## Abstract

Large nonlinear phase shift achieved by exploiting intracavity second order cascaded nonlinear process in a non-phasematched second harmonic generating crystal is transformed into amplitude modulation through soft aperturing the nonlinear cavity mode variation within the laser gain medium to mode-lock a Nd:YVO_{4} laser. The laser delivers stable dual wavelength cw mode-locked pulse train with pulse duration 10.3 ps and average power of 1.84 W and 255 mW at 1064 nm and 532 nm respectively for a pump power of 12 W. A comprehensive theoretical analysis finds the regime of self starting and stable cascaded second order mode-locking, inconformity with the experimental result.

© 2013 Optical Society of America

## 1. Introduction

Nonlinear phase shift, when suitably converted into nonlinear amplitude modulation in a laser cavity, one can achieve mode-locking of laser for the generation of ultrashort pulse train. Kerr-lens mode-locking (KLM) [1,2] which exploits self focusing effect due to positive nonlinear phase distortion in the gain medium itself due to intrinsic third order nonlinear susceptibility (*χ*^{(3)}), has been found to be very efficient and reliable in femtosecond (fs) regime. Although there is only few reports on Kerr-lens mode-locked Nd:YAG laser [3], however, in picosecond (ps) regime KLM has not been popular because the lower intracavity peak power is generally inadequate to drive the required nonlinear loss modulation due to weak intrinsic *χ ^{(}*

^{3)}of the medium. Cascading of two second order nonlinear processes can induce large equivalent third order susceptibility $({\chi}_{eff}^{(3)})$ and thus can mimic the effects of third order nonlinear optical processes in a second order nonlinear optical crystal even at comparatively much lower power [4,5]. Thus cascaded second order nonlinearity demands attention for efficient and stable mode-locking in ps regime. While efficient, stable and reliable mode-locking techniques are available in fs regime, it has always been a challenge to achieve stable passive mode-locking in ps regime due to the limitations in the bandwidth of the popular laser crystals like Nd:YAG or Nd:YVO

_{4}used for ps pulse generation. Pulse width tunable Ti:Sapphire laser when used for passive mode-locking, stable ps pulse train can be extracted. However, exploiting the full bandwidth of rare earth doped crystals for generation of ps pulse train by passive mode-locking suffers from instability. Search for passive mode-locking technique continues paying great concern to simplicity in design and fabrication, efficiency and reliability of the schemes. Mode-locking of solid state lasers in ps regime has been achieved through induced ${\chi}_{eff}^{(3)}$ in two schemes. In one scheme, called nonlinear mirror mode-locking (NLM) [6,7], a second harmonic generation (SHG) crystal is placed near the dichroic output coupler (OC), which reflects the fundamental wave (FW) partially and the second harmonic (SH) beam totally. The FW generates SH in its first pass, and if the SH beam experiences a proper phase shift (-

*π*) with respect to the FW beam, the SH power is almost totally reconverted into FW during the second pass through the nonlinear crystal (NLC). The combination of the nonlinear crystal and the dichroic OC behave as nonlinear mirror having intensity dependent reflection coefficient. Under this condition, laser loss decreases with input intensity and behave as fast saturable absorber. The forward and backward interaction in NLC can be thought of a cascaded second-order interaction and the nonlinear reflectivity thus obtained is due to the negative, imaginary part of ${\chi}_{eff}^{(3)}$. In NLM, choice of the OC is very crucial and the reflectivity of the OC for SH should be greater than the FW to provide nonlinear loss saturation. NLM is a simple technique for ps pulse generation but due to high depth of modulation and large saturation power of NLM, Q-switching interference is usually present and the laser runs in an unstable Q-switched and mode-locked (QML) regime. However, one can either utilize the QML regime for high peak power through active stabilization of the Q-switch envelope [8–10] or can passively suppress the QML instability by inverse loss saturation to achieve cw mode-locking [11]. In the other scheme, called cascaded second order nonlinear mode-locking (CSM) [12,13], the SHG crystal is placed in off phase-matched condition and uses the real part of the induced ${\chi}_{eff}^{(3)}$. It does not provide direct amplitude modulation as in case of NLM, rather forward and return pass through the NLC imprints a nonlinear phase shift (ΔΦ

