The Critical Amount of Optical Feedback for. Coherence Collapse in Semiconductor Lasers. Julius S . Cohen and Daan Lenstra. Abstract-We calculate the ...
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 27, NO. I , JANUARY 1991
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The Critical Amount of Optical Feedback for Coherence Collapse in Semiconductor Lasers Julius S . Cohen and Daan Lenstra
Abstract-We calculate the coherence times and critical feedback rates for a coherence collapsed single-mode semiconductor laser with external optical feedback. We show that, in contrast to what is suggested by certain numerical simulations, coherence collapse need not disappear when the linewidth enhancement parameter would vanish.
I
T has been known for some time that the coherence collapse instability in a single-mode semiconductor laser with delayed optical feedback is enhanced, if not caused, by the relatively large value of the linewidth enhancement parameter a [ 13. In the coherence collapsed state, the linewidth is dramatically broadened to -20 GHz and the coherence time drops to a few ns. Numerical simulations by Schunk and Petermann [2], [3] show that the critical feedback level, above which coherence collapse sets in, increases as CY decreases. This is confirmed by analytical results from the injection-locking model developed by Tromborg er al. [4]. All these results suggest that the critical 0, which would imply feedback level moves to infinity as a the coherence collapse to be absent for (semiconductor) lasers with a very small linewidth enhancement parameter. Although usually semiconductor lasers have CY = 3-5, a recently reported calculation [5] shows that strained layer quantum-well lasers may have much smaller a-values ( a 5 1). Here we present a theoretical study of the critical feedback for coherence collapse, using our self-sustained noise model for coherence collapse [6]. This model has proven to be quite successful in describing the main features in the visibility and the power spectrum of the output light, as well as their dependences on the amount of feedback [6], [7]. Our results confirm that for a + 0, the feedback rate y (for a definition see, e.g., [6]) should be increased for the laser to remain in the state of coherence collapse. This implies that for a given feedback rate, lasers with a small a-parameter show a lesser tendency towards coherence collapse than those with larger values of a. However, we find that the coherence collapse does not disappear when a -+ 0. We show that for CY --+ 0, the character of the power spectrum of the optical field gradually changes from Gaussian to Lorentzian. The coherence time increases as a -+ 0 at a fixed amount of feedback. The discrepancy with the conclusion in [2] may be explained by the fact that in the simulations the amount of feedback was not sufficient for a = 0 to reach coherence collapse. Moreover, we argue that coherence collapse is a dynamically stable state for all values for CY # 0. At a = 0, the coherence time in coherence collapse is tcoh= l / y , provided +
Manuscript received February 15, 1990; revised June 18, 1990. J . S. Cohen is with Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands. D. Lenstra is with the Department of Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands, and the Huygens Laboratory, University of Leiden, The Netherlands. IEEE Log Number 9040996.
that the coherence time is much shorter than the round-trip time 7 of the external cavity, thus tcoh > 1/7. From this we conclude that coherence collapse may exist up to CY = 0. However, we shall show that the stability property ceases to hold at a = 0. For a semiconductor laser operating well above threshold the fringe visibility is in a fair approximation determined by the phase fluctuations [6], viz.
v(t>= 1 ( E ( t ’ ) ~ ( + t t t ) * >
I
= e-(1/2)Aw(r)
(1)
where E ( t ) is the slowly varying envelope of the complex optical electric field and A4+(t) = ( [ @ ( t t ’ ) - @(f’)]’) is the variance of the phase excursions. Integration of the LangKobayashi equations of motion for the coupled variations of power, phase, and carrier density yields an analytic expression for A4+( t ) (see (3.1) in [6]). When the visibility decreases sufficiently fast, we may employ an expansion for small r. We get
+
where y is the feedback rate, w0 an angular frequency close to the frequency of the relaxation oscillation, and hRis its damping rate. The parameters x and y ar? given by x = H I ( 0 ) and y = HI( wo), where the function H I ( U ) is the feedback-induced noise spectrum, cf. [6]. The basic supposition in the theory in [6] is that the driving noise is produced, in fact, by the laser itself through the feedback field. This assumption holds provided the coherence time of the light tcoh is substantially shorter than the external roundtrip time 7. Therefore, in a steady state, the laser brings :bout a noise spectrum that must be identical to the spectrum H I ( U ) of the driving noise. It was shown in [6] that HI( U ) can thus be expressed as
I?, ( U )
=
2
dt cos wt
v(tf
(3)
so that the parameters x and y can be written with (1) as x =2
1-
dt
e-A+d(r)
0
y =2
iom dt cos
oot e-Amm(r).
