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Abstract
In this paper, an analysis of the nonlinear laser operation in an active medium made of a parity time (PT) symmetric structure placed in a Fabry-Perot (FP) resonator is demonstrated for the first time. The FP mirrors’ reflection coefficients and phases, the PT symmetric structure period, primitive cell number, and the gain and loss saturation effects are taken into account in a presented theoretical model. The modified transfer matrix method is used to obtain characteristics of laser output intensity. Numerical results show that the selection of the appropriate phase of the FP resonator’s mirrors makes it possible to obtain different levels of the output intensity. Moreover, for certain value of a ratio of the grating period to the operating wavelength, it is possible to obtain the bistability effect.
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1. Introduction
Parity time (PT) symmetric structures have been analyzed multiple times since the first paper introducing them came out in 1998 [1], where the authors have shown that even non-Hermitian Hamiltonians can exhibit entirely real spectra, as long as they respect the conditions of PT symmetry.
Such one-dimensional structures in optical domain were first investigated in 2007 [2], when the formalism suitable for describing coupled optical PT symmetric systems was introduced. They are built from the same amount of two artificial optical materials with different refractive indices, which satisfy the condition n*(‒z) = n(z) (the asterisk denotes complex conjugate). In general, the refractive index is a complex value n = nRe + jnIm. In the case of the analyzed structure, PT symmetry is created by nRe = nRe(‒z) and nIm = ‒nIm(‒z), where nIm < 0 means gain and nIm > 0 means loss. PT symmetric structures can be arranged in various configurations as: photonic crystals [3], parallel coupling like PT symmetry lattice [4], or optical waveguide networks [5,6]. These structures are examined, because of their very interesting properties such as: beam refraction [7], nonreciprocity of light propagation [7], unidirectional invisibility [8] and coherent perfect absorption [9].
The first PT laser was proposed in 2010 by S.Longhi [10]. It was a DFB structure, consisting of a uniform index grating with two homogeneous and symmetric gain and loss regions. Such device behaved as a laser oscillator and as a coherent perfect absorber (CPA) simultaneously. The next proposed PT laser structure was a system of two PT symmetric coupled dielectric multimode waveguides [11], where single-mode operation was achieved in a broad area multimode laser. PT symmetric coupled two-microring system allowed to achieve single-mode lasing for PT symmetry breaking [12] and allowed to combine the advantages of stable single-mode emission and multimode cavities. Single-mode lasing was experimentally demonstrated in [13], where the pumping conditions of waveguides were changed from entirely pumping to half pumping for PT symmetric stripe lasers. Moreover, single-mode lasing was also observed in the PT microring laser, which consists of bilayer structures arranged periodically in the azimuthal direction on top of the microring resonator [14]. Additionally, application of quantum dots in gain and loss regions, allowed to electrically control the gain and loss levels [15].
In 2016, a new PT DFB/DBR laser structure was proposed, where the real and imaginary parts of the complex refractive index were modulated [16]. Due to the modulated perturbation, such laser structure exhibits much stronger modal discrimination and provides better single-mode operation performance. A similar structure was introduced in [17] for a PT symmetric circular Bragg laser, where mode discrimination and lower gain threshold were observed. The different construction of a PT DFB laser and the experimental implementation were shown in [18]. There were two π/2 phase shifted Bragg gratings: dielectric (gain modulation) and metallic (loss modulation). This construction was compared with laser structures, containing only the index or gain gratings [19], and showed better performances.
The modal discrimination and single-mode operation were obtained in various PT ring laser configurations, such as: a microring laser consisting of two coupled rings and a bypass waveguide (containing a multimode interference coupler, a phase modulator, a semiconductor optical amplifier and a tunable coupler) [20,21]; a fiber ring composed of two cross-coupled fiber cavities with gain and loss [22]; a ternary microring system with equidistantly spaced cavities of loss, gain and neutral resonators [23]; and a single ring laser with an S-bend inside [24].
PT lasers with a multiple-cells-structure were investigated theoretically in [25] and experimentally in [26]. The authors demonstrated that such setup can act as an amplifier or an absorber. In both papers, the PT structures were excited by radiation from both sides and the output signals were outcoupled also from two sides of such structures. The feedback required by the laser was provided by the PT structure itself through modulation of the complex refractive index. However, the reflection of radiation from the ends of these structures was not investigated and the presence of an external resonator was not taken into account.
