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Stokes light induced modulation instability in high power continuous wave fiber amplifiers

Mingjian Yan, Yunhan Zheng, Huisheng Liang, Fangxin Li, Zhigang Han, and Rihong Zhu

Open Access Open Access

Abstract

In this paper, Stokes light induced modulation instability (MI) in high power continuous wave (CW) fiber amplifiers is observed. The investigation shows that the Stokes light generated by inter-modal four wave mixing (IMFWM) and stimulated Raman scattering (SRS) in high power fiber amplifiers can be modulated by the signal light through XPM and cause MI. Then, a sideband will be generated around the second-order Raman frequency shift, which is amplified by SRS and shown as a train of pulses in time domain. It is shown that the frequency shift of the sideband will be influenced by IMFWM and SRS. In addition, the sideband was found to be blue-shifted with the increase of the power, which indicates that the frequency shift of the sideband is mainly depended on MI, while SRS plays the role of amplification.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Modulation instability (MI) refers to a process in which weak perturbations from the steady state grow exponentially as a result of an interplay between the nonlinearity and the group-velocity dispersion. MI exhibits a rich spatiotemporal dynamics and has been studied in many fields such as fluid dynamics [1,2], plasma physics [3–6], and nonlinear optics [7–9]. When MI happens in CW fiber lasers, the laser behaves as a train of short pulses in time domain and the spectrum sidebands are generated in the frequency domain. In recent years, MI has been considered as the physical mechanism behind the generation of supercontinuum and rogue wave [10–16]. In general, MI can be induced by self-phase modulation (SPM) and cross-phase modulation (XPM). SPM-induced MI only occur in the anomalous dispersion regime when one beam is propagating in the fiber [17–20], while XPM-induced MI can occur in the normal dispersion regime when two beams co-propagating in the fiber [21,22]. XPM-induced MI in the fiber was studied theoretically in Refs. [23,24], which showed that the value of group velocity mismatch has a great influence on the instability gain spectrum.

High power fiber lasers (HPFLs) are widely used in many fields due to the excellent performance in beam quality, thermal management, etc. [25,26]. With the increase of the output power, transverse mode instability (TMI) [27] and nonlinear effects [28,29] will occur. When TMI occurs, the output beam profile may exhibit temporal instability, which leads to the degradation of beam quality and limits the further improvement of the power [30–35]. The appearance of nonlinear effects will also lead to the instability of CW in HPFLs. For example, stimulated Brillouin scattering (SBS) will lead to self-pulsing in CW narrow-linewidth fiber lasers, while the high peak pulses will threaten the laser system [36,37]. In recent years, XPM-induced MI in HPFLs has attracted the attention of researchers [38]. It is shown that XPM-induced MI between different transverse modes may occur in normal dispersion regime, and the gain spectrum of MI depends on power distribution and group velocity mismatch. In HPFLs, XPM can also occur between longitudinal modes, resulting in spectrum broadening [39]. When the laser power exceeds a certain value, Stokes light can be generated by IMFWM and SRS [23,24]. In this case, Stokes light induced MI may occur due to the large value of group velocity mismatch between the signal and Stokes light [23,40]. However, Stokes light induced MI in high power CW fiber amplifiers has never been addressed before.

In this paper, we have observed for the first time the phenomenon of Stokes light induced MI in high power fiber amplifiers. Kilowatt level master oscillator power amplifiers (MOPAs) with relatively strong IMFWM and SRS Stokes light were established. Then, the spectral and temporal results are analyzed to illustrate the characteristics of Stokes light induced MI in the MOPAs.

2. Theoretical analysis

In the case of neglecting the fiber loss, co-propagation of two optical beams along the fiber can be described by the coupling equations according to chapter 7 of [41]

