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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 26704–26713
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When does single-mode lasing become a condensation phenomenon?

Baruch Fischer and Rafi Weill  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 26704-26713 (2012)
http://dx.doi.org/10.1364/OE.20.026704


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Abstract

We present a generic route to classical light condensation (LC) in linear photonic mode systems, such as cw lasers, with different grounds from regular Bose-Einstein condensation (BEC). LC is based on weighting the modes in a noisy environment (spontaneous emission, etc.) in a loss-gain scale, rather than in photon energy. It is characterized by a sharp transition from a multi- to single-mode oscillation. The study uses a linear multivariate Langevin formulation which gives a mode occupation hierarchy that functions like Bose-Einstein statistics. Condensation occurs when the spectral filtering has near the lowest-loss mode a power law dependence with exponent smaller than 1. We then discuss how condensation can occur in photon systems, its relation to lasing and the difficulties to observe regular photon-BEC in laser cavities. We raise the possibility that experiments on photon condensation in optical cavities fall in a classical LC or lasing category rather than being a thermal-quantum BEC phenomenon.

© 2012 OSA

1. Introduction

Bose-Einstein condensation (BEC) was predicted in 1924-5, but was experimentally observed seventy years later with atomic particles at ultra-low temperatures [1

1. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of bose-einstein condensation in a dilute atomic vapor,” Science 269(5221), 198–201 (1995). [CrossRef] [PubMed]

,2

2. A. J. Leggett, “Bose-Einstein condensation in the alkali gases: Some fundamental concepts,” Rev. Mod. Phys. 73(2), 307–356 (2001). [CrossRef]

]. The interesting question if and how can BEC occur with non-atomic bosons, such as photons, also attracted attention. A recent paper reported on the observation of photon-BEC in an optical micro-cavity [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

,4

4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

] at a room temperature, and earlier work demonstrated BEC of polaritons [5

5. H. Deng, G. Weihs, C. Santori, J. Bloch, and Y. Yamamoto, “Condensation of semiconductor microcavity exciton polaritons,” Science 298(5591), 199–202 (2002). [CrossRef] [PubMed]

,6

6. R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose-Einstein condensation of microcavity polaritons in a trap,” Science 316(5827), 1007–1010 (2007). [CrossRef] [PubMed]

] and magnons [7

7. S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, “Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping,” Nature 443(7110), 430–433 (2006). [CrossRef] [PubMed]

].

Classical systems can also show condensation effects. We theoretically and experimentally found, for example that the light in actively mode-locked lasers shows in some cases condensation behavior [8

8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

,9

9. R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010). [CrossRef] [PubMed]

]. In the condensate state an optimum short pulse, which corresponds to the lowest pulse mode, becomes dominant in the cavity. It was also noted there [8

8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

] that in higher than 1-dimensional laser mode systems transverse spatial loss modulation or spectral filtering can lead to condensation. We also mention other important theoretical work on condensation phenomena with classical nonlinear waves [10

10. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95(26), 263901 (2005). [CrossRef] [PubMed]

], disordered lasers [11

11. C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101(14), 143901 (2008). [CrossRef] [PubMed]

] and the review in Ref. [12

12. A. Fratalocchi, “Mode-locked lasers: light condensation,” Nat. Photonics 4(8), 502–503 (2010). [CrossRef]

]. The many-body nature of laser modes has been discussed in earlier work. We developed a broad thermodynamic-like approach for mode-locked lasers showing, for example, that mode-locking is a phase transition [13

13. A. Gordon and B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002). [CrossRef] [PubMed]

17

17. A. Gordon, B. Vodonos, V. Smulakovski, and B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express 11(25), 3418–3424 (2003). [CrossRef] [PubMed]

], and it also exhibits critical phenomena [18

18. A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, and B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010). [CrossRef] [PubMed]

,19

19. R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005). [CrossRef] [PubMed]

], in the deep statistical mechanics meaning, where noise takes the role of temperature.

In the present work we deal with a generic mode system with noise, represented by a basic multi-mode cw laser cavity, without nonlinear terms, that nevertheless is found to exhibit a route to condensation which is formally very similar to BEC in a potential-well [20

20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

]. This laser light condensation (LC) phenomenon is based on very different grounds than regular BEC. Its “energy” levels are measured in a loss-gain scale, inherent in laser cavities, where the “ground-state” is the lowest-loss mode, and noise, also inherent in lasers, has the role of temperature. The loss scale gives a mode occupation hierarchy and power spectra that resemble the Bose-Einstein distribution and leads to laser light-condensation (LC). It has special properties compared to regular BEC, as we discuss below. For example, the condensed state can be anywhere in the spectral band, in contrast to BEC where it is always at the lowest energy (frequency) state.

