EDM0'96; 25-26 November 1996; Weetwood Hall, Leeds, UK 96

Locking Characteristics of Semiconductor Lasers with Optical Injection

K.H. Seng, M.K. Haldar and F.V.C. Mendis Department of Electrical Engineering,

National University of Singapore, 10, Kent Ridge Crescent,

Singapore 119260.

Abstract: We have studied the locking characteristics of semiconductor lasers through numerical calculation of the output intensity and change in carrier density of the slave laser during injection locking. We have also obtained the dynamic locking range by examining the roots of the secular determinant of the perturbed system. The lower boundaries of the static and dynamic locking ranges coincide, but the upper boundaries do not. Both the static and dynamic locking ranges are asymmetrical about zero detuning and dependent on injection ratio, linewidth enhancement factor and biasing condition. The upper boundary of the dynamically stable region exhibits an abrupt bend at a very low injection level. Unlike previous work, the locking characteristics at both low and high injection levels have been carefully studied.

I INTRODUCTION

Injection locking is a way of synchronising a free-running slave laser (SL) to a stabilised master laser (ML) when the frequency difference (detuning) between the two is within a well defined range called the locking range. The properties of injection-locked semiconductor lasers are si@icantly influenced by the dependence of the refractive index of the gain medium on the carrier density [l]. This dependence is represented by the linewidth enhancement factor, a-factor. It has been reported that injection locking can achieve single-mode operation, reduce frequency chirping and enhance modulation bandwidth 121. Recently, it has also been linked with frequency conversion using four-wave mixing effects in semiconductor material 131.

Most publications on injection locking have considered low injection levels. For such levels, the optical output of the locked laser differs very little from the free running value and the locking characteristics are described in terms of the detuning and the ratio of the injected power to the locked output power of the SL. Although a few studies have been carried out for high injection levels, the same variables are used for describing the locking characteristics. At high injection levels, the locked power may differ considerably from the free running power and it may be more convenient to describe the locking characteristics in terms of the ratio of injected power (or injected electric field magnitude) to the free running power (or free running electric field magnitude) as these are independent variables. In this paper, which presents a detailed study of static and dynamic characteristics for both low level and high level injection locking, the magnitude of the ratio of the injected electric field to the electric field of the free running SL is taken as the injection ratio.

We begin by deriving the theoretical model of injection locking from the complex, oscillating electric field inside the SL cavity with optical injection and the usual rate equation describing the carrier density. The stationary solution would be studied from a pair of cubic equations, one of which represents the output intensity while the other denotes the change in carrier density due to injection locking. The dynamic locking characteristics are studied by introducing small fluctuations and performing a stability analysis on the model. The simulated results for both the static and dynamic conditions are presented next. We also study the effects on the locking conditions due to variations in a-factor and biasing current.

11 THEORY

The complex electric field of the SL with external optical injection can be described by the following differential equation [4]:

0-7803-3130-3/96/$5.00 0 1996 IEEE

97

In equation (l), E,(t) and El are the field amplitudes of SL and ML respectively, both suitably normalised such that the square of the absolute value gives the photon density; o, and o1 are the angular oscillation frequencies of the free-running SL and ML respectively; $,(t) is the real-valued phase variation of the SL complex field; cc is the linewidth enhancement factor; GN is the gain coefficient; N is the carrier density, No the transparency carrier density and AN the amount the canier density deviates from threshold condition; 2, is the photon lifetime and f d is the longitudinal mode spacing.

Equation (1) can be converted into amplitude-phase representation to obtain two first-order differential equations of Eo(t) and $,(t):

(2 ) d 1 -E,(t) = -G,AN(t)E,(t) +f,,E, cos(A(t)) dt 2

sin ( A ( t )) d 1

-4,(t) = -CZG,AN(t)+ fd - dt 2 E, (t)

(3)

where A(t) = Aot - +,(t) and do = col - oo , A o of which defines the angular frequency detuning between the free running ML and SL.

The equation for carrier density is given by:

N(t) G,(N-N,)E:(t) d -N(t) = J dt 7,

(4)

where J represents the pumping term and 2, is the spontaneous emission lifetime. Equations (2), (3) and (4) form the basic theoretical model in our entire analysis of injection locking.

For studying the injection-locked steady state, the first-order derivatives are set to zero and the condition that both SL and ML must oscillate at similar frequency is imposed:

where h denotes the locked phase of SL relative to that of ML. The following pair of cubic equations in terms of E: and could be obtained, where the former represents the normalised SL output intensity and the latter denotes the change in carrier density due to injection:

4

where 8, is electric field of SL output during steady state injection locking, Ea is the stationary SL field without injection and & is the steady state change in carrier density due to injection.

