Simple inexpensive method of measuring the temporalspreading of a light pulse propagating in a turbidmedium
Giovanni Zaccanti, Piero Bruscaglioni, and Michele Dami
A simple and inexpensive method of measuring statistical parameters related to the time lengthening arisingin the propagation of a light pulse in a turbid medium is presented. The method is based on the repetition ofattenuation measurements of a light beam passing through the turbid medium when the absorption coeffi-cient of the medium surrounding the diffusing particles is varied. The measurements are carried out using acw source and a simple optical receiver with a common photodiode as a detector. The results of twomeasurements are reported together with the results of numerical simulations carried out using the scatteringproperties and geometric parameters corresponding to the experimental situation. Numerical results wereobtained using a Monte Carlo based method. Good agreement between experimental and numerical resultswas found.
1. IntroductionThe scattering effect in the propagation of a light
pulse is twofold: It extracts part of the energy fromthe direction of propagation causing an attenuation ofthe direct beam, and it causes a part of the extractedenergy, even after multiple scattering events, to enterthe finite angular field of view (FOV) of the receiver.So the received pulse, in addition to an attenuation, isalso distorted in shape with respect to the transmittedone. The distortion is due to the fact that the scat-tered received power arrives along trajectories longerthan the ones followed by the unscattered power. Thestudy of this pulse spread is of great interest in manyfields; for example, optical communications throughturbid atmosphere (clouds, fogs, rain) or seawater andlidar applications for precise ranging.
Experimental investigations of this problem haveused two measurement techniques. The first was usedby Bucher and Lerner1 and Mooradian et al.2 3 Theseresearchers reported results of pulse broadening of thereceived power for transmissometric links a few kilo-meters long through actual clouds and fogs. The sec-ond was used by Kuga et al.,4 Elliott,5 and Majumdar6
who reported the results of measurements taken in the
The authors are with University of Florence, Physics Department,3 via S. Marta, I-50139 Florence, Italy.
laboratory on short distances (a few centimeters) usinga picosecond laser pulse.
Measurements in actual clouds showed considerablepulse broadening and delay of the arrival time (up tothe order of 10 ,s). These measurements, especiallythose reported in Ref. 3 where a link between an air-craft over the cloud and a ground-based receiver wasconsidered, required complex and expensive experi-mental setups. Furthermore during these measure-ments it was almost impossible to have sufficientknowledge of the medium (size, shape, numerical den-sity, and homogeneity of the diffusers). For example,during measurements reported in Ref 3 where a meteo-rological aircraft was used to sample the particle sizeand the cloud basis and top, even the optical depth,that is, one of the most important parameters of thelink, was unknown. Thus it is very difficult to use theresults of these measurements for a comparison withtheoretical previsions. So it is preferable to study theproblems by means of measurements on laboratoryscale models7 using well-known diffusing media in con-trolled situations. However the reduction of thesource-receiver distance down to a few centimeterscauses the temporal lengthening to reduce to the orderof picoseconds. Thus direct measurements of tempo-ral lengthening involve the use of picosecond pulses.In recent years, the availability of picosecond lasersources and fast electrooptic devices allowed this typeof measurement to be made.
Kuga et al.4 used a mode-locked Nd:glass laser as a
source and a receiver based on a Kerr shutter and a lowlight level solid state detector system. They reported
measurements on polystyrene spheres at opticaldepths up to 45. Elliott5 used a similar experimentalapparatus with a streak camera detector to measurepulse stretching and delay up to an optical depth of r -170 on emulsions of dielectric liquids. Majumdar 6
also reported results on statistical characteristics ofpicosecond laser pulses propagating through polysty-rene spheres for optical depths up to 10. He used apicosecond laser diode and a high speed photodiodeconnected to a sampling oscilloscope. All these mea-suring techniques require a sophisticated experimen-tal apparatus.
