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Laser-diode-pumped thin-slice c-cut Nd:GdVO4 multimode laser with coherent vector fields Kenju Otsuka Department of Human and Information Science, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan ([email protected]) Received November 24, 2014; revised January 10, 2015; accepted February 8, 2015; posted February 9, 2015 (Doc. ID 227170); published March 17, 2015 Spontaneous formation of coherent vector fields, which results from synchronization of longitudinal modes of orthogonally polarized transverse modes, has been observed in a laser-diode (LD)-pumped thin-slice c-cut Nd:GdVO4 laser with reflective end faces operating under the multi-longitudinal mode oscillation condition. The suppression of antiphase dynamics inherent to multimode solid-state lasers as well as the enhanced self-mixing interference effect as compared with a linearly polarized multimode laser have been identified. © 2015 Optical Society of America OCIS codes: (140.3480) Lasers, diode-pumped; (140.3580) Lasers, solid-state; (260.5430) Polarization; (190.4420) Nonlinear optics, transverse effects in; (270.2500) Fluctuations, relaxations, and noise; (120.7250) Velocimetry. http://dx.doi.org/10.1364/OL.40.001177
The uniaxial vanadate laser crystals such as Nd:YVO4 and Nd:GdVO4 have strong fluorescence anisotropy. In a Nd:GdVO4 crystal, for example, the stimulated emission cross-section along the c axis (π-polarization) for the 4 F3∕2 2 → 4 I11∕2 1 transition is five times larger than that orthogonal to the c axis (σ-polarization) for the 4F 4 3∕2 1 → I11∕2 1 transition [1,2]. Therefore, conventional Nd-doped vanadate crystals are cut along the a axis, i.e., so-called a-cut, yielding the linearly polarized emission parallel to the c axis possessing the larger emission cross-section. While as for passive Q-switching operations, the smaller emission cross-section in the c-cut Nd-doped vanadate crystals have been reported to satisfy good passive Q-switching condition, in which the saturation in the absorber must occur before the gain saturation, yielding higher peak pulse generations [3,4]. Cylindrical vector beams, such as radially or azimuthally polarized beams with generic sharper focus, have been demonstrated in c-cut Nd:YVO4 [5,6], Nd:GdVO4 lasers [7] as well as fiber lasers [8] with external cavity configurations toward applications in nanoscale optical imaging, particle trapping, and particle manipulation [9]. In this Letter, spontaneous formation of polarization vector fields is reported in a thin-slice c-cut Nd:GdVO4 laser with directly coated end mirrors under multilongitudinal operation conditions. By controlling an LD end-pumping beam position on the crystal, synchronization of orthogonally polarized transverse modes has been found to occur. The resultant lasing fields, representing a coherent superposition of multi-longitudinal modes with polarization vector fields, has been demonstrated to induce the suppression of antiphase dynamics and the enhanced self-mixing interference effect as compared with a linearly polarized a-cut thin-slice Nd:GdVO4 laser operating in multi-longitudinal modes. The experimental arrangement is shown in Fig. 1. A collimated lasing beam from the 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 microscope objective lens of different magnifications (i.e., numerical apertures, NA). The laser crystal was a 5 mm2 , 1-mm-thick, 3 at.%-doped c-cut Nd:GdVO4 whose end 0146-9592/15/071177-04$15.00/0
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 patterns were observed with a PbS phototube followed by a TV monitor. Lasing optical spectra were measured with a spectrometer composed of a scanning Fabry–Perot interferometer (SFPI) with a 6.6-MHz resolution and a monochromator with a 0.1-nm resolution. The laser output was detected by an InGaAs photo-detector (bandwidth: DC-125 MHz), and the electric signal was delivered to a spectrum analyzer and a digital oscilloscope to measure power spectra and lasing waveforms. In the case of thin-slice solid-state lasers with coatedend mirrors, input-output characteristics and multimode oscillation properties have been shown to depend on the pump beam focusing condition, using microscope objective lenses with various magnifications, M, due to the mode-matching between pump and lasing mode profiles [10]. Here, a stable resonator condition is achieved due to the thermally induced thin lens localized near M 1 as depicted in the inset of Fig. 1 [11]. In the present study, microscope objective lenses with different magnifications (numerical apertures, NA), M 10 × NA 0.3, 20 × NA 0.5, and 60 × NA 0.85, were used to pump the platelet crystal. The laser exhibited two types of oscillation properties depending on the pump position on the crystal
Fig. 1. Experimental setup. OL: objective lens; BS: beam splitter; SA: spectrum analyzer; DO: digital oscilloscope. © 2015 Optical Society of America
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Fig. 2. Input–output characteristics for NA 0.5 lens focusing. (a) Dual-polarization oscillation. (b) Simultaneous oscillation of orthogonally polarized transverse modes. The crystal position was moved by 150 μm from (a) along the x axis.