In paper [27], unlike the previous one, in the experimental part of such work, authors studied the laser diode with modulations of the real part of the effective index (by dielectric grating) and the imaginary part (by metallic grating) with a relative phase between them. The investigated laser provided very similar spectra on both sides and was only mildly sensitive on the phase between the mentioned gratings. In the theoretical part of such work, the threshold gain of DFB laser, with layers of high and low reflection superimposed on its ends, was determined using the linear transfer matrix tool. Authors showed the influence of the resonator mirrors on the threshold gain, and indicated the optimal resonator mirrors phases for which the threshold gain was the lowest.
Within this paper, an original nonlinear analysis of the laser operation in an active medium made of a PT symmetric structure placed in the Fabry-Perot (FP) resonator is presented. In particular, the influence of the FP mirror phase and reflection on a PT laser output intensity is investigated. Such structure is examined with the help of the modified transfer matrix method (appendix B in [28]), including the gain and loss saturation effects [29,30]. A nonlinear analysis for laser structure in a FP resonator is provided to numerically obtain an output intensity. The obtained characteristics demonstrate that the appropriate selection of the FP mirror phase leads to different levels of the output laser intensities and the bistability effect. Next section presents the theory describing a PT laser in the FP resonator. In section 3, the characteristics showing the nonlinear operation of such laser as a function of the ratio of the grating period to the operating wavelength, the PT primitive cell number and the FP mirror phase and reflection are drawn. Section 4 presents the conclusions.
2. Theoretical analysis
The investigated PT laser structure, composed of the integer number N of the PT primitive cells (composed of a gain layer and a loss layer), placed inside the FP resonator is presented in Fig. 1. An output wave leaves the resonator through a transmission mirror next to the gain or the loss layer, further referred to as setup 1 (Fig. 1(a)) or setup 2 (Fig. 1(b)) respectively. Their output intensities are denoted by Iout(g) and Iout(α), where superscript (g) corresponds to setup 1 and superscript (α) is used for setup 2.
Fig. 1. PT laser’s structure. Laser output from: (a) the gain layer – setup 1; (b) the loss layer – setup 2. (c) Diagram of the n-th PT primitive cell divided into Q sublayers.
Each cell is described by the refractive indices ng or nα for the gain or the loss layer respectively. The indices are expressed accordingly by:
The first parts, taking into account the gain and loss saturation effects, are defined as:
The length of the whole structure L is a product of N primitive cells and a period of a PT structure Λ. Such period equals Λ = a + b, where a and b are widths of the gain and the loss layer respectively. The Fabry-Perot resonator mirrors are described by the reflection and phase coefficients: R1({g,α}) and φ1({g,α}) for a totally reflective mirror, and R2({g,α}) and φ2({g,α}) for a transmission mirror, which reflectivity is lower than unity. To obtain a laser action, the amplitude and phase generation conditions in the resonator have to be met. To satisfy the amplitude condition, the amplitudes of incident and reflected waves on the totally reflective mirror should be equal. Moreover, to satisfy the phase condition, the phase difference of the incident and the reflected wave should be equal to the phase of this mirror.
To investigate the PT laser in the FP resonator, the modified transfer matrix method is used [29]. This method leads to obtaining a longitudinal distribution of the electric field’s complex amplitudes, which allows to numerically calculate the output intensity Iout({g,α}) as a function of the ratio of the grating period to the operating wavelength, the PT primitive cell number and the FP mirror phase and reflection.
In the considered model, the following assumptions are presented: the PT laser operates in a steady-state single mode, the lines of gain and absorption of the PT structure layers are homogenously broadened, and the lasing wavelength is tuned to the center of the gain/absorption line (in this case the spectral shape is unimportant and can be considered as infinitely broadband) making the push-pull effect negligible. For the simplicity of the presented laser model, the Kerr effect (which is weak in III-V compound semiconductors, but dominant in quantum well heterostructure and cannot be neglected [31]) as well as the dependance of the real part of the refractive index on the carrier density (affected by the saturation effects) are not taken into account. Incorporation of these nonlinear effects to the model is left for further work.
The longitudinal distribution of the electric field’s complex amplitudes and the saturation intensities modify the imaginary parts of the refractive indices in each layer. Therefore, each layer of primitive cells is divided into narrow stripes, as it was shown in paper [29], see Fig. 1(c). In each sublayer, the saturable imaginary parts of the refractive indices are:
Calculations of the PT laser output intensity for both setups are performed according to the procedure presented in the appendix. The results of the numerical evaluation of the output intensity for the ratio of the grating period to the operating wavelength, the PT primitive cell number and the FP mirror phase and reflection are presented in the next section.
3. Numerical results
The numerical analysis is done for the PT laser with the refractive index nr = 3.165 for a semiconductor material InP [32,33]. The operating wavelength is λ = 1.55 µm.
