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The influence of parasitic processes on the performance of ultra-broadband noncollinear optical parametric amplifiers (NOPA’s) is investigated for walk-off and non-walk-off compensating configurations. Experimental results with a white-light–seeded NOPA agree well with numerical simulations. The same model shows that 10% of the output energy of an amplified signal can be transferred into a parasitic second harmonic of the signal. These findings are supported by quantitative measurements on a few-cycle NOPA, where a few percent of the signal energy is converted to its second harmonic in the walk-off compensating case. This effect is reduced by an order of magnitude in the non-walk-off compensating configuration. A detailed study of the phase-matching conditions of the most common nonlinear crystals provides guidelines for designing NOPA systems.
©2011 Optical Society of America
1. Introduction
Optical parametric amplifiers (OPA’s) are widely used sources of broadly tunable femtosecond pulses. Critically phase matched borate crystals, such as beta-barium borate (BBO) and lithium borate (LBO), have been employed to generate tunable ultrashort pulses from the visible to the near-infrared wavelength range [1–3]. In particular, a noncollinear OPA (NOPA) allows for ultra-broad gain bandwidth that supports few-cycle pulse durations [4,5]. Signal gain of 103 to 106 can be achieved within millimeter-long crystals as a result of perfect phase matching and large nonlinear coefficients. The lack of energy storage and low residual absorption in the crystals makes it possible to scale OPA’s to several watts of average output power at pulse energies ranging from the sub-μJ level to >100 μJ [6–8].
Remarkable progress has also been achieved in the development of ultrahigh-peak-power OPCPA systems [9]. In particular, the peak power of few-cycle OPCPA has recently been boosted to 10 TW [10]. Furthermore, large-aperture crystals of deuterated potassium dihydrogen phosphate (DKDP) create the potential for ultra-intense optical parametric chirped-pulse amplification (OPCPA) systems (>1023 W/cm2) using kilojoule Nd:glass lasers as pump sources [11,12].
Parasitic nonlinear processes, which have also been observed in optical parametric oscillators [13], can significantly degrade the performance of OPA’s [2,14]. For example, frequency doubling of one or more of the interacting waves can lead to spectral deformation and reduced efficiency. This paper presents a detailed study of parasitic second-harmonic generation of signal and idler waves. The analysis focuses on critically phase matched NOPA’s, but similar issues can occur in quasi-phase-matched materials [15]. Experimental observations are presented for two BBO-based NOPA systems. The first system is a MHz-rate NOPA that is seeded by white-light continuum generated in sapphire—a common choice for ultrafast time-dependent spectroscopy [1,16,17]. The second system has been optimized for amplification of an ultra-broadband width ranging from 650 to 1050 nm to support few-cycle pulse durations. The experimental observations agree well with the results of a 2-D numerical model that was used to evaluate the influence of parasitic processes on the NOPA’s overall efficiency.
To avoid unwanted parasitic processes, different crystals and noncollinear phase-matching configurations can be used. For this purpose, calculations of the phase-matching properties for broadband BBO, LBO, BiBO, and DKDP NOPA’s are presented. These results allow for configurations to be identified that are free from phase-matched parasitic second-harmonic–generation (SHG) processes over the desired signal-wavelength range.
2. Noncollinear phase-matching geometry
NOPA’s achieve broadband gain by having a noncollinear angle α between the pump and signal to match the group velocity of the signal and idler [3,18]. Figure 1 shows a common configuration for BBO, a crystal widely used for visible-wavelength NOPA’s. The pump is extraordinary polarized (e wave) and its Poynting vector (S P) and wave vector (k P) are separated by an angle ρ. For example, ρ = 3.3° and α = 2.6° when the wavelengths of the pump and signal are chosen to λ P = 523 nm and λ S = 800 nm, respectively. All angles are specified internal to the crystal. Pump–signal birefringent walk-off is partially compensated by positioning the signal on the side to which the pump walks off [see Fig. 1(a)]. For negative uniaxial crystals, which are examined in this work, this is away from the crystal axis. In this case, the angle between the pump and signal Poynting vectors is ρ – α = 0.7°. The idler is angularly dispersed (5 × 10–3 deg/nm) and has an average angle relative to the signal of ~7.4°.