surface. The first one is a dual-polarization TEM00 -mode oscillation, featuring the switching among orthogonal polarizations with increasing the pump power. A typical input-output characteristic is shown in Fig. 2(a) for NA 0.5 lens. When the pump position was slightly changed by moving the laser sample along the x axis as indicated by the arrow in Fig. 1 by using a XYZ-axes translation stage with an accuracy of 30 μm, the abrupt switching of the TEM00 -mode pattern in Fig. 2(a) to a linearly-polarized, slightly elongated oval mode 1 shown in Fig. 2(b) was often observed, reflecting the cylindrical symmetry breaking around the pump axis due to a polished surface roughness [11]. The surface quality of the polished crystal was standard 10/5 scratch/dig per MIL-O-13830, i.e., 1-μm width line defects and 50-μm diameter hole defects. An orthogonally polarized mode 2 oscillation took place reproducibly associated with the preceding oval transverse mode oscillation by using population inversions outside the oval mode profile along the laser axis, i.e., transverse spatial hole-burning [12]. Here, the threshold pump power was P th 24 mW, and the slope efficiency was η 54% for NA 0.5 lens. Note that both mode patterns and their polarization directions were found to rotate by α against a crystal rotation of α around the laser axis. It was verified from the separation of two spots against propagation and the observed near-field pattern that two spots emerged from the same position on M2 with a crossing angle of θ 11.4 mrad. This suggests that mode 2 is formed after double roundtrips within the thermally induced cavity depicted in the inset of Fig. 1. As for dual-polarization operations shown in Fig. 2(a), orthogonally polarized modes were found to oscillate at different frequencies, and a beat note in several tens of MHz range was observed. Dual-polarization oscillations (DPO) featuring polarization switching like Fig. 2(a) are not new and have been studied extensively in vertical cavity-surface-emitting laser diodes (VCSEL) [13]. Therefore, let me focus on oscillation properties of the lasing pattern formed of a pair of orthogonally polarized transverse modes shown in Fig. 2(b) in this study. Pump-dependent optical spectra measured by the scanning Fabry–Perot interferometer are shown in Fig. 3,
Fig. 3. Pump-dependent lasing optical spectra measured by the scanning Fabry–Perot interferometer for P 120 mW. The energy diagram indicating π -and σ-polarization transitions is shown in the inset. Spectral intensities of the weaker mode for P 56 and 120 mW are enlarged for identifying modal peaks.
together with the energy diagram [1]. Here, virtual frequency separations of 280 MHz (mode 2–3) and 240 MHz (1–2, 1–4) in the SFPI trace can be reproduced by the free spectral range of 2 GHz and longitudinal mode spacing of Δλ λ2 ∕2 nL 0.258 nm (n 2.1981: refractive index, L 1 mm: cavity length), assuming real oscillation wavelengths of 1063.18, 1065.24, 1065.50, and 1065.76 nm measured by the monochromator. The first lasing mode appeared near 1065.5 nm, which corresponds to σ 1 polarization transition line, 4 F3∕2 1 → 4 I11∕2 1. With increasing the pump, third mode appeared near 1063.2 nm corresponding to the σ 2 -polarization 4 F3∕2 2 → 4 I11∕2 2 transition line, whose relative stimulated emission cross-section to the former one is 0.8 [1,2], and finally four-mode oscillations took place. Modal components separated by Δλ∕2 0.129 nm expected for the double roundtrip mode 2 was suppressed due to the synchronization to the preceding transverse mode 1, i.e., transverse mode-locking, in which every other longitudinal modes separated by Δλ λ2 ∕2 nL were selectively excited as shown below. The four-mode operations were maintained in the entire pump region because of the narrow gain bandwidth of Δλe 1.3 nm [2]. When the pump power was increased above P 100 mW, a “kink” appeared for the stronger transverse mode 1 as indicated by the dashed line in Fig. 2(b), and the weaker transverse mode 2 exhibited four longitudinal mode operations whose lasing frequencies coincided with those of the stronger mode as shown in the lower spectrum for P 120 mW. Here, no beat-note between orthogonally polarized emissions was observed in the power spectra, while, only two modes appeared for the weaker transverse mode 2 for P 56 mW. The frequency locking is considered to take place due to the synchronization of pairing longitudinal modes as depicted by the up and down arrows in Fig. 3, i.e., transverse mode locking [14]. This suggests that a pair of longitudinal modes couples through the field
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Fig. 4. (a) Total output and (b) polarization resolved far-field patterns. P 120 mW (see Media 1). The pair of orthogonally polarized modes are indicated by the frames.
