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Research on laser range profiles based on spatial domain

Yanhui Li, Di Gao, and Hong Liao

Open Access Open Access

Abstract

Information about geometric features and surface material can be obtained by the analysis of the laser range profile (LRP) acquired from the target. Some apparent transforms of the laser range profile that may father obstacle to the target recognition occur when the laser intensity has different spatial distribution. In this paper, a LRP equation is proposed to describe the situation when a single-site radar at an arbitrary location detects the target, and thus simulations of an inclined plate and cone LRP outcome based on the plane wave. As for LRPs based on Gaussian beams, the beam factor is raised. By analysis of the cone LRPs at different intensity distributions, several abnormal intensity-range profiles are found, which may lead to misjudgment for the target. For preparation of the study about LRPs under different weather conditions, a cone LRP at atmosphere turbulence is also simulated.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Laser range profile (LRP) is a relatively new field of research, especially the high resolution imaging applications. Different from Intensity profiles reflecting intensity of each part of the target, laser range profiles (LRPs) as a vital method for target recognition in the optical band, can obtain the 3-D shape and the range information of the target by one pulse [1,2]. The basic idea of LRP is that when the pulse laser illuminates the target, the detector can receive the time-varying scattering reflected signal which contains the information of the target. With analyzing the features of the achieved signal, the details of target structure in the direction of the lidar can be discovered. With benefits such as good range resolution, simple system structure, highly speed of target identification, LRPs establishes the application in the field of target recognition and future intelligent weapons [3,4].

The laser radar emits a pulse which makes backscatter on the target’s surface with its power information, will be accepted by the receiving aperture thus the intensity-range profile of the subject can be obtained [4]. Yet in this process, LRPs would be different due to changes in the laser beam which have influenced the intensity distribution.

In recent years, because of widespread use of laser beams in optical communication, a great deal of attention has been paid by statistical optics scientists to the subject of propagation of laser beams in various random medium [5]. Beam spread and beam wandering are the most perceptible effects of atmospheric turbulence on propagating laser beams [6], which is closely related to the beam radius and the beam center axis. These situations would definitely cause a huge impact on the intensity distribution. Therefore, it is necessary to set up the physic model of these situations and research them.

Some earlier study on the radar ehco has been made. Ove Steinvall’s simulation results show the target shape and reflection characteristics is important for determining the shape and the magnitude of the laser return [7]. Our research group has already accomplished some experimental measurements and theoretical simulation of the laser range profile [8,9]. In their papers, LRPs of cones showing vary postures are simulated and analysis illustrating how the pulse width and the side length make differs in LRPs. The beam center axis occurs in their papers is generally symmetry around the regular target, while the deviation of the beam center axis would lead to a different intensity distribution on the target.

The previous studies are all interested in the target shape or the beam shape. In reality scene the laser might deviate from its original aiming direction, and the intensity distribution on beam radial section would also be changed by the atmosphere. Therefore it is necessary to make research on the spatial distribution of laser intensity. This paper gives a LRP equation based on the laser light irradiating the target at any spatial location. Besides, the Gaussian beam factor is proposed to distinguish the plane wave and the Gaussian beam. By analyzing LRPs of Gaussian beam with different beam radius and changed beam center axis, we find the change of intensity distribution may lead to embarrassments of target recognition.

In the past few decades, much research has been carried out to find a suitable way to reduce the effects of turbulent atmosphere on laser beams [10]. Considering the atmospheric turbulence would also make differs in laser intensity distribution, it is significance to make research on the laser range profile based on spatial domain.

2. Laser one-dimensional range profile

As is shown in Fig. 1, the laser pulse beam is incident on the target. The laser incidence direction is parallel to the Z-axis, and Oxyz is the coordinate system for the target. The whole target is illuminated by a single laser pulse. 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 by the lidar range equation can be written as [11]:

$${P_s} = \frac{{{P_t}}}{{4\pi R_t^2}}\frac{\sigma }{{4\pi R_r^2}}{A_r}{G_r}$$
where, Ps is the received signal power, ${P_t}$ is the transmitter power, ${A_r}$ is the clear aperture of the detector, ${G_r}$ is the gain function, ${R_t}$ and ${R_r}$ are distances from the target to the transmitter and receiver, respectively, and $\sigma $ is the laser scattering cross-section.

 figure: Fig. 1.

Fig. 1. The laser pulse beam radiating the target schematic diagram.

