## Abstract

An 11.8km optical link is established to examine the intensity fluctuation of the laser beam transmission through atmosphere turbulence. Probability density function, fade statistic, and high-frequency spectrum are researched based on the analysis of the experimental data collected in each season of a year, including both weak and strong fluctuation cases. Finally, the daily variation curve of scintillation index is given, compared with the variation of refractive-index structure parameter *C*
^{2}
* _{n}*, which is calculated from the experimental data of angle of arrival. This work provides the experimental results that are helpful to the atmospheric propagation research and the free-space optical communication system design.

©2008 Optical Society of America

## 1. Introduction

Atmospheric turbulence has an important effect on the light waves. The refractive index fluctuations can affect the speed of wave-fronts propagating and produce fluctuations in amplitude and phase of a laser beam propagating through it. Atmospheric refraction of the light energy can cause random temporal and spatial distribution of the light intensity [1–2]. These effects result in optical intensity fluctuations at the receiving plane and severely influence the performance of free-space optical communication systems.

According to Kolmogorov theory [3], if the atmosphere turbulence is homogeneous and isotropic, for weak fluctuation, the probability distribution of received intensity should be lognormal. For strong fluctuation far into the saturation regime, probability distribution is assumed to be governed by the negative exponential distribution. Besides, a model was established by Andrews to describe the behavior of light intensity from weak to strong fluctuation [4, 5].

Experimental observations have been conducted to evaluate turbulence-introduced intensity scintillation and its effects on free-space optical communications [6–15]. The Kolmogorov theory has been certified by many of them, but there are still some experiments that show disagreements with it, especially on high-frequency power spectrum [8, 16–18]. It is not clear what is lying behind the differences. Turbulence intensity, wavelength, temperature gradients, moisture, all these factors can make the results different.

This paper presents an experiment that examines the intensity fluctuation of a laser beam propagating through atmosphere over an 11.8km near horizontal urban path (Fig. 1). The laser link is established between two buildings in Harbin. The optical path extends from the 9th floor (30m above ground level) of a building in Songbei area of Harbin to the roof of the Electronic Information Centre (55m above ground level) in the National University Science Park of Harbin Institute of Technology.

Landscape along the path is complex, mainly including a river and several roads and building arrays. Non-uniform landscape makes the microclimate along the optical path complicated and then affects the measured data. Besides, vibration of the buildings and temperature gradients can also influence the experimental results.

Safety goggles are used to protect our eyes from being burned by the laser. In addition, the laser link is higher than most of buildings near the optical path and beam directions are adjusted only in a small range so that the laser can’t injure other residents’ eyes.

Noting the similarity of our work to the experiment mentioned in Refs. [7, 8], many statements and data analysis methods in this paper are in similar format to them for easy comparison of the results of two experiments, especially in the analysis processes of probability density function and high-frequency spectrum (section 3.1 and 3.3).

## 2. Experimental setup

The experimental setup is shown in Fig. 2. The transmitter is a frequency doubled Nd: YAG laser with the wavelength of 532nm and the maximum output power of 100mW. Full power is transmitted only in some winter days because the smoking caused by city heating system attenuates the light power severely. The original divergence angle of the laser transmitter is 1.3mrad and a beam expander (4mm input diameter and 24mm output diameter) is used to compress the beam width to 250µrad. The laser source and the beam expander are mounted on a 3-dimensional (700mm×450mm) adjustable optics table.

Beam diameter at the receiving plane is about 3–4m, but is hard to determine exactly because of the effects of beam wander and beam spread. A Cassegrain telescope (diameter, D=127mm; focal length, f=1900mm) is placed on the top of the Electronic Information Centre to collect the input light power.

The output of the Cassegrain telescope is divided into two beams by a light splitter. One of the two beams is focused on a Pin photo-detector by a lens (f=78mm) to measure the intensity scintillation, and the other is focused on a CCD camera by its own lens (f=50mm) to measure the fluctuation of angle of arrival (AOA). Either the Pin or the CCD has a 532nm wave filter in front of it to mitigate the influence of the background light. In addition, a light attenuator is used to prevent power saturation of the CCD camera, depending on the received laser power. These devices are mounted on the other 3-dimensional adjustable optics table (Fig. 3). Output signal of the Pin is sampled at the rates of 3000Hz or 5000Hz every 30 seconds by an analog/data (AD) card inserted in a computer.

A temperature and humidity recorder is used to record the temperature and humidity every 3 minutes and a wind velocity recorder is also used to measure wind speed at the receiving side of the link.

