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Time-delay signature suppression of the chaotic signal in a semiconductor laser based on optoelectronic hybrid feedback

Ziwei Xu, Huan Tian, Zhen Zeng, Lingjie Zhang, Yali Zhang, Xinhai Zou, Zhiyao Zhang, Shangjian Zhang, Heping Li, and Yong Liu

Open Access Open Access

Abstract

An approach to generating chaotic signals with low time-delay signatures (TDSs) from a semiconductor laser (SCL) is proposed and demonstrated based on optoelectronic hybrid feedback. Through using a chirped fiber Bragg grating (CFBG) to provide distributed feedback, a chaotic signal with a low TDS is generated from the SCL. With the assistance of the nonlinear optoelectronic feedback provided by a microwave photonic link, the relaxation oscillation effect in the SCL is effectively suppressed, and the periodicity of the oscillation is greatly weakened. Hence, the TDS of the generated chaotic signal from the SCL is further suppressed, and the effective bandwidth is enlarged. Both simulation and experiment are carried out to verify the feasibility of the proposed scheme to suppress the TDS. In the experiment, a chaotic signal with a large effective bandwidth of 12.93 GHz, an extremely high permutation entropy (PE) of 0.9983, and a low TDS of 0.04, is generated by using a CFBG with a dispersion coefficient of 22.33 ps/nm. This TDS value is at the same level as that obtained by using the SCL-based scheme relying solely on distributed feedback in a CFBG with a dispersion coefficient of 2000 ps/nm.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Chaotic signals, which are characterized by noise-like waveforms, large bandwidths and low power spectral densities, have broad application prospects in radar and communication systems due to the benefits of anti-interception, anti-interference and range-Doppler coupling effect suppression ability [1–3]. In general, chaotic signals are generated by using various electrical methods, e.g., oscillation circuits [4–7] and function iterations [8–10], which have limited bandwidths due to the electronic bottleneck. The rapid development of high-resolution radars and broadband secure communications puts forward an ever-increasing bandwidth requirement of chaotic signals.

In recent years, nonlinear feedback systems based on photonics technology, including broadband optoelectronic oscillators (OEOs) [11–13] and chaotic semiconductor lasers (SCLs) [14–17], are regarded as powerful candidates to break through the bandwidth bottleneck of chaotic signal generation. Nevertheless, since the feedback signal is a delayed replica of the output signal, there is an inevitable time-delay signature (TDS) in the autocorrelation diagram of the generated signal, which exposes the system parameter information such as the loop length [18–22]. The TDS is an important factor threatening the security of the system, which must be suppressed as well as possible. In the chaotic signal generation scheme based on broadband OEOs, TDS suppression can be achieved by providing a large loop gain [23] or coupling several OEOs together [24]. The former method places a high requirement on the gain, the saturation output power and the operation bandwidth of the employed electrical amplifier (EA) in the OEO loop. The latter method is too complex and costly to apply in practice. As for the scheme based on chaotic SCLs, cascaded optical injection and coupled optical feedback are generally used to suppress the TDS [25,26]. In the method based on cascaded optical injection, a slight perturbation of operation temperature or injected light intensity will result in the instability of the chaotic state [27]. In the method based on the coupled optical feedback, the bandwidth of the generated chaotic signal is only several gigahertz due to the strong relaxation oscillation effect in the laser cavity [28]. In [29], the TDS of the generated chaotic signal from a SCL has been suppressed to 0.04 through the distributed feedback in a chirped fiber Bragg grating (CFBG) with a dispersion coefficient of about 2000 ps/nm. The experimental results also indicate that a higher dispersion coefficient of the CFBG leads to a lower TDS. Nevertheless, when the grating dispersion of a 10-cm-long CFBG reaches above 2000 ps/nm, the chirp coefficient should be set to be below 0.004 nm/cm, which poses great challenges to the phase masks, and is hard to achieve [32].

In this paper, an approach to suppressing the TDS of the generated chaotic signal from a SCL is proposed and demonstrated based on optoelectronic hybrid feedback. In the proposed scheme, TDS suppression is realized by combining the distributed feedback in a CFBG and the nonlinear optoelectronic feedback provided by a microwave photonic link. Compared with the SCL-based scheme that relies solely on distributed feedback in a CFBG, the proposed scheme can achieve an excellent TDS suppression by using a CFBG with a much smaller dispersion coefficient. In addition, there is no stringent requirement of EA in the microwave photonic link. Both simulation and experiment are implemented to verify the feasibility of the proposed scheme. In the experiment, a chaotic signal with an effective bandwidth of 12.93 GHz, a permutation entropy (PE) of 0.9983 and a TDS of 0.04 is generated. In addition, it is the first time to construct an optoelectronic hybrid feedback model by combining the Lang-Kobayashi equations with the Ikeda equations, which is beneficial for modeling complex optoelectronic hybrid feedback systems.

