1. Introduction

Passive mode-locking of high-power (≥5W), diode-pumped, continuous-wave (CW) solid-state lasers has been achieved using several techniques. Most notably, saturable Bragg reflectors (SBRs) have been used to mode-lock a 60W thin-disk Yb:YAG laser [1], a 20W side-pumped [2] and 23.5W end-pumped [3] Nd:YVO₄ laser, a multi-head 27W Nd:YAG laser [4], and a 5.4W Nd:GdVO₄ laser [5]. Over 5W has also been generated from a nonlinear mirror (NLM) mode-locked Nd:YVO₄ laser [6] and recently 20W has been demonstrated from an additive pulse mode-locked Yb:YAG laser [7]. For mode locked (ML) operation with amplitude modulation, semiconductor-based SBRs have received particular attention due to their operational simplicity and the ability to tailor their properties (i.e., spectral response, modulation depth, and relaxation time) to the characteristics of the laser [8]. However, to achieve CW ML operation with a SBR, the intracavity pulse energy must exceed a critical threshold energy E_c [9], below which Q-switched ML operation that can damage the SBR occurs [2]. This requirement poses a design problem for high-power lasers that often employ large spatial mode areas A in the laser gain medium since E_c is proportional to $\sqrt{F_{L,\mathit{sat}.}A}$ [9,10], where F_L,sat is the saturation fluence of the gain medium. In this paper, we demonstrate that this limitation can be overcome by the insertion of a phase-mismatched second harmonic generation (SHG) crystal in a laser cavity. The phase locking produced by the self-phase modulation (SPM) associated with the cascaded χ⁽²⁾:χ⁽²⁾ process in the SHG crystal [11] assists (or stabilizes [10,12]) the mode locking process initiated by the amplitude modulation in the SBR, thereby (i) significantly reducing the CW ML threshold [12] and (ii) shortening the duration of the generated pulses [13].

The all solid-state, dual-passive mode locking technique is an extension of the scheme used by Agnesi et al. to stabilize the CW ML operation of Nd:BaY₂F₈ [12] - a gain medium with a small stimulated emission cross-section (and hence large F_L,sat.). Similar CW ML schemes employing a combination of amplitude modulation and phase locking via a second-order optical nonlinearity have been demonstrated previously. In particular, G. Cerullo et al. [14] achieved CW ML operation of a diode-pumped Nd:YAG laser by using the inherent phase-locking (i.e., SPM) of cascaded second order nonlinearities in a LiB₃O₅ (LBO) SHG crystal to also induce spatial loss modulation with an intracavity slit aperture. Earlier, a comparable technique with a phase-mismatched potassium titanyl phosphate (KTP) SHG crystal in an anti-resonant ring was employed by Carruthers and Duling [15] to CW ML Nd:YAG. More recently, CW ML operation of Nd:GdVO₄ has been attained using defocusing cascaded Kerr lensing in periodically poled KTP [16]. In a complimentary hybrid scheme, R. Wallenstein et al. [17] used an acousto-optic (AO) modulator and a phase-mismatched LBO crystal to generate ∼10ps pulses from a diode-pumped Nd:YVO₄ laser. Further, we note that L.J. Qian et al. [18] have exploited the negative nonlinear phase shift generated by a phase-mismatched LBO crystal to achieve soliton mode locking under conditions of normal group velocity dispersion (GVD) in a semiconductor saturable-absorber-mirror-assisted Kerr-lens ML femtosecond Cr:Forsterite laser. Here, we present a detailed experimental study of the performance characteristics of the dual-passive CW ML scheme employing a combination of a SBR and a phase-mismatched SHG crystal for a large mode area, high-power, TEM₀₀-mode (M ² < 1.2), diode-pumped, Nd:GdVO₄ laser. In particular, the role of the SHG phase-mismatch is elucidated. The resulting optimization of the mode locking technique for our 76MHz laser cavity generated an extremely stable and robust train of 7.6ps pulses with an average power of greater than 6W at 1063nm

