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Correction and correlation formation of probing field in fiber laser Doppler anemometers and vibrometers
Optics and Lasers in Engineering 32 (2000) 593}604
Correction and correlation formation of probing "eld in "ber laser Doppler anemometers and vibrometers L. Yarovoi *, A. Gnatovskii, N. Medved Training and Science Center **Physical and Chemical Materials Science++ of Kyiv University by Taras Shevtchenko and National Academy of Sciences of Ukraine, 64, Volodimirskaya St., Kyiv 252017, Ukraine Institute of Physics of National Academy of Science of Ukraine, 46, prosp. Nauki, Kyiv 191028, Ukraine
Abstract In this work we o!er a method of "eld formation with given spatial features at the output of multimode "ber sensors using interference phenomena. The method consists of a "eld two-stage correlation transformation at a "ber output. Transformation is realized by a phase modulator and a hologram. The developed method is practically realized in the "ber Doppler anemometer and vibrometer scheme. Features of sensors are discussed in this work. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Correlation optics; Laser Doppler vibrometry and anemometry; Fiber sensors; Correction of laser beams
1. Principle of two-stage correlation formation of periodic 5elds Sensors of physical values using interference phenomena represent one of the practical applications of coherent optics. This class of instruments includes laser Doppler anemometers [1], vibrometers [2], as well as devices using optical photomixing for the selection of useful signal. Note one general particularity of these devices: for e$cient work they require "elds with uniform or required amplitude and phase distribution. So signi"cant di$culties appear, when it is necessary to insert multimode "bers (MMF) into optical channels of sensors because the output "eld of MMF is modulated by random and nonstationary set of speckles [3]. * Correspondence address: P.O. Box 164, Kyiv 03191, Ukraine. Fax: #380-44-266-5108. E-mail address: [email protected], [email protected] (L. Yarovoi) 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 0 5 - 1
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We believe that the usage of the wave front two-stage correlation transformation method [4], or a phase correction method, which allows to convert statistical random "eld of initial beam into "eld with the required spatial distribution of complex amplitude, is the universal solution to these problems. Formation of a "eld with a given periodic amplitude distribution is explored in this work. Results obtained for periodic "elds have self-maintained importance [5] and would serve as a background for studying more complicated nonperiodic distributions on the basis of harmonic analysis. Possibility of using formed periodic "elds for the velocity measurement of objects and #ows is shown with the "ber laser Doppler anemometer (LDA) as an example. Correlating transformation of MMF "eld modulated by specially synthesized spatial modulator 2 is the principle of a wave front correction at MMF output 1 (see Fig. 1). This transformation is realized by hologram 4 to which modulated "eld MMF is written beforehand. We will conduct the calculation of spatial distribution of "eld in the sensor measuring volume with corrector in paraxial approximation. We will show that under the observance of at certain constructive conditions observance the sensor probing "eld is determined "rstly by function of the modulator transmission (concretely by its autocorrelation function), while modulation by MMF speckle structure is vastly smoothened. Let phase transmission u(x , y ) of spatial modulator 2 will be assigned by periodic function: t(x , y )"e\ P" q exp+2p inly ,. L L\
(1)
Fig. 1. 1-"ber, 2-spatial modulator; 3-focusing objective; 4-hologram; (x , y )-plane of the measurements.
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In the considered scheme the phase spatial modulator 2 and focusing objective 3 are located on one plane: at the distance l from "ber end and at the distance l from the hologram 4 plane accordingly. Let e(x, y), be the "eld distribution at MMF end, then [6]: w(x , y )J+[e(x, y)*h (x, y)]*t (x , y )t(x , y ),*h (x , y ), J * J
(2)
where h "(1/ijl)exp+ik(x#y)/2l, is the pulse characteristic of empty space, J t "(1/ijf)exp+!ik(x#y)/2f, is the phase transformation of the objective, f * is * the focal distance of objective 3, k * 2p/j, * * is the convolution operator. We will produce the calculation of integral (1) at condition 1/f"1/l #1/l . In this case objective 3 conjugates MMF end plane with the hologram plane. Writing (1) in explicit form and conducting consequent integrating on x , y , and then on x, y we will obtain in total:
l ikl (x #y ) w(x , y )"! exp l 2fl
l 2pl l ; q e !y # nl , !x L l l k L\
l (2p)l nl ) exp !2ily exp i . l 2k
(3)
This expression describes the "eld in the form of di!raction orders' set being spatial spectrum of modulator. Moreover, each order is an image of MMF end scaled l /l times. In expression (3) phase multiplicands describe a square-law deformation of wave front and of the modulator's various di!raction order direction changing and they are compensated by the hologram. Really, component of the hologram transmission function being responsible for "rst-order di!raction beam formation is determined by relation [6]
ik(x #y ) , t JwH(x , y )exp ! G 2R
(4)
where w (x , y ) is the MMF "eld on the hologram at its recording stage, R is curvature radius of a reference beam. Amplitudes distribution in the "rst di!raction order at the distance z can be presented in the form
ik(x #y ) u(x , y )J [w(x , y )wH(x , y )]exp ! 2R
*hX (x , y ).
