1. Introduction

Periodically poled LiNbO₃ (PPLN) crystals [1] afford efficient and versatile implementations of coherent optical sources and frequency converters based on the quasi phase-matching (QPM) principle, with widespread use for a broad spectrum of applications, encompassing optical telecommunications, sensing, imaging and quantum optics [2–6]. At a local level, the domain reversal process, which underpins PPLN devices, can give rise to intriguing physical effects associated with modifications of the crystal structure arising in poled domains and domain walls. This encompasses exceptionally high strain and electric fields as well as unconventional electrical conduction and nonlinear optical properties [7–11]. Internal electrostatic fields generated by frustrated defects after polarization reversal are also at the origin of refractive index changes of individual poled domains in congruent undoped and MgO-doped LiNbO₃ (LN) and LiTaO₃ (LT) [8,12–14].

Refractive index changes in PPLN structures have so far been observed only in transverse pumping configurations [7–13], whereby the PPLN grating is probed with localized optical fields propagating orthogonally to the direction of the nonlinearity (ferroelectric domain) modulation. So far no evidence has been provided for any refractive index change affecting linear or nonlinear propagation in configurations such as the one illustrated in Fig. 1(a), where PPLN structures are optically pumped collinearly with the grating axis. Therefore standard models for nonlinear interactions in conventional QPM devices consider PPLN structures as purely nonlinear photonic crystals [15–16] and typically account only for a spatial modulation of the optical nonlinearity, assuming the refractive index to be the same in PPLN and unpoled regions [17–18].

 

Fig. 1. (a) Sketch of the OPG experimental set-up. HWP: Half-wave plate, Pol: polarizer, LP: longpass filter, BP: bandpass filter, CCD/PD: camera/photodetector, spherical lenses with focal lengths f₁= 200 mm and f₂ = 20 mm. (b) Conventional OPG with centered (Δx = 250 µm) excitation of the PPLN band and corresponding experimental far-field distributions for (c) pump and (d) collinear signal at 810 nm, at the output of a PPLN band with period Λ = 7.5 µm, $\theta_{x}^{ext}$ and $\theta_{y}^{ext}$ being the horizontal and vertical angular coordinates retrieved by the imaging system. (e),(f) and (g): same as (b), (c) and (d) for OPG pumped at the edge of the PPLN band. Scale bars = 100 µm. The plots show 1D intensity profiles for $\theta_{y}^{ext}=0$ (red lines on CCD images). Different attenuation filters are used in (d) and (g) to avoid saturation of the camera, hence image intensities are not representative of OPG efficiencies.

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Here we demonstrate an interference edge-effect arising at the interface of periodically poled and unpoled regions in LN crystals, which manifests itself in a transverse modulation of the linear throughput of an optical beam propagating along the edge of a PPLN grating. We show how the effect can be attributed to an increase $\Delta{n}$ of the average refractive index in the PPLN region, which in turn can lead to an increase of the nonlinear efficiency of QPM devices pumped close to the PPLN boundary. A systematic study combining theory and experiments on optical parametric generation (OPG) in 5 mol% MgO-doped crystals indicates a PPLN index increase $\Delta{n} = (5.3 \,{\pm}\, 0.05) \times 10^{-5}$. In experiments performed with a pump at 532 nm we observe a 3.6-fold OPG output enhancement of the generated infrared signals obtained when pumping the PPLN grating close to its boundary to unpoled regions, as opposed to the PPLN center as in conventional QPM OPG schemes. The results highlight additional degrees of freedom afforded by the combination of linear and nonlinear engineering in PPLN devices, providing potentially novel pathways to tailor parametric sources for classical and quantum applications.

2. Experiments

The setup used for the optical experiments is shown in Fig. 1(a). The PPLN sample was excited by a frequency-doubled microchip laser (Picophotonics Ltd), delivering 100-ps pulses at a wavelength λ_p = 532 nm with a peak power of 3 kW and a repetition rate of 25 kHz [19]. A half-wave plate and a polarizer were used to align the polarization of the input field along the optic axis of the PPLN crystal (y axis in Fig. 1) and control the incident pump power (P_p). The input beam was focused at the center of a crystal of length L = 2 cm, to lateral (x) and vertical (y) beam waists w_x = 27 µm and w_y = 50 µm, respectively. The sample was kept at constant temperature (T = 78°C) throughout the experiments. The incidence angle (θ_p) and lateral position (Δx) of the pump were adjusted by micromanipulation stages.

