Measurement of mode times of flight in multimodefibers by an interferometric method usingpolychromatic light: theoretical approachand experimental results

Gerald Brun, Isabelle Verrier, Marie Ramos, Jean-Pierre Goure, Paul Ottavi,and Anne-Marie Lambert

The propagation of several modes in an optical fiber is not easy to study. The experiment that wepropose permits us to measure the difference in time propagation between two successive modes of amultimode fiber. The same laser beam is coupled into the fiber to be tested and into the referencesingle-mode fiber. The correlation of output electric fields of the modes propagated by each fiber isrealized by an interferometric system.Key words: Interferometry, correlator, modes, optical fiber, temporal holography, group delay.

r 1996 Optical Society of America

1. Introduction

Most analysis techniques used in instrumentation,in sensor applications, for example, are based oninterferometric methods with monochromatic light.The sensitivity is excellent but dynamic perfor-mances are limited because the interference ordercannot be determined. For this reason, interferom-eters with a large spectral bandwidth of light areattractive. These techniques, which are in the samecategory as modulation of temporal coherence, per-mit direct measurement of an optical path withexcellent precision and good dynamics. This kind ofmeasurement of mode propagation time in opticalfiber realizes amplitude correlation of the electricfields of optical waves propagated by fibers that areset in an interferometric device. A lot of researchhas been done on amplitude correlators,1–13 but ourmethod consists of a modified version of the ones

G. Brun, I. Verrier, M. Ramos, J.-P. Goure, and P. Ottavi arewith the Laboratoire Traitement du Signal et Instrumentation,Unite de Recherche Associee-Centre National de la RechercheScientifique 842, 23 rue du Docteur Paul Michelon, 42023 Saint-Etienne, Cedex 2, France. A.-M. Lambert is with the InstitutUniversitaire de Technologie de Poitiers, 6 allee Jean Monnet,86000 Poitiers, France.Received 25 July 1995; revised manuscript received 26 October

1995.0003-6935@96@071129-06$06.00@0r 1996 Optical Society of America

presented above, because the amplitude correlator isdifferent and has the advantage of directly display-ing in real time the correlation function. Thismethod is called temporal holography. In this pa-per first we present theoretical considerations ofwave propagation in optical fiber and temporal holog-raphy. In Section 3 we present the experimentalresults obtained with some particular conditions ofthe experimental setup, and we compare theoreticaland experimental values.

2. Theoretical Approach

A. Modal Analysis of Light Propagation in Optical Fibers

1. Fiber ParametersLet us consider the step-index optical fiber repre-sented in Fig. 1. The optical profile index is sup-posed to depend only on radius component r, and thedifference between the core index nc value andcladding index ng should be weak. If the core ismade of fused silica, we can calculate with goodprecision its refractive-index value nc at the centralwavelength, l0 5 660 nm, of the explored bandwidth.From Sellmeier’s equation14 we have

nc 5 nc1l02

5 11 1A1l0

2

l02 2 A2

21

A3l02

l02 2 A4

21

A5l02

l02 2 A6

221@2

, 112

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whereA1 50.6961663,A2 50.0684043,A3 50.4079426,A4 5 0.1161414, A5 5 0.8974794, and A6 5 9.896161.Supposing that the numerical aperture 1N.A.2 is

constant over the explored bandwidth, we deduct thecladding index value from

ng 5 ng1l02 5 53nc1l0242 2 1N.A.2261@2. 122

The group index of the core is defined by

Nc 5 Nc1l02 5 nc1l02 2 l01dncdl 2l5l0

, 132

where quantity 1dnc@dl2l5l0can be expressed by

derivation of Sellmeier’s equation15:

3dncdl 4l5l0

5 2l0

nc1l02

3 3 A1A22

1l02 2 A2

2221

A3A42

1l02 2 A4

2221

A5A62

1l02 2 A6

2224 .142

Group indexNg of the cladding is defined by a similarrelation.For one wavelength, each mode of the fiber can be

described by two parameters, l andm, and character-ized by its propagation constant blm; this is related tonormalized propagation constant Blm, which is usu-ally used by most authors:

Blm 51blm@k22 2 ng2

nc2 2 ng2, 152

where k 5 12p@l2 is the propagation constant invacuum.Another parameter usually used is normalized

frequency V defined as

V 52p

laŒnc2 2 ng2, 162

where a is the core radius of the optical fiber.Blm are solutions of the implicit dispersion equa-

tions that can be written, for the first two linearly

Fig. 1. Optical fiber.

