CROSS-SECTIONAL MEASURING OF OPTICAL BEAM

This article deals with problematic of measuring of optical beam in free space optics (FSO). The professional FSO link was created between two buildings standing 1,5 kilometers apart from each other. Signal passing through the atmospheric media between optical heads is affected. This happens due to effects in atmospheric media. This article describes creating of the device for measuring the intensity of optical beam in 2D space and its subsequent rendering into 3D graph.


Introduction
The FSO link consists of two optical heads operating in full duplex mode, both as transmitter and receiver (Fig. 1). These optical heads are situated against each other and must be in direct line of sight. Optical head is usually placed on the roof of buildings, but it can also be placed in windows.
Each optical head has an emitting source, in our case it is a laser diode, which provides conversion of signals from electrical to optical domain. After that, the output signal is shaped by optical lens to desired shape. The resulting beam transmitting from head is of Gaussian intensity distribution.

Gaussian Beam
Intensity distribution of Gaussian beam in the transverse plane corresponds to a circularly symmetric Gaussian function, where the optical axis is also axis of symmetry. Power of Gaussian beam is centered into a narrow cone. Gaussian beam profile is shown on Fig. 2. Radius of Gaussian beam w(z) can be calculated using the equation [2], [3], [9]   where w 0 is half-width at its narrowest point, z is observation plane distance, z 0 is Rayleigh distance. This distance can be described as longitudinal direction of propagation of the waist to the point, where the transverse cross-section area is doubled, z 0 can be calculated [9], [10]  where λ is the wavelength of radiation.
The power of the Gaussian beam is given as the product of half of the maximal intensity and area of a circle with a radius equals to the radius of a central beam [2], [9], [10].

Transmission Media
FSO links are used for wireless data transmission in specific conditions. FSO uses atmosphere as a media for communication. This is associated with the disadvantage of using FSO. In adverse conditions it may be difficult to transfer data, sometimes it is impossible. It is influenced by seasons and actual weather. Most connections are built in areas between buildings; it is always in troposphere layer. Troposphere reaches a height about 10 kilometers above the sea level [19].
Refractive index is one of the basic variables describing the propagation of light in materials. It applies directly to given environment. A number of changes occurring in the atmospheric transmission environment changes refractive index. Local temperature and pressure changes have an effect on refractive index, so that is random function of time and space. Optical beam passing through these changes is influenced and this leads to change its local shape and power of the beam. The main effects influencing transmission of optical signals include [2], [3], [4], [11], and [13]:  extinction of optical intensity due to turbulence in the atmospheric environment,  extinction due to scattering and absorption on molecules and aerosols,  optical intensity fluctuations due to turbulence in the atmospheric environment,  optical intensity fluctuations due to fog, rain, snow, etc.,  short-term disruption of beam caused by flying birds.
These effects can be further divided according to the influence, which they arise (due to turbulence in the atmosphere, the effects of fog, rain and snow, scattering and absorption on molecules). These effects influence each other and always act together. Fluctuations in optical intensity also cause extinction [9], [10].
Mean extinction coefficient on molecules can be calculated as the sum of extinction coefficients [10], where α abs is coefficient of scattering on molecules, α s,m is the coefficient of scattering -Rayleigh scattering, α s,p is the coefficient of scattering -Mie scattering and α fluct is coefficient of decreasing due to fluctuations.

Attenuation by Beam Transmission
This attenuation is important parameter, which has greatest effect on the transferred power through atmospheric media. This attenuation can be calculated by this formula: where L VP is a distance between optical head, L 0 is assistant length. This L 0 can be calculated from diameter of optical transmission system D VOS and the angular width of the transmitted beam φ VS : Substituting into the equation of  12 (10) we get more practical formula for attenuation by beam transmission: (12)

1) Rayleigh Scattering
Scattering can dramatically affect the transfer in atmospheric media. Scattering is not energy loss directly, but rather a change of direction or redistribution of light.
Rayleigh scattering is on molecules of gas or other particles that are smaller than wavelength. It is spectrally dependent on wavelength. Rayleigh scattering is omnidirectional [9], [10].
Rayleigh scattering can be calculated where I Θ is the intensity of light scattered by one particle Θ, I 0 is total intensity of incident radiation, ε 0 permittivity of vacuum (8,85419·10 -12 C 2 ·J -1 ·m -1 ), λ wavelength, r distance of the intensity detector, Θ view angle, F(Θ) in a function of viewing angle and α polarization from Clausius-Mossotti equation.

