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We studied the dynamic behavior of laser-diode-pumped nonidentical thin-slice solid-state lasers, coupled face to face, with orthogonally polarized emissions. When such an incoherent mutual optical coupling was introduced, the coupled lasers exhibited slow fluctuations of transverse-mode patterns, and isolated lasers exhibited stable transversemode-patterns. When one laser exhibited nonorthogonal multi-transverse-mode operations without coupling, simultaneous random bursts of chaotic relaxation oscillations took place in both lasers over time with coupling. A plausible physical interpretation is proposed in terms of the simultaneous excitation of chaotic relaxation oscillations in both lasers through resonances that stem from interference-induced modulation of one laser at a swept beat frequency of the fluctuating nonorthogonal mode pair. Observed instabilities were well reproduced by numerical simulation of the model equation.
©2005 Optical Society of America
1. Introduction
During the past decade the dynamic properties of coupled laser systems have attracted much attention in nonlinear dynamics in the intriguing line of research involving synchronization in coupled chaotic oscillators [1]. Among a variety of coupled laser schemes, class-B lasers that are coherently coupled face-to-face have recently been studied [2–5] as the basis for a modified Lang-Kobayashi model of laser diodes subjected to delayed optical feedback. In such face-to-face systems, the injected field induces chaos; isolated lasers are dynamically stable, and the delay that is the propagation time between the two lasers affects the nonlinear dynamics. Stable localized synchronization of two semiconductor lasers coupled face-to-face operating in periodic and quasi-periodic regimes [2], synchronization of two identical semiconductor lasers coupled face-to-face in a chaotic regime [3], and theoretical considerations [4,5] have been reported. On the other hand, large-amplitude bursting has been observed in nature, in the fields of lasers [6], astrophysics [7], and biology [8,9].
In this paper we propose a novel scheme of orthogonally polarized nonidentical thin-slice solid-state lasers coupled face-to-face with a short delay, in which modulation is introduced into one laser. Here the interference-induced modulation of nonorthogonal transverse modes [10] is considered. We demonstrate synchronized chaotic bursting in two incoherently coupled lasers associated with a slow dynamic deformation of transverse-mode patterns, and we demonstrate experimental dynamic properties of “burst synchronization.” The resonant excitation of large-amplitude bursts in two lasers, which originates from modulation at a slowly swept beating frequency of the fluctuating nonorthogonal mode pair in one laser across relaxation oscillation frequencies, is proposed to explain the synchronized bursting.
This paper is organized as follows. An experimental setup, global and detailed optical spectra of coupled lasers, and spot dancing of transverse patterns are shown in Section. 2. In Section 3, experimental results for the burst synchronization of two lasers are presented, featuring joint time-frequency analysis (JTFA). Fast modulations without bursting are also demonstrated in Section 3 as experimental evidence of the field interference of nonorthogonal transverse modes formed in the thin-slice solid-state lasers. Numerical results that reproduce essential features of observed dynamics are shown in Section 4.
2. Experimental setup, oscillation spectra and destabilization of transverse mode
The experimental setup is shown in Fig. 1. We used two laser-diode-pumped 0.3-mm-thick LiNdP4O12 (LNP) lasers, operating at a 1048-nm wavelength. Both end surfaces of the LNP crystals were coated with dielectric mirrors for the LNP laser wavelength (1% transmission at faced surfaces; high reflection at other surfaces). Each laser was pumped by a laser diode with a microscope objective lens (N.A. =0.25) through a beam-shaping anamorphic prism pair. Defocusing the pump beam onto one laser crystal yielded asymmetric pumping due to aberrations, e.g., astigmatism, to excite nonorthogonal transverse eigenmodes in one laser [10], while TEM00-mode operation in another laser was realized by symmetric pumping. In short, the two lasers are not identical in their transverse mode patterns in the absence of coupling, as is shown in the inset of Fig. 1. With a lens of 20-cm focal length placed between two lasers separated by 80 cm, the beam from one laser was focused on the surface of the other laser. LNP lasers emit linearly polarized light along the pseudoorthorhombic c axis because of the crystal anisotropy. We mounted both crystals such that the c axis of one laser was perpendicular to that of the other to create incoherent mutual optical coupling between the lasers with no beating of their optical fields.
