Recently, the studies of open optical systems containing loss and gain attract increased attention. Although such systems are well-known for many years, the recent trend of non-Hermitian photonics provides the second breath to the investigations of lasers, waveguides, resonators, etc. This is not only due to a different language borrowed from quantum mechanics, but also because of a number of novel phenomena found in loss-gain structures. We name here only a few examples, such as the effects of \({\mathscr {P}}{\mathscr {T}}\) symmetry1,2,3 and exceptional points4,5. These effects include unidirectional invisibility6,7, sensors8,9 and gyroscopes10,11 with enhanced sensitivity, loss-induced12 and asymmetric13 lasing, novel single-mode14,15,16 and vortex17 lasers, coherent perfect absorbers18,19,20,21, and topological bulk-boundary correspondence22,23,24,25.

Disordered photonics is another spotlight of modern research26. It deals with light propagation in the presence of random fluctuations of the medium parameters such as refractive index or unit cell dimensions. The rich physics of such systems rooted in multiple scattering allows to realize a number of unusual features; the scattering properties of single particles and complex structures are well-studied in literature27,28,29. The most prominent feature of disordered systems is the Anderson localization of light30,31,32 appearing as a result of multi-path interference of waves scattering on random inclusions. Multiple scattering can also lead to the peculiar statistical properties of light violating usual diffusion (sub- and superdiffusion) like in optical Levy flights33,34,35. The situation becomes even more complicated when the interplay between disorder and nonlinearity occurs with the subsequent suppression of Anderson localization or promotion of diffusion36,37,38.

There is a recent trend combining together disorder and non-Hermiticity in the random systems with loss and gain. One of the main aims of such combinations is the enhancement of transmission which is usually strongly suppressed due to multiple scattering39. For example, this problem can be solved with the help of the concept of so-called constant-intensity waves in specially designed loss-gain profiles39,40,41,42,43. The non-Hermitian disorder due to random fluctuations of loss and gain can be a source of novel-type localized states44,45,46. Another interesting direction is the effects of asymmetric hopping on many-body localization predicted for dynamics of the quantum particles44,47. Such effects can have an optical analogue realized, e.g., due to nonreciprocity via utilization of magnetic or nonlinear materials. There is also a very active subfield of random lasing obtained in disordered amplifying media48,49,50,51. However, the dynamics of light interaction with structures containing both loss/gain and disorder are still poorly studied.

In this paper, we analyze the propagation of wave fronts through the disordered loss-gain non-Hermitian multilayer structures. The problem is aimed to be as realistic as possible: The fronts are the monochromatic waveforms having finite switching time, the loss and gain are due to resonant media and not merely a phenomenological imaginary part of permittivity, and the disorder can be controlled with external pump and change in time due to gain depletion and loss saturation. We have previously reported the study of short pulse propagation and localization in such media52 with the possibility to slow down or even stop the pulse. Here, we deal with the opposite case of continuous radiation with the emphasis on the transient process of steady-state establishment for the light intensities large enough to saturate the medium and give substantial transmission. This process can have different dynamics depending on the sharpness of the incident wave switching. In particular, we distinguish two regimes, when switching is slow (adiabatic) and fast (non-adiabatic). For these switching regimes, we show that the introduction of disorder changes the characteristic time of the transient process, whereas the resulting intensity of the signal is governed by saturation-limited input intensity. In particular, for the first time we demonstrate how the disorder can be used to switch on and off kinks at the output of the system. Thus, the non-Hermitian approach to disorder-induced control of propagation time, wave-front shape and transmitted intensity proposed in this paper opens new possibilities for ultrafast (picosecond or subnanosecond) multifunctional manipulation of optical signals.


Problem statement

Hereinafter, we consider a host dielectric doped with two-level atoms. Light propagation in such a medium is described by the well-known semiclassical Maxwell-Bloch equations for the dimensionless electric-field amplitude \(\Omega =(\mu /\hbar \omega ) E\) (normalized Rabi frequency), complex amplitude of the atomic polarization \(\rho\), and population difference between the ground and excited states w53,54,55:

$$\begin{aligned} \frac{d\rho }{d\tau }= & {} i l \Omega w + i \rho \delta - \gamma _2 \rho , \end{aligned}$$
$$\begin{aligned} \frac{dw}{d\tau }= & {} 2 i (l^* \Omega ^* \rho - \rho ^* l \Omega ) - \gamma _1 (w-1), \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 \Omega }{\partial \xi ^2}- & {} n_d^2 \frac{\partial ^2 \Omega }{\partial \tau ^2} + 2 i \frac{\partial \Omega }{\partial \xi } + 2 i n_d^2 \frac{\partial \Omega }{\partial \tau } + (n_d^2-1) \Omega = 3 \epsilon l \left( \frac{\partial ^2 \rho }{\partial \tau ^2} - 2 i \frac{\partial \rho }{\partial \tau } - \rho \right) , \end{aligned}$$

