1. Introduction

As an active laser system, forward pulsed laser detection has been successfully applied to various aircraft and precision-guided missiles [1,2] to obtain the azimuth and distance to a target. When an aircraft flies at supersonic speed, a complex flow field is formed that causes the surrounding air density field to change dramatically with time and space, resulting in a drastic change in the refractive index. When a laser beam from a pulsed laser radar system passes through this flow field, there will inevitably be optical distortions, such as offset, intensity loss, and blur, which are called aero-optical effects [3–6]. These effects reduce the pulsed laser echo power, affect the detection accuracy, and invalidate the guidance mode. Therefore, the study of optical distortion and echo power attenuation caused by aero-optical effects is of immense significance for improving the performance and detection accuracy of pulsed laser radar in supersonic settings.

In recent years, with the rapid development of high-speed missiles and aircraft, the study of aero-optical effects has attracted increasing attention. Lee et al. [7] investigated the aero-optical effect caused by supersonic flow on a two-dimensional compressed ramp by experimental and numerical methods and determined the separate contributions of shock waves and boundary waves to light wavefront distortion. Guo et al. [8–10] simulated the supersonic mixing wave flow field above the lateral optical window of a high-speed missile based on large eddy simulation and analyzed the influence of supersonic mixing waves on optical distortion and image degradation via the ray tracing method. Hui et al. [11] adopted a numerical simulation of ray tracing on the optical dome of a high-speed infrared missile and proposed three evaluation parameters to evaluate the changes in temperature, deformation, and refractive index of the optical dome. The results showed that when the missile flight speed exceeds Mach 2, the influence of the aero-optical effect on the imaging quality becomes increasingly significant. Wang et al. [12] employed an angular spectrum propagation model to describe the transmission of light in a supersonic flow field and calculated the optical transfer function and offset of aero-optical imaging. Banakh et al. [13] adopted the phase screen method to investigate the average intensity of the beam passing through the shock wave and the displacement relative to the initial propagation direction. The results showed that the angular displacement of the beam from the initial propagation direction depends only on the altitude at which the shock wave is formed. With an increase in altitude, the impact of the shock wave on the beam decreases. Liu et al. [14] examined the aerodynamic optical transmission characteristics of a supersonic flow field and a heated optical window to obtain the time-varying point spread functions of the flow field and optical window on the flight trajectory. The results showed that light transmission through the aerodynamic heating window has a more significant effect on the imaging quality than that through an aerodynamic flow field.

However, most of the studies focus on aero-optical effects on passive imaging in the lateral optical window of high-speed aircraft, and the research on active detection in forward optical windows is inadequate. For the forward pulsed laser radar, the emitted and received beams are affected by an uneven flow field, making it more difficult to study than passive imaging detection. Additionally, the influence of aero-optical effects on pulsed laser echo waves remains to be investigated. Yao et al. [15] derived an analytical expression of the echo waveform under the influence of a conical shock wave. To the best of the authors’ knowledge, this is the only research on the influence of aero-optical effects on pulsed laser radar echo waves. However, the research simplified the conical shock wave into a single-layer density gradient medium that differed from the shock wave generated in actual flight. Therefore, it is necessary to conduct a more detailed study of the influence of aero-optical effects on the detection performance of forward pulsed laser radar.

In this study, the optical distortion of a laser beam passing through a shock wave and its influence on the echo wave and ranging accuracy in forward pulsed laser radar is investigated for a near-ground high-speed aircraft at Mach 1.5–4. The main contributions are as follows: (1) A novel semi-analytical method combining the traditional laser echo equation with optical distortion parameters is proposed to derive the effect of shock waves on pulsed laser echo wave. (2) A method based on the optimal confidence interval is proposed to evaluate the ranging accuracy of pulsed laser radar. This method replaces the traditional variance and improves the quantitative evaluation of the detection error of pulsed laser radar affected by shock waves. (3) An improved ray tracing method based on inverse distance-weighted (IDW) interpolation with a quadrilateral mesh is proposed. Compared with the method proposed by Liu et al. [6], this method can significantly improve tracking accuracy.

