Bandwidth extension and conversion efficiency improvements beyond phase matching limitations using cavity-enhanced OPCPA

: The conversion efficiency and phase matching bandwidth of ultrafast optical parametric amplification (OPA) are constrained by the dispersion and nonlinear coefficient of the employed crystal as well as pulse shaping effects. In our work we show that an enhancement cavity resonant with the pump seeded at the full repetition rate of the pump laser can automatically reshape the small-signal gain in optical parametric chirped-pulse amplification (OPCPA) to achieve close-to-optimal operation. This new method termed cavity-enhanced OPCPA or C-OPCPA significantly increases both the gain bandwidth and the conversion efficiency, in addition to boosting gain for high-repetition-rate amplification. The goal in C-OPCPA is to arrive at a condition of impedance matching at all temporal coordinates, such that, in the absence of linear losses, all the incident pump power is dissipated in the nonlinear loss element, i.e., converted to signal and idler. The use of a low finesse enhancement cavity resonant with a low average power ( < 1W) and a high repetition rate (78MHz) pump source is shown to achieve more than 50% conversion efficiency into signal and idler from the coupled pump in an optical parametric process, whereas an equivalent amount of pump power in a single-pass configuration leads to negligible conversion. Additionally, the gain

OPCPA scales the performance of OPA by combining chirped pulse amplification with parametric amplification [30]. In OPCPAs, a narrowband pump source with sufficiently long duration and a broadband signal pulse, chirped to match the pump pulse duration, are directed toward a nonlinear crystal, phase matched for parametric amplification. High energy pump pulses can be made sufficiently long so that the peak intensity is below the onset of crystal damage ( Fig. 1(a)). Thus, energy scaling is achieved in this arrangement beyond OPA by distributing pump energy over time. OPCPA systems, however, suffer from limitations due to the Gaussian shaped profiles and time dependent wave-vector mismatches, which result in more optimal conversion at the pump pulse peak as well as degraded conversion efficiency in the leading or trailing ends ( Fig. 1(b)) [31][32][33]. scales OPA by spreading signal over time such that pump intensity over time is kept below damage limit. However, the gain drops off in the wings due to increased wavevector mismatch and a reduction of pump intensity in the tails of the pump/signal distribution. (c) Gain drop off can be mitigated to a certain extent by shaping the pump profiles such that the pump intensity increases in the pulse wings. The pump profile that optimizes conversion at every temporal coordinate is termed the "conformal" pump profile [31][32][33].
To further scale performance, we proposed to coherently combine pump pulses in a low finesse enhancement cavity (which is transparent to signal and idler) containing a nonlinear crystal in which a synchronized signal beam is amplified via OPCPA [34]. This method, entitled cavity-enhanced OPCPA (C-OPCPA), will increase the usable peak pump intensity and is an alternative to average pump power scaling, as the repetition rate is increased. Enhancement cavities have been used in other contexts [35][36][37][38]. In C-OPCPA, however, the cavity supports a natural feedback leading to pump pulse shaping for optimal conversion. Normally in an enhancement cavity, less loss leads to the build-up of more intracavity power. In C-OPCPA, parametric amplification represents a time-varying nonlinear loss element. Thus, temporal regions with less conversion experience greater enhancement of pump power as a result of natural cavity buildup dynamics. This increased pump intensity then drives the initially low conversion parametric process to a more optimal steady state operating point. Thus, the buildup of pump power at temporal coordinates where it is needed can directly compensate for reduced conversion in the wings where large wave-vector mismatch and reduced pump intensity normally limit single pass performance. The intracavity pump thus passively reshapes, converging toward "conformal" profiles as shown in Fig. 1(c) [33]. As we will discuss below, the enhancement cavity here serves as a means to (1) increase the effective pump power and (2) passively reshape the pump pulses for optimal conversion efficiency. This pump pulse self-shaping can extend the signal gain bandwidth to several times the phase-matching bandwidth of the OPA crystal while also maintaining good conversion efficiency. Taken together, we hypothesize that C-OPCPA systems will be able to extend the capabilities of existing nonlinear crystals beyond their material properties. Additionally, the ideas explored here are applicable to nonlinear optics in integrated waveguides where high intensity and dispersion are typically more problematic than in bulk optics [39].
In this work, we demonstrate conversion efficiency and BW well beyond limitations of singlepass OPCPA set by the dispersion and nonlinearity using a C-OPCPA system. Specifically, we show more than 50% conversion efficiency of coupled pump light at a high repetition rate (78 MHz) and an increase in the tuning bandwidth by a factor of 3 to 4 times beyond the phase matching bandwidth under conditions in which an equivalent amount of pump power would lead to negligible conversion in a single-pass configuration. First, we provide a conceptual framework for C-OPCPA in the context of impedance matching (IM) of optical cavities. We then present a detailed experimental study of IM and performance improvements in a C-OPCPA system. Through comparison with the single-pass case, we evaluate the underlying impedance-matched conditions and relate them to the C-OPCPA performance, showing a substantial increase in conversion efficiency and BW. Additionally, we discuss parasitic nonlinear effects observed when the wave-vector mismatch was very large, which we attribute to competition between amplifying phase mismatched and matched signal and mediated by cavity dynamics and self-phase modulation (SPM). Finally, we compare the experimentally demonstrated system to a simulated single-pass configuration with optimal pump power (which exceeded the maximal power our system could provide). We find that C-OPCPA can achieve better conversion efficiency and BW at a fraction of the pump power required to achieve good conversion in single pass. This work thus confirms the essential dynamics expected in C-OPCPA systems, highlighting the performance improvements, and the potential to extend parametric amplification to the high-repetition rate regime.

