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Abstract
Fiber lasers with compact cavity have numerous potential applications in sensing, communications, and medicine. Distributed feedback (DFB) rare-earth doped fiber lasers based on Bragg gratings with a phase shift are the most promising in this aspect. In this paper, we theoretically study such lasers and carry out a complex-frequency analysis of the DFB cavity modes. Our approach is based on the study of poles of open cavity response function and on the laser rate equations. An optimal defect position in the Bragg grating, which maximizes an output power towards one side, was found with this approach. We show that the optimal defect position depends on the pump power. At the pump level close to the lasing threshold, the defect should preferably appear in the middle of the grating to maximize the one-side output power. At higher pumping, the optimal position of the defect becomes asymmetric. We have found specific variables, which allow for determination of optimal defect position for a large variety of DFB laser configurations.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single-frequency fiber-optic laser systems with a narrow (about several kHz and less) lasing bandwidth [1–6] are very important element of modern photonics and fiber optics. They can find wide use in coherent communication lines [7], in quantum communications [8] and in optical sensing systems [9–11]. In this aspect, the most promising are distributed feedback lasers (DFB lasers) [3,12,13]. DFB fiber laser is a short (several centimeters) section of an active fiber. In the core of this fiber a Bragg grating with a structural defect in the form of a phase half-wave shift (π-shift) is formed [14–21]. Due to the presence of the π-shift, a high-Q cavity mode is formed, which favors the generation of a single longitudinal mode. Such lasers exhibit a low level of phase noise and a bandwidth of the emission spectrum as small as 10 kHz or less [14,22], which is especially important for high-precision sounding.
To achieve the highest performance of DFB lasers the accurate choice of the position of the π-shift is required. It is essential to differentiate between Raman lasers and rare-earth-doped (RE-doped) lasers when choosing the laser model. Raman lasers not only show different saturation mechanism but they require taking Kerr nonlinearities into account [23]. For both Raman [23,24] and RE-doped [25–28] lasers theoretical models have been developed. These models are based on the coupled mode theory for forward and backward propagating waves which interact with each other on the gratings [29].
These approaches were implemented to analyze the optimum defect position in a DFB fiber lasers. In was shown theoretically and experimentally that the displacement of the defect from the middle of the grating increases the one-side output power for both Raman [30,31] and Re-doped lasers [32,33]. Moderate asymmetry increases the output through the thinner mirror, but an excessive one reduces the Q-factor of the cavity due to radiation losses, thus an optimal position of the shift exists. DFB fiber lasers were previously studied with coupled mode theory [23–28], which allows to calculate an optimal position for almost every particular DFB fiber laser. However, this approach requires quite complex numerical simulation.
There is another approach for the problem based on the modal decomposition approach. This method is based on the determination of the open cavity modes combined with the laser rate equations [34]. Cavity modes can be calculated with the transfer matrix method [35]. The lasing threshold may be found in the linear approximation when the mode damping becomes equal to zero [36,37]. When the mode damping becomes negative, linear approach is no more valid, and effects of gain saturation and other nonlinear effects should be taken into account, which can be done by the use of laser rate equations.
In this paper we propose a method for the RE-doped fiber DFB laser optimization, which combines modal decomposition described by fiber T-matrix approach and laser rate equations. The proposed method allows fast computing of the optimum position of defect in RE-doped DFB fiber laser. We find parameters, which allow for general description of multiple DFB configurations. With the proposed approach we study the optimal position of the defect in fiber Bragg grating as a function of the pump power. We show that, at a pump level near the lasing threshold, the optimal position of the defect tends to the middle of the grating, while the asymmetry of the optimal position of the defect appears at higher pump power. We calculate the optimal positions of a defect in most common cases of DFB RE doped fiber lasers.
