1.

INTRODUCTION

Laser coherence detection has the advantages of high sensitivity, high measurement accuracy, high directivity, strong anti-interference ability and so on. It is widely used in target detection, synthetic aperture imaging, velocity measurement and other fields[1-4]. With the continuous progress of technology, the target speed is faster and the Doppler frequency is larger, and the requirements for photoelectric conversion and data processing are higher. Therefore, the conventional laser coherence detection method has been unable to meet the higher detection requirements. Dual-frequency laser coherent detection can be used to convert the higher laser frequency to the lower microwave frequency by the method of optical microwave, so as to achieve higher speed target detection[5-7]. Dual frequency laser coherence detection has better advantages in high precision measurement and has become a hot research field[8-10]. The four-optical coherent mixing detection method can realize the velocity measurement in a larger dynamic range by choosing the appropriate frequency difference and detector[11]. It is difficult to distinguish the output signal type of mixing, and using the signal spectrum is one of the effective methods. Due to the influence of target reflection, atmospheric turbulence, light source linewidth and other factors, the spectrum broadening results in the difficulty of resolving the output signal spectrum. At present, many scholars have carried out research on the influence of light source linewidth on laser coherent detection signal[12-15]. There are few researches on the influence of light source linewidth on the signal spectrum of four-optical coherent mixing detection. In this paper, the dynamic range of velocity measurement of the four-optical coherent mixing detection method is analyzed. In view of the difficulty of discriminating the output signal type of mixed frequency, a method of discriminating by using the signal spectrum is proposed. Combined with the statistical theory, the power spectrum function of the coherent component in the four-optical coherent mixing signal is obtained by using the Wiener - Hinchin theorem, and the influence of the line width, frequency difference and speed of the light source on the signal power spectrum is analyzed numerically.

2.

PRINCIPLE OF FOUR-OPTICAL COHERENT MIXING DETECTION

In the four-optical coherent mixing detection method, the assumption of two oscillating light with different frequencies is:

In the formula, U_o1 (r) and U_o2 (r) represent the amplitudes of the oscillations of different frequencies; f_o1 and f_o2 represent the frequencies of the two oscillations; the frequency difference between the two oscillations is Δf_o = f_o2 – f_o1; φ_o1(t) and φ_o2(t) respectively represent the random phase of the two oscillations.

In the four-optical coherent mixing detection, the two-reflection signal light of different frequencies is:

In the formula, U_s1(r) and U_s2(r) respectively represent the amplitude of the two reflected signal light, δ_o1 = 2Vf_o1/c and δ_o2 = 2Vf_o2/c represent the Doppler frequency shift, V represents the radial motion speed of the detection target, and c represents the speed of light, τ_d is the delay time, φ_s1(t) and φ_s2(t) represent the random phase of the two reflected signal light.

Two local oscillator lights and two reflected signal lights are mixed on the detector surface. In this paper, only the difference frequency term after mixing is retained. The theoretical four light coherent mixing signal can be expressed as:

In the formula, I_o, I_s and I_osn (n = 1,2,3,4) respectively represent the amplitude of the signal after mixing, Δφ_o(t), Δφ_s(t) and ϕ_osn(t) respectively represent the random phase of the signal after mixing.

In Formula (3), i_o(t) is the difference frequency signal generated by the coherence of the two local vibrators E_o1 (r, t) and E_o2 (r, t), and its signal frequency is the laser frequency difference Δf; i_s(t) refers to the differential frequency signal generated by the coherence of two reflected signal light, E_s1 (r, t) and E_s2 (r, t). In the remote laser coherent detection, the signal component i_s(t) generated by the coherence of two reflected signal light is very small, so the influence of this signal can be ignored; i_os1(t) and i_os2(t) are differential frequency signals generated coherently by E_o1 (r, t) and E_s1 (r, t), E_o2 (r, t) and E_s2 (r, t), respectively. i_os1(t) and i_os2(t) can be called homodyne coherent signals. i_os3(t) and i_os4(t) are differential frequency signals generated coherently by E_o1 (r, t) and E_s2 (r, t), E_o2(r, t) and E_s1(r, t), respectively. i_os3(t) and i_os4(t) can be called heterodyne coherent signals.

According to Formula (3), the frequencies of heterodyne coherent signals i_os3(t) and i_os4(t) are Δf+δ_o2 and Δf–δ_o1 respectively. When the target moves towards the detection system, the Doppler frequency shift is positive, and when the target is far away from the detection system, the Doppler frequency shift is negative. Therefore, no matter whether the Doppler frequency shift is positive or negative, the frequency of heterodyne coherent signals i_os3(t) and i_os4(t) is always in the form of Δf –|δ₁| or Δf –|δ₂|, that is, the signal frequency and the target speed are decreasing, which is a unique feature of four light coherent mixing detection.

