Bandwidth estimation for multimode optical fibers using thefrequency correlation function of speckle patterns
Behzad Moslehi, Joseph W. Goodman, and Eric G. Rawson
In this paper we present a new method for estimating the bandwidth of multimode optical fibers based onthe frequency correlation function of the speckle patterns generated by the interference of fiber modes.This technique, which does not require a pulse or signal generator, can be utilized to estimate the bandwidthof a multimode fiber using a relatively short length of fiber. By applying this method to a test fiber we ob-tained a bandwidth of -36 MHz km which is in relatively good agreement with the -44-MHz km band-width measured by a conventional pulsed technique.
1. Introduction
Both time- and frequency-domain methods havebeen used to study the intramodal (or chromatic) andintermodal dispersions of optical fibers.1-13 Thetime-domain measurements are concerned with pro-ducing and detecting short optical pulses to observeeither the time delay or the broadening of pulses passingthrough the fiber. Since it is difficult to generate anddetect pulses shorter than 100 psec with an acceptablesignal-to-noise ratio,5 it is necessary to use longerlengths for more accurate measurement of large band-width fibers. There are a few techniques 6 7 for in-creasing the effective length of a fiber by either lettingthe pulse shuttle back and forth between mirrors placedat each fiber end (shuttle pulse measurements) or re-circulating the pulse in the fiber by means of a deflector(circulating-pulse measurement7 ), but the fiber lengthsrequired are still substantial, and the design andalignment of the mirrors in the former and the deflectorin the latter add to the complexity of the systems.Some time-domain methods need a computer to cal-culate the impulse response or its Fourier transform.
The frequency-domain techniques have, in general,advantages over the time-domain techniques in termsof resolution, signal-to-noise ratio, and apparatus sim-plicity.5 On the other hand, some frequency-domaintechniques require a tunable (from dc to a few GHz)signal generator (or light modulator) to sinusoidally
Joseph Goodman is with Stanford University, Department ofElectrical Engineering, Stanford, California 94305; the other authorsare with Xerox Palo Alto Research Center, 3333 Coyote Hill Road,Palo Alto, California 94304.
modulate the intensity of the light source. There arealso interferometric methods'2 (based on the schemedescribed in Ref. 13) for measuring the temporal im-pulse response of short samples of both multimode andsingle-mode fibers. In these technqiues the test fiberis placed in one arm of a Mach-Zehnder interferometerexcited by a broadly tunable laser source, and the in-terference pattern at the output of the interferometeris recorded on a spectrogram; the temporal impulseresponse is then obtained from a holographic recon-struction using another laser source as the referencewave.
In this paper we present a new method for estimatingthe bandwidth of a multimode optical fiber. This newmethod is particularly suitable for measuring band-widths on relatively short lengths of fiber. The specklepatterns of the fiber, generated on its endface at dif-ferent optical frequencies, are used to calculate a fre-quency correlation function (defined later) using a mi-crocomputer. We will show that the 3-dB bandwidthof the fiber is proportional to the 3-dB bandwidth of thespeckle frequency correlation function. Assuming aplanar waveguide model with perfect step profile andequal excitation of all guided modes, we calculate theratio of the two bandwidths. The speckle patternmeasurements, together with the calculated bandwidthratio, are used to estimate the bandwidth of the fiber.
11. Theory
When coherent light is launched into a multimodeoptical fiber, it will propagate in a finite number of fibermodes, each of which travels at a slightly different ve-locity in the fiber. At any point on the fiber end, themodes interfere with one another. This interference,which is due to different velocities of different modes,produces the speckle patterns. 14 The speckle frequencycorrelation function (referred to as the FCF in its nor-
malized form) is defined as the correlation function oftwo speckle patterns at optical frequencies i' and v + AP'.Increasing the frequency difference AP' decreases thecross correlation of the speckle patterns. For a givenAP' the amount of decorrelation depends on the inter-modal dispersion of the fiber.
