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OPTICS LETTERS / Vol. 36, No. 6 / March 15, 2011
Chaotic oscillations associated with the breakup of polarization entangled coherent states in a microchip solid-state laser Kenju Otsuka,1,* Yun-Ting Chen,2 Shu-Chun Chu,2 Chi-Ching Lin,3 and Jing-Yuan Ko4 1
Department of Human and Information Science, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan 2
Department of Physics, National Cheng Kung University, No. 1, University Road, Tainan City 701, Taiwan
3
Department of Physics, National Sun Yat-Sen University, No. 70, Lienhai Road, Kaohsiung 80424, Taiwan 4 Department of Physics, National Kaohsiung Normal University, No. 62, Shenjhong Road, Yanchao Township, Kaohsiung County 824, Taiwan *Corresponding author:
[email protected]‑tokai.ac.jp Received January 10, 2011; revised February 14, 2011; accepted February 14, 2011; posted February 17, 2011 (Doc. ID 140635); published March 11, 2011
We demonstrate the breakup of spatial-polarization entangled lasing patterns, which possess vector phase singularities, and the resultant dynamic instabilities featuring chaotic oscillations. The frequency splitting between a pair of Ince–Gauss (IG) lasing modes, originally forming a coherent entanglement state, and a self-excited additional nonorthogonal IG mode through a new class of transverse effect of self-injection pattern seeding, is shown to result in modal-interference-induced modulation at the beat frequency, leading to chaotic oscillations. © 2011 Optical Society of America OCIS codes: 140.3480, 140.5680, 260.6042, 190.4420, 270.3100.
Much recent research has been focused on singularities in vector fields, i.e., the polarization vectors of paraxial optical beams: vector singularities and Stokes singularities [1] . Vector singularities are isolated such that the orientation of the electric vector of a linearly polarized vector field becomes undefined. The nature of vector singularities has been studied in coherent optical waves with the correlated behavior of spatial structures and polarization states [2–5] . Most recently, a microchip solidstate laser has been used to perform analogous studies of coherent waves, where two orthogonally polarized spatial structures display the formation of a vector singularity, which is the macroscopic analog of the SU (2) generalized coherent states (GCSs) in quantum mechanics [6,7] . We have shown that these spatial and polarization entangled GCSs are formed through the transverse mode locking of a pair of orthogonally polarized Ince–Gauss (IG) lasing modes with a fixed phase difference of ΔΦ ¼ 0 or π through the modal intensity-dependent refractive index change inherent to microchip solid-state lasers [8] . In the context of vector singularities so far, dynamic effects in coherent waves have yet to be studied. This Letter describes nonlinear dynamics, featuring chaotic oscillations, associated with the breakup of a coherent state observed in the model of spatial-polarization entangled lasing patterns (SPEPs) using a microchip Nd:GdVO4 laser with off-axis LD pumping. With the increasing of the pump power, another well-defined IG cavity mode, which does not satisfy the mode-orthogonality condition with IG mode pairs of SPEP, was found to oscillate together with SPEP, suggesting a new class of transverse effect of self-injection pattern seeding. The small frequency splitting of degenerate IG modes, resulting from the failure of transverse mode locking involving nonorthogonally polarized IG modes, was observed associated with the self-excitation of such a disturbing lasing IG mode. The resultant beat note between the nonortho0146-9592/11/060960-03$15.00/0
gonal IG modes modulated the laser, leading to chaotic behaviors. The experimental setup is shown in Fig. 1. A nearly collimated laser diode (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 (NA ¼ 0:25) onto a 6 mm × 6 mm, 1 mm thick, 3 at:% doped c-cut Nd:GdVO4 crystal attached to a plane mirror M1 (transmission at 808 nm > 95% and reflectance at 1064 nm ¼ 99:8%). The output concave mirror M2 (radius of curvature of 1 cm, reflectance at 1064 nm ¼ 99%) was placed 5 mm away from M1 to construct a semiconfocal cavity. Such c-cut Nd:GdVO4 crystals possess high-level transverse isotropy due to the zircon structure with a tetragonal space group. The microchip laser cavity was made in one piece. With the off-axis or azimuthal LD pumping obtained by shifting or tilting the cavity as depicted in Fig. 1, stable single-frequency SPEPs free from dynamic instabilities were observed in the low-pump-power regime. Polarization-resolved patterns and the total lasing pattern without a polarizer of the simplest SPEP state are shown in Fig. 2(a), where the polarization and tilt directions are depicted by the arrows and lines,
