Interferometric Laser Scanner for Direction ... - Semantic Scholar
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Interferometric Laser Scanner for Direction Determination Gennady Kaloshin *,† and Igor Lukin † Received: 30 November 2015; Accepted: 19 January 2016; Published: 21 January 2016 Academic Editor: Markus W. Sigrist V.E. Zuev Institute of Atmospheric Optics SB RAS, Academician Zuev Square 1, Tomsk 634055, Russia; [email protected] * Correspondence: [email protected]; Tel.: +7-382-249-1818; Fax: +7-382-249-2086 † These authors contributed equally to this work.
Abstract: In this paper, we explore the potential capabilities of new laser scanning-based method for direction determination. The method for fully coherent beams is extended to the case when interference pattern is produced in the turbulent atmosphere by two partially coherent sources. The performed theoretical analysis identified the conditions under which stable pattern may form on extended paths of 0.5–10 km in length. We describe a method for selecting laser scanner parameters, ensuring the necessary operability range in the atmosphere for any possible turbulence characteristics. The method is based on analysis of the mean intensity of interference pattern, formed by two partially coherent sources of optical radiation. Visibility of interference pattern is estimated as a function of propagation pathlength, structure parameter of atmospheric turbulence, and spacing of radiation sources, producing the interference pattern. It is shown that, when atmospheric turbulences are moderately strong, the contrast of interference pattern of laser scanner may ensure its applicability at ranges up to 10 km. Keywords: atmospheric turbulence; laser scanning; interference; laser beam
1. Introduction In a previous paper [1], we developed a method according to which, for a synchronous scanning with two laser beams, a wave interference forms in the region of their superposition, with the frequency of resulting oscillation being uniquely related to the direction toward the source. As is well known, the possibility of recording the contrast of interference pattern, formed by optical waves in the atmosphere, is determined both by coherence of the sources of optical radiation, and by random inhomogeneities of refractive index of air along the radiation propagation path [2,3]. In this paper, the method, developed for fully coherent beams, is generalized for the case when interference pattern is formed in the turbulent atmosphere by two partially coherent sources. The analysis performed identified the conditions under which stable interference pattern can be formed in the turbulent atmosphere on extended paths 0.5–10 km in length. In the paper, we describe in detail the method for selecting instrument parameters, ensuring the necessary instrument operability ranges in the atmosphere for any turbulence characteristics possible. The method is based on analysis of the mean intensity of interference pattern, produced by two partially coherent sources of optical radiation. For the chosen instrument parameters of interferometric laser scanning (ILS), we presented the cross sections of distribution of the mean intensity of interference pattern, produced by ILS in the turbulent atmosphere. Visibility of this interference pattern is estimated as a function of propagation pathlength and structure parameter of atmospheric turbulence for a few spacings of radiation sources, producing the interference pattern. ILS may be applied in navigation for determine the direction and the control of changing transient processes in the turbulent atmosphere. In particular, it may be valuable as a laser-sensing Sensors 2016, 16, 130; doi:10.3390/s16010130
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it may be valuable as a laser-sensing instrument in atmospheric optics for the study of intensity fluctuations during the propagation of optical radiation and as applied to an optical refraction, and instrument in atmospheric optics for the study of intensity fluctuations during the propagation of investigation of a mirage detection. optical radiation and as applied to an optical refraction, and investigation of a mirage detection. 2. Method 2. Method 2.1. ILS Specification 2.1. ILS for for Direction Direction Specification Figure 11 shows shows the the optical optical scheme scheme of of ILS, ILS, implementing implementing this Figure this method method for for one-dimensional one-dimensional case. case. Using beam beam splitter splitter and and mirror, mirror, laser laser beam beam with with polarization polarization along along OX OX axis axis is is split split into into two two parallel parallel Using beams with with identical identical apertures. apertures. One passes through through aa beams One of of the the beams, beams, after after being being reflected reflected by by mirror, mirror, passes ˝ polarizer, which turns the polarization plane of the beam through 90, and arrives at piezoelectric polarizer, which turns the polarization plane of the beam through 90 , and arrives at piezoelectric scanner (PS). (PS).The Thesecond secondbeam beam passes through electro-optic modulator (EOM) 1 operating the scanner passes through electro-optic modulator (EOM) 1 operating on theonbasis basis of longitudinal electro-optic effect, through /4 plate (quarter-wave plate), and the second of longitudinal electro-optic effect, through λ/4 plate (quarter-wave plate), and the second (analogous (analogous the first) electro-optic thebeam second beamthrough passes through the polarizer, to the first) to electro-optic modulator.modulator. Then, theThen, second passes the polarizer, which which transmits radiation with polarization along the OY alsoatarrives at the PS mirror. transmits radiation with polarization along the OY axis, andaxis, also and arrives the PS mirror. Sawtooth Sawtooth voltage from sweep-frequency with frequency is to EOM. This voltage from sweep-frequency generator generator (SFG) with(SFG) frequency Ω is applied toapplied EOM. This voltage is voltage to is applied to EOMand 1 directly to π/2 EOMphase 2 viashifter; /2 phase shifter; thus, the voltages at EOM applied EOM 1 directly to EOMand 2 via thus, the voltages at EOM 1 and EOM 21 and EOM 2 have the phaseofdifference of /2. have the phase difference π/2.
Figure 1. 1. The The optical optical scheme scheme of of interferometric interferometric laser laser scanner scanner for Figure for direction direction specification. specification.
A radiative component with polarization along the OY axis is frequency shifted by amount amount Ω relative to the initial laser beam. Thus, two parallel laser beams with identical polarizations arrive at PS represents representsaamirror, mirror,mounted mountedononpiezoceramic piezoceramic plate. The principle work of PS is based PS. PS plate. The principle of of thethe work of PS is based on on the use of reverse piezoelectric effect that can be identified in the deformation of crystal placed the use of reverse piezoelectric effect that can be identified in the deformation of crystal placed into the into the field electric field at the orientation certain orientation of the field and direction of electric electric at the certain of lines of oflines forceof offorce the field and direction of electric axis ofaxis the of the crystal analogously to the method of a beam forming device, as described in previous work [1]. crystal analogously to the method of a beam forming device, as described in previous work [1]. When When control voltage is applied to PS, the parallel laser beams scan in the plane, perpendicular to the control voltage is applied to PS, the parallel laser beams scan in the plane, perpendicular to the plane, plane, containing the axes optical axes beams. of these beams. control forisfeeding is applied containing the optical of these The controlThe voltage for voltage feeding PS appliedPS directly from directly from SFG, which feeds EOM. Then, the scan angle is directly proportional to the frequency SFG, which feeds EOM. Then, the scan angle φ is directly proportional to the frequency Ω. Because of . Because divergence, of diffraction divergence, beams will start overlap at a certain range from PS,will where diffraction the beams willthe start to overlap at atocertain range from PS, where there be there will be anpattern interference pattern with amplitudeofmodulation of radiation with an interference with amplitude modulation radiation with frequency Ω. frequency . The optical signal of POS is accepted by photodetector device (PD) on moving object, consisting of objective, interference and polarization filters, and photodetector with power supply and
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The optical signal of POS is accepted by photodetector device (PD) on moving object, consisting of objective, interference and polarization filters, and photodetector with power supply and frequency meter, which measures the frequency Ω. The onboard photoreceiver device records and extracts the frequency of the recorded signal Ω, which is uniquely related to the direction toward PS. The positions of the laser beams at the point of space Q (Figure 1) are determined by the PS statement controlled with electrical signal of Ω frequency. By means of optical amplification, the change of light beam direction to 1 s of arc is accessible. Hence, setting the direction to within 1 s of the arc is not technically challenging. As such, the accuracy of measurements may exceed at least 1/5000 s of the arc. This unveils wide opportunities in application of the method in both the research and engineering. In particular, it may be valuable in optical refraction spectroscopy, in photothermal deflection spectroscopy for controlling the slowly changing transient processes in gaseous and liquid media and plasma, in atmospheric optics for studying the intensity fluctuations during the propagation of optical radiation, in investigation of mirage detection and reflectance techniques, in geodesy, optical profilometry, ranging, and navigational course detection. At present, this method is used for direction control at high accuracy, with the capabilities of the techniques being limited by the sizes of laser beams. The method of beam forming device (BFD) has essentially the best capabilities in terms of resolution, because the resolution is determined by interference bandwidth. 2.2. Distortions of ILS-Formed Interference Pattern We assume the following scheme of PS and PD of ILS: the sources of laser radiation, located, ( respectively, at points t0, ρ1 u and 0, ρ2 , emit collimated laser beams parallel to each other and to the OX-axis in the direction of positive x values; and PD is located at point Q (Figure 1) with coordinates tx, ρu. We adopt that the quantity ρtr means the vector of spacing of radiative sources, producing the interference pattern. Because the two laser beams that are used to form the interference pattern are obtained by splitting one initial laser beam (see Figure 1), in the initial plane x = 0 the function of second-order mutual coherence of the field of each of these laser beams, representing the Gaussian partially coherent beam, has the following form: ´ ¯ ! ”` ) ˘2 ` ˘2 ı ´ 2 ¯ ` 1 ˘2 Γ2 x “ 0, ρ1 , ρ2 ; ρ j , ρ j “ E02 exp ´ ρ1 ´ ρ j ` ρ2 ´ ρ j { 2 a0 ´ ρ ´ ρ2 {ρ2k (1) where E0 is the initial amplitude at the optical beam axis; a0 is the initial beam radius; ρk is the spatial coherence radius of the initial field; k “ 2 π{λ, λ is the wavelength of the optical radiation in vacuum; ρ1 , ρ2 are the position vectors of the observation points; j “ 1, 2. A similar formula also holds for second-order mutual coherence function of the fields of these two laser beams: ´ ¯ r2 x “ 0, ρ1 , ρ2 ; ρ j , ρ j1 Γ " „ * ´ ¯2 ` ¯ı2 (2) ` 1 ˘2 ˘ ”` 1 ˘ ´ 2 2 2 2 2 “ E0 exp ´ ρ ´ ρ j ` ρ ´ ρ j1 { 2 a 0 ´ ρ ´ ρ j ´ ρ ´ ρ j1 {ρk where j ‰ j1 , and j, j1 “ 1, 2. We consider that, in the formation of the second radiative source, it is admissible to assume that its coherence properties are totally identical to the initial source and are just shifted in space by the amount ρtr . PD is the square-law detector that responds to the power of incoming radiation, the signal of which can be represented as ir px, ρq “ η0 I px, ρq, where η0 is the quantum efficiency coefficient of PD; I px, æq is the instantaneous value of intensity of interference pattern at the detector location, which has the form: I px, ρq “ U1 px, ρq U1˚ px, ρq ` U2 px, ρq U2˚ px, ρq ` 2Re rU1 px, ρq U2˚ px, ρqs
(3)
where Uj px, ρq is the field of optical wave of one source; j “ 1, 2. Suppose that ρ1 “ ´ ρtr {2, and ρ2 “ ρtr {2.
