Phase Noise in Semiconductor Lasers

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298 J O U R N A L OF LIGHTWAVE TECHNOLOGY, VOL. LT-4, NO . 3, MARCH

Phase Noise in Semiconductor LasersCHARLES H . HENRY

(Invited Paper)

Abstract-The subject of phase noise in semiconductor lasers is re-viewed. The description of noise in lasers and those aspects of phasenoise that are relevant to optical communications are emphasized. Thetopics covered include: Langevin forces; laser linewidth above thresh-old and helow threshold; line structure due to relaxation oscillations;phase fluctuations; line narrowing by a passive cavity section and byexternal feedback; coherence collapse due to optical feedback; the shotnoise limits of several schemes of coherent optical communication, andthe linewidth required to approach these ideal limits.

I. INTRODUCTIONA ASCINATING ASPECT of any single-mode laseris the high degree of spectral purity of the laser ra-diation. A related property is the long coherence time ofthe optical field: the time interval during which two com-ponents of the field emitted at different imes from helaser can stably interfere. The behaviorf the laser is sim-ilar to that of other oscillators. Above threshold, the am-plitude of the optical field is nearly fixed, but the phasemay take any value. The wandering of phase can be de-scribed as Brownian motion or phase diffusion [ l ] . Thisrelatively slow wandering of phase determines both thelaser linewidth and the coherence time. Rapid phase fluc-tuations are also of great importance; they introduce er-rors in coherent optical communications and contribute tothe shape of the wings of the laser line.Interest in the linewidth was present from the very be-ginning of laser physics. In their irst paper proposing thelaser, Schawlow and Townes 2] derived a formula for thelinewidth, predicted that the lineshape would be Lorent-zian and that the linewidth would narrow inversely withlaser power, so that the linewidth-power product AvPo isconstant. This formula is only valid below threshold. Lax[3] pointed out that above threshold, the amplitude fluc-tuations of the laser are stabilized andhis is accompaniedby a X 2 reduction inAvPo.We will refer to this correctionas the modified Schawlow-Townes formula. The detailedchange in the linewidth through the threshold region wascalculated by Hempstead and Lax [4]. The extreme nar-rowness of the linewidth f gas lasers, made measurementof the intrinsic linewidth a formidable problem. However,Gerhardt et a l . [5] finally succeededbyusinga500-mfolded interferometer and by operating their He-Ne laser

Manuscript received July 1 , 1985; revised October 16 , 1985.The author is w ith AT&T Laboratories, Murray Hill, NJ 07974.IEEE Log Number 8406714.

at microwatt power levels. Their measurement confirquantitatively the theoretical predictions for linewidcoherence length, including the power dependence ofAvPo reduction calculated by Hempstead and Lax [4]The interest in linewidth and phase noise was renewin the last few years by a burst of activity stimulatedthe availability of single-mode semiconductor lasers tinuously operating at room emperature and he expec-tation that these devices would find many applicationsquiring a high degree of coherence, such as interferand coherent lightwave communications. The first carlinewidth studies of AlGaAs lasers were made y Flemand Mooradian [6]. They observed the Lorentzian shapand the line narrowing nversely with power as expecbut surprisingly, they found that the linewidth was abo50 timesgreater han hatpredicted by the modiSchawlow-Townes formula. This linewidth enhancemwas explained by the author as primarily due to the in the cavity resonance frequency with gain [7]. Thissults in a correction of 1 + a2 o the modified SchawloTownes formula, where the linewidth parameter1 = AA n " is the ratio of the changes in the real and imaginparts of the refractive index with change in carrier number. In AlGaAs and InGaAsP lasers alpha is about[7], 8]-[I21 (one smaller value has been reported [1A similar correction is expected to occurn gas lasers[141, however the correction in this cases expected tof order unity o r less [14] and zero when the cavity mis tuned to the center of the transition line. In a semicductor laser, the laser line occurs at the foot of a stabsorption edge and this causes a to be large [8]. Anditional factor of about X 2 in linewidth results becauin semiconductor asers, he population associated wlaser transition is not fully inverted [6], [7].In addition o he ncreased inewidth, semiconductorlasers depart from the expectations of the classical latheory in another respect. It was found by Daino et[15] that the line shape is not a perfect Lorentzian, buhas satellite peaks far out in the wings that are separfrom the main peak by multiples of the relaxation oslation frequency of the aser. This structure s a conse-quence of the large value of Q! that causes a couplingphase and amplitude fluctuations 161, [171. Vahala e[I81 showed that opposite satellite peaks differ in hedue to the correlation of amplitude and phaseluctuatioThe broad linewidth of the semiconductor laser (A

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HENRY:PHASE NOISE IN SEMICONDUCTORLASERS 299

= 50 - 100 MHz * mW) makes conventional semicon-ductor lasers unacceptable for many applications requir-ingahigh degree of coherence,e.g., high-resolutionspectroscopy, interferometric sensors, and coherent opti-calcommunications.Thebroad inewidthcanbeover-come in a number of ways. Perhaps the most successfulway is to add a passive section to the laser cavity by an-tireflection (AR) coating the laser facet and adding an ex-ternal reflector several tens of centimeters away from thefacet . The laser linewidth narrows approximately as thesquare of he external reflector separation. Using a dif-fraction grating as the external reflector aids single-modeoperation, stability, and allows the laser to be uned overhundreds of angstroms [19]. Wyatt and Devlin [20] havereported stable single-mode operation with a linewidth ofonly 10 kHz with such devices.The aim of this paper s to review phase noise in semi-conductor lasers. No attempt will be made to broadly sur-vey all work in this field, instead we will concentrate inmoredetail on what seemso be the essentials f the subject,especially those aspects that are relevant to optical com-munications. In Section 11, we will develop the equationsgoverning intensity and phase fluctuations in semiconduc-tor lasers. We will take up the linewidth, lineshape, andtime dependence of phase fluctuations in Section 111. Wewill deal with line narrowing by means of a passive sec-tion and by optical feedback in Section IV. We will alsodiscuss the nstability of coherence collapse brought aboutby optical feedback in this section. In Section V, we re-view the detectability limits for several keying schemesused in coherentopticalcommunicationsand he ine-width requirements to reach these limits. brief summaryis given in Section VI.

11. FLUCTUATIONSN S E M I C O N D U C T O R LASERSA . Classical Description of Quantum Noise

Beforedescribingnoise nsemiconductor asers, weshould comment on how to picture the radiation ield of alaser.This is confusingbecause of thecomplimentaryparticle and wave descriptions of light. Phase noise, thesubject of this paper, is a wave phenomenon, while shotnoise, which limits the detectabilityof coherent waves, isa particle phenomenon. We shall think of laser radiationas a classical wave field described by a complex ampli-tude p ( t ) .The shot noise aspects of laser radiation will bethought of as noise associated with the carriers which re-sults from generation and detection of light, because en-ergycanonly be added or removedfrom heradiationfield in quanta of Aw .This viewpoint may not be the only possible one, but tis both simple and consistent with a number f fundamen-tal studies of radiation and lasers:1) Hanbury-Brown and Twiss [21] showed hat whenradiation is divided between two detectors, each detectedsignal consistsof a correlated part proportional to the fluc-tuating light intensity Z(t) = I /3 ( t ) plus uncorrelated shotnoise, which can be thought of as occurring in detection.