It would be therefore interesting to raise the possibility that photon-BEC experiments in optical cavities [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

] can fall in a classical LC or lasing category rather than being a quantum-thermal based photon-BEC phenomenon. We shall argue that LC and photon condensation do not provide a new type of photons (“super-photon”?) or a new “quantum” light state, not more than a single-frequency laser does. It is however a challenging topic that needs further study and experimental work.

2. Mathematical route to laser light condensation (LC)

The ingredients for LC to happen are very simple while having a generic nature: many modes (“particles”), noise (spontaneous emission, thermal etc.) which injects light into those modes, a gain part, and certain conditions on the loss-gain filtering spectrum (loss “potential-well”) as discussed below. A typical example for such system is a cw laser. The equations of motion for the evolution of light modes circulating in the cavity are taken in a most simple form, without dispersion, modulation and nonlinear terms [21

21. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000). [CrossRef]

], or other linear or nonlinear rate equations [22

22. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993), Ch.6.

,23

23. A. E. Siegman, Lasers (University Science Books, 1986).

], but with a noise source [13

13. A. Gordon and B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002). [CrossRef] [PubMed]

19

19. R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005). [CrossRef] [PubMed]

]. We have a multivariate Langevin equation (with shared and variate-dependent coefficients, shown to be a key part for condensation):
damdτ=(gεm)am+Γm,
(1)
where am(τ) are the slowly varying electric field complex amplitudes of the modes (in d-dimensions:amam1,...md), τ is the long term time variable that counts cavity roundtrip frames, and g is a slow saturable gain factor, shared by all modes. It functions as a Lagrange multiplier (and μgε0 as “chemical potential”) for setting the overall cavity power P, andεm is spectral filtering (loss dispersion) due to frequency and mode dependent loss, absorption and gain. Γm is an additive noise term that can originate from spontaneous emission, or any other internal or external sources. It is modeled by a white Gaussian process with covariance 2T:
<Γm(τ)Γn*(τ')>=2Tδ(ττ')δmn, and <Γm(τ)>=0,
where <> denotes average. We describe the modes frequency by Ω=ωω0=(c/n)(|k¯|k0) (where k¯is the wavevector and n is the refractive index), measured with respect to the linewidth center ω0=(ck0/n). The modes discrete relative frequencies are Ωm. With the continuous variable we have am(τ)a(Ω,τ), and εmε(Ω). We note that our study is applicable to discrete modes, as well as to continuous frequencies, alluding to the common dual (but not necessarily the same) terminology and meaning of single-mode and single-frequency lasers.

An important part that we add in this work, compared to the usual laser formulation [22

22. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993), Ch.6.

,23

23. A. E. Siegman, Lasers (University Science Books, 1986).

], is to allow and examine various spectral dependences for ε(Ω), that is usually taken to be parabolic. This is a key point that opens the way for condensation. We show in this work that for certain spectral filtering functions ε(Ω), a linear mode system with noise provides a route to condensation with a transition of noise broadened spectra to a single-frequency oscillation.

We turn to the analysis that leads to LC, which resembles the BEC formalism in a potential-well [20

20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

]. We obtain from Eq. (1), at steady state, the overall power:

P=mpm=m<ama*m>=mTεmgTε0g+T0εNρ(ε)dεε+ε0g
(2)

For the integral at the right hand side of the equation with the continuous net loss variable, we define the density of loss states ρ(ε)that is shown to have a prime role for condensation occurrence. The first term at the right hand side of Eq. (2) gives the power of the lowest loss (ε0) mode, and the second term is the power in all of the higher modes. We can right away notice the resemblance to BEC in a potential-well [20

20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

]. The weight for each spectral component depends on ρ(ε)and a factor 1/(ε+ε0g) that replaces the Bose-Einstein statistics. Here the upper limit of the integral, εN, is set by the spectral filtering (loss “potential-well”) depth. However, it is the density of the low loss-levels at ε0 that determines the integral convergence. We therefore take the spectral filtering functional dependence as a power law around ε0: ε=γ|Ω|η=γ˜|Ω˜|η where Ω˜=Ω/Ω0, γ˜=γΩ0η and Ω0 is the spectral filtering width. It can be shown that it gives the following density of states (modes) in d-dimensions [9