2) Mie Scattering
Mie scattering occurs when the light hits the particle (drop of water, snow, etc.) as large or larger than the wavelength. It is highly variable component of extinction, because the atmosphere is changing very often. This effect is spectrally independent, that means that is not dependent on the wavelength. Mie scattering is directional effect [9], [10].

Turbulence
Atmospheric turbulences affecting beam transmission through the media can be divided into three groups: Dynamic turbulence arise in areas about 5-6 km and do not affect free space optics links. Thermal and mechanical turbulence occurs at low altitudes and affecting FSO links [5], [6], [7].
During the day, when the Earth surface is warmer than the ambient air, the air layer closer to the surface is also warmer. This causes the rising of warm air masses upward, but also diffraction of the optical beam upward. If this rising is sufficiently large, it may result in an effect known as "mirage" (mirroring). The situation is reversed at night and optical beam is bending downwards. Besides these effects, the atmospheric turbulences disturb also the coherence of the propagated beam. Beam distortion caused by turbulence results in a divergence beam, changing the position of the beam center resulting to fluctuations and scintillations [5], [6], [7], [8], [14].
Due to temperature changes and wind speed variations forms local unstable air masses, which are divided into turbulent air bubbles with different sizes [9].
To calculate the attenuation caused by the turbulence we use the following formula based on Rytov approximation [12]: where k represents the wave number, which is dependent on wavelength, C n 2 i s a structural parameter of the refractive index and L is the distance between optical heads. The resulting value α turb represents the mean value of attenuation caused by turbulence [7], [8], [11].
Structural parameter of refractive index used in formulas to calculate the attenuation is a measure for determining the strength of turbulence. The value of this parameter ranges from 10 -14 m 2/3 to 10 -12 m 2/3 depending on strength of the turbulence, as seen in Tab. 1.
Atmospheric turbulence is affecting FSO links with these effects [14]:

Realized FSO Link
Realized link was created using professional MRV optical heads TS5000G (Fig. 3). The distance (Fig. 4)

Measuring Device
To be able to measure the optical beam cross-section it was necessary to construct a measuring device (Fig. 5).
The base of the measuring device has been borrowed from University of Technology in Brno. It is a device manufactured by Festo for movement in axes with stepper motors.
First, it was necessary to connect the basic parts of device. Then it was needed to connect device with a computer. This was done with two crossed serial RS-232 cables, one for each axis.
Then it was using application Festo Configuration Tool (FCT) to set the basic parameters such as type of axis, axis length, motor type, setting the limit sensors, etc. With this application you can also test the functionality and correct behavior of the whole equipment. With possibilities of FCT it was not possible to measure the optical beam cross section, so it was necessary to write the whole new application. It includes motion axes control, but also taking values from optical power meter.

PC Application for Measurement
This application was created in an environment of MATLAB R2009a (Fig. 6). In this application, you can set all parameters that are important for proper measurement. These parameters include the selection of COM ports, the measured wavelength, the required step on axis and time to stabilize the measuring arm [20].

Measurement Method
When you start measuring, power meter will pass through the grid (Fig. 7) from the starting position on horizontal axis. Once power meter arrives at the end of the axis, it comes back to the beginning of the horizontal axis and moves vertical axis one step down. Then again passes through all points of the horizontal axis, as shown in Fig. 7. At each stop removes the value from the power meter and stores it into the matrix. The resulting matrix is rendered into 3D graph and displayed to the user.