(5)
Note that, as it follows from (5), in product wHw phase multiplicands cancel out. Now we shall "nd complex "eld amplitudes distribution in the plane (x , y ) located at the distance z"R from the hologram. Substituting (3) in to (5) and grouping the similar terms in arguments of phase multiplicands we will
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obtain after cancellation:
1 l x #y u(x y )J! exp+ikR , exp ik ijR l R l 2pl l ) q eH !y # nl , !x L l k l L\ l 2pl l x x #y y dx dy . ) e !y # nl , !x exp !ik l l k R (6)
We produce change of the variables in the integrated expression: l l 2pl x "!x ; y "!y # nl (7) l l k and re-group multiplicands of u(x , y ): l x #y 2pinl y l q exp ! u(x y )J exp ik l L R R L\ ikl [x x #y y ] dx dy . ) eH(x , y )e(x , y )exp (8) R l Then we will transform (8) by using a known relation from the Fourier transformation theory [6]:
IK \[eHe]PGH(!K ,!K ) G(!K ,!K ), (9) V W V W where IK \ is an operator of inverse Fourier transformation: IK [e(x, y)]PG(k , k ), V W while K "(k/l )x and K "(k/l )y are spatial frequencies. So, for the "eld V W u(x , y ) we will obtain: x #y 2pinl y l l k q exp ! exp ik u(x y )J L l l R R L\ k l k l ) GH ! x #x ,! y #y R R l l k k ;G ! x ,! y dx dy . (10) l l The integral on the right-hand side of expression (10) is a function of mutual spatial correlation of MMF "eld spectrums at the stages of the hologram recording and reconstruction. The sum being before the integral presents spatial autocorrelation function of the modulator transmission:
l tH t" q exp !2pinly . L R L\
(11)
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Omitting phase factor that does not in#uence upon the "eld distribution formation one can write expression (11) in symbolic form uJ(tH t)(GH G). (12) Let us analyze the obtained expression. The probing "eld at the output of the sensor is the represented structure modulated by interference grating with oscillating factor (tH t). As one can see from (11), this grating is de"ned only by the modulator function of the complex transmission and by constructive parameters of optical scheme, and does not depend on the "eld distribution at MMF end. For example, the modulator with rectangular phase relief, which is depicted in Fig. 2 (broken line), creates the "eld modulated by periodical autocorrelation function of the modulator (see Fig. 3). Periods of spatial harmonics of the probing "eld are determined by the relation R *y " , n"1, 22 . (13) L ll n The in#uence of MMF appears how the noise spatial signal at the sensor output that is expressed in term of the square of mutual correlation function of spatial frequencies' spectrum GH G (see (12)). One can study the nature of this noise by
Fig. 2. e(x, y)-"eld distribution on "ber end (solid line) (y)-depth of phase relief.
Fig. 3. The module of spatial modulator autocorrelation function for case l "R .