Pump pulse self-optimization through impedance matching
The utility of C-OPCPA is its ability to extend the capabilities of existing nonlinear crystals beyond limitations set by their material properties. C-OPCPA overcomes these limitations by treating the parametric process as a nonlinear loss mechanism which can be impedance matched through a cavity's output coupling (OC) ratio. In C-OPCPA, pump pulses are coherently combined in a low-finesse enhancement cavity, as shown in Fig. 2, which is transparent to signal and idler, and contains an OPA crystal in which a synchronized signal beam is amplified. Unconverted pump light after a pass through the nonlinear crystal remains and builds up in the cavity. Thus, the system is an enhancement cavity at the pump wavelength with a nonlinear loss element from the intensity-dependent parametric conversion to signal and idler. The coherent combination of pump pulses allows amplification at high repetition rates and with a relatively low average power threshold, while also boosting the overall conversion efficiency by recycling pump light. We previously showed that such a system has the potential to achieve octave spanning gain [34]. In the following, in order to provide context for the experimental work we review the main concepts and summarize the results of the C-OPCPA operation from our previous study [34]. Then we go beyond that work to demonstrate broad tunability, which is the main aspect that corresponds to the experimental study of bandwidth performance in the later part of this work.
The behavior of a C-OPCPA system can be understood by considering the conversion at each temporal coordinate as an independent, nonlinear loss element and applying expressions for cavity build up and OPA gain as we discuss in Ref. [34]. The built-up intra-cavity power and the resulting power reflected from the input/output coupler can be described as: where R and T are the reflection and transmission coefficients of the input/output coupler and L(t) captures the intra-cavity loss, which includes both linear losses, L linear , and the nonlinear, In C-OPCPA the loss is due to conversion of the pump to signal and idler via parametric amplification. Thus, the loss is nonlinear and time varying. PPLN represents periodically poled lithium niobate as OPA crystal.
time-varying losses, L NL (t), due to the parametric conversion. Since, in OPCPA, a broadband seed is strongly chirped to match the temporal duration of a narrowband pump, each temporal coordinate can be treated as a parametric interaction between locally quasi-monochromatic waves [32]. This "local approximation" limit is valid provided that the signal is sufficiently chirped in which case dispersive propagation effects can be negligible. This is often the case in OPCPA, even with ultrabroad bandwidths. Thus, the temporal behavior of the nonlinear loss in C-OPCPA depends only on the local OPCPA gain, G(t): where l is the crystal length and the last equality holds in the limit of low (<∼20%) pump conversion. The small-signal gain, g(t), has time-dependence through the intra-cavity nonlinear drive, a function of intra-cavity pump intensity, Γ ∼ √︁ I(t), and the local wave-vector mismatch, ∆k(t). Equations (1) and (2) are both satisfied at each temporal coordinate when the equilibrium state of the cavity is reached after multiple round trips. The sum of new and recycled pump power provides sufficient gain to convert the new light that enters the cavity. Thus, temporal coordinates with low incident pump intensity and/or large wave-vector mismatch experience greater intra-cavity enhancement needed to compensate for the initially low small-signal gain. In a linear cavity, when L = T, the cavity is impedance matched; subsequently, the reflected power vanishes, and the intra-cavity power is enhanced by a factor of 1/T. Thus, at IM all the power incident on the cavity is dissipated through the cavity loss. In C-OPCPA, the main loss element is the intracavity conversion to signal and idler. An enhancement cavity with properly selected T and ratio between incident pump and seed powers naturally arrives at equilibrium with conditions close to this, i.e., L NL (t) ≈ T, so that the reflected power is nearly zero and all the incident pump light enters the cavity. The goal in C-OPCPA is to arrive at a condition of IM at all temporal coordinates, such that, in the absence of linear losses, all the incident pump power is converted to signal and idler (nonlinear loss).
