4252{1 R. A Fisher, P. L. Kelley, and T. K. Gustafson, Appl. Phys. Lett. 14, 140 Migus, C.V. Shank, E. P Ippen, and R. L Fork, Quantum. Electron A. Laubereau, Phys. Lett. 29A, 539(1969) H Nakatsuka, D. Grischkowsky, and A C Balant, Phys. Rev. Lett. 47 R.H. Stolen,VRamaswamy, P. Kaiser, and W. Pleibel, Appl. Phys.Lett 33,6991978 191001981} R H Stolen and C Lin, Phys. Rev. A 17, 1148(1978) R. R. Alfano and S. L. Shapiro, Phys, Rev. Lett. 24, 592(1970; 24. 1970 R L. Fork, B.I. Greene, and C v Shank, Appl. Phys. Lett. 38, 671(1981). E B Treacy, J.Quantum Electron. QE5,454(1969) L. F. Mollenauer, R. H Stolen, and J. P. Gordon, Phys. Rev. Lett.45 10951980 A theory of longitudinal modes in semiconductor lasers Kam Y, Lau Ortel Corporation, Alhambra, California 91803 Amnon Yariv California Institute of Technology Pasadena, California 9112 (Received 8 January 1982; accepted for publication 15 February 1982) A theory of longitudinal mode lasing spectrum of semiconductor lasers is developed which takes into account the nonuniform carrier and photon distributions and local gain spectrum shifts inside lasers with low end mirror reflectivities. The theory gives results consistent with observed longitudinal mode behavior in lasers with reduced facet reflectivity PACS numbers: 42.55 Px, 42.60 Fc, 42. 80.Sa The longitudinal mode spectrum of semiconductor la- of the laser cavity imply that the local gain spectrum varies sers has been studied extensively and major observed fea along the length of the cavity. This shift in the gain spectrum tures can be understood in terms of modal competition in a as a function of carrier inversion density is well document- saturation in semiconductor lasers is bam agreed that gain ed 13 When averaged over the entire cavity, the difference in and therefore it should oscillate in a single longitudinal mode which leads to an increase in the number of oscillating modes above threshold. Time and again this was demonstrated in under identical optical power level. Secondly, superlumines- lasers of near-ideal structures5- free from unstable trans- cence 4 takes effect in lasers with a low reflecting mirror verse mode or self-pulsations, which otherwise would render ( < 1%)which results in an effective increase in spontaneous the longitudinal lasing spectrum multimoded. In general emission contribution. Thirdly, at a certain output powe the number of lasing modes increases with increasing spon- level measured at the AR coated facet of the laser(refered to taneous emission and decreases with increasing optical pow- as the exit facet), the average internal optical power density is er.Other subtle aspects of longitudinal mode behavior re- much lower in an AR coated laser when compared to an quire more detailed theories, such as that including electron uncoated laser with identical output power. As mentioned coherence, or transverse waveguiding for satisfactory before, and will be elucidated below, the internal power den explanations sity determines the number of oscillating modes It is well recognized that although ideal in a sense, sin The analysis is based on the normalized forward and gle-mode lasers are not suitable for application in multimode backward photon propagating equations in the laser cavity, fiber systems due to modal noise problems. It is important written for each longitudinal mode that one has a laser that is ideal in every other aspects, such as linear light-current characteristics, single transverse (N-Nomkx *+BxM (la) mode, nonpulsating output, etc, but oscillates in many lon gitudinal modes. It was observed that1, 2 otherwise single mode lasers, when antireflection(AR )coated on one facet, d z -, (N-Nomkx;+ BxA becomes multimoded This letter offers an explanation for where phenomenoi A combination of several effects, all of them involving e nonuniform photon and carrier distributions in an AR WN-Mn)=6/(+8x+x coated laser, is responsible for the multimode behavior. where thex+'s and x 's are the forward and backward prop- First, the different levels of carrier densities in different parts agating photon densities(which are proportional to the light Appl. PI tt40(9),1May1982 0003-6951/82090763-03$01.00 1982 American institute of Physics Downloaded24May2006to131.215.240.9.RedistributionsubjecttoAlplicenseorcopyrightseehttp:/laplaip.org/apl/copyright.jsp