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writing expression (12) in space}frequency domain and taking into account expressions (9) and (11):
l l (14) qe K !2pl n ) eH K !2pl n W W L R R L\ here S(K )"I[;] is the spatial spectrum of the "eld at the sensor output. W Analysis of relation (14) shows that spatial spectrum of "eld at the output of the sensor with correlation transformer has the form of equidistant harmonics. An envelope of each harmonic is proportional to the product of spatial "eld distributions in the plane (x, y) at the stages of recording and hologram reconstruction. It is "nite sizes of "ber end that causes extension of interference "eld spectral components. Because of the light coherence, "elds e(x, y) and e (x, y) at MMF end have a speckle structure. Therefore, spatial harmonics of the "eld at the output of the sensor are modulated by random noise function. For noise reduction it is expedient "rst of all to record the hologram by laser beam or by single mode "ber. In this case in planes (x, y) the "eld distribution will be described by smooth function kind of e "E exp+!(x #y )/r ,, where r is approximately "ber core diameter. 1 S(K )J W 2p
2. Structure of doppler signal in LDA with correlator It is possible to use obtained results for LDA and for laser Doppler vibrometers "ber sensor. Let the particle of the #ow crosses the measuring volume with an interference periodic "eld in the plane (x , y ) at the velocity of v, as it is shown in Fig. 1, and scatters the radiation. The scattered radiation is modulated by periodical function with the frequency being proportional to the velocity <: 2pf "K<, where " K is the spatial frequency of the "eld. Scattered radiation is gathered by receiving "ber and directed to the photodetector. Suppose, the "eld at the sensor output is formed by means of a correlation transformer discussed above. Then, as follows from (12), the "eld intensity distribution in measuring volume is de"ned by the relation: I(y )J(GH G)(GH G)H¹(y ), (15) where ¹(y ) is a modulation function: 2pill y (n!m) $ (16) ¹(y )" qq exp ! L K R L\ Let the #ow to move along axis y , then y "
here
l S (f )"I[I(t)]J qq E f !
E"I[(GH G)(GH G)H]"e eH eHe. is the envelope of photocurrent harmonics.
(17)
(18)
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Fig. 4. The Doppler signal spectrum.
One can determine velocity from (17) as follows: f R (19) <" L , ll n where f is a frequency of the Doppler signal nth harmonic. L Fig. 4 presents the spectrum of the Doppler signal evaluated under relation (17) for di!ractive modulator with rectangular phase transmission and the "eld distribution of multimode "ber as shown in Fig. 2 3. Experimental results Fig. 5a represents the scheme of the experimental set. At the preliminary stage a correcting hologram 7 was recorded. Recording was performed by the radiation of helium}neon laser - 1 in the two wave convergent scheme. Object beam was formed by single-mode "ber 4 with a core diameter of 9 mm. After passing the "ber the radiation was directed to the periodic phase modulator 6 and then to the correcting hologram 7. The hologram was recorded by convergent spherical reference beam which was formed by collimator 8 and objective 9. We installed a similar polarization in interfering waves for raising di!raction e$ciency of the hologram by means of polarizer 10. The hologram recording was realized on holographic plates PFG-03 (Russia). We used di!raction modulator with rectangular phase relief as shown in Fig. 2, whose depth was p approximately. Such modulator has no zero di!raction order to decrease signal/noise ratio. After developing and bleaching the hologram was installed at the same place. We took measures so that the mutual location of corrector elements and the "ber radiate was the same as had been at the hologram recording stage. For studying the synthesized "eld characteristics, hologram 7 was illuminated by object beam, while the reference beam was removed. Formed interference "eld was observed by the horizontal microscope 11, and the "eld intensity distribution in measuring volume was determined by scanning device 12 and registered by plotter 13.
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Fig. 5. (A) scheme of experimental holographic set; (B) measurement velocity set. 1 * laser; 2 * beamsplitter; 3,5,9 * focusing objectives; 4 * single mdoe "ber; 6 * periodic phase modulator; 7 * hologram; 8* collimator; 10 * polarizer; 11 * microscope; 12 * scanning device; 13 * plotter; 14 * rotating glass disk; 15 * multimode "ber; 16 * photomultiplier; 17 * spectrum analyzer.
