Accuracy Test of Laser Station under Conditions of Air Turbulence * *

The application of laser stations to industrial metrology indoors is discussed. In this work, only the analysis of the impact of air turbulence to determine the angle of laser wave propagation is discussed, and other meteorological parameters (pressure, humidity) and particles of pollutants (dust, smoke) are ignored. The main aim of this paper was to test the usefulness of a laser station under turbulent air conditions indoors and to estimate the accuracy of this device with values such as angle and distance measurements. Finally, an experimental setup and a method for measuring the turbulence structure coeffi cient are described. Tests in turbulent air conditions have shown a radical decrease in measurement accuracy. This is demonstrated by both a decrease in power and an increase in the standard deviation of a laser beam, as well as a decrease in the precision of angle and distance measurement. The experimental results show a high correlation between the predicted and measured values.


Introduction
Due to the increasing demand for high-accuracy surveying, new surveying instruments have recently appeared on the market.Recent decades have witnessed an increasing interest in Large-Scale/Large-Volume Dimensional Metrology (LSDM).Many fi elds of application, ranging from construction to shipbuilding and aerospace, have shown increasingly accurate and versatile systems for geometric control.Especially in the last ten years, optical technology has registered a fundamental step forward, alike in terms of metrological performance, versatility, and convenience of use.This diverse stakeholder community is now demanding the dimensional measurement and control of structures and assemblies from several meters up to several kilometers in size as well as to unprecedented levels of accuracy [1][2][3][4], often in non-cooperative environments such as large factory spaces, outdoors, or in underground tunnels.Problems that are common at short range, such as surface form and optical interactions with surfaces, are no longer limiting factors for large-range metrology, where issues such as multi-material thermal control, refractive index correction, lack of accurate traceable commercial equipment, and stable facilities for geodetic calibration are more problematic.Obtaining high-accuracy data could, however, be disturbed by irregular and dynamically changing atmospheric conditions such as temperature variation, turbulence, dust, and smoke.These perturbations are diffi cult to eliminate and have an important impact on the resulting measurements.Determining such changeable meteorological parameters, which are needed to correct the experimental results, is very complex for both outdoor and indoor surveying.Finding a close relationship between the infl uence of external conditions and measurement accuracy is also a complex issue.In this work, we consider only an analysis of the impact of changes in air temperature to determine the angle of laser wave propagation, ignoring other meteorological parameters (e.g., pressure, humidity) and particles of pollutants (e.g., dust, smoke).The eff ect of irregular or even regular variations in atmospheric conditions is diffi cult to eliminate, and reducing their impact on the results of measurement is an intricate problem [5][6][7][8][9].Due to the fact that conditions in indoor spaces are more stable than in outdoor ones, the use of high-accuracy measurements based on laser technology is more convenient in factory halls.The infl uence of meteorological parameters on laser optical bending and ways of taking them into account have been widely discussed in such reports as those of Tatarsky [10], Chiba [11], and Strohben [12].The theoretical considerations have been confi rmed by many experiments conducted in research centers all over the world [2,[13][14][15][16][17][18][19][20].It should be stressed that, in most of these experiments, the laser beam was propagated in the outer layers of the atmosphere.On the other hand, far fewer authors have discussed the problem of laser beam propagation in a closed space [2,6,8,9,15,16].One of the main limitations of the accuracy of optical-length measurements and laser alignment through an uncontrolled atmosphere is the uncertainty in the average refractive coeffi cient over the optical path due to the absence of homogeneity and the turbulence of the atmosphere.Good results can be obtained over short distances at one or more points along the path, but absolute distance measurements with an accuracy of 0.1 ppm or bett er are diffi cult to obtain over distances greater than a few dozen meters.The technology based on the latest achievements of laser physics and electronics has allowed for the development of instruments such as laser stations with accuracy as precise as tenths of a millimeter, which were never att ainable before [21][22][23][24].
The main aim of this paper was to test the usefulness of a Leica TDRA6000 laser station under turbulent air conditions indoors and to estimate the accuracy of this device with values such as the angle and distance measurements declared by the manufacturer.The laser beam was transmitt ed through turbulent hot air (stimulated by a hot fan) that was in a cylindrical tube in the middle of a 30 meter path of a laser beam.
The test measurements, under conditions of forced turbulence indoors, may not be fully adequate as related to the real factory measurements, but they can provide valuable information on the expected impact of unfavorable atmospheric conditions on accuracy measurement with the use of laser stations in Large-Scale/Large-Volume Dimensional Metrology (LSDM).