For all these experimental results, there are no com-parisons with numerical simulations referring to thespecific situations. The only comparison was with theMonte Carlo results reported by Bucher8 in 1973. Thetime spreading results reported by Bucher, however,refer to the case of an infinitely extended receiver witha FOV of 900, which would correspond to the case of afinite receiver with an open FOV, when a plane wave isincident on the turbid medium.
In this paper, we propose a simple measuring meth-od based on the use of a simple, cheap experimentalapparatus. The method, described in Sec. II, is basedon the repetition of attenuation measurements, keep-ing a constant quantity of diffusers in the scatteringcell and every time adding a quantity of dye to increasethe absorption coefficient of the medium. If the scat-tering characteristics (extinction coefficient, albedo,and phase function) of the diffusing spheres remainunchanged when the dye is added, the only effect is anincrease of the absorption coefficient of the medium.This produces an attenuation of the transmitted powerthat depends on the length of the trajectory followedinside the diffusing medium. Thus the received powerdue to multiple scattering that arrives following trajec-tories longer than the unscattered received power, ismore attenuated with respect to the unscattered one.On the basis of these attenuation measurements it ispossible to obtain the statistical parameters describingthe lengthening of the trajectories and hence thebroadening of the pulses. For these measurementsthe source can be a cw laser (a 10 mW He-Ne laser inour experimental setup), and a simple optical receiverwith a common photodiode detector can be used.
Section III reports a brief description of the experi-mental apparatus and Sec. IV reports two examples ofexperimental results obtained with our method at op-tical depths of r 7 and X 31. A comparison with theresults of a Monte Carlo simulation referring to theconsidered diffusers and geometric conditions is alsoreported showing good agreement between the experi-mental data and the theoretical previsions. Conclu-sions are in Sec. V.
II. Proposed Method
A. Theoretical ConsiderationsWhen a light beam (emitted power Pe) propagates in
a diffusing medium, the total received power P is com-posed of both the unscattered received power P0 and
Diffusing Medium
SOURCEEmitted Power
RECEIVERReceived Power P
gPe
Fig. 1. Sketch of the geometric situation. Multiple scatteringtrajectories are indicated. Due to multiple scattering the receivedpulse P is broadened with respect to transmitted pulse Pe. Thefigure refers to a conical beam, but the proposed method of measur-
ing is valid for any shape of emitted light beam.
the scattered received power Pd. Due to multiple scat-tering, the radiation Pd arrives on the receiver follow-ing trajectories longer than those pertaining to theunscattered power (see Fig. 1). This causes the shapeof a received light pulse to be different from that of atransmitted one. We denote with 6 = AL/L the rela-tive lengthening of a trajectory of length L + AL withrespect to the source-receiver distance L, and we de-fine a probability density function f(b) so that f(3)dbrepresents the probability that the received powerreaches the receiver following trajectories of lengthsencompassed between L(1 + ) and L(1 + 6 + d0).
It is clear that the function f(3) also represents theprobability density of the arrival of the received poweron the receiver with a relative temporal lengtheningAt/to with At = AL/c, to = Lic and c is the speed of lightin the medium. (The quantity to represents the timethat the unscattered power takes to reach the receiv-er.) So the function fG() represents the time shape ofthe received pulse when a delta pulse is transmitted.Thus the knowledge of f(6) allows one to obtain thereceived pulse shape corresponding to the emission ofany pulse. The function f(6) in general depends in acomplex way on the geometric situation (source-re-ceiver distance, L, size and FOV of the receiver) and onthe diffusers (size, shape, refractive index, concentra-tion, homogeneity).