To be convinced that the set is working properly we observed "eld distribution when hologram was illuminated by the same single-mode "ber as at the recording stage: e "e. Under this condition, as it follows from relation (18), probing "eld must exhibit narrow autocorrelation pick. Actually, you can see that on photo (Fig. 6) as a bright spot. Fig. 7 represents a photo fragment output "eld (Fig. 7a), the experimental "eld intensity distribution (Fig. 7b) in planes (x , y ) and the calculated "eld intensity according to (15) (Fig. 7c) under condition that the input signal of correlative transformer is of MMF radiation. So you can see that it is possible to produce rather regular grating in spite of random structure of input multimode "eld. Moreover, this periodical structure stays stable when the "ber is being a!ected by variety action. We studied the in#uence of "ber vibrations on the probing "eld. Results are shown in Fig. 8a (stationary state) and Fig. 8b (unsteady state). As it was suggested, periodic structure of the "eld stays unchangeable. Herewith averaging "eld realizations brings about the smoothing of speckle noise signal (see Fig. 8b). For the examination of relation (13) we experimentally researched the probing "eld second spatial harmonic *y period dependency on the reference beam curvature at the hologram recording stage. We installed objective 10 with the possibility of moving along the reference beam axis and several holograms were recorded. Experimentally measured period values were brought into graph in dependence on R (see Fig. 9). On this graph calculating dependency calculated with formula (13) for the scheme parameters: n"2, l "22 mm, l "260 mm, 1/l"0,065 mm is shown by the solid line. It could be seen that there is su$ciently good coincidence of experimental and calculating results. Fiber sensor LDA with correlation "eld formation was researched at a set similar to the preceding one (see Fig. 5b). Rotating glass disk 14 imitated a moving object.
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Fig. 6. Output "eld of two-stage correlative transformer. Case of single-mode "ber "elds autocorrelation.
Fig. 7. Output "eld of two-stage correlative transformer. Input signal is radiation of MMF (a) is photo of "eld fragment (b) is experimental "eld intensity distribution. (c) is calculated "eld intensity distribution.
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Fig. 8. (a) is Output "eld of two-stage correlative transformer. Input signal is radiation of MMF. stationary state. (b) is unsteady state, MMF vibrates.
Fig. 9. Dependence of the second harmonic period *y versus R , dotes * experimental results.
Receipt of scattered radiation was realized by a multimode "ber 15 connected to a photomultiplier 16. Photocurrent of the photomultiplier 16 was analyzed by the spectrum analyzer 17. A photo of Doppler signal spectrum is shown in Fig. 10. Distinction of such sensor regarding to ordinary LDA is the presence of several harmonics. That con"rms preliminary obtained results of calculation shown in Fig. 4. The signal/noise ratio for the second harmonic forms 10 dB. It is important to note that vibrations of "ber transfer do not render an appreciable in#uence upon Doppler spectrum.
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Fig. 10. Photo of Doppler signal spectrum.
4. Discussion and conclusion On the basis of the performed work one can conclude: 1. The method of two-stage phase correlation transformation presented in this paper can be successfully used for transformation statistical random "eld of multimode "ber into "eld with required spatial distribution of complex amplitude. In the case of the periodic modulator the output "eld is also periodic, its concrete spatial distribution is de"ned by the spatial autocorrelation function of modulator. 2. It was shown, that two-stage phase transformer can be successfully applied in laser anemometry and vibrometry. As autocorrelation function of modulator t tH does not depend on the radiation wavelength, that should allow to use semiconductor sources in "ber anemometers and vibrometers. 3. Though in the paper a one-dimensional "eld formation is explored, the obtained results evidently can be generalized for the case of two-dimensional distribution. It also seems to be possible to generalize the proposed method for the case of the "elds formation with a given nonperiodic distribution, as any function may be expanded into the spatial spectral components. Acknowledgements This work was executed under the "nancial support of Science and Technology Center in Ukraine. References [1] Khotyaintsev S, Yarovoi L. A di!erential Doppler velocimeter with a "ber-optic lightgude and a small-size optical emitting probe head. Quantum Electronic (USSR) 1989;16(6):1273}8.
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[2] Lewin A. Compact laser vibrometer for industrial and medical application. Vibration measurements by laser techniques: advances and applications. Ancona, Italy. SPIE 1998;3411:61}7. [3] Diano B, Marchis G, Piassolla S. Speckle and modal noise in optical "bers. Theory and experiment. Optica Acta 1980;28(8):1151}9. [4] Gnatovsky A, Medved N, Shpak M. The use of "ber-optic systems to form synthetic interference "eld. Quantum Electronic (USSR) 1981;8(5):1108}11. [5] Gnatovsky A, Zolochevskaja O, Loginov A, Yarovoy L. The signal discrete representation and processing in holographic correlator. Optical storage, imaging and transmission of information. Kiev, Ukraine: SPIE 1996;3055:186}92. [6] Collier R, Burckhard C, Lin L. Optical holography. New York: Academic press, 1971.
