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Fixation of the optical frequency of slave laser to the frequencies between two adjacent modes by multiple optical injection locking is investigated. Numerical simulation suggests that the frequency pulling among many four wave mixing tones is the origin of the locking mechanism.
©2011 Optical Society of America
1. Introduction
The optical injection locking in lasers is a well-known phenomenon and has been studied by many researchers since it was first demonstrated by Stover [1–3]. Primary concerns about the optical injection locking process were about locking bandwidth, stability, and relative noise characteristics of the slave lasers [4–6]. In the early times injection locking drew attention for its potential in suppressing relaxation oscillation [3]. Other major application of the injection locking is providing local oscillators in coherent optical communication [7]. A relatively new application of the injection locking technique arises with the development of optical comb technology where the optical injection locking is used to extract individual modes from an optical comb [8]. In this case, a master laser has multiple mode output while a slave laser has a single mode output. If the slave laser frequency is tuned close enough to one of the comb component within the locking bandwidth, then the slave laser frequency will lock to the particular comb component. In comparison to a conventional way of using a passive narrow band pass filter to extract individual comb components, this technique is advantageous in two ways: extraction of single mode and amplification of the extracted mode. While it is a rather challenging task to acquire an optical filter with pass bandwidth narrower than GHz to select a single mode from neighboring modes, optical injection locking technique can easily filter out single mode from a comb with several hundred MHz of mode spacing and with ~30 dB of side mode suppression. Although injection locking technique may suffer additional noise from the slave laser, the degree of noises depends on the quality of both master and slave lasers and active stabilization loops have been proven effective to mitigate this problem [7]. In addition, we have found that the optical injection locking technique has a unique advantage over the passive filters. The slave laser can lock not only to one of the mode components but also to frequencies between comb components. These frequencies are bisecting, trisecting point in frequency between two adjacent comb components. Other equipartition points in frequencies can be considered as well, however, as the partition of the mode spacing becomes finer, locking becomes less stable, so that frequency locking only up to the trisecting frequencies has been experimentally observed so far. In this work, the experimental observations of the optical frequency locking of the slave laser at bisecting, trisecting point in frequency between two adjacent comb components are reported. Numerical simulation of a simpler case (only two modes being injected) supports these observations and suggests that the frequency pulling among many four wave mixing (FWM) tones is the origin of the locking mechanism. This effect can be useful in frequency metrology, especially, in doubling or tripling the density of extractable comb from a given optical comb source.
2. Experiment of multiple injection locking
For the multimode injection locking experiment, the master laser has to be carefully chosen so that frequencies and phases of each mode are stable enough to support interactions which will lock the slave laser frequency. Mode-locked lasers are good candidates for the master laser in this sense since mode frequency spacing and the relative phase among modes are almost constant [9]. For the slave laser, predominant single mode operation and frequency tunability are two main qualities required and distributed feedback (DFB) lasers are proven effective candidates [5]. The schematic diagram of the experimental set up is shown in Fig. 1 . The comb source was a mode-locked external cavity semiconductor laser at 10 GHz of repetition rate. The mode-locked laser output was injected into a commercially packaged DFB laser through an optical circulator. The amount of injected optical power was adjusted using a variable optical attenuator. In order to ensure that the polarization of the injected light is matched with that of the DFB laser, a three-element fiber optic polarization controller was inserted. The DFB laser threshold current was ~10 mA and the laser was biased ~3 time of the threshold. The linewidth of the DFB was estimated less than 10 MHz from RF beat tone measurement between two similar DFB lasers and the frequency drift was ~200MHz from the same measurement. The DFB laser frequency was tuned by adjusting both the bias current and the temperature of the cooling unit. The tuning rate was ~5 pm/mA (0.625 GHz/mA). The output of the DFB laser was collected by the same optical circulator and then sent to diagnostics. It should be noted that the optical spectrum analyzer alone is not enough to confirm the injection locking effect due to its limited resolution. In order to measure the frequency differences among the modes from both lasers with higher resolution, the rf spectrum of the photo detector output was monitored. The resolution of the optical spectrum analyzer was 0.02 nm (2.5 GHz at 1550 nm) and the 3 dB bandwidth of the photodetector was 12 GHz.