overlapping within the cavity and goes into synchronized state. Figure 4 shows structural changes of polarizationresolved far-field patterns, together with that of the total output in the regime of four longitudinal-mode operation above the kink in Fig. 2(b). Lasing patterns similar to Fig. 2(b) were achieved with increased thresholds of P th 56 mW and 74 mW for NA 0.3 and 0.85 lens focusing, respectively. The structural change of polarization-resolved patterns, featuring a successive rotation around the laser axis, is obvious. This indicates that the total lasing pattern is formed not by the simple summation of intensities of orthogonally polarized single and double roundtrip transverse modes but the coherent superposition of a pair of orthogonally polarized mode fields, that is transverse mode locking that indicates one-to-one correspondence between spatial structure and polarization state [7,14]. Therefore, it is expected based on the optical spectra indicating synchronization (Fig. 3) and the resultant coherent nature of polarization-resolved lasing patterns (Fig. 4) that relative phases of all the longitudinal modes also tend to become constant. In short, the cooperative longitudinal mode locking is established through the transverse mode locking of orthogonal modes whose beam patterns depend on the longitudinal coordinate. Otherwise, mode-competition fluctuations take place, and polarization-resolved stationary patterns shown in Fig. 4 is not brought about. If such a physical interpretation is correct, the total lasing output field is created by multi-longitudinal modes with coherent vector fields. This phenomenon may manifest itself in the coupled longitudinal and transverse self-organization in lasers due to transverse mode locking that minimizes the free energy of the system [15]. In usual multi-mode operations, the total output field is just the summation of modal fields with uncorrelated amplitudes and phases. In order to examine dynamic properties of the observed lasing fields, power spectra were examined in the free-running condition. The laser output exhibited weak random relaxation oscillations driven by white noise, with mode-competition fluctuations being suppressed as shown in Fig. 5(a). From the Gaussian probability distribution of long-term intensity fluctuations, the fluctuation amplitude was evaluated to be m ΔI∕I av 0.94% (I av : averaged intensity). A similar stabilization
Fig. 5. (a) Long-term evolution of total output SFPI spectra. P 120 mW. (b) Pump-dependent power spectra.
of longitudinal modes was observed in a LiNdP4 O12 laser with an intra-cavity quarter-wave plate [16]. Figure 5(b) shows pump-dependent power spectra for a modal output indicated by the thick arrow in Fig. 3 (i.e., mode 2), which was selected by the monochromator. In two- and three-mode regions below the kink in Fig. 2(b), the modal output showed noise peak(s) at frequencies below the relaxation oscillation frequency, f RO , while these peaks were suppressed for the total output. Here, f RO 1∕2πw − 1∕ττp 1∕2 , where w P∕P th is the relative pump rate, τ 90 μs is the fluorescence lifetime, and τp is the photon lifetime. The observed behavior indicates the antiphase dynamics inherent to N-mode solid-state lasers due to crosssaturation of population inversions [17], where modal outputs exhibit (N − 1) noise peaks below the relaxation oscillation, while these lower frequency components are suppressed for the total output. From the measurement of pump-dependent relaxation oscillation frequencies, τp was determined to be 194 ps. As the pump power was increased beyond the kink in Fig. 2(b), where four-mode operations were also established for the weaker mode, noise peaks below the relaxation oscillation frequencies vanished completely as shown in Fig. 5, although the laser was oscillating in four longitudinal modes as shown in Fig. 3. This implies that all the longitudinal modes exhibit in-phase relaxation oscillations at f RO . This result can be understood if we assume the cooperative locking of all the longitudinal modes through synchronizations of orthogonally polarized pairing transverse modes. To clarify the coherent nature of multimode vector fields, the self-mixing interference effect has been examined in the laser Doppler velocimetry (LDV) scheme [18], by replacing the beam splitter and the IR viewer in Fig. 1 by a glass plate and a rotating Al cylinder with rough surfaces, respectively. Here, a 1-mm-thick, 3 at.%-doped, a-cut linearly polarized multimode Nd:GdVO4 laser, with the same coated end mirror specifications as the c-cut sample, was employed for the LDV experiment to compare the self-mixing interference effect in both multimode lasers. It has been shown that the loss modulation