## 3. Data analysis

The experiment was performed from October 2006 to the end of the year of 2007, lasting more than one year. Table 1 summarizes the date and climate conditions for an initial data set consisting of four weak fluctuation (*σ*
^{2}
* _{I}*<1) and two strong fluctuation (

*σ*

^{2}

*>1) cases. Considering the dynamic range (60dB) of the AD card, data selection is mainly based on whether both high and low irradiance part of the intensity fluctuation can be included.*

_{I}#### 3.1 Probability density function

Usually, scintillation index is used to indicate the strength of the intensity fluctuation. It is defined as the normalized variance of the intensity fluctuations, which can be expressed as

where *I* is light intensity and <·> means ensemble average. For weak fluctuations, scintillation index is equal to the Rytov variance which can be calculated as

where *C*
^{2}
* _{n}* is the refractive-index structure parameter,

*k*=2

*π*/

*λ*is the wave-number,

*λ*is wavelength, and

*L*is the link distance. Probability density function (PDF) of the normalized intensity should be lognormal [3], and it can be described as

For strong fluctuation far into the saturation regime, PDF is assumed to be governed by the negative exponential distribution [19],

Weak fluctuation trials T1–T4 of table 1 are all measured in summer and autumn, because long link distance and larger *C*
^{2}
* _{n}* lead to strong fluctuation in spring and winter. Histograms for T1–T4 and the lognormal fitting curves are shown in Fig. 4, with the light intensity normalized. The results of Fig. 4 are similar to the data analysis figures in Ref. [8].

In Fig. 4, *R*
_{2} (0<*R*
^{2}<1) represents the related coefficient. The fitting result is better if *R*
^{2} is more approximate to 1. For a certain data trial, the interval of [min (*I*), max (*I*)] are divided into n equal areas. The medians of the n areas can make a sequence of *I*=(*I*
_{1}, *I*
_{2} … *I*
_{n}), and frequency counts of *I* in the n areas constitute another sequence of *y*=(*y*
_{1}, *y*
_{2} … *y*
_{n}). After the fitting curve is plotted, each value of *I* will corresponds to a value of *Y* on the fitting curve, then we get the third sequence of *Y*=(*Y*
_{1}, *Y*
_{2} … *Y*
_{n}). Finally, *R* can be defined by

where *DY* and *Dy* are variances of *Y* and *y*, respectively.

As it is shown in Fig. 4, the histograms match the log-normal fitting curve very well for each trial and the related coefficients *R*
^{2} are all above the level of 0.995. Other data trials of weak fluctuation are also analyzed and the results are similar to T1–T4. *R*
^{2} of them are plotted in Fig. 5, the x-axes of which represents scintillation index. The fitting line in Fig. 5 is almost horizontal, which means *R*
^{2} do not vary for different *σ*
^{2}
* _{I}*. So it can be concluded that in weak fluctuation region, the log-normal approximation is equally effective to different trials with different scintillation indices.

Histograms for strong fluctuation trials T5 and T6 are shown in Fig. 6, with the light intensity normalized and the related coefficients *R*
^{2} are also annotated on each graph. The frequency count is evidently governed by the negative exponential distribution in Fig. 6.

Other data trials of strong fluctuation are analyzed too. We found that *R*
^{2} slightly falls down when *σ*
^{2}
* _{I}* rises (Fig. 7). It seems that in strong fluctuation region, the negative exponential approximation is less effective to larger

*σ*

^{2}

*trials than to smaller*

_{I}*σ*

^{2}

*trials. However,*

_{I}*R*

^{2}is still above the level of 0.98 even though

*σ*

^{2}

*is greater than 3, so the negative exponential distribution is still a nice approximation for strong fluctuation trials.*

_{I}#### 3.2 Probability of fade

The probability of fade is defined by the probability that the received light intensity is below some given threshold *I _{T}*. It can be calculated as [3]

where *p*(*I*) is PDF of the intensity. The bit-error-rate (BER) of an on-off-keyed communication system can be given by multiplying the probability of fade with a factor of 0.5. It is customary to express *I _{T}* in decibels, which can be described as the fade margin

The probability of fade for T2, T4 and T5 are presented in Fig. 8. It can be seen in Fig. 8 that the probability of fade falls down rapidly with the increasing of *F _{T}*. Besides, the required

*F*for a given BER is larger for the trial that has a greater σ2I and the BER also rises with the increasing of

_{T}*σ*

^{2}

*for a given*

_{I}*F*. To achieve a BER of 10

_{T}^{-4}, about 13dB fade margin is required for T2 (

*σ*

^{2}

_{I}=0.4991), 20dB is required for T4 (

*σ*

^{2}

*=0.9074), and more than 35dB is required for T5 (*

_{I}*σ*

^{2}

*=1.2249). To achieve a BER of 10*

_{I}^{-5}, the required fade margin will be 15dB, 23dB and more than 40dB for T2, T4 and T5, respectively.

#### 3.3 High-frequency spectrum

Usually, it is accepted that the high-frequency spectrum has a -8/3 power-law dependence [3], but there are also some studies that indicate the high-frequency spectrum to be a -11/3, -14/3, or -17/3 power-law dependence [8, 20, 21].

Power spectra for trials T1, T2, T3 and T5 are plotted in Fig. 9. High-frequency spectra of all these trials have the negative exponential power law dependence, but the power exponents of them are different. The power exponents of high-frequency spectra for T1, T2, T3 and T5 are -11/3, -16/3, -17/3 and -16/3, respectively. It was mentioned that the measured power exponent was -17/3 in reference [8], so it seems that our result is more complicated than the result in reference [8] and any other reports mentioned above.