The remainder of this paper is organized as follows. In Section 2, the pulsed laser radar detection principle and echo wave equation are introduced. In Section 3, the aerodynamic flow field of the aircraft head is calculated to obtain the density distribution that is transformed into the refractive index field through the Gladstone–Dale relationship. In Section 4, a ray-tracing method based on IDW interpolation combined with a quadrilateral mesh is proposed to calculate the optical distortion of the laser beam passing through the shock wave. In Section 5, the propagation model of a laser in a detached shock wave under approximate conditions is established, and the expression of the pulsed laser echo waveform under the influence of the shock wave is derived. In Section 6, the ranging probability distribution function (PDF) and error distribution of the pulsed laser radar are discussed. In Section 7, the results of the flow field, optical distortion, echo waveform, and ranging error under different flight conditions are presented, and the effects of Mach number, detection range, and target tilt angle on the echo are discussed. Section 8 concludes this study.

2. System model

The basic principle of pulsed laser radar is to measure the laser flight time via the threshold time discrimination method to obtain the distance to the target [16].

The characteristics of pulsed laser echo waveforms include the time-domain distribution characteristics and spatial distribution. The temporal distribution feature of a pulsed laser refers to the time-dependent change in instantaneous power at any point in the propagation direction of the laser pulse. It is assumed that the temporal waveform of the laser pulse emitted by the pulsed laser radar follows a Gaussian distribution, defined as follows:

The spatial distribution characteristics of the laser pulse describe the spatial energy distribution of the beam perpendicular to the propagation direction. Assuming that the spatial mode field distribution of the laser is TEM00 (the characteristics of the higher-order laser mode [33,34] will be investigated in the future work), the light intensity distribution follows a Gaussian distribution that is written as [17]

Figure 1 shows the principle of pulsed laser range detection for high-speed aircraft. The pulsed laser echo power of planar targets in the static state can be expressed as follows [18]:

 

Fig. 1. Principle of pulsed laser detection for high speed flow field (The yellow beam denotes the laser beam disturbed by the uneven flow field, and the red one is the undisturbed beam, respectively)

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When the laser beam passes through an uneven flow field, the changing refractive index will distort the optical path, resulting in the attenuation of the echo energy and the broadening of the waveform, thus affecting the detection accuracy of the system.

3. Pulsed laser echo model influenced by shock wave

Equation (3) can only describe the laser echo wave in a static state. For pulsed laser radar in supersonic flight, the beam is distorted by a shock wave with a complex density distribution, making Eq. (3) unsuitable. Therefore, a semi-analytical method for calculating the echo wave is proposed. This method improves the static echo equation by dispersing the transmitting beam into several rays based on geometric optics, tracking the trajectories and energy changes of each ray through ray tracing, and superimposing the results to obtain the final echo wave. Chu et al. [32] proposed a similar simulation method to model the Segmented Planar Imaging Detector for Electro-Optical Reconnaissance (SPIDER), but that work is based on CW wave not pulsed, we improve the method and utilize it to model the echo wave disturbed by the shock wave.

A schematic of the method is shown in Fig. 2. Because the flow field investigated in this study is symmetric along the z-axis, the refractive index varies along the radial axis, and the laser emitter and receiver are small in size, the flow field in front of the emitting and receiving lens can be approximated as symmetric along the y-axis.

 

Fig. 2. Schematic diagram of the proposed echo wave calculation method.