In Ref. [34] we illustrated the above ideas by modeling a C-OPCPA system with a split-step simulation which captures the propagation of the intracavity pump, signal and idler pulses through a nonlinear crystal and the buildup of intracavity pump power as the incident pump is combined through an input/output coupler with the intracavity pump which is retained after passing through the crystal. Representative experimental parameters were used. We reconstructed these results in Fig. 3. However, we expand on Ref. [34] by showing the progression per round trip (or simulation iteration) of the buildup of the intracavity power and the suppression of the reflection starting from an empty cavity. The optimized single-pass OPCPA is included for comparison with identical parameters except for 5-fold-higher pump power, which is required for saturated single-pass gain. In both cases, ∆k = 0 for a 1550-nm signal wavelength, and the peaks of the pump and seed pulses are timed to coincide. Figure 3(a) shows the role of the passive pulse shaping of the intracavity pump, allowing more power to be developed in the pulse wings. IM is perfectly achieved at two temporal coordinates ( Fig. 3(b)). Overall, good IM across the pump pulse temporal profile is achieved, as indicated by the reduced reflection in steady state, shown in Fig. 3(c). The broader fractional conversion profile of the C-OPCPA case compared to the single-pass case ( Fig. 3(d)) shows the advantage of C-OPCPA: in the single-pass case, conversion mostly occurs near the peak of the corresponding Gaussian pump pulse; therefore, gain narrowing limits the total conversion efficiency of incident pump to signal and idler to 43%. The resulting bandwidth of the amplified signal is 52 nm in full width at half-maximum (FWHM), which is slightly larger than the calculated phase-matching bandwidth of 44 nm because of broadening due to amplifier saturation. In comparison, the C-OPCPA case produces a signal pulse with 103 nm bandwidth and 68% total conversion efficiency. Thus, with C-OPCPA, we attain twice the bandwidth and 1.6-fold greater conversion efficiency at 20% of the incident pump power. Our simulation thus illustrates the passive pulse shaping of the intracavity pump profile and IM responsible for the performance improvement of C-OPCPA over the single pass case.
It is important to note that the ability of the cavity to shape the pump pulse to a conformal-like pulse profile and optimize conversion is a consequence of how the parametric interaction changes with wave-vector mismatch. For the single pass case, the peak conversion efficiency for each quasi-CW parametric interaction shifts to longer propagation lengths as the wave-vector mismatch increases (see Fig. 4(a)). This shift can be offset by increasing the pump intensity. When this happens, the peak conversion efficiency with respect to the original pump intensity increases and shifts to shorter propagation lengths [32]. Thus, for a fixed propagation length, pump intensity can be used to compensate for wave-vector mismatch in the parametric interaction. In C-OPCPA, the cavity leverages this by enhancing pump intensity at temporal coordinates where ∆k is large. This reshaping of the intracavity pump intensity then shifts the peak intensity to shorter lengths, effectively aligning the peak conversion to a single propagation length (see Fig. 4(b)). This is not true for all non-linear optical processes. In second harmonic generation (SHG), for example, the peak conversion efficiency shifts to shorter propagation lengths [40] such that for SHG an increase in pump intensity cannot be used to compensate for wave-vector mismatch for a fixed propagation length. In C-OPCPA, however, the cavity can be leveraged to align the peak conversion efficiencies so that it is possible to optimize the conversion across the pulse for a fixed propagation length. This passive optimization of the intracavity pulse shape in combination with impedance matching sets the performance advantage of C-OPCPA systems. While a single pass system could make use of pump pulse shaping to achieve conformal profiles [33], the passive pulse shaping combined with impedance matching in C-OPCPA systems, opens the possibility for perfect conversion at each temporal coordinate across the interacting pulses.
The capability of C-OPCPA to extend gain bandwidth while boosting conversion efficiency could be employed in a wide range of OPCPA systems. A particularly dramatic result can be obtained when it is used in conjunction with a group-velocity matched, degenerate OPA system which already has a phase-matching bandwidth that spans a significant fraction of an octave in single-pass configuration [4]. For example, using cavity parameters similar to those of the example above, but with signal and idler wavelengths equal at 2.06 µm and pump wavelength at 1.03 µm, we have calculated that the C-OPCPA technique can be used to extend the gain bandwidth to 1.3 octaves [34]: noting that the cavity can offset the effect of moderate wave-vector mismatch, the poling period of PPLN can be chosen to bias the wave-vector mismatch versus wavelength such that the maximum deviation of ∆k from zero is minimized over the largest possible wavelength range (see red curve, Fig. 5 (a)).

Fig. 3. Numerical study of C-OPCPA system illustrating impedance matching (IM):
Simulated C-OPCPA with a 10 W, 6 ps, transform-limited pump pulse train at 1037 nm and a 2 µW, 6 ps, chirped signal pulse train at 1550 nm with 100 nm bandwidth, which mix in a 5 mm long PPLN crystal at an 78 MHz repetition rate. Cavity parameters are T = 10% and L linear = 1%. For comparison, optimized single-pass OPCPA is simulated with identical parameters except for a higher pump power level of 50 W. (a) Buildup of the intracavity pump pulse starting with an empty cavity. C-OPCPA recycled pump power accumulates in temporal regions where there is low loss, i.e. less conversion. The accumulation of pump power improves conversion efficiency, particularly in the pulse wings, maintaining gain across the pulse duration. C-OPCPA can achieve a similar shape for the intracavity pump profile through the natural feedback of the interacting pulses in a passive enhancement cavity. Progression of the buildup of intracavity intensity (black curves) is shown. (b) Steady state loss (red) including nonlinear loss from the parametric conversion and linear losses. Static output coupling (OC) value for the cavity (black). The intersection (blue arrows) indicates the impedance matched temporal coordinates. At these coordinates, perfect conversion, i.e., zero reflection, is possible, limited only by nonlinear phase effects and linear losses. (c) Reflection from an initially empty cavity. Progression of the reduction in reflected intensity (black curves) is shown. The final steady state reflection profile is shown in red. The dips correspond to minima in reflected power at the impedance matched temporal coordinates indicated in (b). (d) (Reconstructed from Ref. [34]) Fractional conversion efficiency (i.e. conversion of incident pump to signal plus idler) versus time for the single-pass case compared to the cavity-enhanced case. Fractional intracavity conversion and output coupling minus linear loss are also shown, illustrating impedance matching. The dashed vertical line indicates the length of the crystal in the simulation. In this figure the propagation is continued to show the conversion peak. In (a) for the single pass case, the gaussian pump profile prevents achieving optimal conversion across the chirped signal pulse in a fixed crystal length. In (b) for the C-OPCPA case, passive pulse shaping aligns peak conversion lengths across the chirped signal pulse thus enabling optimal performance in a fixed crystal length. Conversion efficiency in (b) is relative to the intracavity pump power not the incident pump power.
