Frequency-domain nonlinear optics in two-dimensionally patterned quasi-phase-matching media

Advances in the amplification and manipulation of ultrashort laser pulses has led to revolutions in several areas. Examples include chirped pulse amplification for generating high peak-power lasers, power-scalable amplification techniques, pulse shaping via modulation of spatially-dispersed laser pulses, and efficient frequency-mixing in quasi-phase-matched nonlinear crystals to access new spectral regions. In this work, we introduce and demonstrate a new platform for nonlinear optics which has the potential to combine all of these separate functionalities (pulse amplification, frequency transfer, and pulse shaping) into a single monolithic device. Moreover, our approach simultaneously offers solutions to the performance-limiting issues in the conventionally-used techniques, and supports scaling in power and bandwidth of the laser source. The approach is based on two-dimensional patterning of quasi-phase-matching gratings combined with optical parametric interactions involving spatially dispersed laser pulses. Our proof of principle experiment demonstrates this new paradigm via mid-infrared optical parametric chirped pulse amplification of few-cycle pulses.

and temporal/spectral profiles of the interacting ultrashort waveforms remains a demanding problem.
Recently, a complementary approach to OPCPA has been introduced, termed frequency domain optical parametric amplification (FOPA) [13,14]. The seed-pulse is dispersed spatially via a 4-f arrangement analogous to a pulse shaper [15], with amplification occurring at the Fourier plane. By filling this plane with several birefringent phase-matching crystals placed side by side, as in the first experimental demonstration of the FOPA technique [14], the phase-matching condition for different spectral regions can be adjusted separately, thereby relaxing one of the key constraints of conventional OPCPA systems. Moreover, because the seed is spatially chirped, its effective pulse duration can be matched to that of the few-ps pump pulse, allowing for efficient energy transfer. On the other hand, drawbacks to this powerful approach are that the optical path length through each crystal must be precisely matched, the complexity scales with the number of crystals used, and pre-pulses can be introduced by any diffraction from the edges of the crystals.
Here, we introduce and demonstrate a new paradigm for nonlinear-optical devices based on combining spatially dispersed laser pulses with a two-dimensionally patterned quasi-phasematching (QPM) medium. This approach represents an extremely versatile yet experimentally simple platform, allowing for the limitations of existing nonlinear devices, including the above-mentioned drawbacks of the FOPA, to be overcome systematically by lithographic patterning of optimal 2D-QPM gratings. We experimentally demonstrate the approach with a mid-infrared FOPA.
2 Two-dimensional quasi-phase-matching devices In QPM [12,17], the sign of the nonlinear coefficient is periodically or aperiodically inverted, augmenting the phase-matching condition with a term K g to yield |k p − k s − k i − K g | ≈ 0, where k j are the wavevectors of the interacting waves. In periodically poled ferroelectric materials such as LiNbO 3 , a lithography mask defines the QPM grating with high robustness [17][18][19]. Thus, whereas birefringent phase-matching relies only on favorable material properties, QPM media can be freely engineered via lithography. For example, chirped QPM gratings can extend the phase-matching bandwidth well beyond that of periodic QPM gratings [20,21]. Here we take a much more general approach by using fully two-dimensional QPM (2D-QPM) patterns to tailor the nonlinear interactions experienced by spatially-separated spectral components, as illustrated conceptually in Figure 1(a).
First, the QPM period can be varied in the transverse direction such that each spectral component is perfectly phase-matched. Figure 1(b) shows an example of this procedure for a mid-infrared pulse. In general, we can vary the QPM period continuously according to the exact trajectory required by material dispersion, with no inherent bandwidth constraint.
Moreover, the linear properties of the QPM crystal remain homogeneous, so only a single, monolithic, plane-parallel crystal is required. To design a practical QPM grating profile meeting this criterion, we introduce a smooth, nonlinear variation in the QPM period. A corresponding map of QPM period is shown in figure 1(d). To fabricate this design, we introduce the absolute phase of the QPM structure, where the local QPM period is 2π/K g , and the input QPM phase φ QP M (x, 0) is a design degree of freedom (see Methods). Given φ QP M , the nonlinear coefficient satisfies d(x, z) = sgn(cos(φ QP M (x, z))). The QPM phase is imparted to any waves generated during the nonlinear process: therefore, it introduces an opportunity to create an arbitrary phase mask for pulse shaping purposes. We discuss this potential in section 5.
The chosen φ QP M (x, 0) function is shown in figure 1(f), while figure 1(e) shows several of the resulting ferroelectric domains. We emphasize that the domains have no discontinuities, but have a significant curvature. This unique feature strongly contrasts with conventional mechanically-tunable QPM devices, which utilize multiple separate gratings for discrete tuning [22], or straight but "fanned" ferroelectric domains in fan-out devices used for continuous tuning [23].
To confirm that these domain profiles could be fabricated, we inspected the entire width of the devices used, as illustrated in figure