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The actual propagation path of the laser transmission process can be determined as follows:

The incident angle of the deflected beam reaching the target plane is

Because the size of the target is much larger than the beam radius and the beam deflection caused by the shock wave is small in short-range detection, all the emitted beams can be considered as reaching the target plane. In receiving, wave distortion occurs when the beam passes through the shock wave, resulting in the attenuation of the beam intensity on the pupil plane. The Strehl ratio (SR) can be used to characterize the ratio of spot energy with or without the aero-optical effect. The pulsed laser echo wave power can be calculated by substituting Eqs. (4) and (5) into Eq. (3):

Supposing that the size of the target plane is ${l_1} \times {l_2}$, the pulsed laser echo wave power is written as follows:

The energy in the laser spot accounts for 95.45% of the total energy [28]. For convenience, extending the integral range to infinity will not have any significant influence on the results.

Integrating x, the echo wave power can be expressed as

Because $SR$, ${R_L}$, and $\varphi ^{\prime}$ are discrete variables dependent on $y$, the above formula cannot be integrated directly. In geometric optics, the beam can be regarded as multiple beams propagating independently in the flow field, and the echo power can be obtained by superimposing the power of each ray on the entrance pupil surface. Therefore, Eq. (9) can be discretized in the y direction as

Equation (10) is the expression of the pulsed laser echo signal affected by the shock wave.

4. Aerodynamic flow field computation

Owing to the uneven refractive index distribution of the shock wave, the propagation path of the light ray is distorted. The refractive index can be calculated from the density distribution of the external flow field by adopting the Gladstone–Dale relationship. Therefore, it is necessary to model and calculate the flow field to obtain the corresponding density field distribution.

The Reynolds-averaged Navier–Stokes (RANS) solver was used to calculate the steady-state flow-field distribution [19,20]. The head shape of the aircraft is shown in Fig. 3(a). The flight angle of attack is 0°, and the overall structure is z-axis symmetrical, implying that the flow field around the head is also symmetrical relative to the z-axis. Therefore, the flow field in the yoz plane can approximately represent the three-dimensional flow, and its density, pressure, and temperature field distribution can be considered to be symmetrical along the z-axis. Two-dimensional geometric modeling and meshing of the aircraft head along the yoz plane were performed using the pre-processing software ICEM-CFD, and the flow field was calculated using quadrilateral structured grids. The O-type grid topology was adopted for the boundary wave, and the total number of grids was 228096, as shown in Fig. 3(b).

 

Fig. 3. (a) The head structure of high-speed aircraft, (b) Computational mesh used for simulation of flow field

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The RANS solver in FLUENT was used to calculate the flow field. The boundary conditions of the aerodynamic calculations are shown in Table 1. The shear stress transport model was adopted for turbulence, and a second-order upwind scheme was employed to discretize the modified turbulence viscosity equations.

Table 1. Boundary condition of aerodynamic flow

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After obtaining the flow field density distribution of the high-speed aircraft, it must be converted to the refractive index field distribution. According to the Gladstone–Dale relation [21,22],

The pulsed laser radar emits a laser with a wavelength of 905 nm; substituting this into Eq. (12), ${K_{\textrm{GD}}} = 2.25 \times {10^{ - 4}}$.

5. Ray-tracing method and verification

5.1 Ray tracing method based on fourth-order Runge–Kuta

After the refractive index distribution of the external flow field is obtained, the ray tracing method is used to calculate the propagation trajectory and deflection of the rays in the uneven external flow field. The ray equation describes ray propagation in a medium with an arbitrary refractive index distribution. It is one of the basic equations used to study ray propagation in a gradient refractive index medium. In this section, the ray equation is solved to trace the laser beam passing through the flow field outside the aircraft head.

The ray equation is shown as follows:

To obtain the calculation expression of the fourth-order Runge–Kutta method, the substitution $dt = ds/n$ is made; accordingly, the ray equation can be written as [6,23]

In this manner, the ray equation is transformed into a second-order differential equation, and the fourth-order Runge–Kutta method is calculated as follows:

When the initial ray position vector ${\boldsymbol{r}_0}$ and direction vector ${\boldsymbol{T}_0}$ are provided, the next position ${\boldsymbol{r}_1}$ and ${\boldsymbol{T}_1}$ can be obtained using Eq. (8). This calculation is iterated until the entire ray-tracing process is completed. After each step, the refractive index n and refractive index gradient $\nabla n$ of the current position need to be calculated, and the position coordinates of the light ray must be checked to determine whether they are beyond the calculation range. If they are, ray tracing is terminated. The computational mesh range for ray tracing is shown in Fig. 4(a).