98

The pair of cubic equations (6) and (7) are used in our analysis of static locking characteristics. They could be solved for various laser parameters, injection ratios and detuning frequencies. As they are both of cubic order, we can expect each equation to have either 3 real roots or 1 real root with a complex conjugate pair. A complex solution to either of the equation will have no significance in our analysis. Hence, we would be interested in the region where both E: and &are real.

In the study of dynamic locking range, we introduce small fluctuations of 6E, 64 and 6N into equations (2) - (4). Fluctuations of second-order and above are ignored for small signal approximation and by applying Laplace transform to the system of equations, we obtain the following secular determinant [4]:

(8) D(s) = s3 + As2 + Bs + C

where

The dynamic locking range is studied by observing the roots of D(s). The condition for dynamic stability is that all the roots have negative real parts.

lII RESULTS

The laser parameters used for our analysis are given in Table 1. At a particular injection level, each of equations (6) and (7) would have either 3 real roots or 1 real root with a pair of complex conjugates, depending on the detuning frequency. The complex roots are ignored while the real roots are plotted in Fig. 1 for an injection ratio of 0.5. Within a certain detuning range, both E: and AN have 3 real roots, marked (i), (ii) and (iii)? while beyond that, only 1 real root exists, of which is marked (i).

E:(i) and AN(i) are real for all detuning frequencies. However, AN(i) is always positive at around 0.3N* and this should not constitute a solution to an injection-locked state [l]. In the injection-locked state, the MI., should “ help” the SL such that the carrier density of the latter would go below that of threshold condition, resulting in a negative AN. In addition, Ef(i), normalised by E:, is very close to zero for all frequencies of detuning. This is in contrary to the physics of optical injection where it is reasonable to assume that the SL output with optical injection should be at least equal to, if not higher than, that without injection (E;). Hence, solution set (i) cannot represent the injection-locked state.

Solution set (ii) and (iii) are seen to satisfy the abovementioned conditions for injection locking within certain detuning range. Moreover, for both solution set (ii) and (iii), E;/E’, is greater than unity whenever AN is negative. The confusion now lies in the existence of two sets of roots that satisfy the injection-locked conditions. However, we could assume that the SL operates primarily in the mode resulting in the lowest threshold gain [SI, i.e., the most negative AN. From Fig. 1, the static locking range is thus defined as the detuning range where E:(iii) is greater than E: and AN(iii) is negative, which is between -25 GHz and 10 GHz. In general, the static locking range can be defined as where there is an existence of at least one negative real AN.

99

The static locking ranges for a = 3 and 6.6 are depicted in Fig. 2 for injection ratios up to 1.0. The static locking range increases with injection level, but is evidently asymmetrical about zero-detuning. In fact, by considering the negative AN condition during injection locking, it can be easily proven that the static locking range is given by:

We have seen earlier that the upper boundary for static locking range is defined by where E, = E, , hence, the upper limit in equation (9) is simply a linear function of injection ratio. This is also apparent from Figs. 2 and 3 where the upper boundaries of static locking ranges are linear plots. For the lower boundary, linearity is only seen at low injection levels. This is because E, E E, only at low injection levels. At high injection levels, E, differs from E, quite considerably and the lower boundaries of static locking ranges are non-linear. Equation (9) also shows that the asymmetrical characteristic is clearly due to the linewidth enhancement factor pertinent to semiconductor lasers.

Figs. 4 - 7 show the dynamic locking ranges for various injection parameters. The lower boundaries of the static and dynamic locking ranges always coincide whereas the upper boundaries do not. The stable region is asymmetrical about zero-detuning and this characteristic, which is also attributed to the a-factor, has been observed in all experiments reported. Figs. 4 and 5 depict respectively the dynamically stable region at very small injection levels for various a-factor and biasing current. A sharp bend is observed on all the upper boundaries of the stable locking ranges. Other authors have also noticed this bending effect [4]. However, they have observed the entire static locking range to be stable at injection levels below this bend. This observation is different from ours as we have obtained no coincidence between the upper boundaries of static and dynamic locking ranges for all injection ratios. As shown in Fig. 4, the bending occurs at a lower injection level with a larger a-factor and thus the stable locking range is smaller for higher a-factor. In Fig. 5, with a higher biasing current of the SL, the bending occurs at a larger injection ratio and hence the stable region increases.