Correction and correlation formation of probing "eld in "ber laser Doppler anemometers and vibrometers L. Yarovoi *, A. Gnatovskii, N. Medved Training and Science Center **Physical and Chemical Materials Science++ of Kyiv University by Taras Shevtchenko and National Academy of Sciences of Ukraine, 64, Volodimirskaya St., Kyiv 252017, Ukraine Institute of Physics of National Academy of Science of Ukraine, 46, prosp. Nauki, Kyiv 191028, Ukraine
Abstract In this work we o!er a method of "eld formation with given spatial features at the output of multimode "ber sensors using interference phenomena. The method consists of a "eld two-stage correlation transformation at a "ber output. Transformation is realized by a phase modulator and a hologram. The developed method is practically realized in the "ber Doppler anemometer and vibrometer scheme. Features of sensors are discussed in this work. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Correlation optics; Laser Doppler vibrometry and anemometry; Fiber sensors; Correction of laser beams
1. Principle of two-stage correlation formation of periodic 5elds Sensors of physical values using interference phenomena represent one of the practical applications of coherent optics. This class of instruments includes laser Doppler anemometers [1], vibrometers [2], as well as devices using optical photomixing for the selection of useful signal. Note one general particularity of these devices: for e$cient work they require "elds with uniform or required amplitude and phase distribution. So signi"cant di$culties appear, when it is necessary to insert multimode "bers (MMF) into optical channels of sensors because the output "eld of MMF is modulated by random and nonstationary set of speckles [3]. * Correspondence address: P.O. Box 164, Kyiv 03191, Ukraine. Fax: #380-44-266-5108. E-mail address: [email protected], [email protected] (L. Yarovoi) 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 0 5 - 1
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We believe that the usage of the wave front two-stage correlation transformation method [4], or a phase correction method, which allows to convert statistical random "eld of initial beam into "eld with the required spatial distribution of complex amplitude, is the universal solution to these problems. Formation of a "eld with a given periodic amplitude distribution is explored in this work. Results obtained for periodic "elds have self-maintained importance [5] and would serve as a background for studying more complicated nonperiodic distributions on the basis of harmonic analysis. Possibility of using formed periodic "elds for the velocity measurement of objects and #ows is shown with the "ber laser Doppler anemometer (LDA) as an example. Correlating transformation of MMF "eld modulated by specially synthesized spatial modulator 2 is the principle of a wave front correction at MMF output 1 (see Fig. 1). This transformation is realized by hologram 4 to which modulated "eld MMF is written beforehand. We will conduct the calculation of spatial distribution of "eld in the sensor measuring volume with corrector in paraxial approximation. We will show that under the observance of at certain constructive conditions observance the sensor probing "eld is determined "rstly by function of the modulator transmission (concretely by its autocorrelation function), while modulation by MMF speckle structure is vastly smoothened. Let phase transmission u(x , y ) of spatial modulator 2 will be assigned by periodic function: t(x , y )"e\ P" q exp+2p inly ,. L L\
(1)
Fig. 1. 1-"ber, 2-spatial modulator; 3-focusing objective; 4-hologram; (x , y )-plane of the measurements.
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In the considered scheme the phase spatial modulator 2 and focusing objective 3 are located on one plane: at the distance l from "ber end and at the distance l from the hologram 4 plane accordingly. Let e(x, y), be the "eld distribution at MMF end, then [6]: w(x , y )J+[e(x, y)*h (x, y)]*t (x , y )t(x , y ),*h (x , y ), J * J
(2)
where h "(1/ijl)exp+ik(x#y)/2l, is the pulse characteristic of empty space, J t "(1/ijf)exp+!ik(x#y)/2f, is the phase transformation of the objective, f * is * the focal distance of objective 3, k * 2p/j, * * is the convolution operator. We will produce the calculation of integral (1) at condition 1/f"1/l #1/l . In this case objective 3 conjugates MMF end plane with the hologram plane. Writing (1) in explicit form and conducting consequent integrating on x , y , and then on x, y we will obtain in total:
l ikl (x #y ) w(x , y )"! exp l 2fl
l 2pl l ; q e !y # nl , !x L l l k L\
l (2p)l nl ) exp !2ily exp i . l 2k
(3)
This expression describes the "eld in the form of di!raction orders' set being spatial spectrum of modulator. Moreover, each order is an image of MMF end scaled l /l times. In expression (3) phase multiplicands describe a square-law deformation of wave front and of the modulator's various di!raction order direction changing and they are compensated by the hologram. Really, component of the hologram transmission function being responsible for "rst-order di!raction beam formation is determined by relation [6]
ik(x #y ) , t JwH(x , y )exp ! G 2R
(4)
where w (x , y ) is the MMF "eld on the hologram at its recording stage, R is curvature radius of a reference beam. Amplitudes distribution in the "rst di!raction order at the distance z can be presented in the form
ik(x #y ) u(x , y )J [w(x , y )wH(x , y )]exp ! 2R
*hX (x , y ).