2. Thermal-lens-shaped Nd:GdVO₄ laser resonator

A schematic of the high-power, mode-locked oscillator incorporating a 5mm long, 0.5 at.% doped, diode-pumped Nd:GdVO₄ crystal is shown in Fig. 1. The Nd:GdVO₄ laser employs a thermal-lens-shaped (TLS), Brewster-cut, solid-state laser head design whose operational principle has been reported elsewhere [19]. Briefly, the shape of the elliptical pump-induced thermal lens in the gain medium is manipulated in such a manner that it compensates for astigmatism (due to the Brewster crystal geometry, off-axis intracavity focusing elements, etc.) in the laser resonator, thus allowing the generation of a high-quality TEM₀₀-mode output. This manipulation is achieved by (i) employing a combination of a cylindrical and a spherical lens to adjust the spatial profile of the pump radiation emitted from each 40W laser diode array operating at 808nm, (ii) ensuring near one-dimensional thermal conduction (through the top and bottom 5x10mm crystal faces), and (iii) utilizing counter-propagating pumps (with an absorption-length product, αl ≈ 2) for uniform pumping. In a slightly asymmetric 0.5m cavity (the auxiliary cavity with mirrors M₁ and M₂ in Fig. 1) and at a total pump power of 40W (corresponding to an absorbed pump power of ∼32W), the laser generated over 8W of output power with a M ² beam quality factor of less than 1.2 at 1063nm.

For mode locked operation at a 76MHz repetition rate, the cavity is extended on the short 24cm arm of the auxiliary cavity using two mirrors with radii of curvature R ₁ and R ₂ and a flat output coupler (OC) arranged for unity ray transformation on a round trip [20]. Values of R ₁ and R ₂ are chosen so that the reflective intracavity telescope has magnifications M = R ₂/R ₁ of 1.0, 1.5, and 2.0 while keeping the cavity length constant (specifically, R ₁ + R ₂ = 1.5m). As a result, the ∼400μm spot size (half-width 1/e of the field) at M₂ is either reproduced or magnified to 0.6mm or 0.8mm, respectively, at the OC. The mirror M₁ terminating the 26cm long arm of the auxiliary cavity is replaced by a half-wave resonant, InGaAs quantum well SBR with a 1% or 2% modulation depth, a saturation fluence ∼70μJ/cm², and a relaxation time of ≤10ps [21]. The spot size of the intracavity TEM₀₀-mode on the SBR varies decreases from about 300 to 200μm as the laser output power increases (i.e., the pump-induced thermal focal length f _T decreases). To compensate for thermally-induced bowing and thereby match the radius of curvature of the intracavity mode at high laser output powers, the SBR is translated a distance Δz ≈ 3cm towards the gain medium, which reduces the cavity length to 1.97m.

 

Fig. 1. Schematic of the extended-cavity, mode-locked, thermal-lens-shaped (focal length f _T), Brewster Nd:GdVO₄ oscillator: LD = 40W laser diode, SBR = saturable Bragg reflector, OC = output coupler, and the Bismuth triborate (BiBO), β-Barium borate (BBO), and Lithium triborate (LBO) second harmonic crystals are positioned a distance d from the OC. The cavity is extended from the plane-plane auxiliary cavity (dashed mirrors M₁ and M₂) on one end using two mirrors with radii of curvature R ₁ and R ₂ arranged for unity ray transformation on a round trip. The mirror M₁ is replaced by the SBR, which is moved forward by Δz ≈ 3cm to match the thermally-induced bowing of the SBR to the intracavity TEM₀₀-mode.

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In this configuration without the SHG crystal and with an optimum 10% OC, the diode-pumped Nd:GdVO₄ laser achieved CW ML operation with 15(±;1) ps pulses only at the highest ∼8W output power (limited by the proximity of the edge of the first cavity stability region) and only with the 1% modulation depth SBR. This result, which is of course independent of the telescopic cavity magnification, is consistent with the analysis Kärtner et al. [9] which for stable CW ML operation with a SBR requires that the intracavity laser pulse energy exceed a critical energy given by

