Spatial and polarization entanglement of lasing patterns and related dynamic behaviors in laser-diode-pumped solid-state lasers

doi:10.1364/OE.17.021615

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Spatial and polarization entanglement of lasing patterns and related dynamic behaviors in laser-diode-pumped solid-state lasers

K. Otsuka, S.-C. Chu, C.-C. Lin, K. Tokunaga, and T. Ohtomo »View Author Affiliations

Optics Express, Vol. 17, Issue 24, pp. 21615-21627 (2009)
http://dx.doi.org/10.1364/OE.17.021615

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Abstract

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.

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

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]

N. K. Viswanathan and V. V. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 ( 2009). [CrossRef] [PubMed]

]. These features have recently led to significant interest in using optical vector beams for a variety of applications, including super-resolution microscopy [3

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 ( 2006). [CrossRef] [PubMed]

], high-resolution metrology [4

Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 ( 2002). [CrossRef] [PubMed]

], and polarization spectroscopy of single molecules [5

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 ( 2000). [CrossRef] [PubMed]

] 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

Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 ( 2006). [CrossRef] [PubMed]

T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 ( 2007). [CrossRef] [PubMed]

], 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

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 ( 2004). [CrossRef]

], 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

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

M. A. Bandres and J. C. Gutirrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29(2), 144–146 ( 2004). [CrossRef] [PubMed]

–11

S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16506. [CrossRef] [PubMed]

] through transverse mode-locking owing to the intrinsic optical nonlinearity, i.e., light-intensity-dependent refractive-index change, inherent to lasers [12

M. Brambilla, L. A. Lugiato, V. Penna V, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 ( 1991). [CrossRef] [PubMed]

–14

K. Otsuka and S.-C. Chu, “Generation of vortex array beams from a thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34(1), 10–12 ( 2009). [CrossRef] [PubMed]

]. 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

]. 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

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-17-10705. [CrossRef] [PubMed]

]. 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 IG_p,m modes (p, m: mode indices [8

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 ( 2004). [CrossRef]

]) 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

It was found that single-frequency SPEPs originating from orthogonal linearly polarized even- and odd-parity IG_p,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 IG_p,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

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 ( 2004). [CrossRef]

]

\begin{array}{l} I G^{e}_{p, m} (\vec{r}, ε) = C [w_{0} / w (z)] C_{p}^{m} (i ξ, ε) C_{p}^{m} (η, ε) \exp [- r^{2} / w^{2}] \\ \times \exp i [k z + {k r^{2} / 2 R (z)} - (p + 1) ψ_{z} (z)], \end{array}

(1)

\begin{array}{l} I G^{o}_{p, m} (\vec{r}, ε) = S [w_{0} / w (z)] S_{p}^{m} (i ξ, ε) S_{p}^{m} (η, ε) \exp [- r^{2} / w^{2}] \\ \times \exp i [k z + {k r^{2} / 2 R (z)} - (p + 1) ψ_{z} (z)], \end{array}

(2)

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/ z_R ²)^1/2 describes the beam width; z_R = 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. C_p ^m (., ε) and S_p ^m (., ε) are the even and odd Ince polynomials [8

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 ( 2004). [CrossRef]

] 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 + z_R ²/z is the radius of curvature of the phase front, and ψ_z (z) = arctan(z/z_R ). 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 IG_p,m modes obtained from Eqs. (1) and (2). Figure 3(a) shows the typical structure of the IG^e _p,p modes in Figs. 1(b) and 2, i.e., IG_p,p modes have only parabolic nodal lines and no elliptical nodal lines.

Fig. 3 Example analytical patterns of IG modes. (a) IG^e _6,6 , (b) IG^e _7,3 , (c) IG^o _6,4 .

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 ω _p,m = (c/L)[qπ + (p + 1) cos⁻¹ (g₁g₂ )^1/2] (L: cavity length, q: number of half wavelengths along the cavity axis, g_j = 1 - L/R _j: g-parameters of the resonator, where R _j (j = 1, 2) is the radius of curvature of the mirror [8

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 ( 2004). [CrossRef]

]). 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 Δλ_g. (For the present Nd:GdVO₄ laser, Δλ = 0.3 nm, where Δλ_g = 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

K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Springer, 2000) pp. 176–181.

]. 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

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 ( 2000). [CrossRef]

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 α_i, and tilt-angle ϕ_i 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 IG_p,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 IG^e _4,4 and IG^o _4,2 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, IG^o _38,29 and IG^e _38,30.

