Semiconductor laser stabilization by external optical feedback

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 27, NO. 3, MARCH 1991 352 zyx zyxwvutsrqp Semiconductor Laser Stabilization by External Optical Feedback Dag Roar Hjelme, Member, IEEE, Alan Rolf Mickelson, Member, IEEE, and Raymond G . Beausoleil, Member, IEEE Abstract-We report on a general theory describing the effect of external optical feedback on the steady-state noise characteristics of single-mode semiconductor lasers. The theory is valid for arbitrarily strong feedback and arbitrary optical feedback configuration and spectrum. A generalized Langevin rate equation is derived. The equation is, in general, infinite order in d / d t constituting an infinite-order correction to the low-frequency weak-feedback analysis. The general formalism includes relaxation oscillations, and allows us to analyze the effect of feedback on both the laser linewidth, frequency noise, relative intensity noise, and the relaxation oscillation sidebands in the field spectrum. The theory is applied to two important feedback configurations; the laser coupled to a single mirror and the laser coupled to a high-Q cavity. The analysis includes excess low-frequency noise due to temperature fluctuations in the laser chip. zyxw z zyxwvuts I. INTRODUCTION XTERNAL optical feedback has proven to be an effective technique with which to modify intrinsic semiconductor laser properties. During the last decade, many new experiments using semiconductor lasers have been made possible by stabilizing the laser using external optical feedback, and numerous studies of these laser have been reported. Multireflector FabryPerot resonators were used already in the early 1960’s to obtain single-frequency output from generally multimoded lasers [ 11, [2]. The additional reflecting surfaces resulted in a frequency sensitive reflectance, increasing mode discrimination. Soon after the first demonstration of the room temperature semiconductor laser, the same techniques were used to stabilize and tune semiconductor lasers [3]. At this early stage, the effect on singlemode dynamics and linewidth was not considered. Only much later was it found that the addition of a passive external cavity could reduce the linewidth of the laser [4]-[6], and it was soon realized that semiconductor lasers with external optical feedback could become compact and efficient sources for coherent lightwave systems. In general, by using feedback systems one can improve the system performance far beyond the performance of the nonideal elements of the system. From this point of view, the intrinsic frequency noise and drift of the laser can be essentially totally suppressed by a feedback system (optical and/or electrical) that locks the laser to a reference interferometer [7]. A technology based on optical and electronic feedback to diode lasers producing subkilohertz linewidths with broad tunability, will offer relief from complex/expensive dye-laser systems. To understand and effectively design such laser systems, it is necessary E to have a detailed and quantitative knowledge of the physics limiting the performance. It is the purpose of this paper to contribute a general theory describing most of the spectral properties of diode lasers with optical feedback as a function of the system parameters. The dynamical analysis of general laser structures. such as semiconductor lasers with general optical feedback, is considerably more difficult than that of the simple Fabry-Perot cavity laser. The traditional analysis of external optical feedback [8][ 121 has been based on a hybrid cross between a mode approach and a traveling wave approach. Lang et al. [8] modeled the external optical feedback by adding a delay term, K & ( t - T), to the single-mode rate equation for the field. By adding a Langevin noise source [9], [lo], and including carrier density dynamics [ 1I], [ 121 many spectral properties of the laser could be analyzed. This traditional approach works well for weak feedback from a single mirror, and has been shown to be a limit of more accurate models [ 131-[ 151. The earliest generalization of the analysis, applicable to more general feedback, was introduced by Patzak et al. [ 161 who related the frequency derivative of the effective reflectivity to a new time constant. Patzak’s approach is similar to the one presented by Kurokawa [17] for analysis of oscillating electrical circuits. While the generalization of the laser rate equations to generic optical feedback is not obvious, the steady-state analysis is, in general, straightforward albeit complicated. By performing a steady-state round-trip analysis of the complete laser structure, the lasing frequency, photon density, and carrier density are found to be solutions of a set of nonlinear equations for the compound cavity modes [ 181. One approach to the dynamical analysis of general laser structures is to use the excitation coefficients of the compound cavity modes as dynamical variables. It has been shown that for many structures the laser dynamics can be adequately described by a single complex amplitude of a single compound cavity mode [18], [19] and the dynamical equations can be obtained from an analytical continuation of the steady-state equations. However, for many systems-semiconductor lasers with optical feedback in particular-even an analysis based on the compound cavity modes would have to take many modes into account. This is due to the broad noise spectrum observed in semiconductor lasers. The noise spectrum could span many compound cavity mode spacings, restricting an analysis based on one compound cavity mode to the lowfrequency regime. An alternative description of general multielement laser structures is to use the field in each element of the laser as a dynamical variable and derive a set of coupled-cavity equations for these variables [ 131. In the case of weakly coupled cavities, the laser dynamics can adequately be described by one mode from each cavity, but for stronger coupling one needs to include zyxwvutsrqpo zy zyxwvut Manuscript received June 26, 1990. This work was supported by the National Science Foundation Engineering Research Center Program by Grant CDR 8622236. D. R. Hjelme and A. R. Mickelson are with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309. R. G. Beausoleil was with Boeing High Technology Center, Seattle, WA 98124. He is now with Solidite Corporation, Redmond, WA 98052. IEEE Log Number 9142991. 0018-9197/91/0300-0352$01.00 0 1991 IEEE zyxwvutsrqponmlkjihg zyxwvutsrqponmlkjihg zyxw zyxwvutsrqponmlkjihgfedcbaZYXWVU HJELME et al.: SEMICONDUCTOR LASER STABILIZATION 353 zyxwvutsr zyxwvutsrqponm multiple modes from each cavity. A coupled-cavity approach based on coupled equations between one mode in each cavity will then be incomplete. Including other modes soon becomes complicated and undesirable. more general techniques have been Lang et al. [I41 have used a local rate-equation technique and have shown that a -general rate eauation can be derived from an analytical continuation of the steady-state equation [20]. Hjelme et li5i> [211-[231 an based On nient operators describing the external cavity. Tromborg et al. 1241 used a traveling wave description ofthe field in the external cavity and obtained a general rate equation given in terms of a convolution involving the impulse response of the cavity. Any proper analysis that includes coupling of the laser diode to an external optical system, must treat open cavities, i.e., cavities with low quality factors (@factor)' Even for the modes with cavity, the standard approach Of defining equivalent mirror losses uniformly distributed throughout the cavity, underestimates the coupling of spontaneous-emission noise into the laser mode [25]. The purpose Of this paper is to present a theory Of semiconductor lasers with optical feedback. We have earlier reOf ported On a general, accurate Optical feedback, introducing an 'perator that completely describes the feedback effects ["I, and this technique to various feedback geometries [621. we have extended the ysis to include both relaxation oscillations, and frequency and intensity noise spectra [221, [23i. In this paper we expand On Our Previous and a general theory in tems Of convenient operators that easily transform in the linearized noise analysis, to allow analytical results for the various noise spectra' The theory is for strong feedback' However, the linearized analysis used to derive the noise spectra is limited to operating regimes where the laser is operating in a single mode. Hence, we do not consider the "coherence collapse" regime [26]-[30]. Our approach is similar to that of Tromborg er al. [24], however, we include excess low-frequency noise (llf-noise), and derive formulas for the field spectrum including the relaxation oscillation sidebands. Another purpose of this paper is to present a systematic investigation of the spectral properties of the external cavity operated laser diode and the so called "self-locked'' laser diode. Recently, some of the results to be presented in this paper have been reported in papers on "self-locked'' laser diodes [31], [=I. The paper is organized as follows. In Section 11 we derive a generalized ~~~~~~i~ rate equation including both spontaneousemission noise and Iow-frequency noise due to temperature fluctuation, I,, Section 111 we study the steady-state solutions and derive the linearized small-signal rate equation. In Section IV we study the dynamical properties, including the dynamical stability of the steady-state so~utionsin Section IV-A, and mode selection in Section IV-B. In Section IV-C, analytical results for the linewidth, frequency noise, relative intensity noise, and the field spectrum are presented. In Section V we present Some approximate results valid in the weak feedback limit. In Section VI we apply the theory to three cases of interest; the solitary laser, the laser coupled to a single mirror, and the laser coupled to a high-Q cavity. Some conclusions are drawn in Section VII. -1,- ~ l z - G G x & i q c 11 3 I'd1 Fig. 1 . Illustration of the geometry of a laser diode with generic optical feedback. piing to any other optical system, is obtained by integrating the traveling wave equation over the length of the laser, and using the proper boundary One can do this by assuming that the interaction between the foMiard and backward propaInside the active is weak except at the gating cavity, the weak coupling between the forward and backward is due to spatial inhomogeneities. These inpropagating homogeneities can be due to temperature fluctuations and/or locarrier density fluctuations [33]-[35], and are believed to be rise in both intensity noise and part of the origin of the frequency noise at ,ow frequencies, The geometry under consideration here is illustrated in Fig, 1. A one-dimensional model, with the