## Abstract

Evolution of branch points in the distorted optical field is studied when a laser beam propagates through turbulent atmosphere along an uplink path. Two categories of propagation events are mainly explored for the same propagation height: fixed wavelength with change of the turbulence strength and fixed turbulence strength with change of the wavelength. It is shown that, when the beam propagates to a certain height, the density of the branch-points reaches its maximum and such a height changes with the turbulence strength but nearly remains constant with different wavelengths. The relationship between the density of branch-points and the Rytov number is also given. A fitted formula describing the relationship between the density of branch-points and propagation height with different turbulence strength and wavelength is found out. Interestingly, this formula is very similar to the formula used for describing the Blackbody radiation in physics. The results obtained may be helpful for atmospheric optics, astronomy and optical communication.

© 2014 Optical Society of America

## 1. Introduction

When a laser beam propagates through the atmosphere in the presence of strong turbulence, the complex amplitude of the beam will fluctuate with the propagation. Many points with zero amplitude will emerge in the distorted beam, where the phase of the optical field could be undetermined. Usually the points with phase dislocation [1–5] or phase singularity [6,7] are called branch points [1,2,8]. The branch points can cause problems when using classical adaptive optics system to remove atmospheric turbulence blurring [3,5,9–12]. An obvious degradation in adaptive optics correction had been observed as the strength of turbulence increases [13]. The primary reason for the degradation in correction is the presence of branch points in the phase. When branch points induced by atmospheric turbulence are present in a distorted optical field, the phase of the optical wave comprises continuous and discontinuous parts. The least-squares algorithm adopted in classical adaptive optical systems [14–17] is based on an assumption of continuous phase, so the occurrence of branch points will cause the conventional techniques for wavefront reconstruction to fail. In order to improve the compensation ability of the adaptive optical system used in laser propagation through atmosphere, the problem of branch points in the distorted optical fields due to turbulence has attracted more and more attention.

Nye and Berry [1] observed as early as 1974 the branch points due to interference caused by certain types of scattering. From their work we know that branch points are created and annihilated in pairs with opposite rotation or sign and are connected by wave dislocation called branch cuts. Along these branch cuts the phase of the wavefront undergoes a *2π* jump. The branch point is at the origin of the *2π* discontinuity that causes these wave dislocations. Fried and Vaughn [8] explained and quantified the existence of branch points in the phase function when light wave propagating through turbulent atmosphere. Voitsekhovich et al. [18] presented an empirical formula for the branch-points density with different turbulence conditions. Recently, a series of researches on the fundamental nature of the branch-points were reported [19–21]. The branch points have been shown to create in pairs and evolve smoothly in time [22]. The density of branch-points could be used as a characteristic of an atmospheric turbulence simulator [23]. The gradient measurements have been used to estimate the turbulence layers^{’} altitude and strength [24]. The branch-points can be used to estimate the number and velocities of atmospheric layers [25–28]. Sanchez and Oesch [29,30] proposed that branch points indicate the presence of photons with nonzero orbital angular momentum.

Until now, most of the studies about branch points produced in the distorted optical field are conducted for the laser beam propagating through turbulent atmosphere near the ground or parallel to the ground. In the practical engineering, however, especially in laser propagation, communication engineering and astronomical observation, laser beams often propagate through the atmosphere by an uplink or a ground-space path. Therefore, it is necessary to consider the question about branch points in the turbulence distorted optical field when the laser beam propagates through the atmosphere along an uplink path.

The aim of this paper is to investigate the evolution of branch points when laser beams propagate along an uplink atmospheric path. In section 2 we discuss the theory of branch-points detection. Section 3 presents the theoretical calculations of the branch-points density in distorted optical field along an uplink path with two categories of propagation events. A statistical model is proposed to describe the relationship between the density of branch-points and the propagation height. In section 4 we present a summary of our studies.

## 2. Theory of branch points detection

For investigating the evolution of branch points in the distorted laser beam when it propagates through turbulent atmosphere, a basic task is to locate the branch points based on the phase distribution. Here we adopt the method originally proposed by Fried et al. [8,9] for the purpose, in which the branch points are located according to the summation of the phase gradients around a closed loop.

