1852 J. Opt. Soc. Am. A/Vol. 2, No. 11/November 1985

Modal noise arising from slightly nonuniform quantumefficiency of a detector

Tadatoshi Tanifuji

Ibaraki Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, Tokai,Ibaraki-ken 319-11, Japan

Received October 31, 1984; accepted June 10, 1985'

Modal noise caused by the nonuniform quantum efficiency of a detector is analyzed. Laser partition noise is takeninto account in the analysis. The signal-to-noise ratio degrades to about 30 dB in the presence of only a 2% nonuni-form detector quantum efficiency. The signal-to-noise ratio dependence on laser-partition noise is clarified.

INTRODUCTION

Modal noise is a serious factor in the degradation of error rateand signal-to-noise ratio in digital and analog optical trans-mission systems.' Spatial filtering existing in the transmis-sion system, both in splice points2 -5 and at the detector,6

causes signal fluctuation associated with the frequency fluc-tuation of a laser diode. Modal-noise dependences on mis-alignment between spliced fibers have been studied boththeoretically and experimentally, and modal-noise charac-teristics have become considerably clearer. However, receivedsignals also fluctuate when there are no splice points becauseof interference among the various modes.

Nonuniform quantum efficiency of a detector allows in-terference among the nonorthogonal components of the fieldvectors for different modes to cause a phase-dependent de-tector-current fluctuation. A small percent of quantum-efficiency nonuniformity exists, especially in avalanchephotodiodes.7 In addition, when frequency fluctuations ofa laser diode are present, the received signal fluctuates. Thefrequency fluctuations of a laser diode are classified into twotypes: continual frequency change and mode partition. Inpractical cases, laser diodes oscillate in multilongitudinalmodes, and mode partition occurs among different longitu-dinal modes.

In this paper, modal-noise characteristics of graded-indexoptical-fiber transmission systems are analyzed when thereis a slight nonuniform quantum efficiency at the detector inthe presence of laser-partition noise. Modal-noise depen-dence on the V value of graded-index fibers and the nonuni-formity of the detector are clarified.

ESTIMATION OF THE INTERFERENCEEFFECT BETWEEN DIFFERENT MODESARISING FROM THE NONUNIFORMQUANTUM EFFICIENCY OF A DETECTOR

Photocurrent I for a detector is proportional to the integralof the transverse-field vector product of the propagatingmodes at the detector surface and is expressed as

I E outEout*q(r, O)dS, (1)

where 7(r, 0) is the quantum efficiency of a detector. Eut isthe electric-field vector at the receiving end of the fiber andis given by

nEout= E,,v(r, 0) E ail 2 exp j(coit - /3M(i)z)], (2)

AV i1

where 0,3(i) is the phase constant of a mode with a radial-modenumber ,u and an azimuthal-mode number v and ai 1/2

(i = 1,

n) is the ith electric-field amplitude radiated from a laserdiode whose oscillation angular frequency is coi. In the aboveequation it is assumed that modal and chromatic dispersionare negligibly small compared with the oscillation-durationtime of each longitudinal mode radiated from a laser diode.The assumption is valid when modal noise in a graded-indexfiber at the wavelength of 1.3 Atm is considered. All the modesare assumed to be equally excited, to simulate the signal-to-noise ratio in the worst case. Specifically, the mode fieldEM(r, 0) is dependent on xi, as shown later in Eq. (4), but thexi dependence of the EM,(r, 0) has a difference of about 10-4in amplitude for adjacent longitudinal modes of a typicalsemiconductor laser. Therefore wavelength dependence ofthe EM(r, 0) is reasonably neglected. Substituting Eq. (2) intoexpression (1), we obtain

I E 2 f JE,,lVl(r, 0)EM27(r, O )dS r E ai)pllS i=l-

+ E E S E,1,,(r, O)E,,,,,(r, 0)rq(r, N~SAIV1 AMV

X (n X _ ai cos Af, 'A22(~

~i=1 (3)