where \(\tau =\omega t\) and \(\xi =kz\) are the dimensionless time and distance, \(\mu\) is the dipole moment of the quantum transition, \(\hbar\) is the reduced Planck constant, \(\delta =\Delta \omega /\omega =(\omega _0-\omega )/\omega\) is the normalized frequency detuning, \(\omega\) is the carrier frequency, \(\omega _0\) is the frequency of the quantum transition, \(\gamma _{1}=1/(\omega T_{1})\) and \(\gamma _{2}=1/(\omega T_{2})\) are the normalized relaxation rates of population and polarization respectively, and \(T_1\) (\(T_2\)) is the longitudinal (transverse) relaxation time; \(\epsilon = \omega _L / \omega = 4 \pi \mu ^2 C/3 \hbar \omega\) is the light-matter coupling strength with C the density of two-level atoms and \(\omega _L\) the Lorentz frequency; \(l=(n_d^2+2)/3\) is the local-field enhancement factor due to the polarization of the host dielectric with refractive index \(n_d\) by the embedded two-level particles. We numerically solve Eqs. (1)–(3) using the finite-difference approach described in Refs.56,57 well-proven in solving such tasks.

The parameters used for calculations are characteristic, e.g., for semiconductor quantum dots as the active particles. We suppose the exact resonance (\(\delta =0\)). The host refractive index is \(n_d=1.5\). Incident monotonically switching cw field has the central wavelength \(\lambda =0.8\) \(\mu\)m and the envelope as follows,

$$\begin{aligned} \Omega (t)= \frac{\Omega _0}{1+e^{-(t-t_0)/t_p}}, \end{aligned}$$

where \(\Omega _0\) is the amplitude of the resulting cw field (plateau), \(t_p\) is the switching time, \(t_0=5t_p\) is the offset time.

The disorder is introduced to the system through the periodical random variations of the initial population difference \(w_0=w(t=0)\) along the light propagation direction. One can say that the population difference shows which part of two-level atoms in the medium are excited. We use here the two-valued quadratic model of disorder described in Ref.52. In fact, we have the multilayer structure with the initial population difference in the jth layer of the medium corresponding to the distance \((j-1) \delta L < z \le j \delta L\) given by

$$\begin{aligned} w^{(j)}_0 = \text {sgn} (1 - 2 r [(2\zeta _j-1)(r-1)+1]), \end{aligned}$$

where \(\zeta _j\) is the random number uniformly distributed in the range [0; 1], r is the parameter of the disorder strength, \(\text {sgn}\) is the sign function, and \(\delta L=\lambda /4\) is the layer thickness. When \(r=0\), we have the trivial case of purely absorbing medium (all \(w^{(j)}_0=1\), Fig. 1a). For \(r \gtrsim 0.3\), the gain layers with \(w^{(j)}_0=-1\) become possible (Fig. 1b,c). The case of the maximal disorder, \(r=1\), corresponds to the purely amplifying medium (all \(w^{(j)}_0=-1\), Fig. 1d). Thus, the parameter r not only governs deviation from the ordered case of pure loss, but also takes on the role of pumping strength resulting in appearance of gain. In general, the different layers of the structure are under different, randomly distributed pumping and can be lossy or gainy with a certain probability. This can be realized in a side-pumping scheme similar to that utilized in Ref.20 or with the adaptive-pumping approach58. Note that the quadratic model of disorder (Eq. 5) gives essentially the same results as the linear one52, but is more convenient for symmetric representation of gain and loss. The similar linear model was experimentally realized recently in the context of random lasing59.

Non-adiabatic fronts

In this section, we consider the case of non-adiabatically switching field (Eq. 4), when \(t_p \ll T_2\). In particular, we take the relaxation times \(T_1=1\) ns and \(T_2=0.1\) ns and the switching time \(t_p=5\) ps. The Lorentz frequency is \(\omega ^0_L = 10^{10}\) s\(^{-1}\). The full thickness of the medium is \(L=100 \lambda\). The final amplitude is \(\Omega _0=10 \gamma _2\).