2) Using the quantum theory of radiation, Glauber [22]showed hateachmode of th e free-radiation field pos-sesses a continuous setof coherent states. The optical fieldassociated with these states oscillates sinusoidally at themode angular frequency w and approaches the field of aclassical monochromatic wave when the expectation valuefor the energy of the coherent state is many Aw and un-certainties demanded by Heisenbergs principle are neg-ligible. A coherentstate is a pure quantum mechanicalstate, not a mixture, but a state that does not have a def-inite energy. When the energy of a coherent state is mea-sured,destroying he tate,discrete nergies nfiw arefound that have a Poisson distribution. The optical fieldof a single-mode laser can be thought of as a mixture ofcoherent states, that becomes nearly a pure state far abothreshold.3) By using a basis of coherent states, Lax and Louise11[23], [24] showed that, even in the case of nonfree fields,the averages of essentially all measurable laser propertiescan be expressed in terms of averages of a classical (c -number) wave field P(t) associated in a precise mannerwith the quantum field amplitudes. Starting with a fullyquantum mechanical model of a laser, they transformedthe quantum problem into a classical problem f calculat-ing the statistical properties of a fluctuating wave field.They derived he Langevin rate equations and he Fok-ker-Planck equations that P(t) and the probability distri-bution P ( p , t) satisfy. We will discuss the Langevin rateequations in the next section.B . Langevin Rate Equationsquantity, asWe will describe fhe optical field of the laser, a real

E(x , ) = B[P(t)@(x>+ P(t)* +(x>*] (1)where /3 ( t)gives the time dependence,+(x) gives the spa-tial dependence of the optical mode, and B is a constant.It sconvenient tochoose B so that heaverage ntensity( I ) equals the average number of photons in the mode( P ) (electromagnetic energy/Awo). The complex ampli-tude p ( t )can be expressed in terms of two real quantities:intensity Z(t) and phase 4 ( t )

P(t) = ~ ( t ) exp (- wot - i4 ( t ) ) . (2)The behaviorof a semiconductor laser is described y thefield amplitude /3 and carrier number N that controls thegain of the mode. The laser can thus be described as aclassical system having 3 real variables: p , p , and N orI , 4, nd N , where p = /3 + ip.The laser, like many classical systems having severalvariables fluctuating in time, satisfies a set of first-orderordinary differential equations that include random Lan-gevin forces [25]. Consider a system with variables a =(a l ,a 2 , * * ). The Langevin rate equations are

u i = A i @ , t ) + F&), i = 1, 2 , - - * . ( 3 )The Ai are chosen so that ( Fi( t ) )= 0. Without the Lan-gevin force, ( 3 ) is the set of rate equations governing the

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average values (with a small correction when products ofsmall fluctuations are included).The role of the Langevin force is to account for howthe statistical distribution of the variables a ( t ) changes intime. For example, suppose that at t = 0, the variablesare described by ao. After a time t , the fluctuations in thesystem change a. into a distribution of values a describedby probability distribution P (a) .This distribution couldbe found by using a computer to integrate ( 3 ) many timesfrom 0 to t with a suitably chosen random force.In choosing Fi t ) ,we will make the common assump-tion that the system is Markoffian [25], .e. , that the ran-dom forces have no memory and the correlation of theirproducts function is a delta function:

( F i ( t ) ) F , ( u ) ) = 2Dg6(t - U ) (4)where D, s called he diffusion coefficient or diffusionmatrix [25].This name is appropriate because the randomforces cause P(a) to spread in a way analogous to diffu-sion. The spread is limited the drift term y A i , which triesto restore a to the steady-state value.The Markoffian assumption is justified in the caseof thesemiconductor laser for which the major source of fluc-tuations s pontaneousemission.Thisprocess sonlycorrelated foracarrierscattering ime hat soforder

s, a negligibly short time. Only the diffusion coef-ficients are needed to calculate mean square fluctuations.However, for calculations of the line shape and of errorrates, additional assumptions about he Langevin forcesmust be made. These amount to assuming that the Lan-gevin forces have approximately Gaussian amplitude dis-tributions.TheLangevin ateequation or P ( t ) canbederivedeither by using the method of Lax and Louise11 [23], [24]or semiclassically by adding a Langevin force to the waveequation [26], [27]. he resulting equation is

where f l and F p are complex andwo is the cavity resonancefrequency at threshold when the net gain AG is zero. Thelinewidth parameter a determines the change in the cavityangular resonance frequency with net gain. The opticalfield is coupled to the carriers by the dependence of AGon the carrier number N . In the absence of F p and fluc-tuations in N , P increases exponentially with AGt/2 andhas an angular frequency of wo + aAG/2 and I increasesexponentially with AGt.Fora Fabry-Perotcavityofuniformsemiconductormaterial, it is easily established that CY = An /An [7] .When the carrier number changes, the modes shifts in away to keep the real part of the propagation constant k = wn /c constant. This changes the angular frequency byAw = -wugAn l c andchanges hegain G by AG =-2wAn ug/c.The ratio of these two changes is 0112.

The diffusion coefficients relating F p and Fp * are given

tF ,g r t )d t.

E A L 0Fig. 1. The change of complex field amplitude p(t) during a short timThe exp ( - w , t ) time dependence has been removed.

2Dpp* Rwhere R is the spontaneous rate. The physical content(6) and (7) can be seen in Fig. 1 , which shows that dura short time T, the Langevin force changes the complexvalue /3 by F p d t . Equation (6) follows from stationar(that ( F p t + 7) F p t ) ) s independent of t ) and insuthat the change in 0 n Fig. 1 has a random angle incomplex P plane [28, section 1181. Equations (6)andshow hat F p has wo ndependent components and hatthe diffusion coefficient for a product of a component F p in any direction with itself is R f2 . Applying the lawcosines o he riangle in Fig. 1 , it is easily establishethat heaveragechange in I , duringashort ime TRT [7].Therefore it is reasonable to refer to R as the sptaneous emission rate.For a closed cavity, such as an index guided mode whigh reflecting ends, it can be shown [27] hatand

where el/ s the separation of quasi-Fermi levels of conduction ndvalencebandsofhe emiconductor.Evaluations of n,,, for AlGaAs and InGaAsP lasers givevalues of about 2.6 and 1.6, respectively [29]-[31].Tparameter n,,, characterizes the incomplete inversionflevels associated with the lasing transition and goes twhen inversion is complete. Equations (8) and (9) aresentially a statement of the fluctuation-dissipation theo-rem [25], 28, sect. 1231. Theyollowromhe quirement hat nequilibrium,withno ossesfrom hecavity, he spontaneous emissionrate R due to flucttions F p must be equal to the rate of dissipation -G (Authorized licensed use limited to: University of Central Florida. Downloaded on April 27, 2009 at 14:14 from IEEE Xplore. Restrictions apply.