In our method of measurements we consider a fixedgeometrical situation with a fixed concentration ofdiffusers. If we modify (for example, by introducing asmall quantity of dye) the absorption coefficient C
7d Of
the medium (water in our case) in which the diffusersare suspended without modifying the scattering prop-erties of the diffusers [absorption and scattering crosssections, phase scattering function p(O)], we obtain anew situation in which the trajectories the receivedpower follows are identical to those before the absorp-tion coefficient was changed. However, the energyreceived after having followed the trajectories oflength L( + ) is further attenuated by a factorexp[-Td (1 + 6)] where Td is the optical depth due to thedye. So, if P(Or, + rd) is the received power when the
optical depths due to spheres and dye are T, and -rd,respectively, we have
P(r, + Td) = P(r,) J exp[-rd(l + )]f(6)d6
= P(rT) exp(-rd) J exp(-Td6)f(0)d6. (1)
The quantity f ' exp(-rd6)f() is always <1. So theattenuation of the ratio P(,r + rd)/P(rs) is greater thanthat given by the factor exp(-rd) which would be theonly attenuation factor due to the dye if no lengtheningoccurred. Then a measure of P(-r + rd) and P(rT), withTd known, gives the value of (exp(-Td5)), where ( )denotes the average value with respect to :
(exp(-Td6)) = P(r, + rd)/[P(r) exp(-rd)] (2)
Carrying out simple attenuation measurements fordifferent values of led with a cw light source, it is possi-ble to have information on the statistical distributionof 8 and hence on the shape of the received pulse whena short light pulse is transmitted. For example, ifmeasurements are carried out using sufficiently smallvalues of Td, it is possible to use the results to evaluatelower-order moments (n) of the probability densitydistribution f(b). In fact, we have
(exp(-rd6)) = (1 - rdb + Ii-d d2-1/T 33 +...)
= 1 -Td(6) + d/2(32) - 1Td) +.*-- (3)
So, for example, a fitting of the data in a sufficientlyrestricted range of -rd with a second degree polynomialin Ird can provide the values of (8), (2).
From a general point of view the data can be used totest whether a function (8) of an assumed form, forexample, the y or 2-y functions used by Mooradian andGeller,3 fits with the results.
An important point in the previous discussion is thatthe scattering characteristics of the diffusing particlesmust not be substantially modified when the dye isintroduced into the medium in which the particles aresuspended. To study this aspect with reference to ourexperimental situation, we carried out calculations ofthe scattering properties of the turbid medium by tak-ing into account the imaginary part of the refractiveindex n of the medium surrounding the sphereschanged by the addition of dye. For this we extendeda method indicated by Bohren and Gilra9 for the cal-cualtion of the extinction coefficient of spheres in anabsorbing medium to the calculation of scatteringproperties of the medium. The calculations showedthat for the values of the imaginary part of no in whichwe were interested (mnw - 0.35 X 10-5 was the valueneeded to produce ad = 70 m-1 and thus -rd = 7, themaximum value of 'rd used in our measurements, in a10-cm long scattering cell), the scattering properties ofthe diffusing medium were unchanged.
The absorbing medium was a diluted solution of dye.To maintain the concentration of the diffusers un-changed, when the absorbing medium was introducedinto the scattering cell the solution of the dye wasprepared using a sample of water and diffusers withthe same concentration as in the cell.
The remarks made in this section are true for any
geometric situation, in particular for any shape of thetransmitted light beam. However, if the diffusing me-dium is contained in a scattering cell and the receiver isplaced at a certain distance from the exit window, as intypical laboratory situations, the measured elonga-tions are only the ones inside the cell. But the extrapath outside the cell, where there are no diffusers, canbe small (as happens in our geometry), and such thatthe overall lengthening is not substantially differentfrom that measured inside the cell. In principle, how-ever, following the previous line of reasoning, it is alsopossible to measure the lengthening that originates ina certain slab of the medium when the dye is onlyintroduced into the considered slab, i.e., the dye can beused as a probe to study the lengthenings and themultiple scattering problems in general.
The same procedure can be repeated for the situa-tion of a backscattered power receiver. In this case,the results can be used to investigate the multiplescattering effect in the lidar echo.