Fig. 1 Relative position of the wave vectors for the pump (green), signal (red), and idler (black) for critical phase matching in a negative uniaxial crystal. The noncollinear angle between the pump and signal wave vectors is α. The walk-off angle between the pump Poynting vector [S P (blue)] and signal wave vector is ρ. (a) Definition of the walk-off compensating (WC) configuration for negative uniaxial crystals. (b) Non-walk-off compensating (NWC) configuration. The vertical dimension has been expanded 3 × for clarity.
Since the signal and idler are ordinary polarized (o waves), identical phase-matching conditions exist for any orientation of the signal wave vector around the pump wave vector, provided α is kept constant. The largest pump–signal walk-off occurs when the angle between their Poynting vectors is ρ + α = 5.9° [see Fig. 1(b)]. The two extremes are referred to as walk-off compensating (WC) and non-walk-off compensating (NWC), respectively. Note that the NWC configuration has been named as tangential phase matching in the early literature [19].
Figure 2 shows the internal signal angles θ S at which signal wavelengths are phase matched for a BBO NOPA pumped at 523 nm. Curves for several noncollinear angles α are shown (in red) for both the (a) WC and (b) NWC configurations. The NOPA phase matching has, to first order, the broadest bandwidth when the curves are horizontal [3,20]; that is, a choice of internal signal angle achieves perfect phase matching for the broadest range of signal wavelengths. Also shown (in blue) are the phase-matching curves for signal SHG. For the WC case, certain combinations of θ S and α simultaneously phase match both the NOPA and signal SHG processes; therefore, parasitic SHG can occur.
Fig. 2 NOPA phase-matching curves (red) for BBO pumped at 523 nm for several noncollinear angles α: (a) the walk-off compensating (WC) and (b) the non-walk-off compensating (NWC) configurations. The phase-matching curves for signal SHG are shown in blue.
Figure 3 shows a calculation of the phase-matching conditions for idler SHG. The internal angle of the idler wave to the crystal axis, θl, was calculated to compensate the pump wave’s vector component k P, perpendicular to the signal wave vector k S. Two values of α for both the WC and NWC configurations are shown (red curves). The phase-matching curves for idler SHG are shown in blue. In the WC configuration, phase matching is achieved at an idler wavelength of 1265 nm, and the angular deviations between the idler and the SHG phase-matching curve are small, in the wavelength range between 1100 nm and 1500 nm. In contrast, the NWC configuration avoids idler SHG since the phase-matched idler wavelengths (865 nm and 900 nm) are typically not present when pumping at 523 nm.
Fig. 3 The idler wavelengths and internal angles for NOPA phase matching in BBO pumped at 523 nm for two values of α (in red): (a) the WC and (b) the NWC configurations. The phase-matching curves for idler SHG are shown in blue.
3. Experimental observation of parasitic signal SHG
Parasitic processes were experimentally investigated using a BBO-based NOPA pumped at 1 MHz by a fiber chirped-pulse amplifier (FCPA) system. A schematic of the system is shown in Fig. 4 . Pulses from a mode-locked oscillator were stretched from 220 fs to 180 ps, down-selected from 38 MHz to 1 MHz, and amplified by a chain of four ytterbium-doped fiber amplifiers. A large-mode-area photonic-crystal fiber amplifier was used for the final stage, producing 12-μJ pulses centered at 1047 nm [21]. The compressor used two multilayer dielectric gratings (1740 lines/mm) to produce 9-μJ, 250-fs pulses. A portion of the energy (20%) was picked off and weakly focused into a static 4-mm sapphire plate to generate a chirped white-light continuum (WLC) seed [1,16,17]. The rest was doubled in a 2-mm BBO crystal, producing 3.8-μJ pulses at 523 nm for pumping the NOPA.
Fig. 4 (a) Experimental setup. Yb: ytterbium-doped fiber amplifiers; AOM: acousto-optic modulator; WLC: white-light continuum; SHG: second-harmonic generation; NOPA: noncollinear optical parametric amplifier. Near-field profiles of the amplified beams for (b) walk-off compensating (WC) and (c) non-walk-off compensating (NWC) configurations.