Frequency-domain parametric interactions in 2D-QPM media
We next elucidate the parametric process occurring in 2D-QPM media. We focus on 4-f pulse shaper arrangements as the means to introduce spatial chirp. Figure 2(a) shows a simulation of a simplified situation involving a continuous-wave pump and signal, but including the longitudinal variation of the grating. Exponential amplification occurs, followed by depletion of the pump, followed by a rapid change of the QPM period to "turn off" the interaction.
The capability to turn off or modify the parametric interaction in other ways within the nonlinear crystal is unique to structured QPM devices. Moreover, we show in figure 2 in longitudinally-chirped QPM devices using one or multiple QPM gratings [20,24,25], such devices are ultimately limited by coupling between different parts of the spectrum [26]. In contrast, the spatial chirp in the FOPA decouples the spectral components more robustly, enabling greater flexibility.
To reveal the extent of this decoupling for the more subtle case involving real pulsed beams, the complete spatiotemporal profile of the signal must be considered. Our analysis in Supplementary section 2, where we derive this profile, reveals a correspondence between the spatial profile at the input diffraction grating of the 4-f setup and the temporal profile in the Fourier plane. Consequently, the duration of the spatially-chirped signal pulse is where ∆λ is the range of wavelengths involved, and ∆x is the spatial extent of those wavelengths. As well as highlighting an important physical aspect of frequency-domain nonlinear optics, equation (2) allows comparison of τ ef f to the pump duration, which yields design guidelines for efficient operation.
To capture the nonlinear dynamics of the amplification process, we developed a numerical model for nonlinear mixing processes in 2D-QPM media (see Methods). Beyond amplifying the interacting waves, the QPM device can act as an arbitrary phase mask, thereby offering a unique platform for simultaneous gain and pulse shaping. This shaping is provided by the QPM phase φ QP M [figure 1(f)], which is imparted to the generated wave. The idler spectral phase φ i (ν) can be approximated as where k i (ν) is the idler wavevector and φ j (ν) is the spectral phase of wave j. A more general expression for φ i (ν), including additional terms to account for the longitudinal variation of the QPM grating, is given in Supplementary section 3.
There is significant freedom in choosing φ QP M (x, 0), subject only to constraints on the ferroelectric domain angles that can be fabricated. Consequently, very large phases can be imparted onto the idler. Unlike conventional pulse shapers, this phase is fully continuous, and no discrete wrapping between 0 and 2π phase is needed. These properties are illustrated in figure 2(f), which shows the group delay spectra of the signal and idler for two cases. The signal group delay is determined by the dispersion of the material, while the idler group delay is determined by equation (3). The two cases shown correspond to two orientations of a particular QPM grating, thereby emphasizing how the group delay can be modified substantially, over several picoseconds or more, without altering the amplification characteristics.

Experiment
To demonstrate the technique, we implemented the 2D-QPM FOPA as the final stage of a mid-infrared OPCPA system [27,28]  Measured spectra are shown in figure 3(b). The compressed output power after the second diffraction grating was 1.03 W, corresponding to 20.6 µJ pulse energy. Accounting for losses of the diffraction grating (≈ 31 %), the dichroic mirror (≈ 5%), and beam-routing optics, we estimate an average power of ≈ 1.65 W directly after the antireflection-coated 2D-QPM crys-tal, corresponding to 33 µJ pulse energy. We thus infer a quantum efficiency (ratio between output signal photons and input pump photons) of 32%, which is a substantial improvement over our previous OPCPA configuration based on non-collinear power amplification in a conventional PPLN crystal [28].
To compress the output pulses, we adjusted the 1550-nm seed pulses in the OPCPA frontend. The compressed pulses were measured using second-harmonic generation frequency  We assume an initial signal input to crystal FOPA-1 with center frequency ν. In FOPA-1, an idler wave (frequency ν p − ν) is generated. The spectral phase of this idler is related to the QPM phase of FOPA-1 according to equation (3). Next, the FOPA-1 output signal is discarded, while the FOPA-1 idler (frequency ν p − ν) seeds FOPA-2. The idler wave of FOPA-2 is at the original frequency ν, and its phase is related to the QPM phase of FOPA-2.
Therefore, the final signal output at frequency ν of the crystal pair has experienced a phase modulation according to the difference of the QPM phases of the two crystals (derived in Supplementary section 3). By adjusting the QPM phases by design of the crystals, an arbitrary effective phase mask for the signal wave can be achieved, thereby combining amplitude and phase shaping into a single crystal-pair.