The Model
When a laser beam propagates through a turbulent atmosphere, it experiences random fl uctuations in its refractive index.Fluctuations of the refractive index are due to turbulent eddies caused by stochastic variations of temperature.In practice, it is diffi cult to control the composition of the air in optical instruments suffi ciently enough to achieve high accuracy.Under realistic medium conditions, light propagation depends on the concentration of inhomogeneity centers, which change dynamically along the direction of propagation.Laser surveying in an uncontrolled atmosphere requires accurate formulas describing refraction as a function of air perturbation.Due to heterogeneity, the coeffi cient of refraction, which is the main parameter describing the optical properties of the medium, depends on the position of vector r and can be expressed as [8]: where 0 ( ) 1 is the constant refraction coeffi cient for a homogeneous medium dependent on temperature, pressure, and humidity in the air, and 1 ( ) N r  is the correction accounting for the variation in the refraction coeffi cient due to turbulence; the angled brackets  indicate the statistical mean (expected value).Estimation of the uncertainty of laser beam refraction depends on both systematic and random components, as presented in Table 1.Systematic refraction is characterized by regular, slow changes (lasting longer than one minute), and these variations the overlap random fl uctuation caused by air turbulence.The theory of the infl uence of atmospheric turbulence on the propagation of electromagnetic waves has been extensively discussed [10,12].Turbulence as a random process is described by stochastic parameters.Taking into account the random changes in velocity (i.e., its value and direction) of each particle of the medium is a complicated issue, and a complete statistical or stochastic analysis is impossible.The existence of temperature gradients in the atmosphere leads to a scatt ering of the laser beam, and it results in a systematic change in refractive coeffi cient N 0 along the path of propagation.
Variation in N 0 in closed rooms can be connected with the following causes: -daily changes in temperature caused by the production process, -seasonal changes of temperature outside industrial buildings, -heat emitt ed by various machines used in production.
Due to air convection and turbulence, which often occur at the boundary of two diff erent media (e.g., indoor-outdoor), regular changes in component N 0 of refractive index N overlap N 1 , which depends on turbulence.

The Nature of Turbulence
The fl uctuating part of refractive coeffi cient 1 ( ) N r  can be described with the help of the integral Fourier-Stieltjes transformation, considering it as a random vector function [12]: where [ , , ] is a three-dimensional wave vector and d is a random spatial amplitude.
Thus, the covariance function of the fl uctuations in the refractive coeffi cient at points at distance r can be writt en in the form of the following dependence: where the angled  brackets indicate the statistical mean (expected value).
Putt ing equation ( 2) in (3) leads to the spatial form of the following equation [12]: where * is the value of the complex conjugate.
Assuming that the fi eld is homogenous or in a steady state, it cannot have average characteristics that depend on the location in a fi eld where the average value is calculated.This means that the same result should be obtained by moving the sensors (e.g., temperature) from one location to another.Based on this assumption: Due to the double integral in Formula (4), to meet the requirement for homogeneity, the value contained in the brackets   must satisfy the following condition: where δ is the three-dimensional Dirac function and ( ) N k   is the three-dimensional power spectrum of the refractive coeffi cient.
Substituting equation ( 6) into (4) and integrating k (the process of integration can be found in [12]) yields the well-known Fourier transform between the power spectrum density and covariance: On the other hand, the inverse of (7) will have the following form: Much work and many experiments have been devoted to function 10,11,25,26].The most-frequently-used function is the Kolmogorov spectrum, which has the following form: where 2

N
C is a structural constant refractive coeffi cient that acts as a measure of its fl uctuations and , L 0 is the outer scale turbulence and is described as the greatest distance at which fl uctuations in the refractive coeffi cient are correlated, and this is an internal scale that describes the smallest eddies of turbulence.-very strong turbulence [19].