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

E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 ( 1998). [CrossRef]

,18

E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO₂ laser,” Phys. Rev. A 53(6), 4435–4438 ( 1996). [CrossRef] [PubMed]

] 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 HG_1,0 and HG_0,1 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, f_D = 2v/λ, to the laser, which resulted from the interference between a lasing field and a coherent component Doppler-shifted injection field [19

K. Otsuka, “Effects of external perturbations on LiNdP₄O₁₂ lasers,” IEEE J. Quantum Electron. 15(7), 655–663 ( 1979). [CrossRef]

–21

K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 ( 2007). [CrossRef] [PubMed]

], 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 IG_8,8 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 IG_8,8 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 = Σ_i (I_1,i - <I₁ >)(I_2,i - <I₂ >)/[Σ_i (I_1,i - <I₁ >)]^1/2 [Σ_i (I_2,i - <I₂ >)]^1/2. For the phase correlation plot, we used the Hilbert transformation to extract Gabor’s analytical phase [20

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 ( 2002). [CrossRef] [PubMed]

–22

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 ( 2001). [CrossRef] [PubMed]

] from the time series. The analytic signal V_A and its time average <V_A > were calculated by V_A (t) - <V_A (t)> = R_A (t)exp(Φ(t)), where Φ(t) is the analytic phase. Here, V_A (t) = I(t) + iI_H (t), where I(t) is the time series of scalar intensity and I_H is its Hilbert transform [22

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 ( 2001). [CrossRef] [PubMed]

Fig. 9 Intensity fluctuations of odd- and even-party IG_8.8 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., f_D = 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

K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 ( 2002). [CrossRef] [PubMed]

,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

K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 ( 2000). [CrossRef] [PubMed]

,24

R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 ( 2004). [CrossRef]

], 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 IG_8.8 modes, temporal evolution of the analytic phase difference, and amplitude and phase correlation plots when the self-mixing modulation was applied at f_D = 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) IG^o _8,8 mode feedback, (c) IG^e _8,8 mode feedback. f_D = 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

J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945. [CrossRef] [PubMed]

]. 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

D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 ( 2001). [CrossRef] [PubMed]

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, f_D = 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 f_D = 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.

References and links

1.	C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 ( 2007). [CrossRef]
2.	N. K. Viswanathan and V. V. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 ( 2009). [CrossRef] [PubMed]
3.	K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 ( 2006). [CrossRef] [PubMed]
4.	Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 ( 2002). [CrossRef] [PubMed]
5.	B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 ( 2000). [CrossRef] [PubMed]
6.	Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 ( 2006). [CrossRef] [PubMed]
7.	T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 ( 2007). [CrossRef] [PubMed]
8.	M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 ( 2004). [CrossRef]
9.	M. A. Bandres and J. C. Gutirrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29(2), 144–146 ( 2004). [CrossRef] [PubMed]
10.	T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-17-10705. [CrossRef] [PubMed]
11.	S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16506. [CrossRef] [PubMed]
12.	M. Brambilla, L. A. Lugiato, V. Penna V, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 ( 1991). [CrossRef] [PubMed]
13.	J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 ( 1999). [CrossRef] [PubMed]
14.	K. Otsuka and S.-C. Chu, “Generation of vortex array beams from a thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34(1), 10–12 ( 2009). [CrossRef] [PubMed]
15.	K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Springer, 2000) pp. 176–181.
16.	R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 ( 2000). [CrossRef]
17.	E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 ( 1998). [CrossRef]
18.	E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO₂ laser,” Phys. Rev. A 53(6), 4435–4438 ( 1996). [CrossRef] [PubMed]
19.	K. Otsuka, “Effects of external perturbations on LiNdP₄O₁₂ lasers,” IEEE J. Quantum Electron. 15(7), 655–663 ( 1979). [CrossRef]
20.	K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 ( 2002). [CrossRef] [PubMed]
21.	K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 ( 2007). [CrossRef] [PubMed]
22.	D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 ( 2001). [CrossRef] [PubMed]
23.	K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 ( 2000). [CrossRef] [PubMed]
24.	R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 ( 2004). [CrossRef]
25.	J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945. [CrossRef] [PubMed]

