon gas. Processes like isentropic expansion/compression, Joule expansion, as well as aspects associated with beam cleaning/cooling and thermal conduction effects in such chain networks are discussed. Our results can be used to describe in an effortless manner the exceedingly complex dynamics of highly multimoded nonlinear bosonic systems.
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Introduction
In recent years, there has been considerable interest in understanding the rich and complex dynamics of largescale nonlinear multimoded optical systems^{1,2,3,4,5}. To a great extent, what has fueled this effort is the everincreasing quest for high information capacity photonic networks^{6,7,8,9,10} and highpower optical sources^{11}. Similarly, the nonlinear physics of highquality factor multimode microresonators has been intensively investigated in a number of areas, like combgeneration and optomechanics^{12,13,14,15,16}. Yet, what kept us from fully expanding these classical settings, is the sheer complexity behind the nonlinear interactions taking place in such vastly manymoded environments. To appreciate this difficulty, consider for example a multimode optical fiber supporting thousands of modes. In this case, to correctly account for all the nonlinear processes unfolding in this structure, one has to first estimate at least a trillion or so fourwave mixing coefficients, before even attempting to numerically solve the evolution problem—a formidable task which is by itself computationally intensive. Clearly, to overcome these hurdles, a statistical approach has to be deployed that can effectively deal with manybody configurations. In essence, this calls for the development of a theoretical framework, akin to that of statistical mechanics, that has so far allowed us to understand the macroscopic properties of matter, in spite of the fact that on most occasions one has to deal with a great multitude of atoms or molecules—most often exceeding Avogadro’s number^{17}.
Quite recently, an optical thermodynamic theory was put forward, that can selfconsistently describe by means of statistical mechanics the utterly complex processes of energy exchange in nonlinear multimode systems at thermal equilibrium^{18}. In accord with the axioms of thermodynamics^{19}, the entropy of these systems was appropriately defined in terms of other extensive variables and a global equation of state was derived^{18,20}. Moreover, the laws governing isentropic processes were obtained and the prospect for Carnot cycles was suggested. Similar results were derived from the perspective of the grand canonical ensemble along with the expected statistical fluctuations in these systems^{21,22}.
Here, by means of an optical Sackur–Tetrode equation, we investigate the entropic response of largescale weakly nonlinear photonic chain networks. These classical networks may for example involve a large number of evanescently coupled waveguide lattices^{23} that exchange power in both crystalline (Fig. 1a) and other more complex arrangements (Fig. 1b), or could be comprised of arrays of optical resonators (CROWs)^{24} that allow for energy transfer in time—used as building blocks as shown in Fig. 1c. While in this work, we will be primarily dealing with coupled waveguide topologies in order to exemplify these notions, a similar discussion holds for heavily multimode chain cavity configurations and other bosonic systems^{25,26}. In this case, the underlying Sackur–Tetrode equation can be expressed as a function of the electrodynamic momentum flow U, the number of available modes M, and the input optical power \({{{\mathcal{P}}}}\). This analytical formalism developed here is in turn used to study and predict several distinct phenomena that could be of practical importance. These include, for example, the prospect for beam cooling (leading to an improvement in beam quality) as well as Joule expansion and aspects associated with the photocapacity and thermal conductivity of such chain networks. As opposed to previous works that were foundational in nature^{18,20}, the results reported here (applicable to largescale nonlinear tightbinding lattices) establish a tangible theoretical model through which many of these abstract ideas can be conceptually grasped and further developed.
