1. INTRODUCTION

In recent years, photonic integrated circuits have experienced significant advancements, driven by the promise of compactness, efficiency, low cost, and mass production. A wide range of passive and active functionalities, from electro-optical modulation to frequency conversion, has been demonstrated on various platforms such as silicon [1,2], stoichiometric silicon nitride (Si3N4) [3,4], lithium niobate (LN) [5,6], and aluminum nitride [79" target="_self" style="display: inline;">–9]. Many of these functionalities are already available within a standard full-wafer technology at foundries. The control of nonlinear effects in these integrated platforms, either based on the second- or third-order susceptibilities (χ(2) and χ(3), respectively), represents a large area of research, as frequency mixing effects offer a practical solution for light generation. In an effort to access three-wave mixing effects, for communication or quantum applications, the above-mentioned materials are being studied for the demonstration of an efficient and compact χ(2) waveguide. The highest conversion efficiencies (CEs) are typically achieved in devices that employ quasi-phase-matching (QPM) [5,10], which is a technique where, by creating a periodic structure in a nonlinear medium, the phase mismatch between frequencies participating in the targeted three-wave mixing process is compensated, allowing for efficient energy transfer [11]. The efficiency of nonlinear interaction in waveguide devices can be characterized by second-harmonic (SH) generation. Despite the demonstration of CE reaching thousands of %/W, there still have been a limited number of reports on the properties of QPM such as CE bandwidth for continuous wave (CW) or short pulse regimes, along with thermo-optic tuning of QPM. One of the reasons for the limited number of studies is that the waveguide devices typically require complex nanostructuring techniques in order to ensure periodic modulation of nonlinearity for QPM. Recently, we demonstrated that efficient QPM SH generation in Si3N4 waveguides can be achieved by using all-optical poling [12,13], which was also confirmed by other groups [1416" target="_self" style="display: inline;">–16]. Here, an effective nonlinear χ(2) grating builds up spontaneously in the presence of a high-intensity coherent pump and weak SH due to the coherent photogalvanic effect [1719" target="_self" style="display: inline;">–19]. The method is simple to implement and allows inscription of self-organized nonlinear gratings for QPM at a selected pump wavelength in a manner of minutes [20]. This method is therefore useful for studying the influence of dispersion on QPM properties, without the need for fabricating many samples. The method does require the waveguide to be able to withstand high optical intensities with peak power reaching the level of tens of watts, as is the case for the Si3N4 waveguides used in our study. The additional strong points of the platform, such as excellent lithographic control, low propagation loss, and high third-order nonlinearity, have indeed established Si3N4 as a perfect candidate for various applications, including low pulse energy supercontinuum (SC) generation [21–24" target="_self" style="display: inline;">–24], frequency comb generation [3,25], ultralow-loss resonators [26–29" target="_self" style="display: inline;">–29], and entangled photon-pair generation [30,31].

Here, we combine the strengths of all-optical poling and those of the Si3N4 waveguide platform in order to study the QPM properties depending on waveguide dispersion, carefully controlled through altering the waveguide dimensions. Until now, the concept of optimizing QPM by dispersion engineering in integrated photonic waveguides has rarely been employed and most exclusively for group-velocity (GV) matching of short pulses [16,32]. In this work, we show, numerically and experimentally, that the bandwidth of the CE spectrum in a QPM structure can be controlled and significantly enhanced solely by geometric dispersion engineering. By meeting the requirements for maximal CE bandwidth for CW operation, we demonstrate that GV matching for a short-pulsed pump and SH is simultaneously satisfied. The latter provides means for SH generation from a CW signal, i.e., long and ultrashort optical pulses, without sacrificing efficiency of either regime. As a proof-of-concept, we employ this phenomenon to retrieve carrier-envelope-offset frequency (fCEO) of a frequency comb by using an f−2f interferometric technique, where SC and SH of a femtosecond pulse are generated in Si3N4 waveguides. Finally, we address the thermo-optic tuning properties of QPM that are also shown to vary significantly with the waveguide dispersion.

