We report a new graphical tool to analyze optical injection-locked vertical-cavity surface-emitting lasers (VCSELs). It predicts the resonant frequency enhancement and cavity mode behavior for both single- and multi- mode VCSELs under injection locking. Calculations based on this model show excellent agreement with experimental results.
©2012 Optical Society of America
Optical injection locking (OIL) has been shown to be effective to improve the modulation performance of a directly modulated diode laser [1–9]. Resonance frequency greater than 100 GHz was demonstrated experimentally with a vertical-cavity surface-emitting laser (VCSEL) under strong injection locking . This enhancement of resonance frequency also led to multi-Gbps 60 GHz radio-over-fiber (RoF) system demonstration . The directly modulated OIL-VCSELs can be a compact, low power-consumption, cost effective solution for a wide range of RoF and optical communications applications.
There have been many theoretical reports analyzing the enhancement of OIL-VCSEL resonance frequency [1–6]. Most of them provide detailed mathematic descriptions of the resonance frequency and other performance metrics via parametric equations. Although they are useful in predicting OIL-VCSEL performance and often are very comprehensive, they fail to exhibit a coherent and intuitive picture of the locking process and the interplay of various parameters. In this paper, we present a simple graphical tool to visualize and analyze OIL laser phenomena for both single- and multi- mode slave lasers. Excellent agreement with experimental results was obtained. Linewidth enhancement factor (α) and phase difference between the master and the slave laser field (Δϕ) can also be easily visualized and extracted from this model.
2. The ellipse model and theoretical framework
The proposed graphic tool displays the equivalent optical spectra of an OIL laser under various injection conditions, providing a visual reading of the resonance frequency and the inter-dependence of various parameters. The equations are based on steady-state solutions of the standard OIL rate equations . The key control variables are wavelength detuning Δλ and injection ratio Rinj, which are wavelength difference and power ratio between the master and free-running slave lasers, respectively. When the slave laser is locked, the output optical spectrum consists of the master wavelength λmaster, which is now the locked system wavelength, and a side peak of the slave cavity mode λcavity, which is strongly suppressed under optical injection [1–4]. The difference between the two is the resonance frequency . This novel graphic tool exhibits the relationship of the wavelength detuning (Δλ = λmaster-λslave0) and the cavity shift (Δλcav = λcavity-λslave0) under injection locking process.
Here, we keep the slave laser free-running wavelength (λslave0) constant and as the origin of the graph, while varying the master wavelength λmaster as control variable. Based on the OIL rate equations, Δλ and Δλcav are re-expressed as follows:
The relevant parameters are defined in Table 1 .
Eq. (3) can be further simplified to
Given that typical experimental conditions 
we can easily see that Δλcav and Δλ form an ellipse based on Eqs. (1) and (2), with Δϕ as a parameter. Alternatively, an implicit equation of an ellipse can be written to describe the relationship between Δλcav and Δλ:
Where C, D are constants:
Figure 1(a) graphically shows this ellipse as described by Eq. (6) with eccentricity and size of the ellipse being determined by α and Rinj. Each point on the ellipse represents a possible state of the injection-locking system, and corresponds to a specific Δϕ according to Eqs. (1)-(2). However, Δλcav must be greater or equal to 0, and only when Δϕ falls in the range from -π/2 to cot−1α, the slaved laser can reach a steady state [1–4]. This corresponds to the curve bounded by point a and b on the ellipse. Outside this range, the slave laser is unlocked, and returns to the free-running wavelength λslave0, and thus Δλcav returns to zero. The possible state of the system, therefore, is shown by the solid blue trace.
3. How to use the ellipse model
In Fig. 1(b), we show how this graph can be used to provide an intuitive visualization of the optical injection-locking process and spectra. First, let the origin be λslave0, and thus, the x and y axes are relabeled λcavity and λmaster, respectively. We draw a line y = x as an eye guide. Next, we examine some special cases. For each case, we can draw a horizontal line y = λmaster at a certain set value, which intersects y = x and the solid blue curve. The x-coordinates of the two intersecting points represent the optical spectrum of λmaster and λcavity for this case.
For example, in case (i), λmaster = λi is much shorter than λslave0, the line y = λi does not intercept with the ellipse. The slave laser is not locked and stays at the origin λslave0. The y = λi line intercepts with y = x and the solid blue curve at two points, which are the master and free-running slave laser wavelengths. This is indeed the expected optical spectrum.
