Due to their high efficiency, high bandwidth frequency modulation, and compact size, mode-locked diode lasers have attracted greatly attentions as practical and alternative sources for the picosecond and sub-picosecond pulse generation [1]. High repetition rate pulses over GHz are now routinely generated in different types of external-cavity diode lasers (ECDLs) passively mode-locked in various configurations, for example an ECDL with an ultrafast integrated uni-traveling carrier saturable absorber [2], an ECDL consisting of a nonlinear optical feedback mirror [3], and an ECDL composed of a saturable distributed Bragg reflector mirror and a vertical-cavity surface-emitting diode laser [4]. A self-starting passive mode-locking [5] in a conventional Littrow-type ECDL with variable repetition rate from 300 MHz to 1.5 GHz was reported recently by the authors in a GaAs ridge wave-guide ECDL at 795 nm in a gain saturation regime [6]. A theoretical model based on a set of differential equations with time delay was developed to explain the observed passive mode-locking phenomena in semiconductor lasers with internal saturable absorber, predicting bifurcations responsible for the appearance and breakup of a standard mode-locking regime [7]. Recently, a novel interpretation for the origin of self-starting passive mode-locking as a first-order phase transition and its threshold behavior was introduced based on a statistical light-mode dynamics (SLD) theory, in which the effect of spontaneous emission noise was included as a main source term in a Landau-Ginzburg-like theory [8].

In the previous report [6], we have observed a subharmonic (f_rep/2) mode-locking phenomena appeared in advance to the conventional (standard) mode-locking with a repetition rate at integer multiples of f_rep = FSR, i.e., nf_rep, n being an integer and FSR is the free-spectral range of the external cavity, when we increased the injection current over the gain saturation regime. In a standing-wave mode-locked ECDL [9], the repetition rate f_rep of the mode-locked laser is given by f_rep = c/2L, where L is the external cavity length and c is the speed of light. In this paper we report on the results of a comprehensive experimental study for the various nonlinear mode-locking phenomena observed in a self-starting passively mode-locked ECDL, including a standard period-doubled mode-locking, a period-doubling route to chaos, and a total mode-locking between two lowest-order lateral modes of a GaAs ridge wave-guide diode laser.

We first measured the intrinsic feature of frequency locked oscillation due to a coherent coupling or a self-stabilization of lasing frequencies of two lowest-order lateral modes, the TE₀ and TE₁ modes, which is a manifestation of ridge wave-guide diode lasers [10]. We measured the frequency-locked oscillation in the cw regime of our ECDL with a grating mounted in the Littrow configuration and at the diode temperature T = 24.0 °C, in which a GaAs ridge waveguide diode laser (Eagle Yard, EYP-RWL-0790-00100) was used as a gain medium [6]. Figure 1(a) shows the schematic diagram of the experimental set up to measure the polarization states, the spatial mode profiles, and the lasing frequencies of two lateral modes of our ECDL, and also to measure various self-starting passive mode-locking phenomena. We observed that our diode laser emitted always two lowest-order lateral modes and they had different spatial mode profiles and polarization states. The intensity profile of the TE₀ mode has an oval shape with a Gaussian mode profile along the semi-major and semi-minor axes, while the TE₁ mode has a bi-modal structure along its semi-minor axis as can be seen in Fig. 1(b) [10]. The TE₀ mode has a horizontal polarization along the semi-major axes and it is parallel to the polarization axis of the PBS, while the TE₁ mode has a vertical polarization along the semi-minor axis and it is perpendicular to the TE₀ mode. As pointed out in Ref. [10], a ridge wave-guide diode laser may exhibit a self-stabilization of lasing frequencies between the TE₀ and TE₁ modes, when the residual thickness outside the ridge region is below a critical thickness, for example below 0.61 μm in [10]. In our GaAs ridge wave-guide diode laser, we have observed always frequency-locked oscillation due to a coherent coupling between the TE₀ and TE₁ modes, i.e., two modes have the same oscillation frequencies in the whole injection current range above threshold current. Figure 1(c) shows a beat note spectrum between TE₀ and TE₁ modes, indicating the TE₀ and TE₁ modes indeed have the same oscillation frequencies due to the coherent coupling. For this measurement, we have shifted the frequency of the TE₀ mode by 82.364 MHz using an acousto-optic modulator (AOM) and two orthogonal polarization states were mixed after a beam splitter (BS) through a linear polarizer (LP). The beat note spectrum between the TE₀ and TE₁ modes at 82.364 MHz in Fig. 1(c) was measured by a fast photo-detector (PD4).