where ΔR is the total amount of saturable losses (i.e., modulation depth) of the SBR, E_A and E_L = F_L,sat.A are the absorber and laser medium saturation energies, respectively. For Nd:GdVO₄ with an emission cross-section at 1063nm of 7.6x10^-19cm² [22], the 0.64mm² mode area in the gain medium [19] implies E_L ≈ 1.6mJ, while E_A ≈ 40nJ for the smallest 200μm spot size on the SBR. Thus, equation (1) predicts a critical pulse energy of E_c = 0.8μJ, which is almost exactly the single intracavity pulse energy for a 76MHz laser resonator producing 8W of average output power with a 10% OC. Moreover, the ∼15ps pulse duration obtained in this cavity configuration is consistent with the analysis of Paschotta and Keller for a slow saturable absorber [23] operating with an incident pulse fluence only slightly above its saturation fluence. Clearly also in agreement with equation (1) is the observation that the 2% modulation depth SBR could not produce CW ML operation with a 10% OC.

3. Dual-passive mode locking

3.1 Reduction of mode locking threshold

The insertion of a phase-mismatched SHG crystal a distance of d = 35 to 55mm from the OC in the Nd:GdVO₄ laser resonator, allowed CW ML operation to be achieved at output powers below 3W for M < 1.5; that is, the CW mode locking threshold was reduced by more than a factor of two by the presence of an intracavity cascaded χ⁽²⁾:χ⁽²⁾ process [12]. The laser performance was investigated for three anti-reflection coated type I SHG crystals: a 5mm Bismuth borate (BiBO; ee→o) crystal [24] and a 4.9mm β-Barium borate (BBO; oo→e) crystal, both angled-tuned at room temperature, and a 10mm non-critically phase-matched (NCPM) LBO crystal, phase-matched by temperature tuning. The lowest CW ML threshold power was achieved with the BiBO crystal since it possesses the largest effective second-order nonlinearity (i.e., the product of the crystal length and effective SHG coefficient, d_eff.L). However, the most stable long-term (2-3 hour) CW ML operation was observed for BBO, which possesses the largest temperature phase-matching bandwidth [25]. Even the ±0.1°C temperature control on the oven enclosing the 10mm NCPM LBO crystal proved inadequate for stable long-term ML operation [26,27].

 

Fig. 2. Shortest measured pulse duration as a function of the CW mode-locked output power for the TLS Nd:GdVO₄ resonator with M = 2, a 1% modulation depth SBR, a 5.0mm BiBO crystal, and a 13% output coupler; 1/2Δk.L ≈ -π (open circles) and 1/2Δk.L ≈ -2π (filled squares). The horizontal dashed line indicates the 15ps pulse duration obtained at 8W of output power without an intracavity SHG crystal.

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Figure 2 displays the output power dependence of the CW ML pulse durations obtained with the BiBO crystal for the largest telescopic resonator magnification (M = 2), the 1% modulation depth SBR, and a 13% OC. Under these conditions, equation (1) would predict a CW ML threshold output power of over 10W. However, a much lower CW mode locking threshold of only 3.3W, with a pulse duration τ ∼130ps, was achieved at a SHG phase-mismatch 1/2Δk.L around -π(Δk = 2k _ω - k _2ω, where k _ω and k _2ω are the fundamental and second harmonic wave vectors respectively; Section 3.3 justifies the specified sign of Δk). Despite the fact that at 1/2Δk.L ≈ -π there is little inverse saturable absorption due to complete back conversion to the fundamental wave in the SHG process, this result is consistent with the analysis presented in Ref 10, which for the non-solitonic regime predicts reduced critical laser pulse energy

for the onset of CW ML operation, where the inverse saturable absorption coefficient for SHG is given by [12]

In our case for the 5mm BiBO crystal with M = 2, where the beam area in the SHG crystal A_SHG = 1mm², d_eff ≈ 3pm/V [24,28], n ≈ 1.6, and λ_ω = 1063nm (Z ₀ is the vacuum impedance), equations (2) predict E_c = 0.4(±0. 1)μJ - the ±25% error being due to uncertainties in the pulse duration at the CW ML threshold (Fig. 2) and the spot size on the SBR. The theoretical analysis of Schibli et al. [10] is thus in agreement with the experimentally determined value for E_c of 0.33μJ (3.3W output at 76MHz with a 13% OC), in this case with the nonlinearity being due to an approximately equally strong SPM contribution from the cascaded χ⁽²⁾:χ⁽²⁾ process at 1/2Δk.L ≈ -π [11,12].