laser operating in the fundamental transverse mode is assumed. The diode laser has active cavity length L , left facet reflectivity r 2 , and right facet effective reflectivity r e f f ( ~The ) , effective reflectivity includes all coupling to the optical system. The exact form of is dependent on the details of the feedback geometry under consideration, and is in general straightforward to derive. form of r e d W ) , Later is independent of the Our we will derive the effective reflection coefficient for those systems we consider in detail, but for now we will assume that it exists and is known. the laser is biased from a stable low-noise current We and the heat-sink temperature is controlled to avoid drift and longitudinal mode jumps. The total optical field at a point diode cavity will be represented as in the zyxw E(z, t ) = i[E(z, f ) e ' W " r+ E * ( z , r)e-'""'] (1) where W O is the lasing frequency. Following the treatment in [36], the traveling wave equation for the complex field amplitude can be written gE(z, t ) g + F(z, t ) (2a) zyxwvutsrq zyxw 11. GENERALIZED RATE EQUATIONS An accurate treatment of the semiconductor laser that accounts explicitly for the open laser cavity, and thereby the cou- = g(1 - t l ~ / ' )- iq(l - E,~E(*) (2b) where k is the wavenumber, f i g is the group g is the gain, a is the linewidth enhancement factor, and E ( € , ) is the gain (index) compression factor. Typically 1s very small and will be neglected in this study. However, E will be included Osas it plays an important role in determining the cillation damping in laser diodes. F(Z, [) represents the spantaneous-emission contribution to the polarization coupling into the fomard propagating wave. To account for the small spatial inhomogeneities in the cavity, we have to to the backward propagating wave. The coup1ed wave equations Only Order can be written (8; + &)EF = FF - iGkE, (aZ - &)En -Fn + - ak (EF 2k + EB) + i6kEB + azk (EF + E,) 2k - (3a) (3b) where & = k - ( i a r / C g ) i f g is a nonlinear wavenumber operator that accounts for both the linear and nonlinear gain. The 354 zyxwvutsrqponmlkji zyxwvu zyx zyxwvutsrqpon IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21, NO. 3. MARCH 1991 perturbations, 6k and a,k, are assumed to be due to small temperature and camer density fluctuations, and are therefore stochastic variables. To proceed, we will treat the right-hand side of (3) as noise sources. This can be justified by noting that the fluctuations 6k and a, k are assumed small, hence the small fluctuations in the field variables EF(EB) would be second order. The rate equation, including the spontaneous-emission noise source, can be derived by considering the buildup of spontaneous emission [ 2 1 ] . As the laser oscillations build up, the total field is the sum of the field generated by spontaneous transitions, and subsequently amplified by stimulated emission, in all previous round-trips. The rate equation is then obtained from a straight forward integration of the wave equation. A formal solution of (3) can be written E&, t ) The correlation function for FL(r) is not known, however, it should mainly contribute to the low-frequency fluctuations. In the Appendix, we consider temperature fluctuations as the source for the temporal varying spatial inhomogeneities and derive an expression for the power spectrum of FL(t).It is shown that the power spectrum has a 1/f-like frequency dependence, and is proportional to the photon number 1. Thus, the temperature fluctuations will result in a power independent contribution to the laser linewidth. . The operators in equation (6) are, in principle, nonlinear. The wavenumber operator k is assumed to be a function of the carrier number N , and is therefore implicitly a function of time. By assuming that the time derivative of N is negligible compared to the time derivatives of E, the operators in (6) can be made to commute with N , and therefore act only on E. With these approximations, (6) can be written in a rate equation form as zyxwvutsr zyxw zyxwvutsrqponm s: zyxwvutsrqponm = dzf&'- z"F,(z', t) + e-'"EF(O, t) (4) ;(Fe, where FFT(z,I)is equal to the RHS of (3a). A similar solution exists for E s ( z , t). The integral must be interpreted as a stochastic integral, and only its moments and correlations can be related to observable quantities. Using the proper boundary conditions, [a, - EF(O) = r2 EB(0) where we have defined (54 = - r e x ~ ~ ~ r ) [-i(o - wN) + i(G - y) + ;reXi~(t) + F ( t ) + FL(t) 6 = uRg = (1 + ia)GN(N - N J (9) - eG(EI2 (loa) an integration of the equations over the cavity yields where 7 = 2 La,k = 2L/v, is the diode cavity round-trip time, F(t) is the noise source due to spontaneous-emission noise, and FL(t) is the low-frequency noise source due to the temperature fluctuations. Equation (6) describes the fact that the field after one round-trip in the cavity is equal to itself plus the total contribution of spontaneous emission from the cavity. With F = F L = 0, (6) has the form of the steady-state lasing condition obtained from a round-trip analysis of the laser cavity. Note that in (6) teffis an operator obtained by replacing w wo + ia, [21]. Similarly, dispersion in the gain medium can be included by using the same replacement in g (w). For convenience, the field amplitude E(t) is normalized such that IE(t)I2 equals the photon number in the diode cavity. The spontaneous-emission noise source can then be shown to have the correlation function [21] + ( F ( t ) F*(t')) = K2R6(t - f') (7) where R is the spontaneous-emission rate and K 2 is the open cavity correction to the spontaneous-emission rate [ 2 11, [25]. In the remainder of this paper, we will set K = 1, or equivalently include the open cavity correction in R. Equation (7) shows F ( t ) to be 6 correlated. However, in a laser with external optical feedback, the external cavity modifies the spectrum of the integrated spontaneous emission. The resulting spectrum is no longer white, and hence F ( t ) should not be 6 correlated. However, we can show that this is a second order effect that is negligible unless the feedback is very strong [21]. The lowfrequency noise source is written FL(t) = [-! T zyxwvu and wN is the solitary laser cavity resonant frequency closest to the lasing frequency wo, GN = aNG is the differential gain, N,, is the carrier number at which the laser material turns transparent, and rex(wo) is the complex number corresponding to the such that it deoperator r e x ( w O - i d t ) . We have defined rex scribes all deviations from the symmetric uncoated laser diode with facet reflectivity r,. The generalized rate equation (9) is essentially the same as the one introduced in [ 1 5 ] , however, in this paper we have included the spontaneous-emission terms and spatial inhomogeneities. In [I51 it was shown that the operator f,, operating on E(t) were equivalent to a sum of delay terms as used in most standard optical feedback analyses with the notable exception of Tromborg et al. [24]. The operator formalism we use simplifies the understanding of the dynamical properties of the system, with the frequency response of many cavity systems already known. To describe the fluctuations in the high-frequency regime, close to the relaxation oscillation frequency, it is essential to include the carrier number as a dynamical variable. The equation for the total carrier number N ( t ) is found by integrating the equation for the carrier density over the length of the diode cavity. It can be shown that the contribution from the spatial inhomogeneities can be ignored in the carrier density equation, due to the strong damping of low-frequency carrier density fluctuations. With these approximations the carrier number equation can be written as SLdzf(i26k+ 2 O where P is the pump term given by the injection current divided by the electron charge, T, is the carrier recombination time and zyxwvutsrqponmlkj zyxwvutsrqponmlkjihgfed zyxwvutsrqpo zyxwvutsrqponm zyxwvutsrqpon 355 HJELME et al.: SEMICONDUCTOR LASER STABILIZATION G is the real part of G. F N ( t )is a noise source with correlation properties (FN(t) F d t ? ) = (FN(t) F , ( t ? ) + where F[ = E * F EF*. We should note that the open cavity correction does not enter the spontaneous-emission term here as in (9) [21]. 111. STEADY-STATE AND SMALL-SIGNAL EQUATIONS A. Steady-State Analysis In the steady state, the operating frequency, field amplitude, and camer density are found by taking the time average of the rate equations. However, due to the fluctuating phase, the average of the complex field amplitude is zero and the average of the field rate equation would be undetermined. To proceed, we therefore introduce the photon number I and the phase 6, and write E(t) = f i exp (id).Using this definition in the generalized rate equation (9) results in y+ = ia,4 - [ $@ex - E - frequency shift is found to be typically much less than 1 MHz, and therefore negligible for most purposes when the laser is operating far above threshold. Accordingly, in the rest of this paper, this frequency shift will be neglected by setting R = 0 in (15). It is instructive to combine (15a) and (15b) to one equation [Aw - crfAG]' + [iAG]' = (fI'ex12 (16) describing a curve in the Aw - AG plane. As expected, (16) shows that the maximum frequency deviation and excess gain are directly proportional to the "additional losses" rex caused by the external optical system. If lreXl2 is a constant, (16) describes an ellipse [27]. In the general case, the curve will be slightly more complicated. zyxwvutsr zyxwvutsrqponm zyxwvutsr I rex)E + -21 (G - 7 ) + -2 rex + - +EFL. The steady-state equations are found by taking the time average over (13). Special attention must be paid to the third term. In general, the time average of this term cannot be neglected. For the particular case of modulated lasers Schiellerup et al. €371 have shown that this term leads to an average frequency shift. In this paper we are dealing only with unmodulated lasers with narrow spectra and with relaxation oscillation sidebands typically several tens of dB's below the central portion of the line spectrum. Under these conditions, only frequencies close to wo, the lasing frequency, are important for the steady state. We can then use the approximation B. Small Signal Equations As is usual, we assume that the fluctuations of the optical field and carrier number are small perturbations to the steadystate operating points. This allows us to linearize the rate equations. However, the presence of the operator fie, in (9) complicates the dynamical analysis. This operator could, in general, introduce time constants that are long compared to the periods of the relaxation oscillations. For the description of fluctuations in this high-frequency regime, we cannot approximate (9) by a first order differential equation as is usually done. To derive simple