When the quantum efficiency q(r, 0) of the detector isuniform at all areas of the detector, the second term of ex-pression (3) diminishes. In this case no signal fluctuation isobserved as long as the total radiation power of the laser dioderemains constant. However, when ij(r, 0) is not uniform, theintegral in the second term of expression (3) takes a finitevalue. In this case, the received signal fluctuates because oftemporal change of the ai's, even if the total radiation powerof the laser diode remains constant. Thus the fluctuationamplitude depends on both the phase difference between thepropagating modes at each radiation frequency of the laser

0740-3232/85/111852-05$02.00 C 1985 Optical Society of America

Tadatoshi Tanifjui

Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. A 1853

diode and on the longitudinal-mode fluctuations. For thefirst time, the integral in the second term of expression (3) isevaluated for graded-index fibers.

The transverse electric field E$V(r, 0) of graded-index op-tical fibers is given by the following equations 8 :

EsV(r, 0) = FyV(r)cos vO,

FyV(r) = -( +)V! 1/2 1 (2) V(Ž2rexp(2r2/W2),

k= I C a Y' (4)

vl + V2 = A

V - V2 = As

Vl -V2- =-M, (8)

where P(Al, V1, A1, v 2) is given by

p~y y ^) = r Al + Vl)(2A2 + V2 )A E 2 E A ( A(2)8 L /A2 r=O r'=O r r

1X

and

OpV 2 (= n) 2 [l _ 2(2A + v + 1)Vknla

The notation used in the above equations is the fol-lowing:

k = co/c, wave number in free space;c, light velocity in free space;n1, refractive index of fiber core;A, relative-index difference of fiber;L,, Laguerre's polynominal.

We assume that the detector quantum efficiency -(r, 0) isgiven by

77(r, 0) = i70 + Af7 cos(21rN W) cos(MO)(no» >> A-). (5)

In the above equation, N and M represent the period ofradial and azimuthal quantum-efficiency variations, respec-tively. It is possible that nonuniformity of quantum efficiencyfor an actual detector is approximated by a superposition ofthe second term of the above equation with different M andN. For the radial quantum-efficiency variation, the squarefunction of r is assumed in order to obtain an analytical ex-pression for p(Al, vi, 2,u v2) in Eq. (6).

Substituting Eqs. (4) and (5) into the second term of ex-pression (3) and replacing x = 2r2/W 2, we obtain

P(g1, vl, A2, V2 ) = ,f EAlVl(r, O)EA2V2 (r, 0)77(r, W)dS

= An7 f XMl+v2/ 2 LlV1(X)L, 2V2(X)

X exp(-X)cos v10 cos V20

X cos 2rNX cos MO dS. (6)

From this equation it is shown that when only radial nonun-iformity is present, interference between the modes with thesame azimuthal-mode number occurs.

In this case, p(,ul, Vl, , 2 , v2) is given by

A 1(l + Vl)Y 2 + V2\ 1 2 AllM 22P(g1, Vl, A2, V2 ) = - I\Y I r

4 A1 2 r=O r'=O r r

x pr (i + + r + r'+ I (9)

As shown in Eqs. (8), the combination of different modesthat satisfy it decreases as M increases. Therefore the sig-nal-to-noise ratio is considered to improve as M increases.

MODAL-NOISE DEPENDENCE ON LASER-PARTITION NOISE

In this section modal-noise dependence on laser-partitionnoise is analyzed. It is assumed that the total output powerof laser diodes that oscillate with multilongitudinal modes isconstant through time, and only partition noise exists. Re-ceived photocurrent fluctuation arising from laser-partitionnoise is given by differentiating expression (3) by ai as

A = -NAal +-Aa 2aal aa2

+. .. + han.Adan

The time-averaged photocurrent fluctuationby

(AI) is given

(Al) =-(Aa 1) +-(Aa2 )oal d1 a 2

+... + N(Aa').a'

(11)