*) on the FW beam giving nonlinear mode size variation, which in turn can be converted into nonlinear loss modulation with a suitably placed aperture inside the laser cavity. The scheme is analogous to that of KLM but effective in ps regime as it uses large induced ${\chi}_{eff}^{(3)}$ originating from cascaded second order nonlinearity. The placement of the aperture however is critical since it has to be placed accurately at the intra-cavity position where mode-size variation is maximum. Cerullo*

_{NL}*et al.*in their first demonstration of CSM used hard aperture and several dispersive elements which could have significant effect on pulse duration [12]. The nonlinear mode variation was maximized by playing with the resonator stability through changing the positions of the intracavity lenses, which increases the misalignment sensitivity. Theoretical investigation on CSM by the same group was made using operator formalism although the details and complete insight to the mechanism was due [13]. CSM with soft aperture has been demonstrated by Holmgren

*et al.*[14], but the laser is very inefficient as it delivers only 350 mW of output power for a pump power of 16.5 W. The theoretical analysis presented in their report relies on the calculation of diffraction loss and cannot give significant depth in understanding of CSM. The theoretical and experimental study on the nonlinear phase shift and its variation with phase mismatch and intracavity power is of absolute importance for CSM, which, is however, absent in their report and thus cannot provide enough insight to the mechanism and control on the pulse formation. In recent time Iliev et.al. reported experimental study on mode-locking of Nd:YVO

_{4}and Nd:GdVO

_{4}laser, through intracavity SHG in periodically poled crystals and conclusively inferred that the NLM plays significant role in pulse shortening mechanism at perfectly phase-matched condition while the CSM is effective at off-phasematched condition [15,16]. The effect of NLM can indeed be neglected at off-phasematched condition. High power operation of CSM with Nd:YVO

_{4}laser has also been reported very recently where stable mode-locked operation is achieved by controlling the position of the output coupler, however, no theoretical model has been presented [17].

In the present report we demonstrate for the first time, to the best of our knowledge, soft aperture dual wavelength CSM delivering pulse train of width 10.3 ps in both 1064 nm and 532 nm simultaneously from a single laser oscillator. Without using any physical aperture, the nonlinear loss modulation is achieved through reducing mode size with intracavity power at the gain medium and thus increasing effective gain for a short duration through better overlapping between pump and the cavity mode size. Theoretical investigation on the nonlinear phase shift and it’s variation on intracavity intensity and the phase mismatch is carried out to understand the behaviour of intensity dependent refractive index of the nonlinear crystal, arises due to the cascaded second order nonlinear interaction. Finally a direct insight to the CSM mechanism is obtained by calculation of the nonlinear variation of the cavity mode size at the gain medium and comparing the theoretical results with the experiment. The theoretical and experimental work being reported here is carried out at the Dept. of Physics, IIT Kharagpur. The developed dual wavelength picosecond source can find application in soft tissue ablation, photo chemistry, nondegenerate pump probe experiment, biomedical application and micro-structuring. The developed laser can also be used for efficient generation of picosecond UV pulses, which has significant importance for generation of colour centre in dielectric, time resolved photoluminescence and Raman study.

## 2. Experiment

The schematic of the laser layout is shown in Fig. 1. The pump is a 808 nm fiber coupled laser diode array of maximum output power 12 W and is focused to a spot size of 240 µm at the centre of the gain medium which is a 4x4x8 mm^{3}, a-cut, Nd:YVO_{4} crystal having Nd^{3+} concentration of 0.5%. The rear face of the RM is anti-reflection coated at 808 nm and has high reflectivity (>99.5%) at 1064 nm on the other side. The Nd:YVO_{4} crystal is anti-reflection coated on both of its parallel faces for wavelengths 1064 nm and 808 nm and is tilted by an angle of 2° to reduce the effect of undesired satellite cavity. Two concave mirrors M_{1} and M_{2} of radius of curvature 500 mm and 250 mm respectively are used to focus the beam at the OC. The tilting angle of M_{1} and M_{2} is kept as low as possible to avoid cavity astigmatism and thus providing TEM_{00} mode throughout the cavity which has been found to be necessary for stable cw ML operation.

