Instabilities in a semiconductor laser with delayed optoelectronic feedback

Volume 74, number 1,2

OPTICS COMMUNICATIONS

1 December 1989

INSTABILITIES IN A SEMICONDUCTOR LASER WITH DELAYED OPTOELECTRONIC

FEEDBACK G. G I A C O M E L L I ", M. C A L Z A V A R A b a n d F.T. A R E C C H I a,c a Istituto Nazionale di Ottica, Firenze, Italy b CSELT, Torino, Italy ¢ Dipartamenta di Fisica, Universit,~ di Firenze, Italy

Received 12 May 1989

A single mode CW semiconductor laser with an optoelectronic feedback gives a modulated output, with a sharp frequency peak depending on the feedback delay. A theoretical model, with few parameters assigned from the experiment, yields quantitative agreement with observations.

In molecular lasers, i n t r o d u c t i o n o f a feedback resuited in the onset o f deterministic chaos [ 1-3 ]. In the case o f s e m i c o n d u c t o r lasers, external feedback has been p r o v i d e d either by optoelectronic [ 4 - 7 ] or by purely optical means [ 8 - 1 0 ]. Only in this latter case chaos was observed. On the contrary, an optoelectronic feedback has always led to a stable single frequency oscillation. Even though a correlation was established between oscillatory frequency a n d the delay time [ 6 ], no detailed theory has been given so far to relate the observed b e h a v i o r to a model. The role o f delay requires a p r e l i m i n a r y consideration. As well known [ 11] a delay differential equation o f first o r d e r requires an infinity o f initial conditions (all those included in a time interval o f length equal to the delay r ) . However, i f the d y n a m ics includes a filter with a limited bandwidth A v, then the relevant n u m b e r o f degrees o f f r e e d o m will be [12] N=2Avr.

p e r i m e n t with a single m o d e cw laser ( H i t a c h i H L P 1400). Here we report the observations and present the m o d e l equations, whose solutions are in quantitative agreement with the experiments. Precisely, we detect the laser intensity with an avalanche phot o d i o d e ( A P D ) (see fig. 1 ). The detector photocurrent is fed back to the laser junction, being s u m m e d up through a b r o a d b a n d "bias-tee" to the dc p u m p current. The bias-tee low frequency decoupling (200 k H z cut-off) assures that the dc b e h a v i o r is not per-

) BIAS CURRENT BIAS

C~RNER-CUBE

SANPLING OSCILLOSCOPE or

L

/

~

~-/[I

SPECTRUM

II T~V__{__

ANALYZER

~

I

(1)

Thus delay will play an effective role only when N > 1. Therefore in molecular lasers o r in the e x p e r i m e n t o f ref. [4] delay is not included since, e.g. in ref. [4] A v = 15 M H z against a delay in the feedback loop which in o r d i n a r y l a b o r a t o r y c o n d i t i o n d i d not exceed 20 ns. In o r d e r to explore the role o f a delayed optoelectronic feedback, we have p e r f o r m e d an accurate ex-

LLJ

l

[ X ~

APDL\

- -

I

i )<---

\ /

Fig. 1. Experimental set-up. LD: laser diode; APD: avalanche photodiode; the pump current is decoupled from the feedback loop through a bias-tee (here represented as an LC filter). The laser rear facet output is used for monitoring.

0030-4018/89/$03.50 © Elsevier Science Publishers B.V. ( North-Holland )

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feedback gain and delay such that the open loop measurement of the resonance frequency yields a gain of 0 dB and a phase of 0°. The resonance is always at the low frequency side of the natural relaxation peak. As we change the delay T, the resonance disappears, and it appears again for an extra delay AT such that the phase is again 0 °. This phenomenology is satisfactorily reproduced by the following equations, based on a class B laser [ 13 ] with a fast feedback with a low frequency cutoff,

turbed. In the absence of an amplifier in the feedback loop the high frequency cutoff is due to the stray capacitance of the laser diode and it is estimated to be above 2 GHz. The frequency response of the APD is much higher (above 10 GHz). By use ofeq. (1) we see that we have relevant delay effects down to r as small as half a nanosecond. The three control parameters of our experiment are: pump (controlled by the injection current), feedback gain (controlled by the gain of the APD) and delay (adjusted continuously from 2.5 to 6 ns by insertion of an optical delay (translatable corner cube) between laser and APD). Feedback gain and delay are measured at open loop by a network analyzer. For a typical setting of the control parameters the cw output is not qualitatively perturbed. However the spectral distribution (fig. 3) shows the onset of small peaks corresponding to strongly damped excitation by broadband noise (spontaneous emission). For some selected values, a new behavior appears, that is, the output displays a strong modulation with a narrow ( ~ 100 kHz) spectral distribution (figs. 2 and 3c). Once an oscillation has been found it is "robust" against parameter variations. For a fixed pump, these resonance conditions correspond to a

