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Abstract
Synchronization of chaos among a single pair of transverse modes featuring intermode dynamical non-independence, resulting from cross-saturation of modal population inversions and coherent modal field coupling, is explored in a thin-slice solid-state laser with coupled orthogonally polarized transverse modes operating in quasi-locked states. The chaotic signals that are created in one of the pairing modes (sender) by self-mixing modulations are shown to be transferred solely to the partner mode (receiver) among all the other modal fields with different polarizations. The experimental results are reproduced in a simulation of a model equation of coupled laser modes subjected to self-mixing modulations.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Synchronization phenomena of chaotic oscillators [1] are encountered in physical, chemical, and biological systems, as well as in laser systems. In the context of coherently coupled chaotic oscillators, there are a number of different interpretations of chaos synchronization, such as master-slave synchronization and synchronization based on mutually coupled oscillators. In laser physics, a variety of chaos synchronizations have been reported in coherent coupling of solid-state lasers [2–7] and semiconductor laser diodes (LDs) [8–13]. Chaotic oscillations in solid-state lasers are brought about by a sinusoidal pump or loss modulation at the frequency close to the relaxation oscillation frequency and chaos synchronization is established in the form of spatially coupled phase-locked laser arrays [14] or unidirectional injection locking in the master-slave configuration. While, chaotic oscillations in LDs occur by delayed optical feedback, which can be interpreted in terms of delay-driven oscillators [15], and chaotic lasers are coupled unidirectionally or bidirectionally for synchronization. In general, chaos occurs at relaxation oscillation frequencies in the range below few MHz in solid-state lasers and above few GHz in LDs, respectively. Most recently, chaos synchronization in LDs has been investigated intensively toward high bit rate secure optical communications and various schemes of chaos synchronization have been reported [16–18].
Besides, polarization synchronization was reported in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers (VCSELs), where chaos synchronization occurred when the polarization the master laser was perpendicular to that of the free-running slave laser [19,20]. Vectorial chaos synchronizations have been reported in self-pulsating oxide-confined VECSELs proposing a novel encryption scheme, where the phase of the vectorial field is modulated and bit rates are not limited by relaxation oscillation frequency [21].
Different forms of chaos synchronization owing to incoherent coupling through cross-saturation of population inversions among modes have been demonstrated in a modulated multimode laser, depending on the pump power [22]. Chaos synchronizations based on the similar cross-saturation mediated incoherent coupling of orthogonally polarized modes have been reported in a dual-polarization laser [23].
On the other hand, cylindrical polarized vector laser fields have been created from a pair of orthogonally polarized Hermite-Gauss modes, HG0,1 and HG1,0, by their coherent superposition, i.e. transverse mode locking [24,25]. Spatially coherent vector fields forming a higher-order transverse structure were also demonstrated in a uniaxial solid-state laser [26] and such vector fields were shown to be formed by the transverse mode locking of a pair of orthogonally polarized higher-order Ince–Gaussian modes [27].
In this paper, self-organized collective chaos synchronization is reported in the regime of quasi-locked states of different orthogonally polarized transverse eigenmodes in a thin-slice solid-state laser subjected to self-mixing modulations based on the combined effect of coherent modal coupling and incoherent cross-saturation dynamics among modes. A high degree of chaos synchronization, with amplitude correlation coefficients R > 0.99, is found to occur for a particular pair of polarized transverse modal fields, which are embedded in polarization vector fields formed from coupled transverse eigenmodes. Synchronized chaotic relaxation oscillations have been demonstrated using different schemes, i.e., self-mixing modulation of the total output or one of the pairing modes.
2. Experimental results
2.1 Experimental setup
The experimental setup is shown in Fig. 1(a). A nearly collimated lasing beam from a laser diode (wavelength: 808 nm) was passed through an anamorphic prism pair to transform an elliptical beam into a circular one, and it was focused onto a thin-slice laser crystal by a microscopic objective lens of numerical aperture NA = 0.5. The laser crystal was a 3 mm-diameter clear-aperture, 1 mm-thick, 3 at%-doped c-cut Nd:GdVO4 whose end surfaces were directly coated with dielectric mirrors M1 (transmission at 808 nm > 95%; reflectance at 1064 nm = 99.8%) and M2 (reflectance at 1064 nm = 99%). Lasing optical spectra were measured by a multi-wavelength meter (HP-86120B; wavelength range, 700–1650 nm) for obtaining global views and a scanning Fabry-Perot interferometer (SFPI) (Burleigh SAPLUS; 2 GHz free spectral range; 6.6 MHz resolution) for measuring detailed structures. The polarization resolved waveform and power spectrum measurement scheme is boxed by dashed line.