Conclusions
In conclusion, by combining a spatially chirped input wave with a two-dimensionally patterned QPM crystal, we have introduced and demonstrated a new platform for nonlinear optics that has unprecedented flexibility and overcomes the limitations and trade-offs inherent in conventional devices. By using QPM crystals with curved domains fabricated with high fidelity, we have shown that the technique is scalable in bandwidth, since the QPM period trajectory in the crystal can be matched to the input wave, even for extremely broad bandwidths where the QPM period changes significantly and nonlinearly versus position.
The approach is applicable to a wide variety of nonlinear-optical devices, including harmonic generation and optical parametric amplification.
In contrast to conventional ultrafast processes, the 2D-QPM FOPA consists of a con- aperture QPM crystals provides great promise for energy scaling as well [33][34][35]. Moreover, QPM media are available in diverse spectral ranges, for example covering from the ultraviolet via LBGO [36], to the far-infrared via orientation patterned GaAs and GaP [37,38].
Therefore 2D-QPM devices will be applicable for pulse generation, shaping, and amplification across the optical spectrum, from the deep-ultraviolet to far-infrared. We therefore expect that frequency-domain processes enabled by such 2D-QPM media will have broad impact and appeal for many areas of photonics.

Methods
The 2D-QPM grating introduced in section 2 was designed by combining several techniques.
To obtain the transverse variation of the QPM period (direction x), we calculate the position of the spatially chirped spectral components using the grating equation, and apply the Sellmeier relation to find material phase-mismatch ∆k 0 = k p − k s − k i [16] according to figure   1(b). For the longitudinal variation (direction z), we first determine the required effective length according to figure 1(c), and then apply a z-dependent offset in K g ; we construct the offset using hyperbolic tangent functions, in analogy to apodization profiles discussed in the context of chirped QPM media [20,39,40]. The QPM profile for a particular x-slice is shown in figure 2(a), black curve.
Rather than simply removing the QPM grating entirely in the switched-off regions, we maintain a 50-% QPM duty cycle through the whole crystal, and instead switch off the amplification process by rapidly changing the QPM period away from phase-matching (i.e. a "nonlinear chirp" profile). Such a 50-% duty cycle helps suppress photorefractive effects [41][42][43], in addition to the suppression already provided by MgO-doping of the crystal.
The appropriateness of the chosen QPM designs was tested with simulations of the OPA process including pump depletion. The value of K g at the end of the grating is not directly coupled to the OPA gain provided the phase-mismatch ∆k = k p − k s − k i − K g is made large enough. However, nonlinear phase shifts can be introduced by phase-mismatched nonlinear processes [44]: 2D-QPM creates the opportunity to select the phase-mismatch to manipulate these nonlinear phase shifts. Accordingly, we selected the K g profile to yield phase shifts comparable in magnitude to those from the intrinsic χ (3) nonlinearity [45]. This approach, motivated by our recent work on adiabatic excitation of quadratic solitons [40], is another unique capability of 2D-QPM media which we will explore in more detail in future work.
With respect to fabrication, our choice of φ QP M (x, 0) in equation (1)  Our experiments described in section 4 use a mid-infrared OPCPA front-end containing two OPCPA pre-amplifiers [27,28]. A schematic of the system is shown in Supplementary figure 1. The system uses two synchronized lasers for pumping and seeding the pre-amplifiers.
The pump laser has a wavelength of 1064 nm and produces 14-ps pulses (FWHM) at a 50-kHz repetition rate, with 8 W average power. Approximately 5 W is directed to the pre-amplifiers, while the remaining power is directed to a home-built Innoslab-type amplifier, the output of which is used to pump the frequency domain OPA illustrated in figure 3(a). The seed laser is a femtosecond fiber laser with subsequent erbium-doped fiber amplifiers. The laser has a wavelength of 1550 nm and produces 70-fs pulses at an 82 MHz repetition rate, with 250-mW average power. The seed pulses are first spectrally broadened in a dispersion shifted fiber (DCF3, Thorlabs) before being chirped in time with a silicon prism pair and 4-f pulse shaper arrangement. This chirp is transferred to the mid-infrared by the second pre-amplifier, and as such we can optimize the compression of the final amplified mid-infrared pulses by adjusting the dispersion of the infrared seed pulses.
The OPCPA pre-amplifiers are based on longitudinally chirped quasi-phase-matching gratings, implemented in MgO:LiNbO 3 . We use the shorthand aperiodically poled lithium niobate (APPLN) to refer to them in [27]. Both crystals have the same QPM design, and the pump, signal, and idler beams are all collinearly aligned in the crystals. After the first APPLN crystal, we discard the idler wave (mid-infrared output), and route the pump and signal outputs to the second crystal. After the second APPLN crystal, the pump and signal waves are discarded, and we extract the 3400-nm mid-infrared wave to seed the final amplifier (the FOPA). Further information on the system is given in Supplementary section 1.
The chirped QPM gratings used for the pre-amplifiers can in principle be scaled in bandwidth, but careful consideration must be given to several design constraints, described in detail in [26]. These constraints, which relate to favoring the desired OPA process over various unwanted processes, become restrictive when operating in the highly pump depleted regime corresponding to adiabatic frequency conversion. Therefore, the combination of longitudinally chirped QPM devices for convenient and alignment-insensitive pre-amplification to moderate energy levels, followed by the 2D-QPM FOPA for final power amplification, represents a compelling system arrangement which preserves bandwidth scalability, avoids the challenging parasitic processes of highly-saturated chirped QPM devices, and keeps complexity at a minimum since only one FOPA arrangement is required.