Results and discussion
Optical Sackur–Tetrode equation
We begin our discussion by considering a lossless and leakfree, nonlinear array network of M singlemode elements that are coupled via nearestneighbor interactions. In this nonlinear optical chain, the equation of motion is given by \(ida_m/d\xi + \kappa a_{m  1} + \kappa a_{m + 1} + \left {a_m} \right^2a_m = 0\), where a_{m} represents the local mode amplitude at site m and κ stands for the coupling coefficient between two successive elements. These evolution equations can be obtained from the classical Hamiltonian \(\left\langle {H_{{{\mathrm{T}}}}} \right\rangle = \mathop {\sum}\nolimits_m {\kappa \left( {a_{m  1}a_m^ \ast + a_{m + 1}a_m^ \ast } \right) + \left( {1/2} \right)\mathop {\sum}\nolimits_m {\left {a_m} \right^4.} }\) The linear eigenvalues ε_{j} corresponding to the optical supermodes ϕ_{j}〉 of the arrangement shown in Fig. 1a are given by \(\varepsilon _j = 2\kappa {{{\mathrm{cos}}}}\left[ {\pi {\it{j}}/\left( {{\it{M}} + 1} \right)} \right]\) (refs. ^{23,24}). In general, the evolution of this weakly nonlinear system can be described in terms of its supermodes, i.e., \({\psi } \rangle =\mathop{\sum}\limits_{j = 1}^M {c_j( \xi ) {\phi _j} \rangle }\) where the square of the moduli (c_{j}^{2}) of the complex coefficients c_{j} denotes modal occupancies. In this case, the norm in this conservative system \({{{\mathcal{P}}}} = \mathop{\sum}\limits_{j = 1}^M  {c_{j}} ^2\) is preserved, thus implying power/photon conservation in these coupled waveguide/cavity arrangements. In addition, in the weakly nonlinear regime, the expectation value of the total Hamiltonian 〈H_{T}〉 = 〈H_{L}〉 + 〈H_{NL}〉 is dominated by its linear component 〈H_{L}〉 and as a result the internal energy \(U =  \left\langle {\psi \left {H_{{{\mathrm{L}}}}} \right\psi } \right\rangle =  \mathop{\sum}\limits_{j = 1}^M {\varepsilon _j {c_j} ^2}\) remains invariant during evolution, thus establishing the second conservation law associated with this class of structures. In fact, in multimode waveguide settings, the conserved internal energy U so happens to be the longitudinal electrodynamic momentum flow in these systems^{27}. In this respect, the two normalized constants of the motion \({{{\mathcal{P}}}}\) and U are uniquely determined by the initial excitation amplitudes c_{j0}, since \({{{\mathcal{P}}}} = \mathop{\sum}\limits_{j = 1}^M  {c_{j0}} ^2\) and \(U =  \mathop{\sum}\limits_{j = 1}^M \varepsilon _j {c_{j0}} ^2\). At this point, it is important to stress that the sole role of the weak nonlinearity involved (χ^{(2)}, χ^{(3)}, etc.) is to chaotically reshuffle the optical power among modes through multiwave mixing processes^{28}—thus causing the complex amplitudes c_{j} to vary randomly during propagation. In other words, the lack of M conserved quantities or integrability, allows this nonlinear array system to ergodically explore in a fair manner, all its accessible microstates (in its phase space) that lie on the constant energy (U) and power (\({{{\mathcal{P}}}}\)) manifolds. It is this ergodicity that provides the foundation for establishing an entropic thermodynamic theory for the aforementioned multimode chain networks. Moreover, in heavily multimode arrangements, the thermodynamically extensive variables \((U,M,{{{\mathcal{P}}}})\) are related to each other through the optical temperature T (henceforth called temperature) and chemical potential μ associated with the system. This relation is given through a global equation of state, given by \(U  \mu {{{\mathcal{P}}}} = MT\) that explicitly involves the total number of modes M (ref. ^{18}). Here, the optical temperature is associated with the optical field itself and has nothing to do with the actual temperature of the environment the system is embedded in. In general, any system is expected to reach thermal equilibrium by maximizing its entropy. In this case, the thermalized average power levels conveyed by each mode are found to obey a Rayleigh–Jeans distribution^{29,30,31,32}, i.e., \( {c_j} ^2 =  T/(\varepsilon _j + \mu )\).