B. Practical Considerations: Case of High Values of sThe described method of measuring requires a
knowledge of -rd and of the ratio P(r, + Td)/P( 5 ). Tomeasure P(Tr + d) and P(T,) for high values of r, thereare no particular problems. (We define the region ofhigh values of r, as the region in which P0 is an insig-nificant fraction of P.) In typical laboratory geome-tries (L 10 cm, R 1 cm, a 1 where L is the lengthof the scattering cell, R and a are the radius and theFOV semiaperture of the receiver respectively) themeasured attenuation of a light beam, due to the largemultiple scattering contribution, does not overcomethe orders of value comprised between exp(-10) andexp(-15) even when rs is as high as 70.10 We used theprocedure indicated in Ref. 10 to measure the actualvalue of r which can be for this reason substantiallydifferent from the apparent optical depth. (We definethe apparent optical depth a as the natural logarithmof the ratio between power received without and withthe diffusers in the scattering cell.) The value of rd,the optical depth due to the dye in the scattering cell,was evaluated by weighing the quantities of dye, thequantities of water introduced into the scattering cell,and by using the results of a previous calibration. Thecalibration was carried out by taking attenuation mea-surements on weighed samples of dye diluted in purewater, using the same experimental apparatus used tomeasure P(0-8 ) and P(0- + Td).
Because the measured quantities (exp[-rd (1 + )1)are used to obtain information on , an error on Id
causes an error on . However, because L( + ) issubstantially longer than L with high values of rs, weneed only know rd with a precision of a few percent tohave reliable values of . This is easy to obtain withcareful measurements.
C. Practical Considerations: Case of Small Values of sIn the region of small values of (we define this as
the region in which Po is an appreciable fraction of P)the pulse lengthening is small. In this case an evalua-
tion of rd as performed in the preceding case can hardlygive sufficiently accurate results to measure thelengthening. When rs is small we repeated, for a fixedoptical depth, the attenuation measurement with vari-ous values of the receiver FOV using a variable FOVreceiver. In this way it is possible to separate P0 fromP using an extrapolation procedure for a -> 01 which isbased on the consideration that P8 varies with a, and Podoes not vary in our range of a(0.50 < a < 30) due tothe small divergence of the laser beam used. Thus,since P0 follows an exponential attenuation law PO(T +Td) = Po(O) exp[-(0r + Td)I, a direct and accuratemeasurement of the total actual extinction coefficientin the cell is possible in any case. Furthermore, whenPo has been evaluated, we can obtain Pd = P - Po. Soit is possible to study separately the dependence on Id
of the scattered received power.Using this method, we measured lengthening on Pd
down to a few percent of L even when Pd was only -10%of P. The average path lengthening on the total re-ceived power was <1% of L. It can be pointed out thata similar measurement is difficult to obtain using themethods referred to in Sec. I based on the emission oflight pulses. The method can be used when the re-ceived diffused power is an appreciable fraction of thetotal received power. The actual level of , corre-sponding to these situations depends on the geometryof the link and on the size of the suspended particulate.
Ill. Experimental SetupFigure 2 is a block diagram of the experimental setup
proposed to measure the temporal spreading. Weused a 10-mW He-Ne laser (0.6-mrad total divergence,2-mm beam diameter at the scattering cell) and a 10-cm long, 8 cm diam scattering cell (with blackenedlateral walls and antireflecting optical windows). Themain characteristic of our optical receiver was a vari-able FOV. The receiver radius was R = 2 cm and thevalues of the FOV were varied between 0.50 and 3 insteps of 0.50. The detector was a PIN photodiode.(The apparatus is basically the same as that used tomeasure scattered received power in transmissometriclinks.1 1 ,12
The bandpass interference filter F2 (the full width athalf-maximum was 10 nm in our apparatus) in front ofthe receiver was necessary to cancel the IR radiationthat arises when the dye is introduced. A calibratedneutral density filter F was also used to attenuate thebeam when measurements were carried out at smallvalues of I to reduce the dynamic range on which thedetector works.