3.1 Selection of PT laser parameters
Investigation of the PT laser begins with the design of the structure, i.e. the PT grating’s period and phase of the transmission mirror are selected. For this purpose, using the compu-tational algorithm presented in the appendix with accuracy of the calculations Q = 10, the maximal output intensity Iout({g,α}) is analyzed as a function of a ratio of the grating period to the operating wavelength Λ/λ for: setup 1 and setup 2, the primitive cell number N = 10 and four phases of the transmission mirror φ2({g,α}) (0, 0.5π, π, 1.5π). The transmission mirror’s reflectivity equals R2({g,α}) = 0.99. The study of the intensity characteristics as a function of the ratio Λ/λ, makes it possible to observe the behavior of the PT laser structure regardless of the operating wavelength. In the current investigation, the following values have been set: the unsaturated imaginary part of the refractive index ni = 0.1 [14,29,34] (the indicated value facilitates the numerical analysis of the output intensity), the linear distributed losses ngl = nαl = 0 and the saturation intensities Igs = Iαs = 1 W/cm2.
Figure 2 shows the maximal output intensity Iout({g,α}) versus the ratio Λ/λ for setup 1 and setup 2, and four phases of the transmission mirror φ2({g,α}) for the primitive cell number N = 10. The obtained characteristics reveal periodic peaks of the output intensity on the Λ/λ axis resulting from properties of the Bragg grating (caused by Fresnel reflection at the boundaries of the PT structure’s layers) and Bragg resonance condition [29,35], i.e. the grating period of the investigated structure equals to an integer number of half-waves inside each primitive cell. These peaks have the same height regardless of the ratio Λ/λ for fixed values of the phase φ2({g,α}). Therefore, for further analysis, the smallest size (ratio Λ/λ) of the primitive cell is selected, which has peaks of the maximal output intensities in the analyzed characteristics. Moreover, the characteristics for setup 1 and setup 2 are almost the same for the corresponding values of phases φ2({g,α}) differing by π, respectively. Additionally, comparing the maximal output intensity values for all phases shows that the intensity is the highest for only one phase. For setup 1, it equals φ2(g) = 1.5π and for setup 2, it equals φ2(α) = 0.5π.
Fig. 2. Maximal output intensity Iout({g,α}) versus the ratio Λ/λ for setup 1 (a) and setup 2 (b), and various phases of the transmission mirror φ2({g,α}).
The exact arrangement of the first peaks of the maximal output intensity Iout({g,α}) in relation to Λ/λ is shown in Fig. 3, for the same parameters as in Fig. 2. The differences in the collocation of the peaks for all tested phases are shown for a reduced range of the ratio Λ/λ. For both setups, when φ2({g,α}) = 0 and φ2({g,α}) = 1π, the peaks of the maximal output intensity are symmetrically shifted in opposite directions from the highest Iout({g,α}) by the same ratio Λ/λ. When φ2(g) = 0.5π and φ2(α) = 1.5π, the characteristics show two peaks of intensity arranged symmetrically in relation to the highest one. Similar to the previous figure, the intensity Iout({g,α}) is the highest for only one phase, i.e. for setup 1 ‒ φ2(g) = 1.5π and for setup 2 ‒ φ2(α) = 0.5π.
Fig. 3. First peak of the maximal output intensity Iout({g,α}) versus the ratio Λ/λ for setup 1 (a) and setup 2 (b), for various phases of the transmission mirror φ2({g,α}).
In the next step of the investigation, the influence of primitive cells number on the laser output intensity is studied. Figure 4 presents the first peaks of the maximal output intensity Iout({g,α}) versus the ratio of the grating period to the operating wavelength Λ/λ for three values of the primitive cells number N = 10, 50, 100, and for φ2(g) = 1.5π and φ2(α) = 0.5π, respectively. When increasing the number N, the peaks of the maximal output intensity Iout({g,α}) become narrower and tend to overlap with the highest one. Thus, the PT laser structure becomes more selective and requires a more precise selection of Λ/λ to obtain a laser action. Moreover, these peaks have almost the same values for the corresponding N for both setups.
Fig. 4. First peak of the maximal output intensity Iout({g,α}) versus the ratio Λ/λ for setup 1 (a) and setup 2 (b), for various cells numbers N.
Further, the influence of the linear distributed losses and the unsaturated imaginary part of the refractive index on the laser output intensity are investigated. Figure 5 presents the highest peaks of the maximal output intensity as a function of the ratio Λ/λ for: one value of the primitive cells number N = 50, the saturation intensities Igs = Iαs = 1 W/cm2, the transmission mirror reflectivity R2({g,α}) = 0.99, phases of the transmission mirror φ2(g) = 1.5π in the case of setup 1 and φ2(α) = 0.5π for setup 2. Figures 5(a) and 5(b) show the maximal output intensity for four values of the linear distributed losses ngl = nαl (0, 0.0001, 0.001, 0.01), and Figs. 5(c) and 5(d) ‒ for three values of the unsaturated imaginary part of the refractive index ni (0.1, 0.01, 0.001).
