1. Introduction

Picosecond mode-locked lasers are widely used in laser material processing, nonlinear optics and time-resolved spectroscopy [1]. Passive mode-locking (ML) technique based on intracavity nonlinear frequency doubling process has been shown a promising technique to produce a high average power and wide operating range [2]. When the cavity output coupler is chosen to have partial reflectance coating at fundamental wave (FW) and high reflectance (HR) coating at second harmonic wave (SHW), fractional back conversion of SHW into FW arises once the phase control between two waves is appropriated. Therefore, the reflectivity of the FW exhibits intensity-dependent called nonlinear mirror mode-locking (NMML) [3]. Due to the limitation of group velocity mismatched (GVM) between the FW and SHW, the ML pulse duration was usually longer than 10 ps. When the frequency doubling process is operated under phase-mismatched condition, the accumulated nonlinear phase of the FW mimics effective nonlinear index of refraction. In the low SHW conversion regime (pump nondepletion regime), the $n_2^{eff}$ can be approximated by $n_2^{eff}=-\frac{L}{\Delta kL}\frac{4\pi d_{eff}^2}{n_{\omega }^2{n}_{2\omega }\lambda {\epsilon }_{0}c}$ [4], where $\Delta kL$($\Delta kL=(k_{2\omega }-2k_{\omega })\times L$) is the phase-mismatched term in the frequency doubling process, L is the length of nonlinear crystal, $d_{eff}^2$ is the square of effective nonlinear coefficient, λ is the wavelength of FW, c is velocity of light, and n_ω、n_2ω are the refractive index of FW and SHW, respectively. The sign of $n_2^{eff}$ can be further manipulated by the phase-mismatched condition, ∆kL$,$ and the magnitude of $n_2^{eff}$ is at least 1-order larger than the intrinsic n₂ (χ³ process) [5]. Soliton ML pulse has been demonstrated by compensating inherent positive group velocity dispersion (GVD) of laser cavity with χ² induced negative $n_2^{eff}$, creating down chirped effect, and no compensating elements are used [6]. Sub ps pulse duration has also been demonstrated in Yb:YAG laser with soliton-like pulse shaping [7]. Using positive $n_2^{eff}$ scheme, the reported pulse duration was 10.3 ps and time-bandwidth product (TBP) was 1.56 times transform limited value [8]. Since most of laser gain medium and nonlinear crystal are inherent positive Kerr effect [4], combined with positive $n_2^{eff}$ scheme would lead to strong 3-order nonlinearity which is beneficial to ML laser stability, self-starting and self-sustaining [9,10]. The following discussion shows a passive mode-locked Nd:YVO₄ laser using MgO:PPLN with positive $n_2^{eff}$ scheme. Combining large $n_2^{eff}$ and soft aperture effect, the transformed amplitude modulation is achieved and ensured self-starting as well as self-sustaining ML pulse trains. While varying the phase-mismatched conditions, nonlinear phase fulfil an important role in affecting the performance of mode-locked laser.

2. Experimental setup

Figure 1 shows a schematic of cascaded mode-locked Nd:YVO₄ laser by an intracavity second harmonic generation in a 5 mol.% MgO-doped periodically poled lithium niobate (MgO:PPLN) crystal. The intracavity frequency doubler was a MgO:PPLN crystal with 5 mm long and Λ = 6.92 μm grating period phase-matched at 1064 nm when the oven temperature was fixed at 55 °C. A 3 mm long undoped YVO₄ crystal diffusion bonded to a a-cut, 6 mm long 0.4-at.% Nd-doped YVO₄ crystal with an aperture of 3x3 mm² was used as the gain medium and polished with 1 degree wedge to prevent etalon effect. The undoped YVO₄ crystal can help to reduce the thermal effect and operate in the high pump power regime [11]. The incident surfaces of the gain medium were coated with anti-reflection coating (R<1%) at 808, and 1064 nm. To dissipate the heat generation, the laser crystal was placed in a copper block for water cooling at 15 °C. The core radius of the pump diode laser was 100 μm and refocused into the gain medium through a set of 1 to 1 coupling lenses. The V-folded cavity was constructed by the input coupler (M₁), mirror M₂, and output coupler (OC). Two intracavity lenses f₁ (f = 250 mm) and f₂ (f = 125 mm) were used to ensure the beam radius of ~90 μm in the MgO:PPLN crystal and ~300 μm in the gain medium when the thermal focal length of Nd:YVO₄ was ~200 mm. The distance between the gain medium and M₁ was less than 0.5 mm to utilize the spatial hole burning effect [12]. M₁ and M₂ were flat mirrors with HR coating (R>99.8%) at 1064 nm and high transmittance (HT) coating (T>95%) at 808 nm. The OC was chosen to have partial reflectance (R = 78%) at 1064 nm and HR (R>99%) at 532 nm. The HR coating at SHW can utilize part of nonlinear mirror ML effect which was helpful to stabilize the ML pulse [6]. The SHW output power was measured through M₂ which has 50% transmission at 532 nm. The distance between the MgO:PPLN and OC was set around 5 mm. The total cavity length was 805 mm corresponding to 186 MHz ML repetition rate. The soft aperture effect combined with poor spatial overlapping between the pump and cavity beam was beneficial to generate self-starting ML pulse [10].

