2.1. Mie-Scattering Lidar
The schematic diagram of the Mie-scattering lidar is shown in
Figure 1. The collimated laser beam emitted by the transmitting system is expanded by the beam expander and then transmitted to the atmosphere [
27]. Through the combined action of atmospheric Fresnel diffraction, atmospheric extinction and atmospheric turbulence [
28], the backscattered return signal is received by the receiving system and then passes through the spectroscopic system. The return signals of each wavelength are separated and the Mie-scattering signal with a wavelength of 532 nm is selected for our goal. Finally, the return signal is detected, amplified, and displayed by the signal acquisition system.
Table 1 shows the system parameters of the Mie-scattering lidar [
29].
The improved Mie-scattering lidar system is mainly based on the RTS theory, and a suitable aperture diaphragm is added to the original Mie-scattering lidar system to meet the needs of atmospheric turbulence detection. In RTS theory [
25], when light propagates in the atmosphere, if the radius of the scatterer is smaller than the spatial correlation scale
(
, where
is the wavelength of the incident laser light,
L is the laser detection distance), for any large receiving aperture
D, the backscattered light intensity signal can reduce the aperture averaging effect, and there will be some light intensity fluctuations.
The following four conditions are required to be satisfied: (1) the single scattering is approximate and the aerosol particle size is uniform; then, the laser detection distance L, the average radius of aerosol particles , and the incident light wavelength , satisfy ; (2) atmospheric optical thickness 1; (3) the radial light intensity correlation scale () is much larger than the scatterer scale (where is the speed of light, is the pulse width), and the scatterer scale is much larger than the wavelength, that is, ; and (4) under weak fluctuations (), the telescope receiving aperture D, the turbulent coherence length of plane waves , and satisfy (where is the logarithmic light intensity fluctuation variance; , where is the light intensity coherence length derived from the Van–Cittert–Zernike theory, is the effective size of the beam from the light source L, and is the wave number, ). The lidar can be distinguished by two characteristic scales, namely, aerosol spot scale, , and turbulence-related scale, , where F is the focal length of the telescope. Note that , so it generally satisfied for most large-aperture receiving systems ().
Because the spatial scales
and
of the scintillation spot are completely different, three different expressions for the fluctuation of the backscattered light intensity can be obtained by changing the diameter of the telescope’s field of view diaphragm
:
where,
is the variance of the intensity fluctuation of the backscattered light passing through the aperture;
is the variance of the intensity fluctuation of the probe beam. Equation (1) shows that the variance of light intensity fluctuations includes aerosol speckle fluctuations and turbulence fluctuations. Equation (2) shows that the diaphragm smooths the aerosol speckle fluctuations but includes turbulence fluctuations. Equation (3) shows that aerosol speckle fluctuations and turbulence fluctuations are smoothed by the diaphragm, and the total received energy does not fluctuate.
In this paper, the selected telescope aperture
D is 10 inches (254 mm), the repetition frequency
f is 10 Hz, and the adjustable focal length
F is 2500 mm. Considering the experimental operability and the actual laboratory conditions, the selected field of view of the telescope is
. According to the RTS theory, the spatial correlation scale
(
) of the light intensity fluctuation is calculated as shown in
Figure 2. The spatial correlation scale
increases with the increase in the detection distance and incident wavelength. When the detection distance is the maximum value of 3 km and the incident laser wavelength takes the maximum value of 1064 mm,
gets the maximum value of 57.5 mm, which obviously satisfies
. In addition, according to our laboratory’s long-term measurement results of aerosol particle spectral distribution in the Yinchuan area, China, it can be known that the aerosol particle size
. Yinchuan area is located in the junction of the four deserts, namely Badain Jaran, Ulan Buhe, Tengger, and Mu Us deserts, combined with the typical temperate continental semi-arid climate, the aerosol particle size in spring is slightly larger than the average; obviously, all meet the condition
. Therefore, the improved Mie-scattering lidar system can satisfy the RTS theory. By changing the diameter of the field of view diaphragm, the information characterizing atmospheric turbulence can be obtained from the return signal.
Gaussian beam is one of the most commonly used models in lasers. In fact, the light intensity of a Gaussian beam in a plane is perpendicular to the direction of propagation. A Gaussian beam in a homogeneous medium is expressed as:
where,
is the beam radius (i.e.,
),
is the minimum value of
ω(
z) at
z = 0, which is called the light waist radius,
r is the distance between a point in space (
x,
y,
z) and the light source; and
k is the wave number (i.e.,
).
Figure 3 shows the light intensity distribution of the Gaussian beam with a beam waist radius of 20 mm, wavelength
nm, and
. It can be seen that the light spot is a regular circle, and the light spot is at the center.