Assuming a planar waveguide model with perfect stepprofile and equal excitation of all guided modes, thefollowing expression has already been derived for theFCF (see Ref. 15 for details of the proof):
p(Av) [C(y)/y]2 . (1)
In Eq. (1), p(Av) denotes the FCF value at frequencydifference AP,
C(y)= Jo cos(7rt2/2)dt (2)
is the cosine Fresnel integral, and y is a parameter whichdepends on the numerical aperture N.A., length L, andrefractive index n of the waveguide, and is given by thefollowing expression:
y = [2L(N.A.) 2Av/cn]1/2 . (3)
(c is the vacuum velocity of light). At y 1.08 the FCFvalue drops to 0.5; therefore, using Eq. (3), the 3-dB(decorrelation) bandwidth of the FCF is
AP3 dB 0.58[cn/L(N.A.) 21. (4)
According to Eq. (4), the 3-dB decorrelation bandwidth,AP3 dB, is inversely proportional to the length L and thenumerical aperture squared (N.A.)2. In this paper weuse the same simple model to calculate the incoherenttransfer function (i.e., the transfer function valid whenall modes add incoherently) Hinc(f) of the fiber. Recentcalculations by Weierholt,16 to be published elsewhere,confirm that the slab waveguide model is an excellentapproximation to the cylindrical step-index fiber.Consider a planar waveguide of width W, length L, andrefractive index n, with a source on-axis at the input end(see Fig. 1). A detector of width Wd W is centeredat the output end, and x is the dimensional coordinateacross the waveguide. A large number of multiply re-flected images of the source, ranging from - 0 m to +Onm(O... is the complement of the critical angle for totalinternal reflection within the fiber), illuminate the de-tector. From a geometrical optics point of view, thesereflected images are separated by a transverse distanceW. However, for fiber lengths greater than -1 cm,these separate sources are not resolvable by an aperturewith a width equal to that of the waveguide end. It istherefore necessary to consider the geometrical sourcesin resolvable groups [2K + 1 in number, with K =(W/g) tanOm], each group subtending an angle X,/Wwhere ,\, = A/n is the wavelength of light in the guide.With a paraxial approximation, the slant distance fromthe kth source to x is'5
k 2=L+ L k g XX x 2
2W 2 W 2L
For an intensity impulse input Ii(t) = (t), the outputis
KI.(t) = E 6(t - tk),
k=-K(6)
where t, the time taken by the light to travel 1h, is givenby
tk = (k + k(x)]n/c, (7)
and 6 (x) denotes random fluctuations of the fiberlength; we assume that the resulting random phasefluctuations, oh = (2wr'n/c)5h (X), are uniformly dis-tributed over the interval (-7r,7).
The normalized incoherent transfer function, Hinc(f)is the Fourier transform of Eq. (6), normalized by itsvalue at f = 0. Using this and Eqs. (5) and (7), aftersome algebraic manipulations (see the Appendix) it canbe shown that the spatially averaged expected value ofHinc(f) is given by
I (Hinc(f I ~ IC(y)/yI, (8)
where
y = [2L(N.A.) 2 f/cn]1/2(9)
with C(y) defined in Eq. (2) and f the baseband fre-quency. Here, superbars represent ensemble averagesover many different positions of the fiber, while theangle brackets (-) indicate a spatial average of data, asdefined by
fWd/2(/() (1/Wd) fj-xWd x (10)
Comparing Eqs. (1) and (8) we find the following rela-tion between Hinc(f) and the FCF:
I (H j(ff)) 1 2 p(f). (11)
The incoherent transfer function in Eq. (8) drops to halfof its maximum value at y 1.29, therefore its 3-dBbandwidth is
f3 dB 0.83[cn/L(N.A.)2]. (12)
-I A-*_
X L 1 - k Planar waveguidek iDetector
,
-K ,7
Fig. 1. Planar waveguide of length L and width W, illustrating the2K + 1 regions of width XgL/W resolvable by an aperture of width W.The complement of m is the critical angle for total internal
Comparing Eqs. (4) and (12) we conclude that the 3-dBbandwidth of the Hinc(f) is proportional to the 3-dBbandwidth of the FCF, and the bandwidth ratio is aconstant, or
(At dB)inc = constant (-1.43, for the assumed model). (13)(AV3 dB)FCF
Hence, in principle, the fiber bandwidth can be esti-mated by measuring the decorrelation bandwidth of theFCF. It should be mentioned that, although (due to thesimplicity of the model) Eqs. (4) and (12) underestimatethe individual bandwidths, the bandwidth ratio has amore reliable value. More accurate models and betterapproximations would give a more precise value for thebandwidth ratio. In our analysis chromatic dispersionhas been neglected in comparison with intermodaldispersion. The main conclusion here is that thebandwidth of the FCF can be a good measure for esti-mating the bandwidth of multimode optical fibers.
