1. Introduction

Coherent Doppler wind lidar (CDWL) has become an essential technology for atmospheric wind measurements. It has been extensively used in the fields of aviation wind-shear detection [1], weather prediction [2–4], wind power site evaluation [5,6], and ship wake safety assessment [7,8]. A CDWL system generally adopts the monostatic coaxial transceiver (MCT) to guarantee alignment accuracy and better the coherent signal [9]. In the MCT system, the laser pulses with a tail will be reflected by the optical end surface. The end-face reflectance signals are significantly stronger than the atmospheric backscattering signals. The MCT system exhibits a near-field blind zone problem (BZP) for the wind Doppler spectrum that is annihilated by the end-face reflectance crosstalk noise [10]. The blind zone distance can reach tens to hundreds of meters, according to the performance of the laser. However, the near-field detection capability is important for detecting aircraft wake vortex [11], turbine vortex [12], ship-offshore [13], etc. Therefore, solving the BZP problem is of great significance.

Some prescriptions for solving the BZP problem of the CDWL have been considered. Tang et al. [14] localized the crosstalk noise source of the continuous wave MCT CDWL and used the delay fiber, polarization control, and AR coating to reduce the crosstalk noise to some extent. The continuous wave coherent lidar’s effective signal intensity at the laser focal point can be large enough to be identified with the reduced crosstalk noise. However, for the pulse coherent lidar, the blind zone crosstalk noise is much higher than the effective signal level, and the effectiveness of these methods in reducing crosstalk noise is quite limited. Yang et al. [15] proposed to use the pseudo-phase coding technology to improve the distance resolution of the pulse CDWL system and solve the near-field blind zone problem. However the coding system has a lower magnitude of the power spectrum, compared with a non-coding system [16], so the effective detection distance of the coding system is limited. Wang et al. [17] realizes a dual optical receiving direct detecting lidar system to solve the blind zone problem caused by the system geometry factor of the transceiver allograft. Magee et al. [18,19] demonstrated the feasibility of the bistatic coherent lidar system. The bistatic coherent lidar system already has some practical applications. Gatt et al. [20], Weeks et al. [21], Ouisse et al. [22] Belmonte and Kahn [23] have researched coherent receiving arrays but mainly focus on the problem of the random phase scintillation effect caused by turbulence, which affects the effective intermodulation efficiency and thereby limits the maximum SNR of CDWL system. A great solution involves addressing the blind zone problem by incorporating a biaxial receiver system to receive signals while preserving the performance of the pulse-coherent lidar. However, the research on the biaxial receiver system’s performance in the near field, the feasibility, and the specific design of solving the BZP, is lacking.

This paper provides the ABR method that establishes a transceiver separate configuration to avoid the influence of crosstalk light and solve the near-field BZP while retaining the original performance of CDWL. The remaining sections of this paper are structured as follows: Section 2 describes the specific structure and the numerical simulation model of the ABR system. Section 3 presents the analysis of the impact of various ABR parameters such as the focal length, defocus, truncation ratio, and optical axis direction. The result presents that the ABR method for solving the BZP is practical when a sound design is adopted. Section 4 is the summarization.

2. Theory and numerical model

2.1 SNR equation for pulse coherent ABR system

The theoretical discussions regarding laser coherent detection have often been concentrated on the MCT system. Many studies on biaxial systems have mainly been based on the concentric and coaxial transceiver model [20,22]. As Fig. 1(a) shows, a typical MCT system comprises several components. The continuous-wave laser (CW) generates a laser beam, then the beam is divided by a beam splitter (BS) into two segments. One segment undergoes modulation into pulses via the acoustic-optic modulator (AOM), with the pulses subsequently amplified by the erbium-doped fiber amplifiers (EDFA) to increase their average power. The monostatic coaxial transceiver Tx then collimates these pulsed beams before emitting them into space. The circulator (Cir) seperates the emitted light and the backward-scattered light into respective channels. The remaining segment of the CW laser acts as the local oscillator (LO) and is mixed with the backward-scattered light to generate a coherent signal. The balanced photodetector (BPD) converts this optical signal into an analog electrical signal, then the data acquisition system (DAQ) continuously digitizes the analog signal. The digital signal is processed to derive wind speed measurements at various radial distances. The additional auxiliary biaxial receiver (ABR) with the BS, mixer, and BPD attached to it, and the original MCT system excluding mixer and BPD together compose the new ABR system. This augmentation allows the overall system to retain the detection capabilities of a conventional pulsed MCT lidar while also facilitating ABR detection.

