## Abstract

The 3-D target information can be obtained directly by the analysis of the 1-D data of the pulse laser range profile (LRP). In this paper, a new calculational method to generate and process the LRP which is based on the lidar equation is established. The LRPs of different targets are simulated by the new method, and the corresponding profile information is achieved. The effect of the some parameters including the target shape, surface material, incident pulsewidth and incident angle on the LRP is analyzed. In order to validate the theory, the relevant experiment is implemented, and the experimental data agrees well with the simulation result.

© 2010 OSA

## 1. Introduction

The outstanding feature of laser range profile (LRP) is that it can obtain the 3-D shape and the range information of the target by one pulse without optical scanning system. The pulse laser is used for illuminating the targets, and the detector records the scattering signal which is the function of time. With analyzing the features of the achieved signal, the details of target structure in the direction of the radar can be discovered [1,2]. In addition, different targets have different LRPs, which mean that the LRP can be used for identification purposes. Therefore, the LPR with favorable advantages such as good range resolution, simple system structure, highly speed of target identification has been to the research highlight by its immense potency at the field of target identification.

The studies of LRP, similar to the radar range profile, are based on many methods, such as scattering centre, Physics Optical approximation, FDTD. To the smooth targets, Adachi and Uno [3] studied on the reconstruction of range profile of the perfect conductor using a general physical optics and Umashankar [4] obtained the formulation of an inverse range profile of the conductor using the finite-difference time-domain method. Noguchi A [5]. has used a physical optics approximation to reconstruct two dimensional rough surface with Gaussian beam illumination. However, in laser wave band the roughness of a target must be considered. Galdi V [6]. has reconstructed the moderately rough interfaces via quasi-ray Gaussian beams using multifrequency method. But the calculation is very complex and it only acquired the LRPs of the simple targets. L.G. Shirley and G.R. Hallerman of Lincoln Laboratory [7] have discussed the LRPs of the cone, the plate and the cylinder by defining the range-resolved laser radar cross section(RRLRCS) at the diffuse Lambertian case. The expression for the RRLRCS builds on the height function of a target. However the height function is not suitable for the cylinder and for some targets there are two height functions.

In experiment, a research team of TNO defence has succeed in distinguishing the sea-surface targets using the LRP, and their approach can distinguish five ships with no false identification [8–10]. However, they just only compared the received peaks of backscattered waveform with the constructed database of LRPs. In fact, the laser pulse echo carries more information of the targets such as shape, height, surface material. Accordingly, it can be achieved more details by analyzing the data of the LRPs.

Y.H. Li, Z.S. Wu and Y.J. Gong [11] have acquired the laser one-dimensional range profile of a cone using beam scattering by a rough object, but only given the simulation but no experiment. In this paper a novel method of LRP is established, which is based on the electromagnetic Gaussian beams scattering theories and the radar equation. And the expression for LRP of targets with rough surfaces is obtained. As especial cases, the LRPs of the flat plane, the sloped plane, and the circular cone are simulated in detail by the method. Moreover the experiment is set up to get the LRPs for the three different targets, and the achieved results agree with the calculated results. It shows that the method is correct and useful to the application of the LRP. The effect of the some parameters including the target shape and plusewidth on the LRP is analyzed. Those works will be constructive to the backscatter signal analysis and the LRP database conduction.

## 2. Laser Range Profile Theory

As shown in Fig. 1 , the laser pulse beam is incident on the target. The received backscattering pulse beam is different obviously to the different shape of the targets and the different character of surface. The backscattering wave power of a laser pulse is in keeping with the lidar range equation [1]:

*P*is received signal power,

_{s}*P*is transmitter power,

_{i}*σ*is effective target scattering cross-section,

*R*

_{0}is the distance between the target center and the transmit-receive system,

*D*is receiver aperture diameter,

*T*is attenuation factor, and

_{a}*η*is system efficiency.

_{syst}To the Gaussian beam, the amplitude can be written as:

*w*

_{0}is waist radius,

*w*(

*z*) is beam radius, $\overrightarrow{{r}^{\prime}}$ is the position between the point of the target and the target center, and ${g}_{0}(\overrightarrow{{r}^{\prime}})$is the square of the distance between any point of the target and the band center axis. The power is proportional to the square of the amplitude,

Commonly the distance *R*
_{0} between the target center and the transmit-receive system is very large compared to the radial length of the target, *z* = *R*
_{0}.

Then, the Eq. (1) can be written as:

*σ*

^{0}is unit area laser scattering cross-section, and $\sigma ={\displaystyle \int {\sigma}^{0}\text{d}{S}^{\prime}}$. To the Lambert surface, ${\sigma}^{0}=4{\rho}_{r}\mathrm{cos}{\theta}_{i}\mathrm{cos}{\theta}_{r}$, ${\rho}_{r}$ is the hemispherical reflectance, $\mathrm{cos}{\theta}_{i}$and $\mathrm{cos}{\theta}_{r}$are the incident angle and scattering angle, respectively, in the local coordinate of the facet. The Eq. (4) discloses the course of the transmitter energy which is transferred to the targets by transmitter system and is reflected to the received system from the targets. The scattering power of transmitter pulse is determined by the material surface, shape and size of targets, which will enter the received system after attenuated. The received scattering power is restricted by the performance of transceiver system, the transmission path of atmosphere, the laser scattering characteristic of targets. To achieving the LRP of the target, those factors above should be considered integratively.