2. Mode structure of a Bragg grating cavity
Electromagnetic modes of a laser cavity based on a Bragg grating with the defect may be described by T-matrix formalism [38,39]. The T-matrix of a Bragg grating with a defect can be obtained as the product of three T-matrices corresponding to the left (${T_{left}}$) and right (${T_{right}}$) gratings and the defect (${T_{shift}}$): ${T_{full}} = {T_{right}}{T_{shift}}{T_{left}}$, (see Appendix A). Having calculated the matrix ${T_{full}} = \left( {\begin{array}{{cc}} {{T_{11}}}&{{T_{12}}}\\ {{T_{21}}}&{{T_{22}}} \end{array}} \right)$ in this way, we find the complex transmittance t and reflection r coefficients:
As an example, consider a Bragg grating with a structure length of $L = 12$ mm and an amplitude of the refractive index modulation $\Delta n = {10^{ - 4}}$. The resulting transmittance spectrum $T = {|t |^2}$ of the grating without defect reveals a band gap behavior (Fig. 1(a)). Addition of a defect into this grating creates a defect mode manifesting oneself as a transmission resonance (Fig. 1(b)). T-matrix approach to calculation of lasing frequencies of fiber lasers was applied, e.g., in Ref. [11]. There, an experimental frequencies of multimode laser with Fabry-Perot cavity were well described by T-matrix calculations. It is important that the cavity mode frequencies were calculated in the absence of pumping.
Here, we perform an accurate analysis of the complex-frequency electromagnetic modes of the fiber cavity. These frequencies may be found as poles of the transfer function of the system, in particular, of the reflectance and transmittance [40–43]. This procedure gives solutions, which have zero amplitudes for waves traveling towards the cavity and nonzero amplitudes for waves traveling away from the cavity. Such modes may be considered as nontrivial solutions without a source. For an open passive cavity such modes are leaky and, correspondingly, have complex frequencies. The imaginary part of the frequency corresponds to mode damping due to absorption and radiative losses. Gain competes with losses and, thus, changes imaginary part of frequency. In the case of fiber DFB, in accordance with Eq. (1), mode frequencies $\omega = \omega ^{\prime} + i\omega ^{\prime\prime}$ are determined by the condition
These frequencies are illustrated by a plot of $\log ({{{|{{T_{22}}({\omega^{\prime} + i\omega^{\prime\prime}} )} |}^2}} )={-} \log T$, which diverges at the condition (2) (see Fig. 1). The position of the pole determines the wavelength and quality factor of the cavity mode: $\lambda = 2\pi c/\omega ^{\prime}$, $Q = \omega ^{\prime}/({2\omega^{\prime\prime}} )$. Thus, the poles located near the real axes correspond to high-Q modes.
Fig. 1. Transmittance spectra (top figures) and transmittance on a logarithmic scale, $- \log T$, mapped at a complex frequency plane (bottom figures) of the Bragg grating with (a) and without (b) defect. Blue dots correspond to the poles of the transmittance, i.e., to the cavity modes.
Bragg grating without defect exhibits multiple poles located inside the transmission bands (Fig. 1(b)). These poles correspond to Fabry-Perot modes inside the grating. Bragg grating with defect demonstrates similar picture, but an additional pole appears inside the band gap (Fig. 1(a)). This pole corresponds to a defect mode, which creates the resonant peak in transmittance. The latter pole is located near the real axis, therefore, the Q-factor of defect-mode is significantly higher than that of Fabry-Perot modes.
3. Optimizing the position of the defect in the Bragg grating
Let us study a one-side output power of the laser with the cavity considered above as a function of the position of the defect. Lasing occurs at a defect mode, which exists in a band gap. There, the field decay in the Bragg mirror is characterized by a $\sim {e^{ - \kappa {L_i}}}$ factor (see Appendix A), with $\kappa = \pi \Delta n/\lambda$ [39] being a decay rate and ${L_i}$ (i = right, left) the length of the corresponding Bragg mirror. Since the fiber Bragg grating is “weak”, each individual element of the grating reflects a small amount of incident radiation, whereas the reflection at the boundary of the Bragg mirror can be neglected. Therefore, the one-side output power $P_i^{out}$ is proportional to $P_i^{out}\sim {e^{ - 2{L_i}\kappa }}$. The fraction of energy emitted in one direction (say, to the left) is related to the position of the defect as $\frac{{P_{left}^{out}}}{{P_{left}^{out} + P_{right}^{out}}} = \frac{{{e^{ - 2\kappa {L_{left}}}}}}{{{e^{ - 2\kappa {L_{left}}}} + {e^{ - 2\kappa {L_{right}}}}}}$.
