STABILIZING OPTICAL FEEDBACK-INDUCED CHAOS BY SINUSOIDAL MODULATION BEYOND THE RELAXATION FREQUENCY IN SEMICONDUCTOR LASERS
A. H. Bakry, M. Ahmed*
Department of Physics, Faculty of Science, King Abdulaziz University 20803, Jeddah 215S9, Saudi Arabia
Received February 13, 2014
We report on stabilizing the chaotic dynamics of semiconductor lasers under optical feedback (OFB) by means of sinusoidal modulation at frequencies far beyond the relaxation frequency of the laser. The laser is assumed to be coupled to a short external cavity, which is characterized by a resonance frequency spacing higher than the relaxation frequency. The study is based on a time delay rate equation model of OFB, which is suitable for treating the regime of strong OFB and considering multiple reflections in the external cavity. We show that the intensity modulation response of the chaotic laser under strong OFB is enhanced over a narrow frequency band near the doubled relaxation frequency due to a photon-photon resonance. Within this high-frequency band, the sinusoidal modulation may convert the chaotic attractor to a limit cycle, and the small-signal modulation suppresses the relative intensity noise (RIN) to a level only 2 dB higher than the RIN level of the solitary laser.
DOI: 10.7868/S0044451014100010
1. INTRODUCTION
In most of its applications, a semiconductor laser is subjected to an amount of external optical feedback (OFB), such as the back reflection by the reflecting surface in the optical disc system or by the fiber facet in the optical fiber links. OFB may stabilize the laser operation, but on the other extreme, it may cause violent instabilities in the form of chaos, depending on the feedback parameters [1]. The chaotic dynamics under OFB is associated with a state of coherence collapse, which is manifested as significant broadening of the laser line shape and enhanced noise levels [2 5]. Experiments show that the noise level is increased by 20 dB or more as a result of OFB [6]. Intensive research activities have been focused on the control of the chaos dynamics and suppression of the associated noise of lasers [7 18]. Superposition of a high-frequency (HF) current is the most popularly used method to suppress the OFB noise. The OFB noise is well suppressed by-suitable selections of the frequency and amplitude of
E-mail: mostafa.hafez'fflscience.miniauniv.edu.eg. On leave from Department of Physics, Faculty of Science, Minia University, Egypt.
the superposed current. It has been established that the frequency window of the sinusoidal modulation for stabilizing the chaotic dynamics is around the relaxation frequency of the laser. Beyond the relation frequency, the intensity modulation (Of) response drops to lower orders and reaches the dB level at the modulation bandwidth. It was shown in [17] that in the vicinity of the relaxation frequency, chaos is converted into period-1 oscillations or period doubling, depending on the modulation depth. The noise level in the low-frequency regime was predicted to be about 8 dB/'Hz higher than the quantum noise level of the solitary laser [17]. This noise difference occurs mainly because the regime of the relaxation frequency is characterized by a strong coupling between the emitted photons and the injected carriers and by harmonic distortions of the laser signal [19], which contribute to the increase in relative intensity noise (RIN). It is also worth noting that previous studies on controlling chaos by sinusoidal modulation were concerned with intermediate OFB strengths. However, chaos is also induced by strong OFB [5], and hence extending the previous studies to the regime of strong OFB is necessary.
With the external OFB strength varied, the semiconductor laser exhibits a number of chaos regions. The length of the external cavity controls not only the
number and the OFB range of these chaotic regions but also the route to chaos [20,21]. In semiconductor lasers coupled to a short external cavity, the route to chaos is of the period-doubling type [20,21]. The oscillation frequency along these chaos regions increases with the increase in the OFB strength; it starts with the relaxation frequency and ends with the resonance frequency separation of the external cavity in the region of strong OFB [20]. Therefore, OFB has been used to increase the modulation bandwidth of semiconductor laser subjected to strong OFB by a short-distant reflector [5,22,23]. Recent investigations by the authors have shown that strong OFB suppresses the IM response below the ^3 dB level at the modulation frequencies lower than the carrier-photon resonance (relaxation) frequency [24]. This suppression of the IM response is then followed by enhancement of the modulation response over a narrow band of modulation frequencies much higher than the relaxation oscillation. The enhanced modulation response over the millimeter-wave band 54.5 56.5 GHz was reported for a laser diode with an external cavity 0.15 cm in length [24]. This modulation response enhancement was explained as the photon photon resonance between two spectrally neighbored longitudinal modes at frequencies exceeding the relaxation oscillation [24, 25]. It has been shown that within the frequency band of the enhanced IM response, the sinusoidal modulation releases period-1 oscillations with low signal distortion and suppressed noise levels [26]. These findings motivate the authors to examine the possibility of stabilizing the OFB-induced chaotic dynamics and suppressing the associated noise of the laser by modulation the laser far beyond the relaxation frequency.