With increasing laser power above the CW ML threshold of 3.3W, the output pulse duration is observed to shorten since both the SBR and phase-mismatched SHG crystal have a stronger mode-locking performance at higher intracavity pulse fluences and intensities. For the fixed configuration of the cavity in Fig. 2, the pulse duration rapidly drops from ∼130ps to below 30ps at an output power of 4.4W. Thereafter, there is a slower decrease in the ML pulse duration to ∼20ps while the optimum phase-mismatch 1/2Δk.L remains around -π (open circles in Fig. 2). When the output power reaches ∼5W, the CW ML operation with 1/2Δk.L ≈ -2π (the next phase-matching minimum) is preferred (filled squares in Fig. 2), which produces shorter ∼10ps pulse durations at output powers of ∼6W. We note that the measured pulse duration above 5W represents a significant pulse shortening [13] over the ∼15ps obtained without the intracavity SHG crystal at higher output power of 8W.

A similar CW ML performance trend is also observed for the angle-tuned BBO and temperature-tuned NCPM LBO crystals. For instance, with BBO and a 10% OC, the mode locking threshold output power at 1/2Δk.L ≈ -π is 3W - again consistent with equations (2) since the reduction in d_effL is offset by a higher intracavity power (due to the lower output coupling). At CW ML threshold, the pulse durations are again around 100ps, and generally shorter pulse durations are generated as the resonator output coupling is reduced from 15 to 5% (i.e., the intracavity power is increased). However, for the highest CW ML output power, the optimum OC reflectivity is ∼90%.

3.2 Pulse shortening

To further investigate the pulse shortening [13] capabilities of this dual-passive mode locking technique, we varied the beam size in the SHG crystal by operating the CW ML Nd:GdVO₄ laser with different resonator magnifications M while maintaining the same 1.97m cavity length. A reduction in M will decrease the beam area in the SHG crystal, A_SHG, which results in an increase in the strength of the cascaded χ⁽²⁾:χ⁽²⁾ process in the SHG crystal.

Figure 3 shows four representative second harmonic autocorrelation measurements of the shortest mode locked pulse durations obtained near the maximum ∼6W of output power (i.e., at an absorbed diode pump power of 30-35W). The excellent fit of all the data to the autocorrelation of a sech² pulse shape (lines in Fig. 3) is consistent with a passive mode locking process [9,10,23,29]. All the measurements were performed using a conventional 90% reflective OC at 1063nm, although CW ML operation was achieved for flat OCs with transmissivities of up to 20%. Hence, unlike the second harmonic NLM mode locking technique [26,27], a dichroic OC (with high reflectivity for 532nm) is not required. Nonetheless, we found that the best CW ML performance was achieved when the closest face of the SHG crystal is positioned d = 47(±2)mm from the OC surface. As with the NLM technique, this optimum value of d ensures that any residual green radiation reflected from the OC is not in phase with the first second harmonic generation cycle on the second pass through the crystal. This condition, together with the fact that the shortest pulses were routinely obtained for phase-mismatches of 1/2Δk.L ≈ mπ (m = -1, -2, -3 ...), clearly minimizes any perturbative influence of reflected green radiation on the cascaded χ⁽²⁾:χ⁽²⁾ process - essentially allowing an independent nonlinear process to occur on each pass through the SHG crystal.

 

Fig. 3. Measured second-harmonic autocorrelations (data points) of the shortest pulses produced by the dual-passive mode-locked TLS Nd:GdVO₄ laser under different operation conditions; intracavity telescope magnification (M), β-Barium borate (BBO) and Lithium triborate (LBO) doubling crystals, phase mismatch (1/2Δk.L), and saturable Bragg reflector modulation depth (ΔR_SBR ). The data is fit to the autocorrelation function of a sech² pulse (lines).

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The data displayed in Fig. 3 indicate a clear trend: shorter pulse durations are obtained at lower magnifications (M = R ₂/R ₁) of the telescopic cavity, and this is accompanied by a shift of the optimum phase-mismatch of the second harmonic crystal to larger negative values.