Based on the above premises, one can show that the relative entropy associated with a photonic monoatomic chain network (Fig. 1) is given by the following equation (see Supplementary Note 1 for the derivation), i.e.,
Equation (1) is analogous to the entropic Sackur–Tetrode equation—developed more than a century ago (based on an appropriate quantization of the phase space) in order to correctly account for the properties of monoatomic gases^{17,33}. It was first derived in ref. ^{34} in order to account for the small fluctuation entropy of the defocusing nonlinear Schrödinger equation. Like its counterpart in statistical mechanics, the optical entropy of these nonlinear monoatomic chains is extensive with respect to the three other extensive variables \(\left( {U,M,{{{\mathcal{P}}}}} \right)\). In other words, if we let \(\left( {U,M,{{{\mathcal{P}}}}} \right) \to \left( {\lambda U,\lambda M,\lambda {{{\mathcal{P}}}}} \right)\), then from Eq. (1), one directly obtains S → λS. We note that this extensivity in entropy is crucial in developing a selfconsistent thermodynamic formulation—free of Gibbs paradoxes. On the other hand, the optical entropy given above is by nature different from that of an ideal monoatomic gas in standard thermodynamics. This is apparent, since the prefactor in Eq. (1) is now given by the number of modes M (optical volume) as opposed to the optical power \({{{\mathcal{P}}}}\) which here plays the role of N—the number of gas particles. Figure 2a depicts the entropy of a chain system as a function of \(\left( {U,{{{\mathcal{P}}}}} \right)\) as obtained from Eq. (1), when M = 100. The three pertinent equations of state can in turn be derived from (1) by employing the fundamental equation of thermodynamics (see Supplementary Note 2 for the derivation), i.e.,
Equations (2–4), relate the three intensive thermodynamic variables \(\left( {T,\mu ,\hat p} \right)\) to their respective conjugate quantities \(\left( {U,{{{\mathcal{P}}}},M} \right)\). As in statistical mechanics, in this nonlinear multimoded setting, the temperature \(T = \left( {U^2  4\kappa ^2{{{\mathcal{P}}}}^2} \right)/(2MU)\) represents a thermodynamic force responsible for energy transfer ΔU from a hot to a colder subsystem whereas the chemical potential μ dictates the power exchange \({{\Delta }}{{{\mathcal{P}}}}\)—all aiming to maximize the entropy in accord with the second law of thermodynamics. Finally, \(\hat p\) represents the relative optical pressure in this configuration. From Eq. (2), one can readily deduce that as the system attains its lowest possible internal energy \(U_{{{{\mathrm{min}}}}} \to  2\kappa {{{\mathcal{P}}}}\) (as imposed by the band structure of the monoatomic lattice), its temperature tends to zero (T → 0^{+}) and as a result the photon gas becomes a condensate that exclusively occupies the ground state of the lattice. Conversely, when the internal energy is maximized \(U_{{{{\mathrm{max}}}}} \to 2\kappa {{{\mathcal{P}}}}\), the opposite is true (i.e., T → 0^{−}), and hence the highest order mode is now solely occupied (anticondensate)^{35}. The first equation of state (Eq. (2)) directly indicates that the temperature of these lattices is positive for \( 2\kappa {{{\mathcal{P}}}} \, < \, U \, < \, 0\) whereas is negative if the internal energy lies in the domain \(0 \, < \, U \, < \, 2\kappa {{{\mathcal{P}}}}\). On the other hand, when \(U \to 0^ \mp\), the temperature tends to T → ±∞ (as schematically shown in the inset of Fig. 2b), and as a result, equipartition of power takes place among modes, i.e., \( {c_j} ^2 =  T/\mu = {{{\mathcal{P}}}}/M\). In general, negative temperatures correspond to bodies hotter than hot^{17}—a relation that regulates the direction of the energy flow ΔU. While for positive temperatures the lower group of modes is mostly occupied (since \( {c_j} ^2 =  T/(\varepsilon _j + \mu )\)), the opposite is true for negative temperatures where the higherorder modes are populated (Fig. 2b). We note that, the set of the three equations in (2–4) is complete, and is formally equivalent to the global equation of state (\(U  \mu {{{\mathcal{P}}}} = MT\)) as well as to the corresponding Euler equation (\(S = \left( {U  \mu {{{\mathcal{P}}}} + \hat pM} \right)/T\))—a relation that again reaffirms the extensivity of the entropy itself (see Supplementary Note 2). Meanwhile, Fig. 2c depicts energy–temperature curves (U – T) at various power levels, that as we will see dictate the photocapacity of these photonic chain networks.