Fig. 1. (a) Experimental apparatus. LD: laser diode, AP: anamorphic prism pair, OL: objective lens, BS: beam splitter, M: mirror, P: polarizer, MWM: multi-wavelength meter, SFPI: scanning Fabry-Perot interferometer, PD: photo-diode, DO: digital oscilloscope, SA: spectrum analyzer. (b) Input-output characteristics and pump-dependent DPO far-field intensity patterns.
In the case of thin-slice solid-state lasers with coated end mirrors, abbreviated as TS3Ls hereafter, a stable resonator condition is achieved through the thermally induced lensing effect [14], and the input-output characteristics as well as the transverse and longitudinal mode oscillation properties depend directly on the focusing condition (e.g., spot size and shape) of the pump beam on the crystal due to the mode-matching between the pump and lasing mode profiles [28]. In the experiment, the pump-beam diameter was changed by shifting the laser crystal along the z-axis, as depicted in Fig. 1(a). The pump spot size, wp, increased as the laser crystal was shifted away from the pump-beam focus along the z-axis (i.e., z > 0). When wp exceeded about 80 µm, dual-polarization oscillations (DPOs) were observed, starting from TEM00 at the threshold and leading to various transverse modes. When the pump position was precisely changed by moving the laser sample along the x-axis or y-axis with an accuracy of 10 µm and a small tilt of the 1-mm-thick cavity of $|a |\le $ 1.5° was made with an accuracy of 0.3° as depicted in the inset, several DPO transverse modes appeared at fixed wp [29].
2.2 Quasi-locked state in dual-polarized oscillations and stationary characteristics
Typical input-output characteristics are shown in Fig. 1(b). Here, wp = 80 µm, DPO starts at a threshold pump power of Pth = 85 mW in the Hermite-Gaussian HG0,0 modes for orthogonal polarizations. At pump powers P greater than 170 mW, the horizontally polarized mode undergoes successive structural changes, featuring the Ince-Gaussian IGe2,2 mode and ‘doughnut’ mode, while the vertically polarized HG0,0 mode is preserved. The DPO characteristics shown in Fig. 1(b) were reproducibly obtained by adjusting the crystal and pump positions [x, y, z] as well as the tilt angle. Assume that the transverse mode locking occurs in the form of u = u1 + r eiΔϕu2, where u1, u2 are eigenfuctions of pump-dependent DPO eigenmodes shown in Fig. 1(b), where r = ${E_2}/{E_1}$ is the modal field amplitude ratio and Δϕ is their phase difference [27]. Although numerically reconstructed polarization-resolved patterns were resemble to the experimental results, complete transverse mode locking was not established as will be discussed below.
Fig. 2. (a) Polarization-dependent far-field patterns (see Visualization 1). (b) Optical and power spectra of DPO eigenmodes and the total output. P = 172 mW.
The output DPO laser beam was passed through a polarizer, and the polarization-dependent changes in the far-field patterns were observed. Experimental results corresponding to the orthogonally polarized HG0,0 pair are shown in Fig. 2(a). The global oscillation spectrum measured by the multi-wavelength meter indicated a single longitudinal mode at a wavelength of λ = 1065.58 nm, while the adjacent longitudinal mode separated by Δλ= λ2/2nL = 0.258 nm was not observed (L: crystal thickness, n: refractive index).
Detailed optical spectra corresponding to orthogonally polarized HG0,0 mode operation, as measured by the SFPI, are shown on the left of Fig. 2(b). Apart from the complete transverse mode locking of orthogonally polarized modes, as demonstrated in a c-cut Nd: GdVO4 laser with the semi-confocal external cavity, where the pure single-frequency operation occurs [27], peculiar, strongly asymmetric optical mode spectra, each consisting of dominant peaks and weak subsidiary peaks, appear for the orthogonally polarized modes. Despite the degenerate HG0,0 modes in the cold cavity, thermal birefringence in c-cut vanadate crystals [30,31] is evident in TS3L, unlike external cavity lasers whose transverse modes are predominantly determined by the cavity configuration [28]. Moreover, distortion of the transverse mode, presumably due to aberration/astigmatism of a thermal-induced lens, can be seen, e.g., the ‘elongated’ HG0,0 profile in Fig. 2(a). The resulting frequency detuning among orthogonally polarized modes would prevent complete transverse mode locking. In fact, oscillation frequency detuning on the order of 400 MHz remains in the present case.