To relate output power to the electromagnetic energy stored in the cavity, W, let us note that in a source-free cavity the energy decay is described by the imaginary part of frequency, $- dW/dt = 2\omega ^{\prime\prime}W$. Thus, cavity loss equals to $2\omega ^{\prime\prime}W$, where $\omega ^{\prime\prime}$ may be found from T-matrix analysis described above. The latter expression holds both in free cavity and in the presence of source. In fiber lasers, radiative losses usually dominate over the other loss channels such as absorption and scattering into cladding modes. Therefore, the total radiation power is determined by the total decay rate of the cavity: $P_{left}^{out} + P_{right}^{out} = 2\omega ^{\prime\prime}W$. Therefore,
It is convenient to normalize the radiation power in one direction by the same quantity for the system, in which the defect is placed in the middle of the grating (named below as “symmetric cavity”): $P_{left\,\,0}^{out} = P_{right\,\,0}^{out} = P_{\,0}^{out}$, for which Eq. (3) transforms to
where the “0” subscript is associated with the symmetric cavity. As a result, the ratio of the radiation powers in one direction from lasers with asymmetric and symmetric cavities is:The energy stored in the cavity ($W$, ${W_0}$) is determined by the lasing process. This may be analyzed with rate equations (see Appendix B) or with any other laser model. If the pump power is much higher than the lasing threshold the total output power is inverse proportional to the cavity loss [44] leading to $\omega ^{\prime\prime}W \approx {\omega ^{\prime\prime}_0}{W_0}$. Under this assumption optimum defect position defined as the maximum intensity in (5) tends to the edge of the gratings (${L_{left}} = 0$). However, when the defect tends to the gratings edge, radiative loss grows, so that the quality factor of the cavity mode drops significantly and the threshold pump increases. In the case of laser operation near the lasing threshold, the lasing condition may be violated when the defect is shifted and, as a result, the threshold pump grows above the value of an actual pump power.
Let us investigate the dependence of the optimal position of the defect on the pump power ${\cal P}$. Above the lasing threshold, the electromagnetic energy W stored in the cavity grows linearly with increasing pumping power, $W\sim {\cal P} - {{\cal P}_{th}}$ [44]. Let us propose a three-level gain medium (say, Er3+). Laser generation threshold is defined by the sum of two terms: ${{\cal P}_{th}} = {{\cal P}_G} + {{\cal P}_C}$. The first term, ${{\cal P}_G}$, is the power needed to compensate losses in the gain medium itself and make it amplifying. The second term, ${{\cal P}_C}$, is the power needed to compensate cavity losses, and it is proportional to the total cavity loss $\omega ^{\prime\prime}$. In the case of short cavities the pump power may be considered constant along the cavity. Then the value of ${P_G}$ is a characteristic of the gain medium. Thus, it is convenient to separate the cavity loss from the gain loss and introduce variables $P = {\cal P} - {{\cal P}_G}$ and ${P_{th}} = {{\cal P}_{th}} - {{\cal P}_G}$. With these variables, lasers based on different gain media are described in the same way. As shown in Appendix B, the threshold pumping is inversely proportional to the Q factor. Therefore, the thresholds for the lasers with arbitrary and symmetric defect positions are related as ${P_{th}} = ({{Q_0}/Q} ){P_{th0}}$, where the subscript “0” corresponds to the symmetric system. As result, the relation between the energies stored in asymmetric W and symmetric ${W_0}$ systems is
The value of $\omega ^{\prime\prime}$ is taken from the T-matrix calculations, and it implicitly depends on the position of the defect. As a result, a one-side output power normalized by the one-side power of a symmetric laser is equal to:
Note that the value of $\omega ^{\prime\prime}$ is taken from the T-matrix calculations. It implicitly depends on ${L_{left}}$ and ${L_{right}}$.
Let us analyze the results following from Eq. (7). Small shift of the defect from the grating center increases the one-side output power since the radiation is redistributed towards the side with the shorter mirror. This behavior is described by blue and yellow curves in Fig. 2 in the vicinity of ${L_{left}}/L = 0.5$. At larger shift, radiative loss increases, so that the Q-factor of the lasing mode decreases, which leads to a drop in the total output power (see green curve in Fig. 2). Therefore, an optimum defect position exists, which maximizes the one-side output power. This result is consistent with previous works [28,31].
Fig. 2. Lasing output power depending on the position of the defect normalized by that of the symmetric system. The output powers “to the left” and “to the right” vs the defect position are shown by blue and orange curves. The green curve is the total output power. The dashed line shows the symmetric laser output power in one direction. The grating length is $L = 12$ mm, refractive index modulation is $\Delta n = {10^{ - 4}}$, and the pumping power is $P/{P_{th0}} = 10$.