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The second-order mutual coherence function of fields of two partially coherent Gaussian beams of optical radiation for boundary Equations (1) and (2), written with the use of the extended Huygens-Fresnel principle in the case of square-law approximation for the function, which describes the effect of random inhomogeneities of the medium, has the following form [4]: » xUj px, ρq Uj˚1 px, ρqy “
E02 a20 a2 pxq
`
ρ ´ ρj — exp –´ 2 a2 pxq
¯2
´
˘2
ρ ´ ρ j1 ´
2 a2 pxq ´ ¯2 fi ´ ¯ 1 ρ ´ ρ j j δ pxq ffi `i 2 ρ2j ´ ρ2j1 ´ fl 2 a pxq ρ2c pxq
´i
¯ δ pxq ´ 1 ρ ´ ρ ρ j j a2 pxq (4)
˙ 4 2 2 1{2 where a pxq “ a0 1 ` 1` ` a0 {ρ0 is the mean radius of the 3 ˘ “ ‰ ´1 ` laser beam; δ pxq “ Ω0 1 ` a20 {ρ2k is the geometrical factor; δ pxq{ k a2 pxq “ is the difference between curvatures of laser beam wavefronts; ρc pxq ¸ff « ˜ ¸ ff+1{2 #« ˜ 2 2 2 2 a a 4 a0 3 ρ0 2 2 31{2 ρ0 { Ω´ 1 ` 02 ` is the radius of mutual 1 ` Ω´ 1 ` 02 ` 0 0 2 3 ρ0 4 ρ2k ρk ρk coherence of partially coherent laser beams; Ω0 “ k a20 {x is the Fresnel number of emitting aperture; ` ˘´3{5 ρ0 “ 0.3642 Cε2 k2 x is the coherence radius of plane optical wave in the turbulent atmosphere; Cε2 is the structure parameter of atmospheric turbulence; and j, j1 “ 1, 2. Using Equations (3) and (4), we obtain the formula for the mean value of intensity of interference pattern: „
ˆ
2 Ω´ 0
xI px, ρqy “ 2
a20 {ρ2k
„ " „ „ „ * E02 a20 ρ2 ` ρ2tr {4 ρ2tr ρtr ρ δ pxq exp ´ cosh ` exp ´ cos ρ ρ (5) tr a2 pxq a2 pxq a2 pxq ρ2c pxq a2 pxq
Equation (5) can be used to formulate the conditions, restricting the choice of parameters of laser beams and ILS scheme. The linear dimensions lint pxq of the region, where the interference pattern Equation (5) had formed, provided that a pxq yy ρtr , have a value approximately equaling the current laser beam diameter: lint pxq – 2 a pxq (6) The band maxima of interference pattern are at the points ρmax , determined (see Equation (5)) from equation of the form: δ pxq{a2 pxq ρtr ρmax “ 2 n π, where n “ 0, ˘ 1, ˘2, . . .; while the band minima are at points ρmin determined from δ pxq{a2 pxq ρtr ρmin “ p2 n ` 1q π, n “ 0, ˘ 1, ˘2, . . .. Let ρ || ρtr , then ρmax || ρmin || ρtr and ρmax “ 2 n π a2 pxq{ rδ pxq ρtr s, ρmin “ p2 n ` 1q π a2 pxq{ rδ pxq ρtr s, while the width of the interference band ∆lint pxq can be evaluated from the following formula: ∆lint pxq “ 2 |ρmax ´ ρmin | “ 2 π a2 pxq{ rδ pxq ρtr s
(7)
On the other hand, the contrast of distorted interference pattern, determined from the mean intensity (at ρmax « ρmin « ρ), is equal to: ν “
xI px, ρmax qy ´ xI px, ρmin qy – xI px, ρmax qy ` xI px, ρmin qy
" „ *´1 „ ρ2 ρtr ρ cosh 2 exp ´ 2 tr a pxq ρc pxq
(8)
It is evident that, in order for ILS to be operable, at least one complete interference band should be in the field of interference pattern; therefore, Equations (6) and (7) make it possible to formulate the
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Moreover, the contrast of the mean interference pattern near its center (when
cosh tr a 2 x 1 ) will be satisfactory until the coherence radius of optical field at the condition: lint pxq ě ∆lint pxq , which is fulfilled when the range between the sources of laser radiation observation point exceeds the value of transversal shift of the beams; therefore, from Equation (8) we satisfies the condition: obtain the following inequality: ρtr ě π a pxq{δ pxq (9)
ln c x (10) tr interference Moreover, the contrast of the mean pattern near its center (when “ ‰ 2 cosh Summing ρtr ρ{a pxq 1) will beEquations satisfactory until the radius of field form: at the two–conditions, (9) and (10), wecoherence obtain the formula of optical the following observation point exceeds the value of transversal shift of the beams; therefore, from Equation (8) we ax x tr ln c x (11) obtain the following inequality: a (10) ρtr ď ´ ln pνq ρc pxq In order for Equation (11) to be fulfilled, the right-hand side of the inequality should exceed the Summing conditions, Equations (9) andthe (10), we obtain the formula of the following form: left-hand side, two making it possible to formulate condition restricting the initial sizes of the laser beams a0 : a π a pxq{δ pxq ď ρtr ď ´ ln pνq ρc pxq (11) ln k 2 a04 a2 a02 1 1 right-hand 1 02the 1 side to be fulfilled, In order for Equationa0(11) should exceed(12) the 2 2 of the2 inequality 3 0 k 4 k x2 k left-hand side, making it possible to formulate the condition restricting the initial sizes of the laser beams a0 : ˜ ¸ g ˜ ¸ a f 2 2 3. Results f 1 a0 a ´ ln pνq 1 k2 a40 a0 ď 1 ` 2 {e 2 1 ` 02 ` (12) π ρk 3 ρ0 ρk 4 ρ2k x2 3.1. Choice of Parameters of ILS Optical Scheme Figure 2 presents the behavior of implicitly specified function a0 x , defined by Equation (12). 3. Results Values of a0 , lying on the curve that graphically represents the solution of Equation (12), correspond 3.1. Choice of Parameters of ILS Optical Scheme to the condition “ ” in Equation (12). The a0 values, lying below the corresponding curve, Figure 2topresents the behavior of Equation implicitly(12). specified function by Equation (12). 0 pxq, defined All curves in aFigure 2, plotted for different correspond the condition “ ” in Values of a0 , lying on the curve that graphically represents the solution of Equation (12), correspond to parameters of the problem, , k , C 2 , and , are within grey-colored region. Figure 2 shows an the condition ““” in Equation (12). The a0 values, lying below the corresponding curve, correspond assemblage of curves in the form of a nomogram for the initial laser beam radii a0 . This figure shows to the condition “x” in Equation (12). All curves in Figure 2, plotted for different parameters of the 2 two regions (not filled) that allow formation of a stable interference pattern. In filled (grey) in is problem, λ, ρk , Cε , and ν, are within grey-colored region. Figure 2 shows an assemblagearea of curves on the not possible formation of a stable pattern. The effect of the wavelength of optical radiation the form of a nomogram for the initial laser beam radii a0 . This figure shows two regions (not filled) a0 was valueallow of the initial size laser beams estimated by considering the values of formation in the range that formation of of a stable interference pattern. In filled area (grey) is not possible of m to 1.55 m (at the same time, the other parameters were as follows: 2 cm, from 0.51 a stable pattern. The effect of the wavelength of optical radiation λ on the value of the initial size of k 2 a was −13 −2/3 laser beams estimated by considering the values of λ in the range from 0.51 µm to 1.55 0.1, C 0 10 m ). Correspondingly, the contrast of the interference pattern varied fromµm 0.1 2 “ 10´13 m´2{3 ). (at the same time, the other parameters were as follows: ρ “ 2 cm, ν “ 0.1, C 2 k ε −13 −2/3 to 0.5 (at 1.55 m, k 2 cm, C 10 m ); and the structure parameter of the atmospheric Correspondingly, the contrast of the interference pattern ν varied from 0.1 to 0.5 (at λ “ 1.55 µm, 2 −16 m−2/3 1.55 turbulence 2C ´13 m´2from {3 ); and turbulence varied 10the to parameter 10−13 m−2/3 (at atmospheric m, k C2 2varied cm, ρ structure of the k “ 2 cm, Cε “ 10 ε 0.1). from 10´16 m´2{3 to 10´13 m´2{3 (at λ “ 1.55 µm, ρk “ 2 cm, ν “ 0.1).