Fig. 5. Maximal output intensity Iout({g,α}) versus the ratio Λ/λ for setup 1 (a,c) and setup 2 (b,d), for different values of losses ngl = nαl (a,b) and refractive index ni (c,d).
Introducing the linear distributed losses causes a change in the value of Iout({g,α}) peaks. An increase of such losses results in a decrease of the maximal output intensity for the same value of the ratio Λ/λ. However, an increase of the unsaturated imaginary part of the refractive index causes an increase of the maximal output intensity for the same value of the ratio Λ/λ. As a consequence, obtaining a laser action is possible only when the unsaturated imaginary part of the refractive index is greater than the linear distributed losses.
For further analysis, one peak of the maximal output intensity is selected in every setup. The influence of the transmission mirror reflectivity on the laser output intensity is examined to determine values of such reflectivity for which the maximum of laser intensity is obtained.
Figure 6 presents the maximal output intensity Iout({g,α}) and the totally reflective mirror phase φ1({g,α}) versus transmission mirror reflectivity R2({g,α}) for: three values of the primitive cells number N = 10, 50, 100, the ratio of the grating period to the operating wavelength Λ/λ = 0.158, the unsaturated imaginary part of refractive index ni = 0.1, the linear distributed losses ngl = nαl = 0.001, the saturation intensities Igs = Iαs = 1 W/cm2, and for both setups respectively. The characteristics of the output intensity are shown in Figs. 6(a) and 6(c), and the characteristics of the totally reflective mirror phase are shown in Figs. 6(b) and 6(d). Each curve in Figs. 6(a) and 6(c), in the case of both setups, has only one maximum of the output intensity Iout({g,α}). Table 1 presents a summary of these maxima, extended by the values of the maximal output intensities for structures composed of 500 and 1000 primitive cells.
Fig. 6. Output intensity Iout({g,α}) and totally reflective mirror phase φ1({g,α}) versus transmission mirror reflectivity R2({g,α}): (a,b) setup 1 for φ2(g) = 1.5π, (c,d) setup 2 for φ2(g) = 0.5π, for various primitive cells numbers N.
Table 1. Maximal output intensity versus transmission mirror parameters (for details see Fig. 6)
The maximal output intensity characteristics increase with increasing of the number N and each one exhibits a maximum in the presented range of the transmission mirror reflectivity. Moreover, these maxima shift towards lower values of R2({g,α}). Similarly, as in Fig. 4, an increase of the number N causes a rise of the output intensity Iout({g,α}), which corresponds to an increase of the volume of the active medium. Moreover, the values of the output intensity Iout({g,α}) are slightly higher for setup 1, where an output power leaves the PT laser’s structure through the gain layer (see Fig. 1). This is confirmed by the numerical results presented in Table 1. It was experimentally observed that emission from one side of the structure is preferred [19,27]. Additionally, the difference between maxima of the output intensity for both setups increases with the number of primitive cells N. This behavior is caused by the gain/loss layer next to the transmission mirror, which interacts with the increasing intensity of the wave coming from the inside of the laser. Therefore, the output intensity changes with details dictated by the field’s profile and the 1/N relative weight of the layer next to the output mirror.
The totally reflective mirror phase increases slightly with the increase of the transmission mirror reflectivity. Moreover, in order to obtain the laser action, each presented structure, that consists of different numbers of the primitive cells, requires different values of the totally reflective mirror phase. In the case of the smallest investigated primitive cells number N = 10, the laser action can only be obtained for transmission mirror reflectivity greater than 0.2.
Up to this moment, the ratio of the grating period to the operating wavelength, the PT primitive cell number and the FP mirror phase and reflection have been sought for achieving a laser action with the maximal output intensity, i.e. the highest possible output power.
3.2 Influence of FP mirrors on PT laser action
In further research, all possible output intensity Iout({g,α}) levels that can be reached in the PT structure will be investigated. Moreover, the PT structures with phases of the transmission mirror φ2(g) = 1.5π in the case of setup 1 and φ2(α) = 0.5π for setup 2 will be analyzed.