Fig. 1 Schematic diagram of the experimental setup. PC: polarization controller, OC: optical circulator, VA: variable optical attenuator, PD: photodetector, OSA: optical spectrum analyzer, and RFSA: RF spectrum analyzer.
For the DFB lasers used in this experiment, injection locking bandwidth ΔνLock for a single mode injection case follows the typical square root dependence of injected optical power ratio PMaster/PSlave [10].
The proportionality coefficient ν0 was experimentally determined as 68 GHz.
By tuning the DFB laser close enough to one of the longitudinal modes of the mode-locked laser, the DFB laser frequency can be locked. The injected optical power of modes adjacent to the DFB output was 2.7 μW/mode and the fiberized DFB output power was ~2 mW, indicating ΔνLock is ~2.5 GHz ignoring the fiber coupling efficiency between the DFB laser and the pigtail fiber. However, for the multimode injection locking, the measured locking bandwidth was ~250 MHz. Figure 2 shows the optical spectra of master and slave laser. The side mode suppression ratio is ~26 dB. It should be noted that the part of the spectrum corresponding to the mode-locked laser may look flat in this narrow spectral range; however, its full width of half maximum was 1.7 nm.
Fig. 2 Optical spectra of the master and the slave laser. Dotted line: the master laser output, dashed line: free running slave laser output, and solid line: injection locked slave laser output.
Owing to the insufficient optical resolution, it is not clear from this measurement whether the frequency locking has been truly established or not. In order to assure the frequency locking has happened, RF spectrum of the photodetector signal of the slave laser output was measured as the DFB laser frequency is tuned. Figure 3 shows RF spectra of the DFB laser output for unlocked (Fig. 3(a)) and locked (Fig. 3(b)) cases. For the unlocked case, the DFB laser frequency was detuned ~1 GHz away from the one of the master laser mode which the slave laser is intended to lock to. The sharp peak at 10 GHz is due to beatings among mode-locked laser longitudinal modes which are spaced by exact 10 GHz. The other two broad peaks at ~9 and ~11 GHz are due to the beating between the DFB laser mode and two immediately neighboring modes of the targeted mode of the mode-locked laser, respectively. Since the DFB laser frequency suffers hundreds of MHz of frequency drift, these beat tones have quite broad width in comparison to the 10 GHz tone. It should be noted that the potential beat tones at ~1 and 19 GHz are not as prominent due to the photo detector bandwidth limit. On the other hand, when the DFB is locked, these beat tones disappear as can be seen from Fig. 3(b). Therefore RF spectrum of the photodetector signal was used as a tool to confirm an injection locking.
The major distinction between a single mode injection locking and multimode injection locking happens when a targeted slave laser frequency does not coincide with any of injected modes. For example, a bisecting point between two adjacent modes in frequency can be used for the injection locking. Figure 4 and Fig. 5 show optical and RF spectra, respectively. Figure 4 shows that the slave laser frequency has been tuned between two master laser modes. It also shows many new side modes generated by FWM. The side mode suppression ratio is ~18 dB. The injected power was 12.4 μW/mode. Figure 5 (a) shows RF spectrum when the slave laser was not locked. In this case, DFB laser frequency was ~1 GHz detuned from the exact bisecting point in frequency. The 10 GHz peak has the same origin as Fig. 3. The other four broad peaks are ~4, ~6, ~14, and ~16 GHz which are caused by beatings between the DFB laser and four different neighboring modes from the mode-locked laser. Again, once the DFB laser is locked, these broad peaks disappear and very narrow peaks at 5 and 15 GHz appear indicating well defined injection locking.
Fig. 4 Optical spectra of the master and the slave laser. Dotted line: the master laser output, dashed line: free running slave laser output, and solid line: injection locked slave laser output.