Figure Figure2.2.The Thenomogram nomogramfor forchoosing choosingthe theinitial initiallaser laserbeam beamradii radii aa00. .
Based on data presented in Figure 2, the initial laser beam radii a0 can be chosen to be 1…2 mm for pathlengths in the range from 1 m to 10 km. Equation (12) must be solved (using the condition “ ”) with respect to k to determine the minimum acceptable level of the spatial Sensors 2016, 16, 130
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coherence radius of initial field. The data thus obtained are presented in Figure 3 in the form of grey-colored region (in these Based on data in Figure 2, the initial laserfrom beam 0.1 radiitoa00.5, can and be chosen to be 1 . . . 2parameter mm varied the structure of estimates, the contrast ofpresented interference pattern for pathlengths in the 2range from 1 m to 10 km. Equation (12) must be solved (using the condition atmospheric turbulence C varied from 10−16 m−2/3 to 10−13 m−2/3); and any curve, obtained in solution ““”) with respect to ρ to determine the minimum acceptable level of the spatial coherence radius of k
of Equation initial(12), field.falls within this region. The data position thus obtained aregrey-colored presented in Figure 3 inpresented the form ofingrey-colored region (inthat these The shape and of the region, Figure 3, indicate the initial estimates, the contrast of interference pattern ν varied from 0.1 to 0.5, and the structure parameter coherence of optical sources plays the key role (in selecting the parameters of single beams) along of atmospheric turbulence Cε2 varied from 10´16 m´2{3 to 10´13 m´2{3 ); and any curve, obtained in propagation paths up to 300 m in length, while atmospheric turbulence is critical when x 300 m. solution of Equation (12), falls within this region. Figure 3 shows assemblage ofofcurves in the form of apresented nomogram for choosing coherence Thean shape and position the grey-colored region, in Figure 3, indicatethe thatspatial the initial coherence of optical sources plays the key role (in selecting the parameters of single beams) along radius k of the initial field of laser beams that allow the formation of a stable interference pattern. propagation paths up to 300 m in length, while atmospheric turbulence is critical when x > 300 m.
Here also, as in the previous figure, showing the two areas (not filled) that allow formation of a stable Figure 3 shows an assemblage of curves in the form of a nomogram for choosing the spatial coherence interference pattern. Figure 3 demonstrate thatthe theformation laser beam coherence level,pattern. comparable to radius ρk of theData initialinfield of laser beams that allow of a stable interference the initial coherence of 1…2 cm, is reached the(not paths with x of 300 m. Thus, it Here also, as in radius the previous figure, showing the twoon areas filled) that the allowlengths formation a stable interference pattern. Data in Figure 3 demonstrate that the laser beam coherence level, comparable can be further assumed that k 1…2 cm. For laser beams with these parameters, the angular beam width
0 2
to the initial coherence radius of 1 . . . 2 cm, is reached on the paths with the lengths x ě 300 m. 0 in the region, where there already exists the directional diagram equaling Thus, it can be further assumed that ρk “ 1 . . . 2 cm. For laser beams with these parameters, the 2 1 2 angular beam2 width in the region, where there already exists the directional diagram equaling k a0 1 a0 ` k ψ,0 must ˘1{2 not exceed 1’. 2 2 ψ0 – 2{ pk a0 q 1 ` a0 {ρk , must not exceed « 1’.
3. The nomogram for choosing the spatial coherence radius of the initial field of laser beams ρk . Figure 3.Figure The nomogram for choosing the spatial coherence radius of the initial field of laser beams k .
Euqation (11) for known parameters of the optical beams makes it possible to choose the distance between radiative sources ρtr . Figure 4 presents the behavior of the functions:
Euqation (11) for known parameters beams makes it possible to choose the distance g of the optical ¸ ˜ ¸ ˜ f between radiative sources tr . Figure 4 presents k a30 f a20 a20 x2 the behavior 4 a20of the functions: f 1 pxq “ π
f1 x (grey-colored region) and g« f f a (grey-coloredf 2 region) and pxq “ ´ ln pνq e 1 `
x
e1 `
k a03 x x2 k2 a40
1 ˜
k2 a40
x2 k 2 a04
1`
ρ2k
`
3 ρ20
{ 1`
a2 a2 1 02 4 02 k 3 0
a2 4 a20 1 ` 02 ` 3 ρ20 ρk
¸ff «
1 x2 { 3 ρ20 k2 a40
a 2 4 a02 x2 f 2 x ln 1 2 4 1 02 2 3 k a 0 0 k
(13)
ρ2k
a2 1 02 k ˜
a2 1 ` 02 ρk
1 x2 2 2 4 3 0 k a0
¸
(13) 1 ` 4 ρ2k
ff (14)
a2 1 1 02 k 4 2k
(14)
(region in light grey), calculated for different parameter values of the problem. Figure 4 shows nomograms for choosing the values tr , at which the formation of interference pattern is a stable.
of wavelength of the optical radiation ( in the range from 0.51 m to 1.55 m for a0 2 mm, k 2 cm, 0.1, C2 10−16 m−2/3), and the effect of the contrast of interference pattern ( in the range from 0.1 to 0.5 for 1.55 m, a0 2 mm, k 2 cm, C2 10−16 m−2/3), as well as the effect of 2 structureSensors parameter 2016, 16, 130 of atmospheric turbulence C
( C 2
−16 −2/3 to in the range from 10 7 of 10 m
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(region in light grey), calculated for different parameter values of the problem. Figure 4 shows In estimating the dimensions of both regions, presented in Figure 4, we quantified both the effect nomograms for choosing the values ρtr , at which the formation of interference pattern is a stable. of wavelength of the optical radiation ( in the range from 0.51 m to 1.55 m for a0 2 mm, k In estimating the dimensions of both regions, presented in Figure 4, we quantified both the effect −16 m−2/3), and the effect of the contrast of interference pattern ( in the range 0.1, of 2 cm, C2the 10 of wavelength optical radiation λ (λ in the range from 0.51 µm to 1.55 µm for a0 “ 2 mm, 2 “ 10´16 m´2{3 ), and the effect of the contrast −16 interference ρfrom “ 2 cm, ν “ 0.1, C pattern ν (ν in the 0.1 to 0.5 for ε 1.55 m, a0 2 mm, k 2 cm, C2 10of m−2/3), as well as the effect of k 2 ´ 16 ´ 2 { 3 range from 0.1 to 0.5 for λ “ 1.55 µm, a “ 2 mm, ρ “ m ), as well as the effect 22 cm, C 2 ε “ 10 0 −16 −2/3 k structure parameter of atmospheric turbulence C ( C in the range from 10 m to of structure parameter of atmospheric turbulence Cε2 (Cε2 in the range from 10´16 m´2{3 to 10´14 m´2{3 −14 −2/3 10 m for 1.55 m, a 2 mm, k 2 cm, 0.1). for λ “ 1.55 µm, a0 “ 2 mm, ρ0k “ 2 cm, ν “ 0.1).
Figure 4. The nomogram for choosing the spacing between optical axes of laser beams tr . The greycolored region shows the variability range of the function f1 x , and the region in light grey depicts the variability range of the function f 2 x . Figure 4. The Thenomogram nomogramforfor choosing spacing between optical of beams laser beams ρtr . greyThe . The Figure 4. choosing thethe spacing between optical axes axes of laser
tr trvariability Considering that the spacing of radiative that produce pattern must be grey-colored region shows the range ofsources the function f 1 pxq , and theinterference region in light grey colored region shows the variability range of the function f1 x , and the region in light grey depicts . depicts the variability range of the function f 2 pxq smaller f x and than the values of the function f 2 x , data larger than the values of the function the variability range of the function 1 f x . 2
of Figure 4 for pathlengths x from 20 m to 10 km can be used to estimate the admissible values of tr Considering that the spacing ρtr of radiative sources that produce interference pattern must be
Considering that the spacing of radiative sourcesradiative that produce interference pattern mustused be tr that were in the range from cm to 5 cm. is justtrthese spacings sources larger than 1the values of theItfunction f 1 pxq and smallerof than the values of the function f 2 pxq, data of below larger than the values of the function f1 x and smaller than the values of the function f 2 x , data Figure for pathlengths x from 20 m to 10 km can be used to estimate the admissible values of ρtr in to estimate ILS4 operability. x from 20 m to 10spacings km can be radiative used to estimate admissible values of to of for pathlengths tr theFigure range 4from 5 cm. It is just these sourcesthe below tr that were used Figure 5 presents1 cm thetocalculations of the averageofintensity of ILSρinterference pattern (Figure 1) that were used below in the range from 1 cm to 5 cm. It is just these spacings of radiative sources estimate ILS operability. tr −16 m−2/3 and x 10 km for different distances between centers of two identical collimated at C2 to10estimate Figure ILS 5 presents the calculations of the average intensity of ILS interference pattern (Figure 1) at operability. ´2{3 and the 10´16 m x “calculations 10 km for different distances between Figure 5 presents of the intensity of centers ILSainterference patterncollimated (Figure k 1) 2 1.55 m, 2 identical mm, GaussianCε2 “partially coherent beams ( average 0 of two
cm):
2 −16 −2/3 and x beams partially 1.55 µm, distances a0 “ 2 mm, ρk “ 2 centers cm): (a)of ρtrtwo “ 1identical cm, (b) ρcollimated at 1Ccm, 10 km(λfor tr “ 2 cm, 2 cm,10 (c) “trdifferent 3 cm, (b) mtrcoherent (d) between (a) tr Gaussian tr 4 cm, and (e) tr 5 cm. Panels (c) ρ “ 3 cm, (d) ρ “ 4 cm, and (e) ρ “ 5 cm. Panels in Figure 5a–e represent the color tr cm): Gaussian partiallytr coherent beamstr ( 1.55 m, a0 2 mm, k 2 polar Figure 5a–e represent the colorofpolar contour plots (100 shades of two colors). contour plots (100 shades two colors). (a) tr 1 cm, (b) tr 2 cm, (c) tr 3 cm, (d) tr 4 cm, and (e) tr 5 cm. Panels in
Figure 5a–e represent the color polar contour plots (100 shades of two colors).