9. R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010). [CrossRef] [PubMed]

,20

20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

]:
ρ(ε)=Vd(2π)dddkδ(εγ|Ω|η)(Cdηγξ+1)εξ,
(3)
where Cd=Vdbdnk0d1/[(2π)dc],ξ=η11, Vd is the d-dimensional resonator volume, δ(x) is the Dirac delta function, and bd=2,2π,4π for d=1,2,3, respectively. We could also use for the calculation ρ(ε)=(Vd/(2π)ddS/|¯kε|, where dS is a surface element at constant εin the k¯ space.

We now return to Eq. (2), and specifically to the integral at the right hand side. Condensation occurs when the integral converges at ε=0. Then the power population of the light at higher than ε=0 levels stays unchanged (filled) at Pc=T0εNρ(ε)dε/ε. Therefore, additional pumping that increases P beyond Pc must be channeled into the lowest level power p0=T/(ε0g) which starts growing macroscopically. The integral converges when ξ>0, i.e. η<1 (nonsmooth ε(Ω) at Ω=0 and concave, at least at Ω=±0, as shown in the illustrative examples in Fig. 1
Fig. 1 Spectral filtering: (a) Power law function with γ˜=1 and η=0.8. (b), (c) Cut structures (solid line) obtained for example from sections of Gaussian (can be exponential etc.) filtering spectra (combined frequency dependent losses/absorption and gain). They can result from various frequency cutoff mechanisms, such as the far apart longitudinal mode separation in microcavity lasers [3,4]. The cuts lowest loss (ε0) mode at Ω˜=±1 are redefined to be at Ω˜=0.
). At condensation (PPc), the net gain of the lowest mode (“chemical potential) is μgε0=0.

We note that ε(Ω) doesn’t have to be symmetric for condensation as long as the slope of ε(Ω)is smaller than 1 at Ω=0. Examples are the spectral slices in Fig. 1 cut from broad spectra (e.g., Gaussian, exponential) that can result from various reasons, such as boundary-condition caused cutoffs. We also note that except for the prefactor in ρ(ε) the condensation condition doesn’t depend on the dimension d, except when additional spatial or temporal filtering or modulation exists on other dimensions (e.g. transverse) as noted in Ref. [8

8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

].

The noise induced spectrum (mode-population) is given by the integrand of Eq. (2) expressed with the Ω˜=Ω/Ω0 variable:

p(Ω˜)=CdTε(Ω˜)+ε0g.
(4)

For the power law spectral filtering, we have p(Ω˜)=CdT/[γ˜Ω˜η+ε0g]. Figure 2a
Fig. 2 Light spectra, in a semi-logarithmic scale, obtained by Eqs. (4) and 5: (a) For the power law dependence spectral filtering example (Fig. 1a), with γ˜=1 and η=0.8. The various graphs correspond (from bottom to top) to μgε0=0.4,0.2,0.1,0, and respectively to P/Pc=0.23,0.3,0.4,1. In the cut examples (the high and low pass filters in Fig. 1b) the spectra have only the right or left of Ω=0 sides. (b) For the exponential spectral filtering given by Eq. (5), with the above μs, γe=1 and β˜=1.5. In both figures, the three lower graphs correspond to non-condensate states, and the upper one to condensation. At and beyond the condensation transition, μ=0,and the spectrum stays as in the top graph, except for the lowest-loss mode p0 at Ω=0 that grows upon further pumping (not shown here, but given in Fig. 4).
shows spectra for η=0,8 and various overall power values P (or μs), below and at condensation.

More generally, we can have other spectral filtering functions. Then, since the condensation is determined by the exponent near Ω=0, it is required to have for condensation: dεdΩ|Ω=±0<1. Away from Ω=0, ε(Ω) can have any dependence that experiments yield. In many cases there are exponential sections that can be expressed around ε0 as ε(Ω)=γe(eβ|Ω|1). Here the slope at Ω=±0 is η=1, just on the verge of condensation. We can then obtain for the mode density, ρ(ε)=Cd/[β(ε+γe)], and for the spectrum (beyond Ω=0):
p(Ω˜)=CdTγe(eβ˜|Ω˜|1)+(ε0g),
(5)
where β˜=Ω0β. This equation with the exponential term resembles thermal Bose-Einstein distribution and has implications to the discussion below on the LC connection with photon-BEC experiments. Figure 2b shows graphs of this spectrum for various μs. We can see the broad thermal-like exponential dependence.