The nonlinear crystal consisted of a c-cut 5 mol % MgO-doped periodically poled LiNbO₃ (Covesion Ltd) containing five PPLN bands, with periods Λ ranging from 6.9 to 7.7 µm, increasing in steps of 200 nm. Each band was 2 cm-long, 0.5 mm-thick and 0.5 mm-wide. The PPLN bands were laterally separated by 0.2 mm-wide unpoled regions. The output of the nonlinear crystal was imaged on a visible or infrared camera, using a spherical lens in a Fourier configuration, with suitable filters to select pump, signal and idler wavelengths, λ_p = 532 nm, λ_s ≈ 720-840 nm and λ_i ≈ 1480-1590 nm, respectively, at the output.

The standard and edge- pumping OPG configurations considered in the experiments are illustrated by Fig. 1(b) and (e), respectively. The former corresponds to the optimum for collinear OPG excited in the middle of the PPLN band (Δx = 250 µm), which, for Λ = 7.5 µm and at normal incidence (θ_p = 0), yielded the output far-field profiles of Fig. 1(c) and (d), measured at λ_p = 532 nm and λ_s = 810 nm, respectively. In this case, the pump retains its original propagation direction: θ_x^ext = θ_p = 0 and the OPG outputs for the signal (at 810 nm) and idler (at 1550 nm) are both collinear.

When approaching the edge of the PPLN band (Δx = 0) spatial fringes start to appear in the pump throughput, as shown by Fig. 1(f). Furthermore, the pump far-field profile gets skewed to positive angles (θ_x^ext > θ_p = 0), which indicates a redistribution of energy towards the PPLN region during propagation in the crystal. The OPG signal shifts together with the pump to a propagation angle θ_x^ext = 0.2° [Fig. 1(g)]. The parametric signal output power (P_s) in this case is higher than for standard OPG, as apparent from systematic power measurements performed by replacing the CCD camera in Fig. 1 with calibrated photodetectors (Newport 818 UV/DB), whose results are presented in section 4. Although here we focus on experimental results obtained from the PPLN band of period Λ = 7.5 µm, the same edge-enhancement was qualitatively observed on the other PPLN bands as well.

3. Simulations

The OPG efficiency enhancement observed at the PPLN edge may seem counterintuitive, given that only a portion of the pump beam impinges on the ‘nonlinearly active’ (i.e. PPLN) region, as opposed to the case of standard OPG, where the PPLN aperture captures the full extent of the incident pump. To gain further insights into the underlying physical mechanisms, we developed a numerical model for nonlinear beam propagation postulating the existence of an additional (linear) refractive index perturbation at the boundary between the PPLN and unpoled regions. This hypothesis was based on the observation of a diffraction-like pattern in the pump output [Fig. 1(f)] and of its persistence even at low powers. We therefore assumed an index perturbation of the form:

Following the standard approach for first order QPM with an ideal 50% PPLN duty cycle, the nonlinearity profile was also modelled as:

Given the symmetry of the PPLN system under study, which is uniform along the vertical (y) direction, the full dynamics of the three-wave nonlinear interactions in the structure could be fully captured by a reduced 2D model, considering only the longitudinal direction (z-axis) and one transverse axis (x), along which the linear and nonlinear modulations of Eq. (2) and (3) apply. Following standard coupled-mode equation derivations from Maxwell’s equations, in a QPM medium with the χ⁽²⁾ modulation of Eq. (3) [22] and the additional refractive index modulation of Eq. (2), OPG interactions in the PPLN device were described through the following (1 + 1)D equations, written in terms of normalized spatial coordinates $\mathrm{\zeta} = z/L$ and $\xi = x/{w_x}$ [23]:

 

Fig. 2. In-plane propagation plots of : (a) pump and (c) signal for the conventional OPG geometry of Fig. 1(b); (b) pump and (d) signal for the edge-pumped OPG geometry of Fig. 1(e) with Δx = 20 µm. White horizontal lines at x = 0 denote the PPLN edge. Simulations from Eq. (4) with: σ_p = 0.4200, σ_s = 0.6551, σ_i = 1.2836, Γ_p = 24.4507, Γ_s = 16.0957, Γ_i = 8.3551, δ_p = 12.5191, δ_s = 8.2412, δ_i = 4.2779 and Δ = 0.