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When ${P_t}$ has a pulse form $P(t)$, the extended target ${R_t}$ and ${R_r}$ are the distances from the zero point to the transmitter and receiver, respectively. For a single station radar, ${R_t} = {R_r} = {R_0}$. Then, the Eq. (1) can be written as:

$${P_s}(t) = \int {\textrm{d}\sigma } \;\frac{{P(t^{\prime})}}{{4\pi {R_0}^2}}\frac{{{A_r}{G_r}}}{{4\pi {R_0}^2}}$$
where $t^{\prime} = t - ({R_r} + {R_t})/c - 2Z/c$, $\sigma$ is the laser scattering cross-section.

When time ${t_0} = ({R_r} + {R_t})/c$ is chosen as time zero, the pulse propagates to position ${z_t}$ at t time. In this case, Eq. (2) can be obtained as

$${P_s}({z_t}) = \frac{{{A_r}{G_r}}}{{4\pi {R_0}^2 \cdot 4\pi {R_0}^2}}\int {\textrm{d}\sigma } \;P(2{z_t}/c - 2Z/c)$$
Equation (3) is the LRP expression and can be applied to rough targets.

With Eq. (3), LRPs of the sloped-plane and the circular cone based on the plane wave can be obtained:

Figure 2(a) shows LRPs of the sloped-plane with length $l = 0.5\textrm{m}$ when $\alpha = {45^ \circ }$ is the slope angle for the pulse width ${T_0} = 0.1\textrm{ns}$, while Fig. 2(b) shows LRPs of the circular cone with height $h = 0.5\textrm{m}$ when the tangent of the half-cone angle is $\tan \alpha = 0.25$ for the pulse width ${T_0} = 0.1\textrm{ns}$. Besides, the geometric center point of the slope-plane and the cone top point are chosen as the zero point respectively. For intensity distribution of the plane wave is average, LRPs basically show the area of effective target scattering cross-section being covered by the radar distinguish at unit Z. Under this circumstance, for Fig. 2(a) the peak length coincides with the radial size of the sloped-plane while for Fig. 2(b) the rapidly falling edge correspond to the bottom of the circular cone.

 figure: Fig. 2.

Fig. 2. LRPs at normal conditions: (a) LRPs of slope-plane; (b) LRPs of the circular cone

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3. Gaussian beam factor

At most case the incident laser is the Gaussian beam with uncertain incident angle (the angle between the incident direction and normal direction of the target) and position of the beam center axis consequently the LRP equation under more broad conditions is necessary.

To the Gaussian beam, the amplitude can be written as:

$$u(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime},Z) = {E_0}\frac{{{\omega _0}}}{{\omega (Z)}}\exp [ - \frac{{{g_0}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime})}}{{{\omega ^2}(Z)}}]$$
where, ${\omega _0}$ is waist radius, $\omega (Z)$ is beam radius, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime}$is the position between the point of the target and the target center, and ${g_0}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime})$ is the square of the distance between any point of the target and the beam center axis. The power is proportional to the square of the amplitude,
$$P(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r^{\prime}} ) = |u(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime},Z){|^2} = {P_i}\frac{{{\omega _0}^2}}{{{\omega ^2}(Z)}}\exp [ - \frac{{2{g_0}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime})}}{{{\omega ^2}(Z)}}]$$
Define the Gaussian beam far field divergence angle$\phi$:
$$\phi \approx \tan \phi = \frac{{\omega (Z)}}{Z}\textrm{ = }\frac{2}{{{k_0}{\omega _0}}}$$
where, ${k_0}$is the modulus of the incident wave vector. To the Gaussian beam, the LRP equation adds a factor than the case of plane beam,
$$\frac{{{\omega _0}^2}}{{{\omega ^2}(Z)}}\exp \left[ { - \frac{{2{g_0}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r}^{\prime})}}{{{\omega^2}(Z)}}} \right]$$
which, is called the Gaussian beam factor, making consider the intensity distribution.

Obviously, when on the same wavelength of laser, the transmission distance is the same, the beam of different shapes, the laser beam spatial distribution are quite different. Figure 3 shows the Intensity of transverse distance space with the normalized distribution demanding a stable laser power when λ=1.06 µm, Z=1500 m, w0=4000 µm, 3000 µm, 2000 µm, 1000 µm, 500 µm, and the corresponding w(Z) = 1.012 m, 0.506 m, 0.253 m, 0.169 m, 0.127 m, as well as the plane beam (w(Z)= +∞).

 figure: Fig. 3.