2. QPM SPECTRAL RESPONSE

2.1 A. Theoretical Considerations

The CE of a nearly monochromatic optical pump in a QPM device is calculated by [20] where ω is the pump frequency, Lg is the length of the QPM structure, χeff(2) is the effective nonlinearity, nω and n2ω are the effective refractive indices at pump and SH wavelengths, respectively, Aω and A2ω are the effective mode areas at pump and SH wavelengths [5], respectively, and Δβ is the propagation constant mismatch. In Eq. (1), Δβ is expressed as where Λ is the grating period, and β2ω and βω are the propagation constants at the SH and fundamental harmonic, respectively [11]. After all-optical poling, by definition Δβ=0 at the QPM wavelength, which is identical to poling wavelength in case the QPM measurement and poling are done at the same sample temperature. As evident from Eq. (1), the amplitude of CE efficiency is determined by the effective nonlinearity χeff(2) as well as the grating length Lg, while the CE bandwidth can be calculated by sinc-squared function and is determined by Lg and the propagation constant mismatch Δβ around the QPM wavelength. A longer grating length will result in higher efficiency but will also have a reduced conversion bandwidth. Evidently, the Δβ variation near the QPM wavelength has a crucial influence on the conversion bandwidth. In order to maintain a large bandwidth in such a QPM structure without sacrificing efficiency, the change of Δβ relative to the wavelength [i.e., ∂(Δβ)/∂λ] should be equal to zero. This is met when the following equation is satisfied [11]: where λ is the QPM wavelength. We will further refer to the wavelength at which Eq. (3) is satisfied as λ0.

The Δβ dependence on wavelength is determined not only by the refractive index of the employed material but also by the waveguide geometry. In Fig. 1, we plot the expected variation of the QPM response for all-optically poled Si3N4 waveguides. In order to match the subsequent experiments, we ran simulations in the telecommunication band. Here, all the simulations for different waveguide geometry were done using a numerical mode solver. We first consider poling the waveguide at 1.55 μm: after poling, Δβ=0 at this particular wavelength and at the employed light polarization, transverse-electric (TE) or transverse-magnetic (TM), following the all-optical inscription of the self-organized grating with period Λ=λ/[2·(n2ω−nω)]. In Figs. 1(a) and 1(d), we map the Δβ dependence on waveguide width and wavelength, for poling performed on TE or TM polarization, respectively. The height of the waveguide is fixed at 0.75 μm, grating length fixed at Lg=10mm, and effective nonlinearity at χeff(2)=0.1pm/V. The resulting CE according to Eq. (1) is shown in Figs. 1(b) and 1(e) corresponding to TE and TM cases, respectively. As expected, the maximum CE decreases as the cross-section of the waveguide is increased. This occurs because the modes become less confined in larger waveguides causing the light intensity inside the core to drop. The CE maps in Figs. 1(b) and 1(e) show that variation of waveguide geometry can have significant influence on the CE bandwidth. As evident, a broad QPM spectral response can be achieved for TE polarized light in a waveguide with cross-section of around 1.6μm×0.75μm. In contrast, the QPM spectral response for TM polarized light in a waveguide with the same dimensions is significantly narrower. In Figs. 1(c) and 1(f), the CE in waveguides with height 0.75 μm and selected widths of 1.6, 1.8, and 2.0 μm is shown for TE and TM polarized light, respectively. By comparing the CE spectral response when operated with TE light in waveguides with these dimensions, we see that the CE spectral response narrows significantly as the waveguide width is increased from 1.6 to 2.0 μm. The opposite takes place when QPM is induced for the TM case, i.e., the CE spectral bandwidth experiences slight increase in width but remains relatively narrow.

Fig. 1. (a) and (d) Propagation constant mismatch Δβ. (b) and (e) CE dependence on wavelength and waveguide width in 0.75 μm thick Si3N4 waveguide all-optically poled at 1.55 μm. (c) and (f) CE as a function of wavelength along the dashed lines in (b) and (e), respectively. Top and bottom rows consider TE- and TM-polarized light, respectively. Grating length Lg is 10 mm and χeff(2) is 0.1 pm/V.

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The CE spectral bandwidth and shape could also vary considerably for a given waveguide depending on the QPM wavelength, with more dramatic changes when operating near λ0. In Fig. 2, we show the CE dependence on wavelength in waveguides having different QPM conditions and for three cross-sections, namely, 1.6μm×0.75μm, 1.8μm×0.75μm, and 2.0μm×0.75μm. The plots consider TE polarized light. For the case of the 1.6μm×0.75μm cross-section, λ0 is estimated near 1.54 μm using Eq. (3). The same λ0 value is obtained from Fig. 2(a). Increasing or reducing the QPM wavelength around λ0 will cause two peaks to appear in the CE spectrum, which subsequently narrow as this wavelength difference increases. This is also the cause of two local maxima to appear in a plot of CE as a function of wavelength in Fig. 1(c) for QPM in the waveguide with a cross-section of 1.6μm×0.75μm. Consequently, widely dissimilar CE spectra can be obtained from a single waveguide by simply re-poling it at a different wavelength. When the QPM wavelength is far from λ0, as for Figs. 2(b) and 2(c) with λ0 being at 1.595 μm and 1.646 μm, respectively, the CE bandwidth remains narrow. In this case, re-poling the waveguide at different wavelengths would result in a shift in the QPM spectrum without significantly changing its shape.