 

Fig. 1. The total system with the MCT subsystem and the ABR subsystem layout (a) and schematic of target plane and meshing (b).

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As mentioned above, to overcome the BZP, one auxiliary biaxial receiver (ABR) that operates under non-coaxial, non-concentric, and even non-parallel to the MCT main transmitter (Tx) to maximize the overlap function in the blind zone, in which the characters and aims are different from the traditional bistatic coherent lidar [24]. When neglecting the turbulence effect, the SNR function of a coherent lidar system can be written as Eq. (1) [9,20,22,24–27]. The turbulence effect also can be considered [21,28] but it is not this paper’s focus.

Compared to the traditional MCT systems, the ABR offers independent and isolated optical paths for reception, eliminating emission reflection interference. The backscattered signal of coherent wind lidar is mainly generated by the aerosol Mie scattering process. Due to the characteristics of the Mie scattering signal, it is feasible to receive enough energy for the ABR system once the auxiliary receiving telescope is close enough to the transceiver telescope of the MCT system. The primary difference between an ABR system and an MCT system lies in the distribution of the BPLO field on the target plane. For an ABR system, the ${I_{nt}}$ is determined by the transmitting system, and the ${I_{nlo}}$ is influenced by the ABR telescope’s characteristics, including the focal length, aperture size, defocus, position, and orientation relative to the transmitting telescope. The optical axis of the ABR needs an appropriate angular separation and position, and a good modulation of the BPLO field to achieve a comprehensive matching with the transmitted optical field at various distances within the blind zone, rather than just a single-distance emission field.

To evaluate the influence of these ABR characteristics, two coordinate systems, describing the ABR system including the transmitter, are established, as shown in Fig. 1(b). One coordinate system, defined as ABR coordinates, for the fixed frame of the ABR telescope, has its origin at the center of the exit plane of the telescope. The z-axis is aligned with the optical axis of the telescope, and the xy-plane coincides with the exit plane of the telescope's lens. Similarly, the other coordinate system, defined as Tx coordinates, is established for the transmitting telescope. For the sake of convenience, the origin of the Tx coordinate system is set on the y-axis of the ABR coordinate system. In other words, the ABR coordinates of the transmit telescope center is $({{x_0},0,0} )$, where ${x_0}$ represents the distance between the transmitting and ABR telescopes along the x-axis. The target plane is defined by a parallel xy-plane. Due to the axial symmetry of Gaussian beams, the yaw angle $\gamma $ around the z-axis does not affect the field distribution on the target plane. Therefore, only two rotational parameters, the roll angle $\alpha $, and the pitch angle $\beta $, need to be considered. ${X_t}(x,y,z)$, ${Y_t}(x,y,z)$ represents the transverse coordinates of the target point in the Tx coordinate system. ${Z_t}$ represents the axial distance of the target point in the Tx system, where $(x,y,z)$ is the ABR coordinates. The two coordinate transformation is given by Eq. (2),

In the established coordinate systems, the distribution of the normalized transmitted light intensity ${I_{nt}}(x,y,z)$ on any target plane and the distribution of the normalized BPLO intensity ${I_{nlo}}$ can be calculated using Eq. (3) and Eq. (4) [26,27,29],