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H E N R Y : NOISE IN S E MI C O N D U C T O RA S E R S 301

i

I / SLOPE = 36.7 1

0 0. 5 1 .@ 1.5 2. 0 2.5I N V E R S E P O W E R ( mW - ' )

Fig. 2. Linewidth of an AlGaAs laser versus inverse pow er at three tem-peratures. From Welford and M ooradian [30].where ( I ) is heequilibriumphotonnumbergiven by-nsp [27]. (In this case, G, f iwo - el/, and nSp re nega-tive.) In the caseof open resonators, such as encounteredin gain-guided structures or in lasers with low reflectionfacets, relation (8 ) no longer holds and R can be muchgreater than Gn,. This was shown by Petermann [32] forgain-guided structures, and by Ujihara [33] and the author[27] for the case of low reflecting facets. In the case of aconventional semiconductor laser with uncoated cleavedfacets,, this enhancement is only about 13 percent [27].Above laser threshold, the coupling of I and N must betaken into account . In this case, it is more convenient toexpress the equations for the fieldin terms, f I and phase4 . The transformation of (5)-(7) is easily done using (2 )and applying the general procedure for transforming Lan-gevin equations given by Lax [34]. This results in

Z = AGI + R +'F,(t)10). a4 = - AG + F+( t )2 (1 1)

with diffusion coefficientsR2011 = 2RI, 2D++= -I' 2D,, = 0. (12)

Equations (12) can easily be understood with the aid ofFig. 1. The fluctuations in I are due to F , given by theproduct of 2.$1 and the component of F p parallel to andthe fluctuation in 4 is due to F6 given by t h e componentof Fp perpendicular to /3 divided by $I. The lack of cor-relation of these two components of F p leads to D,+= 0.The transformation p, p* to I , r$ leading to (10)-(12) isvalid for Markoffian andom orces;however, F p andFp* are not Markoffian on the time scale of optical fre-quencies. A careful discussion of this transformation,us-tifying (lo)-(12) has been given by Lax [1, sect. VI.The rate equation for the carrier number is

fi = C - S - GI + F N ( t ) 0 3 )where C is the current (carrieds), S is th e spontaneous

emission rate into nonlasing modes (radiative and nonra-diative), and GI is the net rate of stimulated emission. Thegain G is actually the difference between the ratesf emis-sion and absorptionG = R - A. The diffusion coefficientsrelating N with I and q5 are2DNN = C + S + ( R + A ) I ,

2 0 ~ 1 -2RI, 2 0 1 6 = 0. (14)These diffusion coefficients may be obtained directly bythe methods of Lax and Louise11 [23], [24], but are moredifficult to obtain by other methods. The diffusion coeffi-cients for carrier numberN and photon number P may besimplyderived,becausebothquantitiesundergo hotnoise fluctuations (during emission and absorptionevents,N and P changeoppositely by 1). For various ai, fluc-tuating purely from shot noise, the diffusion coefficientscan be determined by inspection of the rate equations1251.The diffusion coefficient 2Dii is just the sum of rates inand rates out and 2 0 , is the negative of the sum the rates

for which i and j both change. For example, 2DNN = C+ S + AP + R ( P + 1)and 2 D N p= - (AP + R ( P +1)). However, the physically useful quantity is intensityI , not photon number P. While ( I ) = ( P ) , Iand P havedifferent distributions and different diffusion coefficients.We can think of the instantaneous distribution of P as aPoissondistributiondetermined heaveragevalue I(t).This leads to th e relation

(f? = ( I 2 >+ ( 1 ) (15)which can be rigorously derived. This relation was usedby McCumber [35], Lax [36], and Shimpe [37] to relatethe diffusion coefficients of P and I . It can be used to de-rive (14).

111. POWER PECTRAF S E M I C O N D U C T O RASERSA. Below Threshold Operation

Laser operation naturally divides into two regimes thatare relatively easy to describe: below threshold operation,in which-AG is large and fluctuations in that alter AG,can be neglected; and above threshold operation, wherefluctuations of I and N are stabilized and may be regardedas small deviations from the steady-state values.The tran-sition region near threshold has been calculated for gaslasers, but not for semiconductor lasers [4], [14]. In thissection, we will consider operation below threshold.The power spectrum is usually measured by passing thelaser radiation hrough a scanning Fabry-Perot nterfer-ometer and measuring the power or by adding a nearlymonochromaticlocalscillator)ield,avingre-quency substantially different from the laser mode, to thelaser field and then measuring the power in the beat spec-trum. In both methods, the measured spectrum is propor-tional to Wp w ) , the spectral density of the Fourier trans-form of p ( t )P (w> =- p(t) exp (iot) t. (16)& --



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It is easily shown by applying theprinciple of stationarity,that is steady-state operation, there is no correlation be-tween different Fourier components of any of the systemvariables o r Langevin forces [28, sect. 1181( P ( W ' ) * P ( 0 ) ) = W p ( W ) 6 (W - W ' ) (17)

and thatm

Wp(w) = s ( P ( t ) * P(O)> exp ( i w0 dt. (18)Below threshold, with AG regarded asconstant, heFourier component of the field amplitude (a )can be cal-culated immediately by writing ( 5 ) in terms of its Fouriercomponents

-m

Fp ( W )P ( w ) = ( A G 1 "2" (19)i W ~ + C Y - - W - -

where F p ( w ) is the Fourier transform of F p , defined ex-actly like (0) n (16). The autocorrelation unctions ofthe Fourier components have he same diffusion coeffi-cients as the time components. For example,

( F ~ ( w ' ) *F ~ ( w ) )R ~ ( w '- w ) . (20)The calculation of ( P ( w ' ) * P ( w ) ) using this relation re-sults in

showing hat the aserbelow threshold has a Lorentzlineshape and a linewidth of-AG RA v = - - - - (22)27r 27rIwhere AG = - R / Z used n he ast equality can be ob-tained by taking the average of (10). When I is related tooptical power,(22) is theSchawlow-Townes inewidthformula. I t is only valid below threshold. One applicationof this formula is to give the noise bandwidth of a laseramplifier.B . Lineshape an d Phase Noise in a SemiconductorLaser Operating Above Threshold

Above hreshold, intensity luctuationsbecomestabi-lized.This stabilization process greatlyncreaseshelinewidth-power product. Assuming that amplitude fluc-tuations in P ( t )are negligible, we can use (2) to expressthe correlation function ( 6 (t)" /3 (0)) as( P@>* P ( O > > = I expexp (i&(t))) (23)

where I is the average intensity and where A 4 ( t ) = 4 ( t )- $ (0).The neglect of fluctuations in the amplitude of Pin (23) is not entirely correct . Vahala et al . [181 showedthat correlations of phase and intensity fluctuations at therelaxation frequency couldexplain about a 20-percent

asymmetry n the structure occurring n the tails oflineshape. For simplici ty, we will neglect this effect.In general (exp iA 4 ( t ) ) ) s difficult to evaluate, bthe case where A 4 ( t ) is a Gaussian variable (one wGaussian probability distribution), it is easily shownintegration that [11(exp ( i A 4 W ) ) = exp 1-i ( A 4 ( t ) * ) I .

