| Royal Swedish Academy of Sciences Physica Scripta Phys. Scr. 91(2016) 013003(17pp) doi:10.1088/0031-8949/91/1/013003

Invited Comment Why the laser linewidth is so narrow: a modern perspective

Alexander Cerjan and A Douglas Stone

Department of Applied Physics, Yale University, New Haven, CT 06520, USA

E-mail: [email protected]

Received 5 August 2015, revised 6 November 2015 Accepted for publication 17 November 2015 Published 8 December 2015

Abstract We review and interpret a modern approach to laser theory, steady-state ab initio laser theory (SALT), which treats lasing and amplification in a unified manner as a non-unitary scattering problem described by a nonlinear scattering matrix. Within the semiclassical version of the theory the laser line has zero width as the lasing mode corresponds to the existence of an eigenvector of theS-matrix with diverging eigenvalue due to the occurrence of a pole of the scattering matrix on the real axis. In this approach the system is infinite from the outset and no distinction is made between cavity modes and modes of the Universe; lasing modes exist both in the cavity and in the external region as solutions satisfying Sommerfeld radiation boundary conditions. We discuss how such solutions can be obtained by a limiting procedure in afinite box with damping according to the limiting absorption principle. When the electromagnetic and matterfields are treated as operators, quantumfluctuations enter the relevant correlation functions and afinite linewidth is obtained, via a generalization of SALT to include noise(N- SALT). N-SALT leads to an analytic formula for the linewidth that is more general than all previous corrected versions of the Schawlow–Townes formula, and can be evaluated simply from knowledge of the semiclassical SALT modes. We derive a simpler version of this formula which emphasizes that the noise is dominated by thefluctuations in the polarization of the gain medium and is controlled by the rate of spontaneous emission. Keywords: laser theory, laser linewidth, microcavities, semiconductor lasers

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1. Introduction some of the weak points in the theory in a classic paper entitled,‘Why is the laser line so narrow? A theory of single The laser has been a critical enabler of the modern discipline quasi-mode laser operation’[ 7]. This work was aimed at a of quantum optics via its highly monochromatic and intense fundamental challenge in laser theory, dealing with the emission properties. The physical understanding of steady- openness of the laser system, which invalidated standard state laser emission has presented challenges since the early Hermitian modal decompositions appropriate for closed sys- days after its invention, asfirst semiclassical[ 1, 2] and then tems. Prior to this work essentially all theories were framed in fully quantum laser theories were developed[ 3–6]. Although terms of‘quasi-modes’ of the type studied by Fox and Li[ 8], these theories were able to capture most of the relevant and were based on phenomenological modal equations with physics, they tended to incorporate some phenomenological cavity damping inserted to represent the outcoupling of the or approximate steps due to the complexity of dealing with modes. The goal of Lang et al was to go beyond these quasi- the full nonlinear and space-dependent semiclassical or mode equations for the laserfield amplitudes to a more fun- quantum laser equations. Over forty years ago, several pio- damental description which included both the cavity and neers of early laser theory, Lang, Scully and Lamb, addressed exterior regions explicitly(i.e. a model of‘the Universe’), and

0031-8949/16/013003+17$33.00 1 © 2016 The Royal Swedish Academy of Sciences Printed in the UK Phys. Scr. 91(2016) 013003 Invited Comment then work back from the modes of the Universe to the quasi- modes of the cavity region alone. Using this approach the authors(a)‘hope(d) to understand the mechanism leading to the extreme monochromaticity of the laser radiation’, and(b) ‘to investigate the sense in which the present calculation, which does not include the cavity dissipation as a pheno- mological loss mechanism, still leads to and implies afluc- tuation–dissipation(FD) theorem’[ 7]. While this approach to understanding the laser equations was groundbreaking at the time, there has been much progress in laser theory in the subsequent years and it seemed to us a good time to re-examine these questions from a modern perspective. One reason for this is that during the past decade a new theoretical framework has been developed, known as the steady-state ab initio laser theory(SALT), which treats the openness of an arbitrary cavity exactly within the semi- classical theory[ 9–12], and thus has provided new insights Figure 1. Schematic of lasers as scattering systems: a singly into lasing phenomena. The resulting nonlinear SALT connected region surrounded by a last scattering surface containing equations for steady-state lasing can be solved efficiently an irregularly shaped cavity defined by a linear dielectric function using a non-Hermitian basis set method and have been shown ec ¹ 1 and a nonlinear gain region. Any incoming signals(blue arrows) are amplified and scattered by the cavity to outgoing states to agree to very high accuracy with brute force simulations of (red arrows); lasing corresponds to outgoing solutions without any the atomic lasing equations[ 13, 14], including evenN-level input, and is only possible at discrete frequencies for a given cavity. models with multiple independent lasing transitions[ 15]. The theory can also be generalized to treat injected signals(I- SALT), as will be relevant below[ 16]. Moreover, recently such as microdisks and microspheres[ 27–29], micro-toroids Pick et al have added Langevin noise to the SALT equations [30], deformed(chaotic) disk cavities[ 31–33], photonic (N-SALT) and, using the SALT steady-state lasing modes as crystal defect-based and surface emitting lasers[ 34–36] and a starting point, calculated the quantum-limited laser line- random lasers[ 37–43]. To describe such devices it was width analytically[ 17]. Their analytic results have very necessary to formulate laser theory in a manner that could recently been confirmed, again by direct simulations of the both describe, and be implemented computationally, for lasing equations, but with appropriate noise terms included cavities with an arbitrary geometry, and to include the full [18]. It should be said that many aspects of the results Maxwell wave equation, not its paraxial approximation. obtained through SALT and N-SALT were anticipated by Moreover, some of the new lasers(e.g. random lasers) are prior theoretical approaches, a partial list of which includes extremely open systems, not even cavities in the usual sense; the work by Spencer and Lamb[ 19], Fu and Haken[ 20–22], nor do they always emit well-behaved narrow beams of light and Henry[ 23]; however, SALT and N-SALT appear to be but rather generate complex emission patterns in two and the most general,flexible and inclusive formalism yet three-dimensions. Thus it was crucial tofind a description developed for steady-state lasing. It treats thefields over the which did not over-simplify the spatial properties of the entire‘Universe’ from the outset, but in a different manner cavities and emission patterns, and which hence treated the from Lang et al, and does not arrive at modal equations with cavity boundary and outcoupling exactly. The solution to this damping in the conventional sense. We will now examine the challenge provided by SALT is to describe the laser as a questions raised by Lang et al from the perspective of SALT scattering system. Any laser consists of two essential ele- and N-SALT. ments: a cavity region, given as a linear dielectric function, fi ec ()x , which describes the scattering and con nement of photons, either injected externally or internally generated, and 2. Lasers as scattering systems a gain region, containing the medium which will be pumped to provide optical gain(sometimes cavity and gain medium essentially coincide). To include all cases the‘cavity’ will 2.1. Motivation and definition of the problem refer simply to a singly connected region large enough to Most early lasers emitted from lasing modes which were enclose all the regions in which ec ()x1 ¹ ; and for simplicity concentrated along some optical axis, e.g.the axis joining the we will assume the gain medium is fully contained within this centers of two mirrors or the geometric or index profile region as depicted infigure 1. defining a waveguide orfiber, and so there was a tendency to The laser so defined can function both as an amplifier of develop laser theories focused on one-dimensional or axially input radiation(when the gain medium is inverted) and as a symmetric equations, and this is overwhelmingly the type of self-organized oscillator/source of radiation in the absence of formulation onefinds in text books[ 24–26]. However, in the input, above thefirst threshold for lasing. From this point of past two decades a great variety of laser systems have been view the laser is a scattering system which both scatters and developed with complex two or three-dimensional cavities, amplifies input radiation, and also, at discrete frequencies,