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intensities) for the ith mode, N is the carrier density, Nom is the carrier density for transparency, k is the gain constant in c =c ±g(N-Nomd cm /unit inversion), Gh is the unsaturated gain in cm B +terms of higher orders in B half of the fraction of spontaneous emission entering the sing mode, z is the distance along the active medium with The gain factor g, differs only slightly (parts in 10)from one z=0 at the center of the laser and g is the gain factor for the mode to another and therefore the distributions xt's are ith mode, which is commonly approximated by a lorentzian almost identical, except for the proportionality factor C with its maximum at the ioth mode which determines the power in the ith mode. The small dif- ferences in the g, s, however, plays a crucial role in determi- g=1/[1+a(i-i2] (ld) nating the Ci's when applying boundary conditions(2) The mode io where the gain maximum occurs varies with Given that the photon distribution of each mode has carrier density and is approximately given by nearly identical shapes, one can derive the following io=[0.095KN]+an arbitrary integer A, constant independent of i Typical values areB=10,a=5X10+, and KNom=200 ∑g[x(z)+x(2]=S[xdz)+x(2],(6 the boundary conditions where SeEP/Po is the ratio of the total power to that in an rbitrarily chosen Oth mode. The photon number in each x;(L/2)=R2,*(L/2);x * (-L/2)=Rx(-L/2), mode is then computed as follows: given a value for S, the n coupled equations(1) can be reduced to two equations for the where L is the length of the laser, typically 250 um; R and Oth mode R2 are the reflectivities of the end mirrors. This boundary value problem involving n coupled nonlinear differential =g01N一Nmxd+BN equations does not lend itself to even easy numerical solu dz tions. Major features can, however be extracted with some KN-Nom)=G, /[1+S(xo*+xo) (8 manipulations and a minimal amount of computations This can then be solved with boundary condition (2)and the Equations (la)and (1b)are averaged over the entire cav- resulting photon and carrier distributions can be used to ty, and the forward and backward photons are summed to compute from Eq (3c)the factors B for each mode. The give the total photon density in the ith mode ratio of the photon number in the ith mode to that in the Oth [A,-B K(N-Nom)]P,=BxN (3a) mode is derived from Eq (3a) where Pi=x *+x;and P 4=1PL2)I(R,R,y12+111-(RR2)21 F-B+(B。-B:M1-Nn/NP 1+R1 B-2∫N=Nm ind P;=(1/L)SP dz= average photon density in the cav- intra-cavity ity, N=(1/L)Sn dz= average carrier density. The factor A, involves the photon density at the mirror facet and is related to the rate of photon loss from the end mirrors. B can be regarded as the overall efficiency of stimulated emis on in the cavity, which is considerably smaller than 1 be cause of the nonperfect overlap of the carriers and photons the photon density is highest near the end facet where the 8 carrier density is lowest because of local gain saturation In the case where both mirrors are sufficiently reflective >20%), the carrier distribution is almost uniform along the length of the cavity and B, approaches g Equation(3a)can be physically interpreted as that the spontaneous emission KBN makes up for the slightly excessive cavity loss A, P,over the stimulated gain KB, N-Nom)P The factors A; and B, for mode i depends on the shape of °00200300a400500 the photon and carrier distributions. Now, the fact is that the Gh +KNo (cm) shape of these distributions depends strongly on the level of pumping and the mirror reflectivities, but only weakly FIG. 1. Output optica id lines) from the exit facet of (a]a commor the mode number This conclusion can be drawn from a di laser with R,=R2=0,3 and(b) an Ar coated laser with R,=0.01 R,=0.3, plotted as a function of G+ +KN.m, which is proportional to the rect integration of Eqs.(la)and(1b), which giv pump current. Appl. Phys. Lett., Vol 40, No 9, 1 May 1982 K. Y Lau and A. Yariv Downloaded24May2006to131.215.240.9.RedistributionsubjecttoAlplicenseorcopyrightseehttp:/laplaip.org/apl/copyright.jsp
