Simulation method of discontinuous optical injection locking of lasers combined with time-frequency analysis

Abstract

The present invention discloses a discontinuous optical injection locking simulation method for a laser combined with time-frequency analysis, which mainly includes: reading an injected optical signal and down-converting it to obtain a baseband signal, using empirical mode decomposition to represent the baseband signal as a superposition of n single-component stationary signals; extracting a stationary signal containing a target frequency component from it; initializing the slave laser parameters; establishing a rate equation group based on the injected optical signal parameters and the stationary signal parameters respectively; using a fourth-order and fifth-order Runge-Kutta algorithm to solve the above two rate equation groups respectively, thereby obtaining the number of carriers and the amplitude and phase of the output light from the slave laser respectively; and then reconstructing the time-frequency distribution and spectrum of the two groups of optical signals; by comparing the time-frequency distribution and spectrum of the two groups of optical signals, analyzing whether the distribution change of the target frequency component during the injection locking process meets the requirements. The present invention can solve the problem that the instantaneous characteristics of the injected signal cannot be analyzed and the frequency component simulation results have no reference value.

Description

Technical Field

[0001] The present invention relates to the field of optical signal injection locking dynamics, and in particular to a discontinuous optical injection locking simulation method for a laser combined with time-frequency analysis. Background Art

[0002] Laser injection locking, an optical frequency and phase synchronization technique based on external light injection into a resonant cavity, has been widely applied in various fields, including dense wavelength division multiplexing, sensing and quantum communication, and arbitrary waveform optical pulse generation. Modeling and simulation analysis of injection locking can effectively simplify the parameterized design of the injection locking process in these applications and facilitate systematic evaluation of the application. Existing injection locking modeling is generally based on a widely accepted injection locking dynamics analysis. This method introduces an injection term into the laser rate equation, decomposing the injected light into the injected term's optical field amplitude and phase difference, respectively. The corresponding injection locking range and steady-state solution are then derived through small-signal analysis. This method demonstrates good simulation results when direct current light or a stationary signal is injected, and the resulting injection locking range and steady-state solution are highly valuable for practical applications. However, when the injected light exhibits a nonstationary modulated data signal, the locking range obtained using this method differs from the actual locking range, and spectrum analysis cannot confirm the specific injection conditions corresponding to the frequency components of the data signal during this process.

[0003] Time-frequency analysis can describe signals from three dimensions: time, frequency, and amplitude. Compared to traditional spectrum analysis, it is more suitable for expressing the instantaneous characteristics of signals. Existing time-frequency analysis has been widely used in audio and image processing, data compression, fault diagnosis, and other fields. However, on the one hand, time-frequency analysis itself is limited by time resolution and frequency resolution, and the scope of application of different algorithms is determined by signal characteristics and computational complexity. On the other hand, in the analysis of laser injection locking, the high-speed signal conversion and narrow frequency locking range place high demands on both time resolution and frequency resolution. The use of rate equations to analyze the injection locking process is also more suitable for single-component signals. Therefore, for the analysis and simulation of injection locking, it is necessary to comprehensively consider the advantages and disadvantages of various time-frequency analysis methods, select an algorithm that can decompose the single-component signal, and combine it with the rate equation for analysis. Summary of the Invention

[0004] In response to the above-mentioned prior art, the present invention provides a laser discontinuous optical injection locking simulation method combined with time-frequency analysis, which can solve the problems of being unable to analyze the instantaneous characteristics of the injection signal and the lack of reference value of the frequency component simulation results.

[0005] In order to solve the above technical problems, the present invention proposes a method for simulating discontinuous optical injection locking of a laser combined with time-frequency analysis, comprising the following steps:

[0006] 1) Read the injected light signal;

[0007] 2) Downconverting the injected optical signal to obtain a baseband signal, and using empirical mode decomposition to represent the baseband signal as a superposition of n single-component stationary signals; extracting the stationary signal containing the target frequency component;

[0008] 3) Initialize the slave laser parameters;

[0009] 4) Establishing a rate equation system based on optical signal parameters;

[0010] 5) Substituting the parameters of the injected optical signal and the parameters of the stationary signal containing the target frequency component described in step 2) into the rate equations described in step 4), thereby forming a rate equation group based on the parameters of the injected optical signal and a rate equation group based on the parameters of the stationary signal;