This trend is consistent with the interpretation that the second harmonic crystal acts as the phase-locker in the mode locking process. In the frequency domain [30], the full cyclical conversion processes occurring in one pass through the nonlinear crystal when 1/2Δk.L ≈ mπ produces a back-converted and correctly phased, fundamental axial-mode spectrum that is about √3 times broader than that of the original incident pulse. Thus, as the magnification M decreases, thereby decreasing the intracavity laser spot size in the nonlinear crystal positioned near the OC, this phase-locking SPM contribution to the cascaded χ⁽²⁾:χ⁽²⁾ process [11] significantly increases in strength. The result is shorter output pulse durations since the bandwidth of the ML spectrum is increased. However, if the nonlinear phase-locking mechanism becomes too strong, multiple pulses per round trip are generated in the diode-pumped CW ML Nd:GdVO₄ laser - in agreement with the analysis of Ref. 10. For example, the onset of multi-pulse operation for the initial optimum phase-mismatch at 1/2Δk.L ≈ -π occurred at more or less twice the CW ML threshold power for all the cavity configurations of Fig. 3. To offset this effect, the strength of the cascaded χ⁽²⁾:χ⁽²⁾ process must be reduced by further detuning the frequency doubling crystal from Δk = 0. Moreover, we found that for M = 1 (Figs. 3(c) and (d)), when the cavity irradiance in the nonlinear crystal is at least a factor of 4 greater than for M = 2, it was also necessary to employ a 2% modulation-depth SBR (i.e., increase E_A in equations (1) and (2)) to obtain stable mode locking with a single intracavity pulse. In this M = 1 case, for both the BBO and LBO crystals, the laser could be mode locked with the 1% modulation-depth SBR from a lower threshold output power of 1.5-2W, but generally exhibited double pulsing per round trip above ∼4W of output power.

The fact that the 4.9mm BBO and 10mm LBO crystals produced similar 7.4ps and 7.6ps output pulse durations for M = 1 and 1/2Δk.L ≈ -4π (Figs. 3(c) and (d)) is also consistent with the interpretation of the dual-passive (amplitude and phase) mode locking mechanism. The factor of two difference in the two crystal lengths compensates very nearly for the difference in their second harmonic nonlinear coefficients [25], thus ensuring approximately the same effective nonlinearity (e.g., d_effL in equation (2b)) in the single-pass, four-fold, forward and backward frequency conversion process.

The effects of group velocity mismatch (GVM) are also expected to be very similar since the values for GVM for the frequency doubling of 1063nm radiation are ∼85fs/mm for BBO [25,31] and 40-50fs/mm for NCPM LBO [25,32]. Thus in both cases, over the characteristic one-quarter crystal length of the nonlinear interaction, the GVM is an insignificant fraction of the generated picosecond CW ML pulse duration. We note that any effects of birefringent beam walk-off are also expected to be negligible due to the relatively large intracavity beam diameter in the SHG crystals.

3.3 The SHG phase-mismatch

To further investigate the role of the SHG phase-mismatch in the dual-passive mode locking technique, we measured the CW ML output pulse duration as a function of the temperature of the 1cm intracavity LBO crystal under the operational conditions of Figs. 3(c) and (d); i.e., M = 1, ΔR _SBR = 2%, and a constant 6W output power with a 10% OC. Specifically, a relative measure of the pulse duration was obtained by monitoring an extra-cavity second harmonic signal produced in the small signal limit (conversion efficiency <5%) by placing the 4.9mm BBO crystal in the unfocused CW ML output beam. Figure 4 shows the data obtained over the region from 1/2Δk.L ≈ -6.5π to 1/2Δk.L ≈ -3.5π where the diode-pumped Nd:GdVO₄ laser exhibited stable single-pulse CW ML output with the shortest ∼10ps pulses. Here, the temperature-dependent Sellmeier equations for LBO reported by Kato [32] together with the observed minima in the single-pass SHG efficiency are used to calibrate Δk to the measured crystal temperature.

The results clearly indicate two important features of the mode locking scheme. First, the fact that the shortest pulses are produced at LBO crystal temperatures less than the NCPM temperature of 148°C confirms that the required phase-mismatch for efficient short-pulse mode locking is indeed negative - as indicated for the results obtained with angle-tuned BiBO and BBO intracavity SHG crystals in Sections 3.1 and 3.2. Second, since the laser output power remained constant (to within ±3%) and the TEM₀₀ output mode size did not vary (constant diode pump current and hence thermal lensing in the gain medium), the shortest ML pulses (corresponding to the highest external second harmonic power) are clearly generated at 1/2Δk.L ≈ mπ, with the minimum pulse duration obtained when m ≈ -4.

