In this paper we report a new method of photonic generation of microwave signals using a multi-transverse mode VCSEL subject to two-frequency optical injection. Numerical simulations show that double injection locking involving two transverse modes can be obtained in these systems. We show that the higher-order transverse mode is excited with a much larger amplitude than that of the fundamental transverse mode. The comparison with the case of a single-transverse mode VCSEL subject to similar two-frequency optical injection shows that multi-transverse mode operation of the VCSEL enhances the performance of the photonic microwave generation system. Broad tuning ranges, beyond the THz region, and narrow linewidths are demonstrated in our system. The maximum frequency of the generated microwave signals can be substantially increased if multimode VCSELs are used instead of single-mode VCSELs.
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Optical injection is a technique that is commonly employed to improve the performance of semiconductor lasers without modifying their design . Attention has been paid recently on the effects of optical injection on a special type of semiconductor laser: the vertical-cavity surface-emitting laser (VCSEL) [2–10]. VCSELs have demonstrated many impressive characteristics, including low threshold current, single-longitudinal mode operation, circular beam profile and easy fabrication in large two-dimensional arrays . Locking of the frequency of the injected (slave) VCSEL to the one of the injecting (master) laser has been shown for reduction of the VCSEL linewidth or for an enhancement of its modulation bandwidth [3–5]. Emission in multiple transverse modes is usually found in solitary VCSELs  as a result of spatial-hole burning effect . The effect of single-frequency optical injection on multi-transverse mode VCSELs has been also recently studied, showing that transverse mode selection or polarization switching can appear in these systems [7–10]. Recent experimental work has also demonstrated single-mode  and multimode  VCSEL-by-VCSEL optical injection locking.
Nonlinear dynamics of optically injected semiconductor lasers can also be used for photonic microwave generation [15–22]. Compared with conventional microwave generation by electronic circuitry with multiple stages of frequency doubling, photonic microwave signal generation have the advantages of lower cost, longer transmission distance, higher speed, low power consumption, less system complexity and the ability to generate tunable microwave signals with higher frequencies , . Photonic microwave generation find applications in radio-over-fiber (RoF) technology that holds great promise for 4G mobile communications systems . Both single-beam optically injected semiconductor lasers [15–19] and dual-beam, or two-frequency, optically injected semiconductor lasers [19–22] have been considered. Tunable narrow-linewidth microwave signals have been generated by using the period-one (P1) oscillations states that appear when the semiconductor laser is subject to a single-beam optical injection scheme . Photonic microwaves from the P1 oscillation of an optically injected edge-emitter semiconductor laser have reached frequencies beyond 100 GHz . The tuning range of the generated microwave signals is limited to several tens of GHz . Recently a very high frequency (121.7 GHz) microwave signal has been generated by using a dual-beam optically injected single-mode DFB semiconductor laser . Simultaneous injection of two laser beams transforms the slave laser into a dual-wavelength laser in such a way that the frequency of the generated microwave signal can be easily tuned by adjusting the frequency spacing between the two master lasers . Double injection locked states are observed when the slave laser is subject to strong optical injection by both master lasers in such a way that stable locking is also observed if only light of one of the master lasers is injected [21–22]. Experimental tuning ranges, limited by the bandwidth of the photodetector, around 20 GHz were demonstrated . Numerical calculations have shown that a wide continuous tuning range of more than 100 GHz is obtained by adjusting the detuning frequency of the two master lasers .
Characterization of the maximum microwave frequency achievable in a photonic microwave system is of interest because it sets the bandwidth limit of the system . For single-beam optically injected semiconductor lasers this limit is given by the free-spectral range (around 100 GHz) because the increase of the detuning frequency necessary to achieve higher microwave frequencies causes the laser to hop to the next mode . The highest frequency that can be generated in a dual-beam optically injected semiconductor laser is limited by the locking range determined by the maximum injection strengths of the two master lasers .