(a) (a)
(b) (b) Figure 5. Cont. Figure 5. Cont. Cont. Figure 5.
in
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(c)
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(d)
(e) Figure 5. The average intensityof of interferometric laser scanning (ILS) interference for different pattern for Figure 5. The average intensity interferometric laser scanning (ILS)pattern interference values of ρtr : (a) 1 cm; (b) 2 cm; (c) 3 cm; (d) 4 cm; and (e) 5 cm. different values of tr : (a) 1 cm; (b) 2 cm; (c) 3 cm; (d) 4 cm; and (e) 5 cm.
Figure 5 shows the five options of the interference pattern where the method is workable in
Figure principle. 5 showsThe the fiveillustrates options the interference pattern where theit represents. method is figure theof real (calculated) interference pattern and that Theworkable in Figure 5 corresponds to the the high real visibility at the weak turbulence and support the method performance: principle. The figure illustrates (calculated) interference pattern and that it represents. The Figure 5 corresponds high visibility at the weak turbulence support 1. For ρtr “ 1to cm,the the interference pattern starts to form at separation of 30 m and from the sources ofthe method performance: the optical radiation, and becomes finally formed at the range of 300 m, comprising three bright 1.
2.
3.
bands, with central band becoming most intense starting from 40 m, and with two sideward symmetricallypattern about thestarts central to band, being For tr 1bands, cm, located the interference form atidentical. separation of 30 m from the sources of 2. For ρtr “ 2 cm, the interference pattern starts to form at the range of 30 m from the sources of the optical optical radiation, and becomes finally formed at the range of 300 m, comprising three bright radiation, and becomes finally formed at the range of 300 m. In this case, the pattern has bands, with central most intense from from 40 m, and two sideward seven intenseband bands;becoming the central band becomes moststarting intense starting 80 m; andwith sideward bandssymmetrically are pairwise identical. bands, located about the central band, being identical. For ρ “ 3 cm, the interference pattern starts form at theat range 30 m from the m sources of the 3. tr For tr 2 cm, the interference pattern startsto to form theofrange of 30 from the sources of optical radiation, and becomes ultimately formed at the range of 300 m, comprising 11 intense optical radiation, and finally formed at thestarting rangefrom of 300 m. In this case, the pattern has bands, with thebecomes central band becoming the brightest 120 m. 4. For ρbands; the central interference pattern starts to form at the range ofstarting 40 m fromfrom the sources of the seven intense the band becomes most intense 80 m; and sideward tr “ 4 cm, optical radiation, and becomes ultimately formed at the range of 400 m, comprising 15 intense bands are pairwise identical. bands, with the central band becoming most intense starting from 170 m. For tr5. 3Forcm, the interference pattern starts to form at the range of 30 m from the sources of ρtr “ 5 cm, the interference pattern starts to form at the range of 40 m from the sources of the radiation,and and becomes ultimately formed formed at the rangeatof the 500 m,range comprising 19 intense the opticaloptical radiation, becomes ultimately of 300 m, comprising bands, with central band becoming most intense starting from 250 m.
4.
11 intense bands, with the central band becoming the brightest starting from 120 m. Contrast For tr3.2. 4 cm, ofthe pattern starts to form at the range of 40 m from the sources of ILSinterference Interference Pattern in the Turbulent Atmosphere
5.
contrast of ILS can be estimated approximate formula, the opticalTheradiation, andinterference becomespattern ultimately formedaccording at the torange of 400 m, comprising obtained from Equation (8). Figure 6 presents estimates of the contrast of ILS interference pattern 15 intense bands, with the central band becoming most intense starting from 170 m. For tr 5 cm, the interference pattern starts to form at the range of 40 m from the sources of
the optical radiation, and becomes ultimately formed at the range of 500 m, comprising 19 intense bands, with central band becoming most intense starting from 250 m.
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(Figure 1) for all values of structure parameter of atmospheric turbulence, realizable in near-water atmospheric layer [5,6]. Figure 6 gives estimates of the visibility of the interference patterns at different levels of turbulence. It was assumed that laser radiation propagated along horizontal path at height of 10 . . . 20 m above water surface. The color (40 colors) contour plots, presented Sensors 2016, 16, 130 9 of 10 in Figure 6a–e, demonstrate the dependence of the contrast of ILS interference pattern (ν) on the laser beam propagation path length (x) and on the value of the structure parameter of atmospheric turbulence ( C 2 ) at the wavelength of the optical radiation 1.55 m for different ranges between turbulence (Cε2 ) at the wavelength of the optical radiation λ “ 1.55 µm for different ranges between a0 “22mm, the centers the centers of of two two identical identical collimated collimated Gaussian Gaussianpartially partiallycoherent coherentbeams beams( (a mm, ρkk “ 22 cm): cm): 0 11cm, cm,(b) (b)ρtrtr“2 2cm, cm,(c)(c)ρtr“ 3 cm, 4 cm, 5 cm. (a) ρtrtr “ 3 cm, (d)(d) ρtr “tr 4cm, andand (e) ρ(e) “ 5 cm. tr tr tr
(a)
(b)
(c)
(d)
(e) Figure 6. The Thecontrast contrastof of interference pattern for different distances between the of centers of Figure 6. ILSILS interference pattern for different distances between the centers emitting tr (b) : (a)2 1cm; cm;(c)(b) 2 cm; cm;and (d)(e) 4 cm; and (e) 5 cm. emitting beam apertures beam apertures ρ : (a) 1 cm; 3 cm; (d)(c)4 3cm; 5 cm. tr
The red-colored regions in Figure 6a–e show the laser beam propagation path lengths x for The red-colored regions in Figure 6a–e show the laser beam propagation path lengths x for different structureparameters parameters of atmospheric the atmospheric turbulence C 2high , when high ofcontrasts of ILS different structure of the turbulence Cε2 , when contrasts ILS interference interference in the turbulent take atmosphere pattern in thepattern turbulent atmosphere place take place 4. Conclusions Conclusions
At higher contrasts of interference pattern, ILS proves to be operable in a narrower range of spacings of of of optical radiation thatthat produce the interference pattern. The closer sources ofthe thesources sources optical radiation produce the interference pattern. Thethe closer the of optical to each other, lower contrast of the recorded interference pattern,pattern, and, at sources ofradiation optical radiation to eachthe other, thethe lower the contrast of the recorded interference last, longer wavelength of the optical the greater ILS operability region. For the and, the at last, the the longer the wavelength of the radiation, optical radiation, the is greater is ILS operability region. wavelength of the optical radiation λ “ 1.55 at spacings of the sources ρtr from1tr cm to 51 cm µm, 1.55 m, at spacings of the sources from For the wavelength of the optical radiation ´15 ´2{−15 3 under conditions of the mean turbulence intensityintensity value of 10 in the m−2/3, often most realized often realized to 5 cmthe under the conditions of the mean turbulence value m of 10 , most in the maritime and coastal atmosphere at midlatitudes, the operability range of the device may reach 10 km and longer, depending on the geometric positions of optical elements. Acknowledgments: The Ministry of Education and Science of the Russian Federation (Project No. 14.604.21.0042) supported this work. Author Contributions: All authors contributed equally to this work. Gennady Kaloshin initiated the project, reviewed the paper and contributed to the interpretation of the results. Igor Lukin developed algorithms for
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maritime and coastal atmosphere at midlatitudes, the operability range of the device may reach 10 km and longer, depending on the geometric positions of optical elements. Acknowledgments: The Ministry of Education and Science of the Russian Federation (Project No. 14.604.21.0042) supported this work. Author Contributions: All authors contributed equally to this work. Gennady Kaloshin initiated the project, reviewed the paper and contributed to the interpretation of the results. Igor Lukin developed algorithms for calculation, processed and interpretation the data. Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations The following abbreviations are used in this manuscript: ILS PS EOM SFG PD BFD
Interferometric laser scanning Piezoelectric scanner Electro-optic modulator Sweep-frequency generator Photodetector Beam forming device
References 1. 2. 3. 4. 5. 6.
Kaloshin, G.A.; Lukin, I.P. An acousto-optical sensor with high angular resolution. Sensors 2012, 12, 3739–3746. [CrossRef] [PubMed] Ishimaru, A. Wave Propagation and Scattering in Random Media; IEEE Press: Piscataway, NJ, USA, 1997. Wolf, E. Introduction to the Theory of Coherence and Polarization of Light; Cambridge University Press: Cambridge, UK, 2007. Andrews, L.C.; Phillips, R.L. Laser Beam Propagation through Random Media, 2nd ed.; SPIE Press: Bellingham, WA, USA, 2005. Pasricha, P.K.; Reddy, B.M. Evaluation of the structure parameter Cn2 over the sea surface. IEEF Proc. 1990, 137, 384–386. Forand, J.L.; Dion, D. Predictive comparisons of marine boundary-layer models. Proc. SPIE 1996, 2828, 129–140. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Interferometric Laser Scanner for Direction Determination Gennady Kaloshin *,† and Igor Lukin † Received: 30 November 2015; Accepted: 19 January 2016; Published: 21 January 2016 Academic Editor: Markus W. Sigrist V.E. Zuev Institute of Atmospheric Optics SB RAS, Academician Zuev Square 1, Tomsk 634055, Russia; [email protected] * Correspondence: [email protected]; Tel.: +7-382-249-1818; Fax: +7-382-249-2086 † These authors contributed equally to this work.