In this paper, we investigate the stability of the chaotic dynamics under modulation frequencies within the frequency band of the enhanced IM response and examine the noise suppression effect. The study is based on an improved time-delay rate equation model of semiconductor lasers augmented by a sinusoidal current signal as well as intrinsic noise sources. The multiple reflections of laser radiation in the external cavity-are taken into account. The present model is applicable to laser cavities supporting single-mode oscillations. Examples include distributed feedback (DFB) lasers, vertical-cavity surface-emitting lasers (VCSELs) with selective oxidation for current and photon confinement [27], and Fabry Perot (FP) lasers with well-controlled transverse structures, in which mode competition induces the side-mode suppression ratio higher than 20 dB [28]. The model can also be generalized to treat OFB in semiconductor lasers with multimode os-
cillations [29]. The noise properties of the modulated signal are determined in terms of RIN and its average value, LF-RIN, in the low-frequency regime. The laser dynamics are analyzed and classified based on the waveform of the laser and its fast Fourier transform (FFT) as well as on the phase portrait of the photon number versus the injected carrier number. We apply the present study to a 1.55 //ni InGaAsP laser coupled to a short external cavity with a relaxation frequency of 4.5 GHz. We show that the modulation response of the chaotic laser due to strong OFB is enhanced over a narrow frequency band beyond 9 GHz, which is twice the relaxation frequency. Within this high-frequency band, the sinusoidal modulation turns out to attract the chaotic at tract or to a stable state with periodic-1 oscillations, and the small-signal modulation could suppress RIN to that of a solitary laser.
In the next section, the time-delay rate equation model of laser dynamics and noise under OFB is introduced. Section 3 introduces the procedures of numerical calculations. Section 4 presents the results of modulation dynamics and noise under OFB. The conclusions of the present work appear in the last section.
2. THEORETICAL MODEL
The proposed laser structure is composed of a laser diode oscillating in a single mode and coupled to an external cavity formed by placing an external reflector of power reflectivity Rcx at a distance Lcx from the front facet of the laser. The laser has a length Ld, a refractive index no and power reflectivities Rj and at the front and back facets. OFB is treated as the time delay of laser light of the electric field due to round trips in the external cavity. The round-trip time, or time delay, is t = 2ncxLcx/c, where /Zg^? IS the refractive index of the external cavity. The present time-delay model of OFB is illustrated by the scheme in Fig. 1. At a time t,
■/Th('xp(içb) ^/Rjcxp(içf)
/il . < Ni»{ i fpex )
Laser cavity
E(f)
E,
Gd pi k it, ■*-;->■•*-
External cavity
t = 2t7i>,.,- /., .,•/<•
E{t-r)-
D
Fig. 1. Scheme of the proposed model of semiconductor lasers under OFB
the boundary conditions of the laser at the back facet (z = 0) and the front facet (z = Ld) are then modified to [301
T7j(+)
laser
(o.t) =
exp {■
E
(+) laser
(LDJ) =
(1)
= y/Rf exp U(t - t)E\Zr{LDA). (2)
Here, and are the forward and backward
propagating components of the electric field inside the laser cavity, and U(t — r) is the time-delay function, which accounts for the OFB due to multiple round trips (reflections) in the external cavity and is given by [30]
U(t — t) = \U\oxp{—j<y?} =
= 1-£(A'„)
p= i
Rf
1 -Rf
p-i
' ' S(t) exp{i/9(f)} v
where S(t) and 9(t) are respectively the photon number and optical phase of the field, S(t — r) and 9(t — r) are the corresponding time-delay values, and p is an index for the round trips. The exponent lot, where lo is the angular frequency of the laser emission, represents the OFB phase due to delay of the optical field in one round trip. The OFB strength is measured by the coefficient r——
/w (1^/W'r^. (4)
with j/ being the coupling efficiency of light injected into the laser cavity. In equations (1) and (2), <f>f and <f>b are the optical phases at the front and back facets. Therefore, the exponentional gain per pass is modified to
1 = s/RjRb |U(t - t)I oxp{(<7 - k)Ld) x
x oxp{—¿(2/?L.d + iff +
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