Figure 7 presents the output intensity Iout({g,α}) and the totally reflective mirror phase φ1({g,α}) versus the number N for: the transmission mirror reflectivity R2({g,α}) = 0.484 (see Table 1 and Fig. 6), the ratio of the grating period to the operating wavelength Λ/λ = 0.158, the unsaturated imaginary part of the refractive index ni = 0.1, the linear distributed losses ngl = nαl = 0.001, the saturation intensities Igs = Iαs = 1 W/cm2, and for both setups respectively. There are characteristics of the output intensity on the left side of Fig. 7, and characteristics of the totally reflective mirror phase on the right side.
Fig. 7. Output intensity Iout({g,α}) (a,c) and totally reflective mirror phase φ1({g,α}) (b,d) versus number N, for setup 1 (a,b) and for setup 2 (c,d).
In general, for a given set of parameters, all possible levels of the output intensity are shown on the characteristics. Only four values of the output intensity (further referred to as the italicized case) were obtained for the investigated range of the primitive cell number N. The first one of them is the maximal output intensity case 1. Successive intensities are getting smaller and have been numbered sequentially (Figs. 7(a) and 7(c)). These intensities are related to the totally reflective mirror phases φ1({g,α}), as shown in Figs. 7(b) and 7(d). Moreover, the phase characteristics φ1({g,α}) for setup 1 and setup 2 differ by π, respectively.
It is very interesting, that in each analyzed output intensity case, all PT laser’s parameters are the same, except the totally reflective mirror phase, which determines achieving the laser action and the level of the output intensity. There is a minimum primitive cell number Nsp (sp stands for “starting point”) that allows for the laser action. An increase of the primitive cell number N causes a rise of the active medium volume and, as a result, the output intensity increases. Moreover, with a rise of the number N, the laser action can be obtained for a larger number of the totally reflective mirror phases that is, different levels of the output intensity can be obtained for the same primitive cell number. This effect was observed for the first time.
The analysis of phase characteristics (Figs. 7(b) and 7(d)) reveals that small changes of the totally reflective mirror phase φ1({g,α}) cause significant changes of the output intensity Iout({g,α}). The obtained results show that the examined structure can be used as an optical switch, where change of the totally reflective mirror phase allows to switch the output intensity.
Finding a reason for such behavior requires an analysis of the field distribution in the laser cavity. Figure 8 shows the longitudinal field distribution of counter running waves in the PT laser’s structure (placed between two mirrors – see Fig. 1) for a structure that consists of N = 100 primitive cells, setup 1 and setup 2, and successive output intensity cases respectively. In setup 1, the blue line represents the amplitudes of the right-going waves |cN| and |aN|, and the red line – the left-going waves |dN| and |bN|. In setup 2, the green line represents the amplitudes of the right-going waves |aN| and |cN|, and the orange line – the left-going waves |bN| and |dN|. This field distribution is obtained for the following parameters: the reflectivity R2({g,α}) = 0.484, the ratio Λ/λ = 0.158, the unsaturated imaginary part of index ni = 0.1, the linear distributed losses ngl = nαl = 0.001, and the saturation intensities Igs = Iαs = 1 W/cm2.
Fig. 8. Longitudinal field distribution of counter running waves |aN|, |bN|, |cN|, |dN|, in a PT laser’s structure consists of N = 100 primitive cells for setup 1 (a,c,e,g) and setup 2 (b,d,f,h), for successive output intensity cases respectively.
The longitudinal field distribution for N = 0 indicates the amplitudes of electromagnetic waves on the totally reflective mirror, and for N = 100 – on the transmission mirror. Therefore, on the totally reflective mirror, the amplitudes |c0| and |d0| are equal to each other for setup 1, see Fig. 1(a), and amplitudes |a0| and |b0| – for setup 2, see Fig. 1(b). Such behavior of the amplitudes indicates the satisfaction of the amplitude generation condition. On the other hand, on the transmission mirror, the difference between the amplitudes |a100| and |b100| for setup 1, and |c100| and |d100| for setup 2, corresponds to the output intensity respectively. The values of output intensity are shown in Fig. 8. The output intensity values for setup 1 are greater than for setup 2 for the corresponding cases.
It can be observed that the longitudinal field distributions shown in Fig. 8 are different for each output intensity case. In the case 1, the amplitudes of counter running waves are amplified as they pass through the entire structure (like in a volume active medium). For other cases, the oscillations of the field distribution appear. As a result, the effective gain area is reduced and the output intensity decreases for the subsequent cases (which differ by the totally reflective mirror phase φ1({g,α})). Therefore, the phase of the totally reflective mirror determines the field distributions inside the PT laser’s resonator. Additionally, the proper setting of the phase of the totally reflective mirror allows to obtain the highest possible output intensity.