Abstract: In this paper, we explore the potential capabilities of new laser scanning-based method for direction determination. The method for fully coherent beams is extended to the case when interference pattern is produced in the turbulent atmosphere by two partially coherent sources. The performed theoretical analysis identified the conditions under which stable pattern may form on extended paths of 0.5–10 km in length. We describe a method for selecting laser scanner parameters, ensuring the necessary operability range in the atmosphere for any possible turbulence characteristics. The method is based on analysis of the mean intensity of interference pattern, formed by two partially coherent sources of optical radiation. Visibility of interference pattern is estimated as a function of propagation pathlength, structure parameter of atmospheric turbulence, and spacing of radiation sources, producing the interference pattern. It is shown that, when atmospheric turbulences are moderately strong, the contrast of interference pattern of laser scanner may ensure its applicability at ranges up to 10 km. Keywords: atmospheric turbulence; laser scanning; interference; laser beam
1. Introduction In a previous paper [1], we developed a method according to which, for a synchronous scanning with two laser beams, a wave interference forms in the region of their superposition, with the frequency of resulting oscillation being uniquely related to the direction toward the source. As is well known, the possibility of recording the contrast of interference pattern, formed by optical waves in the atmosphere, is determined both by coherence of the sources of optical radiation, and by random inhomogeneities of refractive index of air along the radiation propagation path [2,3]. In this paper, the method, developed for fully coherent beams, is generalized for the case when interference pattern is formed in the turbulent atmosphere by two partially coherent sources. The analysis performed identified the conditions under which stable interference pattern can be formed in the turbulent atmosphere on extended paths 0.5–10 km in length. In the paper, we describe in detail the method for selecting instrument parameters, ensuring the necessary instrument operability ranges in the atmosphere for any turbulence characteristics possible. The method is based on analysis of the mean intensity of interference pattern, produced by two partially coherent sources of optical radiation. For the chosen instrument parameters of interferometric laser scanning (ILS), we presented the cross sections of distribution of the mean intensity of interference pattern, produced by ILS in the turbulent atmosphere. Visibility of this interference pattern is estimated as a function of propagation pathlength and structure parameter of atmospheric turbulence for a few spacings of radiation sources, producing the interference pattern. ILS may be applied in navigation for determine the direction and the control of changing transient processes in the turbulent atmosphere. In particular, it may be valuable as a laser-sensing Sensors 2016, 16, 130; doi:10.3390/s16010130
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it may be valuable as a laser-sensing instrument in atmospheric optics for the study of intensity fluctuations during the propagation of optical radiation and as applied to an optical refraction, and instrument in atmospheric optics for the study of intensity fluctuations during the propagation of investigation of a mirage detection. optical radiation and as applied to an optical refraction, and investigation of a mirage detection. 2. Method 2. Method 2.1. ILS Specification 2.1. ILS for for Direction Direction Specification Figure 11 shows shows the the optical optical scheme scheme of of ILS, ILS, implementing implementing this Figure this method method for for one-dimensional one-dimensional case. case. Using beam beam splitter splitter and and mirror, mirror, laser laser beam beam with with polarization polarization along along OX OX axis axis is is split split into into two two parallel parallel Using beams with with identical identical apertures. apertures. One passes through through aa beams One of of the the beams, beams, after after being being reflected reflected by by mirror, mirror, passes ˝ polarizer, which turns the polarization plane of the beam through 90, and arrives at piezoelectric polarizer, which turns the polarization plane of the beam through 90 , and arrives at piezoelectric scanner (PS). (PS).The Thesecond secondbeam beam passes through electro-optic modulator (EOM) 1 operating the scanner passes through electro-optic modulator (EOM) 1 operating on theonbasis basis of longitudinal electro-optic effect, through /4 plate (quarter-wave plate), and the second of longitudinal electro-optic effect, through λ/4 plate (quarter-wave plate), and the second (analogous (analogous the first) electro-optic thebeam second beamthrough passes through the polarizer, to the first) to electro-optic modulator.modulator. Then, theThen, second passes the polarizer, which which transmits radiation with polarization along the OY alsoatarrives at the PS mirror. transmits radiation with polarization along the OY axis, andaxis, also and arrives the PS mirror. Sawtooth Sawtooth voltage from sweep-frequency with frequency is to EOM. This voltage from sweep-frequency generator generator (SFG) with(SFG) frequency Ω is applied toapplied EOM. This voltage is voltage to is applied to EOMand 1 directly to π/2 EOMphase 2 viashifter; /2 phase shifter; thus, the voltages at EOM applied EOM 1 directly to EOMand 2 via thus, the voltages at EOM 1 and EOM 21 and EOM 2 have the phaseofdifference of /2. have the phase difference π/2.
Figure 1. 1. The The optical optical scheme scheme of of interferometric interferometric laser laser scanner scanner for Figure for direction direction specification. specification.
A radiative component with polarization along the OY axis is frequency shifted by amount amount Ω relative to the initial laser beam. Thus, two parallel laser beams with identical polarizations arrive at PS represents representsaamirror, mirror,mounted mountedononpiezoceramic piezoceramic plate. The principle work of PS is based PS. PS plate. The principle of of thethe work of PS is based on on the use of reverse piezoelectric effect that can be identified in the deformation of crystal placed the use of reverse piezoelectric effect that can be identified in the deformation of crystal placed into the into the field electric field at the orientation certain orientation of the field and direction of electric electric at the certain of lines of oflines forceof offorce the field and direction of electric axis ofaxis the of the crystal analogously to the method of a beam forming device, as described in previous work [1]. crystal analogously to the method of a beam forming device, as described in previous work [1]. When When control voltage is applied to PS, the parallel laser beams scan in the plane, perpendicular to the control voltage is applied to PS, the parallel laser beams scan in the plane, perpendicular to the plane, plane, containing the axes optical axes beams. of these beams. control forisfeeding is applied containing the optical of these The controlThe voltage for voltage feeding PS appliedPS directly from directly from SFG, which feeds EOM. Then, the scan angle is directly proportional to the frequency SFG, which feeds EOM. Then, the scan angle φ is directly proportional to the frequency Ω. Because of . Because divergence, of diffraction divergence, beams will start overlap at a certain range from PS,will where diffraction the beams willthe start to overlap at atocertain range from PS, where there be there will be anpattern interference pattern with amplitudeofmodulation of radiation with an interference with amplitude modulation radiation with frequency Ω. frequency . The optical signal of POS is accepted by photodetector device (PD) on moving object, consisting of objective, interference and polarization filters, and photodetector with power supply and
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The optical signal of POS is accepted by photodetector device (PD) on moving object, consisting of objective, interference and polarization filters, and photodetector with power supply and frequency meter, which measures the frequency Ω. The onboard photoreceiver device records and extracts the frequency of the recorded signal Ω, which is uniquely related to the direction toward PS. The positions of the laser beams at the point of space Q (Figure 1) are determined by the PS statement controlled with electrical signal of Ω frequency. By means of optical amplification, the change of light beam direction to 1 s of arc is accessible. Hence, setting the direction to within 1 s of the arc is not technically challenging. As such, the accuracy of measurements may exceed at least 1/5000 s of the arc. This unveils wide opportunities in application of the method in both the research and engineering. In particular, it may be valuable in optical refraction spectroscopy, in photothermal deflection spectroscopy for controlling the slowly changing transient processes in gaseous and liquid media and plasma, in atmospheric optics for studying the intensity fluctuations during the propagation of optical radiation, in investigation of mirage detection and reflectance techniques, in geodesy, optical profilometry, ranging, and navigational course detection. At present, this method is used for direction control at high accuracy, with the capabilities of the techniques being limited by the sizes of laser beams. The method of beam forming device (BFD) has essentially the best capabilities in terms of resolution, because the resolution is determined by interference bandwidth. 2.2. Distortions of ILS-Formed Interference Pattern We assume the following scheme of PS and PD of ILS: the sources of laser radiation, located, ( respectively, at points t0, ρ1 u and 0, ρ2 , emit collimated laser beams parallel to each other and to the OX-axis in the direction of positive x values; and PD is located at point Q (Figure 1) with coordinates tx, ρu. We adopt that the quantity ρtr means the vector of spacing of radiative sources, producing the interference pattern. Because the two laser beams that are used to form the interference pattern are obtained by splitting one initial laser beam (see Figure 1), in the initial plane x = 0 the function of second-order mutual coherence of the field of each of these laser beams, representing the Gaussian partially coherent beam, has the following form: ´ ¯ ! ”` ) ˘2 ` ˘2 ı ´ 2 ¯ ` 1 ˘2 Γ2 x “ 0, ρ1 , ρ2 ; ρ j , ρ j “ E02 exp ´ ρ1 ´ ρ j ` ρ2 ´ ρ j { 2 a0 ´ ρ ´ ρ2 {ρ2k (1) where E0 is the initial amplitude at the optical beam axis; a0 is the initial beam radius; ρk is the spatial coherence radius of the initial field; k “ 2 π{λ, λ is the wavelength of the optical radiation in vacuum; ρ1 , ρ2 are the position vectors of the observation points; j “ 1, 2. A similar formula also holds for second-order mutual coherence function of the fields of these two laser beams: ´ ¯ r2 x “ 0, ρ1 , ρ2 ; ρ j , ρ j1 Γ " „ * ´ ¯2 ` ¯ı2 (2) ` 1 ˘2 ˘ ”` 1 ˘ ´ 2 2 2 2 2 “ E0 exp ´ ρ ´ ρ j ` ρ ´ ρ j1 { 2 a 0 ´ ρ ´ ρ j ´ ρ ´ ρ j1 {ρk where j ‰ j1 , and j, j1 “ 1, 2. We consider that, in the formation of the second radiative source, it is admissible to assume that its coherence properties are totally identical to the initial source and are just shifted in space by the amount ρtr . PD is the square-law detector that responds to the power of incoming radiation, the signal of which can be represented as ir px, ρq “ η0 I px, ρq, where η0 is the quantum efficiency coefficient of PD; I px, æq is the instantaneous value of intensity of interference pattern at the detector location, which has the form: I px, ρq “ U1 px, ρq U1˚ px, ρq ` U2 px, ρq U2˚ px, ρq ` 2Re rU1 px, ρq U2˚ px, ρqs
(3)
where Uj px, ρq is the field of optical wave of one source; j “ 1, 2. Suppose that ρ1 “ ´ ρtr {2, and ρ2 “ ρtr {2.