The optimal position of the defect depends on the length and strength of the grating. Dependence of the one-side output power on the grating length L and on the refractive index modulation $\Delta n$ demonstrates similar behavior: an increase in the reflection coefficient of the Bragg grating, the optimal position of the defect tends to the middle of the grating (Fig. 3). This is due to the fact that the pole position is determined by the product of $\kappa L = \pi ({\Delta nL} )/\lambda$. Thus it is possible to describe DFB lasers with different L and $\Delta n$ by a single variable $\kappa L$. As shown above another variable which determines the optimal defect position is the pump value.
Fig. 3. A normalized one-side output power depending on the position of the defect and the length of the grating (a) and on the strength of the grating (b). The pump power is significantly above the threshold: $P/{P_{th0}} = 10$.
Let us investigate how an optimal defect position depends on the pump power near the lasing threshold, $({P - {P_{th0}}} )/P < < 1$. In this case, the optimal position of the defect tends to the middle of the grating (Fig. 4). This is due to the fact that even a small increase in the cavity loss near the lasing threshold leads to its breakdown. With an increasing pump, $({P - {P_{th0}}} )/P\sim 1$, the maximum one-side output power is reached at a considerable shift of the optimized defect position from the center of the grating. Smaller grating length (or strength) leads to larger shift of the optimized defect position. This effect may be explained by higher sensitivity of the high-Q cavity to small shifts of a defect. When the cavity has high loss, the system is less affected by the small shifts of the defect position.
Fig. 4. A normalized one-side output power depending on the position of the defect and the pump intensity. The calculation was performed for different lengths of the Bragg grating: $L$ = 10 mm (a) and $L$ = 25 mm (b). Refractive index modulation in a fiber grating is $\Delta n = {10^{ - 4}}$.
Now let us find optimal defect position for variety of DFB laser configurations. For that purpose let us describe DFB laser in terms of its normalized pump power, $P/{P_{th0}}$, and the product of grating length and strength, $\kappa L$. The latter quantity may be expressed in terms of transmittance, $T = 1 - {\textrm{tanh}}^2({\kappa L} )$. In particular, the transmittance of the grating is usually measured directly during the fabrication process, which makes it a convenient measure of the grating strength. For the purpose of practical use let us consider the grating transmittance range to be –20 dB to –120 dB and the maximum pump power to be $P/{P_{th0}} = 100$. In this range, optimum defect positions vary from 20% to 49% of the grating’s length (Fig. 5).
Fig. 5. Optimal position of phase shift in the gratings with respect to the edge of the grating for typical grating strengths.
4. Conclusions and discussion
In the present paper, an optimal position of the defect in a Bragg grating in an optical fiber, at which the maximum value of the radiation power in one direction is reached, was investigated. An approach based on the poles of open cavities response function and rate laser equations was developed. Fiber gratings response function was calculated with use of transfer matrix in a slow varying amplitude approximation. The rate equations were used to find the generations threshold and lasing efficiency. An equation describing one-side output power was derived based on the developed approach. An optimal defect position in the Bragg grating, which maximizes the output power towards one side was found.
With an increase in the grating length, or with an increase in its strength, the optimal position of the defect tends to the middle of the grating. The influence of the pump power on the optimal position was investigated. It was shown that the optimal position of the defect depends on the ratio of the operating pump power to the threshold power. At low pump powers, close to the lasing threshold, the optimal position of the defect tends to the middle. With increasing pumping, the deviation of the optimal position increases.
It was shown that a large variety of DFB laser configurations may be described by two parameters: normalized pump power $P/{P_{th0}}$ and the product of gratings length and strength $\kappa L$. For the practical usage of our results we have calculated optimum defect position in a DFB fiber laser in a large range of parameters.
Let us consider limitations of the proposed method. Instead of solving a distributed (spatial) problem, we use a modal approach. The latter is much simpler but has a limited applicability. The limitations of the modal approach are the following.
- 1) In the multimode case, field decomposition in nonlinear Maxwell-Bloch equations is not straightforward, and quite complex approaches were elaborated (see, i.e., [34,45]). In our case, these difficulties do not occur, since the use of single-mode fiber excludes additional transverse modes, and DFB cavity spectrally separates a single longitudinal mode.
- 2) Nonlinear effects change the material properties, so that the modes calculated in a free cavity may differ from lasing modes. Different nonlinearities may occur.