According to the photoelectric characteristics, the photoelectric detector has a certain cut-off response frequency f_c, and the detector can respond only when the frequency of the mixing signal is less than the cut-off response frequency of the detector ^[13]. According to Formula (3), the mixing signal component output by the detector is directly related to the cut-off response frequency f_c of the detector, the frequency shift Δf of the acousto-optic frequency shifter, and the Doppler frequency shift of the moving target. When f_c < Δf ≤ 2f_c, the four light coherent mixing detection system can realize continuous detection of moving targets in a larger dynamic range.

Assuming the laser wavelength λ = 532 nm, the cut-off response frequency f_c = 2 GHz, the laser frequency difference Δf = 4 GHz, and the Doppler frequency shift is positive, the relationship between the d four light coherent mixing detection signal component and the target radial motion speed is obtained through simulation, as shown in Figure 1.

Figure 1

Variation curve of frequency of four-optical coherent mixing detection signal with radial velocity.

It can be seen from the figure that the frequency of signal i_o(t) is a constant invariant. The Doppler frequency shifts δ_o1 and δ_o2 of homodyne coherent signals i_os1(t) and i_os2(t) show a linear increasing relationship with the target

speed; The frequency Δf +δ_o2 of heterodyne coherent signal i_os3(t) also has a linear increasing relationship with the radial velocity of the target. The frequency Δf – δ_o1 of heterodyne coherent signal i_os4(t) shows a decreasing relationship with the radial motion velocity of the target.

It can be seen from Figure 1 that within the range of the cut-off response frequency f_c of the detector, the velocity value corresponding to the longitudinal axis frequency is not unique. Therefore, how to distinguish whether the signals output by the detector are homodyne coherent signals i_os1(t) and i_os2(t) or heterodyne coherent signals i_os4(t) is an important problem in velocity measurement. When the target’s radial motion speed is between 0~532m/s, the output is the homodyne coherent signal components i_os1(t) and i_os2(t). When the target’s radial motion speed is between 532~1064 m/s, the output signal is the heterodyne coherent component signal i_os4(t).

Assuming that the target’s radial velocity is 100m/s and 1000m/s respectively, the spectrum of the output homodyne coherent signal and heterodyne coherent signal is shown in Figure 2. It can be seen from Figure 2 (a) that the spectrum of homodyne coherent signal component output by the detector contains two adjacent spectral peaks, and the theoretical frequency difference between the two adjacent spectral peaks is 2666.67Hz. However, the spectrum of heterodyne coherent signal components has only one peak, as shown in Figure 2 (b)

Figure 2

Signal power spectra at different speeds.(a) Power spectrum of homodyne signal.(b) Power spectrum of heterodyne signal

3.

INFLUENCE OF LIGHT SOURCE LINEWIDTH ON SIGNAL POWER SPECTRUM

According to the above analysis, the frequency spectrum δ₁ and δ₂ of signal components i_os1(t) and i_os2(t) can be distinguished in the frequency domain, which is an important basis for judging homodyne coherent signals. In practice, the signal spectrum is affected by the linewidth of the light source, the laser noise and the phase noise caused by the environment. The signal has certain fluctuations and broadening in the frequency domain, which increases the difficulty of signal spectrum resolution [13] [14]. When the frequency difference Δf of the light source is small or the Doppler frequency difference Δδ is small when the moving speed of the target is small, the resolution of the spectrum δ₁ and δ₂ becomes difficult.

In this paper, the homodyne coherent signals i_os1(t) and i_os2(t) are regarded as random stationary signals, and their autocorrelation functions can be expressed as:

Δϕ(t, τ) = ϕ(t + τ) – ϕ(t) represents the correlation of random phase noise at different times, which can also be understood as the phase change in time τ. According to the signal noise theory, the random phase change of the signal is a zero mean Gaussian random process, so the following relationship can be obtained [15]:

In the formula, Δw_Fw represents the full widtd at half maximum of the laser source, τ_c = 2π/Δw_Fw represents the coherent time, and the relationship between the magnitude of τ_d and τ affects the power spectral function of the random signal.

The autocorrelation functions of i_os1(t) and i_os2(t) can be expressed as:

In the formula, ω_o1 = 2πδ_o1, ω_o2 = 2πδ_o2.