analytical formulas for both the phase and amplitude noise, as well as the field spectrum sidebands we use an expansion of the following form: where ~ ( t and ) + ( t ) are slowly varying amplitude and phase perturbations, respectively, wR is the relaxation oscillation frequency, ER+(t) is the complex amplitude of the relaxation oscillations, and AN@)and NR(t) are the low- and high-frequency camer number perturbations, respectively. The correlation function of F j ( t ) , i = 0 , , - , can be approximated as + zyxwvutsrqp Using Ito calculus [38], we can transform to the I and 4 variables. Assuming that R is approximately the same with and without feedback, the steady-state condition can be written in terms of the frequency shift, Aw, and the excess gain AG as AU E 00 - U, (ip)*/(l AG Go - G, 1 + io)*) (15a) - where w, and G, is the frequency and gain of the solitary laser, respectively, (rex = 0) and Go = (1 ia)GN(No- Ntr).These two equations together with the averaged camer equation, P No/~c GoI = 0, determines the steady-state solutions. It is useful to estimate the additional frequency shift due to the spontaneous-emission factor present in (15). With a typical photon number of lo5 (a few mW output power), the additional + ( F ; ( t )FJL(t')) = R6(t - t') i , j = 0, +, -. (18) Similarly, we can separate both FL(t) and F N ( t )into a low- and high-frequency part. The relaxation oscillation resonance induces low-intensity sidebands at the frequencies wo f wR. Since the values of uR are far larger than those of the laser linewidth, the strong optical carrier component at ooand the weak sidebands at wo f wR are well separated from each other in the frequency domain. Therefore, the relaxation oscillation sidebands can be treated eeparately from the low-frequency fluctuations as indicated in (17). We have not specified the exact value to use for uR,but the analysis to follow is not critically dependent on this value and we can use whatever is most convenient. In what follows, we will therefore choose uR = wR0, the relaxation oscillation frequency of the solitary laser. It should be noted that the field spectrum will be different at +aRand -uR due to the amplitude-phase coupling in the laser. To proceed, we must linearize (9) and (11) in terms of the 356 zyxwvutsrqponmlkjih zyxwvutsrqponm zyxwvutsrqpon zyxwvutsrqpo IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 21, NO. 3, MARCH 1991 perturbations, and then separate the low- and high-frequency fluctuations. To first order, we can do this if we define the new variables A , ( t ) = ER*(t)e-'Q'r'. (19) It should be noted that the power spectra S , calculated from (19) will be the convolution of the true relaxation oscillation sidebands and the central laser line. For narrow laser linewidths, SA+ should be approximately equal to the true field spectrum SE. Linearizing the rate equation, and separating the different frequency components, the resulting equations can be written as + icu)AoAN + iGI21Aop + Fo + FLO @a, f iwR)A+ = i G ~ ( 1+ icU)AoiNR + iG,Z(A+ + A?) + F+ + FL+ @a, - iWR)A- = iGN(l + icu)AOiN: + ;GII(A*, + A-) + F- + FLBN(a,)AN = -2GoAoAop + FN f i ~ ( a+ , iWR)NR = -2GoAoi(A+ + A*_)+ FNR U(a,)Ao(p + i+) = iGN(l (204 (20b) where we have defined the operator A(d, + iw) = 0(a, + iw) + (1 + i a ) w i D N ' ( a , + i o ) - GII. (24) It follows from (23a) that there is a form of "duality" between the amplitude and phase dynamics. A systematic interchange of symbols as follows: P * i+ (254 A-0 (25b) leaves the equations unchanged. Hence, the phase dynamics follows from the amplitude dynamics by replacing H by U and vice versa. Similarly, by comparing (23a) to (23b) and (23c), it follows that ;(A+ A?) has the properties of high-frequency amplitude noise, and :(A+ - A!) has the properties of phase noise. The same conclusions could be reached by arguing that the amplitude spectrum should have Hermitian symmetry, while the phase spectrum skew Hermitian symmetry. These symmetry properties require that x,,,(f) = i [ x ( f ) + x * ( - f ) ] , and xPhase(f)= $ [ x ( f ) - x * ( - f ) ] in agreement with (23). + zyxwvutsrq zyxw (20c) (204 IV. DYNAMICAL PROPERTIES A. Dynamical Stability The dynamical properties of interest includes the dynamical stability, mode selection, and spectral properties. With linearwhere we have defined the operators ized equations, we can most easily evaluate these characteristics in the frequfncy domai!. In the frequency domain, the lin+ i o ) = (a, + iw) - ;[f,,(wo - i t a , + iwl) - rex(wo)] ear operators H , U , and DN becomes complex functions by replacing ia, * w . The transform of the complex conjugates of (214 the operators are just the complex functions evaluated at negative frequencies. The two transformed equations for the low&(a, iwR) = (a, iw) r N (21b) frequency fluctuations can be written in matrix form as and defined r N E ( ( l / r c ) GNI) and GI = -eG. Because it can be shown that any nonlinear gain term in the carrier number equation is negligible compared to the nonlinear gain term in the field equation, we have not considered any such terms. Furwhere the matrix A(iw) is defined as thermore, we have neglected the phase factor e i min the noise sources F i ( t ) . This factor has no effect on the second moments H(iw) U(io) needed for the calculations of the noise spectra. A(iw) = H * ( i w ) U*(-iw) To gain some further insight into the dynamics of the linearized equations, we formally eliminate the camer number from and FTOT is equal to the RHS of (23a). From the identifications the equations by writing done earlier, we can immediately write down the corresponding + + + + 1 A N = d,'(13,)[-2GoAoAop NR = fi,'(a, + FN] + h~)[-2GoAo;(A+ (22a) A?) After some manipulations we can write AGUA~ P + ir<a,) = F, iAo 4 = F+ + FNR]. (22b) zyxw + F~~ + +(I + ~ U ) G ~ A ~ D N I ( ~ , ) F , (234 A@,+ iWR) ;(A+ + A*) + 0(d, + iw,) equations for the high-frequency fluctuations ;(A+ - A i ) + FL+ + ;(I + icU) GNA&N'(~, + iWR)FNR where again FTOT* follows from the RHS of (23b) and (23c). The dynamical stability of the steady-state solutions now follows from (23b) A(a, - iwR) ;(A= F- FL- + A*,) + 0(a, - io,) ;(A- - A*,) ;(I + icU) GNAoDN'(~,- iUR)FER @3c) The system is unstable when one of the zeros of (29) has positive real part. In general, numerical techniques must be used to track the roots. However, in the low-frequency regime (w -+ zyxwvutsrqponmlkjihg zyxwvutsrqponmlkj zyxwvutsrqp zyxw HJELME et al. : SEMICONDUCTOR LASER STABILIZATION 0), we can use the approximation ( 1 4 ) , and find the system to be unstable when where we have neglected a term proportional to G I I r N / w i .The imaginary parts of the roots of ( 3 0 ) give the system resonances. For the case when wR is much less than the spacing between the resonances in the external optical system, the approximation ( 1 4 ) is valid at the relaxation oscillation frequency, and the only significant resonance in the system is the relaxation oscillation, with frequency and damping given by 357 as a result of mode hopping between external cavity modes in analogy with noise-driven transitions of a potential between potential valleys [ 4 1 ] . MBrk et a l . [ 4 1 ] derived the associated potential for mirror feedback. Following their arguments, we can derive a similar potential for arbitrary feedback. At low frequencies and with bias high above threshold, we can neglect a,I and write an equation for 4 a,+ zyxwv = w, - wo + i[Im - Q Re][e-"f,,e'@] + Fm. (32) In the case of weak mirror feedback, there is only one time delay term, and one can derive an equation for the phase difference [ d ( t ) - + ( t - T ) ] . For arbitrary feedback, the operator rex involves many time constants, me,, and no such equation can be derived. However, if the laser is oscillating in one external cavity mode and the mode jump happens in a time interval less than T,,, we can consider the fluctuations on a time scale r << T,, and treat f,, e '@ as constant, and derive an equation for [ + ( t ) - 4 ( 0 ) ] . We have to assume that the laser oscillates (without noise) in one mode for t < 0, then at t = 0 we turn on the noise source. Assuming + ( t 5 0) = 0, we have f , , e i g = rex, and can write zyxwvu zyxwvuts zyxw where wR0 = q and , rI = (1 - Im P / l l + i P I 2 ) G , I . rIis dominated by the nonlinear gain, since we have neglected the spontaneous-emission term. The frequency pulling is mainly due to the changed differential gain and the new coupling between intensity and carriers due to the optical feedback. In the limit where ( 3 1 ) [and ( 1 4 ) ] is valid, one can in fact define new effective differential gain and index constants [ 1 8 ] , [ 1 9 ] , and immediately write down the small-signal response for the case with optical feedback. In the general case, the dynamics are more complex than just a change in effective parameters. Nevertheless, in general, one will observe frequency pulling of both the external cavity resonances and the relaxation oscillation resonance, due to the coupling mechanisms in the laser. In addition to the stability of the steady-state solutions, the sensitivity of these solutions to perturbations in the system parameters is of great importance for practical system design. Fluctuations and drift in the operating frequency of the laser can be caused by perturbation in the injection current or junction temperature. A change in the injection current or junction temperature would induce a change in the solitary laser frequency. Accordingly, it is of interest to evaluate awo/aw,. As we will show later, this derivative is identical to both the inverse of the left-hand side of ( 3 0 ) determining stability, and the square root of the Lorentzian reduction factor. In addition, changes in a particular delay time r, in the external optical system cause the operating frequency of the laser to be pulled according to the value of awo/aT,. Unless we specify rex(w), we cannot evaluate these sensitivity figures. We will come back to these figures when discussing the weak feedback approximation in Section V. With equations for the mode oscillation frequencies and the excess gain and a criterion for dynamical stability, we are in the position to determine the lasing frequency. The solutions of ( 1 5 ) give the mode spectrum induced around the solitary laser mode. However, with many possible steady-state solutions, we have to consider the problem of mode selection. ar4 = - a , ~+ F@ (33) where V(4) is the potential V(4) = -(w, - w0)4 - i[Im - a Re][iI',,(ao)e-'@]]. (34) For weak feedback in an external cavity laser, this potential reduces to the form given by MBrk et a l . [ 4 1 ] . In general, the potential barriers depend on the complex value of r,,(ao)and the frequency deviation from the solitary laser frequency a,. The potential is similar for both the mirror feedback and the self-locked laser, with the narrowest linewidth mode having the largest potential barriers, hence being most stable and therefore being the one to lase. zyxwvutsrqp zyxwvutsrqpo zyxwvut C. Spectral Characteristics 1 ) Noise Spectra: The fluctuation spectra are easily derived from the linear set of equations ( 2 6 ) - ( 2 8 ) . With analytical expressions for the transformed functions, we can find the fluctuation spectra by taking the proper ensemble averages of the absolute value squared of the transformed functions. To keep the analysis simple, we will neglect both the carrier noise source and all off-diagonal correlations as given by ( 1 2 ) . It can be shown that these correlations make only small contributions to the noise spectra [ 3 5 ] . The spectra of interest includes the frequency noise spectrum, S $ ( a ) , and the relative intensity noise spectrum, RIN (a)= 4 S , ( w ) . These spectra can be expressed as B. Mode Selection It has been observed [ 3 9 ] , [ 4 0 ] that a laser with external optical feedback tends to oscillate in the external cavity mode with the lowest linewidth and not in the mode with the lowest threshold gain. This unexpected result has recently been explained . [ l H * ( - i w ) + H(iw)I2 t a $ H * ( - i w ) ID (iw)I 3 - H(iw)I2 (354 . 