Enhancement of efficiency in femtosecond optical parametric oscillators using group-velocity-matching in long nonlinear crystals Open Access

Mid-infrared (mid-IR) femtosecond laser sources are of great interest across many research fields, for applications including frequency comb generation, pump-probe spectroscopy, and materials science.^1–3 In the high repetition rate regime (∼100 MHz), devices based on nonlinear frequency conversion, in particular synchronously-pumped optical parametric oscillators (OPOs), are unrivalled in their ability to generate kilowatt-level pulse trains of quasi-continuous-wave (cw) radiation across 1–12 µm, with pulse durations able to reach as short as a few optical cycles.^4–6 The atmospheric transmission window covering 3–5 µm coincides with onset of the molecular fingerprint region, which contains strong vibrational resonances of hazardous gases such as CH₄, SO₂, and CO. Across this spectral region, OPOs are now mature technology, capable of producing watt-level average power by exploiting nonlinear crystals such as MgO-doped periodically-poled LiNbO₃ (MgO:PPLN) and LiTaO₃ (MgO:PPLT) pumped by well-established Ti:sapphire and Yb fiber lasers.⁷ 

At the same time, femtosecond OPOs cannot generally reach typical conversion efficiencies of their picosecond and cw counterparts, primarily due to more pronounced temporal walk-off effects between the pump, signal, and idler pulses, which propagate with mismatched group velocities through the nonlinear crystal. In the spectral domain, this is manifested as a narrow acceptance window for pump frequencies, which can be efficiently down-converted to the signal and idler output. While this does not pose a major problem for relatively narrowband picosecond lasers, a transform-limited 100 fs pulse at 1064 nm spans a full-width-half-maximum (FWHM) spectral bandwidth of Δλ ∼ 12 nm, far broader than the pump acceptance bandwidth afforded by most crystals longer than a few millimeters. However, exceptions to this general rule arise in several nonlinear crystals, where the dispersion profile is such that the pump and idler group velocities, v_p and v_i, become equal at wavelengths that satisfy phase-matching.⁸ The pump acceptance bandwidth, Δλ_p, can be calculated for a crystal of length, L_c, by using a Taylor expansion of the phase-matching condition, Δk, around the central pump wavelength, λ_p, and retaining terms to the second order,

where Δk₀ = k_p − k_s − k_i − 2π/Λ, and k_p, k_s, k_i, and Λ are the pump, signal, idler, and crystal quasi-phase-matched (QPM) wavevectors, respectively. In a three-wave parametric process, the phase-matching bandwidth is defined by the FWHM of a function proportional to sinc²(ΔkL_c/2), such that the maximum tolerable phase-mismatch is given by

Using Eq. (2), evaluating the derivative in the second term of Eq. (1), and writing the result in terms of the pump-idler group-velocity mismatch (GVM_pi = 1/v_p − 1/v_i), leads to the expression

When GVM_pi is zero, the crystal length has a weaker influence on the pump acceptance bandwidth (⁠$Δλp∝Lc−1/2$ rather than $Δλp∝Lc−1$⁠), relaxing the restriction of narrowband pumping for long crystals. Figure 1 shows the calculated signal parametric gain map for a 42-mm-long MgO:PPLN crystal with four QPM grating periods of Λ = 28.5 µm, 29 µm, 29.5 µm, and 30 µm, calculated using the Sellmeier equations in Ref. 9 at a fixed temperature of 100 °C. The dashed lines represent pump and signal wavelengths corresponding to GVM_pi = 0 for each grating period. It is evident that Δλ_p rapidly increases to ∼14 nm (FWHM) as the signal moves toward the critical wavelength (the inflection points), whereas it is <1 nm elsewhere. Conversely, at the zero-GVM points, the signal parametric gain bandwidth, Δλ_s, becomes narrow, implying that the pump bandwidth is primarily transferred to the idler.

A cw OPO exploiting this principle was reported, using a 8.3-nm linewidth Yb fiber laser at 1064 nm to pump a 50-mm-long MgO:PPLN crystal with a QPM grating period of Λ = 30 µm, at 100 °C, leading to the generation of 5.3 W of broadband mid-IR idler radiation centered at 3454 nm with a FWHM spectrum spanning over 76 nm.⁸ In addition, studies of an ultrafast MgO:PPLN OPO synchronously pumped by spectrally broadened 4 ps pulses at 1047 nm confirmed that when operating in the zero-GVM regime, spectral information is transferred from the pump to idler with high fidelity.¹⁰ Similarly, the broad acceptance bandwidth enabled highly chirped 3 ps pulses from an Yb fiber amplifier to be used to directly pump an OPO based on a 25-mm-long MgO:PPLN crystal, without spectral precompression.¹¹ Finally, it was demonstrated that under group-velocity-matched conditions, a 50-mm-long MgO:PPLN crystal could be pumped with 220 fs pulses from a 5 W Yb fiber laser at 78 MHz to produce up to 0.54 W of idler output tunable across 3342–4229 nm in a single-pass optical parametric generation setup.¹² 