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The second-order mutual coherence function of fields of two partially coherent Gaussian beams of optical radiation for boundary Equations (1) and (2), written with the use of the extended Huygens-Fresnel principle in the case of square-law approximation for the function, which describes the effect of random inhomogeneities of the medium, has the following form [4]: » xUj px, ρq Uj˚1 px, ρqy “
E02 a20 a2 pxq
`
ρ ´ ρj — exp –´ 2 a2 pxq
¯2
´
˘2
ρ ´ ρ j1 ´
2 a2 pxq ´ ¯2 fi ´ ¯ 1 ρ ´ ρ j j δ pxq ffi `i 2 ρ2j ´ ρ2j1 ´ fl 2 a pxq ρ2c pxq
´i
¯ δ pxq ´ 1 ρ ´ ρ ρ j j a2 pxq (4)
˙ 4 2 2 1{2 where a pxq “ a0 1 ` 1` ` a0 {ρ0 is the mean radius of the 3 ˘ “ ‰ ´1 ` laser beam; δ pxq “ Ω0 1 ` a20 {ρ2k is the geometrical factor; δ pxq{ k a2 pxq “ is the difference between curvatures of laser beam wavefronts; ρc pxq ¸ff « ˜ ¸ ff+1{2 #« ˜ 2 2 2 2 a a 4 a0 3 ρ0 2 2 31{2 ρ0 { Ω´ 1 ` 02 ` is the radius of mutual 1 ` Ω´ 1 ` 02 ` 0 0 2 3 ρ0 4 ρ2k ρk ρk coherence of partially coherent laser beams; Ω0 “ k a20 {x is the Fresnel number of emitting aperture; ` ˘´3{5 ρ0 “ 0.3642 Cε2 k2 x is the coherence radius of plane optical wave in the turbulent atmosphere; Cε2 is the structure parameter of atmospheric turbulence; and j, j1 “ 1, 2. Using Equations (3) and (4), we obtain the formula for the mean value of intensity of interference pattern: „
ˆ
2 Ω´ 0
xI px, ρqy “ 2
a20 {ρ2k
„ " „ „ „ * E02 a20 ρ2 ` ρ2tr {4 ρ2tr ρtr ρ δ pxq exp ´ cosh ` exp ´ cos ρ ρ (5) tr a2 pxq a2 pxq a2 pxq ρ2c pxq a2 pxq
Equation (5) can be used to formulate the conditions, restricting the choice of parameters of laser beams and ILS scheme. The linear dimensions lint pxq of the region, where the interference pattern Equation (5) had formed, provided that a pxq yy ρtr , have a value approximately equaling the current laser beam diameter: lint pxq – 2 a pxq (6) The band maxima of interference pattern are at the points ρmax , determined (see Equation (5)) from equation of the form: δ pxq{a2 pxq ρtr ρmax “ 2 n π, where n “ 0, ˘ 1, ˘2, . . .; while the band minima are at points ρmin determined from δ pxq{a2 pxq ρtr ρmin “ p2 n ` 1q π, n “ 0, ˘ 1, ˘2, . . .. Let ρ || ρtr , then ρmax || ρmin || ρtr and ρmax “ 2 n π a2 pxq{ rδ pxq ρtr s, ρmin “ p2 n ` 1q π a2 pxq{ rδ pxq ρtr s, while the width of the interference band ∆lint pxq can be evaluated from the following formula: ∆lint pxq “ 2 |ρmax ´ ρmin | “ 2 π a2 pxq{ rδ pxq ρtr s
(7)
On the other hand, the contrast of distorted interference pattern, determined from the mean intensity (at ρmax « ρmin « ρ), is equal to: ν “
xI px, ρmax qy ´ xI px, ρmin qy – xI px, ρmax qy ` xI px, ρmin qy
" „ *´1 „ ρ2 ρtr ρ cosh 2 exp ´ 2 tr a pxq ρc pxq
(8)
It is evident that, in order for ILS to be operable, at least one complete interference band should be in the field of interference pattern; therefore, Equations (6) and (7) make it possible to formulate the
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Moreover, the contrast of the mean interference pattern near its center (when
cosh tr a 2 x 1 ) will be satisfactory until the coherence radius of optical field at the condition: lint pxq ě ∆lint pxq , which is fulfilled when the range between the sources of laser radiation observation point exceeds the value of transversal shift of the beams; therefore, from Equation (8) we satisfies the condition: obtain the following inequality: ρtr ě π a pxq{δ pxq (9)
ln c x (10) tr interference Moreover, the contrast of the mean pattern near its center (when “ ‰ 2 cosh Summing ρtr ρ{a pxq 1) will beEquations satisfactory until the radius of field form: at the two–conditions, (9) and (10), wecoherence obtain the formula of optical the following observation point exceeds the value of transversal shift of the beams; therefore, from Equation (8) we ax x tr ln c x (11) obtain the following inequality: a (10) ρtr ď ´ ln pνq ρc pxq In order for Equation (11) to be fulfilled, the right-hand side of the inequality should exceed the Summing conditions, Equations (9) andthe (10), we obtain the formula of the following form: left-hand side, two making it possible to formulate condition restricting the initial sizes of the laser beams a0 : a π a pxq{δ pxq ď ρtr ď ´ ln pνq ρc pxq (11) ln k 2 a04 a2 a02 1 1 right-hand 1 02the 1 side to be fulfilled, In order for Equationa0(11) should exceed(12) the 2 2 of the2 inequality 3 0 k 4 k x2 k left-hand side, making it possible to formulate the condition restricting the initial sizes of the laser beams a0 : ˜ ¸ g ˜ ¸ a f 2 2 3. Results f 1 a0 a ´ ln pνq 1 k2 a40 a0 ď 1 ` 2 {e 2 1 ` 02 ` (12) π ρk 3 ρ0 ρk 4 ρ2k x2 3.1. Choice of Parameters of ILS Optical Scheme Figure 2 presents the behavior of implicitly specified function a0 x , defined by Equation (12). 3. Results Values of a0 , lying on the curve that graphically represents the solution of Equation (12), correspond 3.1. Choice of Parameters of ILS Optical Scheme to the condition “ ” in Equation (12). The a0 values, lying below the corresponding curve, Figure 2topresents the behavior of Equation implicitly(12). specified function by Equation (12). 0 pxq, defined All curves in aFigure 2, plotted for different correspond the condition “ ” in Values of a0 , lying on the curve that graphically represents the solution of Equation (12), correspond to parameters of the problem, , k , C 2 , and , are within grey-colored region. Figure 2 shows an the condition ““” in Equation (12). The a0 values, lying below the corresponding curve, correspond assemblage of curves in the form of a nomogram for the initial laser beam radii a0 . This figure shows to the condition “x” in Equation (12). All curves in Figure 2, plotted for different parameters of the 2 two regions (not filled) that allow formation of a stable interference pattern. In filled (grey) in is problem, λ, ρk , Cε , and ν, are within grey-colored region. Figure 2 shows an assemblagearea of curves on the not possible formation of a stable pattern. The effect of the wavelength of optical radiation the form of a nomogram for the initial laser beam radii a0 . This figure shows two regions (not filled) a0 was valueallow of the initial size laser beams estimated by considering the values of formation in the range that formation of of a stable interference pattern. In filled area (grey) is not possible of m to 1.55 m (at the same time, the other parameters were as follows: 2 cm, from 0.51 a stable pattern. The effect of the wavelength of optical radiation λ on the value of the initial size of k 2 a was −13 −2/3 laser beams estimated by considering the values of λ in the range from 0.51 µm to 1.55 0.1, C 0 10 m ). Correspondingly, the contrast of the interference pattern varied fromµm 0.1 2 “ 10´13 m´2{3 ). (at the same time, the other parameters were as follows: ρ “ 2 cm, ν “ 0.1, C 2 k ε −13 −2/3 to 0.5 (at 1.55 m, k 2 cm, C 10 m ); and the structure parameter of the atmospheric Correspondingly, the contrast of the interference pattern ν varied from 0.1 to 0.5 (at λ “ 1.55 µm, 2 −16 m−2/3 1.55 turbulence 2C ´13 m´2from {3 ); and turbulence varied 10the to parameter 10−13 m−2/3 (at atmospheric m, k C2 2varied cm, ρ structure of the k “ 2 cm, Cε “ 10 ε 0.1). from 10´16 m´2{3 to 10´13 m´2{3 (at λ “ 1.55 µm, ρk “ 2 cm, ν “ 0.1).