${r_{lo}}$ and ${r_t}$ are the effective radiuses of the ABR and Tx telescopes. The expressions for the curvature radius ${F_{lo,t}}$, ${e^{ - 2}}$ intensity radius ${w_{lo,t}}$ at the of the ABR and Tx telescope plane, and the Gaussian beam waist radius ${w^{\prime}_{{0_{lo,t}}}}$ at the ABR and Tx fiber end face and its position distance ${s^{\prime}_{lo,t}}$ relative to the ABR and Tx telescope, are given by Eqs. (5–8) [30],

${Z_0} = \pi w_0^2/\lambda $, ${Z^{\prime}_{0_{lo,t}}} = \pi {w^{\prime{2}}_{0_{lo,t}}}^2/\lambda $ represents the Rayleigh length of the Gaussian beam. Parameters denoted by subscript symbols $_{lo,t}$ signify that these parameters are relevant to both LO light and emitter light, and they are governed by the same formula in both instances.

For a fiber-coupled receiver system, when the coupling end face of the fiber is in the focal plane of the telescope, the effective receiving range of the target plane is determined by the field of view (FOV) and the distance to the target plane. The FOV is determined by the focal length of the receiving telescope and the core diameter and numerical aperture of the receiving fiber. Assuming that the angle ${\theta _{re}}$ of FOV (the half conic vertex angle corresponding to the solid angle) is small and the effective receiving area $S$ of the target plane is a circle with a radius ${R_{re}}$, then it can be described as Eq. (9) and Eq. (10),

As Eq. (11) shows, if the telescope is reasonably designed so that the received light can be focused on the end surface of the fiber core, and the image numerical aperture (NA) is smaller than the fiber’s NA. Otherwise, the effective receiving area of the telescope will be truncated for the numerical aperture of the optical fiber,

The situation becomes more complicated when the fiber end surface is out of the focal plane (a.k.a. defocus). The fiber core diameter and position will have a truncating effect on the received area in the same FOV, as Fig. 2 shows. The truncation effect is equivalent to the truncation of a point light source spot by a diaphragm on the telescope plane, where the diaphragm is the fiber optic core and the point light source is the focusing point in the focal plane. The defocusing effect will weaken the actual coherent signal, but considering it will extremely complicate the calculation. Thus the subsequent calculations in this paper ignore the additional truncation effect of the defocus and only consider the effect of the defocus on the modulation of the BPLO Gaussian beam as Eqs. (5–8).

 

Fig. 2. Schematic of target plane and meshing.

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The laser power’s spatiotemporal distribution can be determined by the laser emission time, pulse energy, and pulse width. By accumulating signals within the range of a signal range gate and considering only the local oscillator light’s shot noise, the SNR for individual range gates can be obtained. Excluding factors such as laser energy, atmospheric extinction, and detector efficiency, we can extract the sole influence of the telescope's optical field modulation on the path signal contribution. This influence can be defined as coherent overlap factor (COF) ${\eta _c}$, and is described by Eq. (12),

2.2 Numerical model

The numerical calculation is implemented to quantitatively analyze the influence of different parameters on the effective detection range of the ABR system. The specific flowchart of the numerical simulation is illustrated in Fig. 3. Firstly, the system parameters are initialized based on the physical and mathematical model. This involves dividing the target plane according to the blind zone and further subdividing it into grid points. Subsequently, the coordinates of each grid point and their corresponding intensities of Tx and BPLO light are computed. The COF of each plane is then determined by summing the products of each grid’s area, Tx, and BPLO light intensities within the target plane respectively. Within each distance gate, the COFs from different target planes are accumulated and adjusted by coefficients derived from other parameters to calculate the SNR. Meanwhile, based on the initialized system parameters and the required accuracy for wind speed detection, the SNR and the minimum COF threshold necessary for detection are determined. These two indicators enable the assessment of whether the chosen ABR design is qualified for blind zone detection. Running the analysis with different ABR parameters reveals the relationship between different ABR design parameters and the system performance. This iterative process enables fine-tuning and optimization of the ABR system for enhanced blind zone detection capabilities.