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015 Glass plate
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is introduced to the laser in proportion to the beat signal between lasing and scattered fields, which results from the self-mixing interference phenomenon, where a modulation index is proportional to the fluorescence-to-photon lifetime ratio, K τ∕τp [18]. From the pump-dependent relaxation oscillation frequencies, the photon lifetime for the a-cut laser sample was estimated to be τp 210 ps. Figure 6 shows temporal evolutions of modulated output intensities at f D 2v∕λ (v: speed along the laser axis), where the a-cut laser operated in three longitudinal modes, featuring antiphase dynamics. The Doppler signal-to-noise ratio above the noise level for the c-cut laser was enhanced by ≥10 dB. In usual multimode lasers like the a-cut laser, the selfmixing interference occurs between each longitudinal mode field and the corresponding Doppler-shifted scattered field. No correlation exists between beat signals originating from modal interferences associated with multimode oscillations. Therefore, the modulation amplitude, m, at f D is unchanged for the modal and total output, since the total modulation intensity is the simple summation of modulated modal intensities. This result was verified by numerical simulations of LDV for multimode solid-state lasers with cross-saturation of population inversions [19]. As for multimode lasers with coherent vector fields, beat signals resulting from modal interferences are superimposed constructively, and the self-mixing interference effect is pronounced accordingly, in which the laser acts as a single-mode laser with the same total output power and K value. Finally, let me briefly discuss about nonlinear dynamics, which was observed in association with the failure of transverse mode locking. The crossing angle θ of two spots was increased with changing the pump-focus along the z axis, as depicted in Fig. 7(a), the transverse mode locking was failed and laser exhibited modulation at the beat frequency f B due to the modal interference [20]. With changing the pump power, fast pulsations at f B 14.63 MHz superimposed on periodic oscillations at f RO 541.7 kHz appeared as shown in Figs. 7(b) and 7(c), indicating the subharmonic resonance of f RO f B ∕27.
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The stabilized multi-longitudinal mode oscillations with coherent vector fields has been demonstrated in a c-cut thin-slice Nd:GdVO4 laser. The observed phenomenon is expected in general vector lasers, which are formed of orthogonally polarized transverse modes [9], and it is promising for generating coherent vector fields with increased output powers and producing optical pulses. References 1. F. G. Anderson, P. L. Summers, H. Weidner, P. Hong, and E. E. Peal, Phys. Rev. B 50, 14802 (1994). 2. Y. Sato and T. Taira, IEEE J. Sel. Top. Quantum Electron. 11, 613 (2005). 3. Y.-F. Chen and Y. P. Lan, Appl. Phys. B 74, 415 (2002). 4. K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, IEEE J. Quantum Electron. 47, 638 (2006). 5. Y. Kozawa and S. Sato, Opt. Lett. 30, 3063 (2005). 6. K. Yonezawa, Y. Kozawa, and S. Sato, Opt. Lett. 31, 2151 (2006). 7. K. Otsuka and S.-C. Chu, Opt. Lett. 38, 1434 (2013). 8. R. Zhou, J. W. Haus, P. E. Powers, and Q. Zhan, Opt. Express 18, 10837 (2010). 9. Q. Zhan, Adv. Opt. Photon. 1, 1 (2009) and references therein. 10. Y. Asakawa, R. Kawai, K. Ohki, and K. Otsuka, Jpn. J. Appl. Phys. 38, L515 (1999). 11. Y. Miyasaka, T. Narita, K. Otsuka, H. Makino, and A. Okamoto, Opt. Express 13, 7928 (2005). 12. K. Otsuka, IEEE J. Quantum Electron. 14, 49 (1978). 13. J. Martin-Regalado, J. L. A. Chilla, J. J. Rocca, and P. Brusenbach, Appl. Phys. Lett. 70, 3350 (1997). 14. K. Otsuka, S.-C. Chu, C.-C. Lin, K. Tokunaga, and T. Ohtomo, Opt. Express 17, 21615 (2009). 15. E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, Phys. Rev. A 57, 4899 (1998). 16. K. Otsuka and H. Iwasaki, IEEE J. Quantum Electron. 12, 214 (1976). 17. K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 1692 (1992). 18. K. Otsuka, Sensors 11, 2195 (2011) for review. 19. K. Otsuka, Jpn. J. Appl. Phys. 31, L1546 (1992). 20. K. Otsuka, J.-Y. Ko, T.-S. Lim, and H. Makino, Phys. Rev. Lett. 89, 083903 (2002).