2. Operation principle

Figure 1 shows the schematic diagram of the proposed chaotic source. Continuous-wave (CW) light from a SCL is divided into three parts by using a 1 × 3 optical coupler with a power splitting ratio of 1:1:1. In the upper branch, the CW light is injected into a CFBG via a variable optical attenuator (VOA). Through properly adjusting the VOA to tune the reflection strength, chaotic oscillation is successfully built up in the laser cavity under an obvious period-doubling process. In the lower branch, optical chaotic signal generated by the SCL is detected by using a high-speed photodetector (PD) after propagating through a spool of single-mode fiber (SMF). The electrical chaotic signal from the PD is amplified by using a broadband EA, and is then split into two parts via a power divider. One part of the electrical chaotic signal is used as the output of the chaotic source. The other part is modulated onto the CW light from a tunable laser source (TLS) by using a broadband electro-optic Mach-Zehnder modulator (MZM) biased near its null point, and is sent back to the SCL via an optical isolator (ISO) and the 1 × 3 optical coupler.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the proposed chaotic source. The inset is the reflection spectrum and the delay spectrum of the CFBG. SCL: semiconductor laser; VOA: variable optical attenuator; CFBG: chirped fiber Bragg grating; TLS: tunable laser source; MZM: Mach-Zehnder modulator; SMF: single-mode fiber; PD: photodetector; EA: electrical amplifier; ISO: optical isolator.

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Thanks to the combination of the distributed feedback in the CFBG and the nonlinear optoelectronic feedback provided by the microwave photonic link, the performance of the generated chaotic signal is greatly improved. Firstly, in the upper branch, distributed feedback in the CFBG offers a wavelength-dependent delay, which leads to an irregular separation between the external-cavity modes. On this condition, the TDS of the generated chaotic signal is effectively suppressed. Secondly, in the lower branch, the nonlinear transmission characteristic induced by the electro-optical and the photoelectric conversions increases the complexity of the feedback chaotic signal, which is beneficial for weakening the influence of the relaxation oscillation effect in the SCL. In such a case, the spectrum of the generated chaotic signal is flattened. Thirdly, the microwave photonic link in the lower branch is similar to an OEO without a filter, which is featured by high gain and large bandwidth. Through feeding the strong broadband chaotic signal in the microwave photonic link back into the SCL, the periodicity of the oscillation in the proposed scheme can be greatly weakened. Hence, the TDS of the generated chaotic signal can be further suppressed.

3. Theoretical model and numerical simulation

3.1 Theoretical model

The dynamic process in the SCL can be described by using the Lang-Kobayashi model [30]

$$\frac{{dE(t )}}{{dt}} = \frac{1}{2}({1 + i\alpha } )\left( {{G_s} - \frac{1}{{{\tau_p}}}} \right)E(t )+ \sqrt {2\beta N(t )} \xi + {F_f}({t - {\tau_1}} )+ {F_i}({t - {\tau_2}} )$$
$$\frac{{dN(t )}}{{dt}} = \frac{I}{e} - \frac{{N(t )}}{{{\tau _n}}} - {G_s}{|{E(t )} |^2}$$
where E and N are the complex amplitude of the electrical field and the carrier density in the SCL cavity, respectively. τp and τn represent the lifetime of the photon and the carrier, respectively. α is the linewidth enhancement factor, and β is the spontaneous emission factor. Gs = g(N-N0)/(1+εE2) stands for the gain-saturation effect, where g, N0 and ε are the differential gain, the transparency carrier density and the gain saturation parameter, respectively. ξ is the Gaussian white noise, whose mean value and variance are 0 and 1, respectively. I and e are the bias current and the charge of a single electron, respectively. τ1 and τ2 are the time delay introduced by the CFBG and the microwave photonic link, respectively.