 

Fig. 1 Experimental configuration of a cascaded mode-locked YVO₄/Nd:YVO₄ laser by an intracavity second harmonic generation in a MgO:PPLN crystal. The ML cavity was formed by the flat mirror M₁, M₂ and the output coupler (OC).

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3. Experimental results and discussions

To compare the ML performances of the bonded laser crystal, we varied the phase-mismatched conditions by changing the MgO:PPLN temperature and crystal arrangement by flipping Nd:YVO₄ and YVO₄ sections. The lens, L₂, was fine tuned to keep the same focusing condition of pump beam in Nd:YVO₄ section. L₂ was moved forward, 1~1.5 mm, to the laser crystal when the YVO₄ was switched in the front. To produce the positive cascaded Kerr effect, the temperature of MgO:PPLN was initially fixed at T = 41.5 °C which generated negative phase-mismatched, $\Delta kL\sim -5\pi$. The induced positive kerr lens by the cascaded second-order nonlinearity and soft aperture effect in the gain medium were helpful to stabilize the ML pulses, as depicted in Fig. 2. The filled and open dots represent the crystal arrangement of Nd:YVO₄/YVO₄ and YVO₄/Nd:YVO₄, respectively. When the pumping power was increased to ~10.5 and 12.5 W for two arrangements, the self-starting continuous-wave ML (CWML) pulses were observed. The Q-switched ML (QML) was observed between pump power of 9.2~10.4 W and 11.3~11.9 W for Nd:YVO₄/YVO₄ scheme. Strong thermal lensing induced thermal runaway and unstable cavity when the pump power was higher than 12 W. However, moderate thermal lensing was benefited in low pump power regime which contribute better spatial overlapping between the cavity and pump beam. In the YVO₄/Nd:YVO₄ scheme, YVO₄ was helpful to operate in higher power regime and the QML was observed between the pump power of 11.5 ~12.3 W and 13~15.7 W. The CWML pulses were always self-starting when the pump power was reached threshold and self-sustaining more than several hours.

 

Fig. 2 Measured output power and temporal behavior as a function of incident pump power in Nd:YVO₄/YVO₄ (filled dots) and YVO₄/Nd:YVO₄ (open dots) configurations.

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The temporal characteristic of CWML was monitored by an oscilloscope (1 GHz bandwidth) and RF spectrum analyzer (2 GHz bandwidth) with a fast photodiode (175 ps rise time). Figures 3(a) and 3(b) show a typical time span of 20 ns and 500 s oscilloscope signal which represent a pure CWML without QML background. The radio frequency spectrum showed a signal to noise level of 64 dB for the fundamental harmonic which displayed the pure CWML and no multipulsing, as shown in Fig. 3(c). Figure 3(d) shows high order harmonic beat in the span of 2 GHz. Due to the limited bandwidth of photodiode and spectrum analyzer, high harmonic beat notes decayed slightly. The ML bandwidth was measured by a scanning monochromator with 0.1 nm resolution. The cw Nd:YVO₄/YVO₄ laser spectrum labeled in filled triangle was measured as a reference to calibrate the monochromator, as depicted in Fig. 4. Filled square and open circle dots represent the ML spectrum of Nd:YVO₄/YVO₄ ($\Delta {\lambda }_{FWHM}$ = 0.76 nm) and YVO₄/Nd:YVO₄ ($\Delta {\lambda }_{FWHM}$ = 0.33 nm) configurations, respectively. The red shifted spectrum in Nd:YVO₄/YVO₄ scheme indicates the self-phase modulation induced by the cascaded n₂ effect. The ML pulse duration was monitored by an intensity autocorrelator (APE pulseCheck), as can be seen from the inset of Fig. 4. When a sech² pulse shape was assumed, the FWHM pulse durations were measured to be 2.8 and 5.9 ps in Nd:YVO₄/YVO₄ and YVO₄/Nd:YVO₄ configurations, respectively. The TBP was calculated to be 0.57 for both configurations, corresponding to 1.8 times transform limited. Without group velocity delay compensation, positive chirped pulse was expected. Comparing the performance of two configurations, YVO₄/Nd:YVO₄ scheme has longer pulse duration due to the weak spatial hole burning effect introduced by the 3 mm long YVO₄ section. Although the thermal effect was reduced by YVO₄ and operated in the higher pump power regime, the mode-locking pulse width was broadened by the reduced spatial hole burning effect [12].

