Thermalization of one-dimensional photon gas and thermal lasers in erbium-doped fibers

We demonstrate thermalization and Bose-Einstein (BE) distribution of photons in standard erbium-doped fibers (edf) in a broad spectral range up to ~200nm at the 1550nm wavelength regime. Our measurements were done at a room temperature ~300K and 77K. It is a special demonstration of thermalization of photons in fiber cavities and even in open fibers. They are one-dimensional (1D), meters-long, with low finesse, high loss and small capture fraction of the spontaneous emission. Moreover, we find in the edf cavities coexistence of thermal equilibrium (TE) and lasing without an overall inversion. The experimental results are supported by a theoretical analysis based on the rate equations.


INTRODUCTION
It is commonly accepted that photons in lasers are not in thermal equilibrium (TE), and don't show Bose-Einstein spectral distribution and Bose-Einstein condensation (BEC). We show here thermalization of photons in standard one-dimensional (1D) erbium-doped fiber (edf) cavities, including in the lasing regime, and in open fibers. Photon thermalization and photon-BEC were demonstrated in a dye-filled microcavity [1,2]. They were observed not in the lasing regime but much below it, and required strict conditions [1,2] that included a micron-size cavity with a twodimensional (2D) confinement of lateral modes, very high mirror reflectivities that provided high finesse to trap the photons in a "white wall photon box", and very low losses. They also needed a very high capture of the spontaneously emitted photons in all directions and a cutoff frequency.
All these requirements were crucial for TE and BEC. Insightful theoretical studies discussed the main differences between a low loss quantum-statistics regime that can yield TE and BEC, and a higher losses regime of classical lasers [3,4]. There were also discussions and questions about the nature of photon-BEC in optical cavities, the relation to lasers [3][4][5][6][7][8] and to classical condensation [9][10][11][12][13][14][15][16][17]. In our work we find that for edf systems many restrictions that were required in the microcavity experiment [1,2] are relaxed, and photon TE is obtained in standard one-dimensional (1D) edf cavities, in the lasing regime, and even in open fibers [5]. We note that the lasing occurs where the overall population of the second level is lower than the first one and therefore it can be regarded as thermal lasing without an overall inversion (T-LWI), as can be the situation in the microcavity experiment where the population buildup was regarded as BEC [1,2]. We nevertheless emphasize, that there is inversion at the specific lasing line that is pushed to a high wavelength regime in the edf spectrum due to thermalization where the thermal dependent emission rate is larger than the absorption one. We also stress that thermalization and the needed density of light-mode states (DOS) that we discuss in this paper are the important conditions for obtaining BEC, but we do not give here yet a final conclusion on its observation.

THE ERBIUM-DOPED FIBER (EDF) PLATFORM
Erbium-doped fibers (edf) are widely used as amplifiers in fiber optic communication. Erbium is a "three level" atom system with broad levels due to Stark splittings (Stark manifolds) that can provide gain at the 1550nm wavelength regime, commonly between 1530-1560nm (the C-band) [18]. The pumping from the first to the third level is usually done with wavelengths at ~980nm or , c is the speed of light, and r n is the refractive index. We emphasize that the usual Einstein relation 12 21 BB  (and more generally, where the stimulated emission increases. Therefore, the photon gas starts at the pump input in the fiber with spontaneous and stimulated emissions, and as the pump is depleted after a short distance, the photons propagate at highly non-inverted environment, undergoing spontaneous absorption-emission cycles that lead to thermalization and also stimulated emissions that replicate the spectrum. Therefore, it is possible to reach thermalization in an open fiber as the photons propagate along it, as shown in the experimental part, although most of the photons from spontaneous emission radiate out of the fiber and are lost.  and emission 21  cross-sections of erbium in silica fibers. A bosonic system in thermal equilibrium has a Bose-Einstein (BE) distribution, here the spectrum, k -the Boltzmann constant and  -the chemical potential. g is the light modes degeneracy or their density of states (DOS) which is independent of  for a 1D photon gas with a regular linear dispersion. Usually the photon number is not conserved and 0   , as we have in black-body radiation. In optical cavities and lasers, the photon number is in many cases conserved by pumping that compensates for the unavoidable losses. However, for TE and BE distribution there are additional conditions. In the dye-filled microcavity experiment [1,2], for example, the photon system, that had to meet certain strict conditions, was considered a grand-canonical ensemble with 0   . In our work we find that many of those requirements are relaxed. TE is observed here in regular 1D fiber lasers, in the lasing regime, and even in open 5 fibers without the need of micron-size cavities, high finesse, or a very high capture of the spontaneously emitted photons in all directions and a cutoff frequency [1,2].
We mentioned the possibility of lasing without an overall inversion, but we right away stress that it only means that the total second level population is lower than the first one, and not at the specific lasing wavelength. Such lack of inversion terminology for the overall two states is used in former works [1][2][3][4]16,17]. However, one can correctly argue that for the specific lasing wavelength there is inversion. In our case, it occurs even when the pumping is between the two states at ~1550nm and the photon thermalization spreads the spectrum and transforms photons from low to high wavelengths where the emission cross section is larger than the absorption and compensates for the lower upper-state population. It is therefore a temperature dependent effect that results from the difference between the erbium emission and absorption cross sections, 12 We note again that 1 N and 2 N are the overall levels populations in each of the two broadened levels, and 21 ()  is related to () ij B  , as done in some former works [16,17]. The lower level population at the upper state is simply compensated by the higher emission than absorption rate.
Therefore, even without an overall inversion, the stimulated emission can be larger than the absorption and when it increases beyond the balance with the cavity losses it provides gain and lasing or condensation in the case of very low losses. We show below T-LWI at the high wavelength side of the thermal spectrum, ~1605nm, an unusual wavelength for edf fiber lasers.