 

Fig. 1. (a) Experimental set up. (b) Intensity profiles of two lowest-order lateral modes, TE₀ and TE₁, of a GaAs ridge wave-guide diode laser at 795 nm. (c) Beat note spectrum at 82.364 MHz between cw lasing frequencies of TE₀ and TE₁ modes. MLDL; mode-locked external-cavity diode laser, BS; beam splitter, PBS; polarization beam splitter, AOM; acousto-optic modulator, 1^st; 1^st-order diffracted beam, M; mirror, PDs; fast photo-detectors, LP; linear polarizer, R.B.; resolution bandwidth.

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We then measured the slope efficiency η(16.8 W/A) of the output power of the TE₀ mode at the cw regime, which was 2.2 times larger than that (7.5 W/A) of the TE₁ mode in the whole injection current range as shown in Fig. 2. They were measured at the diode temperature T = 24.0 °C and the threshold current I_th = 40.0 mA. The black lines are only for the eye guidelines and note that there are gain saturation regimes above 110 mA for the TE₀ mode and 140 mA for the TE₁ mode, respectively. The self-starting passive mode-locking phenomena discovered in Ref. [6] and detailed in Fig. 3 below were observed always beyond the gain saturation threshold currents and at the elevated diode temperature, for instance T = 39.5 °C.

Next, we increased the injection current over the gain saturation threshold currents for both modes to the value I = 146.5 mA and the external-cavity length was adjusted to L ₀ = 45.0 cm, i.e, FSR = 333 MHz. In this operation parameters, we observed a self-starting passive mode-locking phenomena as shown in Fig. 3(a) and (b). The data in Fig. 3 and Fig. 4 below were measured by a fast photo-detector at the position of PD1 of Fig. 1. We have checked that the main features of the RF spectra and the pulse train measured at the positions of PD2 and PD3 along the paths of the TE₀ and TE₁ modes, respectively, were completely same as those measured at the position of PD1. That means the TE₀ and TE₁ modes experience the same nonlinear effect that leads to the period-doubled mode-locking, while having the same oscillation frequency due to the coherent coupling effect. This observation indicates that the coherent coupling effect observed in the cw regime in Fig. 2(c) has the same major role for the passive mode-locking of the longitudinal modes belonging to two lowest-order lateral modes. A fast photo-diode with 10 GHz bandwidth (Hammamatsu G4170), a bias-tee (Picosecond Labs 5575A-104), and a low-noise RF amplifier (Miteq, AFS4) connected in series were used to measure the RF spectra and the time domain pulse train. A RF spectrum analyzer (Advantest R3261A) and a 500 MHz digital oscilloscope (Tektronix TDS3054B) with a time resolution of 300 ps were used (not shown in Fig. 1) to record the frequency domain spectra and the time domain pulse trains shown in Figs. 3 and 4.

 

Fig. 2. Output powers of the TE₀ and TE₁ modes as a function of injection current at the diode temperature T = 24.0 °C. η is a slope efficiency measured below the gain saturation regime. Solid lines are for the eye guidelines only.