Figure 9 shows only four power spectra. In fact, more data trials are analyzed for further research of the high-frequency spectrum, totally containing 24 weak fluctuation cases and 11 strong fluctuation cases. Power spectrum is made and power exponent of high-frequency spectrum is calculated for each of these data trials. These power exponents (α) are plotted in Fig. 10. As it is shown in Fig. 10, power exponent varies in a broad range from -8/3 to -21/3. The values of -8/3 and -11/3 can only be observed when *σ*
^{2}
* _{I}* is smaller than 0.4. For weak fluctuation trials, α is in the interval of -21/3 to -13/3 mostly. For strong fluctuation trials,

*α*is in the interval of -21/3 to -15/3. Mean values are -5.48 and -5.78 for weak fluctuation and strong fluctuation trials, respectively, close to the value of -17/3. It is hard to tell what makes the high-frequency power-law complicated. Beam parameter [21], validity of the Kolmogrov model [22], non-uniform wind along the path [23], Turbulence scale and boundary [20, 24], all these factors can affect the experimental results.

#### 3.4 Daily variation of σ^{2}_{I} and C^{2}_{n}

For a Gaussian-beam wave, the variance of AOA can be expressed as [3]

where *L* is the link distance, *D* is the diameter of the receiving aperture, and

$$\Theta =1-\frac{L}{F},$$

$$\Lambda =\frac{2L}{{\mathrm{kW}}^{2}},$$

*F* (*F*≈*L*, if *L* is long enough) and *W* denote the receiver plane phase front radius of curvature and beam radius, respectively. In our experiment, *k* and *L* are on the order of 10^{7} and 10^{4}, respectively, *W* is about 3–4m, and *D*=0.127cm. Based on these parameters, the approximation of *F*≈*L*, *a*≈1 and Λ≈0 can be made, and Eq. (8) can be simplified to

So *C*
^{2}
* _{n}* values can be calculated from the experimental data of AOA by

A CCD was used to measure the variance of AOA. A computer was used to calculate the centroid coordinate of the CCD image and record it at a rate of 1000Hz. The variance of AOA can be calculated from the variance of the centroid coordinate, and then *C*
^{2}
* _{n}* can be calculated by Eq. (11).

The 10 minutes average values of *σ*
^{2}
* _{I}* and

*C*

^{2}

*are presented in Fig. 11 and Fig. 12, respectively, which were measured in a complete diurnal period of 27 March 2007. In Fig. 11 and Fig. 12, both*

_{n}*σ*

^{2}

*and*

_{I}*C*

^{2}

*have greater values in the daytime than at night. In the daytime, two curves have similar profiles. Both of them increase before midday, maximize at noon and then decline in the afternoon. But at night,*

_{n}*C*

^{2}

*appears more stable than*

_{n}*σ*

^{2}

*. The*

_{I}*C*

^{2}

*curve of Fig. 12 is similar to the measurement results in reference [7].*

_{n}We consider that the night deviation of two curves is mainly caused by misalignment of the laser beam. In fact, the scintillation index of a Gaussian beam can be expressed by

where *σ*
^{2}
_{0} is the on-axis scintillation index and *r* is the distance between the receiving point and the beam center. Equation (12) indicates that *σ*
^{2}
* _{I}* increases with square of distance transverse to the optical axis [25], so if the light is not central received, the measured

*σ*

^{2}

*will be larger, but the measured*

_{I}*C*

^{2}

*will not be different because it is independent of*

_{n}*r*. In our experiment, the beam direction will slowly drift with time, especially at night, which means

*r*increases with time. This can explain why two figures are inconsistent with each other in some period of time.

There are multiple potential causes of the beam drift including thermal deformation of the building and bending of the optical line-of-sight induced by large scale vertical dependencies in the atmospheric refractive index.

## 4. Summary and conclusions

A laser transmission experiment was conducted to examine the turbulence-introduced intensity fluctuation over an 11.8km optical path. Data analysis was mainly based on an initial data set which consisted of four weak fluctuation and two strong fluctuation cases. Given the close technical similarity of the experiments and purposes, many of the analysis and figures in this paper are in similar format to reference [7, 8] for easy comparison of the results of two experiments. Histogram of frequency count was made for each case. Log-normal and negative exponential distributions were proved to be nice approximations for weak fluctuation and strong fluctuation, respectively, although the related coefficient slightly declined when *σ*
^{2}
* _{I}* increased in strong fluctuation region. Fade statistics showed that BER increased with the rising of

*σ*

^{2}

*for a fixed fade margin. Power spectra were plotted and power exponent of high-frequency spectrum appeared to vary in a broad range from -8/3 to -21/3. Finally, daily variation of*

_{I}*σ*

^{2}

*was shown, compared with the variation of*

_{I}*C*

^{2}

*. The PDF of intensity and*

_{n}*C*

^{2}

*measurements had the similar results to references [7, 8], but as for power exponent, the condition is more complicated than their reports. This work can benefit the research of laser atmospheric transmission and the free-space optical communication system design.*

_{n}## Acknowledgments

This work is supported by the program of excellent team in Harbin institute of technology.

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