(4)
With (2) we can solve (4) for x and y, leading to self-consistent steady-state solutions for the visibility and the optical spectrum. However, this analysis would be incomplete without some stability consideration, which tells us whether or not the steady-state solution can indeed be reached. To that end we shall build in the appropriate direction of time based upon causality considerations. Suppose that at some step, say n, the laser emits light characterised by the phase excursions AY:, with corresponding pa-
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COHEN AND LENSTRA: COHERENCE COLLAPSE IN SEMICONDUCTOR LASERS
rameters x , and y,. In turn, through the feedback these phase excursions serve as the driving noise force in the following step, thus leading to new spectral noise parameters x , + I and y,, + I and subsequently to a new phase variance A:;+ I ) . We carry out the integrations in ( 4 ) after substitution of ( 2 ) and we obtain
r
r
I' I
.!
\ :!
2.0-;;\
I
I
I
w0 = 20.0 109 rad s-l hR= 2.0ns-l
r
+
where w ( z ) = e-" erfc (iz) = e - ' ? ( 1 2 i / & 1; e ' l d t ) (cf. [8]). We have performed the iteration as explained above using the set of equations (5a) and (5b). We find that the sequence (x,,, y,) converges to a fixed point (x, y ) , as n + 03. This fixed point (x, y ) of the iterative scheme (5) corresponds to a steadystate time-autocorrelation function of the optical field that satisfies the rate equations with effective noise terms (see (2.5) in [6]). Such a steady state must also satisfy the condition tcoh tcoh,the above mentioned condition tcoh> 1 the real part of the argument of w in (5b) is negligible, hence both right-hand sides of (5a) and (5b) become equal, so that ( x , = y,). As a consequence, for a >> 1 we may use w( i z ) = 1 (cf. [SI). Then the stable fixed point of (5) leads to (7)
This solution corresponds to the extreme coherence collapse ) dominated by the t' beas introduced in [9], where A + + ( f is havior. Correspondingly, the optical spectrum has a Gaussianlike line shape. F o r a < < l w e c a n e m p l o y w ( i z ) = ( l / z & ) ( l - 1/2z2), for ( 1 z I + 03) and we find a stable fixed point with
In this regime, the linear term dominates A 6 4 ( t ) ;the corresponding optical spectrum consists of a Lorentzian-like line. At 01 = 0, from (8) we find tcoh l / y , but it is easily verified that the fixed point is not stable, but indifferent, that is, neither stable nor unstable. For arbitrary values of cy and y we have numerically solved the iteration scheme (5). We use the extreme coherence collapse (7) as the initial value and we find that for all cy > 0 the fixed point remains stable as a decreases. The corresponding coherence times are presented in Fig. 1, as a function of y for various values of cy. We find that the asymptote (8) is an excellent ap+
a
Fig. 2. The coherence time of the output light as a function of the linewidth enhancement parameter for various values of the feedback rate y.
proximation when a 5 0.5. Note that as cy -+ 0 the dependence of the coherence time on y approaches the hyperbola tcoh = 1 /y. For various values of y we present tcohas a function of cy in Fig. 2. As expected, and in agreement with numerical simulations by others [2], [3], [lo], we find that smaller values of a lead to longer coherence times. From Fig. 1 it is clear that tcoh may exceed the external round-trip time 7 as y decreases. Then the solution of our iterative scheme violates the assumption tcoh 5 7/2. Hence, we find a critical rate of feedback, yc below which coherence collapse cannot occur, and which is defined by tcoh(?lc) =
7.
(9)
In Fig. 3 we present our prediction for yc as a function of cy, for various values of 7. As a --t 0, yc increases, but remains finite. The stability analysis in [ 111 shows that for strong feedback, the state of coherent feedback is dynamically unstable for all values of a , when wo does not vanish. Thus, the coherence collapse may still occur at cy = 0, provided the external cavity is long enough, so that 7 2 2/y. In all cases, the coherence collapse is dynamically stable, except when cy = 0, in which case the solution is neither stable nor unstable. This means that if a laser with delayed feedback and a # 0 operates in the coherence collapsed state, it will stay there forever, and the presence of small fluctuations will not alter this situation. For a laser with cy = 0, the stability indifference means that the system will be very sensitive to all kinds of external and internal
IEEIE JOURNAL OF QUANTUM ELECTRONICS, VOL. 27, NO. I , JANUARY 1991
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a
Fig. 3. The critical feedback rate as determined by (9) as a function of the linewidth enhancement parameter for various values of the external roundtrip time 7 .
perturbations as well as to initial conditions. This sensitivity has also been observed in the numerical simulations of the complete Lang and Kobayashi equations performed in [lo]. So, we conjecture that any high gain homogeneously broadened gas laser, with a = 0 and sufficiently high feedback, i.e., y >> 1/ 7 , will show coherence collapse with a Lorentzian line shape 2 ~ y In . a traditional low-gain gas laser howof width Au ever, it is difficult to reach the coherence collapse regime as described by the self-sustained noise theory [6], [9] because of the high mirror reflectivities and the long laser cavity length. A final remark must be added here. In maintaining coherence collapse when a -+ 0, we have to increase the amount of feedback. However, in the Lang and Kobayashi equations, on which our theory is based, the multiple external reflections are neglected. This means that our conclusions pertain to situations in which the feedback rate can be increased without violating the condition rR