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One can see in Fig. 3(a) that the RF spectra contains harmonics of the repetition rate of the ECDL, i.e, the optical frequency of the nth-mode of the optical frequency comb is given by f_n = nf_rep + f_ceo, where f_rep = FSR = 333 MHz, n ~ 10⁶, and f_ceo is the carrier envelope offset frequency, and we call this kind of mode-locking as a standard (conventional) mode-locking, since it contains almost the same features observed in a conventional Kerr-lens mode-locked Ti:Sapphire laser [11]. The signal-to-noise (S/N) ratio of the RF spectrum exceeds 50 dB and the line-width of it was measured typically to be below 10 kHz at 3 kHz resolution bandwidth. The corresponding pulse train measured in the time domain is shown in Fig. 3(b). The pulse width of 300 ps was limited by the time resolution of the oscilloscope. Note that the intensities of the RF spectra in the frequency domain and the pulse train in the time domain are almost same (even intensity distribution), indicating a stable mode-locking has been generated.

As contrary to the Figs. 3(a) and (b), we observed a period-doubled mode-locking as shown in Figs. 3(c) and (d) at the exactly same operating parameters of the diode laser, except the cavity length was tuned slightly by a piezo-electric transducer (PZT) to the length L ₁ = L ₀ + 50 nm. In other words, starting from a stable standard mode-locking state, a period-doubled mode-locking has been generated by tuning the cavity length as an amount of 50 nm, which corresponds to 0.8 rad (or 45°) external-cavity phase shift away from the stable mode-locking phase at 795 nm. In this period-doubling mode-locked state, the RF spectrum has harmonics of f_rep/2, i.e., mf_rep/2, m being an integer, a subharmonic mode-locking [6], while the time domain intensity patten experiences period-doubling bifurcation and period-doubling route to chaos as observed in a passively mode-locked fiber ring laser [12] and a Kerr-lens mode-locked Ti:Sapphire laser [13] (see also Figs. 4(c) and (d) below for period-doubling route to chaos).

 

Fig. 3. A RF spectrum in frequency domain (a) and pulse series in time domain (b) of a self-starting passively mode-locked ECDL at the injection current I = 146.5 mA, diode temperature T = 39.5 °C, and the external cavity length L ₀ = 45.0 cm. In (c) and (d), a period-doubled mode-locking is clearly observed, where all the operation parameters of the ECDL are exactly same as in (a) and (b), except the cavity length is tuned by a PZT to the length L ₁ = L ₀ +50 nm. R.B.; resolution bandwidth.

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Although the round trip time of the pulses circulating in the external cavity is still the same, the pulse energy returns back only every two round-trips, forming a so-called period-doubled state as compared with that of Figs. 3(a) and (b). We also note that in the period-doubling mode-locked state, the intensities of the RF spectra in the frequency domain and the pulse train in the time domain were evenly distributed as in the standard mode-locking state. By slightly tilting the angle of back reflection of the external grating from the stable mode-locking and period-doubling mode-locking states, we observed another states of mode-locking, for example a total mode-locking between two lowest-order lateral modes and chaotic pulsations as shown in Fig. 4 below.