[0011] 6) using a fourth-order / fifth-order Runge-Kutta algorithm to solve the two rate equations, respectively, thereby obtaining the number of carriers and the amplitude and phase of the output light of the slave laser in which the injected optical signal participates in locking, and the number of carriers and the amplitude and phase of the output light of the slave laser in which the stationary signal participates in injection locking, and further obtaining the time-frequency distribution and spectrum of the two sets of optical signals obtained after the injected optical signal and the stationary signal light of the target frequency component are respectively injected into the slave laser;

[0012] 7) By comparing the time-frequency distribution and spectrum of the two sets of optical signals, analyze whether the distribution change of the target frequency component during the injection locking process meets the requirements. If not, adjust the parameters of the stationary signal and superimpose it with the other single-component stationary signals in step 2), reconstruct it into a new injected optical signal and return to step 1); if it meets the requirements, end the simulation process.

[0013] Furthermore, the discontinuous optical injection locking simulation method for a laser according to the present invention comprises:

[0014] In step 2), empirical mode decomposition is used to obtain n intrinsic mode functions (IMFs) representing n stationary signals that conform to single-component characteristics. A set of instantaneous frequencies is obtained from each IMF. Based on the target frequency component, the IMF containing the target frequency component is extracted from the n IMFs and used as the stationary signal for the rate equation calculation in the subsequent step. Here, n = 2 to 10.

[0015] The rate equations based on the optical signal parameters established in step 4) are as follows:

[0016]

[0017]

[0018]

[0019] Where A(t) is the amplitude of the light output from the laser, φ(t) is the phase difference between the light output from the laser and the injected light; A i (t) is the optical parameter amplitude of the input light, Δω i (t) is the optical parameter frequency ω of the input light i With the free oscillation frequency ω from the laser s The frequency detuning between the two lasers; N(t) is the number of carriers from the laser; g, N th , κ, α, γ N , γ P , J are gain coefficient, carrier number threshold, coupling coefficient, linewidth enhancement factor, carrier recombination rate, photon decay rate and normalized pump current respectively; when the input light is an injected light signal, the above rate equation group is the rate equation group based on the injected light signal parameters described in step 5); when the input light is a stationary signal of the target frequency component, the above rate equation group is the rate equation group based on the stationary signal parameters described in step 5).

[0020] In step 6), the results of solving the two rate equations include the amplitude A(t) of the output light from the laser, the phase difference φ(t) between the output light from the laser and the injected light, and the carrier number N(t) of the slave laser. Based on the amplitude A(t) of the output light from the laser and the phase difference φ(t) between the output light from the laser and the injected light, calculations and superposition are performed to reconstruct two groups of optical signals, the two groups of optical signals including the output light from the laser injected by the injected light signal and the output light from the laser injected by the stationary signal. The two groups of optical signals are again decomposed using empirical mode decomposition, and the instantaneous frequencies of the two groups of optical signals are obtained. The time-frequency distribution of the two groups of optical signals is obtained based on the amplitude A(t) and instantaneous frequencies of the two groups of optical signals. The two groups of optical signals are discrete Fourier transformed to obtain frequency spectra.

[0021] In step 7), the parameters of the stationary signal are adjusted, including frequency, phase, power and waveform.

[0022] Compared with the prior art, the present invention has the following beneficial effects:

[0023] This invention leverages the advantages of time-frequency analysis in extracting the transient characteristics of non-stationary signals. It uses empirical mode decomposition (EMD) to decompose the injected optical signal into n intrinsic mode functions (IMFs) with the characteristics of a single-component stationary signal, thereby improving the rate equation's simulation of the injected frequency of the non-stationary optical signal. Employing EMD and calculating the instantaneous frequency, this method not only extracts the instantaneous frequency of the injected optical signal but also analyzes the transient characteristics of the injection-locked output light. Comparing the instantaneous frequency distributions of the input and output light helps determine the target frequency's involvement in injection locking. Combining the spectral distribution of the output light with multiple decompositions and reconstructions of the injected optical signal provides an important reference for adjusting the injected optical signal and injection-locking parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 It is a schematic flow chart of the discontinuous optical injection locking simulation method of the laser of the present invention;

[0025] Figure 2 Taking the 5Gbps NRZ code "0111110110" as an example, the baseband signal waveform (three cycles) is obtained by frequency reduction;