 

Fig. 3 Typical temporal and spectral behaviour of ML pulses: (a) Oscilloscope trace in 20 ns time span, (b) Oscilloscope trace in 500 s time span, (c) Common-mode rejection ratio of the first beat note, (d) Harmonic beat notes span in 2 GHz bandwidth.

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Fig. 4 Measured bandwidth of Nd:YVO₄/YVO₄ ML pulse (filled square), YVO₄/Nd:YVO₄ ML pulse (open circle), and cw laser (filled triangle). The autocorrelator traces of Nd:YVO₄/YVO₄ (filled square) and YVO₄/Nd:YVO₄ (open circle) configurations were presented in the inset.

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To further study the ML performances at different phase mismatched conditions, we altered the MgO:PPLN temperature. Figure 5 shows the FW and normalized SHW output power at varied phase-mismatched conditions. Since a stable ML pulse was related to pump power, the pump power was fine-tuned at various crystal temperatures. To keep the same order cascaded χ⁽²⁾ effect at large phase-mismatched condition ($\Delta kL\sim -8\pi$), the ML pump power was found to be slightly higher than crystal temperature closed to phase-matched. 1 W pump power difference was observed when we varied the crystal temperature from 33 to 52 degree. Compare the output performance, the SHW conversion efficiency in Nd:YVO₄/YVO₄ was two times higher than YVO₄/Nd:YVO₄ scheme because of the short pulse duration of FW wave corresponding to high peak power which had better conversion efficiency in second harmonic generation process. More than 100 mW green ML pulses were generated when the $\Delta kL>-5\pi$. However, high output power of SHW wave perturbs and ceases the generation of stable cw ML when the $\Delta kL>-3\pi$ in Nd:YVO₄/YVO₄ scheme. Sometimes CWML pulses with strong cw background were also observed in this regime. In YVO₄/Nd:YVO₄ scheme, the stable CWML pulses can be maintained between −8π$$and -π$$phase-mismatched condition. A slightly longer ML pulse due to weak spatial hole burning help to restrain the generation of SHW and to sustain wide temperature range. The output power of SHW was lower than 60 mW in the whole operation range. The FW power was decreased from 1.6 to 1 W due to the adjustment of pump power to maintain the stable ML. Although the behavior of output power was strongly related to phase-mismatched conditions, the pulse durations were mostly limited by the group velocity mismatched (GVM) and spatial hole burning effects, as shown in Fig. 6. The red dash line shows GVM (8 ps/cm) multiply two times of coherent length at varied crystal temperature which is the shortest pulse duration could be obtained through this cascaded process. Around 4 and 6 ps pulse durations were observed in Nd:YVO₄/YVO₄ and YVO₄/Nd:YVO₄ scheme, respectively. Although the GVM is the limitation of pulse duration in near phase-matched region, it facilitates pulse shortening in negative cascaded ML scheme [13,14]. A round trip cavity GDD is calculated to be few fs level which has little influence on pulse broadening. To generate short pulse duration, adopt short coherent length or low GVM χ⁽²⁾ material, such as LBO (GVM = 0.87 ps/cm for 1064/532 process) crystal could help to reduce this fundamental limitation [7]. Maintaining the same order effective Kerr nonlinearity in LBO crystal, high intracavity intensity and small phase-mismatched are required to compensate the low nonlinear coefficient (d_eff,LBO = 0.85 pm/V). Another constrain might come from the interaction bandwidth including the inherent gain bandwidth or acceptance bandwidth in the second harmonic generation process [15].

 

Fig. 5 Varying phase mismatched condition versus measured FW and normalized SHW output power in Nd:YVO₄/YVO₄ (square dots) and YVO₄/Nd:YVO₄ (circle dots) configuration.