$$\frac{{\partial {A_1}}}{{\partial z}} + \frac{1}{{{v_{g1}}}}\frac{{\partial {A_1}}}{{\partial t}} + \frac{i}{2}{\beta _1}\frac{{{\partial ^2}{A_1}}}{{\partial {t^2}}} = i{\gamma _1}({|{{A_1}} |^2} + 2|{{A_2}} |){A_1},$$
$$\frac{{\partial {A_2}}}{{\partial z}} + \frac{1}{{{v_{g2}}}}\frac{{\partial {A_2}}}{{\partial t}} + \frac{i}{2}{\beta _2}\frac{{{\partial ^2}{A_2}}}{{\partial {t^2}}} = i{\gamma _2}({|{{A_2}} |^2} + 2|{{A_1}} |){A_2},$$
where Aj is the slowly varying amplitude of the beam, V is the group velocity, βj is the group velocity dispersion, and γj is the nonlinear parameter (j=1, 2).For the case of continuous waves, the steady-state solution is obtained:
$$\overline {{A_j}} (z) = \sqrt {{P_j}} \exp (i{\phi _j}),\quad {\phi _j}(z) = {\gamma _j}({P_j} + 2{P_{3 - j}})z,$$
Where Pi is the optical power and ϕj is the n6tfcxonlinear phase shift obtained by the j-th optical field.

Time-dependent perturbations are assumed to test the stability of the steady state

$$\overline {{A_j}} (z,t) = [\sqrt {{P_j}} + {a_j}(z,t)]\exp (i{\phi _j}),$$
where a1 and a2 are the perturbations and satisfy the following coupling equations:
$$\frac{{\partial {a_1}}}{{\partial z}} + \frac{1}{{{v_{g1}}}}\frac{{\partial {a_1}}}{{\partial t}} + \frac{i}{2}{\beta _1}\frac{{{\partial ^2}{a_1}}}{{\partial {t^2}}} = i{\gamma _1}{P_1}({a_1} + a_1^\ast ) + 2i{\gamma _1}\sqrt {({P_1}{P_2})} ({a_2} + a_2^\ast ),$$
$$\frac{{\partial {a_2}}}{{\partial z}} + \frac{1}{{{v_{g2}}}}\frac{{\partial {a_2}}}{{\partial t}} + \frac{i}{2}{\beta _2}\frac{{{\partial ^2}{a_2}}}{{\partial {t^2}}} = i{\gamma _2}{P_2}({a_2} + a_2^\ast ) + 2i{\gamma _2}\sqrt {({P_1}{P_2})} ({a_1} + a_1^\ast ),$$
where K and Ω are the wave number and frequency of the perturbation, respectively. The general solution of Eqs. (5) and (6) is
$${a_j} = {U_j}\cos [kz - {\Omega _j}(t - \frac{z}{{{v_{gj}}}})] + i{V_j}\sin [kz - {\Omega _j}(t - \frac{z}{{{v_{gj}}}})].$$
When K and Ω satisfy the following relation, the equation has a nontrivial solution
$$({k^2} - {h_1})({k^2} - {h_2}) = {C^2},$$
where
$${h_j} = \frac{1}{4}\beta _j^2\Omega _j^2[\Omega _j^2 + {\mathop{\rm sgn}} ({\beta _j})\Omega _{cj}^2],$$
$$C = 2{\Omega _1}{\Omega _2}\sqrt {{\beta _1}{\beta _2}{\gamma _1}{\gamma _2}{P_1}{P_2}} ,$$
$$\Omega _{_{cj}}^2 = \frac{{4{\gamma _1}{P_1}}}{{|{{\beta_j}} |}}.$$
For some values of Ω, the wave number k could have an imaginary part. Then the steady state becomes unstable, and the perturbations increase exponentially along the fiber.

In HPFLs, with the increase of output power, Stokes light with larger frequency shift will be produced due to the influence of SRS and IMFWM. When the XPM-induced MI occur between the signal light and Stokes light, the group velocity mismatch δ = vg2−1- vg1−1 will have a significant impact on the modulation frequency. For the signal wavelength of 1080 nm, assuming that the powers of the signal and Stokes light are P1 = 1000W, P2 = 10W, respectively; γ1= γ2=0.0005 w-1 / m; the group velocity dispersion values of the signal and Stokes light are β1=15.29 fs2/mm and β2=12.15 fs2/mm, respectively. The relationship between the frequency shift of the MI gain peak and the group velocity mismatch δ is shown in Fig. 1(a). It can be seen from the figure that the group velocity mismatch of 1.2ps/m will produce an MI gain peak with 13.5 THz frequency shift. As a result of MI, a modulation sideband may appear around the second-order Raman frequency and will be amplified by SRS (here, the first Stokes light refers to the Stokes light participating in MI, and the second-order Raman frequency refers to the frequency of ∼13.5THz away from the first Stokes). In order to study the influence of signal and Stokes power on the MI, the gain spectra of MI at different powers were simulated. Figure 1(b) shows the gain spectra at different values of P2 when P1=1000W and δ = 1.2 ps/m, while Fig. 1(c) shows the gain spectra at different values of P1 when P2=10W and δ = 1.2 ps/m. It can be seen that the increase of P2 mainly leads to the broadening of gain bandwidth, while the growth of P1 leads to the blue-shift of gain spectrum.

 figure: Fig. 1.