Instead of trying to achieve as broad a bandwidth as possible, an alternative use of this C-OPCPA setup is to achieve wide tunability for broadband pulses while significantly improving the conversion efficiency. Figure 5 illustrates the tuning capability and signal gain bandwidth that can be achieved if the interaction is phase-matched at 1.55 µm and 100-nm seed pulses centered throughout the 1.5 µm to 3.4 µm range are used. In each case, the seed pulse is appropriately stretched to match the pulse duration of the pump, and the PPLN OPA crystal is phase matched at 1.55 µm. These results highlight the ability of the cavity to efficiently amplify the seed pulse when the parametric conversion process is phase mismatched: while the phase-matching bandwidth covers only narrow regions around 1.55 µm and 3.07 µm, efficient conversion of incident pump to signal plus idler (ranging from 27 to 78%), occurs over the full 1.5 µm to 3.4 µm range. The pulses shown in Fig. 5(a) are the amplified seed, which can be either signal or idler depending on the wavelength. Each seed has the same pulse duration and is positively chirped in frequency. We are thus able to represent the pulses in terms of intensity vs wavelength. Near the 3 µm region the slope of the wavevector mismatch flattens as compared to the 1.5 µm region so that the amplification bandwidth is broadened. Thus, the peak intensity is reduced compared to the 1.5 µm region but the width is increased. Figure 5(b) shows the corresponding intracavity pump pulses. The pump pulses are all at a fixed wavelength but have been shifted along the x-axis so that the intracavity pump pulse lines up with the corresponding seed pulse in Fig. 5(a). Thus, Fig. 5 illustrates the action of the cavity, namely, intracavity power builds up to compensate for wavevector mismatch, increasing nonlinear drive so that total converted power to signal and idler is maintained as the seed wavelength is broadly tuned.
These results are possible due to the ability of an increased nonlinear drive to exactly compensate for reductions in gain due to wavevector mismatch when the intracavity conversion is low. Since low intracavity conversion is the natural operating point of the cavity, amplification without phase matching is possible. These results highlight an additional practical capability of C-OPCPA: wide tunability covering a greater-than-octave bandwidth, using a single nonlinear crystal and which does not require temperature tuning to adjust phase matching. Tuning in C-OPCPA occurs as a result of the cavity's response to compensate for wave-vector mismatch. This dynamical tuning can operate much faster than temperature adjustments and may find unique applications that can leverage rapid tunability. For example, the device can potentially be used to efficiently amplify a series of signal pulses where the central wavelength is rapidly varying.

Demonstration of dramatic conversion efficiency improvement in C-OPCPA
As discussed above, reaching a condition closest to IM is the main objective for maximum conversion in C-OPCPA. We show, in the experimental demonstrations below, that an enhancement cavity with properly selected T and ratio between incident pump and seed powers naturally arrives at equilibrium with conditions close to this, i.e., L NL (t) ∼ T, such that the reflected power is nearly zero and all of the incident pump light enters the cavity. Demonstration of the extension of crystal capabilities by evaluating IM will illustrate the utility of the system and is a first step in the realization of C-OPCPA systems. Through a detailed comparison of the single-pass case to C-OPCPA with varying output-coupler values and crystal lengths, we show experimentally in this section how IM leads to dramatic performance improvement.
Specifically, in the study below, we interpret our previous results of a C-OPCPA system that used a 20-mm-long PPLN crystal (Λ=30 µm) [41] with type-0 phase matching, and extend the experimental investigation to a C-OPCPA system with a 10-mm-long PPLN crystal under varying conditions of OC ratio. Importantly, the 10-mm-long crystal provides greater opportunity to investigate BW extension (Section 4), given its improved phase matching bandwidth relative to the 20-mm case. For the 20-mm case we directly compared the single-pass case to the C-OPCPA case by removing the cavity input/output coupler [41]. However, since single-pass conversion was negligible at the available pump powers for the 10-mm-long crystal, the results from the 20-mm-long crystal were scaled appropriately for comparison. Overall, we demonstrated >50% conversion efficiency into signal and idler in a C-OPCPA system utilizing < 1 W of coupled pump power at a 78 MHz repetition rate under conditions that would lead to negligible single-pass conversion.