The next step of the analysis is to examine the dependence of all the achievable output intensities on the ratio of the grating period to the operating wavelength Λ/λ (close to the previously selected value Λ/λ = 0.158). Figure 9 shows the output intensity Iout({g,α}) and the totally reflective mirror phase φ1({g,α}) as a function of the ratio Λ/λ, for all possible output intensity cases (related to the longitudinal field distribution of counter running waves inside the laser cavity, see Fig. 8) achieved for both investigated laser setups.
Fig. 9. Output intensity Iout({g,α}) (a,c) and totally reflective mirror phase φ1({g,α}) (b,d) versus ratio Λ/λ, for successive output intensity cases, for setup 1 (a,b) and for setup 2 (c,d).
The investigated structures have the following parameters: the reflectivity R2({g,α}) = 0.484, the number of primitive cells N = 100, the unsaturated imaginary part of refractive index ni = 0.1, the linear distributed losses ngl = nαl = 0.001, the saturation intensities Igs = Iαs = 1 W/cm2, and the wavelength λ = 1.55 µm.
In Fig. 9, successive output intensity cases are marked with colors. The characteristics of the output intensity Iout({g,α}) are placed on the left side of the figure and are related to the totally reflective mirror phases φ1({g,α}) – placed on the right side.
In general, the ratio Λ/λ strongly affects the output intensity Iout({g,α}). For both setups (Figs. 9(a) and 9(c)), in the case 1 (marked with a black line), a decrease or increase of the ratio Λ/λ from selected value 0.158 causes a rapid decrease of Iout({g,α}). The opposite situation is observed for case 2 (marked with a red line), where changing the ratio Λ/λ causes an increase of Iout({g,α}). For other cases (marked with green and blue lines), a variation of the output intensity Iout({g,α}) is less significant and the values of Iout({g,α}) are much smaller than for previous output intensity cases.
The totally reflective mirror phases φ1({g,α}) (Figs. 9(b) and 9(d)) vary for different output intensity cases. Moreover, the phase condition of the laser action requires a different totally reflective mirror phase for both of the considered setups. It is worth noting that the characteristics of mirror phases φ1(g) and φ1(α) demonstrate a point reflection symmetry (rotation by 180 degrees). Additionally, for the ratio Λ/λ = 0.15798 (close to Λ/λ = 0.158), for both setups, the totally reflective mirror phases’ φ1({g,α}) characteristics intersect for two pairs of output intensity cases: 1 with 3 (marked with black and green lines) and 2 with 4 (marked with red and blue lines), which corresponds to a bistability effect. It means that for one value of the totally reflective mirror phase, the laser can generate two different output intensities, i.e. two different longitudinal field distributions simultaneously fulfill the phase condition of the laser action.
For the selected PT symmetric structure period, primitive cell number and the FP mirrors’ reflection coefficients and phases, changing the output intensity Iout({g,α}) is possible by switching the totally reflective mirror phases φ1({g,α}). Such behavior of the PT laser can be used in the photonic system as a switch.
4. Conclusions
This work shows the output intensity of the PT laser as a function of FP mirrors’ reflection coefficients and phases, the PT symmetric structure period, primitive cell number for two setups of such laser: when the output wave leaves the resonator through a transmission mirror next to the gain layer (setup 1), and – next to the loss layer (setup 2). The study began by analyzing the size of the PT primitive cell, looking for the maximal output intensity. The smallest possible size of the primitive cell, which is determined by the ratio of the grating period to the operating wavelength, was selected for further analysis. The highest output intensity was achieved, when phases of the transmission mirror were 1.5π for setup 1 and 0.5π for setup 2. The maximal output intensity was slightly higher for setup 1 than for setup 2. Introducing the linear distributed losses caused diminishing of the value of the output intensity without changing the optimal size of a PT primitive cell. Moreover, increasing the unsaturated imaginary part of the refractive index caused an increase of the maximal output intensity. Then, the maximal output intensity was analyzed as a function of the transmission mirror reflectivity for both setups and for three values of the primitive cells number. With increasing the number of primitive cells, the transmission mirror reflectivity decreases. The optimal values of such reflection providing the maximal output intensity were determined and used in further analysis.
Next, all possible output intensity levels, that can be reached in the PT structure, were investigated for obtained values of transmission mirror reflectivity. The achieved output intensity levels were related to the totally reflective mirror phases, which were different for both setups. The increase of the number of PT primitive cells resulted in an increase of the number of possible output intensity levels characterized by decreasing values. Therefore, the examined structure can be used as an optical switch, where the totally reflective mirror phase change allows to switch the output intensity. Moreover, the analysis of the output intensity as a function of the ratio of the grating period to the operating wavelength reveals the bistability effect for some totally reflective mirror phases.
The model of a PT laser’s structure with an FP resonator presented in this study may open a door to next-generation laser sources for telecommunication systems and optical computing.