Thus, if Aq5(t) is a Gaussian variable, evaluation offield correlation reduces to evaluating the mean squarA 4 ( t ) .The justification that A 4 (t > s a Gaussian variable ron the fact that above threshold,he deviations of I anfrom he steady-state values are small . In this caseequations obeyed by these deviations and 4 (lo), ((13) are nearly linear and driven by nearly Gaussian Lgevin forces.The solution of a set of linearLangeequations driven by Gaussian forces willalso beGausdistributed [ 2 5 ] .The force Fo is Gaussian. It arises fthe additivecontributions of many independent sourcespontaneous emission throughout the cavity [27] anthe central limit theorem [38], this is sufficient to enthat each component of F p will be a Gaussian variabThe forces F I and F+ are formed by products of comnents of F p and I' To the extent that fluctuationscan be neglected, these forces areaussian variables.force FNarises from shot noise and should have a Poidistribution, but the event rate is so high (about 107that it is also a Gaussian variable.The understanding of the Lorentzian linewidth of a labove threshold, does not require a complete solutio( lo) , (1I) , (13) . The line broadening is primarily dulowrequencyluctuations.Atowrequencies, suppression of intensity fluctuations is extremely gWe can make use of thisy neglecting Z and solve foin (10)

Substituting his expression nto (1 ) , and dropping constant RII term, which only causes a small frequeshift, we arrive at an equation for 66 = F,(t) - - I ( t ) .12 1

The phase change during a time t , A 4 = +( t ) - 4(0given by integrating this equationA+ = 1'F,(t) dt - - F,( t ) dl.

Squaring and averaging this equation with the aid ofand the diffusion coefficients (12) we findR( A $ ( t ) ' ) = - 1 + a2) .2 1

Substitution of (28) into (24) and (23) shows that

21 s t



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1 I 1 I10,000 11,000 12,000 13,0004,000

ENERGY (Cm- )Fig. 3 . Spectrum of the real and imaginary changes in refractive index ofth e GaAs active layer of a buried heterostructure laser when i t is excitedfrom low currents up to threshold; hvI. is the laser energy. From Henryet al . [8].

field autocorrelation function decays exponentially withcorrelation ime T,,,, = 4141 + a2)R. Substitutionof(28) into (23) and using (1 ) results in a Lorentz ineshape,with a full width of IR

4lrzv = - - - ( l + C Y )which differs from the below threshold case by (1 + a)/2. The factor of $ is due to the suppression of intensityfluctuations that contribute to the linewidth below thresh-old.The actor 1 + a2 isdue o he increasedphasechanges brought on by this suppression.Tocompare with heexperiment, it is necessary toreexpress R and I in (29). We can use (8) to write R interms of G and write G = gu, = -In (R,) u g / ( q L )whereR, is the facet power reflectivity, TJ is the quantum effi-ciency, and L is the cavity length. The facet power andintensity are related by 2P0 = TJGZ~V.ith these changes,the linewidth is given by

Equation (30) was confirmed by Welford and Mooradian[30], who measured both the linewidth and the other pa-rameters as a function of temperature (Fig. 2). However,they found an unexpected effect. In addition to the line-widthvarying as P;, theyfoundapower ndependentcomponent that is nearly negligible at room temperature,but is substantial at low temperature. The power indepen-dent component has not been satisfactorily explained asyet 1391.C. Line Structure

High-frequency luctuationsthe wings of the lineshape andshort-time durations. It is justcontribute o structure into the change in q5 duringthese short-time duration

changes in phase that can lead to errors in optical com-munications. To compute ( A 4 ( Q2 ) with no restriction ontime, we have to solve the full quations (lo), (1l ) , (13),but linearized odescribe malloscillationsabout hesteady state, The equations governing the deviationsn 4,I , and N are obtained by expanding I and N as

Z(t) = z + p ( t) N ( t ) = N + n( t ) (31)and expanding S, G, and aAG ass ( t ) s f Nfl(t) (32)G(t) = G + G,n(t) - G,p(t) (33)aAG = aGNn(t) (34)

which results in the linear equations4 aGNn + F,(t) ( 3 5 )p = GNZ~Z + F / ( t ) (36)

= -rNn- Gp + Fv(t ) (37)where hedampingcoefficients = G,Z + R /I and rN= GNZ + S N . Equations (32)-(34) account for t h e changein spontaneousemission atewithcarriernumber, hechange in gain with carrier number and ight ntensity,and the change in mode frequency with carrier number.The change in gain with light intensity (gain saturation)is necessary to account for the large dampingf relaxationoscillations that is observed in index guided lasers and thincrease in damping with light intensity [171, [40]. It isvery likely that gain saturation results from spectral holeburning [41]. Spectral hole burning causes a gain changethat snearlysymmetricabout he laser ine 1411. Thedispersive change in refractive index associated with thischange should be zero at the laser line and for that reasonmakes no contribution o 34). On theotherhand, hechange in gain associated with a change in carrier numberoccurs primarily at higher photon energies wheret resultsin a decrease in absorption. This is illustrated in Fig. 3 ,where the changes in refractive index with carrier densityAn and An for AlGaAs are plotted versus energy [B].The fact that the laser line occurs in the tail of a steepabsorption edge and in the tail f the gain change explainswhy a typically has values of 4-7 in index guided lasersoperated at room temperature.The means square of Aq5(t) is easily found by solving(35)-(37) [171. This is done by writing these equations interms of their Fourier components, solving for ( 4 w ) )and then converting back to ( A 4 ( t ) 2 ) y contour integra-tion. We will only give the result

R(A4(t )2)= - 1 + a2A)2 1+ a2A CO S (36) - exp (-rt ) cos (Q t- 36)]2 r cos 6

(38)where Q GyGG,vI )2 s herelaxationfrequency, r =

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304 JOURNAL OF LIGHTWAVE TECHNOLOGY,VOL.LT-4, NO. 3, MA R C H

n

t ( nsec)Fig . 4. Mean square phase change ( A + ( t ) ' ) versus ime fo r weak damp-ing, moderate damping, and the linear approximation. From Henry [17].

(I?/ f rN ) / 2s the damping rate,A = [( l -+ r,/Q2+ I':X/(G2RZ)]/(l + rhrI'l/GGNI)2

is a constant near that is slightly less than unity and 6 =tan-' (I'iQ) s a small angle [17].The average ( A + ( t ) 2 ) s the sumof a linear term and adamped oscillatory term. This is illustrated in Fig. 4 [17].The linear term gives rise to the linewidth and the corre-lation time, while the oscillatory term causes the satellitesto occur, separated from the main by multiples of the re-laxation oscillation frequency. The spectral shapes arisingfrom the values of (Ad, ( t ) 2 )plotted in Fig. 4 are shownin Fig. 5 . The three curves correspond to weak dampingin which gain saturation is neglected, gain saturation ap-propriate for spectral hole burning n GaAs, and the linearapproximation (28), n which relaxation oscillations areneglected and the lineshape is exactly Lorentzian. Theywere plotted by evaluating (23j, (24), (38) with the aid ofa fast Fourier transform program171. The side peaks tendto be of order 1 percent of the main peak, because of thestrong damping of relaxation oscillations in index guidedlasers. The average (Ad, ( t ) 2 ) as been directly measuredby Diano 1151, and by EichenandMelman 42],whofound an excellent fit of the data on 1.3-pm InGaAsP las-ers using (38) with A = 1. Their data is shown in Fig. 6.Theyalso oundgoodagreementbetween heFouriertransform of exp (-4 (Ar$(t)')) and the spectrum rnea-sured with a scanning Fabry-Perot interferometer.