The third term on the right side of Eq. (1) represents the distributed feedback induced by the CFBG, which can be calculated as

$${F_f}({t - {\tau_1}} )= \frac{{{k_f}}}{{{\tau _{in}}}}\int_{t - T}^t {h({t - t^{\prime}} )E({t^{\prime} - {\tau_1}} )dt^{\prime}}$$
where kf and τin represent the feedback strength and the round-trip time in the laser cavity, respectively. h(t) is the impulse response of the CFBG, which can be obtained by performing inverse Fourier transform on the complex reflection spectrum h(w) of the CFBG [31]. Meanwhile, h(w) can be calculated by using the piecewise-uniform approach [32], where the CFBG is divided into multiple sub-Bragg gratings.

The fourth term on the right side of Eq. (1) is the nonlinear optoelectronic feedback induced by the microwave photonic link. For simplicity, the filtering effect caused by the EA can be described as a 2nd-order bandpass filtering process with a low cut-off frequency of fL and a high cut-off frequency of fH. Hence, the input voltage V(t) of the MZM can be calculated by

$$\left( {1 + \frac{{{f_L}}}{{{f_H}}}} \right)V(\textrm{t} )+ \frac{1}{{2\pi {f_H}}}\frac{{dV(t )}}{{dt}} + 2\pi {f_L}\int_{{t_0}}^t {V({t^{\prime}} )} dt^{\prime} = \gamma GR{|{E({t - {\tau_2}} )} |^2}$$
where G is the controllable net gain of the system. R and γ are the matching resistance and the responsivity of the PD, respectively.

In summary, the proposed chaotic source can be mathematically described by the following differential equations

$$\frac{{du(t )}}{{dt}} = 2\pi {f_L}V(t )$$
$$\frac{{dV(t )}}{{dt}} ={-} 2\pi ({{f_H} + {f_L}} )V(\textrm{t} )- 2\pi {f_H}u(t )+ 2\pi {f_H}GR\gamma {|{E({t - {\tau_2}} )} |^2}$$
$$\begin{aligned} {\kern 1pt} \frac{{dE(t )}}{{dt}} &= \frac{1}{2}({1 + i\alpha } )\left( {{G_s} - \frac{1}{{{\tau_p}}}} \right)E(t )+ \sqrt {2\beta N(t )} \chi \\ &+ \frac{{{k_f}}}{{{\tau _{in}}}}\int_{t - T}^t {h({t - t^{\prime}} )E({t^{\prime} - {\tau_1}} )dt^{\prime}} + \frac{{{k_i}}}{{{\tau _{in}}}}\cos \left( {\frac{m}{2}V({t - {\tau_2}} )+ \frac{\varphi }{2}} \right)\exp ({i2\pi \Delta ft} )\end{aligned}$$
$$\frac{{dN(t )}}{{dt}} = \frac{I}{e} - \frac{{N(t )}}{{{\tau _n}}} - {G_s}{|{E(t )} |^2}$$
where ki and Δf are the injection strength and the detuning frequency, respectively. The last term on the right side of the Eq. (5c) represents the modulation process in the MZM. Thereinto, m and φ are the modulation coefficient and the direct-current (DC) bias-induced phase shift, respectively. In addition, Eqs. (5a) and (5b) are similar to the Ikeda equations which describe the dynamic behaviors in the broadband OEOs. Hence, Eqs. (5a)–(5d) are the combination of the Lang-Kobayashi model and the Ikeda model, which can be used to describe complex optoelectronic hybrid feedback systems.

3.2 Simulation results

Numerical simulations are implemented to verify the feasibility of the proposed scheme to suppress the TDS. The performance of the proposed scheme is compared with those based on either a broadband OEO or a chaotic SCL relying solely on distributed feedback in the CFBG. In the simulation, Eqs. (5a)-(5d) are solved by using the 4th-order Runge-Kutta method, where the time step and the time span are set to be 1 ps and 10 µs, respectively. Table 1 presents the parameters used in the simulation. In addition, the length, the center reflection wavelength and the chirp coefficient of the CFBG are 10 cm, 1550 nm and 0.4 nm/cm, respectively. To model the reflection spectrum and the delay spectrum, the CFBG is divided into 1000 sub-Bragg gratings, each of which has an effective index of 1.46 and an average index change of 5 × 10−4. As a result, the reflection bandwidth and the dispersion coefficient of the CFBG are 12 nm and 83.4 ps/nm, respectively. The integral time T for calculating the distributed feedback is set to be 5 ns, which is much larger than the response time of the CFBG. In the simulation of a broadband OEO and a chaotic SCL relying solely on distributed feedback in the CFBG, the corresponding parameters are identical to those listed above and in Table 1.