Figure 3
Fig. 3 Normalized “chemical potential” μ=(gε0) as a function of the overall normalized laser power P/Pc for γ˜=1, η=0.8 and εN=1.
shows the “chemical potential” (μgε0) dependence on P, for η=0.8. μ is always negative before condensation, and becomes zero at condensation. The condensation is shown in Fig. 4
Fig. 4 Condensation transition: Normalized lowest mode power p0/PC as a function of the overall normalized laser power for γ˜=1, η=0.8 and εN=1. Above the phase transition (PPC) all the excess power goes to p0- the condensation state.
by the power dependence of the normalized low-loss mode p0/Pc. Above the phase transition (PPc) all the excess power goes to the condensation state p0. This is the laser LC route to condensation, characterized by the sharp transition from a multi- to single-mode oscillation.

3. Discussion on lasing, LC, photon-BEC and experiments

LC is formally similar to BEC in having:

  • a. Non-interacting particles - bosons in BEC and light-modes in LC.
  • b. Particle distribution - Bose-Einstein statistics vs. loss-dependent mode weighting in LC.
  • c. A global constraint on the overall bosonic particles number in BEC vs. the overall light-modes power in LC.
  • d. The chemical potential in BEC corresponds to the lowest-mode gain minus loss in LC. In both cases it is negative below condensation, and approaches and becomes zero at and above the transition.
  • e. Certain conditions on the density of states function for bosonic particle and light-modes.
  • f. A similar mathematical route to condensation in BEC and LC.

There are however a few important differences between LC and BEC, besides the basic one that LC is classical and BEC is a quantum-thermal effect. We specify the main ones:

  • a. The states (modes) hierarchy (“energy ruler”) in LC is measured by loss-gain, compared to photon energy (frequency) in BEC.
  • b. It is a general noisy system in LC vs. thermal equilibrium, characterized by temperature, in BEC.
  • c. Laser LC occurs at the lowest-loss state (Ω=0), which can be anywhere in the spectral band, and not necessarily at the lowest photon energy (ω) state, as it is in BEC.
  • d. The spectrum of the oscillating modes in LC (Fig. 2) is dictated by the loss scale. It is “noisization” rather than the BEC thermalization spectrum [4

    4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

    ]. Unlike BEC, the spectra can span over frequencies above, or below, or at both sides of Ω=0. We also note that in LC, below Pc, there is a hole in p(ε) near Ω=0, (|k¯|=k0), resulting from ρ(Ω=0)=0, but not in p(Ω) (in the frequency domain), as seen in Fig. 2. In BEC, prior to condensation, there is a hole in the energy distribution at zero energy (as in black-body radiation spectra), which is always at the lowest possible energy (frequency) [20

    20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

    ].
  • e. Unlike regular BEC [20

    20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

    ], there is no dimensional dependence in LC when the condensation results from spectral filtering, that operates in one dimension (Ω, at the |k¯|k0sphere shell), except for a factor that results from the d-dimensional sphere surface. Nevertheless, when there are in the system other space (or time) dependent loss-gain filtering (or modulation), like in two dimensional transverse mode systems [3

    3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

    ,4

    4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

    ], the condition for condensation can be different [8

    8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

    ].

We elaborate on the LC and BEC connection and the relation to lasers and experiments. Both cases are characterized by a transition to a single-frequency oscillation. Such a transition can also occur in regular lasers, like in ideally homogeneous lasers [23

23. A. E. Siegman, Lasers (University Science Books, 1986).

]. In a way, every single-frequency laser could have been regarded as a condensate state. (Photon densities in single-frequency lasers can be much higher than in the microcavity experiment [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

].) In those lasers the single-mode oscillation is explained by a nonlinear saturation effect or a simple ad hoc assumption that the first mode to reach threshold clamps the gain profile and therefore eliminates the oscillation of all other modes [23

23. A. E. Siegman, Lasers (University Science Books, 1986).

]. In LC, it is embedded in the density of states where high mode states populated by the noise become filled (for certain quasi-continuous mode densities ρ(ε) with exponents η<1), leading to condensation, without an explicit nonlinearity, but with a global constraint, in a similar way to BEC. It is likewise characterized by a negative “chemical potential” μgε0 that becomes zero at condensation (as shown in Fig. 3.) It means that we have a different route to a single-mode oscillation than regular lasers.

