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To provide the underlying physical mechanism for formations of spatial- and polarization-entangled lasing patterns (namely, SPEPs), we performed experiments using a c-cut Nd:GdVO4 microchip laser with off-axis laser-diode pumping. This extends recent work on entangled lasing pattern generation from an isotropic laser, where such a pattern was explained only in terms of generalized coherent states (GCSs) formed by mathematical manipulation. Here, we show that polarization-resolved transverse patterns can be well explained by the transverse mode-locking of distinct orthogonal linearly polarized Ince-Gauss (IG) mode pairs rather than GCSs. Dynamic properties of SPEPs were experimentally examined in both free-running and modulated conditions to identify long-term correlations of IG mode pairs over time. The complete chaos synchronization among IG mode pairs subjected to external perturbation is also demonstrated.
©2009 Optical Society of America
1. Introduction
Optical vector beams have unique features in comparison with homogeneously polarized beams: the pure laser modes have a spatially modulated polarization distribution [1,2]. These features have recently led to significant interest in using optical vector beams for a variety of applications, including super-resolution microscopy [3], high-resolution metrology [4], and polarization spectroscopy of single molecules [5] in addition to unforeseen research areas. On the other hand, Chen et al. have reported a clear indication of spatial and polarization entanglement in high-order Gaussian beams generated from a laser-diode-pumped (LD-pumped) isotropic solid-state laser, which manifests itself in optical vector singularities [6,7], at which the orientation of the electric vector of a linearly polarized vector field becomes undefined. They reconstructed these spatial- and polarization-entangled lasing patterns (called ‘SPEPs’ hereafter) that are close to Laguerre-Gauss (LG) and Ince-Gauss (IG) modes, which are nothing other than orthogonal transverse modes on cylindrical and elliptic coordinates of the laser cavity [8], respectively, in terms of the superposition of many Hermite-Gauss (HG) modes by mathematical manipulations. Here, the entanglement term denotes the correlated behavior of spatial field structures and polarization states and differs from the “entanglement” of constituent modes in the strict quantum mechanical sense. They referred to these patterns as generalized coherent states (GCSs). However, the underlying physical mechanism for coherent superposition of HG modes has yet to be identified [6,7].
In this paper, motivated by the work by Chen et al. [6], we report our examination of the spontaneous entanglement of polarization and spatial structures in lasers by using a microchip c-cut Nd:GdVO4 laser with off-axis LD pumping and provide physics-based interpretations of spatial- and polarization-entangled lasing pattern formations. We show that the polarization entanglement on elliptic coordinates is established by the coherent superposition of orthogonal linearly polarized IG mode pairs selectively excited by off-axis LD pumping [9–11] through transverse mode-locking owing to the intrinsic optical nonlinearity, i.e., light-intensity-dependent refractive-index change, inherent to lasers [12–14]. Dynamic properties of SPEPs are also investigated to identify long-term correlations among IG mode pairs over time in both free-running and modulated conditions. The complete chaos synchronization among IG mode pairs, which indicates robust transverse mode-locking against external perturbations, is demonstrated experimentally.
2. Experimental setup and formation of SPEPs
The experimental setup is shown in Fig. 1 . A nearly collimated LD beam with a wavelength of 808 nm was passed through an anamorphic prism to transform an elliptical beam into a circular one, which was focused by a microscope objective lens (numerical aperture = 0.25) onto a 6-mm-square, 1-mm-thick, 3 at.%-doped c-cut Nd:GdVO4 crystal attached to a plane mirror M1 (transmission at 808 nm > 95%; reflectance at 1064 nm = 99.8%). The output concave mirror M2 (radius of curvature = 1 cm, reflectance at 1048 nm = 99%) was placed 5 mm away from M1 to construct a semi-confocal cavity. Here, c-cut Nd:GdVO4 crystals possess high-level transverse isotropy due to the zircon structure with a tetragonal space group similar to c-cut Nd:YVO4 crystals [6,7]. The microchip laser cavity was made in one piece. The spot size of the pump-beam focus was about 80 µm.