IV. LINENARROWINGA . Ideal Fabry-Perot Cavity with a Passive Section

The linewidth power productof conventional semicon-ductor lasers can be dramatically reducedy adding a pas-sive section to the laser cavity. This is done by 4 R oat-ing one end and adding an external eflector separated byabout 10-20 cm from the semiconductor chip. The spec-trum of such a laser fabricated by Olsson [43] is shownin Fig. 7. The data was taken by beating two similar ex-

Po=lmWI - 3 . l x l o ~

$= I 52ghzcY.5 3

V-Uo ghz)F i g . 5. Power spectrum of the laser line calculated for the three (A$functions of Fig. 4-weak dampin g, intermediate damping, and theear approximation which gives a L orentz line shape. From Henry

2 .0 c /

0.0 I 1 Io l l !0.0 0.4 0.8 1 . 2 1 .6 2.0dns)

F i g . 6 . Direct measurement of (A+(r)') and com parison with a theorfi t using (38). After Eichen and Melman [42].

ternal cavity lasers together. The lineshape is Lorentziin shape and corresponds o a AvP, of about 10 kHmW fo r each laser. This is a reduction of about lo4 fthe laser AvPo = 100 MHz . mW or each isolated laprior to AR coating.This enormous linewidth reductionf a laserwith a passivesection Lp can be readilyunderstood.Thede-crease of linewidth in proportion to i 2 can be undersby consideration of (29), which applies to a uniform lcavity. A long passive section will decrease the averspontaneous emission rate - L i ' and increase the aveintensity Z - L p , for a fixed Po, resulting in a linewidreduction proportional to L i 2 .In [27], the linewidth of a Fabry-Perot cavity with a



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HENRY:PHASENOISE NSEMICONDUCTORLASERS 3050

2 I In I I i0Wv)LTW30n

a - 60

-80 t -1-100 I I-50 50

AV (MH z )Fig. 7. Beat spectrum o f t w o 1 . 5 S y m external cavity lasers having gratingreflectors 15 cm from the laser diode. The center of the spectrum is as1.328 M H z , From the data of Olsson [43].passive section was calculated under he deal assump-tions of a perfect AR coating and loss from the passiveend being the same as from the cleaved uncoated end. Itwas found that the linewidth forinulas (29)-(30) are re-duced by a simple factor given by the square of the ratioof the time spent in the active section to the time spentnboth sections, where the time in each section equals Lh,.This factor is

(39)where a and p refer to the active and passive sections. Theright-hand expression in (39) applies in the case of a longair-filled passive section, where nRa s the group index ofthe semiconductor. For Lp = 15 cm, L, = 300 pm, andn;, = 4, this reduction is about 1.5104,o that a linewidthof 150 Mhz would be reduced to 10 kHz. This sonlyslightly larger than the reduction observedby Olsson [43].Linewidth reduction favors a long external cavity, how-ever, the longitudinal mode spacing decreases as L- andtherefore single-mode operation becomes more difficult toobtain for large L. For a 15-cm external cavity, the lon-gitudinal mode spacing is about 1 GHz or 0.08 A at 1.55pm. All but one of these modes can be suppressedy add-ingfrequencyselectivefilters o heexternalcavity.Abandwidth of a few angstroms can be achieved by usinga diffraction grating s the external reflecting element20],[43]. Additional mode selection can be achieved by plac-ing an etalon in the external cavity [43], by coupled-cav-ity effects due to an imperfect AR coating and due to op-tical nonlinearities [41]. Piezoelectric tuning of the lengthof th e external cavity can be used to select out one modeWIB. Line Narrowing b y External Feedback

The analysis in Section IV-A is restricted to the case ofan ideal passive section with a perfect AR coating of thelaser facet. Herewe review an analysis developed by Kik-uchi and Okoschi [44] and Agrawal [45] that is applicablefor arbitrary laser facet reflectivities and weak or moder-

ate evels of feedback. It is useful to.think of externalfeedback as a form of self-locking. The laser field ( t ) sinjection-locked to a field that was emitted earlier fromthe laser and is reinjected after making a round trip in theexternal cavity. Feedback is described by adding a termKP(t - T ~ )o the equation for the laser field (5) that rep-resents the effect of the field coupled back into the lasercavity after a delay T~i A G2= -io0 +- 1 - io!)

* P ( t ) + K P ( t - 70) f Fp(t). (40)An equation of this type was first used by Lang and Ko -bayashi [46]. It s readily shown by considering steady-state field propagation that

where Ro is the power reflectivity of the external reflectorincluding coupling losses, R, is the facet power reflectiv-ity, and T , is the round-trip time n the semiconductor cav-ity. Equation (40) is not valid for strong levels of feed-back K T , >> 1, since multiple round trips in the externalcavity have been neglected.Equation (40) is more conveniently discussedby chang-ing variables to p ( t )= p ( t ) exp ( - i w t ) . The steady stateis found by keeping6 constant and fluctuations are foundby solving the equation for 0 nder various approxima-tions. The transformed equation is

-iAw +- 1 - ior)2P( t ) ,+ P ( t - T ~ ) xp (iCP) + Fpt( t ) . (42)

where CP = w 0 nd Aw = w - wo. The steady-state op-erating point is found by setting 6nd Fp, o zero. AlsoA G = -2K CO S CP (43)Aw = -K(Q cos CP + sin a). (44)

The different values of CP form an ellipse of solutions AGversus A w , where CP is the steady-state phase angle be-tween the cavity field and the feedback field; see Fig. 8.The additional relation CP = W T ~ ,esults in a discrete setof values for CP, corresponding o he composite cavitymodes. The ellipse forms the locking range over whichself-locking can take place. In practice the locking rangeis restricted to negative values of A G , the portion of therange for which gain reduction takes place. If the lockingrange covers many external cavity modes, the laser willoperate on one o r several modes having the greatest gainreduction. If a grating is used as an externai reflector, onecan tune across the locking range. But in the range of pos-itive AG the laser will choose to operate on a differentlongitudinal mode, for which feedback from the gratingis negligible. For strong feedback, the laser can be belowthreshold except for part of the range with the most neg-



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306 JOURNAL OF LIGHTW.4VE TECHNOLOGY, VOL. LT-4, NO . 3, MA R C HL I I l I I I I ' I I I ~ l 1 ' -@ = 180"

2 - -

l - -

Y -I4 -C J O

-I - ~Aw:w-w0a = 5

-2 - -Au=-aK

I I I I I I l l l ~ , l l-5 0 5

A W ( K . ]Fig. 8. Gain change versus angular frequency change for the steady state s

of operation of a laser with eflective feedba ck. The lasermodes areactually points along he ellipse that become closely spaced when helocking range contains many modes. The angle is th ephase angle be-tween the injected field p ( f - T(,) nd the cavity field @( I ) . rom Henryand Kazarjnov 1.541.ative AG. Decreased values of AG are observed by in-creased ight ntensity 47].Thechange in intensity isproportional to the feedback parameter K .Equation42) is aonlinearifferential-differenceequationcontaining ieldsat wodifferent imes.Thiscomplication has prevented exact solution. The injectedfield suppresses high-frequency phase fluctuations that al-ter the phaseof the laser field relative to the injected field.Only phase changes taking place n a time large comparedto the round-trip time T~ are unsuppressed. The low-fre-quency ehavior an eound by expanding P( t -70) 1441, 1451

/3(t - To)/ = P ( t ) / - T"P'(t)' . (45)This approximation changes the field equation to