In experiments it can be very challenging to relate and identify the different single-frequency oscillation effects, since they are all done in laser cavities. It is possible that experiments in laser cavities fall in LC or a single-mode lasing category, rather than being a thermal photon-BEC phenomenon [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

]. We have shown the role of spectral-filtering, inevitable in lasers, that governs the photon gas “climate”, the “equilibrium” and the spectra in a pumped many-mode cavity with gain. In the dye-filled microcavity experiment [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

,4

4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

], it includes the transverse-modes loss-hierarchy (mode-filtering) due to the difference (even if very weak) in their transverse waveform, mirror-reflectivity, gain, absorption and other losses. The frequency cutoffs due to far apart longitudinal modes in the microcavity can give the needed abrupt spectral filtering profile. However, even without spectral filtering, LC can occur in the lateral domain. The situation can be similar to what we suggested in Ref. [8

8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

] (see there the last paragraph before conclusion), where lateral modulation or confinement, and transverse mode filtering, even if the modes had the same frequency, can cause transversal condensation [8

8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

]. Then the condition on the exponent of the lateral loss profile (“potential-well”) is very much relaxed. In two-dimensions, any finite positive η near Ω=0 gives condensation.

There are also other issues, discussed elsewhere and briefly given here, that need consideration in the microcavity experiments [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

]. The massively lost photon replacement in the cavity by pumping occurs via spectral-loss criterion. It is generally the case in lasers, being non-Hamiltonian with frequency dependent dissipation, not in thermal equilibrium, and not a grand-canonical ensemble. The microcavity is not an exception. The seemingly large photon “keepers” (the mirrors) do not provide better performance than what we usually have in many other lasers. The microcavity mirrors have ultra high reflectivity of 0.999985 [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

,4

4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

], and therefore provide a very high finesse. Nevertheless, since the cavity length is small (~1.5μm), important parameters value stay in the usual regime. The photon loss through the mirrors by itself gives a cavity photon lifetime of ~0.3ns, that is similar or shorter than usual values in lasers. Likewise, the gain coefficient needed for steady state oscillation is relatively large (~0.2/cm). When adding other losses, absorption and scatterings, one obtains even much shorter cavity photon lifetimes and higher gain coefficients. It means that the photon population in the cavity is changing very rapidly. The photon number (power) is kept by the pumping that induces stimulated emission photons which replace in steady state the lost photons.

Broadened mode-spectra resulting from (amplified) spontaneous-emission (is random phase noise) and stimulated-emission, such as those obtained with the transverse-modes in the dye-filled microcavity experiment [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

,4

4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

], basically exist without thermal excitations, at any temperature. Such spectra will not “cool-down” to a single frequency by lowering the temperature, say to ~0°K, if it were not a single-mode lasing but a thermal-BEC effect. (Therefore, it would be crucial to measure in experiments like Ref. 3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

the temperature dependence of the spectrum in a broad temperature range, and not only at or close to room temperatures.) There can be thermal effect in the spectra, since the dye molecules [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

,4

4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

] alone can have within their sub-levels some thermal distribution (with questions and limitations [24

24. D. A. Sawicki and R. S. Knox, “Universal relationship between optical emission and absorption of complex systems: An alternative approach,” Phys. Rev. A 54(6), 4837–4841 (1996). [CrossRef] [PubMed]

]) which affects the photon emission. The microcavity experiment however is not a free dye-molecules system. Therefore the spectral filtering is not only the dye molecules absorption-emission part upon which the Kennard-Stephanov (KS) relation [24

24. D. A. Sawicki and R. S. Knox, “Universal relationship between optical emission and absorption of complex systems: An alternative approach,” Phys. Rev. A 54(6), 4837–4841 (1996). [CrossRef] [PubMed]

] is based. Cavities impose on the photons their own additional rules.

LC can thus explain the microcavity experimental results [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

], including the spectra below and above the collapse to a single frequency oscillation, associating the low mode population near the frequency semi-cutoff as lasing or LC at the lowest-loss transverse mode, and not as photon BEC. We also stress that LC can produce and explain exponentially dependent spectra, similar to the thermal behavior that was attributed to Bose-Einstein distribution [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

,4

4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

], as we have shown above (Eq. (5) and Fig. 2b) for common exponential loss (absorption and gain) spectral filtering functions.