Fig. 1 (a) Experimental setup. LD: laser diode, AP: anamorphic prism pair, P: polarizer, BS: beam splitter, SFP: scanning Fabry-Perot interferometer (free spectral range: 2 GHz; resolution: 6.6 MHz), DO: digital oscilloscope. IR viewer: PbS phototube with a TV monitor. (b) Typical far-field pattern of the SPEPs and the polarization-resolved structural change. Optical spectra for IG mode pairs are also shown. Polarization directions α (i.e., polarizer angles) and tilt directions ϕ of the IG mode pairs are indicated by arrows and dashed lines, respectively. Pump power P = 526 mW.
When the pump-beam position was shifted from the center, i.e., azimuthal and/or off-axis pumping, as depicted by the arrows in Fig. 1(a), complicated lasing patterns appeared in the limited regimes of the pump power, depending on the pump-beam position. It should be noted here that only linearly polarized IG-mode oscillations occur for off-axis pumping in anisotropic laser crystals, e.g., a-cut Nd:GdVO4 and LiNdP4O12 (LNP), which possess strong fluorescence anisotropy [10]. Typical structural changes in emission patterns with respect to the polarizer rotation angle are shown in Fig. 1(b) observed at pump power P = 526 mW, where the polarization direction α is indicated by the arrows and defined as the tilt angle from the crystal’s a-axis, which was set parallel to the x-axis (horizontal direction). A polarization-resolved transverse pattern exhibits successive structural changes with polarizer rotation. From a careful examination of these patterns, we found that all the lasing modes were made up of pairs of two distinct orthogonal linearly polarized IGp,m modes (p, m: mode indices [8]) possessing the same p value (Gouy phase), i.e., same oscillation frequency, depicted by mode numbers 1 and 2 in Fig. 1(b), for example. That implies that the laser exhibits spatial and polarization entanglement in lasing patterns, in which the polarization of the transverse pattern is linear but spatially dependent [6,7].
It was found that single-frequency SPEPs originating from orthogonal linearly polarized even- and odd-parity IGp,p modes were predominantly achieved in the present cavity, where SPEPs with p = 3, 4, 5, 6, 7 and 8 were observed easily and reproducibly by changing the pump position and pump power (i.e., effective gain area). Three examples of SPEPs are shown in Fig. 2 , together with the corresponding optical spectra of lasing patterns. The lasing pattern in Fig. 2(a), which is similar to that in Fig. 1(b), was obtained by shifting the pump position slightly, which caused tilt angle ϕ of the lasing modes to change accordingly. In all cases, single-frequency operations were obtained. These IG mode pairs are considered to be the most suitable modes to be excited simultaneously by the common LD-pump beam targeted on the brightest spot of mode 1 as depicted by the circles in the figures [10,11], which is spatially shared by mode 2. IG mode pairs with larger p were found to be excited in the lower pump-power regime. This may result from the decrease in the effective gain area, which matches the smaller targeting spot for IG mode pairs with larger p.
Fig. 2 Experimental SPEPs formed from even- and odd-parity IGp,p modes, oscillation spectrum, and the associated polarization-resolved patterns. Pump power: (a) 534 mW, (b) 509 mW, (c) 498 mW. α1 ⊥ α2. Polarization directions of patterns in the right column: α1 + 45° for upper patterns; α2 + 45° for lower patterns.
