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Abstract
We study the stability of period-one (P1) oscillations experimentally generated by semiconductor lasers subject to optical injection (OI) and by those subject to optical feedback (OF). With unique advantages of broad frequency tuning range and large sideband rejection ratio, P1 oscillations can be useful in applications such as photonic microwave generation, radio-over-fiber communication, and laser Doppler velocimeter. The stability of the P1 oscillations is critical for these applications, which can be affected by spontaneous emission and fluctuations in both temperature and injection current. Although linewidths of P1 oscillations generated by various schemes have been reported, the mechanisms and roles which each of the OI and the OF play have however not been investigated in detail. To characterize the stability of the P1 oscillations generated by the OI and the OF schemes, we measure the linewidths and linewidth reduction ratios (LRRs) of the P1 oscillations. The OF scheme has a narrowest linewidth of 0.21 ± 0.03 MHz compared to 4.7 ± 0.6 MHz in the OI scheme. In the OF scheme, a much larger region of LRRs higher than 90% is also found. The superior stability of the OF scheme is benefited by the fact that the P1 oscillations in the OF scheme are originated from the undamped relaxation oscillation of a single laser and can be phase-locked to one of its external cavity modes, whereas those in the OI scheme come from two independent lasers which bear no phase relation. Moreover, excess P1 linewidth broadening in the OI scheme caused by fluctuation in injection parameters associated with frequency jitter and relative intensity noise (RIN) is also minimized in the OF scheme.
© 2017 Optical Society of America
1. Introduction
Nonlinear dynamics of period-one (P1) oscillations generated by semiconductor lasers subject to optical injection [1–5] have been closely investigated in recent years for their potential applications in photonic microwave amplification [6], optical frequency conversion [7], radio-over-fiber communication [8–10], and laser Doppler velocimeter [11,12]. These P1 oscillations are developed when an optically injected laser resonates at a frequency corresponding to the detuning between master and red-shifted slave [1,2,5,13]. Without limiting by the bandwidths of the electronic devices as commonly encountered in other photonic microwave generation schemes [14,15], the frequencies of the P1 oscillations are broadly tunable from several gigahertz to tens of gigahertz through adjusting the injection strength and detuning frequency [16, 17]. Moreover, the single-sideband characteristics of the P1 oscillations significantly reduced the microwave power fading effect in transmission [4,8].
Despite the fact that the P1 oscillations have the above-described advantages in photonic microwave generation, their stability are affected inherently by the spontaneous emission noise and also the time-dependent fluctuations in detuning frequency and injection current. These instabilities broaden the linewidths of the P1 oscillations, thereby lowering the performance in the photonic microwave amplification [6], reducing the signal-to-noise ratios in the radio-over-fiber communications [10], and limiting the velocity resolution, accuracy, and maximum detection distance in applications of Doppler velocimeters [11,12].
To generate P1 oscillations with better stability, studies employing external feedback have been reported [18–20]. With optical feedback, optoelectronic feedback, and polarization-rotated feedback in effect, P1 oscillations with reduced linewidths of 50 kHz, 10 kHz, and 3 kHz, respectively, have been demonstrated [18–26]. Although adding feedback on lasers has been shown to stabilize the P1 oscillations, mechanism and contribution the feedback play have not been discussed in detail. Therefore, to better understand each of the roles optical injection and optical feedback play in P1 generations, we investigate the linewidths and the linewidth reduction ratios to quantitatively compare the stability of the P1 oscillations generated by semiconductor lasers subject to optical injection and by those subject to optical feedback.
