Influence Characteristics of Laser Transmission Amplitude Fluctuation Based on Turbulent Medium

In order to study the transmission characteristics of laser in atmospheric turbulent medium and understand the influence degree of various factors on amplitude fluctuation, by means of smooth perturbation method, this paper establishes a theory model of amplitude fluctuation of laser propagation in turbulent medium by using the smooth perturbation method and reflects the amplitude fluctuation degree, carries out specific discussion on each influence factor. The results show that the larger the wavelength, the more stable the amplitude fluctuation. With the increase of laser section radius, the amplitude fluctuates sharply and then decreases slowly after reaching the peak. Transmission distance is the main influence factor of amplitude fluctuation. With the increase of transmission distance, the amplitude fluctuation will become more obvious. The amplitude acquisition can be comprehensively modulated in a specific transmission distance by wavelength and section radius, so as to ensure the stability of the received laser and provide a theoretical basis for the interferometry technology.


Introduction
It is an important method in communication [1] and detection [2] to use the optical wave as carrier to transmit information. If there is no influence of atmospheric turbulence, the echo signal of detected target [3] [4] remains unchanged. The value of echo signal is determined by transmission shape of atmospheric turbulence in reality. It makes the atmospheric attributes change random and continuously [5] [6] [7], thus it influences the received signal inevitably. The change of optical wave attributes caused by medium change should be taken into consideration so as to detect the target accurately. The noise generated by atmosphere [8] [9] [10] can reduce the reception performance of system and can disturb the accuracy of acquired signal. The reception perturbation of signal [11] [12] is more obvious in remote sensing application, thus it should be studied deeply using related theory.
The amplitude is an important attribute of optical wave, which not only includes the intensity information, but also has obvious influence on interference technology [13] [14] [15]. The fluctuation rule of amplitude is an important indicator of propagation of light laser, which can reflect the random magnitude of turbulence fluctuation objectively and can provide beneficial support for processing the received signal. The interference radar has been paid much attention as it has high resolution, high accuracy and can work in all weather. To make the interference information more precise, the amplitude fluctuation rule of light laser propagating in the turbulence medium should be studied, the change values and parameters of amplitude should be analyzed quantitatively, thus the reference laser can be compensated, so as to assure there is interference. This plays an important role in military, transportation, agriculture and manufacture.
To solve the transmission and interference issue of laser in atmospheric turbulence medium, this paper established the theoretical model of amplitude fluctuation by using the concepts in statistics and the smooth perturbation method [16] [17] from the perspective of medium fluctuation of atmospheric turbulence. This paper is of practical significance for studying the interference measurement technology and can provide theoretical basis for interference radar detecting target detection in long distance.

Method
The scalar Helmholtz equation [18] is introduced into the medium fluctuation caused by atmospheric turbulence, thus where S S kz ′ = − is the offset of phase fluctuation caused by atmospheric turbulence, and S kϕ = (5) where S is the phase, ϕ is the optical path function. Let indicate the amplitude fluctuation, where ( , ) A z  is the random amplitude in atmospheric turbulence. The following formula can be obtained by substituting where iS φ χ ′ = + is the turbulence fluctuation. To study the relationship of point coordinates of optical propagation, the Green formula is introduced.
where  and ′  are abscissa of studied point and abscissa of reference point; z and z′ are ordinate of studied point and ordinate of reference point. The following formula can be obtained by solving Equation (7).
To further solve Equation (9), the Fourier transform transition quantities where k is the random wave number of turbulence medium. By combining with Equations (7), (9), (10) and (11), ( , ) z ξ κ can be written as where ( , ) z ε κ ′ is the random dielectric constant under the atmospheric turbulence. The following formula can be obtained by substituting Equation (12) into (10).
The following formula can be obtained by solving conjugate for Equation Thus, the following formula can be obtained.
If using ( , ) z χ  to indicate the frequency fluctuation of amplitude, thus By substituting Equation (12) and its conjugate into Equation (16), the following formula can be obtained.
If the covariance of fluctuation of dielectric constant in the medium satisfies the Gauss's law, the following formula is established.
where 2 ε σ is the dielectric variance; and a is the radius of cross-section of propagating laser. The following formula can be obtained according to the covariance formula.
where ( ) δ κ κ′ + is the unit tensor of covariance; and ( , ) F z χ κ is the two-dimensional spectrum density of amplitude fluctuation.
According to covariance relationship of fluctuation medium, the following formula can be obtained.
Thus the spectrum density can be indicated as the covariance formula of amplitude fluctuation can be obtained as following.
where 0 ( ) J κ is the zero-order Bessel function.
According to the principle of statistical optics [19], 2 (0, ) z χ χ σ ψ = is established, thus the function of amplitude fluctuation can be written as following.
The following formula can be obtained by substituting Equation (16) into Equation (26).
To make the expression more intuitional, let , thus Equation (27) can be written as According to the computational relationship, the following formula can be