Fig. 2. CE dependence on wavelength in waveguides having different QPM wavelengths. The waveguide height is 0.75 μm and widths are (a) 1.6 μm, (b) 1.8 μm, and (c) 2.0 μm. The plots consider TE polarized light; the grating length Lg is 10 mm and χeff(2) is 0.1 pm/V. λ0 indicates the wavelength at which condition by Eq. (3) is satisfied.

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It is important to emphasize that CE estimations, as in Figs. 1 and 2, are extremely sensitive to the refractive index at pump and SH wavelengths. This implies that the material refractive index dependence on wavelength should be determined experimentally with high precision in order to obtain meaningful results through numerical simulations.

2.2 B. Experimental Confirmation

We investigated experimentally such QPM spectral bandwidth dependence on the waveguide geometry. For this purpose, buried Si3N4 waveguides with a height of 0.75 μm and widths of 1.6, 1.8, and 2.0 μm were fabricated according to the photonic Damascene process [26,33]. The 55 mm long waveguides are folded in 11 meanders on a 5mm×5mm chip by using curved sections. The radii of the bends for each waveguide were designed to reduce the influence of mode mixing and had a minimum bend radius of 100 μm. The waveguides have optical propagation loss of ca. 4 dB/m and were all optically poled at 1.55 μm as described elsewhere [20]. We use a PID controller and a Peltier module to maintain the temperature of the chip at 30°C during the poling procedure. During all-optical poling, an effective χ(2) grating is inscribed inside the waveguide ensuring QPM as confirmed previously [13]. In Fig. 3, the measured (dotted) and fitted (dashed–dotted lines) CE dependences on wavelength after poling are shown. The effective nonlinearity χeff(2), grating length Lg, and period Λ were found using a least-squares fit of experimental data by Eq. (1), resulting in relative standard error below 2.2%. Here, the numerically simulated propagation constants at pump and SH wavelengths were provided as functions of wavelength [see Figs. 1(a) and 1(d)] and were kept constant. During poling, we monitored the process such that the CE at the poling wavelength would be of the same order of magnitude in all waveguides and for all polarizations. From Eq. (1), this means that we expect the grating length Lg and strength χeff(2) to be approximately equal in all waveguides. Any change in CE spectral bandwidth could therefore be attributed solely to the difference in dispersion.

Fig. 3. Experimentally measured CE spectra (blue dots) with fits (red dashed line) of waveguides all-optically poled at 1.55 μm with the effective nonlinearity χeff(2), grating length Lg, and period Λ obtained from fit. Waveguide cross-sections are (a) and (d) 1.6μm×0.75μm, (b) and (e) 1.8μm×0.75μm, and (c) and (f) 2.0μm×0.75μm. The polarization is TE for (a)–(c) and TM for (d)–(f) where both polarizations in the waveguides with the same dimensions result in independent parameter sets.

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The experimentally measured QPM wavelength is slightly shifted relative to the 1.55 μm poling wavelength. This is explained by temperature differences in the waveguide during poling performed with a pulsed pump and QPM measurement done with CW light. Despite this slight shift, the QPM spectral responses in Fig. 3 follow the expectations from numerical simulations. Here, the broadest QPM response is obtained in a waveguide with a cross-section of 1.6μm×0.75μm poled using TE-polarized light [see Fig. 3(a)]. The parameters obtained from the fit are included as insets in Fig. 3. The effective nonlinearity χeff(2) and grating length Lg are similar for all waveguides. There is a slight difference in Fig. 3(a), where the grating is estimated to be three times longer than in other waveguides. This could be due to two factors. First, we do not have the full CE spectrum since the data points below 1.535 μm could not be captured due to limitations of the C-band amplifier (Keopsys CEFA-C-PB-HP, output power 37 dBm). Second, errors could arise due to the imprecision of the numerically simulated Δβ. This is particularly impactful since we expect Δβ to be close to zero in a spectrally broad band around the QPM wavelength, and any small error in Δβ could result in increased uncertainty of Lg.