Fig. 1. Experimental setup of two LNP lasers incoherently coupled face-to-face. LD, laser diode; AP, anamorphic prism pair; OL, microscope objective lens; WM, multiwavelength meter; PD, photodiode; SF, scanning Fabry-Perot interferometer; DO, digital oscilloscope; P, polarizer. Far-field patterns of two lasers in the absence of coupling are shown. The associated movie file, indicating stable patterns over times, has a size of 1.43 Mbytes for laser A and 1.42 Mbytes for laser B, respectively.
In the absence of mutual coupling, each laser exhibited stable oscillations without any transverse pattern fluctuation. Optical spectra of the two lasers were simultaneously measured by multiwavelength meters (HP-86120B). Laser A exhibited two adjacent longitudinal modes [a, c] separated by 1.14 nm, and laser B oscillated in a single longitudinal mode b, as is shown in Fig. 2. Detailed oscillation spectra of the two lasers were simultaneously measured by scanning Fabry-Perot interferometers (free spectral range 2 GHz, resolution 6 MHz), and the two-mode oscillation of laser A and the single-mode oscillation of laser B were confirmed, as is shown in Fig. 3. Here the free spectral range of the scanning Fabry-Perot interferometer of 2 GHz frequency is much smaller than the frequency separation of 312 GHz between adjacent longitudinal modes a and c of laser A shown in Fig. 2. As a result, mode c in Fig. 3 appeared as a different interferometer peak, and the frequency separation between modes a and c is meaningless. Multiple external cavity modes separated by 187.5 MHz were not observed. Far-field patterns of both lasers shown in Fig. 1 were measured at the same position by replacing the multiwavelength meter on the left in the figure with a PbS infrared viewer. Therefore the beam diameter of laser B is larger than that of laser A.
Fig. 2. Global optical spectra of two lasers measured by a multiwavelength meter. Polarization directions of the two lasers are orthogonal.
Fig. 3. Detailed oscillation spectra of two lasers measured by scanning Fabry-Perot interferometers with 6-MHz resolution.
Then two lasers were coupled and an oscillation frequency of mode b of laser B was tuned to that of the first lasing mode a of laser A by changing the pump power of laser B as depicted by the arrows in Figs. 2 and 3. When the frequency tuning was complete, as shown in Fig. 4, each laser within the compound cavity exhibited low-frequency transverse-mode fluctuations on millisecond time scales, where the pump powers (threshold pump powers) were 120 mW (50 mW) for laser A and 100 mW (73 mW) for laser B, respectively. Example snapshots of far-field patterns of both lasers are shown in Fig. 5. A video of this figure is available from the associated multimedia. A pronounced fluctuation was observed in laser B. This instability may have resulted from dynamic deformations of transverse modes of the two lasers, through the cross-saturation of population inversions in the transverse direction by mutual light injection with different beam profiles, together with uncorrelated spontaneous emission noise in both lasers. No instability took place in either laser alone by coherent optical feedback from the surface of the other laser. This was confirmed by turning off the pump power to the other laser. This rules out the instability due to coherent delayed optical feedback that is generally observed in laser diodes subject to optical feedback.
Fig. 4. Scanning Fabry-Perot traces when the burst synchronization was the result of tuning the pump power of laser B.
Fig. 5. Snapshots of far-field patterns in lasers coupled face to face. (a) Laser A; the associated movie file has a size of 1.33 Mbytes. (b) Laser B; the associated movie file has a size of 1.94 Mbytes. A stable pattern without spot dancing was observed in each laser when the other laser was turned off.
3. Burst synchronization in lasers coupled face to face
3.1. Global view of burst synchronization
Under this low-frequency transverse mode-competition fluctuation regime, arising from the transverse cross-saturation of population inversions, we investigated the temporal behavior of the two lasers. By using polarizers, we selected each laser’s output. Large-amplitude chaotic bursts arising from relaxation oscillations were observed in both lasers over the course of temporal evolutions. The bursts occurred repetitively at random. Even in burst periods, two-and single-longitudinal-mode oscillations were maintained in laser A and in laser B, respectively.