[0026] Figure 3 Yes Figure 2 The five intrinsic mode functions and one residual signal are obtained by performing empirical mode decomposition on the signal shown;

[0027] Figure 4 Yes Figure 3 The instantaneous frequency distributions calculated from the five natural mode functions are:

[0028] Figure 5 The results of solving the rate equation are as follows: Group I and Group II are the groups where the injected signal light is directly substituted and the group where the stationary signal of the target frequency component is substituted, respectively. From top to bottom, they are: first row: output light amplitude A(t); second row: phase difference φ(t) between the output light and the injected light; third row: number of carriers N(t) from the laser.

[0029] Figure 6 It is a comparison diagram of the instantaneous frequency distribution of the two groups of output light;

[0030] Figure 7 The first row shows the spectrum distribution of the signals substituted into the equations in Group I and Group II, respectively, and the second row shows the spectrum distribution of the two groups of output lights. DETAILED DESCRIPTION

[0031] The design concept of the present invention is to supplement the shortcomings of the laser injection locking rate equation in calculating non-stationary injection signals. A method combining time-frequency analysis and rate equation analysis is used to extract single-component signals from the injected optical signal and perform dynamic analysis based on the rate equation system. By comparing the instantaneous characteristics of the output light, the specific involvement of the target frequency component of the injected optical signal in the injection locking process is inferred, providing an important reference for adjusting injection locking parameters and designing the injected optical signal. First, empirical mode decomposition is performed on the multi-component non-stationary injected optical signal. The instantaneous frequencies of the separated intrinsic mode functions are calculated, and the functions containing the target frequency components are extracted from them. These functions are then inserted into the rate equation to solve the output results. To compare the dynamic analysis method without time-frequency analysis, a control group is included in the simulation, where the non-stationary injected signal light is directly substituted into the rate equation. This not only serves as a supplementary comparison of the simulation differences before time-frequency analysis, but also demonstrates that time-frequency analysis based on empirical mode decomposition and instantaneous frequency calculation still has its own shortcomings in describing stationary signals. Therefore, it is necessary to retain the group where the non-stationary injected signal light is directly substituted into the rate equation for reference. After solving both sets of rate equations, the resulting output light is again subjected to time-frequency analysis and spectral analysis. This time-frequency analysis again employs empirical mode decomposition and instantaneous frequency calculations to maintain consistency in the instantaneous characteristics, thus maintaining the same frequency resolution. The instantaneous characteristics and spectral distribution of the two sets are then compared to determine the involvement of the target frequency component in injection locking. This determines whether various parameters, such as the injection power, frequency detuning, or signal light pattern and chip rate, need to be modified to adjust the target frequency component's contribution to injection locking.

[0032] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but the following embodiments are by no means intended to limit the present invention in any way.

[0033] like Figure 1 As shown, the present invention proposes a method for simulating discontinuous optical injection locking of a laser combined with time-frequency analysis, comprising the following steps:

[0034] (1) Reading the injected optical signal;