Fig. 1. The gain of MI: (a) the relationship between the frequency shift of MI gain peak and the group velocity mismatch; (b) the gain spectra at different values of P2 when P1=1000W, and δ = 1.2 ps/m; (c) the gain spectra at different values of P1 when P2=10W, and δ = 1.2 ps/m.

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3. Experiment

Experimentally, a master oscillator power amplifier (MOPA) was established, of which the scheme is shown in Fig. 2. The system consisted of an oscillator and a power amplifier. The oscillator consisted of high reflectivity fiber Bragg grating (HRFBG), output coupling fiber Bragg grating (OCFBG) and 15m Yb-doped fiber (20/400 µm; cladding absorption 1.20 dB/m near 975 nm) with a numerical aperture (NA) of 0.06/0.46. The HRFBG centered at a wavelength of ∼1080 nm provided a 3-dB spectral bandwidth of about 3 nm with a reflective ratio more than 99%, and the OCFBG centered at a wavelength of ∼1080 nm provided a 3-dB spectral bandwidth of about 0.5 nm with a reflective ratio about 11%. Three 150W 976nm-laser diodes (LDs) were combined via a (6 + 1) ×1 combiner to pump the oscillator. A cladding power stripper (CPS) was spliced after the laser cavity to remove the unabsorbed pump light and the signal light leaking into the inner cladding.

 figure: Fig. 2.

Fig. 2. Schematic of master oscillator power amplifier.

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The power amplifier mainly consisted of a forward (6 + 1) ×1 pump-signal combiner (PSC) (pump fiber, 220/242 µm, NA 0.22; signal fiber, 25/400 µm, NA 0.065/0.46), a counter (6 + 1) ×1 PSC (pump fiber, 220/242 µm, NA 0.22; input signal fiber, 25/400 µm, NA 0.065/0.46; output signal fiber, 25/400 µm, NA 0.11/0.46) and 23m YDF (25/400 µm; NA 0.065/0.46; cladding absorption coefficient 0.57 ± 0.07 dB/m @ 915 nm). Here, the minimum and maximum coiling diameters of the YDF are ∼150 mm and ∼190 mm respectively, which means that there will be more higher-order modes in the amplifier. The output signal fiber with a core NA of 0.11 was used to improve the coupling efficiency of higher-order modes. Several 3×1 pump combiners (PCs) were used to couple 3 976nm-LDs to each arm of the PSCs. Another CPS (fiber: 25/400 µm, NA 0.11/0.46) was spliced after the power amplifier, and then a quartz block head (QBH; fiber: 25/400 µm, NA 0.11/0.46) was utilized to deliver the output signal light of the power amplifier. The length of the passive fiber at the output of the amplifier was ∼8 m (here, the output fiber length refers to the total fiber length of the counter PSC, CPS and QBH.). The output power of the power amplifier was measured by means of a power meter (Spiricon, 10 kW). The Spectrum and temporal characteristics were measured by an optical spectrum analyzer (OSA, YOKOGAWA, AQ6370C) with 0.1nm resolution and an oscilloscope (Tektronix, DPO3032) with 125MHz bandwidth photodetector (Terahertz Technologies, TIA-525I-FC). The beam quality was measured using a beam quality analyzer (Ophir Photonics, M 2 - 200 s).