3. Basic formalism of IG modes
The IG modes propagating along the z-axis of an elliptic coordinate system r = (ξ, η, z), with mode numbers p and m and ellipticity ε, are given by [8]
where the elliptic coordinates are defined in the transverse z plane as x = f(z) cos ξ cos η, y = f(z) sin ξ sin η, and ξ∈[0, ∞], η∈[0, 2π]. Here, f(z) is the semifocal separation of IG modes defined as the Gaussian beam width, i.e., f(z) = f 0 w(z)/w 0, where f 0 and w 0 are the semifocal separation and beam width at the z = 0 plane, respectively; w(z) = w 0 (1 + z 2/ zR 2)1/2 describes the beam width; zR = kw 0 2/2 is the Rayleigh length; and k is the wave number of the beam. The terms C and S are normalization constants, and subscripts e and o refer to even and odd IG modes, respectively. Cpm(., ε) and Spm(., ε) are the even and odd Ince polynomials [8] of order p, degree m, and ellipticity parameter ε, respectively. In Eqs. (1) and (2), r is the radial distance from the central axis of the cavity, R(z) = z + zR 2/z is the radius of curvature of the phase front, and ψz(z) = arctan(z/zR). The parameters ellipticity ε, waist w 0, and semifocal separation f 0 are not independent, but related by ε = 2f 0 2/w 0 2. IG mode patterns can be recognized by two rules: degree m corresponds to the number of hyperbolic nodal lines and (p - m)/2 is the number of elliptic nodal lines. Figure 3 plots some analytical patterns of the IGp,m modes obtained from Eqs. (1) and (2). Figure 3(a) shows the typical structure of the IGep,p modes in Figs. 1(b) and 2, i.e., IGp,p modes have only parabolic nodal lines and no elliptical nodal lines.4. Coherent superposition of IG mode pairs
IG modes with the same longitudinal mode index (i.e., Gouy phase) should be frequency degenerate without any perturbation. Here, the resonant angular frequency in the cold cavity is given by ωp,m = (c/L)[qπ + (p + 1) cos−1 (g1g2)1/2] (L: cavity length, q: number of half wavelengths along the cavity axis, gj = 1 - L/R j: g-parameters of the resonator, where R j (j = 1, 2) is the radius of curvature of the mirror [8]). However, note that this degeneracy is often lifted by the residual astigmatism of the cavity because of the effect of thermal birefringence in the gain medium. Therefore, we assume coherent superposition of orthogonal linearly polarized single-frequency IG mode pairs with a fixed phase difference Δψ, i.e., transverse mode-locking in the quadrature, which is assisted by the intrinsic intensity-dependent saturation-type of optical nonlinearity inherent in lasers [12–14], where the locked pattern intensity is constant and its transverse profile is invariant against propagation.
In microchip solid-state lasers, longitudinal mode spacing determined by the etalon effect of the crystal, Δλ = λ2/2nl (λ: lasing wavelength, n: refractive index, l: crystal thickness), is comparable to the gain bandwidth Δλg. (For the present Nd:GdVO4 laser, Δλ = 0.3 nm, where Δλg = 0.8 nm). Consequently, the oscillation frequency of the first lasing longitudinal mode is detuned from the gain center in general. In such detuned lasers, a change in the light-intensity-dependent refractive index [real part of electric susceptibility Re(χ(3))] is expected through gain saturation [Im(χ(3))]. Such an intensity-dependent refractive index is generic in thin-slice solid-state laser whose lasing frequency is detuned from the gain peak [14,15]. The resultant phase-sensitive interaction among nearly degenerate transverse modes is considered to result in transverse mode locking with a fixed phase difference within the limited lasing intensities of IG mode pairs, i.e., limited pump power region. In the case of orthogonal linear-polarization transverse mode operation, the laser may self-adjust the phase differences to 0 or π such that IG mode pairs create the same longitudinal standing-wave pattern in the cavity independently of the modal intensity ratio. Otherwise, the cavity boundary condition could not be satisfied for orthogonal linearly polarized IG mode pairs with the same oscillation frequency at the same time. In the experiment, SPEP oscillations occurred quite stably and reproducibly, but they appeared within the locking parameter range, i.e., the limited pump power region of approximately ΔP = 20 mW. The similar coherent superposition of orthogonal linearly polarized HG10 and HG01 modes with Δψ = 0 or π, which results in radially or azimuthally polarized doughnut shaped emissions, has been reported in a Nd:YAG laser [16].