Multimode injection locking can be achieved at trisecting point in frequency between two adjacent comb components as well. Figure 6 and 7 show results of the injection locking. The injected power was 12.4 μW/mode. Again, many new side modes were generated by FWM. The side mode suppression ratio is ~10 dB which is less than bisecting case. Once the locking is established, it stays locked for several hours. It should be also noted that this injection locking was less stable than bisecting case as Fig. 7 shows, 10/3, 20/3, 40/3, and 50/3 GHz tones have rather apparent pedestals. Multimode injection locking at quartering points in frequency was never achieved. The numerical model suggests that the finer the division in frequency, the harder it is to achieve stable locking.
Fig. 6 Optical spectra of the master and the slave laser. Dotted line: the master laser output, dashed line: free running slave laser output, and solid line: injection locked slave laser output.
Fig. 7 RF spectra of the slave laser at a trisecting point between two neighboring master laser modes.
3. Numerical modeling of multiple injection locking
The model follows Troger’s approach which is an extension of Lang’s model [11,12]. The complex electric field of the slave laser ε(t) and that of the m-th mode of the mode-locked laser εm(t) are defined as
E(t) and Em(t) are slowly varying amplitudes. It should be noted that ϕ(t) is treated as a function of time while ϕm are considered as constants assuming that the mode-locked laser is noise free. Any information about frequency change will be contained in this ϕ(t) term. Separating the real and imaginary components from a rate equation for the electric field and expressing ε(t) and εm(t) in terms of photon numbers P and Pm where P = V × E2 and Pm = V × Em2, V being mode volume, the equations of motion become where G is the optical gain, γ is the optical loss, τL is the cavity round trip time, and α is the linewidth enhancement parameter. Rsp is the spontaneous emission rate modeled as Rsp = nsp × G, where nsp is the Fermi inversion factor. The optical gain G is approximated by the three parameter logarithmic formwhere Γ is the confinement factor, g0 is the linear gain coefficient, vg is the group velocity, Ntr is the transparency carrier number, Ns is a shift of carrier number, and Ps is the saturation photon number. The rate equation for the carrier number N is modeled aswhere ηi is the internal quantum efficiency, I is the slave laser bias current, γe is the carrier decay rate, and q is the charge of electron.Simulation parameters are summarized in Table 1 . These parameters were chosen to simulate the DFB lasers used in the experiment from their typical ranges because this information was not available from the manufacturer of the devices.
Table 1. Simulation Parameters
Although more than dozens of modes were injected into the DFB laser in our experiment, it is beneficial to simplify the model to understand the locking mechanism of multimode injection locking. Therefore the number of injected modes was limited to two. Nonetheless the results of the simulation match well with the experimental results and it was found that only two modes are enough to understand multiple mode injection locking phenomena qualitatively. The optical spectrum of the slave laser is obtained by Fourier transforming the simulated electric field E(t). The simulated optical spectra are shown in Fig. 8 for locking into the bisecting point in frequency between two injected modes. The bias current to the slave DFB laser was 34 mA, with output power of 1.2 mW, and the injected power was 2.2 μW/mode. The simulated threshold current of the DFB was 10 mA. To assure the injection locking phenomenon, the slave laser frequency was intentionally detuned for 40 MHz from the targeted locking frequency as an initial condition. As the master laser output starts to be injected into the slave laser, the slave laser output electric field was monitored for 20 μs. The slave laser typically reaches to a steady state within a few μs. Figure 8 (a) shows the spectrum of the master laser with each mode are located at + 5 and −5 GHz. The inset shows that the free running slave laser is 40 MHz off from the center. The spectrum after the injection locking has been established is shown in Fig. 8 (b). The inset shows that the slave laser frequency has been pulled to 0 Hz and then stays fixed there. It also shows that many FWM components are generated as we have observed in the experiment (Fig. 4). The side mode suppression is ~20 dB.
