Semiconductor laser stabilization by external optical feedback

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 27, NO. 3, MARCH 1991

352

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.

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

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.

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

0018-9197/91/0300-0352$01.00 0 1991 IEEE

HJELME et al.: SEMICONDUCTOR LASER STABILIZATION

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. 11. GENERALIZED RATE EQUATIONS

An accurate treatment of the semiconductor laser that accounts explicitly for the open laser cavity, and thereby the cou-

353 -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

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

= g(1 - t l ~ / ' )-

iq(l

-

+ F(z, t )

E,~E(*)

(2a)

(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 to inhomogeneities in the cavity, we have 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

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 21, NO. 3. MARCH 1991

354

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 )

=

s:

dzf&'-

z"F,(z',

t)

+ e-'"EF(O,

t)

(4)

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

;(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 wo + ia, in (6) teffis an operator obtained by replacing w [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

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

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

-

rex)E

E

-

I

+ -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

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 is a constant, (16) deby the external optical system. If lreXl2 scribes an ellipse [27]. In the general case, the curve will be slightly more complicated.

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

+

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

IEEE JOURNAL OF QUANTUM ELECTRONICS. VOL. 21, NO. 3, MARCH 1991

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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) , calculated from It should be noted that the power spectra S (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)

(20c) (204

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).

+

PROPERTIES IV. DYNAMICAL 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?)

equations for the high-frequency fluctuations

+ FNR]. (22b)

After some manipulations we can write

AGUA~ P + ir