 

Fig. 3. Numerical modeling flow of transceiver diversity.

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Since the analysis primarily focuses on optimizing the BZP, the ABR telescope parameters (${f_{lo}}$, $d{f_{lo}}$, ${x_0}$, $\alpha $, $\beta $, ${\rho _{locut}}$ and ${R_{lo}}$) are the key variables for analysis. Other parameters related to the transmitter ($\lambda $, P, ${w_0}$, $\Delta T$, ${f_t}$, $d{f_t}$, ${\rho _{tcut}}$), atmosphere (${\beta _s}$, $\sigma $) and receiver ($B$, ${F_h}$, M, ${N_P}$, ${f_s}$) are set to constant values that can be determined by a typical MCT CDWL system and atmosphere environment. The integration computation of the target plane's light intensity can be achieved through finite element method (FEM). In FEM, the path is divided to target planes, and the target plane is divided into grids. As Fig. 1 shows, the target plane grids are uniformly partitioned along both the x-axis and y-axis, with a fixed grid number N in each direction. The horizontal and vertical distances ${D_x},\;{D_y}$ are determined by the minimum value between the telescope's FOV radius ${R_t}$ (the blue circle of Fig. 1) and the effective radius of the BPLO field ${R_{lo}}$ (the yellow circle of Fig. 1). Therefore, the grid partitioning satisfies the condition,

The effective radius of the BPLO field is determined by the energy distribution of an untruncated Gaussian beam propagating from the source to the target plane [19]. To cover the most energy, the effective radius of the field is set to three times the beam waist radius,

The target area is discretized into smaller segments, enabling precise calculations and analysis at different locations within the target plane. In the coordinate system of the receiving telescope, the coordinates of each target grid point are,

For a normalized-energy truncated Gaussian light field, the corresponding ${I_{nt}}$ and ${I_{lo}}$ can be calculated for each coordinate position based on the propagation Eqs. (3), (4). The COF at each corresponding grid point is computed by multiplying the normalized transmitted light intensity and the normalized BPLO intensity. Subsequently, by summing the COF_sim of each target plane, using Eq. (1), the corresponding SNR_sim for each range gate can be obtained. Therefore, the ABR system’s performance for different structural and positional parameter sequences of the ABR telescope can be analyzed.

It should be noted that the SNR of a system involves many parameters, including the characteristics of the laser, the performance of the detector, the effects of the atmospheric environment, and the parameters of the optical components. To make the model practical, these parameters’ typical values are provided in Table 1, which is based on a conventional MCT system. The blind range of the MCT system is set to 90 meters, including the first three range gates with a resolution of 30 meters. To resolve the BZP, the minimum SNR is required for each of these three distance gates. According to the Cramer-Rao lower bound as Eq. (16) [20] shows, the minimum SNR is determined by the wind resolution ${v_{cr}}$,

Table 1. One general MCT system parameters

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This paper mainly focuses on how to solve the BZP by introducing an ABR telescope to form a biaxial transceiver subsystem. With other parameters assumed to be constant, the impact of the ABR telescope parameters on blind zone performance can be analyzed. In the next section, different ABR telescope parameters’ contributions are systematically evaluated, and the optimal design for the ABR telescope that would enable the SNR to meet the minimum requirement for all range gates in the blind zone is determined.

3. Analysis of ABR parameters

The SNR of the ABR is influenced by the geometric position and rotation of the ABR telescope and its optical parameters. To obtain the optimal design of ABR, it is necessary to first determine reasonable ranges of these parameters’ values. The impact of the rotation angle and offset parameters on the COF of the ABR system is relatively straightforward due to the mismatch between the transmitted light field and the BPLO field on the target plane for any ABR telescope.