Behavior of a Laser Beam in a Turbulent Atmosphere
The problem of propagation of an electromagnetic wave in a turbulent atmosphere has been widely discussed [5, 10-12, 17, 18, 20, 26, 28].To bett er understand the problem, such an environment has been approximated by assuming that it behaves like a large number of non-identical lenses (eddies) that diff er in sizes from l 0 to L 0 (Fig. 1).

Fig. 1. Propagation of laser beam in turbulent atmosphere
Source: [8] Concentrating on the situation when the laser beam diameter is smaller than most of the atmospheric heterogeneities (Case B -as can be seen in Figure 2 is justifi ed due to the fact that the typical beam diameter is on the order of millimeters.Also assuming a plane wave front of the laser beam, this condition is fulfi lled for the short propagation distance occurring in factory halls.
Spectral analysis for both a plane and spherical wave front is used to determine the fl uctuation amplitude, phase, and angle of the laser beam scatt ering during propagation.The details of this approach can be found in [12].In order to consider the problem of propagation of a laser beam in a turbulent medium, the following equation must be solved: where E is the scalar wave function of the propagating laser light intensity, 2 k    is the wave number (λ is the wavelength of the laser light), N 1 (r) describes a random local refractive coeffi cient, and Δ is the Laplace operator.
Even if the form of function N 1 (r) is explicitly given, it is only possible to solve the above equation approximately.The classical approach is a perturbation method, which consists of expansion of E into an infi nite series of decreasing elements: where E 0 represents the unperturbed intensity and E 1 is the fi rst correction connected with dissipation.
The infl uence of turbulence on the amplitude and phase of the propagating wave that determine the statistical characteristics of fl uctuations was considered by [10].He assumed that the variations in amplitude are described by the log-normal distribution, and the variance of this distribution has the following form: where L is the propagation distance; i.e., the distance between transmitt er and receiver.

Fig. 2. Action of turbulence components on laser beam (D is diameter of beam, and L is dimension of turbulent eddy)
Source: [8] In practice, the variance in radiation intensity (which is due to the proportionality of the radiation intensity to the square of the amplitude) is also used as follows:

Angular Deviations of the Laser Beam
Assuming that the output of the laser beam is focused and has a diameter of D 0 , the equation determining where the radiation will fall to zero as a result of the divergence of diameter D at a distance L will be [29]: where λ -the wavelength.
If diameter D is smaller than the internal l 0 scale turbulence along the propagation path, the following equation for the angular mean square deviation (variance) of the laser beam at any point in the propagation path can be writt en as such: Details on deriving this equation can be found in [8].
However, the total variance of the arrival angle at the end of path L will be: where L l -the number of elementary segments; therefore:

Test Measurements of the Infl uence of Air Turbulence on Geodetic Laser Survey
Recording of the time data was limited to one minute based on the assumption stated in the introduction of this work.This is because systematic refraction, which in the present work has not been studied, occurs after this period.The total number of measurements consisted of eight independent tests for stimulated air turbulence (using a hot fan).
However, the control measurement was fi rst made in still air.Angular deviations in a laser beam were measured at the end of the propagation path using a Leica TDRA6000 industrial total station with a Leica 1.5" Red Ring refl ector in the "precise" measurement mode.The producer's inspection certifi cate in accordance with DIN 55350-18-4.2.2 specifi es the following accuracy for this device: maximum deviation of distance measurement of ±0.2 mm (standard deviation of single distance measurement σ = 0.1 mm) and standard deviation of angle measurement (Hz and V) σ = 0.15 mgon.
In any case, before the test was recorded, laser beam power by means of a sensor was located at the end of the propagation path to determine the intensity of turbulence on the path of the laser beam and their eff ect on the size of the deviations of the measured angles and distances.