15. D O 0 0

ns

1 D e c e m b e r 1989

ldx --- +X=XY, k dt 1 dY + y_ 1+a(a--XY+BZd) dt 1 dZ --=Z=

1 dX ---

fl dt

(2)

fl dt

Here X is the laser intensity normalized to the saturation value, Y is the difference between carrier population n and its value no at transparency, normalized to the same difference nth--no at threshold; k - 1 and 7-1 are the X and Y lifetime, respectively. The coupling constants have been included in the rescaling parameters (saturation intensity and thresh-

17.00D0

ms

18.000D

ns

I

\

'// \\

F

//

I

L

I

I

r

I

I,

I

Ch. 4 lO. O O m V o l t a / d l v Timebose 200 ps/dlv Ch. 4 P o r a m a t m r s

Rime Time -

Fall Tims P-P V o l t s =

210.2

ps

241.2 56.250 m ~ I t m

t F r e 9.

+ Width

-

-

Prsshoot " RMS V o l t s -

1.37514

417.2

GHz

0.000 ~ps B8.581 mVmlts

OfFset Delay Period

- Width

r

- -73.75 mVolts = I6. O O O 0 n s 727.2 ps

-

Ovsrahoot Outycycls =

310.0

O. O O O 57.37

%ps

Fig. 2. T i m e d e p e n d e n t laser o u t p u t at resonance o b s e r v e d w i t h a digitizing s a m p l i n g oscilloscope ( H P 5 4 1 2 0 , t i m e b a s e 200 p s / d i v ) ; the m o d u l a t i o n d e p t h is ~ 0 . 6 .

98

Volume 74, number 1,2

FREQ:

OPTICS COMMUNICATIONS

O. OM - - 2DOO.DMHz

I I

REF:- IBdBm IDdB/

1 December 1989

FREO: 0 - -

2880 MHz

10dB/div

i

J

ri

F

i

t; - - - i÷ - - ~ (a) RBW: FREO:

(a') 3MHzO VBW: 3 k H z

5WP:******O ^TT:2OdBO

fl. OM - - 2008.BMHz

REF:- 15dBm IgdB/

(b)

FREQ:

FREO:

0

--

2888

MHz

IBdB/div

(b')

D. OM - - 2DDD.DMHz

REF:- IBdBm IDdB/

FREO: 0 - -

2088 MHz

10dB/div

I i

I i

i i

- ~ ~ - ~ ''

~=--, ....... ~

I I

[

(c)

(c')

I i

i i

RBW: 3MHzO VBW: 3 k H z

SWP:******~

^TT:2DdBO

Fig. 3. Experimental power spectrum of the laser output versus theoretical spectrum computed from eq. ( 5 ): (a), (a') without feedback, the broad peak corresponds to the natural relaxation oscillation; (b), (b') with feedback but away from resonance, the small peaks correspond to resonances excited by noise but rapidly decaying; (c), (c') with feedback at resonance, the large narrow peak (30 dB above (a) and (b)) corresponds to a resonance in the delay equations.

o l d p o p u l a t i o n ) . In o r d e r to write the s e m i c o n d u c t o r laser e q u a t i o n s as in eqs. ( 2 ) , that is, in a f o r m s i m ilar to t h a t u s e d in two level systems [ 13 ], it was necessary to i n t r o d u c e in the m o d e l e q u a t i o n s the parameter

nth

a = - - ~ 3 nth

(3)

-- n o

(in a t w o level laser it w o u l d be a = 1 ). T h e p u m p , n o r m a l i z e d to the t h r e s h o l d v a l u e is 99

Volume 74, n u m b e r 1,2

OPTICS COMMUNICATIONS

a+ 1. BZd is the feedback addition to the pump, where B is the low frequency loop gain,

Zd(t) = Z ( t - z )