We mention again that photon densities in single-frequency lasers can be as high as, and higher, than what was calculated for a photon-BEC realization, where it was said that “photon wave packets spatially overlap” [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

]. One may raise questions about that, and argue if there is any new (quantum?) meaning there and in other regular (“classical”?) high photon density systems, such as LC and single-frequency lasers.

We conclude this discussion by noting that although we raised questions on experiments on photon-BEC, we don’t exclude the possibility to observe it. We think however that the whole issue would need further discussion and study, especially of properties that differentiate BEC from single-frequency lasing and LC.

Realization of LC can be obtained in many-mode laser systems. In particular, fiber lasers (one dimensional), such as erbium-doped lasers, are very suitable since they can have many and dense longitudinal modes in their gain bandwidth. For the loss spectrum, it is possible to fabricate synthesized fiber gratings with reflection frequency profiles needed for condensation. So can be used other lasers such as the 2-dimensional microcavity [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

], relying on the transverse modes. In this case, LC condensation can result from either spectral filtering or mode filtering and spatial loss modulation [8

8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

]. Abrupt spectra can be produced by the frequency semi-cutoffs of far apart longitudinal modes in microcavities, while mode filtering and spatial modulation by lateral confinement. LC is then expected to occur in transverse mode systems [3

3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

], as was discussed above.

4. Conclusion

We have presented a classical laser light condensation phenomenon (LC) which is based on a loss hierarchy in a linear mode system with noise and certain spectral filtering. We discussed experimental sides, differences between LC and BEC and difficulties to observe regular photon-BEC in laser cavities. Further experimental study is needed to verify all those issues.

We finally note that the formalism that leads to LC, based on Eq. (1), can have a broad scope. It presents a generic many “particle” Langevin equation with friction coefficients composed of an independent dispersive part εm, and a shared part (g). Besides lasers, we can think of various mechanical or biological particles in fluids which follow this model and can therefore show condensation behavior.

Acknowledgments

This research was supported by the US-Israel Binational Science Foundation (BSF) and the Israel Science Foundation (ISF). I thank Alexander Bekker and Oded Fischer for valuable help.

References and links

1.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of bose-einstein condensation in a dilute atomic vapor,” Science 269(5221), 198–201 (1995). [CrossRef] [PubMed]

2.

A. J. Leggett, “Bose-Einstein condensation in the alkali gases: Some fundamental concepts,” Rev. Mod. Phys. 73(2), 307–356 (2001). [CrossRef]

3.

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468(7323), 545–548 (2010). [CrossRef] [PubMed]

4.

J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys. 6(7), 512–515 (2010). [CrossRef]

5.

H. Deng, G. Weihs, C. Santori, J. Bloch, and Y. Yamamoto, “Condensation of semiconductor microcavity exciton polaritons,” Science 298(5591), 199–202 (2002). [CrossRef] [PubMed]

6.

R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose-Einstein condensation of microcavity polaritons in a trap,” Science 316(5827), 1007–1010 (2007). [CrossRef] [PubMed]

7.

S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, “Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping,” Nature 443(7110), 430–433 (2006). [CrossRef] [PubMed]

8.

R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18(16), 16520–16525 (2010). [CrossRef] [PubMed]

9.

R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104(17), 173901 (2010). [CrossRef] [PubMed]

10.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95(26), 263901 (2005). [CrossRef] [PubMed]

11.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101(14), 143901 (2008). [CrossRef] [PubMed]

12.

A. Fratalocchi, “Mode-locked lasers: light condensation,” Nat. Photonics 4(8), 502–503 (2010). [CrossRef]

13.

A. Gordon and B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett. 89(10), 103901 (2002). [CrossRef] [PubMed]

14.

O. Gat, A. Gordon, and B. Fischer, “Solution of a statistical mechanics model for pulse formation in lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70, 046108 (2004). [CrossRef] [PubMed]

15.

O. Gat, A. Gordon, and B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 151 (2005). [CrossRef]

16.

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett. 93(15), 153901 (2004). [CrossRef] [PubMed]

17.

A. Gordon, B. Vodonos, V. Smulakovski, and B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express 11(25), 3418–3424 (2003). [CrossRef] [PubMed]

18.

A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, and B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett. 105(1), 013905 (2010). [CrossRef] [PubMed]

19.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95(1), 013903 (2005). [CrossRef] [PubMed]

20.

V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A 44(11), 7439–7441 (1991). [CrossRef] [PubMed]

21.