 

Fig. 4. Dependence of the extra-cavity second-harmonic generation signal (data points) as a function of phase-mismatch (i.e., crystal temperature) in the region of stable 6W CW ML operation between 1/2Δk.L ≈ -6.5π and 1/2Δk.L ≈ -3.5π for the 1cm-long NCPM Lithium triborate (LBO) doubling crystal with cavity parameters M = 1 and ΔR _SBR = 2%. Also shown are the SHG (solid line) and SPM (dashed line) contributions for the cascaded χ⁽²⁾:χ⁽²⁾ process from equation (3).

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The requirement that Δk be negative can be understood in terms of the spectral dynamics of the mode locking process. For laser resonators operating without dispersion compensation (e.g., Fig. 1), the long wavelength (red) spectral components of an ultrashort intracavity pulse will preceed the short wavelengths (blue) so that the former are preferentially absorbed by a slow saturable absorber [23]. Although the SBRs have relaxation times of the same order as our CW ML pulse durations [21], they are not ‘fast’ [29] and so will still predominantly deplete the red spectral components on the leading edge of the pulse. In order to compensate for the resultant spectral blue-shifting [23] and hence improve the mode locking process, another intracavity component will need to generate red spectral components mainly on the leading edge of the pulse. For Kerr-like SPM, this requires a positive nonlinear coefficient, which can only be achieved for a negative phase-mismatch in a cascaded χ⁽²⁾:χ⁽²⁾ process [11]. This can also be seen from the expression of the instantaneous field amplitude E _ω(L) after a single pass through the SHG crystal [12]

where ξ ∝ |E _ω(0)|² is the low-depletion SHG efficiency. Clearly, the effective nonlinear Kerr coefficient for the phase-mismatched SHG process (the imaginary part of equation (3) plotted as a dashed line in Fig. 4) is positive only for a negative Δk.

Additional supporting evidence for the above interpretation comes from our attempts to ML the diode-pumped Nd:GdVO₄ laser at a positive phase-mismatch. Only for narrow ranges of positive Δk is CW ML operation possible, and then the pulse duration is typically much greater than 30ps. There is therefore no indication of any pulse shortening due to phase-locking through SPM with a negative effective nonlinear Kerr coefficient: that is, the regime of pure soliton mode locking with positive dispersion and negative nonlinear phase shifts [18] was not attained. Instead, a positive phase-mismatch simply exacerbated the dispersion-generated spectral blue-shift due to reflection from the SBR, thus only allowing ML operation with narrow spectral pulse bandwidths.

The fact that inverse saturable absorption due to the nonlinear cascaded χ⁽²⁾:χ⁽²⁾ process tends to broaden the ML pulse duration [10,12], by enhancing loss for shorter more intense pulses, explains the observed oscillatory dependence of the pulse duration with Δk in Fig.4. When 1/2Δk.L ≈ (m+1/2)π, residual second harmonic radiation exits the SHG crystal, causing loss to the resonator - the resulting inverse saturable absorption loss being proportional to the SHG efficiency. The CW ML laser attempts to compensate for this reduction in resonator quality factor Q by operating with a longer pulse duration, which reduces the SHG efficiency (i.e., cavity loss) at a constant output power. On the other hand, when 1/2Δk.L ≈ mπ, the back conversion to the fundamental in the cascaded χ⁽²⁾:χ⁽²⁾ process is complete on one pass through the SHG crystal, thereby minimizing cavity loss and allowing CW ML operation with a shorter pulse. We note that the same oscillatory behavior was observed by Wallenstein [17] in a hybrid ML Nd:YVO₄ laser with an acousto-optic (AO) modulator and a phase-mismatched LBO crystal. In this case, however, the short ∼10ps pulses could be generated for both positive and negative Δk (i.e., at |1/2Δk.L| ≈ mπ,) since an AO modulator usually operates without an intrinsic spectral dependence to its loss modulation.