 

Fig. 4. (a) Diagram of ray tracing algorithm (the mesh indicates the calculation range), (b)Quadrilateral interpolation of refractive index and refractive index gradient.

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In the tracking process, the refractive index n and refractive index gradient $\nabla n$ at any position must be calculated. The calculation results of the CFD software are based on the discrete data of irregular quadrilateral mesh vertices, and the refractive index and refractive index gradient at any point cannot be directly obtained. Therefore, the data at any point within the grid must be interpolated. Liu et al. [24] proposed an IDW average interpolation method based on a triangular mesh to approximate the data. The principle of IDW is that the relationship between the interpolation point data and the reference point data is inversely proportional to the distance between them [25]. The number of reference points determines the quality of interpolation. The more reference points, the higher the interpolation accuracy is. However, the flow field is divided by a quadrilateral structured mesh that has an improved convergence and accuracy compared with a triangular unstructured mesh [26]. The best solution is to use all four vertices surrounding the interpolation point, instead of the triangular mesh comprising only three vertices. Therefore, the inverse distance-weighted average interpolation method based on quadrilateral meshes was adopted. The specific process is illustrated in Fig. 4(b).

First, for the light position $P = ({{x_m},{y_m}} )$ of the $i\textrm{th}$ step in the ray tracing process, the nearest four grid vertices numbered 1, 2, 3, and 4 are obtained. Thereafter, based on the refractive indexes ${n_{1,2, \ldots 4}}$ and refractive index gradients $\nabla {n_{1,2, \ldots 4}}$ of the four vertices, the refractive index and refractive index gradient of the position P are interpolated as follows:

5.2 Method verification

Owing to the lack of reliable experimental data, it is difficult to verify the ray-tracing method applied to aero-optics. To verify the correctness and effectiveness of the proposed method, a medium is considered with a radial refractive index gradient of [6]

The analytical solution of the propagation trajectory of a ray in this medium is

The relative error of the ray tracing method is defined as

Without losing generality, suppose that ${n_0} = 1.5$, $\alpha = 0.01$, the incident azimuth angle and pitching angle of the light are 0° and 60°, respectively, and the coordinates of the initial incident point are $({{x_0},{y_0},{z_0}} )= ({\cos {{60}^\circ }/\alpha ,0,0} )$; the initial direction cosines are then $({{p_0},{q_0},{L_0}} )= ({{n_0}\sin {0^\circ }\cos {{60}^\circ },{n_0}\cos {0^\circ }\cos {{60}^\circ },{n_0}\sin {{60}^\circ }} )$. Figure 5(a) shows a comparison between the actual ray trajectory and the simulation trajectory when the tracking step $\Delta s$ = 2 mm. It can be observed that the two fit well. Figure 5(b) illustrates the relative error of the ray tracing process. The solid and dotted lines represent the relative error observed based on the method proposed in this paper and that proposed by Liu [6], respectively. When the step size is 2, 1, and 0.5 mm, the relative error of the ray tracing is on the order of ${10^{ - 7}}$, ${10^{ - 8}}$, and ${10^{ - 9}}$, respectively. Compared with Liu’s method, the ray-tracing method based on quadrilateral interpolation exhibits a smaller error and a higher tracing accuracy.

 

Fig. 5. (a) Comparison of actual ray trajectory and simulation trajectory. (b) Relative error of ray tracing method along z axis (dotted lines are the relative error provided by Liu [6]).