Here we report, for the first time to our knowledge, a widely tunable femtosecond OPO based on the principle of group-velocity-matching in a long MgO:PPLN crystal, where we exploit a tunable femtosecond laser to pump at all wavelengths within the maximum spectral acceptance shown in Fig. 1. The large phase-matching bandwidth enables the use of a long crystal (42 mm), well above that used in conventional femtosecond OPOs (∼1 mm). Importantly, the long crystal also provides additional dispersive broadening of the interacting pulses by increasing the effective interaction length from ∼1 mm (for ∼100 fs pulses) to ∼8 mm (for ∼500 fs pulses) due to temporal walk-off, thus leading to enhanced down-conversion efficiency from the pump to OPO output. By deploying this approach, we achieve substantially lower oscillation thresholds and higher conversion efficiencies as compared to conventional femtosecond OPOs. The ability to rapidly tune across the mid-IR using pump wavelength variation is also exploited to perform transmission spectroscopy of methane in the 3.3 µm spectral region. Finally, we explore the possible merits of applying the same technique to alternative nonlinear crystals in the mid-IR.

The experimental setup for the group-velocity-matched femtosecond OPO is shown in Fig. 2. A tunable femtosecond laser system (Radiantis, Blaze), consisting of an integrated Ti:sapphire laser and synchronously-pumped OPO, is used as the pump source, providing transform-limited 80 fs pulses at 80 MHz tunable across 990–1550 nm in the near-IR with up to 500 mW average output power.¹³ For this experiment, only wavelengths across 997–1070 nm were used. The nonlinear crystal for the OPO is 42-mm-long MgO:PPLN, cut along the optical x-axis with all waves polarized along z-axis to access the highest nonlinear coefficient under type-0 (e → ee) quasi-phase-matching. It contains seven separate channels across its transverse aperture containing QPM grating periods increasing from Λ = 28.5 µm to Λ = 31.5 µm in steps of 0.5 µm, and is mounted in an oven with temperature stability of 0.1 °C. The crystal length of 42 mm was determined from calculations of parametric gain (see Fig. 1), as the maximum allowable to accommodate the available pump bandwidth (∼14 nm). The pump beam is focused using a lens, L₁ (f = 100 mm), to a waist radius of w₀ ∼ 22 µm at the center of the crystal, resulting in equal confocal parameters for the pump and resonant signal (b_p ∼ b_s).

Since the antireflection coating is optimized for 1064 nm pumping, single-pass pump transmission losses vary from T ∼ 50% at 1000 nm to T < 1% at 1070 nm, whereas it is T > 99% for all signal wavelengths. The OPO cavity is formed by two concave spherical mirrors, M₁ and M₂ (r = 100 mm), a plane mirror, M₃, and a plane 5% signal output coupler, M₄, mounted on an electronic translation stage for fine control of cavity length synchronization. M₁ is coated for high transmission at the pump wavelength (T > 90% at 1000 nm, T ∼ 70% at 1070 nm) and high reflectivity at the signal (T > 99% across 1300–1500 nm). M₂ is coated for high reflectivity at the signal (T > 99% across 1300–1500 nm) and high transmission at the idler (T > 90% across 3000–5000 nm), ensuring singly-resonant oscillation for the signal. The single-pass idler is collimated by a second lens, L₂ (f = 100 mm), and passes through a long-pass (cut-off < 2.4 µm) filter, F, and a 5-cm-long methane gas cell, before being directed to an optical spectrum analyzer.

After alignment, the OPO was characterized for each MgO:PPLN grating period from Λ = 28.5 µm to Λ = 30 µm by measuring the signal and idler power as a function of pump wavelength and cavity detuning. Figure 3 shows plots of the signal and idler power (triangles) as the cavity length was adjusted, while operating at the optimum pump wavelength for each grating period. Also plotted is the signal central wavelength (circles), extracted from each individual spectrum using a center-of-mass average algorithm. The dotted lines represent signal wavelengths corresponding to points of inflection in Fig. 1. Interestingly, the observed signal remains locked in good agreement with this wavelength over the majority of the synchronization length, during which time the signal and idler powers steadily increase. This is in contrast to standard femtosecond OPOs, where the wavelength continually adjusts according to cavity dispersion in order to match perfect synchronization, but consistent with previous observations near zero-GVM.¹¹ After translating M₄ a critical distance, the wavelength begins to follow the cavity dispersion profile, accompanied by a rapid decrease in output power and an increase in signal bandwidth.