2 Phys. Scr. 91(2016) 013003 Invited Comment scatters specific internally generated modes of outgoing the case of a two-level gain medium it takes the form fi radiation when suf ciently pumped. However, unlike linear 1 D0g^ scattering systems, the laser is subject to nonlinear saturation c w, x = 4p , g () 2 effects which are fundamental to its stability and must be ()w- wa +i g^ () 1 + G() w - wYa ∣()∣x taken into account. ()3 Other nonlinear effects within lasers, such as four-wave mixing, can destabilize their simple harmonic time-response, whereD 0 is the pump parameter(proportional to the but these effects are neglected in SALT, and also(typically) inversion), g^ is the dephasing rate of the polarization, wa is in the earlier laser theories which motivated the work by Lang the atomic resonance frequency, and G()- a wwis a et al[ 7]. Leaving out such effects is equivalent to assuming a Lorentzian centered at wa.D 0 and Y are measured here in appropriate dimensionless units[ 13], and this form assumes stationary inversion density, giving rise to a gain suscept- that there is only one mode, either lasing or amplified, active ibility which does not mix frequencies. This assumption is in the medium.(This is the only case we will consider always valid for single-mode lasing and amplification, which explicitly here, but we will review the more general will be the focus of this work; it is also valid under certain multimode SALT formalism in appendix A). The nonlinear rather general conditions for multimode lasing in microlasers, gain susceptibility represents the polarization of the gain and SALT has been shown to apply quantitatively to such medium and is the source of the amplification and(above cases[ 13–16]. threshold) of laser emission. The lack of any Langevin force Before proceeding, it should be mentioned that the input/ terms in( 2) is a result of the semiclassical approximation, in output theory developed by Collett and Gardiner(and others) which the operators for the electromagnetic and polarization also considers the cavity to be a scattering element[ 44, 45] fields are replaced by their average values, eliminating the and is able to treat the full quantum operator properties of the effects of quantum and thermalfluctuations. Such terms are scatteredfields. This approach has certainly contributed to our taken into account in N-SALT, which will be discussed understanding of open quantum systems and the role played briefly in section 7, but were neglected(intentionally) in the by the reservoirs and noise processes. However, while there work of Lang et al. has been substantial effort expended to describe lasing within Assuming the system is pumped below thefirst lasing this framework[ 46–49], it remains an open problem how to threshold and the inputs are small enough to generate negli- include the strong nonlinearities present in active lasing sys- gible saturation of the gain, this equation defines a linear non- tems in this formalism; whereas at the semiclassical level this unitary scattering(S-)matrix for inputfields. Depending upon is done quite accurately by SALT. the geometry there are different natural basis states describing ()- ()+ fi input and output modes, Ha x)(, Ha x)(, from and to in - nity, i.e.sufficiently far away from the cavity region

2.2. Scattering in the SALT formulation ⎡ ()()- + ⎤ Y=()()xå⎣ca Ha + x da Ha ()x ⎦ Initially we focus on the single-mode lasing case, as was done a by Lang et al, so that the positive frequency component of the ⎡ ⎛ ⎞⎤ ⎢ ()(- +) ⎥ electricfield will be given by: =åca ⎜Ha ()() x + å Sa, b Hb x.⎟ 4 () a ⎣⎢ ⎝ b ⎠⎦⎥ + - w ()()()tExY= ti x,e,() 1 If the pump is strong enough to invert the medium and where the real number,ω, is either the input radiation create gain at the input frequency, theS-matrix will be frequency(for the amplifier) or the unknown frequency of the amplifying, with eigenvalues greater than unity. When func- laser oscillator, which must be self-consistently determined. tioning as an amplifier the boundary conditions are familiar: fi x()isY the unknown spatial lasing(or amplified) mode, not an input wave from in nity interacts with the cavity plus gain assumed to be equal to any passive cavity mode or quasi- medium, scattering according to the appropriate dielectric or mode and is also self-consistently determined; such lasing/ metallic boundary conditions in different regions of the cavity amplifier modes describe the solution over all space, not just and giving rise to an outgoing wave which differs from inside the cavity. passive scattering in that it is amplified and more photonflux Assuming stationary inversion and neglecting quantum emerges from the scattering region than enters it. The case of large inputs leading to a saturated gain and the resulting and thermalfluctuations, we can describe x()forY an arbi- trary laser by a nonlinear wave equation of the form: nonlinearS-matrix can be handled by I-SALT and will be included in the discussion below. ⎡ 2⎤ As noted, In this formulation there is no distinction ´´-+ec pcgYkx4()() = 0, 2 ⎣ () ⎦ between cavity modes and modes of the Universe, every amplified mode exists inside and outside the cavity as a where k cw, and= cg is the complex, frequency-dependent steady-state scattering solution. However, we now consider to saturable gain susceptibility, which can be calculated for an the more challenging and subtle case of laser oscillation in the arbitraryN-level atomic gain medium in steady-state[ 14]. For scattering picture.

3 Phys. Scr. 91(2016) 013003 Invited Comment

have a resonant response near the real values wn, with a fi bandwidth given by gn (assuming well-separated high nesse resonances). The corresponding spatial solutions are referred to by various terms: quasi-modes, quasi-bound states and resonances. Historically the use of multiple terms seems to have arisen from the transition from 3D confinement, as e.g.in a microwave resonator, where the connection to bound states is more apparent, to open, axial mirror-based resona- tors, where the resonances perpendicular to the mirror axis are essentially absent, and the Fox–Li method was developed to determine the highQ resonances. However, there is no fun- damental difference between the various types of outgoing solutions: they all correspond to poles of the passive-cavityS- matrix. Such states also are very well-known from quantum scattering theory(where they are usually referred to as quasi- bound states), and are often considered to be physical states because if an initial condition is given corresponding to the quasi-bound state truncated to the‘cavity region’, it will leak fi out to in nity with a rate given by gn (this property was explicitly demonstrated for the one-dimensional cavity treated by Lang et al). However, in this context we would like to Figure 2. Simulations for a D-shaped laser cavity withR=4μm, emphasize that the true quasi-bound states, including the r= 0.5 R,n 3.5 as shown in the bottom schematic, with a 0 = exponentially growing component at infinity, are not physi- uniform two-level atomic gain medium characterized by la = 1 μm fi cally realizable steady-state solutions and cannot correspond and ^ 10gnm.= (Top) plot of the Lorentzian gain pro le for the atomic medium.(Bottom) plot of the motion of the poles of theS- to the lasing mode. Also, since they occur at the poles of the matrix as the D-shaped cavity is pumped to thefirst lasing threshold. S-matrix for the passive cavity, they can only describe the The pole corresponding to thefirst lasing mode is seen to reach the properties of the passive cavity, independent of the gain real axis(marked with a red circle), and the spatial profile of the medium and spectrum. Thus using‘quasi-mode’ also as a lasing mode at threshold is shown just above. term for the lasing mode in the presence of gain can, and in our view often has, led to confusion. 3. Quasi-modes, resonances and quasi-bound states

Any scattering region offinite extent with a real potential or 4. Threshold lasing modes and constantflux states dielectric function, and hence with a unitaryS-matrix, will have associated with it resonant solutions; i.e. stationary However, as described above, the properties of the laser/ solutions at discrete complex frequencies which are purely amplifier are not governed by a unitaryS-matrix once the outgoing. It is easiest to think of these as evolving out of the complex gain susceptibility is taken into account. This com- bound states of the system at real frequencies, w˜ n, which plex susceptibility violates the Hermiticity of the wave would occur if the region were surrounded by a perfect equation even for the closed cavity and invalidates the stan- lossless reflecting barrier. As the reflectivity of thefictitious dard modal decomposition. But it makes possible purely barrier is reduced from unity to zero these real frequency outgoing solutions at real frequencies, which hence are phy- solutions, corresponding to standing waves, evolve into sical solutions over all of space. The non-equilibrium steady- complex frequency solutions with a substantial outgoing state gain susceptibility depends on the pump level, with its wave component(and no incoming wave from infinity). Due imaginary part changing sign from positive(absorbing) to to the purely outgoing boundary conditions it can be shown negative(amplifying) at the transparency point, at which the that the imaginary part of the frequency must be negative, and upper and lower lasing levels are equally occupied and the theS-matrix, continued to complex frequencies, will become susceptibility vanishes. At this pump value the poles of the fi cavity with gain coincide again with those of the passive in nite with simple poles at wn ign 2,() g n-> 0 . Hence these purely outgoing solutions are discrete and countably cavity. As the pump is increased beyond the transparency infinite, in one to one correspondence with the bound states of point, all of the passive cavity poles continue to exist, but the closed cavity. typically they move up in the complex plane towards the real In the asymptotic region the frequency and wavevector of axis[ 12, 50, 51], at a rate that depends on their position in the the solutions are linked by the dispersion relation and the sign gain spectrum. TheS-matrix of the amplifier can be measured of gn will cause the corresponding spatial solution to grow for real frequencies as the pump increases, and the height of exponentially at infinity and carry a diverging outgoing the resonances corresponding to each pole increases(the photonflux. Hence these solutions are not physically realiz- transmittedflux is greater than the incidentflux due to able, but have physical significance because theS-matrix will amplification), while their width decreases(gain narrowing),