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5.3 Evaluation parameters of aero-optical effect

The optical path difference (OPD) and SR were used to evaluate the aero-optical effects using the ray positions and directions acquired from the ray tracing method. The optical path length (OPL) and OPD are defined as [9]

SR can describe the imaging quality of a ray passing through an inhomogeneous flow field. When the large aperture approximation is satisfied, the SR is expressed as [27]

6. Ranging data distribution and error evaluation

Pulsed laser radar employs the time identification method to obtain the accurate receiving time of the laser echo wave. In this study, the most commonly used constant threshold time identification method is used to identify the echo pulse [29], and the ranging probability distribution characteristics and ranging error are derived to study the influence of the shock wave on the accuracy of pulse laser radar ranging detection.

The optical signal of the pulsed laser echo wave was received by the detector and amplified into a voltage signal by an amplifier:

6.1 Ranging probability distribution characteristics

The noise of a pulsed laser radar receiving system is usually composed of background noise, circuit noise, and amplifier noise [29]. The receiver noise can be described by a continuous zero-mean Gaussian distribution, and the PDF of the noise can be expressed as:

The PDF of the superimposed signal ${V_{sn}}(t )$ consisting of the noise and laser pulse echo signal is defined as

If the laser echo is identified by a constant threshold discriminator, the probability of detecting the pulse between time intervals can be described as [30]

The PDF of pulse laser detection is [31]

According to the relationship between the time of flight and the target distance, the pulsed laser radar ranging PDF is

Substituting Eqs. (25) and (26) into Eq. (30), the PDF of the pulse laser ranging was obtained by the constant threshold method.

6.2 Evaluation of ranging error

The evaluation index of the pulsed laser radar detection ability should also include the ranging accuracy, which is mainly determined by systematic errors and random errors. The law of systematic error is relatively fixed and can be eliminated considerably by technical means such as calibration or compensation algorithms. Because random error in ranging is mainly caused by noise, it is more difficult to eliminate. Therefore, it is necessary to study the influence of shock waves on the ranging accuracy of the pulsed laser radar.

The pulsed laser echo signal no longer obeys a strict Gaussian distribution after the interference of the shock wave. The traditional concept of variance is not suitable for the quantitative description of ranging error; therefore, the optimal confidence interval is used to quantify the ranging error. Given that the confidence degree is $1 - \alpha $, the confidence intervals $[{{r_a},{r_b}} ]$ are defined by

However, there are infinitely many confidence intervals that are determined in this manner. To reasonably evaluate the ranging error, the best confidence intervals should be selected, that is, the shortest interval length. Therefore, the qualification condition, $\rho ({{r_a}} )= \rho ({{r_b}} )$, is added as follows:

The upper and lower bounds of the random error of ranging are ${r_H} - {\bar{r}_{\textrm{sys}}} - {R_r}$ and ${r_L} - {\bar{r}_{\textrm{sys}}} - {R_r}$, respectively.

7. Results and Discussion

7.1 Aerodynamic flow fields under different Mach numbers

Figure 6 shows the average outflow field distribution of the high-speed aircraft at different Mach numbers. There are detached shock waves in the head and shoulder of the aircraft, and the density field on both sides follows an obvious gradient distribution. With an increase in the Mach number, the air in the head is further compressed, and the angle of the shock wave becomes increasingly smaller. The flow field density in front of the head increases with a decrease in the thickness of the shock wave on both sides. In addition, because of the convex platform structure on the shoulder, the flow of air in the z direction is hindered and flows to both vertical sides, where the air is compressed and the detached shock wave is generated again. Because the optical window of the pulsed laser radar is installed on the convex shoulder platform, the emitted and received laser beam passes through two shock waves, resulting in large optical distortion and deflection.

 

Fig. 6. Flow field of aircraft at different Mach numbers. (a) 1.5Ma, (b) 2Ma, (c) 3Ma, (d) 4Ma.