Since significant GVM exists between the pump/idler and signal (GVM_ps = 109 fs/mm for λ_p = 1030 nm, λ_s = 1450 nm), the effective interaction length for the three pulses, L_eff, is determined by temporal walk-off between the pump and signal pulses. As illustrated in Fig. 4, the signal pulse, which has higher group velocity, walks through the pump (and idler) pulse inside the crystal. The position of L_eff within the crystal is dependent on the time delay between pump and signal at the input to the crystal, ΔT, as determined by the OPO cavity synchronous length. It is advantageous to choose a delay which ensures that L_eff is fully contained within the physical crystal length, L_c. Furthermore, both the pump and signal pulses experience broadening due to dispersion in the 42-mm-long crystal. Whereas the resonating signal pulses converge to a steady-state duration of Δτ_s ∼ 500 fs (measured by interferometric autocorrelation), which can be treated as constant over a round-trip, each consecutive pump pulse enters the crystal with an initial duration of Δτ_p0 ∼ 80–100 fs and is significantly dispersion-broadened by the crystal end. Using a simple pulse propagation algorithm which considers group velocity dispersion (GVD) and Kerr nonlinearity,¹⁴ the final pump pulse duration is estimated to be ∼350–400 fs, assuming a sech² pulse shape. A further useful assumption is to consider the pump pulse duration to increase linearly with distance, Δτ_p = Δτ_p0 + a(λ_p)x, where x is the position inside the crystal in units of mm (x = 0 at the crystal entrance, x = L_c at the crystal end), and a(λ_p) is the rate of broadening in fs/mm. This enables general expressions for L_eff to be derived, by using the definition of effective interaction length as the distance over which two pulses are separated by a time delay less than the mean of their respective FWHM durations.¹⁵ The resultant formulas are presented in Table I, together with the range of ΔT for which they are valid, where

Physically, ΔT₁ and ΔT₂ are the minimum and maximum delay, respectively, for which the entire L_eff is utilized within the crystal, and ΔT_M is the maximum delay to achieve a nonzero L_eff. Finally, it can be shown that the maximum value of L_eff occurs when the delay is set to ΔT = ΔT₂, yielding

which, when calculated using realistic values of a = 6.79 fs/mm, GVM_ps = 109 fs/mm, Δτ_p0 = 90 fs, and Δτ_s = 500 fs, results in a value of 7.8 mm. Such a large increase in L_eff compared to the non-group-velocity-matched case (where typically L_eff < 1 mm for ∼100 fs pulses) leads to a correspondingly large increase in parametric gain, which scales exponentially with L_eff in the high-gain regime.¹⁶ In Fig. 5 the calculated value of L_eff is plotted against the signal pulse delay, ΔT, with the corresponding experimentally measured idler output power at the same pump and signal wavelengths plotted on the opposite axis. The experimental peak has been adjusted to coincide with the maximum value of L_eff.

It is evident that the range of delay for which oscillation is sustained corresponds closely with the range for which L_eff occurs entirely within the crystal, and the increasing output power with delay can be explained by an increasing interaction length. This variation in L_eff is a phenomenon unique to the group-velocity-matched OPO, in contrast to conventional femtosecond OPOs that self-adjust the resonating wavelength to ensure that ΔT = 0 at all times.

The zero-GVM condition can be adjusted by variation of crystal temperature; however, this is a slow process. A faster tuning method is to vary the pump wavelength, which leads to a linear variation in the idler wavelength, as the signal wavelength is found to remain fixed at the value which yields the maximum pump acceptance bandwidth. The dependence of the idler power and wavelength on the pump wavelength is plotted in Fig. 6 for Λ = 28.5 µm, where it is seen that a maximum average power of 57 mW is generated for λ_p close to the value calculated in Fig. 1, indicated by the dotted line. The idler central wavelength can be rapidly tuned across 3439–4087 nm at T = 100 °C, extending to 4273 nm at T = 25 °C. Using the Λ = 30 µm grating period at T = 200 °C, the pump tuning window could be shifted to 3132–3420 nm. With a grating period of Λ = 28 µm, it would be possible to reach 4500 nm, a region difficult to access in conventional MgO:PPLN OPOs. Power is limited on the short wavelength side by poor transmission of M₁ for λ_p > 1070 nm. The optimum pump wavelength does not always correspond to the exact value predicted by the Sellmeier equations, likely due to a combination of wavelength dependence of pump power and the optical coatings of M₁ and the crystal. Since the pump wavelength is controllable through software, a random idler wavelength in this range can be selected within less than 10 s.