4 Phys. Scr. 91(2016) 013003 Invited Comment because the effective loss rate is proportional to the imaginary Returning to the fundamental question of modes of the part of the pole frequency, which is decreasing. If there is no Universe versus quasi-modes of the cavity, we summarize the input to the amplifier then there is still no emittedfield and no SALT perspective. indication within semiclassical theory of the approach to • If one treats lasers as unbounded scattering systems from fl lasing(although when quantum uctuations are included the outset, all solutions are defined both inside and fi there will be enhanced noise, ampli ed spontaneous emission, outside the cavity region and there is no distinction near the gain center). If lasing is achieved, one of the poles of between cavity modes and modes of the Universe. theS-matrix reaches the real axis, gain completely balances • If the laser is functioning as an amplifier, the solutions are loss, and we have a zero width resonance on the real axis. simply amplified scattering states determined by the This process is demonstrated infigure 2 for a non-trivial two- boundary conditions in the cavity region and the degree dimensional cavity, a chaotic‘D-shaped’ dielectric cavity of gain at the input frequency. [33], where each pole is seen to move towards the real axis an • If the laser is functioning as a self-organized oscillator the amount proportional to its distance from the center of the gain solutions are purely outgoing states corresponding to an curve and one of the poles experiences enough gain to reach eigenvector of the linear scattering matrix with diverging threshold. eigenvalue due to the existence of a pole on the real axis. Itfirst may seem that we are back at the familiar situation Such solutions require the presence of the saturating of true bound states, which correspond to poles on the real nonlinearity to be physically stable. axis, but this is not so. In the former case there is no photon • For any given cavity and gain susceptibility such purely gain and one simply has a decaying standing wave pattern, outgoing states can only occur at discrete frequencies trapped in some region of space. Here we have a different because they arise from moving the discrete poles of the type of object, which we call a TLM. This type of solution passive cavity onto the real axis. Within semiclassical fl never occurs in Hermitian theories, a purely outgoing solution theory(neglecting quantum and thermal uctuations of fi at real frequency; it reflects the fundamentally non-Hermitian the elds) these solutions have precisely zero linewidth. nature of the electromagnetic wave equations with a complex We reserve the term‘lasing mode’ for these special susceptibility, which is an effective theory in which the matter discrete solutions. degrees of freedom are hidden. Such a solution is character- In summary then, our response to the question‘Why is ized by a steady-state photonflux out to infinity, but with no the laser linewidth so narrow?’ is that indeed this is due to the unphysical growth of intensity outside the cavity and an extreme limit of gain narrowing: only at discrete frequencies emission pattern corresponding to the eigenvector of theS- can the linear gain diverge and generate self-organized matrix with a diverging eigenvalue. It is an example of a CF oscillation. As we will show in the next section, this zero- state: photons are generated within the cavity but are con- width resonance cannot be observed with an input signal, served outside the cavity, hence corresponding to a true i.e.in amplification mode. However, these perfectly mono- physical lasing solution[ 9, 11, 12]. In appendix A we will chromatic purely outgoing solutions can be found by SALT show that each TLM is just one member of a complete and have been verified by brute forcefinite-difference time biorthogonal basis set of CF states which forms a natural basis domain(FDTD) simulations of the laser equations for many fi set for describing the nonlinear lasing solution above examples[ 12, 14, 15]. Such in nitely sharp lines are not of threshold. course obervable in true laser emission due to quantum fl However, we still have a conceptual problem: the linear uctuations as we will discuss in section 7. However, the S-matrix for this amplifier is now infinite at the lasing fre- smallness of these quantum effects lead to the extraordinary quency. Within linear theory any input would be infinitely narrowness of the resulting lines, often many orders of magnitude narrower than the passive cavity resonance amplified, violating conservation of energy. The resolution of linewidths. this dilemma is that infinitesimally above threshold the The effects of quantumfluctuations were intentionally saturating nonlinearity kicks in and stabilizes the solution. In not included in the work of Lang et al, who thus also found a fact the steady-state lasing amplitude rises smoothly from delta function laser line within semiclassical theory. In their zero and has no jump at threshold, just a discontinuous approach theyfind that this extreme narrowing of the passive fi derivative. Even though formally theS-matrix has a pole xed cavity resonance can be regarded as a‘mode-locking effect’ on the real axis above threshold, any input wave at the lasing in which all of the modes of the Universe within the passive frequency simply increases the gain saturation and does not cavity resonance linewidth contribute to one lasing quasi- lead to a runaway of the output. Strictly speaking, any such mode with a single frequency. This point of view emerges input actually forces the pole off the real axis, as we will show from a limiting procedure in which afinite system with with an example in section 5. Hence the nonlinearity is quantized electromagnetic modes is studied and at the end the essential for stablizing the linear CF solution. However, the system size is taken to infinity in the presence of smallfic- actual solution near threshold is accurately described by the titious absorption to damp out reflections from the boundaries single TLM calculated from the linear wave equation with as they tend to infinity. The conditions for this procedure to gain, since the saturation term is not strong enough to change reproduce the correct infinite system behavior are discussed the spatial form of the solution significantly. in section 6 below. We note that Lang et al expressed

5 Phys. Scr. 91(2016) 013003 Invited Comment

Figure 3. Schematic showing the system studied by Lang et al[ 7], which contains two regions, the cavity and the much larger ‘Universe’(which also includes the cavity). Damping is added to the external region to prevent reflections from the boundary at x L=-, and as discussed below, this must be done by a particular limiting procedure in order to obtain the correct result for the laser. dissatisfaction with invoking gain narrowing to explain the laser linewidth, because in their view such arguments(at the time) did not take into account the nonlinearity of the pro- blem. However, the conceptual framework of SALT/I-SALT, outlined here, does take nonlinearity into account, essentially exactly, for single-mode lasing. We will now illustrate the previous concepts with a concrete one-dimensional example, very similar to the one studied by Lang et al, in the process making all of our statements above mathematically precise.