Tables Icon

Table 1. Parameters used in the numerical simulation

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Figure 2(a) and (b) show the optical spectra of the generated chaotic signals from the SCL by using a mirror and the CFBG to achieve feedback, respectively, where the feedback strength is set to be identical in both schemes, i.e. kf = 0.1. It can be seen from Fig. 2(a) that the external-cavity modes of the mirror feedback laser have an identical frequency interval inversely proportional to the time delay, which is not beneficial for suppressing the TDS. Nevertheless, as can be seen in Fig. 2(b), the CFBG provides a wavelength-dependent delay, leading to irregular separation between the external-cavity modes. In addition, Fig. 2(c) shows the autocorrelation diagram (green line) of the chaotic signal generated by the mirror feedback laser, i.e., Fig. 2(a) and that (red line) of the chaotic signal generated by the distributed feedback laser, i.e., Fig. 2(b). It can be seen that the TDS of the chaotic signal generated by the distributed feedback laser is much lower than that generated by the mirror feedback laser, since the periodicity of the optical-field oscillation is destroyed through distributed feedback. Hence, the time delay information has been effectively hidden through distributed feedback. In order to quantify the complexity of the generated chaotic signals, the temporal sequences with a length of 105 are adopted to calculate the PE. The embedding dimension and the embedding delay are two intermediate variables for calculating the PE. In order to get the accurate PE with a time-saving process, the embedding dimension and the embedding delay are set to be 3 and 7, respectively [33]. Figure 2(c) shows the PEs of the generated chaotic signals under different feedback strength. A larger PE corresponds to a higher complex sequence. Hence, the simulation results indicate that the distributed feedback laser outperforms the mirror feedback laser in hiding the time delay information and generating more complex chaotic signal.

 figure: Fig. 2.

Fig. 2. Simulation results of the mirror feedback laser and the distributed feedback laser. (a) Optical spectrum and delay spectrum of the mirror feedback laser working in chaotic state (The inset is the optical spectrum within a span of 400 MHz). (b) Optical spectrum and delay spectrum of the distributed feedback laser working in chaotic state (The inset is the optical spectrum within a span of 400 MHz). (c) Autocorrelation diagrams of the chaotic signals generated by the mirror feedback laser (green line) and the distributed feedback laser (red line). (d) PEs of the generated chaotic signals under different feedback strengths.

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Figure 3 presents the temporal sequences, the power spectra and the autocorrelation diagrams of the chaotic signals generated by using the broadband OEO, the distributed feedback laser and the proposed scheme. The results indicate that the proposed scheme not only suppresses the TDS completely, but also broadens the spectrum and compresses the autocorrelation peak effectively. The PE of the temporal sequence in Fig. 3(c) is calculated to be 0.9999683 (The maximum value is 1), which is larger than those of the temporal sequences in Fig. 3(a) and (b). Figure 4(a) and (b) exhibits the PEs and the TDSs of the generated chaotic signal from the proposed scheme under different combinations of injection strength and feedback strength, respectively. It can be seen that chaotic signals with high PEs and low TDSs can be obtained in a large injection and feedback range, indicating that the proposed scheme is with a strong robustness.

 figure: Fig. 3.

Fig. 3. Simulation results of the chaotic signals generated by using the broadband OEO (upper row), the distributed feedback SCL (middle row) and the proposed scheme (bottom row). (a)-(c) Temporal sequences within 2 ns. (d)-(f) Power spectra. (g)-(i) Autocorrelation diagrams (The insets are the autocorrelation peaks).

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 figure: Fig. 4.

Fig. 4. Simulation results under different combinations of injection strength and feedback strength. (a) PEs. (b) TDSs.