Determination of Turbulence Structure Coeffi cient
Determination of the coeffi cient was based on a power fl uctuation measurement technique using a Coherent PowerMax-USB/RS model UV/VIS laser beam radiometer with a wavelength ranging from 325 nm to 1065 nm.A laser beam with wavelength of 650 nm and a maximum average radiant power of 5 mW was emitt ed from a Leica TDRA6000 industrial laser station, which emerged from a telescope objective having an initial diameter of D 0 = 1 mm [24].Then, the laser beam was transmitt ed through turbulent hot air, which was in a cylindrical tube in the middle of the path of the beam (Fig. 3).The total path length was 30 m.

Verifi cation of Angles and Distances Measurements Accuracy under Conditions of Turbulence
Before performing the proper tests under conditions of forced turbulence, control measurements were performed under stable air conditions.The location of the point was determined by measurement in precision mode on a 1.5" RRR prism at a distance of 30 m while maintaining the horizontal target axis.At the same time, the values of power for the laser beam, emitt ed by the laser station, were registered.They were used to calculate the structural C N coeffi cient, which characterizes the state of the measurement environment using the following formula: The results of observations and the C N calculations are presented in Table 2.

Verifi cation of Angles and Distances Measurements Accuracy under Conditions of Forced Turbulence
The measurements were made thirty times in precision mode (averaging 3 distance readings) aiming at a 1.5" RRR prism using the Automatic Target Recognition function.
The series of readouts was followed by a 15 minute break, after which the measurements were repeated.In total, 8 measurement series were performed, which resulted in 240 observations.Before each set of readings of angles and distances, the power of the laser beam emitt ed by the laser station was recorded.The results of measurements of the laser beam under conditions of forced air turbulence and the calculated C N structural coeffi cient are reported in Table 3.However, Table 4 illustrates a comparison of the predicted angular deviations calculated on the basis of formula (17) with the standard deviations S Hz and S V determined on the basis of measurements performed with a laser station.The examples illustrating deviations of horizontal and vertical angle measurements from the average value for all readouts are presented in Figure 5 and Figure 6.The measurement deviations of the diagonal distance from the average value are presented in Figure 7. of Hz and V angles can increase more in comparison to precision declared by the producer of the equipment.The surveyor must decide then whether further measurements under such conditions will make it possible to achieve required precision.Alternatively the problem of air turbulence within measurement direction can be solved by using such measurement equipment which performs measurements with multiple sampling.On the basis of partial results it calculates average standard deviation of single observation and it stops measurements if limit values have been exceeded.Such solutions are currently used in laser trackers.These options specify a position tolerance when taking stationary measurements.When a point is probed using a stationary measurement, several readings are made within the specifi ed time period and a deviation is calculated.If the deviation exceeds the position tolerance value, the point is refused.The deviation is displayed as the RMS value (Root Mean Square) in the Digital Readout window.For the accurate total stations which are not equipped with such sampling techniques it is necessary to check the strength of atmospheric turbulence.The decision to include the infl uence of turbulence on measurements precision is determined by the result of measurement of standard deviation of the laser beam power and on this basis calculate the turbulence coeffi cient C N.

The value of parameter 2 NC
characterizes the various stages of turbulence:

Fig. 3 .
Fig. 3.The experimental setup for measurement of C N

Fig. 4 .
Fig. 4. Fluctuations in the power of the beam recorded by the radiometer in still air and in stimulated turbulence for test no. 2 of beam intensity, M -average beam power [mW], δ M -standard deviation of beam power, λ -wavelength laser, L -beam propagation path.

Table 1 .
Components of refractive coeffi cient

Table 2 .
Results of control measurements of angle deviations dV, dHz, and distance deviation dD under calm air conditions.Results of standard deviations of measured angles and distances in the table above are close to the values given in the manufacturer's certifi cate.

Table 3 .
Structural coeffi cients calculated on the basis of stimulated turbulence C N for 8 tests

Table 4 .
The predicted and measured angular deviations of the laser beam