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Laser-diode-pumped thin-slice c-cut Nd:GdVO4 multimode laser with coherent vector fields Kenju Otsuka Department of Human and Information Science, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan ([email protected]) Received November 24, 2014; revised January 10, 2015; accepted February 8, 2015; posted February 9, 2015 (Doc. ID 227170); published March 17, 2015 Spontaneous formation of coherent vector fields, which results from synchronization of longitudinal modes of orthogonally polarized transverse modes, has been observed in a laser-diode (LD)-pumped thin-slice c-cut Nd:GdVO4 laser with reflective end faces operating under the multi-longitudinal mode oscillation condition. The suppression of antiphase dynamics inherent to multimode solid-state lasers as well as the enhanced self-mixing interference effect as compared with a linearly polarized multimode laser have been identified. © 2015 Optical Society of America OCIS codes: (140.3480) Lasers, diode-pumped; (140.3580) Lasers, solid-state; (260.5430) Polarization; (190.4420) Nonlinear optics, transverse effects in; (270.2500) Fluctuations, relaxations, and noise; (120.7250) Velocimetry. http://dx.doi.org/10.1364/OL.40.001177
The uniaxial vanadate laser crystals such as Nd:YVO4 and Nd:GdVO4 have strong fluorescence anisotropy. In a Nd:GdVO4 crystal, for example, the stimulated emission cross-section along the c axis (π-polarization) for the 4 F3∕2 2 → 4 I11∕2 1 transition is five times larger than that orthogonal to the c axis (σ-polarization) for the 4F 4 3∕2 1 → I11∕2 1 transition [1,2]. Therefore, conventional Nd-doped vanadate crystals are cut along the a axis, i.e., so-called a-cut, yielding the linearly polarized emission parallel to the c axis possessing the larger emission cross-section. While as for passive Q-switching operations, the smaller emission cross-section in the c-cut Nd-doped vanadate crystals have been reported to satisfy good passive Q-switching condition, in which the saturation in the absorber must occur before the gain saturation, yielding higher peak pulse generations [3,4]. Cylindrical vector beams, such as radially or azimuthally polarized beams with generic sharper focus, have been demonstrated in c-cut Nd:YVO4 [5,6], Nd:GdVO4 lasers [7] as well as fiber lasers [8] with external cavity configurations toward applications in nanoscale optical imaging, particle trapping, and particle manipulation [9]. In this Letter, spontaneous formation of polarization vector fields is reported in a thin-slice c-cut Nd:GdVO4 laser with directly coated end mirrors under multilongitudinal operation conditions. By controlling an LD end-pumping beam position on the crystal, synchronization of orthogonally polarized transverse modes has been found to occur. The resultant lasing fields, representing a coherent superposition of multi-longitudinal modes with polarization vector fields, has been demonstrated to induce the suppression of antiphase dynamics and the enhanced self-mixing interference effect as compared with a linearly polarized a-cut thin-slice Nd:GdVO4 laser operating in multi-longitudinal modes. The experimental arrangement is shown in Fig. 1. A collimated lasing beam from the 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 microscope objective lens of different magnifications (i.e., numerical apertures, NA). The laser crystal was a 5 mm2 , 1-mm-thick, 3 at.%-doped c-cut Nd:GdVO4 whose end 0146-9592/15/071177-04$15.00/0
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 patterns were observed with a PbS phototube followed by a TV monitor. Lasing optical spectra were measured with a spectrometer composed of a scanning Fabry–Perot interferometer (SFPI) with a 6.6-MHz resolution and a monochromator with a 0.1-nm resolution. The laser output was detected by an InGaAs photo-detector (bandwidth: DC-125 MHz), and the electric signal was delivered to a spectrum analyzer and a digital oscilloscope to measure power spectra and lasing waveforms. In the case of thin-slice solid-state lasers with coatedend mirrors, input-output characteristics and multimode oscillation properties have been shown to depend on the pump beam focusing condition, using microscope objective lenses with various magnifications, M, due to the mode-matching between pump and lasing mode profiles [10]. Here, a stable resonator condition is achieved due to the thermally induced thin lens localized near M 1 as depicted in the inset of Fig. 1 [11]. In the present study, microscope objective lenses with different magnifications (numerical apertures, NA), M 10 × NA 0.3, 20 × NA 0.5, and 60 × NA 0.85, were used to pump the platelet crystal. The laser exhibited two types of oscillation properties depending on the pump position on the crystal
Fig. 1. Experimental setup. OL: objective lens; BS: beam splitter; SA: spectrum analyzer; DO: digital oscilloscope. © 2015 Optical Society of America
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Fig. 2. Input–output characteristics for NA 0.5 lens focusing. (a) Dual-polarization oscillation. (b) Simultaneous oscillation of orthogonally polarized transverse modes. The crystal position was moved by 150 μm from (a) along the x axis.