(5)
Note that, as it follows from (5), in product wHw phase multiplicands cancel out. Now we shall "nd complex "eld amplitudes distribution in the plane (x , y ) located at the distance z"R from the hologram. Substituting (3) in to (5) and grouping the similar terms in arguments of phase multiplicands we will
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obtain after cancellation:
1 l x #y u(x y )J! exp+ikR , exp ik ijR l R l 2pl l ) q eH !y # nl , !x L l k l L\ l 2pl l x x #y y dx dy . ) e !y # nl , !x exp !ik l l k R (6)
We produce change of the variables in the integrated expression: l l 2pl x "!x ; y "!y # nl (7) l l k and re-group multiplicands of u(x , y ): l x #y 2pinl y l q exp ! u(x y )J exp ik l L R R L\ ikl [x x #y y ] dx dy . ) eH(x , y )e(x , y )exp (8) R l Then we will transform (8) by using a known relation from the Fourier transformation theory [6]:
IK \[eHe]PGH(!K ,!K ) G(!K ,!K ), (9) V W V W where IK \ is an operator of inverse Fourier transformation: IK [e(x, y)]PG(k , k ), V W while K "(k/l )x and K "(k/l )y are spatial frequencies. So, for the "eld V W u(x , y ) we will obtain: x #y 2pinl y l l k q exp ! exp ik u(x y )J L l l R R L\ k l k l ) GH ! x #x ,! y #y R R l l k k ;G ! x ,! y dx dy . (10) l l The integral on the right-hand side of expression (10) is a function of mutual spatial correlation of MMF "eld spectrums at the stages of the hologram recording and reconstruction. The sum being before the integral presents spatial autocorrelation function of the modulator transmission:
l tH t" q exp !2pinly . L R L\
(11)
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Omitting phase factor that does not in#uence upon the "eld distribution formation one can write expression (11) in symbolic form uJ(tH t)(GH G). (12) Let us analyze the obtained expression. The probing "eld at the output of the sensor is the represented structure modulated by interference grating with oscillating factor (tH t). As one can see from (11), this grating is de"ned only by the modulator function of the complex transmission and by constructive parameters of optical scheme, and does not depend on the "eld distribution at MMF end. For example, the modulator with rectangular phase relief, which is depicted in Fig. 2 (broken line), creates the "eld modulated by periodical autocorrelation function of the modulator (see Fig. 3). Periods of spatial harmonics of the probing "eld are determined by the relation R *y " , n"1, 22 . (13) L ll n The in#uence of MMF appears how the noise spatial signal at the sensor output that is expressed in term of the square of mutual correlation function of spatial frequencies' spectrum GH G (see (12)). One can study the nature of this noise by
Fig. 2. e(x, y)-"eld distribution on "ber end (solid line) (y)-depth of phase relief.
Fig. 3. The module of spatial modulator autocorrelation function for case l "R .