0' = (1 '+ KT0 exp (i+) (46)where AG represents the deviation from the steady stateand terms set to zero by the steady-state condition havebeen dropped. This equation s essentially the same as ( 5 )(with w , removed by transforming to P ' ) except for thecomplex constant dividing the right-hand side. This altersthe effective value for 01 and he Langevin force. Withthese changes, the linewidth can be calculatedn the samemanner as in the derivation of (29). The result found byAgrawal [45] is

Av = A v o[ I + K T ~cos CP - 01 sin +)12 (47)where Av, is the inewidth in theabsenceoffeedbackgiven by (29) and (30) . For a = 0, (47) reduces o heresult derived by Kikuchi and Okoschi [44]. The denom-inator is positive and results in line narrowing all alongthe lower portion of the ellipse for negative d G ( 2 ~os(CP)> 0). Line broadening is possible, but only at very

low feedback levels, when there is a single mode on theellipse. Then adjusting he cavity engthwillmove mode onto different regions of the ellipse for negative and narrowing and broadening can be observed. t higlevels of feedback, there will be several modes or mon the ellipse and lasing will occur for the modef lowgain. The factor which controls the line narrowing isFor K T ~ > 1, narrowing s proportional o ( n g a L Uthe same factor as we found in the analysisn SectionA. The effect of feedback on the linewidth becomes nligible when ~7~ becomes much less than unity. For ex-ample, with L, = 300 hm and L p = 30 cm , K T ~s unfor Ro/R, = lop5. This s hereasonwhyforaccuratelinewidthmeasurements,xternaleflectioneedbackmust be reduced to about 60 dB.Suris and Tager [48] have managed to solve theonear differential-difference equation (42), without makingexpansion (45), but only for the case 01 = 0. They shthat the above analysis (47), predictinga single laser narrowed by feedback, is only valid if the linewidth ofisolated laser Avo is small compared to the external cmode spacing. For example, if A v o = 100 MHz, this cresponds to Lp < 150 cm. Otherwise, the semiconduclaser emission associated with external cavity modestains a number of narrow lines associated with externalcavity modes with an envelope width equal to Avo.C. Coherence Collapse

According to (46), line narrowing should increasewincreasing feedback. However Lenstraet ai. [49] repothat at relatively high-feedback levels, the laser line bcomes enormously broadened. They called this effectherence collapse. This phenomenon was encountered different way by Temkin et ai. [50].They studied the havior of lasers under moderately high levelsof feedbfrom distant reflectors and found that they were unstaSome of their data on this instability is shown in FigCoherent feedback reduces the threshold and increalight intensity of the laser. It takes about 10 round tin the external cavity for the laser to go from initiallyto hestateofcoherent eedback.Thiscorresponds omoving from the origin to the bottomf the ellipse n F8. Just as the steady state is reached, the laser becomesunstable. tsuddenly eturns o he nitialoperation higher threshold and lower intensity in which its not befiting from coherent feedback. Then the build-up prstarts all over again. This instability had been observedpreviously, but had not been explained 51]-[53]. The broadening observed by Lenstra et al. [49] appears tomerely the frequency chirp associated with this cycle.Kazarinov and the author have recently explained thiinstability [54]. It corresponds to a large and rapid phafluctuation in which the laser jumps outof its self-locstate. This instability is puzzling because prior to its on-set, the operating point of the laser reaches a steady sat hebottomof heellipsewith CP = 0 near he onwavelength end of the locking range. Analyses f stabof injection locking show that the short wavelength



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HENRY: PHASE NOISE IN SEMICONDUCTOR LASERS

L =30

Fig . 9. Time dependence of he ntensity of a aser exhibiting nstabilitydue to reflective feedback. Three levels of feedback are shown rangingfrom pow er field reflectivities of -5 dB in the upper figure to -15 dBi n the lower figure. The arrows indicate zero intensity. The steps in in -tensity build up during successive to round trips in the external cavity.Coherence collapse occurs after the steady state is reached. The jitter isdue to the stochastic nature of the instabil ity . From Temkin et al . [ S O ] .

unstable and exhibits high-frequency self-pulsations, butthe long wavelength end of the locking range, near thebottom of the ellipse, is stable and has strongly dampedrelaxationoscillations [ 5 5 ] , [ 5 6 ] , [12]. However,hisconclusion is the result of a conventional linear stabilityanalysis and coherence collapse results from nonlinear ef-fects that are brought aboutby a rare but large fluctuationin spontaneous emission intensity. Our analysis consistedof approximately solving (42), with P ( t - 7,) regardedas constant, without linearizing this equation.Considera argefluctuation nspontaneousemissionwhich decreases the mode intensity. This reduces stimu-lated emission and increases the carrier number. The car-rier number change causes a momentary change in modefrequency that reduces the alignment of the laserield P ( t )and the injected field j3(t - 7,) from @ = 0 to A + . Thisin turn decreases the stimulated emission in the cavity inproportion to cos (A+) . The decrease in stimulated emis-sion increases the carrier number, which causes A + to in-crease further, driving the laser out of its locked + = 0state. There, other forces acting in the laser which try torestore the steady state. These are just the forceshat giverise to relaxation oscillations. For small fluctuations, therestoring orceswin out,butfora arge fluctuation in

1

1 1 I , , , , /0.02 0.01 0.1 0.210.002A W C

Fig. 10. Average nstability ime t, versus the fractional decrease in thethreshold current AC/C, a parameter proportional to feedback parameterK. Th e se tof curves t , ar e for different intensities. Parameter I , is shownin the inset . The inset shows the kink in the light versus c urrent relationresulting from he change in t , in goin g from a to c. From Henry andKazarinov [54].

spontaneous emission, the laser will jump out of lock.The phenomenon has a threshold and can be described byfluctuations that take the system over a barrier [54].An nterestingaspectof hisphenomenon is that forstrongnougheedback,elf-lockingecomeserystrong, the barrierbecomes very high, and theprobabilityof jumping over the barrier becomes negligible; stabilityis restored. This s llustrated in Fig . 10, where he av-erage ime for nstability o occur is plotted versus hedegree of feedback. For strong feedback, the average in-stability time becomes exponentially large. This renewedonsetofstability for strong feedback was observed byTemkin et al. [50]using a laser with an AR-coated facet.Our calculation also showed that the stability is greaternear threshold, where the light intensitys small, than wellabove threshold. This change in stability as the light in-tensity increases give rise to a kinked light versus in-tensity relation that is illustrated in Fig. 10 and was ob-served by Temkin et al. [50].

v. LINEWIDTH REQUIREMENTSN COHERENT O P T I C A LCOMMUNICATIONSA . Shot Noise Limit

Coherentoptical ommunications s urrentlybeingrapidly developed n many aboratories. This nterest isstimulated by severalwell-known dvantages of thismethod: by mixing the weak received field with a localoscillator field, the detected signal can be sufficiently in-creased and filtered so that noise in subsequent amplifi-cation becomes negligible and the physical limit of de-tectability,due o hotnoise,canbeapproached; hecommunication channels can be closeIy spaced, enablinga single optical amplifier to simultaneously amplifymanychannels.The schemes for transmitting digital information are tsame in optical communications as earlier developed formicrowave ommunications [57]. However,noptical