OCIS Codes
(140.3480) Lasers and laser optics : Lasers, diode-pumped
(140.3580) Lasers and laser optics : Lasers, solid-state
(260.5430) Physical optics : Polarization
(270.1670) Quantum optics : Coherent optical effects
(270.3100) Quantum optics : Instabilities and chaos

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: September 15, 2009
Revised Manuscript: November 4, 2009
Manuscript Accepted: November 9, 2009
Published: November 11, 2009

Citation
K. Otsuka, S.-C. Chu, C.-C. Lin, K. Tokunaga, and T. Ohtomo, "Spatial and polarization entanglement of lasing patterns and related dynamic behaviors in laser-diode-pumped solid-state lasers," Opt. Express 17, 21615-21627 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21615

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References

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9(3), 78 (2007). [CrossRef]
N. K. Viswanathan and V. V. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009). [CrossRef] [PubMed]
K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440(7086), 935–939 (2006). [CrossRef] [PubMed]
Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 (2002). [CrossRef] [PubMed]
B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef] [PubMed]
Y. F. Chen, T. H. Lu, and K. F. Huang, “Observation of spatially coherent polarization vector fields and visualization of vector singularities,” Phys. Rev. Lett. 96(3), 033901 (2006). [CrossRef] [PubMed]
T. H. Lu, Y. F. Chen, and K. F. Huang, “Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026614 (2007). [CrossRef] [PubMed]
M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004). [CrossRef]
M. A. Bandres and J. C. Gutirrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29(2), 144–146 (2004). [CrossRef] [PubMed]
T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15(17), 10705–10717 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-15-17-10705 . [CrossRef] [PubMed]
S.-C. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15(25), 16506–16519 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16506 . [CrossRef] [PubMed]
M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43(9), 5114–5120 (1991). [CrossRef] [PubMed]
J. Scheuer and M. Orenstein, “Optical vortices crystals: spontaneous generation in nonlinear semiconductor microcavities,” Science 285(5425), 230–233 (1999). [CrossRef] [PubMed]
K. Otsuka and S.-C. Chu, “Generation of vortex array beams from a thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34(1), 10–12 (2009). [CrossRef] [PubMed]
K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Springer, 2000) pp. 176–181.
R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000). [CrossRef]
E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, “Coupled longitudinal and transverse self-organization in lasers induced by transverse-mode locking,” Phys. Rev. A 57(6), 4899–4904 (1998). [CrossRef]
E. Louvergneaux, D. Hennequin, D. Dangoisse, and P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53(6), 4435–4438 (1996). [CrossRef] [PubMed]
K. Otsuka, “Effects of external perturbations on LiNdP4O12 lasers,” IEEE J. Quantum Electron. 15(7), 655–663 (1979). [CrossRef]
K. Otsuka, T. Ohtomo, A. Yoshioka, and J.-Y. Ko, “Collective chaos synchronization of pairs of modes in a chaotic three-mode laser,” Chaos 12(3), 678–687 (2002). [CrossRef] [PubMed]
K. Otsuka, K. Nemoto, K. Kamikariya, Y. Miyasaka, J.-Y. Ko, and C.-C. Lin, “Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2 ), 026204 (2007). [CrossRef] [PubMed]
D. J. DeShazer, R. Breban, E. Ott, and R. Roy, “Detecting phase synchronization in a chaotic laser array,” Phys. Rev. Lett. 87(4), 044101 (2001). [CrossRef] [PubMed]
K. Otsuka, R. Kawai, S.-L. Hwong, J.-Y. Ko, and J.-L. Chern, “Synchronization of mutually coupled self-mixing modulated lasers,” Phys. Rev. Lett. 84(14), 3049–3052 (2000). [CrossRef] [PubMed]
R. Kawai, J.-S. Lih, J.-Y. Ko, J.-L. Chern, and K. Otsuka, “Chaos Synchronization in a mutually coupled laser array subjected to self-mixing modulation,” J. Opt. B Quantum Semiclassical Opt. 6(7), R19–R32 (2004). [CrossRef]
J.-Y. Ko, C.-C. Lin, K. Otsuka, Y. Miyasaka, K. Kamikariya, K. Nemoto, M. C. Ho, and I. M. Jiang, “Experimental Observations of Dual-Polarization Oscillations in Laser-Diode-Pumped Wide-Aperture Thin-Slice Nd:GdVO(4) Lasers,” Opt. Express 15(3), 945–954 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-945 . [CrossRef] [PubMed]

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