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4. Experimental results

In the experiment, a SCL (7pin-20G-C38) biased at 18 mA (2.48 times its threshold current) is used as the chaotic source, whose output power and center wavelength are 3.42 dBm and 1545.126 nm, respectively. In order to avoid wavelength-drifting effect, a precise temperature controller (TCM1040) is used to maintain the operation temperature of the SCL at 25 °C. The distributed feedback is provided by using a CFBG with a center wavelength, a reflection bandwidth and a dispersion coefficient of 1550 nm, 40 nm and 22.33 ps/nm, respectively. The reflection spectrum and the delay spectrum of the CFBG are shown in the inset of Fig. 1. Through controlling the feedback strength via tuning a VOA, chaotic oscillation is built up in the SCL cavity. After propagating through a spool of SMF (YOFC) with a length of 50 m, the optical chaotic signal is detected by using a PD with an operation bandwidth of 20 GHz. A low-noise amplifier (GT-HLNA-0022 G) with an operation frequency range from 34.25 MHz to 22 GHz and a small-signal gain of 28 dB is employed to amplify the electrical chaotic signal from the PD. Then, an electrical power divider (GTPD-COMB50G) with an operation frequency range from DC to 50 GHz is used to extract the generated electrical chaotic signal. A part of the electrical chaotic signal is also modulated onto the CW light from a TSL (Teraxion PS-TNL) with an output power of 10.03 dBm by using a 25 Gb/s electro-optic MZM (FUJITSU 7938EZ) biased near its null point, which is then sent back to the SCL to achieve nonlinear optoelectronic feedback. In the experiment, there is a detuning frequency of about 8 GHz between the SCL and the TLS, and the frequency of the SCL is larger than that of the TLS.

Figure 5(a) and (b) show the optical spectra and the converted electrical spectra of the SCL under different distributed feedback strengths, respectively. In Fig. 5, the measured curves are intentionally moved in the vertical direction to distinguish the four measurement results, where the feedback strength decreases sequentially from top to bottom. The dynamic behavior of the distributed feedback SCL is similar to the mirror feedback SCL due to the obvious period doubling process. In Fig. 5 (b), the power spectra envelopes bulge near 9.2 GHz and 18.4 GHz, which is attributed to the relaxation oscillation effect in the SCL cavity.

 figure: Fig. 5.

Fig. 5. Experimental results of the SCL under different distributed feedback strengths. The feedback strength decreases sequentially from top to bottom. (a) Optical spectra. (b) Electrical spectra after photoelectric conversion.

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Figure 6 presents the temporal sequences, the amplitude probability distributions and the power spectra of the chaotic signals generated by using the broadband OEO, the distributed feedback laser and the proposed scheme. The chaotic signal generated by using the proposed scheme has a more random temporal sequence and a flatter spectrum. It should be pointed out that the concavity near 17 GHz in the power spectrum is attributed to the frequency response of the PD. PEs with an embedding dimension of 3 and an embedding delay of 7 are calculated based on a temporal sequence length of 105. The calculated PEs of the generated chaotic signals are 0.9915, 0.9956 and 0.9983 for the broadband OEO, the distributed feedback laser and the proposed scheme, respectively, indicating that the proposed scheme generates the highest complex sequence.

 figure: Fig. 6.

Fig. 6. Experimental results of the chaotic signals generated by using the broadband OEO (upper row), the distributed feedback SCL (middle row) and the proposed scheme (bottom row). (a)-(c) Temporal sequences. (d)-(f) Amplitude probability distributions. (g)-(i) Spectra.

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Figure 7(a)-(c) exhibit the autocorrelation diagrams of the generated chaotic signals by using the broadband OEO, the distributed feedback laser and the proposed scheme, respectively. Compared with the results in Fig. 7(a) and (b), the TDS in Fig. 7(c) is largely suppressed. The dispersion coefficient of the CFBG in the experiment is 22.33 ps/nm, which is almost 1/4 of that in the simulation. Hence, it is insufficient to completely suppress the TDS as shown in Fig. 3(i). The residual TDS in Fig. 7(c) is 0.04, which is at the same level as the result in [29] via using a CFBG with a dispersion coefficient of 2000 ps/nm. In addition, the effective bandwidth of the generated chaotic signal by using the proposed scheme is calculated to be 12.93 GHz according to the formula BW = 1/FWHM, where FWHM represents the full-width at half-maximum of the autocorrelation peak.

 figure: Fig. 7.

Fig. 7. Autocorrelation diagrams of the generated chaotic signals by using (a) the broadband OEO, (b) the distributed feedback SCL, and (c) the proposed scheme.