The experimental apparatus, shown schematically in Fig. 6, was adapted from Ref. [41] with several modifications in the parameters for this study, such as the bandwidth, duration, energy of the signal pulses, and OC ratio. An Yb-doped fiber laser (YDFL) centered at 1036 nm seeds the OPCPA pump chain and produces up to 5-W average power and 500 fs pump pulses with a two-stage YDFA. An Er-doped fiber laser (EDFL) centered at 1560 nm serves as the C-OPCPA signal providing ∼ 20 µW of seed power. Both lasers have a 78-MHz repetition rate to match the free spectral range of the enhancement cavity (ring cavity) containing the PPLN crystal, where the finesse is ∼60-120 depending on output couplers (10 or 5%, respectively). PPLN was chosen due to its highest d eff value among nonlinear crystals in this spectral range. An imaging relay enables 55% coupling efficiency of the incident pump beam to the enhancement cavity. The enhancement cavity is locked to the YDFL cavity via the Hänsch-Couillaud detection scheme [42], and the YDFL and EDFL are locked via electronic synchronization [43]. Using this setup, we previously demonstrated a highly efficient C-OPCPA system with a 20-mm-long crystal, and 10% input/output coupler. With only 650 mW of coupled pump power, we are able to achieve nearly 210 mW and 105 mW of signal and the idler power, respectively, corresponding to 55% conversion [41]. The resulting signal spectrum is shown in Fig. 7(a). In contrast, the single-pass configuration with this incident pump power yielded only 1.3 mW of amplified signal, which is 2 orders of magnitude lower. These high conversion efficiency results can be understood in the context of IM. As discussed in Section 2, optimal conversion in C-OPCPA is achieved, when the input/output coupler value, T, matches the power dependent intracavity conversion plus loss. Thus, the incident pump power should be chosen such that the intracavity power matches the power needed to achieve a single-pass conversion of T. Under these conditions for IM, the intracavity power is simply the incident power scaled by 1/T. For the results discussed here, the C-OPCPA has an intracavity pump power of 4W. In the single pass case with 4 W of pump power, we were able to convert 8% of the pump power into signal and idler. Additionally, the cavity optics adds about 2% linear loss. Thus, the total intracavity conversion nearly matches the chosen input/output coupling of 10% and so we are able to achieve good average IM which enables the dramatic conversion efficiency improvement with respect to the coupled incident pump power. Thus, we conclude that even though the intracavity conversion is relatively low, IM leads to large overall conversion efficiency with incident pump power reduced dramatically by a factor of 1/T of the targeted intracavity power level. We further evaluated IM for this case by varying the incident pump power and measuring total conversion and enhancement. This data, shown in Fig. 7(b), illustrates the ability of the cavity to passively maintain high conversion in response to factors that reduce conversion and allows characterization of the IM operating point of the C-OPCPA system. As the pump power increases from zero, the net conversion efficiency increases and saturates at ∼50% for >0.4 W. Accordingly, the enhancement is first high (∼ 26) and then quickly reduces to ∼10 or less for >0.4 W. The cavity naturally employs increased enhancement to offset low incident power, thus maintaining intracavity peak intensity and conversion over a wide range of incident pump power. The passive self-adjustment results in the incident power threshold for amplification being ∼5 times less compared to that of the single-pass amplification as shown in Fig. 7(c). Additionally, IM is maintained over a wide range of pump powers. Conversion efficiency of >50% is observed between 0.4 and 1 W of incident pump power where the enhancement ranges from 11.4 to 7.4. The variation in enhancement across this range is small and close to 10 which is consistent with the IM with a 10% output coupler chosen. The power study thus characterizes the natural feedback between enhancement and conversion needed to maintain optimal IM. This feedback is principally responsible for the improved performance of C-OPCPA systems compared to ordinary systems.
In the next set of experiments, we investigated C-OPCPA performance for a case with even less conversion in single pass, in which we reduce the PPLN crystal length to 10 mm. However, we hypothesize that we can maintain IM and system performance by adjusting the OC ratio. For the same cavity conditions as the 20-mm case (10% output/input coupler), we observed that the enhancement reduces as the incident power increases as expected (Fig. 8(a)) indicative of the natural feedback between enhancement and intracavity conversion. A 10% output coupler with the 10-mm-long crystal, however, does not provide enough enhancement to maintain conversion as indicated by the fact that enhancement is always greater than the impedance matched value of 10, (1/T = 1/0.1), for the range of coupled pump powers accessible by the experimental setup. This indicates insufficient nonlinear drive for IM. However, a 5% output coupler increases the available enhancement and enables the C-OPCPA to be driven into saturation (Fig. 8(b)). Also, good impedance-matching is achieved as the IM enhancement of 1/0.05 = 20 is attained at under 0.5 W. Thus, for a 10-mm-long PPLN, an OC transmission of T=5% was chosen to increase enhancement in order to compensate for reduction in nonlinear drive and maintain optimal IM. In summary, we have shown a 10 times improvement beyond single-pass case for the 20mm-long PPLN with 10% OC and a 20 times improvement for the 10-mm-long one with 5% OC over the corresponding single pass cases. Taken together, the results with the two crystal lengths illustrate the advantages of C-OPCPA. Under proper IM, a small fractional conversion of the intracavity pump power results in a large conversion of the incident pump power thereby extending performance beyond limitations in single pass configuration.