1130 APPLIED OPTICS @ Vol. 35, No. 7 @ 1 March 1996

polarized modes, in the scalar approximation16–19:

J01VŒ1 2 B012

VŒ1 2 B01J11VŒ1 2 B0122

K01VŒB012

VŒB01K11VŒB0125 0

for the LP01 mode, 172

J11VŒ1 2 B112

VŒ1 2 B11J01VŒ1 2 B1121

K11VŒB112

VŒB11K01VŒB1125 0

for the LP11 mode, 182

where J1 is the Bessel function andK1 is theMacDon-ald function 1l 5 0, 12.For a given optical fiber the V value is defined by

working wavelength l 3Eq. 1624, and the numericalresolution of Eqs. 172 and 182 gives the values of Blm ofthe LPlm modes for the considered wavelength. Weare interested in propagation times tlm of each mode,which are defined as

tlm 5L

c

ngNg 1 3Blm 1 1V2 21dBlm

dV 241ncNc 2 ngNg2

Œng2 1 Blm1nc2 2 ng22, 192

where c is the light speed in vacuum, L is the fiberlength, andNg andNc are given by Eqs. 112–142.

2. Numerical ValuesIn our case two fibers are used: One is monomodeand used as a reference at a working wavelength,and the other is bimode when tested. Index Rcharacterizes the reference monomode fiber, andindex T is the multimode tested fiber. In Tables 1and 2 we give numerically useful values to calculatethe mode propagation times in each fiber. It ap-pears that the difference in time propagation be-tween the LP01 and LP11 modes of the multimodefiber is Dt12 5 t2 2 t1 5 736 fms.

B. Reminder of Temporal Holography Theory

Temporal holography is an interferometric methodthat can be compared with the Fourier holographymethod if a spatial variable 1spatial frequency in the

Table 1. Characteristic Values of the Monomode Fiber

Parameters Values

Length L 1.00 mCore index nc 1.45627Cladding index ng 1.45132Core radius aR 1.4 µmNormalized frequency 1V02R 1.5993Normalized propagation constant of the LP01mode B0 5 1B012R

0.2698

Propagation constant of the LP01 modeb0 5 1b012R

1.3829 107 m21

Propagation time of the LP01 mode t0 5 1t012R 4.91882 ns

transformed Fourier space2 is substituted for a tem-poral variable 1temporal frequency in the trans-formed Fourier space2. The goal is to measure thedifference in time propagation between two succes-sive modes of a multimode fiber.The experimental setup 1Fig. 22 is based on four

important elements. A dye laser can be tuned overa wavelength bandwidth range of Dl < 30 nmaround central wavelength l0 5 660 nm. A Mach–Zehnder interferometer consists of monomode andmultimode fibers in each arm. Aspectrometre inter-ferentiel a selection par l’amplitude de la modulation1interferential spectrometer by selection of ampli-tude modulation2 1SISAM2, which is a Michelsoninterferometer in which gratings replace mirrors,7,8combines coherently the beams from the Mach–

Table 2. Characteristic Values of the Multimode Fiber

Parameters Values

Length L 1.00 mCore index nc 1.45627Cladding index ng 1.45132Core radius aT 2.761 µmNormalized frequency 1V02T 2.7417Normalized propagation constant of the LP01mode B1 5 1B012T

0.6052

Normalized propagation constant of the LP11mode B2 5 1B112T

0.0946

Propagation constant of the LP01 modeb1 5 1b012T

1.3848 107 m21

Propagation constant of the LP11 modeb2 5 1b112T

1.3827 107 m21

Propagation time of the LP01 mode t1 5 1t012T 4.922532 nsPropagation time of the LP11 mode t2 5 1t112T 4.92326 ns

Fig. 2. Experimental setup.