2. P1 generated with the OI scheme
2.1. Experimental setup
The experimental setup of the P1 generation with the OI scheme is shown in Fig. 1. A tunable laser (Yenista, Tunics-T100S, O-band) that has a linewidth of 9 ± 1 MHz measured by self-heterodyne interferometry is used as the master laser (ML) and a single-mode distributed-feedback (DFB) semiconductor laser (MITSUBISHI, ML725B11F) with a linewidth of 4.9 ± 0.3 MHz [27–30] is used as the slave laser (SL). All the linewidths shown in this paper are the means and their standard errors calculated from 60 3-dB spectral widths extracted from their corresponding RF spectra taken under 10 kHz bandwidth resolution and 500 ms sweep time. The emitted light of the ML is injected into the SL where the normalized injection strength (ξi), the ratio of the optical field of the injection light to the optical field of the SL output, is adjusted by a variable attenuator and monitored by the power meter (PM). The detuning frequency (Δf) is the frequency difference between the ML and SL. By controlling these injection parameters, the laser can be operated in the P1 dynamical states comprising two optical frequencies, namely the red-shifted frequency of the SL (νSL) and the injected frequency of the ML (νML). Dividing by a 40/60 fiber coupler (FC), the optical spectrum is acquired by an optical spectrum analyzer (OSA) (Advantest Q8384) with 10 pm resolution and the electrical spectrum is detected by a high-speed photodetector (PD) (Newport 1544-A) with 12 GHz bandwidth and analyzed by an electrical spectrum analyzer (ESA) (R&S FSV30) with 30 GHz bandwidth.
Fig. 1 Experimental setup of the P1 generation with the OI scheme. ML: master laser; SL: slave laser; RL1: reference laser; BS: beam splitter; ATT: variable attenuator; PM: power meter; FC: fiber collimator; PD: high-speed photodetector; ESA: electrical spectrum analyzer; OSA: optical spectrum analyzer; ISO: isolator; PC: polarization controller.
To analyze the stability of P1 states, a single-mode DFB semiconductor laser (RL1) with a linewidth of 38 ± 3 MHz was coupled to the output of the SL as a reference laser to measure the optical linewidths of the two optical frequencies constituting P1. When the wavelength of RL1 is adjusted in between those of the ML and SL, the heterodyne signals ML′ and SL′ between the ML and SL to the RL1 can be simultaneously measured, respectively. By computing the difference in linewidths between the ML′, SL′, and RL1, the linewidths of the injected (ΔνML) and the red-shifted (ΔνSL) signals can be individually obtained under different injection parameters [31,32]. Note that, different from the theoretical work [19] that has taken into account only the spontaneous emission noise, in this paper influences from both spontaneous emission noise and frequency jitter on the linewidths are included in the stability assessment.
The state diagram of the OI scheme for different ξi and Δf is shown in Fig. 2. By adjusting ξi and Δf, the laser can be operated in various dynamical states including chaos (CO), period-two (P2), period-one (P1), and stable locking (S). For the P1 states of interest, typical optical and electrical spectra obtained at ξi = 0.24 and Δf = −2.21 GHz are shown in Figs. 3(a) and 3(b), respectively. The output light from the SL contains two optical frequencies as shown in Fig. 3(a), which arise from the red-shifted wavelength of the SL (νSL) and the regenerated signal at the injected wavelength of the ML (νML). The oscillation frequency of the P1 state (fP1) at the beat of 11.75 GHz in the electrical spectrum is also shown in Fig. 3(b).
Fig. 2 State diagram of the OI scheme for different ξi and Δf. S: stably locked region; P1: period-one oscillation; P2: period-two oscillation; CO: chaotic oscillation.
Fig. 3 (a) Optical and (b) electrical spectra of the SL subject to optical injection at (ξi, Δf) = (0.24, −2.21 GHz).
2.2. Oscillation frequencies and linewidths of P1 for different ξi and Δf
Figures 4(a)–4(c) show the oscillation frequencies of the P1 states fP1 as a function of Δf for ξi = 0.13, 0.15, and 0.19, respectively. By increasing Δf, the dynamical state of the SL bifurcates from the S to P1 states after crossing the red dashed lines. For the P1 states, fP1 decreases to a minimum and then increases again when Δf is increased. Under certain injection parameters, as shown in Fig. 4(c), fP1 is relatively insensitive to the change in Δf when the SL is operated near the so-called detuning-insensitive region (DIR) [19,25,33].
Fig. 4 Oscillation frequencies fP1 and linewidths ΔνP1 of P1 for different detuning frequencies at ξi = (a)(d) 0.13, (b)(e) 0.15, and (c)(f) 0.19, respectively. The black dashed and solid lines are the optical linewidths of the ML at free-running (Δν0,ML) and after injection (ΔνML), and the red dashed and solid lines are the optical linewidths of the SL at free-running (Δν0,SL) and under injection (ΔνSL), respectively. S: stable locking state.