Demonstration of bandwidth extension in C-OPCPA
Another goal of this work is to provide a detailed characterization of the gain bandwidth of the C-OPCPA system and compare that to the single-pass case in order to demonstrate the performance improvement enabled by C-OPCPA. Ideally, in order to measure the gain, the seed (signal) source would have either a wide center-wavelength tuning range or produce broadbandwidth pulses compressed to within the intracavity pump pulse duration. However, the seed source in this demonstration experiment had a FWHM bandwidth less than the phase-matching bandwidth for the chosen crystals. Therefore, we interrogated the BW dependence by adjusting the phase-matching condition rather than the signal wavelength. By tuning the phase matched condition over a broad range of wave-vector mismatches and monitoring the phase matched wavelength through the superfluorescence spectrum, we were able to determine and compare the bandwidth performance of the parametric processes in this study. Toward this end, we have taken two approaches to investigate the gain BW for the two cases discussed in the previous section (20 mm thickness with 10%OC and 10 mm thickness with 5%OC).

Bandwidth extension using 20-mm-long PPLN crystal
We compared the single-pass to C-OPCPA bandwidth for the 20-mm case by varying the phase-matching wavelength though temperature tuning the PPLN with the pump and chirped seed temporal overlap fixed at their peaks. Since the phase matching BW is intensity dependent, the single-pass pump intensity was chosen to match the intracavity power at the optimally phase-matched temperature for the C-OPCPA case, allowing a fair comparison. As the phase matching wavelength is tuned, the single-pass conversion efficiency drops off such that the gain-bandwidth follows the usual Gaussian shaped profile as shown in Fig. 9(a). In C-OPCPA, however, the cavity responds to the initially reduced conversion from wave-vector mismatch by increasing the enhancement making available more intracavity pump power and maintaining the nonlinear process. The resulting effect is that the conversion into signal and idler is maintained over the entire tuning range (Fig. 9(b)). Thus, we are able to observe the principle natural feedback dynamics of C-OPCPA; the cavity's ability to compensate wave-vector mismatch with enhancement. We observed non-ideal shifts of the central wavelength of the amplified signal when the wavevector mismatch was large. The central wavelength of the amplified signal in the C-OPCPA setup as determined by the center-of-mass wavelength in measured amplified signal spectrum shifted relative to the input signal spectrum for large wave-vector mismatch in the C-OPCPA setup. We account for this non-ideal behavior in the comparison to the single-pass case. The central wavelength of the amplified signal fixed in the single pass is determined by the overlap of the gain window set pump and the highly chirped seed; however, in C-OPCPA as the temperature is tuned, more optimally phase matched signal wavelengths are brought under the gain window set by the pump pulse due to pump and signal walk-off, causing the central wavelength of the amplified signal spectrum to be shifted ( Fig. 9(b)). This group velocity walk-off can be reduced by limiting the signal bandwidth.
Nevertheless, for the conditions of this experiment we are still able to quantify the BW improvement. Across the range from 1543 nm (T=130) to 1570 nm (T=170), the central wavelength of the amplified signal spectrum matches that of the seed, and we observe the cavity compensating wave-vector mismatch at the desired wavelength and maintaining uniform conversion while the single-pass conversion rolls off. At T=170, beyond the phase matching BW, the single-pass case has <1% of conversion, while the C-OPCPA case has more than 40%. Importantly, at T=200 the amplified signal spectrum is centered at 1571 nm while the phase matching wavelength is at 1593 nm, beyond the phase-matching BW by a factor of 4.2. This result captures the natural feedback mechanism of C-OPCPA; namely, that enhancement can passively offset wavevector mismatch and maintain conversion efficiency over a wider range compared to the single pass case.

Bandwidth extension using 10-mm-long PPLN crystal
We next used a shorter, 10 mm crystal to avoid temporal walk-off effects. Additionally, the crystal had a larger aperture to avoid clipping of the optical beams. Several experimental parameters were adjusted to optimize tuning performance. Namely, the signal bandwidth was made more narrowband, and the 10% OC was replaced with a 5% OC. The signal spectrum is limited in bandwidth to 5 nm by a narrowband filter. This was done to avoid the amplification of out-of-band signal wavelengths. Since the pump source had insufficient power to generate super-fluorescence in single pass, the spectrum from the unseeded cavity with maximum intracavity power was used to determine the phase matching wavelength. Unseeded, maximal conversion occurs when the wave-vector mismatch is zero allowing us to determine the phase matched wavelength from the super-fluorescence.
We again observe the natural feedback mechanism for C-OPCPA in this case, enhancement compensating wave-vector mismatch Fig. 10(a), which is responsible for the BW extension ( Fig. 10(b)). However, we note that although we attempted to constrain signal bandwidth with the intracavity filter, spectral components are generated outside the signal bandwidth from parasitic nonlinear processes with a lower threshold than the intended one. Thus, to isolate the effects of C-OPCPA within the signal band, spectral components outside the bandwidth of the input seed spectrum were numerically filtered for each acquired spectra. This will be discussed in more detail in the next subsection. We achieved a FWHM BW and conversion efficiency of the C-OPCPA system beyond what is expected from the single-pass case. The FWHM bandwidth of the C-OPCPA system, as determined from the spectrally filtered conversion curve ( Fig. 10(b)), is found to be 60 nm. Conversion in the single-pass case was very small and not measurable thus we take the calculated phase matching gain BW of near 20 nm as a comparison. Thus, while single-pass conversion is negligible, C-OPCPA dramatically increases the conversion to more than 60% and extends the tuning range by a factor of 3, well beyond limitations set by the single pass case.