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1173–1185 (2000). [CrossRef]

22.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993), Ch.6.

23.

A. E. Siegman, Lasers (University Science Books, 1986).

24.

D. A. Sawicki and R. S. Knox, “Universal relationship between optical emission and absorption of complex systems: An alternative approach,” Phys. Rev. A 54(6), 4837–4841 (1996). [CrossRef] [PubMed]

OCIS Codes
(270.3430) Quantum optics : Laser theory
(020.1475) Atomic and molecular physics : Bose-Einstein condensates

ToC Category:
Quantum Optics

History
Original Manuscript: July 3, 2012
Revised Manuscript: October 12, 2012
Manuscript Accepted: October 22, 2012
Published: November 12, 2012

Citation
Baruch Fischer and Rafi Weill, "When does single-mode lasing become a condensation phenomenon?," Opt. Express 20, 26704-26713 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26704


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References

  1. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of bose-einstein condensation in a dilute atomic vapor,” Science269(5221), 198–201 (1995). [CrossRef] [PubMed]
  2. A. J. Leggett, “Bose-Einstein condensation in the alkali gases: Some fundamental concepts,” Rev. Mod. Phys.73(2), 307–356 (2001). [CrossRef]
  3. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature468(7323), 545–548 (2010). [CrossRef] [PubMed]
  4. J. Klaers, F. Vewinger, and M. Weitz, “Thermalization of a two-dimensional photonic gas in a white wall photon box,” Nat. Phys.6(7), 512–515 (2010). [CrossRef]
  5. H. Deng, G. Weihs, C. Santori, J. Bloch, and Y. Yamamoto, “Condensation of semiconductor microcavity exciton polaritons,” Science298(5591), 199–202 (2002). [CrossRef] [PubMed]
  6. R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose-Einstein condensation of microcavity polaritons in a trap,” Science316(5827), 1007–1010 (2007). [CrossRef] [PubMed]
  7. S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, “Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping,” Nature443(7110), 430–433 (2006). [CrossRef] [PubMed]
  8. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express18(16), 16520–16525 (2010). [CrossRef] [PubMed]
  9. R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett.104(17), 173901 (2010). [CrossRef] [PubMed]
  10. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett.95(26), 263901 (2005). [CrossRef] [PubMed]
  11. C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett.101(14), 143901 (2008). [CrossRef] [PubMed]
  12. A. Fratalocchi, “Mode-locked lasers: light condensation,” Nat. Photonics4(8), 502–503 (2010). [CrossRef]
  13. A. Gordon and B. Fischer, “Phase transition theory of many-mode ordering and pulse formation in lasers,” Phys. Rev. Lett.89(10), 103901 (2002). [CrossRef] [PubMed]
  14. O. Gat, A. Gordon, and B. Fischer, “Solution of a statistical mechanics model for pulse formation in lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70, 046108 (2004). [CrossRef] [PubMed]
  15. O. Gat, A. Gordon, and B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys.7, 151 (2005). [CrossRef]
  16. B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and B. Fischer, “Formation and annihilation of laser light pulse quanta in a thermodynamic-like pathway,” Phys. Rev. Lett.93(15), 153901 (2004). [CrossRef] [PubMed]
  17. A. Gordon, B. Vodonos, V. Smulakovski, and B. Fischer, “Melting and freezing of light pulses and modes in mode-locked lasers,” Opt. Express11(25), 3418–3424 (2003). [CrossRef] [PubMed]
  18. A. Rosen, R. Weill, B. Levit, V. Smulakovsky, A. Bekker, and B. Fischer, “Experimental observation of critical phenomena in a laser light system,” Phys. Rev. Lett.105(1), 013905 (2010). [CrossRef] [PubMed]
  19. R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett.95(1), 013903 (2005). [CrossRef] [PubMed]
  20. V. Bagnato and D. Kleppner, “Bose-Einstein condensation in low-dimensional traps,” Phys. Rev. A44(11), 7439–7441 (1991). [CrossRef] [PubMed]
  21. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron.6(6), 1173–1185 (2000). [CrossRef]
  22. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993), Ch.6.
  23. A. E. Siegman, Lasers (University Science Books, 1986).
  24. D. A. Sawicki and R. S. Knox, “Universal relationship between optical emission and absorption of complex systems: An alternative approach,” Phys. Rev. A54(6), 4837–4841 (1996). [CrossRef] [PubMed]

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