The seed with 200W was injected into the amplifier. The output power with different pump powers are shown in Fig. 3(a). When the pump power is 3740W, the output power is 3100 W, and the slope efficiency of the amplifier is 77.5%. There was no mode instability in this amplifier, and the laser power was limited by the available pump power. The spectra at different output powers are shown in Fig. 3(b), where, the amplifier was forward pumped at 930, 1300 and 1670 W, bi-direction pumped at 3100 W (the forward and backward pump power is the same). With the increasing of the output power, the spectrum broadened gradually and finally became continuous. The beam quality at 3100 W is shown in Fig. 3(c), M2 x=2.32, M2 y=2.59. The results show that there are many higher-order modes, indicating that IMFWM will occur. Due to IMFWM, there were peaks near 1047 nm and 1117 nm. According to Ref. [39], the peaks near 1047 and 1117 nm should be attributed to the mode combination (21, 01→21, 01). As we can see, a peak appeared near 1117 nm rather than 1135 nm (Raman Stokes peak), indicating that the phase matching condition is satisfied in this case. According to Ref. [42], SRS can be partially suppressed by IMFWM when phase matching condition is satisfied. It is worth noting that with the increasing of the output power, there was a sideband around 1180 nm, of which the peak shifted 13.5 THz from IMFWM Stokes peak at 1117 nm. When the output power was 3100W, the peaks at 1117 and 1180 nm were 18 and 25 dB lower than the signal light, respectively. Figure 3(b) also shows the spectral peak at the 1180 nm wavelength region was blue-shifted ∼4nm when the output power increases from 930W to 3100W.

 figure: Fig. 3.

Fig. 3. Output characteristics when the length of the output passive fiber was ∼8m: (a) output power of the laser as a function of pump power; (b) the spectra at different output powers; (c) the beam quality at 3100 W.

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In order to study the time-domain characteristics of the output laser, the transmission signal of the high reflection in Fig. 2 is measured an oscilloscope (Tektronix, DPO3032) with 125MHz bandwidth photodetector (Terahertz Technologies, TIA-525I-FC). Here, the attenuation of signal was increased with the increase of power to obtain the relative strength of pulses and continuous wave at different powers. Figure 4(a)-(c) shows the time traces at different output powers of the amplifier. When the output power reached 1300W, a train of pulses appeared in the time trace. The peak power and frequency of the pulses increased rapidly with the increase of output power. When a short-pass filter with cutoff wavelength 1150 nm (Edmund Optics, curv_67288) was placed before the patchcord in Fig. 2, the pulses disappeared in the time trace with 1180 nm sideband filtered out in the spectrum, as shown in Fig. 4(d). Combing Fig. 3(b) and Fig. 4, one can conclude that the occurrence and enhancement of the pulses in the time trace keep pace with those of the 1180 nm sideband in the spectrum.

 figure: Fig. 4.

Fig. 4. The temporal characteristics at different output powers: (a)-(c) time traces at the output power of 1300W, 1670W, 3100W, respectively, (d) time trace at 3100W when a short-pass filter (cutoff wavelength 1150 nm) was used, the inset shows the spectra before and after filtering.

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Figure 5(a) shows the spectra of the amplifier at the output powers of 200W, 1670W, and 3100 W respectively, with the ∼8m output passive fiber shortened to ∼2m. The spectral peak at 1117nm was 25 dB lower than that of the signal light when the output power was 3100W, indicating IMFWM and SRS were successfully suppressed after the passive fiber was shortened. Figure 5(a) also shows the disappearance of the spectral peak near 1180 nm compared to the green curve in Fig. 3(b). Thus, the 1180 nm spectral peak was suppressed when the IMFWM and SRS effects of the amplifier was reduced after shortening the passive fiber. The suppression of the 1180 nm spectral peak also induced the disappearance of pulses in time domain after the passive fiber was shortened, as shown by Fig. 5(b).

 figure: Fig. 5.

Fig. 5. (a) The spectra at different output powers when the length of the output passive fiber was ∼2m; (b) The temporal characteristics at the output power of 3100W when the length of the output passive fiber was ∼8m and 2m.