Parasitic nonlinear effects
Thus far, we have experimentally demonstrated gain well beyond the calculated phase matched bandwidth in COPCPA systems; however, at the extremities of the tuning range, non-ideal behavior emerges resulting in the generation of spectral components outside the signal bandwidth range. At the edges of the tuning range, low conversion results in the build of a high level of intra-cavity pump power which becomes available to drive nonlinear processes aside from the desired parametric amplification. This results in the generation of frequency components outside of the signal bandwidth and thus should not be considered in the evaluation of conversion efficiency as the phase matching conditions are tuned. Here, we discuss possible mechanisms and characterize the impact on system performance.
We observe the development of structure on the pump spectrum ( Fig. 11 (a)-(d)) under unseeded and seeded conditions that lead to parasitic nonlinearities that result in new spectral components on the signal ( Fig. 12 (a)-(d)). During unseeded operation, the intracavity pump spectrum is highly structured which can be explained due to the interplay of SPM and cavity filtering. Because of SPM, the spectrum of the pump broadens and effectively loads the cavity with new pump spectrum beyond the initial spectrum. The cavity, which can be thought of as a Fabry-Perot cavity having a transmission bandwidth and a free spectral range, will capture the generated pump spectrum that satisfies the resonant condition which is periodic with the cavity free spectral range. The intra-cavity pump spectrum, therefore, develops side bands as pump wavelengths compatible with the free spectral range and within the reflectance bandwidth accumulate in the cavity. During seeded operation, close to phase-matching there is sufficient intra-cavity conversion to keep the built-up intracavity power low and prevent the development of structure on the pump. At the extreme-case of wave-vector mismatch, the intracavity conversion is low and thus the intracavity power becomes structured and retains the overall shape of the unseeded case. Additionally, under these conditions, a distinct spectral region develops which is not present in the input signal. The effects of parametric amplification using a structured pump spectrum and using a signal wavelength far from the phase matched wavelength may play a role in the generation of new signal spectrum localized closer to the phase matched. To evaluate the impact of the parasitic nonlinearities on the C-OPCPA system performance we consider the total signal power compared to the power in the unamplified signal bandwidth ( Fig. 13(a)), and center of mass wavelength of the amplified/filtered signal as the wave-vector is tuned ( Fig. 13(b)). Overall, the spectral output is robust against the parasitic nonlinear effects for most of the tuning range within a factor of two of the phase-matching bandwidth, from 1.52 µm to 1.57 µm. Toward the long end of this range, from 1.54 µm to 1.57 µm, the cavity maintains the conversion into the signal with little parasitic effects as indicated by the stable center of mass wavelength over this range. The slight shift in the center of mass wavelength can be accounted for by considering that more optimally phase matched signal wavelengths in the signal bandwidth are amplified in the gain window provided by the pump pulse. Thus, the cavity behavior over this range, within a factor of two of the phase-matching bandwidth, is well behaved and can be explained in the context of IM.
However, beyond this range, past 1.57 µm, non-ideal behavior coincides with the intra-cavity pump power increasing from 10 W to 14 W sharply (see Fig. 10(a)) and the center of mass wavelength for the amplified signal deviating from signal range. The jump in intracavity power occurs since the magnitude of the wave-vector mismatch exceeds what the cavity can compensate, and so the intra-cavity pump power simply approaches the maximum possible. This increased intra-cavity pump power drives parasitic nonlinear processes which results in conversion into wavelengths outside the input signal bandwidth and thus results in a shift of the center of mass wavelength of the amplified signal. By numerically filtering the measured output spectrum to eliminate parasitic components from consideration, we can reveal the underlying system performance. While the measured signal power suggests conversion beyond 1.58 µm, the filtered signal power shows the cavity bandwidth drops sharply at 1.57 µm. Thus, although we still measure very large bandwidth improvement, we observe that parasitic effects degrade performance at the extreme ends of the wavelength spectrum. More spectral filtering of the pump may help mitigate these effects by suppressing the initial generation of sidebands.