surface. The first one is a dual-polarization TEM00 -mode oscillation, featuring the switching among orthogonal polarizations with increasing the pump power. A typical input-output characteristic is shown in Fig. 2(a) for NA 0.5 lens. When the pump position was slightly changed by moving the laser sample along the x axis as indicated by the arrow in Fig. 1 by using a XYZ-axes translation stage with an accuracy of 30 μm, the abrupt switching of the TEM00 -mode pattern in Fig. 2(a) to a linearly-polarized, slightly elongated oval mode 1 shown in Fig. 2(b) was often observed, reflecting the cylindrical symmetry breaking around the pump axis due to a polished surface roughness [11]. The surface quality of the polished crystal was standard 10/5 scratch/dig per MIL-O-13830, i.e., 1-μm width line defects and 50-μm diameter hole defects. An orthogonally polarized mode 2 oscillation took place reproducibly associated with the preceding oval transverse mode oscillation by using population inversions outside the oval mode profile along the laser axis, i.e., transverse spatial hole-burning [12]. Here, the threshold pump power was P th 24 mW, and the slope efficiency was η 54% for NA 0.5 lens. Note that both mode patterns and their polarization directions were found to rotate by α against a crystal rotation of α around the laser axis. It was verified from the separation of two spots against propagation and the observed near-field pattern that two spots emerged from the same position on M2 with a crossing angle of θ 11.4 mrad. This suggests that mode 2 is formed after double roundtrips within the thermally induced cavity depicted in the inset of Fig. 1. As for dual-polarization operations shown in Fig. 2(a), orthogonally polarized modes were found to oscillate at different frequencies, and a beat note in several tens of MHz range was observed. Dual-polarization oscillations (DPO) featuring polarization switching like Fig. 2(a) are not new and have been studied extensively in vertical cavity-surface-emitting laser diodes (VCSEL) [13]. Therefore, let me focus on oscillation properties of the lasing pattern formed of a pair of orthogonally polarized transverse modes shown in Fig. 2(b) in this study. Pump-dependent optical spectra measured by the scanning Fabry–Perot interferometer are shown in Fig. 3,
Fig. 3. Pump-dependent lasing optical spectra measured by the scanning Fabry–Perot interferometer for P 120 mW. The energy diagram indicating π -and σ-polarization transitions is shown in the inset. Spectral intensities of the weaker mode for P 56 and 120 mW are enlarged for identifying modal peaks.
together with the energy diagram [1]. Here, virtual frequency separations of 280 MHz (mode 2–3) and 240 MHz (1–2, 1–4) in the SFPI trace can be reproduced by the free spectral range of 2 GHz and longitudinal mode spacing of Δλ λ2 ∕2 nL 0.258 nm (n 2.1981: refractive index, L 1 mm: cavity length), assuming real oscillation wavelengths of 1063.18, 1065.24, 1065.50, and 1065.76 nm measured by the monochromator. The first lasing mode appeared near 1065.5 nm, which corresponds to σ 1 polarization transition line, 4 F3∕2 1 → 4 I11∕2 1. With increasing the pump, third mode appeared near 1063.2 nm corresponding to the σ 2 -polarization 4 F3∕2 2 → 4 I11∕2 2 transition line, whose relative stimulated emission cross-section to the former one is 0.8 [1,2], and finally four-mode oscillations took place. Modal components separated by Δλ∕2 0.129 nm expected for the double roundtrip mode 2 was suppressed due to the synchronization to the preceding transverse mode 1, i.e., transverse mode-locking, in which every other longitudinal modes separated by Δλ λ2 ∕2 nL were selectively excited as shown below. The four-mode operations were maintained in the entire pump region because of the narrow gain bandwidth of Δλe 1.3 nm [2]. When the pump power was increased above P 100 mW, a “kink” appeared for the stronger transverse mode 1 as indicated by the dashed line in Fig. 2(b), and the weaker transverse mode 2 exhibited four longitudinal mode operations whose lasing frequencies coincided with those of the stronger mode as shown in the lower spectrum for P 120 mW. Here, no beat-note between orthogonally polarized emissions was observed in the power spectra, while, only two modes appeared for the weaker transverse mode 2 for P 56 mW. The frequency locking is considered to take place due to the synchronization of pairing longitudinal modes as depicted by the up and down arrows in Fig. 3, i.e., transverse mode locking [14]. This suggests that a pair of longitudinal modes couples through the field
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Fig. 4. (a) Total output and (b) polarization resolved far-field patterns. P 120 mW (see Media 1). The pair of orthogonally polarized modes are indicated by the frames.