Thermodynamic processes
The above results can be used to predict the outcome of more complex processes like, for example, that associated with Joule expansion of the photon gas in such nonlinear heavily multimoded environments. This prospect is shown schematically in Fig. 3, where as an example, χ^{(3)} nonlinear array supporting M modes suddenly expands to four times its size (M → 4M), while all the lattice parameters are kept the same. In this microcanonical scenario, both the internal energy U and power \({{{\mathcal{P}}}}\) remain constant during this abrupt transition. From Eqs. (2–3), one can quickly deduce that after Joule expansion, the temperature is reduced to onefourth of its original value (T → T/4), while the chemical potential μ is entirely unaffected. This photonic response is in stark contrast to the Joule expansion behavior expected from ideal monoatomic gases where the temperature is constant whereas the chemical potential substantially changes. On the other hand, even in this case, the absolute entropy of the system always increases in response to this irreversible expansion (for details see Supplementary Note 3). In a similar vein, isentropic processes (S = const) can also be examined. These effects can be readily produced if, for example, a Kerr nonlinear multimode array adiabatically expands/contracts (in which case the mode occupancies c_{j}^{2} are unaltered) while the individual singlemode elements remain the same (only the coupling coefficient κ adiabatically changes). Since M and \({{{\mathcal{P}}}}\) are invariant, an isentropic transition can only occur provided that U/κ = const (see Supplementary Note 4 for the derivation). This last relation can be justified given that \(U = \mathop{\sum}\limits_{j = 1}^M \varepsilon _j {c_j} ^2\) and ε_{j} ∝ κ. From the first two equations of state (Eqs. (2–3)), one can quickly conclude that U/T = const and μ/T = const. These latter isentropic relations, applicable for the photon gas in multimode chain networks, are to some extent reminiscent of their counterparts in the theory of monatomic gases, i.e., pV^{5/3} = const and TV^{2/3} = const (ref. ^{17}). Nevertheless, there is a fundamental difference between these two physical settings, in the sense that in the former case there is a physical change in the lattice structure (and hence in the width of the density of states) while in the latter the gaseous substance stays the same.