Figure Figure2.2.The Thenomogram nomogramfor forchoosing choosingthe theinitial initiallaser laserbeam beamradii radii aa00. .
Based on data presented in Figure 2, the initial laser beam radii a0 can be chosen to be 1…2 mm for pathlengths in the range from 1 m to 10 km. Equation (12) must be solved (using the condition “ ”) with respect to k to determine the minimum acceptable level of the spatial Sensors 2016, 16, 130
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coherence radius of initial field. The data thus obtained are presented in Figure 3 in the form of grey-colored region (in these Based on data in Figure 2, the initial laserfrom beam 0.1 radiitoa00.5, can and be chosen to be 1 . . . 2parameter mm varied the structure of estimates, the contrast ofpresented interference pattern for pathlengths in the 2range from 1 m to 10 km. Equation (12) must be solved (using the condition atmospheric turbulence C varied from 10−16 m−2/3 to 10−13 m−2/3); and any curve, obtained in solution ““”) with respect to ρ to determine the minimum acceptable level of the spatial coherence radius of k
of Equation initial(12), field.falls within this region. The data position thus obtained aregrey-colored presented in Figure 3 inpresented the form ofingrey-colored region (inthat these The shape and of the region, Figure 3, indicate the initial estimates, the contrast of interference pattern ν varied from 0.1 to 0.5, and the structure parameter coherence of optical sources plays the key role (in selecting the parameters of single beams) along of atmospheric turbulence Cε2 varied from 10´16 m´2{3 to 10´13 m´2{3 ); and any curve, obtained in propagation paths up to 300 m in length, while atmospheric turbulence is critical when x 300 m. solution of Equation (12), falls within this region. Figure 3 shows assemblage ofofcurves in the form of apresented nomogram for choosing coherence Thean shape and position the grey-colored region, in Figure 3, indicatethe thatspatial the initial coherence of optical sources plays the key role (in selecting the parameters of single beams) along radius k of the initial field of laser beams that allow the formation of a stable interference pattern. propagation paths up to 300 m in length, while atmospheric turbulence is critical when x > 300 m.
Here also, as in the previous figure, showing the two areas (not filled) that allow formation of a stable Figure 3 shows an assemblage of curves in the form of a nomogram for choosing the spatial coherence interference pattern. Figure 3 demonstrate thatthe theformation laser beam coherence level,pattern. comparable to radius ρk of theData initialinfield of laser beams that allow of a stable interference the initial coherence of 1…2 cm, is reached the(not paths with x of 300 m. Thus, it Here also, as in radius the previous figure, showing the twoon areas filled) that the allowlengths formation a stable interference pattern. Data in Figure 3 demonstrate that the laser beam coherence level, comparable can be further assumed that k 1…2 cm. For laser beams with these parameters, the angular beam width
0 2
to the initial coherence radius of 1 . . . 2 cm, is reached on the paths with the lengths x ě 300 m. 0 in the region, where there already exists the directional diagram equaling Thus, it can be further assumed that ρk “ 1 . . . 2 cm. For laser beams with these parameters, the 2 1 2 angular beam2 width in the region, where there already exists the directional diagram equaling k a0 1 a0 ` k ψ,0 must ˘1{2 not exceed 1’. 2 2 ψ0 – 2{ pk a0 q 1 ` a0 {ρk , must not exceed « 1’.
3. The nomogram for choosing the spatial coherence radius of the initial field of laser beams ρk . Figure 3.Figure The nomogram for choosing the spatial coherence radius of the initial field of laser beams k .
Euqation (11) for known parameters of the optical beams makes it possible to choose the distance between radiative sources ρtr . Figure 4 presents the behavior of the functions:
Euqation (11) for known parameters beams makes it possible to choose the distance g of the optical ¸ ˜ ¸ ˜ f between radiative sources tr . Figure 4 presents k a30 f a20 a20 x2 the behavior 4 a20of the functions: f 1 pxq “ π
f1 x (grey-colored region) and g« f f a (grey-coloredf 2 region) and pxq “ ´ ln pνq e 1 `
x
e1 `
k a03 x x2 k2 a40
1 ˜
k2 a40
x2 k 2 a04
1`
ρ2k
`
3 ρ20
{ 1`
a2 a2 1 02 4 02 k 3 0
a2 4 a20 1 ` 02 ` 3 ρ20 ρk
¸ff «
1 x2 { 3 ρ20 k2 a40
a 2 4 a02 x2 f 2 x ln 1 2 4 1 02 2 3 k a 0 0 k
(13)
ρ2k
a2 1 02 k ˜
a2 1 ` 02 ρk
1 x2 2 2 4 3 0 k a0
¸
(13) 1 ` 4 ρ2k
ff (14)
a2 1 1 02 k 4 2k
(14)
(region in light grey), calculated for different parameter values of the problem. Figure 4 shows nomograms for choosing the values tr , at which the formation of interference pattern is a stable.
of wavelength of the optical radiation ( in the range from 0.51 m to 1.55 m for a0 2 mm, k 2 cm, 0.1, C2 10−16 m−2/3), and the effect of the contrast of interference pattern ( in the range from 0.1 to 0.5 for 1.55 m, a0 2 mm, k 2 cm, C2 10−16 m−2/3), as well as the effect of 2 structureSensors parameter 2016, 16, 130 of atmospheric turbulence C
( C 2
−16 −2/3 to in the range from 10 7 of 10 m
for 2016, 16,1.55 10−14 m−2/3Sensors 130 m, a0 2 mm, k 2 cm, 0.1).
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(region in light grey), calculated for different parameter values of the problem. Figure 4 shows In estimating the dimensions of both regions, presented in Figure 4, we quantified both the effect nomograms for choosing the values ρtr , at which the formation of interference pattern is a stable. of wavelength of the optical radiation ( in the range from 0.51 m to 1.55 m for a0 2 mm, k In estimating the dimensions of both regions, presented in Figure 4, we quantified both the effect −16 m−2/3), and the effect of the contrast of interference pattern ( in the range 0.1, of 2 cm, C2the 10 of wavelength optical radiation λ (λ in the range from 0.51 µm to 1.55 µm for a0 “ 2 mm, 2 “ 10´16 m´2{3 ), and the effect of the contrast −16 interference ρfrom “ 2 cm, ν “ 0.1, C pattern ν (ν in the 0.1 to 0.5 for ε 1.55 m, a0 2 mm, k 2 cm, C2 10of m−2/3), as well as the effect of k 2 ´ 16 ´ 2 { 3 range from 0.1 to 0.5 for λ “ 1.55 µm, a “ 2 mm, ρ “ m ), as well as the effect 22 cm, C 2 ε “ 10 0 −16 −2/3 k structure parameter of atmospheric turbulence C ( C in the range from 10 m to of structure parameter of atmospheric turbulence Cε2 (Cε2 in the range from 10´16 m´2{3 to 10´14 m´2{3 −14 −2/3 10 m for 1.55 m, a 2 mm, k 2 cm, 0.1). for λ “ 1.55 µm, a0 “ 2 mm, ρ0k “ 2 cm, ν “ 0.1).
Figure 4. The nomogram for choosing the spacing between optical axes of laser beams tr . The greycolored region shows the variability range of the function f1 x , and the region in light grey depicts the variability range of the function f 2 x . Figure 4. The Thenomogram nomogramforfor choosing spacing between optical of beams laser beams ρtr . greyThe . The Figure 4. choosing thethe spacing between optical axes axes of laser
tr trvariability Considering that the spacing of radiative that produce pattern must be grey-colored region shows the range ofsources the function f 1 pxq , and theinterference region in light grey colored region shows the variability range of the function f1 x , and the region in light grey depicts . depicts the variability range of the function f 2 pxq smaller f x and than the values of the function f 2 x , data larger than the values of the function the variability range of the function 1 f x . 2
of Figure 4 for pathlengths x from 20 m to 10 km can be used to estimate the admissible values of tr Considering that the spacing ρtr of radiative sources that produce interference pattern must be
Considering that the spacing of radiative sourcesradiative that produce interference pattern mustused be tr that were in the range from cm to 5 cm. is justtrthese spacings sources larger than 1the values of theItfunction f 1 pxq and smallerof than the values of the function f 2 pxq, data of below larger than the values of the function f1 x and smaller than the values of the function f 2 x , data Figure for pathlengths x from 20 m to 10 km can be used to estimate the admissible values of ρtr in to estimate ILS4 operability. x from 20 m to 10spacings km can be radiative used to estimate admissible values of to of for pathlengths tr theFigure range 4from 5 cm. It is just these sourcesthe below tr that were used Figure 5 presents1 cm thetocalculations of the averageofintensity of ILSρinterference pattern (Figure 1) that were used below in the range from 1 cm to 5 cm. It is just these spacings of radiative sources estimate ILS operability. tr −16 m−2/3 and x 10 km for different distances between centers of two identical collimated at C2 to10estimate Figure ILS 5 presents the calculations of the average intensity of ILS interference pattern (Figure 1) at operability. ´2{3 and the 10´16 m x “calculations 10 km for different distances between Figure 5 presents of the intensity of centers ILSainterference patterncollimated (Figure k 1) 2 1.55 m, 2 identical mm, GaussianCε2 “partially coherent beams ( average 0 of two
cm):
2 −16 −2/3 and x beams partially 1.55 µm, distances a0 “ 2 mm, ρk “ 2 centers cm): (a)of ρtrtwo “ 1identical cm, (b) ρcollimated at 1Ccm, 10 km(λfor tr “ 2 cm, 2 cm,10 (c) “trdifferent 3 cm, (b) mtrcoherent (d) between (a) tr Gaussian tr 4 cm, and (e) tr 5 cm. Panels (c) ρ “ 3 cm, (d) ρ “ 4 cm, and (e) ρ “ 5 cm. Panels in Figure 5a–e represent the color tr cm): Gaussian partiallytr coherent beamstr ( 1.55 m, a0 2 mm, k 2 polar Figure 5a–e represent the colorofpolar contour plots (100 shades of two colors). contour plots (100 shades two colors). (a) tr 1 cm, (b) tr 2 cm, (c) tr 3 cm, (d) tr 4 cm, and (e) tr 5 cm. Panels in
Figure 5a–e represent the color polar contour plots (100 shades of two colors).