Appendix
In order to determine the output intensity of the PT laser for setup 1 (see Fig. 1(a)), the procedure scheme is shown below. For setup 2 (see Fig. 1(b)), the procedure is similar and the appropriate values will be given in parentheses. Equations with (a) refer to setup 1, and with (b) – setup 2. Steps of the procedure are the following:
- 1. Select the parameters of the examined laser structure: the number of primitive cells N, the number of sublayers Q (see Fig. 1(c)), the period of the PT structure Λ, the unsaturated imaginary part of the refractive index ni, the linear distributed losses ngl = nαl, the value of output intensity Iout(g) (Iout(α)), the values of saturation intensities Igs and Iαs, and the output mirror parameters: reflectivity R2(g) (R2(α)) and phase φ2(g) (φ2(α)) from ranges (0, 1) and (0, 2π), respectively.
- 2. Calculate the amplitudes aN (cN) and bN (dN) in the N-th gain layer next to the transmission mirror using the following formulas:
Equation (4(a) is correct for any value of arg(aN(Q)), Eq. (4(b)) is correct for any value of arg(cN(Q)). They are assumed to be 0 for the purpose of these calculations.
- 3. Using the amplitudes aN(Q) (cN(Q)) and bN(Q) (dN(Q)), determine the amplitudes aN(Q-1) (cN(Q-1)) and bN(Q-1) (dN(Q-1)) in the adjacent sublayer with the equations:$$\left[ {\begin{array}{c} {a_n^{({i - 1} )}}\\ {b_n^{({i - 1} )}} \end{array}} \right] = \left[ {\begin{array}{cc} {exp [{ - j{k_0}({{n_r} - jn_{gi}^{(i )} + j{n_{gl}}} ){a / Q}} ]}&0\\ 0&{exp [{j{k_0}({{n_r} - jn_{gi}^{(i )} + j{n_{gl}}} ){a / Q}} ]} \end{array}} \right]\left[ {\begin{array}{c} {a_n^{(i )}}\\ {b_n^{(i )}} \end{array}} \right], $$$$\left[ {\begin{array}{c} {c_n^{({i - 1} )}}\\ {d_n^{({i - 1} )}} \end{array}} \right] = \left[ {\begin{array}{cc} {exp [{ - j{k_0}({{n_r} + jn_{\alpha i}^{(i )} + j{n_{\alpha l}}} ){b / Q}} ]}&0\\ 0&{exp [{j{k_0}({{n_r} + jn_{\alpha i}^{(i )} + j{n_{\alpha l}}} ){b / Q}} ]} \end{array}} \right]\left[ {\begin{array}{c} {c_n^{(i )}}\\ {d_n^{(i )}} \end{array}} \right], $$where i = Q, n = N, k0 is the wave number in vacuum corresponding to the studied wavelength.
- 4. In the same way, using Eq. (5) for i = Q and n = N, continue calculating the amplitudes aN(Q-2) (cN(Q-2)) and bN(Q-2) (dN(Q-2)) in the next sublayer, until the amplitudes aN(0) (cN(0)) and bN(0) (dN(0)) on the left border of the gain (loss) layer are obtained. This way, the distribution of wave amplitudes in the mentioned layer will be iteratively calculated.
- 5. Using amplitudes aN(0) (cN(0)) and bN(0) (dN(0)), determine the amplitudes cN(Q) (aN(Q) and dN(Q) (bN(Q)) in the loss (gain) layer next to the border of the gain (loss) layer with the following equations for n = N:$$\left[ {\begin{array}{c} {c_n^{(Q )}}\\ {d_n^{(Q )}} \end{array}} \right] = \left[ {\begin{array}{cc} {{{({{n_\alpha } + {n_g}} )} / {({2{n_\alpha }} )}}}&{{{({{n_\alpha } - {n_g}} )} / {({2{n_\alpha }} )}}}\\ {{{({{n_\alpha } - {n_g}} )} / {({2{n_\alpha }} )}}}&{{{({{n_\alpha } + {n_g}} )} / {({2{n_\alpha }} )}}} \end{array}} \right]\left[ {\begin{array}{c} {a_n^{(0 )}}\\ {b_n^{(0 )}} \end{array}} \right], $$$$\left[ {\begin{array}{c} {a_n^{(Q )}}\\ {b_n^{(Q )}} \end{array}} \right] = \left[ {\begin{array}{cc} {{{({{n_g} + {n_\alpha }} )} / {({2{n_g}} )}}}&{{{({{n_g} - {n_\alpha }} )} / {({2{n_g}} )}}}\\ {{{({{n_g} - {n_\alpha }} )} / {({2{n_g}} )}}}&{{{({{n_g} + {n_\alpha }} )} / {({2{n_g}} )}}} \end{array}} \right]\left[ {\begin{array}{c} {c_n^{(0 )}}\\ {d_n^{(0 )}} \end{array}} \right]. $$