Figures 4(d)–4(f) show the corresponding linewidths of the P1 (ΔνP1) as a function of Δf at ξi = 0.13, 0.15, and 0.19, respectively. As shown, the linewidths of the νML (ΔνML), the νSL (ΔνSL), and the corresponding P1 oscillations (ΔνP1) vary significantly under different injection parameters. In most conditions, ΔνP1 is broader than the sum of the optical linewidths of the SL and ML at free-running (Δν0,SL+Δν0,ML) showing that excess broadening from frequency jitter and relative intensity noise (RIN) occurs in the P1 oscillation. As can be seen in Fig. 4(e), a minimum ΔνP1 of 4.7 ± 0.6 MHz is obtained at ξi = 0.1512 and Δf = 2.75 GHz. Note that this minimum linewidth is narrower than the optical linewidth of the SL at free-running (Δν0,SL = 4.9 ± 0.3 MHz), showing that good phase correlation between the νML and the νSL is achieved through optical injection in this particular condition.
3. P1 generated with the OF scheme
3.1. Experimental setup
The experimental setup of the P1 generation with the OF scheme is shown in Fig. 5. The laser diode (LD) used in the OF scheme is the same one as the SL used in the OI scheme for better comparison. The output of the LD is fed back to the laser cavity through a partially reflecting mirror, where the distance between the LD and the mirror determines the external cavity length (Lext). The normalized feedback strength ξf which is the ratio between the optical field of the feedback light and the optical field of the LD output is adjusted via a variable attenuator. By controlling the feedback parameters, P1 oscillations originated from the undamped relaxation oscillations can be generated. Same instruments as those used in the OI scheme are used to analyze the optical and electrical signals.
Fig. 5 Experimental setup of the P1 generation with the OF scheme. LD: laser diode; BS: beam splitter; ATT: variable attenuator; PM: power meter; FC: fiber collimator; PD: high-speed photodetector; ESA: electrical spectrum analyzer; OSA: optical spectrum analyzer; ISO: isolator; RL2: reference laser; PC: polarization controller.
To analyze the stability of P1 oscillations in the OF scheme, a reference laser (RL2) (Yenista, Tunics-T100S, O-band) same as the ML used in the OI scheme was coupled to the output of the LD to measure the optical linewidths of the primary wavelength (PW) of the center mode and the secondary wavelength (SW) of the adjacent sideband from the amplitude modulation that constitutes P1. By adjusting the wavelength of RL2 in between the wavelengths of PW and the SW, the heterodyne signals PW′ and SW′ of the PW and SW with RL2 can be simultaneously measured, respectively. By computing the difference in linewidths between the PW′, SW′, and RL2, the optical linewidths of the PW (ΔνPW) and SW (ΔνSW) can be individually obtained under different feedback parameters. Note that the linewidths measured here are from the P1 oscillations induced and affected by solely the OF, not the linewidths of P1 oscillations generated by the OI and stabilized by the OF [19] .
Figure 6 shows the state diagram of the OF scheme for different ξf and Lext. By adjusting ξf and Lext, the LD can be operated in various dynamical states such as chaos oscillation (CO), quasi-period oscillation (QP), period-one oscillation (P1), and stable (S) states. As in the OI case, we focus on the characteristics of the P1 oscillations. Typical optical and electrical spectra of the P1 oscillations obtained at (ξf, Lext)=(0.014, 109 mm) are shown in Figs. 7(a) and 7(b), respectively. The center mode and the adjacent sideband associated with the amplitude modulation are marked as the PW and the SW in Fig. 7(a), respectively. As can be seen in Fig. 7(b), the P1 frequency fP1 at their beat is around the relaxation oscillation frequency of 7.5 GHz.
Fig. 6 State diagram of the OF scheme for different ξf and Lext. CO: chaotic oscillation; QP: quasi-period oscillation; P1: period-one oscillation; S: stable states.
Fig. 7 (a) Optical and (b) electrical spectra of the LD subject to optical feedback at (ξf, Lext) = (0.014, 109 mm).