358 .. . .. zyxwvutsrqponml zyxw zyx zyxwv IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21, NO. 3, MARCH 1991 tion function. However, due to the amplitude-phase coupling and the complicated form of the noise spectra, no simple analytical result is possible. Within the low-frequency regime, the camer number adiabatically follows the field amplitude fluctuations and provides gain saturation damping of the amplitude fluctuations. Hence, the noise contribution to the field spectrum comes from the frequency fluctuations only. Neglecting the amplitude fluctuations, the autocorrelation function is given by zyxwvutsrq zyxwvut zyxwvutsrq zyxwvutsrqpon (%)’I. where we have defined the function D(iw) = U(iw) H*(-iw) + U*(-iw) H(iw) and U, H, and DN are defined in ( 2 1 ) and ( 2 4 ) by letting a, + 0. SL(w) is the power spectrum of the temperature related fluctuations and is defined in the Appendix. Since SL(w) is proportional to I , the low-frequency contribution to s, and RIN is power independent. The amplitude and phase are and we can define the cross-correlation spectrum [ 4 2 ] SP4(4 = i [ ( & ( w ) i J * ( w ) ) - ~ ( 7= ) Aiexp [-; ([w+ = A; exp [-? 1 (36) 7) - Im 7) I r a d( S4(w) (&*(U)P(w))l - [U(iw) H*(iw) 1 + U*(iw) H(iw)] I’ U ( i w ) ] [ H ( - i w ) - H*(iw)] + [U(-iw) + U*(io)][H*(-io)- H(io)] ID(iw)I I [U*(-iw) - U(iw)][H(-iw) + H*(iw)] + [U(-iw) - U*(io)][H*(-io) ’ ID This correlation causes the field spectrum to be slightly asymmetric. Away from the line center, the field spectrum can be approximated by the spectra of A , ( t ) , SA+(W) zyxwv (39) w7/2 In general, we cannot solve the integral in (39)9 but we can consider some limiting cases. If Sb (a)is approximately constant in the frequency range of interest, the Fourier transform + U(-io) H*(-iw)] + CY$ w 1 2 ) ] 1. + H(iw)] (37) of (39) results in a Lorentzian line with linewidth AwL = sb(o) = Awo[l - Im (IliawRe r e x ] - ’ rex = Awo(~w,~/~o0)-’. (40) = (A+(w) 3 S,-(-o) = ( A J - w ) AT(-w)) = R[lU*[-i(w + = (AT(w) AT (U)) wR)] - H*[-i(w + ID(i(w + SdO) + CY; * -i(o ‘+ ((U*[-i(w + IU[i(w + wR)] + H [ i ( w + wR)]I’ + %)I2 + wR1)) + ( ~ [ i ( +w wR)] + ~ [ i ( w+ oR)1)l2 1 wR)]I2 + wR)l - H * ( - i [ o ID(i(w OR)] - H*(-i[o resulting in analytical formulas for the relaxation oscillation sidebands. In (38), w is the frequency deviation from the relaxation oscillation frequency wR. 2) Field Spectrum and Linewidrh: In general, the field spectrum is given by the Fourier transform of the field autocorrela- + WR])) lD(i(o + WR)I2 - (U[i(w + 4’ 1 + wR)] + H [ i ( o + wR)])12 (38b) We call the factor (aw,/aw,,)’ the Lorentzian linewidth reduction factor, to differentiate it from the smaller linewidth reduction seen in the presence of excess low-frequency noise [see (42)]. zyxw zyxwvutsrqponmlkj zyxwvutsrqponmlkjihgf zyxwvutsrqponml zyxwvut 359 HJELME er al. : SEMICONDUCTOR LASER STABILIZATION For excess low-frequency noise we can consider two limits. For frequencies where the phase modulation index (0 = 6 w / w ) is comparable to unity, the “side-band power” is significant and the line is broad compared to the Lorentzian line, and with wings falling off faster than the Lorentzian [43]. For a low phase modulation index, very little power is in the sidebands, and the wings fall off according to the phase noise spectrum [see (44)]. For l/f-noise, the integral in (39) would not converge. However, this is only due to the integration down to zero frequency. Any practical measurements would have a finite measurement time, and hence the lower integration limit would be wmin and the integral converge. Similar conclusions can be reached if one uses a self-heterodyne technique with finite delay time to measure the field spectrum. Since l/f-noise is seen in most noise measurements, it is of interest to learn more about the resulting field spectrum even if we cannot solve the integral in (39). The laser linewidth will be approximately equal to the Fourier frequency at which the unity phase modulation index occurs [7], or equivalently, at the frequency at which the rms phase deviation reaches one [see (47)]. Halford [44] has used a definition of the linewidth consistent with this, that can be evaluated directly from the phase noise spectrum. For the case of white frequency noise, the rms square phase noise for all frequencies larger than the linewidth is 1 /T rad2. In general, we can write this as nm 2 3 Aw . _ dw I 2T T S,(w) - = - rad2 zyxwvuts zyxwvut This agrees, within a constant factor, with the linewidth of the proposed Lorentzian to the power 3 / 2 [31], [44]. It is clear from (40) and (42) that the important quantity for linewidth reduction is the frequency derivative of the effective losses in the external optical system, the same quantity that determines the dynamical stability of the steady-state solutions. As expected, the linewidth and stability are closely related. Consistent with the discussion in the Appendix, we can assume a frequency noise spectrum of the form + 1) (45) where S,, So, and S, are given in (35a), (35b), and (37), respectively. It should be noted that S,, s,, and s,, were derived under the assumption that the equations where valid only in the low-frequency regime, while (45) shows that they could be used all the way up to the high-frequency regime. In fact, it seems as if these equations could be used all the way up to high frequencies since the only assumption needed is that [4(t - 7) $(t)] << ~ / 2 which , is the case if 7 << 7,, where 7, is the laser’s coherence length. The first thing to note from (45) is the asymmetry due to the amplitude-phase coupling introduced by both the CY factor and the optical feedback. This asymmetry does not show up in the central part of the field spectrum when we neglect the amplitude fluctuations. Towards the line center the phase noise will dominate, and the field spectrum will be given by (41) and use it as a definition for the linewidth of a laser with an arbitrary phase noise spectrum. In the case of dominating 1 /f noise, the linewidth then follows as: S&(w) = Am,(: from a Lorentzian for small Lorentzian linewidth reductions, to a 1/f-noise dominated line shape for stronger feedback. In the high-frequency regime, we have already approximated the field spectrum with SA, (38). It is very instructive to rewrite the two spectra as (43) where Aw, is the Lorentzian linewidth and w, is the comer frequency below which the 1 /f-noise dominates. The linewidth can now be written as which for flat frequency noise spectrum will coincide with the tails of the Lorentzian line. For the case of dominating 1/f-noise, the field spectra falls off like 1 / w 3 , rather than 1/U’ as for the Lorentzian, consistent with the line shapes proposed by Halford [44] and Laurent er al. [31]. Some remarks are due on the general form of the field spectrum and the validity of the approximation (46). It follows from (46) that the square rms phase deviation (47) determines the power ratio between the optical “camer” having width 2w and the side-band power [7], [45]. An optical feedback system will, in general, reduce the phase noise over only a limited bandwidth. The resulting linewidth could be vary narrow, nevertheless the side-band power could still be significant. Such a feedback system would capture only a small amount of the power into a sharp spectral line. For frequencies lower or equal to the linewidth, we cannot find an analytical form for the line shape. However, for the frequency noise spectra under consideration (no fine structure at these low frequencies), the central portion of the line cannot deviate much from a Lorentzian or a Lorentzian to the power 3/2. zyxwvut V, WEAK FEEDBACKAPPROXIMATION If we assume that Po, = Awo(~ws/C300)-2, then for feedback such that (aw,/aw,)’ << Aw0/(2w,) the line shape is close to a Lorentzian with the linewidth reduced by the Lorentzian linewidth reduction factor. On the other hand, if (aw,/13w,)~ >> Aw0/(2w,), the line deviates strongly from a Lorentzian with the linewidth reduced by the square root of the Lorentzian linewidth reduction factor. Consequently, the line shape changes Even for the simplest feedback geometries, the general formulas derived in the previous sections are quite complex. Therefore, it is of great interest to find simple approximate formulas for the important case of weak optical feedback. We assume that a feedback element with a general complex amplitude reflection coefficient given by r3(w) exp [ -i43(w)] is located a distance Le, from the laser. If Ir31 << 1, r2, then the feedback 360 iS zyxwvut zyxw zyxw zyxwvutsrqponmlkjihg zyxwvu zyxwv zyxw VOL. IEEE JOURNAL OF QUANTUM ELECTRONICS. weak, and the effective rear facet reflectivity can be written (48) where r L 2 L e , / c . Therefore, the “additional loss” caused by the external cavity is given by + wherewehavedefinedD, = ( ? r / T d F d ) [ r 3 ( W o w ) e-iIorr+d3(wo+w)1 - r3(wo)]. To keep the analysis simple, we have neglected the gain saturation coefficient G,and used aT = -1. From (53), it follows that the frequency noise will exhibit strong reduction across a bandwidth where I D 3 / w ( is large. For the limit w 0, we can write + (+ S,(O) = - 1 21 WO = - - - W, 7rdl + CYZ = Go - G, = r, sin (worL + 4, + tan-’ 2* -- r3 cos ( ~ a) + ~43).7 (50a) ~ (50b) Fd The linewidth reduction and stability are governed by Often the term containing the time constant ( 7 L + a4,/aw0) dominates the linewidth reduction. In particular, a4,/aw0 can be very large if r3 is the reflection from a high-Q cavity with very long storage times. To gain some further insight, we consider the specific case of operating the laser on an external cavity resonance. On resonance, r 3 ( ~ 0 w ) = symmetric in U , and 4 3 ( ~ 0+ U ) = antisymmetric in w . If, in addition, we choose the operating frequency such that worL + tan-’ a = 2 n a , n E I , the sensitivity of the operating frequency of the laser to variations in the injection current, junction temperature, or the distance L , is governed by + awo -I [ J’ 3 ~ 4 ~ (54a) ) 2s?)