2.2. Numerical Simulation of Gaussian Beam Transmission in Atmosphere
When the laser is transmitted in the atmosphere, it is subjected to the dual action of the turbid atmosphere and continuous turbulence and then diffracted in the atmospheric space, resulting in atmospheric scattering and absorption and a series of turbulent effects caused by turbulent refraction. In the phase screen model, the light propagation in atmospheric turbulence mainly starts from the light propagation equation. Assuming that the transmission path of the light beam is composed of several thin phase screens distributed in a vacuum environment, and when the phase change caused by the refractive index of the medium is particularly small, the process of light propagation in a vacuum environment and the phase modulation of the medium can be regarded as completely independent of each other and completed simultaneously [
1]. That is, the multi-layer phase screen model of light propagation in a continuous random medium is shown in
Figure 4, in which the circles or ellipses represent random media in the atmosphere, such as atmospheric molecules, aerosols, etc. The phase disturbance is firstly generated by a thin phase screen with negligible thickness. After propagating through the vacuum of distance
, the phase disturbance of the previous phase screen is superimposed on this phase screen to repeat the phase modulation, with the above process repeating until the end of the transmission.
Assuming that the light travels along the
direction, the light field can be expressed as
. When the laser passes through the atmospheric turbulence with a refractive index of
, under the condition of paraxial approximation, the light propagation satisfies the parabolic equation:
where,
,
is the wavelength of the light wave,
is the wave number, and
is the refractive index fluctuation.
Let the transmission distance
L of the light wave be divided into
segments according to
equal intervals, so the head and tail coordinates of the
i-th segment are
and
, respectively, and the
i-th phase screen is set at the endpoint
of each segment. For each phase screen, the grid spacing of
is equally divided into
grids, and the above formula is solved by the step-by-step method. It can be known that the field distribution of the
I + 1-th phase screen is:
where,
and
are the inverse Fourier transform and Fourier transform,
is the field distribution of the
i-th phase screen,
is the distance between the phase screen, and
and
are the phase space wave number with the unit of
, which are related to the scale of turbulence.
The most important problem in the numerical simulation method of light propagation is to construct the appropriate phase screen, so that it can accurately reflect the change in the atmospheric turbulent refractive index as much as possible [
30], that is, to choose the appropriate refractive index fluctuation
. We use the spatial spectrum model of atmospheric turbulence to obtain a random field of the phase space, and then use Fourier transform to get the spatial distribution of the two-dimensional phase. The specific process is as follows: generate a complex Gaussian random number matrix
, select an atmospheric turbulence power density spectral function
to filter it to obtain a random field in the complex space, and perform inverse Fourier transform on the discrete complex random field to obtain the real phase distribution in space.
Atmospheric turbulence refractive index spectral function
and atmospheric turbulence phase spectral function
using Kolmogorov spectrum [
2] are written by:
where,
is the atmospheric refractive index structural constant when the laser travels along the horizontal path, and the unit is
;
is the spatial frequency;
is the turbulent outer scale, and
is the turbulent flow inner scale, which determines the largest and small eddies in the turbulent eddy.
The complex space random field obtained after filtering the complex Gaussian random number matrix
is given by:
where,
and
are Gaussian random numbers whose real and imaginary parts have a mean of 0 and a variance of 1. The real and imaginary parts of the phase field obtained by Equation (10) are independent of each other, and both satisfy the spatial–spectral distribution.
From the above, the field distribution in the continuous space can be obtained. However, the numerical simulation needs to generate a discrete random field, that is, it needs to perform discrete Fourier transform on it. Taking a phase screen as an example, as shown in
Figure 5, the square phase screen with side length
L is divided into
N equal parts to form
square grids with a spacing of
. Then, the discrete Fourier transform of Equation (10) can be obtained:
where,
is the wavenumber interval of the phase space, according to the sampling theorem,
.
In order to obtain the phase distribution in real space, the inverse Fourier transform of Equation (12) can be given by:
2.3. Low-Frequency Sampling Optimization
For the simulated turbulent phase screen generated by the Fourier transform method, the range of the minimum wave number interval
of the phase screen does not include
and
, so it has the disadvantage of insufficient low frequency [
31]. We use the method of adding sub-harmonics to perform low-frequency compensation on the phase screen by the Fourier transform method. In essence, the high-frequency phase screen simulated by the FFT method is interpolated and fitted to improve the statistical characteristics of low frequency. The specific process is as follows: firstly, divide the square area surrounded by the first sampling point of the low-frequency part limited to the high-frequency part into nine areas with equal area; then, set the sampling points contained in the area (including the edge part) to zero; and finally, take eight small areas outside the center as new sampling points to form a
-order (
) harmonic network.