3.2. Oscillation frequencies and linewidths of P1 for different ξf and Lext
Figures 8(a)–8(c) show the fP1 as a function of Lext at ξf = 0.011, 0.015, and 0.017, respectively. As Lext increases, fP1 decreases along with the decreasing of the external cavity frequency. For about every 20 mm corresponding to the relaxation oscillation frequency of fr ∼ 7.5 GHz, mode-hopping occurs where fP1 changes abruptly with a frequency difference corresponding to the mode spacing c/2Lext. As can be seen, since the P1 oscillations are originated from the undamped relaxation oscillation in the OF scheme, fP1 are found to be bounded around fr. While changing the bias current or altering the laser threshold through proper control of the external parameters could both change fr and thus tune fP1, in practice the former is found to be more effective than the latter given that the feedback from external coupling is usually weak and the change in fr is relatively limited.
Fig. 8 Oscillation frequencies fP1 and linewidths ΔνP1 (blue dots and curves) of P1 for different Lext at ξf = (a)(d) 0.011, (b)(e) 0.015, and (c)(f) 0.017, respectively. The black and red curves are the linewidths of the PW (ΔνPW) and the SW (ΔνSW) and black dashed line is the linewidth of the SL at free-running (Δν0), respectively.
Figures 8(d)–8(f) show the P1 linewidths ΔνP1 (blue curves) as a function of Lext at ξf = 0.011, 0.015, and 0.017, respectively. As can be seen, ΔνP1 varies with the period as in the fP1. When Lext is a multiple of c/2 fr, LD constructively couples with the external cavity [23]. By phase-locking to one of the external cavity modes, the P1 oscillations are more stable and have linewidths ΔνP1 even lower than Δν0 at one of its local minimum. On the contrary, when LD is operated in the vicinity of the mode-hopping regions where two external cavity modes have similar intensity levels and compete with each other, the P1 oscillations are relatively unstable and have linewidths broadened to one of its local maximum. Aside from the apparent periodicity, ΔνP1 in general decreases as ξf becomes stronger or Lext becomes longer.
While SW is merely the sideband from relaxation oscillation that constitutes P1, ΔνSW is expected to have the similar value of ΔνP1. However, under the influence of frequency jitter, as can be seen in Figs. 8(d)–8(f), discrepancy between ΔνSW and ΔνP1 increases when ΔνP1 decreases. Their difference is in the broadening of ΔνPW, which indicates the fact that the optical linewidth of SW seen in the spectrum is not only originated from ΔνP1 but also largely contributed by the frequency jitter influencing on the LD.
4. Stability of P1 generated with OI and OF schemes
4.1. Frequency, linewidth, and power
In the OI scheme, the generation of P1 oscillations arise from the injection between two independent lasers, namely the ML and SL. When the injection current and operation temperature fluctuate, the wavelengths of the injection and the red-shifted signals from the respective ML and SL vary independently and frequency jitter-induced linewidth broadening occurs. Note that, although the ML and the SL bear no phase relation at free-running, depending on the injection parameters, the phases of the two optical signals νML and νSL in the optical spectrum of the SL constituting the P1 oscillations can still be locked with different levels through nonlinear coupling under optical injection. Therefore, the linewidths of the P1 oscillations can vary in a broad range depending on the injection parameters chosen.
Figures 9(a)–9(c) show the frequency, linewidth, and power of the P1 oscillations generated for different ξi and Δf in the OI scheme, respectively. As can be seen in Fig. 9(a), large tunability of fP1 is found. A detuning insensitive region (DIR) where the fP1 is least sensitive to the variations of the injection parameters is shown [19,33]. In Fig. 9(b), a minimum-linewidth point (MLP) where ΔνP1 has its minimum is also marked. As shown, the MLP does not overlap with the DIR in the injection parameter space. This displacement shows that the P1 oscillations in the DIR does not necessary have the minimal linewidths that are least sensitive to the spontaneous emission noise and environmental fluctuation. Moreover, from the MLP, the linewidth in generally increases when ξi and Δf increase. These results agree well with those described theoretically in [19].
Fig. 9 Mappings of the P1 frequency, linewidth, and power for different injection and feedback parameters generated with the (a)–(c) OI and (d)–(f) OF schemes, respectively. DIR: detuning-insensitive region; MLP: minimum-linewidth point; Hopf: Hopf bifurcation.