Figure 4(a-b) shows the transmitter intensity and the BPLO of the ABR, and Fig. 4(c) is the corresponding target plane’s COF that is determined by the transmitted and BPLO fields’ intensity distributions and relative positions on the target plane itself. The white crosshair depicted in these figures represents the projection of the xy-axis of the principal coordinate system, also known as the ABR coordinate system, along the z-axis in the direction of the target plane. Assuming that the telescope truncation effect is ignored and the target plane transmitted and BPLO fields are both non-truncated Gaussian optical fields. The normalized radius of the effective receiving region ($n{R_{re}}$), the normalized radius of the BPLO optical field ($nw{0_r}$), and the normalized center distance between the transmitter and the BPLO optical fields ($n{x_0}$) are normalized by the ${e^{ - 2}}$ intensity radius ($w{0_t}$) of the transmitted Gaussian optical field. The nCOF of the ABR system is normalized by the COF of the MCT system. Figure 4(d-f) presents the nCOF changing with respect to $n{x_0}$ and $nw{0_r}$. When the receiving area is limited as in Fig. 4(e) case ($n{R_{re}} = 1$), both the center distance and the radius of the BPLO light field should not be too large to obtain a high nCOF. The approximation can be obtained $\textrm{nCOF} > - 20\textrm{dB}$ (as a typical example) in the triangular region $n{x_0} - 3 + 0.3nw{0_r} < 0$, $n{x_0} > 0$, $nw{0_r} > 0$. As the $nw{0_r}$ and $n{x_0}$ tend to zero, the nCOF value increases. When the receiving area is smaller like in Fig. 4(d) case ($n{R_{re}} = 0.1$), the $\textrm{nCOF} > - 20\textrm{dB}$ triangular region will be closer to the origin. If the effective receiving area is large enough as Fig. 4(f) case ($n{R_{re}} = \textrm{10}$), the area expands into a scalloped shape, allowing for larger values of $nw{0_r}$ and $n{x_0}$. However, for real coherent lidar systems especially those with fiber receivers, the effective receiving area is limited in the near-field region by the focal length of the ABR telescope and the NA of the receiver fiber. As in Fig. 4(h), within the target distance of 90 m, to make the target plane have a large receiving area, the receiving telescope is required to have a small focal length. A small focal length also causes the $nw{0_r}$ value to increase as Fig. 4(g) suggests. An appropriate focal length should balance the $n{R_e}$ and the $nw{0_r}$ to get higher nCOF. Since the parameter values after normalization under different parameters still have a large span, the unit dB is used here to express its value, i.e., the value of the color bars of Fig. 4(d-f), Fig. 4(g) and Fig. 4(f) are expressed as $10\lg (\textrm{nCOF})$, $10\lg (nw{0_r})$, and $10\lg (n{R_{re}})$.

 

Fig. 4. Grid distribution of target plane intrinsic light (a), emitted light (b), and coherent light field (c). The profile of the ABR system’s nCOF under different values of $nw{0_r}$ and $nx0$ when $n{R_{re}} = 0.1$(d), $n{R_{re}} = 1$(e), and $n{R_{re}} = 10$(f). The profile of $nw{0_r}$ (g) and $n{R_{re}}$ (h) under different focal lengths and target planes’ distance.

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Practical systems also need to consider the truncation effect of the telescope aperture, and the telescope structure will also determine the lower bound of the $n{x_0}$. The analysis of non-truncated Gaussian light can only be used as a reference.

3.1 Analysis of ABR offset distance and rotation angles

The offset distance between the Tx telescope and ABR telescope will determine the normalized center distance $nx0$ between the emitted light field and the BPLO light field. Neglecting the physical limit of real telescopes’ structure and assuming the two optical axes of the telescopes are parallel, ${x_0}$, which is the distance between the transmitting and ABR telescopes along the x-axis, is set from 0 to 100 mm. As shown in Fig. 5(a), the smaller the offset distance of the ABR system the larger the COF. For small offset distances, the COF of the ABR system within the blind zone still can be larger than the MNCOF (the minimum of the needed COF to identify the effective signal). However, the ABR and Tx telescopes’ structure radius determines the minimum offset distance. Taking the general system in Table 1 as an example, the minimum offset distance ${x_0}$ is about 80 mm. Unfortunately, in such cases, the ABR system COFs are smaller than the MNCOF within the blind zone. As a result, it has to rotate the ABR telescope and create an unparalleled ABR and Tx optical axis to enhance the COF.