Typical output waveforms of the two lasers indicating burst synchronization and magnified views are shown in Figs. 6(a) and 6(b). Here, “burst synchronization” means weak synchronization, where only the two lasers exhibit simultaneous bursting with correlated envelopes, whereas individual oscillations within an envelope need not be synchronized in amplitude or phase, as is shown later. It should be noted that these waveforms measured with low sampling rates show coarse-grained temporal evolutions.
Fig. 6. Global view of burst synchronization. (a) Long-term time series indicating synchronized bursting dynamics. (b) Magnified view of burst synchronization. (c) JTFA of (b). Pump powers 120 mW (laser A), 100 mW (laser B).
The results of the JTFA of Fig. 6(b) are shown in Fig. 6(c). From this global view of the JTFA signal, a swept frequency component is clearly seen in both lasers. This component decreases at first, leading to a burst (broadened bright zone in a power spectrum) as the frequency approaches the relaxation oscillation frequency of laser A (a horizontal bright line), as is shown below. The burst disappears as the swept frequency component separates from the relaxation oscillation frequency. Such a sweeping frequency component is not resulted from the beating of two lasers because the interference of orthogonally polarized optical fields never occurs. The beating of orthogonal transverse modes in both lasers is also ruled out because the entire beam was focused onto the detector.
3.2. Resonant excitation of chaotic relaxation oscillations
Similar modulations of lasers at frequencies higher than the intrinsic relaxation oscillation frequency have been demonstrated to appear experimentally and theoretically through interference of a pair of nonorthogonal transverse eigenmodes in thin-slice solid-state lasers with asymmetric end pumping, in which the avoided crossing of the eigenfrequencies of pair modes was observed when the control parameter was changed [10,11].
In the present face-to-face scheme, dynamic deformation of transverse modes occurs, indicating transverse pattern fluctuations; deformation comes about through cross-saturation of population inversions in the transverse direction of two thin-slice lasers when beams with different profiles are injected. When such instability takes place, as demonstrated in the video, the eigenfrequency difference (i.e., beat frequency) between nonorthogonal eigenmodes in the isolated laser B is not kept constant in time. If a pair of unstable nonorthogonal modes, whose beat frequency changes over times, is excited, laser B can be modulated at swept beat frequency f B. As laser B is modulated, intensity modulation is also brought about in laser A at beat frequency f B though incoherent coupling. It was demonstrated in the asymmetric pumping experiment that chaotic relaxation oscillation took place when the beat frequency approached the relaxation oscillation frequency of the laser [10,11] through resonances. Similarly, when the beat frequency is near the relaxation oscillation frequency as shown in Fig. 6(c), synchronized bursting is considered to have appeared in both lasers through incoherent coupling as shown in Figs. 6(a) and 6(b).
Detailed waveforms of the two lasers measured at 100 MS/s, the highest sampling rate of the digital oscilloscope (Tektronix 420A, 200-MHz bandwidth) that we used, and magnified views during bursting (region B) are shown in Fig. 7. It is obvious that the envelopes of both lasers exhibit similar variations and that individual spiking oscillations within the synchronized envelope of the burst are not correlated in either amplitude or phase. Correlation plots of the amplitudes and phases of the two lasers in different regions, A, B, and C, are shown in Figs. 8(a)-8(c), respectively. Here the phase is calculated from the Gabor’s analytic signal with Hilbert transformation of the experimental time series [12]. Phase correlations are seen in region A, and tailing (i.e., relaxation oscillation) in region C, but no correlation exists in the spiking region in B.
We carried out detailed JTFA by using the experimental time series. The JTFA patterns for both lasers corresponding to the time series shown in Fig. 7(a) are shown in Fig. 9. When the beat frequency f B in laser B approaches the relaxation oscillation frequency of laser A, f RO,A (of laser B, >f RO,B), a large-amplitude chaotic burst is excited simultaneously in both lasers through resonant excitation of chaotic spiking oscillations in both lasers through incoherent coupling. Depending on the sweep rate of f B, a synchronized burst was found to build up around the time when f B≃2f RO,A and f B≃f RO,B.