However, it will be shown later that the observed orthogonally polarized transverse modes are not independent, and they form partially coherent fields, namely, a quasi-locked state, through mutual nonlinear interaction of closely spaced orthogonally polarized HG0,0 modes. The power spectra of the orthogonally polarized eigenmodes are shown on the right of Fig. 2(b), together with that for the total output. Here, each power spectrum is the average spectrum of 100 power spectra, which were obtained for the minimum spectral frame update interval of 160 µs. The f1-peak corresponds to the fundamental relaxation oscillation noise, while the f2-peak arises through transverse cross-saturation of population inversions of orthogonally polarized modes [32]. Note that the lower frequency peak at f2 is suppressed in the total output. This implies inherent antiphase dynamics featuring intermode dynamical non-independence in multimode lasers, where the total output is self-organized to behave as a single mode laser that exhibits a noise peak only at f1 and there are no peaks below f1 [33,34].
Figure 3(a) shows noise intensities at f1 and f2 as a function of the polarizer angle, θ. It should be noted that noise intensities at the lower relaxation oscillation frequencies, f2, which result from the cross-saturation dynamics of modal population inversions, rapidly decrease as θ approaches two critical angles ${\pm} $θc, which are symmetric with respect to the x and y axes. The corresponding power spectra are shown in Fig. 3(b). It is apparent that the f2 peak due to the inherent antiphase dynamics is strongly suppressed below the measurement-system noise level for a pair of polarized transverse fields, similar to the case of the total output spectrum in the right column of Fig. 2(b). While, complete suppression of the frequency noise peaks below f1 does not occur for such a pair of transverse modes with different polarizations merely through the incoherent effect of cross-saturation of population inversions in multimode lasers including DPOs [32–34]. Consequently, this phenomenon is considered to result from coherent phase-sensitive interactions of modal fields, reflecting the influence of transverse mode-locking.
Fig. 3. (a) Polarization-dependent relaxation oscillation intensity. (b) Power spectra for a pair of transverse modes. (c) Noise-driven modal and total output fluctuations. Pump power, P = 172 mW.
On the other hand, random modal relaxation-oscillation fluctuations of a pair of polarized modal outputs, which are driven by mode-partitioned spontaneous emission noise, and the total output were found to exhibit a high degree of synchronization. Part of the output beam was divided into three beams for simultaneous measurement of the two signals in different polarizations and the total output. It is apparent that noise-driven relaxation oscillations of a pair of polarized modal outputs and the total output are synchronized, where their amplitude correlation coefficient given by R = Σi (I1, i - <I1>)(I2, i - <I2>)/[Σi (I2, i - <I2>)]1/2[Σi (I1, i - <I1>)]1/2 was as large as 0.992. Here, simultaneous measurements were carried out by three InGaAs photodiodes (New Focus 1811, DC-125 MHz) followed by a digital oscilloscope (Tektronics TDS 3052, DC-500 MHz). Zoomed in views of waveforms are shown in Fig. 3(c).
2.3 Chaos synchronization in vector lasers subjected to self-mixing modulation
Now, a highly sensitive self-mixing modulation experiment [6] owing to the large fluorescence-to-photon lifetime ratio [35] was carried out to investigate the response of the vector laser, where 50% of the total beam was focused onto a rotating Al-cylinder as shown in Fig. 1(a). Here, the effective interference is expected among each linearly polarized component of randomly polarized scattered fields and the corresponding polarization resolved lasing field pattern with the same polarization [36].
When the Doppler-shift frequency, fD = 2v/λ(v: moving speed along the laser axis), was tuned around the relaxation oscillation frequency, chaotic relaxation oscillations were easily brought about. The antiphase chaotic relaxation oscillations of DPO eigenmodes, together the corresponding power spectra, are shown in Fig. 4(a). The chaotic attractor of each mode inherits the antiphase dynamic character of the corresponding stationary state shown in Fig. 2(b), and the unsynchronized chaotic relaxation oscillations are obvious. While, the generic nature of a single pair of polarized transverse fields is preserved such that the pair behaved as a coherent mode, exhibiting a high degree of chaos synchronization as for a single pair of transverse modes. Results are shown in Fig. 4(b), together with the corresponding power spectrum without f2 peak and correlation plots indicating R = 0.993. Chaos synchronization was readily suppressed when θ was shifted slightly from θc.
Fig. 4. (a) Chaotic oscillation of DPO eigenmodes. (b) Synchronized chaotic oscillation of a single pair of polarized transverse modes. P = 172 mW.
When complete transverse mode-locking is established, the vector laser behaves as an ‘all-in-one’ single mode laser and all polarization-resolved chaotic waveforms are automatically synchronized [27]. In the quasi-locked states, on the contrary, only a single pair of polarized transverse fields is found to exhibit chaos synchronization.
Moreover, a high degree of chaos synchronization occurred between the part of the total output that did not pass through the polarizer and either of the paired modes. This implies that the chaotic regime inherits the self-organized synchronization of noise-driven relaxation oscillations among a pair of transverse modal outputs and the total output without self-mixing modulations.