- 6. Next, similarly to point 3 of this procedure, calculate the amplitudes cN(Q-1) (aN(Q-1)) and dN(Q-1) (bN(Q-1)) in the adjacent sublayer of the loss (gain) layer with equation for i = Q and n = N:$$\left[ {\begin{array}{c} {c_n^{({i - 1} )}}\\ {d_n^{({i - 1} )}} \end{array}} \right] = \left[ {\begin{array}{cc} {exp [{ - j{k_0}({{n_r} + jn_{\alpha i}^{(i )} + j{n_{\alpha l}}} ){b / Q}} ]}&0\\ 0&{exp [{j{k_0}({{n_r} + jn_{\alpha i}^{(i )} + j{n_{\alpha l}}} ){b / Q}} ]} \end{array}} \right]\left[ {\begin{array}{c} {c_n^{(i )}}\\ {d_n^{(i )}} \end{array}} \right], $$$$\left[ {\begin{array}{c} {a_n^{({i - 1} )}}\\ {b_n^{({i - 1} )}} \end{array}} \right] = \left[ {\begin{array}{cc} {exp [{ - j{k_0}({{n_r} - jn_{gi}^{(i )} + j{n_{gl}}} ){a / Q}} ]}&0\\ 0&{exp [{j{k_0}({{n_r} - jn_{gi}^{(i )} + j{n_{gl}}} ){a / Q}} ]} \end{array}} \right]\left[ {\begin{array}{c} {a_n^{(i )}}\\ {b_n^{(i )}} \end{array}} \right]. $$
- 7. Using Eq. (7) for i = Q and n = N, successively determine the amplitudes cN(Q-2) (aN(Q-2)) and dN(Q-2) (bN(Q-2)) in the following sublayers until the amplitudes cN(0) (aN(0)) and dN(0) (cN(0)) are obtained on the left layer boundary.
- 8. If there is another gain (loss) layer to the left of the loss (gain) layer, using amplitudes cN(0) (aN(0)) and dN(0) (cN(0)), calculate the amplitudes aN-1(Q) (cN-1(Q)) and bN-1(Q) (dN-1(Q)) in the gain (loss) layer next to the border of the loss (gain) layer with equation for n = N:$$\left[ {\begin{array}{c} {a_{n - 1}^{(Q )}}\\ {b_{n - 1}^{(Q )}} \end{array}} \right] = \left[ {\begin{array}{cc} {{{({{n_g} + {n_\alpha }} )} / {({2{n_g}} )}}}&{{{({{n_g} - {n_\alpha }} )} / {({2{n_g}} )}}}\\ {{{({{n_g} - {n_\alpha }} )} / {({2{n_g}} )}}}&{{{({{n_g} + {n_\alpha }} )} / {({2{n_g}} )}}} \end{array}} \right]\left[ {\begin{array}{c} {c_n^{(0 )}}\\ {d_n^{(0 )}} \end{array}} \right], $$$$\left[ {\begin{array}{c} {c_{n - 1}^{(Q )}}\\ {d_{n - 1}^{(Q )}} \end{array}} \right] = \left[ {\begin{array}{cc} {{{({{n_\alpha } + {n_g}} )} / {({2{n_\alpha }} )}}}&{{{({{n_\alpha } - {n_g}} )} / {({2{n_\alpha }} )}}}\\ {{{({{n_\alpha } - {n_g}} )} / {({2{n_\alpha }} )}}}&{{{({{n_\alpha } + {n_g}} )} / {({2{n_\alpha }} )}}} \end{array}} \right]\left[ {\begin{array}{c} {a_n^{(0 )}}\\ {b_n^{(0 )}} \end{array}} \right]. $$
- 9. Repeat the calculations from points 3 to 8, until the left side of the first primitive cell (N = 1) is reached, i.e. the totally reflective mirror is adjacent to the left side of the loss (gain) layer.
- 10. Check the amplitude and phase conditions on this mirror. In the case of the amplitude condition, the wave amplitudes should satisfy equation:
However, in the case of the phase condition, the wave phases should meet:
If the amplitude and phase conditions are not met, procedure is performed for another value of the output intensity Iout(g) (Iout(α)).
This procedure is used to calculate the output intensity for different parameters of the investigated laser structure.
Funding
Politechnika Warszawska.
Acknowledgments
The authors wish to thank Miss Urszula Wyszyńska for checking linguistic correctness of the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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