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308 JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. LT-4, NO . 3, MA R C H

communications, shot noise rather than thermal noiseim-itsdetectabilityand phase noise, which isnegligible inthe microwave case, hinders attainment of the fundamen-tal limits . The physical imit of shot noise s not easilyreached. It seems that the closer a modulation method isto the shot noise limit in the absence of phase noise, themore sensitive it is to phase noise. In this section, we illfirst consider the shot noise limit, which can in principlebe reached using homodyne phase shift keying (PSK). Inthe next section, we will consider the more practical het-erodynechemesf differential phasehifteying(DPSK), frequency shift keying (FSK) and amplitude shiftkeying (ASK), and the linewidth requirements that lasersused in these methods must satisfy.Incoherent ightwavedetect ion, he incidentopticalsignal field /3.A(t) s mixed witha local oscillator field Pe(t)and then detected by a quantum detector, such as a p-i-ndiode. We can model the detector by the simple equationfor the number of carriers n generated during detection

h = - - +T d V d I P A $- f ie( + F,!(t) (48)

where T[, s the inverse detector response time, F , is aLangevin force describing the shot noise fluctuations, andqdI/3,., + is the average rate of opticalgeneration ofcarriers at the detector. For frequencies less than the de-tector response time, we can neglect h in (48). The de-tected signals s ( t ) = n/T , is given by

s ( t ) = 2 T d ( l A f B ) l i 2 cos ( W A R + d A B ) + F,~(t) (49)where wAB = wA - W B and dAB = +A - + B. The Langevinforce is due to shot noise of generation of carriers. Thediffusion coefficient of this force will be given by the av-erage rate of generation

2 D r m = q d r B (50)where generation due to I,., has been neglected.In principle, the shot noise limit can be reached i n ho-modyne (PSK). In this case, wA = wR nd the bits of and 0 are generated by altering d A y T. t is assumedthat aside from this modulation, and +* are kept equal.(A method for doing this is by means of a phase-lockedloop [-58].) The signal integrated over one bit time T isgiven by anaveragevalue ( s ) = 2 q c l ( ~ A ~ B ) 2 ~lus afluctuation part A s = F,,(t) dt . The meansquare of Asis (As) = 2D,, T = rd , T. The number ofphotocarriersgenerated in each bit will be Poisson distributed. Sincethe average number of detected carriers per bit is largecompared o unity, hisdistribution is closelyapproxi-mated by a Gaussian andF,,(t) can be regarded as a Gauss-ian variable.The signalassociated with I and 0 will be two Gaussiandistributions centered at th e two val-ues of (s). The error probability PE is found by integrat-ing the tails of the distribution that extend beyond zero.This is an error function, but aside from a prefactor oforder unity, it is

D P S K

P C + P sI INT CO

F S K

ASKP C + P s

r I N T COM

Fig. 11. Schematic diagram of the methods of detection for: diffephase hiftkeying (DPSK ). frequencyphase-shift keying (FSKamplitude hiftkeying ASK). The components reopticaldetecto(DET), handpass amplifiers w,,~. w ,, w 2 , time delay T , integratorfor t ime T , and decision circuit or comparator (COMP).

7 - D P S KAb* = A u,-m 6 -aI

4 -

9 I.100 200 300 40 0 50 0 6RgA VA

Fig. 12 . Averagebi tenergy P , (photons)versus he atioof hit alaser linewidth, necessary to achieve an error rate of lo-. From J1581.

= exp (-2Ps)where PB = qZAT the average number of detected siphotons in each bit reaching the detector. Setting PIOp9, we find P R = 10. This is the shot noise limitminimum detectable power limited by shot noise [-5B. Heterodyne Detection

Ideally, the homodyne PSK scheme is more sensthan other methods. However, it requires phase ockithe local oscillator to the received field. This isa comprocedure and one hat is very sensitive to phase [ 5 8 ) .The heterodyne methods DPSK, F S K , and ASmore resistant o phase noise and nearly as sensitiPSK. The detection methods for the 3 schemes we discuss are sketched in Fig. 11 . The minimum valuPs for these schemes are not easily calculated owinthe nonlinear nature of the detection process that invsquaring or taking a product of the detected signals. Hever , these limits have been worked out using the samethod as for he microwave case 1-57], [ 5 8 ] , butwAuthorized licensed use limited to: University of Central Florida. Downloaded on April 27, 2009 at 14:14 from IEEE Xplore. Restrictions apply.


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HENRY: NOISE IN SEMICONDUCTOR LASERS 309

shotnoisereplacing hermalnoise. Here, we will imitourselves to a qualitative discussion.In the PSK case, we saw that PE depends exponentiallyon the squaredaveragesignaldivided by theaveragedmean square noise. This is generally true for all the de-tectionschemes.Allheterodynedetectionschemes mixP A and PB that differ in angular frequency by @ A B . In thiscase, the noise is unchanged, but the signal is sinusoidal(49) and heaveragesquaredsignal shalf hatof hehomodyne case. This increases PB by 3 dB.The most sensitive of the three heterodyne schemes isDSPK.In hisscheme,phase smodulatedas nPSK.After detection, the detected signal is divided, one half isdelayed by the bit time T , and the two channels are mul-tiplied together. The resulting signal is proportional to cos[ $ A B ( t ) - $ A B ( t - T ) ] ,which changes sign when the twosuccessive bits differ. Ideally, this detection method suf-fers only a 3-dB loss in sensitivity due to heterodyne de-tection.The signal does not depend on absolute phase so that itwill not be affected by slow phase drifts. There will be apower penalty (increase in P B ) , however, if the angularargument of the cosine, which is normally 0 or T, hangesdue to phase diffusion during the bit time T . If the prob-ability of a phase change $A - q iB of greater han n / 2exceeds l ow9 ,unacceptable errors will occur regardlessof t h e signal power. We can readily estimate he linewidthat which this will occur. The phase changeA$,., - A$B isa Gaussian variable. If we assume that both of the lasersgenerating fieldsA and B have strongly damped relaxationoscillations, then the mean square change f $ A - bB dur-ing time T can be related to the linewidths A vA and AvBby ( 2 8 ) and ( 2 9 ) : ( A $ : ) + ( A $ ; ) = 2 n T ( A v A+ AYE) .The error probability is found by integrating a Gaussiandistribution with this mean square value to find the areaof th e tail beyond n/2. The error probability is given bythe Gaussian distribution evaluated at a 1 2 , aside from aprefactor of order unity. This results in

where RB = l iT is the bit rate [ 5 9 ] . If we set AvA = AvAandequate P E o we find RB/A\vA = 210. This s heminimum allowable bit-rate linewidth ratio for DPSK re-gardless of power. Salz [58] finds that to nearly reach theideal limit of P B ,occurring in the absence of phase noise(3 dBabove heshotnoise imit, P E = 20) RB/AvA =500-600 is needed. His results are shown in Fig. 12.The linewidth requirements for FSK and ASK are lessstringent than for DPSK, but this is purchased at the priceof reduced sensitivity in the absence of phase noise andincreasedbandwidth of theelectronics. nFSK, uA ismodulated and wA B switches between two values, w 1 andw2. The detected signal is divided and each half is passedthrough a narrow-band amplifier. Each signal is then en-velope detected bysquaringand ntegrating.The wosignals are then compared to determine whether the bit is

1 or 0. Passing hesignal hrough wobandpassfilters doubles the mean square noise compared to DPSK,introducing an additional 3-dB increase in P B [57], 58]In ASK, the signal field is modulated on and off. Afterdetection, the signal is passed through a bandpass ampli-fier and then envelope-detected by squaring. In this case,

the noise is the same as in DSPK, but the signal changein going from 1 to 0 is reduced by a factor of 2 .This change increases P B by 6 dB (P s = 8 0 ) .An approximate analysis of the effect of phase noisenFSK and ASK has been made by Garrett and Jacobsen[60], [61]. They assume that the effect of phase noise isto cause average frequency shifts A j . When averaged overa bit time

(PB = 40).