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7.2 Effect of different Mach numbers on optical distortion

To quantitatively analyze the optical distortion of the transmitting and receiving beams in the optical window of the pulsed laser radar at different Mach numbers, 100 light rays emitted parallel to the z-axis were tracked, and the aero-optical effect evaluation parameters at different Mach numbers were calculated for a flight at 100 m altitude and 300 K temperature. As shown in Fig. 7(a), the change in the OPD is related to the Mach number and y-axis position. For Mach 1.5 and 2, the OPD increases gradually with an increase in the y-axis coordinate. However, the OPD at Mach 1.5 initially changes quickly and subsequently changes gradually, and its minimum and maximum values are -2.08 and 5.8 µm, respectively. In contrast, the OPD at Mach 2 changes relatively constantly, and its minimum and maximum values are -0.5 and 6.8 µm, respectively. It should be noted that the negative value on the longitudinal axis only indicates that the phase plane of the emitted light moves backward relative to its initial position, and the absolute value of the OPD is usually considered. This implies that the OPD at Mach 3–4 decreases with an increase in the y-axis coordinates, and the change rate of Mach 4 is greater than that of Mach 3. The minimum and maximum OPD at Mach 4 are -13.9 and 0.38 µm, respectively. This shows that for Mach 3–4, the density gradient near the optical axis is large, and the light distortion is significant. Furthermore, the OPD increases with an increase in the Mach number near the optical axis, while the opposite trend is observed at a position far from the optical axis.

 

Fig. 7. (a) OPD distribution of the emitted beam along the optical window at different Mach numbers. (b) SR distribution of receiving beams along the optical window under different Mach numbers.

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Figure 7(b) shows the SR distribution of the received beam along the optical window at different Mach numbers. It can be seen that SR decreases with an increase in the Mach number. When the flight speed is Mach 4, the average SR value is of the order of 10⁻², which shows that the energy loss of the laser echo spot at a high Mach number is extremely large. Moreover, the SR fluctuates vertically along the optical window, and the amplitude varies with distance. This is because the density distribution in the shock wave is uneven and the optical grid is not sufficiently fine in the instantaneous flow field.

7.3 Effect of shock wave on pulse laser echo waveform

To systematically investigate the effect of shock waves on pulse laser echo waveforms, the echo waveforms under different Mach numbers, detection ranges, and target angles were compared. The simulation parameters were as follows: $D = 0.02$ m, ${P_0} = 20$ W, ${\eta _{\textrm{sys}}} = 0.64$, $\tau = 6$ ns, $\phi = 20$ mrad, $\rho = 0.3$, $\lambda = 905$ nm, and $u(\lambda )= 0.27$.

7.3.1 Effect of Mach numbers on echo waveform

To examine the influence of Mach numbers on the echo waveform, the tilt angle of the target plane was fixed at 40° and the detection range was 50 m. By comparing five groups of data under different Mach numbers, as shown in Fig. 8, the amplitude and peak position of the echo waveform were found to be considerably different. With an increase in the Mach number, the corresponding maximum power decreases significantly. For instance, when the flight speed is Mach 1.5, the peak power is 0.088 µW, and when the flight speed is Mach 4, the peak power decreases to 1.45 × 10⁻³ µW, only 1/60.7 of the former. At Mach 4, the echo waveform is essentially invisible owing to the interference of the aero-optical effect. In addition, with an increase in the Mach number, the peak position of the echo shifts to the right. From the static state to Mach 4, the peak shifts by 1 ns due to the deflection of the ray as it passes through the shock wave, resulting in a longer propagation range. In addition, it is noted that all the echo waveforms approximately follow a Gaussian distribution, and the pulse widths exhibit a slight broadening with an increase in the Mach number.

 

Fig. 8. Echo waveform of different Mach numbers when $\beta = 40^\circ $, ${R_r} = 50\textrm{m}$.