overlapping within the cavity and goes into synchronized state. Figure 4 shows structural changes of polarizationresolved far-field patterns, together with that of the total output in the regime of four longitudinal-mode operation above the kink in Fig. 2(b). Lasing patterns similar to Fig. 2(b) were achieved with increased thresholds of P th 56 mW and 74 mW for NA 0.3 and 0.85 lens focusing, respectively. The structural change of polarization-resolved patterns, featuring a successive rotation around the laser axis, is obvious. This indicates that the total lasing pattern is formed not by the simple summation of intensities of orthogonally polarized single and double roundtrip transverse modes but the coherent superposition of a pair of orthogonally polarized mode fields, that is transverse mode locking that indicates one-to-one correspondence between spatial structure and polarization state [7,14]. Therefore, it is expected based on the optical spectra indicating synchronization (Fig. 3) and the resultant coherent nature of polarization-resolved lasing patterns (Fig. 4) that relative phases of all the longitudinal modes also tend to become constant. In short, the cooperative longitudinal mode locking is established through the transverse mode locking of orthogonal modes whose beam patterns depend on the longitudinal coordinate. Otherwise, mode-competition fluctuations take place, and polarization-resolved stationary patterns shown in Fig. 4 is not brought about. If such a physical interpretation is correct, the total lasing output field is created by multi-longitudinal modes with coherent vector fields. This phenomenon may manifest itself in the coupled longitudinal and transverse self-organization in lasers due to transverse mode locking that minimizes the free energy of the system [15]. In usual multi-mode operations, the total output field is just the summation of modal fields with uncorrelated amplitudes and phases. In order to examine dynamic properties of the observed lasing fields, power spectra were examined in the free-running condition. The laser output exhibited weak random relaxation oscillations driven by white noise, with mode-competition fluctuations being suppressed as shown in Fig. 5(a). From the Gaussian probability distribution of long-term intensity fluctuations, the fluctuation amplitude was evaluated to be m ΔI∕I av 0.94% (I av : averaged intensity). A similar stabilization
Fig. 5. (a) Long-term evolution of total output SFPI spectra. P 120 mW. (b) Pump-dependent power spectra.
of longitudinal modes was observed in a LiNdP4 O12 laser with an intra-cavity quarter-wave plate [16]. Figure 5(b) shows pump-dependent power spectra for a modal output indicated by the thick arrow in Fig. 3 (i.e., mode 2), which was selected by the monochromator. In two- and three-mode regions below the kink in Fig. 2(b), the modal output showed noise peak(s) at frequencies below the relaxation oscillation frequency, f RO , while these peaks were suppressed for the total output. Here, f RO 1∕2πw − 1∕ττp 1∕2 , where w P∕P th is the relative pump rate, τ 90 μs is the fluorescence lifetime, and τp is the photon lifetime. The observed behavior indicates the antiphase dynamics inherent to N-mode solid-state lasers due to crosssaturation of population inversions [17], where modal outputs exhibit (N − 1) noise peaks below the relaxation oscillation, while these lower frequency components are suppressed for the total output. From the measurement of pump-dependent relaxation oscillation frequencies, τp was determined to be 194 ps. As the pump power was increased beyond the kink in Fig. 2(b), where four-mode operations were also established for the weaker mode, noise peaks below the relaxation oscillation frequencies vanished completely as shown in Fig. 5, although the laser was oscillating in four longitudinal modes as shown in Fig. 3. This implies that all the longitudinal modes exhibit in-phase relaxation oscillations at f RO . This result can be understood if we assume the cooperative locking of all the longitudinal modes through synchronizations of orthogonally polarized pairing transverse modes. To clarify the coherent nature of multimode vector fields, the self-mixing interference effect has been examined in the laser Doppler velocimetry (LDV) scheme [18], by replacing the beam splitter and the IR viewer in Fig. 1 by a glass plate and a rotating Al cylinder with rough surfaces, respectively. Here, a 1-mm-thick, 3 at.%-doped, a-cut linearly polarized multimode Nd:GdVO4 laser, with the same coated end mirror specifications as the c-cut sample, was employed for the LDV experiment to compare the self-mixing interference effect in both multimode lasers. It has been shown that the loss modulation
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015 Glass plate
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Fig. 6. Self-mixing LDV. (a) Modulated intensity waveforms. (b) Total output power spectra. P 100 mW.