The variable FOV receiver enabled us to separate Poand Pd in the region of moderate values of I (when Po isand appreciable fraction of P). So it is possible tomeasure the true value of r even when a substantialamount of scattered power is received. With this ap-paratus we measured temporal lenthening correspond-ing to a few picoseconds using a 10-cm long scatteringcell, but the same apparatus, using the same opticaldepths in a shorter cell, can also measure temporallenthening briefer than one picosecond. In fact, a
SOURCE ~~SCATTERING OPTICALWUHUL FCELLF 2 R CIE
Fig. 2. Block diagram of the experimental setup. In our measure-ments the source was a cw 10-mW He-Ne laser. F is a calibratedneutral density filter, F2 is an interference bandpass filter. Theoptical receiver (radius R and angular FOV semiaperture a) uses aphotodiode as a detector. In the scattering cell (filled with bidis-tilled water) the diffusing particles (polystyrene spheres in our case)are introduced and measurements are repeated with the addition of
an absorbing medium (a blue dye).
scaling down of all geometry by a K factor, with acorresponding multiplication of the concentration ofspheres and dye by the same factor, causes a K factorreduction of the scattered power path lengths but noreduction of its attenuation.
The simplicity of the experimental apparatus en-ables one to measure a large range of optical depthsand values for the radius and FOV of the receiver. Theshape of the transmitted light beam can also be easilyvaried.
IV. Examples of Experimental ResultsWe carried out measurements using 0.33-,um poly-
styrene spheres (practically monodispersions) sus-pended in bidistilled water as diffusers and brilliantcresyl blue Certistain (supplied by Merck) as dye. Tocheck that the introduction of the dye does not modifythe scattering properties of the spheres we carried outseveral preliminary measurements of extinction coeffi-cient and scattering phase function p(O) for 0 < 2.250on polystyrene spheres and dye both separately and ona mixture. The technique indicated in Ref. 13 wasused. As a result we saw that, with dye only, even at anoptical depth of the order of 10, the dye does not makeany appreciable contribution to the scattered receivedpower. Furthermore, when a mixture of spheres anddye was used with rI < 1, the introduction of the dyeonly modified the absorption coefficient of the medi-um leaving the optical properties of the suspendedspheres unchanged.
Figure 3 reports an example of measurements athigh optical depth. For all the data of the figure, theconcentration of diffusing spheres was fixed and corre-sponded to -r- = 31.0 while the optical depth rd, due tothe dye, varied from 0 to 1.5. It is to be pointed outthat a strong effect of multiple scattering reduced thereceived power for r, = 31.0 and a = 10 by a factor of 1.7X 106 with respect to that received in clear water condi-tions. This corresponded to an apparent opticaldepth of 14.3. The attenuation factor was found tovary proportionally to 1/a2 for the values of a used. Asmentioned in Sec. III, the use of the calibrated neutralfilter when low values of Ir are measured allowed us notto exceed three decades for the dynamic range of thedetector. The figure shows the ratio P(0- + Id)/P(18)
VS Td and +, 0, X represent the experimental resultspertaining to a = 30, 20, and 10, respectively, whereasthe continuous line represents the function exp(-Td).The ratio, which in the absence of lengthening would
Fig. 3. Example of measurement at a high value of optical depth (r,= 31.0). The ratio P(rs + Td)/P(Ts) between the total receivedpower, when the optical depth due to the dye is rd and when no dye ispresent, is reported vs rd. The marks refer to the experimentalresults (+, 0, X correspond to a = 30, 20, 1°, respectively), whereasthe continuous line represents exp(-Td). The differences betweenthe experimental results and the continuous line are ascribed to
extra path attenuation.
be exp(--rd), actually decreases more and more show-ing high lengthenings. The differences between P(0r+ Id)/P(Id) and exp(-rd) are very large and cannot besubstantially modified by a small inaccuracy in theevaluation of rd. The results pertaining to a = 1 areonly reported for Id • 0.73 because for larger values ofId the received power with a < 1 was too small to beaccurately measured with our apparatus. The resultsin Fig. 3 show that the lengthening did not depend on ain the range of a investigated for our geometric situa-tion.