After grating angle adjustment, we then deliberately increased the injection current by 0.8 mA to the value I = 147.3 mA and the external cavity length by 1.09 μm to the value L ₂ = L ₀ + 1.14 μm from the operational parameters in Figs. 3(c) and (d). In these parameters, we observed a total mode-locking phenomena as shown in Figs. 4(a) and (b). Note that the pulse train in Figs. 4(b) and (d) were displayed up to 100 ns. A total mode-locked state between two lowest-order lateral modes has been observed previously in a Kerr-lens mode-locked Ti:Sapphire laser [13]. As one can see, the RF spectra and the pulse train in a total mode-locked state have sub-structures in that the RF spectra consist of two series of harmonics of f_rep/2 with different intensities and the pulse train composed of two series of period-doubled pulse trains with different intensities as well. This feature can be understood by considering that our ECDL operates in a standing-wave cavity [9, 13] and two lowest-order lateral modes are coupled coherently and resulted in the generation of period-doubled mode-locking. Note that due to the factor of 2.2 difference in the slope efficiencies of the TE₀ and TE₁ modes as in Fig. 2(a), the RF intensities and pulse intensities corresponding to the outputs of the two different lateral modes should have different intensities. Since the pulses in the external cavity travel alternatively along two lateral modes in this parameter regime [13], the complete round-trip time may take twice long as compared to the case of the period-doubled mode-locked state. Thus, in each round-trip, two kinds of output pulses shifted by T in time would be emitted and the corresponding RF spectra could be shifted by f_rep/4, resulting in a full agreement with the experimental observations in Figs. 4(a) and (b). Therefore, in the total mode-locked state, the RF spectra and the pulse train associated with the TE₀ and TE₁ modes have the exactly same features as those of the period-doubling mode-locked state, except the RF spectra and the pulse train of the TE₁ mode are now shifted by f_ref/4 in frequency and by T in time, respectively. This observation supports our conclusion that the mode-locked pulses in the total mode-locked state indeed travel alternatively through two lowest-order lateral modes [13] and experience simultaneously a period doubled mode-locking in each of them. We observed that once the two distinct mode-locking phenomena, i.e., the period-doubled mode-locking due to the coherent coupling effect and the total mode-locking, were self-triggered at specific values of the external-cavity phases, they oscillated stably over two hours.

 

Fig. 4. In (a) and (b), all the operation parameters of the ECDL are same as in Fig. 3, except now the injection current I = 147.3 mA, the cavity length L ₂ = L ₀ +1.14 μm, and the feedback angle of the grating is deliberately adjusted to see the total mode-locking between the lateral TE₀ and TE₁ modes (see text). A chaotic pulsation shown in (c) and (d) are obtained at the same operation parameters as in (a) and (b), except the cavity length is slightly tuned by 50 nm from L ₂ and returned back to L ₂ quickly, showing an optical bi-stability.

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Finally, by keeping all the operation parameters of the ECDL as same as in the Figs. 4(a) and (b), we changed now the external cavity length by 50 nm from L ₂ and returned quickly back to L ₂. By doing this, we observed chaotic pulsations in the time domain as shown in Fig. 4(d), having an irregular intensity distribution in the pulse train up to 17 pulses in 100 ns, showing a possible route to chaos via the period-doubled mode-locking and also an optical bi-stability of the passively mode-locked ECDL. The corresponding RF spectra shown in Fig. 4(c), however, looks quite similar to those of the total mode-locking state in Fig. 4(a), except some harmonic components had irregular intensities. Thus, we can understand that the observed route to chaos via a period-doubled mode-locking follows the same route to chaos observed previously in a passively mode-locked fiber laser with dispersion management ring cavity [12]. We note, however, that in Ref. [12] only time domain chaotic pulsations was observed.

In conclusion, we have observed various nonlinear mode-locking phenomena in a self-starting passively mode-locked ECDL employing a GaAs ridge wave-guide diode laser as a gain medium at 795 nm and it was operated in a gain saturation regime: such as a standard mode-locking, a period-doubled mode-locking in each of two lowest-order lateral modes experiencing a coherent coupling, a total mode-locking [13], and a period-doubling route to chaos. Our ECDL system, which emits ps pulse train in the time domain and has a narrow line-width of the optical frequency comb in the frequency domain, consists of only a gain medium and an external reflection grating, thus it is very compact, inexpensive, and robust. We anticipate that there are great potentials of future applications of the current system not only for the experimental and theoretical study of nonlinear mode-locking phenomena, but also for the practical applications in the frequency metrology [11], in the optical coherence tomography, and in the optical telecommunications. Currently, we are developing a theoretical model based on the SLD theory [8] to explain the observed nonlinear mode-locking phenomena in our passively mode-locked ECDL system and the results will be published elsewhere.