A KTP crystal of length 9 mm, cut for type-II SHG (θ = 90°, φ = 23.5°) of 1.064 µm, is placed near the OC. The OC is chosen to have reflectivity of 96% at 1064 nm and 20% at 532 nm such that maximizing intracavity power the cascaded effect can be enhanced as well as optimum SH power can be coupled out . The different arms of the Z-shaped passive cavity of length 81.7 cm are optimized to be 250 mm, 405 mm and 162 mm respectively which gives a cavity mode size of 442 µm at the gain medium and 213 µm at the centre of the KTP crystal, as determined by the ABCD matrix formalism. No mode-locking is observed when the KTP crystal is set at exact phase matched condition (φ = 23.5°) corresponding to $\Delta kL=({k}_{2\omega}^{e}-{k}_{\omega}^{o}-{k}_{\omega}^{e})L=0$, where *k _{i}* are the wave numbers and

*L*is the length of the KTP crystal. As the crystal is detuned from phase matching by an angle more than 1° in the direction of increasing φ, mode-locking is found to set in but initially with a smaller depth of modulation of the mode-locked pulses. Depth of modulation of the mode-locked pulses is found to increase as the phase mismatch angle (Δφ) is continued to increase in the same direction and full depth of modulation is achieved around Δφ ≈1.5°. As the crystal is further detuned, the depth of modulation decreases and when the phase mismatch angle increases over 2°, mode-locking finally disappears. However, no mode-locking is observed for any phase mismatch in the other side of the phase matching angle. Most stable operation with full depth of modulation of the mode-locked pulses is found to occur when the crystal is set approximately by an angle of 25°20

^{′}. Finally the laser is optimized for stable mode-locked operation by varying the separation between KTP and the OC and the distance is finally kept $\sim 3\text{mm}$for optimum operation.

## 3. Results

The laser generates mode-locked pulse train simultaneously at both 1064 nm and 532 nm with pulse repetition rate 178 MHz. In Fig. 2(a), we show a typical measured trace of non-collinear back ground free intensity autocorrelation, employing a 3 mm long BBO crystal, for 1064 nm pulse. The FWHM of the autocorrelation trace is measured to be 15.9 ps, corresponding to a pulse width of 10.3 ps for assumed *sec*h^{2} pulse shape. The spectrum of the pulse is shown in Fig. 2(b) as measured by a grating based spectrometer having resolution of 0.08 nm. The spectral bandwidth is measured to be 0.18 nm which is greater than 0.11 nm, the transform limited bandwidth corresponding to 10 ps pulse. The ML sets in for input pump power of 9 W and maximum mode-locked output power achieved are 1.84 W and 255 mW at 1064 nm and 532 nm respectively corresponding to the input pump power of 12W (Fig. 3).

Owing to the high effective nonlinear coefficient of KTP (3.5 pm/V) and the intra-cavity effect, power obtained at 532 nm and at 1064 nm, proves the laser to be an efficient dual wavelength ps source. More over the output power can be scaled up since the ML scheme is not restricted by the input pump power. The ML regime is found to be highly stable in both short and long time scale and is never affected by the Q-switching instability. The stability of the mode-locking is evident from oscilloscope trace of the mode-locked pulse train in different time scale (Fig. 4). The output power fluctuation is less than 1% and does not drop even after running for hours. The pulse amplitude fluctuation as measured in oscilloscope is less than 5% rms.