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Fig. 1. (Color online) Experimental setup. 2011 Optical Society of A merica
March 15, 2011 / Vol. 36, No. 6 / OPTICS LETTERS
respectively (Media 1). These patterns were reproduced well numerically in the form of IGe ð2; 2Þ þ weiΔΦ IGo ð2; 2Þ with an elliptic parameter ε ¼ 10, a weighting ratio w ¼ 0:7, and a phase difference ΔΦ ¼ π. Pure singlefrequency stable operation was observed as shown in the optical spectrum and the power spectrum of the laser output in Fig. 2(b), where only intrinsic relaxation oscillation noise appeared at f RO . With increasing pump power, an additional IG mode (s), which is nonorthogonally polarized with respect to each IG mode forming an SPEP in the lower pump-power region, appears resulting from the increased gain area. Example polarization-resolved lasing patterns are shown in Fig. 3(a) together with the lasing pattern without a polarizer. Here, in addition to the orthogonally polarized IGe ð5; 1Þ- and IGo ð5; 1Þ-mode pair indicated by modes 1 and 2, which form the SPEP state in the low-pump-power regime, a well-defined IGo ð5; 1Þ mode with a larger elliptic parameter, ε ¼ 10, which is linearly polarized along the direction of α ¼ 65° indicated by mode 3, becomes tangible. Note that the present coexisting IG mode exhibits nonorthogonal polarization with respect to other IG modes, as shown in Fig. 3(a). Nearly degenerate singlelongitudinal mode operation was observed with a scanning Fabry–Perot interferometer (free-spectral range: 2 GHz) within a frequency resolution of a few MHz, as shown in the right of Fig. 3(a). This is reasonable because the cavity resonance frequencies of the involved IGe;o ðp; mÞ modes with the common p are degenerate [9] . However, an extremely small frequency splitting seems to be occurring, as indicated by the arrows in the magnified spectrum, contrary to the single-frequency operation in pure SPEP shown in the magnified spectrum of Fig. 2(b). Figure 3(b) shows another example lasing pattern observed by changing the pump position, in which additional IG modes which are nonorthogonally polarized to the pair of IG modes, joined in lasing. In this case, in addition to the pair of orthogonally polarized IGe ð3; 1Þ modes 1 and 2, linearly polarized well-defined IGe ð3; 3Þ modes 3 and 4 go into oscillation. Note that
Fig. 2. (Color online) (a) Polarization-resolved lasing patterns of a pure SPEP state formed by IGð2; 2Þ modes with even and odd parities. (b) Optical spectrum and power spectrum. Pump power, P ¼ 490 mW (Media 1).
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Fig. 3. (Color online) Polarization-resolved patterns together with the total lasing pattern without a polarizer and observed optical spectrum with a magnified view. (a) Pump power, P ¼ 673 mW and (b) P ¼ 721 mW. Polarization directions α are depicted by arrows, and tilt angles φ are shown by solid and dashed lines.
small frequency splitting can be also identified in the magnified optical spectrum shown in the right of Fig. 3(b). To confirm the emergence of well-defined linearly polarized IG modes, we carried out numerical simulations. The theoretically reconstructed SPEPs are shown in Figs. 4(a) and 4(b). Numerically reconstructed patterns resemble those of the experimental results in Figs. 3(a) and 3(b), respectively, while the numerical “distorted” pattern at α ¼ 65° seems to be dominated by the IGo ð5; 1Þ mode shown in an upper figure of the right column of Fig. 4(a) in the experiment. Similarly, numerically reconstructed “twisted” polarization-resolved patterns seem to be dominated by IGe ð3; 3Þ modes in the experiment in the case of Fig. 4(b). These patterns are nothing other than IG modes as reproduced theoretically in the lower figures of the right column of Fig. 4.
Fig. 4. (Color online) Theoretical SPEP patterns: (a) IGe ð5; 1Þ þ 0:8ei0 IGo ð5; 1Þ, ε ¼ 1:5 and (b) IGe ð3; 1Þþ 0:45ei0 IGe ð3; 1Þ, ε ¼ 10. The right column indicates self-excited mode(s) instead of polarization-resolved pattern(s) in the dashed box(es) (upper pattern) and the corresponding numerically reproduced IG mode(s) (lower pattern).
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Fig. 5. (Color online) Intensity waveform and the corresponding power spectrum. (a) Chaotic spiking oscillation corresponding to Fig. 3(a). (b) Chaotic relaxation oscillation corresponding to Fig. 3(b).