The data shown in Fig. 3 were analyzed to obtain apossible form of the density probability function f(8)introduced in Sec. II. Two models of f(6) were consid-ered and used in Eq. (2) for comparison with the data.Figure 4 shows the results of this comparison: thequantity (exp(-Id8)) [Eq. (2)] is reported vs rd. Themarks correspond to the measured data in Fig. 3 for a= 30. The continuous line is a best fit of the dataobtained by assuming a function f(8) = a28 exp(-a8).This is the y function considered by Mooradian andGeller.3 The dashed line is a best fit by means of anexponential function f(8) = b exp(-b8). The values ofthe best parameters a and b of the two functions were a= 1.256 and b = 0.448. Note that the y function gave avery good fit with the data, substantially better thanthe exponential function. A x2 test, based on statisti-cal errors of 5% on the measured ratios P(0s + Id)/P(Is),gave a 99% significance level for the y function.
As described in Sec. II.A the function f(8) also repre-sents the time shape of the received pulse when a veryshort light pulse (ideally a delta function of time) istransmitted through the scattering cell. So the knowl-edge of the function f(8) enables us to draw the shape ofthe received pulse. Figure 5 shows the curve corre-
25 _I
0. 00 l
Ea a Ea t
6 ' Ed Fig. 4. The marks represent the same results reported in Fig. 3,pertainingtoa = 3°. Thecontinuousanddashedlinesrepresentthequantity (exp(-T-d3) [Eq. (2)] best fit with the results, which wascalculated by using a y function and an exponential function, respec-tively, for the probability function f(b). The y function fits the
results very well.
.500 l l l
f (6)
375
250
125
0. 000
6 = AL/L= A t/t 0Fig. 5. Plot of the y function that best fits with the results in Fig. 4.The f(b) function has the same shape as the received pulse when a
delta pulse is transmitted.
sponding to the shape of the received pulse as obtainedfrom the y function that fits with the results in Fig. 4.(The total received power can be derived from themeasured apparent optical depth). From Fig. 5 weobtain the value of 0.80 to for the arrival time delay ofthe peak intensity and the value of 1.93 for the half-power pulse width (to is the time necessary for theunscattered power to cross the diffusing medium andin our geometry is -440 ps). These results are aboutthree times greater than those reported by Kuga et al.4
for 0.481-,gm spheres at Ir = 31. This difference can beattributed to the different scattering properties of the
two types of particle: the asymmetry parameter of thescattering function is 0.703 for the 0.33-Am spheresand 0.849 for 0.481-Am spheres.
As stated in Sec. I, a Monte Carlo simulation wasalso carried out to evaluate the attenuation factor andsome statistical moments for the lengthening in thesame situation as in the measurements. We used thesame geometric situation (including the blackenedwalls of the scattering cell) and optical depth. Thephase scattering function used was evaluated by Mietheory taking into account nominal data for the diffus-ing spheres. Table I reports the comparison betweenthe first three statistical moments (6) (2) (63) asobtained from Monte Carlo calculations and from the-y function obtained from experimental results ((6n) =
(n + 1)Ilan for the y function). The calculated andmeasured apparent optical depth ra pertaining to a =10 is also reported.
Figure 6 shows results pertaining to a small value ofI. The +, 0 and X refer to experimental results per-taining to a = 30,20,10, respectively. We followed theprocedure described in Sec. II.C., that is, the attenua-tion measurements carried out with different values ofthe receiver FOV were used to obtain the unscatteredreceived power Po[Po(r) = Po(r = 0) exp(-r)] by ex-trapolation, so that an accurate measurement of I waspossible both with spheres alone and with spheres plusdye. The ratio between the scattered received powerPd and the unscattered one P0, was thus analyzed. Ifno elongation occurred, Pd/PO would remain constantwhen the dye is introduced without changing the con-centration of spheres. In Fig. 6 the marks for I < 7.06correspond to measurements taken with differentquantities of spheres without dye, while the marks forr > 7.06 correspond to a fixed concentration of spheres(corresponding toIr8 = 7.06) with the addition of differ-ent quantities of dye. It can be seen that passing fromI = 7.06 to I = 14 we have a reduction (due to extraattenuation) of -30% on Pd/Po. The reduction wasabout the same for all the used values of a, showinglengthenings nearly independent of a in our geometryfor the values of a investigated.