Zehnder interferometer. A beam splitter, set 45 degfrom the beams diffracted by the gratings, adds thesignals. A lens behind the interferometer imagesthe grating signals onto a linear photodiode arraywith storage and data processing.The two gratings act as variable delay lines as a

function of the emitted wavelength; the delay intro-duced varies linearly across the diffracted beamtransverse profile. Then the information at the endof the fibers is optically correlated when opticalwavelength bandwidth Dl is explored by the laser.In this way the temporal variable 1the time propaga-tion difference between the modes of the multimodefiber2 is changed in a spatial variable measured onthe recorder. The signal that we characterize isissued from the multimode fiber. We compare thissignal to the one issued from the monomode fiber,which is a time reference and compensates forchromatic dispersion of the signals. At the end ofthe correlator the signal can be represented in spaceas in Fig. 3 where all the wavelengths in the exploredbandwidth,Dl, are considered. For one wavelengthwe obtain interference fringes that are characterizedby a specific interfringe; for another wavelength weobtain interference fringes with different inter-fringes 1Fig. 42. When all the wavelengths are con-sidered in the explored bandwidth, different pat-terns are superimposed and there are some particularareaswhere all the extrema of light intensity coincide.In this way the final light-intensity profile shows aflat tint 1continuous background2 on which morecontrasting spots appear; one can observe as manyspots as modes in the multimode fiber.

Fig. 3. Correlation between signals issued from the two fibers.

Fig. 4. Superposition of interferences fringes for two differentwavelengths.

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A theoretical description of this experience isbased on the modal theory of light propagation infibers. This model is given in Ref. 20. At the endof calculation, we can write the final intensity as

E c 5 1a02s02 1 a1

2s12 1 a22s222G2Dn

1 2a1a2s1s2G2Dn cos1Db12L2sin c1pDt12Dn2

1 2a0a1s0s1G2Dn cos1Db01L2sin c3p1QX 1 Dt012Dn4

1 2a0a2s0s2G2Dn cos1Db02L2sin c3p1QX 1 Dt022Dn4,

1102

where si is the far-field amplitude distribution associ-ated with each mode where i 5 0, 1, 2 are the indicesthat designate the mode of the monomode fiber, thefundamental mode, and the second mode of themultimode fiber, respectively;

ai 5 eeR2

s1x, y2si*1x, y2dxdy

is the coupling coefficient between the initial ampli-tude laser-beam distribution s1x, y2 and the ampli-tude distribution si1x, y2 of eachmode; Dbi j 5 bj 2 bi isthe difference in the propagation constant of eachmode 1i, j 5 0, 1, 22; Dti j 5 tj 2 ti is the difference indelay time of each mode; G 5 G1n2 is the continuumtemporal frequency spectrum. Q5 212N02@1n0 cos u082is a conversion parameter between the difference intime propagation separating two successive modes ofthe multimode fiber and their corresponding spatialdimensions shown on the detector; u08 is the angle ofthe diffracted beam by the gratings and is defined bythe grating formula, sin u08 5 sin u 2 N0l0; u0 is theangle of the incident beam onto the grating forcentral wavelength l0; N0 is the groove number permillimeter of the grating; n0 is the central frequencyof the explored bandwidth corresponding to l0.Figure 5 shows a theoretical curve that gives the

light intensity collected onto the linear detector as afunction of position X onto it. A spatial gap, X0 51Dt02@Q2 2 1Dt01@Q2 5 1t2 2 t0 2 t1 1 t 02@Q 5 Dt12@Q,appears between the two spots representing thefundamental and the second mode of the multimodefiber. By measuring this gap, we can finally deter-

Fig. 5. Theoretical light intensity onto the detector.

1132 APPLIED OPTICS @ Vol. 35, No. 7 @ 1 March 1996

mine the difference in propagation time betweenthese two successive modes. The spots observed inFig. 5 correspond to the maxima of the last twosin c31sin x2@x4 functions in Eq. 1102. These functionsare affected by either positive or negative terms; inthis case these terms are negative, and that is whythe spots are dark. Numerical values of the theoreti-cal model are issued from the geometric parametersof the experimental setup and are given in Table 3.Finally we obtain the theoretical spatial separationbetween the two modes of the multimode fiber:X0 5 0.5 mm.

3. Experimental Results

A. Experimental Output Signal

The experimental setup described above permitsrecording the video signal issued from the detector1the CCD array2 when 61 storages are realized for 61wavelength values symmetrically chosen around thecentral wavelength, l0 5 660 nm, with an entirebandwidth range of Dl 5 20 nm 3one pattern for eachone third of a nanometer 1Fig. 624.21 Two darkerspots separated by distance X08 stand out from thecontinuous Gaussian background. These spots cor-respond to the two modes of the optical fiber to betested, as predicted by theory.SeparationX08 is given byX08 5 n3 x0 µm, where n

is the pixel number between the two spots read onthe CCD array and x0 is the size of one pixel.Spatial quantity X08, resulting from the difference intime propagation between the two modes, is mea-sured on the CCD array with g transversal magnifi-cation of the real quantity X0 directly connected toDt12. Indeed the output lens permits adjusting the