A major advantage of utilizing a long nonlinear crystal is the improvement in conversion efficiency, which we verified by performing power scaling measurements, a typical example of which is shown in Fig. 7. We observed oscillation thresholds as low as 5 mW, even with significant transmission losses for the pump. Signal and idler slope efficiencies were found to be 11.1% and 15.4%, respectively, where the pump power was corrected for transmission loss through M₁ only. With 419 mW at 1023 nm incident on the crystal, the pump depletion was determined to be 78%, and 61 mW of idler power at 3369 nm was measured, equivalent to a quantum conversion efficiency of 47.9%. Using the Manley-Rowe relation and the pump depletion value, it is calculated that 71 mW of idler is produced, and residual losses contributed by M₂, L₂, and the long-pass IR filter account for the discrepancy. Parasitic visible light generated by non-phase-matched sum-frequency mixing was also observed; however, the measured power contained in these beams was <10 mW.

For comparison, a typical femtosecond OPO based on a 1-mm-long MgO:PPLN crystal has a threshold of ∼50 mW and an idler slope efficiency of ∼12%, meaning the group-velocity-matched oscillator provides a twofold improvement in output idler power, at a pump power of 419 mW.⁷ The signal power could be increased to 76 mW at the expense of the idler by increasing the output coupling from 5% to ∼13%. Due to the large parametric gain, it is likely that the OPO could support far higher signal output coupling; however, we were unable to characterize this due to a lack of suitable output couplers.

Also shown in the inset to Fig. 7 is the pump spectrum recorded with the OPO on (shaded area) and off (dotted line), indicating that depletion occurred over the entire spectral bandwidth. We recorded a maximum idler power of 65 mW at 3610 nm, with 20 mW available at 3231 nm and 12 mW at 4180 nm. Furthermore, we found that the OPO operates with high long-term passive power stability, exhibiting rms fluctuations of 0.81% and 0.76%, measured over 1 h for the signal and idler, respectively, as shown in Fig. 8. This is a result of the OPO preferentially operating at the group-velocity-matched wavelengths, even when the cavity length is subject to significant perturbation (as presented in Fig. 3).

Pulse durations of the pump and signal, determined using interferometric autocorrelation, are shown in Figs. 9(a) and 9(c). The corresponding spectra are shown in (b) and (d), with the calculated phase-matching bandwidths also plotted as dotted lines. When calculating the pump acceptance bandwidth in Fig. 9(b), a sech² signal pulse of 3 nm bandwidth at 1459 nm was assumed, and, likewise, to calculate the signal gain bandwidth in (d), the measured pump bandwidth of 13.8 nm was used. It can be seen that the pump bandwidth is almost as broad as the phase-matching bandwidth, indicating that a crystal length of 42 mm is very close to theoretical maximum length for such short pump pulses, even in the presence of zero-GVM. Lying in the range ∼400–600 fs, signal pulses are significantly broader than the pump, due to a combination of pump-signal GVM, self-phase-modulation (SPM), and group-delay dispersion (GDD = 14 793 fs² at 1300 nm, 7463 fs² at 1600 nm).

Despite this, it is clear that the signal spectrum is constrained by the gain bandwidth imposed by the 42-mm crystal length, resulting in pulses close to the transform limit (ΔνΔτ ∼ 0.315). It is important to note that this performance is obtained in the absence of any dispersion compensation within the OPO cavity.

Without access to a suitable mid-IR autocorrelator, we were unable to measure the idler pulse durations. However, an estimation can be made based on a convolution integral between the pump and signal fields, which yields a duration in the range 440–630 fs. The idler spectrum was measured using a He-Ne calibrated Fourier-transform wavemeter with resolution of ∼3 nm. As shown in Fig. 10(a), the FWHM was measured to be within the range 140–180 nm, dependent on the pump bandwidth. Note that the idler spectrum at the shortest wavelength exhibits strong asymmetry, due to the corresponding pump wavelength containing similar SPM-induced features.¹⁰ Using pump and signal data from Figs. 9(b) and 9(d), in conjunction with the Manley-Rowe relation, we calculate an idler bandwidth of 151 nm, in strong agreement with the measured value of 164 nm from a simultaneously recorded idler spectrum. The lack of idler spectral broadening is indicative of negligible GVM between the pump and idler waves, as expected from the calculation (GVM_pi = 0.7 fs/mm for λ_p = 1035 nm, λ_i = 3561 nm).