In conjunction with the appearance of disturbing IG mode(s), chaotic oscillations were found to take place. The intensity waveform and power spectrum corresponding to Figs. 3(a) and 3(b) are shown in Figs. 5(a) and 5(b), respectively. The laser showed chaotic spiking oscillations consisting of Q-switching-type spikes in the case of Fig. 5(a), while chaotic relaxation oscillations took place as for Fig. 5(b). Let us provide a probable physical interpretation into the observed dynamic instabilities. Fist of all, it should be noted that the well-defined IG mode always emerged by dominating a resembling polarization-resolved partial field pattern of the SPEP possessing the same polarization state as that of the well-defined IG mode, as shown in the right column of Fig. 4. This suggests that the welldefined IG mode, which satisfies the lasing boundary condition of the optical cavity, is self-excited by the resembling partial field possessing the strong spatial overlapping with the cavity IG mode, in which the partial field acts as a self-injection seeding pattern. A t this moment, a part of energy is considered to be transferred from the SPEP to the excited IG lasing mode. Such a self-excited IG mode tends to coexist with the SPEP state. However, the transverse mode locking of three IG modes (i.e., orthogonally-polarized IG mode pair of SPEP and self-excited IG mode) cannot be established because the polarization state of a self-excited IG mode is not orthogonal to that of each IG mode forming SPEP. Instead, a small frequency splitting is considered to take place among SPEP and the self-excited disturbing IG mode, assisted by the inherent modal-intensity (polarization) dependent refractive index change [8] . In short, the self-induced anisotropy, i.e., symmetry-breaking, results in the splitting of empty-cavity IG modes with the common p value. A similar symmetry-breaking mediated-avoided crossing of polarization eigenfrequencies has been reported in vertical-cavity semiconductor lasers [10] .
On the other hand, the mode-orthogonality generally holds in the IG mode family [9] . However, such mode orthogonality is violated between each IG mode forming SPEP and the self-excited IG mode(s), because their spatial coordinates are tilted to each other, as depicted by the solid and dashed lines in Figs. 3(a) and 3(b). It is well known that the laser exhibits chaotic relaxation oscillations when the modulation is applied around the relaxation oscillation frequency of the laser, f RO ¼ ð1=2πÞ½ ðp − 1Þ=ττp &1=2 p ¼ P=P th where P th , threshold pump power; τ, fluorescence lifetime; τp , photon lifetime) or its harmonic frequencies, f m ¼ n × f RO (n: integer), while chaotic spiking oscillations take place with increasing the modulation amplitude [11,12] . In the present experiment, the gain (stimulated emission) modulation is considered to brought about at a beat frequency f m ¼ transf B through modal interference of nonorthogonal R R ~ SPEP E ~ 'D dxdyþ verse modes in the form of BN 0 ð x y E ~ SPEP and E ~ D denote the SPEP and the c:c:Þ, where E self-excited disturbing IG fields, B is the stimulated emission coefficient, and N 0 is the population inversion density [13] . A s a result, the laser exhibits chaotic oscillations if we assume a small frequency splitting f B on the order of f RO among nonorthogonal transverse modes. In summary, dynamic instabilities associated with the breakup of coherent waves have been demonstrated in a microchip solid-state laser resulting from the competition between a pair of “gain-guided” IG modes excited by off-axis pumping and a “geometry-guided” IG mode excited through self-injection pattern seeding. The small splitting of polarization eigenfrequencies was shown to induce strong chaotic oscillations through modalinterference-induced modulation at the beat frequency. References 1. I. Freund, Opt. Commun. 199, 47 (2001). 2. L. Gil, Phys. Rev. Lett. 70, 162 (1993). 3. T. Erdogan, O. K ing, W. Wicks, D . G. Hall, E. H. A nderson, and M. J . Rooks, A ppl. Phys. Lett. 60, 1921 (1992). 4. Y . F. Chen, K . F. Huang, H. C. Lai, and Y . P. Lan, Phys. Rev. Lett. 90, 053904 (2003). 5. I. V. Veshneva, A . I. K onukhov, L. A . Melnikov, and M. V. Ryabinina, J . Opt. B 3, S209 (2001). 6. Y . F. Chen, T. H. Lu, and K . F. Huang, Phys. Rev. Lett. 96, 033901 (2006). 7. T. H. Lu, Y . F. Chen, and K . F. Huang, Phys. Rev. E 75, 026614 (2007). 8. K . Otsuka, S.-C. Chu, C.-C. Lin, K . Tokunaga, and T. Ohtomo, Opt. Express 17, 21615 (2009). 9. M. A . Bandres and J . C. Gutié rrez-Vega, J . Opt. Soc. A m. B 21, 873 (2004). 10. J . P. Woerdman, A . K . J ansen van D oorn, and M. P. van Exter, Laser Phys. 7, 63 (1997). 11. K . Otsuka, Nonlinear Dynamics in Op tical Comp lex Systems (Springer, 2000). 12. K . Otsuka, J .-Y . K o, and T. K ubota, Opt. Lett. 26, 638 (2001). 13. K . Otsuka, J .-Y . K o, T.-S. Lim, and H. Makino, Phys. Rev. Lett. 89, 083903 (2002).