The autocorrelation function of a random signal is an even function. According to the Wiener Sinchen theorem, the autocorrelation function of a random stationary signal and the power spectral density function are Fourier change pairs. Fourier transform formula of signal autocorrelation function:

Take Formula (8) into Formula (9), and get the power spectrum functions of signals i_os1(t) and i_os2(t) by rounding off the negative frequency components through integral operation:

According to the above formula, the power spectrum of homodyne coherent signals i_os1(t) and i_os2(t) is related to laser linewidth Δw_FW, delay time τ_d, Doppler frequency shift δ₁ and δ₂.

4.

NUMERICAL SIMULATION ANALYSIS

The power spectra of homodyne coherent signals i_os1(t) and i_os2(t) in four optical coherent mixing are numerically analyzed and discussed by using formula (10).

In the numerical analysis, if the laser wavelength λ =532 nm and τ_d = 0.1τ_c when the target moving speed is 50 m/s and the light source linewidth Δw_FW is 3 kHz, 5 kHz and 10 kHz respectively, the numerical simulation results of power spectrum under different light source frequency differences are shown in Table 1.

Table 1

V =50 m/s, the numerical calculation results of the power spectrum under different light source frequency differences

According to Table (1),when the light source linewidth Δw_FW = 3 kHz and the laser frequency difference is 1.7GHz, the theoretical Doppler frequency difference is 566.7Hz, and the power spectra of signal i_os1(t) and i_os2(t) can be distinguished. When the linewidth of the light source Δw_FW = 5 kHz and the laser frequency difference is 3GHz, the theoretical Doppler frequency difference is 1000Hz, and the power spectra of signals i_os1(t) and i_os2(t) can be distinguished. When the linewidth of the light source Δw_FW =10 kHz and the laser frequency difference is 6GHz, the theoretical Doppler frequency difference is 2000Hz, and the power spectra of signals i_os1(t) and i_os2(t) can be distinguished.

When the motion speed is 50 m/s and 100 m/s respectively, the frequency difference of the light source is F. The numerical calculation results are shown in Table 2.

Table 2

Δf = 4 GHz, the numerical calculation results of the power spectrum under different light source line

When the motion speed V=50 m/s, the theoretical Doppler frequency difference is Δδ = 1333.3 Hz. From the numerical calculation results, when the linewidth of the light source is less than or equal to 7 kHz, the peak frequency difference of the power spectrum of signals i_os1(t) and i_os2(t) can be extracted, that is, the power spectrum of signals i_os1(t) and i_os2(t) can be distinguished; When the linewidth of the light source is greater than 7 kHz, the peak frequency difference of the power spectrum of signals i_os1(t) and i_os2(t) cannot be extracted, that is, the power spectrum of signals i_os1(t) and i_os2(t) cannot be distinguished.

When the motion speed V=100 m/s, the theoretical Doppler frequency difference is Δδ = 2666.67 Hz. From the numerical calculation results, when the linewidth of the light source is less than or equal to 14 kHz, the power spectrum of signals i_os1(t) and i_os2(t) can be distinguished; When the linewidth of the light source is greater than 14 kHz,the power spectrum of signals i_os1(t) and i_os2(t) cannot be distinguished.

From the above numerical analysis results, From the above analysis results, it can be concluded that the condition that the power spectrum of signal i_os1(t) and i_os2(t) can be distinguished is that the theoretical Doppler frequency difference Δδ is greater than 1/5 of the light source linewidth Δw_FW. At the same time, when the linewidth of the light source and the moving speed of the target are fixed, the smaller the laser frequency difference, the smaller the theoretical Doppler frequency difference, the greater the difficulty of power spectrum resolution of the homodyne coherent signals i_os1(t) and i_os2(t), and the greater the error of spectrum resolution.

5.

CONCLUSION

In the four light coherent mixing detection, in view of the difficulty in distinguishing the mixing output signal, this paper proposes a method to distinguish by using the power spectrum, obtains the zero difference coherent signal power spectrum model by using the statistical theory, and analyzes the relationship between the light source linewidth, frequency difference, motion speed and the signal power spectrum. Through theoretical analysis, the limit condition that the power spectrum of homodyne coherent signals with different frequencies can be resolved is that the Doppler frequency difference Δδ is greater than 1/5 of the light source linewidth f_FW. The research results in this paper show that the four light coherent mixing detection method can achieve a wider range of moving target detection, and theoretically prove that there is a resolution limit for two homodyne coherent signals in the frequency domain. But this research content also needs strict experimental verification, and needs to explore more effective mixed frequency signal discrimination methods.