 

Fig. 5. The target planes COF within the blind zone for different offset distances ${x_0}$(a).the target planes COF within the blind zone for different optical axis intersection distances ${L_p}$ when pitch angles $\beta $ are set to 1° (b), 3’ (c), and 0 (d).

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When the ABR and Tx optical axes intersect at a roll angle ($\alpha $), the normalized center distance $nx0$ can be small within the target planes which are close to the optical axis intersection. Precise alignment of the optical axis direction is very difficult, thus a real ABR system will also introduce a pitch angle ($\beta $) except the roll angle ($\alpha $). The pitch angle will extend the normalized center distance $nx0$ as the target plane distance increases. The analysis for the influence of the pitch and roll angles is also based on the parameters in Table 1 and the offset distance is set to ${x_0} = {r_{lo}} + {r_t} + 5mm$ (5 mm taking into account the thickness of the telescopes’ structure). The pitch angle ($\beta $) is set to three values, 1°, 3’, and 0. The roll angle ($\alpha $) can be computed by dividing the optical axis intersection distance (${L_P}$) by the offset distance ${x_0}$ In the research, the optical axis intersection is assigned to cover the blind zone using the values of 20 m, 45 m, 70 m, and 95 m. Figure 5(b-d) displays the analysis results. When the roll angle is 1°, the COF is smaller than the MNCOF for any target plane within the blind zone, as shown in Fig. 5(b). When the roll angle is set to a smaller value of 3’ (a typical value considering general telescopes’ pointing accuracy), the COF of partial target planes is larger than the MNCOF, as shown in Fig. 5(c). These target planes are close to the optical axis intersection and only cover a small portion of the blind zone. The maximum COF occurs when the target plane distance is smaller than the optical axis intersection distance (${L_P}$). The maximum COF would decrease as the ${L_P}$ increases while the shape of the COF becomes wider. Therefore, reasonable ${L_P}$ should make the whole blind zone planes with enough COF. As Fig. 5(d) case with zero roll angle, the maximum COF reaches the same level as the MCT system when the target plane distance is equal to ${L_P}$. The system blind zone coverage will be smaller but the near field blind zone still exists. It means that the effective detection of near field and far field within the blind zone is challenging without considering optimized telescopes but using the same telescope as the transmitter.

3.2 Analysis of ABR telescope parameters

As introduced in the prior part, the ABR telescope parameters play a significant role in the BPLO light field distribution and consequently influent the blind zone range gates SNRs. The minimum SNR determines whether all the range gates’ signals within the blind zone can be effectively recognized. For a case system given by Table 1, assume a goal function that is the minimum SNR of the first three range gates in the blind zone as shown in Eq. (17),

$SN{R_1}$, $SN{R_2}$, and $SN{R_3}$ denote the SNR of the first, second, and third blind zone distance gates of the case system respectively.

Considering the actual telescope aperture and its truncation effects, to get larger COFs the aperture of the receiving telescope should be small to achieve small $nx0$ in the whole blind zone. It is easy to get a small aperture if the telescope has a large truncation ratio or a small focal length. A small focal length can contribute to a large FOV, which allows more options under a given MNCOF. However, a large truncation causes a small truncation efficiency that means much BPLO energy will be lost. A small focal length will cause Rayleigh distance to scaled-down then the beam will diverge into a larger spot at far distances. To get the effect of each parameter and a roughly optimized design of the ABR telescope, three ABR telescope parameters are considered and analyzed in the order of focal length-truncation ratio-biased focus.