To confirm such a conjectured transverse phase-locking phenomenon, we carried out a numerical reconstruction of observed polarization-dependent emission patterns. Lasing mode patterns were obtained numerically by the coherent superposition of IG mode pairs indicated by 1 and 2, in the form of IG 1 + w exp(i Δψ)IG 2 (w: mode weight ratio). We used Ince-Gauss functions, IG 1 and IG 2, that were estimated from the observed IG mode pairs to reproduce the experimental lasing patterns and polarization-resolved patterns, assuming a proper phase difference between the two IG modes. Here, we assumed experimental values of the mode weight ratio w determined from optical spectra, polarization direction αi, and tilt-angle ϕi of IG mode pairs (i = 1,2) indicated in Figs. 2(a), 2(b), and 2(c), where ϕ1 and ϕ2 are not identical. Theoretical results corresponding to Figs. 2(a), 2(b), and 2(c) are shown in Figs. 4(a) , 4(b), and 4(c), respectively. The phase differences were assumed to be Δψ = π for (a) and (b) and 0 for (c), where the mode weight ratios are described in each figure. The experimental results are reproduced by the theoretical results remarkably well. Polarization-resolved emission patterns in Fig. 1(b) were also well reproduced by the theoretical reconstruction, assuming Δψ = 0.
Fig. 4 Theoretically reconstructed patterns corresponding to Fig. 2. Polarization directions α and tilt angles ϕ were set identical to those in Fig. 2.
IG mode pairs besides the even- and odd-parity IGp,p modes in Fig. 2 occasionally formed SPEPs, as shown in Figs. 5(a) and 6(a) , and the theoretically reconstructed patterns are shown in Figs. 5(b) and 6(b), assuming Δψ = π and 0, respectively.
Fig. 5 (a) Experimental SPEPs formed from the IGe4,4 and IGo4,2 modes, oscillation spectrum, and associated polarization-resolved patterns. P = 530 mW. (b) Theoretically reconstructed patterns.
Fig. 6 Typical example of single-frequency SPEPs and polarization-resolved patterns observed in the larger c-cut Nd:GdVO4 laser cavity. (a) Experimental result. Pump power, P = 91 mW. α1 = 55°, ϕ1 = 70°, α2 = 65°, ϕ2 = - 20°. (b) Theoretically reconstructed patterns.
Here, SPEPs consisting of higher-order IG mode pairs with large p and m values like Fig. 6(a) appeared in larger laser cavities, which possess a larger beam diameter of the fundamental TEM00 mode. With increasing lasing beam diameter, the targeting spot diameter increased relative to the LD pump-beam focus. As a result, complicated SPEPs are considered to be excited by off-axis LD pumping. The SPEP shown in Fig. 6(a) was observed by inserting the same c-cut Nd:GdVO4 crystal into a laser cavity consisting of a flat mirror M1 and a concave mirror M2 with a 10-cm radius of curvature separated by 1 cm.
The observed single-frequency SPEP and its polarization-resolved patterns in Fig. 6(a) were well reconstructed by the transverse mode-locking of IG mode pairs indicated in the figure. Similar polarization-resolved patterns shown in Fig. 1 of ref. 6 were fitted by the GCSs [6] given by Eq. (7) in that paper; however, the detailed structures of these patterns have been shown to be more precisely reconstructed by orthogonal linearly polarized IG modes, IGo38,29 and IGe38,30.
In fact, phase differences of 0 and π, which are expected to result in stationary transverse lasing patterns in the laser cavity as mentioned before, reproduced the experimental patterns. On the other hand, in usual phase locking phenomena among transverse modes, except for orthogonally polarized modes [16], the optical phases of constituent modes possessing similar amplitudes, which belong to the same degenerate family, have been shown to be given by Δψ = ± π/2 such that the emission energy of the system is maximized [17,18] and the mode profile is maintained against propagation. For example, the doughnut mode (vortex), which is nothing other than the LG01 mode of the stable resonator, is formed as the stationary locked pattern for the HG1,0 and HG0,1 modes.