In this work we propose a new method of photonic generation of microwave signals using a multi-transverse mode VCSEL subject to two-frequency optical injection. The dynamics of the two transverse modes of a VCSEL is found using the model of  extended to consider dual-beam optical injection. In contrast to the experimental results of  in which optical injection is orthogonally polarized to the linear polarization of the transverse modes, we now consider the same linear polarization for both optical injections and transverse modes. Our calculations show that double injection locking involving the two transverse modes can be obtained in these systems. We show that the higher-order transverse mode is excited with a much larger amplitude than that of the fundamental transverse mode. The comparison with the case of a similar single-transverse mode VCSEL subject to the same two-frequency optical injection shows that the extra degree of freedom given by the multi-transverse mode operation of the VCSEL is useful for enhancing the performance of the photonic microwave generation system. In fact we obtain that the amplitude of the total power generated by multi-transverse mode VCSELs is much larger than that obtained with a similar single-transverse mode VCSEL subject to the same two-frequency optical injection. This shows that while for single-frequency injection, excitation of a second mode is detrimental for photonic generation of microwave signals , for two-frequency injection, excitation of a second mode is beneficial. Wide tuning ranges and narrow linewidths are also demonstrated in our system.
2. Theoretical model
The theoretical model is based on a spatially dependent dynamical model of a multi-transverse mode VCSEL subject to single-frequency optical injection . In this paper we extend it for considering a two-frequency optical injection. The simulated cylindrically symmetric weak index-guided structure, together with the complete details of the model can be found in . The model describes the complex amplitudes of two transverse modes with their two possible linear polarizations. Subscripts x and y will be used to denote the two orthogonal linear polarization directions. The appropriate transverse modes of the structure are the LPmn modes. Here we treat the case of VCSELs that can operate in the fundamental (LP01) and in the first order (LP11) transverse modes. Subscripts 0,1 will be used to denote the LP01 and LP11 modes, respectively. The equations describing the polarization and transverse mode behavior of the VCSEL with a two-frequency injected optical field read as 23–24], respectively. κr is the relative loss of the LP11 mode with respect to the LP01 mode. It determines the value of the injection current at which the LP11 mode begins lasing. I(r) represents a uniform current injection over a circular disc of 6 μm radius, and then I(r) = I if r<6 μm, and I(r) = 0, elsewhere. The normal gain normalized to the threshold gain, gij (i = 0,1, j = x, y), and gijk (i = 0,1; jk = xy,yx) are given by
In the two-frequency injection scheme the VCSEL is optically injected by a master laser 1 (ML1) and a master laser 2 (ML2), with optical frequencies ω1 and ω2, respectively. Two frequency detunings, Δω1 = ω1-ωth and Δω2 = ω2-ωth, with respect to the central frequency between the two polarizations of the fundamental mode ωth = (ω0x + ω0y)/2, appear in the Eqs. The injection terms only appear in the Eqs. for E0x and E1x, hence linearly polarized optical injection in the x-direction is considered for both master lasers. In this way if the solitary VCSEL emits only in the x-linear polarization, as it will be considered in this work, the directions of the polarizations of optical injections and VCSEL are parallel. Optical injection terms are also characterized by the injection strengths, kim, and the VCSEL roundtrip time, τin = 2L/vg, where vg is the group velocity. The injection strength kim (i = 0,1, m = 1,2) is given by25]. In this work we choose the kim values in relation to what should be expected when light from a multi-transverse mode VCSEL is injected into a similar device. k01 (k12) is related to the spatial overlap integral between the profiles of the fundamental modes (the higher-order modes) of master and slave lasers. Then a reasonable choice is k01 = k12 = ks, where ks is the injection strength between similar transverse modes. k02 (k11) is related to the spatial overlap integral between the profiles of the fundamental mode of the slave and the higher order mode of the master (higher-order mode of the slave and the fundamental mode of the master) VCSEL. In this way we choose k02 = k11 = kc, with kc ≤ ks, where kc is a “crossed” injection strength between transverse modes with different order. The frequency splitting between the orthogonal polarizations of the LP01 mode, 2γp0/(2π), between the orthogonal polarizations of the LP11 mode, 2γp1/(2π), and between the two transverse modes with the same polarization, γptr/(2π), are obtained from the calculation of the waveguide modes via the Helmholtz equation [23–24]. We have chosen the values of refractive indexes ncore,x, ncore,y and ncladd in such a way that 2γp0/(2π) = 10 GHz and γptr/(2π) = 63 GHz. Spontaneous emission noise processes are modeled by the terms ξ ± taken as complex Gaussian white noise sources of zero mean and delta-correlated in time. In the noise terms we consider integrated over the active region carrier distributions, and . The rest of the parameters that appear in the Eqs. and their meaning are specified in the Table 1 . Time and space integration steps of 0.01 ps and 0.12 microns, respectively, have been used. The boundary conditions for the carrier distribution are taken as N(∞,t) = 0, n(∞,t) = 0. The initial conditions correspond to a below threshold stationary solution, i.e to I = 0.1Ith, where Ith is the threshold current.