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writing expression (12) in space}frequency domain and taking into account expressions (9) and (11):
l l (14) qe K !2pl n ) eH K !2pl n W W L R R L\ here S(K )"I[;] is the spatial spectrum of the "eld at the sensor output. W Analysis of relation (14) shows that spatial spectrum of "eld at the output of the sensor with correlation transformer has the form of equidistant harmonics. An envelope of each harmonic is proportional to the product of spatial "eld distributions in the plane (x, y) at the stages of recording and hologram reconstruction. It is "nite sizes of "ber end that causes extension of interference "eld spectral components. Because of the light coherence, "elds e(x, y) and e (x, y) at MMF end have a speckle structure. Therefore, spatial harmonics of the "eld at the output of the sensor are modulated by random noise function. For noise reduction it is expedient "rst of all to record the hologram by laser beam or by single mode "ber. In this case in planes (x, y) the "eld distribution will be described by smooth function kind of e "E exp+!(x #y )/r ,, where r is approximately "ber core diameter. 1 S(K )J W 2p
2. Structure of doppler signal in LDA with correlator It is possible to use obtained results for LDA and for laser Doppler vibrometers "ber sensor. Let the particle of the #ow crosses the measuring volume with an interference periodic "eld in the plane (x , y ) at the velocity of v, as it is shown in Fig. 1, and scatters the radiation. The scattered radiation is modulated by periodical function with the frequency being proportional to the velocity <: 2pf "K<, where " K is the spatial frequency of the "eld. Scattered radiation is gathered by receiving "ber and directed to the photodetector. Suppose, the "eld at the sensor output is formed by means of a correlation transformer discussed above. Then, as follows from (12), the "eld intensity distribution in measuring volume is de"ned by the relation: I(y )J(GH G)(GH G)H¹(y ), (15) where ¹(y ) is a modulation function: 2pill y (n!m) $ (16) ¹(y )" qq exp ! L K R L\ Let the #ow to move along axis y , then y "
here
l S (f )"I[I(t)]J qq E f !
E"I[(GH G)(GH G)H]"e eH eHe. is the envelope of photocurrent harmonics.
(17)
(18)
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Fig. 4. The Doppler signal spectrum.
One can determine velocity from (17) as follows: f R (19) <" L , ll n where f is a frequency of the Doppler signal nth harmonic. L Fig. 4 presents the spectrum of the Doppler signal evaluated under relation (17) for di!ractive modulator with rectangular phase transmission and the "eld distribution of multimode "ber as shown in Fig. 2 3. Experimental results Fig. 5a represents the scheme of the experimental set. At the preliminary stage a correcting hologram 7 was recorded. Recording was performed by the radiation of helium}neon laser - 1 in the two wave convergent scheme. Object beam was formed by single-mode "ber 4 with a core diameter of 9 mm. After passing the "ber the radiation was directed to the periodic phase modulator 6 and then to the correcting hologram 7. The hologram was recorded by convergent spherical reference beam which was formed by collimator 8 and objective 9. We installed a similar polarization in interfering waves for raising di!raction e$ciency of the hologram by means of polarizer 10. The hologram recording was realized on holographic plates PFG-03 (Russia). We used di!raction modulator with rectangular phase relief as shown in Fig. 2, whose depth was p approximately. Such modulator has no zero di!raction order to decrease signal/noise ratio. After developing and bleaching the hologram was installed at the same place. We took measures so that the mutual location of corrector elements and the "ber radiate was the same as had been at the hologram recording stage. For studying the synthesized "eld characteristics, hologram 7 was illuminated by object beam, while the reference beam was removed. Formed interference "eld was observed by the horizontal microscope 11, and the "eld intensity distribution in measuring volume was determined by scanning device 12 and registered by plotter 13.
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Fig. 5. (A) scheme of experimental holographic set; (B) measurement velocity set. 1 * laser; 2 * beamsplitter; 3,5,9 * focusing objectives; 4 * single mdoe "ber; 6 * periodic phase modulator; 7 * hologram; 8* collimator; 10 * polarizer; 11 * microscope; 12 * scanning device; 13 * plotter; 14 * rotating glass disk; 15 * multimode "ber; 16 * photomultiplier; 17 * spectrum analyzer.