is introduced to the laser in proportion to the beat signal between lasing and scattered fields, which results from the self-mixing interference phenomenon, where a modulation index is proportional to the fluorescence-to-photon lifetime ratio, K τ∕τp [18]. From the pump-dependent relaxation oscillation frequencies, the photon lifetime for the a-cut laser sample was estimated to be τp 210 ps. Figure 6 shows temporal evolutions of modulated output intensities at f D 2v∕λ (v: speed along the laser axis), where the a-cut laser operated in three longitudinal modes, featuring antiphase dynamics. The Doppler signal-to-noise ratio above the noise level for the c-cut laser was enhanced by ≥10 dB. In usual multimode lasers like the a-cut laser, the selfmixing interference occurs between each longitudinal mode field and the corresponding Doppler-shifted scattered field. No correlation exists between beat signals originating from modal interferences associated with multimode oscillations. Therefore, the modulation amplitude, m, at f D is unchanged for the modal and total output, since the total modulation intensity is the simple summation of modulated modal intensities. This result was verified by numerical simulations of LDV for multimode solid-state lasers with cross-saturation of population inversions [19]. As for multimode lasers with coherent vector fields, beat signals resulting from modal interferences are superimposed constructively, and the self-mixing interference effect is pronounced accordingly, in which the laser acts as a single-mode laser with the same total output power and K value. Finally, let me briefly discuss about nonlinear dynamics, which was observed in association with the failure of transverse mode locking. The crossing angle θ of two spots was increased with changing the pump-focus along the z axis, as depicted in Fig. 7(a), the transverse mode locking was failed and laser exhibited modulation at the beat frequency f B due to the modal interference [20]. With changing the pump power, fast pulsations at f B 14.63 MHz superimposed on periodic oscillations at f RO 541.7 kHz appeared as shown in Figs. 7(b) and 7(c), indicating the subharmonic resonance of f RO f B ∕27.
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The stabilized multi-longitudinal mode oscillations with coherent vector fields has been demonstrated in a c-cut thin-slice Nd:GdVO4 laser. The observed phenomenon is expected in general vector lasers, which are formed of orthogonally polarized transverse modes [9], and it is promising for generating coherent vector fields with increased output powers and producing optical pulses. References 1. F. G. Anderson, P. L. Summers, H. Weidner, P. Hong, and E. E. Peal, Phys. Rev. B 50, 14802 (1994). 2. Y. Sato and T. Taira, IEEE J. Sel. Top. Quantum Electron. 11, 613 (2005). 3. Y.-F. Chen and Y. P. Lan, Appl. Phys. B 74, 415 (2002). 4. K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, IEEE J. Quantum Electron. 47, 638 (2006). 5. Y. Kozawa and S. Sato, Opt. Lett. 30, 3063 (2005). 6. K. Yonezawa, Y. Kozawa, and S. Sato, Opt. Lett. 31, 2151 (2006). 7. K. Otsuka and S.-C. Chu, Opt. Lett. 38, 1434 (2013). 8. R. Zhou, J. W. Haus, P. E. Powers, and Q. Zhan, Opt. Express 18, 10837 (2010). 9. Q. Zhan, Adv. Opt. Photon. 1, 1 (2009) and references therein. 10. Y. Asakawa, R. Kawai, K. Ohki, and K. Otsuka, Jpn. J. Appl. Phys. 38, L515 (1999). 11. Y. Miyasaka, T. Narita, K. Otsuka, H. Makino, and A. Okamoto, Opt. Express 13, 7928 (2005). 12. K. Otsuka, IEEE J. Quantum Electron. 14, 49 (1978). 13. J. Martin-Regalado, J. L. A. Chilla, J. J. Rocca, and P. Brusenbach, Appl. Phys. Lett. 70, 3350 (1997). 14. K. Otsuka, S.-C. Chu, C.-C. Lin, K. Tokunaga, and T. Ohtomo, Opt. Express 17, 21615 (2009). 15. E. Louvergneaux, G. Slekys, D. Dangoisse, and P. Glorieux, Phys. Rev. A 57, 4899 (1998). 16. K. Otsuka and H. Iwasaki, IEEE J. Quantum Electron. 12, 214 (1976). 17. K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. A 46, 1692 (1992). 18. K. Otsuka, Sensors 11, 2195 (2011) for review. 19. K. Otsuka, Jpn. J. Appl. Phys. 31, L1546 (1992). 20. K. Otsuka, J.-Y. Ko, T.-S. Lim, and H. Makino, Phys. Rev. Lett. 89, 083903 (2002).