Suppose the complex amplitude of the turbulence-distorted beam at the detect plane is *U(**r** _{p,q})*, in which

*r**is the position vector of the sampling point*

_{p,q}*(p, q)*(

*p, q*are the sequence number of the points in x and y direction, respectively). Its phase can be determined simply by

*Re{ U(*

*r**and*

_{p,q})}*Im{ U(*

*r**represent the real and the imaginary parts of the complex amplitude, respectively. The spatial gradient of the phase distribution can be defined as*

_{p,q})}*{…}*denotes taking a principal-value,

_{pv}*d*is the sampling interval,

*l**and*

_{x}

*l**are unit vectors parallel to x- and y-axis, respectively. Equation (2) can be also expressed as the function of the complex amplitude by substituting Eq. (1) into Eq. (2), that is [9],*

_{y}According to the method proposed by Fried, we can judge if a point be a branch point by calculating the following contour integral over a closed loop surrounding it, which can be expressed as

*C*denotes the closed loop contour and

*d*

**is a vector parallel to the tangent of the contour**

*r**C*. Using the discrete form of the phase gradient described above, the discrete form of Eq. (4) can be written as

*C*is set to be a square whose four corners are at

*r*

_{p,q}_{,}

*r*

_{p + 1,q}_{,}

*r*

_{p + 1,q + 1}_{,}and

*r**. Because of the opposite direction between the integral orientation and vectors*

_{p,q + 1}

*l**,*

_{x}

*l**, two minuses appear in Eq. (5). If*

_{y}*S*in Eq. (5) equals to

_{p,q}*+ 2π*, it means that a positive branch point exists in the contour, while a negative branch point will appears in the contour if

*S*in Eq. (5) is equal to

_{p,q}*-2π.*If the result of Eq. (5) equals to

*0*, it means that no branch point is enclosed in the contour. Thus, the location of branch points and their number in distorted optical field can be detected by Eq. (5). Because branch points are created and annihilated in pairs with opposite rotation or sign, we could calculate the density of positive branch points instead of calculating the density of all the branch-points.

## 3. Simulation results

Our simulations are based on the numerical solution of the parabolic equation that describes the propagation of monochromatic light waves through turbulent atmosphere:

*= A + iΦ*denotes the complex wave field,

*r**= (x,y)*is the transverse position vector,

*▽*,

^{2}= ∂^{2}/∂x^{2}+ ∂^{2}/∂y^{2}*k*is the wave number related to the wave length

*λ*by

*k = 2π/λ*, and

*n*denotes the refractive-index fluctuations.

Numerical simulation results presented in this paper are based on the Hufnagel-Valley (H-V) vertical atmospheric model, which is very well known and often cited. The H-V model has a very simple form and is described as:

*h*is the height above ground level in kilometers,

*w*is an upper level wind speed, given as the Root Mean Square wind speed [m/s] average over the 5~20 km above ground level range. C

_{n}

^{2}(0) is the refractive index structure parameter at the ground level, which could be used to represent the atmospheric turbulence strength on the surface level. The vertical profile for the H-V model is shown in Fig. 1 with C

_{n}

^{2}(0) = 1.7 × 10

^{−14}m

^{-2/3}and w = 21 m/s .

The density of branch-points is calculated under the following conditions: a collimated plane wave propagates through the turbulent atmosphere along a ground-space path; the receive telescope has a diameter of 0.6m; the propagation height is 20km. The transverse grid parameters are: 256 × 256 grid size, 0.0075m grid step. 80 phase screens are used along the propagation path [18,31]. The average densities of the branch-points are obtained when 80 propagation events are performed for all the cases. Because the Rytov number has been proved to be a fundamental scaling parameter for propagation through turbulence [32], the density of branch-points versus the Rytov number is also given. Rytov number is a value calculated for the log-amplitude variance of a complex optical field after propagation through the turbulent atmosphere by use of the Rytov approximation. For a plane wave propagating along an uplink path, the Rytov number can be calculated by [33]