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7.3.2 Effect of target plane tilt angle on echo wave

To examine the influence of the target plane tilt angle on the echo wave, the detection range was set to 50 m as the Mach number ranged from 1.5 to 4. By comparing the echo waveforms corresponding to the target plane tilt angles of 0°, 20°, 40°, 60°, and 80° (Fig. 9(a)–9(e)), it is observed that for the same Mach number, the echo wave amplitude decreases with an increase in the target tilt angle. For instance, at 3 Mach, the echo peaks at 0° and 80° are 2.74 × 10⁻² µW and 1.95 × 10⁻³ µW, respectively, and the difference is more than 10 times. Furthermore, the echo peak position shifted to the right for all tilt angles except 0°, and the degree of right shift increased with an increase in the tilt angle. This is because an increase in the tilt angle extends the optical path of the deflected laser beam, increasing the time at which it is received by the system. Moreover, the echo wave pulse widths at different tilt angles also change, as shown in Fig. 9(f). With an increase in the target tilt angle, the pulse width broadening also increases, although the growth is small between 0° and 60°, not exceeding 10%. However, when the tilt angle exceeds 60°, the pulse width broadening increases rapidly. Simultaneously, the Mach number was also found to influence the pulse width broadening; the larger the Mach number, the larger the broadening was.

 

Fig. 9. Echo waveform of different tilt angles when the detection range is 50 m : (a) 0Ma. (b) 1.5Ma. (c) 2Ma. (d) 3Ma. (e) 4Ma. (f) pulse width of different target tilt angle.

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7.4 Evaluation of aero-optical effects on ranging accuracy

To study the influence of different Mach numbers on the ranging PDF, the simulation parameters were set as follows: $M = 9$, ${A_f} = 1.7 \times {10^6}$, $\sigma = 50$ mV, and ${V_{\textrm{th}}} = 0.2$ V. The detection range was set at 50 m. The ranging PDFs at different Mach numbers and target tilt angles are shown in Fig. 10(a) and (b). The ranging PDF follows a negative skewness distribution, with a gentle ascending edge and steep descending edge. With an increase in the Mach number, the peak values of the ranging PDF decrease continuously, the ranging distribution moves to the right, and the waveform broadens, implying that the random error of ranging increases. Particularly at Mach 4, the PDF shift to the right is obvious, and the broadening is at its maximum. With an increase in the target tilt angle, the PDF peak also decreases, the PDF range remains constant, the peak shifts to left (especially at angles of 60°–80°), and the waveform pulse width broadens.

 

Fig. 10. Ranging PDF distribution at 50 m detection range under different conditions: (a) target tilt angle is 40°, different Mach number, (b) Mach 3, different target tilt angles

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Generally, the error distribution of ranging data can be measured with a given confidence of $1 - \alpha = 95\%$. Figure 11(a) shows the variation in the ranging error with respect to the target tilt angles at different Mach numbers. The systematic error and random error increase with an increase in the target tilt angle. Additionally, at the same target tilt angle, the systematic error decreases with an increase in the Mach number, and the random error increases gradually, corresponding with the results in Fig. 10(a). The variation in the ranging error with the detection range at different Mach numbers is shown in Fig. 11(b). The systematic error decreases with an increase in the detection range. However, the random error exhibits the opposite trend, increasing with an increase in detection range.

 

Fig. 11. Ranging error under different conditions :(a) detection range 50 m, different target tilt angles; (b) target tilt angle 40°, different detection ranges.

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8. Conclusion

In this study, the influence mechanism of aero-optical effects on forward pulsed laser radar was investigated for the first time. The flow field outside the aircraft head was modeled using the RANS method, and an improved ray tracing method based on the IDW interpolation with a quadrilateral mesh was proposed to obtain ray directions and positions. A novel semi-analytical method was proposed to derive the pulsed laser echo equation affected by shock waves, and the echo waveforms with different Mach numbers and target plane tilt angles were obtained by numerical calculations. Furthermore, the effect of shock waves on the ranging accuracy of forward pulsed laser radar was evaluated based on the optimal confidence interval. The ranging PDF distributions and ranging errors with different Mach numbers, detection ranges, and target plane tilt angles were calculated. The main conclusions drawn from this study are as follows: