DIGITAL CORRELATION OF FIRST ORDER SPACE TIME IN A FLUCTUATING MEDIUM Budi Santoso Center For Partnership in Nuclear Technolog, National Nuclear Energ Agenc (BATAN) Puspiptek, Serpong ABSTRACT DIGITAL CORRELATION OF FIRST ORDER SPACE TIME IN A FLUCTUATING MEDIUM. The stud of fluctuating medium has been of great interest through the use of the correlation techniques A laser beam is known to form a coherent beam which can be made to propagate within the fluctuating medium. This will allow the stud of the outgoing beam using digital correlation technique. Based on the power spectrum, the integral transformation of the correlation function, one can obtain for instance the radius and mass of the particles eecuting Brownian motion in the dispersed solution. To correlate the laser beam directl ma not allow the detection of signals b electronic means. A method of digitizing the light signals b means of light beat heterodne technique is therefore adopted. The temporal and special correlation functions can be measured. Kewords: fluctuating medium, Brownian motion, laser INTRODUCTION Correlation techniques have plaed an important role in the stud of fluctuating media. In particular laser light is a good probe due to its coherent properties. However the light frequencies range between 0 3 and 0 5, make it impossible to make the correlation directl, because no such electronic devices could handle such high frequencies. With the rapid development in light beating techniques [3,4], this digital correlation method is available. Since the correlation of the scattered light carries significant information, about the nature of the fluctuating media, one can applies this method directl. An eample is the measurement of the dnamic viscosit, radius and mass of molecules undergoing Brownian motions, etc.[5]. Correlation techniques can also be applied to stud the properties of normal fluid, liquid near critical point, stable plasma, laminar flow etc. [6]. The studies ma be too complicated if the are carried out b conventional methods. THEORETICAL ANALYSIS The correlation function of electric field E ( R,, E ( R, at different positions R, R undergoing fluctuations can be written as [7] * Γ R, R, τ ) =< E ( R, E ( R, t + ) > () ( τ 8
The brackets indicate the time average and τ is the time dela at R taken to be later then R. The above epression is called the mutual correlation,or mutual coherent function of them light signals from R and R. This is a more general form of the first order correlation function, for the special case when R = R he function reduces to self correlation function. Written as * ( τ Γ τ ) =< E ( R, E ( R, t + ) > () If τ is to be zero, then Γ (0) is the intensit of light at R. Most of the studies are based on information s of Γ ( τ ) or Γ (0). Little has been made on the studies based on Γ ( τ ), since it involves the problem of measuring such quantit, and then the problem of interpretation of the data obtained. The purpose of this stud is to show how Γ ( τ ) can be measured digitall, so that the corresponding power spectrum that contained phsical information can be obtained. Using a beam splitter, a helium neon laser beam with frequenc ω0 can be split into two beams. The first beam is modulated b an ultrasonic modulator of frequenc Ω so that the frequenc becomes ω = ω0 + Ω. The beam is then epanded b a beam epander (BE) into.5 mm in diameter, and is made to pass through a quarter wave plate (QP) as a reference beam. The ais of the plate is positioned in such a wa it makes an 0 angle β to he direction of the linearl polarized beam. The ais of the plate is chosen to be the direction -ais as reference of analsis, while -ais 0 0 is perpendicular to this ais, anti clock wisel. In the case β = 45, the electric field component of the outgoing beam will be circularl polarized otherwise it is elliptical. In general therefore the electric field of this reference beam taken to be zero (on the laser beam ais, R = 0 ) can be written as E = E ep[ i( ω t + δ )] + E ˆ ep[ i( ω t + π / + )] (3) ˆ δ ˆ and ŷ being the normal vectors along the and aes respectivel, while δ is a constant phase factor. The second split beam is also modulated b an ultrasonic modulator of frequenc Ω so that the electric field of the beam after epander and passed though a quarter wave plate (QP) can be written as F = F ep[ i( ω t + η)] + F ˆ ep[ i( ω t + π / + )] (4) ˆ η 8
Before the beam passes a fluctuating medium. Here again η is a phase factor, ω = ω0 + Ω. After the beam passes through a fluctuating medium the field eperiences fluctuations, let it be indicated b G, satisfing ~ G = A( ξ, F (5) ~ ~ where A( ξ, is a comple function satisfing A ( = Aep( iφ) and where A = A ~ ( ξ, φ = arg( A). The variable ξ measures the distance of a particular measurement position from the ais of the laser beam. We could correlate the reference beam, and the fluctuating beam either ˆ or ŷ components and can be shown to satisf the following I J J = E + AF + E F A(0, cos[( ω ω ) t + φ(0, + ρ] = E + AF + E F A( ξ, cos[( ω ω ) t + φ( + σ ] = E + AF + E F A( ξ, sin[( ω ω ) t + φ( + σ ] ρ and σ are some phase constants, which if there is no polarization changes, their difference is alwas zero for r=0 [7]. The components along ˆ and ŷ are made b adjusting the polarizer in the eperimental set up. If the correlation is for the same component ( e.g. ˆ and ˆ ) it is called the auto correlation, otherwise it is called cross correlation. ( ˆ and ŷ ). We have the auto correlation function C ( τ ) =< I K (0, J ( t + τ ) >= < A(0, A( t + τ )cos[( ω ω ) τ + φ( t + ς ) + ρ σ ] > and the cross correlation function C ( τ ) =< I K (0, J ( t + τ ) >= < A(0, A( t + τ )sin[( ω ω ) τ + φ( t + ς ) + ρ σ ] > ρ σ should be made zero or π b adjusting the aes of the polarizer or/and the quarter wave plates. K, K are constants namel K = E E F F and K = EE F F. In practice these constants can be determined (in arbitrar uni b measuring C and C in the absence of (7) (8) 83