To be convinced that the set is working properly we observed "eld distribution when hologram was illuminated by the same single-mode "ber as at the recording stage: e "e. Under this condition, as it follows from relation (18), probing "eld must exhibit narrow autocorrelation pick. Actually, you can see that on photo (Fig. 6) as a bright spot. Fig. 7 represents a photo fragment output "eld (Fig. 7a), the experimental "eld intensity distribution (Fig. 7b) in planes (x , y ) and the calculated "eld intensity according to (15) (Fig. 7c) under condition that the input signal of correlative transformer is of MMF radiation. So you can see that it is possible to produce rather regular grating in spite of random structure of input multimode "eld. Moreover, this periodical structure stays stable when the "ber is being a!ected by variety action. We studied the in#uence of "ber vibrations on the probing "eld. Results are shown in Fig. 8a (stationary state) and Fig. 8b (unsteady state). As it was suggested, periodic structure of the "eld stays unchangeable. Herewith averaging "eld realizations brings about the smoothing of speckle noise signal (see Fig. 8b). For the examination of relation (13) we experimentally researched the probing "eld second spatial harmonic *y period dependency on the reference beam curvature at the hologram recording stage. We installed objective 10 with the possibility of moving along the reference beam axis and several holograms were recorded. Experimentally measured period values were brought into graph in dependence on R (see Fig. 9). On this graph calculating dependency calculated with formula (13) for the scheme parameters: n"2, l "22 mm, l "260 mm, 1/l"0,065 mm is shown by the solid line. It could be seen that there is su$ciently good coincidence of experimental and calculating results. Fiber sensor LDA with correlation "eld formation was researched at a set similar to the preceding one (see Fig. 5b). Rotating glass disk 14 imitated a moving object.
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Fig. 6. Output "eld of two-stage correlative transformer. Case of single-mode "ber "elds autocorrelation.
Fig. 7. Output "eld of two-stage correlative transformer. Input signal is radiation of MMF (a) is photo of "eld fragment (b) is experimental "eld intensity distribution. (c) is calculated "eld intensity distribution.
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Fig. 8. (a) is Output "eld of two-stage correlative transformer. Input signal is radiation of MMF. stationary state. (b) is unsteady state, MMF vibrates.
Fig. 9. Dependence of the second harmonic period *y versus R , dotes * experimental results.
Receipt of scattered radiation was realized by a multimode "ber 15 connected to a photomultiplier 16. Photocurrent of the photomultiplier 16 was analyzed by the spectrum analyzer 17. A photo of Doppler signal spectrum is shown in Fig. 10. Distinction of such sensor regarding to ordinary LDA is the presence of several harmonics. That con"rms preliminary obtained results of calculation shown in Fig. 4. The signal/noise ratio for the second harmonic forms 10 dB. It is important to note that vibrations of "ber transfer do not render an appreciable in#uence upon Doppler spectrum.
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Fig. 10. Photo of Doppler signal spectrum.
4. Discussion and conclusion On the basis of the performed work one can conclude: 1. The method of two-stage phase correlation transformation presented in this paper can be successfully used for transformation statistical random "eld of multimode "ber into "eld with required spatial distribution of complex amplitude. In the case of the periodic modulator the output "eld is also periodic, its concrete spatial distribution is de"ned by the spatial autocorrelation function of modulator. 2. It was shown, that two-stage phase transformer can be successfully applied in laser anemometry and vibrometry. As autocorrelation function of modulator t tH does not depend on the radiation wavelength, that should allow to use semiconductor sources in "ber anemometers and vibrometers. 3. Though in the paper a one-dimensional "eld formation is explored, the obtained results evidently can be generalized for the case of two-dimensional distribution. It also seems to be possible to generalize the proposed method for the case of the "elds formation with a given nonperiodic distribution, as any function may be expanded into the spatial spectral components. Acknowledgements This work was executed under the "nancial support of Science and Technology Center in Ukraine. References [1] Khotyaintsev S, Yarovoi L. A di!erential Doppler velocimeter with a "ber-optic lightgude and a small-size optical emitting probe head. Quantum Electronic (USSR) 1989;16(6):1273}8.
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[2] Lewin A. Compact laser vibrometer for industrial and medical application. Vibration measurements by laser techniques: advances and applications. Ancona, Italy. SPIE 1998;3411:61}7. [3] Diano B, Marchis G, Piassolla S. Speckle and modal noise in optical "bers. Theory and experiment. Optica Acta 1980;28(8):1151}9. [4] Gnatovsky A, Medved N, Shpak M. The use of "ber-optic systems to form synthetic interference "eld. Quantum Electronic (USSR) 1981;8(5):1108}11. [5] Gnatovsky A, Zolochevskaja O, Loginov A, Yarovoy L. The signal discrete representation and processing in holographic correlator. Optical storage, imaging and transmission of information. Kiev, Ukraine: SPIE 1996;3055:186}92. [6] Collier R, Burckhard C, Lin L. Optical holography. New York: Academic press, 1971.