Figure 4. (Top) plot of the transmitted intensity of an injected signal with frequency win, into a passive, two-sided cavity with partially reflecting mirrors(R= 0.8) on both ends, as indicated in the 5. Lasing modes as zero-width resonances: a simple schematic.(Bottom) plot of the poles, wp of the scattering matrix of example the same cavity, corresponding to purely outgoing solutions at complex frequency. The real frequency resonances of the cavity, The example studied by Lang et al was a one-dimensional where complete transmission is seen, are related to the poles of the cavity. cavity of lengthL+l, with perfectly reflecting mirrors at x l L,=-and a partially reflecting delta function mirror at x=0, as shown infigure 3. The‘cavity’ is the region []l0, with a width given by the passive cavity decay rate 2 and the‘Universe’ is the region, []L,0 ,- where in treating gc = -c()() L R2ln, whereL is the length of the cavity, and lasing a small uniform absorption is added over all space, the R is the reflectivity of the mirrors at the cavity boundary. The limit L ¥ is taken, and lasing emission is in a single incident and reflected waves on the left side of the cavity direction towards x=-¥ . For such a system the passive satisfy a continuity boundary condition, with both incoming cavity will have a reflection coefficient of modulus unity at all and outgoing components, whereas the transmitted wave frequencies, and resonant reflection will be distinguished only leaving the right side of the cavity also is continuous but in by a long delay time in reflection and an associated build-up addition is purely outgoing. offield intensity in the cavity. We prefer to make a small Any purely outgoing monochromatic wave is described modification of the Lang et al example and study a symmetric mathematically by the Sommerfeld radiation condition[ 52], two sided partially transmitting(Fabry–Perot) cavity with d1- limx2 i ky¶- x 0, = 5 laser emission to x=¥ , and with passive cavity reso- ∣ ∣()()∣x ∣ () nances distinguished by enhanced transmission, reaching ∣x ∣¥ unity at the center, and with width, g . As with the Lang  c in which x= x x , w = ck, andd is the dimensionality of et al example, the electromagnetic part of the problem will be ˆ ∣ ∣ the system. In one-dimension the external waves are simple described by the scalar Helmholtz equation with gain. How- plane waves(and not e.g. cylindrical or spherical waves) and ever, unlike Lang et al, no limiting procedure is applied; our the Sommerfeld condition can be imposed anywhere outside system is infinite from the start and nofictitious absorption is the boundary; thus at the boundary on the right side of the added to avoid reflections from perfect mirrors far away from cavity thefield must be an outgoing plane wave, satisfying the cavity. Hence, consistent with our scattering approach, we start from continuous solutions for the passive cavity+ ⎡ x ky¶- xi⎤ 0. = 6 ⎣ () ()⎦x= L () ‘Universe’, and will impose outgoing boundary conditions at infinity on the lasing solutions. We begin by reviewing the properties of such a one- As noted, lasing would correspond to a solution satisfy- dimensional optical cavity. Infigure 4 we plot the transmitted ing a purely outgoing boundary condition on both sides of the intensity of an incident signal through the passive cavity at cavity(or in all directions in higher dimension). If we attempt frequency win. As expected we observe a series of resonances to impose this boundary condition on the passive cavity we of the cavity, spaced by the free spectral range, D = cLp, find that no solutions are possible at real frequencies, but

6 Phys. Scr. 91(2016) 013003 Invited Comment there are discrete periodically spaced solutions at

p gc wm = +m D - i ()7 2cL 2 as shown infigure 4. The discreteness of the frequencies is a direct consequence of the Sommerfeld condition which can only be satisfied when there is constructive interference in the cavity. The fact that all the resonance frequencies have the same imaginary part is not generic, but is particular to one- dimensional cavities of this type. As noted above, the presence of these purely outgoing(pole) solutions leads to resonant transmission on the real axis centered at Re[wm ] with width gc. Applying the Sommerfeld condition to the outgoing solutions e kxi, and the relation w = ck implies that g+x2 ∣ ()xe∣Y~c at ¥, the unphysical divergence we alluded to above. In order tofind normalizable physical solutions which are purely outgoing(i.e. lasing solutions) we need to add the effect of the gain medium to the cavity dielectric function in fi the form of cg as de ned in( 3) above. For this simple one- dimensional scalar model(similar to Lang et al), below threshold, the electricfield satisfies Figure 5. (Top) plot of the transmitted intensity of an injected signal as a function of frequency, win, illustrating line narrowing with ⎡ 2 2⎤ ⎣¶x +()()nk⎦ Y x0, = 8 () increased gain. Iin=I0 = 0.01, and the two-sided cavity has partially reflecting mirrors(R= 0.8) at both ends,(see schematic). with the Sommerfeld boundary condition, where now the The pump on the cavity is increased from the transparency point, D = 0 (red), to just above thefirst lasing threshold, D = 0.0040 index of refraction n ec pcg=+n 1 n4i2, and = the - 0 0 (blue), calculated using I-SALT[ 16]. As can be seen, the width of gain nonlinearity is neglected, because we are below thresh- the cavity resonance narrows during this process. The slight shift of old and these are not yet physically allowed solutions(they the resonance is due to line pulling from central atomic transition fi are simply modi ed active cavity resonances). Note that the frequency, wa= 60, with g^= 2. At the transparency point this solutions inside the cavity here take the form system is identical to that shown infigure 4, and the outermost red fi A nx B nxcos, i.e.they are sin trigonometric() ( functions )+ curve seen here is identical to that from gure 4.(Bottom) plot of the of a complex argument. This will remain true at threshold and motion of the pole of the associated cavity resonance, wp, through is a key point which holds more generally: the electricfields the complex plane as the pump is increased. This is also calculated using I-SALT, and thus includes the saturation of the gain medium for the physical system with gain cannot be purely real; if they due to the injected signal, which prevents the pole from reaching the were real they would conserve photonflux and describe true real axis even though thefinal pump value is above thefirst non- standing waves, whereas lasing or amplified modes must have interacting modal threshold. For the specific realization of the fi running wave components. Any model whichfinds real motion of the pole shown, the injected eld is taken at w=in 60.25, spatial solutions for lasing or amplified modes has made an which corresponds to the peak in the blue curve. The pole is also seen to drift due to line pulling. Frequencies and rates are reported in approximation. Such approximations have often been made in units ofc/L. laser theory, including by Lang et al[ 7, 53–55](see further discussion in section 6). When we add the gain medium to the cavity and pump As the pole rises towards the real axis the amplified above the transparency point ()D0 >the imaginary part of resonance narrows. Due to the frequency-dependent satur- fi cg corresponds to ampli cation. This has an immediate effect, ation of the gain medium, as already noted, injected signals causing the poles to move upwards, as discussed earlier and with different frequencies will induce somewhat different as shown for this specific cavity infigure 5. While the pole trajectories for the pole in order to maximize the effective solutions are still not physically realizable, we can observe a gain[ 16]. These are essentially line-pulling or pushing effects change in the cavity properties via the behavior of the of the gain medium, which for a single frequency are always transmission resonances at real frequencies. To calculate this towards the atomic transition frequency. In the active cavity effect accurately with afinite input wave we must now use the with dispersion, the precise width of the transmission reso- full saturable gain susceptibility of( 3). The simulations nance can no longer be defined as the distance of the pole shown here are performed using I-SALT which provides a from the real axis, as probing different frequencies will yield nearly exact description of the steady-state single-frequency slightly different distances of the pole from the real axis. response of an active optical cavity to injected signals and any These line-pulling effects lead to the mild asymmetry in the self-generated lasing signals[ 16], including the spatial series of curves shown infigure 5. degrees of freedom. SALT and I-SALT are reviewed in more The bluest(and narrowest) curve shown infigure 5 detail in appendix A. corresponds to a pump value just above the non-interacting