As λmaster increases, y = λmaster starts to intersect the ellipse, but the intercepts have negative x value, and the slave laser is still unlocked. The locking starts when λmaster increases further to y = λii, which intercepts with the ellipse at point a (case (ii)). The y = λii line shows what we expect to observe: the intercept with y = x is the master wavelength, whereas the intercept with x = 0 is the slave, now as cavity mode. Here we see that the cavity mode can never be bluer than the free-running slave laser wavelength.
As λmaster increases further, the slave stays locked (e.g. case (iii)). In each case, we can obtain the cavity mode from the intercept between y = λmaster and the ellipse, with the master being at the intercept between y = λmaster and y = x. When the slave laser is locked at point b, the slave laser’s cavity mode is actually on the blue side of the master laser’s wavelength. This continues until, the master wavelength increases beyond point b. This chart thus provides an intuitive insight onto the wavelength relationship between λmaster and λcavity for each unlocked state and locked states.
4. Analysis of OIL VCSELs based on the ellipse model
The ellipse is changed by varying the injection ratio and linewidth enhancement factor. Each possible state on the ellipse can be set by detuning. In the following sections, these parameters are modeled and investigated, allowing for extraction of trends in dynamics with respect to each. In addition, the behaviors of multi-mode OIL-VCSELs would also be analyzed with the ellipse model.
4.1 Wavelength detuning (Δλ)
The ellipse model can be used best to visualize the master laser wavelength and slave laser’s cavity mode wavelength. Figure 2(a) shows three different steady states under different detuning: blue detuning (a, Δλ<0), zero detuning (b, Δλ = 0), and red detuning (c, Δλ>0). Based on the ellipse model, it can be seen that the frequency spacing between master laser and slave laser’s cavity mode decreases with increasing wavelength of the master laser. This frequency spacing is approximately the resonance frequency of the slave laser. Hence the resonance frequency can be directly extracted from the ellipse model. To verify this, the frequency responses are simulated with small signal analysis [1–4], plotted in Fig. 2(b), and compared with that from the ellipse. For example, the spacing between the master and slave laser cavity mode is 0.93 nm under condition a (Δλ = −0.66 nm) shown in Fig. 2(a), in agreement with resonance frequency 116 GHz showing in blue curve in Fig. 2(b). The same holds true for other cases. This verifies the detuning dependence of the resonance frequency extracted from the ellipse graphical model.
4.2 Injection ratio (Rinj)
Besides wavelength detuning Δλ, the injection ratio Rinj is another important control parameter in OIL experiment. It expresses itself into the size of the ellipse. From Eq. (6), the area of the ellipse can be expressed in the following:
It is seen that the area of the ellipse is proportional to Rinj. This is illustrated in Fig. 3 showing the evolution of the ellipses under different Rinj. This essentially provides a full map to visualize the cavity behaviors under different OIL conditions. For example, to study the OIL dependence on injection ratio Rinj, one can draw a horizontal line at a specific Δλ value to intercept the different ellipses. It is immediately seen that λcavity red shifts with increasing Rinj.
4.3 Linewidth enhancement factor (α)
The linewidth enhancement factor is determined by the laser gain material and has large impact on the behaviors of OIL-laser: it determines the asymmetricity of the locking range. In the ellipse model, it exclusively determines the eccentricity e by the following formula:
Thus the ellipse model provides an experimental method to measure α by measuring its eccentricity:
Figure 4 shows the ellipses with different eccentricity corresponding to different α. The asymmetricity of the locking range is also illustrated: the blue locking range LRb stays the same under different α, while the red locking range LRr is proportional to.
4.4 Multimode slave VCSELs
Multimode VCSELs, either multi-transverse-mode or single-transverse-mode with two polarization modes, have interesting cavity mode behavior and greatly enhanced frequency response under OIL [8, 10]. However, conventional analytical treatment and numerical simulation on multimode OIL-laser is cumbersome. The ellipse model, on the other hand, provides a much simpler way to visualize this process. Figure 5 shows the ellipse model for a VCSEL with two transverse modes at 2 nm apart. The origin is set to be the free-running wavelength of the 1st order mode, and an ellipse (in blue) is set up. A second ellipse (in red) is set up for the fundamental mode, with its center shifts by 2 nm in both x and y axes. In this example, Rinj = 19 dB. The two ellipses are large enough so that they have an overlap.