The BBO crystal used in the NOPA was 4 mm long and was cut at an angle such that the seed beam was approximately perpendicular to the crystal face (27.1° and 21.6° for the WC and NWC geometries, respectively). The pump and seed were aligned in the horizontal plane and were focused into the NOPA crystal with ~120-μm beam waists (half-width at 1/e 2). The pump–signal walk-off lengths, defined as the distance required for a change in separation equal to their beam waists, were 8.2 mm for WC and 0.96 mm for NWC. The impact of pump–signal walk-off is clearly seen in the near-field beam profiles [Figs. 4(a) and 4(b)]. In the WC case, the near field is approximately Gaussian. In contrast, in the NWC case, there is a large asymmetry along the direction of walk-off (x–z plane). A detailed understanding of these profiles requires a full 4-D model that is beyond the scope of this paper. Such a treatment would need to include diffraction of the three waves and their relative walk-off in space and time. Despite significant differences in the beam profiles, the maximum signal power obtained for each configuration was roughly equal at ~850 nJ, when the seed delay and phase matching were optimized for 780 nm.
Amplified signal spectra for the two walk-off configurations are shown in Fig. 5 . For the WC configuration, parasitic signal SHG was phase matched in the center of the gain band [Fig. 5(a)]. This resulted in a narrow spectral line at 412 nm (shown in blue), a large notch in the center of the signal spectrum at 824 nm, and smaller notches on either side at 788 nm and 865 nm. The side notches are due to non-degenerate sum–frequency generation: (788 nm)–1 + (865 nm)–1 = (412 nm)–1. No parasitic signal SHG was observed for the NWC configuration, and the signal spectrum had a smooth Gaussian-like shape [Fig. 5(b)].
Fig. 5 Signal spectra for (a) the WC and (b) the NWC configurations. The pump angle θ P and noncollinear angle α were similar in both cases. In the WC configuration, the signal angle θ S of 28.2° made it possible for parasitic second-harmonic generation to produce a narrow spectral line at 410 nm, leaving notches in the signal spectrum in the 780- to 870-nm range.
Second-harmonic generation from the idler was also observed in the WC configuration. Figure 6 shows a measurement of its spectrum. Like the idler, which is angularly dispersed because of noncollinear phase matching, the idler second harmonic (SH) is angularly dispersed, producing a visible rainbow at the output of the amplifier that extends from 550 nm to 700 nm. This observation fits well to the phase-matching calculations shown in Fig. 3. A rough estimate of the idler SH energy is 0.1 to 0.4 nJ, which is less than 1% of the signal energy. Errors in this measurement result from calibration uncertainties for the broadband edge filter used to separate the idler and its second harmonic.
Fig. 6 Spectra measured from idler SHG for the WC configuration. The spectrum spans much of the visible spectrum, as shown by the color bar above.
4. Numerical model for parasitic signal and idler SHG
The growth of parasitic signal and idler SHG was investigated using a 2-D numerical model. The equations of motion for the relevant optical fields were integrated using a split-step code that employed a fourth-order Runge–Kutta algorithm [22]. The nonlinear propagation was modeled using Eqs. (1) for the field amplitudes in the time domain, Aj(z, t), where j = S, I, P, SSH and ISH for the signal, idler, pump, signal SH, and idler SH fields, respectively.
A constant nonlinear coefficient deff was assumed to be 2.16 pm/V [23]. Linear propagation effects such as dispersion and phase matching were included in the frequency domain using conjugate field amplitudes aj(z,Ω), calculated from the time-domain amplitude using
Ω is defined as the optical-frequency difference relative to the field’s center frequency ωj. The linear part of the split-step propagation, for step size h, was given bywhere nj is the frequency-dependent refractive index calculated using Sellmeier equations for BBO [24]. The second term in the exponent is added so that calculations are done in the frame of the signal pulse traveling with a group velocity v S.Figure 7 compares measured and simulated signal spectra for the WC configuration. Results are plotted for a range of signal angles θ S. Similar features are seen in both, despite the fact that the simulations use a simple 2-D model that does not include effects of spatial walk-off and intensity variation over the pump, signal, and idler beams. In both cases, notchesare seen in the signal spectra corresponding to the generation of signal SHG, and the notches shift to longer signal wavelengths as the signal angle is reduced, consistent with the SHG phase-matching curves shown in Fig. 2. The calculated spectral features of parasitic signal SHG agree well with the experimental results.
Fig. 7 Comparison of experimental and numerical simulations for a range of signal angles in the WC configuration.