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Figure 2 shows the principle results of this study: the longitudinal mode spectrum of the two lasers in Fig. 1 is 3mw shown at identical output powers of 3-mw/facet. These re- sults are consistent with that observed in Refs. 11 and 12. at higher output power levels(> 7-8 m W/facet)though, both lasers give essentially single-mode output, although the un- coated laser spectrum is somewhat"purer. "Lasers with in- creased facet reflectivity exhibit spectrum of even higher pu rity, due principally to the higher intracavity optical power and the smaller variation in the gain spectrum along the length of the cavity. This research was supported by the Defence Advanced 8 Research Project Agency H. Statz and G deMars, Quantum Electronics, edited by C. H. Towns FIG. 2. Lasing spectrum of the two lasers in Fig. I at 3-m Columbia University, New York, 1960 D. Renner and J. E. Carroll, Electron. Lett. 14, 781( 1978) P. Brosson, w.w. Ruble, N. B. Patel, and J. E Ripper, IEEEJ Quantum Given a pump level (designated by the unsaturated gain Gh), the laser length and facet reflectivities, one can self-consis- 'H. Namizaki, IEEE J Quantum Electron QE.11, 427(1975). tently compute from Eqs. (7)-9)the total power ratio S and KAiki, M. Nakamura, TKuroda,].Umeda,.Ito, and MMaeda the power in each mode IEEE J Quantum Electron Figure 1 shows the computed optical power output (all J. Katz, S. Margalit, D. P, wilt, P. C. Chen, and A, Ya modes)from the exit facet as a function of(Gh +KN Let37,987(19 laser with R,=R2=0.3 and(b)an AR coated laser with R, 706// d Y Suematsu, J Appl. Phys. 52, 2653(1 (which is proportional to the pump current)for(a)a common KSeki,T.Kamiya, and HYanai,IEEEJQuantum QE-17 R.G. Plumb and J. P. Curtis, Electron Lett reduced to 1%o. Also shown is the average intracavity optical Ettenberg,DBotez, DB.Gilbert,Jc and H V,Kowger power in both cases. At comparable output power levels, the IEEE J Quantum Electron. QE-17,2211 laser manifests the large number of longitudinal modes, evi- York, 1978), Part A p 17 eterostructy ACademic, New dent from Eq (9) i4L. W. Casperson, J. Appl. Phys. 48, 258(1977) Femtosecond interferometry for nonlinear optics J-M. Halbout and C L. Tang Materials Science Center, Cornell University, Ithaca, New York 14853 A technique of time-resolved interferometry capable of observing nonlinear optical phenomena lown to 70 fs is presented. We report the direct observation of the rotational contributions to the onlinear refractive index of molecular liquids using this technique. The subpicosecond dynamics of such a nonlinearity in CS, are investigated. This technique can be readily extended to solids, in particular laser glasses PACS numbers: 42. 65.-k. 07.60.Fs a wide variety of nonlinear optical effects arises from tronic distribution around a fixed nuclear configuration of the nonlinear polarization P, third order in the electric the molecules gives rise to the"purely electronic"contribu field. There are two intrinsic contributions to this third-or- tion. The second contribution is of"nuclear"origin, arising der nonlinearity. First, the nonlinear distortion of the elec- from the optical field induced motions of the nuclei of the Appl. Phys. Lett. 40(9), 1 May 1982 00036951/82/09076503501 1982 American institute of Physics Downloaded24May2006to131.215.240.9.RedistributionsubjecttoAlplicenseorcopyrightseehttp:/laplaip.org/apl/copyright.jsp
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