fluctuating medium. In this case C = C / K and C = C / K are the required normalized auto an cross correlation functions. The following rearrangement can be made ~ ~ * C + ic =< A(0, A ( t + τ ) > ep[ i( ω ω ) τ ] (9) Where the last term on the right hand side has been taken out from the time average, since it is not function of t.the term in the bracket is the normalized first order space-time mutual correlation functions (SPMCF) given b ~ ~ ~ γ ( τ ) =< A (0, A * ( t + τ ) >= γ ( τ )ep[ iθ ( τ )] (0) ( τ ) = C + C with γ and θ ( τ ) = tan ( C / C ) ( ω ω ) τ Phsical information contained in the power spectrum that can be derived b Fourier transforming the SPMCF ~ γ ( ω) = ~ γ ( τ )ep[ i ( ωτ )] dτ () This ~ γ ( τ ) represents Γ ( τ ), where the sub-indices have been represented b ξ. EXPERIMENTAL SET UP The eperimental set up is based on then optical heterodne technique where the beat signals are separated into two components, the ˆ and ŷ components, called the phase quadrature technique [8]. A Mach-Zender heterodne interferometer is set up completed with the electronic measuring sstem. In this eperiment the fluctuating medium is artificiall made b blowing thermal air across the beam path. The initial beam coming from Helium-Neon laser is chosen such that the direction of electric field polarization is along a reference ais. The signals detected b two detectors D and D (see figure -) are filtered to suppress unwanted frequencies. The filters allow to pass the frequenc carrier ranging from 0 to 70 khz. In this eperiment, is chosen to be 54.935 khz. In selecting the sampling period, the correlation can be observed b modulating the signals in the range of 00 t0 00 Hz. The correlation functions observed are shown in fig, while in figure 3 we have the first order space time correlation functions. It is not clear et what the phsical interpretation of these functions. 84
RESULTS AN DISCUSSIONS Measurements have been made for ξ =,,3,4, mm.. This limitation is due to the size of the detector aperture having the diameter mm, and the beam epansion which has.5 mm in diameter. The Helium-Neon laser has the maimum output of mw. The beam intensit decreases with respect to ξ. should follow a Gaussian profile I ( ξ) = I 0 ep( µξ ), where I, 0 µ are constants. There seems to be an anomal at ξ = mm b showing a stronger time and spatial correlation function deca compared to other different location. The reason is not understood et. The thermal fluctuating medium is made b blowing thermal air across the beam path. The temperature of the medium is around 65 0 C. There seems to be a significant different in the function obtained, if the temperature is reduced to room temperature. The deca pattern of γ ( τ ) is thus dictated b two variables at least namel the temperature and the wind velocit of the fluctuating air turbulence. Figure 3 shows the eperimental measurements of γ ( τ ) as function of ξ and τ. In the case τ = 0 the function γ (0) is called mutual correlation function (MCF). This laborator simulation eperiment can be etended to the measurement of atmospheric air turbulence, in which the deca pattern of MCF is due to the aerosol and molecular turbulences. The function obtained 3 does not follow the pattern M ( ξ) ep( βξ 5 / ) as shown in reference [9], because the have different condition too. CONCLUSIONS Using digital correlation techniques, the space-time correlation function can be measured successfull, its modulus and phase. The technique can be applied for other studies such as in the measurement of microemulsions, an important studies in micro-biolog, pharmaceutical, and others. In particular, the measurement of this simulated atmospheric turbulence can be etended to the real air atmospheric turbulence in an effort to understand the nature of atmospheric behavior. REFERENCES. OHTSUKA, Y., et all, J Appl Phsics, (98).. DEGIORGIO, V., Coherent Optical Engineering, Edited b FT Arrechi, and V Degiorgio, North Holland Publ Comp. (977). 3. FANTE R.L., Proc IEEE, 63, (975). 4. OHTSKA, Y, J., Appl Phsics, 7 (978). 85
5. SANTOSO, B., Measurement of Micro Emulsion Radii, GIGA (005). 6. CROSSIGNANI B, et all, Statistical Properties of Scattered Light, Academic Press (975). 7. BORN M, WOLF, E., Principles Optics, Pergamon Press (975). 8. LEADER JC., Proc Soc Photo Opt Instr Eng, W.C Carter (979). 9. Lutomirski, RF, Proc Soc Photo Opt Instr Eng, W.C Carter (979). 0. FORRESTER AT, and VAN BUEREN HG., Optics Comm, (97). 86
Fig.. Diagram of the eperimental set up Fig. a. Auto correlation with no turbulence Vertical ais is C (0, τ ) and horizontal ais is τ maimum 9 msec. Fig. b. Auto correlation with turbulence. 87
Fig. c. Cross correlation with turbulence Vertical ais is C (0, τ ) and horizontal ais is τ maimum 9 msec. Fig. d. Cross correlation with turbulence. Fig. 3a. First Order Space time Correlation. Fig. 3b. Phases of First Order Space-time Correlation. 88