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The spatial evolution of the OPG pump and signal beams in the simulations exhibits a very good qualitative agreement with the experimental observations reported in Fig. 1. The propagation plot of Fig. 2(b) clearly highlights pump spatial modulation effects arising from constructive interference [bright region in Fig. 2(b)] in grazing reflection at the refractive index discontinuity occurring at the edge of the PPLN region (Δx = 0, white line), which entail a redistribution of power towards the PPLN region (x > 0) and a modulation of the pump transverse profile. This linear effect is accompanied by an enhancement of the nonlinear gain, apparent from a comparative analysis of the OPG signal intensity plots of Fig. 2(c) and (d). Consistently with the experimental observations of Fig. 1(d) and (g), the OPG signal does not undergo any significant distortion, retaining the clean beam profile of conventional OPG. This is justified by the exponential scaling laws of OPG in the high-gain regime (see also discussion of Fig. 5), which nonlinearly amplify the contribution to signal emission from the main peak in the pump intensity distribution, obscuring the effect of smaller side-modulations.

4. Quantitative analyses

We investigated further the response at the PPLN edge using as a reference the optimal conditions for standard collinear OPG in the PPLN structure. We then analyzed, through theory and experiments, the impact of the lateral position (Δx), propagation angle (θ_p) and power (P_p) of the incident pump in a systematic way comparing the conventional and edge-pumped cases.

We started by conducting systematic numerical simulations and measurements for different Δx, to map the far-field distribution of the pump output (Fig. 3) in a purely linear regime (Γ_q= 0). By fitting the output angular distributions of theory and experiments, with $\Delta{n}$ as a free parameter for the former, we could infer a value of $\Delta{n} = (5.3\, \pm \, 0.05) \times 10^{-5}$, consistent across all sets of measurements, with an accuracy of 10⁻⁶. As shown by Fig. 3, simulations performed with this value faithfully reproduced the modulations of the main peaks in the experimental far field profiles, matching very well their amplitude and angular position in the central portion of the beam, with some discrepancy appearing on the tails, due to a not perfectly Gaussian shape of the input pump and artifacts at the edges of our optical system in the experiments.

 

Fig. 3. Angular distributions of the pump (λ_p= 532 nm) throughput in the edge pumping configuration (Fig. 1(e)), as retrieved from measurements (red dashed line) and simulations performed with w_x = 27 µm, $\Delta{x} = 1\, \mu{m}$, $\Delta{n} = (5.3 \,{\pm}\, 0.05) \times 10^{-5}$ (blue solid line).

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The $\Delta{n}$ inferred from the linear analysis is consistent with the order of magnitude of previous estimates of an index contrast of 4.26 × 10⁻⁵ in MgO-LN poled domains, obtained by other groups and with very different methods [13]. Compared to the latter, our configuration offers the advantage of accurately retrieving in-situ index changes of PPLN structures, with only minor adjustments to the conventional QPM geometry, rather than ad-hoc transverse pumping experiments. Furthermore, it provides the first evidence of an impact of the refractive index changes of poled regions on nonlinear processes at the PPLN edge, as detailed in what follows.

In light of the results of Fig. 2, the enhanced nonlinear response is qualitatively explained as a consequence of diffraction at the PPLN index discontinuity (Δn), giving rise to local intensity enhancements of the pump field in the PPLN area, which in turn benefits the parametric gain and hence the signal output, due to its nonlinear dependence on the local pump intensity.