Fig. 8 Simulated optical spectra of two mode injection: (a) free running slave laser (solid line), master laser (dotted line), (b) slave laser output after injection locking. (Insets are zoom-in of slave laser output)
The simulated results for injection locking into a trisecting point between two master laser modes are shown in Fig. 9 . This time, two master laser modes are located a −10/3 and + 20/3 GHz. The slave laser was again detuned by 40 MHz off from the target frequency. The injected power level was the same as Fig. 8. The inset of Fig. 9 (b) clearly indicated that the injection locking has been established. It also shows that many FWM tones are generated and matches well with the measure spectrum (Fig. 6). The side mode suppression is ~10 dB.
Fig. 9 Simulated optical spectra of two mode injection: (a) free running slave laser (solid line), master laser (dotted line), (b) slave laser output after injection locking. (Insets are zoom-in of slave laser output).
Figure 10 shows how the slave laser frequency varies depending on the injected power level. For the injection locking to the bisecting and trisecting points in frequency, the slave laser frequencies are pulled to zero offset frequency and stay locked while for the injection locking to a quartering point, the slave laser frequency is continuously pulled to a nearest injected mode without locking. This result is consistent with our experiences where stable injection locking to bisecting and trisecting points was achieved while the injection locking to a quartering point never succeeded.
Fig. 10 Simulated slave laser frequency offset from injection locking frequency vs. injected optical power. Solid line: locking to the bisecting point, dotted line: locking to the trisecting point, and dashed line: locking to the quartering point.
In order to understand the mechanism behind the multiple mode injection locking effects, a series of optical spectra were calculated for various injected optical power level (Fig. 11 ).
Fig. 11 Simulated optical spectra of two mode injection for various injection power level: (a) PMaster/PSlave = 1.87 × 10−4, (b) PMaster/PSlave = 8.60 × 10−4, (c) PMaster/PSlave = 2.78 × 10−3, and (d) PMaster/PSlave = 3.29 × 10−3. (Insets are zoom-in of slave laser output).
In this simulation, initial offset frequency δf of the slave laser from the bisecting point between two injected modes is set at 100 MHz. At a very low injection power level, degenerated FWM from modes at -f0 and δf generates tones at −2f0 - δf and f0 + 2δf while degenerated FWM from modes at δf and f0 generates tones at -f0 + 2δf and 2f0 - δf where -f0 and f0 are the master laser mode frequencies, - 5 and + 5 GHz, respectively. In addition, FWM from -f0, δf, and f0 generate -δf ( = -f0 + f0 - δf) tone. Beside these strong tones, there are many other tones generated by FWM for different combination of participating modes [13]. As the injection power level increases, two effects take place; generation of more FWM tones, and pulling among those tones. Figure 11 (c) shows this effect clearly. From the inset plot, we can see that mode spacing has been reduced from initial 200 MHz to ~100 MHz and the slave laser frequency has been pulled to ~50 MHz. This frequency pulling has the same origin as the single mode injection locking i.e. the slave laser is pulled to the tone at -δf. This effect continues until injection locking happens (Fig. 11 (d)). For the injection locking to a trisecting point, the same mechanism follows as well.
4. Conclusions
The optical injection locking of DFB laser by multiple mode injection has been investigated. It was found that the slave laser can lock to equipartition points in frequencies between two adjacent modes as well as each individual mode of the master laser. Those points are bisecting and trisecting points in frequency between two master laser modes. The locking bandwidths in these cases are normally much narrower than single mode injection locking case. Numerical simulation shows that FWM plays an important role in the multiple mode injection locking mechanism. First, FWM generates many mixing signals and pulling among these mixing tones leads to the injection locking. The injection locking to a quartering point was never observed in the experiment, which is also consistent with the numerical simulation result. In this case, pulling from the nearest injected mode overwhelms this multimode injection locking effect. Our observation suggests that as the slave laser locking frequency is switched from one of the maser laser modes to those equipartition points, the locking stability and the side mode suppression ratio degrades. The stability test of this system is believed to provide more solid explanation of this limit and expected to be done in the future. The multiple mode injection locking effect will be useful in metrology application. For a given optical comb source, this effect allows to extract more frequency components than the original comb can provide, thus enabling finer frequency scale.
Acknowledgments
This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (MEST) (NRF 2010-0009754).
References and links
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