Features of mode selection in a combined Fabry – Perot cavity with distributed feedback of counter-propagating waves

. An analysis is made of the possibility of isolating relatively high-Q modes or groups of such modes in low-Q combined Fabry-Perot cavities with distributed feedback of counter-propagating waves in order to ensure resonant interaction of the electromagnetic field with the polarization oscillations of an amplifying or absorbing medium filling the cavity. We considered two-level active media with homogeneous or inhomogeneous broadening of a spectral line, which can be both smaller or larger than the photon bandgap of a cavity. Particular attention is paid to the change in the well-known spectrum of polariton modes which takes place due to the transition from an absorbing to an amplifying medium under conditions that allow the realization of spontaneous or laser generation of the so-called superradiant modes.


Introduction
Distributed feedback (DFB) of counter-propagating waves, due to various periodic structures (photonic crystals) and accompanying the reflection of waves from the mirrors of Fabry-Perot cavities (FP), is widely used in various physical research and modern technologies, for example, semiconductor ones [1][2][3]. For example, it allows one to control the quality factor of modes in relatively low-Q cavities, which are required for the unique lasing regimes available in the presence of highly amplifying active media (both classical, such as electron beams, and quantum, such as semiconductor heterostructures) [4,5]. In this regard, the properties of the combined Fabry-Perot cavity with distributed feedback (FP-DFB) are not yet sufficiently studied, and in this paper we consider the features of the mode spectra of such a cavity both in the absence of an active medium (the so-called cold modes) and in the presence of an absorbing or amplifying medium with homogeneous or inhomogeneous broadening of the spectral line (the so-called hot modes).

Dispersion of the cold and hot modes
We confine ourselves to the analysis of a one-dimensional (1D) laser model with a twolevel active medium uniformly filling a low-Q combined Fabry-Perot cavity, along which / c  is the light velocity in a matrix of the active medium. Characteristic mode spectra, i.e. dependences of the mode growth rates  on frequency  , are found by solving the linearized Maxwell -Bloch equations [4,5] for a given population inversion of the working levels of active centers, p n const  : The shape of these spectra is largely dictated by the so-called cooperative frequency   is the inhomogeneous broadening of spectral line of the working transition of an active medium.
High-frequency oscillations of the electromagnetic field in an empty cavity are characterized by the photon lifetime E T , which determines the Q-factor of cold modes (see an example in Fig. 1, triangles). Since all modes have the same Q-factor in a pure Fabry-Perot cavity (Fig. 1a), a distributed feedback of counter-propagating waves may be used for the selection of various laser modes (Fig. 1b). A set of modes with different Q-factors may be created in such combined cavities. In the presence of an inverted active medium these cold modes turn into hot modes with different growth rates that allow one to obtain generation regimes corresponding to various dynamic classes of lasers. Referring for a detailed description of such mode selection to the review [5], we will formulate shortly the obtained qualitative conclusions. A single-mode monochromatic generation of the highestquality modes with In the sufficiently low-Q cavities, in which the lifetime of the oscillations of the highfrequency dipole moments of the active centers, 2 T , exceeds the lifetime of the photons, E T , the polarization oscillations of the active medium influence a lot the spectra and structure of the modes. In this case, the coupled waves of the electromagnetic field and oscillations of the dipoles of the active medium arise (polarization waves, or polaritons) [4][5][6][7] which have a dispersion different from the dispersion of electromagnetic waves if the inhomogeneous broadening of the spectral line of a medium does not exceed the cooperative frequency (cf. Figs. 2a and 2b, as the typical examples). Unlike the previously studied polariton modes for an absorbing medium with negative inversion, in this paper special attention is paid to polaritons in an inverted active medium (for example, heterolasers in which a polariton is formed by an exciton and a photon), where the dispersion becomes qualitatively different. Along with mode growth rates, it substantially depends on not only both indicated relaxation times E T and 2 T , but also on the ratio of the cooperative frequency in a medium with positive inversion and the following three frequencies: the intermode interval, / L  , the bandgap of the FP-DFB cavity, 2 , and the inhomogeneous broadening, 2 . Below two important examples are discussed to show how filling of a cavity with a resonant active medium changes the spectrum of modes (from cold to hot ones) depending on the magnitude of the inhomogeneous broadening of the spectral line of an active medium. If the inhomogeneous broadening of the spectral line of a medium is larger than the cooperative frequency, the lasing takes place for some of the most high-quality electromagnetic modes with the highest growth rates (Fig. 2a) [4,5]. This is true both for the class B lasers with low cooperative frequencies, and for the class C and D lasers with fairly high values of c  . The opposite situation appears at high densities or sufficiently large dipole moments of active centers, when the cooperative frequency becomes comparable to or larger than the polarization relaxation rate. In this case the ensemble of radiating active centers exhibits its collective properties and the polariton modes are lasing (Fig. 2b). In the latter case, the maximum growth rate of mode reaches the value c  (under a complete inversion of the populations of energy levels of an active medium).
If the inhomogeneous broadening dominates, 2 1     , the dispersion of electromagnetic modes is weakly changed by active centers, while the maximum growth rate decreases to the so-called effective cooperative frequency [4,5],  It is obvious that for obtaining collective spontaneous emission with the highest-power pulses (without strong oscillations) the active samples are needed with length on the order of the optimal length ~2 1 L   and ~2 L determined in the cases of strongly inhomogeneous and homogeneous broadenings by the effective cooperative length c B  and the cooperative length c B , respectively. In the above estimates we assume that the inversion specified by pumping ( p n ) is on the order of unity. Otherwise, it is necessary to take into account that maximal growth rates are obtained by the multiplication of /