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An amplifier with central wavelength of 1090nm was also built to study the Stokes light induced MI. The structure of the amplifier was similar to that in Fig. 2 (bi-direction pumped, while the forward and backward pump power is the same). The gain fiber length was 15m and the minimum coiling diameter was 120mm, which increases the loss of higher-order modes. Here, the length and coiling diameter of the gain fiber was changed to study the influence of gain fiber length and transverse modes on MI. In addition, the NA of output fiber core was changed to 0.065. Limited by mode instability, the maximum output power is 2500 W. The beam quality of the amplifier at 2500W is shown in Fig. 6(a), M2 x=1.64, M2 y=1.75. Compared Fig. 6(a) with Fig. 3(c), it can be seen that the 1080 nm amplifier excites more higher-order modes. According to Ref. [43], the fiber system exciting more high order mode content exhibits lower beam quality but higher mode instability threshold. Therefore, the mode instability threshold of 1080 nm amplifier is higher. The spectrum of the amplifier at 2500W is shown in Fig. 6(b). It can be seen that there was an obvious Raman Stokes peak near 1146 nm, which is 23 dB lower than the signal peak. Due to the relatively small coiling diameter of the gain fiber, the IMFWM Stokes peak was not obvious in the spectrum compared to those in Fig. 3(b). Further, a sideband appeared near 1200nm (37 dB lower than the signal peak), of which the peak shifted 11.6 THz from Raman Stokes peak at 1146 nm. Figure 6(c) shows the time trace of the amplifier at 2500W, which shows similar pulse characteristic as that in Fig. 4(c).

 figure: Fig. 6.

Fig. 6. The output characteristics of the 1090nm-amplifer at the power of 2500 W: (a) beam quality; (b) spectrum; (c) time trace.

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4. Discussion

According to the experimental results, both the 1080 nm-amplifier and the 1090 nm-amplifier exhibit pulses in time domain and a sideband around the second-order Raman frequency (refers to 1180nm-component in 1080 nm-amplifier and 1200 nm component in 1090 nm-amplifier) in spectral domain. In addition, the occurrence and enhancement of the pulses keep pace with those of the sideband, according to the experimental results of the 1080 nm amplifier. This phenomenon can be attributed to Stokes light induced MI, with instability induced by different Stokes light for the two amplifiers.

For the 1080nm-amplifier, due to the large coiling diameter of YDF (the minimum diameter is 150 mm), the loss of high-order modes is low. As a result, IMFWM occurred first and SRS was partially suppressed. The group velocity mismatch δ between the 1080 nm signal light and the 1117 nm IMFWM Stokes light is mainly composed of two portions: one portion is caused by the intermodal dispersion between LP21 and LP01 modes, which is about 0.35 ps/m according to Ref. [44]; the other one is caused by the waveguide dispersion between the signal light and the Stokes light, which is 0.79 ps/m. When the MI occurs between the LP21 mode in the signal light and the LP01 mode in the Stokes light, the group velocity mismatch can reach 1.14ps/m. According to Fig. 1(a), δ = 1.14ps/m will result in a gain peak with ∼13 THz frequency shift, which is close to the Raman gain peak, and amplified by SRS. As a result, a sideband with pulses appeared around 1180 nm (∼13THz away from 1117 nm). It should be noted that in general, MI will lead to symmetrical sidebands, but we only observed sidebands near 1180 nm in our experiment, which is attributed to SRS. According to the experimental results in Figs. 3(b) and 5(a), Stokes light induced MI are mainly generated in the output passive fiber, and it can be suppressed by shortening the output passive fiber. In addition, it can be seen from Fig. 3(b) that the sideband around 1180 nm blue-shifted ∼4 nm as the output power increases from 930W to 3100W, this is due to the frequency shift of MI gain spectrum caused by the increase of power, which is consistent with the simulation results in Fig. 1(c).

For the 1090 nm-amplifier, the minimum coiling diameter of the gain fiber was 120 mm, the high-order modes were suppressed. In this case, only the LP01 mode is considered, and the group velocity mismatch between 1090 and 1146 nm is about 1.03 ps/m. According to Fig. 1(a), the frequency shift of the gain peak is about 11.6 THz, which is consistent with the result in Fig. 6(b).

From the experimental results, it can be seen that the Stokes light induced MI phenomenon occurred at 1300W in the 1080nm-amplifier, while the similar phenomenon was observed at 2500W in 1090nm-amplifier. The difference of trigger threshold should be attributed to the frequency shift of modulation sideband. According to the simulation and experimental results, the frequency shift of the sideband in the 1080nm-amplifier (the sideband frequency shift is∼13.5 THz) is closer to Raman gain peak than that of 1090 nm-amplifier (the sideband frequency shift is 11.6THz). Therefore, the Stokes light induced MI showed a lower threshold in the1080nm-amplifier. Compared with 1080 nm amplifier, the gain fiber of 1090 nm amplifier was shortened by 8 m, while the MI did not seem to be suppressed, indicating that the MI mainly occurred in the output passive fiber in our experiments. However, we believe that other parameters such as spectral bandwidth, seed power and pump wavelength will also affect the threshold of MI, which needs further study.