## 4. Theoretical analysis of CSM

To gain a deep insight to the CSM mechanism and optimize it appropriately, it is absolutely necessary to calculate the nonlinear phase shift suffered by the FW beam after round trip through the KTP crystal. The FW beam, in it’s forward pass through KTP generate SH under type-II interaction. If the phase difference$\Delta \vartheta ={\vartheta}_{2\omega}-2{\vartheta}_{\omega}$, accumulated between the SH and FW beam due to propagation between the KTP and the OC is properly adjusted the SH, in its return pass through KTP will be completely down converted to FW. These two cascaded processes can theoretically be modeled using the coupled amplitude equation given by

*A*and

_{FW}*A*are the field amplitudes of fundamental and second harmonic respectively,

_{SH}*n*and

_{ω}*n*are the refractive indices of FW and SH and

_{2ω}*d*is the effective nonlinear coefficient for the interaction. Since the sample length (

_{eff}*L*= 9 mm) is much smaller than the Rayleigh length (~130 mm), we can neglect diffraction effects and treat every point on the beam cross section as independent of others. So the plane wave approximation is valid enough for the beam propagation through the KTP crystal only. Normalizing the field amplitudes to the initial fundamental amplitude ${A}_{FW}(0)$ and setting $\xi =z/L$, we get the following dimensionless equations:

*, represented by the color code (in units of*

_{NL}*π*), as a function of intensity and phase mismatch, $\Delta kL$ (in units of

*π*).

We see that the phase shift changes sign at the phase-matching angle. Thus the cascaded interaction gives rise to a self-focussing or defocussing nature similar to that of a Kerr medium, with an intensity dependent refractive index given by:

where,*I*is the intracavity fundamental intensity at KTP crystal. For large nonlinear phase shift, when ΔΦ

*varies linearly with the intensity [4], ${n}_{2}^{eff}$ can be calculated from (assuming nonlinear phase shift of ΔΦ*

_{NL}_{NL}/2 per pass):It is clear that for $\Delta kL<0$, the nonlinear phase shift is positive and the KTP crystal behaves as a self-focussing medium with positive ${n}_{2}^{eff}$ and on the other side, for $\Delta kL>0$, we have a negative phase shift giving rise to a self-defocussing behaviour with negative ${n}_{2}^{eff}$.

Finally we apply the formalism for closed form Gaussian beam analysis of a resonator with an intracavity Kerr medium [18,19] to calculate the nonlinear mode size variation at the laser gain medium with the intracavity peak power. Our analysis shows that mode size at the laser gain medium decreases with an increase in the peak power when the cascaded nonlinear process makes to behave the KTP crystal as a self-focussing medium corresponding to a positive nonlinear phase shift. Figure 6 shows the mode size variation at the laser gain medium as a function of intracavity peak power for different values of Δ*kL*. Comparing Fig. 5 and 6, it is evident that for Δ*kL* = -*π*, the nonlinear phase shift and also the self-focussing power of the KTP reaches maximum, resulting in a maximum mode size variation at the laser gain medium. Thus at Δ*kL* = -*π*, corresponding to the KTP crystal angle φ = 25°5′38^{″}, the driving force for the CSM, analogous to the KLM [20,21], would reach a maximum value. At this optimum value of Δ*kL*, one should expect a self starting and self sustained mode-locking and it agrees remarkably well with our experimental findings. In fact the driving force corresponding to the optimum value of Δ*kL* is so strong that one does not even need to control the cavity elements to work at the limit of geometrical cavity instability, rather can safely work in the stable cavity regime.