Beam selfcleaning and cooling
The analytical formalism (Eqs. (2–4)) developed here, can now be utilized to predict the response of such nonlinear chain systems in settings of practical interest like those allowing for beam selfcleaning^{36,37,38,39} and cooling. Figure 4a schematically shows such a nonlinear multicore optical arrangement involving M = 30 sites. In this configuration, the power \(({{{\mathcal{P}}}} = 2)\) is injected in the \({\hat{{{\boldsymbol{y}}}}}\) polarization and is uniformly distributed (with random phases) between the supermodes having eigenvalues ε_{j} in the range of −0.5κ ≤ ε_{j} ≤ 1.2κ, in which case U = −0.77, as shown in Fig. 4b. For these initial conditions, Eqs. (2–4) predict the following equilibrium parameters T = 0.33, μ = −5.38. While at the input (z = 0) the intensity distribution displays considerable disorder (Fig. 4c), this speckle pattern disappears (Fig. 4d) after the system attains thermal equilibrium at the output (z = L). What facilitates this beam selfcleaning effect has to do with the fact that during thermalization, the power is reshuffled in such a way so as to favor the lower order modes (for T > 0) in the Rayleigh–Jeans distribution (Fig. 4b)—a process that removes the initial speckle in the beam. Yet, while the output beam at (z = L) seems to improve in the nearfield, its multimode farfield is still highly divergent and thus is of poor quality as shown in Fig. 4e. Interestingly, the thermodynamic formulation developed here suggests that this situation can be greatly improved if in this same array structure, a cooler beam is launched—having a perpendicular polarization \({\hat{{{\boldsymbol{x}}}}}\) (Fig. 4f). In this latter arrangement, each waveguide site is assumed to be substantially birefringent, thus prohibiting any power exchange between the two beams. Instead, the two wavefronts only interact through crossphase modulation^{24}. In the example provided in Fig. 4f, the internal energy and power corresponding to each polarization is \(U_x =  3.95,{{{\mathcal{P}}}}_{\it{x}} = 2,U_y =  0.77,{{{\mathcal{P}}}}_{\it{y}} = 2\). Based on these initial conditions, Eq. (2) indicates that on its own, each polarization would have settled to an equilibrium temperature of T_{x} = 1.7 × 10^{−3}, T_{y} = 0.33. On the other hand, once interacting together in a canonicallike arrangement, the two beams reach the same temperature T = 0.075 which can be exactly predicted following the procedure in Supplementary Note 5. In other words, the \({\hat{{{\boldsymbol{y}}}}}\) polarized beam can be considerably cooled after exchanging energy ΔU with its \({\hat{{{\boldsymbol{x}}}}}\) polarized counterpart. This in turn leads to a ~3fold improvement in the farfield of the original beam (Fig. 4g).
Heat capacity and thermal conductivity of nonlinear chain networks
The formalism developed here, allows one to define a photocapacity C_{M} = ∂U/∂T associated with such nonlinear multimode photonic systems—a property that is in every respect analogous to that of the heat capacity pertaining to various phases of matter. Like its counterpart (heat capacity), this quantity critically depends on the density of states and hence it is characteristic of the multimode optical system itself. In this respect, it is formally possible to assign a C_{M} capacity to any possible manymode arrangement, irrespective of whether it is discrete or continuous. For the chain networks considered here, the photocapacity can be explicitly obtained from Eq. (2) and it is given by \(C_M = M  \left T \rightM^2\left( {T^2M^2 + 4\kappa ^2{{{\mathcal{P}}}}^2} \right)^{  1/2}\). Figure 5a shows the C_{M} function (blue curve) for such a chain configuration, involving M = 200 sites while conveying a power of \({{{\mathcal{P}}}} = 20\). In this case, the photocapacity function is an even function of temperature—as one should expect from the symmetry in the density of states. The photocapacity curves corresponding to two different types of Lieb lattices are also displayed, as an example, in Fig. 5a for the same parameters (M = 200, \({{{\mathcal{P}}}} = 20\)). Note that the C_{M}(T) function for the Lieb2 lattice with crosscouplings is in this case highly asymmetric—a feature attributed to the band structure of this specific system.