Figure 2 shows the results of transmitted and reflected intensities calculations for different values of the disorder strength r. The initial population difference used in calculations is the same as in Fig. 1. Note that we consider here a single realization of disorder, since the observed features of interest for us are the same for different realizations at a certain level of disorder r: the specific oscillations of output intensity can differ, but the time needed for steady-state establishment and the final intensity are essentially the same for every realization. In the ordered system (\(r=0\), see Fig. 2a), which is the uniform resonantly absorbing medium, the time needed for the front to pass through (\(\sim 150 t_p\)) is much longer due to dispersion than the free propagation time (\(L n_d/c \sim 0.08 t_p\)) and, in fact, is governed by the relaxation time \(T_2=20 t_p\). The stationary response of the medium is established after its saturation and is seen from \(\sim 300 t_p\) on.

Increasing disorder strength r results in larger number of gain layers. As a result of stimulated emission in these layers, saturation needs less time and the stationary level is established faster. This is especially obvious for \(r=0.5\) (Fig. 2c). For even larger r, the number of gain layers becomes so large that amplified emission in the form of powerful bursts happens in the very first instants of time effectively returning most two-level particles to the ground level. Therefore, the propagation time of the front through the highly-amplifying system is similar to that in the case of purely absorbing medium (compare Fig. 2d and a).

Figure 1
figure 1

The example of initial population difference distributions for different disorder strengths r.

Figure 2
figure 2

Intensity profiles for the transmitted and reflected light in the case of incident non-adiabatic front. Different panels show the results for different disorder strengths r.

Figure 3
figure 3

Dynamics of population difference at the entrance of non-adiabatic front in the medium with different disorder strengths r.

Figure 4
figure 4

Distributions of population difference along the medium with different disorder strengths r at the time instant \(t=20 t_p\). The medium is excited by the non-adiabatic wave front.

Figure 5
figure 5

Intensity profiles for the transmitted and reflected light in the case of incident adiabatic front. Different panels show the results for different disorder strengths r.

The difference in the response time can be illustrated with the dynamics of population difference at the entrance of the structure shown in Fig. 3. This figure demonstrates the initial stage of medium saturation with the oscillations converging to the very low (almost zero) value. These are the well-known Rabi oscillations with the frequency given by the so-called Rabi frequency and, hence, dependent on the incident radiation amplitude. Note that for these oscillations to appear, the Rabi frequency should be larger than the medium relaxation rate that is easily satisfied in our calculations (\(\Omega _0=10 \gamma _2\)). It is seen that although the dynamics for the amplifying (\(r=1\)) and the absorbing medium (\(r=0\)) start from absolutely different levels (\(w=-1\) and \(w=1\), respectively), the oscillations of population difference very closely follow each other. On the contrary, for \(r=0.5\), we also start from \(w=-1\) (the first layer with gain), but the subsequent dynamics strongly differs from those for \(r=0\) and \(r=1\). This confirms that similarity of the transmitted intensity profiles in Fig. 2d,a is not accidental.

An additional corroboration of this conclusion is given in Fig. 4, which shows the distribution of population difference for different disorder strengths r at the time instant \(t=20 t_p\) corresponding to the initial stage of radiation interaction with the medium (before the steady-state is established). It is seen that we have the random variations at \(r=0.4\) (Fig. 4b) and \(r=0.5\) (Fig. 4c) which can be treated as the variations of population difference around zero value. In other words, the medium can be considered as saturated on average. The light-matter interaction is comparatively weak in this case (there is no loss and gain on average) resulting in the increased speed of signal propagation, especially for \(r=0.5\). On the contrary, the distributions for the purely absorbing (r = 0, Fig. 4a) and purely amplifying media (r = 1, Fig. 4d) are very similar, except for some local excitation due to random wanderings of light inside the medium. This confirms the rapid relaxation of amplifying medium due to spontaneous emission, so that the incident wave front propagates further in such effectively de-inverted medium giving the response analogous to that for \(r=0\).

We see from Fig. 2 that the stationary level of transmission is around \(51.5 \%\) and reflection is only about \(1.5 \%\). The rest (almost half the energy of the wave) is absorbed by the saturated medium. How realistic is it? Let us estimate the level of stationary population difference necessary for this value of absorption. In the steady-state approximation, the two-level medium can be described with the effective dielectric permittivity as follows62

$$\begin{aligned} \varepsilon _{eff} = \varepsilon '_{eff} + \varepsilon ''_{eff} = n_d^2 + \frac{K (-\delta + i \gamma _2)}{1 + |\Omega |^2/\Omega _{sat}^2}, \end{aligned}$$