7 Phys. Scr. 91(2016) 013003 Invited Comment through gain-narrowing of an amplified input signal, for the reasons we described above: gain saturation forces all poles off the real axis. Hence this effect does require correct treat- ment of the nonlinearity as stated by Lang et al. However, we have shown here that it is possible to treat the nonlinearity exactly at each stage and explain the zero width of the laser line quantitatively. The model we have treated here is essentially the same as that treated by Lang et al except for the pedagogical differ- ence of choosing a two-sided cavity in which one can study transmission resonances; in the next section we will study exactly the model of Lang et al by a different but equivalent method. As noted, in the standard SALT approach the system is infinite from the outset and there is no dissipation outside the cavity, which is in our view the most natural model for a real laser system. In contrast, Lang et al place their cavity in a much larger box which eventually will be taken to infinite size, and add dissipation external to the cavity, intended to damp unphysical reflections from the boundaries of the larger fictional box. We will discuss the effects of this difference in the next section. Figure 6. (Top) plot of the transmitted intensity of an injected signal as a function of frequency win, into the active, two-sided cavity with partially reflecting mirrors(R= 0.8) on both ends, as shown in figure 5. The pump on the cavity is heldfixed just above thefirst 6. Lasing modes and the limiting absorption lasing threshold, D0 = 0.0040, while the incident intensity is principle(LAP) decreased from Iin=I0 = 0.01 (blue), to Iin = 0 (red), where the cavity is operating as a free-running laser, calculated using I-SALT While it is clear that lasing modes should satisfy the purely [16]. As can be seen, the width of the cavity resonance continues to outgoing(Sommerfeld) boundary condition, this leads to a fi narrow during this process, nally resulting in the lasing signal with rather unfamiliar description of the spatial behavior of the infinitesimal width. Note, the outermost blue curve shown here is the same as that shown infigure 5.(Bottom) plot of the motion of the lasing modes in terms of functions that are not power pole of the associated cavity resonance, wp, through the complex orthogonal, but instead satisfy a biorthogonality condition plane as the injected signal is turned off for the specific realization of [25, 56, 57]. Basis functions of this type, the CF states[ 9–12], w=in 60.25. This is also calculated using I-SALT, and thus includes are central to the standard computational algorithm of SALT. the saturation of the gain medium due to the injected signal, which The approach taken by Lang et al of beginning with afinite prevents the pole from reaching the real axis until the injected signal system and adding damping to remove the effect of the is turned off. For different values of win (not shown) the pole is able to reach the real axis before the injected signal is fully turned off. boundaries of the system is an alternative which has evolved into the well-developed computational technique of sur- rounding a cavity or region of emitters with a perfectly first lasing threshold. This may seem initially puzzling; was matched(damping) layer(PML). When done correctly this not the lasing threshold the point at which the pole reaches the can reproduce the imposition of Sommerfeld conditions to real axis? That definition only applies when there is no input arbitrary accuracy. In fact, newer computational methods for signal. With an input signal, even at the lasing frequency, the solving the SALT equations[ 58] make use of PMLs and do saturation of the gain medium prevents the pole from reaching not impose the Sommerfeld boundary conditions in the the real axis, keeping the responsefinite, and the amplifed straightforward manner of matching to outgoing solutions at a resonance retains a non-zero width. last scattering surface as did the earlier methods of satisfying To see the true zero-width semiclassical lasing solution the SALT equations[ 11]. The PML approach gives essen- we slowly turn off the incident signal with the pump held tially perfect agreement with the Sommerfeld approach above thefirst lasing threshold. With this protocol we see that to SALT. the transmission resonance width continues to narrow, as However, as noted above, the correct lasing solutions shown infigure 6, while the pole continues to move towards must be complex functions in space, so as to represent the the real axis. As the injected signal is turned off, the pole non-zero photonflux, which grows inside the cavity and reaches the real axis and the cavity begins to undergo self- remains constant outside. For the simple 1D cavities studied organized lasing action, shown as the reddest curve in the top here and by Lang et al, this means that the spatial solutions panel offigure 6. This curve has precisely zero linewidth; no inside a constant-index cavity must be trigonometric func- radiation comes out of the cavity except at this single fre- tions of complex argument, corresponding to exponential quency without input waves(assuming effectively zero growth towards the mirrors as well as oscillation. However, in temperature in the cavity and neglecting quantumfluctuations the work of Lang et al theyfind a lasing mode of the form ai w-t until section 7). This effect can never be fully achieved E ()[(x, t»- C0 e() sin k0 x l)] for 0

8 Phys. Scr. 91(2016) 013003 Invited Comment with thefinite damping, and carefully take the limit L ¥ prior to g  0; this requirement is known as the LAP. From a practical computational point of view the LAP means that one needs to solve the non-Hermitian spatial problem with Dirichlet(or other linear homogeous) boundary conditions andfinite damping, while maintaining the condition gkL1 . The solutions so obtained have only exponentially small sensitivity to the presence of the boundary and repro- duce to high accuracy those obtained by actually imposing directly the Sommerfeld conditions. Hence they are sig- nificantly different from the Hermitian solutions calculated before the damping is added(or when the limits are taken in the opposite order), which can carry no netflux as they are bound states. The modern PML techniques are a means for introducing non-uniform absorption which is impedance matched so that reflections are negligible even whenL is not very large and one reproduces the Sommerfeld solutions with a much smaller and manageable computational cell. But the principle is basically the same: in order to get the correct spatial solutions one must treat the non-Hermitian problem fi Figure 7. Plot of the resonances of the nite system with external with damping from the outset. damping depicted infigure 3 for three different values ofL, , = L l25 To illustrate this explicitly we briefly analyze the Lang L l100, and=L l200, from= top to bottom, except that the optical confinement is instead provided by a passive cavity dielectric of et al system from this point of view. The system is the one- n=6 in the cavity, rather than a partially reflecting mirror. In the sided cavity between []l0,attached to the‘Universe’ between external region an absorption of 0.003gis= present. In(c) the []L,0 ,- as shown infigure 3, with perfect mirrors(Dirichlet isolated solutions of the problem with damping are compared to the boundary conditions), E l E L0(), where ( )E=-x t,(is =) resonances of the same cavity subject to the Sommerfeld boundary the electricfield of the system. Before adding the gain and condition on an infinite system without damping, given by( 7)(red circles). Perfect agreement is found as discussed in the text.(Note considering lasing, we explicitly include the weakly absorb- that due to the presence of the passive cavity dielectric constant the ing material uniformly in the Universe, and focus on the right side of( 7) must be multiplied by n1in making the region external to the cavity. The wave equation in this region comparison). The green crosses shown in(c) show the locations of is written as the resonances when uniform gain is added to the cavity tofind the TLM, n i60.005. As can be=- seen, the poles corresponding to ⎡ 2 1+ ig 2⎤ ¶x - ¶ t E() x, t = 0, - L x < 0. ()9 resonances of the cavity have moved towards the real axis,with the ⎣⎢ c2 ⎦⎥ rightmost one reaching the lasing threshold, while those corresp- onding to the resonances of the Universe have stayed relatively We expand thefield inside of the system in terms of a set of fixed, as discussed in the text. This is thus an alternative method to basis functions,U (x), so that E x t= U x wt-i, and k ()()åk k , e implement the Sommerfeld conditions for lasing modes. solve for the form of these basis functions in the Universe region as

ikx-g kx -ikx + g kx U x= ce2 + c e2 , is the difference between gain and loss,ω is the lasing fre- k() 1, k 2,k - L x  0, ()10 quency,C 0 is a constant, and k0 = W c is the real wavevector corresponding to the passive cavity resonance frequency,Ω. where we have expanded the exponent for g  1. The Since the spatial part of this function is real, it does not coefficients c k1,and c k2,and the quantized values ofk are set fl capture the photon ux through the partially transmitting by the boundary conditions Uk (-L), Uk ()l0= , and the mirror and is only an approximation to the complex sine wave matching conditions atx=0, and will vary withL,l and the solution which is correct for this one-sided cavity. transparency of the mirror at the origin. In our view the origin of this problem is the neglect of the Wefirst apply the LAP to obtain the Sommerfeld solu- LAP[ 59–61], which gives the correct mathematical proce- tion for this problem, taking the limit L ¥ prior to g  0. dure for recovering the Sommerfeld radiation condition from Performing the limits in this order on the solution ofU k(x), afinite system. Lang et al solved the spatial part of the pro- (10), onefinds that to satisfy the Dirichlet boundary condi- fi fi blem for niteL, nding the quantized Hermitian basis states tion, Uk ()L0-=, we must choose c k1,= 0, as the term for the whole passive system(‘modes of the Universe’) and multiplying this coefficient is diverging as -L -¥ for then addressing the lasing problem within this Hermitian finiteγ. Then, with the spatial limit taken, we can safely basis set. Effectively the spatial part of the problem isfixed at reduce the absorption in the problem to zero, recovering finiteL, and a small absorption,γ, is subsequently added U x= c -ikx - ¥ xe, 0, 11 everywhere in the Universe so as to eliminate the effect of k() 2, k () reflection from thefictitious perfect mirror at x L=-. For finding that all of the solutions of our problem are purely this procedure to work one needs to solve the spatial problem outgoing from the cavity, consistent with the Sommerfeld