In the case of TE polarization, the QPM spectrum becomes narrower with the increase of waveguide width from 1.6 to 2.0 μm [Figs. 3(a)–3(c)]. The opposite occurs when the same waveguides are poled using a TM-polarized pump [Figs. 3(d)–3(f)]. These experimental results perfectly confirm that dispersion engineering of QPM can be employed to enable both efficient narrowband and broadband responses in the same waveguide by leveraging the two modes of polarization. Most importantly, reaching broadband response does not come at the price of efficiency since long gratings can still be used.

3. GV MATCHING AND fCEO Retrieval

As shown previously, the largest bandwidth of QPM is achieved when Eq. (3) is satisfied. Meeting the requirement by Eq. (3) also ensures GV matching taking place in the waveguide, which, together with QPM, is key requirements for efficient SH generation of pulsed light [34]. The presence of GV matching was confirmed by calculation of the group index ngω dependence on wavelength (see Appendix A). We demonstrate the benefit of simultaneous phase and GV matching by fCEO detection proof-of-principle experiment. For this purpose, an f−2f interferometer (for details, see Appendix B) was implemented [35,36]. Here, a beam from a femtosecond pulse source is split and delivered to two Si3N4 waveguides. One of the waveguides, having dimensions of 2.65μm×0.78μm, is dispersion-engineered to generate a TE dispersive wave (DW) at around 0.78 μm wavelength where the SH of the pump is located (for details on DW engineering, see Appendix C). The other beam is delivered to a waveguide with a cross-section of 1.6μm×0.75μm, which has a broad CE spectral response after all-optical poling, as shown previously. The spectra of obtained SC and SH from each chip are shown in Fig. 4(a). Both beams leaving the chips are collected and delivered to a high-speed detector that is connected to a frequency analyzer. The retrieved beat notes are displayed in Fig. 4(b). Here, the frep corresponds to the repetition rate of the laser at 100 MHz. The fCEO is detected with a signal-to-noise ratio (SNR) at 32.7 dB with the resolution bandwidth being 50 kHz when the average output powers of the pump and SH at the output waveguide are 4.2 dBm and −25.1dBm, respectively. Moreover, the SNR would not reduce during several days of operation, suggesting that such power coupled inside the waveguide would not erase the inscribed nonlinear grating. The obtained SNR is sufficient for ensuring frequency comb stabilization without phase slips; yet, it can be further enlarged by increasing the power coupled to the waveguide in which SH generation is taking place. However, there is an optimal coupled power before SC generation takes place in the poled waveguide that results in the erasure of the grating [13].

Fig. 4. Carrier-envelope-offset frequency measurement employing an all-optically poled Si3N4 waveguide. (a) Spectra of SC generated in a 2.65μm×0.78μm dispersion-engineered waveguide and SH of femtosecond pulses in all-optically poled 1.6μm×0.75μm waveguide measured by an optical spectrum analyzer (Yokogawa AQ6374). (b) Measured beat notes between SC and SH. The resolution bandwidth of frequency analyzer (Rohde & Schwarz FSUP Signal Source Analyzer) is 50 kHz.

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4. DEPENDENCE OF DISPERSION ENGINEERED QPM ON TEMPERATURE

QPM wavelength variation is enabled by the thermo-optic effect, as shown previously in LN [37] and Si3N4 [20] integrated photonics platforms. The change of the waveguide temperature causes alteration of phase-matching condition and, consequently, the shift of QPM wavelength. The QPM wavelength variation magnitude and sign will also depend on the waveguide geometry and can be estimated from the derivative of Δβ with respect to the pump wavelength λ,∂(Δβ)/∂λ. In the vicinity of the poling wavelength, if ∂(Δβ)/∂λ>0, then the QPM wavelength will blueshift with the increase of temperature and vice versa. Also, the shift in QPM wavelength will be greater for smaller absolute ∂(Δβ)/∂λ. The change in QPM wavelength ΔλQPM with small variation of temperature can be approximated as (for illustrative explanation, see Appendix D) In Fig. 5(a), we plot the ∂(Δβ)/∂λ dependence on waveguide width in a 0.75 μm thick Si3N4 waveguide all-optically poled at a QPM wavelength of 1.55 μm at a temperature of 30°C. Data in Fig. 5(a) can also be used to estimate the geometrical parameters of the waveguide core at which the CE spectral response would be the broadest, meaning whenever ∂(Δβ)/∂λ is near 0. For example, for the case of a 0.75 μm thick Si3N4 waveguide, the broadest CE for TE polarized light will be present in waveguides with widths of 0.8 and 1.595 μm; for TM, the waveguide width will be around 0.82 μm.