In the numerical simulation, we found that two polarization-resolved patterns in the right column in Figs. 2, 5(a), and 6(a) were reversed when Δψ = 0 was replaced by π, and vice versa. For Δψ = ± π/2, on the other hand, patterns in the right column were found to be identical and to repeat at every 90° rotation of the polarizer, unlike the experimental observations. Numerical results corresponding to Fig. 5(a) are shown in Fig. 7 . Therefore, we conclude that the observed polarization-entangled stationary lasing patterns resulted from transverse mode-locking among orthogonal linearly polarized IG modes with Δψ = 0, π, in which each polarization state is assigned to a well-defined IG mode pattern.
Fig. 7 Effect of phase difference ΔΨ on polarization-resolved transverse structures corresponding to Fig. 5(a). (a) Δψ = 0. (b) ± π/2.
5. Dynamic properties of SPEPs
This section reports on an experimental investigation of the dynamic properties of SPEPs in free-running as well as in modulated conditions to identify the long-term correlations of IG mode pairs over time. The experimental setup is shown in Fig. 8 . Part of the SPEP beam (~4%) reflected by a glass plate impinged on a rotating cylinder to apply a self-mixing modulation at a Doppler-shift frequency, fD = 2v/λ, to the laser, which resulted from the interference between a lasing field and a coherent component Doppler-shifted injection field [19–21], where v is the cylinder’s speed along the lasing beam axis. The remaining beam was used for simultaneous measurements of temporal evolutions of orthogonal linearly polarized IG mode pairs.
Fig. 8 Experimental setup for measuring temporal correlations among IG mode pairs of odd- and even IG8,8 shown in the inset. PD: photodiode (bandwidth, DC−125 MHz); DO: digital oscilloscope (bandwidth, DC−200 MHz).
We investigated random fluctuations in the modal intensities of IG pairs (even and odd IG8,8 modes), I1 and I2, around the relaxation-oscillation frequency f1, temporal evolution of analytic phase difference, amplitude, and phase correlations under the free-running condition. Results are shown in Fig. 9 together with far-field patterns and modal output power spectra. The data length and time interval for correlation plots were 1 ms and 0.16 µs, respectively, and the number of data points was 6250. Here, the correlation coefficient is defined as R = Σi (I1,i - <I1>)(I2,i - <I2>)/[Σi (I1,i - <I1>)]1/2 [Σi (I2,i - <I2>)]1/2. For the phase correlation plot, we used the Hilbert transformation to extract Gabor’s analytical phase [20–22] from the time series. The analytic signal VA and its time average <VA> were calculated by VA(t) - <VA(t)> = RA(t)exp(Φ(t)), where Φ(t) is the analytic phase. Here, VA(t) = I(t) + iIH(t), where I(t) is the time series of scalar intensity and IH is its Hilbert transform [22].
Fig. 9 Intensity fluctuations of odd- and even-party IG8.8 modes, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots under the free-running condition. P = 478 mW.
The amplitude correlation coefficient of random modal relaxation-oscillation fluctuations, which are induced predominantly by mode-partitioned spontaneous emission noise, was as large as R = 0.893. The phases of noise-driven modal relaxation-oscillation fluctuations were synchronized to some extent, although the phase synchronous state was occasionally interrupted by an abrupt jump of ΔΦ = Φ1 − Φ2 = 2π × n (n: integer) as indicated by the red arrow, for example. In short, the SPEP behaved like an “all-in-one” coherent mode, whose output power spectrum exhibited a single relaxation-oscillation frequency peak at f1, as shown in the inset of Fig. 9.