9 Phys. Scr. 91(2016) 013003 Invited Comment Finally, we can make contact with the SALT approach by adding gain(taken to be uniform for convenience) in the cavity region while leaving the damping present outside the cavity. The added gain has a much bigger effect on the cavity resonance solutions which have high amplitude there, than on the external standing wave solutions. For large enough gain one of the cavity resonances can reach the real axis and will then correspond to a TLM for uniform gain, as shown in figure 7(c) (green crosses). This is the essence of the SALT solution method adopted in[ 58] except in that work a PML Figure 8. Plot of the modal amplitudes for a small region near the damping layer is used for higher computational efficiency and cavity-Universe boundary for two modes, one corresponding to a the saturating nonlinearity is taken into account. Thus the resonance of the cavity(top), and the other corresponding to a Lang et al method agrees with the SALT approach if care is resonance of the external Universe(bottom). As can be seen, the cavity mode is a nearly pure running wave after it exits the cavity. taken to apply the limits in the right order. radiation condition,( 5). However, if these two limits are taken 7. Quantum limited laser linewidth: an in the opposite order, setting the absorption to zero before alternative form letting the length increase to infinity, wefind that Since this paper is devoted to the‘SALT perspective’ on the U˜k()(()) x= b k sin k x + L , - L x 0,() 12 laser linewidth, we include here a section on thefinite laser linewidth due to quantumfluctuations, which has recently which is exactly the result of Lang et al(outside the cavity been addressed within two different generalizations of SALT, region). With this result the matching to the cavity allows for one based on input–output theory[ 50, 51] and another based real sine wave solutions, whereas with the outgoing solutions on nonlinear perturbation theory around the SALT solution only complex sine waves are allowed in the cavity. The [17]. The fundamental origin of afinite quantum linewidth for complex sines provide the outgoingflux at the partially the laser is the presence of spontaneous emission, which leads transmitting mirror which is needed for continuity with the to random jumps in the time-dependent phase of the output field outside the cavity. field. The resulting phase diffusion is linear in time and the These conclusions can be further reinforced by directly phase diffusion coefficient determines the intrinsic linewidth. solving for the eigenfunctions of thefinite system studied by The basic physical understanding of this goes back to the Lang et al, shown in the three panels offigure 7. AsL is seminal work of Schawlow–Townes(ST)[ 62], which was increased, the modes of thisfinite system separate into two subsequently substantially extended by Lax, Henry, Peter- sets. One set consists of discrete isolated solutions with mann and many others. Traditionally, the ST linewidth is smaller imaginary frequency(damping); the number of such written in terms of the cavity decay rate, gc, as solutions does not change with increasingL. The second set 2 w0 g c has larger damping, and gets denser and denser, forming a dwST = nsp,() 13 psuedo-continuum asL is increased. Thefirst set corresponds 2P precisely to the complex resonance frequencies of the passive where w0 is the lasing frequency andP is the total output cavity, as can be confirmed by plotting the known resonances power. Here we have chosen to include in the ST formula the fi of the cavity given by( 7), shown in gure 7(c). The spatial ‘incomplete’ inversion factor,n sp, which is sometimes treated functions corresponding to the passive resonances are iden- as a correction to it( nsp  1 measures the degree of inversion tical to the resonant solutions of the infinite system without at the laser frequency and tends to unity when the lasing damping inside the cavity, as there is no loss there, and the transition is completely inverted, the limit assumed by boundary condition at the mirror is the same. Outside the Schawlow and Townes in their original work)[ 63]. mirror these solutions experience a damping not present in the As discussed extensively above, semiclassical laser the- infinite system and hence decay slowly as they propagate ory neglects quantum effects such as spontaneous emission outwards. On the scale shown in the top panel offigure 8 one and leads to a delta function linewidth. In order to go beyond sees only that the outgoing boundary condition is satisfied and this one needs to treat all threefields, the electromagnetic not the weak decay. In contrast, the spatial solutions field, the polarization, and the inversion as quantum operators corresponding to the dense set of damped modes of the and consider their higher moments. The laser linewidth will Universe can be seen in the bottom panel offigure 8; they be obtained from the correlation function of the electricfield form standing waves outside the cavity, with small leakage near the lasing frequency. It is well known that the effects of into the cavity, and do not satisfy the purely outgoing con- these quantumfluctuations can be captured by adding ditions. Hence simply discarding the latter and keeping the appropriate Langevin(random) noise terms to the equations first set is equivalent tofinding the Sommerfeld solutions of motion for the threefields[ 64]. These noise terms will be of the passive problem without damping, in the vicinity of related to the dissipation and amplification in the system by the cavity. the appropriate FD relations, and are then typically combined

10 Phys. Scr. 91(2016) 013003 Invited Comment tofind the total effective noise upon the electricfield is the inversion of theαth atom, which would relax to the fi ¨ ¨ steady-state value ofd 0 in the absence of an electric eld at ⎣⎡e´´E()(), t +c , txEx⎦⎤ =- 4p + ˜tot,() 14 P F fi the rate g. We have also simpli ed by assuming the lasing in which F˜tot ()x, t is the total effective Langevin noise driving signal to be on resonance with the atomic transition, w0w a=, thefield, and contains contributions from the noise in all three and that the dipole matrix element andfield are co- fields. This was one of the questions raised in the original linear, ·qqY0 =Y0. work by Lang et al. It is now understood that semiclassical Thefirst term in( 17) corresponds to the contribution theory violates these FD relations but they are restored by from thermal and zero pointfluctuations from outside the including the Langevin terms cavity to thefield inside the cavity; as such it is controlled by fi In this section we will rst provide a new interpretation the decay rate of the cavity, gc. The second and third terms of the ST linewidth, arguing that it is more natural to express stem from spontaneous emission events leading to the this quantity in terms of the rate of spontaneous emission dephasing of the atomic polarization, and these terms depend rather than the passive cavity decay rate, as the former is on the parameters of the gain medium. For optical fre- directly connected to the physical process which yields the quencies, the number of thermal photons present is negligible, phase diffusion. Subsequently, we will provide an overview so the directfieldfluctuations do not have any effect on of the recent work by Pick et al[ 17], which provides the most resulting linewidth. Moreover, for most lasers(class A and B) complete treatment of the quantum limited laser linewidth [65], g  g^, so that only the second term in( 17) remains, known, and has been shown to agree quantitatively withfirst 1 á † ¢ ñ =d - ¢ 2 + principles FDTD simulations[ 18]. Ftot () t Ftot t t t å∣ga ∣()()1 da . 18 ()() g^ a

7.1. Novel interpretation of ST linewidth Assuming that the atoms are uniformly distributed over the cavity, the sum over the atoms in the cavity can be converted It seems initially puzzling that the noise due to spontaneous into an integral emission in the gain medium should lead to the standard ST formula, which is expressed only in terms of the passive ¢ 2 † 2∣g ∣ áFtot () t Ftot t¢ ñ =d t - t¢ cavity linewidth, gc, the total output power, and the lasing ()() g frequency. One would expect the spontaneous emission rate ^ to appear as a fundamental quantity, not the cavity decay rate; ⎛ N2 () x ⎞ 2 ´ ò ⎜ ⎟D()()() xY0 xd x , 19 and here we will show that the ST linewidth can(and perhaps ⎝ N2()() x- N 1 x ⎠ should) be written in an equivalent form where this is the case. where N2 ()x (N1(x)) is the density of atoms in the upper ¢ The total Langevin force, Ftot () t , in units of the quantized (lower) lasing level, and ga gY0() xa =. Equation( 19) now fi expresses the Langevin force in terms of the incomplete eld amplitude operator, F˜totp wYF tot=()() t 0 xi2, has a direct contribution from the amplitudefluctuations of the inversion factor[ 63], fi fl eld, Fa () t , due to quantum uctuations from outside the N2 () x cavity, but also has indirect contributions from Langevin nsp () x = .() 20 N2()() x- N 1 x forces driving the atomic polarization and inversion,F p and Fd respectively[ 64], Traditionally at this step the atomic populations are ⎛ 1 ⎞ approximated to be uniform in space which clamps the Ftot()() t= F a t + ⎜ ⎟ åga Fp ()() t , 15 ⎝ w0- w a + i g^ ⎠ a inversion to its value at the lasing threshold, » = ()thr [64]. In this approximation, the integral where the summation runs over all of the atoms in the cavity D()x D0 D0 ()thr in( 19) becomes just D nsp, and can be related back to the at locationsx α and 0 cavity decay rate by 2pw 0 thr 2 gaxi q ·()Y0= a ,() 16 D() g¢  0 gc = ,() 21 g 2 is the dipole coupling element for the quantized electricfield. ^ q is the dipole matrix element for the lasing transition, and which states that at the lasing threshold, gain must equal loss. fi 2 0()x Yis the normalized mode pro le, ò ∣Y0= ∣ xd 1, of the Using this approximation contributes one factor of gc in the electricfield at frequency w0. The correlation function for the standard ST linewidth formula,( 13). However, this is a rather total effective Langevin force can then be calculated[ 64] as, crude approximation. Due to spatial hole-burning the atomic populations generally vary substantially in space, and as a † ¢ ¢ áFtot ()( t Ftot ()() tñ =d t - t gcn th consequence the overall inversion is not clamped at the 2 g ⎡ g ⎤⎞ threshold value. +∣a ∣ g 1 +d + d- d , 17 å 2 ⎢ ^ ()(a 0 a )⎥⎟ () Using( 21) obscures the fact that the physical origin of a g^ ⎣ 2 ⎦⎠ the noise is spontaneous emission in the gain medium. It then in whichnth is the number of thermal photons at the lasing is more natural to relate the linewidth to the rate of sponta- frequency, g^ is the dephasing rate of the polarization, andd α neous emission, gsp, which can be calculated from the rate of