Result and Discussion
It can be seen from Equation (30)     With the increase of radius of cross-section of laser, the amplitude fluctuation increases greatly and then reduces gradually. Until the radius of cross-section is very big, the amplitude fluctuation reaches to be stable and is close to be 0. There is a peak of amplitude fluctuation when the radius of cross-section is between 4 mm and 10 mm, indicating the atmospheric turbulence medium has a significant influence on transmission characteristic of cross-section in this range. This is not beneficial for receiving echo signal of optical information, and the interference compensation optical path is not easy to be realized. It can be seen from Figure 2 that with the increase of transmission distance, the amplitude fluctuation peak comes later, but the curves have the same stable trend finally. Therefore, the radius of cross-section of laser should increase as much as possible when satisfying related conditions. This not only can make the transmission flux sufficient, but also can make the amplitude fluctuation not obvious, so as to acquire effective laser.
When the wavelength is 632.8 nm, and the radius of cross-section of laser is 5 mm, 10 mm and 40 mm respectively, the relationships between transmission distance and amplitude fluctuation are shown in Figure 3.   When the laser transmits and detects in long distance, the radius of cross-section should be selected appropriately, so as to reduce the amplitude fluctuation and to assure the stability of information reception.
When the laser transmits in the real atmosphere, the light intensity on crosssection has change as influenced by atmospheric turbulence, which can reflect the amplitude fluctuation. The test was conducted on a fine day. When the transmission distance of incident laser is 300 m and 1000 m, respectively, the obtained light spots are shown in Figure 5 [20].
It can be seen from Figure 5 that the transmission distance of laser has a significant influence on amplitude fluctuation of light spot. During the transmission process, the light spot of initial laser is no longer regular, which has obvious speckle phenomenon. The influence of atmospheric turbulence on amplitude

Conclusions
This paper studied the influences of optical wave attributes and transmission distance on amplitude fluctuation in turbulence medium. The study shows that the amplitude fluctuation reduces with the increase of wavelength. The amplitude fluctuation is even close to be ideal in the near-infrared field. With the increase of radius of cross-section of laser, the amplitude fluctuation has a peak then reduces slowly. The transmission distance has the most obvious influence on amplitude fluctuation. With the increase of transmission distance, the amplitude fluctuation grows more serious, but it can be adjusted by selecting appropriate radius of cross-section. It is an effective solution for assuring to acquire amplitude effectively to select the appropriate wavelength and radius of cross-section under the specific transmission distance.
In the practical transmission of laser, the energy loss increases with the increase of transmission distance as the atmosphere absorbs and scatters the energy of laser. Thus the amplitude acquired at last has more obvious fluctuation.
Except adjusting the attributes of laser itself, the received information also can be improved by adjusting the phase parameters of reference optical path in the interference measurement field. We will study the parameters of reference optical path further in the subsequent work, so as to apply the transmission of laser in turbulence medium in the interference measurement and to provide theoretical basis for interference measurement technology in long distance.