When the incident pulsewidth *T*
_{0} is the full width *T* at half maximum of the pulse, the laser range equation can be written as:

Equation (5) is the formula of the LRP. where, $R=-{\overrightarrow{r}}^{\prime}\cdot {\widehat{k}}_{0}$, ${\widehat{k}}_{0}$is the unit vector of the incident wave, $\Delta =cT/2$ is the radar distinguishing unit, ${R}^{\prime}$ is the aggregate of the points in the target surface with the same radial component *R.*
${R}_{1}$ is the radial coordinate of any point of the target. Equation (5) shows that the backscattering power${P}_{s}(t)$ is the mean value between range ${R}_{1}-\Delta /2$ and range${R}_{1}+\Delta /2$ at any time *t*. Then, the backscattering signal contains the information of the girdle band in the target surface which radial width is $cT/2$.

## 3. Laser Range Profile Simulation and Analysis

Consider the incident laser beam is assumed to have a Gaussian wave front which is large enough to contain the entire target. The laser beam center is incident from the *z*-axis which is defined as the central axis of the target. The incident waveform is a Gaussian function, ${u}_{i}(t)={E}_{0}\mathrm{exp}(-{t}^{2}/{T}_{0}^{2}+i\omega t)$, the corresponding incident power is ${P}_{i}(t)={\left|{E}_{0}\right|}^{2}\mathrm{exp}(-2{t}^{2}/{T}_{0}^{2})$. The target distance *R*
_{0} is 50*m* in the following cases.

#### 3.1 Flat plane

When the flat plane area is *S* and the hemispherical reflectance of the Lambert flat plane is *ρ _{r}* which depends on the character of the material in the target surface. The LRP equation of the flat plane can be written as:

*t*= 2

*z/c*.

As it shows in Fig. 2
, the LRPs of the plane is simulated when *ρ _{r}* of the Lambert flat plane is 0.9, 0.6, and 0.3. Where, the value of the incident power is normalized to unity and the relative backscattering power is plotted along the vertical axis. It can be seen that the backscattering pulse is still a Gaussian-shaped pulse and its pulsewidth is the same as the incident pulse. In addition, the backscattering power is proportional to

*ρ*.

_{r}At this case, the backscattering signal shape can be decided whether the target is a flat plane or not by comparing to the incident pulse shape. The surface roughness can be inversed calculation by backscattering power.

#### 3.2 Sloped-plane

The side length of the square sloped-plane is *l* and the incident angle (the angle between the incident direction and normal direction of the sloped-plane) is *θ*. The LRP equation of the sloped plane can be written as:

Figure 3(a)
shows the LRPs of the sloped-plane when *θ* = 60° and *T*
_{0} = 300ps for *l* = 0.03m, 0.1m, 0.2m and 0.3m. As it can be seen, when *l* increases, the backscattering power increases and the top of the LRPs become flatter. This is because that the incident pulse intensity increases and the backscattering power increases with *l* increasing when the beam can cover the targets. When *T*
_{0} is fixed, Δ will keeps invariant, which means the ability of distinguishing the radial range is invariant. When the radial size of the target is much bigger than Δ, the pulse top is flat and it shows the scattering cross section of each part of the target is same. The peak length coincides with the radial size of the sloped-plane. Figure 3(b) shows the LRPs of the sloped-plane when *θ* = 60° and *l* = 30mm for *T*
_{0} = 300ps, 100ps, 50ps and 30ps. As it can be seen, with the decreased pulsewidth, the backscattering power decreases and the backscattering pulse can reflect the target profile shape exactly. This is because the smaller the incident pulsewidth, the lower the pulse power at the given pulse peak power. With the *T*
_{0} decreasing Δ decreases, the ability of distinguishing the radial range increases and the LRPs can reflect the target profile shape.

It can be found that the backscattering pulse is wider than the incident pulse, and the broadening coincides with the radial range of the target. When Δ is much smaller than the radial size of the target, the scattering pulse can reflect the sloped-plane profile shape. The peak length coincides with the radial size of the sloped-plane. At this case, comparing the shape of the backscattering signal with the incident pulse shape can decide whether the target is a sloped-plane or not.