111. Experiments
In the FCF experiments a speckle intensity distri-bution I(x,v) is first detected, then the frequency of thesource is changed by Ar' to allow detection of the in-tensity I(x,r + Ar'). The detected data are used tocalculate a frequency covariance function, first byperforming spatial averaging over the detected data,then by repeating the experiment many times, and ineffect ensemble averaging the spatially averaged results.Thus the covariance function of interest can bewritten
The FCF is defined as the covariance function in itsnormalized form, i.e.,
(A'- (AP)c(O) = (15)
Figure 2 is a schematic of the experimental apparatusused to measure the FCF. This apparatus is basicallythe same as the one used in Ref. 15. The light sourcewas a single-longitudinal-mode laser diode (MitsubishiML-3001) with a nominal wavelength of 0.848 gin. Thesource frequency was tuned over a range of up to 30 GHzby varying the excitation current of the laser. AFabry-Perot interferometer with a 7.5-GHz free spectralrange and a finesse of -50 was used to monitor the fre-
JIm Fig. 2. Apparatus used to measure the frequencylay correlation function for two different types of
step-index fiber.
_pPrinter! Plotter
1.0
FIBER 2 FIBER 1L=3.3m L=3m
~10.5
o I iI I I I I0.1 1 10 100
Av(GHz)
Fig. 3. Frequency correlation function (FCF) curves obtained fortwo different types of step-index fiber (using the system shown in Fig.2): (a) fiber 1, Siecor fiber with N.A. 0.28, 100-gim core diam, andL 3 m; (b) fiber 2, Corning Fat Fiber with N.A. 0.31, 100-,gm core
diam, and L 3.3 m.
30
2 20 100 Bandwidth * Lengthroduct (MHz km)
0 40
C~ 10
EE
_0 2 4 6 B 10 12Length (m)
Fig. 4. Minimum required source tunability for step-index fibersas a function of fiber length with fiber bandwidth as a parameter.
1.0 _ A s (a)
FIBER 2 SPECKLEL = 3.3 m FOCUSSED
INCREASINGLY -<1 0.5- b MISFOCUSSED
_ ~ ~~~CSPECKLEd
I I i ill I I I I0.1 1 10 100
Av(GHz)
Fig. 5. Frequency correlation function (FCF) curves of fiber 2 forfour different focusing cases: (a) focused speckle pattern; (b), (c),and (d) misfocused speckle patterns (in the order of increasing degree
quency difference AP. With this system the frequencydifference could be set to better than 0.1 GHz. Thespeckle patterns on the fiber end were imaged onto a256-detector linear array. The output power of thelaser was monitored by a separate detector for use innormalizations. Two different types of step-index fiberwith the following nominal specifications were mea-sured:Fiber 1: Siecor fiber with N.A. 0.28, 100-gtm core
diam, and L 3 m.Fiber 2: Corning Fat Fiber, N.A. 0.31, 100-gm core
diam, and L 3.3 m.Figure 3 shows the results of the measurements made
on these test fibers using the system shown in Fig. 2.From this figure, the 3-dB decorrelation bandwidths offiber I and fiber 2 are -8.3 and 2.25 GHz, respectively.Assuming linear extrapolation (no mode coupling), the3-dB decorrelation bandwidth for 1 km of the fiberswould be -25 MHz * km for fiber 1 and 7.5 MHz km forfiber 2. In Fig. 4 we have plotted a set of curves showingthe necessary source tunability for step-index fibers asa function of fiber length, with fiber bandwidth as aparameter. From these curves it is clear that, formeasuring the bandwidth of large bandwidth fibers,either a higher source tunability or a longer length (orboth) is required.
Speckle focusing effects on the FCF curves were alsoinvestigated by varying the position of the imaging lensbetween the end of the fiber and the detector array (seeFig. 2). Figure 5 shows the FCF curves of fiber 2 forfour different focusing cases: one focused speckle caseand three misfocused speckle cases. As can be seen, fora speckle pattern out-of-focus on the detector, the FCFcurve shifts to higher frequencies. Thus, for a givenfrequency difference AP, a misfocused speckle patterngives a higher correlation (and hence a higher band-width) than that of a focused pattern.
By sending short pulses into a 500-m length of fiber1, its pulse broadening was measured (fiber 2 was notmeasured). A Hamamatsu laser light pulser was usedto generate the input pulses. Figure 6 shows photo-graphs of the output pulses emerging from the ends ofthe two different lengths of fiber 1. Figure 6(a) is fora short fiber length (L 0.7 i) and Fig. 6(b) is for a longfiber length (L 500 in). The full width half-maximumpulse widths, TFWHM, of the output pulses are -0.5 and-4 nsec, respectively. Therefore, for L 500 in,TFWHM of the impulse response is also -4 nsec. Thisyields a 3-dB bandwidth of fA dB 0.35/TFWHM 44MHz km. From Eq. (13) and using the measuredspeckle decorrelation 3-dB bandwidths (AV 3 dB 25MHz km for fiber 1 and 7.5 MHz km for fiber 2), weestimate 3-dB bandwidths of -36 MHz km for fiber 1and 11 MHz km for fiber 2; such low bandwidths arecharacteristic of step-index fibers. Considering thesimple model and various assumptions used in theanalysis, the agreement (36 MHz km vs 44 MHz * km)is fairly good. Profile distortion, unequal excitation ofthe guided modes, and mode mixing could account forthe remaining discrepancy in the estimated and mea-sured bandwidths. In addition, by using only one light
source and the same launch conditions in both the FCFand conventional bandwidth measurements, we believethat better agreement might be achieved. The pro-posed technique can also be applied to graded-indexfibers, for which some measurements have been re-ported,' 5 but the corresponding analysis has not beencarried out.