- 3) Kerr nonlinearity in glass is very low, so that this kind of nonlinearity may influence dynamics of high-power fiber lasers and Raman lasers. Let us estimate parameter limitations imposed by Kerr nonlinearity on the lasing mode properties. One may expect some influence of Kerr nonlinearity on the DFB laser performance, if the refractive index change leads to significant change in phase, $\Delta \phi \sim \pi$. In this case, cavity mode will change. Conversely, for a cavity with an effective length ${L_{eff}}$, one may neglect Kerr nonlinearity at $\Delta \phi = 2\pi \Delta n{L_{eff}}/\lambda \ll \pi$. In a DFB fiber laser, cavity mode confinement is determined by the grating strength, ${L_{eff}}\sim 1/\kappa$. We can rewrite the $\Delta \phi$ estimation as
Now let us estimate maximum the pump power typical of RE-doped DFB laser with the efficiency of 10% and cavity quality factor of ${10^5}$. With the parameters ${n_2} = 3 \times {10^{ - 20}}{\textrm{m}^\textrm{2}}\textrm{/W}$ [31], ${S_{eff}} = {10^{ - 10}}{\textrm{m}^\textrm{2}}$, $\lambda = 1.5 \times {10^{ - 6}}\textrm{m}$, and $\kappa = 2 \times {10^2}{\textrm{m}^{{-1}}}$ [46] we get
Note that typical RE-doped DFB lasers efficiencies are about 1% and typical pump powers are 1 W or less [17,22,46–48].
- 1) Change in the permittivity of the gain medium due to pump is usually very low [40,43]. Especially, this is true for the rare-earth based media, where the gain does not exceed 1 cm−1, and corresponding change in the permittivity due to gain is below ${10^{ - 4}}.$ Therefore, the field distribution in the modes can be considered independent of gain and, particularly, can be calculated in a gain-free cavity.
- 2) Pump depletion in RE-doped DFB lasers was discussed in Ref. [46]. There, it was shown that the pump depletion can be neglected at strong enough grating, $\kappa > k^{\prime\prime}$, where $\kappa $ is the grating strength and $k^{\prime\prime}$ is the pump inverse propagation length determined by an absorption in the gain medium. Therefore, our approach is well established for short cavity lasers, in which a typical absorption length of RE-doped gain is about several centimeters, while a typical short cavity allows longitudinal electromagnetic field confinement on a sub-centimeter scale.
Appendix A. T-matrix approach for optical fibers
Consider a monochromatic electromagnetic wave traveling along the fiber core. Let the electric field described by the transverse distribution ${\textbf e}({x,y} )$ have longitudinal component of the wavevector $\beta $ and frequency $\omega $. Since the parameters of the fiber weakly vary along the fiber, we can introduce the slowly varying field amplitude $A(z )$. As a result, the electric field dependence of the traveling wave can be written as
When dealing with Bragg gratings, it is necessary to describe interaction of the two waves traveling in the opposite directions. Let us define these waves as: ${E_a} = A(z )\textrm{exp} ({ - i\omega t - i\beta z} )$, ${E_b} = B(z )\textrm{exp} ({ - i\omega t + i\beta z} )$. In the presence of the periodic modulation of the fiber core refractive index, these waves interact with each other [39]:
With the account of the boundary conditions one can find complex transmission t and r coefficients for the traveling wave amplitudes:
As a result,
With these coefficients one can find total reflected ($R = {|r |^2}$) and transmitted ($T = {|t |^2}$) energy.
Let us consider Т-matrix of the total Bragg grating with the phase shift. This matrix is a product of three Т-matrixes: ${T_{full}} = {T_{left}}{T_{shift}}{T_{right}}$, where ${T_{left}}$ and ${T_{right}}$ correspond to the left and right gratings. These matrixes can be evaluated as described above.
The Т-matrix of phase shift (${T_{shift}}$) corresponds to the forward and backward waves traveling without any interaction. As a result, off-diagonal terms of the matrix are equal to zero. A wave travelling the defect gets the phase shift. To define this phase shift, note that the resonance wavelength is ${\lambda _{Bragg}} = 2\Lambda n$. Therefore, the phase shift of the traveling wave equals 2π at the length of $2\Lambda $. As a result, $\Lambda /2$ shifts correspond to π/2 shift of the traveling wave, which means
With the obtained total Т-matrix one can find poles of the transfer function of the system. The position of the pole determines the wavelength and quality factor of the cavity mode: $\lambda = 2\pi c/\omega ^{\prime}$, $Q = \omega ^{\prime}/({2\omega^{\prime\prime}} )$.