Finally, we demonstrate the potential of our OPO source for spectroscopy, by performing transmission measurements through a 5-cm-long gas cell containing CH₄/N₂ partial pressures of 10%/90%, respectively. The broadband nature of the idler enables multiple vibrational modes to be probed simultaneously, as shown in Fig. 10(b), where data recorded from the spectrum analyzer at a fixed central idler wavelength (blue) is compared to the HITRAN database (red). The blue plot contains data from a single OPO spectrum at a fixed wavelength. Scanning the pump wavelength enables the full mid-IR region of interest to be covered in a few seconds.

It is interesting to note that group-velocity-matching can also be realized in several other QPM nonlinear crystals across the near- and mid-IR. Table II shows example idler wavelengths and required QPM grating periods for periodically-poled MgO-doped stoichiometric LiTaO₃ (MgO:PPsLT), and orientation-patterned GaP (OP-GaP), with MgO:PPLN presented for comparison. Also calculated is the effective pump-signal interaction length (assuming 90 fs pump and 500 fs signal pulses), and estimated single-pass parametric gain when pumped by a 500 mW, 90 fs laser at 80 MHz. For each crystal, a physical length of L_c = 42 mm was assumed and an estimate for the temporal broadening factor a was determined using the final pump pulse duration calculated using numerical pulse propagation. The maximum gain was calculated in the high-gain limit considering the decreasing peak pump intensity along the crystal,¹⁶ 

where the exponent was determined using the modified Γ coefficient defined in Eq. (24) of Ref. 17. All refractive indices were evaluated at room temperature using the relevant Sellmeier equations from Refs. 9, 18, and 19. The final column highlights the percentage improvement in gain compared to a 1-mm-long crystal. The reduction in gain due to decreasing pump intensity toward the end of the crystal is more than compensated by the increase in the effective interaction length, ensuring a net increase in gain toward longer pump-signal delays.

MgO:PPsLT presents similar mid-IR wavelengths to MgO:PPLN, but zero-GVM can be achieved at shorter pump wavelengths, lying in the Ti:sapphire gain bandwidth. Additionally, smaller GVM between pump and signal leads to greater enhancements in gain. OP-GaP is rapidly becoming the crystal of choice for deep mid-IR frequency conversion; however, early samples suffer from high linear absorption.^20,21 When pumped at the Er and Tm/Ho wavelengths at 1.55 µm and 1.95 µm, respectively, the large gain enhancement could greatly improve conversion efficiencies at challenging and sought-after wavelengths.

In summary, we have demonstrated an efficient femtosecond OPO tunable across 3132–4273 nm in the mid-IR and 1392–1568 nm in the near-IR, based on the concept of group-velocity-matching between pump and idler pulses in a 42-mm-long MgO:PPLN crystal. The OPO delivers up to 65 mW of mid-IR average power at 80 MHz repetition rate in 440–630 fs pulses, with FWHM bandwidths of 140–180 nm, depending on the pump bandwidth. By using dispersive broadening of the resonant signal to ensure interaction of all three waves over an extended effective interaction length (∼8 mm), we achieve quantum conversion efficiencies up to 47.9% into the mid-IR. The low pump threshold of ∼5 mW and high slope efficiency will open up the possibility of direct pumping with low-power femtosecond fiber oscillators, for example, to generate tunable mid-IR seed pulses for optical parametric amplifiers.

The OPO has been shown to preferentially operate at the zero-GVM condition over a large cavity synchronization length, leading to high passive power stability. Signal pulses have been measured to be transform-limited without intracavity dispersion compensation, due to restricted gain bandwidth imposed by the 42-mm-long crystal, which also facilitates effective parametric transfer to the idler. The wavelength flexibility of the pump has enabled optimization of output power for each grating period, also allowing rapid tuning across mid-IR spectral regions, which we have exploited to perform broadband spectroscopy of methane. We have also derived equations for the effective interaction length, which support the observed variation in output power as a function of cavity delay. The OPO conversion efficiency could be further improved using optimized optical coatings, and alternative cavity design in order to achieve the ideal focusing parameter of ξ = 2.84.²³ Further experiments will study pumping alternative nonlinear crystals at group-velocity-matched spectral regions in the near-IR in order to achieve boosted efficiency at challenging wavelengths in the deep mid-IR.

Ministerio de Ciencia, Innovación y Universidades (MICINN) (No. TEC2015-68234-R); European Commission (EC) (Mid-Tech, No. H2020-MSCA-ITN-2014); Generalitat de Catalunya (CERCA Programme); Severo Ochoa Programme for Centres of Excellence in R&D (Grant No. SEV-2015-0522) and the European Social Fund (No. BES-2016-079359); Fundació Privada Cellex.