Figures 9(d)–9(f) show the frequency, linewidth, and power of the P1 oscillations generated for different Lext and ξf in the OF scheme, respectively. When injection current and temperature fluctuate, the optical linewidths of PW and SW are also broadened by frequency jitter. Nevertheless, since the P1 oscillations in the OF scheme are originated from the undamped relaxation oscillation of a single laser and can be phase-locked to one of its external cavity modes, they are dissociated from the frequency jitter and are in general more stable than those generated with the OI scheme. Larger bluish regions with smaller ΔνP1 can be seen in Fig. 9(e) compared to those in Fig. 9(b). As for the minimum linewidths, the minimal ΔνP1 in the OF scheme is about 0.21 ± 0.03 MHz, which is more than an order lower than the minimal ΔνP1 of 4.7 ± 0.6 MHz in the OI scheme. As shown in Figs. 9(e) and 9(f), P1 oscillations generated by the OF scheme can have both narrow linewidths and higher power at the same time. Moreover, the frequency, linewidth, and power of the P1 oscillations generated by the OF are found to vary periodically with the Lext. Despite the apparent periodicity shown in Fig. 9(e), the linewidths tend to decrease gradually as the feedback becomes stronger or the external cavity becomes longer. Similar tendency is also found in the theoretical work where the linewidths of the P1 oscillations from the OI scheme decrease as the feedback strength or the cavity length of the added OF increases [19].
4.2. Linewidth reduction ratio
To further show the stabilization of the P1 oscillations generated by the OI and the OF schemes, we calculate their linewidth reduction ratio (LRR) under different injection and feedback parameters in Figs. 10(a) and 10(b), respectively. In the OI scheme, LRR is defined as the reduction in ΔνP1 normalized to the sum of the optical linewidths of the two optical signals constituting P1 (LRR = (ΔνML + ΔνSL − ΔνP1)/(ΔνML + ΔνSL)). In the OF scheme, LRR is defined as the reduction in ΔνP1 normalized to the optical linewidth of SW influenced by both amplitude modulation and frequency jitter (LRR = (ΔνSW − ΔνP1)/ΔνSW). In the OI scheme, LRR = 1 indicates the two optical signals constituting P1 are perfectly phase locked while LRR = 0 means the signals are completely out of phase and there is no reduction in ΔνP1. In the OF scheme, LRR = 1 indicates the linewidth broadening from the amplitude modulation is much smaller than that contributed by frequency jitter while LRR = 0 means the former well exceeds the latter. As can be seen, LRRs higher than 90% can be easily obtained in the OF scheme where a significant reddish region covering most of the feedback parameter space is shown in Fig. 10(b). Together with Figs. 9(b) and 9(e), these results show that the P1 oscillations generated by the OF scheme are in general more stable and have narrower linewidths in a greater region compared to those generated by the OI scheme.
Fig. 10 Mappings of linewidth reduction ratios for the P1 oscillations generated with the (a) OI and (b) OF schemes.
5. Conclusion
In conclusion, we studied the stability of P1 oscillations generated by semiconductor lasers subject to either OI or OF experimentally. Different from those configurations where the OF is added to stabilize the P1 oscillations generated already by OI, in this paper the individual contributions of the OI and the OF on the P1 linewidths are investigated. The ΔνP1 as low as 0.21 ± 0.03 MHz and LRR as high as 98% are obtained in the OF scheme, as compared with respective figures of 4.7 ± 0.6 MHz and 95% in the OI scheme. This advantage arises because the P1 oscillations in the OF scheme are originated from the undamped relaxation oscillation of a single laser and can be phase-locked to one of its external cavity modes, whereas those in the OI scheme come from two independent lasers which bear no phase relation. Moreover, excess P1 linewidth broadening in the OI scheme caused by fluctuation in injection parameters associated with frequency jitter and RIN is also minimized in the OF scheme. Not only the narrowest ΔνP1 is obtained in the OF scheme, generating P1 oscillations with the OF scheme also has the advantage of having significant regions of stable P1 oscillations in the parameter space so that choosing/maintaining the operation parameters for stable P1 generation becomes less critical. While adding feedback to the laser in general shows the above-mentioned advantages, it is still worth mentioning that the OI scheme by contrast can have extended frequency tuning range and higher RF power in the P1 oscillations generated.
Funding
Ministry of Science and Technology, Taiwan (MOST 103-2112-M-007-019-MY3 and 106-2112-M-007-003-MY3); National Tsing Hua University, Taiwan (106N539CE1).
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