Similar to Fig. 1(b), we analyze the cavity behavior when λmaster is swept from blue to red. When λmaster<λi, the slave laser is unlocked. At λmaster = λi, 1st order mode gets locked. As λmaster moves from λi to λiii, the 1st order mode remains locked in Fig. 5(a). Its cavity wavelength red shifts and follows the x-coordinate of the intercept between y = λmaster and the blue ellipse. The fundamental mode, on the other hand, also red shifts, keeping a constant 2 nm to the red side of the 1st order mode. To visualize this, the portion of the blue ellipse, bounded by point a and b, is replicated and red shifted by 2 nm. When it intercepts with y = λmaster, its x-coordinate represents the cavity wavelength of the fundamental mode. The case of y = λmaster = λii is a typical example at this stage. As λmaster moves beyond λiii, it falls outside the locking range of the 1st order mode, and both 1st order mode and fundamental mode tends to shift back to its free-running wavelength. However, in this particular example, y = λmaster = λiii is inside the locking range of the fundamental mode. The locking thus switches from 1st order mode to fundamental mode in Fig. 5(b). Correspondingly, the portion of the red ellipse, bounded by point c and d, is replicated and blue shift by 2 nm to represent the 1st order mode. The case of y = λmaster = λiv is a representative at this stage. When λmaster moves beyond λv, both modes are unlocked, and their cavities move back to free-running wavelength. The ellipse diagram provides an intuition to simplify the analysis of multimode OIL-laser, and its application applies to OIL-lasers with any number of modes.
5. Comparison with experimental results
To verify the theoretical modeling, a 1.55 μm slave single-mode VCSEL (SM VCSEL) is optically injection-locked by a master DFB laser, with various locking conditions. This SM VCSEL has a 5 μm aperture, 0.6 mA lasing threshold. The experimental setup is similar to that in Ref. 8. The VCSEL is biased at 4.8 mA to yield a −3.0 dBm free-running output power. A continuous-wave (CW) DFB laser is used as the master laser and the injection ratio is fixed at 8 dB. Polarization controller (PC) is used to match the master polarization with the VCSEL preferred polarization. In our experiment, the single-transverse-mode VCSEL emits two polarization of light, separated by 0.4 nm in wavelength. Light coming from the master laser was polarized at an angle ~45° with respect to both polarizations of the VCSEL . In OIL-VCSEL experiment, master wavelength is usually swept from blue to red to obtain stable locking condition. In general, there is a hysteresis and the locking range is much smaller if the detuning is reversed [12, 13]. In our experiment, with wavelength detuning Δλ changing from −0.21 nm to 0.99 nm, four regimes are observed in the experiment: four wave mixing (FWM: Δλ = −0.21~0.11 nm), locking on the 1st polarization mode (1st P: Δλ = 0.11~0.44 nm), locking on the 2nd polarization mode (2nd P: Δλ = 0.44~0.92 nm), unlocked (Δλ = 0.92~0.99 nm). Figure 6(a) shows the typical optical spectrum and small signal frequency response in these four regimes, all obtained in experiment. λ and λcavity is extracted from the measured spectrum data for each polarization mode and is fitted nicely into ellipses, shown in Fig. 6(b). The four wave mixing regime is observed at near the blue edge of the locking range of the 1st polarization mode. With increasing detuning, stable locking state is observed to follow the blue ellipse for the 1st polarization mode and subsequently the red ellipse for the 2nd polarization mode. The RF modulation resonance matches well with the cavity modes and can be read from the ellipse chart easily. From experimental result, the linewidth enhancement factor is extracted to be ~4.0 based on ellipse model measurement.
6. Discussion and summary
In summary, we established a very new ellipse model to describe the cavity mode behavior of OIL-VCSELs. This graphical tool provides an intuitive visualization of the impact of wavelength detuning, injection ratio, and linewidth enhancement factors on OIL-VCSELs. The ellipse model is also applicable to multimode VCSELs and shows excellent agreement with experiment. It will be an effective tool in the further analysis of OIL-VCSELs.
The authors wish to acknowledge the support of US Department of Defense National Security Science and Engineering Faculty Fellowship N00244-09-1-0013, and the National Basic Research Program of China (973 Program 2012CB315606), and the National Natural Science Foundation of China (NSFC) under grant 61071084, and the State Key Laboratory of Advanced Optical Communication Systems and Networks, China.
References and links
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