11 Phys. Scr. 91(2016) 013003 Invited Comment stimulated emission, gSE [15], as 7.2. N-SALT laser linewidth 2 gSE ()x 2∣∣g¢ 2 The previous derivation of the laser linewidth is useful for its gsp ()x = = Y0()()x , 22 2np g^ relative simplicity, but it fails to take into account a few well known corrections, such as those stemming from relaxation wheren is the number of photons in the steady-state lasing p oscillations of the gain medium(α factor)[ 23, 67], the field, and is related to the intensity by I p w=n2. Thus, 0 0 p openness of the cavity(Petermann factor)[ 56, 57, 68–70], we can rewrite( 19) in terms of the rate of spontaneous and dispersion(bad-cavity factor)[ 63, 71–74]. To include emission, keeping the full space dependence of the problem, these effects, Pick et al instead begin with the wave equation, as (14), and assume that the known SALT lasing mode solutions † ¢ ¢ to the semiclassical problem remains relatively unchanged in áFtot () t Ftot tñ =d t - t ò gsp()()()()x D x n sp x xd. 23 ()() the presence of noise[ 17], expanding the electricfield as

fi wm-ti As noted above, the coef cient of the Langevin noise ()()()tå am tExY m= x,e.() 28 determines the laser linewidth by standard manipulations. m Using the more correct form of this coefficient above leads to a quantum-limited linewidth, Although the total noise source F˜tot ()x, t is broadband, only the noise near the lasing frequency is strongly amplified, and ò gsp()()()x D x n sp xd x thus the total noise source is also expanded as dw = .() 24 2n p iw-t ˜tot ()()t= å m tFxm F x,,e.() 29 Typically, the next step is to relate this to the total output m power of the cavity via This allows for one to write down a set of coupled mode PSTwg c= n, p 25() equations for the modal amplitudes

2 2 which is also only an approximate relation, neglecting spatial a˙m()=å Cmn a n t - a n0 am ()()() t+ fm t, 30 variations of the intensity. Using both of the approximations n () mentioned above and some other simplifications valid for highQ cavities(and neglecting the amplitude contribution that are dependent upon the projections of spontaneous contained in theα factor), one then arrives at the standard ST emission noise onto the corresponding lasing modes,f μ, and linewidth form,( 13). are coupled through the self-saturation and cross-saturation However, the actual power emitted can be calculated coefficients Cmn. These coefficients are related to the spatially using the true spatially varyingfields using Poynting’s theo- dependent parameters of the system through the Green’s rem[ 66], as function of( 14), such that for class A lasers

¶e w I0w 0 2 2()m 2 e=- Y -iwm 2 Ym dx P Im[]∣()0∣()x d x . 26 ò ¶ a 2p òc C n Cmn = ,() 31 ¶ 2 2 ⎣⎡w e() w ⎦⎤ Ymdx Using the specific form of the imaginary part of the dielectric òC ¶w wm function for a two-level gain medium(and using real units, not SALT units), the linewidth can then be written as iYm·() m tF d x òC fm () t = .() 32 w ¶ 2 2 0 ⎣⎡w e() w ⎦⎤ Ymdx dw = gsp()()()x D x n sp xd x ò ¶w wm 2P ò C ´ g x D xd x . 27 ò sp ()() () This same method is also applicable to class B lasers, however the modal response is no longer instantaneous and (32) becomes an integro-differential equation in time with Here we have derived a more physically intuitive and contributions from the recent history of the medium. The accurate form of the laser linewidth formula in which the main effect of the delayed response is to introduce relaxation critical role of spontaneous emission in the gain medium leaps oscillations into the spectrum. out:the linewidth is determined by the spatially varying, The total phase diffusion of the modal amplitudes can then * ¢ ¢ mode-dependent spontaneous emission rate and by the be calculated by considering áfm t fn tñ ~dmn d t - t()), ( ) ( steady-state saturated inversion in the presence of spatial which can be determined from the auto-correlation of the total hole-burning. Only if one makes the approximation of Langevin noise, and is given by the FD theorem in terms of ò gspD xd g c»does this reduce to the standard ST formula, Im[]c. The resulting N-SALT linewidth for a single mode and in making this substitution one tends to hide the true laser has the same general features as the simpler result of( 27), origin of the linewidth, which is spontaneous emission. but it now contains all of the known corrections discussed