Table 3. Geometric Parameters of the Experimental Setup

Parameters Values

Groove number per millimeterof the gratings

N0 5 150 grooves@mm

Central frequency of theexplored bandwidth

n0 5 4.5454 1014 Hz

Central wavelength of theexplored bandwidth

l0 5 660 nm

Light-frequency area Dn 5 1.38 1013 HzLight-wavelength area Dl 5 20 nmAngle of the incident ray on thegrating for the central wave-length

u0 5 82.5°

Angle of the refracted ray withregard to the grating normal

u08 5 63.18°

Parameter of conversion Q 5 1.4628 1029 s m21

Delay time of the fundamentalmode of the monomode fiber

t0 5 4.9139 ns

Delay time of the fundamentalmode of the multimode fiber

t1 5 4.9176 ns

Delay time of the second mode ofthe multimode fiber

t2 5 4.9183 ns

Delay-time difference betweenthe fundamental and thesecond mode of the multimodefiber

Dt12 5 736 fs

entire signal size by the CCD detector’s size.Considering optogeometrical conversion factorQ, wegive the value of the delay-time difference betweenthe two modes by

Dt12 5 QX05 QX08

g5 Q

nx01µm2

g, 1112

where g 5 1.35.The measured pixel number in Fig. 6 is n 5 68

pixels, and the size of the pixel is x0 5 10 µm for theCCD used. Then Dt12 5 7.37 3 10213 s 5 737 fs.

B. Comparison between Theoretical and ExperimentalResults

Good agreement is found between the theoreticalmodel 1Fig. 52 and the experimental results 1Fig. 62 asshown in Fig. 7. However, the experimental curvehas small irregularities that may be explained by thediscrete sum of a finite number of curves 161measure-ments2 realized to obtain an experimental signal,whereas the theoretical signal is calculated withintegration over the entire wavelength bandwidthrange.The limits of this system are imposed by spectral

bandwidth Dl of the laser source and also by thepixel dimension. The limit from the explored spec-tral bandwidth corresponds to the minimal thick-ness of the spots imposed by the width of the sincfunction in Eq. 1102. This spatial width is dX 5Xmax 2 Xmin 5 1@1QDn2 and leads to a limitation that

Fig. 6. Experimental results.

Fig. 7. Comparison between theoretical and experimental re-sults.

can be expressed as

QdX 5 1dDt1221 5l0

2

cDl5 73 fs. 1122

Then delay times of less than 73 fs cannot bemeasured in this case. But this time resolutionmay be improved by an increase in the spectralbandwidth of the source and perhaps by a decreasein the central wavelength, l0. As to the detectorresolution, the pixel width is x0 5 10 µm, andbecause three pixels are needed to separate thesespots correctly, the weakest detectable temporalvariation is, taking into account the conversionfactorQ,

1dDt1222 53Qx0

g5 33 fs , 1dDt1221. 1132

The minimum detectable temporal variation corre-sponds to the ability of the detector to separate twospots that are infinitely thin but temporally veryclose 133 fs2. This detector resolution can also beimproved by an increase in the transversal magnifi-cation of the lens at the end of the setup. However,the resolution of our system is limited by the ex-plored spectral bandwidth and not by the detector.Therefore this method leads to measuring times asshort as 1 ps or less with a separation of ,70 fms.

4. Conclusion

We have proposed results of a new method formeasuring the modal group delay difference in mul-timode fibers. The method rests on the use of areal-time amplitude cross-correlator with wide-wavelength bandwidth signals derived from a SI-SAM device.Our system can have a very good temporal resolu-

tion, as short as 73 fs 1although it is used with cwlight2 and allows measurements to be carried out onvery short fiber samples. For the same resolution,femtosecond laser pulses are currently required.These short laser pulses carry weak energy, but theirintensity is often sufficient to induce nonlinear ef-fects that disturb the measurements.This new experiment is attractive because of its

many applications. In this way, very short timescan be measured with great precision. For ex-ample, diopter separation can be measured directlywith this method.

The authors are grateful to A. Barthelemy and C.Froehly from the Institut de Recherches en Commu-nications Optique et Microondes Laboratory 1Limo-ges, France2 for fruitful discussions and collabora-tion.

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