Fig. 3. Intensity distribution of Gaussian beam with the beam radius change.

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Moreover, the incident angle and the position of the beam center axis should be added to the LRP equation.

 figure: Fig. 4.

Fig. 4. The Gaussian beam scatter schematic diagram of the rough target.

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The target surface equation can be written as:

$$f(x,y,z) = 0$$
To simplify the situation, the target is supposed to be a convex quadric body (Fig. 4). With a target coordinate system$Oxyz$ and incident coordinate system$OXYZ$, when $\theta $ is the zenith angle, the transformation between the coordinate system XYZ and a target object coordinate system xyz can be written as:
$$\left( {\begin{array}{{c}} x\\ y\\ z \end{array}} \right) = \left( {\begin{array}{{ccc}} 1&0&0\\ 0&{\cos \theta }&{ - \sin \theta }\\ 0&{\sin \theta }&{\cos \theta } \end{array}} \right)\left( {\begin{array}{{c}} X\\ Y\\ Z \end{array}} \right)$$
Equation (8) and Eq. (9) can be combined as:
$$f(x(X,Y,Z),y(X,Y,Z),z(X,Y,Z)) = 0$$
To calculate the backscattering between the target and the incident laser, the unit laser scattering cross-section can be written as,
$$d\sigma (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime}) = 4\pi {f_r}(\beta ){\cos ^2}\beta dS\textrm{ = }4\pi {f_r}(\beta )\cos \beta dXdY$$
where, ${f_r}(\beta )$ is the scattering coefficient, $\cos \beta$ is cosine of the scattering angle, which can be further written as,
$$\cos \beta \textrm{ = }\frac{{ - {f_Z}}}{{\sqrt {f_X^2 + f_Y^2 + f_Z^2} }}\textrm{ = }\frac{{\sin \theta {f_y} - \cos \theta {f_z}}}{{\sqrt {f_x^2 + f_y^2 + f_z^2} }}$$
The position of the laser source is $O^{\prime}(0,0,0)$, where ${R_0}$ means the side range between the laser source and the target center point. Furthermore, the beam center axis whose location is defined as $X = {X_0},Y = {Y_0}$, is parallel to the incident direction(the positive direction of Z axis).

So far, combining the radar range equation, LRPs equation with any incident angle to the target can be written as,

$${P_r}({\textrm{Z}_0}) = \frac{{{A_r}{G_r}}}{{4\pi {R_0}^2 \cdot 4\pi {R_0}^2}}\int_{{Z_0} - \Delta /2}^{{Z_0} + \Delta /2} {dZ^{\prime}\int_{{C_0}} {{P_i}} } \frac{{{\omega _0}^2}}{{{\omega ^2}(Z)}}\exp [ - \frac{{2{g_0}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime})}}{{{\omega ^2}(Z)}}]d\sigma (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime})$$
$({C_0}:f(x(X,Y,Z^{\prime}),y(X,Y,Z^{\prime}),z(X,Y,Z^{\prime})) = 0$, ${g_0}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ^{\prime}) = {(X - {X_0})^2} + {(Y - {Y_0})^2}$, $\cos \beta \gt 0)$

where, $\Delta \textrm{ = }c{T_0}/2$ is the radar resolution unit.

4. Influences on LRPs by the intensity distribution

According to the illustration above, there are influences on LRPs by the Gaussian beam factor. To calculate the effect more exactly, simulations of the circular cone LRP based on the Gaussian beam has been made. Additionally, combining the surface equation of the circular cone to Eq. (12), the LRP equation of the circular cone can be made. The circular cone height (h) is 0.5m, bottom radius is $r = h\tan \alpha $, The incident angle ($\theta $) is 0°.

Besides, the Gaussian incident pulse adopted can be written as,

$${u_i}(t) = {E_0}\exp ( - {t^2}/T_0^2 + i\omega t)$$
Corresponding to Eq. (14), the pulse power can be written as,
$${P_i}(t) = {E_0}^2\exp ( - 2{t^2}/{T_0}^2)$$
when the laser scans from the top to the bottom of the circular, the target cross section increases while the light intensity decreases, both of the two factors make contribution to the backscattering power and decide on the trend of the LRPs. As shown in Fig. 5(a), when the beam radius is as small as $\omega (Z) = r$, the intensity is focused nearby the beam center axis thus the peak position is near the balance point where the two factors’ contribution neutralize rather than correspond to the bottom of the circular cone. On the other hand, when the beam radius keeps increase until it is more than $\omega (Z) = 4r$, the intensity distribution turns to be average and the LRP is close to the LRP of the plane wave ($\omega (Z) = \infty $). Another two incident angles $\theta = 15^\circ $and $\theta = 45^\circ $are adopt in Fig. 5(b) and Fig. 5(c).

 figure: Fig. 5.