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Fig. 6 Temperature tuned minimum GVM-allowed (simulation, dash line) and measured ML pulse durations of Nd:YVO₄/YVO₄ (filled square) as well as YVO₄/Nd:YVO₄ (open circle) scheme.

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Figures 7 and 8 show the measured ML bandwidth and TBP divided ideal product of a sech² shape at varied phase-mismatched conditions. As expected, the Nd:YVO₄/YVO₄ scheme has wider ML bandwidth than YVO₄/Nd:YVO₄ scheme due to the strong spatial hole burning effect. Largest ML bandwidth, $\Delta {\lambda }_{FWHM}$ = 0.9 nm, was recorded at $\Delta kL\sim -4\pi$. To explain the broaden bandwidth where the SHW conversion efficiency was more than 5% (pump depletion regime), simply plane wave coupled wave equations were used to calculate nonlinear phase [16]. Average pump intensity, I_ave = P_peak/(πω²), and effective nonlinear coefficient of MgO:PPLN, d_eff = 14 pm/V, are used in the calculation. Adopting the measured average FW power, pulse repetition and pulse durations at varied phase mismatched conditions, the single-pass nonlinear phase shift, ϕ_nl, can be calculated by Runge–Kutta method [17]. New frequency components from self-phase modulation (SPM) can be estimated as Δω~2 × ϕ_nl/Δt [18], where Δω is the new frequency from SPM and $\text{Δ}t$ is the ML pulse duration. Open stars and diamonds represent the broaden bandwidth from the SPM process. In Nd:YVO₄/YVO₄ scheme, the increased tendency of ML bandwidth between −7.5π <ΔkL< −4π was quite matched the calculation model. The cascaded Kerr coefficient, $n_2^{eff}$, can be also calculated from $n_2^{eff}=\lambda {\varphi }_{nl}/(2\pi I_{ave}L)$ [5] and found in the range of 2 × 10⁻¹⁷ (∆kL~-8π) ~5 × 10⁻¹⁷ m²/W (∆kL~-2π). The large effective nonlinear index of refraction combined with soft aperture effect tolerances the ML operating conditions, usually take ∆kL~-π [8]. Since the nonlinear refractive index of Nd:YVO₄ (n₂ = 1 × 10⁻¹⁸ m²/W) [19] and MgO:LN (n₂ = 2 × 10⁻¹⁹ m²/W) [20] are at least one-order smaller than $n_2^{eff}$in this report, cascaded Kerr effect is still dominated the mode-locking process. Soft aperture effect was also studied by changing the pump beam focusing conditions. Stable and self-starting ML pulse can be supported when the ratio between cavity and pump beam size in the gain medium is larger than 2. Although the slight increment of TBP is due to the flattened spectral gain by spatial hole burning effect, the nonlinear phase further aggravates the chirping effect when ∆kL was reached −4π, as shown in Fig. 8. However, the increased tendency of bandwidth was reversed when ∆kL> −4π. We speculated the cascaded ${\chi }^{\left(3\right)}$ process was restrained due to the diminished FW power which was converted to SHW. In YVO₄/Nd:YVO₄ scheme, the ML bandwidth and TBP were preserved around 0.3 nm and 0.567, respectively. Comparing with Nd:YVO₄/YVO₄ scheme, almost two times longer pulse duration and reduced pump intensity lead to smaller nonlinear phase effect which does not dominate the ML bandwidth. To examine the relation between ML power stability and nonlinear phase, 10 minutes power stability measurement was also conducted while changing the crystal temperature and crystal arrangement. The r. m. s. values were all less than 1% in most cases expecting the temperature closed to the phase-matched condition (∆kL> −2π). No obvious relation was observed between ML power stability and nonlinear phase.

 

Fig. 7 Measured ML and calculated SPM bandwidth at varied phase-mismatched conditions.

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Fig. 8 Calculated time-bandwidth product divided ideal product of a sech² shape at varied phase-mismatched conditions.

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4. Conclusion

We have presented the influence of nonlinear phase in a passive mode-locked picosecond laser at 1064 nm based on the positive cascaded second-order Kerr lens. Adopting mismatched MgO:PPLN crystal in an intracavity second-harmonic process and soft aperture configuration, self-starting and self-sustaining CW ML pulses were achieved at a pump power higher than 11 W. The output ML pulse trains have a repetition of 186 MHz, more than 1 W output power and pulse duration of ~3 ps which is about 3 times shorter than the published result [8]. The ML bandwidth and TBP can be further manipulated by nonlinear phase which create new frequency component in SPM process.