3.3. Fast modulation without bursting: evidence of modal interference
The physical interpretation of modulations in terms of the beating of a pair of nonorthogonal transverse modes in laser B requires experimental evidence because such a mode pair cannot be distinguished in the oscillation mode spectra measured by the scanning Fabry-Perot interferometers of 6 MHz resolution that we used. Therefore, we increased the pump power of laser B to 110 mW to increase the frequency separation of nonorthogonal modes as demonstrated in Refs. 10 and 11. As expected, a pair of nonorthogonal modes [3, 4] separated by f B (~40 MHz) appeared in optical spectrum of laser B as shown in Fig. 10, while global optical spectra shown in Fig. 2 were maintained in both lasers.
Fig. 8. Correlation plots of wave forms in regions A, B, and C in Fig. 7(a). Left, amplitude; right, phase.
Unlike chaotic bursting, however, intensity modulation at frequencies much larger than the relaxation oscillation frequency in laser B did not induce modulation in laser A when laser A was oscillating in the TEM00 mode. Instead, simultaneous fast modulations of both lasers were observed only when a nonorthogonal mode pair [1, 2] was also excited in laser A, as shown Fig. 10. A long-term time series for the two lasers and their JTFA patterns are shown in Fig. 11. This result is reasonable because cross-saturation dynamics cannot respond to injected light intensity variations much faster than the relaxation oscillation frequency, i.e., the response time of the laser. In this case swept beat frequencies of both lasers f B were kept sufficiently higher than relaxation oscillation frequencies (~1 MHz), and chaotic bursting was not observed over time.
Fig.11. Last modulations without bursting observed in the optical spectra shown in Fig. 10.(a) Long-term evolution of waveforms. Magnified views are shown in the insets. (b) JTFA patterns of (a).
4. Numerical simulation of excitation of chaotic burst through resonances
Let us investigate the observed burst synchronization numerically by using the model of lasers incoherently coupled face-to-face and subject to interference-induced modulation. To extract essential modulation dynamics, we employ spatial averages of the transverse electric field and population inversions similar to Refs. 10 and 11. The phenomenological laser equation is given by introducing the effect of modal interference-induced modulation [10,11] and incoherent coupling as follows:
The dynamic effect of coherent feedback to each laser with a short delay is neglected, as justified by the experimental observation mentioned in Section 2. Here w i is the relative pump power normalized by the first-mode threshold, N i is the normalized population inversion density of the ith mode, and E i is the spatial average of normalized field amplitude. We assume two longitudinal modes a (i=1), c (i=2) in laser A and a pair of nonorthogonal modes b (i=[3, 4]) in laser B. K is the normalized fluorescence lifetime ratio, τκ (κ=1/2τp, the damping rate of the optical cavity; τp is the photon lifetime). Tme is scaled by κ, β is the cross-saturation coefficient of population inversions of modes in each laser, γ is the cross-saturation coefficient of population inversions of modes in different lasers through incoherent coupling, g 3,4 is the field interference coefficient among nonorthogonal modes in laser B, ϕ3,4 is the phase difference between nonorthogonal modes, and ΔΩ3,4=Δω3,4/κ is the normalized angular frequency difference between nonorthogonal modes. We assume Gaussian white noise in the frequency, Dξ(t) in Eq. (9), where D is the noise strength, <ξ(t)>=0, and <ξ(t)ξ(t′)>=δ(t-t′).