5. Impact of signal SHG on NOPA efficiency
One benefit of an accurate model is the ability to evaluate the impact of parasitic processes by comparing results calculated either with or without the process. Furthermore, the axial evolution of each field within the crystal can be calculated to show the exchange of energy between the fields. The parameters shown in Table 1 were chosen for a case study of a BBO NOPA, corresponding approximately to the experimental configuration in Fig. 4.
Table 1. Input Parameters for Signal SHG Case Study
Figure 8 shows amplified signal spectra for cases where the signal SHG was either included or suppressed in the model. The sharp notch and signal SH are evident in Fig. 8(a). Some spectral modulation is also seen in Fig. 8(b), even though SHG was suppressed in the simulation and the experimentally measured spectra were smooth [see Fig. 5(b)]. In this case the modulation is due to reconversion of the signal. The 2-D nature of model, which only considers one value of peak temporal intensity for the pump and signal, means that averaging of reconversion effects over the Gaussian beam were not included and the spectral structure was not averaged away.
The impact of signal SHG on peak amplifier efficiency is shown in Fig. 9 . Simulations predict that 10% of the signal pulse energy is lost to SHG at the optimum crystal length of 2.3 mm. In contrast, simulations of idler SHG show that a small increase in efficiency of a few percent can be expected when idler SHG is present. Similar effects have been observed in pulsed optical parametric oscillators (OPO’s); absorption of the idler removes it from the three-wave interaction, increasing the threshold at which the signal is reconverted to the pump [25].
Fig. 9 Simulated signal energy along the crystal, (a) with and (b) without signal SHG. In both cases signal reconversion means that the length for optimum efficiency is 2.3 mm. When SHG is included in the simulations, 10% of the signal energy at this length is lost to the parasitic process.
6. Parasitic processes in few-cycle NOPA
NOPA’s can support the extremely large optical bandwidth required by few-cycle pulses. Parasitic nonlinearities may occur over a significant wavelength range of the signal and idler. To investigate parasitic SHG effects over a larger spectral bandwidth, measurements with a few-cycle NOPA were performed. The experimental setup of the system is shown in Fig. 10 and is similar to the one presented in Refs [5,7]. A Ti:sapphire oscillator (Femtolasers Rainbow) delivering few-cycle pulses is used for seeding both the NOPA and a state-of-the-art FCPA system. The FCPA system delivers 900-μJ, 650-fs pump pulses that are frequency doubled in a 1-mm BBO crystal with an efficiency of 56%, resulting in 500-μJ pump pulses for the NOPA. Focusing to 100 GW/cm2 provides a gain of 5 × 104. The signal pulses traverse a pulse-shaping device and a prism stretcher before being imaged into the BBO crystal. The noncollinear angle is 2.5° to achieve broadband phase matching. This is slightly lower than the angle for group-velocity matching of signal and idler (2.6°). As a result, the phase matching is improved in the spectral wings, while a slight phase mismatch is generated in the center. This configuration leads to a larger effective gain bandwidth, therefore shorter pulses at the output of the system [20]. Because of the high pump intensity, the 0.5-nJ oscillator pulses can be amplified to 25 μJ in a single stage in both the WC and NWC configurations.
Quantitative measurements of the parasitic process were made using a bandpass filter (Newport BG.40) to separate the signal’s fundamental and SH pulses. Their spectra for the WC and NWC configurations are shown in Figs. 11(a) and 11(b), respectively. The SH energy for the WC geometry measured after the filter is 1.1 μJ, corresponding to ~4% of the energy of the amplified signal. The measured spectra clearly show phase-matched SHG of 860-nm signal to 430 nm. This agrees with the phase-matching calculations for the NOPA configuration. The amplified signal spectrum shows modulation that may originate not only from signal SHG, but also from sum–frequency generation (similar to the side notches shown in Fig. 5).
Fig. 11 Measured spectra for (a) the WC and (b) the NWC configurations containing the signal and its second harmonic. Note that both the fundamental and the second-harmonic spectra have been normalized to their peak value, independently.
In contrast, the parasitic signal SHG is significantly reduced in the NWC configuration and the spectrum of the amplified signal has less modulation. For the same amplified pulse energy, the signal SH power after the BG.40 filter is only 37 nJ, corresponding to ~0.1% of the amplified signal energy. Phase-matching calculations show that no signal wavelengths are perfectly phase matched in this configuration, resulting in inefficient signal SHG.