3. Two-frequency optically injected single-mode VCSELs
In this section, we present the results corresponding to the two-frequency optical injection on a VCSEL emitting only in the fundamental transverse mode. We have chosen a value κr = 10 to assure LP01,x mode operation. Results are given in terms of the separation between the frequencies of the two master lasers, Δf = (ω2-ω1)/(2π), the frequency detuning of ML1 with respect to the frequency of the LP01,x mode, Δν, and the injection strength κs. We show in Fig. 1(a) the dynamical evolution of a free-running single-transverse mode VCSEL when switched-on at time t>0 with a I = 1.8Ith bias current value. The VCSEL emits in the steady state in the LP01,x mode with a single peaked RF spectrum appearing at the relaxation oscillation frequency. Spectra are calculated after a transient time of 14 ns.
Figure 1(b) shows the results for an injection strength of κs = 10−2, Δf = 100 GHz, and with the frequency of ML1 just at the LP01,x mode frequency. An almost sinusoidal time trace is obtained for the power of LP01,x mode, that is the only mode with nonnegligible power. The frequency of this sinusoidal modulation is Δf. The optical spectrum consists of two well defined peaks at the ML1 and ML2 optical frequencies and the RF spectrum has a very narrow peak at Δf, the frequency separation between ML2 and ML1. This situation is similar to the double injection locking that has been obtained using a single mode DFB laser subject to dual-beam injection  and in simulations using a single-mode rate equation model .
4. Two-frequency optically injected multi-transverse mode VCSELs
We now present the results corresponding to two-frequency optical injection on a multi-transverse mode VCSEL. This laser is exactly the same than the single-transverse mode VCSEL but changing the parameter κr to a smaller value (κr = 1.022) for which both transverse modes have rather similar losses. Results obtained for the solitary multimode VCSEL (κs = κc = 0) are plotted in Fig. 2(a) . In this case the VCSEL shows a steady-state in which both transverse modes, LP01 and LP11, are excited in the x-direction. The steady-state total power in Fig. 2(a) is 1.7, a similar value to the value, 2.1, that was obtained for Fig. 1(a). The power in the LP11,x mode is only slightly larger than that of LP01,x. Fig. 2(a) shows that the x-polarized RF spectrum has two peaks that appear due to the multi-transverse mode character of the VCSEL . The multimode character of the system is also clear from the optical spectra of Fig. 2(a). The separation between LP01,x and LP11,x, similar to the separation between LP01,y and LP11,y, is around 63 GHz.