We next explore the possibility for energy ΔU transport phenomena occurring when a hot and a cold optical nonlinear multimode system are linked together under nonequilibrium conditions. To some extent, these processes have much in common with heat conduction problems in solids. Figure 5b shows such an arrangement where a square multicore photonic lattice L_{1} with M_{1} = 900 sites is connected to a graphenelike system L_{2} (M_{2} = 896) via a chain bridge L_{3} involving M_{3} = 140 evanescently coupled waveguides. The bridge L_{3} is appropriately designed using birefringent elements so as to allow for energy exchange ΔU between the two lattices L_{1} and L_{2} while prohibiting any power transfer \({{\Delta }}{{{\mathcal{P}}}}\). System L_{1} together with the link array L_{3} are initially brought to thermal equilibrium where both share a temperature of T_{1} = 0.3 when conveying a total power of \({{{\mathcal{P}}}}_{1 + 3} = 65\) at a total internal energy U_{1+3} = −60. Similarly, lattice L_{2} (before is connected to the bridge) is kept at T_{2} = 0.1 when \({{{\mathcal{P}}}}_2 = 40\) and U_{2} = −46. Once L_{2} is put in contact with the L_{3} bridge, a nonequilibrium thermodynamical process ensues during which the entropy of the combined system starts to increase as expected from the second law of thermodynamics (Fig. 5c). During this nonequilibrium stage, internal energy ΔU starts to flow through the bridge from hot to cold (L_{1} → L_{2}). In this case, the local temperature along this bridge (obtained after projecting on the local eigenfunctions) is found to linearly decrease from T_{1} to T_{2} in full accord to Fick’s law of heat diffusion^{40}, J = −k∇T. This in turn allows one to express this energy transfer through an effective photoconduction coefficient K as ΔU/Δz = −K(T_{1} − T_{2}). From our simulations, we estimate that here the photoconduction coefficient of the L_{3} chain is K ≈ 3.2 × 10^{−5}.
Conclusion
By harnessing concepts from statistical mechanics, we have developed an optical thermodynamic theory that explicitly provides the Sackur–Tetrode entropy associated with large nonlinear photonic chain networks. Based on these premises, the temperature and chemical potential of such systems can be obtained in closed form as a function of the initial excitation conditions. The formalism developed here is general and can be readily deployed to describe a number of thermodynamic processes, like isentropic expansion/compression, Joule expansion, as well as aspects pertaining to beam selfcleaning and cooling in such chain networks. Of interest would be to explore similar possibilities in quasiconversive systems under nonequilibrium conditions. Our results not only provide a platform to understand and predict in a systematic way the utterly complex dynamics of heavily multimoded nonlinear optical systems that are nowadays of importance in areas ranging from highpower photonic settings to cold atom condensates and optomechanics. In addition, they could be of practical importance in designing highbrightness multimode optical sources.
Methods
Simulations
Throughout this study, simulations were conducted by numerically solving the discrete nonlinear Schrödinger equations, using 4th order Runge–Kutta methods. Ensemble averages were obtained by performing a number of simulations with the same initial modal amplitudes but different random relative phases. The modal occupancies c_{j}^{2} obtained were then averaged over several ensembles. More information concerning these results can be found in the Supplementary Methods.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding author upon reasonable request.
Change history
04 October 2023
A Correction to this paper has been published: https://doi.org/10.1038/s4200502301400z
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Acknowledgements
This work was supported by the Office of Naval Research (ONR) through the grants (N000141812347), and MURI (N000142012789), the National Science Foundation (NSF) (EECS1711230, CBET 1805200, ECCS 2000538, ECCS 2011171), the Army Research Office (ARO) (W911NF1710481), the Air Force Office of Scientific Research (AFOSR) (FA95501410037, FA95502010322), the US–Israel Binational Science Foundation (BSF; 2016381) and the Qatar National Research Fund (QNRF) (NPRP90201006). This work was partially supported by an MPS Simons collaboration (Simons grant 733682). P.S.J. thanks the Polish Ministry of Science and Higher Education for Mobility Plus scholarship (1654/MOB/V/2017/0).
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D.N.C. and M.K. conceived the idea. F.O.W. developed the theory. F.O.W., P.S.J., and M.P. conducted the simulations and data analysis. All the authors contributed to the writing of the original draft, review, and editing.
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Wu, F.O., Jung, P.S., Parto, M. et al. Entropic thermodynamics of nonlinear photonic chain networks. Commun Phys 3, 216 (2020). https://doi.org/10.1038/s42005020004841
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DOI: https://doi.org/10.1038/s42005020004841
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