Examples of numerical results indicating burst synchronization are shown in Fig. 12. Here we assume w
1=w
2=2.421 for laser A and w
3=w
4=1.655 for laser B. Other parameters are β=2/3, γ=0.1, and g=0.016. To allow us to extract the essential feature of the observed behaviors, a beat frequency ΔΩB/2π=ΔΩ3,4/2π=f
B/κ was artificially swept linearly in time and D=5×10-5, as shown in Fig. 12(a). Note that synchronized chaotic bursting builds up from sustained relaxation oscillations, indicated by 
The essential feature of burst synchronization has been well reproduced by the numerical simulation. To be specific, the peculiar nature of the observed burst synchronization is reproduced remarkably well by the simulation, in which laser A exhibits pulsations around the larger DC level, as indicated by the green left arrow, whose envelope is well correlated with that of lasers B, while laser B does spiking pulsations. It is interesting to note that the envelope correlation among two synchronized bursts was found to be lost when a common beat signal was introduced into both lasers simultaneously. The observed phenomenon would be interpreted in terms of “slow passage” dynamics [13,14] in nonlinear coupled-element systems, i.e., the response of coupled nonlinear oscillators subjected to a parameter swept to one element through resonances.
By introducing the higher frequency beating in both lasers, we could successfully reproduce the experimental wave forms shown in Fig. 11. Numerical time series and JTFA patterns for both lasers are shown in Figs. 13(a) and 13(b), where pairs of nonorthogonal modes i=[1, 2] and [3, 4] in Fig. 10 are assumed for lasers A and B, respectively. Here the term g 1,2 E 1 E 2 cosϕ1,2 was added to Eqs. (5)–(6), and a phase equation for ϕ1,2 similar to Eq. (9) was introduced. The beat frequency f B/κ was swept in time for both lasers, and D=5×10-5. It was confirmed that no modulation occurs in laser A (B) when field-interference-induced modulation is introduced only into laser B (A). This parallels the experimental observations.
Fig. 12. Numerical results indicating burst synchronization. The normalized beat frequency f B/κ in laser B was swept as shown in the upper figure. As for parameters, see the text. Normalized intensities of lasers A and B are given by + and +, respectively.
Fig. 13. Numerical result indicating fast modulation without bursting. (a) Numerical time series. (b) JTFA patterns of (a).
5. Dynamic characterization of burst synchronization
Finally, we examine an envelope synchronization measure to characterize the burst synchronization according to the method proposed by DeShazer et al. in their paper on synchronized bursting in two fiber lasers [15]. We introduce a threshold intensity I th and generate a new symbolic time series S(t), where S(t)≡1 for I(t) >I th and S(t)≡-1 for I(t) <I th. We define the level of burst synchronization, LBS, by the following cross-correlation:
where S A and S B are the burst time series of lasers A and B. N is the total number of sampled values. By definition, M=1 implies perfect envelope synchronization, M=-1 indicates complete antisynchronization, and there exists no relation on average between the bursts of two lasers for M=0 [15].
We calculated a time evolution of the level of burst synchronization (i.e., local level of burst synchronization; local LBS) by using a window width of 2048 data points and a moving window step of 256 data points. Examples of an experimental time series of burst synchronization, a symbolic time series, S A(t), S B(t), and the calculated local LBS, M(t), are shown in Figs. 14(a), 14(b), and 14(c), respectively. Note that the level of burst synchronization is found to change in a time corresponding to the time series: In the burst regions a, b, c, local LBS exhibits intermediate, low, and high values, respectively. The (global) level of burst synchronization for the whole time series of Fig. 14(a) was M=0.20.
Fig. 14. Level of burst synchronization. (a) Experimental time series. (b) Symbolic time series. (c) Evolution of the local level of burst synchronization.
6. Summary
In summary, the burst synchronization phenomenon induced by transverse-mode competition and field-interference-induced modulation has been demonstrated in laser-diode-pumped orthogonally-polarized thin-slice solid-state lasers incoherently coupled face-to-face.
The origin of synchronized bursting has been attributed to the interference of nonorthogonal modes formed in one laser, transverse cross-saturation of population inversions featuring slow spot dancing, and resonant excitation of chaotic pulsations over time in both lasers through incoherent coupling. Observed phenomena have been reproduced by numerical simulation of the model equation.
Observed dynamics should provide insight into the synchronization of general bursting phenomena in coupled nonlinear systems through a parameter swept across resonances.
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