Figure 2(b) shows the results of the multi-transverse mode VCSEL subject to two-frequency optical injection. The levels of the injection strengths, Δf, and Δν are equal to those considered in the single-transverse mode case (Fig. 1(b)). In this way we can compare the performance of microwave generating systems using single and multi-transverse mode VCSELs. Similarly to Fig. 1(b), Fig. 2(b) also shows the double injection locking phenomenon. Again, two well defined peaks at the frequencies of ML1 and ML2 are observed in the x-polarized optical spectrum resulting in a RF spectrum with a narrow peak at the Δf frequency. But Fig. 2(b) also shows an interesting and novel feature when compared to Fig. 1(b): the high-order transverse mode LP11,x is excited with a much larger amplitude than that of the LP01,x mode. The power of both transverse modes oscillate with a phase difference near π/2. In this way the variation of the total power is nearly sinusoidal, as shown in the time traces and RF spectrum of Fig. 2(b). Interestingly, the amplitude of the total power is much larger than the one obtained with the equivalent single-transverse mode VCSEL case illustrated in Fig. 1(b). This shows that the amplitude of the microwave signal generated by two-frequency optical injection is enhanced if multi-transverse mode operation in the VCSEL is considered instead single-transverse mode operation.
5. Comparison between single and multi-transverse mode cases
In this section we make a detailed comparison between the two-frequency optical injection in single and multi-transverse mode VCSELs. Figure 3(a) shows the results of the multi-transverse mode VCSEL subject to two-frequency optical injection when increasing Δf and with similar conditions to those of Fig. 2(b). Sinusoidal time traces are obtained for the power of LP01,x, LP11,x modes and total power. The amplitude of the oscillations decreases as Δf increases. We have also included the time trace corresponding to the single-transverse mode VCSEL to show that the enhancement due to multi-mode operation is maintained. We show in Fig. 4(a) the peak-to-peak amplitude of the total power as a function of Δf. Results for single and multi-transverse mode VCSELs, Asm and Amm respectively, are included. Also results for two injection strength levels are shown. All the cases reported in this fig. have an almost sinusoidal variation of the total power. Peak-to-peak amplitudes decrease as Δf increases. An increase of the injection strength κs leads to larger peak-to-peak amplitudes for both, single and multi-transverse mode cases. This is also illustrated in Fig. 3(b) in which κs is increased with respect to Fig. 3(a). For all the values of Δf and κs the amplitude of the oscillation obtained with multi-transverse mode VCSELs is larger than that obtained for single-transverse mode VCSELs as it is shown in Fig. 4(a). This enhancement is quantified in Fig. 4(b), in which the ratio between peak-to-peak amplitudes of the total power obtained for multi and single-mode devices, Amm/Asm, is plotted as a function of Δf. A maximum value of 2.6 is obtained for κs = 10−2 at Δf = 63.4 GHz, value that is very near the transverse mode separation. A not so well defined maximum is obtained for κs = 3 10−2 at Δf = 100 GHz. These values are near the 63 GHz transverse mode separation indicating that maximum enhancement is obtained when the optical frequencies of ML1 and ML2 are close to the frequencies of the free-running LP01,x, and LP11,x modes, respectively.
Figure 4(a) also shows that the largest variations of peak-to-peak amplitudes with Δf occur in the microwave region (<300 GHz). In Fig. 4 we have also considered a Δf range that goes beyond the microwave range. In this region the generated radiation has appreciable amplitude that increases as κs is increased. The amplitude of this modulation slightly decreases when Δf is in the THz region. An example of the dynamical evolution obtained for Δf = 500 GHz, a value larger than those of the microwave region, is shown in Fig. 5(a) . Single and multimode VCSELs are able to respond at 500 GHz with appreciable amplitudes. The response of multimode VCSELs is such that the power of both transverse modes oscillates nearly in phase.
Figure 6(a) shows the dependence of Amm and Asm on the injection strength κs. Results are also given in terms of the injection ratio defined as the ratio of the injected optical power by the m-master laser, (Pinj,0m + Pinj,1m) versus the output power of the free-running multimode VCSEL. Both Amm and Asm increase with κs . Double injection locking is observed when κs is larger than 2.5 10−3 and 2.9 10−3 for single and multimode VCSELs, respectively.
The curve for multi-transverse mode VCSELs has a small step near κs = 10−2 that is due to a sudden change of the phase difference between the power time series of both transverse modes: that phase difference is near π if κs ≤9 10−3 and changes to a value near π/2 at κs = 10−2. Above the κs value for which the step in Amm is observed, the ratio Amm/Asm, has an almost constant value of 2.6, as it is shown in Fig. 6(b).