(4)

is the delayed detector photocurrent, and Z is coupled to the intensity X (detector efficiency included in B) through the third equation in (2) which acts as a high pass filter with a low frequency cutoff. Eqs. (2) may look oversimplified as compared with more elaborated versions of semiconductor laser equations [ 14 ]. In fact we have not included a detuning. Detuning plays a main role in inducing a phase dynamics. Phase variations are crucial in the case of optical feedback, however they play no role for optoelectronic feedback which is intensity sensitive. Thus, inclusion o f a detuning term in eqs. (2) would not alter significantly the main results. For the same reasons, eqs. (2), which assume a single longitudinal mode, are still adequate to describe the behavior of the laser in the oscillatory regime in which, due to the dependence of the cavity refractive index on the carrier density, the emission spectrum hecomes multi-frequency. The fixed points of eqs. (2) coincide with the two fixed points (OFF and O N ) of the laser without feedback. A linear analysis around the ON point yields for fl--,0 the following equation d2X dt 2

+ 2F ~

+ •2X= ff22BXd,

in the real system there are stray capacitances due to the laser mounting which depress the high frequency part of the spectrum, and these capacitances were not taken into account in the model. In table l, the 7th column shows the calculated frequency as compared to the measured one, for different delays (see 4th column) which provided the same resonance. Notice that the resonance condition eq. (6) is equivalent to the zero phase condition observed in the experiment. Since eq. (6a) is periodic in toz the same resonanc e frequency can be obtained with an infinite number of z-values, separated by a constant interval Az=2~z/oz In column 5 the values of 1/Az are reported, in good agreement with f-values from column 6. Our model is at variance with the heuristic proposal of refs. [ 5-7 ] that the resonances would be an integer multiple of t/z. Indeed the last column of table 1 shows that this latter conjecture yields a bad agreement as compared with that (6th column) coming from our model. Such an optoelectronic system shows comparatively pure power spectra, and good intensity and tunability, thus, it may be conveniently used as a high 1.0

O.B

(5)

where X~(t)=x(t-z) is the delayed intensity. The resonances of eq. (5) correspond to 2F~o tan(coz)= o~2_~22,

Bsin(coz)<0

~'-~4B2= (~¢~2--0)2) 2-~ (2FCO) 2 ,

1 D e c e m b e r 1989

0.6

(6a) (6b)

era 0.4

k

where

~2=(kTaa),

2F=7(l+aa).

Eqs. (5), (6) depend only on four measurable parameters, that is the laser relaxation frequency £2 and its half width F, the loop gain B and the delay z. Introducing the values form the experiment of fig. 3c (see table l, column 1 to 4) we obtain the theoretical spectrum of fig. 3d', which displays a quantitative agreement with fig. 3c, with the exception of the high frequency side of the natural relaxation peak. In fact, lO0

0.2

0"8

' 0 5 ' t O ' I 5 ' 2 0 ' 2 5 ' 3 0

tau

(ns)

Fig. 4. (z, B ) p a r a m e t e r space. C u r v e s are the loci o f H o p f b i f u r cations o f eq. ( 5 ) . For a fixed gain B we h a v e a p e r i o d i c i t y in the delay z (see eq. ( 6 a ) ) . The o v e r l a p p i n g o f instability regions suggests the possibility of o b s e r v i n g coupling between different reso n a m modes.

Volume 74, number 1,2

OPTICS COMMUNICATIONS

1 December 1989

Table 1 Comparison between experimental data and theoretical models. The first four columns report the experimental data corresponding to self-oscillation, whose frequency f i s in the sixth column. The seventh column shows the frequency f ' given by our model, while the eighth column reports the integer multiplets of 1/z according to the heuristic model of refs. [5-7]. In the fifth column we give the reciprocal of the delay interval for which we obtain again self-oscillation at the same frequency. Notice that each 1/Az matches quite accurately its corresponding frequency.

o/2n

F/2n

(MHz)

(MHz)

1189 1189

95 95

1496 1496 1496 1781 1781 1781 1781 1781 t781

B

f

f'

kit

(MHz)

(MHz)

(MHz)

943

1053 1053

1088 1053

1132 1078

3.55 4.29 5.00

1351 1408

1376 1376 t376

1382 1378 1381

1408 1399 1400

2.92 3.50 4.08 4.70 5.28 5.87

1724 1724 1613 1724 1695

1690 1690 1690 1690 1690 1690

1692 1686 1695 1686 1691 1691

1714 1712 1716 1702 1704 1704

r (ns)

l/Az (MHz)