The goal function of the focal length is analyzed based on the parameters in Table 1. The optimization of focal length is based on the design mentioned in section 3.1 with the offset distance ${x_0} = {r_{lo}} + {r_t} + 5mm$, the pitch angle $\beta = 3^{\prime}$, and the optical axis intersection distance ${L_p} = 50\textrm{ }m$. As shown in Fig. 6(a), we can see that as the focal length increases, the three blind range gates’ SNRs exhibit an initial increase followed by a decrease. Within a narrow range of small focal lengths, all three blind range gates can achieve higher SNR. On either side of the extreme point of the focal length, the SNRs for the blind zone distance gates drop rapidly. As shown in Fig. 6(a, d), the nCOF of all the target planes within the blind zone can be higher than about −30 dB the MCT system’s COF normalized MNCOF when the focal length meets that goal function value is always bigger than the needed SNR (about −37 dB). For this case system, when the focal length is set at 7 mm, the goal function reaches its extremum, and the nCOF of all the target planes within the blind zone can exceed −15 dB.

 

Fig. 6. The first three blind zone range gates’ SNRs of the ABR telescope with different focal length values(a), truncation ratio values (b), and defocus values(c); The first three blind zone range gates’ nCOF of the telescope with different focal length values(d), truncation ratio values (e), and defocus values(f).

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After getting the optimized focal length of 7 mm, analysis and optimization of the truncation ratio are obtained. The results are displayed in Fig. 6(b, e). Small truncation ratios can strengthen SNRs within the blind zones. But when truncation ratios are smaller or higher enough all the blind zone distance gates’ SNR, the target planes’ nCOF will rapidly decrease. Smaller truncation ratios will sharply increase telescope aperture, make the transmitting and receiving telescopes more widely space, and lead to the signal across the path, especially in the near target plane, decreasing dramatically. The smallest effective truncation ratio has been limited to 0.68 (black dashed line) by fiber optical NA. After reaching the smallest ratio, increasing the telescope size will no longer mean a smaller truncation ratio. The goal function reaches its extremum −25 dB when the truncation ratio is equal to 0.75, and the nCOF of all the target planes within the whole blind zone can be higher than −15 dB.

The defocus will change both the waist radius and position of the BPLO Gaussian beam. A large defocus will make the BPLO light field focus on a small dot or severely diverge. Ignoring the defocus’ impact on the receiving range, an appropriate defocus may make the BPLO light field distribute better to deal with the BZP. Using the focal length of 7 mm and the truncation ratio of 0.75 from the prior section, we get Fig. 6(c, f) for the defocus effect on SNR and nCOF. The result suggests that defocus could improve the signals, but only those with close ranges. In the case system, mild defocus is only slightly beneficial to the first range gate’s SNR. The goal function represents the minimum SNR, in most cases the SNR of the furthest range gate. Thus moving away from zero defocus, the goal function values exhibit a rapid exponential decrease in dB form. However, within a defocus range of −0.2 mm to 0.2 mm, the goal is still higher than the needed SNR, and the nCOF of all the blind zone target planes can be bigger than −30 dB. With designated parameters, ABR systems are competent for solving the BZP. To get better signals, the defocus should be zero. If defocus is unavoidable, small defocuses like ±0.05 mm, also don't weaken the signal much.

The above analysis provides both qualitative and quantitative conclusions for solving the BZP by using the ABR system. For a typical CDWL system in Table 1, a set of design parameters that can completely overcome the BZP is obtained. The analysis proves the feasibility of the BZP-solving ABR system and also clarifies the influence trends of all the geometric positions, rotation, and optical parameters of the ABR telescope. Reducing system offset distance, pitch angle, and defocus as much as possible plus picking optional values for focal length and truncation ratio can improve the ABR system performance.