Figure 7(a) shows the dependence of Amm and Asm on the injection strength κc for two different values of κs. Both Amm and Asm increase with κc. Note that all amplitudes vanish when κc→0. For the single-mode VCSEL with double injection this means that the VCSEL recovers the locked state with single-frequency optical injection characterized by a constant value of the power. For the two-mode VCSEL the situation can be explained as follows. If only injection from ML1 (ML2) with no coupling to the LP11,x (LP01,x) mode is considered, for instance κ01 = 3 10−2, κ12 = κc = 0 (κ12 = 3 10−2, κ01 = κc = 0) the optical spectrum of the VCSEL consists on a single peak at the ML1 (ML2) frequency, the power of LP01,x (LP11,x) is constant and the RF spectrum is flat. If injection from ML1 and ML2 is considered such that κ01 = κ12 = 3 10−2, κc = 0, the optical spectrum of the VCSEL consists on two peaks at the ML1 and ML2 frequencies, the powers of both modes, LP01,x and LP11,x, are constant and the RF spectrum keeps on being flat. A nonzero value of κc is enough for the RF spectrum to develop a peak and for both modes to oscillate at the Δf frequency. This shows that the reason for the transverse modes power oscillations is the nonzero value of the κc injection strength.
Figure 7(a) shows that for large values of κs, Amm > Asm. However, for κs = 10−2, Amm≤Asm until the curve for multi-transverse mode VCSELs has a small step near κc /κs = 0.42. This step is again due to a sudden change of the phase difference between the power time series of both transverse modes: that phase difference is near π if κc/κs ≤0.4 and changes to a value near π/2 at κc/κs = 0.45. Figure 7(b) shows the ratio Amm/Asm corresponding to data of Fig. 7(a). Larger enhancement of Amm with respect to Asm can be obtained when κc is small providing that κs is large enough. This situation is illustrated in Fig. 5(b) for which a value of Amm/Asm = 4.3 is reached.
The dependence of Amm and Asm on the bias current of the VCSEL is analyzed in Fig. 8(a) . Both quantities increase with the bias current. However the ratio Amm/Asm slightly decrease in a linear way with I. For instance, when κs = 3 10−2, it goes from 2.6 at I/Ith = 1.2 to 2.2 at I/Ith = 8. We have also analyzed the effect of the frequency detuning, Δν, on the dynamics of the system. Dynamics obtained for the multimode case when Δν = 5 GHz and Δν = −5 GHz are very similar to those obtained when Δν = 0 GHz, providing that the system is in the double injection locked regime. This indicates that there is an appreciable range of Δν for which the enhancement of the amplitude of the generated microwave signal due to high-order transverse mode excitation does not depend on the Δν value. The relative phase between the power time series corresponding to the LP01,x and LP11,x modes in the double-injection locked multi-transverse mode VCSEL can change depending on the injection conditions. Comparison between Fig. 3(a) and Fig. 3(b) indicates that it decreases as the value of the κs injection strength increases.
We have also calculated the RF linewidths of the signals generated by double-injection locking of multi-transverse mode VCSELs. In this paper most spectra have been calculated with a temporal window of 1.05 μs and with a maximum frequency of 2000 GHz. Then linewidths are measured with a 0.95 MHz frequency resolution. We have considered in our calculations a finite linewidth of the optical spectrum of both master lasers. This effect has been included by considering phase fluctuations in the two external signals as in Ref . Equal linewidths and independent phase fluctuations have been considered for both master lasers. 3-dB RF linewidths are shown in Fig. 8(b) as a function of the linewidth of the optical spectrum of one of the MLs. Narrow linewidths, in the MHz range, are demonstrated like in [21–22]. RF linewidth increases with the injection linewidth. The RF linewidth is almost twice the injection linewidth of the ML if this ≥ 1 MHz. In all the RF spectra the obtained sidemode suppression ratio has been more than 50 dB, like in . The second largest peak is the first harmonic corresponding to the peak that appears at Δf.