Measurement During Daylight
At the first measurements it was found that the ambient sunlight greatly influences the measurement, even if the power meter detector was overshadowed against direct sunlight with black tube. Several measurements were done at wavelengths 830-860 nm with different steps, but the measurement has always been very influenced by parasitic light. The taken values of optical power during the day with a clear sky were between 40-60 W. The values taken at night were about 1 W. So values taken in daylight were unusable and did not testify about the shape of the measured beam. For measurement was used power meter Thorlabs PM 120 and silicon detector S120B with measurement range from 50 nW to 50 mW, operating at λ between 400 nm and 1100 nm.

Measurement on April 21 During Night
Graph in Fig. 8 was measured on April 21, 2011 in following conditions: temperature 18,7 °C, humidity 44 %, no wind, pressure 1015,6 hPa. This graph was measured using two measurements when the equipment was moved after first measurement and resulting values were combined. Thereby was reached axis size 1200 mm. You can see a large optical power in the areas around the center location of the optical head (right side of the graph) and decreasing power values away from head. In all measurements it was used all 3 transmitting lasers (2x 40 mW, 1x 60 mW). For small distances (less than 300 m) it is possible to turn some lasers off.

Measurement on April 25 During Night
Graph in Fig. 9 was measured on April 25, 2011 in following conditions: temperature 11,7 °C, humidity 77 %, southeast wind 0,03 m/s, pressure 1017,8 hPa. Graph was measured by three independent measurements and then values were properly combined. This allowed capturing a larger beam width (3x 600 = 1800 mm), on which we can see the top of the beam with high optical power and sloping edge of the beam on both sides. In this measurement was taken approximately 61 % of beam width. On the graph we can observe many peaks, which are caused by measuring time, which is about 30 minutes and also by changing atmospheric media.

LOWESS Smoothing of Measured Graph
Graph in Fig. 9 is still influenced by the effects of atmospheric environment, so it was suitable to smooth this graph, as you can see in Fig. 10. Graph was interspersed in MATLAB R2009a with quadratic polynomial function LOWESS (Locally Weighted Scatterplot Smoothing), to highlight a shape of optical beam. This function uses locally weighted regression proposed by Cleveland in 1979 [15].
The following is a sketch of the LOWESS algorithm [16], [17], [18]:  count the number of points in the neighborhood. This is number of points times LOWESS fraction rounded to the nearest integer. This number is called q,  use tricube function to generate a weighted least squares fit. The weight given to point (x k , y k ) is: where d i is distance from x i to its q th nearest neighbor, T is tri-cube weight function [15], [16].
 Compute the residuals of this fit.
 Compute a bi-square weight function of the residuals.
 Use the horizontal weight multiplied by the vertical weights in a weighted least squares fit.
Smoothing was calculated with a range of 50 %, when the R-Square value was 0,7948. The R-Square is a parameter indicates how close the regression to real data values is. When the R-Square value is 1, model is consistent with the data.

Smoothing with Gaussian Function
For smoothing graph of optical power was also used Gaussian function (Fig. 11). This function should better correspond with the real beam, because the original signal should be Gaussian-shaped beam. For smoothing was used this function: (20)

Conclusion and Future Work
This article summarizes the creation of the measuring device, including PC application. This device can measure optical power of beam cross-section with desired accuracy. The disadvantage of this device may be relatively small measuring area, which is particularly true for connections with large distances, where because of the influence of beam divergence does not measure the whole shape of optical beam. In this time it is ordered horizontal axis of length 2 meters solving this problem for future work.
This will allow larger scale of measuring device without having to push or other manipulations during measurement. It is also planned to automation of equipment to measure the statistics of optical beam in various weather conditions. This will include automating of steps necessary for measurements with built-in PC, operated via remote control using VNC software application. The measurements will be repeated in time periods (e.g. every day) and the results will be statistically processed.
Consideration is also creation of a completely new program in the LabVIEW environment that would better satisfy requirements for reliability and ease of modifiability of the program for measuring. One of the essential objectives is to ensure protection of equipment against weather conditions, which will include the creation of such covers and seal, to avoid any intrusion of water and other particles into the device.