The idler SHG was also observed in the WC configuration and spanned from 550 nm to 750 nm. This is supported by the calculation of the phase-matching conditions shown in Fig. 3. As expected, no idler SHG was measured in the NWC configuration.
7. Parasitic phase-matching curves for other common nonlinear crystals
Parasitic SHG processes are not limited to BBO-based NOPA’s. The phase-matching curves for signal and idler SHG are shown in Fig. 12 for other common nonlinear crystals: LBO, BiBO, and DKDP. In all cases, the noncollinear angle was chosen for group-velocity matching of the signal and idler. The resulting signal angle within the crystal is plotted in dashed lines for both configurations, together with the corresponding SHG phase-matching curves, in Figs. 12(a), (c), and (e), respectively. The internal idler angles for idler SHG are plotted in Figs. 12(b), (d), and (f) (dashed lines). The SHG phase-matched signal and idler wavelengths for each crystal type and configuration are given in Table 2 .
Fig. 12 Phase-matching curves for signal and idler SHG for other common NOPA crystals: [(a),(b)] DKDP; [(c),(d)] LBO; and [(e),(f)] BiBO.
Table 2. SHG Phase-matched Signal and Idler Wavelengths for BBO, LBO, BiBO, and DKDP in both WC and NWC Configurations for NOPA’s Pumped by the Second Harmonic of Yb-based Systems (⌊P = 510 to 525 nm)
The resulting idler wavelengths, which are phase matched for second-harmonic generation, and the corresponding signal wavelengths are displayed in Table 2. The last row of Table 2 represents the usable parasitic SHG-free signal wavelength range around an 800-nm central wavelength. It has to be taken into account for the design of a broadband OPCPA system. In summary, the NWC configuration offers the largest parasitic-SHG-free bandwidth for all considered crystals. Furthermore, BBO in the NWC configuration seems to be most suitable for ultra-broadband NOPA’s since it is free of parasitic-SHG phase matching until 1129 nm. Deviations from the assumed noncollinear angles and pump wavelength will shift the SHG phase-matched wavelength slightly, although the general behavior remains the same.
8. Summary and conclusions
A detailed analysis of parasitic nonlinear effects, namely second-harmonic generation of the signal and idler waves, on broadband high-repetition-rate NOPA’s is presented. Results from an experimental study of a white-light–seeded NOPA agree well with 2-D numerical simulations. Modeling shows that second-harmonic generation of the signal can reduce the NOPA output energy by 10%.
A quantitative analysis was carried out for an ultra-broadband, few-cycle NOPA. For the WC configuration, a few percent of the output is converted into signal SH while, the effect is reduced by at least an order of magnitude for the NWC configuration. Strong spectral modulation is observed in the WC configuration that originates from SHG and SFG of the spectral components of the signal and idler. Such modulation can significantly reduce the pulse quality of the output pulses and can be avoided by using the NWC configuration. A detailed phase-matching analysis for the most common nonlinear crystals is presented as a guide for designing NOPA systems.
The NOPA application dictates which configuration is best suited. For a laser front end, for example, a symmetric near-field profile and good spectral uniformity across the beam are required. In the case of low pump-pulse energy (<10 μJ) and peak power (~10 MW), the WC geometry is the best choice, provided parasitic SHG can be avoided over the amplifier’s wavelength range. For spectroscopic applications, where the bandwidth and tuning range of the signal are primary requirements, the NWC configuration is a better choice, provided that lower gain is acceptable or that the poor beam symmetry and spectral nonuniformity across the beam are tolerable. In the case of an ultra-broadband few-cycle NOPA, the NWC configuration must be used to avoid power loss and spectral modulation on the ultra-broad spectrum. Distortions in the amplified beam can be avoided in this case by weakly focusing the high-peak-power pump laser.
Acknowledgment
This work has been supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article. This work was also supported by the German Federal Ministry of Education and Research (BMBF) with project 03ZIK455 ‘onCOOPtics,’ the Helmholtz Institute Jena, and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement 240460. S. Hädrich acknowledges financial support by the Carl Zeiss Stiftung Germany.
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