6. Discussion and conclusions
Peak-to-peak amplitudes in this work have been given in the arbitrary units used in the spin-flip model . An idea of the magnitude of the power corresponding to the peak-to-peak amplitudes obtained in this work can be obtained by taking typical values of the power emitted by a VCSEL with 12 μm diameter like the one simulated in this paper. For instance, for a typical value of the total power~0.4 mW for I/Ith = 1.8, Amm (mW) = 0.235Amm (a.u.). For the conditions of Fig. 2(b) this corresponds to a 1.1 mW peak-to-peak amplitude for 1.17 mW of power injected by each ML.
The model that we have used includes the polarization degree of freedom of transverse modes. In our case the role played by y-polarized transverse modes has been negligible because we have considered an optical injection that is parallel to the x-polarization of the solitary VCSEL. Preliminary work using an extension of this model to consider orthogonally polarized optical injection indicates that the results obtained in this work are maintained. This result is of interest to show that the proposed microwave signal generation system is independent on the polarization of the master lasers. In our calculations we have assumed that the κc parameter that represents the “crossed” injection strength between transverse modes of different order is equal for LP01 injected into LP11 and for LP11 injected into LP01 mode. Since this is a simplification in our model, future work is intended to know if the results obtained in this work are maintained when asymmetric “crossed” injection strengths are considered. Also in our calculations we have considered a high value of the γs parameter such that the results of the simulations should be the same if the carrier difference n is set equal to 0 (i.e., if integration of the equations with only the total carrier density is performed). Future work is also planned to investigate the influence of smaller values of the γs parameter, and hence of the spin of the carriers on our results.
Figure 4 shows that a typical increase in the amplitude obtained with multimode VCSELs with respect to single-mode VCSELs is between 2 and 3. We discuss now the corresponding increase in the maximum frequency of the generated microwave signals. Assuming that amplitudes larger than a value A are required, we calculate the maximum value of the microwave frequency obtained with the single-mode VCSEL, Δfmax,sm, such that Asm >A if Δf <Δfmax,sm. We also calculate the corresponding quantity for the multimode VCSEL, Δfmax,mm, given by Amm >A if Δf <Δfmax,mm. Figure 9 shows the results for A = 4 a.u. as a function of the injection strength or the injection ratio. The maximum frequency increases significantly when using multimode VCSELs instead of single-mode VCSELs. For instance the maximum frequency increases from 193 GHz to 578 GHz for an injection ratio of 14 dB.
Summarizing, in this work we have made a theoretical study of the dynamical properties of single and multi-transverse mode VCSELs when they are subject to two-frequency optical injection. We have focused our analysis in the double injection locking observed at large injection strengths, useful for photonic microwave signal generation. Numerical simulations of single and multi-transverse mode VCSELs have shown that the double injection locking can be obtained when these devices are subject to two-frequency optical injection. For the case of multimode VCSELs we have shown that the response of multi-transverse mode VCSELs under two-frequency optical injection is larger than that obtained with similar single-transverse mode VCSELs. In this way high-order transverse mode excitation enhances the performance of the photonic microwave generation system based on two-frequency optical injection. This shows that while for single-frequency injection, excitation of a second mode is detrimental for generating microwave signals , for two-frequency injection, excitation of a second mode is beneficial. Narrow linewidths are demonstrated in our system. Broad tuning ranges, beyond the THz region, have been determined in terms of the characteristics of the optical injection and of the transverse mode properties of the VCSEL. The maximum frequency of the generated microwave signals can be substantially increased if multimode VCSELs are used instead of single-mode VCSELs.
The authors acknowledge support from the Ministerio de Ciencia e Innovación under project TEC2009-14581-C02-02. A. Q. was supported in part by the Fondo Social Europeo (FSE) under the programme Junta de Ampliación de Estudios (JAE-predoc).
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