5. Conclusion

In conclusion, Stokes light induced MI has been observed in high power continuous wave fiber amplifiers. The experimental results showed that the Stokes light can be modulated by the signal through XPM, where the Stokes light was internally produced through IMFWM and SRS. In agreement with the simulation results, a sideband appeared around the second-order Raman shift under the influence of MI, while pulses appear in the time domain. Further, the sideband caused by Stokes light induced MI was found to be blue-shifted with the increase of the power, which is consistent with the simulation results. The observation of MI in this paper may be helpful to study the generation of supercontinuum and rogue wave in high power CW fiber lasers.

Funding

National Natural Science Foundation of China (61875087).

Acknowledgments

The authors would like to thank Junbo Li for measuring the beam quality.

Disclosures

The authors declare no conflicts of interest.

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Figures (6)
Fig. 1.
Fig. 1. The gain of MI: (a) the relationship between the frequency shift of MI gain peak and the group velocity mismatch; (b) the gain spectra at different values of P2 when P1=1000W, and δ = 1.2 ps/m; (c) the gain spectra at different values of P1 when P2=10W, and δ = 1.2 ps/m.

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Fig. 2.
Fig. 2. Schematic of master oscillator power amplifier.

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Fig. 3.
Fig. 3. Output characteristics when the length of the output passive fiber was ∼8m: (a) output power of the laser as a function of pump power; (b) the spectra at different output powers; (c) the beam quality at 3100 W.

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Fig. 4.
Fig. 4. The temporal characteristics at different output powers: (a)-(c) time traces at the output power of 1300W, 1670W, 3100W, respectively, (d) time trace at 3100W when a short-pass filter (cutoff wavelength 1150 nm) was used, the inset shows the spectra before and after filtering.

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Fig. 5.
Fig. 5. (a) The spectra at different output powers when the length of the output passive fiber was ∼2m; (b) The temporal characteristics at the output power of 3100W when the length of the output passive fiber was ∼8m and 2m.

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Fig. 6.
Fig. 6. The output characteristics of the 1090nm-amplifer at the power of 2500 W: (a) beam quality; (b) spectrum; (c) time trace.

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Equations (11)

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∂ A 1 ∂ z + 1 v g 1 ∂ A 1 ∂ t + i 2 β 1 ∂ 2 A 1 ∂ t 2 = i γ 1 ( | A 1 | 2 + 2 | A 2 | ) A 1 ,
∂ A 2 ∂ z + 1 v g 2 ∂ A 2 ∂ t + i 2 β 2 ∂ 2 A 2 ∂ t 2 = i γ 2 ( | A 2 | 2 + 2 | A 1 | ) A 2 ,
A j ¯ ( z ) = P j exp ⁡ ( i ϕ j ) , ϕ j ( z ) = γ j ( P j + 2 P 3 − j ) z ,
A j ¯ ( z , t ) = [ P j + a j ( z , t ) ] exp ⁡ ( i ϕ j ) ,
∂ a 1 ∂ z + 1 v g 1 ∂ a 1 ∂ t + i 2 β 1 ∂ 2 a 1 ∂ t 2 = i γ 1 P 1 ( a 1 + a 1 ∗ ) + 2 i γ 1 ( P 1 P 2 ) ( a 2 + a 2 ∗ ) ,
∂ a 2 ∂ z + 1 v g 2 ∂ a 2 ∂ t + i 2 β 2 ∂ 2 a 2 ∂ t 2 = i γ 2 P 2 ( a 2 + a 2 ∗ ) + 2 i γ 2 ( P 1 P 2 ) ( a 1 + a 1 ∗ ) ,
a j = U j cos ⁡ [ k z − Ω j ( t − z v g j ) ] + i V j sin ⁡ [ k z − Ω j ( t − z v g j ) ] .
( k 2 − h 1 ) ( k 2 − h 2 ) = C 2 ,
h j = 1 4 β j 2 Ω j 2 [ Ω j 2 + sgn ( β j ) Ω c j 2 ] ,
C = 2 Ω 1 Ω 2 β 1 β 2 γ 1 γ 2 P 1 P 2 ,
Ω c j 2 = 4 γ 1 P 1 | β j | .
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