Polariton modes of an active medium with homogeneous broadening
Let us discuss in more detail the features of the spectrum of polariton modes in a medium with a small inhomogeneous broadening of a spectral line, 1  , and a long relaxation time of the polarization, 2 E TT  , when the dispersion profiles of wave inan absorbing and amplifying media are qualitatively different [5,6]. In the case of an amplifying medium placed in a cavity with sufficiently high-quality modes, it turns out that only the corresponding polariton (but not electromagnetic) modes can have positive growth rates, i.e. be unstable, as it has already been demonstrated in Fig. 2b. According to [4,5], these modes are formed by the so-called polarization waves, the instability of which, due to their negative energy in the inverted active medium, can lead to collective spontaneous emission or lasing of one or several superradiant modes.
A typical example is shown in Fig. 3b for a Fabry-Perot cavity filled with an inhomogeneous broadened active medium, 0.1  . For a medium in the ground state (Fig.  3a,  1 p n  ), the linear dispersion profile of an empty cavity (Fig. 2a, circles; Fig. 3a, dots) splits into two branches of the polariton and electromagnetic modes. Upon excitation of the active medium by pumping ( 1 p n  ), the flat parts of these branches transform into an asymmetric curve of the polariton modes (Fig. 3b, circles), and the parts that close to the bisector are not changed practically and form a branch of damped electromagnetic modes (not shown in Fig. 3). When the threshold is slightly exceeded, stationary lasing will occur at the central mode ( 0  ), which is nearest to the center of the spectral line of the active medium, i.e. , and has a maximum growth rate. In Fig. 4, the typical spectral features of a combined FP-DFB cavity are compared to that of a pure Fabry-Perot cavity. The latter has a symmetric linear spectrum of cold modes (Fig. 4a), for which, in a certain range of pump powers, most of the hot modes around the highest-Q central one have a decay rate or a very low growth rate, that ensures stationary single-mode lasing in a reasonably wide range of inhomogeneous broadening of the spectral line of an active medium, up to the value ~0.1  . The combined FP-DFB cavity of the same Q-factor has an asymmetric spectrum of cold modes of various Q-factors with a bandgap of order 2 .
 In such a situation, a monochromatic lasing may be obtained by means of a non-zero detuning of the Bragg resonance from the frequency of the working transition, 0  , the proper choice of which ensures amplification of a mode with the highest Q-factor at the edge of the bandgap (Fig. 4b) and excludes noticeable excitation of the neighboring modes. a b Fig. 4. Spectra of the hot modes of (a) the pure Fabry-Perot cavity with length 2 L  , reflection factor 0.37 R  , 0  and (b) the combined FP-DFB cavity of the same length and Q-factor, caused by the combination of the following parameters: reflection factor 0.

Electromagnetic modes of an active medium with strong inhomogeneous broadening
In the case of the large inhomogeneous broadening, 2 1     , of a spectral line tuned to the Bragg resonance,    , the well-marked superfluorescence or superradiant lasing require a low-Q Bragg cavity with ~1 b or a related combined cavity (in the case of finite reflections R from the ends of a sample) with the considerable difference between the growth rates of the most unstable hot modes. By excluding cavities that are too long, Then, the mode selection preventing the dephasing of dipole oscillations of active centers during collective spontaneous emission is optimal if, for spectral regions with enhanced interaction of active centers with the electromagnetic field of this mode, the formally calculated cooperative frequency of active centers from a separated spectral region singled out by a particular cold mode will be about the relaxation rate 1

Conclusions
The analysis of the hot modes of the combined FP-DFB cavity filled with a resonant medium with a homogeneous or inhomogeneous broadening of a spectral line shows the possibility of efficient selection of the modes responsible for the interaction of the polarization oscillations of the medium with the electromagnetic field and allows one to judge about various lasing regimes, including the superradiant lasing, which depend on the properties of the active medium, pump level, phases ratio and reflection factors, formed by both distributed feedback inside the cavity and local mirrors on its edges. The obtained results are important for the development of new quantum and classical oscillators based on the active media demonstrating coherent properties.