Figure 4 plots the OPG output signal power (P_s) as a function of the lateral position of the incident pump beam in the crystal (Δx), for a fixed pump excitation power P_p = 8.0 mW. Moreover, θ_p = 0 and λ_p = 810 nm, which is the optimum for conventional OPG at the center of the PPLN. The PPLN corresponds to 0 < Δx < 500 µm and the unpoled area to Δx < 0. Simulations (shaded region) and measurements (markers) clearly indicate the existence of a value close to the PPLN edge, Δx ≈ 20 µm, where the OPG output is maximized (keeping all other parameters fixed). There the OPG efficiency is almost twice the value obtained well inside the PPLN. On the other hand, when Δx < 0, P_s is rapidly decreasing as most of the pump power is fed into the unpoled region.

 

Fig. 4. Average signal power (P_s) as a function of the distance between the pump beam and the edge (Δx), for P_p= 8.0 mW and θ_p = 0. Markers (with error bars): experimental data. Solid line: simulations (the shaded region denotes the signal power standard deviation, determined on the basis of 50 simulation runs per data point with randomized initial conditions, accounting for initiation of the process by vacuum noise fluctuations).

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For a more quantitative analysis of the OPG enhancement, Fig. 5 plots the output signal power (P_s) as a function of the input pump power (P_p) for standard OPG (red circles, Δx = 250 µm) and for OPG at the PPLN edge (blue asterisks, Δx = 20 µm). Figure 5(a) shows the data in linear scales, while Fig. 5(b) plots them in a logarithmic [ln(P_s)] vs square root (√P_p) scale, in light of the expected input-output power dependence of conventional phase-matched OPG in the undepleted pump and high-gain approximation, which follows an exponential law of the form [26]:

 

Fig. 5. Evolution of the OPG signal output power (P_s) with the input pump power (P_p), for θ_p = 0 and λ_p = 810 nm, plotted in: (a) linear and (b) logarithmic scale [corresponding to the OPG law of Eq. (5)]. Markers: experimental data. Solid lines: numerical fits based on Eq. (5). The red circles and blue stars correspond to excitation in the middle and at the edge of the PPLN band, yielding the gain coefficients g_mid = 259.9 /√W and and g_edge = 268.7 /√W, respectively.

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The rescaling of the axes adopted in Fig. 5(b) translates the exponential dependence of Eq. (5) into a linear one and enables a direct retrieval of the gain coefficient from the experimental data (markers in Fig. 5), through linear fits (solid lines) of the form: ln(P_s) = A₀ + g√P_p, where A₀ is constant and g is the fitting parameter. In what follows, the extrapolated gain coefficients for OPG excited in the middle and at the edge of the PPLN band will be denoted as g_mid and g_edge, respectively. The good agreement between the linear fit and the experimental data in Fig. 5(b) confirms the validity of the undepleted-pump high-gain approximation in both standard and edge-enhanced OPG. The gain coefficients retrieved from the fits in the two cases are g_mid= 259.9 /√W and g_edge= 268.7 /√W, respectively, with a gain factor enhancement amounting to Δg = g_edge− g_mid= 8.8 /√W. At the highest pump power level, P_p = 8.5 mW, the signal power featured a 2.2-fold enhancement at the PPLN edge.

The data in Fig. 5 are recorded with a bandpass filter which selects the signal at 810 nm. Removal of the filter, to retrieve the full spectral extent of signal emission in the near infrared, resulted in Δg = 10.8 /√W. This value could be further increased by adjusting the pump incidence angle to θ_p = −0.042°. Under optimal excitation conditions, pumping the PPLN close to the edge yielded a 3.6-fold enhancement in the signal output power with respect to standard OPG.

5. Conclusions

In conclusion, we observed an enhancement of the OPG efficiency at the edge of periodically poled MgO:LiNbO₃, achieved as a result of the refractive index increase in the PPLN region. A 3.6-fold increase of the signal output was experimentally obtained for an excitation power of 8.1 mW in the green. A systematic analysis of the linear and nonlinear response of the PPLN device as a function of the pump incidence angle, power and position yielded very good agreement between theory and experiments and enabled to infer a refractive index difference of 5.3 × 10⁻⁵ between the PPLN and unpoled regions of the crystal. The observed effects may prove useful in tailoring coherent parametric sources relevant for a variety of classical and quantum applications of QPM media, by engineering the interplay of linear and nonlinear effects at PPLN boundaries.