(a) (a)
(b) (b) Figure 5. Cont. Figure 5. Cont. Cont. Figure 5.
in
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(c)
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(d)
(e) Figure 5. The average intensityof of interferometric laser scanning (ILS) interference for different pattern for Figure 5. The average intensity interferometric laser scanning (ILS)pattern interference values of ρtr : (a) 1 cm; (b) 2 cm; (c) 3 cm; (d) 4 cm; and (e) 5 cm. different values of tr : (a) 1 cm; (b) 2 cm; (c) 3 cm; (d) 4 cm; and (e) 5 cm.
Figure 5 shows the five options of the interference pattern where the method is workable in
Figure principle. 5 showsThe the fiveillustrates options the interference pattern where theit represents. method is figure theof real (calculated) interference pattern and that Theworkable in Figure 5 corresponds to the the high real visibility at the weak turbulence and support the method performance: principle. The figure illustrates (calculated) interference pattern and that it represents. The Figure 5 corresponds high visibility at the weak turbulence support 1. For ρtr “ 1to cm,the the interference pattern starts to form at separation of 30 m and from the sources ofthe method performance: the optical radiation, and becomes finally formed at the range of 300 m, comprising three bright 1.
2.
3.
bands, with central band becoming most intense starting from 40 m, and with two sideward symmetricallypattern about thestarts central to band, being For tr 1bands, cm, located the interference form atidentical. separation of 30 m from the sources of 2. For ρtr “ 2 cm, the interference pattern starts to form at the range of 30 m from the sources of the optical optical radiation, and becomes finally formed at the range of 300 m, comprising three bright radiation, and becomes finally formed at the range of 300 m. In this case, the pattern has bands, with central most intense from from 40 m, and two sideward seven intenseband bands;becoming the central band becomes moststarting intense starting 80 m; andwith sideward bandssymmetrically are pairwise identical. bands, located about the central band, being identical. For ρ “ 3 cm, the interference pattern starts form at theat range 30 m from the m sources of the 3. tr For tr 2 cm, the interference pattern startsto to form theofrange of 30 from the sources of optical radiation, and becomes ultimately formed at the range of 300 m, comprising 11 intense optical radiation, and finally formed at thestarting rangefrom of 300 m. In this case, the pattern has bands, with thebecomes central band becoming the brightest 120 m. 4. For ρbands; the central interference pattern starts to form at the range ofstarting 40 m fromfrom the sources of the seven intense the band becomes most intense 80 m; and sideward tr “ 4 cm, optical radiation, and becomes ultimately formed at the range of 400 m, comprising 15 intense bands are pairwise identical. bands, with the central band becoming most intense starting from 170 m. For tr5. 3Forcm, the interference pattern starts to form at the range of 30 m from the sources of ρtr “ 5 cm, the interference pattern starts to form at the range of 40 m from the sources of the radiation,and and becomes ultimately formed formed at the rangeatof the 500 m,range comprising 19 intense the opticaloptical radiation, becomes ultimately of 300 m, comprising bands, with central band becoming most intense starting from 250 m.
4.
11 intense bands, with the central band becoming the brightest starting from 120 m. Contrast For tr3.2. 4 cm, ofthe pattern starts to form at the range of 40 m from the sources of ILSinterference Interference Pattern in the Turbulent Atmosphere
5.
contrast of ILS can be estimated approximate formula, the opticalTheradiation, andinterference becomespattern ultimately formedaccording at the torange of 400 m, comprising obtained from Equation (8). Figure 6 presents estimates of the contrast of ILS interference pattern 15 intense bands, with the central band becoming most intense starting from 170 m. For tr 5 cm, the interference pattern starts to form at the range of 40 m from the sources of
the optical radiation, and becomes ultimately formed at the range of 500 m, comprising 19 intense bands, with central band becoming most intense starting from 250 m.
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(Figure 1) for all values of structure parameter of atmospheric turbulence, realizable in near-water atmospheric layer [5,6]. Figure 6 gives estimates of the visibility of the interference patterns at different levels of turbulence. It was assumed that laser radiation propagated along horizontal path at height of 10 . . . 20 m above water surface. The color (40 colors) contour plots, presented Sensors 2016, 16, 130 9 of 10 in Figure 6a–e, demonstrate the dependence of the contrast of ILS interference pattern (ν) on the laser beam propagation path length (x) and on the value of the structure parameter of atmospheric turbulence ( C 2 ) at the wavelength of the optical radiation 1.55 m for different ranges between turbulence (Cε2 ) at the wavelength of the optical radiation λ “ 1.55 µm for different ranges between a0 “22mm, the centers the centers of of two two identical identical collimated collimated Gaussian Gaussianpartially partiallycoherent coherentbeams beams( (a mm, ρkk “ 22 cm): cm): 0 11cm, cm,(b) (b)ρtrtr“2 2cm, cm,(c)(c)ρtr“ 3 cm, 4 cm, 5 cm. (a) ρtrtr “ 3 cm, (d)(d) ρtr “tr 4cm, andand (e) ρ(e) “ 5 cm. tr tr tr
(a)
(b)
(c)
(d)
(e) Figure 6. The Thecontrast contrastof of interference pattern for different distances between the of centers of Figure 6. ILSILS interference pattern for different distances between the centers emitting tr (b) : (a)2 1cm; cm;(c)(b) 2 cm; cm;and (d)(e) 4 cm; and (e) 5 cm. emitting beam apertures beam apertures ρ : (a) 1 cm; 3 cm; (d)(c)4 3cm; 5 cm. tr
The red-colored regions in Figure 6a–e show the laser beam propagation path lengths x for The red-colored regions in Figure 6a–e show the laser beam propagation path lengths x for different structureparameters parameters of atmospheric the atmospheric turbulence C 2high , when high ofcontrasts of ILS different structure of the turbulence Cε2 , when contrasts ILS interference interference in the turbulent take atmosphere pattern in thepattern turbulent atmosphere place take place 4. Conclusions Conclusions
At higher contrasts of interference pattern, ILS proves to be operable in a narrower range of spacings of of of optical radiation thatthat produce the interference pattern. The closer sources ofthe thesources sources optical radiation produce the interference pattern. Thethe closer the of optical to each other, lower contrast of the recorded interference pattern,pattern, and, at sources ofradiation optical radiation to eachthe other, thethe lower the contrast of the recorded interference last, longer wavelength of the optical the greater ILS operability region. For the and, the at last, the the longer the wavelength of the radiation, optical radiation, the is greater is ILS operability region. wavelength of the optical radiation λ “ 1.55 at spacings of the sources ρtr from1tr cm to 51 cm µm, 1.55 m, at spacings of the sources from For the wavelength of the optical radiation ´15 ´2{−15 3 under conditions of the mean turbulence intensityintensity value of 10 in the m−2/3, often most realized often realized to 5 cmthe under the conditions of the mean turbulence value m of 10 , most in the maritime and coastal atmosphere at midlatitudes, the operability range of the device may reach 10 km and longer, depending on the geometric positions of optical elements. Acknowledgments: The Ministry of Education and Science of the Russian Federation (Project No. 14.604.21.0042) supported this work. Author Contributions: All authors contributed equally to this work. Gennady Kaloshin initiated the project, reviewed the paper and contributed to the interpretation of the results. Igor Lukin developed algorithms for
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maritime and coastal atmosphere at midlatitudes, the operability range of the device may reach 10 km and longer, depending on the geometric positions of optical elements. Acknowledgments: The Ministry of Education and Science of the Russian Federation (Project No. 14.604.21.0042) supported this work. Author Contributions: All authors contributed equally to this work. Gennady Kaloshin initiated the project, reviewed the paper and contributed to the interpretation of the results. Igor Lukin developed algorithms for calculation, processed and interpretation the data. Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations The following abbreviations are used in this manuscript: ILS PS EOM SFG PD BFD
Interferometric laser scanning Piezoelectric scanner Electro-optic modulator Sweep-frequency generator Photodetector Beam forming device
References 1. 2. 3. 4. 5. 6.
Kaloshin, G.A.; Lukin, I.P. An acousto-optical sensor with high angular resolution. Sensors 2012, 12, 3739–3746. [CrossRef] [PubMed] Ishimaru, A. Wave Propagation and Scattering in Random Media; IEEE Press: Piscataway, NJ, USA, 1997. Wolf, E. Introduction to the Theory of Coherence and Polarization of Light; Cambridge University Press: Cambridge, UK, 2007. Andrews, L.C.; Phillips, R.L. Laser Beam Propagation through Random Media, 2nd ed.; SPIE Press: Bellingham, WA, USA, 2005. Pasricha, P.K.; Reddy, B.M. Evaluation of the structure parameter Cn2 over the sea surface. IEEF Proc. 1990, 137, 384–386. Forand, J.L.; Dion, D. Predictive comparisons of marine boundary-layer models. Proc. SPIE 1996, 2828, 129–140. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