[TIFd The frequency noise is reduced by the Lorentzian linewidth reduction factor as expected. The RIN, on the other hand, is reduced by a factor (1 a’)at low frequencies independent of the feedback level (we have-assumed the feedback to be strong enough to dominate in the U-operator). For a 5, this corresponds to a 14 dB relative intensity noise reduction at low frequencies. Because D, will in general be periodic, it follows from (53) and (46) that the field spectrum will show strong external cavity modes if the period of D, is smaller or comparable to the solitary laser linewidth. At the external cavity resonances ID,( = 0, and the frequency noise spectrum, and the field spectrum would be equal to the solitary laser spectra. Away from the resonances the spectra would be given by (53). As a result, the field spectrum contains a number of narrow lines associated with each cavity mode, with an envelope width equal to Awo [lo]. + - Fd AG(w) - -(rL+ ~ zyxwvuts where Fd E ( 7 r r z ) / ( l - r : ) is the finesse of the laser diode cavity. The corresponding frequency shift and excess gain are given by AW E 21, NO. 3, MARCH 1991 -~7 FdTd r3(rL + E)] VI. APPLICATION OF FORMALISM To check the formalism presented in the previous sections, we apply it to the well-known case of a Fabry-Perot cavity semiconductor laser. Moreover, by considering this case, we have a reference with which to compare the optical feedback systems considered in the next sections. The noise spectra are found by simply letting reff= r, in the formulas derived in the previous sections. The expressions for the frequency noise, relative intensity noise, and cross-correlation noise can be written: S,(w) = q 2 1( 1 (524 It is apparent from ( 5 2 ) that the “effective” time constant Teff = rL a&/aw0 is a very important parameter for the feedback system. To find simple expressions for the noise spectra we have to assume that the frequencies under consideration are sufficiently low and the feedback sufficiently high that rex-term dominates in U(w) [(21)]. The noise spectra can then be approximated as THE A . Solitary Laser Diode + zyxw y)+ + .;y) (1 + w2 ID3D, + r; +w i m 1 2 (53b) Spm(w)= R 21 -- (1 -> + 2SL(W) (Yo; W Without thermal noise, the formulas for the relative intensity noise and the frequency noise spectrum reduce to the wellknown formulas in the literature [46], [47]. It follows from these results that the “comer frequency” in the relative intensity noise and frequency noise differs by a factor a$, i.e., w , , ~ = ~ zyxwvutsrqponmlkjihg zyxw HJELME et al.: SEMICONDUCTOR LASER STABILIZATION 361 zyxwvutsrq zyxwvutsrqpo zyxwvuts zyxwvutsr zyxwvutsr However, since cyT is close to unity, the two frequencies will be close. To keep the number of parameters to a minimum, we normalize all frequencies, linewidths, and damping constants to the relaxation oscillation frequency wR. We have used the following parameters; A o ~ / w R= 0.001, o,/oR = 0.001, 01 / O R = io-', r N / W R = 0.05, G I I / w R = 0.15, W R T d = 0.1, a! = 5 , and oT = -0.9. Fig. 2(a) shows the relative intensity noise RIN and the frequency noise Sd,showing excess low-frequency noise and the characteristic relaxation oscillation peak. It is important to note that the relative strength/damping of the relaxation oscillations in RIN are very sensitive to the value of the gain nonlinearity GI, RIN (wR)/RIN (0) f&/[rN(rN + G11)]2. We have chosen typical values of r N and G,I from the literature. Fig. 2(b) shows the various components of the field spectrum. The figure clearly shows how SA, approaches the tails of the Lorentzian line at low frequencies, while showing relaxation oscillation resonances at higher frequencies. The crosscorrelation spectrum S , is also shown to peak at the relaxation oscillation frequency. It is of interest to derive simple analytical results for the strength and asymmetry of the relaxation oscillation sidebands. Neglecting, for the moment, the asymmetry in the field spectrum, we find the amplitude of the relaxation oscillation sidebands to be SE(OR) = cy2R/81'i. Normalized to the Lorentzian line this results in zyx 0 i -m -10 E-20 -30 Normalized Frequency corresponding to relaxation oscillation sidebands typically 30 dB below the central Lorentzian line. The asymmetry in the , (wR) field spectrum is due to the cross-correlation spectrum S a~rN/4ioRr;. The relative asymmetry can be approximated as Fig. 2. Field and noise spectra for the solitary laser diode. (a) Normalized frequency and relative intensity noise; S, ( U )/ A U ~(solid line), and N N ( U ) / [ ( ~ R ~ ; ) / ( I U(dashed ;)] line). (b) Normalized field spectra, SA*(@) (solid line) and S L ~ ~ (dashed ~ ~ ~ line), ~ ~ and ~ normalized ( U ) cross-correlation spectrum SPQ(o)(dotted line). All spectra in (b) are normalized to 41/Aw0. For typical laser parameters this results in approximately 10% asymmetry [42], [48]. Fig. 3. Illustration of the geometry used for a laser diode with mirror feedback. B. Laser Coupled to a Single Mirror In this section we consider the simplest form of optical feedback; external feedback from a mirror placed a distance Le, from the laser. External cavity operation is attractive for a number of applications requiring narrow (sub-MHz) linewidth. Monolithic integrated semiconductor lasers are clearly advantageous, however, no monolithic laser has been reported with linewidths in the sub-MHz region. So thus far, perhaps, the easiest way to achieve narrow linewidths is to use external cavity operation. Several authors [4]-[61, [81-[301, [371, [391-1411, 1491 have considered the operating characteristics of the external cavity operated laser. Here, we apply the general formalism developed in the previous sections to this cavity configuration, to study both the excess low-frequency noise, relaxation oscillations, and field spectrum. The geometry under consideration here is shown in Fig. 3. Here r2 and r, are the amplitude reflection coefficients at the left laser facet and the external mirror respectively. The coefficient r, includes all losses due to imperfect coupling between the laser diode mode and the external cavity mode. We define the feedback level as the value of r, in dB (feedback level = 20 log r 3 ) . The effective reflection coefficient reff is the reflectivity seen looking into the external cavity and takes the form where rL = 2 L , , / c is the external cavity round-trip time. We consider only the case of an uncoated symmetric laser with facet reflectivity r2. To treat a laser with one coated facet with reflectivity r, (the facet facing the external cavity), we would replace r, with r, in (58) only, and keep the formula for rex unchanged. This procedure avoids any possible singularities when r, + 0 and lets rex contain all the effects due to the deviation from the symmetric uncoated cavity. 1 ) Numerical Calculation of Noise Spectra: As one would expect, we find that the effect of the external optical feedback on the laser fluctuation spectra is markedly different if the external cavity free-spectral range Aue, = 27r/rL is larger or smaller than the relaxation oscillation frequency oR.We therefore consider two cases, one with Aoex < uRand one with Awe, 362 zyxwvut zyxwvu zyx IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21, NO. 3. MARCH 1991 zyxwvuts zyxwvut > wR. For each cavity we consider various feedback levels and operating frequencies. First, consider a relatively short cavity with Awe, = 3wR with feedback level varying from -30 to -60 dB. The fluctuation spectra for an operating frequency such that 0 ~ + 7 tan-’ ~ a! = 2n7r is shown in Fig. 4(a)-(c). Since our approximation for the field spectrum is divergent at the origin and not valid for frequencies less than the linewidth, we have artificially set the value at the origin to 0 dB. For weak feedback we have only a slight narrowing of the line, while the strength of the relaxation oscillation sidebands keeps growing until the feedback reaches about -55 dB. Increasing the feedback beyond -55 dB starts to dampen the relaxation oscillations and to pull the oscillation frequency towards higher frequencies. At the same time the linewidth continues to decrease. However, for such a short cavity, even at -30 dB feedback, the linewidth reduction is only modest, approximately one order of magnitude. It is clear from Fig. 4 that this feedback system has a very broad effective bandwidth, suppressing the noise to frequencies much larger than the relaxation oscillation frequency. Fig. 5(a)-(c) shows the variation of the spectra with deviation of the operating frequency away from the weak feedback optimum considered in Fig. 4. For the feedback level considered in Fig. 5, -50 dB, the relaxation oscillation resonances spike at a normalized offset frequency of about f 0 . 2 . For weaker feedback, the strongest resonances are observed for zero offset frequency. For stronger feedback, the resonances become strongly damped for offset frequencies around zero, and strong resonances are observed only for normalized offset frequencies close to $0.5. Hence, for stronger feedback we observe both a strong narrowing of the line and strong damping of the relaxation oscillations as we approach zero offset frequency. In fact, the minimum linewidth and maximum damping of the relaxation oscillations for stronger feedback, are for frequencies slightly more negative than the optimum weak feedback frequency. To avoid any external cavity resonances and achieve strong damping of the relaxation oscillations one would use a short cavity. However, too short a cavity will not provide much linewidth reduction unless the feedback is very high. This can be difficult to achieve in practice due to the small active region of a laser diode. Next, consider a relatively long cavity with Awe, = 0 . 