Figs. 6 and 7 illustrate the stable locking ranges for high injection levels. With all other variables unchanged, the dynamically stable regions always increase with higher injection ratio. In Fig. 6, the dynamic locking range is seen to increase with higher a-factor. In Fig. 7, the dynamic locking range reduces as biasing current is increased. Comparing Figs. 6 and 7 with Figs. 4 and 5 , an interesting phenomenon can be seen, i.e., although the dynamic locking range is increased with higher a-factor or smaller biasing current at high injection levels, it is reduced by the same effects at low injection levels.

IV CONCLUSIONS

We have studied the characteristics of the injection-locked semiconductor by analysing the normalised output intensity, E: , and change in d e r density, AN, of the SL. In the injection-locked steady state, E: is expected to be at least equal to the output intensity without optical injection, EL, and AN should have a negative real value. The upper boundary of the static locking range is always linear but the lower boundary is only linear at low injection levels. The static locking range increases with higher injection ratio, larger a- factor and higher biasing current of the SL. It is also asymmetrical about zerodetuning and this is due to the a-factor. The lower boundaries of the static and dynamic locking ranges always coincide but coincidence is not observed for the upper boundaries at all injection levels. The dynamically stable region also increases with injection level and is asymmetrical about zero-detuning, the latter of which is again attributed to the a-factor. The upper boundary of the stable locking range exhibits a sharp bend at a very small injection ratio. Interestingly, we could increase the dynamic locking range at high injection levels by increasing the a-factor or decreasing the biasing current although the dynamic locking range decreases with higher a-factor or lower biasing current at low injection levels.

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REFERENCES

[l]

[2]

[3]

[4]

Roy Lang, "Injection locking properties of a semiconductor laser", IEEE Journal of Quantum Electronics, 1982, vol. QE-18, no. 6, pp 976-983. J. Wang, M.K. Haldar, L. Li and F.V.C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking", IEEE Photonics Technology Letters, 1996, vol. 8, no. 1, pp 34-36. G.P. Bava, P. Debemardi and G. Osella, "THz frequency conversion in injection locked semiconductor laser oscillators", IEE Proceedings - Optoelectronics, 1996, vol. 143, no. 1, pp 41-48. F. Mogensen, H. Olesen and G. Jacobsen, "Locking conditions and stability properties for a semiconductor laser with external light injection", IEEE Journal of Quantum Electronics, 1985, vol. QE-21, no. 7, pp 784-793. C.H. Henry, N.A. Olsson and N.K. Dutta, "Locking range and stability of injection locked 1.54pm InGaAsP semiconductor lasers", IEEE Journal of Quantum Electronics, 1985, vol. QE-21, no. 8,

[5 ]

pp 1152-1 156.

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..... .............................................................. - ............. : ....................... :

_ ........... > .................................................................. 0 ,

02-.. . . . . . :... ........... : .............. L... . . . . . . : .................... :

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4 0 - 4 0 - 2 0 0 20 40 en Oscwkq. GHz

(a)

-t" % " " ' ' o 0.1 02 03 a4 0.6 OB 0.7 0.8 os I W&nW. El-

Fig. 2: Static locking range for various a-factor, i/Ith = 1.3. The upper boundaries are similar for all x-factor but the lower boundaries are different.

"F

-80' ' ' " ' ' * " ' 0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

higtm ratio. ElEcm

Fig. 1: Locking characteristics for 0: = 3 , VI, = 1.3 and EIE, = 0.5. (a) normalised SL output power (b) normalised change in carrier density.

Fig. 3: Static locking range for various biasing current, a = 3. The lower boundary deviates further from zerodetuning as bias is increased.

2-02- 0. F-0.3.

0 -0.4 -

-0.5

-0.6

-0.7

-

-

-

Longitudinal mode spacing

Gain coefficient

Carrier density at transparency

a-parameter (CSP-laser)

a-parameter (BH-laser)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x lo* Inpaion Rto. E1&8

f d 125 GHZ

GN No 1.1 x m-3

a 3 .O

a 6.6

1.1 x IO-’* m3 s”

4‘ I 0 0.W 0.01 0.016 0.m am5

hksc&l Rtlo. Emor

Fig. 4: Bending effect of upper boundary of dynamic locking range for various a - factor.

Fig. 5 : Bending effect of upper boundary of dynamic locking range for various biasing current, a = 3.

I

-. .. a-6.6 -.

0 0.1 0.2 03 0.4 05 0.6 0.7 0.8 09 1 h m rslio. E 1 h

(.I.IIIIIII 6oo 0.1 02 oa 0.4 os as 0.7 ae 0.9 1

ilwiul nb. EllEW

Fig. 6: Dynamic locking range for various a-factor. Fig. 7: Dynarmc locking range for various biasing current, a = 3

Table 1 : Laser parameters and their numerical values

Spontaneous emission lifetime

Photon lifetime I ZP I2PS

Normalised bias current