As shown in
Figure 6, the sampling interval of the
p-th harmonic network becomes
, that is, the original high-frequency sampling area is replaced by
low-frequency sampling small areas. According to the Kolmogorov spectrum, the subharmonic low-frequency compensation screen can be expressed as:
Therefore, the phase screen expression after sub-harmonic compensation is written by:
The phase structure function is a quantity that describes the statistical properties of the atmospheric turbulence phase screen, and the accuracy of the simulation is generally judged by a large number of phase screens. The phase structure function is defined by [
32]:
When the laser is transmitted in the atmospheric turbulence whose spectral model takes the Kolmogorov spectrum, the theoretical expression of the structure function is given by [
32]:
For the inner scale, taking infinity (
) and infinitely small, there are:
where
is the modified Bessel function of the third kind, and
is the Gamma function.
From a statistical point of view, when giving a large number of random phase screen samples, its structure function will gradually converge to its expected structure function. In order to reduce the calculation amount, the expected structure function can be calculated directly [
17]. The specific process is as follows: perform inverse Fourier transform on the phase spectral function
to obtain the auto correlation function
, and then substituting it into Equation (17) can get its structure function. The auto correlation functions of the high-frequency phase screen and low-frequency compensation can be written by:
where,
and
are the spatial filter functions,
,
, and
is the atmospheric coherence length, which is taken as 0.185.
2.4. Scintillation Index of Laser Return Signal
For the RTS theory, the light intensity change of the lidar return spot is the research focus. Under weak turbulence conditions, the scintillation index
is one of the commonly used physical quantities to describe atmospheric turbulence.
represents the normalized light intensity fluctuation difference, which is defined as:
where, < > represents the ensemble mean of the light intensity, and
represents the radial distance from the optical axis of the Gaussian beam.
It is known that the Gaussian beam will produce spot drift after transmission in atmospheric turbulence. In order to obtain its scintillation index
through several light spots with a transmission distance of
, one need to judge the drift distance
of the center of mass of the light spot [
33]. The spot centroid is defined as:
Usually, its first moment is used to represent the coordinates of the centroid of the spot, namely [
1]:
In the simulation, ten light spots with a transmission distance of are taken. For each light spot, if , the center of mass of the drifted light spot is selected as the center of the circle. In addition to this, take as the radius to make a circle, and count the light intensity of points in the circle. If , the center of mass of the spot after the drift is also taken as the circle center. Then, and are used as the radii to make a circle, respectively. Through counting the light intensity of N nodes in the ring, the scintillation index can be calculated by the definition formula . Finally, ten statistical averages are performed to obtain the final scintillation index at a certain distance.
For a Gaussian beam, according to the Rytov theory, the spot scintillation index of the beam is defined as:
where,
,
,
.
2.5. Inversion of Turbulent Profile
The refractive index structure constant
can directly describe the fluctuation of the refractive index and is one of the most intuitive physical quantities to measure the intensity of atmospheric optical turbulence. The atmospheric turbulence intensity profile is the variation of atmospheric refractive index structure constant
with height [
33]. In this paper, the refractive index structure constant
can be calculated from the correlation relationship between the known spherical wave scintillation index
and the atmospheric refractive structure constant
in the horizontal and vertical directions, so as to obtain the atmospheric turbulence intensity profile.
Kolmogorov believes that atmospheric turbulence is composed of turbulent eddies with great difference and different scales. Under the large Reynolds number
Re (
, where
is the characteristic velocity of the fluid,
is the overall characteristic scale of the fluid, and
ν is the molecular kinematic viscosity coefficient), the turbulent eddies of different scales coexist. After the cascade process, the small-scale turbulence finally reaches a statistical equilibrium process, that is, local isotropic turbulence [
2]. The three-dimensional power spectrum function of the locally isotropic turbulent refractive index fluctuation is expressed as:
where,
is the atmospheric turbulence power spectrum function,
κ is the spatial frequency,
is the turbulent inner scale, that is, the scale that determines the smallest vortex in the turbulent vortex, and
is the inner scale correction model factor.
The Kolmogorov atmospheric turbulence power spectrum is an ideal power spectrum model, which does not consider the inner scale effect of turbulence. It considers that the outer scale
L0 is infinite and the inner scale
lo is zero [
34]. At this time,
. So the three-dimensional power spectrum function is expressed as:
Under the condition of weak fluctuation (
), the scintillation index satisfies:
where,
is the logarithmic amplitude fluctuation variance,
2π/λ is the wave number, and
is the spatial frequency.
In the vertical direction, according to the Kolmogorov spectrum, the relationship between the spherical wave scintillation index
and the refractive index structure constant
is given by:
The atmospheric turbulence intensity profile can be inverted by the iterative method. The entire detection range 0~
L km is evenly divided into several parts, and the corresponding length interval of each part is
. Assuming that the refractive index structure constant
in each segment is a constant, then the
in each segment can be directly extracted from the integral equation. Here, taking Equation (32) as an example, firstly, integral the range from zero to
, the corresponding spherical wave scintillation index
,
within zero to
can be obtained. Then, by iterating in turn, the curve of
changing with the detection height under the vertical detection path can be obtained. Therefore, using the hierarchical iteration method, Equation (32) can be converted to Equation (33).