3.3 Simultaneous analysis of all the parameters

Sequential optimization mentioned in context could give a reasonable design for solving typical MCT systems’ BZP. Getting the optimum parameter sequence and all parameters’ combinations that could solve the BZP is of greater importance. By exhausting all the values of the suitable range, the simultaneous analysis of all the parameters can be done, and the result is as Fig. 7. By exhausting all the values of the suitable range, the simultaneous analysis of all the parameters can be done. In Fig. 7, different columns represent different optical axis intersection distance values and different rows represent different pitch angle values. The horizontal coordinates of all the subgraphs represent the focal length, and the vertical coordinates represent the truncation ratio. The color bar represents the goal function value. For the general MCT system as Table 1 shows, the suitable range of all the parameter sequences of the ABR system is set as Eqs. (18–22). The offset distance is still determined by ${x_0} = {r_{lo}} + {r_t} + 5mm$.

 

Fig. 7. The goal of the ABR with different sequences of four parameters ${f_{lo}}$ (the horizontal coordinates of each subfigure), ${\rho _{locut}}$ (the vertical coordinates of each subfigure), $\beta $ (different rows of the subplot correspond to different values as indicated in purple on the right-hand side), ${L_P}$ (different columns of the graph correspond to different values as indicated in red above).

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Simultaneous analysis results reveal that multiple ABR parameters’ combinations could overcome the BZP. When picking a small pitch angle and ${L_p}$ in the range of half to one times the maximum blind zone distance like 60 m, there are higher values of the optimum ABR system’s goal function and more available options for the focal lengths and truncation ratios. For example, with the same optical axis intersection distance ${L_p} = 60\textrm{ }m$ (the second column of Fig. 7), the highest focal length bound of −37 dB is higher than 100 mm when the pitch angle is 1’ or 0, while that is 7 mm when the pitch angle is 6’. The 1’ pitch angle system also has a higher maximum goal value of −20 dB than the 6’ system whose maximum goal value is −30 dB. Similarly, we can observe from the second row that the larger usable design parameters profile and higher maximum goal value can be acquired when the optical axis intersection distance is equal to 60 m. For designs with small focal lengths, corresponding truncation ratio values usually cover a wide range from 0.1 to 3. Notably, when a large focal length is allowed and adopted, the available truncation ratio will have fewer options and the system performance will decline. The system with the best performance in the research settlements has 0 pitch angle, 60 m optical axis intersection distance, 7 mm focal length, and 0.75 truncation ratio, and its goal function could reach −18 dB. From the perspective of telescope design, the telescope with a small focal length of about 7 mm and a truncation ratio of about 0.75 not only can get a higher goal but can also accommodate poorer angular alignment accuracy. For the case system, even with the pitch angle larger than 3’ and the optical axis intersection distance from less than 30 m to more than 90 m, the telescope design can still meet the requirement of solving BZP.

4. Conclusion

This paper introduces the ABR method that uses one ABR telescope specialized to receive the blind zone signals to solve the general MCT system BZP while maintaining the MCT system’s long-range detection capability. Based on the general SNR equation of coherent wind lidar, a numerical model of the transceiver system is established. Based on this numerical model, the ABR system is analyzed to solve the BZP problem. For a typical pulsed CDWL system, the ABR method is proven feasible to solve the BZP problem. To get better signals in the blind zone, the auxiliary telescope should be as close as possible to the MCT system telescope, and the accuracy of the pitch angle alignment of the telescope should be ensured at a small level like 1`. Among all design parameters, the accuracy of pitch angle control and the choice of the ABR focal length have demonstrated great importance. An appropriately small focal length of the ABR telescope like 7 mm can enhance the signal of the blind zone range gates, loosen the tolerance of pitch angle accuracy, and expand the range of choices for the optical axis intersection distance and truncation ratio. Besides, an auxiliary telescope with a small focal length is effective for reducing the difficulty of guaranteeing the pitch angle control precision, and it also reduces the space requirements of the ABR system. Matching with a suitable truncation ratio and optical axis intersection distance, the ABR system could eliminate the blind zone. This shows that the ABR system method is easy to implement, the pulsed-CDWL system without blind zone can be achieved by only adding a coherent coupler, a balanced detector, and an appropriately placed small-size ABR telescope.