IV. Conclusion
In summary, we have presented a new method forestimating the bandwidth of a multimode optical fiberusing the frequency correlation function (FCF) of thespeckle patterns generated on the endface of the fiber.The bandwidth estimation based on this method is inrelatively good agreement with the conventionalbandwidth measurements. This technique does notrequire a pulse generator or light intensity modulatorwhich is usually needed in conventional bandwidthmeasurements. Moreover, lack of intensity modulationeliminates the problem of phase modulation of thesource, which could cause errors in the bandwidthmeasurements. In the speckle method, the, requiredfrequency change (which is constantly monitored) canbe simply achieved by varying the excitation current of
(a)
(b)
Fig. 6. Output pulse waveforms of fiber 1: (a) L 0.7 m; (b) L t
the laser diode source. This system, which requires amicrocomputer for the FCF calculations, is applicableto both step-index and graded-index multimode fibersfor bandwidth estimation using a short length of testfiber. Since the speckle method uses a relatively shortfiber length for bandwidth measurement, it is of greaterimportance when the multimode fibers being measuredhave larger bandwidths and the source wavelength isnear the dispersion-free wavelength of the fiber material(-1.3 gm). Improvement of the model should resultin more accurate bandwidth estimates.
We would like to thank R. E. Norton and M. D. Baileyfor valuable assistance with the experimental mea-surements.
Appendix
The incoherent transfer function of the waveguideshown in Fig. 1 is the Fourier transform of Eq. (6),i.e.,
1 KHi.0 (f) = (2 j exp(-i27rthf), (Al)
(2K + 1) k=-K
where t is given by Eq. (7), the factor 1/(2K + 1) infront of the summation is a normalization factor, andK is given by the integer part of W/X\g tanOm. UsingEqs. (5) and (7), we can rewrite Eq. (Al) as
Hic(f) = exp [ i (-I L +
K . (2-rnIk2
X2L k2 gX . [ f
X =- exp-i I-I 2g -IwI exp I (x )-k =-K c \ 2 W L vI(A2)
where oh (x) = (27rn/c)3 k(x)v are the random phasefluctuations. In the following, we will find an expres-sion for (Hinc(f)); superbars and angle brackets indicateensemble and spatial averages, respectively, as ex-plained before. Taking the ensemble average of bothsides of Eq. (A2), we get the following expression for thenormalized incoherent transfer function:
Hine(f) ex1 p II~L + -I/Ij sinc(2K +1) [(c 2L ke)
K f- [ X k (A3)XZ [t k~c 2W2 WI'X =Kexp g
In the above we used the fact that
exp [iq Oh) = f sinOrf/) ~sinc L' (A4)P P ] (7rflv) 0(4
We also assumed that the random phase fluctuations,k (x), are uniformly distributed over the interval
(-7,7r).Assuming typical values for the parameters involved
in the above equations, it can be shown that the spatialaverage of the ensemble-averaged incoherent transferfunction, given in Eq. (A3), is approximately equal toits value at x = 0, or
1 [. 27rnL\ 1 If
(2K + 1) [ CJ (Hinc(f)) ~ + 1)Kexp I- l fL sinc -e [ (2rL)( f - (A5)
Now, we allow K to become large and pass from a dis-crete summation to a continuous integral expression inEq. (A5). The result is
(HijCf)) ;~ exp [-i f(L ) l sinc (
X f 2 exp[-i47rLf(N.A.)2S2/cn]ds,v-1/2
(A6)
which can be expressed in terms of a Fresnel integral.The result is
I (H 0in(f)) I I [sinc (t)I [C(y)/y] (A7)
where C(y), the cosine Fresnel integral, and its argu-ment y are given in Eqs. (2) and (9), respectively. Wefurther simplify the result by making the approximationthat sinc(f/v) 1 (f and v are the baseband and opticalfrequencies, respectively); therefore the end result is
k7T~.TT)1 -IC(Y)/yI. (A8)
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