[0035] (2) The injected optical signal is down-converted to obtain a baseband signal, and the baseband signal is represented as a superposition of n single-component stationary signals by empirical mode decomposition; a stationary signal containing the target frequency component is extracted from it; n intrinsic mode functions are obtained by empirical mode decomposition to represent n stationary signals that meet the characteristics of a single component; a set of instantaneous frequencies is obtained from each intrinsic mode function, and according to the target frequency component, the intrinsic mode function containing the target frequency component is extracted from the n intrinsic mode functions, and this is used as the stationary signal for the subsequent step of rate equation calculation. In the present invention, n = 2 to 10. In this example, n is 5. In order to facilitate subsequent calculations, the time sampling rate is set for the data, and data interpolation and amplitude normalization are performed. Taking the injected optical signal of 5Gbps non-return-to-zero code with an optical carrier center wavelength of 1552.7407nm and a code type of "0111110110" as an example, the signal is received by a photodetector and down-converted, recorded, and the signal is interpolated at a time sampling rate of 1ps, and the amplitude is normalized to obtain the following: Figure 2 The waveform shown in the figure shows the normalized amplitude on the vertical axis and the time on the horizontal axis. Considering the frequency resolution of the spectrum distribution in subsequent calculations and in order to reduce the impact of the marginal effect on the analysis results in the subsequent time-frequency analysis, it is necessary to repeat the optical signal data for multiple cycles, but this will undoubtedly increase the amount of calculation. Therefore, after comprehensive consideration, Figure 2 The signal light waveform in the image lasts for three cycles. In order to extract the stationary signal containing the target frequency component, Figure 2 The injected light signal in the is subjected to empirical mode decomposition, specifically: for the input signal s(t), the maximum and minimum points are obtained; the upper and lower envelopes of the signal are constructed by interpolating the cubic spline function for the maximum and minimum points, and the mean function m(t) of the upper and lower envelopes is calculated; it is determined whether h(t) = s(t) - m(t) is an intrinsic mode function. If not, the above steps are continued for h(t) until h(t) meets the intrinsic mode function conditions; h(t) that meets the intrinsic mode function conditions is subtracted from the original signal, that is: r(t) = s(t) - h(t), and r(t) is regarded as the "original signal" s(t) in the subsequent steps, and the intrinsic mode function is continued to be screened. In this way, the initial injected light signal is decomposed into several intrinsic mode functions and a residual signal; Figure 2 The results of empirical mode decomposition are as follows: Figure 3 As shown, the vertical axis is the normalized amplitude and the horizontal axis is time; the results of calculating the instantaneous frequency are as follows Figure 4 As shown, the vertical axis is the instantaneous frequency, the horizontal axis is the time, and the waveform grayscale is the instantaneous amplitude. Figure 3 The residual signal in does not meet the characteristics of a stationary signal, so its instantaneous frequency is not calculated.

[0036] (3) Initialize the simulated laser parameters. In this example, g = 4.7·10 4 s -1 、N th =2·10 7 ,κ=2.25·10 11 s -1 ,α=5,γ N =1·10 9 s -1 , γ P =5·10 11 s -1 , J=4.1386·10 17 s -1 ;

[0037] (4) Establishing a rate equation group based on optical signal parameters, as shown below;

[0038]

[0039]

[0040]

[0041] Where A(t) is the amplitude of the light output from the laser, φ(t) is the phase difference between the light output from the laser and the injected light; A i (t) is the optical parameter amplitude of the input light, Δω i (t) is the optical parameter frequency ω of the input light i With the free oscillation frequency ω from the laser s The frequency detuning between the two lasers; N(t) is the number of carriers from the laser; g, N th , κ, α, γ N , γ P , J are gain coefficient, carrier number threshold, coupling coefficient, linewidth enhancement factor, carrier recombination rate, photon decay rate and normalized pump current respectively; when the input light is an injected light signal, the above rate equation group is the rate equation group based on the injected light signal parameters described in step (5); when the input light is a stationary signal of the target frequency component, the above rate equation group is the rate equation group based on the stationary signal parameters described in step (5).

[0042] (5) Substitute the parameters of the injected optical signal and the parameters of the stationary signal containing the target frequency component described in step (2) into the rate equations established in step (4), respectively, to form two sets of rate equations based on the injected optical signal parameters and the stationary signal parameters. The two sets of rate equations are solved using the fourth-order and fifth-order Runge-Kutta algorithms, that is, the fourth-order method is used to provide candidate solutions for the equations, and the fifth-order method is used to control the error, so as to obtain the laser dynamic parameters obtained by the corresponding optical signal parameters participating in the injection locking within the simulation duration period with an adaptive step size, that is, the output light amplitude, output light phase, and carrier number. Due to the use of the adaptive step size algorithm, the solution of different substitution objects has good adaptability in terms of simulation calculation amount and accuracy, but this also makes the results not have the same sampling points and fixed frequency resolution. In order to facilitate subsequent analysis and signal reconstruction, after the equations are solved, the solution results will be resampled with the initial sampling interval and time, and the signal will be extracted or interpolated.

[0043] Taking the target frequency component of 10GHz as an example, Figure 4 It can be seen that there is only one natural mode function containing the instantaneous frequency of 10 GHz, which is IMF1. Therefore, only the parameters of IMF1 are substituted into the equations to be solved. Figure 5 Taking an injection power ratio of -9.6dBm and a center frequency difference of 15.5GHz as an example, the results of solving two groups of rate equations after substituting parameters are shown. Group I shows the solution results when the injected optical signal is directly substituted, while Group II shows the solution results when the IMF1 parameters are substituted. From top to bottom, the output light amplitude, output light phase, and carrier number obtained from solving the rate equations are shown. From this, we can make a preliminary judgment that the results in Group I show the characteristics of steady-state locking, which is consistent with the experimental phenomenon, while the results in Group II show the characteristics of non-steady-state locking, which means that the IMF1 containing the target frequency component cannot achieve locking from the laser by injecting it alone.