Figure 2(a) shows the density of branch-points versus propagation height above ground level for the given wavelength when the turbulence strength parameter C_{n}^{2}(0) is set to be 2.736 × 10^{−14}m^{-2/3}, 3.420 × 10^{−14}m^{-2/3}, 4.104 × 10^{−14}m^{-2/3}, 4.788 × 10^{−14}m^{-2/3}, and 5.472 × 10^{−14}m^{-2/3}, respectively. From Fig. 2(a) it can be seen that the density of branch-points changes with the turbulence and propagation height, which can be divided into four regions with different behaviors of the branch-points density. The boundaries of the regions are indicated by thin dotted lines (shown in Figs. 1, 2(a) and 3(a)). The first region is near the ground (0≤h≤1km) with rapidly dropped turbulence strength, in which the density of branch-points starts at zero and increases nonlinearly with the increasing of the height. The second one is a region (1km≤h≤5km) with the declining turbulence strength, in which the density of branch-points begins to grow rapidly, almost linearly. We know that branch points are created in distorted optical field under the combined influences of turbulence effect and diffraction, and the evolution of the branch-points density is mainly determined by the rates of branch-points pair creation and annihilation. The probability of annihilation would increase as the separation distance between the pair of branch-points decreases [21,22]. The turbulence in the second region is strong enough and the turbulence effect is dominant; the creation rate of branch points is far more than the annihilation rate; the separation distance of the branch-points pair is relatively far and it is difficult to annihilate with each other; so there is a rapid growth of the branch-points density. In the third region (5km≤h≤12km), there is a fluctuation of the turbulence strength. The rapid growth of the branch-points density is replaced by a slower and nonlinear increase at the outset of this region, and then the density of branch-points begins to decrease gradually after reaching its maximum. It means that in this region the creation rate of branch-points decreases and the annihilation rate increase with the propagation height. In this region, the density of branch-points is very high and the separation distance between the pair of branch-points would be decreased, so the annihilation probability of the branch-points pair would be increased. When the creation rate of branch-points is equal to the annihilation rate, the branch-points density will reach a peak. In the fourth region (h≥12km), the turbulence strength continues to decrease and the density of the branch-points comes down slowly because of the main diffraction effect and weak turbulence effect. The creation rate of branch-points is less than the annihilation rate in this region.

It is worth noting that the propagation height corresponding to the peak density of the branch points changes with the turbulence strength. If the turbulence strength is stronger, there will be more branch points created in the distorted optical field, the peak density of the branch-points will be higher, and the height at which the density of the branch-points reaches its peak will be lower. These evolution properties of the branch-points density maybe reveal the mechanism of the branch-points creation and annihilation when a beam propagates through turbulent atmosphere. The rapidly rising density in the second region described above indicates that the creation of the branch-points because of the atmosphere turbulence is dominant in this stage. The height with peak densities corresponds to the positions where the creation and the annihilation of the branch points are in equilibrium. While the slowly setting densities in the fourth region maybe tell us the annihilation mechanism of the branch points because of the diffraction in uplink atmosphere with only the weaker turbulence.

As a comparison, we also draw the relationship between the branch-point density and the Rytov number in Fig. 2(b), which is obtained by simply replacing the propagation height of Fig. 2(a) with the corresponding Rytov number calculated according to Eq. (8). It can be seen that the evolution of the density could also be divided into four regions with different evolution behaviors. In the first region, the density of branch-points is quite low, almost equal to zero. In the second region, the density of branch-points increases nonlinearly with the increasing of the Rytov number at the outset of this region and then rises very rapidly, almost linearly. The density of branch-points reaches its maximum and then decreases rapidly in the third region. In the fourth region, the density of branch-points decreases more quickly with the slightly increasing of the Rytov number because of the very weak turbulence strength in large propagation height. The most notable may be the fact that the peak density of the branch-points increases linearly with the corresponding Rytov number.