Let us assume that each longitudinal mode partition isapproximated by a statistical process whose average value iszero and whose root mean square remains a finite value, asshown in Fig. 1. Under this condition the time-averagedfluctuation of the photocurrent also becomes zero. Thereforethe variance of I is calculated using Eq. (10) with the resultbeing

(10)

X(v + r + r')

(v + 1) ... (v + r)(v + 1) ... (v + r')[1 + (27rN)2](V+r+r'+l)/2

In this equation (Q'+) indicates the combination selectingthe v group number from the total number of ,u + v. On theother hand, when only azimuthal nonuniformity exists, in-terference among the following modes takes place:

cos[(v + r + r' + 1)tan-l(2wN)]. (7)

r2 = (,I2) = F d (A\a,2)i=1 dTa)

+ 57 Y-i(dI)(dI'(AaOAaj)1i=l j,. T aj Taj

Tadatoshi Tanifuji

1854 J. Opt. Soc. Am. A/Vol. 2, No. 11/November 1985

l~in

1-/

a/

1.0

HI __ I_

t

Fig. 1. Statistical model of laser-partition noise; time-average am-plitude of each logitudinal mode is assumed to be 1/n.

where

ci -A;1 E P(Al, Pl, A2, PO s "VU1V2P2(i)Zaai lP1 A2P2(i = 1, n).

(12)

Using the relation 7,=1 Alai = 0, Eq. (12) is transformedinto

j[( raa) (O) j(AcaiAaj). (13)i=l j~d ha a aiI

As shown in this equation, 0a2 becomes zero if 1/Oact = OI/

0aj1 (i i j). This means that if the phase difference betweenmodes is identical for each longitudinal mode of a laser diode,no signal fluctuation is observed. Thus the fluctuation am-plitude depends on the phase difference between modes foreach longitudinal mode at the output end of the fiber.Fluctuation amplitude also depends on the laser-partitionnoise characteristic, that is, (aiActaj) in Eq. (13). An exactestimate of laser-partition noise requires a knowledge of in-ternal laser parameters, which are difficult to measure.Therefore we assume that partition noise takes the extremeform of mode hopping with the instantaneous power whollyon one of the n modes, as shown in Fig. 1. In this case, thestandard deviations of (/a, 2) and (lai Aaj) are given by

-0.5 -10 1 1

NFig. 2. p as a function of radial quantum-efficiency fluctuation pe-riod M calculated by Eq. (7).

3 illustrates the azimuthal quantum-efficiency variation de-pendence of p calculated from Eq. (9). As shown in the figure,p decreases gradually as the difference between Al1 and .2 in-creases.

Figure 4 shows the signal-to-noise ratio when azimuthalquantum-efficiency nonuniformity exists at the detector. Inthe calculation, it is assumed that laser diodes oscillate withtwo longitudinal modes with only the existence of partitionnoise. The signal-to-noise ratio changes rapidly with length.Therefore, the minimum value of this ratio is estimated.

It is found that as the variation period increases, the sig-nal-to-noise ratio improves. However, for an M smaller than2, the ratio degrades to about 30 dB, implying that the re-ceived signal fluctuates about 0.3 dB. Figures 5(a)-5(c) showplots of the spatial form of the detector-surface quantum ef-ficiency forM = 1 to 3 for A- = 0.1. In the actual transmis-sion system, an avalanche photodiode is used. The nonuni-formity of the quantum efficiency reaches a few percent. 7

This causes a signal fluctuation such as misalignment at theoptical-fiber connector.

and

(Atai ajc) = - - (Aai2)n

(i = 1, 2,... n). (14)

Using Eqs. (13) and (14), the signal-to-noise ratio is givenby

SNR = (I)/a. (15)

NUMERICAL RESULTS

First, p(mi, Vl, 1.2, V2) given by Eqs. (7) and (9) was estimated.In the calculation, the core diameter and the refractive-indexdifference of a graded-index fiber are assumed to be 50 gm and1%, respectively. Figure 2 shows the radial quantum-effi-ciency variation dependence of p at a wavelength of 1.3 Am.It became clear that for an N smaller than 10, the field vectorsof each mode approximately form an orthogonal set. Theninterference among the different modes gradually increasesand takes its maximum at a certain value of N. The maxi-mizing value is different for different pairs of modes. Figure

1.0

0.8

0.6

P Ii= lVi~

0.4 - M=2

0.2 M M=4

CO. 0 1/2 3 4 5

-0.2 - 1 M=4

-0.4

-0.6

-0.8 -

-1.0 _

Fig. 3. p between different modes calculated by Eq. (8).