2 5 0 ~ ~ with feedback level varying from -30 to -60 dB. The fluctuation spectra at the optimum weak feedback frequency, tan-’ a! = 2 n a , is shown in Fig. 6(a)-(c). External cavity modes starts to be noticeable already at low feedback levels. A splitting and peaking of the relaxation oscillations happens at approximately -50 dB. For stronger feedback, the external cavity modes become more pronounced, and finally dominating over the relaxation oscillations at -30 dB. Fig. 7(a)-(c) shows the variation of the spectra with deviation of the operating frequency away from the optimum weak feedback frequency considered in Fig. 6. The feedback level is -50 dB. The variations seen in Fig. 7 are very similar to the ones seen in Fig. 5 for the short cavity. For weaker feedback ( I -55 dB), the relaxation oscillations peak while the linewidth is narrowest as we approach zero “offset” frequency. For stronger feedback (-30 dB), the relaxation oscillations damp while the external cavity modes become very sharp at the zero “offset” frequency. The main drawback of the longer cavity is the narrower effective bandwidth of the feedback system resulting in the strong external cavity modes. By increasing the feedback level the external cavity modes will eventually grow to become comparable to the zyxwvuts + zyxwvutsrq (C) Fig. 4. Field and noise spectra for a laser diode with mirror feedback from a short cavity with free-spectral range Awex = 3wR, operating frequency such that w o ~ + L tan-’ CY = 2 n ~ and , feedback level varying from -30 to -60 dB. (a) Normalized frequency noise spectrum. (b) Normalized relative intensity noise spectrum. (c) Normalized field spectrum clipped at 0 dB. Normalization constants are the same as in Fig. 2.-In (a) and (b) the normalized frequency varies from to 10, and in (c) one frequency unit is equal to oR. main mode and the laser goes unstable [26]-[30]. However, the present theory is a perturbation approach, and hence, not valid in this operating regime. A cavity with Awe, = wR is a somewhat unique case, since zyxw zyxwvutsrqponmlkj zyxwvutsrqponmlkjihgfed HJELME er al.: SEMICONDUCTOR LASER STABILIZATION 363 zyxwvutsrqpo zyxwvu zyxwvu (C) Fig. 5 . Field and noise spectra as a function of operating frequency. Same cavity as in Fig. 4 with -50 dB feedback. The offset frequency is defined as the deviation from the operating frequency considered in Fig. 4, normalized to Awex. (C) the relaxation oscillation resonances and the external cavity resonances overlap. The resulting field spectrum is shown in Fig. 8. For intermediate feedback levels there is a competition between these resonances and the relaxation oscillation resonances split. For stronger feedback, the external cavity resonances at f w R dominate. Fig. 6. Field and noise spectra for a laser diode with mirror feedback from a long cavity with free-spectral range Awe, = 0.25wR, operating frequency such that O ~ T tan-’ ~ a = 2 n a , and feedback level varying from -30 to -60 dB. (a) Normalized frequency noise spectrum. (b) Normalized relative intensity noise spectrum. (c) Normalized field spectrum clipped at 0 dB. Normalization constants are the same as in Fig. 2. In (a) and (b) the normalized frequency varies from to 2, and in (c) one frequency unit is equal to wR. + . I . 364 I .. zyxwvutsrqpon zyxwvu zyxwvu zyxw IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21. NO. 3, MARCH 1991 zyxwvu zyx zyxwvut Fig. 8. Field spectrum for a laser diode with mirror feedback from a cavity with free-spectral range Amex = w R , operating frequency such that o07‘ + tan-’ a = 2n*, and feedback level varying from -30 to -60 dB. One frequency unit is equal to wR. Normalization constant is the same as in Fig. 2. level. From these observations, one could try to find an optimum choice for the external cavity length and feedback level, however, there are a number of trade offs. Depending on the application, either long or short cavities could be used. It seems that a good compromise is a relative short cavity with free-spectral range slightly larger than the relaxation oscillation frequency. For most lasers, this corresponds to a cavity of a few centimeters. 2) Some Approximate Results: Even for the simple case of mirror feedback, the general formulas are quite complex. To easily compare this feedback configuration to the “self-locked” laser diode considered in the next section, it is of great interest to find simple approximate formulas for the case of weak feedback. Furthermore, the simplicity of the weak feedback results allow simple expressions to be derived for the sensitivity figures discussed in Section IV-A. In the weak feedback limit, the phase and gain conditions are well known. To show the similarities between optical feedback and injection locking one can combine the two steady-state equations to one equation describing an ellipse in the A u - AG plane [27], [Au - a!AGl2 + [+AGl2 = [ T ~ ~ / T ~ FThe ~]*. linewidth reduction, and hence the stability is given by au,/auo cos ( 0 ~ 7 ~tan-’ a).From this = 1 we see that the stable solutions correspond to the lower part of the ellipse. Because there will always be a solution with AG I 0, only the section with AG I0 is of interest for operating the laser. This leads to an asymmetric “locking” range, the range over which the laser “locks” to one external cavity resonance, as in the injection locking case [46]. The total locking range is given by (C) Fig. 7. Field and noise spectra as a function of operating frequency. Same cavity as in Fig. 6 with -50 dB feedback. The offset frequency is defined as the deviation from the operating frequency considered in Fig. 6, normalized to Awex. To summarize, we see that the short cavities have the advantage of a large effective bandwidth. For weak feedback, this bandwidth is less than the relaxation oscillation frequency and the resonances are not strongly affected. For strong feedback, the effective bandwidth grows and becomes larger than the relaxation frequency and strong damping of the resonances result. The cost of using a long cavity is a narrower effective bandwidth resulting in many strong external cavity resonances. The only advantage of the longer cavity is the resulting narrow linewidth compared to the shorter cavity with comparable feedback zyxw zyxwv + + Aulock= Au, + Aw- a (1 + m) ?rr, (59) 7d Fd where A u + and A u - are the locking ranges on the positive and negative side of Au = 0 axis in the Au - AG-plane. For long external cavities, there are many possible steady-state solutions along the ellipse. However, according to the discussion in Section IV-B, the laser will operate close to A u = 0 with maximum linewidth reduction. Maximum stability and linewidth reduc- zyxwvutsrqponmlkjihg zyxwvutsrqponmlkji zyxwvutsrqp zyxw zyxwvuts HJELME et al. : SEMICONDUCTOR LASER STABILIZATION tion are achieved when worL + tan-’ = 2n*, n d , corresponding to zero frequency shift, Aw = 0, and an excess gain of A G = d ( ~ r ~ / r ~ F ~ ) ( la’)). / ( l The maximum Lorentzian linewidth reduction is given by + Variations in the laser operating frequency due to perturbations in the injection current or junction temperature are reduced by the amount a w o / a w , = ( m ( 1 r r 3 r L ) / ( F d r d ) ) - ’It. should be noted that, even though it appears that the frequency stability is better for longer cavities, the number of possible cavity modes also increases, making the laser susceptible to mode jumps. The sensitivity of the operating frequency to external cavity resonance variations or equivalently cavity length variations is given by where Awex = 27r/rL is the cavity free-spectral range. In other words, a length change of X / 2 will tune the laser frequency an amount equal to the free-spectral range. Cavity length perturbations could easily be due to air temperature and pressure variations, that could set the ultimate limit on the stability of a real system. At the “optimum” operating frequency, wrL tan-’ a = 2 nr, the frequency noise spectrum can be approximated as + 365 achieve substantial linewidth reduction are those that include some form of optical feedback or use very fast electronic feedback (requiring bandwidth larger than the free-running laser linewidth) [ 5 9 ] - [ 6 1 ] . The system of interest in this section consists of a semiconductor laser with optical coupling to a separate Fabry-Perot reference cavity [ 6 2 ] , [ 6 3 ] , [ 3 1 ] , [ 3 2 ] . The success of the present stabilization scheme relies on having optical feedback occur only at the resonances of the reference cavity. With the appropriate geometry, the laser self-locks to the cavity resonance. The reference cavity serves two purposes, it provides the necessary optical feedback that reduces the linewidth, and it provides the center frequency stabilization to the cavity resonance. The stability and tunability of the optical self-locking system make it a potentially very attractive source for many applications that earlier required dye lasers. Recently, self-locking laser systems have been used for laser cooling of atomic vapor [U], and high resolution spectroscopy [ 6 5 ] . The basic frequency stabilization system is illustrated in Fig. 9. The confocal Fabry-Perot cavity (CFP) is operated off-axis and has two reflected beams (labeled type I and 11). It is important to note that the two beams have very different characteristics. The beam of type I, the reflection mode, has a power minimum at the cavity resonance. In contrast, the beam of type 11, the transmission mode, has the desired characteristic of a power maximum on resonance. The configuration in Fig. 8 is one of many possible configurations resulting in resonant optical feedback. The optical locking of the laser to the reference cavity is observed directly with a photodiode monitoring the power transmitted through the cavity. The transmitted power versus laser frequency will be of the same type as the type I1 reflected power. As is evident from the linewidth formulas in Section IV-C-2, the slope of the reflection coefficient as a function of frequency curve is a very important parameter for determining linewidth. Clearly, the slopes of the frequency curves of the resonant feedback are going to be orders of magnitude steeper than they would be for the simple mirror feedback considered in the previous section. The effective reflection coefficient is straightforward to derive and is found to be zyxwvutsrq zyxwvutsrqp zyxwvutsrq zyxwvuts -- which obviously is valid only for w r L / 2 < T . Thus, we have a significant noise reduction for frequencies w < min [r3AwLD/2A , w e x ] ,where AWL^ is the laser diode free spectral range. Hence, for short extemal cavities the effective bandwidth of the feedback system iz directly proportional to the feedback level. A bandwidth larger than wR requires feedback larger than approximately - 35 dB. C . Laser Coupled to a High-Q Cavity In the previous section we discussed the external cavity operated semiconductor laser. In that feedback configuration, large linewidth reductions are possible. However, for long-term frequency stability and broad frequency tunability, the system leaves much to be desired. The closely spaced external cavity modes, the small-mode discrimination, and the lack of a sharp frequency reference combined with the laser’s sensitivity to both junction temperature and injection current drift/fluctuation makes it very difficult to control the operating frequency in a system with mirror feedback. Due to its high efficiency, reliability, compact size and low cost, the semiconductor laser remains an attractive component