It is interesting to note that the very shortest pulses are produced a slightly greater phase-mismatches than 1/2Δk.L = mπ. This shift towards Δk = 0 is most pronounced for m = -6 (Fig. 4), where the larger phase-mismatch has already considerably reduced the strength of the inverse saturable absorption. As a result, the Nd:GdVO₄ laser can tolerate a little residual loss to SHG in favor of an increase in the strength of the phase-locking due to SPM in the cascaded χ⁽²⁾:χ⁽²⁾ process. The positive gradient of the imaginary part of equation (3) around 1/2Δk.L ≈ mπ (dashed line in Fig. 4) then ensures a shift towards zero Δk in the optimum phase-mismatch generating the shortest pulses. For m = -4, this shift is considerably smaller due to the increase in the strength of the inverse saturable absorption from SHG.

We also note that the dependence of the gradient of the imaginary part of equation (3) on Δk around 1/2Δk.L ≈ mπ could serve to further stabilize the shortest CW ML pulses against the dispersion-generated frequency-dependent reflectivity of the SBR [23]. Since LBO, in common with BBO and BiBO, has positive group velocity dispersion for SHG around 1μm, the red spectral components of the pulse will experience a slightly higher effective nonlinear Kerr coefficient than the blue: specifically, for the red spectral components, Δk is less negative (i.e., shifted towards Δk = 0 in Fig. 4). The resulting frequency dependence of the imaginary part of equation (3) may partially balance the preferential absorption of leading red components of the pulse spectrum by the SBR in the dual-passive CW ML scheme.

4. Summary

We have performed a detailed investigation into the operational characteristics of a simple all solid-state, dual-passive, CW mode locking technique that employs a phase-mismatched (1/2Δk.L ≈ mπ) SHG crystal as an axial-mode phase locker and a SBR the primary amplitude modulator. We show that this quadrature mode locking technique, first demonstrated by Agnesi et al. to stabilize the CW ML operation of Nd:BaY₂F₈ [12], can be readily extended to lasers operating with large mode areas, and hence large laser gain medium saturation energies. Specifically, the technique is shown to generate extremely stable and robust 7-8ps sech² pulses at 76MHz from a 6-7W, diode-pumped, Brewster Nd:GdVO₄ oscillator incorporating a thermal-lens-shaped laser head for astigmatism-compensated, ∼1mm², fundamental TEM₀₀-mode operation.

The inclusion of the phase-locking cascaded χ⁽²⁾:χ⁽²⁾ nonlinearity in the laser cavity is shown to reduce the CW ML threshold by more than a factor of two and produces significant pulse shortening. Of particular importance is the reduction in the threshold for CW ML operation since the resultant suppression of the Q-switched ML regime significantly reduces the likelihood of intracavity optical damage, especially to semiconductor-based saturable absorbers. Together with the improved passive mode locking (i.e., shorter pulse generation), the ability to CW ML reliably, high-power, large mode area solid-state lasers by this technique should benefit many applications; in particular, nonlinear frequency conversion by harmonic generation and optical parametric conversion. Indeed, the 6W, 7.6ps output from the studied 76MHz Nd:GdVO₄ laser can be readily frequency doubled to produce 3.5W of green radiation at 532nm using a 2cm NCPM LBO crystal.

The stability and performance of our picosecond CW ML Nd:GdVO₄ laser is dependent upon the characteristics and alignment (i.e., Δk.L) of the intracavity SHG crystal: (i) crystals with large temperature phase-matching bandwidths are preferred; (ii) crystals with a large second-order nonlinearity are preferred to allow for efficient multiple frequency back conversions in one pass; (iii) operation with a negative phase-mismatch is strongly preferred since a positive effective SPM coefficient counteracts the frequency-dependent reflectivity of the SBR in a dispersive laser cavity; but (iv) an excessively strong second-order nonlinearity results in the generation of multiple pulses per cavity round trip - in agreement with the analysis of Ref 10. The shortest CW ML pulses are generated in this non-soliton mode locking regime when the phase-mismatch 1/2Δk.L ≈ mπ, which corresponds to the case where the inverse saturable absorption is minimized so that only SPM contributes through phase-locking to the mode locking process.