In the same figure (continuous curves) the results ofa numerical simulation using a Monte Carlo-basedcode (Semoc code)' 4 are also reported. We underlinethat the Monte Carlo procedure was used for the calcu-lation of one point only for each value of a (in theparticular case of the figure for r8 = 7). The otherpoints of the continuous curves were deduced from the
Table 1. Comparison Between Calculated and Experimental Results forr = 31.0; Particle Size 0.33 jm
Monte Carlo Experimentalresults results"
(a) 1.8 1.6(32) 3.9 3.8(33) 10.2 12.3Ta 13.8 14.3
a The experimental results pertaining to the statistical momentsrefer to the -y function obtained with the best fit (Fig. 4).
Pd /Po
100
050
0. 000
Fig. 6. Measurement example at a small value of optical depth.The ratio between the scattered (Pd) and unscattered (P0) receivedpower is reported vs r for a = 30 20, 1° (+, 0, X, respectively). Datapertaining to T S 7.06 refer to different concentrations of spheresonly; those pertaining to r > 7.06 refer to a fixed concentration ofspheres and different concentrations of dye. The continuous curvesrepresent the results of a numerical calculation based on the Monte
Carlo method.
results of the data at r = 7, by using the scalingrelationships introduced in Ref. 15. These relation-ships involve calculated values of the statistical mo-ments of the path elongation of scattered radiationreceived at a particular value of r8 (here at rs = 7). Theagreement between theoretical and experimental datais good especially with regard to the down slope of thecurves for > 7.06 which is related to the above-mentioned statistical moments of path elongation.
The results Pd/Po in Fig. 6 pertaining to a = 30 andcorresponding to the measurements when dye wasused are also reported in Fig. 7 normalized to the valueof Pd/PO corresponding to the maximum concentrationof spheres and no dye. The continuous line representsthe second degree polynomial [Eq. (3)] that fits withthe experimental results. The obtained values of thecoefficients of the polynomial give (3 ) = 0.045 and (62)
= 0.0011 corresponding to a mean temporal lentheningof -20 ps. The results obtained by the Monte Carlosimulation are () = 0.050 and (32) = 0.007.
We thus measured an average path lengthening ofthe order of 5% of L on the only scattered receivedpower (for P0 there is no lengthening) that is only 12%of the total received power when a = 30 and 1% of Pfor a = 10.
The measurements for small values of I, when thelengthening on the total received power is very small(as in the case just considered), might be of little inter-est from a practical point of view, because the length-ening in this case is practically insignificant. Howev-er, from a general point of view, the results can beuseful in understanding the problem of the propaga-tion of a light beam in multiple scattering conditionsand for a comparison with results of the analytic treat-ment of the problem.
Fig. 7. The same data as in Fig. 6 pertaining to aas the ratio Pd(Ts + Td)/Pd(r.,) exp(Td) vs Td with Tis equal to Pd(Td + T)/Po(rd + T) divided by Pdl7.06. The continuous line represents the second
that best fits with the results.
V. ConclusionsWe have presented a simple and inex
od of measuring the temporal lengtherpulse propagating in turbid media. 'Ibased on attenuation measurements carcw laser source. Preliminary results shcmeasurements of temporal lengthenintained both in the case of high values creceived power is almost totally due to ssmall values of . We believe that timethod can be useful to investigate bot]problem of light pulse communication aproblem of propagation in multiple scations.
The experimental results refer to the a thin beam (where beam diameter andsmall with respect to the receiver's FOV). However analogous measuremepeated for any general shape of the lighdrical, conical, Gaussian, plane wave, iscetc.) so that the experimental method cautility.
A comparison with the results of asimulation referring to the consideredgeometric conditions, is also reportedagreement between the experimental datoretical previsions.
This work was supported by M.P.I.-40% funds.The authors wish to express their appreciation to NedaBianchini for her help in preparing the typescript.
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