Comparison of experimental data to numerically simulated, depletion-optimized single-pass OPA
We have shown experimentally that C-OPCPA improves conversion efficiency and bandwidth by comparing measured C-OPCPA results to a single-pass case with a fixed pump power. The single-pass power level chosen was representative of the average intracavity power in the corresponding C-OPCPA case. Conversion efficiency in single pass OPA however is dependent on the level of incident pump power [32,44] and thus in the previous section in principle we could have achieved better single-pass performance however we were limited by the maximum available pump power. In this section, we numerically determine the optimal single pass pump intensity for the range of wave-vector mismatch achieved in the measurements and compare this to the corresponding experimentally measured C-OPCPA. This comparison enables us to evaluate C-OPCPA with the best case single pass scenario for a given ∆k. We calculated the pump intensity required for the depletion optimization. To avoid the need to optimize the time delay between signal and pump pulses, long, chirped signal pulses were chosen such that in the gain window defined by the pump pulse duration (1 ps) the signal had spectral components spanning the bandwidth range of the signal source used in the experiment (2.5 nm). As expected, conversion efficiency is cyclic with increasing pump intensity (Fig. 14). Increased cycles of saturation and back conversion result in pump and signal pulses that are highly structured in intensity and phase. Thus, only the first maximum in conversion efficiency corresponds to a realistic single-pass parametric amplifier, and thus corresponds to the best-case performance single-pass amplifier. Additionally, our simulations validate that the value of the depletion-optimizing pump increases while the conversion efficiency decreases with increasing wave-vector mismatch (see Fig. 14).

Fig. 14. Simulated conversion vs pump power:
Conversion to signal and idler as incident pump power is increased for single pass OPA corresponding to the experimental conditions with a 10 mm PPLN. The black curve corresponds to a phase-matched wavelength of λ PM = 1.55 µm (i.e ∆k = 0) and the red curve corresponds to λ PM = 1.6 µm. This simulation illustrates that an increase in wavevector mismatch can be offset by increasing the incident pump power, but the absolute conversion efficiency drops.
To establish the performance of C-OPCPA relative to the depletion optimized case, we ran simulations for the range of wave-vector mismatches corresponding to the experiments; and we compared simulated depletion-optimized conversion efficiency to the experimental C-OPCPA conversion efficiency (Fig. 15(a)) and simulated depletion-optimizing pump power and intracavity C-OPCPA pump power, (Fig. 15(b)). We compare total conversion efficiency and bandwidth. Conversion in the depletion-optimized single-pass case is peaked at 20% with a pump power higher than the intracavity pump power in the C-OPCPA which was already enhanced by a factor of 9 compared to the incident coupled pump power. Alternatively, the C-OPCPA case has a conversion efficiency of 60% with considerably lower incident pump power. As the phase-matched wavelength is tuned, the depletion-optimizing pump requires even more power, while the C-OPCPA passively increases to larger power levels through enhancement with a fixed incident pump power. The intracavity power increases by 50% at the end of the tuning range to compensate for up to 3 × 10 3 µm −1 of wave-vector mismatch. Under these conditions, the depletion-optimized single-pass case has roughly one third the conversion and twice the bandwidth as the experimental C-OPCPA case. The experimental C-OPCPA case, however, has roughly a 30% larger gain-bandwidth product with a fixed pump intensity and a substantially smaller average power.
The results of this section illustrate that C-OPCPA performance is based on passively adjusting enhancement, whereas the depletion-optimized single-pass case requires a large, variable-power pump source to reach optimal performance. This represents a key advantage of C-OPCPA over single-pass OPCPA in terms of pump power utilization. In C-OPCPA high conversion is achieved through IM which in principle can be perfectly matched to have full conversion, whereas the depletion-optimized single-pass case faces conversion limits set by phase-matching and drops with increase ∆k. Taken together this section shows that C-OPCPA not only compensates ∆k, but can also achieve a greater degree of conversion efficiency through the dynamics of intracavityconversion and enhancement. Thus, C-OPCPA extends parametric amplification beyond what is possible in a single pass case and represents a significate performance improvement.

Conclusion
In conclusion, we have experimentally demonstrated the basic operating principle of C-OPCPA in which parametric amplification is viewed as a loss element in an enhancement cavity. Optimal performance is achieved by varying the output coupler value to achieve conditions closest to impedance matching. The natural feedback between conversion efficiency/loss and intracavity pump power enhancement leads to a self-optimizing system that maximizes conversion efficiency in the presence of low pump power and/or wavevector mismatch. By varying the output coupler to achieve optimal impedance matching we were able to experimentally demonstrate 55% conversion efficiency with a crystal length of L=20 mm and OC of 10%, and more than 50% for a crystal length of L=10 mm and OC of 5%. With only 650 mW of pump power, a single pass parametric amplifier, under similar conditions resulted in only 0.52% conversion for the 20 mm long crystal and negligible conversion for the 10 mm long crystal. Additionally, by using narrow band sources and adjusting the phase matching conditions, we were able to demonstrate a 4.2 times improvement in bandwidth beyond the phase-matching bandwidth for the L=20 mm/10% OC case and a 3 times improvement for the L=10 mm/5% OC case. Further comparisons to detailed single pass simulations show that the tuning response with the C-OPCPA system presented in this work performs more effectively than a single pass system that uses an optimized pump power at each signal wavelength however the C-OPCPA system achieves the same level of performance with a fixed power pump source at approximately1/20th the power level.
The C-OPCPA system presented in this work was a proof-of-principle experiment demonstrating the key features of the system, namely, that enhancement can passively be used to compensate wave-vector mismatch through impedance matching. Although these results are encouraging, continued development of C-OPCPA sources need to address the limitations and technical challenges observed in the measurements presented here. Namely, limitations due to group velocity walk-off, linear losses, parasitic nonlinear losses, spatial effects, damage threshold, and cavity locking dynamics need to be considered carefully in future implementations.