Let us note that the substitution of $A(z )= A(0 )\textrm{exp} [{({ - i\Delta \beta + \tilde{\kappa }} )z} ]$, $B(z )= B(0 )\textrm{exp} [{({i\Delta \beta + \tilde{\kappa }} )z} ]$ into Eq. (11) gives $\tilde{\kappa } ={\pm} \sqrt {{\kappa ^2} - \Delta {\beta ^2}} $. Thus, the field dependence in the Bragg grating is exponential, as is stated in Section 3. The exponential field dependence inside the cavity is also demonstrated in numerical models, where nonlinear effects were taken into account [28].
Appendix B. Laser rate equations
Energy dissipation rate $\gamma$ is determined by the imaginary part of the mode frequency, $\gamma = 2\omega ^{\prime\prime}$. Equations for the laser with three-level gain medium may be written in a form [44,49]
where $n = W/\hbar \omega $ is a photon number in the lasing mode, D is an average population inversion of the working transition of the three-level gain medium (the upper level is supposed to be unpopulated because of fast 3→2 relaxation), which can assume the values in a region $- 1 \le D \le 1$, $N$ is the number of gain atoms in the cavity, $\Omega = \frac{{4\pi \omega {T_2}}}{\hbar }\frac{{\int {{{|{({{{\textbf d}_{12}}{{\textbf E}^ \ast }} )} |}^2}C({\textbf r} )dV} }}{{\int {\varepsilon {{|{\textbf E} |}^2}dV} }}$ is the interaction parameter of the electromagnetic mode with the gain medium, ${T_2}$ is a transverse relaxation time of the gain medium, C is a concentration of Er ions, ${\Gamma _1} = 1/{T_1}$ is an inverse longitudinal relaxation time. The pumping rate ${\Gamma _P} = \frac{{2{\sigma _P}\lambda {n_G}}}{{\hbar c}}P$ is proportional to the pumping power P, with ${\sigma _P}$, $\lambda $ and ${n_G}$ being the absorption cross section, transition wavelength and refractive index of the gain medium, respectively.The solution of Eqs. (14) and (15) above the lasing threshold is defined by the conditions $\dot{n} = 0$ and $\dot{D} = 0$:
The threshold pumping rate $\Gamma _P^{th}$ is determined by the condition $n = 0$:
Note that $\gamma /\Omega < < 1$, so that $\Gamma _P^{th} \approx {\Gamma _1}({1 + 2\gamma /\Omega } )$. Threshold pump $\Gamma _P^{th}$ is a sum of a constant part, ${\Gamma _1}$, and of the part $2{\Gamma _1}\gamma /\Omega $ proportional to the dissipation rate $\gamma$.
The energy stored in the cavity linearly depends on the pump rate and is proportional to the quality factor of the cavity $Q\sim 1/\gamma $: $W = \frac{{\hbar \omega N}}{\gamma }[{({1 - \gamma /\Omega } )({{\Gamma _P} - \Gamma _P^{th}} )} ]\approx \frac{{\hbar \omega N}}{\gamma }({{\Gamma _P} - \Gamma _P^{th}} )$.
Let us briefly consider the rate equations for the laser with four-level gain medium, which may be written in a form [44,49]
The stationary solution of these equations is $D = \gamma /\Omega $, $n = N[{{\Gamma _P}({{\gamma^{ - 1}} - {\Omega ^{ - 1}}} )- {\Gamma _1}({{\Omega ^{ - 1}}} )} ]$. In contrast to the three-level gain, the threshold pump of the laser based on the four-level gain is proportional to the dissipation rate $\gamma$: $\Gamma _P^{th} = {\Gamma _1}\frac{\gamma }{{\Omega - \gamma }} \approx {\Gamma _1}\left( {\frac{\gamma }{\Omega } + {{\left( {\frac{\gamma }{\Omega }} \right)}^2}} \right) \approx {\Gamma _1}\frac{\gamma }{\Omega }$. The energy stored in the cavity linearly depends on pump rate and inversely proportional to the dissipation rate of the cavity mode $W \approx \frac{{\hbar \omega N}}{\gamma }({{\Gamma _P} - \Gamma _P^{th}} )$ similar to that of laser with the three-level gain. This brings us to the idea of the similar description for three-level and four-level gain by redefining of the pump rate for the three-level gain (${\Gamma _P} \to {\Gamma _P} - {\Gamma _1}$ and $\Gamma _P^{th} \to \Gamma _P^{th} - {\Gamma _1}$).
Funding
Russian Science Foundation (20-72-10057).
Disclosures
The authors declare no conflicts of interest.
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