for frequency-stabilized sources. Since the first reported frequency stabilized semiconductor laser in 1970 [ 5 0 ] , several frequency-stabilization techniques have been developed. Among these techniques are locking to Fabry-Perot interference fringes and to atomic or molecular transition lines using electronic servo control with feedback to the injection current [ 5 0 ] - [ 5 3 ]or temperature [ 5 4 ] . Other systems use optical feedback [ 5 5 ] - [ 5 6 ] or a hybrid method using electronic as well as optical feedback [ 5 7 ] - [ 5 8 ] . Due to the spectral characteristics of the semiconductor laser frequency noise, the only systems which can where rL = 2 L e x / c ,Lex is the distance between the laser diode and the CFP cavity, and rCFP= r3 exp ( -i+3) is the reflection coefficient seen at the input port of the reference cavity. r3(w) and +3(w) are given by r,=& , 1 where wq is the resonant angular frequency of mode 4 of the resonator, AwFSR= 2 m / 4 L , is the free-spectral range, L, is the distance between the mirrors, and K is the power feedback ratio. The coefficient of finesse, E, and the finesse F, are related by F, = while the ratio of the free-spectral range to the full width at half-maximum resolution of the cavity AwFWHM is given by A w F S R / A w F w H M = F,. TW, 366 IEEE JOURNAL OF QUANTUM ELECTRONICS, zyxwvu zyxw VOL. 21, NO. 3, MARCH 1991 zyxwvutsrqp zyxwvutsrqp Fig. 9. Illustration of the resonant optical feedback system. I ) Numerical Calculations of Noise Spectra: As for mirror feedback, it is found that the effect of the coupling to the reference cavity on the noise spectra is quite different if the reference cavity free-spectral range AwFSR is larger or smaller than the relaxation oscillation frequency wR. We therefore consider two cases, one with ~ W F S R> wR and one with AwFSR < wR. For each cavity we consider various feedback levels and operating frequencies. First, consider a cavity with relatively large free-spectral range AwFsR = 5wR, finesse F, = 500, and feedback varying from -30 to -60 dB. Fig. 10(a)-(c) shows the resulting spectra as a function of feedback level, at an operating frequency such that 0 ~ 7 tan-' ~ a = 2 n ~ The . overall behavior is very similar to the case of feedback from a mirror close to the laser. However, the linewidth reduction for the same feedback level is much larger, and the relaxation oscillations are less damped. The narrower linewidth (same feedback level) is mainly due to the greatly increased effective time constant of the reference cavity as compared to the mirror feedback cavity. Similarly, the behavior of the noise spectra as a function of the operating frequency as shown in Fig. 1 l(a)-(c), is similar to the mirror feedback with Awex < wR. In Fig. 11, the feedback level is -50 dB. For stronger feedback, we note a stronger asymmetry, as compared to mirror feedback, around the optimum weak feedback operating frequency for the strong feedback case. Feedback from this short cavity seems to be an almost ideal feedback system. The system has a broad effective bandwidth, and simultaneously provides very good linewidth reduction. Next, consider a cavity with AOFSR = 0.25wR, finesse F, = 25. As demonstrated in Fig. 12(a)-(c), the behavior of the noise spectra as a function of feedback level, at an operating frequency such that + tan-' a = 2n7r, is almost the same as for the short cavity. Only relatively weak resonances are noticeable at the cavity resonances away from the line center. The same can be said about the spectra as a function the offset frequency as shown in Fig. 13(a)-(c). This long cavity provides excellent linewidth reduction. However, the narrow effective bandwidth of the feedback system can result in strong external cavity resonances and only a modest reduction of the power in the tails of the field spectrum. In Fig. 13 we have chosen 7 J 7 , = 1. Changing this ratio can enhance the cavity resonances in the noise spectra. To summarize, we see that for this system, as for mirror feedback, a cavity with large free-spectral range has the advantage of large effective bandwidth. Furthermore, with a high finesse cavity, a short cavity can also provide large linewidth reduction, unlike mirror feedback from a short cavity. In addition to the reduction of the frequency noise, the main purpose the feedback system was designed to serve, Figs. 9 to 12 show a large reduction of the amplitude noise. Reduction of more than 10 dB can occur, in agreement with experimental observations [63] and the discussion in Section V. 2) Some Approximate Results: Approximate results for the steady-state condition in the weak feedback limit were already given in Section V. The possible solutions describe the curve [Aw - a ; A G l 2 + [ l A G ] ' = [ T ~ ~ ( W ) / T ~ inFthe ~ /Aw ', - AG zyxwvutsrqponmlkji + zyxw (C) Fig. 10. Field and noise spectra for a laser diode coupled to a high-Q cavity with free-spectral range AwFSR = 5wR, rL = r,, finesse F, = 500, operating frequency such that worL + tan-' CY = 2nir, and feedback level varying from -30 to -60 db. (a) Normalized frequency noise spectrum. (b) Normalized relative intensity noise spectrum. (b) Normalized field spectrum clipped at 0 dB. Normalization constants are the same as in Fig. 2. In (a) and (b) the normalized frequency varies from to 10, and in (c) one frequency unit is equal to wR. zyxwvutsrqponmlkj zyxwvutsrqponmlkjihgfed zyxwvutsrqponmlk zyxw HJELME er al. : SEMICONDUCTOR LASER STABILIZATION 367 (C) Fig. 11. Field and noise spectra as a function of operating frequency. Same cavity as in Fig 9 with -50 dB feedback The offset frequency IS defined as the deviation from the operating frequency considered in Fig 9, normalized to AwFWHM zyxwvuts zyxwvutsr zyxwvu zyx plane, the same as for mirror feedback. However, with r d w ) varying from a very small value far from the cavity resonance, to & at resonance, the curve Starts close to the origin (i.e., approximately at the solitary laser solution) for frequencies far from the cavity resonance, and sweeps out a curve close to an ellipse for frequencies close to the resonance. The resulting (C) Fig 12 Field and noise spectra for a laser diode coupled to a high-Q cavity with free-spectral range A W =~ O.25wR, ~ ~ T~ = 7 , . finesse F, = 25, operating frequency such that wo7L + tan-' (Y = 2 n a , and feedback level valylng from -30 to -60 dB. (a) Normalized frequency noise spectrum (b) Normalized relative intensity noise spectrum (b) Normalized field spectrum clipped at 0 dB Normalization constants are the same as in Fig 2. In (a) and (b) the normalized frequency vanes from to 2, and in (c) one frequency unit is equal to w R . 368 I ” I .. zyxwvutsrqpo zyxwv zyxwvuts zyxwv zyxwvu zyx zyxwvuts zyxw IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21, NO. 3, MARCH 1991 The linewidth reduction and stability are governed by the quantity aw,/awo. For wo E wq this may be written . cos (worL + tan-’ a) (66) where r, = 4L,/c. Consequently, maximum stability is achieved when worL tan-’ a = 2 n r , n d . Under this condition, and in the high-resolution limit T di Fcrc/Fdrd << 1 , we find for the Lorentzian linewidth reduction factor + (a) zyxwvutsrqpon Fluctuations in the operating frequency of the laser caused by perturbations in the laser temperature or injection current are dramatically reduced by the amount a w o / a w , 1/[JK(1 (Y2)(FcT,/Fdrd)],where the inequality applies in the high-resolution limit. In practice, for a Fabry-Perot resonator with a resolution of 5 MHz, and a power feedback ratio of - 3 0 dB, the influence of temperature and current fluctuations can be decreased by as high as a factor of 400. In addition, changes in the distance L between the external resonator cause the operating frequency of the laser to be pulled according to the expression + duo dL (b) I zyxw -2A W F W H M (68) in other words, a length change of h / 2 will ~ tune the laser frequency by an amount equal to the resolution of the stabilizing cavity. When worL tan-’ a = 2n7r and wo = wq, the frequency noise spectrum can be approximated as + I (69) Thus, we have a significant frequency noise reduction for frequencies w < min [ a & A w L D / 2 F d , A W F R S / ~From ]. (70) it follows that for frequencies larger than A W F W H M the / ~ , phase noise spectrum is approximately constant, S, ( U ) = Awo(Fdrd)’/[?r’K(l a’)]. Hence, we have flat sidebands in the field spectrum, and the suppression of the side-band power is only a function of the feedback level. The rms phase deviation is approximately [from (47)] + (C) Fig. 13. Field and noise spectra as a function of operating frequency. Same cavity as in Fig. 1 1 with -50 dB feedback. The offset frequency is defined as the deviation from the operating frequency considered in Fig. 1 1 , normalized to AwFWHM. locking range depends on the exact phase relations in the feedback system, however, the main contribution to the locking range can be written As for the mirror feedback, the locking range is determined by the feedback level and a-factor. Consequently, for the feedback system to capture most of the laser power into a sharp spectral &function, the rms phase deviation should be much less than one. For typical parameters, this requires K > -60 dB. Furthermore, from Figs. 10 to 13, we see that typically the relaxation oscillations peak at a feedback level between -50 and -60 dB. Hence, to achieve good system performance one should typically use more than -50 dB feedback. Comparing the “self-locked’’ diode to the external cavity operated laser, we see that for weak feedback, in many ways, they zyxwvutsrqponmlkjihg zyxwvutsrqponmlk zyxw zyxwvutsrqponm HJELME et al. : SEMICONDUCTOR LASER STABILIZATION 369 zyxwvu zyxwv zyxwvut behave the same as if we had replaced rLfor the mirror feedback with F,r, for the cavity feedback. It is interesting to note that the “effective” bandwidth of the feedback system is independent of the resolution of the cavity. That means that there is not necessarily a trade-off between the requirement of frequency stability (requiring very high resolution cavities) and excess noise in the tails of the field power spectrum. The only tradeoff is the requirement of a reference cavity with AwFsR larger than wR, to ensure a large effective feedback system bandwidth and low power in the tails. This is very different from the external cavity operated laser, where one cannot obtain a sharp reference frequency without lengthening the cavity and introducing strong cavity modes and a large fraction of the power in the tails. The feedback from a high-Q cavity can react to both fast and slow variations in the laser, since it has a long storage time Fcrc as well as a relative short round-trip time 7., icantly to the spectrum, so to keep the analysis simple, we consider only this first term. We include temperature fluctuations in all three spatial dimensions by