Fig. 5. (a) ∂(Δβ)/∂λ as a function of waveguide width in a 0.75 μm thick Si3N4 waveguide all-optically poled at 1.55 μm. When the ∂(Δβ)/∂λ is positive in the waveguide around poling wavelength, the QPM wavelength will blueshift with the increase of temperature, and vice versa. (b) and (c) Measurements (points) and fits (lines) of CE spectra at different temperatures in poled 1.8μm×0.75μm waveguide for operation using TE or TM polarized light, respectively. (d) Extracted QPM wavelength as well as temperature dependence and linear fits. Tuning slopes Δλ/ΔT calculated from linear fits are 691±38pm/K and −237±12pm/K for TE and TM polarized light, respectively.

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From Fig. 5(a) and Eq. (4), we can expect that, in a 1.8μm×0.75μm waveguide, the QPM wavelength should redshift (blueshift) with the increase of temperature if TE (TM) polarized light is used. In order to confirm this, we all-optically poled a waveguide with a cross-section of 1.8μm×0.75μm using 1.55 μm light polarized at TE and TM and measured the QPM spectrum with a CW wave for different temperature settings. The SH spectra at several temperatures in the range from 20°C to 40°C for TE and TM polarized light are plotted in Figs. 5(b) and 5(c), respectively. The CE spectra were fitted according to Eq. (1) and normalized to maximal value of the fit function. From location of CE maxima at different chip temperatures, the QPM wavelengths are extracted and plotted in Fig. 5(d). We experimentally confirm that, in case QPM is induced for TE polarized light, the QPM wavelength is redshifted with the increase of temperature. The slope of this change is 691±38pm/K. When the waveguide is all-optically poled using TM polarized light, the QPM wavelength is blueshifted with increase of temperature and changes with −237±12pm/K.

5. CONCLUSIONS

Implementing QPM in waveguiding structures allows for extra degrees of freedom. Here, the possibility to vary the waveguide geometry is leveraged to adjust and control the QPM properties in Si3N4 waveguides. Experimentally, the QPM inside the waveguides is achieved using an all-optical poling technique that is simple to implement and does not require traditionally applied complex nanostructuring techniques. Through optimization of waveguide core dimensions, we modify the dispersion of the waveguides to achieve broadband and efficient frequency conversion in waveguides with QPM. The CE bandwidth can be increased by minimizing gradients of propagation constant mismatch Δβ with respect to the wavelength. Meeting the mentioned condition in a waveguide with QPM results not only in broadband phase-matching but also in GV matching. The latter provides the possibility of processing large bandwidth optical signals such as ultrashort pulses. We employ this phenomenon to retrieve the fCEO using an f−2f interferometer technique as a proof-of-principle experiment. The obtained SNR, the measurement with resolution bandwidth being 50 kHz is 32.7 dB, sufficient for frequency comb phase stabilization. Finally, we show that the waveguide dispersion determines the QPM wavelength variation magnitude and sign with temperature. The absolute QPM wavelength variation with temperature is larger for small |∂(Δβ)/∂λ|, while the sign of QPM wavelength change with temperature is determined by that of ∂(Δβ)/∂λ near the QPM wavelength. In case the slope is positive, the QPM wavelength will blueshift with increase of the temperature and vice versa. This is experimentally confirmed by measuring the QPM wavelength variation due to thermo-optic effect in a waveguide with a cross-section of 1.8μm×0.75μm at TE and TM polarizations, respectively. In this case, the Δβ gradients with respect to wavelength vary both in sign and magnitude for each polarization. For poling with TE polarized light, the QPM wavelength redshifts with the increase of temperature with a slope of 691 pm/K. For poling with TM polarized light, the QPM wavelength blueshifts with −237pm/K. While the experimental validations were carried out in all optically poled Si3N4 waveguides, it is important to note that the obtained rules and properties are general and can be applied for any waveguide platform and any poling mechanism.

6 Acknowledgment

Acknowledgment. We acknowledge J. He and T. Liu for assistance in the sample test.