where \(\Omega _{sat}^2 = \gamma _1 (\gamma _2^2 + \delta ^2)/4 l^2 \gamma _2\) is the saturation intensity, \(K=3 \omega _L l^2/\omega (\gamma _2^2 + \delta ^2)\). In the exact resonance (\(\delta =0\)), we have \(\varepsilon '_{eff} = n_d^2\) and \(\varepsilon ''_{eff} = 3 \omega _L T_2 w_{eff}\), where \(w_{eff}=(1 + |\Omega |^2/\Omega _{sat}^2)^{-1}\) is the sought-for effective population difference. Since \(\varepsilon _{eff} = (n + i \kappa )^2\), we can easily connect \(w_{eff}\) with the effective absorption coefficient \(\kappa\), which, in turn, can be linked to the transmission as \(T=\exp (-4 \pi \kappa L / \lambda )\). For the parameters used in our calculations, one should take \(w_{eff} \approx 5.5 \cdot 10^{-4}\) to reach the transmission of \(50 \%\). This value of effective population difference is close to zero (i.e. the medium is indeed saturated) and has the same order of magnitude as the stationary population difference obtained in our numerical calculations.

Note that in our estimation, we have neglected reflection which is indeed very low as seen in Fig. 2. This can be easily explained with a simple calculation of transmittion and reflection of light from a uniform layer with the effective permittivity having small imaginary part. Finally, we see from the expression \(w_{eff}=(1 + |\Omega |^2/\Omega _{sat}^2)^{-1}\) that it should depend on the incident wave intensity: increasing intensity, we can make absorption smaller due to saturation. In other words, the low-intensity waves are almost entirely absorbed, whereas the high-intensity ones are mostly transmitted. The effect of disorder on the propagation time can be conveniently observed at the intermediate intensities, not very low and not very high (e.g., \(\Omega _0=10 \gamma _2\) as in Fig. 2).

The features discussed in this section are also valid for two interacting wave fronts as shown in Supplementary Information.

Adiabatic fronts

In this section, we consider the case of adiabatically switching field (Eq. 4), when \(t_p \gg T_2\). In particular, we take the relaxation times \(T_1=1\) ns and \(T_2=0.1\) ps and the switching time \(t_p=30 T_2\). The Lorentz frequency is \(\omega ^0_L = 10^{11}\) s\(^{-1}\). The full thickness of the medium is \(L=200 \lambda\), which is long enough for a kink to form and can be traversed after \(Ln_d/c \sim 0.27 t_p\) in the case of dispersion-free medium. The final amplitude is \(\Omega _0=0.3 \gamma _2\), so that there are no Rabi oscillations and the light-matter interaction is quasi-stationary in this case60.

It is known that the adiabatically switching waveform (Eq. 4) undergoes self-steepening resulting in the kink (shock wave) formation after some distance passed through the resonantly absorbing medium6163. Such a kink is seen in Fig. 5a for the disorder strength \(r=0\), when all the layers are the same absorbing medium. For increased disorder, we still obtain the kink at the exit (Fig. 5b at \(r=0.4\)), although the system is now non-uniform and contains both loss and gain layers. Note the increased speed of this kink. For \(r=0.5\), when both loss and gain are equally probable, the kink formation is totally suppressed as shown in Fig. 5c. Moreover, the transmitted intensity grows ahead of the incident intensity (at early times \(t<4 t_p\)). These features are due to the large number of gain layers providing the proper amplification of the signal and fast saturation of the medium, so that the wave can almost freely propagate through the structure in later times. For even stronger disorders, the portion of gain layers becomes so large that even tiny impinging radiation rapidly stimulates a powerful burst of energy as seen in Fig. 5d for \(r=1\) (uniform gain medium). After this burst, the most part of the particles return to the ground state, so that the medium remains only weakly excited and gets saturated by the incident wave front. Although the kink is not formed in this case, the transmitted profile is closer to a kink than for \(r=0.5\), manifesting a characteristic offset time between input and output signals.

Thus, disorder gives us an opportunity to control the shape of the output wave switching on and off kink formation. As shown in Supplementary Information, a similar effect disorder has on a pair of interacting adiabatic fronts.


In summary, we have proposed a novel approach for wave-front velocity and shape governing in the multilayer structure with disorderly distributed resonant loss and gain. This concept involves functionalities of both non-Hermitian system and disordered multilayer providing several new optical effects. The two types of fronts were considered—the adiabatic (slowly switching) and non-adiabatic (rapidly switching) ones. These two cases have different dynamic features—kinks and Rabi oscillations, respectively,—which condition the propagation characteristics of the wave fronts. Introduction of disorder governed by external pumping results in a number of easily noticeable novel features, such as kink suppression and propagation time shortening. These disorder-induced effects together with the transmitted intensity determined by the medium saturation give us a tool to control transmission, shape and propagation time of the waves with the finite switching time. Such possibilities are extremely important for manipulation of light produced by realistic laser sources and can be used in optical switching and data processing among other applications.