[0044] (6) The fourth-order and fifth-order Runge-Kutta algorithms are used to solve the above two rate equations respectively, thereby obtaining the number of carriers and the amplitude and phase of the output light of the slave laser in which the injected light signal participates in locking, and the number of carriers and the amplitude and phase of the output light of the slave laser in which the stationary signal participates in injection locking, and then obtaining the time-frequency distribution and spectrum of the two groups of optical signals obtained after the injected light signal and the stationary signal light of the target frequency component are respectively injected into the slave laser.

[0045] The results of solving the two rate equations include the amplitude A(t) of the output light from the laser, the phase difference φ(t) between the output light from the laser and the injected light, and the number of carriers N(t) from the laser. Based on the amplitude A(t) of the output light from the laser and the phase difference φ(t) between the output light from the laser and the injected light, two sets of optical signals are reconstructed by calculation and superposition. The two sets of optical signals include the output light from the laser injected by the injected light signal and the output light from the laser injected by the stationary signal. The two sets of optical signals are decomposed again using the empirical mode, and the instantaneous frequency of the two sets of optical signals is obtained. The time-frequency distribution of the two sets of optical signals is obtained based on the amplitude A(t) and instantaneous frequency of the two sets of optical signals. The two sets of optical signals are discrete Fourier transformed to obtain the spectrum. In this example, the output optical signals from the laser are respectively established based on the amplitude and phase information of the two sets of solutions, and their time-frequency distribution is calculated using the same time-frequency analysis method as before, and then the spectrum of the two sets of optical signals is calculated.

[0046] (7) By comparing the time-frequency distribution and spectrum of the two groups of optical signals, analyze whether the distribution change of the target frequency component during the injection locking process meets the requirements. If it does not meet the requirements, adjust the parameters of the stationary signal and superimpose it with the other single-component stationary signals in step (2), reconstruct it into a new injection optical signal and return to step (1); if it meets the requirements, end the simulation process. Among them, the parameters of the stationary signal to be adjusted include frequency, phase, power and waveform. Analyzing the solution results in the example, we get the following: Figure 6 The time-frequency distribution shown is the same as Figure 7 The spectrum comparison shown in the figure (the first row is the input optical spectrum, and the second row is the output optical spectrum) shows that Group I and Group II respectively analyze the solution results of two sets of rate equations. The output optical spectrum of Group I shows a subcarrier near 20.5 GHz. Its time-frequency distribution intuitively shows the occurrence position of this subcarrier in time, corresponding to the "1" code distribution of the injected optical signal. However, the instantaneous frequency "jitter" during the duration of the "1" code cannot be physically explained. While the solution results of Group II show from the waveform that the injection of the intrinsic mode function IMF1 alone cannot lock the slave laser, the time-frequency distribution of Group II shows that the instantaneous frequency of the unstable subcarrier in the output light still shows a regular distribution in time: from the time the code element changes from "0" to "1" to the time the code element changes back to "0", the change of the instantaneous frequency of Group II always shows a gradual to drastic change. Figure 4From the time-frequency distribution of IMF1, we can infer that the impact of the target frequency component of 10 GHz on injection locking is mainly located on the rising edge of the code element from "0" to "1", and the longer the duration of "1", the longer the duration of this impact. If we want to establish an injection signal dominated by the target frequency component of 10 GHz, we can keep the other intrinsic mode functions unchanged and adjust IMF1 from the waveform, amplitude, and frequency. Under the premise of ensuring that it still meets the definition of the intrinsic mode function, we can increase the proportion of the target frequency in its time-frequency distribution. Then, we can reconstruct it with the other intrinsic mode functions into a new injection optical signal and conduct a new round of simulation analysis.

[0047] Although the present invention has been described above in conjunction with the accompanying drawings, the present invention is not limited to the above-mentioned specific embodiments. The above-mentioned specific embodiments are merely illustrative and not restrictive. Under the guidance of the present invention, ordinary technicians in this field can make many variations without departing from the purpose of the present invention, and these are all protected by the present invention.