When the self-mixing modulation was applied to the SPEP at the relaxation-oscillation frequency, i.e., fD = f1, large-amplitude chaotic relaxation oscillations appeared for IG mode pairs. Results are shown in Fig. 10 . It should be noted that the modulation amplitudes of the SPEP laser were greatly enhanced compared with those of anisotropic lasers [20,21] because the scattered light fields from the rotating cylinder can effectively interfere with the SPEP lasing field, while only a small fraction of scattered light fields can interfere with the linearly polarized lasing field from anisotropic lasers. The amplitude as well as phase correlations of chaotic relaxation-oscillation fluctuations were found to be drastically improved compared with Fig. 9 because the noise effect was substantially masked by periodic perturbations. The observed behavior is similar to the chaos synchronization phenomenon in a two-laser array subjected to self-mixing modulation [23,24], which occurs when two lasers are phase locked with Δψ = π through spatial field coupling.
Fig. 10 Synchronized chaotic oscillations with SPEP (i.e., total) beam feedback: Intensity fluctuations of odd- and even-party IG8.8 modes, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots when the self-mixing modulation was applied at fD = f1 = 200 kHz. P = 478 mW.
Such a complete chaos synchronization was found to take place for any polarization-resolved patterns independently of the mutual angle between their polarizing axes, α, besides α1 ⊥ α2, even when the polarization-resolved beam of an arbitrary polarizing angle impinged on the cylinder as a result of a polarizer being inserted as depicted in Fig. 8. Typical examples are shown in Fig. 11 . It thus appears that the observed dynamic state can be referred to as spatial and polarization entanglement of transverse modes in chaotic lasers.
Fig. 11 (a) Total beam (SPEP) feedback. (α: polarization direction of an observed pattern), (b) IGo8,8 mode feedback, (c) IGe8,8 mode feedback. fD = f1 = 200 kHz. P = 478 mW.
As the pump power was increased beyond the lock-in range, chaos synchronizations failed. Results for the free-running case are shown in Fig. 12 . Note that an additional transverse mode with a different oscillation frequency, depicted by a red circle, appeared and the two-frequency operation arose, leading to the failure of transverse mode locking. Here, the lower-frequency relaxation oscillation component arose in modal power spectra at f2 = 311 kHz accordingly, reflecting the three-dimensional cross-saturation of transverse modes [25]. In this case, the locking time of temporal phase synchronous states was found to decrease as compared with Fig. 9, featuring intermittent phase slipping, where a gradual change of 2π × n (n: integer) of the analytic phase difference, Φ1 - Φ2, took place as shown by the blue arrow, for example [22].
Fig. 12 Failure of transverse mode locking: Polarization-resolved modal intensity fluctuations, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots of noise-driven random relaxation-oscillation fluctuations when the pump power was increased to P = 526 mW.
Unsynchronized oscillations due to the self-mixing modulation at the relaxation oscillation frequency, fD = f1 = 480 kHz, which were observed in the absence of transverse mode locking, are shown in Fig. 13 .
Fig. 13 Unsynchronized chaotic relaxation oscillations with SPEP (total) beam feedback: polarization-resolved modal intensity fluctuations, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots when the self-mixing modulation was applied at fD = f1 = 480 kHz. P = 526 mW.
6. Conclusion
In summary, we have generated stable single-frequency spatial- and polarization-entangled transverse lasing patterns from a microchip solid-state laser with off-axis LD pumping. We have shown experimentally and theoretically that these lasing patterns are formed by spontaneous transverse mode-locking among orthogonal linearly polarized Ince-Gauss mode pairs excited by tuning the LD pump-beam position on the crystal and the pump power (i.e., modal intensities), assisted by the intrinsic optical nonlinearity inherent in laser media. The dynamic behaviors and long-term correlations of polarization-resolved lasing patterns have been investigated experimentally under free-running and modulated conditions. Complete chaos synchronizations of modal large-amplitude relaxation oscillations have been demonstrated. The physics of such spatial and polarization entanglements in transverse modes of lasers clarified here provides an insight into the formation of laser beams with polarization vector singularities toward applications of laser vector beams.
Acknowledgements
The work of S.-C. C. was supported in part by a grant from the National Science Council of Taiwan, R.O.C., under contract no. NSC 96-2112-M-006-019-MY3.
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