Figure 3 presents the evolution of branch-points density as a function of the propagation height and the Rytov number for the given turbulence strength when the wavelength of the laser beam is taken as λ = 0.6μm, λ = 0.82μm, λ = 1.06μm, and λ = 1.319μm, respectively. We can see from Fig. 3 that, under the same propagating parameters, the shorter wavelength is taken, the more number of the branch points will be generated. The main reason may be that the branch-points generation is a result of multiple scattering of the light waves on the inhomogeneous atmosphere. The scattering will become stronger with the decrease of the wavelength, leading to an increase of the branch-points density. We can also see from Fig. 3(a) that the height corresponding to the peak density of the branch-points nearly remains constant with different wavelengths, while the Rytov number corresponding to the peak density of branch-points is getting smaller with the increased wavelength (as shown in Fig. 3(b)).

For further revealing the relationship between the density of branch-points and the propagation height with different turbulence strengths and wavelengths, we make a non-linear fitting to the simulated data shown in Figs. 2(a) and 3(a). We find that the relationship can be well fitted by the following model

where*A*,

*B*and

*n*are parameters changed with propagation conditions. It is interesting that this formula is very similar to the formula used to describe Blackbody radiation in physics.

The fitting data are shown in Tables 1 and 2. We can see from the fitted data that, the fitted parameter *A* decreases with the increase of C_{n}^{2}(0) and λ, while the parameter *B* will decrease with the increase of C_{n}^{2}(0) and increase with the increase of λ. Further analysis to the fitted dada indicates that *B* is nearly proportional to (λ/C_{n}^{2}(0))^{2/3}.

It can be seen from the fitted curves shown in Fig. 2(a) that the relationship between the density of branch-points and the propagation heights with different turbulence strengths can be well fitted with the model of Eq. (9) when the wavelength is set to be 1.06μm. If the turbulence strength is weaker, the fitted result will be better. For the simulated data presented here, the correlation coefficients of R^{2} are all larger than 0.99. For example, the correlation coefficient of the fitting is equal to 0.99716 for the situation of C_{n}^{2}(0) = 2.736 × 10^{−14}m^{-2/3}.

Figure 3(a) also shows the fitted curves between the branch-points density and the propagation height with different wavelengths when the turbulence strength parameter is set to be 4.104 × 10^{−14}m^{-2/3}. From the fitted curves we see that the fitted result will be better with the increase of the wavelength. For example, when the wavelength is equal to 1.319μm, the correlation coefficient is equal to 0.9978, while the correlation coefficient is just 0.96762 when the wavelength is reduced to 0.6μm.

## 4. Conclusions

Numerical simulations to the evolution of branch points are carried out for a collimated plane wave propagating in atmospheric turbulence along a ground-space path. Two categories of propagation events are explored. In the first category, the wavelength is fixed with the turbulence strength changing. In the second one, the turbulence strength is fixed with the wavelength changing. The results for the density of branch-points reveal that the evolution could be divided into four interesting regions with the increase of the propagation height and the Rytov number, respectively. The first region is near the ground with rapidly dropped turbulence strength, where the density of branch-points seems to be small. The second one is a region with the turbulence strength declining, in which the density of branch-points begins to grow rapidly, almost linearly. In the third region, the density of branch-points reaches its maximum. In the fourth region, the turbulence strength is very weak and the density of branch-points comes down with the increase of the propagation height and the Rytov number, respectively. The relationship between the density of branch-points and propagation height with different turbulence strength and wavelength can be well fitted with a model which is similar to the formula used for description of Blackbody radiation in physics. The evolution properties of the branch-points density maybe reveal the mechanism of the branch-points creation and annihilation when a beam propagates through turbulent atmosphere. The results obtained may be useful in atmospheric optics, astronomy and optical communication.

## Acknowledgments

Xiao-Lu Ge thanks Prof. Chengyu Fan for his support in the numerical calculation. The work is supported in part by National Natural Science Foundation of China (NNSFC) under Grant Nos. 11074152, 10934003 and 11104165 as well as the Research Foundation for the Doctoral Program of Higher Education of China under Grant No. 20113704110002.

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