2.4 Sender-receiver relationship in polarization dependent self-mixing modulation
Such a dynamical non-independence of a single pair of transverse modal outputs and the total output in the chaotic regime is inherent to the quasi-locked states; a high-degree of chaos synchronization between a single pair of transverse modal outputs and the total output occurred when one mode was subjected to self-mixing modulation, where part of the polarized output along +θc (or -θc) was focused on the rotating cylinder (see the set-up enclosed in the dashed line in Fig. 1(a)). The example shown in Figs. 5(a)-(b) indicates a high degree of chaos synchronization, with a correlation coefficient of R = 0.995. In other words, the modulated mode acts as an information ‘sender’, while the partner mode acts as a ‘receiver’ and the chaotic signal is transferred to the partner mode, which is embedded in the transverse vector lasing fields, forming a coherent field together with the modulated mode. Figure 5(c) shows the change in the amplitude correlation coefficient, R, between a pair of transverse vector fields when the polarization direction of the probe beam (i.e., receiver) was varied. This sender-receiver type of synchronization shows potential for a secure metrology system, where a self-mixing signal created by the output beam toward a target [35], which is polarized along ${\pm} $θc, can be accurately retrieved solely by the output beam polarized along ${\mp} $θc.
Fig. 5. (a) Synchronized chaos for LDV modulation of one of the paring modes (i.e., the sender) and (b) correlation plots. (c) Dependence of R on polarization direction of the receiver. P = 172 mW.
Essentially the same synchronization between the single pair of transverse modes shown in Figs. 4 and 5 occurred for quasi-locked states among the IGe2,2 – HG0,0 as well as the TEM0,1*– HG0,0 modes in Fig. 1(b), for both total and one-beam feedback from the rotating cylinder. Figure 6 shows the results for the +θc-polarized beam feedback, together with polarization resolved far-field patterns.
Fig. 6. Synchronized chaos. (a)-(c) IG22,2 – HG0,0, P = 180 mW (see Visualization 2). (d)-(f) TEM0,1* – HG0,0, P = 200 mW (see Visualization 3).
On the basis of repeated experiments, the critical angles are found to be approximately θc ${\cong} $ arctan(${\pm} $1/r), where the field components of the orthogonally polarized eigenmodes along θc coincide. If we introduce coherent field coupling among modes, such a pair of transverse modes formed by the superposition of orthogonally polarized eigenmode fields with the equal amplitude is expected to synchronize and behave just like a single-mode.
3. Numerical simulations
Numerical simulations were conducted for comparison with the experimental results. The simulation used the following two-mode laser equations, which include incoherent modal cross-saturation and coherent field coupling in the short delay regime, i.e., fD $\tau_{\textrm{D}}\,{\ll}\,1/\textrm{f}_1$ 1/f1.
The numerical results are shown in Fig. 7, where w = 1.53, K = 3.37 × 105, fD = 2.37 × 105 and ɛ = 8.3 × 10−6. As for the pair of transverse eigenmodes, the free-running modal power spectra exhibit an f2-peak representing antiphase dynamics, as shown in Fig. 7(a), whereas this peak disappears for the single pair of modes through synchronization under an increased modal field coupling, which corresponds to the increased spatial overlapping of the pair of transverse modes polarized along θc expected from polarization resolved far-field patterns in Figs. 2(a), 6(a) and 6(d), as shown in Fig. 7(b).
Fig. 7. Numerical results. (a), (c) g1 = 1, g2 = 0.70, β1,2 = 0.46, β2,1 = 0.75, η = 0.001. (b), (d) g1 = g2 = 0.9, β1,2 = β2,1 = 0.6, η = 0.008. (a), (b) Free-running, m1 = m2 = 0. (c), (d) Total modulation, m1 = m2 = 0.001. (e) One-mode modulation, m1 = 0.001, m2 = 0.
When the self-mixing modulation is applied to the total (m1 = m2) or one mode (m2 = 0), a high degree of chaos synchronization occurs between the single pair of modes in both cases, as shown in Figs. 7(d)-(e), while synchronization fails for the pair of transverse eigenmodes as shown in Fig. 7(c).
4. Summary
In summary, synchronized chaotic relaxation oscillations were found to occur between a single pair of polarized transverse mode fields in the quasi-locked states of coupled orthogonally polarized transverse modes in a thin-slice solid-state laser subjected to self-mixing modulations. The sender-receiver relationship of the chaotic signals hidden in the lasing polarization vector fields occurs between a single pair of transverse fields. This relationship inherits the dynamical non-independence in noise-driven modal intensity fluctuations. The experimental results were well reproduced by numerical simulations of a model of two-mode lasers subjected to self-mixing modulations, which included incoherent cross-saturation and coherent field coupling among modes.
Disclosures
The authors declare no conflicts of interest.
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