## 5. Discussion

CSM is found to be an efficient technique for the generation of stable picosecond pulse, never being affected by the Q-switching instability. That is probably due to the inverse saturation effect. When the pulse formation starts, the peak intensity increases, and the cavity mode size deceases at the laser gain medium. This increases the gain through better overlapping between pump and the cavity mode. However, as the pulse get stronger, the cavity mode size at the laser gain medium can even become smaller than the pump mode, which may cause to increase the resonator loss giving rise to inverse loss saturation at higher intensity. Inverse loss saturation resist the mode-locking from suffering the Q-switching instability, but simultaneously broadens the pulse width [11,22,23]. This may be the reason for not getting the bandwidth limited pulse width in CSM. The dynamics of pulse formation and the stability analysis of CSM will be reported in our next communication which will hopefully be able to quantify the stability regime of CSM. However, another significant contribution to the broadening of pulsewidth is certainly the group velocity mismatch (GVM) between the FW and SH in KTP [13]. Some times under the action of the cascaded second order interaction KTP may exhibit quadratic polarization switching, which may contribute in mode-locking mechanism [24]. If there is any appreciable polarization change in the KTP, the output beam would contain that signature. We analyze the output beam with a polarizer and find that the 98% of the total power of 1064nm is with vertical polarization. We also calculate the phase retardation in the KTP due to cascaded process, in line of Ref [25], with the parameters used in the present experiment and it reveals that insignificant change in polarization may only occur. These make us to infer that the change in the polarization is too small to contribute in the mechanism of mode-locking for the present scheme. Some times mode-locking based on intracavity SHG may contain contribution from both NLM and CSM mechanism depending on the the choice of output coupler and the phase-matching condition. In the present scheme, the KTP is set at off-phasematched condition and for a stable mode-locking with full depth of modulation the phase mismatch angle is found to be 1.5°, which is also consistent with the theoretical prediction. In such off-phasematched condition the pulse shortening mechanism due to the NLM can be neglected [15,16] and the achieved mode-locking is thus only due to the nonlinear lensing produced from second order cascading. The OC that has been used in this experiment has a reflectivity greater for the fundamental wavelength (*R _{1064}* = 96%) than the second harmonic (

*R*= 20%). In this case, the reflectivity of the nonlinear mirror comprising of SHG crystal and dichroic OC, decreases and thus the nonlinear saturable loss increases with the intracavity peak power and as such it cannot help NLM. However, the increase in nonlinear loss with the intracavity peak power may contribute to the inverse saturation effect. The reflectivity of the OC plays an important role in determining the duration of the mode-locked pulse. The pulse width is found to decrease with the increase in overall reflectivity of the OC [15,16]. In the present report our aim is to design the CSM laser in such a way that the maximum output power at the SH can be obtained in addition to the fundamental but without compromising the mode-locking stability and thus 80% of the SH is coupled out. This is another reason for not achieving the bandwidth limited pulse width. One may get a shorter pulse by increasing the reflectivity at the SH and thus compromising with the output power of the SH radiation.

_{532}## 6. Conclusion

Second order cascaded nonlinear interaction in an intracavity nonphasematchesd type-II SHG KTP crystal produces large nonlinear phase shift on the FW wave and induces large $\mathrm{Re}({\chi}_{eff}^{(3)})$to make the KTP crystal to behave as a Kerr media. For a positive ΔΦ* _{NL}*, KTP exhibits self-focussing behaviour, which under closed form Gaussian beam analysis shows the cavity mode at the gain medium to decrease with the power and gives direct nonlinear loss modulation through better overlapping between pump and the cavity mode resulting in an efficient, stable CSM. The developed soft aperture CSM laser generates dual colour picosecond pulses of width 10.3 ps at 1064 nm and 532 nm simultaneously from a single resonator. The laser delivers average power of 1.84 W and 255 mW at 1064 nm and 532 nm respectively corresponding to a pulse repetition rate of 178 MHz. The theoretical analysis shows the ΔΦ

*and ${n}_{2}^{eff}$to reach a maximum for Δ*

_{NL}*kL*= -π, which in turn provides strong CSM driving force through maximum mode size variation with power. The theoretical analysis agrees well with the experimental observations and finds the regime of stable, self starting and self sustained CSM.

## Acknowledgments

S. Mukhopadhyay gratefully acknowledges Department of Higher Education, Govt. of West Bengal and Indian Institute of Technology Kharagpur for necessary support. S. Mondal acknowledges CSIR, Govt. of India for maintenance fellowship and P. K. Datta acknowledges DRDO (ERIP), Govt. of India, and DST, Govt. of India, for financial support. Prof. S. K. Bhakta is acknowledged for useful discussion.

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