including an average over the transverse dimensions of the laser mode. Hence, we can use the approximation VII. CONCLUSION In summary, we have provided a formalism for calculating any spectral function of a semiconductor laser with external optical feedback. We have derived analytical expressions for the relative intensity noise, the frequency noise, and the field spectrum including relaxation oscillations and the amplitude-phase cross-correlation spectrum. Low-frequency temperature fluctuations are shown to introduce a l/f-noise component in the laser noise spectra. This excess low-frequency noise results in a weaker linewidth reduction as compared to the white noise case. The linewidth reduction is shown to change from a square law dependence on the storage time in the external cavity at low feedback levels, to a linear dependence at stronger feedback. General weak feedback approximations show the low-frequency frequency noise to be strongly reduced according to the Lorentzian linewidth reduction factor, while the relative intensity noise reduction achieves a maximum value of (1 + 01 ’), that is, typically 10-15 dB. In addition to a large linewidth reduction factor, it is shown that a large “effective” bandwidth of the feedback system is important for the system to capture most of the laser emission into a sharp spectral line. The effective bandwidth is shown to be approximately equal to the square root of the power feedback ratio times the laser diode cavity free-spectral range. Typically, more than -40 dB feedback is required to achieve a bandwidth equal to the relaxation oscillation frequency. We have applied the formalism to both the solitary laser diode and two optical feedback configurations; the simple mirror feedback and the feedback from a high-Q cavity. The external cavity operated laser with short cavity is shown to provide modest linewidth reduction and stability, and excellent relaxation oscillation damping. For the “self-locked’’ laser with a cavity with large free-spectral range, we have the very fortunate case of both wide effective bandwidth, large linewidth reduction, and very good frequency stability. Such a system with more than -50 dB feedback is able to capture most of the laser emission in a sharp spectral delta function, with very little power distributed over the sidebands. where x is the susceptibility, and T ( x , y, z, t ) represents the temperature fluctuation in the laser diode. The power spectrum can now be expressed as FL(r) = -E TAT 11 1: dx dy dz 2 i 6 k ( x , y , z, r) (A.l) AT where AT is the area of the laser mode. Assuming that the fluctuations 6 k ( z , i ) are due to temperature fluctuations, we can write A7 zyxwvu APPENDIX In this appendix we will derive the power spectrum of the low-frequency noise source FL(t)defined in (8). We can show that, generally, only the first term in (8) will contribute signif- - 1: 1: dz dz’ ( T ( x , y , z , U ) T*(x’, y ’ , z’, U)>. (‘4.3) The temperature fluctuations should satisfy a diffusion equation with a Langevin source such that + e. a,T = D ~ V ’ T (A.4) With the proper boundary conditions stated, the diffusion equation can be solved using Fourier analysis techniques. Lang er al. [35] have solved the diffusion equation and found the resulting low-frequency laser fluctuations. By restricting the dominant source of the temperature fluctuations to be in the active region of the laser only, they used numerical integration and found l/f-like behavior over only a limited frequency range. Using arguments presented by Voss and Clarke [66], one can in fact argue that temperature fluctuation sources located in only a small transverse area, will not result in 1/f-noise over many decades. We find that for a case where the sources for the fluctuations exist in a transverse area much larger than the active region, the resulting power spectrum has 1 lf-like behavior over several decades. Under these assumptions, we can approximate the power spectrum of the temperature fluctuations in the active region as (A.5) Assuming that the temperature correlation function is approximately constant over the transverse dimensions of the laser . 370 .. zyxwvutsrqpo zyxwvu zyxwvutsrq . .. IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21, NO. 3, MARCH 1991 mode, and that 0 is a white noise source, we find I - [4] M. W.Flemming and A. Mooradian, “Spectral characteristics of external-cavity controlled semiconductor lasers,” IEEE J. Quantum Electron., vol. QE-17, pp. 44-59, 1981. [5] S. Saito and Y. Yamamoto, “Direct observation of Lorentzian lineshape of semiconductor laser and linewidth reduction with external grating feedback,” Electron. Lett., vol. 17, pp. 325-327, 1981. [6] L. Goldberg, H. 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Kurokawa, “Some basic characteristics of broadband negative resistance oscillator circuits,” Bell Syst. Tech. J . , vol. 48, pp. 1937-1955, 1969. [181 R . J . Lang and A. Yariv, “Analysis of the dynamical response of multielement semiconductor lasers,” IEEE J. Quantum Electron., vol. QE-21, pp. 1683-1688, 1985. [I91 -, “Semiclassical theory of noise in multielement semiconductor lasers,’’ IEEE J. Quantum Electron., vol. QE-22, pp. 436449, 1986. [20] -, “An exact formulation of coupled-mode theory for coupled cavity lasers,” IEEE J. Quantum Electron., vol. QE-24, pp. 6672, 1988. [21] D. R . Hjelme, “Temporal and spectral properties of semiconductor lasers,” Ph.D. dissertation, Univ. Colorado, Boulder, 1988. [22] D. R . Hjelme, A. R . Mickelson, R. G. Beausoleil, J. A. McGarvey, and R. L. Hagman, “Semiconductor laser stabilization by external optical feedback,” Laser Diode Technology and Applications, L. Figueroa, Ed., Proc. SPIE, vol. 1043, pp. 167174, 1989. [23] -, “Semiconductor laser stabilization by external optical feedback,” Tech. Dig. CLE0’89. Washington, DC: Opt. Soc. Amer., 1989, vol. 11, pp. 300-301. [24] B. Tromborg, H. Olesen, X. Pan, and S. Saito, “Transmission line description of optical feedback and injection locking for Fabry-Perot and DFB lasers,” IEEE J. Quantum. Electron., vol. QE-23, pp. 1857-1889, 1987. [25] C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Techno/., vol. LT-4, pp. 288-297, 1986. [26] D. Lenstra, B. H. Verbeek, and A. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron., vol. QE-21, pp. 674-679, 1985. [27] C. H. Henry and R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” ZEEE J. Quantum Electron., vol. QE-22, pp. 294-301, 1986. zyxwvutsrqp zyxwvutsrqp zyxwvutsrq Hence, &(U) shows a 1 /f-dependence for frequencies larger thanf, = w,/2?r. For typical numbers ( L = 300 pm, D , = 0.24 cm2/s [35]) we havef, 100 Hz. The 1/$-dependence when f goes to zero, assures that the total power in the fluctuations are finite (the integral of &(U) converge). To simplify the use of S,(w) we use the following analytical approximation: WI (A.8) where we have defined aT = Re aTx/Im a,x. Since all the parameters in (A.8) are not known, we rewrite (A.8) in a form where only known or measurable parameters appear 0, - where U, is the comer frequency where the flikker ( 1 /f)-frequency noise is equal to the white component of the frequency noise in the solitary laser diode. We take the value aT = -0.9 from Lang et al. [35]. zyxwvutsrqp ACKNOWLEDGMENT The authors gratefully acknowledge stimulating discussions with L. Hollberg. REFERENCES D. A. Kleinman and P. P. 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Opt. Soc. Amer. B , vol. 5 , pp. 1225-1227, 1988. 1651 A. Hemmerich, D. H. McIntyre, D. Schropp, Jr., D. Meschede, and T . Hansch, “Optically stabilized narrow linewidth semiconductor laser for high resolution spectroscopy,” Opt. Commun., vol. 75, pp. 118-122, 1990. 1661 R. F. Voss and J. Clarke, “Flikker ( I / f ) noise: Equilibrium temperature and resistance fluctuations,” Phys. Rev. B , vol. 13, pp. 556-573, 1976. zyx zyxwvutsrqp Dag Roar Hjelme (S’86-M’87) was born in Valldal, Norway, on March 25. 1959. He received the M.S. degree in electrical engineering from the Norwegian Institute of Technology, Trondheim, Norway, in 1982, and the Ph.D. degree in electrical engineering from the University of Colorado, Boulder, i n 1988. From 1983 to 1984, he was with the Norwegian Institute of Technology, Division of Physical Electronics, working on fiber optics and integrated optics. He is currently a Postdoctoral Research Associate with the Department of Electrical and Computer Engineering, University of Colorado. His current research interests include spectral and dynamic properties of semiconductor lasers, ultrafast optics, and electrooptic sampling. 312 zyxwvutsrqponmlkjih zyxw zyxwvutsrqpo Alan Rolf Mickelson (S’72-M’78) was born in Westport, CT, on May 2, 1950. He received the B.S.E.E. degree from the University of Texas, El Paso, in 1973, and the M.S. and Ph.D. degrees from the California Institute of Technology, Pasadena, in 1974 and 1978, respectively. Following a postdoctoral period at Caltech in 1980, he joined the Electronics Research Laboratory of the Norwegian Institute of Technology, Trondheim, Norway, at first as an NTNF Postdoctoral Fellow, and later as a staff scientist. His research in Norway primarily concerned characterization of optical fibers and fiber compatible components and devices. In 1984 he joined the Department of Electrical and Computer Engineering at the University of Colorado, Boulder, where he became an Associate Professor in 1986. His research presently involves semiconductor laser characterization, integrated optic device fabrication and characterization, and fiber system characterization. IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 27, NO. 3, MARCH 1991 Raymond G. Beausoleil (S’86-M’86) was born on May 25, 1958 in Waterbury, CI. He received the B.S. degree in physics from the California Institute of Technology, Pasadena, in 1980, and the M.S. and Ph.D. degrees in physics from Stanford University, Stanford, CA, in 1984 and 1987, respectively. His doctoral work involved the measurement of the hydrogen ground-state Lamb shift using high-resolution two-photon laser spectroscopy. From 1980 to 1982, he was with the Optical Sciences Group, NASA Jet Propulsion Laboratory, Pasadena,. CA, where he worked on laser spectroscopy and computer-aided optical design. From 1986 to 1989, he was a research scientist with the Boeing High Technology Center, Bellevue, WA, where he conducted research and development on semiconductor laser frequency control and laser radar systems. In 1989, he joined Solidite Corporation, Redmond, WA, where he is currently the Director of Technical Operations. He is the principle investigator of research and development projects on high efficiency nonlinear optical frequency conversion, diode-pumped solidstate lasers, and Ti:Sapphire lasers. Dr. Beausoleil is a member of the American Physical Society, the Optical Society of America, and the Society of Sigma Xi.