12 Phys. Scr. 91(2016) 013003 Invited Comment above, and can be written as[ 17], for systems such as random lasers the difference can be pronounced due to strong line-pulling effects from the gain w0 2 dwNSALT- =1 + a˜ medium in such a lowQ cavity. Using the proper lasing mode 2P () profile also affects theα factor due to its dependence uponC. w2 Im e YY2 dx Im e N2 ()x 2d x 0 ò[][]0 ò D x 0 Thus, despite the fact that the N-SALT linewidth prediction ´ () , 33 2 () and numerical confirmation have yielded no qualitatively new 2 ⎛ w0 de ⎞ Y ⎜e + ⎟dx physical effects in the behavior of the laser linewidth, the ò 0 2 ∣w0 ⎝ dw ⎠ N-SALT result represents a significant development in laser where Y0 is the normalized SALT lasing mode, and physics, unifying these previously disparate theories. ˜ [][]CCImais= a generalization Re of the Lax–Henryα factor. Very recently this formula has been quantitatively confirmed usingfirst-principles numerical simulations[ 18] 8. Conclusion for laser parameters where it predicts significant departures from the standard(corrected) form of the ST formula. Similar In summary we have presented a scattering framework for to the discussion in section 7.1, note that in( 33) the cavity semiclassical laser theory and showed how precisely it leads decay rate does not naturally appear, while the imaginary part to zero laser linewidth, due to the discreteness of the poles of of the response of the gain medium plays a critical role. the scattering matrix. This approach treats the full system as The most striking feature of the N-SALT linewidth infinite from the beginning and does not introduce afinite formula is that it contains all of the different linewidth cor- ‘Universe’ and impose a limiting procedure and damping to rections due openness, incomplete inversion and gain dis- recover the infinite system result, although such an approach persion in an interconnected form. Prior to the development is a valid alternative method if the LAP is observed. We were of SALT-based linewidth theories[ 17, 50, 51], the most inspired to analyze this question by the classic work of Lang general form of the laser linewidth known was based upon the et al which introduced the notion of‘modes of the Universe’. original ST result multiplied by all of these corrections Our scattering point of view underlies recently developed fi independently(the incomplete inversion factor,n sp, Peter- ab initio laser theory(SALT), which leads to ef cient com- mann factor,K, the bad-cavity factor,B), as well as theα putational tools for calculating steady-state lasing properties. factor correction: A generalization of the ab initio theory to treat noise allows accurate calculation of the intrinsic linewidth due to quantum 2 w0 g c 2 fluctuations when one goes beyond semiclassical laser theory. dwST nsp KB 1a=+ .() 34 2P () We emphasize that the quantum-limited linewidth depends To illustrate this difference, consider the Petermann factor directly on the properties of the gain medium in the active originally derived as[ 68], lasing state, and it may be useful to write it in a form which does not involve the passive cavity decay rate, but rather 2 2 emphasizes the spontaneous emission rate. f0 ()xd x òC K = ,() 35 2 f0 ()xd x òC Acknowledgments fi in which f0 is the passive cavity mode pro le. These factors of the mode profile are all present in( 33)(corrected to the We acknowledge partial support by NSF grant DMR- 1307632. ADS thanks Marlan Scully for drawing our atten- lasing mode profiles Y0), but do not exist in isolation. Rather, thefield profiles serve as the weighting function in the tion to the work of Lang et al. We thank Steven Johnson and integrals over the other correction factors within the cavity, Adi Pick for very useful conversations about the laser line- such as the bad-cavity factor[ 63, 71–74], width, noise effects, Perfectly Matched Layers and the lim- iting absorption principle. We would also like to thank Li Ge 2 ⎛ ⎞2 for stimulating discussions on methods forfinding the poles ⎜ g^ ⎟ 1 B = = ,() 36 of theS-matrix for two-dimensional cavities. ⎜ gc ⎟ w0 ¶ e ⎜ g^ + ⎟ 1 + 2 ¶w ⎝ 2 ⎠ w0 as well as what would otherwise be the cavity decay rate, Appendix A. Review of SALT gc w0 Im[] e e». Hence all the space dependence which arises from nonlinear spatial hole-burning is taken into In this appendix we will provide a brief review of SALT to account within the line width formula for thefirst time. The assist in understanding the discussion in section 4, and the interconnected nature of these effects means that except for results presented in section 5. However, the interested reader certain limits, we cannot consider these effects to be separate. is referred to works dedicated to the SALT and the CF basis The second improvement that N-SALT provides over [9, 11, 12], and its elaborations to treat amplified modes[ 16], previous theories is its use of the lasing mode profile rather and complex gain media[ 14, 15]. The SALT equationsfind than the passive cavity mode profile. While the difference the steady-state solutions to the semiclassical Maxwell–Bloch between these two profiles is negligible for high-Q cavities, equations, for which the wave equation was given above in

13 Phys. Scr. 91(2016) 013003 Invited Comment

(2). For this brief review we will only consider the simplest in terms ofD and Ys, and inserted into the wave equation as case of two-level atomic gain media, which leads to the Bloch ⎡ ⎛ 2 ⎞ ⎤ equations for the evolution of the atomic polarization: 4p∣ q ∣()D x 2 ⎢´´-⎜ec ()x + ⎟ks ⎥Ys ()x inversion are ⎣⎢ ⎝  ()ws - wa + i g^ ⎠ ⎦⎥ + + id = 0, ¶tPg () x,t = -g^ + i wa Pg - (·)()qE q*, A.1 ()  ()A.5 2 - + tD()(), tg¶=- D Dxx0 -g g -·(). A.2 -PPEwhich explicitly identifies the gain due to the inversion of the () i () atomic medium, and in whichθ has been taken to be collinear to the electricfield. Likewise, using( A.2), we can solve For an electricfield with a single frequency component, directly for the inversion as or operating in the stable multi-mode regime, we can expand fi D x the electric eld over itsN constituent frequency components D = 0 () ()x N ,() A.6 2 as 4 ∣q ∣ 2 1 +2 å GsY s N  g^ g s + - w ti ()()tExåYs= s x,() e , A.3 s in which the denominator contains the effects of gain- where E+ is the positive frequency component of the electric saturation due to the electricfield within the cavity, and fi + G =g2 w - w2 + g2 is the Lorentzian gain curve. eld, E = E2Re]. In[ this expansion we make no distinc- s^ (() s a ^) tion between lasing modes, which are associated with the Together,( A.5) and( A.6) form the fundamental SALT poles of the scattering matrix, and amplified modes due to equations(for the two level gain case):N self-consistent, injected signals, which are externally generated. differential equations for the unknown spatial profiles of theN The critical assumption in SALT is the stationary modes in the electricfield atN unknown frequencies, coupled inversion approximation(SIA), which states that ¶tD0= , through the nonlinear spatial hole-burning in the inversion. even when multiple frequencies in the electricfield are pre- These are to be solved with outgoing Sommerfeld sent; i.e. the populations do not respond to beating terms boundary conditions at infinity, which can only be satisfied between the modes, that would lead to four wave mixing and at discrete frequencies. The SALT equations also reveal the frequency comb generation. This assumption requires that the natural units for the electricfield and atomic inversion, 2 non-radiative relaxation time of the gain medium is much ESALT ∣q ∣() g^ g=E2, and DDSALT (∣∣p) q g^= 4. slower than these beating terms, g D, whereΔ is the The numerical solution of the SALT equations is per- fi minimum frequency difference D =ws - w n present in the formed by expanding the mode pro le functions over a basis set of frequencies comprising the electricfield, ,w and is { s } ()s wY= satisfied by nearly all microcavity lasers. In the parlance s ()()åaxn n s f x; ,() A.7 n developed for the study of temporal chaotic laser behavior, the SIA is valid for class A and B lasers, but not class C and then using a nonlinear solver tofind the expansion fi ()s [65, 75]. For class B lasers, which include most semi- coef cients an . There are two different basis sets that have conductor based devices, the mode spacing must also be well been used to solve the SALT equations. Thefirst, and more separated from the relaxation oscillation frequency, so as to developed, is based on a set of CF states[ 9, 12], which are avoid resonantly driving relaxationfluctuations which could defined as destabilize the multimode solution. The relaxation oscillation ⎡ 2⎤ fi ⎣-´´+ec ()()() + hn Fxx⎦ n k w = u x;() 0, A.8 frequency is wRO kg^~[75], where k gc=2 is the eld () decay rate of the cavity. For microcavities without an injected and satisfy the outgoing boundary condition at frequencyω at signal, kD , the mode spacing from the free spectral FSR the surface of last scattering. Here, hn ()w and un()x; ware the range of the cavity, so wRO FSRgD< FSR, andD as such, eigenvalue and eigenstates of the CF equation, and D = DFxxis the profile of the pump. A key aspect the SIA is still valid when g < DFSR [16]. Furthermore, 0()( 0 ) SALT has been rigorously tested using direct FDTD simu- of the SALT approach is that for any given frequency lations of the Maxwell–Bloch equations, with excellent (wavevector) the CF states form a complete biorthogonal quantitative agreement found[ 13–16]. basis for an arbitrary purely outgoing solution; hence they can As the SIA removes any time dependence from the expand any solution to the SALT equations. inversion, the polarization must now contain exactly the same Moreover, by observing the similarities between( A.8) fi frequency components as the electricfield and( A.5), it is clear that just above the rst lasing threshold where the spatial hole-burning in the inversion is negligible, N + - ws ti the TLM is given by a single CF basis state at the lasing g ()()tPx= å s px,() e . A.4 s frequency satisfying fi Using the multimode expansion for the electric eld,( A.3), g^D0 the polarization equation,( A.1), can be solved independently h() w = .() A.9 w- wa + i g^ for each frequency component, providing an expression for ps

14 Phys. Scr. 91(2016) 013003 Invited Comment externallyfixed, and the mixed boundary condition can be satisfied at any frequency.

Appendix B. Partially reflecting mirrors in SALT

Traditionally, SALT has been used to simulate semiconductor based microlasers, in which the optical confinement for one- dimensional systems is provided by Fresnel reflection due to the dielectric mismatch across the cleaved facets of the device. However, for even the materials with the largest index of refraction, the optical confinement due to Fresnel reflection is weak, R 0.36

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