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5. Conclusion

In summary, we have proposed and demonstrated a scheme to generate chaotic signals with low TDSs from a SCL based on the combination of distributed feedback in a CFBG and nonlinear optoelectronic feedback provided by a microwave photonic link. In this scheme, a CFBG with a large dispersion coefficient is not required as that in the conventional SCL-based scheme relying solely on distributed feedback in a CFBG. In addition, the stringent requirement of EA is eased compared with the broadband OEO scheme. Both the simulation and the experimental results indicate that the proposed scheme can generate a chaotic signal with a lower TDS, a larger effective bandwidth and a flatter spectrum than those generated by using the broadband OEO and the distributed feedback SCL. In the experiment, a chaotic signal with a large effective bandwidth of 12.93 GHz, an extremely high PE of 0.9983 and a low TDS of 0.04 was generated. Therefore, the proposed scheme is beneficial for generating complex broadband chaotic signals for high-resolution radar and broadband secure communication applications.

Funding

National Natural Science Foundation of China (61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).

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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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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Figures (7)
Fig. 1.
Fig. 1. Schematic diagram of the proposed chaotic source. The inset is the reflection spectrum and the delay spectrum of the CFBG. SCL: semiconductor laser; VOA: variable optical attenuator; CFBG: chirped fiber Bragg grating; TLS: tunable laser source; MZM: Mach-Zehnder modulator; SMF: single-mode fiber; PD: photodetector; EA: electrical amplifier; ISO: optical isolator.

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Fig. 2.
Fig. 2. Simulation results of the mirror feedback laser and the distributed feedback laser. (a) Optical spectrum and delay spectrum of the mirror feedback laser working in chaotic state (The inset is the optical spectrum within a span of 400 MHz). (b) Optical spectrum and delay spectrum of the distributed feedback laser working in chaotic state (The inset is the optical spectrum within a span of 400 MHz). (c) Autocorrelation diagrams of the chaotic signals generated by the mirror feedback laser (green line) and the distributed feedback laser (red line). (d) PEs of the generated chaotic signals under different feedback strengths.

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Fig. 3.
Fig. 3. Simulation results of the chaotic signals generated by using the broadband OEO (upper row), the distributed feedback SCL (middle row) and the proposed scheme (bottom row). (a)-(c) Temporal sequences within 2 ns. (d)-(f) Power spectra. (g)-(i) Autocorrelation diagrams (The insets are the autocorrelation peaks).

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Fig. 4.
Fig. 4. Simulation results under different combinations of injection strength and feedback strength. (a) PEs. (b) TDSs.

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Fig. 5.
Fig. 5. Experimental results of the SCL under different distributed feedback strengths. The feedback strength decreases sequentially from top to bottom. (a) Optical spectra. (b) Electrical spectra after photoelectric conversion.

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Fig. 6.
Fig. 6. Experimental results of the chaotic signals generated by using the broadband OEO (upper row), the distributed feedback SCL (middle row) and the proposed scheme (bottom row). (a)-(c) Temporal sequences. (d)-(f) Amplitude probability distributions. (g)-(i) Spectra.

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Fig. 7.
Fig. 7. Autocorrelation diagrams of the generated chaotic signals by using (a) the broadband OEO, (b) the distributed feedback SCL, and (c) the proposed scheme.

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Tables (1)

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Table 1. Parameters used in the numerical simulation

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Equations (8)

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d E ( t ) d t = 1 2 ( 1 + i α ) ( G s − 1 τ p ) E ( t ) + 2 β N ( t ) ξ + F f ( t − τ 1 ) + F i ( t − τ 2 )
d N ( t ) d t = I e − N ( t ) τ n − G s | E ( t ) | 2
F f ( t − τ 1 ) = k f τ i n ∫ t − T t h ( t − t ′ ) E ( t ′ − τ 1 ) d t ′
( 1 + f L f H ) V ( t ) + 1 2 π f H d V ( t ) d t + 2 π f L ∫ t 0 t V ( t ′ ) d t ′ = γ G R | E ( t − τ 2 ) | 2
d u ( t ) d t = 2 π f L V ( t )
d V ( t ) d t = − 2 π ( f H + f L ) V ( t ) − 2 π f H u ( t ) + 2 π f H G R γ | E ( t − τ 2 ) | 2
d E ( t ) d t = 1 2 ( 1 + i α ) ( G s − 1 τ p ) E ( t ) + 2 β N ( t ) χ + k f τ i n ∫ t − T t h ( t − t ′ ) E ( t ′ − τ 1 ) d t ′ + k i τ i n cos ⁡ ( m 2 V ( t − τ 2 ) + φ 2 ) exp ⁡ ( i 2 π Δ f t )
d N ( t ) d t = I e − N ( t ) τ n − G s | E ( t ) | 2
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