Af = A$ A - A 6 B23rT (53)Af has a Gaussian distribution for A j . They calculate theeffect of phase noise on error rates by calculating PE as afunction of Afand convoluting this with the Gaussian dis-tribution.Their analysis shows that FSK and ASK are much lesssensitive to phase noise than DSPK. For example, FSKhas a minimum linewidth requirement similar to DPSK.If the two frequencies n FSK are separated by m RB,wherern = 1 - 3 , a value of Af exceeding m R B / 2 will cause anerror. This will occur if A$A - A+B = a m . In the caseof DPSK, an error occurs when the corresponding quan-tity is ~ 1 2 ,herefore, for m > 1 / 2 , the linewidth require-ments are greater for DSPK than for FSK. The error ratefor this frequency change is derived in the same way as( 5 2 ) and is given by

Comparing hese woequations, we see hatminimumlinewidth for FSK is 4m2 greater than for DSPK.VI. S U M M A R Y

Noise in lasers can be rigorously described in terms offluctuations of a complex classical wave field ( t ) that hasboth ntensity andphasefluctuations.Below hreshold,both fluctuations contribute equally to the broadening ofthe laser line, which is Lorentziann shape and has a idthgiven by the Schawlow-Townes formula. Above thresh-old, he suppression of ow-frequency ntensity fluctua-tions by changes in carrier number changes causes an ad-ditionalbroadening of ( 1 + a 2 ) / 2 , where CY = 4-7 forAlGaAsandInGaAsP asers at room temperature. Thisresults n AuP, = 6 0 - 1 2 0 MHz - mW.This ineshapeabovehresholds pproximatelyLorentzian,buthassmall side peaks separated from the line center by the re-laxation oscillation frequency.Enormous line narrowing can be achieved y extendingthe lasercavity with a passive section. Linewidths of only10 kHz have been achieved while still maintaining single-



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30 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-4, NO. 3, MARCH

modeoperation by AR coatingonefacetandadding apassive section about 15 cm long. Line narrowing by op-tical feedback is a consequence of self-locking of the op-tical field by the delayed field returning after external re-flect ion. The instability of coherence collapse can occurin which a arge and rapid phase fluctuation causes helaser to jump outof the self-locked state. This instabilityis suppressed by very strong optical feedback.Thedetection sensitivityndigital oherentopticalcommunications is limited by shot noise occurring whenlight is detected and by phase noise. While the linewidthis due to low-frequency phase fluctuations, errors in op -tical communications are caused by phase changes occur-ring during one bit time. The minimum number of pho-tonserit PB dependsnhemethod of digitalmodulation. The shot noise limit of PB = 10 for an errorprobability of lop9can be achieved by homodyne PSK,but this scheme is extremely sensitive to phase noise. Het-erodyne DPSK has a P , 3 dB more than the shot noiselimit, which can be approached for R B I A v A = 500. Thelinewidth requirements for heterodyne FSK and ASK arean order of magnitude less restrictive on linewidth thanDPSK and have PB that are 6 and 9 dB greater than theshot noise limit, respectively.

ACKNOWLEDGMENTThe author wishes to thank N . A. Olsson, J. Salz, M .Lax, R. Schimpe, and G. P. Agrawal for stimulating dis-cussions.

REFERENCESM .Lax,Classicalnoise V: Noise i n self-sustainedoscillators,A . L. Schawlow and C. H. Townes. Infrared and optical masers ,M .Lax, Quantum noiseV: hase oise in a omogeneouslybroadened maser , i n Physics and Quantum Electronics, P. L. Kel-ley , B. Lax, and P. E. Tannenwald , Eds. New York: McGraw-Hill ,R . D. Hempstead an dM .Lax,ClassicalNoiseVI: Noise in self-sustained oscillators near threshold. Phys. Rev. , vol . 161, pp. 350-366,1967.H. Gerhardt , H . Well ing, and A . Guttner, Measurements of the laserline width due to quantum phase and quantum amplitude noise aboveand below thresho ld, 2. Physik, vol . 253, pp . 113-126, 1972.M . W. Flemin gan d A . Mooradian,Fundamental inebroadeningof single-mode (GaA1)As diode asers, A p p l . Phys. Lett., vol. 38,pp . 511 , 1981.C. H . Hen ry ,Theory of he inewidth of semiconductor asers,IEEE J . Quantum Electron. , vol. QE-18. p p . 259-264, 1982.C . H . Hen ry , R . A. Logan.an d K . A. Bertness,Spectraldepen-dence of the cha nge in the refractive index due to carrier injection inGaAs asers . J . A ppl . Ph ys . , vol. 5 2 , pp . 4457-4461, 1981.C. H. Harder , K . Vahala . and A . Yariv, Measurement of the line-width enhancement factor CY of semiconductor asers, Appl. Phys.L e f t . . vol . 4 2 , p p. 328-330, 1983.1. D. Henning and J . V. Coll ins , Measurement of the semiconductor929,1983.laser linewidth broadening factor, Electron. Lett . , vol. 19, pp. 927-R. Schimpe, B. S tegmuller , and W . Harth , FM noise of index-guidedGaAlAs diode lasers , Electron. Letf . , vol. 2 0 , pp. 206-208, 1984.C. H. Hen ry , N . A. Olsson, and N. K. Dutta , Lockin g rangean dstability of injection locked 1.54-p InGaAsP semiconductor lasers ,ZEEE J . Quantum Electron. , vol. QE-21, pp . 1152-1 156, 1985.K.Kikuchiand T.Okoshi ,Estim ation of inewidthenhancementfactor a of AlC aAs lasers by correlation measurement between FMand AM noises , IEEE J . Quantum Electron. , vol. QE-21, pp. 669-673,1985.

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*

Charles H . Henry wa sborn n Chicago, IL, in1937. He received the M.S. degree in physics fromthe University of Chicago, Chicago, IL, in 1959,and the Ph.D . degr ee in physics from the Univer-sity of Illinois, Urbana , in 196 5.Since 1965, he has been a member of the staffin he Semic onductorElectronics ResearchDe -partment, AT&T Bell Laboratories , Murray Hil l ,NJ. From 1971 to 1976, he served as head of thisdepartmen t. H is research is primarily on the phys-ics associated with light emittine device technol-v1985.

an dechniques. New York: McGraw-Hill ,966, h..Am erican Association forheAdvancement of Science..~ ogy. He is the author of over 80 published papers.[57] M. chwartz , W . R . Bennett, and S . Stein , CommunicationSystems D r.Henry is a Fellow of theAmericanPhysicalSociety and of the

Phase Noise in Semiconductor Lasers

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