Fig. 5. LRPs of circular cone with the beam radius change at different incident angles: (a) $\theta = {0^ \circ }$ (b) $\theta = {15^ \circ }$ (c) $\theta = {45^ \circ }$with the center of the cone on the beam center axis.

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LRPs in off-axis situations are also worth of study cause that the target center point is regualarly not on the beam center axis. So several simulations of that have been made.

In Fig. 6(a) d represents for the distance between the circular cone center axis and the beam center axis. When the beam center axis deviate with d increases at $\omega (Z)\textrm{ = }r$, the total intensity of the Z section decrease and the intensity distribution make relatively weaker influences to the LRP than the target cross section, as a result we find the peak is lower and the peak position move toward to the position correspond to the circular cone bottom. Moreover, with d increases the rate of the intensity attenuation becomes lower and lower while the scattering cross section keeps a stable increase resulted in a faster and faster increase of the backscattering power, the LRP shape changes from convex to concave.

 figure: Fig. 6.

Fig. 6. LRPs of circular cone with the beam center axis deviate when $\theta \textrm{ = }{0^ \circ }$: (a) $\omega (Z)\textrm{ = }r$ (c) $\omega (Z)\textrm{ = 2}r$(c) $\omega (Z)\textrm{ = 4}r$.

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Figure 6(b) shows LRPs when $\omega (Z)\textrm{ = }2r$ $\theta \textrm{ = }{0^ \circ }$ with the beam center axis deviate. Comparing Fig. 6(b) to Fig. 6(a), we find that the peak position would no longer change and correspond to the bottom of the circular cone, which indicates the peak position is only controlled by the scattering section when the beam radius is big enough to ignore the intensity change caused by moving the beam center axis.

As is shown in Fig. 6(c), the peak position as well as the trend of the line are highly consistent as a result of the average intensity distribution in general, which can be verified by comparing the LRP when $\omega (Z)\textrm{ = 4}r$ and the LRP when $\omega (Z)\textrm{ = }\infty $ in Fig. 5(a)

In summary, the intensity distribution changes when the beam radius is different, which lead to diverse LRPs of the same target. When the beam radius is 4 times bigger than the target radius, the intensity distribution of the Gaussian beam is near to that of the Plane wave, so LRPs are of good description of the target profile shape. When the beam radius is as big as the target radius, laser intensity is highly concentrated on the beam center, so LRPs affected by this intensity distribution are of diverse shapes, which makes it hard to identify the profile shape of the target. Besides, LRPs with two more beam aiming angle are given in the following part of this article.

Considering the incident angle is $\theta \ne {\textrm{0}^ \circ }$, the circular cone is no longer symmetry about the target center axis thus we should distinguish the right deviation and the left deviation so the right is settled as the positive direction. Obviously shown in Fig. 7(a), there are still changes in the peak position and the concavity when the beam radius is small enough.

 figure: Fig. 7.

Fig. 7. LRPs of circular cone with the beam center axis deviate when $\theta \textrm{ = }{15^ \circ }$: (a) $\omega (Z)\textrm{ = }r$ (c) $\omega (Z)\textrm{ = 2}r$(c) $\omega (Z)\textrm{ = 4}r$.

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From Fig. 7(b), the LRPs are quite different on both sides of the center axis as in shown with dotted line. In this case, the intensity distribution remains a main factor of the backscattering power changing when the beam center axis on the left side of the center axis (d<0), while the cross section area dominants when the beam center axis on the left side of the center axis (d>0).Moreover, similar patterns can be found in Fig. 7(c).

From Figs. 8(a), (b) and (c), similar conclusions could be found. Besides, the LRPs in different intensity distributions show obvious disparity, which may lead to hard work on target recognition.

 figure: Fig. 8.

Fig. 8. LRPs of circular cone with the beam center axis deviate when $\theta \textrm{ = }{45^ \circ }$: (a) $\omega (Z)\textrm{ = }r$ (c) $\omega (Z)\textrm{ = 2}r$(c) $\omega (Z)\textrm{ = 4}r$.