EXPERIMENT AND RESULTS
We turn to the experiment and the measurements with edf ring cavities and open fibers and after that we discuss the theoretical results. We used edf with various erbium concentrations and gain figures: 30m of 11dB/m, 300m of 1dB/m and 30m and 100m of 30dB/m. It is important to emphasize right away that the results were quite similar for the four different fibers, and we 6 show here only a few spectra. The pumping was at a 980nm wavelength except for one case that it was with amplified spontaneous emission (ASE) of edf at the 1550nm region. The ring edf cavity had regular laser losses mostly from connectors and the output coupler of 10%. It is therefore a low finesse cavity, but since it is (30-300)m long it has a relatively long photon     Figure 2 gives experimental spectra for an edf ring cavity measured at a room temperature, ~300K, and Fig. 3 at a liquid Nitrogen temperature, 77K. They show broad thermal spectra (the straight line parts) which nicely match the BE distribution at a broad wavelength region for both temperatures. At 300K for low pumping, the BE band reaches ~90nm and for higher pumping it spans over ~150nm. The lasing peaks are at the right side of the BE spectral band. At 77K the logarithmic BE slope is 300 / 77 3.9  times higher then the 300K slope, as the theory predicts, but the spectral range is only ~50nm and the lasing is slightly shifted to a lower wavelength. We don't elaborate here on the effect of the chemical potential that can be disregarded in most of the band, except for the edge, as seen in the experimental spectra, but becomes zero (equals to the edge frequency) upon condensation. It is more important for the BEC study that we will discuss in a future paper on BEC. Figure 4 shows experimental spectra for an open 100m, 30dB/m edf, at a room temperature. In this experiment the BE spectral band reaches ~200nm and in the cavity case it is accompanied by a sharp oscillation line at the right side, that is again T-LWI. Here we used a different kind of pumping directly from the first to the second level by ASE of an edf fiber with a spectrum centered around 1550nm. It is a kind of "white" light pumping that doesn't generate inversion of the overall broad two levels but can give oscillation and lasing!! In the laser case this pumping was inside the cavity. It is striking to see the broad BE spectral spreading out of the narrow spectrum pumping. The thermalization process spreads and transforms the photons from low to higher wavelengths. Thus, TE can be reached not only in close cavities, but also in open fibers, as the photons propagate and become thermlized. In the distributed rate equation model that we calculated but don't show here, we could see how the photon thermalization develops along the propagation in the fiber as the distance is larger than p l .
The TE is obtained significantly below an overall inversion, but when the pumping was increased in the cavity case, lasing started without an overall inversion at an unusually long wavelength of ~1605nm, as seen in Fig. 2. It is the high wavelength side of the broad TE band.
We observed in some cases bistability (seen in Fig. 2) and hysteresis. Our calculation showed that it can be attributed to a satuarable absorber mechanism at the fiber section where the pump is very much depleted, and due to strong backwards ASE of the edf, but it can be eliminated by a moderate pump variation along the fiber. We also note that one has to be careful about the meaning of oscillation without an overall inversion, as we noted above), when attributing it to lasing. We will report on this topic in the future.

THEORETICAL ANALYSIS
The theoretical analysis is based on the lumped or the distributed rate equations models, given in the Methods part. We show in Fig. 5 the theoretical spectrum of the thermalization effect on the light that propagates along an (100m, 30dB/m) open fiber. We can see the similarity to the experimental result in Fig. 4. Figure 6a shows broad BE spectra (>120nm) calculated by the  The drastic difference between the thermal lasers and regular lasers spectra can be seen by comparing Figs. 6a and 6b. As mentioned above, the thermal lasers with broad BE spectra have low cavity losses. We used in Fig. 6a   that implies its depletion after a short distance in the edf. It shows that the upper state population quickly falls to a small value. In the case with the oscillation, the population ratio stabilizes at a finite value well below an overall inversion (~0.18 for the experiment parameters). We note that besides the input and output measurements of the pump and the signal, and therefore the inversion, their variations along the fiber are obtained by the theoretical analysis, as done in former works, such as in the microcavity experiment, [1,2,6-8].

THE RATE EQUATIONS MODEL
We elaborate here on the theoretical base of the lumped and distributed rate equations. The lumped model gives the basic results shown in the theoretical figures that nicely describe the experimental spectra. The second distributed rate equations model allows following the pump and signal propagation along the fiber (z-dependence).

A. The lumped rate equation
We first describe the lumped model that gives the basic results. As in the dye microcavity system [7][8], the photon population and its dependence on frequency is governed by a rate equation where the erbium atoms are modeled as broadened two level systems:

(4)
The chemical potential is defied by