#### 3.3 Circular cone

The LRPs of the circular cone with the bottom radius *a*,the half-cone angle *α* and the height *h* are studied. The laser is incidence from the central axis on the circular cone top. The LRP equation of the circular cone can be written as:

Figure 4(a)
shows the LRPs of the circular cone when *α* = 11.3° and *T*
_{0} = 300ps for *h* = 0.1m, 0.2m and 0.3m. Figure 4(b) shows the LRPs of the circular cone when *α* = 11.3° and *h* = 0.1m for *T*
_{0} = 300ps, 100ps and 50ps. It can be seen that the LRPs rise slowly but fall quickly, the peak position is near the bottom position of the circular cone and the LRP width is the height of the circular cone. In Fig. 4(a), when it keeps *α* invariant, the enlarged *h* increases the scattering cross section, which causes the backscattering power increase. In Fig. 4(b) it can be seen that the narrow incident pulsewidth causes the decrease of the peak power of the LRPs. It is because when the pulsewidth is narrow down, the backscattering power in one radar cross section, section, which causes the backscattering power increase. In Fig. 4(b) it can be seen that the narrow incident pulsewidth causes the decrease of the peak power of the LRPs. It is because when the pulsewidth is narrow down, the backscattering power in one radar distinguishing unit is weaken, so that the LRP intensity is decreased. While, the LRP shape is close to the exact circular cone profile and the imaging effect will be better.

In addition, the range profile in the microwave band is very sensitive to the variety in the incident angle. The LRP has a similar character. At this case, the LRP of the circular cone can be written as:

Where, $t=2\left({R}_{0}-{z}_{0}\right)/c$, ${t}^{\prime}=2\left({R}_{0}-{\widehat{k}}_{s}\cdot {\overrightarrow{r}}^{\prime}\right)/c$, $-h/2\le {z}_{0}\le h/2$, $0\le {\theta}_{0}\le \pi /2$ is the incident angle, ${\overrightarrow{r}}^{\prime}=((h/2-z)\mathrm{tan}\alpha \mathrm{cos}\phi ,(h/2-z)\mathrm{tan}\alpha \mathrm{sin}\phi ,z)$, ${\widehat{k}}_{s}=(\mathrm{sin}{\theta}_{0},0,\mathrm{cos}{\theta}_{0})$,is the unit vector of the backscattering wave, $\mathrm{cos}{\phi}_{0}\ge -\mathrm{cot}{\theta}_{0}\cdot \mathrm{tan}\alpha $.

Figure 5
shows the LRPs of three different incident angles *θ*
_{0}. It can be seen that the peak value and position are all different. When incident pulse wave is parallel to the axis of the circular cone (${\theta}_{0}=0$), the radial length of the LRP is just the length of the circular cone. When incident pulse waveband deviates ${\theta}_{0}$ from the axis of the circular cone, the radial length of the LRP is narrowed to $h\mathrm{cos}{\theta}_{0}-a\mathrm{sin}{\theta}_{0}$.

## 4. Experiment setup

Based on the analysis above, the LRP signal can be obtained from the theory and simulation. Then the experiment system of Fig. 6 is set up and is used to measure the backscattering signal of the different targets of the flat plane, the sloped-plane and the circular cone. After extending and collimating, the laser beam covers the target fully and the convergent backscattering light enters the detector. When the signal goes through amplifier and enters the oscilloscope, the data can be extracted and processed. The vertical distance from laser to target is 50m, the laser is mode-locked Ti: sapphire laser of central wavelength 0.8μm and pulsewidth 300ps, the detector bandwidth is 10GHz and the oscilloscope bandwidth is 11GHz.

#### 4.1 Flat plane

*ρ _{r}* of the Lambert flat plane is 0.9 and 0.6. Figure 7
shows that theoretical calculation and experimental data is complete match. Backscattering pulse is still a Gaussian-shaped pulse with the same pulsewidth, and the backscattering power is proportional to

*ρ*.

_{r}#### 4.2 Sloped-plane

The side length *l* of the sloped-plane is 30mm and the incident angle *θ* is 30° and 60°. Figure 8
shows that theoretical calculation is consistent with the experimental results. The backscattering pulse is more width than the incident pulse, the broadening coincide with the radial range of the target. And the scattering power decrease with increasing incident angle. It is because the larger the incident angle is, the smaller the scattering cross section is.

#### 4.3 Circular cone

The circular cone height *h* is 0.1m, bottom radius *a* is 0.02m and the material is aluminium and teflon. The laser is incidence from the central axis on the circular cone top. Figure 9
shows that theoretical calculation is close approximately to the experimental data. The backscattering pulse is more width than the incident pulse and the broadening coincide with the radial range of the target. Meanwhile, the scattering power is different with the different material and increases with increasing target reflectance.

## 5. Conclusion

In this work the LRP equation is acquired and the LRP signal of simulation and experiment are achieved. The backscattering simulations and experiments are done for the flat plane, the sloped-plane and the circular cone. The backscattering waveforms received from different profile, roughness and incident angle are shown. The method remains to be tested with experimental data, and can be expanded to handle more complex geometries. This paper offers theoretical basis and simulation method for the abstraction and identification of the target feature in laser wave band.

## Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 60771038 and the Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences Under Grant No. SKLST200906.

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