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5. Research on laser range profiles in atmosphere turbulence based on spatial domain

The amplitude and phase of a laser would have random fluctuations when it transmits in random media (such as atmospheric turbulence) [12,13]. We only study on spatial domain without considering the time factor, thus the intensity in atmosphere turbulence can be written as [14,15],

$$I(i,j) = {I_0}(i,j)\exp [4\chi (\rho ,0)]$$
where ${I_0}(i,j)$ is the backscattering power of a scattering cross section unit without atmosphere turbulence, the intensity fluctuation factor is indicated by the index, $\chi (\rho ,0)$ is a Gaussian random variable with mean as $- {\delta _x}^2$ and variance as ${\delta _x}^2$, ${\delta _x}^2$ is the logarithm amplitude variance determined by the atmosphere turbulence distribution. Sample results of the beam intensity distribution on a radial section are shown in Fig. 9 and are described below.

 figure: Fig. 9.

Fig. 9. Intensity distribution of Gaussian beam

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The intensity flicker caused by turbulence make the beam indistinct, which means the intensity distribution is no more consequent but undulate. From Fig. 9 the Gaussian beam amplitude fluctuation is more and more intense with the $\delta _x^2$ increasing, which could result in LRPs of terrible quality. For a more intuitive display, LRPs of Gaussian beam under atmosphere turbulence circumstances are simulated.

As is shown in Fig. 10, the LRPs of the circular cone varies in amplitude when the logarithm amplitude variance changes. The LRP of cone is simulated with a radar-target distance 5km, the beam aiming angle at 0, and the deviation between beam center and target center at 0. When $\delta _x^2\textrm{ = }0$, it means there are no turbulence effects on LRPs, the LRP is of good shape. When the $\delta _x^2$ increases, the radar echo intensity would be with a wavy motion. When $\delta _x^2$ is large enough as $\delta _x^2\textrm{ = }0.5$, the LRP of the cone is too fluctuant to recognize its profile shape.

 figure: Fig. 10.

Fig. 10. LRPs of circular cone in atmosphere turbulence

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6. Conclusion

The spatial intensity distribution shows strong influences on target recognition by the LRP, which may even lead to some misjudgments of the target. For the radar imaging and target recognition, our study is significant. There is a great quantity of factors making differences on the LRP except for the spatial intensity. For further study, our group will do some works on how the atmosphere turbulence affects radar imaging.

Funding

National Natural Science Foundation of China (61431010, 61475123); the Higher Education Discipline Innovation Project (B17035).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

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4. Y. Mou, Z.-s. Wu, Z.-j. Li, and G. Zhang, “Geometric detection based on one-dimensional laser range profiles of dynamic conical target,” Appl. Opt. 53(35), 8335–8341 (2014). [CrossRef]  

5. M. Yousefi, S. Golmohammady, A. Mashal, and F. D. Kashani, “Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(11), 1982–1992 (2015). [CrossRef]  

6. M. Charnotskii, “Optimal beam focusing through turbulence,” J. Opt. Soc. Am. A 32(11), 1943–1951 (2015). [CrossRef]  

7. O. Steinvall, “Effects of target shape and reflection on laser radar cross sections,” Appl. Opt. 39(24), 4381–4391 (2000). [CrossRef]  

8. Y. H. Li and Z. S. Wu, “Targets Recognition Using Subnanosecond Pulse Laser Range Profiles,” Opt. Express 18(16), 16788–16796 (2010). [CrossRef]  

9. L. Yanhui, W. Zhensen, G. Yanjun, and Z. Geng, “Analytical model of a laser range profile from rough convex quadric bodies of revolution,” J. Opt. Soc. Am. A 29(7), 1383–1388 (2012). [CrossRef]  

10. T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017). [CrossRef]  

11. R. Schoemaker and K. Benoist, “Characterisation of small targets in a maritime environment by means of laser range profiling,” Proc. SPIE 8037, 803705 (2011). [CrossRef]  

12. Z.-S. Wu and Y.-Q. Li, “Scattering of a partially coherent Gaussian–Schell beam from a diffuse target in slant atmospheric turbulence,” J. Opt. Soc. Am. A 28(7), 1531–1539 (2011). [CrossRef]  

13. Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014). [CrossRef]  

14. H. Ahlberg, S. Lundqvist, D. Letalick, I. Renhorn, and O. Steinvall, “Imaging Q-switched CO2 laser radar with heterodyne detection: design and evaluation,” Appl. Opt. 25(17), 2891 (1986). [CrossRef]  

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Figures (10)
Fig. 1.
Fig. 1. The laser pulse beam radiating the target schematic diagram.