0.227 0.227

2.65 3.71

80 80 80

0.162 0.162 0.162

50 50 50 50 50 50

0.103 0.103 0.103 0.103 0.103 0.103

f r e q u e n c y source, e.g., as a clock in a fast signal reg e n e r a t o r in digital d a t a t r a n s m i s s i o n , a n d in all optical d a t a processing. It d o e s n o t r e q u i r e e l e c t r o n i c circuitry, a n d can be " i n p r i n c i p l e " c o m p l e t e l y built in an i n t e g r a t e d optics device. As a c o n c l u s i o n , o u r m o d e l c o n s i d e r a t i o n s describe a c c u r a t e l y the e x p e r i m e n t a l results, y i e l d i n g an a g r e e m e n t o f a few parts in 103 w h e r e a s p r e v i o u s m o d e l s give a rough a g r e e m e n t w i t h i n a few percent. T h e i n t r o d u c t o r y c o n s i d e r a t i o n s (see eq. ( 1 ) ) h a v e s h o w n that o u r d y n a m i c s i m p l i e s m o r e t h a n o n e degree o f f r e e d o m . T h u s the a v a i l a b l e p h a s e space has the right d i m e n s i o n to a c c o u n t for c h a o t i c p h e n o m ena. Since h o w e v e r no a m p l i f i c a t i o n was a d d e d on the f e e d b a c k loop, we d i d n o t e x p l o r e the n o n l i n e a r d o m a i n sufficiently far f r o m the o n s e t o f the instability to o v e r c o m e t h e t h r e s h o l d for chaos. T h i s is also s h o w n by the stability d i a g r a m in the p a r a m e ters space B - r (fig. 4). A n o p e r a t i n g c o n d i t i o n as that i n d i c a t e d by the a r r o w shows the c o e x i s t e n c e o f t w o i n d e p e n d e n t frequencies. I f a sufficient a m p l i f i c a t i o n p e r m i t s to increase t h e B v a l u e for t h e s a m e r, the system s h o u l d be f o r c e d far b e y o n d the instability threshold, a n d a c o m p e t i t i o n b e t w e e n t w o ( o r m o r e ) f r e q u e n c i e s s h o u l d occur. W e p l a n to e x p l o r e this n o n l i n e a r d y n a m i c s in a successive work.

We t h a n k P. C a m b i n i , G. G i u s f r e d i a n d P. Salieri for useful discussions.

References [ 1 ] F.T. Arecchi, W. Gadomski and R. Meucci, Phys. Rev. A 34 (1986) 1617. [2 ] F.T. Arecchi, R. Meucci and W. Gadomski, Phys. Rev. Lett. 58 (1987) 2205. [3] F.T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci and J.A. Roversi, J. Opt. Soc. Am. B 5 (1988) 1153. [4] S. Machida and Y..Yamamoto, Optics Comm. 57 (1986) 290. [5] T.L. Paoli and J.E. Ripper, J. Quantum Electron. QE6 (1970) 335. [ 6 ] K.Y. Lau and A. Yariv, Appl. Phys. Lett. 45 (1984) 124. [7] P. Paulus, R. Langenhorst and D. J~iger, Electron. Lett. 23 (1987) 471. [ 8 ] T. Mukai and K. Otsuka, Phys. Rev. Lett. 55 ( 1985 ) 1711. [9] Y. Cho and T. Umeda, Optics Comm. 59 (1986) 131. [ l0 ] C. Dente, P.S. Durkin, K.A. Wilson and C.E. Moeller, IEEE J. Quantum Electron. QE24 (1988) 2441. [ 11 ] N. Minorsky, Nonlinear oscillations (Van Nostrand Reinhold Ed., Princeton, 1962 ). [ 12] C.E. Shannon and W. Weaver, The mathematical theory of communication (University of Illinois Press, 1949). [ 13 ] F.T. Arecchi, in: Instabilities and chaos in quantum optics, eds. ET. Areeehi and R.G. Harrison (Springer Verlag, 1987) p. 9. [14]G.P. Agrawal and N.K. Dutta, Long wavelength semiconductor lasers (Van Nostrand, 1986).

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