Claims

1. A method for simulating discontinuous optical injection locking of a laser combined with time-frequency analysis, characterized in that: The following steps are involved: 1) Read the injected light signal; 2) Downconverting the injected optical signal to obtain a baseband signal, and using empirical mode decomposition to represent the baseband signal as a superposition of n single-component stationary signals; extracting a stationary signal containing a target frequency component therefrom; 3) Initialize the slave laser parameters; 4) Establishing a rate equation system based on optical signal parameters; 5) Substituting the parameters of the injected optical signal and the parameters of the stationary signal containing the target frequency component described in step 2) into the rate equations described in step 4), thereby forming a rate equation group based on the parameters of the injected optical signal and a rate equation group based on the parameters of the stationary signal; 6) using a fourth-order / fifth-order Runge-Kutta algorithm to solve the two rate equations, respectively, thereby obtaining the number of carriers and the amplitude and phase of the output light of the slave laser in which the injected optical signal participates in locking, and the number of carriers and the amplitude and phase of the output light of the slave laser in which the stationary signal participates in injection locking, and further obtaining the time-frequency distribution and spectrum of the two sets of optical signals obtained after the injected optical signal and the stationary signal light of the target frequency component are respectively injected into the slave laser; 7) By comparing the time-frequency distribution and spectrum of the two sets of optical signals, analyze whether the distribution change of the target frequency component during the injection locking process meets the requirements. If not, adjust the parameters of the stationary signal and superimpose it with the other single-component stationary signals in step 2), reconstruct it into a new injected optical signal and return to step 1); if it meets the requirements, end the simulation process.

2. The method for simulating discontinuous optical injection locking of a laser according to claim 1, wherein: In step 7), the parameters of the stationary signal are adjusted, including frequency, phase, power and waveform.

3. The method for simulating discontinuous optical injection locking of a laser according to claim 1, wherein: In step 2), empirical mode decomposition is used to obtain n intrinsic mode functions to represent n stationary signals that meet the characteristics of a single component; a set of instantaneous frequencies is obtained from each intrinsic mode function, and according to the target frequency component, the intrinsic mode function containing the target frequency component is extracted from the n intrinsic mode functions, and this is used as the stationary signal participating in the rate equation calculation in the subsequent step.

4. The method for simulating discontinuous optical injection locking of a laser according to claim 3, wherein: n＝2～10。 5. The method for simulating discontinuous optical injection locking of a laser according to claim 1, wherein: The rate equations based on the optical signal parameters established in step 4) are as follows: Where A(t) is the amplitude of the light output from the laser, φ(t) is the phase difference between the light output from the laser and the injected light; A i (t) is the optical parameter amplitude of the input light, Δω i (t) is the optical parameter frequency ω of the input light i With the free oscillation frequency ω from the laser s The frequency detuning between the two lasers; N(t) is the number of carriers from the laser; g, N th , κ, α, γ N , γ P , J are gain coefficient, carrier number threshold, coupling coefficient, linewidth enhancement factor, carrier recombination rate, photon decay rate and normalized pump current respectively; When the input light is an injected light signal, the above-mentioned rate equation group is the rate equation group based on the injected light signal parameters described in step 5); when the input light is a stationary signal of the target frequency component, the above-mentioned rate equation group is the rate equation group based on the stationary signal parameters described in step 5).

6. The method for simulating discontinuous optical injection locking of a laser according to claim 1, wherein: In step 6), the results of solving the two rate equations include the amplitude A(t) of the light output from the laser, the phase difference φ(t) between the light output from the laser and the injected light, and the carrier number N(t) from the laser; Reconstructing two sets of optical signals by calculating and superimposing the amplitude A(t) of the output light from the slave laser and the phase difference φ(t) between the output light from the slave laser and the injected light, the two sets of optical signals including the output light from the slave laser injected by the injected light signal and the output light from the slave laser injected by the stationary signal; The two optical signals are decomposed again using empirical mode decomposition, and their instantaneous frequencies are calculated. The time-frequency distributions of the two optical signals are obtained based on their amplitudes A(t) and instantaneous frequencies. Discrete Fourier transforms are performed on the two optical signals to obtain their spectra.

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