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Fig. 2.
Fig. 2. LRPs at normal conditions: (a) LRPs of slope-plane; (b) LRPs of the circular cone

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Fig. 3.
Fig. 3. Intensity distribution of Gaussian beam with the beam radius change.

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Fig. 4.
Fig. 4. The Gaussian beam scatter schematic diagram of the rough target.

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Fig. 5.
Fig. 5. LRPs of circular cone with the beam radius change at different incident angles: (a) $\theta = {0^ \circ }$ (b) $\theta = {15^ \circ }$ (c) $\theta = {45^ \circ }$with the center of the cone on the beam center axis.

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Fig. 6.
Fig. 6. LRPs of circular cone with the beam center axis deviate when $\theta \textrm{ = }{0^ \circ }$: (a) $\omega (Z)\textrm{ = }r$ (c) $\omega (Z)\textrm{ = 2}r$(c) $\omega (Z)\textrm{ = 4}r$.

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Fig. 7.
Fig. 7. LRPs of circular cone with the beam center axis deviate when $\theta \textrm{ = }{15^ \circ }$: (a) $\omega (Z)\textrm{ = }r$ (c) $\omega (Z)\textrm{ = 2}r$(c) $\omega (Z)\textrm{ = 4}r$.

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Fig. 8.
Fig. 8. LRPs of circular cone with the beam center axis deviate when $\theta \textrm{ = }{45^ \circ }$: (a) $\omega (Z)\textrm{ = }r$ (c) $\omega (Z)\textrm{ = 2}r$(c) $\omega (Z)\textrm{ = 4}r$.

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Fig. 9.
Fig. 9. Intensity distribution of Gaussian beam

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Fig. 10.
Fig. 10. LRPs of circular cone in atmosphere turbulence

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Equations (16)

Equations on this page are rendered with MathJax. Learn more.

P s = P t 4 π R t 2 σ 4 π R r 2 A r G r
P s ( t ) = ∫ d σ P ( t ′ ) 4 π R 0 2 A r G r 4 π R 0 2
P s ( z t ) = A r G r 4 π R 0 2 ⋅ 4 π R 0 2 ∫ d σ P ( 2 z t / c − 2 Z / c )
u ( r ⇀ ′ , Z ) = E 0 ω 0 ω ( Z ) exp ⁡ [ − g 0 ( r ⇀ ′ ) ω 2 ( Z ) ]
P ( r ⇀ ′ ) = | u ( r ⇀ ′ , Z ) | 2 = P i ω 0 2 ω 2 ( Z ) exp ⁡ [ − 2 g 0 ( r ⇀ ′ ) ω 2 ( Z ) ]
ϕ ≈ tan ⁡ ϕ = ω ( Z ) Z  =  2 k 0 ω 0
ω 0 2 ω 2 ( Z ) exp ⁡ [ − 2 g 0 ( r ⇀ ′ ) ω 2 ( Z ) ]
f ( x , y , z ) = 0
( x y z ) = ( 1 0 0 0 cos ⁡ θ − sin ⁡ θ 0 sin ⁡ θ cos ⁡ θ ) ( X Y Z )
f ( x ( X , Y , Z ) , y ( X , Y , Z ) , z ( X , Y , Z ) ) = 0
d σ ( r ⇀ ′ ) = 4 π f r ( β ) cos 2 β d S  =  4 π f r ( β ) cos ⁡ β d X d Y
cos ⁡ β  =  − f Z f X 2 + f Y 2 + f Z 2  =  sin ⁡ θ f y − cos ⁡ θ f z f x 2 + f y 2 + f z 2
P r ( Z 0 ) = A r G r 4 π R 0 2 ⋅ 4 π R 0 2 ∫ Z 0 − Δ / 2 Z 0 + Δ / 2 d Z ′ ∫ C 0 P i ω 0 2 ω 2 ( Z ) exp ⁡ [ − 2 g 0 ( r ⇀ ′ ) ω 2 ( Z ) ] d σ ( r ⇀ ′ )
u i ( t ) = E 0 exp ⁡ ( − t 2 / T 0 2 + i ω t )
P i ( t ) = E 0 2 exp ⁡ ( − 2 t 2 / T 0 2 )
I ( i , j ) = I 0 ( i , j ) exp ⁡ [ 4 χ ( ρ , 0 ) ]
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