-

Tadatoshi Tanifjui

(Aa,2) = I [n - 1n2

(i = 1,2 .... n)

Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. A 1855

50n= 2

,?= 0.02

6=0.5%

z 40C/,

30r2 4 6 8 10

MFig. 4. Signal-to-noise ratio as a function of azimuthal quantum-efficiency variation period M.

50

0

1 40

30

2 4 6 8 10

nFig. 6. Signal-to-noise ratio dependence on the number of longitu-dinal modes of a laser diodes.

M=1 M=2(a) (b)

M=3(C)

Fig. 5. Spatial form of the detector-surface quantum efficiency for (a) M = 1, (b) M = 2, and (c) M = 3 for Aqj = 0.1.

Figure 6 shows the signal-to-noise ratio dependence on thenumber of longitudinal modes. As shown in the figure, theratio improves slightly as the number of longitudinal modesincreases, indicating that modal noise is not suppressed onlywhen the number of longitudinal modes is increased.Mode-partition noise must also be suppressed to improve thesignal-to-noise ratio. This ratio is also calculated to be below70 dB when nonuniformity of radial quantum efficiency ispresent. This is because interference takes place only be-tween modes with the same azimuthal mode number. In theactual case, azimuthal nonuniformity, rather than radialnonuniformity, occurs as shown in Fig. 2 of Ref. 7. Thereforeit is important to reduce azimuthal nonuniformity to suppressmodal noise.

CONCLUSION

Modal noise caused by nonuniform quantum efficiency inassociation with laser-partition noise has been analyzed. Itwas found that the signal-to-noise ratio degrades to about 30dB in the presence of only 2% azimuthal nonuniformity of

detector quantum efficiency. Radial quantum-efficiencynonuniformity does not degrade the signal-to-noise ratio.Furthermore, the signal-to-noise ratio is not dependent on thenumber of longitudinal modes in the presence of mode par-tition.

ACKNOWLEDGMENTS

The author wishes to thank K. Kojima, N. Uchida, and Y.Negishi for their encouragement.

REFERENCES

1. R. E. Epworth, "The phenomenon of modal noise in analog anddigital optical fiber systems," in Prbceedings of Fourth Conferenceon Optical Communication (Institute for Applied Physics, SwissFederal Institute of Technology, Zurich, 1978), pp. 492-501.

2. H. Olsen, "Dependence on modal noise on source coherence andfiber length," Electron. Lett. 16, 217-218 (1980).

3. G. E. Rawson, J. W. Goodman, and R. E. Norton, "Frequency de-pendence of modal noise in multimode optical fibers," J. Opt. Soc.Am. 70, 968-976 (1980).

M=21)? = 0.02

- =0.5%_ - - a= 1.0 %

~~~~~ I

Tadatoshi Tanifuji

I

1856 J. Opt. Soc. Am. A/Vol. 2, No. 11/November 1985

4. K. 0. Hill, Y. Tremblay, and S. Kawasaki, "Modal noise in multi-mode fiber links: theory and experiment," Opt. Lett. 5,270-272(1980).

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Tadatoshi Tanifjui

T. Iwakami, and K. Kobayashi, "Low noise and high temperatureInP/InGaAsP/InGaAs avalanche photo diodes with a planarstructure grown by vapor phase epitaxy," in Proceedings of TenthEuropean Conference on Optical Communication (SEL ResearchCenter, Stuttgart, 1984), pp. 220-221.

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